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Mathematical Applications and Modelling is the second in the series of the yearbooks of the Association of Mathematics Educators in Singapore. The book is unique as it addresses a focused theme on mathematics education. The objective is to illustrate the diversity within the theme and present research that translates into classroom pedagogies.The book,...Until the mid-twentieth century, topological studies were focused on the theory of suitable structures on sets of points. The concept of open set exploited since the twenties offered an expression of the geometric intuition of a 'realistic' place (spot, grain) of non-trivial extent. Imitating the behaviour of open sets and their relations led... more...
This is a unique book related to the theory of functions of a-bounded type in the half-plane of the complex plane, which is constructed by application of the Liouville integro-differential operator.
In addition, the book contains improvements of several results such as the Phragmen-Lindelof Principle and Nevanlinna Factorization in the Half-Plane,...
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Program Resources
Founded in 1888 to further mathematical research and scholarship, the American Mathematical Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.
The Mathematical Association of America (MAA) is the world's largest organization devoted to the interests of collegiate mathematics. Members of the MAA receive many valuable benefits for modest dues. These benefits are designed to stimulate interest in mathematics by providing expository books and articles on contemporary mathematics and on recent developments at the frontiers of mathematical research, and by exchanging information about important events in the mathematical world. A major emphasis of the MAA is the teaching of mathematics at the collegiate level, but anyone who is interested in mathematics is welcome to join.
To ensure the strongest interactions between mathematics and other scientific and technological communities, it remains the policy of SIAM to advance the application of mathematics and computational science to engineering, industry, science, and society; promote research that will lead to effective new mathematical and computational methods and techniques for science, engineering, industry, and society; and provide media for the exchange of information and ideas among mathematicians, engineers, and scientists.
The American Statistical Association (ASA) is a scientific and educational society founded in 1839 with the following mission: To promote excellence in the application of statistical science across the wealth of human endeavor.
Michigan Council of Teachers of Mathematics is organized to encourage an active interest in mathematics and its teachings and to work toward the improvement of mathematics education programs in Michigan.
The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students.
Kappa Mu Epsilon an honor society in Mathematics promotes the interest of mathematics among undergraduate students. The chapters' members are selected from students of mathematics and other closely related fields who have maintained standards of scholarship, have professional merit, and have attained academic distinction.
Math Horizons is intended primarily for undergraduates interested in mathematics. Its purpose is to introduce students to the world of mathematics outside the classroom including stories of mathematical people, the history of an idea or circle of ideas, applications, fiction, folklore, traditions, institutions, humor, puzzles, games, book reviews, student math club activities, and career opportunities and advice. Get a copy in the departmental office!
Plus magazine opens a door to the world of maths, with all its beauty and applications, by providing articles from the top mathematicians and science writers on topics as diverse as art, medicine, cosmology and sport.
Pi in the Sky is a semi-annual periodical designated for high school students in Alberta and British Columbia, Canada, with the purpose of promoting mathematics, establishing direct contact with teachers and students, increasing the involvement of high school students in mathematical activities, and promoting careers in mathematical sciences.
MathWorld is a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers. Like the vibrant and constantly evolving discipline of mathematics, this site is continuously updated to include new material and incorporate new discoveries.
The Math Forum is a leading center for mathematics and mathematics education on the Internet. The Math Forum's mission is to provide resources, materials, activities, person-to-person interactions, and educational products and services that enrich and support teaching and learning in an increasingly technological world.
SIGCSE provides a forum for problems common among educators working to develop, implement and/or evaluate computing programs, curricula, and courses, as well as syllabi, laboratories, and other elements or teaching and pedagogy.
Women and Mathematics Information Server provides information about the Women and Mathematics Network. It also provides resource information to Program Directors, Teachers, and others trying to students, especially girls, pursue Mathematics or the related Mathematical Sciences.
ACM, the world's largest educational and scientific computing society, delivers resources that advance computing as a science and a profession. ACM provides the computing field's premier Digital Library and serves its members and the computing profession with leading-edge publications, conferences, and career resources.
Through its members, the IEEE (Eye-triple-E) is a leading authority in technical areas ranging from computer engineering, biomedical technology and telecommunications, to electric power, aerospace and consumer electronics, among others.
The Consortium for Computing Sciences in Colleges is a non-profit organization focused on promoting quality computer-oriented curricula as well as effective use of computing in smaller institutions of higher learning which are typically non-research in orientation. It supports activities which assist faculty in making appropriate judgments concerning computing resources and educational applications of computer technology.
The Computing Research Association (CRA) seeks to strengthen research and advanced education in computing and allied fields. It does this by working to influence policy that impacts computing research, encouraging the development of human resources, contributing to the cohesiveness of the professional community and collecting and disseminating information about the importance and the state of computing research.
The Association for Women in Computing is a non-profit professional organization for women and men who have an interest in information and technology. The Association is dedicated to the advancement of women in the technology fields.
The Liberal Arts Computer Science Consortium is an organization of computer scientists from quality liberal arts schools. The group is dedicated to supporting undergraduate computer science through active curriculum development and scholarly activity in the field of computer science education.
Communications of the ACM is a vehicle for ACM members to communicate their research findings and ideas, Every month Communications brings its readers the latest in technology trends as written by the very creators and innovators of those technologies.
Computer, the flagship publication of the IEEE Computer Society, publishes peer-reviewed technical content that covers all aspects of computer science, computer engineering, technology, and applications. The articles selected for publication are edited to enhance readability for the general Computer reader. Computer is a resource that practitioners, researchers, and managers can rely on to provide timely information about current research developments, trends, best practices, and changes in the profession.
The Charles Babbage Institute (CBI) is an historical archives and research center of the University of Minnesota. CBI is dedicated to promoting study of the history of information technology and information processing and their impact on society. CBI preserves relevant historical documentation in all media, conducts and fosters research in history and archival methods, offers graduate fellowships, and sponsors symposia, conferences, and publications.
2013-14 Academic Year Colloquium Schedule
September 5, 2013
A degree in mathematics or computer science is excellent preparation for graduate school in areas such as mathematics, statistics, computer science, engineering, finance, and law. Come learn about graduate school and options you will have to further your education after graduation.
October 3, 2013
Algebraic geometry has been at the corner of much of mathematics for hundreds of years. Its applications range from number theory to modern phyiscs. Yet, it begins quite humbly with the study of conic sections: circles, ellipsys, hyperbolas, and parabolas. What is algebraic geometry and how did it grwo beyond the scope of these familiar curves to become one of the most important branches of methematics today?
October 10, 2013
Color, texture, pattern - there's more than first meets the eye in my crocheting. Come for a hands-on experience and new insights into finding mathematics in unlikely objects and expressing math concepts in artful ways. Invite your knitting and crocheting friends, bring someone who claims "I can't do math," attend with a classmate who thinks math is too abstract to be interesting, bring along that education major or art student. Build a bridge between math and their world: enjoy this event together.
October 31, 2013
We live in an era where there has never been greater access to information. Being able to sift through and analyze this information to understand what is "noise" and what can actually lead to valuable insights has become a highly demanded commodity. In turn, so to have Data Analysts. For profit-seeking companies, the realization of business objectives through reporting of data to analyze trends, creating predictive models for forecasting and optimizing business processes for enhanced performance has become pivotal for sustainable success. In this talk, we will provide an introduction to data analytics and we will review how our employer, EY, uses data anayltics to build a better working world.
November 7, 2013
A research problem concerning the Colin de Verdiere number of a graph recently led me on a journey that provides a great example of the interconnected nature of mathematics. We'll take a relaxing cruise through some of the topics involoved, including ideas from Analysis, Algebra, Geometry, and Graph Theory, see how they all fit together, and talk about some of the mysteries that remain. Only knowledge of arithmetic is needed.
While games are ordinarily thought of as a means for entertainment and distraction, they are also inherently useful to accomplish all manner of other purposes. Among other thigns, games-both digital and physical-can be used to teach, modify behaviors, infulence opinions, and improve physical and mental health. I will share some of the major heuristics that are useful in designing games for "serious" purposes, as well more general knowledge of game design and the game industry. Additionally, I will share my experiences as a graduate student in the serious game design MA program at Michigan State University.
November 21, 2013
Title:
Adjusting Child Support Payments in Michigan
Speaker:
Michael A. Jones Associate Editor Mathematical Reviews Ann Arbor, MI
Abstract:
Michigan uses an unusual formula in the calculation of child support payments. For divorced parents in Michigan, the base monetary support each parent is expected to contribute to raising their child is adjusted according to the number of (over)nights spent with the parents. Curiously, this adjustment is based on a rational polynomial function parameterized by k that describes the amount of money that A must pay B, where B must pay A if the result is negative. In the 2004 Michigan Child Support Formula Manual, k=2, meaning the polynomials are quadratic; while k=3 (for cubic polynomials) in both the 2008 and 2013 editions. In this talk, we will brainstorm and collaborate in using calculus to examine this function, explain the effect of changing k, and point out an alternative form that stretches and translates a simpler function. This talk is based on joint work with Jennifer Wilson (New School University, NY).
Location:
Palenske 227
Time:
3:30 PM
February 6, 2014
Anaerobic digestion is a biochemical process in which organic matter is broken down to biogas and various byproducts in a oxygen-free environment. When used in waste treatment facilities, the biogas i scaptured before it excapes into the atmosphere. It can then be used as renewable energy either by combusting the gas to produce electrical energy or by extracting the methane and using it as a natural gas fuel. In industrial applications anaerobic digestion appears to be difficutl to control and reactors often experience break-down resulting in little or no biogas production. In this talk we describe a model for anaerobic digestion and illustarte how qualitative and numberical analysis give guidelines for how to control the system to (1) stablize and (2) optimize biogas production. At the same time the model explains various possible pitfalls in industial installations.
Location:
Palenske 227
Time:
3:30 PM
February 13, 2014
Question: What do sums of powers have to do with approximations of factorials? Answer: Integration by parts. No, really? In this talk we will see how a clever use of standard calculus techniques leads to the Euler-Maclaurin formula, a powerful way of connecting sums to integrals, and how this formula solves several classic problems.
Location:
Palenske 227
Time:
3:30 PM
March 3, 2014
The Theory of differentiation is well-known to any student who has taken calculus. However, it make sense of a non-integer order derivative takes considerable more work. Tools are needed from complex analysis, harmonic analysis and linear algebra to understand a half derivative. In this talk, we will begin by investigating what it means to take the square root of a matrix, and viewing a derivatie as a "really large matrix" we can begin to make sense of a half derivative. With these simple tools, we can make sense of even crazier object such as derivatives of imaginary order!
Location:
Palenske 227
Time:
3:30 PM
March 6, 2014
Partial geometries were first described in 1963 by R.C. Bose. They are finite point line geometires specified by three parameters that are defined by a set of four basic axioms. Each partial geometry has a strongly regular point graph. While some very simple shapes can be understood as partial geometries, the number of proper ones is actually limited. In this talk we will define both the geometries and the graphs and explore some connections between them. We will also look at how we can use a group of automorphisms acting on the geometry to clasify it as one of three types. Finally we will see how this work enables us to generate a list of parameters for potential partial geometries and how we are beginning to investigate these possiblilities.
Location:
Palenske 227
Time:
3:30 PM
April 3, 2014
Erdos asked: when does the base 3 expansion of a power of 2 omit the digit 2? His conjectured answer is that this only happens for 1, 4, and 256, but this conjecture is still open, and has proven to be very elusive. There underlies a deep relationship between the primes 2 and 3. Our attempt to understand this relationship has led to interesting connections among symbolic dynamical systems, graph theory, p-adic analysis, number theory, and fractal geometry. Despite the awesome variety of mathematics involved, linear algebra should be sufficient background knowledge for this talk. I report on joint work with Jeff Lagarias of the University of Michigan and Artem Bolshakov of the University of Texas of Dallas.
Location:
Palenske 227
Time:
3:30 PM
April 10, 2014
Patterns appear everywhere in the world around us from zebra stripes, to hexagonal honeycombs, to spiral arrangements of sunflower seeds, to the periodic ups and downs of a population size due to seasonal migration. Similar patterns also arise in experiments done in many disciplines, such s physics, chemistry, and biology. One goal in studying pattern formation is to understand why and how these patterns are created. Another goal is to determine whether similar patterns from vastly different systems can be described and understood through similar mathematical model equations. This talk will describe how a pattern can be represented mathematically and how basic knowledge of functions and derivatives can help determine when and where the patterns will exist. Ananlytical and numerical results will be compared with experimental observations. Finally, the connection between the underlying pattern and the observation of a single, isolated pulse, called an oscillon, will be described.
Location:
Palenske 227
Time:
3:30 PM
April 17, 2014
Operations Research in an area of applied math that deals with analyzing and optimizing many different systems: industial, nonprofit, government, healthcare, etc. It operates at the intersection of math, engineering, statistics, computer science, and business. We will talk about common focus areas like minimizing waiting times for important public services, and scheduling staff in an optimal way. The methods are incredibly powerful--optimization decisions can often involve hundreds of thousands of variables, and sometimes millions or billions.
Location:
Palenske 227
Time:
3:30 PM
2012-13 Academic Year Colloquium Schedule
September 6, 2012
Title:
Point of View: Scientific Imagination in the Renaissance (Program 3 from The Day the Universe Changed)
September 13, 2012
Rubik's Slide is a puzzle which consists of a $3 \times 3$ grid of squares that is reminiscent of a face of the well-known cube. Each square may be lit one of two colors or remain unlit. The goal is to use a series of moves, which we view as permutations, to change a given initial arrangement to a given final arrangement. Each play of the game has different initial and final arrangements. To examine the puzzle, we use a simpler $2 \times 2$ version of the puzzle to introduce a graph-theoretic approach, which views the set of all possible puzzle positions as the vertices of a (Cayley) graph. For the easy setting of the puzzle, the size of the graph depends on the initial coloring of the grid. We determine the size of the graph for all possible arrangements of play and determine the associated god's number (the most moves needed to solve the puzzle from any arrangement in the graph). We provide a Hamiltonian path through the graph of all puzzle arrangements that describes a sequence of moves that will solve the easy puzzle for any initial and final arrangements. Further, we use a computer program to determine an upper bound for god's number associated to the graph representing the medium and hard versions of the puzzle.
September 20, 2012 27, 2012
Pattern, repetition, and symmetry play important roles in the aesthetics of imagery. Tessellations use patterns of repeated geometric shapes to cover the plane. Uniform tessellations use regular polygons to cover the plane with no gaps or overlaps. The polygons in such tessellations can be decorated in such a way to give rise to interesting visual patterns. The inherent symmetry of regular polygons gives rise to tessellations containing symmetry patterns. Example symmetric tessellation patterns will be presented. An explanation of algorithmic techniques for constructing uniform tessellations will also be presented.
October 4, 2012
We present a model for the optimal design of an online auction/store by a seller. The framework we use is a stochastic optimal control problem. In our setting, the seller wishes to maximize her average wealth level, where she can control her price per unit via her reputation level. The corresponding Hamilton-Jacobi-Bellmann equation is analyzed for an introductory case, and pulsing advertising strategies are recovered for resource allocation.
While visiting the Calculus and Physical Sciences Tutorial Lab at Grand Rapids Community College, a question was posed: for what values of $n$ will the sum of the first $n$ positive integers be a perfect square? A thorough investigation of the problem and the introduction of the concept of an isosceles "almost" right triangle yielded a number of interesting results. One of the results involves a sequence of rational numbers that converges to $\sqrt{2}$, yielding some excellent approximations.
October 25, 2012
Optimization is the branch of mathematics concerned with optimal performance---finding the best way to complete a task. It has been put to good use in a great number of diverse disciplines: advertising, agriculture, biology, business, economics, engineering, manufacturing, medicine, telecommunications, and transportation (to name but a few). In this lecture, we will showcase its amazing utility by demonstrating its applicability in the area of visual art, which at first glance would seem to have no use for it whatsoever! We will begin by describing how to use integer programming to construct a portrait out of complete sets of double nine dominoes. We will then describe how high quality solutions to certain large-scale traveling salesman problems can lead to beautiful continuous line drawings. We will conclude by presenting other examples of Opt Art---art constructed with the assistance of mathematical optimization techniques.
November 1, 2012
A Skolem sequence of order $t$ is a sequence $2t$ integers such that each integer between 1 and $t$ appears twice and two instances of the integer $k$ are $k$ apart. For example, 5242354311 is a Skolem sequence of order 5. These sequences, and their generalizations, are very interesting from a combinatorial point of view and have many applications. In this talk, we will discuss Skolem sequences and some generalizations: extended, Langford, and near-Skolem sequences. We will also discuss a few applications of these sequences, including integer partitioning and graph decompositions.
November 8, 2012
George Hart, the designer of the sculpture Comet!, which hangs in the science complex atrium, will return to Albion for a hands-on workshop on mathematical sculpture. During his visit to Albion, he will lead participants in a hands on construction of a brand new never seen geometrical sculpture. During the workshop, the mathematical ideas behind the sculpture will be explained and participants will build their own personal sculpture with playing cards. For other examples of his work, see georgehart.com.
November 29, 2012
The Adjusted Winner procedure is a fair division procedure used to divide contested items between two people so that the allocation satisfies three desirable properties (efficiency, equitability, and envy-freeness). After reviewing these properties and the procedure, I'll explain how the procedure is related to cake cutting. Further, exploiting information and manipulating the Adjusted Winner procedure is an example of the game of Chicken. This talk combines ideas from two previously published papers: Michael A. Jones and Stanley F. Cohen, Fairness: How to Achieve It and How to Optimize in a Fair-Division Procedure, Mathematics Teacher 94 (3) 2004: 170-174. and Michael A. Jones, Equitable, Envy-free, and Efficient Cake Cutting for Two People and Its Application to Divisible Goods, Mathematics Magazine 75 (4) 2002: 275-283.
December 6, 2012
Oftentimes, multiple different yo-yo tricks can be done sequentially before the yo-yo returns to the user's hand. Tricks can be done like that due to the fact that some tricks end where others begin, and vice versa. If we take these common start/end points to be nodes on a directed graph, all sorts of possibilities for mathematical examination open up. In this talk, we will look at how interesting parts of graphs (such as cycles) translate into yo-yo trick combos, and also how real-world restrictions on yo-yo trick combos affect what we can do with the graphs.
December 6, 2012
This presentation intends to cover the basics of what a fractal is. Since fractals don't tend to have integer dimensions like we are used to this will include how to determine the dimension of fractals. We will also discuss some simpler fractals that are easy to conceptualize many of these will come from a group of fractals known as the polygaskets. The polygaskets are fractals that are based on recursively using a polygon shape to create them. A prime example of these is Sierpinski's triangle which is a fractal based off of a triangle.
January 31, 2013
Title:
Necessity and Scope in the Logic of Quantification
Speaker:
Jeremy Kirby Associate Professor Philosophy Albion College Albion, MI
Abstract:
When I say "Eight is necessarily greater that seven," I state something that is true. In contrast, when I say "The number of planets is necessarily greater than seven," I say something that is false. (We can conceive of a smaller solar system, indeed at times the number of planets is revised.) Furthermore, the locutions "eight" and "the number of planets" seem to pick out the same thing? How can it be both true and false of the same thing that it is necessarily greater than seven?
February 7, 2013
This talk will describe the mathematics behind Google's page rank algorithm. We will see how Google sets up and solves an eigenvector problem to decide which of the web pages containing your search terms are most relevant. The talk will also touch briefly on graph theory and computational complexity.
February 14, 2013
In this colloquium, we will see how to construct not only the trivial owl bundle on a goat (and a bonus nontrivial owl bundle), but a fish tank that can mirror-reverse your fish. Fiber bundles are more than just something you should be eating for breakfast every day. They can be used to describe and construct forces of nature in this universe and the next. They are also good for hours of pure topological fun.
February 21, 2013
Modern computer graphics cards have GPUs (graphic processing units) that can do several hundred million calculations per second. I will demonstrate my new algorithms that exploit this power to create and animate Escher-like tessellations (tilings) of the plane in real time. Besides being fun, the animations dramatically illustrate the geometry behind the tessellations. I will also discuss how parametric equations, symmetry groups, homogenous coordinates, linear algebra, computational geometry, computer graphics, and data structures all come together to create the algorithms behind the animations. TesselManiac is my third major tessellation program, my previous programs include TesselMania and Tessellation Exploration.
February 28, 2013
A degree in mathematics or computer science is excellent preparation for employment in areas such as teaching, actuarial science, software development, engineering, and finance. Come learn about career opportunities awaiting you after graduation. Slides from the talk are available at
April 11, 2013
In mathematics, chaos can be defined as a deterministic dynamical system which has aperiodic long-term behavior and exhibits sensitive dependence on initial conditions. Surprisingly, such systems can be coupled together and made to synchronize. If their communication is delayed, this chaotic behavior can also be broken and stable periodic behaviors will emerge from the coupled system. Join me as we study the basics of chaotic systems and explore some examples of the synchronization of chaos (with and without delay).
Guided by the predictions of a discrete-time mathematical model, we induced a sequence of bifurcations (dynamic changes) in laboratory insect populations by manipulating one of the biological parameters in the system. In particular, we were able to induce chaotic dynamics. The data from these 8-year-long time series show the fine structure of the deterministic chaotic attractor as well as lattice effects (dynamic effects arising from the fact that organisms come in discrete units). We show that "chaos" is manifest in discrete-state noisy biological systems as a tapestry of patterns that come from the deterministic chaotic attractor and the lattice attractors, all woven together by stochasticity.
2010-11 Academic Year Colloquium Schedule
September 2, 2010
For over 30 years people around the world have been captivated by the Rubik's cube. Why is it so popular? What makes it a good puzzle? This talk will cover the history and design of the cube, explore some mathematics related to the cube, discuss solving the cube, and explore some possible and impossible patterns. I will bring several cubes for the audience to play with after the talk.
September 9, 2010 23, 2010
Is there a way to predict when and where such failure occurs? In this talk I will discuss some recent research directed at providing answers to these critical real-world problems. After a brief tutorial on the basic math, physics, and metallurgy required to attempt to answer such questions, I will review prior work that used a well characterized patch of Titanium Aluminum (TiAl) to evaluate the utility of a scalar fracture initiation parameter (fip) to predict the relative resistance of grain boundaries to microcracking when subjected to stress. I will then discuss new research that has generalized the idea of a scalar fip to a physically motivated damage tensorD that measures the amount of physical damage that accumulates at stressed grain boundaries as they evolve through space and time. Local lattice curvature near the grain boundary, local elastic and plastic stress evolution, and accumulated dislocation content at the grain boundary are among the quantities considered. Then, using data generated from a three dimensional, nonlinear, crystal plasticity finite element simulation of the same experimental TiAl region, the ability of this the tensor D to predict the location of "weak" grain boundary locations where micro-cracking is likely to occur.
This work is funded by the NSF Materials World Network Grant DMR-0710570, the Deutsche Forschungsgemeinschaft (DFG) Grant EI 681/2-1, and the Department of Mathematics and Computer Science at Albion College.Location:Palenske 227Time:3:10CitationClick for BibTeX citation
September 30, 2010
If it is easy to verify the solution to a problem, is it easy to solve that problem? This is the famous P vs NP problem. There are other important open questions we can ask. Is a uniformly randomm instance of a hard to solve problem still hard to solve? Are there specific structures in the solution space to a problem that will prevent certain algorithm techniques from working? This talk explores what is currently known about these questions, and we will use the well-known problems 3-SAT and factoring as examples. The talk will also introduce some new work in defining a random problem model that has many of the properties of 3-SAT but for which we can prove behavior that we observe experimentally but not yet prove for 3-SAT.
October 7, 2010
In grade school, students learn a standard set of Euclidean triangles. Among this set, the usual 45-45-90 and 30-60-90 triangles are the only right triangles with rational angles and side lengths each containing at most one square root. Are there any other such right triangles? We answer this question and present an elegant complement, called Ailles' rectangle, that deserves to be in every geometry teacher's toolkit.
October 21, 2010
The Four Color Theorem is a simple and believable statement: at most four colors are needed to color any map drawn in the plane or on a sphere so that no two regions sharing a boundary receive the same color. It might be surprising to find out that mathematicians searched for a proof of this statement for over a century until finally finding one in 1976. In this talk, we'll consider the "proof" given by Alfred Kempe, a proof published in 1879 and thought to be correct until an error was found in 1890. You're invited to look carefully at Kempe's proof and see if you can do what many 19th century mathematicians could not do—find the flaw.
October 28, 2010
We'll explore an algorithm that takes $n$ points in $\mathbb{R}^2$ or $\mathbb{R}^3$ and produces a piecewise-linear spiral that uses the given points as its initial nodes. We generate further points in the spiral by repeatedly taking a convex combination of $m \le n$ (existing) points at a time. In particular, let $P_0,\ldots,P_{n-1}$ be the initial points, and let $0\le t_1, t_2, \ldots, t_m \le 1$ be fixed parameters with $t_1+t_2+\cdots+t_m=1$. Produce more points by using the formula $P_{k+n} = t_1P_k + t_2P_{k+1}+ \cdots + t_mP_{k+m-1}$ for each $k\ge 0$.
We can then ask a lot of questions: Where does the spiral end up?, How long is it? When and how can we arrange things so that the segment lengths are a geometric series? What is the general behavior of the spiral as it approaches its limit? The tools we'll use will come from linear algebra, complex analysis, infinite series, and linear recurrences. We'll also talk a bit about how this problem evolved from a Problem of the Week to several REU projects and papers (including one in the Spring 2010 $\Pi$ME Journal).
November 4, 2010
Deal or No Deal was a prime time game show on the National Broadcasting Corporation network in which a Contestant selects one of 26 suitcases. Inside each suitcase is a different dollar amount; all 26 dollar amounts are known beforehand. In a series of rounds, the Contestant is asked to "deal" (in which she accepts a monetary offer from a Banker) or to "no deal" (in which she has to open a specified number of suitcases, thereby revealing the dollar amounts inside the suitcases). The game ends when either she accepts an offer or, after opening all of the suitcases except the one she selected at the outset, she receives the monetary amount in her selected suitcase.
Because each suitcase may contain any of the fixed monetary amounts, selecting a suitcase is analogous to a lottery in which each value has an equal likelihood of being selected. Assuming the Banker's offer is based on a utility function that describes the Contestant's utility or value for money and incorporates the Contestant's view toward the risk of participating in the lottery, the Banker makes an offer so that the Contestant is indifferent between accepting the Banker's offer and continuing to play the game.
In this talk, I will introduce the basics of utility theory and will explain how the Banker could use a utility function to determine an offer. I will demonstrate how data from televised episodes may be used to recover the utility function. Further, I will examine a paradoxical offer from NBC's online version of the game.
A forthcoming paper of the same name is co-authored with Jennifer Wilson, New School University, New York.
November 11, 2010
Have you ever wondered if you can study mathematics and/or computer science off-campus? Either during the summer or during the academic year? Each year a number of high-quality academic opportunities are availableto Albion College students. Options include research/study internships at
academic institutions both within the United States and abroad,
numerous federal government agencies, and
a number of government scientific laboratories.
In this presentation we will tour a new portion of the Albion College Math/CS website that illustrates these various opportunities as well as provide adviceon how to apply, deadlines, any other pertinent information.
November 18, 2010
A popular form of folk dance is English country dance. In one simple English country dance, four couples dance as two groups of two couples. As the dance progresses, each couple moves to a new position and dances with another couple. Can you have such a dance where each couple dances in each of the four positions with each of the other three couples? What are other mathematical restrictions on such dances?
December 2, 2010
Students Mathematics and Computer Science Albion College Albion, Michigan, USA
Abstract:
Robert Calvert, "Decoding the Enigma" Through my talk I will talk about the enigma's build, the main people involved in decoding it, and the methods used in decoding it.
Cassie Labadie, "Incorporating Mathematical Museum Exhibits into Classrooms" How do you make learning math fun? Studies show that learning through traditional means, such as lecture and taking notes, does not make the information the students are gaining commit to memory. We will take a look at the importance of creative pedagogical practices in the classroom, and how you apply these to a math classroom. We will be focusing on different mathematical museums and museum exhibits that can be implemented in the classroom, and how you change both simple and complicated exhibits into fun learning experiences for students in the classroom.
Culver Redd, "Meaningful Play: How Games Can Be Productive In Our Society" During this past October, I attended a conference at Michigan State University called Meaningful Play. This talk will disseminate my experience of this conference. Meaningful Play was held to display the potential for games to be used to enhance education, general learning, academic study, and many other aspects of our lives, as well as to examine the current state of the industry that creates games for these purposes. The results, ideas, and opinions expressed at this conference are, I believe, extremely valuable to students—particularly those with interest in computer science—as they detail the forefront of a quickly growing aspect of computer science, as well as one possible future for the educational systems of America and the world.
January 27, 2011
Decision analysis is a procedure for identifying, clearly representing, and formally assessing important aspects of a decision involving uncertainty. The procedure, developed by operations research and business professors, now is widely used in research evaluating medical treatments. Greg Saltzman, Professor of Economics and Management at Albion College, taught a course in 2008 and 2010 at the University of Michigan School of Public Health for medical researchers, "Cost Utility and Decision Analysis." He will present an introduction to decision analysis during his talk.
February 3, 2011
Suppose G is a finite graph. Two players play a game on G as follows: one player takes n markers (which represent "cops") and assigns each one to a vertex of G; then the second player takes one marker (representing a "robber") and assigns it to a vertex of G. The players then alternate turns, each moving any number of his or her markers to adjacent vertices each turn. If a cop is moved to the same vertex as the robber, the cop player wins. If the robber player can always avoid such an outcome the robber player wins. Certainly the cop player can win on a given graph G if sufficiently many cops are at his disposal. But what is the fewest number of cops needed to guarantee that the robber can always be captured? This was a topic at a summer Research Experience for Undergraduates (REU) at Michigan State University in the 2010. The investigations of several of the participants will also be highlighted.
February 10, 2011
Title:
Fractals : Hunting The Hidden Dimension
Speaker:
NOVA DVD
Abstract: This video takes viewers on a fascinating quest with a group of pioneering mathematicians determined to decipher the rules that govern fractal geometry.
February 17, 2011
Archimedes (c. 287 BCE--c. 212 BCE) used polygons inscribed within and circumscribed about a circle to approximate pi. In this talk, we will extend his work by approximating the areas of circular sectors. This is done by adjoining parabolic segments to triangular subregions of his inscribed regular polygons. While much of the mathematics would have been familiar to Archimedes, the calculations involved quickly outstrip the computational power of ancient Greece, and so Mathematica is used to facilitate calculations. The method allows us to derive recurrence relations that can be used to approximate pi more accurately.
Why is copper soft and ductile while rock salt is hard and brittle? One would guess that the mechanical behavior of crystalline materials is inextricably linked to how their atoms are bonded, but just as important is how their atoms are arranged in crystal structures. Plastic (permanent) deformation is achieved through the motion of crystal defects, while failure through fracture results from the rupture of atomic bonds. In order to fully understand and optimize mechanical behavior of materials, it is therefore necessary to understand the arrangement of atoms. But how can we determine the positions of atoms in a material? Atomic arrangements are typically studied using diffraction techniques (x-ray, electron, neutron) by implementing Bragg's Law and Structure Factor calculations to determine not only the size and shape of the unit cell, but also the atom positions and types within the unit cell. Armed with this information, it is possible to understand the details of mechanical behavior, in particular the anisotropic nature of plastic deformation. This talk will review and build on these concepts to illustrate how the macroscopic deformation and fracture behavior, and the ultimate performance of planes, trains, and automobiles, is a function of the crystallographic orientation distribution in both single and polycrystalline materials. Examples of the role of non-random crystallographic orientation distribution in the anisotropic behavior of a number of materials, including FeAl, TiAl, and Ti will be presented. The implications of this anisotropic behavior will be discussed.
Optimal prediction is about a decade old now, but has fast become one of the most exciting new areas in Optimal Stopping. The original paper by Graversen, Peskir, and Shiryaev showed, in an elegantly simple way, that one could compute the best time to stop a Brownian motion "as close as possible" to its ultimate maximum over a finite time interval. Since then, researchers have worked to extend this idea to other diffusions, different measures of "close", and to financial applications. In this talk, we review the original approach, extensions, and current research including the recent application to infinite horizon prediction The area is rich with potential for new research, and it is hoped that young mathematicians will be encouraged to read more on the subject after this talk.
April 7, 2011
In this talk we will consider the mathematical problem associated with special linear deformations of an incompressible and nonlinear elastic cube. We will discover that the problem admits a wide variety of different solutions, depending on the magnitude and direction of external isotropic forces. To understand why certain solutions are preferred by nature, we will then study an associated energy minimization problem that leads to a selection criterion to determine the optimal deformed state of the cube. Finally, we will connect the mathematical appearences of these multiple solutions, natural and mathematical stability, and the fundamentals of bifurcation theory.
April 21, 2011
I will show how optimal moves in the combinatorial games Nim, Wythoff, and Euclid are related to binomial representations of integers, the Fibonacci numbers, and a proof that the positive rational numbers are countable, respectively.
April 28, 2011
Parallel computing is a method in which many calculations can be carried out simultaneously. Not every algorithm or problem can gain an increase in speed from being executed in parallel. In recent years the bottlenecks of output of a single computer processor has increase a demand for multicore processors. At the same time our trusty graphics processors have helped in such acts of massive computation. Using these techniques there are global computation projects that you can use your very own equipment at home in order to help better understand illness and disease by simulating problems millions of time in order to exceed what was previously understood. Other applications include deep oil exploration and finances.
During the Big Bang, a point of infinite curvature of spacetime, the basic rules for how the universe behaves break down. While general relativity accurately describes the universe and the effects of gravity on the larger scale, it struggles with points of singularity such as the Big Bang. It needs to be infused with quantum mechanics, the rules of behavior for the very small, to explain the likes of the Big Bang and black holes. With quantum mechanics applied to general relativity, there arise ambiguities in the ordering of factors in the definitive equation for the state of the universe. To find out the proper ordering of factors, we turn to a computer model of the universe that has arguably made a good choice using a completely different methodology, simplified to the toy model of one space and one time dimension. We do this by comparing our varied possibilities to the computer model to try and ascertain hints to how our universe behaves in the realm of the very small.
The development of a software program can often be a long and difficult task. I was recently part of a development team for the creation of an application to control the iRobot Create, which I call Zoomba. This application remote controls the speed, direction, and movement of the iRobot Create via an Android phone. This talk will discuss the design, implementation, coding, and testing of our application as well as give an overview of the software development process in general. I will also give a brief demonstration of how the application works.
April 28, 2011
Fantasy sports have been extremely popular among sports fans for many years now, and starting this summer I was lucky enough to get a job with one of these companies. Running a fantasy sports website does not mean manually inputting stats in for each player and calculating each teams results, but instead writing software that will automatically handle all of this. Working at OnRoto Fantasy Sports, I had to learn the computer languages of Perl, C, and SQL to be able to write scripts to improve the website at OnRoto. I will focus on a few projects that I have completed over the past year including the new mobile site that is in the late development stages that I am currently working on.
April 28, 2011
I will discuss the company MCP asset management from the the outside as well as the inside. I will discuss the decisions that are made by the workers of the company to make money as well as the decisions made by the company to choose quality clients. These decisions include people who they choose to allow to invest their money with as well as who they should accept money from. Both of these decisions involve who the company thinks is reliable and using legal means to acquire their money. Many possible clients and investors use questionable means to acquire and grow their money. Overall this is a dog eat dog market that can eat a company up quickly if they do not do reliable research on clients and investors.
April 28, 2011
My Colloquium talk involves using binary programming to convert image files to pixel art. I created a model for choosing what values should be used in creating a smaller image based on the larger image. In order to get data for the image I used a program called Gimp to save it in a format that I could use and create a binary value matrix to base my function on. I used the program MPL to minimize the function that I created. Unfortunately I needed to split the problem into 4 problems because when I made the model I needed more variables to convert the image than I had. I took the result matrices and combined them and used the resulting matrix in Mathematica to create an image.
April 28, 2011
The oral talk will show the current situation and comparison of elementary school math education in China and US. I will focus on the differences and similarities in two different math education system by collecting data and information on the history of elementary school math education, the materials they are using for math study, teaching methods to inspire students' interests in math. At the same time, I will show the importance of math education that affects students' life. And finally, I will talk about the pros and cons of two different math education systems and effects on students' math ability.
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Presenting Fourier methods as a set of tools for solving applied problems, this introduction also gives the student advanced theoretical treatments that are a part of pure mathematics.
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books.google.com - The,... making, models and algorithms
Decision making, models and algorithms: a first course
The, and challenges in this book provide a unique presentation of the subject matter.
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Mathematics 1
Covers the parts of calculus and algebra which have proved fundamental to all of mathematics and its applications. It is the first of a pair of courses, MATH1110 and MATH1120, designed to cover a range of mathematical topics of importance to students in the Sciences, Engineering or Commerce.
In algebra, students learn concepts and symbolic manipulation when calculating with large numbers of variables. In calculus, they learn concepts used when working with continuously changing variables. Both ways of thinking are essential in the mathematics met by students in the Sciences, Engineering and Commerce.
This course has a prerequisite, either being - band 5 or better in the NSW HSC course "Mathematics" (Commonly known as "2 unit mathematics") or; - completion of extension 1 or extension 2 mathematics in the NSW HSC or; - a pass in MATH1002 or; - a mark of 10/20 or better in an invigilated sitting of the MATH1110 Math Placement Test. Only one attempt at this invigilated quiz is permitted.
Not to be counted for credit with MATH1210.
Available in 2014
At the successful completion fo this course students will have 1. gained the necessary background to study further university level mathematics as required in their program of study. 2. gained mathematical knowledge and skills in the areas of calculus, functions, vectors and complex numbers. 3. improved their analytical ability, in particular their skills at problem-solving.
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Microsoft Mathematics provides a graphing calculator that plots in 2D and 3D, step-by-step equation solving, and useful tools to help students with math and science studies.
Microsoft Mathematics provides a set of mathematical tools that help students get school work done quickly and easily. With Microsoft Mathematics, students can learn to solve equations step-by-step while gaining a better understanding of fundamental concepts in pre-algebra, algebra, trigonometry, physics, chemistry, and calculus.
Microsoft Mathematics includes a full-featured graphing calculator that's designed to work just like a handheld calculator. Additional math tools help you evaluate triangles, convert from one system of units to another, and solve systems of equations.
The Step-by-Step Equation Solver
Students can use this to learn how to solve difficult math problems.
Graphing calculator
Its full features and large two-dimensional and enhanced three-dimensional color graphs can better illustrate problems and concepts.
Formulas and Equations Library
Students will find more than 100 commonly used equations and formulae to help identify and apply equations.
Triangle Solver
This graphing tool explains triangles and their parts.
Unit Conversion tool
Students can use this handy tool to quickly and easily convert units of measure, including length, area, volume, weight, temperature, pressure, energy, power, velocity, and time.
With the Microsoft Mathematics Add-in for Word and OneNote, you can perform mathematical calculations and plot graphs in your Word documents and OneNote notebooks. The add-in also provides an extensive collection of mathematical symbols and structures to display clearly formatted mathematical expressions. You can also quickly insert commonly used expressions and math structures by using the Equation gallery. The Microsoft Mathematics Add-in can help you with the following tasks:
Compute standard mathematical functions, such as roots and logarithms
Compute trigonometric functions, such as sine and cosine
Find derivatives and integrals, limits, and sums and products of series
Perform matrix operations, such as inverses, addition, and multiplication
Perform operations on complex numbers
Plot 2-D graphs in Cartesian and polar coordinates
Plot 3-D graphs in Cartesian, cylindrical, and spherical coordinates
Solve equations and inequalities
Calculate statistical functions, such as mode and variance, on lists of numbers
One of the controversial issues that keeps coming up in computer science education is the role of the IDE – Integrated Development Environment. Some people believe that students should learn using command line instructions so that they become more aware of the roles of compiling and linking. I'm going to jump right in and say that insisting on command lines in a first course makes about as much sense as requiring that new car drivers start their cars with a hand crank.
Now I'm not saying that compiling is not an important concept – it is. And so is linking. But in this day and age it doesn't have the importance it did back in the old days when we often had to manually determine how and when the different parts of a program were loaded into memory at run time. You don't need that in a first course. In fact I am not sure where it does fit into the curriculum.
The first course of programming or computer science should be about success and for building a foundation for the future. I understand the argument that compiling and linking is part of that foundation and 20 years ago that was true. I don't think that it is true today. Today we have so many varied ways of building software and integrating it into something that the computer understands and can present to users that focusing on just one doesn't make as much sense early on. We have web pages, Internet APIs, mobile devices, PCs, and more. It's all a little bit different.
Returning to the success part of a first course. An IDE helps make beginners successful by making things they don't need right away invisible and allows students to focus on logic, problem solving and the specifics of the first programming languages. IDEs also support powerful debugging tools which makes fixing problems faster and easier. Yes I know you can get a listing of errors and print it all out from a command line tool. In fact it can be a whole lot like back in the punch card days! Oh yeah that will excite students.
So other than hiding compiling and linking what are the knocks against IDEs? Well one is that they are too GUI focused and this focus takes away from the focus on algorithms, problems solving and programming basics. This is an easy argument to make because students, especially youngers ones in high school and earlier, do often get caught up in the GUI. I see this as a problem as much of the instructor as the tool though. By providing templates or pre-built GUI code an instructor can help keep the focus away from the GUI. At the same time today's students are used to a real GUI program rather than white letters on a black "console" background. I would also argue that I/O is easier in many ways with, say, Windows Forms objects than parsing input and output strings on a command line program. This makes processing more data more easy which leads to better testing and prevents a lot of the early problems with I/O that beset many beginners.
The other big complaint about IDEs, especially professional ones like Visual Studio and Eclipse, is that they are too complicated. I taught HS CS for a number of years using professional IDEs – some for C/C++ from Borland but mostly Visual Studio for C++, J++ (a Java replacement), C# and of course Visual basic. Students generally adapted to the complexity very quickly and easily. With Visual Studio the similarities to other Microsoft products like Office made this even easier. But I'll accept that it may be a bit complicated from many high school students (and their instructors) but please don't insult college students by saying it is too complex for them Seriously? What sort of people are you recruiting?
Of course there are simplified IDEs to use if you really find Visual Studio too complex. Take Small Basic for example. Are there really command line compilers that are most simple then this:
There are also some strong advantages to IDEs such are powerful built-in debugging tools. I have used these in Visual Studio to show how recursion looks, how variables change, and how decision statements are not always what they seem (think = vs. == in C++). IntelliSense or other auto completion technologies (as seen in Visual Studio and Small Basic) allow for almost unlimited exploration of language and library options.
If you want to program in Visual Studio, C# or other dot net languages there are command line compilers available. They come standard with the .NET Framework and your Windows computer probably already has then installed. But are they really the way to go? Not for me. Give Visual Studio and/or Small Basic a try and see how they work for you.
I've been wondering lately what it is about loops (in computer programs) that is so hard for students to get their heads around. A college professor was telling me (back a while ago but it stuck with me) that they had assigned a program to print out the words to The Twelve Days of Christmas and had explicitly asked student to use loops. And when you think about it this is a natural for loops because of all the repetition. A good number (or bad depending on point of view) had actually submitted solutions without loops. Some of these students had previously taken and passed Advanced Placement Computer Science in high school! What's up with that?
Clearly it is not lazy. using loops is the way lazy people would do it. (Though I like to think that there is a fine line between lazy and efficient at times) No, students had used cut and paste and a lot of typing to do this all inline.
Now loops are something we do without thinking all the time. Climbing stairs is a while loop. Think about it – we repeat the same step motions until we get to the top or bottom of the stairs. We check, usually with our eyes, to see when we are there and then change our motions. We've all seen what happens when people don't check haven't we? Blind people climbing familiar steps memorize the number of steps and effectively execute a for/next loop to do the same thing. Eating is the same. We keep putting food in our mouth until either we are full or we run out of food which ever comes first.
And yet all too often students fail to see how programming syntax allows them to do the same things in a program. I don't get it.
Loops of course are all the same in programming. Oh the syntax is different for different types of loops and in different programming languages but basically they have the same components.
Setting initial conditions
Changing conditions
Checking to see if the condition has changed such that the loop should terminate
In the middle somewhere useful work happens – giving the benefit of the doubt.
Here is a Small Basic example
For i = 1 To 10 ' Set initial conditions
t = t + i ' pretend this is useful
EndFor ' change the value of i and see if we are done
!--crlf-->!--crlf-->!--crlf-->
Here is a C# while loop
TwoWord = "ABC"; // Set an initial condition
do
{
TwoWord = Console.ReadLine(); // Change the condition
}
while (TwoWord.CompareTo("exit") == 1); // See if the ending condition is met
!--crlf-->!--crlf-->!--crlf-->!--crlf-->!--crlf-->!--crlf-->
I think students get how loops work. Really I do. I think if you asked a student to explain the components of a loop they could do it. But translating that knowledge to a specific problem seems to be harder. I don't know why. I hear this sort of story (students struggling with using loops) from teachers all the time. There are other aspects of programming that students also struggle with the application. Arrays for example seem to be a struggle for many. Far too often I hear stories of students having variables like A1, A2, … A10 when a simple array would be much more efficient.
Concepts are easier than applications of those concepts. Problem solving of new problems is harder than applying old algorithms to well known problems. What's the answer? How do you deal with this? Or don't you see it with your students? If not, what is your secret?
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More About
This Textbook
Overview
Business Math, Tenth Edition unlocks the world of math by showing how it is used in the business world. Written in a conversational style, the book covers essential topics such as banking, interest, insurance, taxes, depreciation, inventory, and financial statements. It carefully explains common business practices such as markup, markdown, and cash discounts—showing students how these tools work in small business or personal finance. Authors encourage self-starters from the beginning, with the review of basic math, annotated examples, stop and check exercises, skill builders and application exercises. This edition includes updated problem sets, new trends and laws, a revised financial statements chapter and the one-of-a-kind MyMathLab
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Links Thread, For Your Mathematics Students: MathWords.COM in Links, Downloads and Scripts; Link: Mathwords A to Z
...This website is designed for math students who need an easy-to-use, easy-to-understand math resource all ...
For Your Mathematics Students: MathWords.COM
...This website is designed for math students who need an easy-to-use, easy-to-understand math resource all in one place. It is a comprehensive listing of formulas and definitions from beginning algebra to calculus. The explanations are readable for average math students, and over a thousand illustrations and examples are provided...
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Number theory is concerned with the properties of the natural numbers: 1,2,3,.... During the seventeenth and eighteenth centuries, number theory became established through the work of Fermat, Euler and Gauss. With the hand calculators and computers of today, the results of extensive numerical work are instantly available and mathematicians may traverse the road leading to their discoveries with comparative ease. Now in its second edition, this book consists of a sequence of exercises that will lead readers from quite simple number work to the point where they can prove algebraically the classical results of elementary number theory for themselves. A modern high school course in mathematics is sufficient background for the whole book which, as a whole, is designed to be used as an undergraduate course in number theory to be pursued by independent study without supporting lectures.
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'... admirably suitable for those meeting number theory for the first time and for unsupported individual study.' Nick Lord, The Mathematical Gazette
Book Description
Now in its second edition, this book consists of a sequence of exercises that will lead readers from simple number work to the point where they can prove algebraically the classical results of elementary number theory for themselves. A high-school course in mathematics is the only requirement.
Most Helpful Customer Reviews
This book is a carefully sequenced set of problems along with answers and a few comments. Burn uses those problems to introduce important number theory ideas. I enjoyed working through the problems to learn more about number theory. Most problems are accessible to those with a good high school mathematics background.
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Utah Mathematics Core
Secondary Mathematics I
Students in Secondary Mathematics I will deepen and extend understanding of linear relationships, in part by contrasting them with exponential phenomenon, and in part by applying linear models to data that exhibit a linear trend. Students will use properties and theorems involving congruent figures to deepen and extend understanding of geometric knowledge. Algebraic and geometric ideas are tied together. Students will experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.
All Secondary I Core Curriculum should be included in the Secondary I Honors Curriculum.
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Geometry for Dummies
9780470089460
ISBN:
0470089466
Edition: 2 Pub Date: 2008 Publisher: Wiley & Sons, Incorporated, John
Summary: Get un-stumped in a hurry! Proofs made easierMake friends with lines and angles, theorems and postulates - and prove itThe proof is in the pudding - and the parallelogram, and sometimes the rhombus. With this friendly guide, youa?ll soon be devouring proofs with relish. Youa?ll find out how a proofa?s chain of logic works and discover some basic secrets for getting past rough spots. Before you know it, youa?ll be pro...ving triangles congruent, calculating circumference, using formulas, and serving up pi.Discover how to: Identify lines, angles, and planes Calculate the area of a triangle Figure the volume and surface area of a pyramid Bisect angles and construct perpendicular lines Work with 3D shapes
Ryan, Mark is the author of Geometry for Dummies, published 2008 under ISBN 9780470089460 and 0470089466. One hundred eighty one Geometry for Dummies textbooks are available for sale on ValoreBooks.com, sixty three used from the cheapest price of $4.31, or buy new starting at $8.19.[read more]
0470089466 Your purchase benefits those with developmental disabilities to live a better quality of life. Your purchase benefits those with developmental disabilities to live [more]
0470089466 Your purchase benefits those with developmental disabilities to live a better quality of life. Your purchase benefits those with developmental disabilities to live a better quality of life. some wear on edges and corners minimal cover wear minimal stains on edges minimal wear on pages
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More About
This Textbook
Overview
Addressing individual learning styles, Tom Carson presents targeted learning strategies and a complete study system to guide students to success. Carson's Study System, presented in the "To the Student" section at the front of the text, adapts to the way each student learns, and targeted learning strategies are presented throughout the book to guide students to success. Tom speaks to students in everyday language and walks them through the concepts, explaining not only how to do the math, but also where the concepts come from and why they work.
Related Subjects
Meet the Author
Tom Carson's first teaching experience was teaching guitar as an undergraduate student studying electrical engineering. That experience helped him to realize that his true gift and passion are for teaching. He earned his MAT in mathematics at the University of South Carolina. In addition to teaching at Midlands Technical College, Columbia State Community College, and Franklin Classical School, Tom has served on the faculty council and has been a board member of the South Carolina Association of Developmental Educators (SCADE). Ever the teacher, Tom teaches outside the classroom by presenting at conferences such as NADE, AMATYC, and ICTCM on topics such as Combating Innumeracy, Writing in Mathematics, and Implementing a Study System
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More About
This Textbook
Overview
Perfect for the one-term course, Essentials of Precalculus with Calculus Previews, Fifth Edition provides a complete, yet concise, introduction to precalculus concepts, focusing on important topics that will be of direct and immediate use in most calculus courses. Consistent with Professor Zill's eloquent writing style , this full-color text offers numerous exercise sets and examples to aid in student comprehension, while graphs and figures throughout serve to illuminate key concepts. The exercise sets include engaging problems that focus on algebra, graphing, and function theory, the sub-text of many calculus problems. The authors are careful to use calculus terminology in an informal and accessible way to facilitate the students successful transition into future calculus courses. With an outstanding collection of student and instructor resources, Essentials of Precalculus with Calculus Previews offers a complete teaching and learning package.
Key Features:
• Available with WebAssign Online Homework and Grading System
• Vibrant four-color design illuminates key concepts and improves students' comprehension of graphs and figures.
• Translating Words into Functions section illustrates how to translate a verbal description into a symbolic representation of a function and demonstrates these translations with actual calculus problems.
• Chapter Review Exercises include problems that focus on the algebra, graphing, and function theory, the sub-text of so many calculus problems. Review questions include conceptual fill--in-the-blank and true/false, as well as numerous thought-provoking exercises.
• The Calculus Preview found at the end of each chapter offers students a glimpse of a single calculus concept along with the algebraic, logarithmic, and trigonometric manipulations that are necessary for the successful completion on typical problems related to that concept.
• Provides a complete teaching and learning program with numerous student and instructor resources, including the Student Resource Manual, WebAssign Access, Complete eLearning Center, and • Complete Instructor Solutions Manual.
• Includes a new section on simple harmonic motion in Chapter 4.
• A new section of parametric equations, as well as a new calculus preview of 3-space, has been added to Chapter 6.
• Rotation of polar graphs is now discussed in Section 6.6
• The discussion of the hyperbolic functions in Section 5.4 has been expanded.
• Numerous new problems have been added throughout the text.
• The final exam at the end of the text has been
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Riverdale Pk, MD Chemistry a course involving college-level algebra concepts and trigonometric concepts. It serves as a preparation for calculus and includes the following topics:
1. FUNCTIONS AND MODELS
Linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, inverse trigonometric;
Properties of functions;
Transformations of functions;
Inverse of functions;
2
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Math Competition LinksAmerican Mathematics Contest 8 (Middle School) The AMC 8 is a 25 question, 40 minute multiple choice examination in junior high school (middle school) mathematics designed to promote the development and enhancement of problem solving skills. The examination provides an opportunity to apply the concepts taught at the junior high level to problems that not only range from easy to difficult but also cover a wide range of applications.
American Mathematics Contest 10 (Secondary Grades) The AMC 10 is a 25-question, 75-minute multiple choice examination in secondary school mathematics containing problems which can be understood and solved with pre-calculus concepts. Calculators are allowed. The main purpose of the AMC 10 is to spur interest in mathematics and to develop talent through the excitement of solving challenging problems in a timed multiple-choice format. The problems range from the very easy to the extremely difficult.
American Mathematics Contest 12 (Secondary Grades) The AMC 12 is a 25-question, 75-minute multiple choice examination in secondary school mathematics containing problems which can be understood and solved with pre-calculus concepts. Calculators are allowed. The main purpose of the AMC 12 is to spur interest in mathematics and to develop talent through solving challenging problems in a timed multiple-choice format. Because the AMC 12 covers such a broad spectrum of knowledge and ability there is a wide range of scores. The National Honor Roll cutoff score, 100 out of 150 possible points, is typically attained or surpassed by fewer than 3% of all participants. The AMC 12 is one in a series of examinations (followed in the United States by the American Invitational Examination and the USA Mathematical Olympiad) that culminate in participation in the International Mathematical Olympiad, the most prestigious and difficult secondary mathematics examination in the world.
The Mandelbrot Competition (Secondary Grades) In those ten years the contest has grown to two divisions encompassing students from across the United States as well as from several foreign countries. Nearly half of the competitors in the USA Math Olympiad in the last couple of years have been Mandelbrot competitors. The Mandelbrot Competition is split into two divisions: Division A for more advanced problem solvers and Division B for less experienced students.
Mathcounts (Grades 7-8) Each year, more than 500,000 students participate in MATHCOUNTS at the school level. Those who do tell us that their experience as a "mathlete" is often one of the most memorable and fun experiences of their middle school years.
Math Problems of the Week (Grades K-12) The Problem of the Week is an educational web site that originates at the University of Mississippi. All the prizes are generously donated by CASIO electronics. All contest winners are chosen randomly from the pool of contestants that successfully solve that week's problem.
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Mathematics Level 1
Introduction
Good with numbers? The Mathematics Level 1 Subject Test assesses the knowledge you've gained from three years of college-preparatory mathematics, including two years of algebra and one year of geometry. If you've excelled in these courses, taking the test can support your high school grades, indicate an interest in pursuing math-based programs of study (science, technology, engineering, economics, etc.), and differentiate you in the admission process.
Additional Things to Know
If you have taken trigonometry or elementary functions (precalculus) or both, received grades of B or better in these courses, and are comfortable knowing when and how to use a scientific or graphing calculator, you should select the Level 2 test. If you are sufficiently prepared to take Level 2, but elect to take Level 1 in hopes of receiving a higher score, you may not do as well as you expect. You may want to consider taking the test that covers the topics you learned most recently, since the material will be fresh in your mind. You should also consider the requirements of the colleges and programs you are interested in.
Areas of Overlap on Math Level 1 and Math Level 2
The content of Level 1 has some overlap with Level 2, especially in the following areas:
Elementary algebra
Three-dimensional geometry
Coordinate geometry
Statistics
Basic trigonometry
How Test Content Differs
Although some questions may be appropriate for both tests, the emphasis for Level 2 is on more-advanced content. The tests differ significantly in the following areas:
Number and Operations. Level 1 measures a more basic understanding of the topics than Level 2. For example, Level 1 covers the arithmetic of complex numbers, but Level 2 also covers graphical and other properties of complex numbers. Level 2 also includes series and vectors.
Geometry and Measurement. A significant percentage of the questions on Level 1 is devoted to plane Euclidean geometry and measurement, which is not tested directly on Level 2. On Level 2, the concepts learned in plane geometry are applied in the questions on coordinate geometry and three-dimensional geometry. The trigonometry questions on Level 1 are primarily limited to right triangle trigonometry (sine, cosine, tangent) and the fundamental relationships among the trigonometric ratios. Level 2 includes questions about ellipses, hyperbolas, polar coordinates and coordinates in three dimensions. The trigonometry questions on Level 2 place more emphasis on the properties and graphs of trigonometric functions, the inverse trigonometric functions, trigonometric equations and identities, and the laws of sines and cosines.
Data Analysis, Statistics and Probability. Both Level 1 and Level 2 include mean, median, mode, range, interquartile range, data interpretation and probability. Level 2 also includes standard deviation. Both include least-squares linear regression, but Level 2 also includes quadratic and exponential regression.
Seek advice from your high school math teacher if you are still unsure of which test to take. Keep in mind you can choose to take either test on test day, regardless of what test you registered for.
Please note that these testsThink about how you are going to solve the question before picking up your calculator. It may be that you only need the calculator for the final step or two and can do the rest in your test book or in your head. Don't waste time by using the calculator more than necessary. For about half of the questions, there's no advantage, or perhaps even a disadvantage, to using a calculator. For the other half of the questions, a calculator may be useful or necessary.
Read the question carefully so that you know what you are being asked to do. Sometimes a result that you may get from your calculator is NOT the final answer. If an answer you get is not one of the choices in the question, it may be that you didn't answer the question being asked. You should read the question again. It may also be that you rounded at an intermediate step in solving the problem, and that's why your answer doesn't match any of the choices in the question.
The answer choices are often rounded, so the answer you get might not match the answer in the test book. Since the choices are rounded, plugging the choices into the problem might not produce an exact answer.
Don't round any intermediate calculations. For example, if you get a result from the calculator for the first step of a solution, keep the result in the calculator and use it for the second step. If you round the result from the first step and the answer choices are close to each other, you might have a problem.
Bring a calculator that you are used to using. It may be a scientific or a graphing calculator, but if you're comfortable with both, bring a graphing calculator. The most important consideration is your comfort level with the calculator. Test day is not the time to start learning how to use a new calculator, even if it has more capabilities.
Verify that your calculator is in good working condition before you take the test. You may bring batteries and a backup calculator to the test center. Remember, no substitute calculators or batteries will be available at the test center. You can't share calculators with other test takers.
If you are taking the Mathematics Level 1 test, make sure your calculator is in degree mode ahead of time so you won't have to worry about it during the test.
If your calculator malfunctions at the test center, and you don't have a backup calculator, you must tell your test supervisor when the malfunction occurs. You can choose to cancel your scores on the test.
If you are using a calculator with large characters (one inch high or more) or a calculator with a raised display that might be visible to other test takers, you will be seated at the discretion of the test supervisor.
You may not use your calculator for sharing or exchanging, or removing part of a test book or any notes relating to the test from the test room. Such action may be grounds for dismissal or cancellation of scores or both. You do not have to clear your calculator's memory before or after taking the test.
Figures that accompany problems are intended to provide information useful in solving the problems. They are drawn as accurately as possible except when it is stated in a particular problem that the figure is not drawn to scale. Even when figures are not drawn to scale, the relative positions of points and angles may be assumed to be in the order shown. Also, line segments that extend through points and appear to lie on the same line may be assumed to be on the same line. The text "Note: Figure not drawn to scale." is included on the figure when degree measures may not be accurately shown and specific lengths may not be drawn proportionally
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Algebra II Workbook for Dummies (For Dummies)
by Mary Jane Sterling Publisher Comments
From radical problems to rational functions -- solve equations with ease Do you have a grasp of Algebra II terms and concepts, but can't seem to work your way through problems? No fear -- this hands-on guide focuses on helping you solve the many types of... (read more)
Geometry for Dummies (For Dummies)
by Mark Ryan Publisher Comments
Learning geometry doesn't have to hurt. With a little bit of friendly guidance, it can even be fun! Geometry For Dummies, 2nd Edition, helps you make friends with lines, angles, theorems and postulates. It eases you into all the principles and... (read more)
Algebra I for Dummies (For Dummies)
by Mary Jane Sterling Publisher Comments
The pain-free way to ace Algebra I Does the word polynomial make your hair stand on end? Let this friendly guide show you the easy way to tackle algebra. You'll get plain-English explanations of the basics — and the tougher stuff — in terms... (read more)
Algebra II for Dummies (For Dummies)
by Mary Jane Sterling Publisher Comments
Besides being an important area of math for everyday use, algebra is a passport to studying subjects like calculus, trigonometry, number theory, and geometry, just to name a few. To understand algebra is to possess the power to grow your skills and... (read more)
Algebra Workbook for Dummies (For Dummies)
by Mary Jane Sterling Publisher Comments
From signed numbers to story problems — calculate equations with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or worse yet, not know where to begin? No fear — this hands-on-guide focuses on... (read more)
The Humongous Book of Trigonometry Problems
by W. Michael Kelley About the Author
W. Michael Kelley is a former award-winning calculus teacher and the author of The Complete Idiot's Guide to Calculus, The Complete Idiot's Guide to Precalculus, and The Complete Idiot's Guide to Algebra.... (read more)
Geometry Workbook for Dummies (For Dummies)
by Mark Ryan Publisher Comments
Geometry is one of the oldest mathematical subjects in history. Unfortunately, few geometry study guides offer clear explanations, causing many people to get tripped up or lost when trying to solve a proof—even when they know the terms and concepts... (read more)
Functions Modeling Change: A Preparation for Calculus
by Eric Connally Publisher Comments
With its combination of concepts and skill-building, as well as the focus on functions, the new third edition better prepares readers for calculus. In order to make this complex subject more engaging, the authors incorporate the rule of four, superior... (read more)
Algebra I (Cliffs Study Solver)
by Mary Jane Sterling Publisher Comments
Algebra I is "the" fundamental subject that is the stepping stone to all other math. It is a core subject required of all students in order to graduate from high school.... (read more)
Barron's E-Z Precalculus (Barron's E-Z)
by Lawrence Leff Publisher Comments
An experienced math teacher breaks down precalculus into a series of easy-to-follow lessons designed for self-teaching and rapid learning. The book features a generous number of step-by-step demonstration examples as well as numerous tables, graphs, and... (read more)
Mathematics Made Simple 5TH Edition
by Abraham Sperling Publisher Comments
Brushing up on math has never been easier! Just about everyone can use some extra help improving or remembering basic math skills. Finally, all the information you need to master the basics, once and for all, is at your fingertips. Featuring several... (read more)
Mathematics-Grade 4:
by School Specialty Pub Publisher Comments
With Mathematics: A Step-By-Step Approach, Grade 4 Homework Booklet students will love building their mathematics skills while completing the fun activities in this great book Divided into four steps: addition and subtraction, multiplication, division... (read more)
Rapid Math Tricks & Tips: 30 Days to Number Power
by Edward H Julius Publisher Comments
Demonstrates a slew of time-saving tips and tricks for performing common math calculations. Contains sample problems for each trick, leading the reader through step-by-step. Features two mid-terms and a final exam to test your progress plus hundreds
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Understanding Intermediate Algebra: A - 6th edition
Summary: You'll have the confidence and knowledge to succeed in this course and any subsequent math course you take with UNDERSTANDING INTERMEDIATE ALGEBRA: A COURSE FOR COLLEGE STUDENTS. Hirsch and Goodman's gradual introduction of concepts, rules, and definitions through a wealth of illustrative examples (both numerical and algebraic) will help you compare and contrast related ideas and understand the sometimes-subtle distinctions among a variety of situations.
0495109029
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Basic College Mathematics With Early Integers - 2nd edition
Summary: The Bittinger Series changed the face of developmental education with the introduction of objective-based worktexts that presented math one concept at a time. This approach allowed students to understand the rationale behind each concept before practicing the associated skills and then moving on to the next topic. With this revision, Marv Bittinger continues to focus on building success through conceptual understanding, while also supporting students with quality applications, exerci...show moreses, and new review and study materials to help them apply and retain their knowledge951613417-5-0
$15.28 +$3.99 s/h
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Penntext Downingtown, PA
With CD! This is an INSTRUCTOR COPY. May have minimal notes/highlighting, minimal wear/tear. Please contact us if you have any Questions166.38 +$3.99 s/h
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The Best Textbooks Ypsilanti, MI
2010
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Mastering Essential Math Skills
This exercise book is an excellent resource to practice and review math skills you´ll need to establish a strong foundation and smooth transition into Algebra and other higher math courses. Workbooks are available for 4th – 5th grade and middle school / high school.
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Abstract:
Many visually impaired people have difficulty learning mathematics. The lack of higher math skills limits educational and job opportunities for many visually impaired people since numerous professions are requiring more technical skills. The objective of the Screen Reader Architecture Math System (SRAMS) is to design, develop, and test a computer based system which uses successful screen reading architecture to enable visually impaired students to independently manipulate math expressions in order to solve math problems. The rules of mathematics will be analyzed and combined with screen reading navigational logic to create a specialized system to assist the visually impaired student in interacting with math expressions and problems. This research is being conducted by Automated Functions, Inc. (AFI) who is a leader in the field of text and graphic based screen reader technology. AFI is expert in the development of portable low cost hardware, which will be used to create a low cost battery powered SRAMS unit. SRAMS will provide tools which make it easier for the visually impaired student to succeed in higher math (i.e., Algebra 1, Algebra 2, Trigonometry, Calculus). This system will provide the means for thousands of visually impaired people to succeed in higher math and therefore be qualified to fill the ever- increasing technical job and educational positions.
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Here's the perfect self-teaching guide to help anyone master differential equations, a common stumbling block for students looking to progress to advanced topics in both science and math
Covers First Order Equations, Second Order Equations and Higher, Properties, Solutions, Series Solutions, Fourier Series and Orthogonal Systems, Partial Differential Equations and Boundary Value Problems, Numerical Techniques, and more.
Perfect for a student going on to advanced analytical work in mathematics, engineering, and other fields of mathematical science.
About the author
Steven Krantz, Ph.D., is Chairman of the Mathematics Department at Washington University in St. Louis. An award-winning teacher and author, Dr. Krantz has written more than 45 books on mathematics, including Calculus Demystified, another popular title in this series. He lives in St. Louis, Missouri.
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A simple, visual guide to helping children understand maths with Carol Vorderman
Reduce the stress of studying maths and help your child with their homework, following Help Your Kids with Maths a unique visual guide which will demystify the subject for everyone.
Updated to include the latest changes to the UK National Curriculum... more...
When not used on a daily basis, basic math concepts are difficult to recall and use. When people plan to return to school, they must take entrance and placement exams with a significant math portion. Idiot's Guides: Basic Math and Pre-Algebra helps readers get back up to speed and relearn the primary concepts of mathematics, geometry, and pre-algebra... more...
This book contains the lectures presented at a conference held at Princeton University in May 1991 in honor of Elias M. Stein's sixtieth birthday. The lectures deal with Fourier analysis and its applications. The contributors to the volume are W. Beckner, A. Boggess, J. Bourgain, A. Carbery, M. Christ, R. R. Coifman, S. Dobyinsky, C. Fefferman, R.... more...
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Find a Garnet Valley can guide, but students must do the work.Introduction to the basics of symbolic representation and manipulation of variables. Liberal use of concrete examples. May require review of arithmetic concepts, including fractions and decimals
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Basic College Mathematics with Early IntegersNormal 0 false false false MicrosoftInternetExplorer4 Basic College Mathematics with Early Integersis a new addition to the Martin-Gay worktext series. This text is designed for a 1-semester basic math courses in which anearlyintroduction ofintegersis desired. Integers are introduced in chapter 2, and students continue to work with them throughout the text. This gives students ample opportunity to practice operations with integers and to become comfortable with them, prior to being introduced to algebra in chapter 7, Equations. The Whole Numbers; Integers and Introduction to Variables; Fractions; Decimals; Ratio, Proportion, and Measurement; Percent; Statistics and Probability; Equations; Geometry; Tables; The Bigger Picture; Exponents and Polynomials For all readers interested in basic college mathematics.
Tools to Help Students Succeed
ix
Additional Resources to Help You Succeed
xi
Preface
xiii
Applications Index
xxiii
The Whole Numbers
1
(110)
Tips for Success in Mathematics
2
(5)
Place Value and Names for Numbers
7
(9)
Adding Whole Numbers and Perimeter
16
(12)
Subtracting Whole Numbers
28
(11)
Rounding and Estimating
39
(8)
Multiplying Whole Numbers and Area
47
(13)
Dividing Whole Numbers
60
(16)
Integrated Review--Operations on Whole Numbers
74
(2)
An Introduction to Problem Solving
76
(11)
Exponents, Square Roots, and Order of Operations
87
(24)
Group Activity: Modeling Subtraction of Whole Numbers
97
(1)
Vocabulary Check
98
(1)
Highlights
98
(4)
Review
102
(7)
Test
109
(2)
Integers and Introduction to Variables
111
(66)
Introduction to Variables and Algebraic Expressions
112
(8)
Introduction to Integers
120
(10)
Adding Integers
130
(8)
Subtracting Integers
138
(10)
Integrated Review--Integers
146
(2)
Multiplying and Dividing Integers
148
(9)
Order of Operations
157
(20)
Group Activity: Magic Squares
164
(1)
Vocabulary Check
165
(1)
Highlights
165
(3)
Review
168
(5)
Test
173
(2)
Cumulative Review
175
(2)
Fractions
177
(107)
Introduction to Fractions and Mixed Numbers
178
(12)
Factors and Simplest Form
190
(13)
Multiplying and Dividing Fractions
203
(11)
Adding and Subtracting Like Fractions and Least Common Denominator
214
(15)
Integrated Review---Summary on Fractions and Operations on Fractions
227
(2)
Adding and Subtracting Unlike Fractions
229
(12)
Complex Fractions, Order of Operations, and Mixed Numbers
241
(9)
Operations on Mixed Numbers
250
(34)
Group Activity
267
(1)
Vocabulary Check
268
(1)
Highlights
268
(6)
Review
274
(6)
Test
280
(2)
Cumulative Review
282
(2)
Decimals
284
(79)
Introduction to Decimals
285
(12)
Adding and Subtracting Decimals
297
(12)
Multiplying Decimals and Circumference of a Circle
309
(9)
Dividing Decimals
318
(13)
Integrated Review---Operations on Decimals
329
(2)
Fractions, Decimals, and Order of Operations
331
(10)
Square Roots and the Pythagorean Theorem
341
(22)
Group Activity: Maintaining a Checking Account
349
(1)
Vocabulary Check
350
(1)
Highlights
350
(3)
Review
353
(5)
Test
358
(2)
Cumulative Review
360
(3)
Ratio, Proportion, and Measurement
363
(84)
Ratios
364
(10)
Proportions
374
(9)
Proportions and Problem Solving
383
(12)
Integrated Review---Ratio and Proportion
393
(2)
Length: U.S. and Metric Systems
395
(11)
Weight and Mass: U.S. and Metric Systems
406
(11)
Capacity: U.S. and Metric Systems
417
(9)
Conversions Between the U.S. and Metric Systems
426
(21)
Group Activity: Consumer Price Index
433
(1)
Vocabulary Check
434
(1)
Highlights
434
(4)
Review
438
(5)
Test
443
(2)
Cumulative Review
445
(2)
Percent
447
(66)
Percents, Decimals, and Fractions
448
(11)
Solving Percent Problems Using Equations
459
(8)
Solving Percent Problems Using Proportions
467
(10)
Integrated Review---Percent and Percent Problems
475
(2)
Applications of Percent
477
(10)
Percent and Problem Solving: Sales Tax, Commission, and Discount
487
(8)
Percent and Problem Solving: Interest
495
(18)
Group Activity: How Much Can You Afford for a House?
501
(1)
Vocabulary Check
502
(1)
Highlights
502
(3)
Review
505
(4)
Test
509
(2)
Cumulative Review
511
(2)
Statistics and Probability
513
(49)
Reading Pictographs, Bar Graphs, Histograms, and Line Graphs
514
(12)
Reading Circle Graphs
526
(9)
Integrated Review--Reading Graphs
533
(2)
Mean, Median, and Mode
535
(6)
Counting and Introduction to Probability
541
(21)
Group Activity
547
(1)
Vocabulary Check
548
(1)
Highlights
548
(3)
Review
551
(5)
Test
556
(4)
Cumulative Review
560
(2)
Introduction to Algebra
562
(54)
Variable Expressions
563
(9)
Solving Equations: The Addition Property
572
(7)
Solving Equations: The Multiplication Property
579
(7)
Integrated Review---Expressions and Equations
585
(1)
Solving Equations Using Addition and Multiplication Properties
586
(8)
Equations and Problem Solving
594
(22)
Group Activity: Modeling Equation Solving with Addition and Subtraction
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Saxon Math programs are designed and structured for immediate, measurable and long-lasting results. By employing a proven method of incremental development and continual review strategies, each piece of supplementary curriculum provides time to practice, process and learn beyond mastery.
This Math Intermediate 3 Written Practice Workbook contains exercises designed to refresh students' memories, deepen understanding of concepts, shift gears between different types of problems, and see how different math topics are related. (Workbook reprints exercises from the text with space for the student to show their work).
Product:
Saxon Math Intermediate 3: Written Practice Workbook
Author:
Hake
Prepared by:
Saxon Publishers
Edition Description:
Student
Binding Type:
Paperback
Media Type:
Book
Minimum Age:
8
Maximum Age:
8
Minimum Grade:
3rd Grade
Maximum Grade:
3rd Grade
Number of Pages:
240
Weight:
0.89 pounds
Length:
10.9 inches
Width:
8.3 inches
Height:
0.41 inches
Publisher:
Saxon Publishers
Publication Date:
March 2007
Subject:
Math
Curriculum Name:
Saxon
Learning Style:
Auditory, Kinesthetic, Visual
Teaching Method:
Traditional
There are currently no reviews for Saxon Math Intermediate 3: Written Practice Workbook.
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We used four other math curricula in the course of 15 years of homeschooling. I looked at Saxon Math when our oldest was in the 5th grade and didn't think I could do it. I was wrong. After years of struggling with other math programs I made the switch for our two smallest in 1st grade. We have used the Saxon Math for three years now and it has made a big difference in how we look at Math. I would highly recommend this curriculum. There is no DVD from Saxon for k-3, but starting in 4/5 you can purchase one. There is a company called Destination Math that has an online program that runs right with the Saxon Math. Just for the record, Math was not my best subject when I was in school. So those first years of teaching were a struggle for me to try to teach something I had a hard time understanding. Saxon Math has helped me while I have been teaching. It's very easy to follow and one of the few subjects that I actually follow very closely by the teacher's book.
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How much math do I need to know For MCAT ? (No calculator? No problem! )
A very common question I hear from my MCAT students is that "How much math do I need to know ?" On Test Day, no calculators allowed. The following tips will help you all identify what math skills you'll need.
MCAT Math:
The ability to perform arithmetic calculations, including proportion, ratio, percentage, and estimation of square root.
An understanding of fundamental topics in the following areas (at the level of second-year high school algebra coursework): exponentials and logarithms (natural and base ten); scientific notation; quadratic and simultaneous equations; graphic representations
of data and functions including terminology (abscissa, ordinate), slope or rate of change, reciprocals, and various scales (arithmetic, semi-log, and log-log).
The knowledge of the definitions of the basic trigonometric functions (sine, cosine, tangent); sin and cos values of of 0º, 90º, and 180º; the relationships between the lengths of sides of right triangles containing angles of 30º, 45º, and 60º; the inverse
trigonometric functions (arcsin, arccos, arctan).
The use of metric units; the ability to balance equations containing physical units. Conversion factors between metric and British systems will be provided when needed.
An understanding of relative magnitude of experimental error and of the effect of propagation of error; an understanding of reasonable estimates and the significant digits of a measurement.
The ability to calculate at an elementary level the mathematical probability of an event.
An understanding of vector addition, vector subtraction, and right-hand rule is required. Dot and cross products are not required.
The ability to calculate the arithmetic mean (average) and range of a set of numerical data; an understanding of the standard deviation as a measure of variability; an understanding of the general concepts of statistical association and correlation. Calculation
of statistics such as standard deviations and correlation coefficients is not required.
An understanding of calculus is not required.
This information is base on the AAMC publication on summer 2014. I do not anticipate that it changes for the new MCAT in 2015.
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+ By Mikhail Kulesh
The micro Mathematics is a powerful calculation software for your smartphone or tablet. It is primarily intended for the verification, validation, documentation and re-use of mathematical calculations.
This app is an enhanced mathematical calculator oriented around a worksheet, in which equations and expressions are created and manipulated in the same graphical format in which they are presented. You can calculate any formula you want and show the result in the text format and as a function plot.
This version supports the following types of objects that can be inserted in the worksheet, edited and calculated:
- A constant that can be presented as a number of an expression. There are built-in constants pi and e. - An interval variable that changes from given minimum value up to given maximum value with given step. - A function of one argument. The function expression can contain any other functions defined in the worksheet or built-in functions and operators in combination. - A graph for a function defined in the worksheet. In this version, only one function can be plotted within one graph object, but the number of graph objects is not limited. The plot parameters like a line width and line color, label number and label color, plot size can be adjusted. - A text field that contains an arbitrary text.
The app contains one worksheet that can be freely edited, several pages with examples and a documentation page. The whole worksheet and any example page can be stored on SD card (SD writing permission is needed).
This version has following mathematical limitations: it does not support complex numbers, functions of many arguments, special functions, vectors, matrices, integration, differentiation and many other things from high-level mathematics. This is the first version and it supports school-level of mathematical calculations only.
If you need more mathematics (like functions of many arguments, plots for several functions, contour plot, summation and product operations, derivative and definite integrals, logical operators), please look at micro Mathematics Plus version.
Keywords: mathematics, calculator, plotter, Mathcad®
Languages: English, Russian
P.S. In 1989, the author was impressed by Mathcad on DOS. Like Mathcad, this app enables live editing of typeset mathematical notation, combined with its automatic computations. Note that the files saved from Mathcad can NOT be opened in this app.
Mathcad® is a registered trademark of PTC Inc. or its subsidiaries in the U.S. and in other countries.
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This is a free online textbook designed for the Advanced Algebra instructor. According to the author, he "developed a set of...
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This is a free online textbook designed for the Advanced Algebra instructor. According to the author, he "developed a set of in-class assignments, homework and lesson plans, that work for me and for other people who have tried them. The complete set comprises three separate books that work together:•The Homework and Activities Book contains in-class and homework assignments that are given to the students day-by-day." "•The" target=״_blank״ Concepts Book provides conceptual explanations, and is intended as a reference or review guide for students; it is not used when teaching the class." •The" target=״_blank״ Teacher's Guide provides detailed lesson plans; it is your guide to how the author "envisioned these materials being used when I created them (and how I use them myself) " target=״_blank״ Instructors should note that this book probably contains more information than you will be able to cover in a single school year."
This is a free textbook by Boundless that is offered by Amazon for reading on a Kindle. Anybody can read Kindle books—even...
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This is a free textbook by Algebra textbook is a college-level, introductory textbook that covers the important subject of Algebra -- one of the basic building blocks of studies in higher mathematics. Boundless works with subject matter experts to select the best open educational resources available on the web, review the content for quality, and create introductory, college-level textbooks designed to meet the study needs of university students.This textbook covers:The Building Blocks of Algebra -- Real Numbers, Exponents, Scientific Notation, Order of Operations, Working with Polynomials, Factoring, Rational Expressions, Radical Notation and Exponents, Basics of Equation SolvingGraphs, Functions, and Models -- Graphing, Functions: An Introduction, Modeling Equations of Lines, Functions Revisited, Algebra of Functions, TransformationsFunctions, Equations, and Inequalities -- Linear Equations and Functions, Complex Numbers, Quadratic Equations, Functions, and Applications, Graphs of Quadratic Functions, Further Equation Solving, Working with Linear InequalitiesPolynomial and Rational Functions -- Polynomial Functions and Models, Graphing Polynomial Functions, Polynomial Division; The Remainder and Factor Theorems, Zeroes of Polynomial Functions and Their Theorems, Rational Functions, Inequalities, Variation and Problem SolvingExponents and Logarithms -- Inverse Functions, Graphing Exponential Functions, Graphing Logarithmic Functions, Properties of Logarithmic Functions, Growth and Decay; Compound InterestSystems of Equations and Matrices -- Systems of Equations in Two Variables, Systems of Equations in Three Variables, Matrices, Matrix Operations, Inverses of Matrices, Determinants and Cramer's Rule, Systems of Inequalities and Linear Programming, Partial FractionsConic Sections -- The Parabola, The Circle and the Ellipse, The Hyperbola, Nonlinear Systems of Equations and InequalitiesSequences, Series and Combinatorics -- Sequences and Series, Arithmetic Sequences and Series, Geometric Sequences and Series, Mathematical Inductions, Combinatorics, The Binomial Theorem, Probability'
" Algebra for College Students is designed to be used as an intermediate level text for students who have had some prior...
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" Algebra for College Students is designed to be used as an intermediate level text for students who have had some prior exposure to beginning algebra in either high school or college. This text explains the why's of algebra, rather than simply expecting students to imitate examples.״Please note that this site will try to sell supplements and you must create an account. However, there is no charge for the download of the textbook. As noted on the website, "Free access to the online book. Includes StudyBreak Ads (advertising placed in natural subject breaks)."
This is a free, online textbook that is also part of an online course. According to the author, "Analysis is the study of...
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This is a free, online textbook that is also part of an online course. According to the author, "Analysis is the study of limits. Anything in mathematics which has limits in it uses ideas of analysis. Some of the examples which will be important in this course are sequences, infinite series, derivatives of functions, and integrals. As you know from calculus, limits are the basis of understanding integration and differentiation, and, as you also know from calculus, these things are the basis of everything in the world you could ever need to know.״
This is a free, online textbook offered in conjunction with MIT's OpenCourseWare. "Over the last 100 years, the mathematical...
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This is a free, online textbook offered in conjunction with MIT's OpenCourseWare. "Over the last 100 years, the mathematical tools employed by physicists have expanded considerably, from differential calculus, vector algebra and geometry, to advanced linear algebra, tensors, Hilbert space, spinors, Group theory and many others. These sophisticated tools provide powerful machinery for describing the physical world, however, their physical interpretation is often not intuitive. These course notes represent Prof. Tisza's attempt at bringing conceptual clarity and unity to the application and interpretation of these advanced mathematical tools. In particular, there is an emphasis on the unifying role that Group theory plays in classical, relativistic, and quantum physics. Prof. Tisza revisits many elementary problems with an advanced treatment in order to help develop the geometrical intuition for the algebraic machinery that may carry over to more advanced problems.״
This is a free, online textbook that is a wikibook. "This book will help you learn how to do mathematics using Algebra. It...
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This is a free, online textbook that is a wikibook. "This book will help you learn how to do mathematics using Algebra. It has chapters (parts of the book) with lessons (parts of the chapter about one idea). A lesson has five parts: 1.Vocabulary - gives special words you need for the lesson. 2.Lesson - gives a new idea and how to use this idea. 3.Example Problems - gives the steps to do problems using the new idea. 4.Practice Games - gives places for amusement where you do problems. 5.Practice Problems - You do problems.״
This is a free, onlne textbook. According to the authors, "We are two college mathematics professors who grew weary of...
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This is a free, onlne textbook. According to the authors, "We are two college mathematics professors who grew weary of forcing our students to purchase expensive college algebra textbooks whose mathematical content has slowly degraded over the years. Our solution? Write our own. The twist? We made our college algebra book free and we distribute it as a .pdf file under the Creative Commons License. What's more, the LaTeX source code is also available under the same license.״
״Here are my online notes for my Algebra course that I teach here at Lamar University, although I have to admit that it's...
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״Here are my online notes for my Algebra course that I teach here at Lamar University, although I have to admit that it's been years since I last taught this course. At this point in my career I mostly teach Calculus and Differential Equations.Despite the fact that these are my "class notes", they should be accessible to anyone wanting to learn Algebra or needing a refresher for algebra. I've tried to make the notes as self contained as possible and do not reference any book. However, they do assume that you've has some exposure to the basics of algebra at some point prior to this. While there is some review of exponents, factoring and graphing it is assumed that not a lot of review will be needed to remind you how these topics work.״
״Here are my online notes for my differential equations course that I teach here at Lamar University. Despite the fact that...
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״Here are my online notes for my differential equations course that I teach here at Lamar University. Despite the fact that these are my "class notes", they should be accessible to anyone wanting to learn how to solve differential equations or needing a refresher on differential equations.I've tried to make these notes as self contained as possible and so all the information needed to read through them is either from a Calculus or Algebra class or contained in other sections of the notes.״
״At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a...
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״At first blush one might think that of all areas of mathematics certainly arithmetic should be the simplest, but it is a surprisingly deep subject. We assume that students have some familiarity with basic set theory, and calculus. But very little of this nature will be needed. To a great extent the book is self-contained. It requires only a certain amount of mathematical maturity. And, hopefully, the student's level of mathematical maturity will increase as the course progresses. Before the course is over students will be introduced to the symbolic programming language Maple which is an excellent tool for exploring number theoretic questions.״
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Description: This book aims to be an accessible introduction into the design and analysis of efficient algorithms. Throughout the book, we will explain only the most basic techniques, and we will give intuition for and an introduction to the rigorous mathematical methods needed to describe and analyze them.
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Mathematics > Mensuration
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This book presents a unified treatise of the theory of measure and integration. In the setting of a general measure space, every concept is defined precisely and every theorem is presented with a clear and complete proof with all the relevant details. Counter-examples are provided to show that certain conditions... more
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Do numbers leave you feeling -- well, numb? This compendium of fascinating data will bring figures to life while answering lots of questions you've wondered about. Discover which is higher in sodium: fast food french fries, the hamburger, or cherry pie; w... more
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"The Abyss of Time: An Architect's History of the Golden Section" explores the nature of the famous Fibonacci Series, its ability to produce the Golden Section proportions that regularly appear in ancient architecture, and other topics. (Architecture)... more
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This book is designed to be an introduction to the theories of measurement and meaningfulness, and not a comprehensive study of those topics. A major theme of this book is the psychophysical measurement of subjective intensity. This has been a subject of intense interest in psychology from the very beginning... more
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The 15-kilometer bike race follows Perimeter Path around the park. Mike's big brother and sister say he's too young to race. They say he shouldn't even try. But Mike knows that if he just gets a chance (and maybe a little help from a friend) he'll be able to make... more
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How do you eat a fraction? How many ways can you make 100? Young readers will learning math and reading when they discover these engaging books that explore beginning math skills in a grade-appropriate text format. The full-color, photo-illustrated books make a perfect addition to your math collection. This series... more
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This book presents the theory of integration over surfaces in abstract topological vector space. Applications of the theory in different fields, such as infinite dimensional distributions and differential equations (including boundary value problems), stochastic processes, approximation of functions, and calculus of variation on a Banach space, are treated in detail.... more
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Seeking to build on the momentum generated by the groundbreaking 2002 conference, 145 delegates from 42 countries gathered for the Second IUPAP International Conference on Women in Physics. During three days of energizing and inspirational discussions, posters, and networking, attendees assessed the progress that had been made in the intervening... more
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Measures are the subject of this unusual book, in which Robert Tavernor offers a fascinating account of the various measuring systems human beings have devised over two millennia. Tavernor urges us to look beyond the notion that measuring is strictly a scientific activity, divorced from human concerns. Instead, he sets... more
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-- Kids learn basic math concepts as they make crafts, play games, and work puzzles. -- The crafts have clear instructions with detailed illustrations, and text boxes on each page explain which math skills are being learned. -- Hands-on activities use basic household items. -- Youngsters learn sorting, ordering, matching,... more
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Math Monsters is an amusing and enlightening series that introduces young readers to mathematical concepts through the humorous antics of Addison, Mina, Multiplex, Split, and their friends. The books are based on the Math Monsters public television series, developed in cooperation with the National Council of Teachers of Mathematics (NCTM)... more
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For many years, this elementary treatise on advanced Euclidean geometry has been the standard textbook in this area of classical mathematics; no other book has covered the subject quite as well. It explores the geometry of the triangle and the circle, concentrating on extensions of Euclidean theory, and examining in detail many relatively recent theorems. Several hundred theorems and corollaries are formulated and proved completely; numerous others remain unproved, to be used by students as exercises. The author makes liberal use of circular inversion, the theory of pole and polar, and many other modern and powerful geometrical tools throughout the book. In particular, the method of "directed angles" offers not only a powerful method of proof but also furnishes the shortest and most elegant form of statement for several common theorems. This accessible text requires no more extensive preparation than high school geometry and trigonometry.
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The book assumes that you are familiar with simple geometrical concepts like congruence of triangles, parallelograms, circles and the most elementary theorems and constructions as can be found in Kiselev's book Kiselev's Geometry / Book I. Planimetry.Read more ›
If you enjoyed proofs in high school geometry, this will be a pleasure to work through. Without being bogged down by calculations or coordinates, this book presents some of the more famous (to contest participants) and advanced theorems of Euclidean geometry. This should be required for anyone working towards olympiad level geometry and any math team coaches. Be warned that, though still a minority in terms of the type of problems, this book has no constructions. For that, I recommend (albeit reluctantly due to a lack of better alternatives) Altshiller-Court's College Geometry. Previous reviewers are correct to say that most of the problems are just proofs of stated theorems, but that is what most higher math textbooks do. You'll be surprised to find, though, that even just those proofs are sufficient to cement the knowledge found in this cheap and small book in your head. For a variety of applications and more interesting exercises, a next step with this material (though more of a lateral move in terms of content and downwards move in terms of difficulty) is Coexeter's Geometry Revisited. This book assumes a STRONG background in high school geometry.
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Trade in The Heart of Mathematics: An Invitation to Effective Thinking for an Amazon Gift Card of up to £1.36, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more
Book Description--This text refers to an out of print or unavailable edition of this title.More About the Authors
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From the reviews of the second edition: "In this book, the reader will see that mathematics is a network of intriguing ideas a ] . It is a really nice book. a ] This attractive book contains a lot of well-known and interesting problems a ] . It is really true that the ideas presented in this book are some of the most fascinating and beautiful ones around. a ] The reader will have something to explore, to learn, to think, to enjoy, and to add new aspects to his view of everything." (Valentina DagienA, Zentralblatt MATH, Vol. 1065, 2005)
--This text refers to an out of print or unavailable edition of this title.
This is a fantastic introduction to all things mathematical. The emphasis is on mathematical *thinking* rather than methods, and numerous puzzles and props allow the maths to be really understood.
But... the book is an introduction to the methods of a professional mathematician, so the topics aren't algebra, trigonometry, calculus. Rather there are loads of interesting and challenging examples from knot theory, topology, chaos theory, equivalence classes, number theory, voting maths. Comes with 3D glasses, non transitive dice, CD-Rom etc.
Well worth the not-insubstantial price. Perhaps accompanied with the other expensive but excellent book Mathematics from the Birth Of Numbers, which is far more 'traditional' (lots of algebra, proof as well as history of maths).
The Heart of Mathematics is one of the best texts designed to improve mathematical thinking I have come across. It is packed with interesting problems and resources. I am not sure that I would have paid the new price, but the used copy that I purchased was much cheaper and in very good condition. I have only had the book for few weeks, but have already used a number of the tasks in my lessons (KS3 and 4). It has worked particularly well with getting some of the lowest ability students motivated and interested. They recently came to a lesson armed with problems that they had researched themselves to challenge me. This is a group that would normally be reluctant to do any work at all!
The Heart of Mathematics is an unconventional math survey aimed primarily at social science and humanities students. While students in "soft" majors are the primary intended audience, math majors and others who have already progressed beyond the introductory level are likely to find this book of great interest as well. The book gives readers a good feel for the variety of problems that mathematicians tackle. In fact, one of the book's great strengths is the range of topics it covers, from number theory and games, to topology, to chaos and fractals. It does this with little use of conventional mathematical notation or jargon, and the level of presentation is so elementary that the book can be "read" just as any non-technical book can be read. At the same time, the authors go to great lengths to encourage reader participation. Many hands-on demonstrations and experiments are provided, and the end-of-chapter exercises ask readers to discuss the material with others and write about their experiences. The topics presented are fascinating. I read this book on my vacation and found several passages to read to my wife and daughter almost every day. (This provided a lot of amusement for everyone when my 12-year-old daughter would solve problems in a few seconds that I had been pondering without much success.) The book's subtitle is "An Invitation to Effective Thinking," and the authors present problem-solving strategies that can be applied to problems within and outside the field of mathematics. While readers will no doubt be familiar with many of them already, it is difficult for me to imagine anyone who would not benefit from at least some of the strategies presented. The authors' writing is very informal with a lot of corny humor - possibly too much for a lot of people - but at the same time you do get a sense of the authors as good guys who know some important things and want to share the wealth. In summary, this is a most unusual and stimulating book. Highly recommended.
70 of 71 people found the following review helpful
5.0 out of 5 starsMath Appreciation, not computation or rigor23 Sep 2004
By Ted R. Shoemaker - Published on Amazon.com
Format:Hardcover|Verified Purchase
_The_Heart_Of_Mathematics_ was never intended as a traditional textbook to teach you how to calculate. If that is what you are looking for, you need a different book. Its value -- and this is the best book of its kind that I have found -- is in helping the reader gain an appreciation for mathematics. Its title could well have been _Math_Appreciation_ . It was most likely intended as a way of satisfying the "math requirement" for non-math majors who feel allergic to math.
I have read comments from several people debating the merits of this book. Perhaps it would help to inject an analogy into the conversation. Suppose you wanted to learn (or teach) music. One teacher chooses to teach her students how to play the piano; another has her students listen to CDs of great performances; another teaches his students how to read music; and another teaches the biographies of Beethoven and Mozart. Which of these teachers is right? Which kind of music do you want to learn?
The question itself is mistaken, if you think that it has exactly one correct answer. The best answer is: ALL OF THEM. The problem here is not in what any one of these approaches will teach, but in what it omits.
Now, translating back from the metaphor: I want my children to learn how to compute AND how to love math. Which is right? Both of them.
This book shows you how to have fun with math. If you or your students end up learning something, and wanting to learn more -- that's the idea.
102 of 107 people found the following review helpful
5.0 out of 5 starsWithout Any Doubt, One of the Best Math Books I've Ever Read28 May 2001
By Rawitat Pulam - Published on Amazon.com
Format:Hardcover
After browsing this book for a while in a book store somewhere near Tokyo station, I decided to buy it immediately, even it's price will about almost doubled the price I would get if I buy from internet. (I bought it for about $115). But, it turned out to be one of the best decision I'd ever made in my life as a book-worm. (Well, I do love reading :-) This book will change the way you look at the world, the way you look at yourself, the way you think, and so on. It is cleary one of the best books on Math I've ever read so far (I do love Math as well :) Instead of throwing you the formulas, the authors lead you to the Mathematical Thinking. How to think this, how to solve that. What Mathematic really is and what are its applications, and why it had been used that way. All explained through simple but fun-to-think example/problems. Using non-technical language, and almost no formula(!). In fact, almost everyone with little math background can enjoy this book. It doesn't seem to require anything more than middle school math. So, you can really see the pictures, the very big pictures, clearly in your mind, which is really importand. With the wide range of content, from Prime Numbers to Chaos, Fractals, Infinity, 4th Dimension, and more. The concept of each is well-explained, in a manner I mentioned earlier. This book will definitely help you to understand the making of "modern" science better. One more thing is: Mathematic is not something that far from our everyday life as some might think. It's the alternative way to view our everyday life. The way with reason, pattern, logic, relation betweening things, and more. (Well, I realized this before start reading this book. But for those who haven't realize, this book will open the door for you). Math can also be fun. (In fact, it IS .. That's why I love it :-) And this book tell you why, and how to find and feel those "fun" and enjoy Math. Without any complicated computation, and limited only by your imagination.
46 of 48 people found the following review helpful
4.0 out of 5 starsGreat adult self ed resource29 July 2004
By Julie Brennan - Published on Amazon.com
Format:Hardcover|Verified Purchase
The disturbing reviews indeed completely miss the point. The goal of this book is not to turn you into a mathematician. It is to help you appreciate what mathematics is.
I am planning on using this text for an adult self ed study group this fall. The goal is not to try to prove Cantor's method. You explore it and gain some understanding, but it isn't a mastery course that you come out of passing a test for, unless you are sitting in a classroom designed with that in mind, and the larger audience for this book is not in that narrow context. If you come out of it learning how to think mathematically, learning different ways to approach solving problems, learning that there is fun, beauty, art, order and sense to math, if you begin to *see* math in the world you live in, in nature, in ways you never noticed before - that is the goal. It is also threaded with history and the human drama that created math.
Both negative reviews were so poorly written and clearly missed the point that I dismissed them, but others I've recommended the book to have been confused, so I felt the need to respond.
I have also watched the video/DVD series these two authors put out through the Teaching Company, the Joy of Thinking, and I love what they are doing. Is every lecture perfect and resonating with everybody? No, but most resonate with most people. It certainly opened my eyes to things I never understood. Much of this book covers the same type of material.
Some people will find it more interesting than others, that is the nature of personal preference certainly. But the negative feedback indicates the book is flawed based on specific use in college classroom context, and it appears the reviewers did not understand the purpose of the book.
The four vs. five stars reflects the fact this is a first ed and could be just little more user friendly for lay people vs. college course users. I look forward to seeing the 2nd edition.
35 of 37 people found the following review helpful
5.0 out of 5 starsGets to the heart of mathematics!28 May 2004
By A Customer - Published on Amazon.com
Format:Hardcover
I disagree strongly with the previous reviewer who found the text "disturbing." That person and his/her student friend missed the point entirely! This is not a textbook aimed at the traditional recipes for solving sets of mathematical problems. Rather, it is a survey of mathematical thought from ancient to modern times and the astonishing aspect is that it is within the grasp of all students to comprehend it! For example, we don't just learn the Pythagorean formula for right triangles and apply it to specific problems. We discover with hands on clarity WHY Pythagoras' theorem is true! What could be more elegant that Euclid's easily understood proof that there are infinitely many prime numbers? Moreover, we get to see those abstract notions put to great use in encryption without which even amazon.com would not be the great success that it is! All of this is comprehensible to any student willing to read the text and to participate in classroom discussion. The authors nurture creative thinking throughout keeping students alert to and on the lookout for patterns while encouraging them to try new methods of attacking problems. This is how REAL mathematics works! Also, they make it clear that mathematics is not a closed subject having solved all number problems. They provide many examples of problems that took centuries to solve (Fermat's Last Theorem) along with some that have yet to be cracked (Goldbach's Conjecture). Things really start to get interesting when the text delves into the nature of infinity. The authors set this up very cleverly, first, with an early introduction of a simple and innocent looking game which is eventually used as a stepping stone into Cantor's proof and, second, with a highly visual analogy of numbers on a conveyor belt used to compare the cardinality of sets. Finally, they treat the student to an infinity of infinities! The student cannot help but grasp the essence of the great ideas and appreciate the thinking that yielded such marvelous concepts. The text introduces many more areas of fascinating mathematics some which were touched on in earlier reviews here. I particularly enjoyed the discussions of the fixed point theorem as well probability and statistics in the final chapter where the student sees the need to question statistical data (polls). The student will acquire an appreciation of both the power and limitations of statistical inference. Will the student leave the course laden with mathematical techniques and skills that will allow them to solve systems of partial differential equations or to model nonequilibrium chemical processes or to design the first interstellar space probe? Of course not. They will leave the course as better thinkers and with a much greater appreciation of mathematics!
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Saxon Math 87 Solutions Manual (7th Grade) Third Edition
Full step by step solutions to all lesson and investigation problems,
full step by step solutions to the 23 cumulative tests, answers to
supplemental practice problems, and facts practice problems.
Grade Level: 7
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Calculus Essentials For Dummies [NOOK Book] ...
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This Book critical concepts taught in a typical two-semester high school calculus class or a college level Calculus I course, from limits and differentiation to integration and infinite series. This guide is also a perfect reference for parents who need to review critical calculus concepts as they help high school students with homework assignments, as well as for adult learners headed back into the classroom who just needRelated Subjects
Meet the Author
Mark Ryan is the owner of The Math Center in Chicago, Illinois, where he teaches students in all levels of mathematics, from pre-algebra to calculus. He is the author of Calculus For Dummies and Geometry 26, 2013
Not a Good Book. Not Recommended at all.
1. Spends too much time on social relations about students to professors and other students rather than focus on the mathematics.
2. Formulas are small and difficult to see, very distracting.
3. Worst of all, e-book only goes to page 74 (attempts to turn the page and advance will not work, gives an error message that the activity reader is not responding). so the last 1/3 of the book is really not available.
4. There is some calculus value, but overall not a good book.Why??!!
Jeez this is a ccaucus book
1 out of 14 people found this review helpful.
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0387950699
9780387950693
Complex Analysis:The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing Ph.D. qualifying exams in complex analysis.
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Rent Complex Analysis 1st edition today, or search our site for Theodore W. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Springer.
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PREFACEThe study of real analysis is indispensible for a prospective graduate student of pure orapplied mathematics. It also has great value for any undergraduate student who wishesto go beyond the routine manipulations of formulas to solve standard problems, becauseit develops the ability to think deductively, analyze mathematical situations, and extendideas to a new context. In recent years, mathematics has become valuable in many areas,including economics and management science as well as the physical sciences, engineering,and computer science. Our goal is to provide an accessible, reasonably paced textbook inthe fundamental concepts and techniques of real analysis for students in these areas. Thisbook is designed for students who have studied calculus as it is traditionally presented inthe United States. While students find this book challenging, our experience is that seriousstudents at this level are fully capable of mastering the material presented here. The first two editions of this book were very well received, and we have taken painsto maintain the same spirit and user-friendly approach. In preparing this edition, we haveexamined every section and set of exercises, streamlined some arguments, provided a fewnew examples, moved certain topics to new locations, and made revisions. Except for thenew Chapter 10, which deals with the generalized Riemann integral, we have not addedmuch new material. While there is more material than can be covered in one semester,instructors may wish to use certain topics as honors projects or extra credit assignments. It is desirable that the student have had some exposure to proofs, but we do not assumethat to be the case. To provide some help for students in analyzing proofs of theorems,we include an appendix on "Logic and Proofs" that discusses topics such as implications,quantifiers, negations, contrapositives, and different types of proofs. We have kept thediscussion informal to avoid becoming mired in the technical details of formal logic. Wefeel that it is a more useful experience to learn how to construct proofs by first watchingand then doing than by reading about techniques of proof. We have adopted a medium level of generality consistently throughout the book: wepresent results that are general enough to cover cases that actually arise, but we do not strivefor maximum generality. In the main, we proceed from the particular to the general. Thuswe consider continuous functions on open and closed intervals in detail, but we are carefulto present proofs that can readily be adapted to a more general situation. (In Chapter 1 1we take particular advantage of the approach.) We believe that it is important to providestudents with many examples to aid them in their understanding, and we have compiledrather extensive lists of exercises to challenge them. While we do leave routine proofs asexercises, we do not try to attain brevity by relegating difficult proofs to the exercises.However, in some of the later sections, we do break down a moderately difficult exerciseinto a sequence of steps. In Chapter 1 we present a brief summary of the notions and notations for sets andfunctions that we use. A discussion of Mathematical Induction is also given, since inductiveproofs arise frequently. We also include a short section on finite, countable and infinite sets.We recommend that this chapter be covered quickly, or used as background material,returning later as necessary. v
vi PREFACE Chapter 2 presents the properties of the real number system lR. The first two sectionsdeal with the Algebraic and Order Properties and provide some practice in writing proofsof elementary results. The crucial Completeness Property is given in Section 2.3 as theSupremum Property, and its ramifications are discussed throughout the remainder of thischapter. In Chapter 3 we give a thorough treatment of sequences in IR and the associated limitconcepts. The material is of the greatest importance; fortunately, students find it rathernatural although it takes some time for them to become fully accustomed to the use of €.In the new Section 3.7, we give a brief introduction to infinite series, so that this importanttopic will not be omitted due to a shortage of time. Chapter 4 on limits of functions and Chapter 5 on continuous functions constitutethe heart of the book. Our discussion of limits and continuity relies heavily on the use ofsequences, and the closely parallel approach of these chapters reinforces the understandingof these essential topics. The fundamental properties of continuous functions (on intervals)are discussed in Section 5.3 and 5.4. The notion of a "gauge" is introduced in Section 5.5and used to give alternative proofs of these properties. Monotone functions are discussedin Section 5.6. The basic theory of the derivative is given in the first part of Chapter 6. This importantmaterial is standard, except that we have used a result of Caratheodory to give simplerproofs of the Chain Rule and the Inversion Theorem. The remainder of this chapter consistsof applications of the Mean Value Theorem and may be explored as time permits. Chapter 7, dealing with the Riemann integral, has been completely revised in thisedition. Rather than introducing upper and lower integrals (as we did in the previouseditions), we here define the integral as a limit of Riemann sums. This has the advantage thatit is consistent with the students first exposure to the integral in calculus and in applications;since it is not dependent on order properties, it permits immediate generalization to complexand vector-valued functions that students may encounter in later courses. Contrary topopular opinion, this limit approach is no more difficult than the order approach. It also isconsistent with the generalized Riemann integral that is discussed in detail in Chapter 10.Section 7.4 gives a brief discussion of the familiar numerical methods of calculating theintegral of continuous functions. Sequences of functions and uniform convergence are discussed in the first two sections of Chapter 8, and the basic transcendental functions are put on a firm foundation inSection 8.3 and 8.4 by using uniform convergence. Chapter 9 completes our discussion ofinfinite series. Chapters 8 and 9 are intrinsically important, and they also show how thematerial in the earlier chapters can be applied. Chapter 10 is completely new; it is a presentation of the generalized Riemann integral(sometimes called the "Henstock-Kurzweil" or the "gauge" integral). It will be new to manyreaders, and we think they will be amazed that such an apparently minor modification ofthe definition of the Riemann integral can lead to an integral that is more general than theLebesgue integral. We believe that this relatively new approach to integration theory is bothaccessible and exciting to anyone who has studied the basic Riemann integral. The final Chapter 1 1 deals with topological concepts. Earlier proofs given for intervalsare extended to a more abstract setting. For example, the concept of compactness is givenproper emphasis and metric spaces are introduced. This chapter will be very useful forstudents continuing to graduate courses in mathematics. Throughout the book we have paid more attention to topics from numerical analysisand approximation theory than is usual. We have done so because of the importance ofthese areas, and to show that real analysis is not merely an exercise in abstract thought.
PREFACE vii We have provided rather lengthy lists of exercises, some easy and some challenging.We have provided "hints" for many of these exercises, to help students get started toward asolution or to check their "answer". More complete solutions of almost every exercise aregiven in a separate Instructor s Manual, which is available to teachers upon request to thepublisher. It is a satisfying experience to see how the mathematical maturity of the studentsincreases and how the students gradually learn to work comfortably with concepts thatinitially seemed so mysterious. But there is no doubt that a lot of hard work is required onthe part of both the students and the teachers. In order to enrich the historical perspective of the book, we include brief biographicalsketches of some famous mathematicians who contributed to this area. We are particularlyindebted to Dr. Patrick Muldowney for providing us with his photograph of ProfessorsHenstock and Kurzweil. We also thank John Wiley & Sons for obtaining photographs ofthe other mathematicians. We have received many helpful comments from colleagues at a wide variety of institutions who have taught from earlier editions and liked the book enough to expresstheir opinions about how to improve it. We appreciate their remarks and suggestions, eventhough we did not always follow their advice. We thank them for communicating with usand wish them well in their endeavors to impart the challenge and excitement of learningreal analysis and "real" mathematics. It is our hope that they will find this new edition evenmore helpful than the earlier ones.February 24, 1999 Robert G. BartleYpsilanti and Urbana Donald R. Sherbert THE GREEK ALPHABET A ex Alpha N v Nu fJ � � Xi � B Beta r Gamma 0 0 Omicron 7r y � /) Delta Il Pi E e Epsilon P Rho a p Z � Zeta I; Sigma H 1/ Eta T r Tau e () Theta 1 v Upsilon I Iota <I> ({J Phi K K Kappa X X Chi A ).. Lambda II 1/1 Psi M J.i- Mu Q w Omega
To our wives, Carolyn and Janice, with our appreciation for their patience, support, and love.
CHAPTER 1 PRELIMINARIESIn this initial chapter we will present the background needed for the study of real analysis.Section 1.1 consists of a brief survey of set operations and functions, two vital tools for allof mathematics. In it we establish the notation and state the basic definitions and propertiesthat will be used throughout the book. We will regard the word "set" as synonymous withthe words "class", "collection", and "family", and we will not define these terms or give alist of axioms for set theory. This approach, often referred to as "naive" set theory, is quiteadequate for working with sets in the context of real analysis. Section 1 .2 is concerned with a special method of proof called Mathematical Induction.It is related to the fundamental properties of the natural number system and, though it isrestricted to proving particular types of statements, it is important and used frequently. Aninformal discussion of the different types of proofs that are used in mathematics, such ascontrapositives and proofs by contradiction, can be found in Appendix A. In Section 1 .3 we apply some of the tools presented in the first two sections of thischapter to a discussion of what it means for a set to be finite or infinite. Careful definitionsare given and some basic consequences of these definitions are derived. The importantresult that the set of rational numbers is countably infinite is established. In addition to introducing basic concepts and establishing terminology and notation,this chapter also provides the reader with some initial experience in working with precisedefinitions and writing proofs. The careful study of real analysis unavoidably entails thereading and writing of proofs, and like any skill, it is necessary to practice. This chapter isa starting point.Section 1.1 Sets and FunctionsTo the reader: In this section we give a brief review of the terminology and notation thatwill be used in this text. We suggest that you look through quickly and come back laterwhen you need to recall the meaning of a term or a symbol. x If an element is in a set A, we write X EA xand say that is a member of A, or that x belongs to A . If x is not in A, we write x ¢ A.If�very element of a set A also belongs to a set B , we say that A is a subset of B and write orWe say that a set A is a proper subset of a set B if A � B, but there is at least one elementof B that is not in A. In this case we sometimes write A C B. 1
2 CHAPTER 1 PRELIMINARlES1.1.1 Definition Two sets A and B are said to be equal. and we write A = B. if theycontain the same elements. Thus. to prove that the sets A and B are equal. we must show that A � B and B � A. A set is normally defined by either listing its elements explicitly. or by specifying aproperty that determines the elements of the set. If P denotes a property that is meaningfuland unambiguous for elements of a set S. then we write {x E S P(x)} :for the set of all elements x in S for which the property P is true. If the set S is understoodfrom the context. then it is often omitted in this notation. Several special sets are used throughout this book. and they are denoted by standardsymbols. (ytle will use the symbol : = to mean that the symbol on the left is being definedby the symbol on the right.)• The set of natural numbers N := {I. 2. 3 }. • . . .• The set of integers Z : = to. 1. -1.2, -2, · · . },• The set ofrational numbers Q : = {min : m, n E Z and n =I- OJ.• The set of real numbers R The set lR of real numbers is of fundamental importance for us and will be discussedat length in Chapter 2.1.1.2 Examples (a) The set {x E N : x2 - 3x + 2 = O}consists of those natural numbers satisfying the stated equation. Since the only solutions ofthis quadratic equation are x = 1 and x = 2, we can denote this set more simply by {I, 2}.(b) A natural number n is even if it has the form n = 2k for some k E N. The set of evennatural numbers can be written {2k : k E N},which is less cumbersome than {n E N : n = 2k, k E N}. Similarly, the set of odd naturalnumbers can be written {2k - 1 k E N}. : oSet OperationsWe now define the methods of obtaining new sets from given ones. Note that these setoperations are based on the meaning of the words "or", "and", and "not". For the union,it is important to be aware of the fact that the word "or" is used in the inclusive sense,allowing the possibility that x may belong to both sets. In legal terminology, this inclusivesense is sometimes indicated by "andlor".1.1.3 Definition (a) The union of sets A and B is the set AU B := {x : x E A or x E B} .
1.1 SETS AND FUNCTIONS 3(b) The intersection of the sets A and B is the set AnB := {x : x E A and x E B} .(c) The complement of B relative to A is the set AB := {x : x E A and x rt B} . A U B !IIID AB � Figure 1.1.1 (a) A U B (b) A n B (c) AB The set that has no elements is called the empty set and is denoted by the symbol 0.Two sets A and B are said to be disjoint if they have no elements in common; this can beexpressed by writing A n B = 0. To illustrate the method of proving set equalities, we will next establish one of theDeMorgan laws for three sets. The proof of the other one is left as an exercise.1.1.4 Theorem If A, B, C are sets, then(a) A(B U C) = (AB) n (AC),(b) A(B n C) = (AB) U (AC) .Proof. To prove (a), we will show that every element in A (B U C) is contained in both(AB) and (A C), and conversely. If x is in A(B U C), then x is in A, but x is not in B U C . Hence x is in A, but xis neither in B nor in C . Therefore, x is in A but not B, and x is in A but not C . Thus,x E AB and x E AC, which shows that x E (A B) n (AC). Conversely, if x E (AB) n (AC), then x E (AB) and x E (AC). Hence x E Aand both x rt B and x rt C . Therefore, x E A and x rt (B U C), so that x E A (B U C) . Since the sets (AB) n (A C) and A (B U C) contain the same elements, they areequal by Definition 1.1.1. Q.E.D. There are times when it is desirable to form unions and intersections of more than twosets. For a finite collection of sets {A I A 2 , .. . , An }, their union is the set A consisting ofall elements that belong to of the sets Ak , and their intersection consists of all at least oneel�ments that belong to of the sets Ak • all This is extended to an infinite collection of sets {A I A2 , ••• , An . . .} as follows. Theirunion is the set of elements that belong to of the sets An In this case we at least onewrite 00 U An {x : x E A n for some n E N} . n =1 :=
4 CHAPTER 1 PRELIMINARIESSimilarly, their intersection is the set of elements that belong to all of these sets An In thiscase we write 00 n An := {x : x E An for all n E N} . n=lCartesian ProductsIn order to discuss functions, we define the Cartesian product of two sets.1.1.5 Definition If A and B are nonempty sets, then the Cartesian product A x B of Aand B is the set of all ordered pairs (a, b) with a E A and b E B. That is, A x B := {(a, b) : a E A, bE B} . Thus if A = {l, 2, 3} and B = {I, 5}, then the set A x B is the set whose elements arethe ordered pairs (1, 1), (1, 5), (2, 1), (2, 5), (3, 1), (3, 5).We may visualize the set A x B as the set of six points in the plane with the coordinatesthat we have just listed. We often draw a diagram (such as Figure 1.1.2) to indicate the Cartesian product oftwo sets A and B . However, it should be realized that this diagram may be a simplification.For example, if A := {x E IR : 1 .:::: x .:::: 2} and B : = {y E IR : 0 .:::: y .:::: 1 or 2 .:::: y .:::: 3},then instead of a rectangle, we should have a drawing such as Figure 1.1 . 3 . We will now discuss the fundamental notion of afunction or a mapping. To the mathematician of the early nineteenth century, the word "function" meant adefinite formula, such as f (x) := x2 + 3x - 5, which associates to each real number xanother number f(x) . (Here, f(O) = -5, f(1) = - 1, f(5) = 35 .) This understandingexcluded the case of different formulas on different intervals, so that functions could notbe defined "in pieces". 3 AxB 2 B 1 (a , b) AxB I I b ------ I I I a A 2 Figure 1.1.2 Figure 1.1.3
1.1 SETS AND FUNCTIONS 5 As mathematics developed, it became clear that a more general definition of "function"would be useful. It also became evident that it is important to make a clear distinctionbetween the function itself and the values of the function. A revised definition might be: A function f from a set A into a set B is a rule of correspondence that assigns to each element x in A a uniquely determined element f (x) in B .But however suggestive this revised definition might be, there is the difficulty o f interpretingthe phrase "rule of correspondence". In order to clarify this, we will express the definitionentirely in terms of sets; in effect, we will define a function to be its graph. W hile this hasthe disadvantage of being somewhat artificial, it has the advantage of being unambiguousand clearer.1.1.6 Definition Let A and B be sets. Then a function from A to B is a set f of orderedpairs in A x B such that for each a E A there exists a unique E B withb (a, b) E f. (Inother words, if (a, b) E f and (a, b)E f, then b = b.) The set A of first elements of a function f is called the domain of f and is oftendenoted by D(f). The set of all second elements in f is called the range of f and isoften denoted by R(f). Note that, although D(f) = A, we only have R(f) � B. (SeeFigure 1.1.4.) The essential condition that: (a, b) E f (a, b) E f implies that b= b andis sometimes called the vertical line test. In geometrical terms it says every vertical linex = a with a E A intersects the graph of f exactly once. The notation f : A--o;B f i s a function from Ais often used to indicate that into B. W e will also say that f is amapping of A into B, or that f maps A into B . If (a, b) is an element in f, it is customaryto write b = f(a) o r sometimes 1-+ a b. r; r�EI �------�--� _________ A=DV) ___ a ________ �.1 Figure 1.1.4 A function as a graph
6 CHAPTER 1 PRELIMINARIESIf b = f(a), we often refer to b as the value of f at a, or as the image of a under f.Transformations and MachinesAside from using graphs, we can visualize a function as a transformation of the set DU) =A into the set RU) � B. In this phraseology, when E (a, b) f, we think of as taking f athe element from A and "transforming" or "mapping" it into an element = in b f(a)RU) � B . We often draw a diagram, such as Figure 1 . 1.5, even when the sets A and B arenot subsets of the plane. b =!(a) R(f) Figure 1.1.5 A function as a transfonnation There is another way of visualizing a function: namely, as a machine that acceptselements of DU) = A as inputs and produces corresponding elements of RU) � B asoutputs. f, If we take an element x E D U) and put it into then out comes the correspondingvaluef(x). f, If we put a different element y E DU) into then out comes ey ) which may for may not differ from f(x). If we try to insert something that does not belong to DU) f, finto we find that it is not accepted, for can operate only on elements from DU). (SeeFigure 1.1.6.) This last visualization makes clear the distinction between and f f (x): the first is the f xmachine itself, and the second is the output of the machine when is the input. Whereasno one is likely to confuse a meat grinder with ground meat, enough people have confusedfunctions with their values that it is worth distinguishing between them notationally. x t ! !(x ) Figure 1.1.6 A function as a machine
1.1 SETS AND FUNCTIONS 7Direct and Inverse ImagesLet f:A � B be a function with domain DC!) = A and range RC!) � B.1.1.7 Definition If E is a subset of A, then the direct image of E under f is the subsetf(E) of B given by f(E) := {f(x) : x E E } . ,If H is a subset of B, then the inverse image of H under f is the subset f - (H) of Agiven by f- (H) := {x E A : f(x) E H } .Remark The notation f- (H) used in this connection has its disadvantages. However,we will use it since it is the standard notation. E y, Thus, if we are given a set � A, then a point E B is in the direct image f(E) x, Eif and only if there exists at least one point E such that Y, = f(x ,). Similarly, given H x2a set � B, then a point is in the inverse image f - (H) if and only if Y : = 2 f(x2 )belongs to H. (See Figure 1 . 1 .7.)1.1.8 Examples (a) Let f : IR � IR be defined by f(x) :=x2 • Then the direct imageof the set E := {x : 0 � x � 2} is the set f(E) = {y : 0 � Y � 4}. If G := {y : 0 � Y � 4}, then the inverse image of G is the set f - (G) = {x : -2 �x � 2}. Thus, in this case, we see that f - C!(E» =1= E. On the other hand, we have f (J - (G ») = G. But if H : = {y : -1 � Y � I}, thenwe have f (J - (H») = {y : 0 � y � I } =1= H. A sketch of the graph of f may help to visualize these sets.(b) Let f : A � B, and let G, H be subsets of B . We will show that f- (G n H) � f - (G) n f- (H).For, if x E f - (G n H), then f(x) E G n H, so that f(x) E G and f(x) E H. But thisimplies that x E f - (G) and x E f - (H), whence x E f - (G) n f - (H). Thus the statedimplication is proved. [The opposite inclusion is also true, so that we actually have setequality between these sets; see Exercise 13.] 0 Further facts about direct and inverse images are given in the exercises. E f � H Figure 1.1.7 Direct and inverse images
8 CHAPTER 1 PRELIMINARIESSpecial Types of FunctionsThe following definitions identify some very important types of functions.1.1.9 Definition Let I : A -+ B be a function from A to B .(a) The function I i s said to be injective (or to b e one-one) if whenever x , i= x2 , then I (x,) i= l (x2 ) · If I is an injective function, we also say that I is an injection.(b) The function I is said to be surjective (or to map A onto B) if I (A) = B; that is, if the range R(f) = B. If I is a surjective function, we also say that I is a surjection.(c) If I is both injective and surjective, then I is said to be bijective. If I is bijective, we also say that I is a bijection.• In order to prove that a function I is injective, we must establish that: for all x" x2 in A , if I (x,) = l (x2 ) , then x, = x2• To do this we assume that I (x,) = l (x2) and show that x, = x2 . [In other words, the graph of I satisfies the first horizontal line test: Every horizontal b b line y = with E B intersects the graph I in at most one point.]• To prove that a function I is surjective, we must show that for any b E B there exists at least one x E A such that I (x) = b. [In other words, the graph of I satisfies the second horizontal line test: Every horizontal b b line y = with E B intersects the graph I in at least one point.]1.1.10 Example Let A := {x E lR : x i= l } and define / (x) := 2x/(x -l) for all x A. ETo show that I is injective, we take x, and x2 in A and assume that I (x,) = l (x2). Thuswe have 2x, 2x2 -- = --, x, - 1 x2 - 1which implies that x, (x2 - 1) = x2 (x, - 1), and hence x, = x2 . Therefore I is injective. 1) To determine the range of I, we solve the equation y = 2x/(x - for x in tenus ofy. We obtain x = y / (y - 2), which is meaningful for y i= 2. Thus the range of I is the setB : = { y E lR : y i= 2}. Thus, I is a bijection of A onto B. 0Inverse FunctionsIf I is a function from A into B, then I is a special subset of A x B (namely, one passingthe vertical line test.) The set of ordered pairs in B x A obtained by interchanging themembers of ordered pairs in I is not generally a function. (That is, the set I may not passboth of thehorizontal line tests.) However, if I is a bijection, then this interchange doeslead to a function, called the "inverse function" of I.1.1.11 Definition If I : A -+ B is a bijection of A onto B, then g := feb, a) E B x A: (a, b) E f}is a function on B into A. This function is called the inverse function of I, and is denotedby 1-. The function 1- is also called the inverse of I .
1 . 1 SETS AND FUNCTIONS 9 We can also express the connection between I and its inverse I - I by noting thatD(f) R(f- I ) = and R(f) D(f - = I) and that b = I(a) if and only if a = I - I (b).For example, we saw in Example 1.1.10 that the function I(x) : = x 2x I -is a bijection of A := {x E JR : x 1= I} onto the set B := {y E JR : y 1= 2}. The functioninverse to I is given by y I - I (y) := y - for y E B. -2 IRemark We introduced the notation I - (H) in Definition 1.1.7. It makes sense even if I does not have an inverse function. However, if the inverse function I - I does exist, then I- I (H) is the direct image of the set H � B under I- I .Composition of FunctionsIt often happens that we want to "compose" two functions I, g I (x) by first finding and gthen applying to get g (f (x)); however, this is possible only when I (x) belongs to the g.domain of In order to be able to do this for all I(x), we must assume that the range ofI g. is contained in the domain of (See Figure 1.1.8.)1.1.12 Definition If I A : � B and g : B �C, and if R(f) � D(g) = B, then thecomposite function g o I (note the orderl) is the function from A into C defined by (g 0 f)(x) := g(f(x)) for all x E A .1.1.13 Examples (a) The order of the composition must be carefully noted. For, let Iand g be the functions whose values at x E JR are given by I(x) := 2x and g(x) := 3x 2 1. -Since D(g) = JR and R(f) � JR = D(g), then the domain D(g 0 f) is also equal to JR , andthe composite function g 0 I is given by (g 0 f)(x) = 3 (2x) 2 1 = 12x 2 1. - - B A c I � gol Figure 1.1.8 The composition of f and g
10 CHAPTER 1 PRELIMINARIESOn the other hand, the domain of the composite function l o g is also JR, but (f g)(x) = 2(3x 2 - 1) = 6x 2 2. 0 -Thus, in this case, we have g I =1= l o g. 0(b) In considering g I, some care must be exercised to be sure that the range of 1 is 0contained in the domain of g. For example, if I(x) := l - x 2 and g(x ) : = ./X,then, since D(g) = {x : x � O}, the composite function g o I is given by the formula (g f)(x) = J1=7 0only for x E D(f) that satisfy I(x) � 0; that is, for x satisfying - 1 x 1 . � � We note that if we reverse the order, then the composition l o g is given by the formula (f o g)(x) = l - x,but only for those x in the domain D(g) = {x : x � O}. o We now give the relationship between composite functions and inverse images. Theproof is left as an instructive exercise.1.1.14 Theorem Let I:A --+ B and --+ g : B C be functions and let H be a subset ofC. Then we haveNote the reversal in the order of the functions.Restrictions of Functions _____________________If I : A --+ B is a function and if A l A, we can define a function II : A l --+ B by C II (x) := I(x) for x E A I •The function II is called the restriction of I to A I . Sometimes it is denoted by II = I I A I It may seem strange to the reader that one would ever choose to throw away a part of afunction, but there are some good reasons for doing so. For example, if I : JR --+ JR is thesquaring function: for x E JR, Ithen is not injective, so it cannot have an inverse function. However, if we restrict to Ithe set A l := {x : x �O}, then the restriction IIA I is a bijection of onto Al Therefore, A Ithis restriction has an inverse function, which is the positive square root function. (Sketcha graph.) Similarly, the trigonometric functions S(x) := x sin and C(x) := x cos are not injectiveon all of R However, by making suitable restrictions of these functions, one can obtainthe inverse sine and the inverse cosine functions that the reader has undoubtedly alreadyencountered.
12 CHAPTER 1 PRELIMINARIES19. Prove that if I : A � B is bijective and g : B � C is bijective, then the composite g 0 I is a bijective map of A onto C.20. Let I : A � B and g : B � C be functions. (a) Show that if g 0 I is injective, then I is injective. (b) Show that if g o I is surjective, then g is surjective.21. Prove Theorem 1.1.14.22. Let I, g be functions such that (g 0 f)(x) = x for all x E D(f) and (f 0 g)(y) = y for all y E D(g). Prove that g = 1-.Section 1.2 Mathematical InductionMathematical Induction is a powerful method of proof that is frequently used to establishthe validity of statements that are given in terms of the natural numbers. Although its utilityis restricted to this rather special context, Mathematical Induction is an indispensable toolin all branches of mathematics. Since many induction proofs follow the same formal linesof argument, we will often state only that a result follows from Mathematical Inductionand leave it to the reader to provide the necessary details. In this section, we will state theprinciple and give several examples to illustrate how inductive proofs proceed. We shall assume familiarity with the set of natural numbers: N := {l, 2, 3, . . . },with the usual arithmetic operations of addition and multiplication, and with the meaningof a natural number being less than another one. We will also assume the followingfundamental property of N.1.2.1 Well-Ordering Property of N Every nonempty subset of N has a least element. A more detailed statement of this property is as follows: If S is a subset of N and if m mS =f. 0, then there exists E S such that .:::: k for all k E S. On the basis of the Well-Ordering Property, we shall derive a version of the Principleof Mathematical Induction that is expressed in terms of subsets of N.1.2.2 Principle of Mathematical Induction Let S be a subset of N that possesses thetwo properties:(1) The number 1 E S.(2) For every k E N, if k E S, then k + 1 E S. Then we have S = N.Proof. Suppose to the contrary that S =f. N. Then the set NS is not empty, so by the m.Well-Ordering Principle it has a least element Since 1 E S by hypothesis (1), we know mthat > 1 . But this implies that m 1 is also a natural number. Since - m m 1 < and - msince is the least element in N such that m rt S, we conclude that m 1E - S. We now apply hypothesis (2) to the element k := m 1 in S, to infer that k + 1 = -(m + m 1 ) 1 = belongs to S. But this statement contradicts the fact that - m rt s. m Sincewas obtained from the assumption that NS is not empty, we have obtained a contradiction.Therefore we must have S = N. Q.E.D.
1.2 MATHEMATICAL INDUCTION 13 The Principle of Mathematical Induction is often set forth in the framework of properties or statements about natural numbers. If P(n) is a meaningful statement about n E N,then P(n) may be true for some values of n and false for others. For example, if PI (n) isthe statement: "n 2 = n", then PI (1) is true while PI (n) is false for all n > 1 , n E N. Onthe other hand, if P2 (n) is the statement: "n 2 > then P2 (1) is false, while P2 (n) is true I",for all n > 1, n E N. In this context, the Principle of Mathematical Induction can be formulated as follows.For each n E N, let P (n) be a statement about n. Suppose that:(I) P(1) is true.(2) For every k E N, if P (k) is true, then P (k + 1) is true. Then P(n) is true for all n E N. The connection with the preceding version of Mathematical Induction, given in 1 .2.2,is made by letting S : = {n E N : P(n) is true}. Then the conditions (1) and (2) of 1 .2.2correspond exactly to the conditions (1) and (2), respectively. The conclusion that S = Nin 1 .2.2 corresponds to the conclusion that P(n) is true for all n E N. In (2) the assumption "if P(k) is true" is called the induction hypothesis. In establishing (2), we are not concerned with the actual truth or falsity of P(k), but only withthe validity of the implication "if P(k), then P(k + I)". For example, if we consider thestatements P(n): "n = n + 5", then (2) is logically correct, for we can simply add 1 toboth sides of P(k) to obtain P(k + 1). However, since the statement P (1): "1 = is false, 6"we cannot use Mathematical Induction to conclude that n = n + 5 for all n E N. It may happen that statements P (n) are false for certain natural numbers but then aretrue for all n ::: no for some particular no. The Principle of Mathematical Induction can bemodified to deal with this situation. We will formulate the modified principle, but leave itsverification as an exercise. (See Exercise 12.)1.2.3 Principle of Mathematical Induction (second version) Let no E N and let P (n)be a statement for each natural number n ::: no. Suppose that:(1) The statement P (no) is true.(2) For all k ::: no the truth of P(k) implies the truth of P(k + 1). Then P (n) is true for all n ::: nO" Sometimes the number n o in (1) is called the base, since it serves as the starting point,and the implication in (2), which can be written P(k) =} P(k + 1), is called the bridge,since it connects the case k to the case k + 1 . The following examples illustrate how Mathematical Induction is used to prove assertions about natural numbers.1.2.4 Examples (a) For each n E N, the sum of the first n natural numbers is given by 1 + 2 + . . . + n = !n(n + 1). To prove this formula, we let S be the set of all n E N for which the formula i s true.We must verify that conditions ( 1) and (2) of 1 .2.2 are satisfied. If n = 1, then we have1 = ! . 1 . (1 + 1) so that 1 E S, and (1) is satisfied. Next, we assume that k E S and wishto infer from this assumption that k + 1 E S. Indeed, if k E S, then 1 + 2 + . . . + k = !k(k + 1).
16 CHAPTER 1 PRELIMINARIES 6. Prove that n3 + 5n is divisible by 6 for all n E N. 7. Prove that 52n- 1 is divisible by 8 for all n E N. 8. Prove that 5" - 4n - 1 is divisible by 16 for all n E N. 9. Prove that n3 + (n + 1)3 + (n + 2)3 is divisible by 9 for all n E N.10. Conjecture a formula for the sum 1/1 . 3 + 1/3 · 5 + . . . + 1/(2n - 1) (2n + 1), and prove your conjecture by using Mathematical Induction.1 1. Conjecture a formula for the sum of the first n odd natural numbers 1 + 3 + . . . + (2n - 1), and prove your formula by using Mathematical Induction.12. Prove the Principle of Mathematical Induction 1 .2.3 (second version).13. Prove that n < 2" for all n E N.14. Prove that 2" < n ! for all n � 4, n E N.15. 2n Prove that 2n - 3 :::: 2 - for all n � 5, n E N.16. 2 Find all natural numbers n such that n < 2" . Prove your assertion.17. Find the largest natural number m such that n 3 - n is divisible by m for all n E No Prove your assertion.18. Prove that 1/0 + 1/v2 + . . . + I/Jn > In for all n E N.19. Let S be a subset of N such that (a) 2k E S for all k E N, and (b) if k E S and k � 2, then k - 1 E S. Prove that S = N. xn x220. Let the numbers be defined as follows: X l := 1, := 2, and x"+2 �(xn+ l xn ) := + for all xn n E N. Use the Principle of Strong Induction (1.2.5) to show that 1 :::: :::: 2 for all n E N.Section 1.3 Finite and Infinite SetsWhen we count the elements in a set, we say "one, two, three, . . . ", stopping when wehave exhausted the set. From a mathematical perspective, what we are doing is defining abijective mapping between the set and a portion of the set of natural numbers. If the set issuch that the counting does not terminate, such as the set of natural numbers itself, then wedescribe the set as being infinite. The notions of "finite" and "infinite" are extremely primitive, and it is very likelythat the reader has never examined these notions very carefully. In this section we willdefine these terms precisely and establish a few basic results and state some other importantresults that seem obvious but whose proofs are a bit tricky. These proofs can be found inAppendix B and can be read later.1.3.1 Definition (a) The empty set 0 is said to have elements. 0(b) If n E N, n a set S is said to have elements if there exists a bijection from the set Nn := . , n} {l, 2, . . onto S.(c) A set S is said to be finite if it is either empty or it has n elements for some n E N.(d) A set S is said to be infinite if it is not finite. Since the inverse of a bijection is a bijection, it is easy to see that a set S has nelements if and only if there is a bijection from S onto the set { I , 2, . . . , n}. Also, since the ncomposition of two bijections is a bijection, we see that a set Sj has elements if and only
1 .3 FINITE AND INFINITE SETS 17 Sl S nif there is a bijection from onto another set that has elements. Further, a set is Tl Tl 2 T2finite if and only if there is a bijection from onto another set that is finite. It is now necessary to establish some basic properties of finite sets to be sure that thedefinitions do not lead to conclusions that conflict with our experience of counting. From nthe definitions, it is not entirely clear that a finite set might not have elements for morethan one n. value of Also it is conceivably possible that the set := N {I, } 2, 3, . . . might bea finite set according to this definition. The reader will be relieved that these possibilitiesdo not occur, as the next two theorems state. The proofs of these assertions, which use the Nfundamental properties of described in Section 1 .2, are given in Appendix B.1.3.2 Uniqueness Theorem If S is a finite set, then the number of elements in S is aunique number in N.1.3.3 Theorem The set N ofnatural numbers is an infinite set. The next result gives some elementary properties of finite and infinite sets. A m B1.3.4 Theorem (a) If is a set with elements and is a set with elements and ifnAnB = 0, then AUB m n has + elements. A(b) If is a set with mENelements and C�A is a set with 1 element, then is a A Cset withm 1 elements. -(c) If C is an infinite set and B is a finite set, then CB is an infinite set.Proof. (a) Let I be a bijection of Nm onto A, and let be a bijection of Nn onto gB. We define h on Nm+n by h(i) := I(i) for i = 1" " , m and h(i) := g(i - m) fori = m + 1 , , m + n. We leave it as an exercise to show that h is a bijection from Nm+n "onto A U B. The proofs of parts (b) and (c) are left to the reader, see Exercise 2. Q.E.D. It may seem "obvious" that a subset of a finite set is also finite, but the assertion mustbe deduced from the definitions. This and the corresponding statement for infinite sets areestablished next.1.3.5 Theorem Suppose that S and T are sets and that T � S. S T(a) If is a finite set, then is a finite set. T S(b) If is an infinite set, then is an infinite set.Proof. (a) If T = 0, we already know that T is a finite set. Thus we may suppose thatT #- 0. The proof is by induction on the number of elements in S. If S has 1 element, then the only nonempty subset T of S must coincide with S , so Tis a finite set. Suppose that every nonempty subset of a set with k elements is finite. Now let S bea set having k + 1 elements (so there exists a bijection I of Nk+ l onto S), and let T � S.IfJ (k + 1) rt T, we can consider T to be a subset of Sl := S{f(k + I)}, which has kelements by Theorem 1 .3.4(b). Hence, by the induction hypothesis, T is a finite set. On the other hand, if I (k + 1 ) E T, then Tl := T{f(k + I)} is a subset of Sl SinceSl has k elements, the induction hypothesis implies that Tl is a finite set. But this impliesthat T = Tl U {f(k + I)} is also a finite set.(b) This assertion is the contrapositive of the assertion in (a). (See Appendix A for adiscussion of the contrapositive.) Q.E.D.
18 CHAPTER 1 PRELIMINARIESCountable SetsWe now introduce an important type of infinite set.1.3.6 Definition (a) A set S is said to be denumerable (or countably infinite) if there exists a bijection of onto N S. S(b) A set is said to be countable if it is either finite or denumerable. S(c) A set is said to be uncountable if it is not countable. S From the properties of bijections, it is clear that is denumerable if and only if thereexists a bijection of onto S N. Also a set S1 is denumerable if and only if there exists a S1 S2bijection from onto a set that is denumerable. Further, a set is countable if and T1 T1 T2only if there exists a bijection from onto a set that is countable. Finally, an infinitecountable set is denumerable.1.3.7 Examples (a) The set E := {2n : n E N} even of natural numbers is denumerable, N Esince the mapping f : � defined by f (n) : = 2n for n E N, is a bijection of onto N E. Similarly, the set 0 : = {2n - 1 : n E N} odd of natural numbers is denumerable. all(b) The set Z of integers is denumerable. To construct a bijection of onto Z, we map 1 onto 0, we map the set of even natural N Nnumbers onto the set of positive integers, and we map the set of odd natural numbersonto the negative integers. This mapping can be displayed by the enumeration: Z= {O, 1 , - 1 , 2, -2, 3, -3, · · . }.(c) The union of two disjoint denumerable sets is denumerable. Indeed, if = A {a1 , a2 , a3 , · · ·} and B = {b1 , b2 , b3 , · · . }, we can enumerate the elements of AU B as: o1.3.8 Theorem The set N x N is denumerable.Informal Proof. Recall that x N N consists of all ordered pairs (m, n), where m, n E N.We can enumerate these pairs as: ( 1 , 1), ( 1 , 2) , (2, 1), (1, 3), (1, 4), · · · , (2, 2), (3, 1),according to increasing sum m + n, and increasing m. (See Figure 1. 3 .1.) Q.E.D. The enumeration just described is an instance of a "diagonal procedure", since wemove along diagonals that each contain finitely many terms as illustrated in Figure 1.3.1.While this argument is satisfying in that it shows exactly what the bijection of x � N N Nshould do, it is not a "formal proof , since it doesn t define this bijection precisely. (SeeAppendix B for a more formal proof.) As we have remarked, the construction of an explicit bijection between sets is oftencomplicated. The next two results are useful in establishing the countability of sets, sincethey do not involve showing that certain mappings are bijections. The first result may seemintuitively clear, but its proof is rather technical; it will be given in Appendix B.
1.3 FINITE AND INFINITE SETS 19 • • • ( 1 ,4 ) ( 2,4) Figure 1.3.1 The set N x N1.3.9 Theorem Suppose that S and T are sets and that T � S.(a) If S is a countable set, then T is a countable set.(b) If T is an uncountable set, then S is an uncountable set.1.3.10 Theorem The following statements are equivalent:(a) S is a countable set.(b) There exists a surjection of onto S. N(c) There exists an injection of S into N.Proof. h (a) ::::} (b) If S is finite, there exists a bijection of some set Nn onto S and wedefine H on N by H(k) := {���� for k = n., . . . , n, for k > 1Then H is a surjection of N onto S. If S is denumerable, there exists a bijection H of N onto S, which is also a surjectionof N onto S.(b) ::::}(c) If H is a surjection of N onto S, we define HI : S -+ N by letting HI (s) bethe least element in the set H - I (s) : = {n E N : H(n) = s}. To see that HI is an injectionof S into N, note that if s, t E S and nst : = HI (s) = HI (t), then s = H(nst) = t.(c) ::::}(a) If HI is an injection of S into N, then it is a bijection of S onto HI (S) � N.By Theorem 1 .3.9(a), HI (S) is countable, whence the set S is countable. Q.E.D.1.3.11 Theorem The setQ of all rational numbers is denumerable.Proof. The idea of the proof is to observe that the set Q+ of positive rational numbers iscontained in the enumeration: I 1 2 1 2 3 1 T 2 T 3 2 T 4which is another "diagonal mapping" (see Figure 1 .3.2). However, this mapping is not aninjection, since the different fractions ! and � represent the same rational number. N N To proceed more formally, note that since x is countable (by Theorem 1 .3.8),it follows from Theorem 1 .3.10(b) that there exists a surjection f of onto x N If N N.
20 CHAPTER 1 PRELIMINARIES 3 4 3 3 2 3 4 4 4 4 4 Figure 1.3.2 The set Q+g : x -+ N N Q+ is the mapping that sends the ordered pair (m, n) into the rational number having a representation m / n,then g is a surjection onto Q+ . Therefore, the compositiong 0 f is a surjection of N Q+ , onto and Theorem 1 .3.10 implies that Q+ is a countable set. Similarly, the set Q- of all negative rational numbers is countable. It follows as inExample 1 .3.7(b) that the set = Q Q- U {O} U Q+ is countable. Since contains Q it N,must be a denumerable set. Q.E.D. The next result is concerned with unions of sets. In view of Theorem 1 .3. 10, we neednot be worried about possible overlapping of the sets. Also, we do not have to construct abijection.1.3.12 Theorem If Am is a countable set for each m E N, then the union A : = U:= I Amis countable.Proof. For each m E N, let rpm be a surjection of N onto Am We define 1{1 N N -+ A : xby 1{I(m, n) := rpm (n).We claim that 1{1 i s a surjection. Indeed, if a E A, then there exists a least m E N such thata E Am whence there exists a least n E N such that a = rpm (n). Therefore, a = 1{I(m, n). x Since N N is countable, it follows from Theorem 1 .3. 10 that there exists a surjectionf:N -+ N N whence 1{1 f is a surjection of N onto A. Now apply Theorem 1 .3.10 x 0again to conclude that A is countable. Q.E.D.Remark A less formal (but more intuitive) way to see the truth of Theorem 1 .3.12 is toenumerate the elements of Am m E N, as: A l = {a ll a 12 , a n . . . }, A2 = {a21 , a22 , a23 . . . }, A 3 = {a3 1 , a32 , a33 . . . },We then enumerate this array using the "diagonal procedure":as was displayed in Figure 1 .3. 1 .
1.3 FINITE AND INFINITE SETS 21 The argument that the set Q of rational numbers is countable was first given in 1874by Georg Cantor (1845-1918). He was the first mathematician to examine the concept ofinfinite set in rigorous detail. In contrast to the countability of Q, he also proved the set lR.of real numbers is an uncountable set. (This result will be established in Section 2.5.) In a series of important papers, Cantor developed an extensive theory of infinite sets andtransfinite arithmetic. Some of his results were quite surprising and generated considerablecontroversy among mathematicians of that era. In a 1877 letter to his colleague RichardDedekind, he wrote, after proving an unexpected theorem, "I see it, but I do not believe it". We close this section with one of Cantor s more remarkable theorems.1.3.13 Cantors Theorem If A is any set, then there is no surjection of A onto the setpeA) of all subsets of A.Proof. Suppose that q; : -+ A peA) i s a surjection. Since q; (a) i s a subset o f A, either abelongs to q;(a) or it does not belong to this set. We let D := {a E A : a ¢ q; (a)}.Since D is a subset of A, if q; is a surjection, then D = q; (ao ) for some ao E A. We must have either ao E D or ao ¢ D. If ao E D, then since D = q; (ao ) we musthave ao E q;(ao ), contrary to the definition of D. Similarly, if ao ¢ D, then ao ¢ q; (ao ) sothat ao E D, which is also a contradiction. Therefore, q; cannot be a surjection. Q.E.D. Cantor s Theorem implies that there is an unending progression of larger and largersets. In particular, it implies that the collection p eN) of all subsets of the natural numbersN is uncountable.Exercises for Section 1.3 1. Prove that a nonempty set Tl is finite if and only if there is a bijection from Tl onto a finite set Tz . 2. Prove parts (b) and (c) of Theorem 1.3.4. 3. Let S := { I , 2} and T := {a, h, c}. (a) Determine the number of different injections from S into T. (b) Determine the number of different surjections from T onto S. 4. Exhibit a bijection between N and the set of all odd integers greater than 13. 5. Give an explicit definition of the bijection f from N onto Z described in Example 1.3.7(b). 6. Exhibit a bijection between N and a pr<lper subset of itself. 7. Prove that a set Tl is denumerable if and only if there is a bijection from Tl onto a denumerable set Tz . 8. Give an example of a countable collection of finite sets whose union is not finite. 9. Prove in detail that if S and T are denumerable, then S U T is denumerable.10. Determine the number of elements in P(S), the collection of all subsets of S, for each of the following sets: (a) S := {I, 2}, (b) S := { l , 2, 3}, (c) S := { l , 2, 3, 4}. Be sure to include the empty set and the set S itself in P(S).1 1 . Use Mathematical Induction to prove that if the set S has n elements, then P (S) has 2n elements.12. Prove that the collection F(N) of all finite subsets of N is countable.
CHAPTER 2 THE REAL NUMBERSIn this chapter we will discuss the essential properties of the real number system RAlthough it is possible to give a formal construction of this system on the basis of a moreprimitive set (such as the set N of natural numbers or the set Q of rational numbers), wehave chosen not to do so. Instead, we exhibit a list of fundamental properties associatedwith the real numbers and show how further properties can be deduced from them. Thiskind of activity is much more useful in learning the tools of analysis than examining thelogical difficulties of constructing a model for R The real number system can be described as a "complete ordered field", and wewill discuss that description in considerable detail. In Section 2.1, we first introduce the"algebraic" properties-{)ften called the "field" properties in abstract algebra-that arebased on the two operations of addition and multiplication. We continue the section withthe introduction of the "order" properties of JR and we derive some consequences of theseproperties and illustrate their use in working with inequalities. The notion of absolute value,which is based on the order properties, is discussed in Section 2.2. In Section 2.3, we make the final step by adding the crucial "completeness" property tothe algebraic and order properties of R It is this property, which was not fully understooduntil the late nineteenth century, that underlies the theory of limits and continuity andessentially all that follows in this book. The rigorous development of real analysis wouldnot be possible without this essential property. In Section 2.4, we apply the Completeness Property to derive several fundamentalresults concerning JR, including the Archimedean Property, the existence of square roots,and the density of rational numbers in R We establish, in Section 2.5, the Nested IntervalProperty and use it to prove the uncountability of R We also discuss its relation to binaryand decimal representations of real numbers. Part of the purpose of Sections 2.1 and 2.2 is to provide examples of proofs ofelementary theorems from explicitly stated assumptions. Students can thus gain experiencein writing formal proofs before encountering the more subtle and complicated argumentsrelated to the Completeness Property and its consequences. However, students who havepreviously studied the axiomatic method and the technique of proofs (perhaps in a courseon abstract algebra) can move to Section 2.3 after a cursory look at the earlier sections. Abrief discussion of logic and types of proofs can be found in Appendix A at the back of thebook.Section 2.1 The Algebraic and Order Properties of lRWe begin with a brief discussion of the "algebraic structure" of the real number system. Wewill give a short list of basic properties of addition and multiplication from which all otheralgebraic properties can be derived as theorems. In the terminology of abstract algebra, thesystem of real numbers is a "field" with respect to addition and multiplication. The basic22
2.1 THE ALGEBRAIC AND ORDER PROPERTIES OF lR 23properties listed in 2.1.1 are known as the field axioms. A binary operation associates witheach pair (a, b) a unique element B(a, b), but we will use the conventional notations ofa+b and a.b when discussing the properties of addition and multiplication.2.1.1 Algebraic Properties of JR On the set JR of real numbers there are two binary +operations, denoted by and . and called addition and multiplication, respectively. Theseoperations satisfy the following properties:(AI) a+b b+a = for all a, b in lR (commutative property of addition);(A2) (a + b) + c a + (b + c) for all a, b, c in JR (associative property of addition); =(A3) there exists an element 0 in JR such that 0 + a = a and a + 0 = a for all a in lR (existence ofa zero element);(A4) for each a in JR there exists an element - a in JR such that a + (-a) = 0 and (-a) + a = 0 (existence of negative elements);(MI) a · b = b · a for all a, b in JR (commutative property of multiplication);(M2) (a · b) . c = a . (b · c) for all a, b, c in JR (associative property ofmultiplication);(M3) there exists an element 1 in JR distinct from 0 such that 1 . a = a and a . 1 = a for all a in JR (existence of a unit element);(M4) for each a =1= 0 in lR there exists an element 1 Ia in JR such that a . ( l I a) = 1 and (lla) . a = 1 (existence of reciprocals);(D) a . (b + c) = (a . b) + (a . c) and (b + c) . a = (b · a) + (c . a) for all a, b, c in JR (distributive property of multiplication over addition). These properties should be familiar to the reader. The first four are concerned withaddition, the next four with multiplication, and the last one connects the two operations.The point of the list is that all the familiar techniques of algebra can be derived from thesenine properties, in much the same spirit that the theorems of Euclidean geometry can bededuced from the five basic axioms stated by Euclid in his Elements. Since this task moreproperly belongs to a course in abstract algebra, we will not carry it out here. However, toexhibit the spirit of the endeavor, we will sample a few results and their proofs. 0 1, We first establish the basic fact that the elements and whose existence were assertedin(A3) and (M3), 0 are in fact unique. We also show that multiplication by always resultsinO.2.1.2 Theorem (a) If and are elements in JR with z a z + a = a, then z = O.(b) lfu and are elements in JR with b =1= 0 = then u . b b, u = 1.(c) If E JR, then a = o.a.0Proof. (A3), (A4), (A2), the hypothesis z + a = a, and (A4), we get (a) Using Z = z + O = z + (a + (-a » = (z + a) + (-a) = a + (-a) = O.(b) Using (M3), (M4), (M2), the assumed equality u . b = b, and (M4) again, we get u = u . 1 = u . (b . (lIb» = (u . b) . (lIb) = b . (lIb) = 1 .(c) We have (why?) a + a 0 = a . 1 + a 0 = a . ( l + 0) = a . 1 = a. · ·Therefore, we conclude from (a) that a · 0 = O. Q.E.D.
24 CHAPTER 2 THE REAL NUMBERS We next establish two important properties of multiplication: the uniqueness of reciprocals and the fact that a product of two numbers is zero only when one of the factors iszero.2.1.3 Theorem (a) If i= 0 and b in JR. are such that a a . b = I, then b = Ila .(b) b = 0, then either = 0 or b = o. lia · a (a) Using (M3), (M4), (M2), the hypothesis a . b = I, and (M3), we have 4Proof. b = 1 . b = « l la) . a) . b = (lla) . (a . b) = (1Ia) . 1 = Ila .(b) It suffices to assume a i= 0 and prove that b = O. (Why?) We multiply a . b by Iia andapply (M2), (M4) and (M3) to get (lla) . (a . b) = ((1la) . a) . b = 1 . b = b.Since a . b = 0, by 2.1.2(c) this also equals (1Ia) . (a . b) = (1Ia) · 0 = o.Thus we have b = O. Q.E.D. These theorems represent a small sample of the algebraic properties of the real numbersystem. Some additional consequences of the field properties are given in the exercises. The operation of subtraction is defined by a - b := a + ( - b) for a, b in lR.. Similarly,division is defined for a, b in JR. with b i= 0 by alb : = a . (lIb). In the following, wewill use this customary notation for subtraction and division, and we will use all thefamiliar properties of these operations. We will ordinarily drop the use of the dot to indicatemultiplication and write ab for a . b. Similarly, we will use the usual notation for exponentsand write a2 for aa, a 3 for (a 2 )a ; and, in general, we define an+ 1 : = (an )a for n E N. Weagree to adopt the convention that a l = a. Further, if a i= 0, we write aD = 1 and a - I for1 Ia, and if n E N, we will write a -n for (I Iat , when it is convenient to do so. In general,we will freely apply all the usual techniques of algebra without further elaboration.Rational and Irrational NumbersWe regard the set N of natural numbers as a subset of JR., by identifying the natural numbern E N with the n-fold sum of the unit element 1 E lR.. Similarly, we identify 0 E Z with thezero element of 0 E JR., and we identify the n-fold sum of -I with the integer -no Thus,we consider N and Z to be subsets of lR.. Elements of JR. that can be written in the form bIa where a, b E Z and a i= 0 are calledrational numbers. The set of all rational numbers in JR. will be denoted by the standardnotation Q. The sum and product of two rational numbers is again a rational number (provethis), and moreover, the field properties listed at the beginning of this section can be shownto hold for Q. The fact that there are elements in JR. that are not in Q is not immediately apparent.In the sixth century B .C. the ancient Greek society of Pythagoreans discovered that thediagonal of a square with unit sides could not be expressed as a ratio of integers. In viewof the Pythagorean Theorem for right triangles, this implies that the square of no rationalnumber can equal 2. This discovery had a profound impact on the development of Greekmathematics. One consequence is that elements of JR. that are not in Q became knownas irrational numbers, meaning that they are not ratios of integers. Although the word
2. 1 THE ALGEBRAIC AND ORDER PROPERTIES OF R 25"irrational" in modem English usage has a quite different meaning, we shall adopt thestandard mathematical usage of this term. We will now prove that there does not exist a rational number whose square is 2. In theproof we use the notions of even and odd numbers. Recall that a natural number is even ifit has the form 2n for some n E N, and it is odd if it has the form 2n - 1 for some n E N.Every natural number is either even or odd, and no natural number is both even and odd.2.1.4 Theorem There does not exist a rational number r such that r 2 = 2.Proof. Suppose, on the contrary, that p and q are integers such that (p jq) 2 = 2. Wemay assume that p and q are positive and have no common integer factors other than 1.(Why?) Since p2 = 2q 2 , we see that p2 is even. This implies that p is also even (becauseif p = 2n - 1 is odd, then its square p2 = 2(2n 2 - 2n + 1) - 1 is also odd). Therefore,since p and q do not have 2 as a common factor, then q must be an odd natural number. Since p is even, then p = 2m for somem E N, andhence 4m 2 = 2q 2 , so that 2m 2 = q 2 .Therefore, q 2 is even, and it follows from the argument in the preceding paragraph that qis an even natural number. Since the hypothesis that (pjq) 2 = 2 leads to the contradictory conclusion that q isboth even and odd, it must be false. Q.E.D.The Order Properties of JR.The "order properties" of JR. refer to the notions of positivity and inequalities between realnumbers. As with the algebraic structure of the system of real numbers, we proceed byisolating three basic properties from which all other order properties and calculations withinequalities can be deduced. The simplest way to do this is to identify a special subset of JR.by using the notion of "positivity".2.1.5 The Order Properties of JR. There is a nonempty subset lP of JR., called the set ofpositive real numbers, that satisfies the following properties:(i) If a, b belong to lP, then a + b belongs to lP.(ii) If a, b belong to lP, then ab belongs to lP.(iii) If a belongs to JR., then exactly one of the following holds: a E lP, a = 0, -a E lP. The first two conditions ensure the compatibility of order with the operations of addition and multiplication, respectively. Condition 2. 1.5 (iii) is usually called the TrichotomyProperty, since it divides JR. into three distinct types of elements. It states that the set{-a : a E P } of negative real numbers has no elements in common with the set lP ofpositive real numbers, and, m�eover, the set JR. is the union of three disjoint sets. If a E lP, we write a > 0 and say that a is a positive (or a strictly positive) real number.If a E lP U {O}, we write a � 0 and say that a is a nonnegative real number. Similarly, if-a E lP, we write a < 0 and say that a is a negative (or a strictly negative) real number.If .-a E lP U {OJ, we write a � 0 and say that a is a nonpositive real number. The notion of inequality between two real numbers will now be defined in terms of theset lP of positive elements.2.1.6 Definition Let a, b be elements of R(a) If a - b E lP, then we write a > b or b < a.(b) If a - b E lP U {OJ, then we write a � b or b � a.
26 CHAPTER 2 THE REAL NUMBERS The Trichotomy Property 2. 1 .5 (iii) implies that for a, b E lR exactly one of the following will hold: a > b, a = b, a < b.Therefore, if both a S b and b S a, then a = b . For notational convenience, we will write a<b<cto mean that both a < b and b < c are satisfied. The other "double" inequalities a S b < c,a S b S c, and a < b c are defined in a similar manner. s To illustrate how the basic Order Properties are used to derive the "rules of inequalities",we will now establish several results that the reader has used in earlier mathematics courses.2.1.7 Theorem Let a, b, c be any elements of R(a) If a > b and b > c, then a > c.(b) If a > b, then a + c > b + c.(c) Ifa > b and c > 0, then ca > cb. Ifa > b andc < 0, then ca < cb.Proof. (a) If a - b E lP and b - c E lP, then 2. 1.5(i) implies that (a - b) + (b - c) =a - c belongs to lP. Hence a > c.(b) If a - b E lP, then (a + c) - (b + c) = a - b is in lP. Thus a +c > b + c.(c) If a - b E lP and c E lP, then ca - cb = c(a - b) is in lP by 2. 1.5(ii). Thus ca > cbwhen c > O. On the other hand, if c < 0, then -c E lP, so that cb - ca = (-c) (a - b) is in lP. Thuscb > ca when c < O. Q.E.D. It is natural to expect that the natural numbers are positive real numbers. This propertyis derived from the basic properties of order. The key observation is that the square of anynonzero real number is positive.2.1.8 Theorem (a) Ifa E lR nd a i= 0, then a 2 > o. a(b) I > o.(c) Ifn E N, then n > o.Proof. (a) By the Trichotomy Property, if a i= 0, then either a E lP or -a E lP. If a E lP,then by 2.1.5(ii), a 2 = a . a E lP. Also, if -a E lP, then a2 = (-a)( -a) E lP. We concludethat if a i= 0, then a 2 > O.(b) Since 1 = 1 2 , it follows from (a) that 1 > O.(c) We use Mathematical Induction. The assertion for n = 1 is true by (b). If we suppose theassertion is true for the natural number k, then k E lP, and since 1 E lP, we have k + 1 E lPby 2. 1.5(i). Therefore, the assertion is true for all natural numbers. Q.E.D. It is worth noting that no smallest positive real number can exist. This follows byobserving that if a > 0, then since ! > 0 (why?), we have that 0 < !a < a.
2.1 THE ALGEBRAIC AND ORDER PROPERTIES OF lR 27Thus if it is claimed that a is the smallest positive real number, we can exhibit a smallerpositive number !a. This observation leads to the next result, which will be used frequently as a method ofproof. For instance, to prove that a number a :::: 0 is actually equal to zero, we see that itsuffices to show that a is smaller than an arbitrary positive number.2.1.9 Theorem If a E lR is such that 0 � a for every > 0, then a = O. < £ £Proof. Suppose to the contrary that a > O. Then if we take : = !a, we have 0 £0 < £0 < a.Therefore, it is false that a for every > 0 and we conclude that a = O. < £ £ Q.E.D.Remark It is an exercise to show that if a E lR is such that 0 � a � £ for every £ > 0,then a = O. The product of two positive numbers is positive. However, the positivity of a productof two numbers does not imply that each factor is positive. The correct conclusion is givenin the next theorem. It is an important tool in working with inequalities.2.1.10 Theorem If ab > 0, then either(i) a > 0 and b > 0, or(ii) a 0 and b O. < <Proo f. First we note that ab > 0 implies that a "I 0 and b "I O. (Why?) From the Trichotomy Property, either a > 0 or a O. If a > 0, then Ija > 0 (why?), and therefore <b = Oja)(ab) > O. Similarly, if a 0, then Ija 0, so that b = Oja)(ab) O. Q.E.D. < < <2.1.11 Corollary If ab 0, then either <(i) a 0 and b > 0, or <(ii) a > 0 and b O. <Inequalities ____________________________We now show how the Order Properties presented in this section can be used to "solve"certain inequalities. The reader should justify each of the steps.2.1.12 Examples (a) Determine the set A of all real numbers x such that 2x + 3 � 6 . We note that we havet xEA 2x + 3 � 6 2x � 3 X�3 2Therefore A = { x E lR x � �}.: :(b) Determine the set B : = { x E lR x 2 + x > 2 } . We rewrite the inequality so that Theorem 2.1.10 can be applied. Note that xEB x2 + X 2 > 0 ¢::=> - (x - l)(x + 2) > O.Therefore, we either have (i) x - I > 0 and x + 2 > 0, or we have (ii) x - I 0 and <x + 2 O. In case (i) we must have both x > 1 and x > -2, which is satisfied if and only < <==>tThe symbol should be read "if and only if.
28 CHAPTER 2 THE REAL NUMBERSif x > 1 . In case (ii) we must have both x < 1 and x < -2, which is satisfied if and onlyif x <: -2. We conclude that B = {x E lR. : x > I} U {x E lR. : x < -2}.(c) Detennine the set { C := x E lR. : 2x+ 21 < x + I} . We note that 2x + 1 x-I x E C {:::::::} -- - 1 < 0 {:::::::} -- < 0. x+2 x+2Therefore we have either (i) x - I < 0 and x + 2 > 0, or (ii) x - I > 0 and x + 2 < o.(Why?) In case (i) we must have both x < 1 and x > -2, which is satisfied if and only if-2 < x < 1 . In case (ii), we must have both x > 1 and x < -2, which is never satisfied. We conclude that C = {x E lR. : -2 < x < 1 }. D The following examples illustrate the use of the Order Properties of lR. in establishingcertain inequalities. The reader should verify the steps in the arguments by identifying theproperties that are employed. It should be noted that the existence of square roots of positive numbers has not yetbeen established; however, we assume the existence of these roots for the purpose of theseexamples. (The existence of square roots will be discussed in Section 2.4.)2.1.13 Examples (a) Let a ::: 0 and b ::: O. Then(1)We consider the case where a > 0 and b > 0, leaving the case a = 0 to the reader. It followsfrom 2. 1 .5(i) that a + b > O. Since b2 - a2 = (b - a)(b + a), it follows from 2.1.7(c) thatb - a > 0 implies that b2 - a2 > O. Also, it follows from 2.1 . 10 that b2 - a2 > 0 impliesthat b - a > O. If a > 0 and b > 0, then ya > 0 and ,.jb > O. Since a = (ya) 2 and b = (.Jb) 2 , thesecond implication is a consequence of the first one when a and b are replaced by ya and,.jb, respectively. We also leave it to the reader to show that if a ::: 0 and b ::: 0, then(1)(b) If a and b are positive real numbers, then their arithmetic mean is 4 (a + b) and theirgeometric mean is .j(J}. The Arithmetic-Geometric Mean Inequality for a, b is(2)with equality occurring if and only if a = b. To prove this, note that if a > 0, b > 0, and a =1= b, then ya > 0, ,.jb > 0 and ya =1=,.jb. (Why?) Therefore it follows from 2.1 .8(a) that (ya - ,.jb) 2 > O. Expanding thissquare, we obtain a - 2M + b > 0,whence it follows that M < 4 (a + b) .
30 CHAPTER 2 THE REAL NUMBERS 6. Use the argument in the proof of Theorem 2.1.4 to show that there does not exist a rational number s such that sZ = 6. 7. Modify the proof of Theorem 2. 1.4 to show that there does not exist a rational number such that = 3. tZ t 8. (a) Show that if x, y are rational numbers, then x + y and xy are rational numbers. (b) Prove that if x is a rational number and y is an irrational number, then x + y is an irrational number. If, in addition, x =1= 0, then show that xy is an irrational number. 9. Let K := {s + : s, (a) If x l Xz Et./i t E Q}. E K, then xl + Show that K satisfies the following: XzK and xlxZ K. E E (b) If x =1= 0 and x K , then l /x K.E (Thus the set K is a subfield of lR. With the order inherited from R, the set K is an ordered field Q that lies between and R).10. (a) If a < b and e .:::: d, prove that a + e < + d. b (b) If 0 < a < b and 0 .:::: e .:::: d, prove that 0 .:::: ae .:::: d. b1 1 . (a) Show that if a > 0, then l /a > 0 and l/(l/a) = a. b, (b) Show that if a < then a < � (a + < b) b.12. Let a, b, e, d be numbers satisfying 0 < a < b and e < d < O. Give an example where ae < bd, and one where bd < ae.13. = If a, b E R, show that a Z bZ 0 if and only if a 0 and b O. + = =14. If 0 .:::: a < b, show that aZ .:::: ab < bZ• Show by example that it does not follow that aZ < ab < bZ•15. If 0 < a < b, show that (a) a < ,;aij < b, and (b) l /b < l /a.16. Find all real numbers x that satisfy the following inequalities. (a) x Z > 3x 4, + (b) 1 < XZ < 4, (c) l /x < x, (d) l /x < xZ•17. 8 8 Prove the following form of Theorem 2.1.9: If a E R is such that 0 .:::: a .:::: for every > 0, then a O. =18. 8 8. Let a, b E R, and suppose that for every > 0 we have a :::: b + Show that a .:::: b.19. Prove that [ � (a + b)] z � (aZ + bZ) for all a, b E Show that equality holds if and only if .:::: lR. a = b.20. (a) If O < e < l , show that O < ez < e < 1 . (b) If 1 < e, show that 1 < e < eZ •21. (a) Prove there is no n (b) EN such that 0 < n < 1. (Use the Well-Ordering Property of Prove that no natural number can be both even and odd. N.)22. (a) If e > 1, show that en 2: e for all nE N, and that en > e for n > 1. E N, (b) If 0 < e < 1, show that en .:::: e for all n and that en < e for n > 1.23. If a > 0, b > 0 and n E N, show that a < b if and only if an < bn . [Hint: Use Mathematical Induction].24. (a) If e > 1 and m, E N, show that em > en if and only if m > n. n (b) If 0 < e < 1 and m, n E N, show that em < en if and only if m > n. then e1l m < e l ln if and only if m > n.25. Assuming the existence of roots, show that if e > 1,26. Use Mathematical Induction to show that if a R and m, n, E then am+n = am an and E N, (a m ) " = a mn .
2.2 ABSOLUTE VALUE AND THE REAL LINE 31Section 2.2 Absolute Value and the Real LineFrom the Trichotomy Property 2. 1 .5(iii), we are assured that if a E JR and a i= 0, thenexactly one of the numbers a and -a is positive. The absolute value of a i= ° is defined tobe the positive one of these two numbers. The absolute value of ° is defined to be 0.2.2.1 Definition The absolute value of a real number a, denoted by lal, is defined by lal := {� if a > 0, if a = 0, -a if a < 0. For example, 1 5 1 = 5 and 1 - 8 1 = 8. We see from the definition that l a l 2: ° forall a E JR, and that la l = ° if and only if a = 0. Also 1 - al = lal for all a E JR. Someadditional properties are as follows.2.2.2 Theorem (a) labl = la Ilbl for alla, b E R(b) la l 2 = a 2 f all or aE R(c) Ifc 2: 0, then lal � c if and only if -c � a � c.(d) -Ia l � a � lal for all aE RProof. (a) If either a or b is 0, then both sides are equal to 0. There are four other casesto consider. If a > 0, b > 0, then ab > 0, so that labl = ab = lallbl. If a > 0, b < 0, thenab < 0, so that labl = -ab = a(-b) = lallbl. The remaining cases are treated similarly.(b) Since a2 2: 0, we have a2 = la2 1 = l aal = lall a l = lal 2 •(c) If lal � c, then we have both a � c and -a � c (why?), which is equivalent to -c �a � c. Conversely, if -c � a � c, then we have both a � c and -a � c (why?), so thatlal � c.(d) Take c = lal in part (c). Q.E.D. The following important inequality will be used frequently.2.2.3 Triangle Inequality If a, b E JR, then la + bl � l al + Ib l.Proof. From 2.2.2(d), we have -I a l � a � lal and - Ibl � b � Ibl. On adding theseinequalities, we obtain - (Ial + Ibl) � a + b � lal + Ibl ·Hence, by 2.2.2(c) we have la + bl � lal + Ibl. Q.E.D. It can be shown that equality occurs in the Triangle Inequality if and only if ab > 0,which is equivalent to saying that a and b have the same sign. (See Exercise 2.) There are many useful variations of the Triangle Inequality. Here are two.2.i.4 Corollary If a, b E JR, then(a) i lal - Ibl i � la - bl,(b) l a - bl � lal + Ib l .Proof. (a) We write a = a - b + b and then apply the Triangle Inequality to get lal =I(a - b) + b l � la - b l + I b l. Now subtract I bl to get la l - I bl � la - bl. Similarly, from
32 CHAPTER 2 THE REAL NUMBERSIbl = Ib - a + a l ::s Ib - a l + l a l, we obtain -Ia - bl = - Ib - a l ::s l a l - Ibl · If we combine these two inequalities, using 2.2.2(c), we get the inequality in (a).(b) Replace b in the Triangle Inequality by -b to get la - b I ::s l a I + I - b I. Since I - b I =Ibl we obtain the inequality in (b). Q.Ep. A straightforward application of Mathematical Induction extends the Triangle Inequality to any finite number of elements of lR.2.2.5 Corollary If a i a2 , . . . , an are any real numbers, then l a l + a2 + . . . + an I ::s l a l l + l a2 1 + . . . + I an I · The following examples illustrate how the properties of absolute value can be used.2.2.6 Examples (a) Determine the set A of x E lR such that 12x + 31 < 7. From a modification of 2.2.2(c) for the case of strict inequality, we see that x E A ifand only if -7 < 2x + 3 < 7, which is satisfied if and only if -10 < 2x < 4. Dividing by2, we conclude that A = {x E lR : - 5 < x < 2}.(b) Determine the set B := {x E lR : Ix - 1 1 < Ix l}. One method is to consider cases so that the absolute value symbols can be removed.Here we take the cases (i) x � 1 , (ii) O ::s x < 1 , (iii) x < O.(Why did we choose these three cases?) In case (i) the inequality becomes x - I < x,which is satisfied without further restriction. Therefore all x such that x � 1 belong to theset B. In case (ii), the inequality becomes -(x - 1) < x, which requires that x > !. Thus,this case contributes all x such that ! < x < 1 to the set B. In case (iii), the inequalitybecomes -(x - 1) < -x, which is equivalent to 1 < O. Since this statement is false, novalue of x from case (iii) satisfies the inequality. Forming the union of the three cases, we :conclude that B = { x E lR x > !}. There is a second method of determining the set B based on the fact that a < b ifand only if a 2 < b2 when both a � 0 and b � O. (See 2.1 . 13(a).) Thus, the inequality Ix - 1 1 < Ix I is equivalent to the inequality Ix - 1 1 2 < Ix 1 2 . Since l a 1 2 = a 2 for any a by2.2.2(b), we can expand the square to obtain x 2 - 2x + 1 < x 2, which simplifies to x > !.Thus, we again find that B = {x E lR : x > ! }. This method of squaring can sometimes beused to advantage, but often a case analysis cannot be avoided when dealing with absolutevalues. 2(c) Let the function f be defined by f(x) := (2x + 3x + 1)/(2x - 1) for 2 ::s x ::s 3.Find a constant M such that I f (x) I ::s M for all x satisfying 2 ::s x ::s 3. We consider separately the numerator and denominator of 2 1 2x + 3x + 1 1 If(x) 1 = 12x - 1 1From the Triangle Inequality, we obtain 12x 2 + 3x + 1 1 ::s 21xl 2 + 31x l + 1 ::s 2 . 32 + 3 · 3 + 1 = 28since Ix I ::s 3 for the x under consideration. Also, 12x - 1 1 � 2 1 x I - 1 � 2 . 2 - 1 = 3since Ix l � 2 for the x under consideration. Thus, 1/12x - 1 1 ::s 1/3 for x � 2. (Why?)Therefore, for 2 :s x ::s 3 we have I f(x) 1 :s 28/3. Hence we can take M = 28/3. (Note
2.2 ABSOLUTE VALUE AND THE REAL LINE 33that we have found one such constant M; evidently any number H > 28/3 will also satisfyIf(x)1 ::::: H. 1t is also possible that 28/3 is not the smallest possible choice for M.) DThe Real LineA convenient and familiar geometric interpretation of the real number system is the realline. In this interpretation, the absolute value la I of an element a in lEt is regarded as thedistance from a to the origin O. More generally, the distance between elements a and b inlEt is la - bl. (See Figure 2.2.1 .) We will later need precise language to discuss the notion of one real number being"close to" another. If a is a given real number, then saying that a real number x is "close to" ashould mean that the distance I x a I between them is "small". A context in which this idea -can be discussed is provided by the terminology of neighborhoods, which we now define. -4 -3 -2 -1 o 2 3 4 �IE-- I (-2 ) - (3 ) 1 = 5 -��I Figure 2.2.1 The distance between a = -2 and = 3 b2.2.7 Definition Let a E lEt and £ > O. Then the £-neighborhood of a is the set V/a) :={x E lEt: Ix - al < £}. For a E lEt, the statement that x belongs to V, (a) is equivalent to either of the statements(see Figure 2.2.2) -£ < x a < £ -<==> a £ < X < a + £. - - -- -- -+ -- -- O -- -- _r -- -- � -- -- -- -- � � -- -- -- -- -- a - e a a + e Figure 2.2.2 An s-neighborhood of a2.2.8 Theorem Let a E lEt. Ifx belongs to the neighborhood V, (a) for every £ > 0, thenx = a.Proof. If a particular x satisfies I x - a I < £ for every E: > 0, then it follows from 2. 1 .9that Ix - a l = 0, and hence x = a. Q.E.D.2.2.9 Examples (a) Let U : = {x : 0 < x < 1}. If a E U, then let £ be the smaller ofthe two numbers a and 1 - a. Then it is an exercise to show that V/a) is contained in U .Thus each element of U has some £-neighborhood of it contained in U .(b) If I := {x : 0 ::::: x ::::: 1}, then for any £ > 0, the £-neighborhood V, (0) of 0 containspoints not in I, and so V, (0) is not contained in I. For example, the number x, := -£/2 isin V, (0) but not in I .(c) If Ix - al < £ and I y - bl < £, then the Triangle Inequality implies that I (x + y) - (a + b) 1 = I (x - a) + (y - b) 1 ::::: Ix - al + Iy - bl < 2£.
2.3 THE COMPLETENESS PROPERTY OF lR 35order properties described in the preceding sections, but we have seen that -J2 cannot berepresented as a rational number; therefore -J2 does not belong to Q. This observationshows the necessity of an additional property to characterize the real number system. Thisadditional property, the Completeness (or the Supremum) Property, is an essential propertyof R, and we will say that R is a complete ordered field. It is this special property thatpermits us to define and develop the various limiting procedures that will be discussed inthe chapters that follow. There are several different ways to describe the Completeness Property. We choose togive what is probably the most efficient approach by assuming that each nonempty boundedsubset of R has a supremum.Suprema and Infima _______________________________________________We now introduce the notions of upper bound and lower bound for a set of real numbers.These ideas will be of utmost importance in later sections.2.3.1 Definition Let S be a nonempty subset of R(a) The set S is said to be bounded above if there exists a number U E R such that s ::s U for all s E S. Each such number u is called an upper bound of S.(b) The set S is said to be bounded below if there exists a number W E R such that w ::s s for all S E S. Each such number w is called a lower bound of S.(c) A set is said to be bounded if it is both bounded above and bounded below. A set is said to be unbounded if it is not bounded. For example, the set S : = {x E R : x < 2} is bounded above; the number 2 and anynumber larger than 2 is an upper bound of S. This set has no lower bounds, so that the setis not bounded below. Thus it is unbounded (even though it is bounded above). If a set has one upper bound, then it has infinitely many upper bounds, because if uis an upper bound of S, then the numbers u + 1 , u + 2, . . . are also upper bounds of S.(A similar observation is valid for lower bounds.) In the set of upper bounds of S and the set of lower bounds of S, we single out theirleast and greatest elements, respectively, for special attention in the following definition.(See Figure 2.3.1.) Figure 2.3.1 inf S and sup S2.3.2 Definition Let S be a nonempty subset of R(a) If S is bounded above, then a number u is said to be a supremum (or a least upper bound) of S if it satisfies the conditions: (1) u is an upper bound of S, and (2) if v is any upper bound of S, then u ::s v.
36 CHAPTER 2 THE REAL NUMBERS(b) If S is bounded below, then a number w is said to be an infimum (or a greatest lower bouud) of S if it satisfies the conditions: (1) w is a lower bound of S, and (2 ) if t is any lower bound of S, then t :s w. It is not difficult to see that there can be only one supremum of a given subset S of R(Then we can refer to the supremum of a set instead of a supremum.) For, suppose thatu j and u2 are both suprema of S. If uj < u2 then the hypothesis that u2 is a supremumimplies that uj cannot be an upper bound of S. Similarly, we see that u 2 < u is notpossible. Therefore, we must have u = u 2 . A similar argument can be given to show thatthe infimum of a set is uniquely determined. If the supremum or the infimum of a set S exists, we will denote them by sup S and inf S.We also observe that if u is an arbitrary upper bound of a nonempty set S, then sup S :s u.This is because sup S is the least of the upper bounds of S. First of all, it needs to be emphasized that in order for a nonempty set S in JR. to havea supremum, it must have an upper bound. Thus, not every subset of JR. has a supremum;similarly, not every subset of JR. has an infimum. Indeed, there are four possibilities for anonempty subset S of R it can (i) have both a supremum and an infimum, (ii) have a supremum but no infimum, (iii) have a infimum but no supremum, (iv) have neither a supremum nor an infimum. We also wish to stress that in order to show that u = sup S for some nonempty subset Sof JR., we need to show that both (1) and (2) of Definition 2.3.2(a) hold. It will be instructiveto reformulate these statements. First the reader should see that the following two statementsabout a number u and a set S are equivalent: (1) u is an upper bound of S, (1) s :S u for all s E S.Also, the following statements about an upper bound u of a set S are equivalent: (2) if v is any upper bound of S, then u :s v, (2) if z < u, then z is not an upper bound of S, (2") if z < u, then there exists Sz E S such that z < sz (2") if e > 0, then there exists se E S such that u e < se -Therefore, we can state two alternate formulations for the supremum.2.3.3 Lemma A number u is the supremum of a nonempty subset S of JR. if and only ifu satisfies the conditions:(1) s :s u for all s E S,(2) if v < u, then there exists s E S such that v < s. We leave it to the reader to write out the details of the proof.2.3.4 Lemma An upper bound u of a nonempty set S in JR. is the supremum of S if andonly if for every e > 0 there exists an Se E S such that u e < se -
2.3 THE COMPLETENESS PROPERTY OF lR 37Proof. If u is an upper bound of S that satisfies the stated condition and if v < u, then we Ssput e := U - v. Then e > 0, so there exists E S such that v = U - e < ss . Therefore, vis not an upper bound of S, and we conclude that U = sup S . Conversely, suppose that U = sup S and let e > O. Since U - e < u, then U - e is not san upper bound of S. Therefore, some element s of S must be greater than U - e ; that is, sU - e < s . (See Figure 2. 3 .2.) Q.E.D. U -E SE U t_ t I - III I """ _ �------�V---JS Figure 2.3.2 u = sup S It is important to realize that the supremum of a set may or may not be an elementof the set. Sometimes it is and sometimes it is not, depending on the particular set. Weconsider a few examples.2.3.5 Examples (a) If a nonempty set Sl has a finite number of elements, then it canbe shown that Sl has a largest element U and a least element w. Then U = sup Sl andw = inf Sl and they are both members of Sl . (This is clear if Sl has only one element, andit can be proved by induction on the number of elements in Sl ; see Exercises 1 1 and 12.)(b) The set S2 := {x : 0 .::: x .::: I} clearly has 1 for an upper bound. We prove that 1 isits supremum as follows. If v < 1, there exists an element s E S2 such that v < s . (Nameone such element s . ) Therefore v is not an upper bound of S2 and, since v is an arbitrarynumber v < 1, we conclude that sup S2 = l . 1t is similarly shown that inf S2 = O. Note thatboth the supremum and the infimum of S2 are contained in S2 .(c) The set S3 := {x : 0 < x < I} clearly has 1 for an upper bound. Using the sameargument as given in (b), we see that sup S3 = 1. In this case, the set S3 does not containits supremum. Similarly, inf S3 = 0 is not contained in S3 . DThe Completeness Property of RIt is not possible to prove on the basis of the field and order properties of R that werediscussed in Section 2.1 that every nonempty subset of R that is bounded above has asupremum in R However, it is a deep and fundamental property of the real number systemthat this is indeed the case. We will make frequent and essential use of this property,especially in our discussion of limiting processes. The following statement concerning theexistence of suprema is our final assumption about R Thus, we say that R is a completeorderedfield.2.3.6 The Completeness Property of R Every nonempty set of real numbers that hasan upper bound also has a supremum in R This property is also called the Supremum Property of R The analogous propertyfor infima can be deduced from the Completeness Property as follows. Suppose that S isa nonempty subset of R that is bounded below. Then the nonempty set S := { -s : s E S}is bounded above, and the Supremum Property implies that U := sup S exists in R Thereader should verify in detail that -u is the infimum of S.
38 CHAPTER 2 THE REAL NUMBERSExercises for Section 2.3 1. Let Sl := {x E R : x ::: O}. Show in detail that the set Sl has lower bounds, but no upper bounds. Show that inf Sl = O. 2. Let Sz = {x E R : x > O}. Does Sz have lower bounds? Does Sz have upper bounds? Does inf Sz exist? Does sup Sz exist? Prove your statements. 3. Let S3 = {lIn : n E N}. Show that SUP S3 = 1 and inf S3 ::: O. (It will follow from the Archi- medean Property in Section 2.4 that inf S3 = 0.) 4. Let S4 := { I - (-I) n ln : n E N}. Find inf S4 and sup S4 5. Let S be a nonempty subset of R that is bounded below. Prove that inf S = - sup{ -s: S E S}. 6. If a set S � R contains one of its upper bounds, show that this upper bound is the supremum of S. 7. Let S � R be nonempty. Show that U E R is an upper bound of S if and only if the conditions t E R and t > u imply that t � S. 8 . Let S � R be nonempty. Show that if u = sup S, then for every number n E N the number u - 1 I n is not an upper bound of S, but the number u + 1 I n is an upper bound of S. (The converse is also true; see Exercise 2.4.3.) 9. Show that if A and B are bounded subsets of R, then A U B is a bounded set. Show that sup(A U B) = sup{sup A , sup B}.10. Let S be a bounded set in R and let So be a nonempty subset of S. Show that inf S :s inf So :s sup So :s sup S.1 1 . Let S � R and suppose that s* := sup S belongs to S. If u � S, show that sup(S U {u}) = sup{s*, u}.12. Show that a nonempty finite set S � R contains its supremum. [Hint: Use Mathematical Induc- tion and the preceding exercise.]13. Show that the assertions (1) and (I) before Lemma 2.3.3 are equivalent.14. Show that the assertions (2), (2), (2"), and (2111) before Lemma 2.3.3 are equivalent.15. Write out the details of the proof of Lemma 2.3.3.Section 2.4 Applications of the Supremum PropertyWe will now discuss how to work with suprema and infima. We will also give some veryimportant applications of these concepts to derive fundamental properties of R We beginwith examples that illustrate useful techniques in applying the ideas of supremum andinfimum.2.4.1 Example (a) It is an important fact that taking suprema and infima of sets is compatible with the algebraic properties of JR. As an example, we present here the compatibilityof taking suprema and addition. Let S be a nonempty subset of JR that is bounded above, and let a be any number inR Define the set a + S : = {a + s : S E S}. We will prove that sup(a + S) = a + sup S.
2.4 APPLICATIONS OF THE SUPREMUM PROPERTY 39 If we let u := sup S, then x ::::: u for all X E S, so that a + x ::::: a + u . Therefore, a + uis an upper bound for the set a + S; consequently, we have sup(a + S) ::::: a + u . v v Now if is any upper bound of the set a + S, then a + x ::::: for all X E S. Con vsequently x ::::: - a for all X E S, so that v - a is an upper bound of S. Therefore, vu = sup S ::::: - a, which gives us a + u ::::: v. Since v is any upper bound of a + S, vwe can replace by sup(a + S) to get a + u ::::: sup(a + S). Combining these inequalities, we conclude that sup(a + S) = a + u = a + sup S.For similar relationships between the suprema and infima of sets and the operations ofaddition and multiplication, see the exercises.(b) If the suprema or infima of two sets are involved, it is often necessary to establishresults in two stages, working with one set at a time. Here is an example. Suppose that A and B are nonempty subsets of lR that satisfy the property: for all a E A and all b E B.We will prove that sup A ::::: inf B.For, given b E B, we have a ::::: b for all a E A. This means that b is an upper bound of A, sothat sup A ::::: b. Next, since the last inequality holds for all b E B, we see that the numbersup A is a lower bound for the set B. Therefore, we conclude that sup A ::::: inf B. 0FunctionsThe idea of upper bound and lower bound is applied to functions by considering therange of a function. Given a function f : D -+ lR, we say that f is bounded above ifthe set feD) = {f(x) : x E D} is bounded above in lR; that is, there exists B E lR suchthat f(x) ::::: B for all x E D. Similarly, the function f is bounded below if the set feD)is bounded below. We say that f is bounded if it is bounded above and below; this isequivalent to saying that there exists B E lR such that I f (x) I ::::: B for all x E D. The following example illustrates how to work with suprema and infima of functions.2.4.2 Example Suppose that f and g are real-valued functions with common domainD � R We assume that f and g are bounded.(a) If f(x) ::::: g(x) for all x E D, then sup f(D) ::::: sup g(D), which is sometimes written: sup f(x) ::::: sup g(x). XED xED We first note that f(x) ::::: g(x) ::::: sup g(D), which implies that the number sup g(D)is an upper bound for feD). Therefore, sup feD) ::::: sup g(D).(b) We note that the hypothesis f(x) ::::: g(x) for all x E D in part (a) does not imply anyrelation between sup feD) and inf g(D). For example, if f(x) : = x 2 and g(x) : = x with D = {x : 0 ::::: x ::::: I}, then f(x) :::::g(x) for all x E D. However, we see that sup f(D) = l andinf g(D) = O. Since sup g(D) =1, the conclusion of (a) holds.
40 CHAPTER 2 THE REAL NUMBERS(c) If f (x ) :::: g(y) for all x, y E D, then we may conclude that sup f(D) :::: inf g(D),which we may write as: sup f(x) :::: inf g(y). XED yED(Note that the functions in (b) do not satisfy this hypothesis.) The proof proceeds in two stages as in Example 2.4.1 (b). The reader should write outthe details of the argument. 0 Further relationships between suprema and infima of functions are given in the exercises.The Archimedean PropertyBecause of your familiarity with the set JR and the customary picture of the real line, it mayseem obvious that the set N of natural numbers is not bounded in R How can we prove this"obvious" fact? In fact, we cannot do so by using only the Algebraic and Order Propertiesgiven in Section 2.1 . Indeed, we must use the Completeness Property of JR as well as theInductive Property of N (that is, if n E N, then n + 1 E N). The absence of upper bounds for N means that given any real number x there exists anatural number n (depending on x) such that x < n.2.4.3 Archimedean Property If x E JR, then there exists nx E N such that x < nx Proof. If the assertion is false, then n :::: x for all n E N; therefore, x is an upper bound ofN. Therefore, by the Completeness Property, the nonempty set N has a supremum U E RSubtracting 1 from u gives a number u 1 which is smaller than the supremum u of N . -Therefore u 1 is not an upper bound of N, so there exists m E N with u 1 < m. Adding - -1 gives u < m + 1 , and since m + 1 E N, this inequality contradicts the fact that u is anupper bound of N. Q.E.D.2.4.4 Corollary If S := { l i n : n E N}, then inf S = O.Proof. Since S =f. 0 is bounded below by 0, it has an infimum and we let w := inf S. 1t isclear that w ::: O. For any e > 0, the Archimedean Property implies that there exists n E Nsuch that l ie < n, which implies l in < e. Therefore we have 0 :::: w :::: l in < e.But since e > 0 is arbitrary, it follows from Theorem 2.1 .9 that w = O. Q.E.D.2.4.5 Corollary 1ft > 0, there exists nt E N such that 0 < lint < t .Proof. Since inf{l/n : n E N} = 0 and t > 0, then t is not a lower bound for the set{ l In : n E N}. Thus there exists nt E N such that 0 < l int < t . Q.E.D.2.4.6 Corollary If y > 0, there exists n y E N such that n y - 1 :::: y < n y •
2.4 APPLICATIONS OF THE SUPREMUM PROPERTY 41Proof. The Archimedean Property ensures that the subset : = {m Ey E N : < m} N y ofis not empty. By the Well-Ordering Property ny .by Then ny - 1 does not belong to Ey, 1.2.1, Ey has a least element, which we denote and hence we have ny - 1 < ny. :S y Q.E.D. Collectively, the CorollariesProperty of R 2 .4.4-2.4.6 are sometimes referred to as the ArchimedeanThe Existence of ,J2The importance of the Supremum Property lies in the fact that it guarantees the existence ofreal numbers under certain hypotheses. We shall make use of it in this way many times. Atthe moment, we shall illustrate this use by proving the existence of a positive real numberx 4) x2 2; x 2. such that = that is, the positive square root of It was shown earlier (see Theorem2.1 . that such an cannot be a rational number; thus, we will be deriving the existenceof at least one irrational number.2.4.7 Theorem There exists a positive real number such that x x2 = 2. Let S := {s JR.: 0 :S S, S 2 E t > 2,< 2}. t2 > 41 E Since S, the set is not empty. Also, S isProof. 2,bounded above by because if then so that t ¢ S. Therefore the Supremum xProperty implies that the set S has a supremum in JR., and we let := sup S. Note thatx > We will prove that x2 2 by ruling out the other two possibilities: x2 < 2 and x2 > 2 1. = First assume that x2 < 2. We will show that this assumption contradicts the fact that .x sup S by finding an n E N such that x 1 In E S, thus implying that x is not an upper = +bound for S. To see how to choose n, note that lln2 :S lin so that (x -1n ) 2 x2 2x n1 x2 -1n (2x 1) . + = n+ - + 2" :S + +Hence if we can choose n so that 1 -(2x + 1) < 2 _x2, nthen we get (x lln)2 < x2 (2 - x2) 2. By assumption we have 2 - x2 > 0, so that + + =(2 -x2)/(2x + 1) > O. Hence the Archimedean Property (Corollary 2.4.5) can be used toobtain n E N such that -1n < ---1 . 2 -x2 2x +These steps can be reversed to show that for this choice of n we have x 1 In E S, which x +contradicts the fact that is an upper bound of S. Therefore we cannot have x2 < 2.x - Now isassumeanthat x2 > 2. We willcontradictingis then possible tosupndS.mToEdo this, notethat 1 Im also upper bound of S, show that it the fact that x fi = N such that (x - � y x2 - � + �2 > x2 2x = mHence if we can choose m so that 2x - < x2 - 2, m
42 CHAPTER 2 THE REAL NUMBERSthen (x - I/m) 2 > X 2 - (X2 - 2) = 2. Now by assumption we have x 2 - 2 > 0, so that(x 2 - 2)/2x > 0. Hence, by the Archimedean Property, there exists m E N such that 1 -< x2 - 2 . -- m 2xThese steps can be reversed to show that for this choice of m we have (x - 1 Im) 2 > 2. Now Sif E S, then s 2 < 2 < (x - I/m) 2 , whence it follows from 2. 1 . 13(a) that s < x - 1 1m .This implies that x - I I m is an upper bound for S, which contradicts the fact that x = sup S.Therefore we cannot have x 2 > 2. Since the possibilities x 2 < 2 and x 2 > 2 have been excluded, we must have x 2 = 2. Q.E.D. By slightly modifying the preceding argument, the reader can show that if a > 0, thenthere is a unique b > ° such that b2 = a. We call b the positive square root of a and denoteit by b = Ja or b = a 1 /2 . A slightly more complicated argument involving the binomialtheorem can be formulated to establish the existence of a unique positive nth root of a,denoted by :.ya or a 1 /n , for each n E NoRemark If in the proof of Theorem 2.4.7 we replace the set S by the set of rational :numbers T : = {r E Q ° :::: r, r 2 < 2}, the argument then gives the conclusion that y :=sup T satisfies i = 2. Since we have seen in Theorem 2. 1 .4 that y cannot be a rationalnumber, it follows that the set T that consists of rational numbers does not have a supremumbelonging to the set Q. Thus the ordered field Q of rational numbers does not possess theCompleteness Property.Density of Rational Numbers in lR.We now know that there exists at least one irrational real number, namely .j2. Actuallythere are "more" irrational numbers than rational numbers in the sense that the set ofrational numbers is countable (as shown in Section 1 .3), while the set of irrational numbersis uncountable (see Section 2.5). However, we next show that in spite of this apparentdisparity, the set of rational numbers is "dense" in lR. in the sense that given any two realnumbers there is a rational number between them (in fact, there are infinitely many suchrational numbers).2.4.8 The Density Theorem If x and y are any real numbers with x < y, then thereexists a rational number r E Q such that x < r < y.Proof. It is no loss of generality (why?) to assume that x > 0. Since y - x > 0, itfollows from Corollary 2.4.5 that there exists n E N such that 1 / n < y - x. Therefore,we have nx + I < ny. If we apply Corollary 2.4.6 to nx > 0, we obtain m E N withm 1 :::: nx < m. Therefore, m :::: nx + 1 < ny, whence nx < m < ny. Thus, the rational -number r : = mi n satisfies x < r < y. Q.E.D. To round out the discussion of the interlacing of rational and irrational numbers, wehave the same "betweenness property" for the set of irrational numbers.2.4.9 Corollary If x and y are real numbers with x < y, then there exists an irrationalnumber z such that x < z < y.
44 CHAPTER 2 THE REAL NUMBERS Prove that sup{g(y) : y E Y} :s inf{f(x) : x E X} We sometimes express this by writing y x x y sup inf hex, y) :s inf sup h (x, y). Note that Exercises 8 and 9 show that the inequality may be either an equality or a strict inequality.1 1 . Let X and Y be nonempty sets and let h : X x Y -+ JR have bounded range in JR. Let F: X -+ JR and G : Y -+ JR be defined by F(x) := sup{h(x, y) : y E Y}, G(y) : = sup{h(x, y) : x E X}. Establish the Principle of the Iterated Suprema: sup{h(x, y) : x E X, Y E Y} = sup{F(x) : x E X} = sup{G(y) : y E Y} We sometimes express this in symbols by x.y x y y x sup h (x, y) = sup sup h(x, y) = sup sup h(x, y).12. Given any x E JR, show that there exists a unique n E Z such that n 1 :s x < n. -13. If y > 0, show that there exists n E N such that 1/2 < y. n14. Modify the argument in Theorem 2.4.7 to show that there exists a positive real number y such that l = 3.15. Modify the argument in Theorem 2.4.7 to show that if a > 0, then there exists a positive real number z such that z?- = a .16. Modify the argument in Theorem 2.4.7 to show that there exists a positive real number u such that u3 = 2.17. Complete the proof of the Density Theorem 2.4.8 by removing the assumption that x > O.18. If u > 0 is any real number and x < y, show that there exists a rational number r such that x < ru < y. (Hence the set {ru: r E Q} is dense in JR.)Section 2.5 IntervalsThe Order Relation on lR. detennines a natural collection of subsets called "intervals". Thenotations and tenninology for these special sets will be familiar from earlier courses. Ifa, b E lR. satisfy a < b, then the open interval detennined by a and b is the set (a, b) := {x E lR. : a < x < b}.The points a and b are called the endpoints of the interval; however, the endpoints are notincluded in an open interval. If both endpoints are adjoined to this open interval, then weobtain the closed interval detennined by a and b; namely, the set [a, b] := {x E lR. : a :s x :s b}.The two half-open (or half-closed) intervals detennined by a and b are [a, b), whichincludes the endpoint a, and (a, b], which includes the endpoint b. Each of these four intervals is bounded and has length defined by b - a. If a = b, thecorresponding open interval is the empty set (a, a) = 0, whereas the corresponding closedinterval is the singleton set [a, a] = {a}.
2.5 INTERVALS 45 There are five types of unbounded intervals for which the symbols 00 (or +00) and -ooare used as notational convenience in place of the endpoints. The infinite open intervalsare the sets of the form (a, 00) := {x E JR. : x > a} and ( - 00, b) := {x E JR. : x < b}.The first set has no upper bounds and the second one has no lower bounds. Adjoiningendpoints gives us the infinite closed intervals: [a, 00) := {x E JR. : a :S x} and (- 00, b] := {x E JR. : x :S b}.It is often convenient to think of the entire set JR. as an infinite interval; in this case, we write(-00, 00) := R No point is an endpoint of (-00, 00).Warning It must be emphasized that 00 and - 00 are not elements of JR., but only convenient symbols.Characterization of IntervalsAn obvious property of intervals is that if two points x, y with x < y belong to an interval I,then any point lying between them also belongs to I . That is, if x < t < y, then the point tbelongs to the same interval as x and y. In other words, if x and y belong to an interval I,then the interval [x, y] is contained in I . We now show that a subset of JR. possessing thisproperty must be an interval.2.5.1 Characterization Theorem If S is a subset of JR. that contains at least two pointsand has the property(1) if x, Y E S and x < y, then [x, y] S; S,then S is an interva1.Proof. There are four cases to consider: (i) S is bounded, (ii) S is bounded above butnot below, (iii) S is bounded below but not above, and (iv) S is neither bounded above norbelow. Case (i): Let a := inf S and b := sup S. Then S S; [a, b] and we will show that(a, b) S; S. If a < z < b, then z is not a lower bound of S, so there exists X E S with x < z. Also,z is not an upper bound of S, so there exists y E S with z < y. Therefore z E [x, y], soproperty (1) implies that z E S. Since z is an arbitrary element of (a, b), we conclude that(a, b) S; S. Now if a E S and b E S, then S = [a, b] . (Why?) If a 1. S and b f- S, then S = (a, b).The other possibilities lead to either S = (a, b] or S = [a, b). Case (ii): Let b := sup S. Then S S; (-00, b] and we will show that (-00, b) S; S. For,if z < b, then there exist x , Y E S such that z E [x, y] S; S. (Why?) Therefore (-00, b) S; S.If b E S, then S = ( - 00, b], and if b 1. S, then S = (-00, b). Cases (iii) and (iv) are left as exercises. Q.E.D.Nested IntervalsWe say that a sequence of intervals In n E N, is nested if the following chain of inclusionsholds (see Figure 2.5.1):
46 CHAPTER 2 THE REAL NUMBERS ( ( Figure 2.5.1 Nested intervals For example, if In := [0, lin] for n E N, then In :;2 In+ 1 for each n E N so that thissequence of intervals is nested. In this case, the element ° belongs to all In and theArchimedean Property 2.4.5 can be used to show that ° is the only such common point.(Prove this.) We denote this by writing n::: 1 In = { O} . It is important to realize that, in general, a nested sequence of intervals need nothave a common point. For example, if in := (0, lin) for n E N, then this sequence ofintervals is nested, but there is no common point, since for every given x > 0, there exists(why?) m E N such that 1 1 m < x so that x ¢ 1m Similarly, the sequence of intervalsKn := (n, (0) , n E N, is nested but has no common point. (Why?) However, it is an important property of lR that every nested sequence of closed, boundedintervals does have a common point, as we will now prove. Notice that the completenessof lR plays an essential role in establishing this property.2.5.2 Nested Intervals Property If In = n [an bn J E N, is a nested sequence of closed bounded intervals, then there exists a number � E lR such that � E In for all E N. nProof. Since the intervals are nested, we have In S; II for all E N, so that an Sn for blall E No Hence, the nonempty set {an : E N} is bounded above, and we let � be its n nsupremum. Clearly an S � for all E N. n We claim also that � S n for all This is established by showing that for any particular b n.n, the number n is an upper bound for the set {ak : k E N} . We consider two cases. (i) If bn S k, then since In :;2 Ik , we have ak S k S bn . (ii) If k < then since Ik ;2 In we have b n,ak S an S bn · (See Figure 2.5.2.) Thus, we conclude that ak S bn for all k, so that n is an bupper bound of the set {ak : k E N} . Hence, � S bn for each E N. Since an S � S bn for nall we have � E In for all E N. n, n Q.E.D. I� E ---- -- 4 --------��I Figure 2.5.2 If k < n, then In � Ik
2.5 INTERVALS 472.5.3 Theorem IfIn := [an bn ], n E fir, is a nested sequence ofclosed, boundedintervalssuch that the lengths bn - an of In satisfy inf{bn - an : n E fir} = 0,then the number � contained in In for all n E fir is unique.Proof. If rJ := inf{bn : n E fir}, then an argument similar to the proof of 2.5.2 can be usedto show that an .::: for all n, and hence that � .::: In fact, it is an exercise (see Exercise 1 1.10) to show that x E In for all n E if and only if � .::: x .::: If we have inf{bn - an : n E fir 1.N} = 0, then for any e > 0, there exists an m E N such that ° .::: - � .::: bm - am < e . 1Since this holds for all e > 0, it follows from Theorem 2.1 .9 that - � = 0. Therefore, we 1conclude that � = is the only point that belongs to In for every n E N. 1 Q.E.D.The Uncountability of IR _____________________The concept of a countable set was discussed in Section 1 .3 and the countability of the setQ of rational numbers was established there. We will now use the Nested Interval Propertyto prove that the set IR is an uncountable set. The proof was given by Georg Cantor in1874 in the first of his papers on infinite sets. He later published a proof that used decimalrepresentations of real numbers, and that proof will be given later in this section.2.5.4 Theorem The set IR ofreal numbers is not countable.Proof. We will prove that the unit interval I := [0, 1] is an uncountable set. This impliesthat the set IR is an uncountable set, for if IR were countable, then the subset I would alsobe countable. (See Theorem 1 .3.9(a).) The proof is by contradiction. If we assume that I is countable, then we can enumeratethe set as 1 = {x, x2 " , xn }. We first select a closed subinterval 1 of I such that • • .x ¢ 1, then select a closed subinterval 12 of 1 such that x2 ¢ 12 , and so on. In this way,we obtain nonempty closed intervals 1 ;2 12 ;2 . . . ;2 In ;2 . . .such that In S; I and Xn ¢ In for all n . The Nested Intervals Property 2.5.2 implies thatthere exists a point � E I such that � E In for all n. Therefore � =j:. xn for all n E N, so theenumeration of I is not a complete listing of the elements of I, as claimed. Hence, I is anuncountable set. Q.E.D. The fact that the set IR of real numbers is uncountable can be combined with the factthat the set Q of rational numbers is countable to conclude that the set IRQ of irrationalnumbers is uncountable. Indeed, since the union of two countable sets is countable (see l .3.7(c)), if IRQ is countable, then since IR = Q U (IRQ), we conclude that IR is also acountable set, which is a contradiction. Therefore, the set of irrational numbers IRQ is anuncountable set.tBinary RepresentationsWe will digress briefly to discuss informally the binary (and decimal) representations of realnumbers. It will suffice to consider real numbers between ° and 1 , since the representationsfor other real numbers can then be obtained by adding a positive or negative number.tThe remainder of this section can be omitted on a first reading.
48 CHAPTER 2 THE REAL NUMBERS If x E [0, 1], we will use a repeated bisection procedure to associate a sequence (an ) ofOs and I s as follows. If x i= 4 belongs to the left subinterval [0, 4 ] we take a l := 0, whileif x belongs to the right subinterval [4, 1] we take a l = 1. If x = & then we may take alto be either ° or 1. In any case, we have al < x < al + 1 -- - 2 -- 2We now bisect the interval [4al 4 (a l + 1) ]. If x is not the bisection point and belongsto the left subinterval we take a2 := 0, and if x belongs to the right subinterval we takea2 := 1. If x = ! or x = �, we can take a2 to be either ° or 1. In any case, we have aj + - < x < aj + a2 + 1 a2 - 2 22 - - - 22 . 2 --We continue this bisection procedure, assigning at the nth stage the value an := ° if x is notthe bisection point and lies in the left subinterval, and assigning the value an := 1 if x liesin the right subinterval. In this way we obtain a sequence (an ) of Os or Is that correspondto a nested sequence of intervals containing the point x. For each n, we have the inequality aj a2 an aj a2 . an + 1(2) 2" + 22 + . . . + 2n :s x :s 2" + 22 + . . + � .If x is the bisection point at the nth stage, then x = m /2n with m odd. In this case, we maychoose either the left or the right subinterval; however, once this subinterval is chosen, thenall subsequent subintervals in the bisection procedure are determined. [For instance, if wechoose the left subinterval so that an = 0, then x is the right endpoint of all subsequentsubintervals, and hence ak = 1 for all k � n + 1. On the other hand, if we choose the rightsubinterval so that an = 1, then x is the left endpoint of all subsequent subintervals, andhence ak = ° for all k � n + 1. For example, if x = �, then the two possible sequences forx are 1 , 0, 1 , 1, 1 , · " and 1 , 1 , 0, 0, 0" . . .] To summarize: If x E [0, 1], then there exists a sequence (an ) of Os and I s such thatinequality (2) holdsfor all n E N. In this case we write(3)and call (3) a binary representation of x. This representation is unique except whenx = m /2n for m odd, in which case x has the two representations x = (.aja2 an _ l lOOO · . )2 = (.aja2 an _jOl1 1 . . )2 • • • • • •one ending in Os and the other ending in 1 s. Conversely, each sequence of Os and Is is the binary representation of a unique realnumber inn [0, 1]. The inequality corresponding to (2) determines a closed interval withlength 1 /2 and the sequence of these intervals is nested. Therefore, Theorem 2.5.3 impliesthat there exists a unique real number x satisfying (2) for every n E N. Consequently, x hasthe binary representation (.ala2 an . ) 2 • • .Remark The concept of binary representation is extremely important in this era of digitalcomputers. A number is entered in a digital computer on "bits", and each bit can be put inone of two states--either it will pass current or it will not. These two states correspond tothe values 1 and 0, respectively. Thus, the binary representation of a number can be storedin a digital computer on a string of bits. Of course, in actual practice, since only finitelymany bits can be stored, the binary representations must be truncated. If n binary digits
2.5 INTERVALS 49 nare used for a number x E [0, 1], then the accuracy is at most 1 /2 . For example, to assurefour-decimal accuracy, it is necessary to use at least 15 binary digits (or 15 bits).Decimal RepresentationsDecimal representations of real numbers are similar to binary representations, except thatwe subdivide intervals into ten equal subintervals instead of two. Thus, given x E [0, 1], if we subdivide [0, 1 ] into ten equal subintervals, then x belongsto a subinterval [b l /lO, (b l + 1)/10] for some integer bl in {O, 1 , . . . , 9}. Proceeding as inthe binary case, we obtain a sequence (bn) of integers with 0 � bn � 9 for all n E N suchthat x satisfies(4)In this case we say that x has a decimal representation given by x = .bl b2 · · . bn • • • •If x :::: I and if B E N is such that B � x < B + 1, then x = B.b l b2 bn • • • where the "decimal representation of x - B E [0, 1] is as above. Negative numbers are treated similarly. The fact that each decimal determines a unique real number follows from Theorem n2.5.3, since each decimal specifies a nested sequence of intervals with lengths l/lO • The decimal representation of x E [0, 1] is unique except nwhen x is a subdivisionpoint atnsome stage, which can be seen to occur when x = m/lO for some m, n E N, 1 �m � IO . (We may also assume that m is not divisible by 10.) When x is a subdivisionpoint at the nth stage, one choice for bn corresponds to selecting the left subinterval, whichcauses all subsequent digits to be 9, and the other choice corresponds to selecting theright subinterval, which causes all subsequent digits to be O. [For example, if x = ! thenx = .4999 · · · = .5000 · · · , and if y = 38/100 then y = .37999 · · · = .38000 · · · .]Periodic DecimalsA decimal B.b l b2 • • . bn . . • is said to be periodic (or to be repeating), if there existk, n E Nsuch that bn = bn+m for all n :::: k. In this case, the block of digits bk bk+ 1 . . . bk+m _ 1 isrepeated once the kth digit is reached. The smallest number m with this property is calledthe period of the decimal. For example, 19/88 = .2159090 · · · 90 · · · has period m = 2with repeating block 90 starting at k = 4. A terminating decimal is a periodic decimalwhere the repeated block is simply the digit O. We will give an informal proof of the assertion: A positive real number is rational ifand only if its decimal representation is periodic. For, suppose that x = p/q where p, q E N have no common integer factors. Forconvenience we will also suppose that 0 < p < q. We note that the process of "longdivision" of q into p gives the decimal representation of p/q . Each step in the divisionprocess produces a remainder that is an integer from 0 to q - 1 . Therefore, after at most qsteps, some remainder will occur a second time and, at that point, the digits in the quotientWill begin to repeat themselves in cycles. Hence, the decimal representation of such arational number is periodic. Conversely, if a decimal is periodic, then it represents a rational number. The idea of theproof is best illustrated by an example. Suppose that x = 7.31414 · · · 14 · · · . We multiplyby a power of 10 to move the decimal point to the first repeating block; here obtaining lOx = 73. 1414 · · · . We now multiply by a power of 10 to move one block to the leftof the decimal point; here getting 1000x = 7314. 1414 · . . We now subtract to obtain an
CHAPTER 3 SEQUENCES AND SERIESNow that the foundations of the real number system lR have been laid, we are preparedto pursue questions of a more analytic nature, and we will begin with a study of theconvergence of sequences. Some of the early results may be familiar to the reader fromcalculus, but the presentation here is intended to be rigorous and will lead to certain moreprofound theorems than are usually discussed in earlier courses. We will first introduce the meaning of the convergence of a sequence of real numbersand establish some basic, but useful, results about convergent sequences. We then presentsome deeper results concerning the convergence of sequences. These include the MonotoneConvergence Theorem, the Bolzano-Weierstrass Theorem, and the Cauchy Criterion forconvergence of sequences. It is important for the reader to learn both the theorems and howthe theorems apply to special sequences. Because of the linear limitations inherent in a book it is necessary to decide whereto locate the subject of infinite series. It would be reasonable to follow this chapter witha full discussion of infinite series, but this would delay the important topics of continuity,differentiation, and integration. Consequently, we have decided to compromise. A briefintroduction to infinite series is given in Section 3.7 at the end of this chapter, and a moreextensive treatment is given later in Chapter 9. Thus readers who want a fuller discussionof series at this point can move to Chapter 9 after completing this chapter. Augustin-Louis Cauchy Augustin-Louis Cauchy (1789-1857) was born in Paris just after the start of the French Revolution. His father was a lawyer in the Paris police de partment, and the family was forced to flee during the Reign of Terror. As a result, Cauchys early years were difficult and he developed strong anti revolutionary and pro-royalist feelings. After returning to Paris, Cauchys father became secretary to the newly-formed Senate, which included the mathematicians Laplace and Lagrange. They were impressed by young Cauchys mathematical talent and helped him begin his career. He entered the Ecole Polytechnique in 1805 and soon established a reputation as an excep tional mathematician. In 1815, the year royalty was restored, he was appointed to the faculty of the Ecole Polytechnique, but his strong political views and his uncompromising standards in mathematics often resulted in bad relations with his colleagues. After the July revolution of 1830, Cauchy refused to sign the new loyalty oath and left France for eight years in self-imposed exile. In 1838, he accepted a minor teaching post in Paris, and in 1848 Napoleon III reinstated him to his former position at the Ecole Polytechnique, where he remained until his death. Cauchy was amazingly versatile and prolific, making substantial contributions to many areas, including real and complex analysis, number theory, differential equations, mathematical physics and probability. He published eight books and 789 papers, and his collected works fill 26 volumes. He was one of the most important mathematicians in the first half of the nineteenth century.52
3.1 SEQUENCES AND THEIR LIMITS 53Section 3.1 Sequences and Their LimitsA sequence in a set S is a function whose domain is the set N of natural numbers, andwhose range is contained in the set S. In this chapter, we will be concerned with sequencesin JR and will discuss what we mean by the convergence of these sequences. ,3.1.1 Definition A sequence of real numbers (or a sequence in JR) is a function definedon the set N = {I 2, . . . } of natural numbers whose range is contained in the set JR of realnumbers. In other words, a sequence in JR assigns to each natural number n = 1 , 2, . . . a uniquelydetermined real number. If X : N -+ JR is a sequence, we will usually denote the value of Xat n by the symbol xn rather than using the function notation X (n). The values xn are alsocalled the terms or the elements of the sequence. We will denote this sequence by thenotations X, (Xn : n E N).Of course, we will often use other letters, such as Y = (Yk ) Z = (Zj) and so on, to denotesequences. We purposely use parentheses to emphasize that the ordering induced by the naturalorder of N is a matter of importance. Thus, we distinguish notation ally between the sequence (xn : n E N), whose infinitely many terms have an ordering, and the set of values{xn : n E N} in the nrange of the sequence which are not ordered. For example, the sequence X := « _ l) : n E N) has infinitely many terms that alternate between - 1 and 1 , nwhereas the set of values {(_l) : n E N } is equal to the set { - I , I } , which has only twoelements. Sequences are often defined by giving a formula for the nth term xn Frequently, it isconvenient to list the terms of a sequence in order, stopping when the rule of formationseems evident. For example, we may define the sequence of reciprocals of the even numbersby writing X := ( �, � , �, � , . . -) ,though a more satisfactory method is to specify the formula for the general term and write X := (;n : n E N)or more simply X = (1/2n) . Another way of defining a sequence is to specify the value of Xl and give a formulafor xn+ l (n ::=: 1) in terms of xn More generally, we may specify Xl and give a formulafor obtaining xn +l from x l x2 " , xn Sequences defined in this manner are said to beinductively (or recursively) defined.3.1.2 Examples (a) If E JR, the sequence B := b . . . ), all of whose terms equal (b, b, b,b, is called the constant sequence Thus the constant sequence 1 is the sequence b.(1, 1 , 1 , . . . ), and the constant sequence ° is the sequence (0, 0, 0, . . . ).
54 CHAPTER 3 SEQUENCES AND SERIES(b) If b E �, then B := (bn ) is the sequence B = (b, b2 , b3 , • • • , bn , • • •) . In particular, ifb = �, then we obtain the sequence (�n .• n E N) = (�2 4� �8 . . . , �n . . .) . 2 2 (c) The sequence of (2n : n E N) of even natural numbers can be defined inductively byor by the definition Yl := 2, Yn+ l := Yl + Yn •(d) The celebrated Fibonacci sequence F := Un ) is given by the inductive definition 11 : = I, 12 := 1 , In+ 1 := In - 1 + In (n ::: 2).Thus each term past the second is the sum of its two immediate predecessors. The first tenterms of F are seen to be (1, 1 , 2, 3, 5, 8, 13, 21, 34, 55" , .) . DThe Limit of a SequenceThere are a number of different limit concepts in real analysis. The notion of limit of asequence is the most basic, and it will be the focus of this chapter.3.1.3 Definition A sequence X = (xn ) � in is said to converge to x E �, x or is said tobe a limit of if for every e > 0 there exists a natural number K (e) such that for all (xn ),n ::: K (e), the terms satisfy xn IXn - xl <e. If a sequence has a limit, we say that the sequence is convergent; ifit has no limit, wesay that the sequence is divergent.Note The notation K (e) is used to emphasize that the choice of K depends on the valueof e. However, it is often convenient to write K instead of K (e). In most cases, a "small"value of will usually require a "large" value of K to guarantee that the distance e IXn - xlbetween and is less than e for all n ::: K K (e). xn x = x, we will use the notation When a sequence has limit lim X = x orWe will sometimes use the symbolism xn -+ x, which indicates the intuitive idea that thevalues xn "approach" the number x as n -+ 00 .3.1.4 Uniqueness of Limits A sequence in lR. can have at most one limit.Proof. x x" Suppose that and are both limits of For each e > 0 there exist K such (xn ).that IXn - x i <e/2 for all n ::: K, and there exists K " such that e/2 for all IXn - x"I <n ::: K " . We let K be the larger of K and K " . Then for n ::: K we apply the TriangleInequality to get l x - x"I = lx - xn + xn - x"I :s lx - xn l + IXn - x"I < e/2 + e/2 = e.Since e > 0 is an arbitrary positive number, we conclude that x - x" = O. Q.E.D.
3.1 SEQUENCES AND THEIR LIMITS 55 For x E lR and e > 0, recall that the e-neighborhood of x is the set V/x) := {u E lR : lu - x l < e}.(See Section 2.2.) Since u E V/x) is equivalent to l u - x l < e, the definition of convergence of a sequence can be formulated in terms of neighborhoods. We give several differentways of saying that a sequence xn converges to x in the following theorem.3.1.5 Theorem LetX = (xn ) be a sequence ofreal numbers, andletx E R The followingstatements are equivalent.(a) X converges to x.(b) For every e there exists a natural number K such that for all n ::: K, the terms xn > 0,satisfy IXn- x l < e.(c) For every e there exists a natural number K such that for all n ::: K , the terms xn > 0,satisf x - e < x < + e. y n X(d) For every e -neighborhood Vg (x) of x, there exists a natural number K such that forall n ::: K, the terms xn belong to Vg (x).Proof. The equivalence of (a) and (b) is just the definition. The equivalence of (b), (c),and (d) follows from the following implications:lu - x l < e -¢:=> -e < u x < e -¢:=> x - e < u < x + e -¢:=> u E V/x). - Q.E.D. With the language of neighborhoods, one can describe the convergence of the sequenceX = (xn ) to the number x by saying: for each e-neighborhood Vg (x) of x, all but a finitenumber of terms of X belong to Vg (x). The finite number of terms that may not belong tothe e-neighborhood are the terms x l x2 • • • , XK - 1 .Remark The definition of the limit of a sequence of real numbers is used to verify that aproposed value x is indeed the limit. It does not provide a means for initially determiningwhat that value of x might be. Later results will contribute to this end, but quite often it isnecessary in practice to arrive at a conjectured value of the limit by direct calculation of anumber of terms of the sequence. Computers can be helpful in this respect, but since theycan calculate only a finite number of terms of a sequence, such computations do not in anyway constitute a proof of the value of the limit. The following examples illustrate how the definition is applied to prove that a sequencehas a particular limit. In each case, a positive e is given and we are required to find a K,depending on e, as required by the definition.3.1.6 Examples (a) lim(l/n) = o. . If e > 0 is given, then l /e > O. By the Archimedean Property 2.4.5, there is a natural number K = K(e) such that l/K < e. Then, if n ::: K, we have l/n � l/K < e.Consequently, if n ::: K, then � - 0 � < e. =Therefore, we can assert that the sequence ( l/n) converges to O.
56 CHAPTER 3 SEQUENCES AND SERIES(b) Let £ > 0 be 1» To find K, we first note that if N, then lim(1/(n2 + given. = o. n E 1 1 <1 n2 + 1 < - - -. -- n2 nNow choose K such that 1/K < £, as in (a) above. Then n � K implies that l/n < £, andtherefore 1_1+ 1 - 0 1 n2 1+ 1 < .!. < e . n2 _ __ n =Hence, we have shown that the limit of the sequence is zero. 3n + 2 )(c) lim ( n + l 3. = Given £ > 0, we want to obtain the inequality 3n + 2(1) 1 n + l _ 31 < £when is sufficiently large. We first simplify the expression on the left: 1 3n + 2 _ 3 1 1 3n + 2 - 3n - 3 1 1 2 1 _ < .!. . n = 1 n+l = n+l = n+l n+l nNowforthe inequalitywe also havesatisfied,£then the inequality (1)Therefore theiflimitKof<thethen if any3n � K, l/n < £ is l/n < and hence (1) holds. holds. Thus 1/ £,sequence is . n(d) If 0 < b < 1, then lim(b ) = o.we see that use elementary properties of the natural logarithm function. If £ > 0 is given, We will bn < £ In b < In £ {::::::} n {::::::} n > In £jIn b.that Klastexample,b,ifb reversed because.0In1 bbn<<0.)£ for allifnwe choose KweKhaveInnumber) such(The > inequality then we will have 0 < Thus � K. Thus to be a lim(bn O. For In £/ In is 8, and if £ is given, then we wouldneed > .01/ In.8 = � =. =20.6377. Thus K 2 1 would be an appropriate choice for £ .01 . = = 0in mind the connection between the £notionthe K is to think of aitsequence, one way toK £Remark The K £ Game ( ) the and of convergence of as a game called the keep In ( )Game. B challenges Player A asserts that a certain number x is thevalue for a£ sequence (xnAPlayerrespond to the challenge by coming up with a value specifisuchlimit of > O. Player ). In this game, this assertion by giving Player A a cmust value of that works,Kthen thatfor which Playerall ofn > K . If Player A can always findBacan give aKspecific value of £he>wins,Xnand the<sequence I -x I £ forcannot respondHowever, if PlayerPlayer B wins, and we conclude that the sequence doesis convergent. adequately, then 0 Anot converge to x. In order produce that a sequence>X0 such ) does not converge to the number K isis enough to can showaone numbern£0 satisfyingnthat � Kmatter whatIXnatural number (This tochosen, one find particular K (x no = n such that n -x I � £0. x, it Kwill be discussed in more detail in Section 3.4.) K
3.1 SEQUENCES AND THEIR LIMITS 573.1.7 Example The sequence (0, 2, 0, 2, . . . , 0, 2, . . . ) does not converge to thenumber O. If Player A asserts that 0 is the limit of the sequence, he will lose the K (e) Gamewhen Player B gives him a value of e < 2. To be definite, let Player B give Player Athe value eo = 1 . Then no matter what value Player A chooses for K, his response willnot be adequate, for Player B will respond by selecting an even number n > K. Then thecorresponding value is xn = 2 so that I Xn - 01 = 2 > 1 = eo. Thus the number 0 is not thelimit of the sequence. 0Tails of SequencesIt is important to realize that the convergence (or divergence) of a sequence X = (xn )depends only on the "ultimate behavior" of the terms. By this we mean that if, for anynatural number m, we drop the first m terms of the sequence, then the resulting sequenceXm converges if and only if the original sequence converges, and in this case, the limits arethe same. We will state this formally after we introduce the idea of a "tail" of a sequence.3.1.8 Definition If X = (X l x2 , · · · , xn , · · ·) is a sequence of real numbers and if m is agiven natural number, then the m-tail of X is the sequence Xm := (xm +n : n E N) = (xm + ! xm+2 . . . ) For example, the 3-tail of the sequence X = (2, 4, 6, 8, 10, . . . , 2n, . . . ), is the sequence X3 = (8, 10, 12, . . . , 2n + 6, . . . ).3.1.9 Theorem Let X = (xn : n E N) be a sequence ofreal numbers and letm E N. Thenthe m -tail Xm = (xm+n : n E N) of X converges if and only if X converges. In this case,lim Xm = lim X.Proof. We note that for any p E N, the pth term of Xm is the (p + m)th term of X.Similarly, if q > m , then the qth term of X is the (q - m)th term of Xm . Assume X converges to x . Then given any e > 0, if the terms of X for n 2: K (e)satisfy I Xn - X I , < e, then the terms of Xm for k 2: K (e) - m satisfy I Xk - X I < e. Thus wecan take Km (e) = K (e) - m, so that Xm also converges to x. Conversely, if the terms of Xm for k 2: Km (e) satisfy I Xk - x l < e, then the terms ofX for n 2: K(e) + m satisfy I Xn - x l < e. Thus we can take K (e) = Km (e) + m . Therefore, X converges to X if and only if Xm converges to X . Q.E.D. We shall sometimes say that a sequence X ultimately has a certain property if sometail of X has this property. For example, we say that the sequence (3, 4, 5, 5, 5, . . . , 5, . . . )is "ultimately constant". On the other hand, the sequence (3, 5, 3, 5, . . . , 3, 5, . . . ) is notultimately constant. The notion of convergence can be stated using this terminology: A sequence X converges to x if and only if the terms of X are ultimately in every e-neighborhoodof x. Other instances of this "ultimate terminology" will be noted below.Further Examples ________________________In establishing that a number x is the limit of a sequence (xn ), we often try to simplifythe difference I Xn - x l before considering an e > 0 and finding a K(e) as required by thedefinition of limit. This was done in some of the earlier examples. The next result is a moreformal statement of this idea, and the examples that follow make use of this approach.
3.2 LIMIT THEOREMS 63 The proof of (b) is now completed by taking Y to be the sequence (l/zn ) and using thefact that X . Y = (xn /zn ) converges to x(l/z) = x/z. Q.E.D. Some of the results of Theorem 3.2.3 can be extended, by Mathematical Induction, to afinite number of convergent sequences. For example, if A = (an ) B = (bn ), . . . , Z = (zn ) are convergent sequences of real numbers, then their sum A + B + . . . + Z = (an + bn +. . . + zn ) is a convergent sequence and(1) lim(an + bn + . . . + zn ) = lim(an ) + lim(bn ) + . . . + lim(zn )Also their product A B · . . Z := (an bn . . . zn ) is a convergent sequence and .(2)Hence, if k E N and if A = (an ) is a convergent sequence, then(3)We leave the proofs of these assertions to the reader.3.2.4 Theorem If X = (xn ) is a convergent sequence of real numbers and if xn 0 for 2:alln E N,then x = lim(xn ) 2: O.Proof. Suppose the conclusion is not true and that x < 0; then B : = -x is positive. SinceX converges to x, there is a natural number K such that x B < xn < X + B - for all n 2: K.In particular, we have xK < X + B = X + (-x) = O. But this contradicts the hypothesisthat xn 2: 0 for all n E N. Therefore, this contradiction implies that x 2: O. Q.E.D. We now give a useful result that is formally stronger than Theorem 3.2.4.3.2.5 Theorem If X (xn ) and Y = (Yn ) are convergent sequences ofreal numbers and = n E N, then lim(xn ) ::::: lim(yn ).ifxn ::::: Yn for allProof. Let zn := Yn - xn so that Z := (zn ) = Y X and zn 0 for all n E N. It follows - 2:from Theorems 3.2.4 and 3.2.3 that Q.E.D. The next result asserts that if all the terms of a convergent sequence satisfy an inequalityof the form a ::::: xn ::::: b, then the limit of the sequence satisfies the same ineqUality. Thusif the sequence is convergent, one may "pass to the limit" in an inequality of this type.3.2.6 Theorem If X = (xn ) is a convergent sequence and if a ::::: xn ::::: b for all n E N,then ::::: lim(xn ) a ;£ b.Proof. Let Y be the constant sequence (b, b, b, . . . ). Theorem 3.2.5 implies that lim X :::::lim Y = b. Similarly one shows that a ::::: lim X . Q.E.D. The next result asserts that if a sequence Y is squeezed between two sequences thatconverge to the same limit, then it must also converge to this limit.
68 CHAPTER 3 SEQUENCES AND SERIES16. (a) Give an example of a convergent sequence (xn) ofpositive numbers with lim(xn + 1 Ixn) = l . (b) Give an example of a divergent sequence with this property. (Thus, this property cannot be used as a test for convergence.)1 7. Let X = (xn) be a sequence of positive real numbers such that lim(xn + 1 lxn) = L > l . Show that X is not a bounded sequence and hence is not convergent.1 8. Discuss the convergence of the following sequences, where a , satisfy 0 < a < 1, > l . n b b (a) (n2a ), (b) Wlnn ),2 (c) Win!), (d) (n !/n ) n19. Let (xn) be a sequence of positive real numbers such that lim(x� / ) = L < l . Show that there n exists a number r with 0 < r < 1 such that 0 < xn < r for all sufficiently large n E Use this to show that lim(xn) = O. N. n20. (a) Give an example of a convergent sequence (xn) of positive numbers with lim(x� / ) = l . n (b) Give an example of a divergent sequence (xI!) of positive numbers with lim(x,: / ) = l . (Thus, this property cannot be used as a test for convergence.)2l. Suppose that (xn) is a convergent sequence and (Yn) is such that for any [; > 0 there exists M such that IXn - YI! I < [; for all n 2: M. Does it follow that (y,) is convergent?22. Show that if (xn) and (Yn) are convergent sequences, then the sequences (un) and (vn) defined by un := max{xn , Yn} and Vn := min{xn , Yn } are also convergent. (See Exercise 2.2. 16.)23. Show that if (xn), (yn), (zn) are convergent sequences, then the sequence (w,) defined by wn := mid{xn , Yn zn } is also convergent. (See Exercise 2.2. 17.)Section 3.3 Monotone SequencesUntil now, we have obtained several methods of showing that a sequence X xn ) of realnumbers is convergent: = (always) difficult touse Definition x.1.3 or multiple of3.1.5 terms in aThis is often (but not (i) We can do. Theorem directly. 3to converge tocanand employIXTheorem 3.1.10. the (ii) We 0, dominate n I by a - sequence an ) known (to beemploy Theorems 3.1.9,tails, aalgebraic combinations, absolute values, or squareknown (iii) We canbyidentify X as sequence obtained from other sequences that are roots, convergent taking 3.2.3, 3.2.9,and (iv) We can "squeeze" X between two 3.2.10. that converge to the same limit and or sequencesuse Theorem 3.2.7. the "ratio test" of Theorem 3 (v) We can use .2.11.Exceptoffor (iii), allandthesethen verifyrequire that we already know (or at least suspect) thevalueThere are manyof we methods that our suspicionisisnocorrect. candidate for the limit the limit, instances, however, in which there obviousof a sequence, even though a preliminary analysisresultssuggest thatused to show issequencethis and the next two sections, we shall establish ismay known. The convergenceintroduce In that can be method we a likely. inthis section istoeven thoughappliesscopethe limitthatnot monotone in the following sense. isis convergent more restricted value of than the methods we give in the next two, but it the inmuch easier employ. It to sequences are
74 CHAPTER 3 SEQUENCES AND SERIES Leonhard Euler Leonhard Euler (1707-1783) was born near Basel, Switzerland. His clergy man father hoped that his son would follow him into the ministry, but when Euler entered the University of Basel at age 14, his mathematical talent was noted by Johann Bernoulli, who became his mentor. In 1727, Euler went to Russia to join Johanns son, Daniel, at the new St. Petersburg Academy. There he met and married Katharina Gsell, the daughter of a Swiss artist. During their long marriage they had 13 children, but only five survived childhood. In 1741, Euler accepted an offer from Frederick the Great to join the Berlin Academy, where he stayed for 25 years. During this period he wrote landmark books on calculus and a steady stream of papers. In response to a request for instruction in science from the Princess of Anhalt-Dessau; he wrote a multi-volume work on science that became famous under the title Letters to a German Princess. In 1766, he returned to Russia at the invitation of Catherine the Great. His eyesight had deteriorated over the years, and soon after his return to Russia he became totally blind. Incredibly, his blindness made little impact on his mathematical output, for he wrote several books and over 400 papers while blind. He remained busy and active until the day of his death. Eulers productivity was remarkable: he wrote textbooks on physics, algebra, calculus, real and complex analysis, analytic and differential geometry, and the calculus of variations. He also wrote hundreds of original papers, many of which won prizes. A current edition of his collected works consists of 74 volumes.Exercises for Section 3.3 1 . Let X l := 8 and xn + l := � xn + 2 for n E N. Show that (xn) is bounded and monotone. Find the limit. 2. Let Xl > 1 and xn +l := 2 - Ijxn for n E N. Show that (xn) is bounded and monotone. Find the limit. 3. Let x, � 2 and xn +l := 1 + Fn-=-I for n E N. Show that (xn) is decreasing and bounded below by 2. Find the limit.4. Let X l := 1 and xn + l := .)2 + xn for n E N. Show that (xn) converges and find the limit. 5. Let Y, := .jp, where p > 0, and Yn + l := .)p + Yn for n E N. Show that (Y,) converges and find the limit. [Hint: One upper bound is 1 + 2.jp.] 6. Let a > 0 and let z, > O. Define zn+l := .)a + zn for n E N. Show that (z,) converges and find the limit. 7. Let x, := a > 0 and xn +l := xn + Ijxn for n E N. Deterrnine if (xn) converges or diverges. 8. Let (an ) be an increasing sequence, (b ) a decreasing sequence, and assume that a < b for all n E N. Show that lim(an) � lim(bn)� and thereby deduce the Nested Intervals Property 42 from the Monotone Convergence Theorem 3.3.2. 9. Let A be an infinite subset of lR that is bounded above and let u := sup A. Show there exists an increasing sequence (xn) with xn E A for all n E N such that u = lim(xn).10. Let (xn) be a bounded sequence, and for each n E N let sn := SUP{xk : k � n} and tn := inf{xk : k � n}. Prove that (sn) and (tn) are monotone and convergent. Also prove that if lim(sn) = lim(tn), then (xn) is convergent. [One calls lim(sn) the limit superior of (xn), and lim(tn) the limit inferior of (xn).]
76 CHAPTER 3 SEQUENCES AND SERIESA tail of a sequence (see 3.1 .8) is a special type of subsequence. In fact, the m-tailcorresponds to the sequence of indices n l = m + 1 , n 2 = m + 2" " , n k = m + k, · · · .But, clearly, not every subsequence of a given sequence need be a tail of the sequence. Subsequences of convergent sequences also converge to the same limit, as we nowshow.3.4.2 Theorem If a sequence X = (xn) of real numbers converges to a real number x,then any subsequence X = (xn ) of X also converges to X . kProof. Let c > 0 be given and let K(c) be such that if n ?: K(c), then IXn - x l < c.Since n l < n 2 < . . . < nk < . . . is an increasing sequence of natural numbers, it is easilyproved (by Induction) that nk ?: k. Hence, if k ?: K(c), we also have nk ?: k ?: K(c) sothat Ixnk - x I < c. Therefore the subsequence (xn k ) also converges to x. Q.E.D. n3.4.3 Example (a) lim(b ) = 0 if 0 <lb < 1 . n We have already seen, in Example 3.1 . 1 1 (b), that if 0 < b < 1 and if xn : = b , thenit follows from Bernoulli s Inequality that lim(xn ) = O. Alternatively, we see that since n 1 no < b < 1 , then xn+ 1 = b + < b = xn so that the sequence (xn) is decreasing. It isalso clear that 0 :::: xn :::: 1 , so it follows from the Monotone Convergence Theorem 3.3.2that the sequence is convergent. Let x : = limxn . Since (x2n) is a subsequence of (xn)it follows fromnTheorem 3.4.2 that x = lim(x2n) . Moreover, it follows from the relation nx2n = b2 = (b )2 = x� and Theorem 3.2.3 that x = lim(x2n) = (lim(xn) f = x2 .Therefore we must either have x = 0 or x = 1 . Since the sequence (xn) is decreasing andbounded above by b < 1 , we deduce that x = O.(b) lim(cl /n ) = 1 for c > 1 . This limit has been obtained in Example 3.1. 1 1 (c) for c > 0, using a rather ingeniousnargument. We give here an alternative approach for the case c > 1 . Note that if zn := c l / ,then zn > 1 and zn+ 1 < zn for all n E N. (Why?) Thus by the Monotone ConvergenceTheorem, the limit Z : = lim(zn ) exists. By Theorem 3.4.2, it follows that Z = lim(z2n) Inaddition, it follows from the relation n ln z2n = C I /2 = (c / ) I /2 = z�/2and Theorem 3.2.10 that 1 Z = lim(z2n ) = (lim(zn ») /2 = ZI /2 .Therefore we have � = z whence it follows that either z = 0 or z = 1 . Since zn > 1 for alln E N, we deduce that z = 1 . We leave it as an exercise to the reader to consider the case 0 < c < 1. 0 The following result is based on a careful negation of the definition of lim(xn) = x. Itleads to a convenient way to establish the divergence of a sequence.3.4.4 Theorem Let X = (xn) be a sequence of real numbers. Then the following areequivalent:(i) The sequence X = (xn) does not converge to x ER
3.4 SUBSEQUENCES AND THE BOLZANO-WEIERSTRASS THEOREM 77(ii) There exists an Co > 0 such that for any k E N, there exists nk E N such that nk :::: kand IXn - x l :::: Co k(iii) There exists an Co > 0 and a subsequence X = (xn ) of X such that IXn - x I :::: Co k kfor all k E N.Proof. (i) => (ii) If (xn) does not converge to x, then for some Co > 0 it is impossible tofind a natural number k such that for all n :::: k the terms xn satisfy IXn - x I < Co- That is,for each k E N it is not true that for all n :::: k the inequality IXn - x I < Co holds. In otherwords, for each k E N there exists a natural number n.k :::: k such that IXn k - x I :::: Co - (ii) => (iii) Let Co be as in (ii) and let n l E N be such that n I :::: I and l,xn I - x I :::: CoNow let n 2 E N be such that n 2 > n l and IXn - x l :::: co; let n 3 E N be such that n 3 > n 2 2and Ixn 3 - x I :::: Co Continue in this way to obtain a subsequence X = (xnk ) of X suchthat IXnk - x l :::: Co for all k E N. (iii) => (i) Suppose X = (xn ) has a subsequence X = (xn k ) satisfying the conditionin (iii). Then X cannot converge to x; for if it did, then, by Theorem 3.4.2, the subsequenceX would also converge to x. But this is impossible, since none of the terms of X belongsto the co-neighborhood of x. Q.E.D. Since all subsequences of a convergent sequence must converge to the same limit,we have part (i) in the following result. Part (ii) follows from the fact that a convergentsequence is bounded.3.4.5 Divergence Criteria If a sequence X = (xn ) of real numbers has either of thefollowing properties, then X is divergent.(i) X has two convergent subsequences X = (xn k ) and X" = (xrk ) whose limits are notequal.(ii) X is unbounded. n3.4.6 Examples (a)The sequence X := « _ I )) is divergent. The subsequence X : = « - 1 ) 2n ) = ( 1 , I , . . . ) converges to 1, and the subsequence n lX" := « _ 1) 2 - ) = (- 1 , - 1 , · · ·) converges to - 1 . Therefore, we conclude from Theorem 3.4.5(i) that X is divergent.(b) The sequence (1, !, 3, !, . . .) is divergent. This is the sequence Y = (Yn), where Yn = n if n is odd, and Yn = l/n if n is even.It can easily be seen that Y is not bounded. Hence, by Theorem 3.4.5(ii), the sequence isdivergent.(c) The sequence S := (sin n) is divergent. This sequence is not so easy to handle. In discussing it we must, of course, make useof elementary properties of the sine function. We recall that sin(rr /6) = ! = sin(5rr /6)and that sinx > ! for x in the interval II := (rr /6, 5rr /6). Since the length of II is 5rr /6 -rr16 = 2rr/3 > 2, there are at least two natural numbers lying inside II ; we let n l be thefirst such number. Similarly, for each k E N, sinx > ! for x in the interval Ik := (rr/6 + 2rr(k - 1), 5rr/6 + 2rr(k - l) ) .Since the length of Ik is greater than 2, there are at least two natural numbers lying insideIk ; we let n k be the first one. The subsequence S := (sin n k ) of S obtained in this way hasthe property that all of its values lie in the interval [!, 1].
78 CHAPTER 3 SEQUENCES AND SERIES Similarly, if k E N and k is the interval (7n /6 + 2n(k - 1), l In /6 + 2n(k - 1)) , k :=then it is seen that sinx < -! for all x E k and the length of k is greater than 2. Let mkbe the first natural number lying in k . Then the subsequence S" := (sinmk ) of S has theproperty that all of its values lie in the interval [- 1, -!]. Given any real number c, it is readily seen that at least one of the subsequences Sand Sil lies entirely outside of the !-neighborhood of c. Therefore c cannot be a limit of S.Since c E lR is arbitrary, we deduce that S is divergent. DThe Existence of Monotone SubsequencesWhile not every sequence is a monotone sequence, we will now show that every sequencehas a monotone subsequence.3.4.7 Monotone Subsequence Theorem IfX = (xn) is a sequence ofreal numbers, thenthere is a subsequence of X that is monotone.Proof. For the purpose of this proof, we will say that the mth term xm is a "peak" ifxm � xn for all n such that n � m. (That is, xm is never exceeded by any term that followsit in the sequence.) Note that, in a decreasing sequence, every term is a peak, while in anincreasing sequence, no term is a peak. We will consider two cases, depending on whether X has infinitely many, or finitelymany, peaks. Case 1: X has infinitely many peaks. In this case, we list the peaks by increasing 2subscripts: xm ] , xm , · · · , xmk , . . . . Since each term is a peak, we have xm ] > xm2 - · · · - Xmk - . . . . - > > >Therefore, the subsequence (xmk ) of peaks is a decreasing subsequence of X. Case 2: X has a finite number (possibly zero) of peaks. Let these peaks be listed by 1increasing subscripts: xm 1 , xm , . . . , Xm Let s : = mr + I be the first index beyond the last • 2 rpeak. Since xS1 is not a peak, there exists s2 > s 1 such that xS ] < xS2 . Since xS2 is not a peak, S3 s3there exists > s2 such that x82 < x . Continuing in this way, we obtain an increasingsubsequence (xsk ) of X. Q.E.D. It is not difficult to see that a given sequence may have one subsequence that isincreasing, and another subsequence that is decreasing.The Bolzano-Weierstrass TheoremWe will now use the Monotone Subsequence Theorem to prove the Bolzano-WeierstrassTheorem, which states that every bounded sequence has a convergent subsequence. Becauseof the importance of this theorem we will also give a second proof of it based on the NestedInterval Property.3.4.8 The Bolzano-Weierstrass Theorem A bounded sequence of real numbers has aconvergent subsequence.First Proof. It follows from the Monotone Subsequence Theorem that if X = (xn) isa bounded sequence, then it has a subsequence X = (xn k ) that is monotone. Since this
3.4 SUBSEQUENCES AND THE BOLZANO-WEIERSTRASS THEOREM 79that the subsequencebounded, it follows from the Monotone Convergence Theorem 3.3.2subsequence is also is convergent. Q.E.D.interval II [a,Since the setnof values {xn : n E N} is bounded, this set is contained in anSecond Proof. b]. We take 1. We:now bisect Itwo parts: lequal subintervals I{ and Ir, and divide the set of indices := :={n E N n > I } into I into two A l := {n E N : n > nl xn E I{}, BI = {n E N : n > n l xn E In. If.2.1.)isIfinfinite, finitetake 1then BI{ mustlet ninfibeite, and we take 1 numberlet nA Ibe(See1 Al we : = and A l is a set, 2 I be 2 n the smallest natural I{ and in the :=smallest now bisect 1 into Btwo equal subintervals I� and I;, and2 divide the set {n E N : natural number in I 2 Wen > n2 } into two parts: 2IfnA2 is infinite,Bwemust be3 infinite, and we3 takethe smallestand let n number smallest naturalafi ite set,inthen 2 take 1 I� and let n be 13 I; natural3 be the in A2 . If A2 is := :=number continue in this way to obtain a sequence of nested intervals I 1 B2 • We a subsequence (x of X such that xn E I for k E No Since the2length of IIk is ;2 ;2 I ;2 . . . ;2. . . and n) k kequalI tofor all ka)/2N.-Moreover, since xTheoremkboth belong to is ,awe have common point E (b - k l , it follows from E and 2.5.3 that there (unique) I� k nk � k IXn k - � I :::: (b - a)/2k- l ,whence it follows that the subsequence (xnk ) of X converges to �. Q.E.D.because there is3.4.8 isthatversion ofcalled dealsBolzano-WeierstrassinTheoremjor sequences, Theorem anothersometimes it that the with bounded sets IR (see Exercise 1 1 .2.6). It is readily seen bounded sequence can have various subsequences that convergeto different limits-1, otherasubsequencesexample, the sequenceand_it1)n ) subsequences thatthat converge to or even diverge. For that converge to 1, « has has subsequences +diverge. X be a sequence of real numbers and let X be a subsequence of X. Then X is a Letof X, then is also a subsequence hassequence initits own right, and so it of X.subsequences. We note that if X" is a subsequence X3.4.9 Theorem Let = (xn) be a bounded sequence ofreal numbers and let x IR have Xthe property that every convergent subsequence of converges to x. Then the sequence E Xconverges to x.If.X doesSuppose M >)0ofisXa such that the sequence X sothat thereI exist for > 0nandN.aProof. notXconverge to x, boundTheorem 3.4.4 implies that IXn M Co all E subsequence (xnk = then for :::: (1) IXnk - x l � Co forall k E N.WeierstrassaTheorem implies X, the numberconvergent asubsequenceXX". Sincethe Bolzanoa Since X is subsequence of that X has a M is also bound for . Hence X" is alsocsubsequence of X,ofitx,converges to x by1).hypothesis. Thus, its terms ultimately belong to the o-neighborhood contradicting ( Q.E.D.
80 CHAPTER 3 SEQUENCES AND SERIESExercises for Section 3.4 1 . Give an example of an unbounded sequence that has a convergent subsequence. 2. Use the method of Example 3.4.3(b) to show that if 0 < c < then lim(c l lli ) = 1 . 1, 3 . Let Un ) be the Fibonacci sequence of Example 3.1.2(d), and let Xli := 1n+ I/1n Given that lim(xn) = L exists, determine the value of L. 4. Show that the following sequences are divergent. (a) ( 1 - (_l)n + l/n) , (b) (sin nn/4). Let = 5. X (xn ) and Y = ( ,)be given sequences, and let the "shuffled" sequence Z = (zn )be zl xI Zz Y YI, " , zZn _ 1 xn zZn defined by : = := := := Y" Show that Z is convergent if and , · · ·· only if both and Y are convergent and lim = lim Y. X X 6. Let X := n lin for n E N. < x" if and only if 0 + n (a) Show that x < and infer that the inequality is valid l/n)" n, n n+1 for 2: 3. (See Example 3.3.6.) Conclude that (xn) is ultimately decreasing and that X := exists. lim(xn ) (b) Use the fact that the subsequence also converges to x to conclude that = 1 . (xz) x 7 . Establish the convergence and find the limits of the following sequences: (a) ( (1 + l/nZZ)nZ} (b) ((1 + 1/2n)n ) , (d) (0 + 2/n) n . (c) (0 + 1/n ) n ) , 2 ) 8. Determine the)limits of the following. ( 2 3) (a) 3n) 1 / " , (b) (0 + 1/2n) n . = 9. Suppose that every subsequence of X (x,) has a subsequence that converges to O. Show that = lim X O.10. Let (x.) be a bounded sequence and for each n E N let sn := sup{xk : k n} and S := inf{sli } 2: Show that there exists a subsequence of (xn ) that converges to S. 2: (1 1 . Suppose that x" 0 for all n E N and that lim _1)" xJ exists. Show that (Xli ) converges.12. Show that if (xn ) is unbounded, then there exists a subsequence (x ) such that lim(1/x ) = O. � �13. If xn := (_I) n /n, find the subsequence of (x.) that is constructed in the second proof of the := Bolzano-Weierstrass Theorem 3.4.8, when we take II [- 1 , 1]. S S14. Let (xn ) be a bounded sequence and let := sup{xn : n E N). Show that if � {xn : n E N}, then there is a subsequence of (x,, ) that converges to s.15. Let (In ) be a nested sequence of closed bounded intervals. For each n E N, let xn E In Use the Bolzano-Weierstrass Theorem to give a proof of the Nested Intervals Property 2. 5 . 2 .16. Give an example to show that Theorem 3.4.9 fails ifthe hypothesis that X is a bounded sequence is dropped.Section 3.5 The Cauchy CriterionThe Monotone Convergence Theorem is extraordinarily useful and important, but it has thesignificant drawback that it applies only to sequences that are monotone. It is important forus to have a condition implying the convergence of a sequence that does not require us toknow the value of the limit in advance, and is not restricted to monotone sequences. TheCauchy Criterion, which will be established in this section, is such a condition.
3.5 THE CAUCHY CRITERION 813.5.1 Definition A sequence X = (xn ) of real numbers is said to be a Cauchy sequenceif for every e > 0 there exists a natural number H(e) such that for all natural numbersn, m ::: H(e), the terms xn xm satisfy IXn - xm I < e . The significance of the concept of Cauchy sequence lies in the main theorem of thissection, which asserts that a sequence of real numbers is convergent if and only if it is aCauchy sequence. This will give us a method of proving a sequence converges withoutknowing the limit of the sequence. However, we will first highlight the definition of Cauchy sequence in the followingexamples.3.5.2 Examples (a) The sequence (l/n) is a Cauchy sequence. If e > 0 is given, we choose a natural number H = H(e) such that H > 2/e. Thenif m , n ::: H, we have 1/ n S 1/ H < e /2 and similarly 1/ m < e /2. Therefore, it followsthat if m , n ::: H, then I� � I - �n + � < 2 + 2 n - < m m � � = e.Since e > 0 is arbitrary, we conclude that (l / n ) is a Cauchy sequence.(b) The sequence (1 + (_ l)n ) is not a Cauchy sequence. The negation of the definition of Cauchy sequence is: There exists Co > 0 such that forevery H there exist at least one n > H and at least one m > H such that IXn - xm I ::: Co .For the terms xn := 1 + ( - I t we observe that if n is even, then xn = 2 and xn+ l = o. If ,we take Co = 2, then for any H we can choose an even number n > H and let m := n + 1to get IXn - xn+ l l = 2 = Co ·We conclude that (xn ) is not a Cauchy sequence. DRemark We emphasize that to prove a sequence (xn ) is a Cauchy sequence, we maynot assume a relationship between m and n, since the required inequality IXn - xm I < emust hold for all n, m ::: H (c). But to prove a sequence is not a Cauchy sequence, we mayspecify a relation between n and m as long as arbitrarily large values of n and m can bechosen so that IXn - xm I ::: Co · Our goal is to show that the Cauchy sequences are precisely the convergent sequences.We first prove that a convergent sequence is a Cauchy sequence.3.5.3 Lemma If X = (xn ) is a convergent sequence ofreal numbers, then X is a Cauchysequence.Proof. If x := lim X, then given e > 0 there is a natural number K(e/2) such that if K(e/2) then IXn - x l < e/2. Thus, if H(e) K(e/2) and if n, ::: H(e), then wen- ::: := mhave IXn - xm l = I (xn - x) + (x - xm ) 1 S IXn - xl + IXm - x l < e/2 + 8/2 = 8.Since 8 > 0 is arbitrary, it follows that (xn ) is a Cauchy sequence. Q.E.D.
82 CHAPTER 3 SEQUENCES AND SERIES In order to establish that a Cauchy sequence is convergent, we will need the followingresult. (See Theorem 3.2.2.)3.5.4 Lemma A Cauchy sequence ofreal numbers is bounded.Proof. Let X := (xn ) be a Cauchy sequence and let 8 := 1. If H := H(l) and n ::: H,then IXn - XH I < 1. Hence, by the Triangle Inequality, we have IXn I ::: IxH I + 1 for alln ::: H. If we set M := sup { lX I I , Ix2 1 , " " IxH_ I I , IXH I + I } ,then it follows that IXn I M for all n E N. ::: Q.E.D. We now present the important Cauchy Convergence Criterion.3.5.5 Cauchy Convergence Criterion A sequence ofreal numbers is convergent if andonly if it is a Cauchy sequence.Proof. We have seen, in Lemma 3.5.3, that a convergent sequence is a Cauchy sequence. Conversely, let X = (xn) be a Cauchy sequence; we will show that X is convergent tosome real number. First we observe from Lemma 3.5.4 that the sequence X is bounded. kTherefore, by the Bolzano-Weierstrass Theorem 3.4.8, there is a subsequence X = (xn )of X that converges to some real number x*. We shall complete the proof by showing thatX converges to x* . Since X = (xn) is a Cauchy sequence, given 8 > 0 there is a natural number H(8/2)such that if n, m ::: H(8/2) then(1) kSince the subsequence X = (xn ) converges to x*, there is a natural number K ::: H (8/2)belonging to the set {n p n2 } such that , • • • IXK - x* 1 < 8/2.Since K ::: H(8/2), it follows from (1) with m = K that IXn - xK I < 8/2 for n ::: H (8/2).Therefore, if n ::: H (8/2), we have IXn - x* 1 = I (xn - xK ) + (xK - x*) 1 ::: IXn - xK I + IXK - x* 1 < 8/2 + 8/2 = 8.Since 8 > 0 is arbitrary, we infer that lim(xn) = x*. Therefore the sequence X is convergent. Q.E.D. We will nqw give some examples of applications of the Cauchy Criterion.3.5.6 Examples (a) Let X = (xn) be defined by and for n > 2.
86 CHAPTER 3 SEQUENCES AND SERIESExercises for Section 3.5 1. Give an example of a bounded sequence that is not a Cauchy sequence. ( ) ) 2. Show directly from the definition that the following are Cauchy sequences. (a) n+1 n , 2! ( (b) 1 + � + . . . + � . n! (n + : ) 3. Show directly from the definition that the following are not Cauchy sequences. n (a) ( _ I)n ) , (b) (_ ) . (c) (in n ) . 4. Show directly from the definition that if (xn ) and (Yn ) are Cauchy sequences, then (xn + Yn ) and (xn yn ) are Cauchy sequences. 5. If xn := yn, show that (xn ) satisfies lim IXn+ 1 - xn I = 0, but that it is not a Cauchy sequence. 6. Let p be a given natural number. Give an example of a sequence (xn ) that is not a Cauchy sequence, but that satisfies lim Ixn +p - xn I = O. 7. Let (xn ) be a Cauchy sequence such that xn is an integer for every n E N. Show that (xn ) is ultimately constant. 8. Show directly that a bounded, monotone increasing sequence is a Cauchy sequence. 9. If 0 < r < 1 and IXn +1 - xn I < rn for all n E N, show that (xn ) is a Cauchy sequence.10. If Xl < Xz are arbitrary real numbers and xn := 1 (Xn _Z + xn - l ) for n > 2, show that (xn ) is convergent. What is its limit?1 1. If Yl < Yz are arbitrary real numbers and Yn := bn - l + � Yn -z for n > 2, show that (Yn ) is convergent. What is its limit? -l12. If X l > 0 and xn + l := (2 + xn ) for n ?: 1 , show that (xn ) is a contractive sequence. Find the limit.13. If X l := 2 and xn + l := 2 + l /xn for n ?: 1 , show that (xn ) is a contractive sequence. What is its limit?14. The polynomial equation X3 - 5x + 1 = 0 has a root r with 0 < r < 1. Use an appropriate contractive sequence to calculate r within 10-4 .Section 3.6 Properly Divergent SequencesFor certain purposes±oo".convenient to define what is meant for a sequence (xn ) of realnumbers to "tend to it is3.6.1 Definition Let (xn ) be a sequence of real numbers.(i) We say that (xn ) tends to exists a natural number K (ex) such that if nlim(xn(ex), then xnif>for every ex E � there and write 2: K ) +00, = +00, ex.(ii) We say that (xn ) tends to exists a natural number K ([3) such that if nlim(xn([3), then xnif<f�r every [3 E � there and write 2: K ) -00, = -00, [3. We say that (xn ) is properly divergent in case we have either lim(xn ) or = +00lim(xn ) = -00.
3.6 PROPERLY DIVERGENT SEQUENCES 87venient notationshould lim(xn)expressions.using thethat have beentrue wheninlim(xn)as=a±oo. The reader in the realize that we are Results symbols +00 proved purely sectionsfor conventional limits above = L (for L E �) may not remain and -00 earlier con3.6.2 fact, if a � limen) = +00. InExamples E (a)is given, let K (a) be any natural number such that K (a) > a.(b) lim(n2 ) = +00. If K (a) is a natural number such that K (a) > a, if n ::: K (a) then we haven2 ::: n > a. and n(c) If e > 1, then lim(e ) = +00. Let e = 1 + b, where b > O. If a E � is given, let K (a) be a natural number such thatK (a) > a/b . If n ::: K (a) it follows from Bernoulli s Inequality that en = (1 + b)n ::: 1 + nb > 1 + a > a.Therefore lim(en ) = +00. oseenMonotone sequences are particularly simple.2in regard to theirsequence is convergent in the Monotone Convergence Theorem 3.3 that a monotone convergence. We haveif and only if it is bounded. The next result is a reformulation of that result.3.6.3 Theorem A monotone sequence of real numbers is properly divergent if and onlyif it is unbounded.(a) If (xn) is an unbounded increasing sequence, then lim(xn) = +00.(b) If (xn) is an unbounded decreasing sequence, then lim(xn) -00. =Proof. (a) Suppose that (xn ) is an increasing sequence. We know that if (xn ) is bounded,then ait < ,xn a But since(xn) is is increasing,then for any a Exn�for all existsn(a). )Since suchthattrary, convergent.that lim(xn)unbounded, we have a < there n n (a E N a is is If (xn) ()arbi it follows . = +00. ::: Part (b) is proved in a similar fashion. Q.E.D.properly following "comparisonimplicitly used it in Example in 6showing that a sequence is The divergent. [In fact, we theorem" is frequently used 3. .2(c).]3.6.4 Theorem Let(xn) (Yn) and be two sequences of real numbers and suppose that(1) for all n E N.(a) If lim(xn) = +00, then lim(yn) = +00.(b) If lim(Yn) -00, then lim(xn) = - 00. =Proof. (a) If lim(xn) = +00, and if a E � is given, then there exists a natural numberK(a) such that if n ::: K(a), then a < xn. In view of (1), it follows that a < Yn for alln ::: K (a). Since a is arbitrary, it follows that lim(yn) = +00. The proof of (b) is similar. Q.E.D.there exists m ETheoremthat6xn remains truenif:::condition (1) is ultimately true; that is, ifRemarks (a) N such 3. .4 for all YnS m.
88 CHAPTER 3 SEQUENCES AND SERIES(b) If condition (1) of Theorem 3.6.4 holds and if lim(yn ) +00, it does not follow =that lim(xn-00.+00. Similarly, if (3.6.4 to show that a sequence tends to +00not follow thatlim(Yn ) ) In using Theorem 1 ) holds and if lim(xn ) -00, it does [respectively, = = =less] thanneed to show that the terms ofterms of a sequence that is known to[respectively,-00] we or equal to the corresponding the sequence are ultimately greater tend to +qo[respectively, -00].comparisonit theorem" is often more convenient toinequality such as (1), the following "limit Since is sometimes difficult to establish an use than Theorem 3.6.4.3.6.5 Theoremthat for some Let(xn ) and (Yn ) L E JR, L > 0, be two sequences ofpositive real numbers we have and suppose(2)Then lim(xn) +00 and lim(yn ) +00. = if only if =Proof. If (2) holds, there exists K E N such that for all n :::: K.a slight modification Y < xn < 3 L) n for all :::: details conclusionHence we have G L) ofn Theoremn.6.4.YWe leaventhe K. The to the reader.now follows from Q.E.D.However,reader can some partial results that can benot hold if either L cases, L will be The there are show that the conclusion need established in these 0 or as +00. = =seen in the exercises.Exercises for Section 3.6 1 . Show that if (xn ) is an unbounded sequence, then there exists a properly divergent subsequence. 2. Give examples of properly divergent sequences (xn ) and (Yn ) with Yn =1= 0 for all n E N such that: (a) (xn /yn ) is convergent, (b) (xn /yn ) is properly divergent. 3. Show that if xn > 0 for all n E N, then lim(xn ) = 0 if and only if lim(1/xn ) = +00. 4. Establish the proper divergence of the following sequences. (a) (JiI), (b) (In"TI), (c) (vn=!), (d) (n/vlITI). 5. Is the sequence (n sin n) properly divergent? 6. Let (xn ) be properly divergent and let (Yn ) be such that lim(xn yn ) belongs to JR. Show that (Yn ) converges to O. 7. Let (xn ) and (Yn ) be sequences of positive numbers such that lim(xn /yn ) = O. (a) Show that if lim(xn ) = +00, then lim(yn ) = +00. (b) Show that if (Yn ) is bounded, then lim(xn ) = O. 8. Investigate the convergence or the divergence of the following sequences: (a) (Jn 2 + 2) , (b) (Jil/ (n 2 + 1)), (c) (Jn 2 + 1/JiI) , (d) (sin Jil). 9. Let (xn ) and (Yn ) be sequences of positive numbers such that lim(xn /yn ) = +00, (a) Show that if lim(yn ) = +00, then lim(xn ) = +00. (b) Show that if (xn ) is bounded, then lim(yn ) = O.to. Show that if lim(an /n) = L, where L > 0, then lim(an ) = +00.
3.7 INTRODUCTION TO INFINITE SERIES 89Section 3.7 Introduction to Infinite SeriesWe will nowdiscussedbriefmore detail in to infinite9, seriesbecause of its importance, wetopicthat will befewgive a in introduction Chapterbe seen toofbereal numbers.consequenceswill but immediate This is a ofestablish awe have met in thisThese results willtheorems results here. infinite series is sometimes "defined" to be "an expression of In elementary texts, an chapter.the form"(1)However, thisthis array of symbols, which calls forisana infinite no particularadditions to wecan attach to "definition" lacks clarity, since there priori number of value that beperformed.3.7.1 Definition If X := (xn ) is a sequence in �, then the infinite series (or simply theseries) generated by X is the sequence S := (S ) defined by k Sj := xj s2 := Sj + x2 (= xj + x2 )The numbers xn are called the terms of the series and the numbers sk are called the partialthe sum thisthe valueIfof thisexists, we this that this series exist, we say that the serieslimitsums of or series. lim S series. If say limit does not is convergent and call this S isdivergent. It is convenient to use symbols such as(2) or orto denote both the infinite series S generated by the sequence X = (xn ) and also to denotethe valueexhibitingcaseinfinite series whose convergence orindivergencebeisregarded merely asa way of lim S, in an this limit exists. Thus the symbols (2) may to be investigated.understood that double use of these notationsthatofnot lead to mustisbe established. it isIn practice, this the convergence (or divergence) the series any confusion, provided does Just as a sequence may be indexed such its first element not xj, but is O or Xsor x99 we will denote the series having these numbers as their first element by the xsymbols or or NIt shouldby snoted that when the first term in the series is x then the first partial sum isdenoted be N.In nonmathematical language, guard words are interchangeable; however, in mathematics,.Warning The reader should these against confusing the words "sequence" and "series"
3.7 INTRODUCTION TO INFINITE SERIES 93the converges.(s(It)isoffar fromsums converges, limit of this seriesalternatingtoharmonic series(7) sequence n partial obvious that the proving that the is equal In 2.) 0Comparison TestsOur first test showsa convergentterms ofthennonnegative seriesconvergent. by the corresponding terms of that if the series, a the first series is are dominated3.7.7 Comparison Test Let X := (xn ) and Y := (Yn ) be real sequences and suppose thatfor some KEN we have(8) for n � K.(a) Then the convergence of L Yn implies the convergence of L Xn .(b) The divergence of L xn imples the divergence of L Yn Proof. (a) Suppose that L Yn converges and, given e let M(e) E N be such that if > 0,m > n � M(e), then Yn + l + . . . + Ym < e.If m > sup{K, M(e)}, then it follows that 0 .::: xn+ l + . . . + xm .::: Yn+l + . . . + Ym < e,from which the convergence of L xn follows.(b) This statement is the contrapositive of (a). Q.E.D.quently very is sometimes difficult to establish the inequalities (8), the next result is fre Since it useful.3.7.8 Limit Comparison Test Suppose that X := (xn ) and Y := (Yn ) are strictlypositivesequences and suppose that the following limit exists in lR:(9) r := lim (�:) .(a) If r =I 0 then L xn is convergent if and only if L Yn is convergent.(b) Ifr = 0 and if L Yn is convergent, then L Xn is convergent.Proof. (a) It follows from (9) and Exercise 3.1.17 that there exists K E N such that!r xn /Yn .::: 2r for n � K, whence .::: for n � K.If we apply the Comparison Test 3.7.7 twice, we obtain the assertion in (a).(b) If r = 0, then there exists K E N such that for n K, �so that Theorem 3.7 .7(a) applies. Q.E.D.Remark The Comparison Tests 3. 7 .7 and 3.7 . 8 depend on having a stock of series thatone knowsthis be convergent (or divergent). The reader will find that the p-series is oftenuseful for to purpose.
3.7 INTRODUCTION TO INFINITE SERIES 95Exercises for Section 3.7 1. Let L an be a given series and let L bn be the series in which the terms are the same and in the same order as in L an except that the terms for which an = 0 have been omitted. Show that L an converges to A if and only if L bn converges to A. 2. Show that the convergence of a series is not affected by changing afinite number of its terms. (Of course, the value of the sum may be changed.) 00 00 3. By using partial fractions, show that 1 1 � 1 (a) = 1, (b) L = - > 0, if a > O. (n + 1)(n + 2) n =O (ex + n)(ex + n + 1) ex 00 � n(n + 1)(n 1 1 (c) = + 2) 4 4. If L xn and L Yn are convergent, show that L(Xn + Yn) is convergent. 5. Can you give an example of a convergent series L xn and a divergent series L Yn such that L(xn + Yn) is convergent? Explain. 00 6. (a) Show that the series L cos n is divergent. 00 n =1 (b) Show that the series L (cos n) / n 2 is convergent. 00 ( _ 1) n =1 n 7. Use an argument similar to that in Example 3.7.6(f) to show that the series convergent. � ..;n is 8. If L an with an > 0 is convergent, then is L a� always convergent? Either prove it or give a counterexample. 9. If L an with an > 0 is convergent, then is L ;a;. always convergent? Either prove it or give a counterexample.10. If L an with an > 0 is convergent, then is L Janan+ 1 always convergent? Either prove it or give a counterexample.11. If L an with an > 0 is convergent, and if bn := (al + . . . + an)/n for n E N, then show that L bn is always divergent. 0012. Let L a(n) be such that (a(n)) is a decreasing sequence of strictly positive numbers. �f s(n) n =1 n denotes the nth partial sum, show (by grouping the terms in s (2 ) in two different ways) that n n n n n n � (a(1) + 20(2) + . . . + 2 a(2 )) :s s(2 ) :s (a (1) + 2a(2) + . . . + 2 - 1 a (2 - I )) + a(2 ). 00 00 n n Use these inequalities to show that L a (n) converges if and only if L 2 a(2 ) converges. n =1 n =1 00 This result is often called the Cauchy Condensation Test; it is very powerful.13. Use the Cauchy Condensation Test to discuss the p-series L (1/nP) for p > O. n =114. Use the Cauchy Condensation Test to establish the divergence of the series: 1 1 (a) L - (b) " n ln n � n(ln n)(1n ln n) (c) L --,,----.,--------: . : 1 n(ln- n)(ln In n)(ln In In n)15. Show that if c > 1, then the following series are convergent: 1 1 (a) " -- , (b ) " . � n(ln n)C � n(ln n)(lnln n)C
CHAPTER 4 LIMITSwhich systematic use is made of various limitingto of a sequencearea ofnumbers. In this"Mathematical analysis" is generally understood concepts. In the preceding chapter we refer to that mathematics inchapter one willthese basic limiting concepts:limitlimita function. of realstudied we of encounter the notion of the the of The rudimentary notion of a limiting process struggled in the 1680s as Isaac Newton (1646-1716) emerged the other and their creative(1642-1727) and GottfriedsLeibniz was initially unknown towith the creation of the Calculus. Thoughquite different, both realized the need to fonnulate a notion of function and theinsightsquantities being "closeworkone another. Newton used the word "fluent" to denote aidea of were each person to"relationship between variables, and inby any given difference,iabut neverhe discussed nor in"to which they approach nearer thandiminishedwork Princip Leibniz introduced thelimits his major in 1687 go beyond,effect attaintoto,indicate quantities are depended in infinitum". and he invented "infinites"function" till the a quantity that on a variable, tennimallybecame standard terminology, handling the concept of a limit. The tenn "function"soonnewsmall" numbers as a way of and Leibniz also introduced the tenn "calculus" forthis In 1748, Leonhard Euler (1707-1783) published his two-volume treatise Introductio in method of calculation.Analysin Infinitorum, in which he discussed power series, the exponential and logarithmicfunctions, the trigonometric functions, and the three-volume Institutiones followedIntegralistutiones Calculi Differentialis in 1755 and many related topics. This was Calculi by Instithe1768-70.ofoflimitworksvery intuitive and its loosenessotherto acalculus forproblems.era, Butin concept These was remained the standard textbooks on number of many years.descriptions the to provide the basis for rigorous proofs.mathematicians of the Verbal limit concept were proposed by led butnoneInwas adequate 1821, Augustin-Louis Cauchy (1789-1857) published his lectures on analysis in hisCours dAnalyse, which set the standard for mathematical exposition for many years. Hediscourse. Hebut thericoncept ofitions still remained elusive. ofanwith greater mathematicalwas concernedfonnulated and nin manyand presentedthe level precision in care than hispredecessors, with gor defi limit ways raised arguments In early chapter he gave thefollowing definition: If the successive values attributed to the same variable approach indefinitely a fixcalled thesuch thatall thefinally differ from it by as little as one wishes, this latter is ed value, limit of they others.strassTheisfithe one weHefonnulating precise language andofrigorous proofs, and his defiWeier (1815-1897). inuse today. on a precise definition limit were taken by Karl nition nal steps insistedof limit96
4.1 LIMITS OF FUNCTIONS 97 Gottfried Leibniz Gottfried Wilhelm Leibniz (1646-- 1 716) was born in Leipzig, Germany. He was six years old when his father, a professor of philosophy, died and left his son the key to his library and a life of books and learning. Leibniz entered the University of Leipzig at age 15, graduated at age 17, and received a Doctor of Law degree from the University of Altdorf four years later. He wrote on legal matters, but was more interested in philosophy. He also developed original theories about language and the nature of the universe. In 1672, he went to Paris as a diplomat for four years. While there he began to study mathematics with the Dutch mathematician Christiaan Huygens. His travels to London to visit the Royal Academy further stimulated his interest in mathematics. His background in philosophy led him to very original, though not always rigorous, results. Unaware of Newtons s unpublished wo.k, Leibniz published papers in the l680s that pre sented a method of finding areas that is known today as the Fundamental Theorem of Calculus. He coined the term "calculus" and invented the dy /dx and elongated S notations that are used today. Unfortunately, some followers of Newton accused Leibniz of plagiarism, resulting in a dispute that lasted until Leibniz s death. Their approaches to calculus were quite different and it is now evident that their discoveries were made independently. Leibniz is now renowned for his work in philosophy, but his mathematical fame rests on his creation of the calculus.Section 4.1 Limits of Functionsidea ofsectionclosewillf(but different from)atcthenotion ofis that thetovaluesaftechnicalintuitiveIn this the function introduceathe important point c the limit of a function. The close toL when x is we having limit L (x) are to of "close to" and thisBut it is necessary in have8-8 definitionway ofworking with the idea . is accomplished the givenbelow.necessary thatforedthe defined atpoints closea to cIttoneedf notthepoint nced at the point c, isitthe In order be idea of thepoints of function at abe defi to be meaningful,but is limit nearshould for the following definition. c. make study interesting. This it be definf at enoughreasonthere exists at least Let A S; JR.. EA A, x =ftccEsuchisthatclusterc lpoint of A if for every 8 > 04.1.1 Definition one point x point JR. a x - < 8. Ia cluster point of theisdistinctiffrominc8-neighborhood neighborhoods8,asc follows:cAcontainsc at This definition set A every the language of V8(c) (c - 8) of point isleast one point of A rephrased = + .Note The point c may or may not be a member of A, but even if it is in A, it is ignoredbe points in V8(c) n A distinctcluster cpointorder fornot, sincecluster point of A. that therewhen deciding whether it is a from in of A or c to be a we explicitly require� := Forgives a neighborhood of 1then the point noispoints cluster point of A, since choosing ! example, if A {l, 2}, that contains 1 not a of A distinct from 1 . The same is :=true for the point 2, so we see that A has no cluster points.4.1.2 Theorem A number cE JR. is a cluster point of a subset of A JR. if and only if thereexists a sequence (an ) A in such that lim(an ) c an =ft c n E N. = and for all
98 CHAPTER 4 LIMITScontainslim(ais) aonec.pointpoint A distinct fromanyThen aN E A,(lln)-neighborhoodc I V<I /nI(c)nProof. at least =cluster an in of A, then for c. n E n the an =1= c, and Ian - I If cimplies n if there exists a sequence (a ) in A{c} with lim(a ) = c, then for any8V8>(c)Conversely, K such thata iffor n K, K, whichEbelong to A and are distinct from c.<?od 0 there exists of c contains the points n n thenn an V8(c). Thereforenthe 8-neighborh ::: ::: Q.E.D.the set. next examples emphasize that a cluster point of a set may or may not belong to The4.1.3 Examples (a) FortheopenintervalAI := (0, 1), every point of the closed interval [0,1] is a cluster point of A I. Note that the points 0,1 are cluster points of Ai but do notbelong to A I . All the points of A l are cluster points of AI.(b) A finite set has no cluster points .(c) The infinite set N has no cluster points.in A4Theaset A4 := {lIn :A4.E N} has only the point 0 as a cluster point. None of the points(d) is cluster point of nfollows Ifrom[0, 1], then the set As 2.4.8I that every point in Ithearational point of As . I. It(e) If := the Density Theorem := n Q consists of all is cluster numbers in 0at a cluster point ofthis brief detour, we now return to the concept of the limit of a function Having made its domain.The Definition of the Limit _____________________Wenote that inthe precise definition of the limit of a function Inat aatpoint cnot. is importantto exclude c fromdefinition, it isin the determination ofis defi ed c or . It In any case, now state thiswe consideration immaterial whether I the limit.a real number L is saidA beJR,limitlet cIbeataccluster point ofeA>. For a functionaI8 : >A 0 such4.1.4 Definition Let � to a and of if, giveneany 0 there exists -+ JR,that if x E A and 0 < Ix cl < 8, then I / (x) - L I < . -Remarks (a) Since the value of 8 usually depends on e, we will sometimes write 8(e)instead ofinequality 0 < Ix dependence. to saying x =1= c(b) The 8 to emphasize this cl is equivalent - . If L is a limit of I at c, then we also say that I converges to L at c. We often write L = lim I(x ) or L = lim I. x----+ c x�cpoints dosay that "/(x)move anywhere.asTheapproaches coo. (But it should be noted that theWe also not actually approaches L x symbolism ) I (x) -+ L as x -+ cis also used sometimes to express the fact that I has limit L at c. If thefirlimit of Iisatthatdoesvalueexist, we say that uniquely determined. This uniqueness Our of the definitionthe limit, butofmustlimitdeduced at c.is not part st result c of not L the be is I diverges .
4.1 LIMITS OF FUNCTIONS 99limit at e. If f A JR e4.1.5 Theorem : -+ A, f and if is a cluster point of then can have only oneProof. Suppose that numbers L and L satisfy Definition 4.1.4. For any s > 0, there exists8(s/2)8(s/2) such thatxifExAEandand<0Ix< Ixel <el8(s/2), thenthen -xL I <LI < s/2. Nowexists > such that if A 0 o < 8(s/2), -If(x) s/2.Alsothere If( ) - -let 8 := that 8(s/2)}. Then if x E A and 0 < Ix - el < 8, the Triangle Inequalityimplies inf{8(s/2), IL - L I IL - f(x)1 If(x) - LI < s/2 s/2 = s. :::: + +Since s > 0 is arbitrary, we conclude that L - L 0, so that L = L. = Q.E.D.Figure 4.1.1.) We observe that because nicely described in terms of neighborhoods. (See The definition of limit can be very Vo (e) = (e - 8, e + 8) = {x : Ix - el < 8},the inequality 0Vo<(e)Ixof-e.elSimilarly, the inequality that xLIi=<esandequivalent to sayingneighborhood to the s-neighborhood V,(L) sayingIn this way, we obtain the following < 8 is equivalent to is x belongs to the 8- If(x) - of L.that f(Thebelongs should write out a detailed argument to establish the theorem.result. x) reader y G iven v,, (L) �+-----7f c -------r--���+---� x There exists Vo(c) Figure 4.1.1 The limit of f at c is L.4.1.6 Theorem Let f : A -+ JR and let e be a cluster point of A. Then the followingstatements are equivalent.(i) lim f(x) = L. x ..... c(li) Given any s -neighborhood V,(L) of there exists a 8 -neighborhood Vo(e) ofe such L,that if x i= e is any point in Vo (e) n then f(x ) belongs V/L). A, We now give some examples that illustrate how the definition of limit is applied. Examples (a) lim b b. x ..... c4.1.7 = To be more explicit, let f(x) b for all x E JR. We want to show that lim f(x) b. := x ..... c =If s > 0 is given, we let 8 1. (In fact, any strictlyblpositive s. will serve theis purpose.)Then if 0 < Ix - elDefinition have I thatx)lim bl = I b=-b. 0 < 8 Since s > 0 arbitrary, :=we conclude from < 1, we 4.1.4 f( - f(x) x ..... c =
102 CHAPTER 4 LIMITSan eo(ii) (i). [The proofLis such that no matter what a-neighborhood of then there exists -neighborhood Veo ( ) a contrapositive argument.] If (i) is not true, c we pick, there =>will be at least one number x8 in n V8(c) with x8 =1= c such that !(x8) ¢ Ve (L). Hence A ofor every n E N, the (l/n)-neighborhood of c contains a number xn such that o< I Xn - c l < l/n andbut such that for all n E N.We converge thatLthe sequencewe nhave shownconverges to notbut thethen (ii) is(f(xtruedoesnot conclude to implies (i). (x ) in {c} that if (i) is c, true, sequence not n )) . We Aconclude that (ii) . Therefore Q.E.D.be establishedseeourusingnext section thatpropertiesthe)basic limit properties ofconverges to a We shall byin the corresponding many of for convergent sequences. functions canwe know from work with sequences that if (xn is any sequence that For example, 2that the c, then ; converges to c • Therefore, c2number function(xhe)x) X 2 has limit lim h x)by the• sequential criterion, we can conclude := ..... x c e =Divergence CriteriaIt is often important to be able to show (i) that a certain number is not the limit of a functionis aa consequence of (the proof of) Theorem 4.1.8 .aWe leaveathe details offollowing resultat point, or (ii) that the function does not have limit at point. The its proof as animportant exercise.4.1.9 Divergence Criteria Let A £ JR, let ! : A � JR and let c E JR be a cluster pointofA.(a) Ifin A with L E JR, xn =1= c n E not then ! does for all N L c have limit at if and only if there exists a sequence such that the sequence (xn ) converges to but c the sequence (xn)(f(xn )) not(b) does converge to The function ! does not L. c (x ) have a limit a t if and only if there exists a sequence (xn)in A with xn =1= c n E NR for all such that the sequence n c converges to but the sequence(f(xn )) not does converge in We now give some applications of this result to show how it can be used.4.1.10 Example (a) lim(1/x) does not exist in JR. x .....o As in Example 4.1 .Examplecp(x) 7(d)l/x for xdown if c 0 since we cannot obtainO.a 7(d), let 4.1. breaks > O. However, here we consider c :=The argument that inin of that example. Indeed, if we take the sequence (x ) with x =bound suchEasN,given lim(x 0, but cp(x 1/0/n) n. As we know, nthe sequence = nit) is not bounded. Hence, by Theorem 4.1 .9n(b), := l/n for n then (2) n ) = = = (cp(xn » ) = (n) is not convergent in JR, sincex lim.0(1/x) does not exist in R ....(b) lim0 sgn(x) does not exist. x ..... sgn(x) { +10 for x > 0, Let the signum function sgn be defined by := -1 for x < 0, =x for O.
104 CHAPTER 4 LIMITSExercises for Section 4.1 1 . Determine a condition on Ix - 1 1 that will assure that: (a) Ix2 - 1 1 < �, (b) Ix2 - 1 1 < 1/10-3 , (c) Ix 2 - 1 1 < l / n for a given n E N, (d) Ix 3 - 1 1 < l/n for a given n E N. < 2. Determine a condition on Ix - 41 that will assure that: (a) I JX - 2 1 < �, (b) I JX - 21 10-2 . 3. Let e be a cluster point of A S; R and let I: A --+ R. Prove that lim I (x) = L if and only if lim I/(x) - L I = x .... c o. x -+c 4, Let I : R --+ R and let e E R. Show that lim I (x) = L if and only if x�o I (x + e) = L. lim x�c 5. Let I := (0, a) where a > 0, and let g (x) := x 2 for x E I . For any points x, c E I, show that I g (x) - e2 1 ::: 2a lx - e l . Use this inequality to prove that lim x2 = e2 for any e E I . x-+c 6. Let I be an interval in R, let I : I --+ R, and let e E l. Suppose there exist constants K and L such that I /(x) - L I ::: K lx - el for x E I. Show that lim I (x) = L. x-+c 7. Show that lim x3 3 = e for any e E R. x-+c 8. Show that x .... c JX = Jc for any e > O. lim 9. Use either the 8-8 definition of limit or the Sequential Criterion for limits, to establish the following limits. (a) . hm x-+ 2 -- - 1 , -1 1 X = (b) . hm x-+ I -- = - , x 1 +x 1 2 lim - 0, = x2 1. x2 - X + 1 (c) (d) 1m - . x-+o Ixl x .... I X + 1 210. Use the definition of limit to show that (a) lim (x 2 + 4x) = 1 2, X "" 2 (b) . hm x+5 2x + 3 x -+ - I -- = 4.1 1 . Show that the following limits do not exist. 1 . 1 (a) r1m - (x > 0), (b) hm - (x > 0), x-+o x2 x-+o JX (c) lim (x+ sgn(x» , (d) lim sin(1/x 2 ). x-+o x-+o1 2. Suppose the function I : R --+ R has limit L at 0, and let a > O. If g : R --+ R is defined by g (x) : = I (ax) for x E R, show that lim g(x) = L. � x--+o13. Let e E R and let I : R --+ R be such that ETc I (x) = L. ( f (a) Show that if L = 0, then lim I (x) = O. x-+c (b) Show by example that if L "# 0, then I may not have a limit at e.14. Let I : R --+ R be defined by setting I (x) := x if x is rational, and I (x) = 0 if x is irrational. (a) Show that I has a limit at x = O. (b) Use a sequential argument to show that if e "# 0, then I does not have a limit at e.15. Let I : R --+ R, let I be an open interval in R, and let e E l. If II is the restriction of I to I, show that II has a limit at e if and only if I has a limit at e, and that the limits are equal.16. Let I : R --+ R, let J be a closed interval in R, and let e E J. If 1 is the restriction of I to J , 2 show that if I has a limit at e then 1 has a limit at c. Show by example that it does not follow that if 1 has a limit at e, then I has a limit at e. 2 2
106 CHAPTER 4 LIMITS(b) Ifh : A --+ R ifh (x ) 0 =I- for all x E A, and iflim h = H =I- 0, then x->c lim (L) = 1::.. . h H x->cProof. One proof of this theorem is exactly similar to that of Theorem 3.2.3 . Alternatively,it can be proved by that x =I-use for Theorems 3.c2=3 lim 4.1.8It. For example, Theorembe4any8sequence in A such making c of n E N, and . and(xn) . follows from (xn) .1. let nthatOn the other hand, Definition 4.2.3 implies that for n E N.Therefore an application of Theorem 3.2.3 yields lim ((fg) (xn ») = lim (J(xn)g (xn») = [lim (J(xn »)] [lim (g (xn »)] = LM.Consequently, it follows from Theorem 4.1. 8 that lim(fg) = lim ((fg)(xn ») = LM. x->c The other parts of this theorem are proved in a similar manner. We leave the details tothe reader. Q.E.D.Remarks (1) We note that, in part (b), the additional assumption that H = lim h =I- 0 ismade. If this assumption is not satisfied, then the limit x->c - 1. lm I(x) h(x) x->cmay or may not exist. But even if this limit does exist, we cannot use Theorem 4.2.4(b) toevaluate it.(2) Let A S; IR., and let 11 12 , , In be functions on A to IR., and let c be a cluster point of . • •A. If for k = 1, . . , n, .then it follows from Theorem 4.2.4 by an Induction argument that L 1 L2 + Ln = lim(f1 f2 +...+ .t:n ) + +...+ , x->candIn particular, we deduce that if L = lim I and n E N, then x->c
SOME EXTENSIONS OF THE LIMIT CONCEPT 111 8. Let n E N be such that n � 3. Derive the inequality _x2 :": xn :": x 2 for -1 < < the fact that lim x2 = 0 to show that lim xn = O. x 1. Then use x�o x�o 9. Let f, g be defined on A to JR and let c be a cluster point of A. (a) Show that if both lim f and lim(f + g) exist, then lim g exists. x--+c x--+c x--+c (b) If lim f and lim fg exist, does it follow that lim g exists? x--+c x--+c x--+c10. Give examples of functions f and g such that f and g do not have limits at a point c, but such that both f + g and fg have limits at c.1 1 . Determine whether the following limits exist in R (a) lim sin(1 !x2 ) (x i= 0), (b) lim x sin(1 !x 2 ) (x i= 0), x--+o x--+o (c) lim sgn sin(1 !x) (x i= 0), (d) lim .;x sin(1 !x 2 ) (x > 0). x--+o x--+o x .... 012. Let f: JR --+ JR be such that f(x + y ) = f(x) + fey) for all x , y in R Assume that lim f = L exists. Prove that L = 0, and then prove that f has a limit at every point c E R [Hint: First note that f(2x) = f(x) + f(x) = 2f(x) for x E JR. Also note that f(x) = f(x - c) + f(c) for x, c in R]13. Let A <;; JR, let f: A --+ JR and let c E JR be a cluster point of A. If lim f exists, and if If I x .... c denotes the function defined for x E A by If I (x) := If(x) l , prove that lim If I = I lim f l14. Let A <;; JR, let f : A --+ JR, and let c E JR be a cluster point of A. In addition, suppose that x--+c x--+c f (x) � 0 for all x E A, and let IT be the function defined for x E A by (x) := (IT ) .fT(X). If x--+c f exists, prove that x--+c IT = Jx--+c f · lim lim limSection 4.3 Some Extensions of the Limit ConcepttIn this section, we shall present three types of extensions of the notion of a limit of afunction that often occur. Since all the ideas here are closely parallel to ones we havealready encountered, this section can be read easily.One-sided LimitsThere are times when a function I may not possess a limit at a point c, yet a limit doesexist when the function is restricted to an interval on one side of the cluster point c . For example, the signum function considered in Example 4. 1 . lO(b), and illustratedin Figure 4.1 .2, has no limit at c = O. However, if we restrict the signum function to theinterviu (0, 00), the resulting function has a limit of 1 at c = 0: Similarly, if we restrict thesignum function to the interval (-00, 0), the resulting function has a limit of - 1 at c = O.These are elementary examples of right-hand and left-hand limits at c = O.4.3.1 Definition Let A E lR and let I: A R --+(i) If c E lR is a cluster point of the set A n (c, 00) = {x E A: x > c}, then we say that L E lR is a right-hand limit of I at c and we write x ..... x ..... lim / = L or lim I(x) = L c+ c+tThis section can be largely omitted on a first reading of this chapter.
112 CHAPTER 4 LIMITS if given any e>0 there exists a 8 = 8 0 such that for all x E with 0 (e) > A < x - c 8, then I f (x) - L I < < e.(ii) If c E JR is a cluster point of the set ( - 00 , c) = {x E An A: < x c}, then we say that L E JR is a left-hand limit of f at c and we write lim f = L or lim f (x) = L x .....c- x .....c- if given any e > 0 there exists a 8 > 0 such that for all x E A with 0 < c - x < 8, then I f (x) - LI < e.Notes ( 1 ) The limits lim f and lim f are called one-sided limits of f at c. It is x -+c+ x -+c-possible that neither one-sided limit may exist. Also, one of them may exist without theother existing. Similarly, as is the case for f (x) := sgn(x) at = 0, they may both exist cand be different.(2) If is an interval with left endpoint c, then it is readily seen that f: A � JR has a Alimit at c if and only if it has a right-hand limit at c. Moreover, in this case the limit lim f x -+cand the right-hand limit lim f are equal. (A similar situation occurs for the left-hand limit x -+c+ Awhen is an interval with right endpoint c.) The reader can show that f can have only one right-hand (respectively, left-hand)limit at a point. There are results analogous to those established in Sections 4.1 and 4.2 fortwo-sided limits. In particular, the existence of one-sided limits can be reduced to sequentialconsiderations.4.3.2 Theorem Let A £ JR, let f � JR, and let c E JR be a cluster point of :A An(c, 00). Then the following statements are equivalent:(i) lim f = L. x -+c+(ii) For every sequence (xn) that converges to c such thatxn E A and xn > c for all n E N, the sequence (J (xn ) ) converges to L . We leave the proof of this result (and the formulation and proof of the analogous resultfor left-hand limits) to the reader. We will not take the space to write out the formulationsof the one-sided version of the other results in Sections 4.1 and 4.2. The following result relates the notion of the limit of a function to one-sided limits.We leave its proof as an exercise.4.3.3 Theorem Let A £ JR, let f: A � JR, and let c E JR be a clusterpoint of both of thesets A n (c, 00) and A n ( 00 c) . Then xlim.c f = L if and only ifxlim+ f = L = xlim- f. - , .... ---+c ---+c4.3.4 Examples (a) Let f (x) := sgn(x) . We have seen i n Example 4. 1 . 1 O(b) that sgn does not have a limit at O . It i s clear that lim sgn(x) = + 1 and that lim sgn(x) = - 1 . Since these one-sided limits are different,x -+ o+ x -+ o-it also follows from Theorem 4.3.3 that sgn(x) does not have a limit at O.(b) Let g (x) := e 1 /x for x =1= O. (See Figure 4.3 . 1 .)
SOME EXTENSIONS OF THE LIMIT CONCEPT 115 The function g does not tend to either 00 or -oo as x � O. For, ifa > o then g(x) < afor all x < 0, so that g does not tend to 00 as x � O. Similarly, if f3 < 0 then g(x) > f3 forall x > 0, so that g does not tend to -00 as x � O. D While many of the results in Sections 4. 1 and 4.2 have extensions to this limitingnotion, not all of them do since ±oo are not real numbers. The following result is ananalogue of the Squeeze Theorem 4.2.7. (See also Theorem 3.6.4.)4.3.7Theorem Let A � JR, let I, g: A � R and let c E JR be a cluster point of A.Suppose that I(x) :s g(x) for all x E A, x =f. c .(a) If x---+c I = 00, then x-+c g = 00. lim lim(b) If x-+c g = -00, then x-+c I = -00. lim limProof. (a) If x�c I = 00 and a E JR is given, then there exists 8(a) > 0 such that if lim0 < I x - c l < 8 (a) and x E A, then /(x) > a. But since /(x) :s g(x) for all x E A,x =f. c,it follows that if 0 < Ix - cl < 8 (a) and x E A, then g(x) > a. Therefore x�c g = 00. lim The proof of (b) is similar. Q.E.D. The function g(x) = l/x considered in Example 4.3.6(b) suggests that it might beuseful to consider one-sided infinite limits. We will define only right-hand infinite limits.4.3.8 Definition Let A � JR and let I : A � R If c E JR is a cluster point of the setA n (c, 00) = {x E A: x > c}, then we say that I tends to 00 [respectively, -00] asx � c+, and we write lim 1 = 00 x---+ c+ [respectivelY, lim 1 = -ooJ x-+ c + if for every a E JR there is 8 = 8(a) > 0 such that for all x E A with 0 < x - c < 8, thenI(x) > a [respectively, I(x) < a].4.3.9 Examples (a) Let g(x) := l /x for x =f. O. We have noted in Example 4.3.6(b)that lim g does not exist. However, it is an easy exercise to show that x�o lim (l/x) = 00 and lim (l/x) = -00. X�O+ x�o-(b) It was seen in Example 4.3.4(b) that the function g(x) := e 1 /x for x =f. 0 is not boundedon any interval (0, 8), 8 > O. Hence the right-hand limit of e1/x as x � 0 + does not existin the sense of Definition 4.3.1 (i). However, since x l /x < e1/ for x > 0, xit is readily seen that lim e1/ = 00 in the sense of Definition 4.3.8. D x�o+Limits at Infinity _________________________It is also desirable to define the notion of the limit of a function as x � 00. The definitionas x � -00 is similar.
116 CHAPTER 4 LIMITS4.3.10 Definition Let A S; JR and let f: A ---+ R Suppose that (a, 00) S; A for somea E R We say that L E JR is a limit of f as x ---+ 00, and write x--+oo x--+oo lim f = L or lim f (x) = L ,if given any c>0 there exists K = K (c) > a such that for any x > K, thenI f (x) - L I < c. The reader should note the close resemblance between 4.3.10 and the definition of alimit of a sequence. We leave it to the reader to show that the limits of f as x ---+ ±oo are unique wheneverthey exist. We also have sequential criteria for these limits; we shall only state the criterionas x ---+ 00. This uses the notion of the limit of a properly divergent sequence (see Definition3 .6.1).4.3.11 Theorem Let A S; JR, let f : A ---+ JR, and suppose that (a, 00) S; A for somea E R Then the following statements are equivalent: x--+oo(i) L = lim f·(ii) For every sequence (xn ) in A n (a, 00) such that lini(xn ) = 00, the sequence (J(xn »)converges to L . We leave it to the reader tQ prove this theorem and to formulate and prove the companionresult concerning the limit as x ---+ -00.4.3.12 Examples (a) Let g(x) : = l /x for x =1= o. x ----+ oo x-+ - oo It is an elementary exercise to show that lim ( l /x) = 0 = lim ( l /x). (See Figure4.3.4.)(b) Let f (x) := l /x 2 for x =1= o. X-H�) x----+ -oo The reader may show that lim ( l /x 2 ) = 0 = lim ( l /x 2 ). (See Figure 4.3.3.) Oneway to do this is to show that if :::: 1 then 0 :s 1/x 2 :s 1/x. In view of part (a), this implies x x --+ oothat lim (1/x 2 ) = O. 0 Just as it is convenient to be able to say that f (x) ---+ ±oo as x for JR, it is---+ c cEconvenient to have the corresponding notion as x ---+ ±oo. We will treat the case wherex ---+ 00.4.3.13 Definition Let A S; JR and let f: A ---+ R Suppose that (a, 00) S; A for somea E A. We say that f tends to 00 [respectively, - 00] as x ---+ 00, and write x--+oo lim f = oo [respectively, lim f = ] x --+ oo - 00if given any aE JR there exists K = K(a) > a such that for any x > K, then f(x) > IX[respectively, f (x) < a]. As before there is a sequential criterion for this limit.4.3.14 Theorem Let A E JR, let f: A ---+ JR, and suppose that (a, 00) S; A for somea E R Then the following statements are equivalent:
CHAPTER 5 CONTINUOUS FUN CTIONSWe now begin the study of the most important class of functions that arises in real analysis:the class of continuous functions. The term "continuous" has been used since the time ofNewton to refer to the motion of bodies or to describe an unbroken curve, but it was notmade precise until the nineteenth century. Work of Bernhard Bolzano in 1 8 17 and AugustinLouis Cauchy in 1 821 identified continuity as a very significant property of functions andproposed definitions, but since the concept is tied to that of limit, it was the careful work ofKarl Weierstrass in the 1870s that brought proper understanding to the idea of continuity. We will first define the notions of continuity at a point and continuity on a set, and thenshow that various combinations of continuous functions give rise to continuous functions.Then in Section 5.3 we establish the fundamental properties that make continuous functionsso important. For instance, we will prove that a continuous function on a closed boundedinterval must attain a maximum and a minimum value. We also prove that a continuousfunction must take on every value intermediate to any two values it attains. These propertiesand others are not possessed by general functions, as various examples illustrate, and thusthey distinguish continuous functions as a very special class of functions. In Section 5.4 we introduce the very important notion of uniform continuity. Thedistinction between continuity and uniform continuity is somewhat subtle and was not fullyappreciated until the work of Weierstrass and the mathematicians of his era, but it proved to Karl Weierstrass Karl Weierstrass (=WeierstraB) ( 1 8 15-1 897) was born in Westphalia, Ger many. His father, a customs officer in a salt works, insisted that he study law and public finance at the University of Bonn, but he had more interest in drinking and fencing, and left Bonn without receiving a diploma. He then enrolled in the Academy of Munster where he studied mathematics with Christoph Gudermann. From 1841-1854 he taught at various gymnasia in Prussia. Despite the fact that he had no contact with the mathematical world during this time, he worked hard on mathematical research and was able to publish a few papers, one of which attracted considerable attention. Indeed, the University of Konigsberg gave him an honorary doctoral degree for this work in 1 855. The next year, he secured positions at the Industrial Institute of Berlin and the University of Berlin. He remained at Berlin until his death. A methodical and painstaking scholar, Weierstrass distrusted intuition and worked to put everything on a firm and logical foundation. He did fundamental work on the foundations of anthmetic and analysis, on complex analysis, the calculus of variations, and algebraic geometry. Due to his meticulous preparation, he was an extremely popular lecturer; it was not unusual for him to speak about advanced mathematical topics to audiences of more than 250. Among his auditors are counted Georg Cantor, Sonya Kovalevsky, Gosta Mittag-Leffler, Max Planck, Otto Holder, David Hilbert, and Oskar Bolza (who had many American doctoral students). Through his writings and his lectures, Weierstrass had a profound influence on contemporary mathematics. 119
120 CHAPTER 5 CONTINUOUS FUNCTIONS be very significant in applications. We present one application to the idea of approximating continuous functions by more elementary functions (such as polynomials). The notion of a "gauge" is introduced in Section 5.5 and is used to provide an alter native method of proving the fundamental properties of continuous functions. The main significance of this concept, however, is in the area of integration theory where gauges are essential in defining the generalized Riemann integral. This will be discussed in Chapter 10. Monotone functions are an important class of functions with strong continuity proper ties and they are discussed in Section 5.6. Section 5.1 Continuous Functions In this section, which is very similar to Section 4. 1 , we will define what it means to say that a function is continuous at a point, or on a set. This notion of continuity is one of the central concepts of mathematical analysis, and it will be used in almost all of the following material in this book. Consequently, it is essential that the reader master it. 5.1.1 Definition Let A � lit let f A : -+ lit and let c E A. We say that f is continuous x at c if, given any number c > 0 there exists 8 > 0 such that if is any point of A satisfying Ix cl < 8, then I f(x) - f(c) 1 < c . - If f fails to be continuous at c , then we say that f i s discontinuous at c. As with the definition of limit, the definition of continuity at a point can be formulated very nicely in terms of neighborhoods. This is done in the next result. We leave the verification as an important exercise for the reader. See Figure 5 . 1 . 1 . v6(c) Figure 5.1.1 Given V£ (f(c», a neighborhood Vs (c) is to be determined.t< 5.1.2 Theorem A function f A : -+ lR. is continuous at a point c E A if and o.nly ifgiven any c -neighborhood V£ (f (c» of f (c) there exists a 8 -neighborhood V� (c) of c such that if x x is any point of A n V� (c), then f( ) belongs to Ve (f(c» , that is, f(A n V8 (c» � V/f (c». Remark (1) If c E A is a cluster point of A, then a comparison of Definitions 4. 1 .4 and 5 . 1 . 1 show that f is continuous at c if and only if (1) f (c) = lim f(x). x---+ c
5.1 CONTINUOUS FUNCTIONS 121 cThus, if is a cluster point of A, then three conditions must hold for f to be continuousatc: (i) f c must be defined at (so that f(c) makes sense), x---> c f c (ii) the limit of at must exist in JR (so that lim f(x) makes sense), and (iii) these two values must be equal. c(2) If E A is not a cluster point of A, then there exists a neighborhood of such V8(c) cthat A n V/c) = {c}. f Thus we conclude that a function is automatically continuous at a cpoint E A that is not a cluster point of A. Such points are often called "isolated points"of A. They are of little practical interest to us, since they have no relation to a limitingprocess. Since continuity is automatic for such points, we generally test for continuity onlyat cluster points. Thus we regard condition (I) as being characteristic for continuity at c. A slight modification of the proof of Theorem 4. 1.8 for limits yields the followingsequential version of continuity at a point.5.1.3 Sequential Criterion for Continuity A function f : A -+ JR is continuous at thepointc E A if and only if for every sequence (xn ) in A that converges to c, the sequence(J(xn » ) converges to f(c). The following Discontinuity Criterion is a consequence of the last theorem. It shouldbe compared with the Divergence Criterion 4.1 .9(a) with L = f(c). Its proof should bewritten out in detail by the reader.5.1.4 Discontinuity Criterion Let A s; JR, let :A f -+ JR, and let c E A. Then f is cdiscontinuous at if and only if there exists a sequence (xn ) in A such that (xn ) convergesto but the sequence ( c, ) does not converge to J(xn » f(c). So far we have discussed continuity at a point. To talk about the continuity of a functionon a set, we will simply require that the function be continuous at each point of the set. Westate this formally in the next definition.5.1.5 Definition Let A S; JR and let f: B A -+ R If is a subset of A, we say that f is B fcontinuous on the set if is continuous at every point of B.5.1.6 Examples (a) The constant function f(x) := b is continuous on R x ---> c It was seen in Example 4.1 .7(a) that if E JR, then limc Since f(x) = b. f(c) = b, x---> cwe have lim f(x) = f(c), f and thus is continuous at every point E R Therefore c f iscontinuous on R(b) g(x) := x is continuous on R x ---> c c It was seen in Example 4.1 .7(b) that if E JR, then we have lim Since g = c. g(c) = C, g cthen is continuous at every point E R Thus g is continuous on R(c) h(x) := x 2is continuous on R c It was seen in Example 4. 1 .7(c) that if E JR, then we have lim Since x ---> c h = c2 • h(c)= c2 , then h is continuous at every point c E R Thus h is continuous on R(d) cp(x) := I/x is continuous on A := {x E JR: x > O}.
122 CHAPTER 5 CONTINUOUS FUNCTIONS c It was seen in Example 4.1 .7(d) that if E A, then we have lim rp = 1/c. Since x-+c I crp (c) = 1 c, this shows that rp is continuous at every point E A. Thus rp is continuouson A.(e) rp (x) := 1/x is not continuous at x = O. Indeed, if rp(x) = 1 /x for x > 0, then rp is not defined for x = 0, so it cannot becontinuous there. Alternatively, it was seen in Example 4. 1 . l O(a) that lim rp does not exist x-+o xin ffi., so rp cannot be continuous at = O.(f) The signum function sgn is not continuous at O. The signum function was defined in Example 4. 1 . 1 O(b), where it was also shown that Rlim sgn(x) does not exist in Therefore sgn is not continuous at x = 0 (even though sgn 0x-+ois defined). It is an exercise to show that sgn is continuous at every point c =1= o. f(g) Let A := ffi. and let be Dirichlet s "discontinuous function" defined by f (x) := g if xis rational, if x is irrational.We claim that is f not continuous at any point of R(This function was introduced in 1 829by P. G. L. Dirichlet.) c Indeed, if is a rational number, let (xn ) be a sequence of irrational numbers thatconverges to c. (Corollary 2,.1.9 to the Density Theorem 2.4.8 assures us that such asequence does exist.) Since = 0 for all E N, we have lim ( j (xn ) ) = 0, while f(xn ) nf(c) f = 1 . Therefore is not continuous at the rational number c. b On the other hand, if is an irrational number, let (Yn ) be a sequence of rational b.numbers that converge to (The Density Theorem 2.4.8 assures us that such a sequencedoes exist.) Since = 1 for all E N, we have lim ( j ( n ) ) = 1, while f(Yn ) n y = O. feb) fTherefore is not continuous at the irrational number b. Since every real number is either rational or irrational, we deduce that is not fcontinuous at any point in R >(h) Let A : = {x E R x O}. For any irrational number x > 0 we define = O. For hex)a rational number in A of the form min, with natural numbers m , having no common nfactors except 1 , we define h(mln) := l in. (Sometimes we also define := 1 .) h(O) 0.8 0.6 0.4 - 0.2 - � . . . . . . . . .. . . . . . . . . , t! • • • o 0.5 1.5 2 Figure 5.1.2 Thomaes function.
5.1 CONTINUOUS FUNCTIONS 123 We claim that h is continuous at every irrational number in A, and is discontinuousat every rational number in A. (This function was introduced in 1875 by K. J. Thomae.) Indeed, if a > 0 is rational, let (xn ) be a sequence of irrational numbers in A thatconverges to a. Then lim (h(xn ») 0, while h (a ) > O. Hence h is discontinuous at a. = On the other hand, if b is an irrational number and 8 > 0, then (by the ArchimedeanProperty) there is a natural number n o such that l / no < 8 . There are only a finite number of rationals with denominator less than no in the interval (b - 1 , b + 1). (Why?)Hence 8 > 0 can be chosen so small that the neighborhood (b - 8 , b + 8) contains norational numbers with denominator less than no- It then follows that for Ix - bl < 8 , x EA, we have Ih(x) - h(b) 1 = Ih(x)1 :s: l / n o < 8. Thus h is continuous at the irrationalnumber b. Consequently, we deduce that Thomaes function h is continuous precisely at theirrational points in A. D5.1.7 Remarks (a) Sometimes a function f : A -+ IR is not continuous at a point cbecause it is not defined at this point. However, if the function f has a limit L at the pointc and if we define F on A U {c} -+ IR by F(x) := (L f(x) for x = c, for x E A ,then F is continuous at c. To see this, one needs to check that x-+c F = L, but this follows lim(why?), since x-+c f = L. lim(b) If a function g : A -+ IR does not have a limit at c, then there is no way that we canobtain a function G : A U {c} -+ IR that is continuous at c by defining G (x) := ( e g(x) for x = c, for x E A .To see this, observe that if x�c G exists and equals lim e, then x�c g must also exist and limequal e .5.1.8 Examples (a) The function g(x) : = sin(1 /x) for x #- O (see Figure 4.1 .3) doesnot have a limit at x = 0 (see Example 4. 1.1O(c» . Thus there is no value that we can assignat x = 0 to obtain a continuous extension of g at x = o.(b) Let f(x) = x sin(1/x) for x #- o. (See Figure 5.1.3.) Since f is not defined at x = 0,the function f cannot be continuous at this point. However, it was seen in Example 4.2.8(f)that x-+o (x sin (1 /x») = O. Therefore it follows from Remark 5.1.7 (a) that if we define limF : IR -+ IR by F(x) := ( 0 x sin(1 /x) for x = 0, for x #- 0,then F is continuous at x = O. D
5.2 COMBINATIONS OF CONTINUOUS FUNCTIONS 12514. Let A := (0, 00) and let k : A -+ lR be defined as follows. For x E A, x irrational, we define k(x) = 0; for x E A rational and of the form x = m i n with natural numbers m, n having no common factors except 1, we define k(x) := n. Prove that k is unbounded on every open interval in A. Conclude that k is not continuous at any point of A. (See Example 5 . 1 .6(h).)15. Let f : (0, 1) -+ lR be bounded but such that lim f does not exist. Show that there are two 0 X "" sequences (xn ) and (Yn ) in (0, 1) with lim(xn ) = 0 = lim(yn ), but such that lim (J(xn ) ) and lim (f (yn ) ) exist but are not equal.Section 5.2 Combinations of Continuous Functions I gLet A � JR and let and be functions that are defined on A to JR and let E R In bby 1 + In addition, if : A g, I - g, Ig, bl ·then we defined the quotient function denoted by h -+fI h.Definition 4.2.3 we defined the sum, difference, product, and multiple functions denoted JR is such that for all E A, hex) =1= 0 x The next result is similar to Theorem 4.2.4, from which it follows. I g5.2.1 Theorem Let A � JR, let and be functions on A to JR, and let b E R Suppose c I gthat E A and that and are continuous at c.(a) Then I + g, I - g, Ig, and bl are continuous at c. : -+(b) If h A JR is continuous at c E A and if h (x) =1= 0 for all x E A, then the quotientfI h is continuous at c.Proof. If c E A is not a cluster point of A, then the conclusion is automatic. Hence we cassume that is a cluster point of A. g(a) Since I and are continuous at c, then x ..... c x --+ c I(c) = lim I and g(c) = lim g.Hence it follows from Theorem 4.2.4(a) that x --+ c g)(c) = I(c) + g(c) = lim(f + g). (f +Therefore 1 + g is continuous at c. The remaining assertions in part (a) are proved in asimilar fashion. x --+ c c(b) Since E A , then h(c) =1= O. But since h(c) = lim h, it follows from Theorem 4.2.4(b)that I (c) = -c) = xlim. I = lim ( I ) I ( - .... h (c) lim h x..... h c c _ _ x --+ c . h hTherefore 1/ is continuous at c. Q.E.D. The next result is an immediate consequence of Theorem 5.2. 1 , applied to every pointof A. However, since it is an extremely important result, we shall state it formally.
126 CHAPTER 5 CONTINUOUS FUNCTIONS5.2.2 Theorem A S; �, let f and g be continuous on A to �, and let b E R Let(a) The functions f + g, f - g, fg, and bf are continuous on A.(b) If h : A --+ � is continuous on A an d hex) 1= 0 for x E A, then the quotient fl h iscontinuous on A.Remark To define quotients, i t i s sometimes more convenient to proceed as follows. If : --+cP A �, let E : cP A l := {x A (x) 1= OJ. We can define the quotient on the set flcp Alby(1) (L) (x) := f (x) cp cp (x) cpIf is continuous at a point E c AI it is clear that the restriction of cP to CPI is also Al c.continuous at Therefore it follows from Theorem S.2. 1 (b) applied to that CPI flcplis continuous at E c A. Since (fjcp)(x) = (flcpl )(x) for E x Al it follows that is f/cpcontinuous at E c A I f Similarly, if and cP are continuous on then the function A, flcp,defined on Al by ( 1), is continuous on AI 5.2.3 Examples (a) Polynomial functions. If p is a polynomial function, so that p(x) = an xn + an_ I xn - 1 + . . . + a l x + ao for x �, then it follows from Example 4.2.5(f) that p(c) = lim p for any c E �. Thusall Ea polynomialfunction is continuous on R x-+c(b) Rational functions. q If p and are polynomial functions on �, then there are at most a finite numbera I . . . , am of real roots of If fj. q. x {a I . . . , am } then q(x) 1= 0 so that we can define therational function by r p(x) r(x) := q (x) -It was seen in Example 4.2.S(g) that if q(c) 1= 0, then p (c) . p (x) . r (c) = -- = hm -- = hm r (x) . q (c) q (x) x-+c x-+cIn other words, r is continuous at c. Since c is any real number that is not a root of q, weinfer that a rational function is continuous at every real numberfor which it is defined.(c) We shall show that the sine function sin is continuous on R To do so we make use of the following properties of the sine and cosine functions.(See Section 8.4.) For all x, y, z E � we have: S I z l , I cos z l 1 , I sin z l s sin x - sin y = 2 sin [! (x - y)] cos U (x + y)] . cHence if E �, then we have S 2 · !Ix - cl . 1 = Ix - c l . I sinx - sin c lTherefore sin is continuous at c. Since c E � is arbitrary, it follows that sin is continuouson R(d) The cosine function is continuous on R
5.2 COMBINATIONS OF CONTINUOUS FUNCTIONS 127 We make use of the following properties of the sine and cosine functions. For allx, y, z E � we have: I sin z I ::s Iz I , I sin z I ::s 1, cos x - cos y = -2 ! ! sin[ (x + y)] sin[ (x - y)].Hence if C E �, then we have I cos x - cos cl ::s 2 · 1 · !I c - x l = Ix - c l · c. cTherefore cos is continuous at Since E � is arbitrary, it follows that cos is continuouson R (Alternatively, we could use the relation cos x = sin(x + 7r/2).)(e) The functions tan, cot, sec, csc are continuous where they are defined. For example, the cotangent function is defined by cos x cotx := -- . sm xprovided sin x i= 0 (that is, provided x i= n7r, n E Z). Since sin and cos are continuouson �, it follows (see the Remark before Example 5.2.3) that the function cot is continuouson its domain. The other trigonometric functions are treated similarly. D5.2.4 Theorem Let A S; �, let I : A -+ �, and let I I I be defined by I I I (x) := /(x)1 Ifor x E A . c(a) If I is continuous at a point E A, then I I I is continuous at c.(b) IfI is continuous on A, then I I I is continuous on A.Proof. This is an immediate consequence of Exercise 4.2.13. Q.E.D.5.2.5 Theorem Let A S; �, let I : A -+ �, and let I (x) :::: 0 for all x E A. We let v1be defined for x E A by (v1) (x) : = JI (x) .(a) If I is continuous at a point E A, then v1 is continuous at c c.(b) If I is continuous on A, then v1 is continuous on A.Proof. This is an immediate consequence of Exercise 4.2.14. Q.E.D.Composition of Continuous FunctionsWe now show that if the function I : A -+ � is continuous at a point and if g : B -+ �c b I(c),is continuous at = then the composition g o I is continuous at c. In order to assurethat g o I is defined on all of A, we also need to assume that I(A) S; B.5.2.6 Theorem Let A, B S; R and let I : A -+ � and g: B -+ � be functions such thatI(A) S; B. If I is continuous at a point C E A and g is continuous at = E B, then b I(c)the composition g 0 I : A -+ � is continuous at c.Proof. Let W be an e-neighborhood of g (b). Since g is continuous at b, there is a 8-neighborhood V of b = I(c) such that if y E B n V then g(y) E W. Since I is continuousat c, there is a y-neighborhood U of c such that if x E A n U , then I (x) E V. (SeeFigure 5.2.1.) Since I(A) S; B, it follows that if x E A n U, then I (x) E B n V so thatg o I(x) = g (f(x» E W. But since W is an arbitrary e-neighborhood of g (b), this impliesthat g o I is continuous at c . Q.E.D.
128 CHAPTER 5 CONTINUOUS FUNCTIONS v b = fCc) u W c g( b) f g A B c Figure 5.2.1 The composition of I and g .5.2.7 Theorem Let A , B � �, let f : A ---+ � be continuous on A, and let g: B ---+ � becontinuous on B . f (A) � B, then the composite function g o f : A ---+ � is continuous Ifon A .Proof. The theorem follows immediately from the preceding result, if f and g are continuous at every point of A and B, respectively. Q.E.D. Theorems 5.2.6 and 5.2.7 are very useful in establishing that certain functions arecontinuous. They can be used in many situations where it would be difficult to apply thedefinition of continuity directly.5.2.8 Examples (a) Let g 1 (x) := Ix I for x E R It follows from the Triangle Inequalitythat I g 1 (x) g 1 (c) 1 s Ix- - clfor all x , c E R Hence g 1 is continuous at c E R If f: A ---+ � is any function that iscontinuous on A, then Theorem 5.2.7 implies that g 1 0 f = I f I i s continuous on A . Thisgives another proof of Theorem 5.2.4.(b) Let g2 (x) := ,JX for x ::: It follows from Theorems 3.2.10 and 5 . 1 .3 that g2 is O.continuous at any number c ::: If f; A ---+ � is continuous on A and if f (x) ::: 0 for O.all x E A, then it follows from Theorem 5.2.7 that g2 0 f = J1 is continuous on A. Thisgives another proof of Theorem 5.2.5.(c) Let g (X ) ; = sinx for x E R We have seen in Example 5.2.3(c) that g is continuous 3 3on R If ; A ---+ � is continuous on A , then it follows from Theorem 5.2.7 that g 0 f is fcontinuous on A. 3 In particular, if (x) ; = 1 /x for x 0, then the function g (x) ; = sin(l /x) is contin f =1=uous at every point c [We have seen, in Example 5.1 .8(a), that g cannot be defined =1= O.at 0 in order to become continuous at that point.] 0Exercises for Section 5.21 . Determine the points of continuity of the following functions and state which theorems are used in each case. 2 x + 2x + 1 (a) I(x) := (x E JR), (b) g (x) := Jx + Jx (x 2:: 0), x2 + 1 ./1 + I sin x l (c) h(x) := (x =I 0), (d) k(x) := cos 11� (x E JR). 1 + x2 x
5.3 CONTINUOUS FUNCTIONS ON INTERVALS 1292. Show that if f : A --+ � is continuous on A � � and if n E 11, then the function r defined by r ex) = (f(x)t for x E A, is continuous on A.3. Give an example of functions f and g that are both discontinuous at a point c in � such that (a) the sum f + g is continuous at c, (b) the product fg is continuous at c.4. Let x � [x] denote the greatest integer function (see Exercise 5 . 1 .4). Determine the points of continuity of the function f(x) := x - [x], X E R 5. Let g be defined on � by g(l) := 0, and g(x) := 2 if x =1= 1, and let f(x) := x + 1 for all x Show that lim g o f =1= (g 0 f)(0). Why doesnt this contradict Theorem 5.2.6? ER x-->o x--> c6. Let f, g be defined on � and let c E R Suppose that lim f = b and that g is continuous at b. Show that lim g o f = g(b). (Compare this result with Theorem 5.2.7 and the preceding x-->c exercise.)7. Give an example of a function f : [0, 1] --+ R that is discontinuous at every point of [0, 1] but such that If I i s continuous on [0, 1 ] . 8. Let f, g be continuous from � to �, and suppose that fer) = g(r) for all rational numbers r. Is it true that f(x) = g(x) for all x E �? n 9. Let h: � --+ � be continuous on � satisfying h(m/2 ) = 0 for all m E Z, n E N. Show that hex) = o for all x E R10. Let f: � --+ � be continuous on �, and let P := {x E � : f(x) > O}. If c E P, show that there exists a neighborhood Vo (c) � P.1 1 . If f and g are continuous on �, let S := {x E � : f(x) 2: g(x)}. If (sn ) � S and lim(sn ) = s, show that S E S.1 2. A function f : � --+ � is said to be additive if f (x + y) = f (x) + f (y) for all x, y in �. Prove that if f is continuous at some point xo then it is continuous at every point of R (See Exercise 4.2. 1 2.)13. Suppose that f is a continuous additive function on R If c := f (1), show that we have f(x) = cx for all x E R [Hint: First show that if r is a rational number, then fer) = cr.]14. Let g : � --+ � satisfy the relation g(x + y) = g(x)g(y) for all x, y in R Show that if g is continuous at x = 0, then g is continuous at every point of R Also if we have g(a) = 0 for some a E �, then g(x) = 0 for all x E R15. Let f, g : � --+ � be continuous at a point c, and let hex) := sup {f(x), g(x)} for x E R (I ) Show that hex) = � (x) + g(x» + � I f(x) - g(x) 1 for all x E R Use this to show that h is continuous at c.Section 5.3 Continuous Functions on IntervalsFunctions that are continuous on intervals have a number of very important properties thatare not possessed by general continuous functions. In this section, we will establish somedeep results that are of conside;;able importance and that will be applied later. Alternativeproofs of these results will be given in Section 5.5.5.3.1 Definition A function f : A -+ is said to be bounded on A if there exists a lRconstant > such that I f(x) 1 ::s for all x E A. M 0 M In other words, a function is bounded on a set if its range is a bounded set in R Tosay that a function is not bounded on a given set is to say that no particular number can
130 CHAPTER 5 CONTINUOUS FUNCTIONS Iserve as a bound for its range. In exact language, a function is not bounded on the setA if given any M > 0, there exists a point M E A such that x M ) > M. We often say I I (x I Ithat is unbounded on A in this case. I For example, the function defined on the interval A := (0, 00) by := l / is I(x) xnot bounded on A because for any M > ° we can take the point xM := 1/(M + 1) in 11to get M ) = 1 / M = M + 1 > M. This example shows that continuous functions need I (x xnot be bounded. In the next theorem, however, we show that continuous functions on acertain type of interval are necessarily bounded.5.3.2 Boundedness Theoremt Let I := [a, b] be a closed bounded interval and letI: I -+ 1R. be continuous on I. Then I is bounded on I.Proof. Suppose that I i s not bounded on I. Then, for any n E N there i s a number xn E Isuch that I I (xn ) I > n. Since I is bounded, the sequence X (xn ) is bounded. Therefore, :=the Bolzano-Weierstrass Theorem 3.4.8 implies that there is a subsequence X (xn ) of X = rthat converges to a number x. Since I is closed and the elements of X belong to I, it followsfrom Theorem 3.2.6 that x E I. Then I is continuous at x, so that (J (xn ) ) converges toI(x). We then conclude from Theorem 3.2.2 that the convergent sequence (J(xn) ) mustbe bounded. But this is a contradiction since for r E N.Therefore the supposition that the continuous function I is not bounded on the closedbounded interval I leads to a contradiction. Q.E.D. To show that each hypothesis of the Boundedness Theorem is needed, we can constructexamples that show the conclusion fails if any one of the hypotheses is relaxed. (i) The interval must be bounded. The function := I(x) x x for in the unbounded,closed interval A := [0, 00) is continuous but not bounded on A. (ii) The interval must be closed. The function g(x) x x := l / for in the half-openinterval B : = (0, 1] is continuous but not bounded on B. h (iii) The function must be continuous. The function defined on the closed intervalC : = [0, 1 ] by hex):= l / forx x E (0, 1] and h(O) : = 1 is discontinuous and unboundedon C.The Maximum-Minimum Theorem I5.3.3 Definition Let A � 1R. and let : A -+ R We say that I has an absolute maxi x*mum on A if there is a point E A such that I(x*) � I(x) for all x E A.We say thatI has an absolute minimum on A i f there i s a point x* E A such that for all x E A.We say that x* is an absolute maximum point for I on A, and that x is an absoluteminimum point for I on A, if they exist. *tThis theorem, as well as 5.3.4, is true for an arbitrary closed bounded set. For these developments, see Sections1 1 .2 and 1 1 .3.
5.3 CONTINUOUS FUNCTIONS ON INTERVALS 131 We note that a continuous function on a set A does not necessarily have an absolutemaximum or an absolute minimum on the set. For example, f(x) := 1/x has neither anabsolute maximum nor an absolute minimum on the set A := (0, 00). (See Figure 5 .3.1). f fThere can be no absolute maximum for on A since is not bounded above on A, and fthere is no point at which attains the value 0 = {f (x) : x inf E A}. The same function hasneither an absolute maximum nor an absolute minimum when it is restricted to the set (0, 1),while it has both an absolute maximum and an absolute minimum when i t i s restricted tothe set [1, 2].In addition, f (x) = 1 Ix has an absolute maximum but no absolute minimumwhen restricted to the set [1, 00), but no absolute maximum and no absolute minimumwhen restricted to the set (1, 00). It is readily seen that if a function has an absolute maximum point, then this pointis not necessarily uniquely determined. For example, the function g(x) := x 2 defined forx E := [-1, +1] A has the two points x = ±1giving the absolute maximum on A, and x =0the single point yielding its absolute minimum on A. (See Figure 5 .3 .2.) To pick anextreme example, the constant function hex) := 1 x lR for E is such that every of point lRis both an absolute maximum and an absolute minimum point for h. --I---------L...--+ x --------����---- x 2 - 1 > = Figure 5.3.1 The function Figure 5.3.2 The function f(x) = 1 jx (x 0). g(x) x2 (Ixl S 1).5.3.4 Maximum-Minimum Theorem Let I := [a, b] b e a closed bounded interval andletf lR : I � be continuous on I. Then f has an absolute maximum and an absoluteminimum on I.Proof. Consider the nonempty set f (I) := {f (x) : x E I} of values of f on I. In Theorem5.3.2 it was established that f(l) is a bounded subset of lR. Let s* := sup f(l) and s* :=inf f(l). We claim that there exist points x* and x i n I such that s* = f(x*) and s = * *f(x* ). We will establish the existence of the point x*, leaving the proof of the existence ofx* to the reader. Since s* = sup f(l), if n E N, then the number s* - lin is not an upper bound of theset f(l). Consequently there exists a number xn E I such that(1) 1 s* - - < f(xn ) :s s* for all n E No nSince I is bounded, the sequence X := (xn ) is bounded. Therefore, by the BolzanoWeierstrass Theorem 3.4.8, there is a subsequence X = (xn ) of X that converges to some rnumber x*. Since the elements of X belong to I = [a, b], it follows from Theorem 3 . 2.6
132 CHAPTER 5 CONTINUOUS FUNCTIONS ( )that x* E I . Therefore f is continuous at x* so that lim f (xn ) = f(x*). Since it follows rfrom ( 1) that r 1 nr r s* - - < f(xn ) ::::: s* for all E N, )we conclude from the Squeeze Theorem 3.2.7 that lim(f(xn ) = s*. Therefore we have f(x*) = lim ( J (xn) ) = s* = sup f (l) .We conclude that x* is an absolute maximum point of f on I. Q.E.D. The next result is the theoretical basis for locating roots of a continuous function bymeans of sign changes of the function. The proof also provides an algorithm, known asthe Bisection Method, for the calculation of roots to a specified degree of accuracy andcan be readily programmed for a computer. It is a standard tool for finding solutions ofequations of the form f (x) = 0, where f is a continuous function. An alternative proof ofthe theorem is indicated in Exercise 1 1.5.3.5 Location of Roots Theorem Let I = [a, b] and let f : I -+ lR be continuous onI . f(a) < 0 < feb), or f(a) > 0 > feb), then there exists a number c E (a , b) such If ifthat f(c) = o.Proof. We assume that f(a) < 0 < feb). We will generate a sequence of intervals bysuccessive bisections. Let Il := [a p b l ] where a l := a , b l := b, and let P I be the midpoint ! + =1=P I : = (a l b l ) If f(P I ) = 0, we take c := P I and we are done. If f(P I ) 0, then eitherf(P I ) > O or f(P I ) < O. If f(P I ) > 0, then we set a2 := a l , b2 := p l while if f(P I ) < 0,then we set a2 := P I b2 := b l · In either case, we let 12 := [a2 , b2 ]; then we have 12 C IIand f(a2 ) < 0, f(b2 ) > O. We continue the bisection process. Suppose that the intervals 11 12 , . . . , Ik havebeen obtained by successive bisection in the same manner. Then we have f(ak ) < 0and f (bk ) > 0, and we set Pk := (ak bk ) . If f(Pk ) = 0, we take c := Pk and we are ! +done. If f(Pk ) > 0, we set ak+ 1 := ak , bk+ 1 : = Pk while if f(Pk ) < 0, we set ak+ 1 := Pk bH I := bk · In either case, we let Ik+ 1 := [aH I , bHI ] ; then IH I C Ik and f(ak+ ) < 0,f (bk+ l ) > O. If the process terminates by locating a point Pn such that f (pn ) = 0, then we are done.If the process does not terminate, then we Qbtain a nested sequence of closed boundedintervals In := [an bn ] such that for every n E N we have andFurthermore, since the intervals are obtained by repeated bisection, the length of In is n lequal to bn - an = (b - a)/2 - • It follows from the Nested Intervals Property 2.5.2 thatthere exists a point c that belongs to In for all n E N. Since an ::::: c ::::: bn for all n E N, we n l n l <have 0 - C - an - bn - an = (b - a)/2 - , and 0 - bn c - bn - an = (b - a)/2 - . < < < -Hence, it follows that lim(an ) = c = lim(bn ) . Since f is continuous at c, we have ( ) lim J (an ) = f(c) = lim J (bn ) .( )The fact that f(an ) < 0 for all n E N implies that f(c) = lim J (an) ( ) ::::: o. Also, the fact ( )that f (bn ) :::: 0 for all n E N implies that f (c) = lim J (bn ) :::: O. Thus, we conclude thatf(c) = O. Consequently, c is a root of f. Q.E.D.
5.3 CONTINUOUS FUNCTIONS ON INTERVALS 133 The following example illustrates how the Bisection Method for finding roots is appliedin a systematic fashion.5.3.6 Example The equation = e x f(x) x - 2 0= c has a root in the interval [0, 1], fbecause is continuous on this interval and = f(O) - 2 0 f(1) - 2 > O. < and = e Weconstruct the following table, where the sign of f( Pn ) determines the interval at the next Pnstep. The far right column is an upper bound on the error when is used to approximate c,the root because we have i Pn - c i ::: ! (bn - an ) 1/2n . =We will find an approximation Pn with error less than 10 -2 . n an bn Pn f(Pn ) ! (bn - an ) 1 0 .5 - 1.176 . 5 2 .5 1 .75 -.412 .25 3 .75 1 .875 +.099 .125 4 .75 .875 . 8125 -.169 .0625 5 .8125 .875 . 84375 -.0382 .03125 6 .84375 .875 .859375 +.0296 .015625 7 .84375 .859375 .8515625 .0078125We have stopped at n = � 7, obtaining c P7 . 8515625 with error less than .0078125 . =This is the first step in which the error is less than 10-2 • The decimal place values of P7 pastthe second place cannot be taken seriously, but we can conclude that . 843 c .860. D < <Bolzanos TheoremThe next result is a generalization of the Location of Roots Theorem. It assures us that acontinuous function on an interval takes on (at least once) any number that lies betweentwo of its values.5.3.7 Bolzanos Intermediate Value Theorem Let I be an interval and let f : I -+ lR.be continuous on I. If a, b E l and if k E lR. satisfies f(a) k feb), < < then there exists apoint cE a b such that f(c) k. I between and =Proof. Suppose that a b and let g(x) f(x) k; then g(a) 0 g(b). By the < := - < <Location of Roots Theorem 5.3.5 there exists a point c with a c b such that 0 = < <g(c) f(c) - k. Therefore f(c) k . = = If b a, let hex) < := k - f(x) so that h(b) 0 h(a). Therefore there exists a point < <c with b c a such that 0 h(c) k - f(c), whence f(c) k. < < = = = Q.E.D.5.3.8 Corollary Let I [a, b] be a closed, bounded interval and let f I -+ lR. be = :continuous on I. If k E lR. is any number satisf ying inf f(l) ::: k ::: sup f(l),then there exists a number c E I such that f (c) k. =Proof. It follows from the Maximum-Minimum Theorem 5 . 3.4 that there are points c*and c* in I such that inf f(l) f(c ) ::: k ::: f(c*) sup f(l). = = *The conclusion now follows from Bolzano s Theorem 5.3.7. Q.E.D.
134 CHAPTER 5 CONTINUOUS FUNCTIONS The next theorem summarizes the main results of this section. It states that the imageof a closed bounded interval under a continuous function is also a closed bounded interval.The endpoints of the image interval are the absolute minimum and absolute maximumvalues of the function, and the statement that all values between the absolute minimumand the absolute maximum values belong to the image is a way of describing Bolzano osIntermediate Value Theorem.5.3.9 Theorem Let I be a closed bounded interval and let f : I ---+ � be continuouson I . Then the set f (I) : = {f (x) : x E I } is a closed bounded interval.Proof. Ifm := f(l) we let inf and M sup := f(l), then we know from the Maximum mMinimum Theorem 5.3.4 that and M belong to f(l). Moreover, we have S; M]. f(l) [m, k [m,If is any element of M], then it follows from the preceding corollary that there exists eEla point k = f(c). such that Hence, k E f(l) and we conclude that M] S; [m, f(l). f(l)Therefore, [m, is the interval M]. Q.E.D.Warning If := [a, b] I is an interval and I ---+ is continuous on I, we have proved f: �that f(l) [m, M]. is the interval We have not proved (and it is not always true) that f(l)is the interval [f(a), feb)]. (See Figure 5.3.3.) M feb) f(a) m x a x. x· b Figure 5.3.3 1 (1) = [m, M]. The preceding theorem is a "preservation" theorem in the sense that it states thatthe continuous image of a closed bounded interval is a set of the same type. The nexttheorem extends this result to general intervals. However, it should be noted that althoughthe continuous image of an interval is shown to be an interval, it is not true that the imageinterval necessarily has the sameformas the domain interval. For example, the continuousimage of an open interval need not be an open interval, and the continuous image of anunbounded closed interval need not be a closed interval. Indeed, if f(x) := 1/(x2 + 1)for x E �, f � then is continuous on [see Example 5.2.3(b)]. It is easy to see that ifII := ( - 1, 1), f(ll ) = then G , ] which is not an open interval. Also, if 1, 12 := [0, 00),then f(l2 ) = (0, 1], which is not a closed interval. (See Figure 5.3.4.)
5.3 CONTINUOUS FUNCTIONS ON INTERVALS 135 -1 Figure 5.3.4 Graph of I (x) = I/(x 2 + 1) (x E lR). To prove the Preservation of Intervals Theorem 5.3.10, we will use Theorem 2.5.1characterizing intervals.5.3.10 Preservation of Intervals Theorem Let I be an interval and let f : I -+ � becontinuous on I . Then the set f (l) is an interval.Proof. Let a, fJ E f(l) with a fJ; then there exist POiIlts a, b E l such that a = f(a) <and fJ feb). Further, it follows from Bolzano s Intermediate Value Theorem 5.3.7 that =if k E (a, fJ) then there exists a number e E l with k f(c) E f(l). Therefore [a, fJ] � =f(l), showing that f(l) possesses property (1) of Theorem 2.5.1. Therefore f(l) is aninterval. Q.E.D.Exercises for Section 5.3 1. Let 1 : = [a, b] and let I : 1 --+ lR be a continuous function such that I (x) > Prove that there exists a number ct > 0 such that I (x) 2: ct for all x E I. 0 for each x in I . 2. Let 1 := [a, b] and let I : 1 --+ lR and g : 1 --+ lR be continuous functions on I. Show that the set E := (x E 1 : I(x) = g (x)} has the property that if (xn ) s::: Exo then Xo E E . and xn --+ 3. Let 1 : = [a, b] and let I : 1 --+ lR be a continuous function on 1 such that for each x in 1 there exists y in 1 such that I /(y) 1 � � I /(x) l . Prove there exists a point c in 1 such that I (c) = o. 4. Show that every polynomial of odd degree with real coefficients has at least one real root. 5. Show that the polynomial p(x) := x4 + 7x3 - 9 has at least two real roots. Use a calculator to locate these roots to within two decimal places. 6. Let I be continuous on the interval [0, 1] to lR and such that 1(0) = I (1). Prove that there exists apoint c in [0, �] such that /(c) = I (c + D. [Hint: Consider g (x) = I (x) - I (x + D .] Conclude that there are, at any time, antipodal points on the earths equator that have the same temperature. 7. Show that the equation x = cos x has a solution in the interval [0, 7l 12]. Use the Bisection Method and a calculator to find an approximate solution of this equation, with error less than 10-3 . 8. Show that the function I(x) := 2 ln x + .;x - 2 has root in the interval [1, 2]. Use the Bisection 2 Method and a calculator to find the root with error less than 10- . 9. (a) The function I(x) := (x - l ) (x - 2)(x - 3)(x - 4)(x - 5) has five roots in the interval [0, 7]. If the Bisection Method is applied on this interval, which of the roots is located? (b) Same question for g (x) := (x - 2)(x - 3)(x - 4)(x - 5)(x - 6) on the interval [0, 7].10. If the Bisection Method is used on an interval of length 1 to find Pn with error Ipn - cl < 10-5, determine the least value of n that will assure this accuracy.
136 CHAPTER 5 CONTINUOUS FUNCTIONS1 1 . Let 1 [a, b], let f : 1 --+ R be continuous on I, and assume that f(a) < 0, f(b) > 0. Let := W := {x E 1 : f(x) < OJ, and let w := sup W. Prove that f (w) = 0. (This provides an alter native proof of Theorem 5.3.5.)12. Let 1 := [0, Jr/2] and let f : 1 --+ R be defined by f (x) := sup{x 2 , cos x } for x E I. Show there exists an absolute minimum point Xo E 1 for f on I. Show that Xo is a solution to tHe equation cos x = x 2 • x---+ - oo x---+oo1 3 . Suppose that f : R --+ R is continuous on R and that lim f = ° and lim f = 0. Prove that f is bounded on R and attains either a maximum or minimum on R. Give an example to show that both a maximum and a minimum need not be attained.14. Let f : R --+ R be continuous on R and let {3 E R. Show that if Xo E R is such that f(xo ) < {3, then there exists a 8-neighborhood U of X o such that f (x) < {3 for all x E U.15. Examine which open [respectively, closed] intervals are mapped by f(x) := x2 for x E R onto open [respectively, closed] intervals.16. Examine the mapping of open [respectively, closed] intervals under the functions g (x) := 1/(x 2 + 1) and h (x) := x 3 for x E R.17. If f : [0, 1] --+ R is continuous and has only rational [respectively, irrational] values, must f be constant? Prove your assertion.18. Let 1 := [a, b] and let f : 1 --+ R be a (not necessarily continuous) function with the property that for every x E I, the function f is bounded on a neighborhood V8 (x) of x (in the sense of x Definition 4.2. 1). Prove that f is bounded on I.19. Let J := (a, b) and let g : J --+ R be a continuous function with the property that for every x E J , the function g is bounded on a neighborhood V8 (x) of x. Show by example that g is not x necessarily bounded on J .Section 5.4 Uniform ContinuityLet A � JR and let I:A -+ JR. Definition 5.1.1 states that the following statements areequivalent: (i) I is continuous at every pointuE A; (ii) given s > and0 UE u) 0 x A, there is a 8 (s, > such that for all such that A xEand Ix - ul u), < 8 (s, then I/ (x) - l(u)1 < s. The point we wish to emphasize here is that 8 depends, in general, on both s > and 0uE u I A. The fact that 8 depends on is a reflection of the fact that the function may changeits values rapidly near certain points and slowly near other points. [For example, considerI(x) : = sinO/x) for x 0; > see Figure 4.1.3.] I Now it often happens that the function is such that the number 8 can be chosen to beindependent of the point uE s. A and to depend only on For example, if I (x) := 2x forall xE JR, then II (x) - l(u)1 2 1x - ul ,=and so we can choose 8 (s, u) s/2 for aIl s > 0, u E R (Why?) := On the other hand if g(x) := I /x for x E A := {x E R x > O } , then(1) g(x) - g(u) u ux x- = -- .
5.4 UNIFORM CONTINUITY 137 uIf E A is given and if we take(2)then if Ix - ul 8 (e, u), we have Ix - ul 4u so that 4u x � u, whence it follows < < < <that 1/x 2/u. Thus, if Ix - u l 4u, the equality (1) yields the inequality < <(3) Ig(x) - g(u)1 S (2/u2 ) Ix - ul .Consequently, if Ix - ul 8 (e, u), then (2) and (3) imply that < Ig(x) - g(u)1 (2/u 2 ) (4u2 e) = e. <We have seen that the selection of 8 (e, u) by the formula (2) "works" in the sense that itenables us to give a value of 8 that will ensure that I g (x) - g(u)1 e when Ix - ul 8< <and x, u E A. We note that the value of 8 (e, u) given in (2) certainly depends on the pointu E A. If we wish to consider all u E A, formula (2) does not lead to one value 8 (e) > 0that will "work" simultaneously for all u > 0, since inf{8 (e, u) : u > O} O. = An alert reader will have observed that there are other selections that can be madefor 8. (For example we could also take 8, (e, u) inf { �u, �u 2 e}, as the reader can show; :=however, we still have inf { 8 , (e, u): u > O} 0.) In fact, there is no way of choosing one =value of 8 that will "work" for all u > 0 for the function g(x) 1/x, as we shall see. = The situation is exhibited graphically in Figures 5.4.1 and 5.4 .2 where, for a givene-neighborhood �(4) about 4 f(2) and �(2) about 2 = f(4), the corresponding max =imum values of 8 are seen to be considerably different. As u tends to 0, the permissiblevalues of 8 tend to O. VE�:r.�====���� 1 � ------�-" ----�.x --���-------- --�x -- -- � � �2 -�1 1) - neighborhood 2 1) -neighborhood Figure 5.4.1 g (x) = l /x (x > 0) . Figure 5.4.2 g (x) = l /x (x > 0) .5.4.1 Definition Let A � IR and let A """"* R We say that is uniformly continuous f: fon A if for each e > there is a 8 (e) > such that if 0 0 x, u E A are any numbers satisfyingIx - ul < 8 (e), then If(x) - f(u) 1 < e. It is clear that if f is uniformly continuous on A, then it is continuous at every point ofA. In general, however, the converse does not hold, as is shown by the function g (x) = 1/xon the set A := {x E IR : x > O}. It is useful to formulate a condition equivalent to saying that f is not uniformlycontinuous on A. We give such criteria in the next result, leaving the proof to the reader asan exercise.
138 CHAPTER 5 CONTINUOUS FUNCTIONS5.4.2 Nonuniform Continuity Criteria Let A S; lR and let I : A --+ R Then the following statements are equivalent:(i) I is not uniformly continuous on A.(ii) U There exists an 80 > 0 such that for every 8 > 0 there are points x8 8 in A such that IX8 - u81 8 and I I(x8) - l(u8)1 ::: 80 , <(iii) There exists an 80 > 0 and two sequences (xn ) and (u n ) in A such that lim(xn - un ) = 0 and II(xn ) - l(un )1 ::: 80 for all n E N. We can apply this result to show that g(x) := 1 /x is not uniformly continuous on A :={x E lR : x > OJ. For, if xn := l i n and un := 1 / (n + 1), then we have lim(xn - un ) 0, =but Ig(xn ) - g(u n )1 I for all n E N. = We now present an important result that assures that a continuous function on a closedbounded interval I is uniformly continuous on I. Other proofs of this theorem are given inSections 5 .5 and 1 1 .3.5.4.3 Uniform Continuity Theorem Let I be a closed bounded interval and let I : I --+lR be continuous on I. Then I is uniformly continuous on I.Proof. If I i s not uniformly continuous on I then, by the preceding result, there exists80 > 0 and two sequences (xn ) and (un ) in I such that I Xn - u n l lin and II(xn ) - <l(un )1 ::: 80 for all n E N. Since I is bounded, the sequence (xn ) is bounded; by theBolzano-Weierstrass Theorem 3.4.8 there is a subsequence (xn ) of (xn ) that converges to kan element z. Since I is closed, the limit z belongs to I, by Theorem 3.2.6. It is clear thatthe corresponding subsequence (u n ) also converges to z, since k I z, Now if is continuous at the point then both of the sequences (J (xnk )) and (J (unk »)must converge to I(z). But this is not possible sincefor all nE N. I Thus the hypothesis that is not uniformly continuous on the closed bounded Iinterval I implies that is not continuous at some point zE I. Consequently, if is I Icontinuous at every point of I, then is uniformly continuous on I. Q.E.D.Lipschitz FunctionsIf a uniformly continuous function is given on a set that is not a closed bounded interval,then it is sometimes difficult to establish its uniform continuity. However, there is a condition that frequently occurs that is sufficient to guarantee uniform continuity.5.4.4 Definition Let A S; lR and let I : A --+ R If there exists a constant K > 0 suchthat(4) II (x) - l(u) 1 .::: K Ix - ulfor all x, u EA , then I is said to be a Lipschitz function (or to satisfy a Lipschitzcondition) on A.
5 .4 UNIFORM CONTINUITY 139 The condition (4) that a function f:I I --+ lR. on an interval is a Lipschitz functioncan be interpreted geometrically as follows. If we write the condition as f(x) - f(u) � K, I x-u I x, u E I, x ::j:. u,then the quantity inside the absolute values is the slope of a line segment joining the points(x, f (x)) (u, f (u) ) and f . Thus a function satisfies a Lipschitz condition if and only if theslopes of all 1ine segments joining two points on the graph of y = f (x) over are bounded Iby some number K.5.4.5 Theorem If f : A --+ lR. is a Lipschitz function, then f is uniformly continuouson A.Proof. If condition (4) is satisfied, then given 8 > 0, we can take 8 := 8j K. If x, u E Asatisfy Ix - u I < 8, then 8 If(x) - f(u) 1 K · K = 8. <Therefore f is uniformly continuous on A. Q.E.D.5.4.6 Examples (a) If f(x) := x 2 on A := [0, b], where b > 0, then If(x) - f(u)1 = Ix + ul lx - ul � 2b lx - ulfor all x, u in [0, b]. Thus f satisfies (4) with K := 2b on A, and therefore f is uniformlycontinuous on A. Of course, since f is continuous and A is a closed bounded interval, thiscan also be deduced from the Uniform Continuity Theorem. (Note that f does not satisfya Lipschitz condition on the interval [0, 00).)(b) Not every uniformly continuous function is a Lipschitz function. Let g(x) := ,JX for x in the closed bounded interval I := [0, 2]. Since g is continuouson I, it follows from the Uniform Continuity Theorem 5.4.3 that g is uniformly continuouson I. However, there is no number K > ° such that Ig(x)1 � Klxl for all x E I. (Whynot?) Therefore, g is not a Lipschitz function on I.(c) The Uniform Continuity Theorem and Theorem 5 .4 . 5 can sometimes be combined toestablish the uniform continuity of a function on a set. We consider g(x) := ,JX on the set A := [0, 00). The uniform continuity of g onthe interval I := [0, 2] follows from the Uniform Continuity Theorem as noted in (b). IfJ : = [1, 00), then if both x, u are in J, we have Ig (x) - g(u) 1 = I JX - JUI = ; + � � � Ix - u l · x- uThus g is a Lipschitz function on J with constant K �, and hence by Theorem 5.4.5, =g is uniformly continuous on [1, 00). Since A = I U J, it follows [by taking 8 (8) :=inf { 1, 8[ (8), 8 (8) }] that g is uniformly continuous on A . We leave the details to the Jreader. 0The Continuous Extension TheoremWe have seen examples of functions that are continuous but not uniformly continuous onopen intervals; for example, the function = f(x) Ijx on the interval (0, 1). On the otherhand, by the Uniform Continuity Theorem, a function that is continuous on a closed boundedinterval is always uniformly continuous. So the question arises: Under what conditions is a
140 CHAPTER 5 CONTINUOUS FUNCTIONSfunction unifonnly continuous on a bounded open interval? The answer reveals the strengthof uniform continuity, for it will be shown that a function on (a, b) is uniformly continuousif and only if it can be defined at the endpoints to produce a function that is continuous onthe closed interval. We first establish a result that is of interest in itself.5.4.7 Theorem If I A --* : lR is uniformly continuous on a subset A of lR and if (xn ) isa Cauchy sequence in A, then (J (xn ») is a Cauchy sequence in lR.Proof. Let (xn ) be a Cauchy sequence in A, and let e > 0 be given. First choose 8 > 0 Usuch that if x, in A satisfy Ix - u l < 8, then I/(x) - l(u) 1 < e. Since (xn ) is a Cauchysequence, there exists H(8) such that IXn - xm I < 8 for all n, m > H (8). By the choice of8, this implies that for n, m > H (8), we have I/(xn ) - l(xm ) 1 < e. Therefore the sequence(f(xn » is a Cauchy sequence. Q.E.D. The preceding result gives us an alternative way of seeing that I(x) := 11x is notuniformly continuous on (0, 1). We note that the sequence given by Xn := lin in (0, 1) isa Cauchy sequence, but the image sequence, where I(xn ) = n, is not a Cauchy sequence.5.4.8 Continuous Extension Theorem A function I is uniformly continuous on theinterval if and only if it can be defined at the endpoints a and such that the ex (a, b) btended function is continuous on [a, b].Proof. ( {:::) This direction is trivial. (=}) Suppose I is uniformly continuous on (a, b). We shall show how to extend I x cto a; the argument for b is similar. This is done by showing that lim I (x) = L exists, and .....this is accomplished by using the sequential criterion for limits. If (xn ) is a sequence in(a, b) with lim(xn ) = a, then it is a Cauchy sequence, and by the preceding theorem, thesequence (J(xn ») is also a Cauchy sequence, and so is convergent by Theorem 3 . 5 . 5 . Thusthe limit lim(J(xn ») = L exists. If (u n ) is any other sequence in (a, b) that converges to a,then lim(un - xn ) = a - a = 0, so by the uniform continuity of I we have lim(J(un ») = lim(J(u n ) - I(xn ») + lim(J(xn ») = O + L = L.Since we get the same value L for every sequence converging to a, we infer from thesequential criterion for limits that I has limit L at a. If we define I (a) := L, then I iscontinuous at a. The same argument applies to b, so we conclude that I has a continuousextension to the interval [a, b]. Q.E.D. Since the limit of I(x) := sin(l lx) at 0 does not exist, we infer from the ContinuousExtension Theorem that the function is not uniformly continuous on (0, b] for any b > O. X ""On the other hand, since lim x sin(l Ix) = 0 exists, the function g(x) := x sin(l Ix) is 0unifonnly continuous on (0, b] for all b > O.ApproximationtIn many applications it is important to be able to approximate continuous functions byfunctions of an elementary nature. Although there are a variety of definitions that can beused to make the word "approximate" more precise, one ofthe most natural (as well as one oftTbe rest of this section can be omitted on a first reading of this chapter.
5.4 UNIFORM CONTINUITY 141the most important) is to require that, at every point of the given domain, the approximatingfunction shall not differ from the given function by more than the preassigned error.5.4.9 Definition Let I S; lR be an interval and let s : I -+ R Then s is called a stepfunction if it has only a finite number of distinct values, each value being assumed on oneor more intervals in I . For example, the function s :[-2, 4] -+ lR defined by 0, -2 .::: x < - 1 , 1, - 1 .::: x .::: 0, 1 2 0 < < !, X s ex) : = 3, ! .::: x < l , -2, 1 .::: x .::: 3, 2, 3 < x .::: 4,is a step function. (See Figure 5.4.3.) y H ____ �__� � L- -- -- -- x -- -- � -L J- - Figure 5.4.3 Graph of y = sex). We will now show that a continuous function on a closed bounded interval I can beapproximated arbitrarily closely by step functions.5.4.10 Theorem Let I be a closed bounded interval and let I : I -+ lR be continuous onI . If £ > 0, then there exists a step function se : I -+ lR such that I I (x) - s /x) I < £ for Eall x I.Proof. Since (by the Uniform Continuity Theorem 5 .4.3) the function I is uniformly 0 0continuous, it follows that given £ > there is a number 8 (£) > such that if x , yE I - yland Ix < 8 (£), then I I (x) - l(y) 1 < £. Let I := [a, b] and let mEN be sufficiently h (b - a)/mlarge so that := < 8 (£). We now divide I = [a, b] m into disjoint intervals h; [a, a h], kof length namely, II : = + and I : = + (a (k - )h, a kh] k 2, . . . , m. l + for = k hSince the length of each subinterval I is < 8 (£), the difference between any two values kof I in I is less than £ . We now define(5) se (x) := I(a + kh) for x E lk k = 1 , . . . , m,
142 CHAPTER 5 CONTINUOUS FUNCTIONS S8so that is constant on each interval Ik . (In fact the value of S8 on Ik is the value of f at Ik .the right endpoint of See Figure 5 .4.4.) Consequently if x E lk then If(x) - s8 (x) 1 = I f (x) - I (a + kh) 1 8 . <Therefore we have I f(x) - s/x)1 8 for all x E I. < Q.E.D. Figure 5.4.4 Approximation by step functions. Note that the proof of the preceding theorem establishes somewhat more than wasannounced in the statement of the theorem. In fact, we have proved the following, moreprecise, assertion.5.4.11 Corollary Let I :=[a , b] be a closed bounded interval and let I lR be : I --+continuous on I. If 8 > 0, there exists a natural number m such that if we divide into I Ik h := S8 sm disjoint intervals having length (b - a)/m, then the step function defined inequation (5) satisfies I f (x) - s (x) I 8 < for all x E I. Step functions are extremely elementary in character, but they are not continuous(except in trivial cases). Since it is often desirable to approximate continuous functions byelementary continuous functions, we now shall show that we can approximate continuousfunctions by continuous piecewise linear functions. I5.4.12 Definition Let := [a , b] be an interval. Then a function g : I --+ lR is said to be I Ipiecewise linear on if is the union of a finite number of disjoint intervals ,1 m II . . . g Iksuch that the restriction of to each interval is a linear function.Remark It is evident that in order for a piecewise linear function to be continuous g I, gon the line segments that form the graph of must meet at the endpoints of adjacentsubintervals Ik , Ik+ 1 (k = 1, . . . ,m 1). -5.4.13 Theorem Let I be a closed bounded interval and let f : I --+ lR be continuous on sI. If 8 > 0, then there exists a continuous piecewise linear function g : I --+ lR such thatIf(x) - g8(x) 1 8 for all x E I . <Proof. Since f is uniformly continuous on I := [a , b], there is a number 8(8) > 0 suchthat if x, y E I and I x y l 8(8), then I/(x) - l (y) 1 8. Let m E N be sufficiently - < <
5.4 UNIFORM CONTINUITY 143 h (b - a)/mlarge so that := < 8 (8). Divide I = [a, b] m into disjoint intervals of lengthh; namely let II = + and let Ik = ( + [a, a h], + ] for = a (k - l)h, a kh k 2, . . . , m. On g,each interval Ik we define to be the linear function joining the points (a + (k - l)h, I(a + (k - l)h) ) and (a + kh, I(a + kh) ) .Then g, is a continuous piecewise linear function on I . Since, for x E Ik the value I (x) iswithin 8 of I(a + (k - l)h) and I(a + kh), it is an exercise to show that II(x) - g,(x)1 <8 for all x E Ik ; therefore this inequality holds for all x E I . (See Figure 5.4.5 .) Q.E.D . .., y = fix) + E .... , I - Y = g,(x) , ,/ y = fix) - E - " " " _ , ,1 y = fix) :I I , I I I I I "" _ ... .... I r - - - .... I I : I I I I I I I Figure 5.4.5 Approximation by piecewise linear function. We shall close this section by stating the important theorem of Weierstrass concerningthe approximation of continuous functions by polynomial functions. As would be expected,in order to obtain an approximation within an arbitrarily preassigned 8 > we must be 0,prepared to use polynomials of arbitrarily high degree.5.4.14 Weierstrass Approximation Theorem Let I = and let : I -+ JR be a [a, b] Icontinuous function. If 8 ° is given, then there exists a polynomial function pe such >that I I (x) - e Cx) I p < 8 for all E I . x There are a number of proofs of this result. Unfortunately, all of them are ratherintricate, or employ results that are not yet at our disposal. One of the most elementaryproofs is based on the following theorem, due to Serge BernsteIn, for continuous functionson [0, 1]. Given I : [0, 1] -+ JR, BernsteIn defined the sequence of polynomials:(6) The polynomial function Bn is called the nth I; Bernstein polynomial for it is a polynomial n I of degree at most and its coefficients depend on the values of the function at the + n 1 equally spaced points 0, lin, 21n, . . . , kin, . . . , 1, and on the binomial coefficients (n) n! n(n - 1) . . . .(n. - k + 1) . = = k k!(n - k) 1 · 2 ·k
144 CHAPTER 5 CONTINUOUS FUNCTIONS5.4.15 BernsteIns Approximation Theorem Let f : [0, 1] -+ JR be continuous and lete > 0. There exists an n, E N such that ifn ::: nt then we have If(x) - Bn (x)1 for all < ex E [0, 1] . The proof of Bernstein s Approximation Theorem is given in [ERA, pp. 169-172]. The Weierstrass Approximation Theorem 5.4.14 can be derived from the BernsteinApproximation Theorem 5.4.15 by a change of variable. Specifically, we replacef : [a, b] JR -+ by a function : F [0, 1] -+ JR, defined by F(t) := f (a + (b - a)t) t E [0, 1] . forThe function F can be approximated by Bernstein polynomials for F on the interval [0, 1],which can then yield polynomials on [a, b] that approximate f.Exercises for Section 5.4 1 . Show that the function f(x) := l /x i s uniformly continuous o n the set A := [a, (0), where a is a positive constant. 2. Show that the function f(x) := l /x 2 is uniformly continuous on A := [ 1 , (0), but that it is not unifonnly continuous on B := (0, (0). 3. Use the Nonunifonn Continuity Criterion 5.4.2 to show that the following functions are not uniformly continuous on the given sets. (a) f (x) := x 2 , A := [0, (0). (b) g(x) := sin(1/x), B := (0, (0). + 4. Show that the function f(x) := 1/(1 x 2 ) for x E JR is uniformly continuous on JR. 5. Show that if f and g are uniformly continuous on a subset A of R, then f + g is unifonnly continuous on A. 6. Show that if f and g are uniformly continuous on A � JR and if they are both bounded on A, then their product fg is uniformly continuous on A. 7. If f(x) := x and g (x) := sin x, show that both f and g are uniformly continuous on JR, but that their product fg is not unifonnly continuous on R. 8. Prove that if f and g are each unifonnly continuous on JR, then the composite function f o g is unifonnly continuous on R. 9. If f is uniformly continuous on A � JR, and I f(x) 1 � k > ° for all x E A, show that l /f is uniformly continuous on A.10. Prove that if f is unifonnly continuous on a bounded subset A of JR, then f is bounded on A.1 1 . If g(x) := JX for x E [0, 1], show that there does not exist a constant K such that lg(x ) l ::: Klx l for all x E [0, 1]. Conclude that the unifonnly continuous g is not a Lipschitz function on [0, 1].12. Show that if f is continuous on [0, (0) and unifonnly continuous on [a, (0) for some positive constant a, then f is uniformly continuous on [0, (0).13. Let A � JR and suppose that f : A --+ JR has the following property: for each e > ° there exists a function gE : A --+ JR such that gE is uniformly continuous on A and If(x) - gE (x) 1 < e for all x E A. Prove that f is unifonnly continuous on A.14. A function f: JR --+ R is said to be periodic on JR if there exists a number p > ° such that f (x + p) = f (x) for all x E R. Prove that a continuous periodic function on JR is bounded and uniformly continuous on R.
5.5 CONTINUITY AND GAUGES 14515. If lo(x) 1 x[Hin[0, Thecalculate the first fewasserts that polynomials for 10 • Show that they := for E coincide with Ia- t: 1], Binomial Theorem BernsteIn (a + b)n t (n)akbn-k .] = =O k16. If II (x) x for x E [0, 1], calculate the first := kfew BernsteIn polynomials for II . Show that they coincide with II.17. If 12 (x) x2 lfor x [0,l1], calculate the first few BernsteIn polynomials for 12 . Show that := E Bn (x) (1 - /n)x2 + ( /n)x. =Section 5.5 Continuity and GaugesWe will now introduce some concepts that will be used later--especially in Chapters 7 10and on integration theory. However, we wish to introduce the notion of a "gauge" nowbecause of its connection with the study of continuous functions. We first define the notionof a tagged partition of an interval.5.5.1 Definition A partition of an interval l := [a, b] is a collection P = {Ii . . . , In } ofnon-overlapping closed intervals whose union is [a, b]. We ordinarily denote the intervals Ii [Xi -I Xi]by : = where a = Xo < . . . < Xi -I < Xi < . . . < Xn = b .The points Xi (i 0, = . . . , n ) are called the partition points of P. If a point has been ti Ii i . tichosen from each interval for = 1 , . . , n, then the points are called the tags and theset of ordered pairs p = {(I" tl ), . • • , (In tn ) }is called a tagged partition of I. (The dot signifies that the partition is tagged.) The "fineness" of a partition P refers to the lengths of the subintervals in P. Instead ofrequiring that all subintervals have length less than some specific quantity, it is often useful Iito allow varying degrees of fineness for different subintervals in P. This is accomplishedby the use of a "gauge", which we now define.5.5.2 Definition A gauge on I is a strictly positive function defined on I. If a is a gaugeon I, then a (tagged) partition P is said to be a-fine if(1) for i 1, = . . . , n. We note that the notion of a-fineness requires that the partition be tagged, so we do notneed to say "tagged partition" in this case. Xi ! Ii Figure 5.5.1 Inclusion (1). I A gauge a on an interval assigns an interval + [t - a(t), t a(t)] to each point t E l . PThe a-fineness of a partition requires that each subinterval ( of is contained in the P tiinterval determined by the gauge a and the tag for that subinterval. This is indicated
146 CHAPTER 5 CONTINUOUS FUNCTIONSby the inclusions in (I); see Figure 5 .5 . 1. Note that the length of the subintervals is alsocontrolled by the gauge and the tags; the next lemma reflects that control.5.5 3 Lemma : If a partition P of! : = [a, b] is 8 -fine and x E I, then there exists a tag tjin P such that Ix - tj I :::: 8 (t).Proof. If x E I, there exists a subinterval [xi I Xi] from - P that contains x. Since P is8-fine, then(2)whence it follows that Ix - ti I :::: 8 (t). Q.E.D. In the theory of Riemann integration, we will use gauges 8 that are constant functionsto control the fineness of the partition; in the theory of the generalized Riemann integral,the use of nonconstant gauges is essential. But nonconstant gauge functions arise quitenaturally in connection with continuous functions. For, let I: I -+ lR. be continuous onI and let c > 0 be given. Then, for each point t E l there exists 8, (t) > 0 such that < <if Ix - t l 8, (t) and x E I , then I/(x) - l(t) 1 c. Since 8, is defined and is strictlypositive on I , the function 8, is a gauge on I . Later in this section, we will use the relationsbetween gauges and continuity to give alternative proofs of the fundamental properties ofcontinuous functions discussed in Sections and 5.3 5 .4 .5.5.4 Examples (a) If 8 and y are gauges on I := [a, b] and if 0 < 8 (x) :::: y (x) for allx E I , then every partition P that is 8-fine is also y-fine. This follows immediately fromthe inequalities ti - y et) :::: ti - 8 (t) andwhich imply that ti E [ti - 8 (t) , ti + 8 (t) ] S; [tj - y et) , tj + y et) ] for i = I, · · · , n.(b) If 8 1 and 82 are gauges on I := [a, b] and if 8 (x) := min{8 1 (x), 82 (x) } for all x E I,then 8 is also a gauge on I . Moreover, since 8 (x) :::: 8 1 (x), then every 8-fine partition is8 1 -fine. Similarly, every 8-fine partition is also 82 -fine.(c) Suppose that 8 is defined on I := 8 (x) := { to [0, I ] by �x if x if 0 = 0, < x 1. ::::Then 8 is a gauge on [0, 1 ] . If 0 t :::: then [ t - 8 (t), t + 8 (t)] [� t, � t ] , which does < 1, = O. Pnot contain the point Thus, if is a 8-fine partition of I, then the only subinterval in Pthat contains 0 must have the point 0 as its tag. { to(d) Let y be defined on I := [0, 1 ] by if x = 0 or x = I, y (x) := �x if 0 < x :::: � , � (1 - x) if � < x < 1.Then y is a gauge on I , and it is an exercise to show that the subintervals in any y-fine 1partition that contain the points 0 or must have these points as tags. 0
5.5 CONTINUITY AND GAUGES 147Existence of 8-Fine Partitions ____________________In view of the above examples, it is not obvious that an arbitrary gauge 8 admits a 8-fine JRpartition. We now use the Supremum Property of to establish the existence of 8-finepartitions. In the Exercises, we will sketch a proof based on the Nested Intervals Theorem2.5.2.5.5.5 Theorem If 8 is a gauge defined on the interval [a, b], then there exists a 8 -finepartition of [a, b].Proof. Let E denote the set of all points x E [a, blsuch that there exists a 8-fine partitionof the subinterval [a, x]. The set E is not empty, since the pair ([a, x], a) is a 8-fine partitionof the interval [a, x] when x E [a, a + 8(a)] and x :s b. Since E S; [a, b], the set E is also We claim that u E E. Since u - 8 (u) u = sup E, there exists v E E such that u -bounded. Let u := sup E so that a u :s b. We will show that u E E and that u = b. <8(u) v u. Let PI be a 8-fine partition of [a, v] and let 12 := PI U ([v, u], u). Then < < <l2 is a 8-fine partition of [a, u], so that u E E. If u b, let E [a, b] be such that u < W u + 8(u). If Q I is a 8-fine partition < W <of [a, u], we let Q2 := Q I U ([u, w], u). Then Q2 is a 8-fine partition of [a, w], whencew E E. But this contradicts the supposition that u is an upper bound of E. Therefore u = b. Q.E.D.Some Applications ________________________Following R. A. Gordon (see his Monthly article), we will now show that some of the majortheorems in the two preceding sections can be proved by using gauges.Alternate Proof of Theorem 5.3.2: Boundedness Theorem. Since I is continuous on/ , then for each t E / there exists 8(t) 0 such that if x E / and Ix - tl :s 8Ct), then >let K := max{ I /Ct )1 : i = 1, " , n }. By Lemma 5 . 5 . 3, given any x E / there exists i withI / (x) - 1(0 1 :s 1 . Thus 8 is a gauge on / . Let {(Ii ti ) }7= 1 be a 8-fine partition of / and iIx - ti I :s 8Ct), whence I/(x)l :s I/(x) - ICti )1 + I/(t)1 :s 1 + K.Since x E / is arbitrary, then I is bounded b y I + K on / . Q.E.D.Alternate Proof of Theorem 5.3.4: Maximum-Minimum Theorem. We will prove the x*.existence of Let M sup{f(x) : x E I } and suppose that I(x) M for all x E / . := < ISince is continuous on /, for each t E / there exists 8(t) 0 such that if x E / and >Ix - tl Ct) :s 8 , then I(x) ! (M + I(t)). Thus 8 is a gauge on / , and if {(Ii ti ) }7= 1 is a <8-fine partition of /, we let - M := I max{M + I CtI ), . . . , M + ICtn )} 2By Lemma 5.5.3, given any x E /, there exists i with I x - ti I :s 8(t) , whence I(x) 4 (M + I(ti )) :s M. <5ince x E / is arbitrary, then M « M) is an upper bound for I on /, contrary to thedefinition of M as the supremum of I. Q.E.D.Alternate Proofof Theorem 5.3.5: Location ofRoots Theorem. We assume that I (t) =1= 0for all t E /. Since I is continuous at t, Exercise 5 . 1.7 implies that there exists 8(t) 0 >such that if x E / and I x - tl :s 8(t), then I(x) 0 if I(t) 0, and I(x) 0 if I(t) O. < < > >Then 8 is a gauge on / and we let {(Ii ti )}7= 1 be a 8-fine partition. Note that for each i,
5.6 MONOTONE AND INVERSE FUNCTIONS 149 8. Let a be a gauge on I := [a, b] and suppose that I does not have a a-fine partition. (a) Let e := ! (a + b). Show that at least one of the intervals [a, e] and [e, b] does not have a a-fine partition. n (b) Construct a nested sequence (In ) of subintervals with the length of In equal to (b - a)/2 such that In does not have a a-fine partition. (c) Let � E n� l In and let p E N be such that (b - a)/2P < a (� ). Show that Ip � [� - a (� ) , � + a (� )], so the pair (lp, �) is a a-fine partition of Ip 9. Let I := [a, b] and let f : I -+ R be a (not necessarily continuous) function. We say that f is "locally bounded" at e E l if there exists a (e) > 0 such that f is bounded on I n [e - a (e), e + a (e)]. Prove that if f is locally bounded at every point of I, then f is bounded on I.10. Let I := [a, b] and f : I -+ R We say that f is "locally increasing" at e E l if there ex ists a(e) > 0 such that f is increasing on I n [e - a(e), e + aCe)]. Prove that if f is locally increasing at every point of I, then f is increasing on I.Section 5.6 Monotone and Inverse FunctionsRecall that if A S; JR, JR then a function f : A -+ is said to be increasing on A if wheneverxl x2 E A and Xl ::::: X2 then f (Xl ) ::::: f (x2 ). The function f is said to be strictly increasingon A if whenever x l X2 E A and X l < x2 then f (x l ) < f (x2 ) . Similarly, g : A -+ is JRsaid to be decreasing on A if whenever x l x2 E A and X l ::::: x2 then g (x l ) 2: g (x2 ) . Thefunction g is said to be strictly decreasing on A if whenever x l x2 E A and X l < x2 theng(x l ) > g (x2) · If a function is either increasing or decreasing on A, we say that it is monotone on A. Iff is either strictly increasing or strictly decreasing on A, we say that f is strictly monotoneon A. JR We note that if f : A -+ is increasing on A then g := - f is decreasing on A; JRsimilarly if cp : A -+ is decreasing on A then 1/1 : = - cp is increasing on A. In this section, we will be concerned with monotone functions that are defined on aninterval J S; R We will discuss increasing functions explicitly, but it is clear that there arecorresponding results for decreasing functions. These results can either be obtained directlyfrom the results for increasing functions or proved by similar arguments. Monotone functions are not necessarily continuous. For example, if f (x) := for 0XE [0, 1]and f(x) : = for X E 1 (1, 2],then f is increasing on [0, 2], but fails to be 1.continuous at X = However, the next result shows that a monotone function always hasboth one-sided limits (see Definition 4.3.1) JR in at every point that is not an endpoint of itsdomain.5.6.1 Theorem Let J S; JR be an interval and let f J -+ JR be increasing on J. Suppose :that c E J is not an endpoint of J. Then(i) lim f sup{f(x) : X E J, X < c}, = x-+c- x ..... c+(ii) lim f inf{f (x) : X E J, X > c}. =Proof. (i) First note that if X E J and X < c, then f (x) ::::: f (c). Hence the set {f(x) :X E J, X < c}, which is nonvoid since c is not an endpoint of J, is bounded above by f (c).Thus the indicated supremum exists; we denote it by L. If e > 0 is given, then L - e is not Can upper bound of this set. Hence there exists Ye E J, Ye < such that L - e < f (Ye ) ::::: L.
150 CHAPTER 5 CONTINUOUS FUNCTIONSSince I is increasing, we deduce that ifoe := e - ye and ifO < e - Y < 0e , then Ye < Y < eso that L - e < I (ye ) ::: I (y) ::: L.Therefore I I (y) - L I < e when 0 < e - Y < °e Since e > 0 i s arbitrary we infer that (.i)holds. The proof of (ii) is similar. Q.E.D. The next result gives criteria for the continuity of an increasing function I at a point ethat is not an endpoint of the interval on which I is defined.5.6.2 Corollary Let I � lR be an interval and let I : I -+ lR be increasing on I. Supposethat e E l is not an endpoint of I. Then the following statements are equivalent.(a) I is continuous at e.(b) lim 1 = I (e) = X4C+ I· lim X4C-(c) sup{f(x) : x E I, x < e} = I (e) = inf{f (x) : x E I, x > e}. This follows easily from Theorems 5.6.1 and 4.3.3. We leave the details to the reader. Let I be an interval and let I : I -+ lR be an increasing function. If a is the leftendpoint of I , it is an exercise to show that I is continuous at a if and only if I (a) = inf {f(x) : x E I, a < x}or if and only if I (a) = lim I. Similar conditions apply at a right endpoint, and for x "*a +decreasing functions. If I : I -+ lR is increasing on I and if e is not an endpoint of I, we define the jump ofI at e to be lI (e) := lim I - lim I. (See Figure 5.6. 1.)It follows from Theorem 5 .5.1 x-+c+ x-+c-that lI (e) = inf{f (x) : x E I, x > e} - sup{f(x) : x E I, x < e}for an increasing function. If the left endpoint a of I belongs to I, we define the jump of IfI at a to be II (a) : = lim I - I (a). the right endpoint b of I belongs to I , we define x",*a+the jump of I at b to be lI (b) := I (b) - lim I· x",*b- c Figure 5.6.1 The jump of f at c.5.6.3 Theorem Let I � lR be an interval and let I : I -+ lR be increasing on I. If e E l,then I is continuous at e ifand only if lI (e) = O.
5.6 MONOTONE AND INVERSE FUNCTIONS 151Proof. If c is not an endpoint, this follows immediately from Corollary 5.6.2. If eEl isthe left endpoint of I, then ! is continuous at c if and only if ! (c) = lim !, which is x---+c+equivalent to j (c) = O. Similar remarks apply to the case of a right endpoint. f Q.E.D. We now show that there can be at most a countable set of points at which a monotonefunction is discontinuous.5.6.4 Theorem Let I £ IR be an interval and let ! I IR be monotone on I. Then the : -+set ofpoints D £ I at which ! is discontinuous is a countable set.Proof. We shall suppose that ! is increasing on I . It follows from Theorem 5.6.3 that = :D {x E I jf (x) =1= O}. We shall consider the case that I [a, b] is a closed bounded :=interval, leaving the case of an arbitrary interval to the reader. We first note that since ! is increasing, then j (c) 2: 0 for all e E l. Moreover, if fa ::::: Xl < . . . < Xn ::::: b, then (why?) we have(1)whence it follows that jf (x l ) + . . . + j/xn ) ::::: ! (b) - ! (a).(See Figure 5.6.2.) Consequently there can be at most points in I = [a, b] where j/x) 2: k(f(b) - !(a»/ We conclude that there is at most one point X I where j (x) = k. E f!(b) - ! (a); there are at most two points in I where j (x) 2: (f(b) - ! (a» /2; at most fthree points in I where j (x) 2: (f(b) - ! (a» /3, and so on. Therefore there is at most a fcountable set of points X where j (x) > O. But since every point in must be included in D f Dthis set, we deduce that is a countable set. Q.E.D.
152 CHAPTER 5 CONTINUOUS FUNCTIONS Theorem 5.6.4 has some useful applications. For example, it was seen in Exercise5.2.12 that if h : ffi. � ffi. satisfies the identity(2) hex + y) hex) + hey) for all x, y E R =and if h is continuous at a single point xo then h is continuous at point of R ThuS, everyif h is a monotone function satisfying (2), then h must be continuous on R [It followsfrom this that hex) Cx for all x E ffi., where C = h(1).] :=Inverse Functions _________________________We shall now consider the existence of inverses for functions that are continuous on an I S; Rinterval We recall (see Section 1.1) that a function I:I � ffi. has an inverse Ifunction if and only if is injective ( one-one); that is, = x, y I x y E and =f:. imply thatI(x) I(y). =f:. We note that a strictly monotone function is injective and so has an inverse. I I ffi.In the next theorem, we show that if : � is a strictly monotone continuous function, I gthen has an inverse function on J := 1(1) that is strictly monotone and continuous I g, Ion J. In particular, if is strictly increasing then so is and if is strictly decreasing g.then so is5.6.5 Continuous Inverse Theorem Let I S; ffi. be an interval and let I I ffi. be : �strictly monotone and continuous on I. Then the function inverse to I is strictly monotone gand continuous on J : = 1(1).Proof. I I We consider the case that is strictly increasing, leaving the case that is strictlydecreasing to the reader. I I Since is continuous and is an interval, it follows from the Preservation of IntervalsTheorem 5.3.10that J := l(l) I is an interval. Moreover, since is strictly increasing onI, g ffi. I it is injective on I ; therefore the function : J � inverse to exists. We claim g Yt Y2 Ethat is strictly increasing. Indeed, if Yt Y2 Y t I(xt) Y2 J with < then = and =l(x2 ) x l x2 I. for some xt x2 ; E We must have < xt � x2 otherwise which implies thatYt I(xt) � l(x2 ) Y2 = = Yt Y2 contrary to the hypothesis that < Therefore we haveg(Yt) xt x2 g(Y2 ) = < = Yt Y2 Since and J Y t Y2 are arbitrary elements of with < we gconclude that is strictly increasing on J. g It remains to show that is continuous on J. However, this is a consequence of the factthatg(J) I = g c is an interval. Indeed, if is discontinuous at a point E J, then the jump g c g g.of at is nonzero so that lim < lim If we choose any number =f:. y->c- y->c+ x g(c) satisfying g x lim < < limx�c- g, x x-+c+ x g(y) then has the property that =f:. YE for any J. (See Figure5.6.3.) x I, I Hence � which contradicts the fact that is an interval. Therefore we conclude gthat is continuous on J . Q.E.D.The nth Root FunctionWe will apply the Continuous Inverse Theorem 5.6.5 to the nth power function. We need nto distinguish two cases: (i) even, and (ii) odd. n n (i) even. In order to obtain a function that is strictly monotone, we restrict ourattention to the interval l : = [0, 00). Thus, let I(x) xn xn I. n := for E (See Figure 5.6.4.) Wehave seen (in Exercise 2.1.23) = < = x Y , I (x) x y I (y); that if ° � < then I thereforeis strictly increasing on I . Moreover, it follows from Example 5 .2.3( a) that I is continuous I.on Therefore, by the Preservation of Intervals Theorem 5.3.10, l(l) J := is an interval.
5.6 MONOTONE AND INVERSE FUNCTIONS 155 y 0 < r <1 r-------�MF- r= O �------�-- x Figure 5.6.8 Graphs of x -+ xr (x 2: 0).Exercises for Section 5.61. If I := [a, b] is an interval and I I -+ JR is an increasing function, then the point a [respectively, : b] is aanisabsolute minimumminimum pointmaximum]I.point for I on I. If I is strictly increasing, then the only absolute [respectively, for I on2. If I and g are increasing functions on an interval I s:; JR, show that I + g is an increasing function on I. If I is also strictly increasing on I, then I g is strictly increasing on I. +3. Show that both I(x) := x and g(x) x - I are strictly increasing on I [0, 1], but that their := := product Ig is not increasing on I.4. Show that if I and g are positive increasing functions on an interval I, then their product Ig is increasing on I.5. Show that if I := [a, b] and I : I -+ JR is increasing on I , then I is continuous at a if and only if I(a) inf{f(x) x E (a, b]). = :6. Let I s:; JR be an interval and let I : I -+ JR be increasing on I. Suppose that e E l is not an endpoint of I. Show that I is continuous at c if and only if there exists a sequence (xn ) in I such that xn < c for n = 3, 5, . . . ; xn > c for n 2, 4, 6, . . . ; and such that c lim(xn ) and I, = = I(c) lim (J(xn »). =7. Let I s:; JR be an interval and let I : I -+ JR be increasing on I. If c is not an endpoint of I, show• that the jump jf (c) of I at c is given by inf{f(y) - I(x) : x < c < y, x, y E l}.8. Let I, g be increasing on an interval I s:; JR and let I(x) > g(x) for all x E I. If Y E l(l) n g (l), show that I I (y) < g-I (y). [Hint: First interpret this statement geometrically.] -9. Let I := [0, 1] and let I I -+ JR be defined by I(x) x for x rational, and I(x) 1 - x for : := x irrational. Show that I is injective on I and that I (f (x» x for all x E I. (Hence I is its := = own inverse function!) Show that I is continuous only at the point x �. =
156 CHAPTER 5 CONTINUOUS FUNCTIONS10. Let I := [a, b] and let : I --+ R be continuous on I. If has an absolute maximum [respec I c II tively, minimum] at an interior point of I, show that is not injective on I.11. Let I(x) := x for x E [0, 1], and I(x) := 1 + x for x E ( 1 , 2]. Show that I and r l are strictly increasing. Are I and 1 1 continuous at every point? -12. : Let I [0, 1] --+ R be a continuous function that does not take on any of its values twice and with 1(0) < 1(1). Show that I is strictly increasing on [0, 1].13. : Let h [0, 1] --+ Rbe a function that takes on each of its values exactly twice. Show that h If c < c are the points where h attains its supremum, cannot be continuous at every point. show that c [Hint: 1 0, c = 1 . Now examine the points 2where h attains its infimum.] 214. Let x E 1x > 0. Show that if E Z, E N, and R, = m, p n, q mq = np , then (x l /n ) m = (x 1/q )p .15. If x E x > 0, and if r, S E Q, show that x rX s x r R, = H XSxr and (xr )s xrs (xs ) . = = =
CHAPTER 6 DIFFERENTIATIONPrior to the seventeenth century, a curve was generally described as a locus of pointssatisfying some geometric condition, and tangent lines were obtained through geometricconstruction. This viewpoint changed dramatically with the creation of analytic geometryin the 1630s by Rene Descartes ( 1596-1650) and Pierre de Fermat ( 1601-1665). In thisnew setting geometric problems were recast in terms of algebraic expressions, and newclasses of curves were defined by algebraic rather than geometric conditions. The conceptof derivative evolved in this new context. The problem of finding tangent lines and theseemingly unrelated problem of finding maximum or minimum values were first seen tohave a connection by Fermat in the 1630s. And the relation between tangent lines to curvesand the velocity of a moving particle was discovered in the late 1660s by Isaac Newton.Newton s theory of "fluxions", which was based on an intuitive idea of limit, would befamiliar to any modem student of differential calculus once some changes in terminologyand notation were made. But the vital observation, made by Newton and, independently, byGottfried Leibniz in the 1680s, was that areas under curves could be calculated by reversingthe differentiation process. This exciting technique, one that solved previously difficult areaproblems with ease, sparked enormous interest among the mathematicians of the era andled to a coherent theory that became known as the differential and integral calculus. Isaac Newton Isaac Newton (1642-1727) was born in Woolsthorpe, in Lincolnshire, Eng land, on Christmas Day; his father, a farmer, had died three months earlier. His mother remarried when he was three years old and he was sent to live with his grandmother. He returned to his mother at age eleven, only to be sent to boarding school in Grantham the next year. Fortunately, a perceptive teacher noticed his mathematical talent and, in 1661 , Newton entered Trinity College at Cambridge University, where he studied with Isaac Barrow. When the bubonic plague struck in 1665-1666, leaving dead nearly 70,000 persons in London, the university closed and Newton spent two years back in Woolsthorpe. It was during this period that he formulated his basic ideas concerning optics, gravitation, and his method of "fluxions", later called "calculus". He returned to Cambridge in 1 667 and was appointed Lucasian Professor in 1669. His theories of universal gravitation and planetary motion were published to world acclaim in 1687 under the title Philosophite Naturalis Principia Mathematica. ,However, he neglected to publish his method of inverse tangents for finding areas and other work in calculus, and this led to a controversy over priority with Leibniz. Following an illness, he retired from Cambridge University and in 1696 was appointed War den of the British mint. However, he maintained contact with advances in science and mathematics and served as President of the Royal Society from 1703 until his death in 1727. At his funeral, Newton was eulogized as "the greatest genius that ever existed". His place of burial in Westminster Abbey is a popular tourist site. 157
158 CHAPTER 6 DIFFERENTIATION In this chapter we will develop the theory of differentiation. Integration theory, includ ing the fundamental theorem that relates differentiation and integration, will be the subjectof the next chapter. We will assume that the reader is already familiar with the geometricaland physical interpretations of the derivative of a function as described in introductorycalculus courses. Consequently, we will concentrate on the mathematical aspects of thederivative and not go into its applications in geometry, physics, economics, and so on. The first section is devoted to a presentation of the basic results concerning the differentiation of functions. In Section 6.2 we discuss the fundamental Mean Value Theoremand some of its applications. In Section 6.3 the important L Hospital Rules are presentedfor the calculation of certain types of "indeterminate" limits. In Section 6.4 we give a brief discussion of Taylor s Theorem and a few of itsapplications-for example, to convex functions and to Newton s Method for the locationof roots.Section 6.1 The DerivativeIn this section we will present some of the elementary properties of the derivative. We beginwith the definition of the derivative of a function.6.1.1 Definition LetifxEI L 0 satisfies < I �� Ix -cl f c f I 0�, be an interval, let : -+ and let We say that areal number is the derivative of at if given any e > there exists 8 (e) > such that < 8 (e), then e E l. 0(1) [ f(X)x - cf(c) L [ - _ < e.In this case we say that f c, is differentiable atf (c) and we write for L. In other words, the derivative of f c at is given by the limit(2) f(c) f(xX) - f(c) = x -+ c lim C c -provided this limit exists. (We allow the possibility that may be the endpoint of theinterval.) cNote It is possible to define the derivative of a function having a domain more generalthan an interval (since the point need only be an element of the domain and also a clusterpoint of the domain) but the significance of the concept is most naturally apparent forfunctions defined on intervals. Consequently we shall limit our attention to such functions.f (c). f I f� x. f. Whenever the derivative of : -+ exists at a point e E l, its value is denoted by In this way we obtain a function whose domain is a subset of the domain ofexample, if f(x) x2 x E �,In working with the function for f, c � it is convenient to regard it also as a function of For then at any in we have f, (c) f(xx) - cf(c) - x2X - c2 := = x -+ c lim = x -+ c lim --- = C lim(x c) = 2c x-+c + . f f(x) = 2x x E R -Thus, in this case, the function is defined on all of � and for
6.1 THE DERIVATIVE 159 f We now show that continuity of at a point c is a necessary (but not sufficient)condition for the existence of the derivative at c.6.1.2 Theorem If f: I JR has a derivative at c E I, then f is continuous at c . -+Proof. For all x E I, x =1= c, we have f(x) - f(c) = ( f(X) - c » ) (x - c). x - f(CSince f (c) exists, we may apply Theorem 4.2.4 conceming the limit of a product toconclude that hm(f(x) - f(c» = hm ( f(X) - C ) ( hm (x - c)) . x-+c . x--+c X- f(C» . x--+c = f(c) · O = o.Therefore, lim f(x) = f(c) so that f is continuous at c. Q.E.D. x-->c The continuity of f: I JR at a point does not assure the existence of the derivative -+at that point. For example, if f (x) := I x I for x E JR, then for x =1= 0 we have (f (x) -f (0» I (x - 0) = Ix IIx which is equal to 1 if x > 0, and equal to - 1 if x < O. Thus the limitat 0 does not exist [see Example 4. 1 . lO(b)] , and therefore the function is not differentiableat O. Hence, continuity at a point c is not a sufficient condition for the derivative to existat c.Remark By taking simple algebraic combinations of functions of the form t-+ I x x - cl,it is not difficult to construct continuous functions that do not have a derivative at a finite (oreven a countable) number of points. In 1 872, Karl Weierstrass astounded the mathematicalworld by giving an example of a function that is continuous at every point but whosederivative does not exist anywhere. Such a function defied geometric intuition about curvesand tangent lines, and consequently spurred much deeper investigations into the concepts fof real analysis. It can be shown that the function defined by the series f(x) := L 21n cos (3n x) 00 n =Ohas the stated property. A very interesting historical discussion of this and other examples ofcontinu�)Us, nondifferentiable functions is given in Kline, p. 955-966, and also in Hawkins,p.44-46. A detailed proof for a slightly different example can be found in Appendix E. There are a number of basic properties of the derivative that are very useful in thecalculation of the derivatives of various combinations of functions. We now provide thejustification of some of these properties, which will be familiar to the reader from earliercourses.6.1.3 Theorem Let I S; JR be an interval, let c E I, and let f : I JR and I JR -+ g: -+ c.be functions that are differentiable at Then: a E JR, then the function af is differentiable at c, and(a) If(3) (a!) (c) = af(c).
6.1 THE DERIVATNE 161(b) The function fd2 . . fn is differentiable at c, and(8) (fd2 " fn )(c) = f{(c)f2 (c) " , fn (c) + fl (c)f�(c) · · · fn (c) + . . . + fl (c)f2 (c) . . . f�(c). An important special case of the extended product rule (8) occurs if the functions areequal, that is, fl = f2 = . . . = fn = f. Then (8) becomes(9) (r)(c) = n(f(c»n- I f(c).In particular, if we take f(x) := x, then we find the derivative of g(x) := x n to be g (x) =nxn -I , n E N. The formula is extended to include negative integers by applying the QuotientRule 6.1.3(d).Notation If I S; JR is an interval and f : I --+ JR, we have introduced the notation to f Idenote the function whose domain is a subset of and whose value at a point is the cderivative f(c) f c. of at There are other notations that are sometimes used for for f;example, one sometimes writes Df for f. Thus one can write formulas (4) and in the (5)form: D(f + g) = Df + Dg, D(fg) = (Df) . g + f · (Dg). xWhen is the "independent variable", it is common practice in elementary courses to writedf/dx f. for Thus formula (5) is sometimes written in the form d ( df ) ( dg ) dx (J(x)g(x») = dx (x) g(x) + f(x) dx (x) .This last notation, due to Leibniz, has certain advantages. However, it also has certaindisadvantages and must be used with some care.The Chain Rule ________________________We now tum to the theorem on the differentiation of composite functions known as the"Chain Rule". It provides a formula for finding the derivative of a composite function gof gin terms of the derivatives of and f. We first establish the following theorem concerning the derivative of a function at apoint that gives us a very nice method for proving the Chain Rule. It will also be used toderive the formula for differentiating inverse functions.6.1.5 Caratheodorys Theorem Let f be defined on an interval I containing the point c.Then f is differentiable at c if and only ifthere exists a function cp on I that is continuousat c and satisfies(10) f(x) - f(c) = cp(x)(x - c) for x E I.In this case, we have cp(c) = f(c). cp(x) := {�roof. (::::}) If f(c) exists, we can define cp by f(x) - f(c) for x =1= c, x E I, x -c f(c) for x = c.The continuity of cp follows from the fact that lim cp (x) = f (c). If x = c, then both sides x->-cof (10) equal 0, while if x =1= c, then multiplication of cp(x) by x - c gives (10) for all otherx E I.
6. 1 THE DERIVATIVE 1636.1.7 Examples (a) If f :n I � is differentiable on I and g(y) yn for y E � and := - l , it follows from the Chain Rule 6.1 .6 that -+n E N, then since g (y) ny = (g f) (x) = g (J (x») . f (x) for x E I. 0Therefore we have (r)ex) n (J (x) r- 1 f (x) for all x E I as was seen in (9). =(b) Suppose that f I : � is differentiable on I and that f (x) =j:. ° and f (x) =j:. ° for -+x E I. If hey) I / y for y =j:. 0, then it is an exercise to show that h(y) - Ill for := =y E �, =j:. 0. Therefore we have Y (�)(X) (h f)(x) h(J(x»)f(x) :::;: - (J(x»)2 for x E I. f = 0 = f(x)(c) The absolute value function g(x ) :=Ixl is differentiable at all x =j:. ° and has derivativeg (x) sgn(x) for x =j:. 0. (The signum function is defined in Example 4.I .1O(b).) Though =sgn is defined everywhere, it is not equal to g at x = ° since g (0) does not exist. Now if f is a differentiable function, then the Chain Rule implies that the functiong0 f = I f I is also differentiable at all points x where f (x) =j:. 0, and its derivative is givenby I fl(x) = sgn (f(x» . f(x ) { �j��X) if f(x) > 0, = if f(x) 0. <If f is differentiable at a point c with f (c) = 0, then it is an exercise to show that I f I isdifferentiable at c if and only if f (c) = 0. (See Exercise 7.) For example, if f(x) x2 - 1 for x E �, then the derivative of its absolute value :=I fl(x) Ix 2 - 1 1 is equaI to If I (x) sgn(x2 - 1) . (2x) for x =j:. 1 , - 1 . See Figure 6. 1 . 1 = =for a graph of If I · y -2 -1 2 Figure 6.1.1 The function Ifl (x ) = Ix 2 - 1 1 .(el) It will be proved later that if Sex) := sinx and C(x) := cos x for all x E �, then S(x) = cos x = C(x) and C(x) = - sinx = -Sex)for all x E R If we use these facts together with the definitions sin x 1 tanx := -- , sec x := -- , cos x cos x
6. 1 THE DERIVATIVE 165However, it is necessary to deduce the differentiability of the inverse function g from theassumed differentiability of f before such a calculation can be performed. This is nicelyaccomplished by using Caratheodory s Theorem.6.1.8 Theorem Let I be an interval in JR and let f : I -+ JR be strictly monotone andcontinuous on I. Let J : = f(l) and let g : J -+ JR be the strictly monotone and continuousfunction inverse to f. If f is differentiable at e E l and f (e) =j:. 0, then g is differentiableat d := fee) and I I 1(12) g (d) = fee) f (g(d» =Proof. Given e E JR, we obtain from Caratbeodorys Theorem 6.1.5 a function q; on Iwith properties that q; is continuous at e, f(x) - fee) q;(x)(x - e) for x E I, and q;(e) = =fee). Since q;(e) =j:. 0 by hypothesis, there exists a neighborhood V (e - 8 , e + 8) such:=that q;(x) =j:. 0 for all x E V n I. (See Theorem 4 .2.9 .) If U f(V n I), then the inverse :=function g satisfies f (g (y ») y for all y E U , so that = y - d f(g(y ») - fee) q; (g(y ») . (g(y) - g(d»). = =Since q;(g(y» =j:. 0 for y E U , we can divide to get 1 g(y) - g(d) = q;(g(y») . (y d). -Since the function I/(q; g) is continuous at d, we apply Theorem 6.1.5 to conclude that 0g(d) exists and g(d) = I/q;(g(d») I/q;(e) Ilf(e). = = Q.E.D.Note The hypothesis, made in Theorem 6.1.8, that fee) =j:. 0 is essential. In fact, iffee) 0, then the inverse function g is never differentiable at d fee), since the assumed = =existence of g(d) would lead to 1 f(e)g(d) 0, which is impossible. The functionf(x) := x 3 with e 0 is such an example. = = =6.1.9 Theorem Let I be an interval and let f I JR be strictly monotone on I. Let : -+J f(l) and let g J JR be the function inverse to f. Iff is differentiable on I and := : -+! (x) =j:. 0 for x E I, then g is differentiable on J and I(13) g = f 1o g Proof. If f is differentiable on I, then Theorem 6.1 .2 implies that f is continuous on I,and by the Continuous Inverse Theorem 5 . 6.5, the inverse function g is continuous on J.Equation (13) now follows from Theorem 6.1.8. Q.E.D.�emark If f and g are the functions of Theorem 6.1.9, and if x E I and y E J are relatedby y = f(x) and x = g(y), then equation (13) can be written in the form I I g (y) (f O 1g)(y) , = yE J, (g f)(x) f1(X) , x E I. or 0 =It can also be written in the form g (y) 1 If (x), provided that it is kept in mind that x =and y are related by y f(x) and x g(y). = =
168 CHAPTER 6 DIFFERENTIATIONSection 6.2 The Mean Value TheoremThe Mean Value Theorem, which relates the values of a function to values of its derivative,is one of the most useful results in real analysis. In this section we will establish thisimportant theorem and sample some of its many consequences. We begin by looking at the relationship between the relative extrema of a functionand the values of its derivative. Recall that the function f : I -+ lR is said to have arelative maximum [respectively, relative minimum] at e E l if there exists a neighborhoodV : = Vo(e) of e such that f(x) :s fee) [respectively, fee) :s f(x)] for all x in V n I. Wesay that f has a relative extremum at e E l if it has either a relative maximum or a relativeminimum at e. The next result provides the theoretical justification for the familiar process of findingpoints at which f has relative extrema by examining the zeros of the derivative. However,it must be realized that this procedure applies only to interior points of the interval. Forexample, if f(x) : = x on the interval l := [0, 1], then the endpoint x = 0 yields the uniquerelative minimum and the endpoint x = 1 yields the unique maximum of f on I, but neitherpoint is a zero of the derivative of f.6.2.1 Interior Extremum Theorem Let e be an interior point ofthe interval I at whichf: I -+ lR has a relative extremum. the derivative of f at e exists, then !, (c) O. If =Proof. We will prove the result only for the case that f has a relative maximum at e; theproof for the case of a relative minimum is similar. > If f (c) 0, then by Theorem 4.2.9 there exists a neighborhood V S; I of e such that > f(x) - fee) 0 for x E V, x =1= e. x-e >If x E V and x e, then we have f(x) - fee) (x - c) . f(x) - e > = fee) O. x-But this contradicts the hypothesis that f has a relative maximum at e. Thus we cannot >have f(e) O. Similarly (how?), we cannot have f(e) O. Therefore we must have <f(e) O. = Q.E.D.6.2.2 Corollary Let f : I lR be continuous on an interval I and suppose that f has a e -+relative extremum at an interior point e of Then either the derivative of f at does not I.exist, or it is equal to zero. We note that if f(x) : = Ixl on r := [-1 , 1], then f has an interior minimum at x = 0;however, the derivative of f fails to exist at x = o.6.2.3 Rolles Theorem Suppose that f is continuous on a closed interval I := [a, b), thatthe derivative f exists at every point of the open interval (a, b), and that f(a) = feb) = O. eThen there exists at least one point in (a, b) such that f(e) = O.Proof. If f vanishes identically on I, then any e in (a, b) will satisfy the conclusion ofthe theorem. Hence we suppose that f does not vanish identically; replacing f by -fif necessary, we may suppose that f assumes some positive values. By the Maximum >Minimum Theorem 5.3.4, the function f attains the value sup{f(x) : x E I} 0 at somepoint e in I. Since f(a) = feb) = 0, the point e must lie in (a, b); therefore f(e) exists.
6.2 THE MEAN VALUE THEOREM 169 Figure 6.2.1 Rolle s TheoremSince f has a relative maximum at e, we conclude from the Interior Extremum Theorem6.2.1 that f(e) = O. (See Figure 6.2.1.) Q.E.D. As a consequence of Rolle s Theorem, we obtain the fundamental Mean ValueTheorem.6.2.4 Mean Value Theorem Suppose that f is continuous on a closed interval I :=[a, b] and that f has a derivative in the open interval (a, b). Then there exists at least one ,point e in (a, b) such that feb) - f(a) = f(e)(b - a).Proof. Consider the function ({J defined on I by feb) - f(a) (x - a). ((J(x) := f(x) - f(a) - b-a[The function is simply the difference of f and the function whose graph is the line ({Jsegment joining the points (a, f(a)) and (b, feb)); see Figure 6.2.2.] The hypotheses of a x c b Figure 6.2.2 The Mean Value Theorem
170 CHAPTER 6 DIFFERENTIATIONRolle s Theorem are satisfied by cp since cp is continuous on [a, b], differentiable on (a, b),and cp(a) = q;(b) = O. Therefore, there exists a point c in (a, b) such that 0 = q;(C) = f(C) feb) - a . b - f(a) _Hence, feb) - f(a) = f(C)(b - a). Q.E.D.Remark The geometric view of the Mean Value Theorem is that there is some point onthe curve y = f (x) at which the tangent line is parallel to the line segment through thepoints (a, f(a)) and (b, feb)). Thus it is easy to remember the statement of the MeanValue Theorem by drawing appropriate diagrams. While this should not be discouraged,it tends to suggest that its importance is geometrical in nature, which is quite misleading.In fact the Mean Value Theorem is a wolf in sheep s clothing and is the FundamentalTheorem of Differential Calculus. In the remainder of this section, we will present some ofthe consequences of this result. Other applications will be given later. The Mean Value Theorem permits one to draw conclusions about the nature of afunction f from information about its derivative f. The following results are obtained inthis manner.6.2.5 Theorem Suppose that f is continuous on the closed interval [ := [a, b], that fis differentiable on the open interval (a, b), and that fex) = 0 for x E (a, b). Then f isconstant on [ .Proof. > We will show that f(x) = f(a) for all x E [ . Indeed, if x E [ , x a, is given,we apply the Mean Value Theorem to f on the closed interval [a, x]. We obtain a point c(depending on x) between a and x such that f(x) - f(a) = f (c)(x - a). Since f (c) = 0(by hypothesis), we deduce that f(x) - f(a) = O. Hence, f(x) = f(a) for any x E [ . Q.E.D.6.2.6 Corollary Suppose that f and g are continuous on [ := [a, b], that they are differentiable on (a, b), and that f (x) = g (x) for all x E (a, b). Then there exists a constantC such that f = g + C on [ . : --+ Recall that a function f [ lR. is said to be increasing on the interval [ if whenever Ix x2 in [ satisfy x I < x2 then f (x I ) :s f (x2 ). Also recall that f is decreasing on [ if thefunction -f is increasing on [ .6.2.7 Theorem Let f: [ --+ lR. be differentiable on the interval [ . Then:(a) f is increasing on [ if and only if f (x) :::: 0 for all x E [ .(b) f is decreasing on [ if and only if f (x) :s 0 for all x E [ .Proof. (a) Suppose that fex) :::: 0 for all x E If x l x2 in satisfy X l < x2 then we [. [apply the Mean Value Theorem to f on the closed interval J := [X l X2 ] to obtain a pointc in (xl x2 ) such that
6.2 THE MEAN VALUE THEOREM 171Since f ee) ::: 0 and x2 - x I > 0, it follows that f (x2 ) - f (x l ) ::: o. (Why?) Hence,f(x l ) ::::: f (x2 ) and, since x I < x2 are arbitrary points in I, we conclude that f is increasing on I . For the converse assertion, we suppose that f i s differentiable and increasing on I .Thus, for any point x i= e i n I, we have (J (x) - f ) / (x - ::: (Why?) Hence, by (c) c) o.Theorem 4.2.6 we conclude that f (x) - fee) ::: f ee) = lim o. x -+c x -e (b) The proof of part (b) is similar and will be omitted. Q.E.D. A function f is said to be strictly increasing on an interval I if for any points x I x2 inI such that x I < x2 we have f (x l ) < f (x2 ) . An argument along the same lines of the proof of Theorem 6.2.7 can be made to show that a function having a strictly positive derivative onan interval is strictly increasing there. (See Exercise 13.) However, the converse assertionis not true, since a strictly increasing differentiable function may have a derivative thatvanishes at certain points. For example, the function f : -+ defined by f (x) := x 3 is JR JR JR, O.strictly increasing on but f (0) = The situation for strictly decreasing functions issimilar.Remark It is reasonable to define a function to be increasing at a point if there is aneighborhood of the point on which the function is increasing. One might suppose that,if the derivative is strictly positive at a point, then the function is increasing at this point.However, this supposition is false; indeed, the differentiable function defined by g (x) : = { Xo 22 + x sin(1/x) if x i= 0, if x = 0, 1,is such that g (0) = yet it can be shown that g is not increasing in any neighborhood of O.x = (See Exercise 10.) We next obtain a sufficient condition for a function to have a relative extremum at aninterior point of an interval.6.2.8 First Derivative Test for Extrema Let f be continuous on the interval I : = [a, b]and lete be an interiorpoint of I. Assume that f is differentiable on (a, c) and (e, Then: b).(a) If there is a neighborhood (e - 8 , e + 8) � I such that f ex) ::: 0 for e - 8 < x < eand f (x) ::::: 0 for e < x < c + 8, then f has a relative maximum at e.(b) If there is a neighborhood (e - 8 , e + 8) � I such that f (x) :s 0 for e - 8 < x < eand f (x) ::: 0 for e < x < e + 8, then f has a relative minimum at e.Proof. c, (a) If x E (e - 8 , ) then it follows from the Mean Value Theorem that there c (eexists a point ex E (x , ) such that fee) - f (x) = - x)f (ex ) Since f (ex ) ::: 0 weinfer that f(x) ::::: fee) for x E (e - 8, c). Similarly, it follows (how?) that f (x) ::::: fee)for x E (e, e + 8). Therefore f (x) ::::: fee) for all x E (e - 8,e + 8) so that f has a relativemaximum at e. (b) The proof is similar. Q.E.D.Remark The converse of the First Derivative Test is 6.2.8 not true. For example, thereexists a differentiable function f : -+ with absolute minimum at x = 0 but such that JR JR
172 CHAPTER 6 DIFFERENTIATIONI takes on both positive and negative values on both sides of (and arbitrarily close to)x = O. (See Exercise 9.)Further Applications of the Mean Value Theorem _____________We will continue giving other types of applications of the Mean Value Theorem; in doingso we will draw more freely than before on the past experience of the reader and his or herknowledge concerning the derivatives of certain well-known functions.6.2.9 Examples (a) Rolles Theorem can be used for the location of roots of a function.For, if a function g can be identified as the derivative of a function I, then between any tworoots of I there is at least one root of g . For example, let g(x) : = cos x, then g is known tobe the derivative of I(x) := sinx. Hence, between any two roots of sinx there is at leastone root of cos x. On the other hand, g (x) = - sin x = - I (x), so another application ofRolle s Theorem tells us that between any two roots of cos there is at least one root of sin.Therefore, we conclude that the roots of sin and cos interlace each other. This conclusionis probably not news to the reader; however, the same type of argument can be applied tothe Bessel functions In of order n = 0, 1, 2, . . . by using the relations [x n In (x)] = x n In - 1 (x), [xn - 1 In (x)] = _x -n In+1 (x) for x > O.The details of this argument should be supplied by the reader.(b) We can apply the Mean Value Theorem for approximate calculations and to obtainerror estimates. For example, suppose it is desired to evaluate JT05 We employ the Mean .Value Theorem with I(x) : = yx, a = 100, b = 105, to obtainfor some number c with 100 < c < 105. Since 10 < JC < JT05 < .JI2I = 1 1, we canassert that 5 5 2(1 1) .JI05 - 10 < -- , -- < 2(10)whence it follows that 10.2272 < ,JI05 < 10 . 2500 . This estimate may not be as sharp asdesired. It is clear that the estimate JC < .J1 05 < .JI2I was wasteful and can be improvedby making use of our conclusion that JT05 < 10 . 2500. Thus, JC < 10.2500 and we easilydetermine that 5 0.2439 < 2(10.2500) < .JiOs - 10.Our improved estimate is 10.2439 < ,J105 < 10.2500. oInequalitiesOne very important use of the Mean Value Theorem is to obtain certain inequalities.Whenever information concerning the range of the derivative of a function is available, thisinformation can be used to deduce certain properties of the function itself. The followingexamples illustrate the valuable role that the Mean Value Theorem plays in this respect.
6.2 THE MEAN VALUE THEOREM 1736.2.10 Examples (a) The exponential function I (x) := eX has the derivative I (x) =eX for all x E R Thus I (x) >1 for x > 0, and I (x) < 1 for x < 0. From these relationships, we will derive the inequality(1) for x E Rwith equality occurring if and only if x = 0. > If x = 0, we have equality with both sides equal to 1 . If x 0, we apply the MeanValue Theorem to the function I on the interval [0, x]. Then for some c with 0 < c < xwe have eX - eO eC (x - 0). = > >Since eO = 1 and eC 1, this becomes eX 1 x so that we have eX - > 1 + x for x O. >A similar argument establishes the same strict inequality for x < O. Thus the inequality (1)holds for all x, and equality occurs only if x = O.(b) The function g (x) : = sin x has the derivative g (x) = cos x for all x E R On the basisof the fact that - 1 :s cos x :s 1 for all x E JR, we will show that(2) -x :s sinx :s x for all x :::: o.Indeed, if we apply the Mean Value Theorem to g on the interval [0, x], where x > 0, weobtain sinx - sin O = (cos c)(x - 0)for some c between 0 and x . Since sin 0 = 0 and - 1 :s cos c :s 1 , we have -x :s sin x :s x.Since equality holds at x = 0, the inequality (2) is established. >(c) (Bernoulli s inequality) If a 1 , then(3) (1 + x)" :::: 1 + ax for all x > -1,with equality if and only if x = O. This inequality was established earlier, in Example 2.1 . 13(c), for positive integervalues of a by using Mathematical Induction. We now derive the more general version byemploying the Mean Value Theorem. > If h ex) := (1 + x)" then h(x) = a(I + x)", - l for all x - 1 . [For rational a thisderivative was established in Example 6.1 .10(c). The extension to irrational will be dis >cussed in Section 8.3.] If x 0, we infer from the Mean Value Theorem applied to h onthe interval [0, x] that there exists c with ° < c < x such that h (x) - h (0) = h (c) (x - 0).Thus, we have l (1 + x)" - 1 = a(I + c)",- x. > >Since c 0 and a - I > 0, it follows that (1 + c)", - l 1 and hence that (1 + x)" >1 + ax. If - 1 < x < 0, a similar use of the Mean Value Theorem on the interval [x, 0]leads to the same strict inequality. Since the case x = 0 results in equality, we conclude >that (3) is valid for all x - 1 with equality if and only if x = O.(d) Let a be a real number satisfying 0 < a < 1 and let g(x) = ax - x" for x :::: o.Then g(x) = a(l X ", - l ), so that g(x) < 0 for 0 < x < 1 and g(x) 0 for x 1 . - > >Consequently, if x ::: 0, then g(x) :::: g ( I ) and g(x) = g(I) if and only if x = 1 . Therefore,if x :::: 0 and 0 < a < 1, then we have x" :s ax + (1 - a).
174 CHAPTER 6 DIFFERENTIATIONIf a b > O.and if we let x = a /b and multiply by b, we obtain the inequality > 0 and aCt b 1 -Ct :s aa + (1 - a)b,where equality holds if and only if a b. = DThe Intermediate Value Property of Derivatives _____________We conclude this section with an interesting result, often referred to as Darbouxs Theorem.It states that if a function f is differentiable at every point of an interval I , then the functionf has the Intermediate Value Property. This means that if f takes on values A and B, thenit also takes on all values between A and B. The reader will recognize this property as one ofthe important consequences of continuity as established in Theorem 5 . 3 .7. It is remarkablethat derivatives, which need not be continuous functions, also possess this property.6.2.11 Lemma ---* Let I � JR. be an interval, let f : I JR., let c E I, and assume that fhas a derivative at c. Then:(a) If f (c) > 0, then there is a number 8 > 0 such that f (x) > f (c) for x E I such thatc < x < c + 8.(b) If f(c) < 0 , then there is a number 8 > 0 such that f (x) > f (c) for x E I such thatc - 8 < x < c.Proof. (a) Since · f (x) - f (c) = f(c) > 0, 11m x-->c -CXit follows from Theorem 4.2.9 that there is a number 8 > 0 such that if x E I and 0 <Ix - c l < 8, then ----- f (x) - f (c) > 0. x-cIf x E I also satisfies x > c, then we have f (x) - f (c) o. f (x) - f (c) = (x - c) . > x-cHence, if x E I and c < x < c + 8, then f (x) > f (c). The proof of (b) is similar. Q.E.D.6.2.12 Darbouxs Theorem If f is differentiable on I [a , b] and if k is a number = (a)between f and f then (b), there is at least one point c in (a, b) such that f (C) = k.Proof. Suppose that fea) < k < f(b). We define g on I by g(x) kx - f (x) for :=x E I . Since g is continuous, it attains a maximum value on I . Since g (a) k - f (a) > 0, =it follows from Lemma 6.2.1 1(a) that the maximum of g does not occur at x a. Similarly, =since g (b) k - f (b) < 0, it follows from Lemma 6.2. 1 1(b) that the maximum does not =occur at x b. Therefore, g attains its maximum at some c in (a, b). Then from Theorem =6.2. 1 we have 0 = g(C) = k - f(C) . Hence, f(C) = k. Q.E.D. ---*6.2.13 Example g (x ) := { I The function g: [-1 , 1] 0 for for JR. defined by 0 < x :s 1 , x = 0, -1 for - 1 :s x < 0,
6.2 THE MEAN VALUE THEOREM 175(which is a restriction of the signum function) clearly fails to satisfy the intermediate valueproperty on the interval [- 1 , 1]. Therefore, by Darboux s Theorem, there does not exist afunction f such that f (x ) = g (x ) for all x E [- 1 , 1]. In other words, g is not the derivativeon [ - 1 , 1] of any function. 0Exercises for Section 6.2 1. For each of the following functions on JR to JR, find points of relative extrema, the intervals on which the function is increasing, and those on which "it is decreasing: (a) I(x) := x Z - 3x + 5, (b) g(x) := 3x - 4xz , (c) h(x) := x 3 - 3x - 4, (d) k(x) := X4 + 2x z - 4. 2. Find the points of relative extrema, the intervals on which the following functions are increasing, and those on which they are decreasing: (a) I(x) := x + l /x for x i= 0, (b) g (x) := x/(x z + 1) for x E JR, (c) h(x) := Jx - 2 Jx + 2 for x > 0, (d) k(x) := 2x + l /x z for x i= o. 3. Find the points of relative extrema of the following functions on the specified domain: (a) I(x) := Ix z - 1 1 for 4 :::: x :::: 4, - (b) g(x) := 1 - (x - l )z/3 for 0 :::: x :::: 2, (c) h(x) := x lx z - 121 for -2 :::: x :::: 3, (d) k(x) := x(x - 8) 1 /3 for 0 :::: x :::: 9. 4. Let ai az " , an be real numbers and let I be defined on JR by n I(x) := � )ai - x) z for x E lR. i= 1 Find the unique point of relative minimum for I.(5. Let a > > 0 and let n E N satisfy n � 2. Prove that b l /n n a l /n - b l /n < (a - b) l /n . [Hint: Show x that I(x) := - (x - 1)I/ is decreasing for x � 1, and evaluate I at 1 and a/b.] 6. Use the Mean Value Theorem to prove that I sinx - sin yl :::: I x - yl for all x, y in lR. Use the Mean Value Theorem to prove that (x - 1)/x < lnx < x - I for x > 1 . [Hint: Use the fact that D ln x = l /x for x > 0.] 7. 8. b b). Let I: [a, ] --+ JR be continuous on [a, b] and differentiable in (a, Show that if x_ I (x) = lim a A, then I(a) exists and equals A. [Hint: Use the definition of I(a) and the Mean Value Theorem.] 9. Let I : JR --+ JR be defined by I(x) := 2X4 + x4 sin(1/x) for x i= 0 and 1(0) := O. Show that I has an absolute minimum at x = 0, but that its derivative has both positive and negative values in every neighborhood of O.10. Let g : JR --+ JR be defined by g(x) := x + 2x z sin(1/x) for x i= 0 and g(O) := O. Show that g (0) = 1, but in every neighborhood of 0 the derivative g (x) takes on both positive and negative values. Thus g is not monotonic in any neighborhood of O.11. Give an example of a uniformly continuous function on [0, 1] that is differentiable on (0, 1) but whose derivative is not bounded on (0, 1).12. Ifh(x) := 0 for x < 0 and h (x) := 1 for x � 0, prove there does not exist a function I : JR --+ JR such that I (x) = h (x ) for all x E lR. Give examples of two functions, not differing by a constant, whose derivatives equal h(x) for all x i= O.13. Let I be an interval and let I : I --+ JR be differentiable on I. Show that if I is positive on I, then I is strictly increasing on I.14. Let I be an interval and let I : I --+ JR be differentiable on I. Show that if the derivative I is never 0 on I, then either I (x) > 0 for all x E I or I (x) < 0 for all x E I.
176 CHAPTER 6 DIFFERENTIATION f f15. Let I be an interval. Prove that if is differentiable on I and if the derivative is bounded on f I, then satisfies a Lipschitz condition on I. (See Definition 5.4.4.) f: f x) b x - f(x»)1 e b.16. Let [0, 00) --+ lR be differentiable on (0, 00) and assume that --+ as --+ 00. x -+ oo (a) Show that for any h > 0, we have lim + h) (f(x h= (b) Show that if f(x) x --+ a as --+ 00, then b = O. x-+oo (c) Show that lim (f(x)lx) b. =17. Let f, g be differentiable on and suppose that f(O) g(O) and f ex) g(X) for all x :::: O. lR = � Show that f(x) g(x) for all x :::: O. � Let I := [a, b] and let f : I e be differentiable at c E I. Show that for every > 0 there --+ lR < <18. exists 8 > 0 such that if 0 Ix - y l 8 and a x c y b, then � � � � I f(x) - yf ey) - f(c)1 < e. - x A differentiable function f : I lR is said to be on I := [a, b] if for e < <19. --+ uniformly differentiable every > 0 there exists 8 > 0 such that if 0 I x - y I 8 and x, y E I, then I f(x) - yf ey) - f(x)1 < e. - x Show that if f is uniformly differentiable on I, then f is continuous on I.20. Suppose that f [0, 2] : --+ lRis continuous on [0, 2] and differentiable on (0, 2), and that f(O) = 0, f( l ) = 1 , f(2) = 1 . (a) Show that there exists c l E (0, 1) such that f(C l ) = 1 . (b) Show that there exists c2 E (1, 2) such that f(C2 ) = o. (c) Show that there exists c E (0, 2) such that f (c) 1 /3. =Section 6.3 LHospitals RulesThe Marquis Guillame Fran�ois L Hospital (1661-1704) was the author of the first calculusbook, LAnalyse des injinimentpetits, published in 1696. He studied the then new differentialcalculus from Joharm Bernoulli (1667-1748), first when Bernoulli visited L Hospital scountry estate and subsequentty through a series of letters. The book was the result ofL Hospital s studies. The limit theorem that became known as L Hospital s Rule firstappeared in this book, though in fact it was discovered by Bernoulli. The initial theorem was refined and ext.ended, and the various results are collectivelyreferred to as L Hospital s (or L Hopita1 s) Rules. In this section we establish the most basicof these results and indicate how others can be derived.Indeterminate FormsIn the preceding chapters we have often been concerned with methods of evaluating limits.It was shown in Theorem 4.2.4(b) that if A := lim f(x) and B := x-+c g(x), and if B i= 0, lim x�cthen f(x) = � lim x-+c g(x) B .However, if B = 0, then no conclusion was deduced. It will be seen in Exercise 2 that ifB = ° and A i= 0, then the limit is infinite (when it exists). The case A = 0, B = ° has not been covered previously. In this case, the limit of thequotient fig is said to be "indeterminate". We will see that in this case the limit may
6.3 LHOSPITALS RULES 177not exist or may be any real value, depending on the particular functions f and g. Thesymbolism % is used to refer to this situation. For example, if a is any real number, andif we define f(x) : = ax and g(x) : = x, then . f(x) = lim ax = lIm a = a. 11m . . o -- x --+ g(x) x--+O - X x--+OThus the indeterminate form % can lead to any real number a as a limit. Other indeterminate forms are represented by the symbols 00 / 00, 0 . 00, 00 , 1 00 , 000 ,and 00 - 00. These notations correspond to the indicated limiting behavior and juxtaposition of the functions f and g. Our attention will be focused on the indeterminate forms 0/0and 00/00. The other indeterminate cases are usually reduced to the form % or 00/00 bytaking logarithms, exponentials, or algebraic manipulations.A Preliminary ResultTo show that the use of differentiation in this context is a natural and not surprisingdevelopment, we first establish an elementary result that is based simply on the definitionof the derivative. f g6.3.1 Theorem Let and be defined on [a, b], let f(a) g(a) 0, and let g(x) ::j:. 0 = =for a < x < b. f g If and are differentiable at a and if g (a) ::j:. 0, then the limit of fig ata exists and is equal to f (a)/ g (a). Thus- f(x) f(a) x --+a+ g(x) g(a) . lim =Proof. Since f(a) = g(a) 0, we can write the quotient f(x)/g(x) for a < x < b as =follows: f(x) - f(a) f(x) f(x) - f(a) x -a = g(x) g(x) - g(a) g(x) - g(a) . x -aApplying Theorem 4.2.4(b), we obtain f(x) - f(a) __x_ xlim+ -,--:,---a-,-,.- - f (a) · f ( ) _--+a _ x - -=c _ x--lim+ g(x) lim g(x) - g(a) g(a) +a = Q.E.D. x--+a+ X - aWarning The hypothesis that f(a) = g(a) = 0 is essential here. For example, if f(x) :=x + and g(x) 2x + 3 for x E JR, then 17 := lim f(x) 17 , f(O) 1 x --+O g(x) 3 while g(O) 2 = = - The preceding result enables us to deal with limits such as x2 + x = 2 0 + lim . 1 1 - O x --+ sin 2x 2 cos 0 2To handle limits where f and g are not differentiable at the point a, we need a more generalversion of the Mean Value Theorem due to Cauchy.
178 CHAPTER 6 DIFFERENTIATION6.3.2 Cauchy Mean Value Theorem Let f and g be continuous on [a, b] and differentiable on (a, b), and assume that g(x) =1= 0 for all x in (a, b). Then there exists e in (a, b)such that feb) - f(a) = -,- . f(e) g(b) - g(a) g (e)Proof. As in the proof of the Mean Value Theorem, we introduce a function to whichRolles Theorem will apply. First we note that since g(x) =1= 0 for all x in (a, b), it followsfrom Rolles Theorem that g(a) =1= g(b). For x in [a, b], we now define h(x) := feb) - I (a) (g(x) - g(a») - (I(x) - f(a»). g(b) _ g(a)Then h is continuous on [a, b], differentiable on (a, b), and h(a) = h(b) = O . Therefore,it follows from Rolles Theorem 6.2.3 that there exists a point e in (a, b) such that f(a) 0 = h(e) = feb) - g(a) g(e) - f(e). g(b) -Since g(e) =1= 0, we obtain the desired result by dividing by (e). g Q.E.D.Remarks The preceding theorem has a geometric interpretation that is similar to that ofthe Mean Value Theorem 6.2.4. The functions f and g can be viewed as determining a curvein the plane by means of the parametric equations x = f(t), Y = get) where a ::: t ::: b.Then the conclusion of the theorem is that there exists a point (f(e), g(e» on the curve forsome e in (a, b) such that the slope g(e)/f(e) of the line tangent to the curve at that pointis equal to the slope of the line segment joining the endpoints of the curve. Note that if g(x) = x, then the Cauchy Mean Value Theorem reduces to the MeanValue Theorem 6.2.4.LHospitals Rule, IWe will now establish the first of LHospitals Rules. For convenience, we will considerright-hand limits at a point a; left-hand limits, and two-sided limits are treated in exactly thesame way. In fact, the theorem even allows the possibility that a = -00. The reader shouldobserve that, in contrast with Theorem 6.3.1, the following result does not assume thedifferentiability of the functions at the point a. The result asserts that the limiting behaviorof f(x)/g(x) as x � a + is the same as the limiting behavior of f(x)/g(x) as x � a+,including the case where this limit is infinite. An important hypothesis here is that both fand g approach 0 as x � a +.6.3.3 LHospitals Rule, I Let -00 ::: a < b ::: 00 and let f, g be differentiable on (a, b)such that g(x) =1= 0 for all x E (a, b). Suppose that(1) f(x) = 0 = x-+a+ g(x). lim x-+a+ lim(a) . f(x) hm . f(x) Ifx-+a+ -,- = L E JR, then x-+a+ -- = L. hm g (x) g (x)(b) hm If x-+a+ f(x) . f(x) . -,- = L E {-oo, oo}, then hm -- = L. g (x) x .....a+ g (x)
6.4 TAYLOR S THEOREM 1876.4.4 Theorem Let I be an interval, let Xo be an interiorpoint ofI , and let n ::: 2. Supposethat the derivatives I, I ", . . . , I(n) exist and are continuous in a neighborhood of Xo andthat I (xo) = . . . = l (n - I J cxo) = 0, but I(n ) (xo) =1= 0. >(i) Ifn is even and I(n) (xo) 0, then I has a relative minimum at xo(ii) Ifn is even and I(n ) (xo) < 0, then I has a relative maximum at xo(iii) Ifn is odd, then I has neither a relative minimum nor relative maximum at xoProof. Applying Taylor s Theorem at xo we find that for x E I we have . I(n) (c) I(x) = Pn I (x) + Rn _ 1 (x) = I(xo) + --- (x - xo) n , - n ,.where c is some point between Xo and x. Since I(n) is continuous, if I(n) (xo) =1= 0, thenthere exists an interval U containing Xo such that I(n ) (x) will have the same sign as I(n ) (xo)for x E U. If x E U, then the point c also belongs to U and consequently I(n) (c) andI(n) (xo) will have the same sign. > > (i) Ifn is even and l (n)(xo) O, then forx E U we have l (n) (c) ° and (x _ xo)n :::° so that Rn _1 (x) ::: 0. Hence, I(x) ::: I(xo) for x E U, and therefore I has a relativeminimum at Xo- (ii) If n is even and I(n) (xo) < 0, then it follows that Rn _1 (x) :::: ° for x E U, so thatI (x) :::: I (xo) for x E U. Therefore, I has a relative maximum at xo > (iii) If n is odd, then (x - xo) n is positive if x Xo and negative if x < xo Consequently, if x E U, then Rn _1 (x) will have opposite signs to the left and to the right of xoTherefore, I has neither a relative minimum nor a relative maximum at xo Q.E.D.Convex Functions ________________________The notion of convexity plays an important role in a number of areas, particularly in themodem theory of optimization. We shall briefly look at convex functions of one real variableand their relation to differentiation. The basic results, when appropriately modified, can beextended to higher dimensional spaces. Definition Let I S; JR be an interval. A function I : I � JR is said to be convex 6.4.5 Ion I if for any t satisfying ° :::: t :::: 1 and any points x x2 in I , we have 1 (( 1 - t)x I + tx2 ) :::: ( 1 - t)l(x l ) + tl (x2 ) · Note that if XI < x2 then as t ranges from ° to 1, the point ( 1 - t)x I + tX2 traverses Ithe interval from x l to x2 • Thus if I is convex on I and if x x2 E I, then the chord joiningany two points (X I I(x l » and (x2 l(x2 » on the graph of I lies above the graph of I·(See Figure 6.4.l .) : A convex function need not be differentiable at every point, as the example I (x) = Ix I,X E JR, reveals. However, it can be shown that if I is an open interval and if I: I �JR is convex on I, then the left and right derivatives of I exist at every point of I.As a consequence, it follows that a convex function on an open interval is necessarilycontinuous. We will not verify the preceding assertions, nor will we develop many otherinteresting properties of convex functions. Rather, we will restrict ourselves to establishingthe connection between a convex function I and its second derivative I", assuming thatI" exists.
6.4 TAYLORS THEOREM 189Newtons MethodIt is often desirable to estimate a solution of an equation with a high degree of accuracy. TheBisection Method, used in the proof of the Location of Roots Theorem 5.3.5, provides oneestimation procedure, but it has the disadvantage of converging to a solution rather slowly.A method that often results in much more rapid convergence is based on the geometricidea of successively approximating a curve by tangent lines. The method is named after itsdiscoverer, Isaac Newton . r r. Let I be a differentiable function that has a zero at and let x I be an initial estimate ofThe line tangent to the graph at (x l l(x l )) has the equation y = I(x l ) + I(x l )(x - x l )and crosses the x-axis at the point x2 := x I - I (Xll)) . I(x -, -(See Figure 6.4.2.) If we replace x l by the second estimate x2 then we obtain a point x3 and so on. At the nth iteration we get the point xn+ l from the point xn by the formula xn+ l := xn - I (xnn)) I(xUnder suitable hypotheses, the sequence (xn ) will converge rapidly to a root of the equationI (x) = 0, as we now show. The key tool in establishing the rapid rate of convergence isTaylor s Theorem. y ---r----�._��--._--� x Figure 6.4.2 Newtons Method6.4.7 Newtons Method Let I := [a, b] I: and let I � lR be twice differentiable on 1 .Suppose that I (a) I (b) M < 0 and that there are constants m , such that I I (x) I :::: m > 0and I I" (x) I M ::: for all xE I and let K : = M12m. Then there exists a subinterval 1*containing a zero r of such that for any I Xl E 1* the sequence (xn ) defined by(5) I (x xn+ l := xn - I(xnn)) for all n E N, r.belongs to 1* and (xn ) converges to Moreover(6) for all n E N.
6.4 TAYLOR S THEOREM 191IK en+ 1 1 < 1 O-2m so that the number of significant digits in Ken has been doubled. Because of this doubling, the sequence generated by Newton s Method is said to converge"quadratically".(b) In practice, when Newtons Method is programmed for a computer, one often makes aninitial guess x 1 and lets the computer run. If x l is poorly chosen, or if the root is too near theendpoint of I , the procedure may not converge to a zero of f . Two possible difficulties areillustrated in Figures 6.4.3 and 6.4.4. One familiar strategy is to use the Bisection Methodto arrive at a fairly close estimate of the root and then to switch to Newton s Method forthe coup de grace. --------��r_��---+ x Figure 6.4.3 xn -+ 00 . Figure 6.4.4 xn oscillates between x 1 and x2 •Exercises for Section 6.4 1. Let f(x) cos ax for x E where a =1= O. Find f(n)(x) for n E N, x E R := 3 R 2. Let g(x) := Ix 1 for x E R Find g(x) and gl/ (x) for x E and gll (X) for x =1= O. Show that R, gil (0) does not exist. 3. Use Induction to prove Leibnizs rule for the nth derivative of a product: (jg)(nx) t (nk)f(n-kx)g(kX). = 4. Show that if x > 0, then 1 + 1x - k x2 k=O :5 JI+X 1 + x. :5 1 5. Use the preceding exercise to approximate Jf.2 and ./2. What is the best accuracy you can be sure of, using this inequality? ./2 6. Use Taylors Theorem with n =2 to obtain more accurate approximations for Jf.2 and . 3 1 i2 3 mate Vf2 and 3J2. I 7. If x > 0 show that 1(1 + x) / - (1 + x - x )1 (5/81)x . Use this inequality to approxi :5 8. If f(x) := eX, show that the remainder term in Taylors Theorem converges to zero as n -+ 00, for each fixed Xo and x. [Hint: See Theorem 3.2.1 1.]• 9. If g(x) sinx, show that the remainder term in TaylorS Theorem converges to zero as n := -+ 00 for each fixed Xo and x.10. Let h(x) e- 1/x2 for x =1= 0 and h(O) O. Show that h(n) (0) = 0 for all E N. Conclude that := := n the remainder term in Taylors Theorem for Xo 0 does not converge to zero as n for = -+ 00 x =1= O. [Hint: By L Hospitals Rule, xlim h(x) /xk = 0 for any k E N. Use Exercise 3 to calculate --+O h(n) (x) for x =1= 0.]
192 CHAPTER 6 DIFFERENTIATION1 1. If x E [0, 1] and n E N, show that Iln( 1 + X) - (x - x22 + x3 + " + (_ l )n- l xnn ) l < nxn++ ll . 3 Use this to approximate In 1.5 with an error less than 0.01. Less than 0.001.12. We wish to approximate sin by a polynomial on [- 1 , 1] so that the error is less than 0.001 . Show that we have I in x - (x - x; + ::0)1 < 5�0 for Ix l 1 . S � e13. Calculate correct to decimal places. 714. Determine whether or not x = is a point of relative extremum of the following functions: ° (a) f(x) := x3 + 2, (b) g(x) := sinx -x, (c) h(x) := sin x + iX3, (d) k(x) := cos x - 1 + �x2.15. Let f be continuous on [a, b] and assume the second derivative j" exists on (a, b). Suppose that the graph of f and the line segment joining the points (a, f(a» and (b, f(b» intersect at a point (xo f(xo» where a < Xo < b. Show that there exists a point c E (a, b) such that j" (c) = 0.16. Let I � lR be an open interval, let f : I lR be differentiable on I, and suppose j" (a) exists -+ at a E I. Show that j"( ) - 1" f(a + h) - 2f(a) + f(a - h) a - h� h2 Give an example where this limit exists, but the function does not have a second derivative at a.17. Suppose that I � is an open interval and that j" (x) for all x E I. If e E l, show that the lR 2: ° part of the graph of f on I is never below the tangent line to the graph at (c, f (c» .18. Let I � lR be an interval and let e E l. Suppose that f and g are defined on I and that the derivatives f(n) , g(n) exist and are continuous on I. If f(k) (c) = and g(kc) = for ° ° k = 0, 1 , " , n - 1, but g(n) (c) =f:. 0, show that h f(x) f(n) (c) .m -- = -- . x-+c g(x) g(n) (c)19. Show that the function f(x) : = x3 - 2x - 5 has a zero r in the interval I : = [2, 2. 2 ]. If X l : = 2 and if we 2define the sequence (xn ) using the Newton procedure, show that IXn+ 1 - rl � (0. ) IXn - 1 . Show that x4 is accurate to within six decimal places. 7 r20. Approximate the real zeros of g(x) := X4 -x - 3.21. Approximate the real zeros of h(x) := x3 - x - I. Apply Newtons Method starting with the initial choices (a) X := 2, (b) Xl := 0, (c) X l := -2. Explain what happens.22. The equatiQn lnx =l x - 2 has two solutions. Approximate them using Newtons Method. What happens if Xl := � is the initial point?23. The function f(x) = 8x3 - 8x2 + 1 has two zeros in [0, 1]. Approximate them, using Newtons Method, with the starting points (a) X l := k, (b) X l := �. Explain what happens.24. Approximate the solution of the equation X = cos x, accurate to within six decimals.
CHAPTER 7 THE RIEMANN INTEGRALWe have already mentioned the developments, during the 1630s, by Fermat and Descartesleading to analytic geometry and the theory of the derivative. However, the subject weknow as calculus did not begin to take shape until the late 1 660s when Isaac Newtoncreated his theory of "fluxions" and invented the method of "inverse tangents" to find areasunder curves. The reversal of the process for finding tangent lines to find areas was alsodiscovered in the 1680s by Gottfried Leibniz, who was unaware of Newton s unpublishedwork and who arrived at the discovery by a very different route. Leibniz introduced theterminology "calculus differentialis" and "calculus integralis", since finding tangent linesinvolved differences and finding areas involved summations. Thus, they had discovered thatintegration, being a process of summation, was inverse to the operation of differentiation. During a century and a half of development and refinement of techniques, calculusconsisted of these paired operations and their applications, primarily to physical problems.In the 1 850s, Bernhard Riemann adopted a new and different viewpoint. He separated theconcept of integration from its companion, differentiation, and examined the motivatingsummation and limit process of finding areas by itself. He broadened the scope by considering all functions on an interval for which this process of "integration" could be defined:the class of "integrable" functions. The Fundamental Theorem of Calculus became a resultthat held only for a restricted set of integrable functions. The viewpoint of Riemann ledothers to invent other integration theories, the most significant being Lebesgue s theory ofintegration. But there have been some advances made in more recent times that extend even Bernard Riemann (Georg Friedrich) Bernard Riemann (1826-1866), the son of a poor Lutheran minister, was born near Hanover, Germany. To please his father, he enrolled (1846) at the University of GOttingen as a student of theology and philosophy, but soon switched to mathemtics. He interrupted his studies at Gottingen to C. F. study at Berlin under G. J. Jacobi, P. G. J. Dirichlet, and G. Eisenstein, but returned to Gottingen in 1 849 to complete his thesis under Gauss. His thesis dealt with what are now called "Riemann surfaces". Gauss was so enthusiastic about Riemanns work that he arranged for him to become a privatdozent at Gottingen in 1 854. On admission as a privatdozent, Riemann was required to prove himself by delivering a probationary lecture before the entire faculty. As tradition dictated, he submitted three topics, the first two of which he was well prepared to discuss. To Riemanns surprise, Gauss chose that he should lecture on the third topic: "On the hypotheses that underlie the foundations of geometry". After its publication, this lecture had a profound effect on modem geometry. Despite the fact that Riemann contracted tuberculosis and died at the age of 39, he made major contributions in many areas: the foundations of geometry, number theory, real and complex analysis, topology, and mathematical physics. 193
194 CHAPTER 7 THE RIEMANN INTEGRALthe Lebesgue theory to a considerable extent. We will give a brief introduction to theseresults in Chapter 1 0. We begin by defining the concept of Riemann integrability of real-valued functions JR,defined on a closed bounded interval of using the Riemann sums familiar to the readerfrom calculus. This method has the advantage that it extends immediately to the case af nfunctions whose values are complex numbers, or vectors in the space JR • In Section 7.2,we will establish the Riemann integrability of several important classes of functions: stepfunctions, continuous functions, and monotone functions. However, we will also see thatthere are functions that arenot Riemann integrable. The Fundamental Theorem of Calculusis the principal result in Section 7.3.We will present it in a form that is slightly moregeneral than is customary and does not require the function to be a derivative at everypoint of the interval. A number of important consequences of the Fundamental Theoremare also given. In Section 7.3 we also give a statement of the definitive Lebesgue Criterionfor Riemann integrability. This famous result is usually not given in books at this level,since its proof (given in Appendix C) is somewhat complicated. However, its statement iswell within the reach of students, who will also comprehend the power of this result. Thefinal section presents several methods of approximating integrals, a subject that has becomeincreasingly important during this era of high-speed computers. While the proofs of theseresults are not particularly difficult, we defer them to Appendix D. An interesting history of integration theory, including a chapter on the Riemann integral, is given in the book by Hawkins cited in the References.Section 7.1 Riemann IntegralWe will follow the procedure commonly used in calculus courses and define the Riemannintegral as a kind of limit of the Riemann sums as the norm of the partitions tend to O.Since we assume that the reader !s familiar-at least informally-with the integral from acalculus course, we will not provide
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Regular Session
5/20/13 - 8/23/13
MAT 30 Algebra I
Topics include a review of fundamentals, real numbers, algebraic expressions, first degree equations in one variable, polynomials, factoring, rational expressions, graphing, square roots, quadratic equations, and exponents. Problem solving and informal geometry will be integrated throughout the course. Credits for this course do not fulfill degree requirements. Prerequisite: A grade of C, not C- or higher in MAT 9, MAT 20, or an appropriate score on the UMA placement test.
WEB 03619 McAleer 3
MAT 125 Analytical Geometry and Introductory Calculus I
An introduction to calculus for all students. Differential calculus of the algebraic, trigonometric, exponential, and logarithmic functions. In addition, the definite integral and the fundamental theorem of calculus are studied. Graphing calculators are recommended due to the exploratory, geometric, and intuitive approach which emphasizes an understanding of the basic concepts of function, limit, derivative, and integral. Prerequisite: MAT124
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In this applet, the user applies Euler's Method to modeling population growth using the Malthus exponential model and the...
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In this applet, the user applies Euler's Method to modeling population growth using the Malthus exponential model and the Verhulst constrained growth model. After finding the Euler solution, the user can "check" the solution with the Adaptive Euler Approximation or with a slope field. Also, the user can enter an exact solution obtained from separating variables (or whatever) and again check the Euler solution graphically.
This site has has interactive explanations and simulations of math from alegrbra to trigonometry. Just click the...
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This site has has interactive explanations and simulations of math from alegrbra to trigonometry. Just click the "interactive" tab on the top left menu and you can choose different simulations. It includes, the complete definition of parabolas, reaching beyond the ability to graph into the realm of why the graph appears as it does. It also has vivid descriptions of angles including circle angles for geometry. It also has calculators for principal nth roots, gdc, matrices, and prime factorization. It's definitely worth checking out. Quote from site: "A parabola is actually a locus of a point and a line. The point is called the focus and the line the directrix. That means that all points on a parabola are equidistant from the focus and the directrix. To change the equation and the graph of the interactive parabola below just click and drag either the point A, which is the focus, or point B, which controls the directrix." This is an interactive site that allows people to change the graph to understand why directrix and focus dictate parabolic graphs web page shows links to lectures for a course on Numerical Methods in Engineering taught by the author in the Spring...
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This web page shows links to lectures for a course on Numerical Methods in Engineering taught by the author in the Spring Semester of 2009. Click on the lecture links for class notes, Matlab scripts and functions, and assignments. Subjects covered: vectors and matrices in Matlab, graphics in Matlab, programming, numerical linear algebra, solution to equations, numerical integration, data fitting, and ordinary differential equationsThe ability to quantify the uncertainty in our models of nature is fundamental to many inference problems in Science and...
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The ability to quantify the uncertainty in our models of nature is fundamental to many inference problems in Science and Engineering. In this course, we study advanced methods to represent, sample, update and propagate uncertainty. This is a "hands on" course: Methodology will be coupled with applications. The course will include lectures, invited talks, discussions, reviews and projects and will meet once a week to discuss a method and its applications.
This course focuses on dynamic optimization methods, both in discrete and in continuous time. We approach these problems from...
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This course focuses on dynamic optimization methods, both in discrete and in continuous time. We approach these problems from a dynamic programming and optimal control perspective. We also study the dynamic systems that come from the solutions to these problems. The course will illustrate how these techniques are useful in various applications, drawing on many economic examples. However, the focus will remain on gaining a general command of the tools so that they can be applied later in other classes.
This course introduces students to the theory, algorithms, and applications of optimization. The optimization methodologies...
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This course introduces students to the theory, algorithms, and applications of optimization. The optimization methodologies include linear programming, network optimization, integer programming, and decision trees. Applications to logistics, manufacturing, transportation, marketing, project management, and finance. Includes a team project in which students select and solve a problem in practice.
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Wolframalpha is damn near magical. It could almost pass the Turing test if you restrict your inquiries to those with computable answers. Still, it's not quite suitable as a student's calculator--any more than google.com is suitable as a textbook.
The biggest issue is a lack of persistence of results (i.e. using the output from three inputs ago in a new calculation). It also tends to give multiple outputs for a given input--which can create confusion. There's also a speed issue.
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Start reading chapter2. Learn how to produce random numbers on your spreadsheet. On the Google spreadsheet the functions that produces uniformly distributed pseudo-random numbers are called RAND for random real numbers, and RANDBETWEEN for random integers. You can find them in the Help menu if you look for "function list" then select the category "math" in the "select an option" drop-down list. See the category "statistical" for other statistical functions.
Homework. Do exercise 5 p. 91.
4th Week: Feb. 11, 13
Read 2.1, 2.2, 2.3
Wednesday's class will be in SBS B110. A colloquium speaker is coming to talk to us about a project to improve precalculus classes at Cal State Northridge by collecting data about how teaching practices affect student learning and using the data to make changes. She will then switch topics and talk about geometry.
Homework: Chapter 3 Project 3.9 Impalas. You can do this on a spreadsheet but I think it will be easier to get it right if you use Sage or MATLAB or another general package that can handle math. If you'd like to use Sage here is sage worksheet that may help get you started: Modeling project 3.9 Impalas ( I'll explain it in class.
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The book is the second edition of an interesting collection of 120 problems presented by accomplished and renowned researchers on probability theory. The book is presented at an advanced measure theoretic level. The choice of the problems reflect the authors' research interests and is evident from their acknowledgement in the introduction: "…we do not view this set of exercises as being 'the' good companion to a course in probability theory …, but rather we have tried to present some perhaps not so classical aspects...". The collection, nonetheless, provides a profound insight into some of the more difficult techniques that have their place in the arsenal of a modern probabilist. In conjunction with the detailed solutions and relevant references, the book is a valuable educational resource for a beginning researcher in probability and mathematical statistics.
There are six chapters each containing a set of problems. The themes logically evolve from fundamentals and distributional aspects to theory of stochastic processes. From a general perspective the problems are, as admitted by the authors, of two different sorts: the 'stripping' exercises and the 'dressing' exercises. The former attempt to bring advanced reasoning needed to pursue a theoretical investigation to the level accessible to a student with a modest exposure in fundamentals of measure theory based probability. The latter set of exercises aim towards the opposite direction where some basic facts that are typically one-dimensional in their initial scope are embedded in an infinite-dimensional framework. Essentially all problems are accompanied by their complete solutions. Two exercises (2.7 and 6.29) that contain some unsolved questions (marked in the text by circles) are presented as challenge problems to the readers.
Several simple but extremely useful additions facilitate efficient usage of the text. These include a short list of frequently used notation and comments following individual problems that provide additional insight. Pointers to additional citation is also a useful resource for readers interested in further contextual learning.
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University serves up new algebra-free math class
A new mathematics course is on the menu at the University of Louisiana at Lafayette this fall: MATH 102. Freshmen who major in non-science areas mayNationally, universities are moving away from formula-based teaching, toward concept-oriented, practical applications of mathematics, said Myers, who is also director of freshman mathematics at the University.
She said this course will likely appeal to the majority of students enrolled in the College of the Arts and the College of Liberal Arts.
In the past, college algebra was the first mathematics course taken by all undergraduates. There are two paths students follow upon completion of their initial math course. One path is taken primarily by business and science-oriented majors, who are required to take advanced mathematics courses. The other path is for non-science majors. The courses they take emphasize applied mathematics.
"Creating a college algebra alternative designed specifically for non-science majors will enable us to better serve these two different populations," said Myers. Course topics in MATH 102 include traditional concepts, such as linear and exponential functions, as well as topics designed to increase students' ability to reason quantitatively. The course emphasizes critical thinking.
"Our primary goal is to make students better educated. As
determining which of two possible financial situations is most advantageous
applying deductive and inductive reasoning, such as understanding the error of statements such as "All services not available in all areas."
reading and interpreting graphs, particularly recognizing graphs which have been designed to intentionally mislead
computing the consequences of not paying off a "no-money down, two-years interest free" purchase during the interest-free period
comparing a flat-rate subscription movie site to a service which charges an initial fee plus per-movie charge
using proportional reasoning to determine which size of product is the best deal
recognizing when it is appropriate to use the phrase "growing exponentially."
"Students will be able to use the reasoning and mathematical skills they learn in this course to enhance their decision-making skills, both personally and professionally," Myers added.
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ALGEBRA+TRIGONOMETRY - 3rd edition
Summary: The new 3rd edition of Cynthia Young's Algebra & Trigonometry continues to bridge the gap between in-class work and homework by helping readers overcome common learning barriers and build confidence in their ability to do mathematics. The text features truly unique, strong pedagogy and is written in a clear, single voice that speaks directly to students and mirrors how instructors communicate in lectures. In this revision, Young enables readers to become independent, success...show moreful learners by including hundreds of additional exercises, more opportunities to use technology, and a new themed modeling project that empowers them to apply what they have learned in the classroom to the world outside the classroom. The seamlessly integrated digital and print resources to accompany Algebra & Trigonometry 3e offer additional tools to help users experience success611150470648031 used book - free tracking number with every order. ?book used -used
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Mathematics of Egypt, Mesopotamia... - 07 edition
Summary: In recent decades it has become obvious that mathematics has always been a worldwide activity. But this is the first book to provide a substantial collection of English translations of key mathematical texts from the five most important ancient and medieval non-Western mathematical cultures, and to put them into full historical and mathematical context.The Mathematics of Egypt, Mesopotamia, China, India, and Islamgives English readers a firsthand understanding and appreciation of the...show morese cultures' important contributions to world mathematics. The five section authors--Annette Imhausen (Egypt), Eleanor Robson (Mesopotamia), Joseph Dauben (China), Kim Plofker (India), and J. Lennart Berggren (Islam)--are experts in their fields. Each author has selected key texts and in many cases provided new translations. The authors have also written substantial section introductions that give an overview of each mathematical culture and explanatory notes that put each selection into context. This authoritative commentary allows readers to understand the sometimes unfamiliar mathematics of these civilizations and the purpose and significance of each text. Addressing a critical gap in the mathematics literature in English, this book is an essential resource for anyone with at least an undergraduate degree in mathematics who wants to learn about non-Western mathematical developments and how they helped shape and enrich world mathematics. The book is also an indispensable guide for mathematics teachers who want to use non-Western mathematical ideas in the classroom. ...show less
Edition/Copyright:07 Cover: Publisher:Princeton University Press Year Published: 2007 International: No
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Precalculus : Functions and Graphs - 11th edition
ISBN13:978-0495108375 ISBN10: 0495108375 This edition has also been released as: ISBN13: 978-0495385042 ISBN10: 0495385042
Summary: Clear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications have made this text popular among students year after year. This latest edition of Swokowski and Cole's PRECALCULUS: FUNCTIONS AND GRAPHS retains these features. The problems have been consistently praised for being at just the right level for precalculus students like you. The book also provides calculator examples, including specific key...show morestrokes that show you how to use various graphing calculators to solve problems more quickly. Perhaps most important-this book effectively prepares you for further courses in mathematicsHelp save a tree. Buy all your used books from Green Earth Books. Read. Recycle and Reuse.
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Matrices, Vectors, and Their Operations
Basic definitions and notations
Matrix addition and scalar-matrix multiplication
Matrix multiplication
Partitioned matrices
The "trace" of a square matrix
Some special matrices
The Rank of a Matrix
Rank and nullity of a matrix
Bases for the four fundamental subspaces
Rank and inverse
Rank factorization
The rank-normal form
Rank of a partitioned matrix
Bases for the fundamental subspaces using the rank normal form
Complementary Subspaces
Sum of subspaces
The dimension of the sum of subspaces
Direct sums and complements
Projectors
Revisiting Linear Equations
Introduction
Null spaces and the general solution of linear systems
Rank and linear systems
Generalized inverse of a matrix
Generalized inverses and linear systems
The Moore-Penrose inverse
Determinants
Definitions
Some basic properties of determinants
Determinant of products
Computing determinants
The determinant of the transpose of a matrix — revisited
Determinants of partitioned matrices
Cofactors and expansion theorems
The minor and the rank of a matrix
The Cauchy-Binet formula
The Laplace expansion
Singular Value and Jordan Decompositions
Singular value decomposition (SVD)
The SVD and the four fundamental subspaces
SVD and linear systems
SVD, data compression and principal components
Computing the SVD
The Jordan canonical form
Implications of the Jordan canonical form
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Math and YOU : The Power and Use of MathematicsAs the world around us changes and information comes at warp speed, it is more important than ever to be quantitatively literate. Yet most U.S. students leave high school with quantitative skills far below what they need and what employers are seeking, and virtually every college finds that many students need remedial mathematics. Based on the latest educational research, Math & YOUhelps students develop the quantitative skills needed to be successful in school and the workplace, using real data, problems based on everyday situations, and activities built around topics that are recognizable and relevant. With this approach, students become comfortable with quantitative ideas and proficient in applying them. In addition, to support the printed text, Math & YOUprovides an online eBook accompanied by additional teaching aids, all part of a robust companion Web site . Hardcover edition available upon request. Ask your local W.H. Freeman representative. Math & YOU Hallmarks Confidence with Mathematics. One of the goals of the Math & YOUprogram is to help students become comfortable with quantitative ideas and proficient in applying them. Students routinely quantify, interpret, and check information such as comparing the total compensation of two job offers, or comparing and analyzing a budget Cultural Appreciation. Math & YOUprovides examples and exercises that help student to understand the nature of mathematics and its importance for comprehending issues in the public realm. Logical Thinking. The Math & YOUprogram develops habits of inquiry, prepares students to look for appropriate information, and exposes them to arguments so that they can analyze and reason to get at the real issues. Making Decisions. One of the main threads of the Math & YOUprogram is to help students develop the habit of using mathematics to make decisions in everyday life. One of the goals of the text is for students to see that mathematics is a powerful tool for living. Mathematics in Context.The Math & YOUprogram helps students to learn to use mathematical tools in specific settings where the context provides meaning. Number Sense. The Math & YOUprogram begins with a chapter that reviews the meaning of numbers, estimation and measuring. Throughout the rest of the program students develop intuition, confidence, and common sense for employing numbers. Practice Skills. Throughout the Math & YOUprogram students encounter quantitative problems that they are likely to encounter at home or work. This helps students become adept at using elementary mathematics in a wide variety of common situations.
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Mathematics for Elementary Teachers
9780470105832
ISBN:
0470105836
Edition: 8 Pub Date: 2008 Publisher: Wiley, John & Sons, Incorporated
Summary: Now in its eighth edition, this book masterfully integrates skills, concepts, and activities to motivate learning. It emphasizes the relevance of mathematics to help readers learn the importance of the information being covered.
Burger, William F. is the author of Mathematics for Elementary Teachers, published 2008 under ISBN 9780470105832 and 0470105836. Two hundred seventy nine Mathematics for Elementary T...eachers textbooks are available for sale on ValoreBooks.com, one hundred twenty eight used from the cheapest price of $0.01, or buy new starting at $36
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Description of Saxon Math Intermediate 3: Power Up Workbook by CurtisWith a focus on continual practice and assessment, Saxon Math programs build foundational concepts and critical thinking skills through real world problem-solving and applications.
This Power Up Workbook includes "Facts Practice," "Jump Start," "Mental Math" and "Problem Solving" sections. Works in tandem with the Intermediate 3 student textbook – text sometimes refers students to problems within the workbook and workbook sometimes refers students to instructions within the text in order to complete exercises.
Product:
Saxon Math Intermediate 3: Power Up Workbook
Author:
Curtis Hake
Vendor:
Saxon/Harcourt Pusblishers
Edition Description:
Student
Binding Type:
Paperback
Media Type:
Book
Minimum Grade:
3rd Grade
Weight:
0.4 pounds
Length:
10.6 inches
Width:
7.9 inches
Height:
0.2 inches
Publisher:
Hmh Supplemental Power Up Workbook.
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For an excellent web site with some great discussion of study skills check out the following site by Martin Greenhow. The site has some occasional comments pertaining to the school Dr. Greenhow teaches out but is non the less a great site that goes into much greater detail that I do here.
Before I get into the tips for how to study math let me
first say that everyone studies differently and there is no one right way to
study for a math class. There are a lot
of tips in this document and there is a pretty good chance that you will not
agree with all of them or find that you can't do all of them due to time
constraints. There is nothing wrong with
that. We all study differently and all
that anyone can ask of us is that we do the best that we can. It is my intent with these tips to help you
do the best that you can given the time that you've got to work with.
Now, I figure that there are two groups of people here
reading this document, those that are happy with their grade, but are interested
in what I've got to say and those that are not happy with their grade and want
some ideas on how to improve. Here are a
couple of quick comments for each of these groups.
If you have a study routine that you are happy with and you
are getting the grade you want from your math class you may find this an
interesting read. There is, of course,
no reason to change your study habits if you've been successful with them in
the past. However, you might benefit
from a comparison of your study habits to the tips presented here.
If you are not happy with your grade in your math class and
you are looking for ways to improve your grade there are a couple of general
comments that I need to get out of the way before proceeding with the
tips. Most people who are doing poorly
in a math class fall into three main categories.
The first category consists of the largest group of students
and these are students that just do not have good study habits and/or don't
really understand how to study for a math class. Students in this category should find these
tips helpful and while you may not be able to follow all of them hopefully you
will be able to follow enough of them to improve your study skills.
The next category is the people who spend hours each day
studying and still don't do well. Most
of the people in this category suffer from inefficient study habits and
hopefully this set of notes will help you to study more efficiently and not
waste time.
The final category is those people who simply aren't
spending enough time studying. Students
are in this category for a variety of reasons.
Some students have job and/or family commitments that prevent them from
spending the time needed to be successful in a math class. To be honest there isn't a whole lot that I
can do for you if that is your case other than hopefully you will become a more
efficient in your studies after you are through reading this. The vast majority of the students in this
category unfortunately, don't realize that they are in this category. Many don't realize how much time you need to
spend on studying in order to be successful in a math class. Hopefully reading this document will help you
to realize that you do need to study more.
Many simply aren't willing to make the time to study as there are other
things in their lives that are more important to them. While that is a decision that you will have
to make, realize that eventually you will have to take the time if you want to
pass your math course.
Now, with all of that out of the way let's get into the
tips. I've tried to break down the hints
and advice here into specific areas such as general study tips, doing
homework, studying for exams, etc. However, there are three broad, general areas
that all of these tips will fall into.
Math is Not a Spectator Sport
You cannot learn mathematics by
just going to class and watching the instructor lecture and work problems. In order to learn mathematics you must be
actively involved in the learning process.
You've got to attend class and pay attention while in class. You've got to take a good set of notes. You've got to work homework problems, even if
the instructor doesn't assign any.
You've got to study on a regular schedule, not just the night before
exams. In other words you need to be
involved in the learning process.
The reality is that most people
really need to work to pass a math class, and in general they need to work
harder at math classes than they do with their other classes. If all that you're willing to do is spend a
couple of hours studying before each exam then you will find that passing most
math classes will be very difficult.
If you aren't willing to be
actively involved in the process of learning mathematics, both inside and
outside of the class room, then you will have trouble passing any math
class.
Work
to Understand the Principles
You can pass a history class by
simply memorizing a set of dates, names and events. You will find, however, that in order to pass
a math class you will need to do more than just memorize a set of
formulas. While there is certainly a
fair amount of memorization of formulas in a math class you need to do
more. You need to understand how to USE
the formulas and that is often far different from just memorizing them.
Some formulas have restrictions on
them that you need to know in order to correctly use them. For instance, in order to use the quadratic
formula you must have the quadratic in standard form first. You need to remember this or you will often
get the wrong answer!
Other formulas are very general
and require you to identify the parts in the problem that correspond to parts
in the formula. If you don't understand
how the formula works and the principle behind it, it can often be very
difficult to use the formula. For
example, in a calculus course it's not terribly difficult to memorize the
formula for integration by parts for integrals.
However, if you don't understand how to actually use the formula and
identify the appropriate parts of the integral you will find the memorized formula
worthless.
Mathematics
is Cumulative
You've always got to remember that
mathematics courses are cumulative.
Almost everything you do in a math class will depend on subjects that
you've previously learned. This goes
beyond just knowing the previous sections in your current class to needing to
remember material from previous classes.
You will find a college algebra
class to be very difficult without the knowledge that you learned in your high
school algebra class. You can't do a
calculus class without first taking (and understanding) an Algebra and a
Trigonometry class.
So, with these three main ideas in mind let's proceed with
some more specific tips to studying for a math class. Note as well that several of the tips show up
in multiple sections since they are either super important tips or simply can
fall under several general topics.
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0618318836
9780618318834
Beginning Algebra with Arithmetic Review:Beginning Algebra with Arithmetic Review is an activity-based text designed to help students gain a strong conceptual foundation by building connections among topics of mathematics in a meaningful, real-world context. Guided by the instructor and prompted by the text activities, students explore, make conjectures, interpret, discuss, and ultimately find meaning and understanding in basic mathematical principles. Based on the 18 modules of the award-winning Maricopa Mathematics Modules, this text embodies a thorough implementation of the AMATYC Crossroads Standards and provides a seamless transition from Basic Mathematics through Introductory Algebra.
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completely revised comprehensive drafting book for high school includes solid drafting instruction, board drafting techniques, and computer aided drafting techniques. Each chapter provides a large number of practice problems, "Tech Math" incorporating math skills needed for the covered topics, and "Success on the Job" employability skills needed on the job.
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A Quick Note: The original review was posted on 02/26/2009 and is being transferred here to my new site. The prices may have changed. Please click the links below for current pricing. And: please read my disclaimer on the receipt of this item. One last note: I have an additional review on a MathTutor product on my blog. You can view it here. Thanks ;)
Math Tutor DVD
Does teaching math cause you to grimace and say "I don't know how they got it, but this is the answer"? Or do your kids have to go over and over a lesson before it clicks or does it never really click? Well not anymore.
I would like to introduce you to the Math Tutor DVD series created by Jason Gibson. The DVD's available from Math Tutor range from basic math to Physics to Statistics. Each "course" has 2 DVD's and at least 7 hours of tutoring. There are also worksheet CD's that can be purchased for some of the courses. Jason Gibson is the instructor for each lesson and offers students of all ages an easy, no-nonsense approach to math problems. He does not give lectures but gives instruction by talking you through the problems. Mr. Gibson is highly qualified to teach math as he has an MS degree in both Electrical Engineering and Physics and has a history of teaching even his friends. Mr. Gibson wrote "I have a passion for making 'complex' subjects easy to understand". Watching him in these videos verifies this statement. You can tell he enjoys what he's doing and genuinely wants his students to learn.
MathTutor offer students the opportunity to improve their skills in the comfort of their own home. Mr. Gibson's techniques allow the students to build confidence along the way. The videos aren't fancy and colorful (there's a whiteboard and Mr. Gibson), but they are helpful to anyone needing math tutoring.
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North Lauderdale, FL CalculusThe student will learn how to create slides and add animations to give life to the presentation. After learning the interface and how powerpoints work, the student will be ready to give a presentation like no other. Elementary Math includes the building blocks that teach the student how to solve real world problems
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The purpose of this study was to examine common algebra-related misconceptions and errors of middle school students. In recent years, success in Algebra I is often considered the mathematics gateway to graduation from high school and successThis research alignment study compares content assessed on course finals from Kentucky public universities in highest level remedial mathematics courses and content assessed on college placement examinations. These assessments are used to determine...
In this study, the author examined the relationship of probability misconceptions
to algebra, geometry, and rational number misconceptions and investigated the potential
of probability instruction as an intervention to address misconceptions in all the effects of different instructional
types used in a mathematics intervention setting. In recent years, school staff have
implemented mathematics intervention programs to aid struggling students and...
Optimal trajectory generation is an essential part for robotic explorers to execute the total exploration of deep oceans or outer space planets while curiosity of human and technology advancements of society both require robots to search for support of the national movement to improve mathematics instruction and assessment, states and districts are looking for the best tools to measure student progress toward proficiency. There is a national dialogue about how to use 8th-grade
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History of Mathematics An Introduction
9780072885231
ISBN:
0072885238
Publisher: McGraw-Hill Higher Education
Summary: This text is designed for the junior/senior mathematics major who intends to teach mathematics in high school or college. It concentrates on the history of those topics typically covered in an undergraduate curriculum or in elementary schools or high schools. At least one year of calculus is a prerequisite for this course. This book contains enough material for a 2 semester course but it is flexible enough to be used... in the more common 1 semester course.
Burton, David M. is the author of History of Mathematics An Introduction, published under ISBN 9780072885231 and 0072885238. Eighteen History of Mathematics An Introduction textbooks are available for sale on ValoreBooks.com, thirteen used from the cheapest price of $0.98, or buy new starting at $99
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Linear Algebra
9780817642945
ISBN:
0817642943
Edition: 2 Pub Date: 2004 Publisher: Birkhauser Boston
Summary: From a review of the first edition: "A logical development of the subject . . . all the important theorems and results are discussed in terms of simple worked examples. The student's understanding . . . is tested by problems at the end of each subsection, and every chapter ends with exercises." a?CURRENT SCIENCE A cornerstone of undergraduate mathematics, science, and engineering, this clear and rigorous presentation... of the fundamentals of linear algebra is unique in its emphasis and integration of computational skills and mathematical abstractions. The power and utility of this beautiful subject is demonstrated, in particular, in its focus on linear recurrence, difference and differential equations that affect applications in physics, computer science, and economics. Key topics and features: a? Linear equations, matrices, determinants, vector spaces, complex vector spaces, inner products, Jordan canonical forms, and quadratic forms a? Rich selection of examples and explanations, as well as a wide range of exercises at the end of every section a? Selected answers and hints a? Excellent index This second edition includes substantial revisions, new material on minimal polynomials and diagonalization, as well as a variety of new applications. The text will serve theoretical and applied courses and is ideal for self-study. With its important approach to linear algebra as a coherent part of mathematics and as a vital component of the natural and social sciences, Linear Algebra, Second Edition will challenge and benefit a broad audience.
Kwak, Jin Ho is the author of Linear Algebra, published 2004 under ISBN 9780817642945 and 0817642943. Seven hundred thirty four Linear Algebra textbooks are available for sale on ValoreBooks.com, fifty four used from the cheapest price of $68.86, or buy new starting at $35.38
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MaGI Project for Precalculus (Math 1011)
Project MaGI [Ma(thematics) G(ateway) I(ntervention)], supported by a grant from CUNY and completed in academic year 2013, was aimed at reducing withdrawal rates in Math 1011 via two interventions:
Administration of a Precalculus Readiness Diagnostic administered in the first week of classes (Fall 2013 and Spring 2014), with aggressive advisement into Math 1021 for students whose performance indicated weak preparation.
Coordination of the content and pacing of the Math 1011 sections. This was accomplished through the introduction of weekly recitation sections, one session for each lecture section. Recitations were conducted by a CUNY Mathematics doctoral student and a peer mentor. The Project Director and Course Coordinator (Professor Anthony Clement) designed weekly problem sets for the recitations. The activities were designed to reinforce the conceptual material presented in lecture with a combination of standard exercises and open ended problems for discussion and guided discovery learning. Additionally, monthly coordination meetings were held for all of the instructional staff (lecture instructors, recitation leaders, and peer mentors) to discuss content, pacing, and pedagogy.
Recitations were designed in collaboration with the Brooklyn College Learning Center. The Director, Richard Vento, identified and trained the peer mentors. Since most of the recitations were held in that facility, Math 1011 students had the opportunity to become acquainted with the services offered for courses in a variety of College departments.
Problem Sets
During each weekly recitation session, students would work on the following problem sets which the instructor would then go over and answer any inquiries students had regarding these problems.
Schedules
Precalculus Standard Curriculum
The Precalculus Standard Curriculum (PDF) provides a listing of the topics that are normally covered in Precalculus. Please keep in mind that each professor has a unique teaching style and hence may cover certain topics in more or less detail. The standard curriculum mainly serves as a guide as to what topics you should anticipate learning in Precalculus.
Math 1011 Final Exams
Below are previous Math 1011 final exams. You can use these to help prepare for your upcoming final or to practice solving problems to reinforce your math knowledge and boost your confidence.
Precalculus Help Resources
Brooklyn College Learning Center
The Learning Center, located in 1300 Boylan Hall, offers Brooklyn College students FREE peer tutoring in courses across the curriculum in a comfortable, supportive environment well stocked with computers and reference materials for student use. The Center is open every weekday, some evenings, and weekends.
The Learning Center offers Precalculus tutors (as well as tutors for other mathematical subject areas). Please take advantage of this valuable resource as these tutors can assist you with understanding the course material, help you prepare for an upcoming exam, and answer any other Precalculus inquiries you may have.
To find out more information about the Learning Center, please visit their website or call them at 718.951.5821.
Brooklyn College Library: Math - Study Help Subject Guide Page
The "Math - Study Help" Subject Guide page offers a comprehensive list of links to free online resources for various mathematical subject areas, such as Precalculus. The resources being linked to on this page can of great value in supplementing your course material and help you reinforce concepts, view examples, solve practice problems, and boost your confidence.
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Real and Complex Analysis - 3rd edition
Summary: In this advanced text, the basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. ...show moreProofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject
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SpeQ mathematics -
Educational/Mathematics ... SpeQ is ...
3.
Visual mathematics -
Educational/Mathematics ... Visual mathematics is a highly interactive visualization software (containing -at least- 67 modules) addressed to High school, College and University students. This is a very powerful tool that helps to learn and solve problems by the hundreds in a very short time. Included areas: Arithmetic, Algebra, Geometry, Trigonometry, Analytic Geometry and miscellaneous.Visual mathematics, a member of the Virtual Dynamics mathematics Virtual Laboratory, is an Intuitively-Easy-To-Use software.Visual ...
4.
mathematics Tools -
Educational/Other ... mathematics Tools is a tools that help people in solving Mathematical problems such as: - Solving quadratic equation and cubic equation - Solving System of equations (2 or 3 unknowns) - Working in the Base Number System - Practice: Calculate the value of PI using the Probability - Explore the Fibonacci Number - Quiz Calculate It Quickly: help children in calculating - Calculate the Greatest Common Divisor or Least Common Multiple and lots of features will be update in the next version. ...
5.
Aviaion mathematics -
Utilities/Other Utilities ... Project avmath implements a computational solution to the aviation wind triangle and is targeted towards learners that have completed pre-calculus and are headed for calculus. ...
6.
mathematics Worksheet -
Educational/Teaching Tools7.
mathematics Quiz -
Educational/Mathematics ... This program is designed for students aged 7 - 9 in Primary 1 - 3. There are over 1500 challenging Maths quizzes and problem sums to practise on. Topics include Addition, Subtraction, Multiplication, Division, Length, Weight, Time, Money, Fractions, Graphs, Permeter, Area, Volume, Geometry, etc. Questions are modelled closely to primary education curriculum. All test papers comes with model answers, fully automated marking system and performance report card. This personal e-tutor is also able to ...
Higher mathematics Quiz -
Educational/Mathematics ... This program is designed for students aged 10 - 12 in Primary 4 - 6. There are over 2000 challenging Maths quizzes and problem sums to practise on. Topics include Whole numbers, Decimals, Fractions, Area, Perimeter, Volume, Geometry, Percentage, Ratio, Proportions, Rate, Speed and Algebra, etc. Questions are modelled closely to primary education curriculum. All test papers comes with model answers, fully automated marking system and performance report card. This personal e-tutor is also able to ...
10.
Web Components for mathematics -
Utilities/Other Utilities ... A framework of configurable mathematical software components written in the Java language, meant to be used on instructional Web pages. This project will take the original version (called JCM) and modify it to use Swing and JavaBeans. ...
Archimedean Academy Mathematics
From Short Description
1.
schleichfahrt reloaded -
Utilities/Other Utilities ... Remake of the original game "archimedean dynasty" (aka Schleichfahrt). Currently the original game is needed but later new game content can be added and could substitute the old ones (already impl). State: story-mode finished, fight under dev ...
2.
GLM -
Programming/Components & Libraries ... GLM or OpenGL mathematics is a C++ mathematics library created for 3D software based on the OpenGL Shading Language (GLSL) specification.GLM provides classes and functions designed and implemented with the same naming conventions and functionalities than GLSL so that when a programmer knows GLSL, he knows GLM as well which makes it really easy to use.This project is not limited by GLSL features. An extension system, based on the GLSL extension conventions, provides extended capabilities: matrix ...
MathCast Equation Editor <br>Manages numerous equations in equation lists<br>Equations can be exported into pictures (PNG, BMP)<br>Equations can be copied to the clipboard and pasted into documents<br>Equations can be exported to MathML and be incorporated to web p ...
5.
Math Flight -
Educational/Mathematics ... Learning mathematics can be a challenge for anyone. Math Flight can help you master it with three fun activities to choose from! With lots of graphics and sound effects, your interest in learning math should never decline. Help is always available in an audio and visual format with the simple click of a button. A great utility for teachers and parents is the ability to track all users progress in the statistics menu. This educational software is great for children and even adults wanting to ...
6.
Open Math Edit ...
GEUP -
Educational/Science ... GEUP is an excellent tool to enter into the world of mathematics and geometry in a totally interactive and visual way. It is a perfect application for teaching maths at any level.Thanks to GEUP you can, among other things:* Construct mathematic models of the real world.* Define and work with complex geometric shapes, like cones, functions, etc, as well as simple geometric shapes.* Calculate terms like parallelism, perpendicular, alignment or equidistance.* Multilingual support. ...
9.
UberSmart Math Facts -
Educational/Mathematics ... Is your son or daughter struggling with mathematics? Do you get frustrated when they continue to miss the same problem over and over again? Do you feel it is time-consuming to teach them? Just think about it - what if you could get your hands on a software program that helps them learn the basic math facts quickly, easily, and thoroughly? I mean really learn the math facts. A program that your child will enjoy using, requires little supervision, and yet is very effective? That is precisely what ...
10.
Mathpad -
Educational/Mathematics ... Mathpad is an easy to use text editor for mathematics. You can mix together ordinary text and any mathematical expression. Ideal for math teachers to create quizzes, tests and handouts. Also, you can save the formatted text as an image. With most equation editors you choose a template with the mouse, type a few keystrokes and repeat the process. Mathpad doesn't work this way. Mainly, you just type characers at the keyboard. Most people find this to be easier and faster.Mathpad has some unusual ...
Archimedean Academy Mathematics
From Long Description
1.
Magic academy Deluxe -
Games/Arcade ... Magic academy Deluxe transports you to a world full of spells, magic, witchcraft and mystery. In the purest Harry Potter style, Magic academy Deluxe places you in the role of Annie, a young student learning the magic arts at the academy of Magic, who must investigate the strange disappearance of her older sister: Irene.The mechanics of Magic academy Deluxe are based on finding hidden objects on the screen full of details. You must use all your skill and eyes over 21 phases full of exceptional ...
2.
Math Games Level 1 -
Educational/Mathematics ... Undoubtedly, mathematics is one of the most important subjects taught in school. It is thus unfortunate that some students lack elementary mathematical skills. An inadequate grasp of simple every-day mathematics can negatively affect a person's life. ...
3.
mathematics Worksheet Factory -
Educational/Other4.
MatheAss -
Multimedia & Design/Other Related Tools ... MatheAss (former Math-Assist) is a computer program for the numerical solution of many problems in school mathematics. It finds wide use in Germany in high school mathematics, for schools in the federal state of Hessen (Germany) exists a state license, which allows all secondary schools using MatheAss. "MatheAss" means Mathematical Assistant and that's exactly what this program is designed to be. It's for secondary level or high school students and teachers and anybody else who has ...
5.
Cooking academy for Mac OS -
Games/Other Games ... Grab your oven mitts and don your Chef's Hat! Cooking academy Cooking academy! ...
6.
GEUP 3D -
Multimedia & Design/Graphic & Design ... GEUP 3D is an interactive solid geometry software for math calculation and visualization. It allows to create dynamic and general constructions/applications visually by defining math elements. GEUP 3D allows the modification of the construction visually (directly in screen) and it calculates each one of its particular cases in real time. GEUP 3D is the equivalent of GEUP 4 in the 3D space.With GEUP 3D you can study graphically a general problem and obtain dynamically its particular cases. This ...
7.
InterReg -
Educational/Mathematics ... Interpolation and Regression are fundamental and important calculations in mathematics. Mr. Newton and Mr. Gauss were engaged in-depth with numerical solutions for these problems. Today, there are improved algorithms, that can solve such tasks. InterReg allows you to do such complex calculations just with some point-and-click. So this program is not only for mathematics and engineers. The applications are many and reach from education over science to productive use in your company: in example a ...
8.
Cooking academy 2 for Mac OS -
Games/Other Games ... Welcome to the World Culinary Workshop! Cooking academy 2 will put you in the kitchens of restaurants from all corners of the globe. From Chinese BBQ Pork Buns, to Mexican Tamales, to Japanese Sushi, make your way through 60 different recipes from eight different countries! Learn interesting trivia about food while mastering all new skills and mini-games including food processors, mixers, raiding the fridge and much more!Unlock new recipes and trophies as you complete recipes and exams in each ...
9.
Maths Toolkit -
Educational/Science ... Maths Toolkit is a tool designed jointly by Intel and the British mathematics Association, whose aim is to make it easier for students to learn the various fields of mathematics, and principally focussed on cartesian maps.Maths Toolkit is divided into different three parts:- Co-ordinates and graphs: it works to test the graphic representation of any function.- Graph creation: it generates various types of graphs from a numerical data relation.- 2D shape creation: it lets you draw shapes and ...
10.
Charmed Words 1.006-0 -
Games/Other Games ... Charmed Words places you in a beautiful fantasy world where Goblins, Dark Genii, Daemons and worse use powerful spells in their quest to incinerate your beloved academy
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Advanced Algebra II: Conceptual Explanations
Advanced Algebra II: Conceptual Explanations•The Homework and Activities Book contains in-class and homework assignments that are given to the students day-by-day." " target="_blank" "•The...
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•The Teacher's Guide provides detailed lesson plans; it is your guide to how the author "envisioned these materials being used when I created them (and how I use them myself) Instructors should note that this book probably contains more information than you will be able to cover in a single school year."
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Algebra seems mysterious to me. I really don't "get" what an equation represents. Why do we do the same thing to both sides?
This tutorial is a conceptual journey through the basics of algebra. It is made for someone just beginning their algebra adventure. But even folks who feel pretty good that they know how to manipulate equations might pick up a new intuition or two.
This tutorial is a survey of the major themes in basic algebra in five videos! From basic equations to graphing to systems, it has it all. Great for someone looking for a gentle, but broad understanding of the use of algebra. Also great for anyone unsure of which gym plan they should pick!
Like the "Why of algebra" and "Super Yoga plans" tutorials, we'll introduce you to the most fundamental ideas of what equations mean and how to solve them. We'll then do a bunch of examples to make sure you're comfortable with things like 3x – 7 = 8. So relax, grab a cup of hot chocolate, and be on your way to becoming an algebra rockstar.
And, by the way, in any of the "example" videos, try to solve the problem on your own before seeing how Sal does it. It makes the learning better!
You've been through "Equation examples for beginners" and are feeling good. Well, this tutorial continues that journey by addressing equations that are just a bit more fancy. By the end of this tutorial, you really will have some of the core algebraic tools in your toolkit!
Not every linear equation has exactly one solution. Some have no solutions and others might have an infinite number of solutions. This tutorial will give you the intuition on when these different situations arise.
You feel comfortable solving for an unknown. But life is all about stepping outside of your comfort zone--it's the only way you can grow! This tutorial takes solving equations to another level by making things a little more abstract. You will now solve for a variable, but it will be in terms of other variables. Don't worry, we think you'll find it quite therapeutic once you get the hang of it.
You know that converting a fraction into a decimal can sometimes result in a repeating decimal. For example: 2/3 = 0.666666..., and 1/7 = 0.142857142857...
But how do you convert a repeating decimal into a fraction? As we'll see in this tutorial, a little bit of algebra magic can do the trick!
In 72 years, Sal will be 3 times as old as he is today (although he might not be... um... capable of doing much). How old is Sal today?
These classic questions have plagued philosophers through the ages. Actually, they haven't. But they have plagued algebra students! Even though few people ask questions like this in the real-world, these are strangely enjoyable problems.
You feel good about your rapidly developing equation-solving ability. Now you're ready to fully flex your brain.
In this tutorial, we'll explore equations that don't look so simple at first, but that, with a bit of skill, we can turn into equations that don't cause any stress! Have fun!
When solving equations, there is a natural hunger to figure out what an unknown is equal to. This is especially the case if we want to evaluate an expression that the unknown is part of. This tutorial exposes us to a class of solvable problems that challenges this hunger and forces us to be the thinking human beings that we are!
In case you're curious, these types of problems are known to show up on standardized exams to see if you are really a thinking human (as opposed to a robot possum).
Some of Sal's oldest (and roughest) videos on algebra. Great tutorial if you want to see what Khan Academy was like around 2006. You might also like it if you feel like Sal has lost his magic now that he doesn't use the cheapest possible equipment to make the videos.
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Search Results
About this title: Here is a Spanish-language edition of a popular classroom aid, originally published in English by Barron's. In an easy, informal way, the author helps students detect fascinating patterns in math word problems, then solve them. Problems involve whole numbers, fractions, percentages, ratios and proportions, equations, geometry, and algebra. Touching on fundamentals of statistics and probabilities, students discover how math can often predict future outcomes and events. A final chapter instructs students on seeking out
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The aim of this paper is to analyse different types of representations used by grade 7 and 8 students in solving a word problem. The problem was offered to finalists of a mathematics contest held outside the classroom. The analysis of the products of different students show the key role of the representations used to solve the problem and how they are critical in the development of a sustainable process for informal learning of formal methods of solving systems of linear equations.
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This mobile computing platform for web and handheld devices performs matrix operations like multiplication/inverses; plots graphs of functions and operations on polynomials; manipulates complex numbers; and evaluates other math, science, and engineering calculations, such as Fast Fourier Transforms (FFT). Also available from
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In view of the pervasive roles that quantitative analysis plays throughout our society, a basic familiarity with the disciplines of mathematics has become an integral part of a liberal arts education. As a college for women, Mills recognizes the importance of encouraging women to study mathematics, and of providing them with the high-quality instruction they need to succeed in these disciplines. Encouraging mathematical literacy is part of the College's continued effort to increase the analytical competence of its women graduates.
Mathematics is an excellent field both for lifetime intellectual interest and for career preparation. Women are becoming increasingly prominent in the field. Recent presidents of both the American Mathematical Society and the Mathematical Association of America have been women. Mathematics also serves as an excellent basis for business, finance, engineering, sciences, teaching, actuarial work, and fields that need highly developed analytical skills, such as law.
Small, interactively taught classes provide students with an ideal environment for learning mathematics. The cross-registration program with UC Berkeley enables outstanding students to take advantage of a wide range of mathematics courses not usually available at a small college.
Note: The basic calculus sequence (MATH 047–MATH 048) begins in the fall. Students who need additional preparation before taking calculus should enroll in Pre-Calculus (MATH 003) along with a workshop (MATH 003L) in the fall; this course is prerequisite for, and leads directly into, first-semester Calculus I (MATH 047) in the spring. To determine which basic sequence is appropriate, the student should take the self-placement quizzes offered by the department and consult with mathematics advisors. Students who plan to do further work in mathematics, science, or engineering are advised to continue the calculus sequence by taking Linear Algebra (MATH 050) and Multivariable Calculus (MATH 049).
Before declaring a major in mathematics, a student must have completed Calculus I (MATH 047), Calculus II (MATH 048), and Linear Algebra (MATH 050). The grade in each of these courses should be at least a B-. Some exceptions may be allowed upon the recommendation of the department. Students required to declare a major before completing these courses may provisionally declare the mathematics major. The provisional declaration will be revoked if the student does not earn at least a B- in MATH 047, MATH 048, and MATH 050. Proficiency in basic logical and problem-solving skills, as determined by the instructor, is required for enrollment in advanced courses.
Mills also offers an accelerated degree program: BA/MA in mathematics that enables women to obtain both a BA and an MA in mathematics in five years.
With Mills unique Bachelor's to Master's Accelerated Degree Programs you can earn two academic degrees in five years—increasing your career options after college. Click a link below for courses you can be taking now to prepare for your Bachelor's to Master's Accelerated Degree. Education/Teaching Credential Computer Science Mathematics
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Free Delivery Worldwide : Worksheets for Classroom or Lab Practice for Integrated Arithmetic and Basic Algebra : Paperback : Pearson Education (US) : 9780321759245 : 0321759249 : 06 Jan 2012 : Worksheets for Classroom or Lab Practice offer extra practice exercises for every section of the text, with ample space for students to show their work. These lab- and classroom- friendly workbooks also list the learning objectives and key vocabulary terms for every text section, along with vocabulary practice problems and new â Getting Readyâ exercises to correspond to those in the text...
Free Delivery Worldwide : Worksheets with the Math Coach for Basic College Mathematics : Paperback : Pearson Education (US) : 9780321748522 : 0321748522 : 16 Jan 2012 : These worksheets provide extra practice exercise for every section of the text with ample space for students to show their work on the practice exercises and Math Coach problems.
p>This Saxon Math Homeschool 7/6 Tests and Worksheets book is part of the Saxon Math 7/6 curriculum for 6th grade students, and provides supplemental facts practice tests for each lesson, as well as 23 cumulative tests that cover every 5-10 lessons. The included activity sheets are designed to be used with the activities given in the (sold- separately) student worktext. A testing schedule and five optional, reproducible, recording forms are also provided; three forms allow students to record their work on the daily lessons, mixed practice exercises, and tests, while the remaining two forms help teachers track and analyze student performance. Solutions to all problems are in the (sold- separately) Solutions Manual. 241 newsprint-like, perforated pages, softcover. Three-hole-punched ...
Less
By Debra Schwiesow of Creighton University. Perfect for homework or small group assignments, these worksheets help students apply their knowledge and experience to their work as parents, counselors, caretakers, and teachers. Easy, hands-on activities guide students to observe, record, and analyze the behavior of children they encounter in the real world; also, the worksheets correspond with the Lessons in Observation videos on the premium website where applicable. One individual and one small group activity
This pack of WriteShop Primary Book B Activity Set Worksheets is designed to accompany the sold-separately WriteShop Primary Teacher's Guide BBuy Early Phonics 4 Worksheets (c,s,a,t,m,n) by Ian Mitch and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones!
This pack of WriteShop Primary Book A Activity Set Worksheets is designed to accompany the sold-separately WriteShop Primary Teacher's Guide A
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Quantitative Reasoning 1 and 2 (BBA only)
Course Description
(as of March 1, 2012)
Quantitative Reasoning 1: Introductory Quantitative Reasoning
This course reviews the fundamentals of elementary and intermediate algebra with applications to business and social science. Topics include: using percents, reading and constructing graphs, Venn diagrams, developing quantitative literacy skills, organizing and analyzing data, counting techniques, and elementary probability. Students are also exposed to using technology as graphical and computational aids to solving problems.
Quantitative Reasoning 2: Advanced Quantitative Reasoning
A continuation and extension of topics and themes developed in the introductory Quantitative Reasoning course, with special emphasis on data visualization techniques. Probability theory and statistical analysis methods are introduced. This course prepares students for higher‐level courses in Design and Management, including Financial Management and Business Models and Planning.
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...Thomas and received an A in the course. Linear Algebra is the study of matrices and their properties. The applications for linear algebra are far reaching whether you want to continue studying advanced algebra or computer science
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Numerical Computing in C#
In this lesson I will show how to numerically solve algebraic and ordinary differential equations, and perform numerical integration with Simpson method. I will start with the solution of algebraic equations. The secant method is one of the simplest methods for solving algebraic equations. It is usually used as a part of a larger algorithm to improve convergence. As in any numerical algorithm, we need to check that the method is converging to a given precision in a certain number of steps. This is a precaution to avoid an infinite loop.
Our second example is a Simpson integration algorithm. The Simpson algorithm is more precise the naive integration algorithm I have used there. The basic idea of the Simpson algorithm is to sample the integrand in a number of points to get a better estimate of its variations in a given interval.
Finally, let me show a simple code for solving first order ordinary differential equations. The code uses a Runge-Kutta method. The simplest method to solve ODE is to do a Taylor expansion, which is called Euler's method. Euler's method approximates the solution with the series of consecutive secants. The error in Euler's method is O(h) on every step of size h. The Runge-Kutta method has an error O(h^4) Runge-Kutta methods with a variable step size are often used in practice since they converge faster than fixed size methods
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Participants selected instances in which physics can be used to illustrate and explain mathematics. They developed brief lessons for use at the high school level which use equipment common to most high school science based on these applications of physics to mathematics.
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Key Biscayne Algebra lesson builds on what was learned before. If you don't understand one of the foundational steps, you will get lost later on. Kelvin uses multiple techniques to help students understand the basics of Algebra
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Spring 2010 MTH 44 Mathematical Applications (3 units)
Section 4473 Class Begins January 25, 2010
Course Description: This course will cover the topics of percents, US/Metric units of measurement, Geometry, and the arithmetic of signed numbers, as well as basic introductions to Statistics and Algebra. Students will: learn how to convert between fractions, decimals and percents; explore formulas for area and perimeter of geometric shapes; analyze statistical graphs and measurements; and solve basic algebraic equations. Application problems will be presented and solved in each of the main topic areas with an emphasis that prepares students for Elementary Algebra 9 to 12
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Product Details
Stock #
14661
ISBN #
Published
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Product Description
"On the Foundations of Mathematics." In this chapter originally published in NCTM's First Yearbook (1926), Eliakim Hastings Moore, the president of the American Mathematical Society (AMS), proposes a series of reforms related to mathematics learning and teaching$9.00
Customers Who Bought This Also Bought...
Mathematics curriculum remains a central issue in efforts to improve mathematics learning opportunities for students. NCTM's 72nd Yearbook takes a look at the advancements that have been made as well as the challenges that remain.
NCTM's Sixty-seventh Yearbook reports on the impact of technology, furnishes a rich context in which to observe the use of technology to help students better understand mathematics, and gives us all a glimpse of what the future might hold.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
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Calculus: Concepts and Contexts
Presents a streamlined approach to teaching calculus that focus on major concepts and supports those with precise definitions, patient explanations, ...Show synopsisPresents a streamlined approach to teaching calculus that focus on major concepts and supports those with precise definitions, patient explanations, and graded problemsFair. Noticeable wear, but still very usable. Interior is free...Fair. Noticeable wear, but still very usable. Interior is free from markings. Minor damp-staining on a few
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New and Published Books – Page 4
Applications, Models, and Computing
In the traditional curriculum, students rarely study nonlinear differential equations and nonlinear systems due to the difficulty or impossibility of computing explicit solutions manually. Although the theory associated with nonlinear systems is advanced, generating a numerical solution with a...
Based on the author's well-established courses, Group Theory for the Standard Model of Particle Physics and Beyond explores the use of symmetries through descriptions of the techniques of Lie groups and Lie algebras. The text develops the models, theoretical framework, and mathematical tools to...
Through numerous illustrative examples and comments, Applied Functional Analysis, Second Edition demonstrates the rigor of logic and systematic, mathematical thinking. It presents the mathematical foundations that lead to classical results in functional analysis. More specifically, the text...
Proofs, Structures and Applications, Third Edition
Taking an approach to the subject that is suitable for a broad readership, Discrete Mathematics: Proofs, Structures, and Applications, Third Edition provides a rigorous yet accessible exposition of discrete mathematics, including the core mathematical foundation of computer science. The approach is...
Master the tools of MATLAB through hands-on examplesShows How to Solve Math Problems Using MATLAB
The mathematical software MATLAB® integrates computation, visualization, and programming to produce a powerful tool for a number of different tasks in mathematics. Focusing on the MATLAB toolboxesBrings Readers Up to Speed in This Important and Rapidly Growing Area
Supported by many examples in mathematics, physics, economics, engineering, and other disciplines, Essentials of Topology with Applications provides a clear, insightful, and thorough introduction to the basics of modern topology....
Partial Differential Equations, Fourier Series, and Special Functions
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Modern Tools to Perform Numerical DifferentiationThe original direct differential quadrature (DQ) method has been known to fail for problems with strong nonlinearity and material discontinuity as well as for problems involving singularity, irregularity, and multiple scales. But now researchers in...
Through several case study problems from industrial and scientific research laboratory applications, Mathematical and Experimental Modeling of Physical and Biological Processes provides students with a fundamental understanding of how mathematics is applied to problems in science and engineering....
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Geometric Algebra for Computer Graphics
Synopses & Reviews
Publisher Comments:
Geometric algebra (a Clifford Algebra) has been applied to different branches of physics for a long time but is now being adopted by the computer graphics community and is providing exciting new ways of solving 3D geometric problems.John Vince (author of numerous books including 'Geometry for Computer Graphics' and 'Vector Analysis for Computer Graphics') has tackled this complex subject in his usual inimitable style, and provided an accessible and very readable introduction.As well as putting geometric algebra into its historical context, John tackles complex numbers and quaternions; the nature of wedge product and geometric product; reflections and rotations (showing how geometric algebra can offer a powerful way of describing orientations of objects and virtual cameras); and how to implement lines, planes, volumes and intersections. Introductory chapters also look at algebraic axioms, vector algebra and geometric conventions and the book closes with a chapter on how the algebra is applied to computer graphics
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Find a South River Calculus took discrete mathematics, which covered logic and finite differences. In real analysis and topology, there were lots of proofs that used manipulation of quantifiers (such as epsilon-delta definitions). I took an advanced course in probability, which covered topics such as two-dimensional prob
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Ever since the launch of Sputnik, Westerners knew that Russians did science differently. Here is evidence of how they do geometry.
As a quick look at the detailed MathSciNet review MR2264644 (2007i:53001) (or simply the table of contents ) will tell the reader how much and what kind of math it actually contains, I will here choose to focus on the philosophy behind this extraordinary book.
Here are the basic foundational assumptions of this book, (I slightly paraphrase from the authors' own preface):
Geometry is a bridge between pure mathematics and the natural sciences, especially physics, and it is important and useful to relate to the beginning student how these are intertwined.
The study of global properties of geometric objects leads to far-reaching developments in topology.
The geometric theory of Hamiltonian systems leads to the development of symplectic and Poisson geometry.
Geometry of complex and algebraic manifolds unifies Riemannian geometry with complex analysis, algebra and number theory.
Thus the authors believe there is a need for an introductory text that brings the beginning student of geometry up to speed on quite a few different fronts. There are many beginning texts out there which focus on one of these many aspects, but Novikov and Taimanov are up for the bigger challenges: To write a comprehensive text that will take a beginner all the way to the frontlines, to provide the student a smooth passage through various fundamental concepts and perspectives, and to integrate seamlessly many physical applications throughout the text. This is not a text only in complex geometry, symplectic geometry, or Poisson geometry; it is a text which incorporates all of differential geometry.
A question begs an answer: Who should be the lucky beginner student in this differential geometry course of a lifetime? The language used by the authors is simple, fluent and very understandable, even after translation, and the mathematical prerequisites seem at a first glance to be minimal. However, the calculus background required includes the implicit function theorem, there is substantial use of basic linear algebra, and unless one is willing to skip the chapters on complex geometry, one does need to have some exposure to functions of a complex variable. To me, therefore, it makes perfect sense that this book is published in the Graduate Studies series of the AMS. Very motivated undergraduates, especially those who are starting graduate school in mathematics soon may of course look in, but it must be clear to anyone that the book is going to require some mathematical muscle.
To me the ideal audience for this book is a beginning mathematics graduate student who is hoping to learn geometry, but who also is interested in learning some (mathematical) physics. Knowing something about physics at the beginning is a plus; in my opinion, Arnold's Mathematical Methods of Classical Mechanics (Springer GTM) would provide a more than sufficient preparation for this book, both in substance and in tone. However, the lack of any previous exposure to physics can easily be compensated by the desire to learn about it. It is important to note here that the integration of physics into the main flow of the text is quite smooth, and this adds so much to the text that reading around this material, even though it would still be possible, would be a real waste and a pity.
Between the two of them, the authors have already written many textbooks in geometry before, some of which have been translated into English (see for instance [DFN1–DFN3] and [NF]), but this may be their ultimate masterpiece. Highly recommended for anyone interested in learning geometry, except possibly those who may have physics anxiety (which must exist if math anxiety does).
Gizem Karaali is assistant professor of Mathematics at Pomona College. She believes she may have physics infatuation, a disorder which should be on the opposite side of the spectrum from physics anxiety but possibly could be just as harmful.
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My recent explorations of Calculus and its teaching revealed the following observations.
1) Calculus is the first course where students consider a new kind of problems: infinite problems. I call a problem infinite if it does not allow a finite solution. Infinite problems (as opposed to finite problems) require a different thinking process.
2) A typical way of teaching Calculus avoids the thinking needed for infinite problems in two simple steps. First, students are given a list of standard infinite problems with the answers to memorize. Second, any other problem of the course is reduced to one of the standard problems in a finite sequence of steps.
3) The concept of directed set plays the central role in explorations of problems involving limits (and therefore in any Calculus problem).
4) Mathematica online would be extremely useful for teaching and learning Calculus. It would allow to focus on the new thinking while any finite computational process could be delegated to Mathematica.
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Find a DickinsonThe course typically provides a survey of mathematical techniques used in the working world. The purpose of the couse is to give students valuable, practical experience at organizing information and then analyzing it. As a Texas-certified math teacher, I have found many of my students like to uGet the Help you need. I can explain the material in ways that make sense.I hold a B.E.E from the University of Minnesota, a 5 year, 167 semester hour curriculum. I have been active in Engineering for 50 years
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math review for the GRE, GMAT, and SAT. This math refresher workbook is designed to clearly and concisely state the basic math rules and principles of arithmetic, algebra, and geometry which a student needs to master. This is accomplished through a series of carefully sequenced practice sets designed to build a student's math skills step-by-step. The workbook emphasizes basic concepts and problem solving skills. Strategies for specific question types on the GRE, GMAT, and SAT are the focus of the Lighthouse Review self study programs.
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About the Author
The workbook was designed by graduates from University of Texas at Austin.
Most Helpful Customer Reviews
I'd recommend this book for anyone who needs a good math refresher. This book is in an easy to use format, starting with easy material and progressing to the more difficult from there. Each section contains a brief (but good) explanation followed by a fair number of practice questions. Don't look for any long-winded explanations in this book, it is a refresher, just as they say. Also, it is printed on good stock, so it holds up nicely. I took mine cross country with no problems, unlike the review books printed on newprint which didn't fare as well. The biggest problem I encountered with this book is that it does not contain any test specific questions such as the GMAT's data sufficiency questions or the quantitative comparisons on the GRE. If you are looking for practice questions, this is not the book for you. If you want a math refresher, this is the book for you. If you are particularly strong in math, you probably won't find this book very helpful, unless you want to do basic math problems, out of test format. I used this book as a math refresher, then went on to one of the other review books for specific question practice. That combination seemed to work for me.
I am horrible at math but needed to brush up for my second shot at the GMAT. This book did an exceptional job of reintroducing math concepts I hadn't seen in years. There were plenty of practice problems with simple and practical explanations. The first time I took the GMAT, I used Kaplan's math prep book. Although it gave me practice answering GMAT type questions, the reality was I really needed to first was brush up on my math fundamentals. This book covers arithmatic, algebra, and geometry in 37 lessons. A great buy for anyone taking an ETS test!
I was faced with the frightening prospect of taking GREs again, some thirty years after the first time I took them, since I'm now applying to grad school for a second Master's degree. I expected that I'd have no problems with the written portion of it, but frankly the math portion terrified me.
This book provided succinct and clear explanations of the basic principles I needed to do well on the math section. I am sure I would have scored in the 10th percentile without it; with it, I got a very respectable (for a humanities major) 650.
I have since passed it along to my daughter, who was preparing for her SATs. Her comments mirrored mine. I look forward to seeing how her scores improved over her first administration of the test.
It has been 20 years since i got my undergrad degree. I needed the GREs to get into grad school and was told I needed a 550 on each portion. I was most worried about the math section so I bought this book. Wow. I forgot how much I had forgotten. After cruising through this book, I destroyed the math section on the GREs! There were some problems that were identical to the ones I had practiced on. My only regret...that I did not buy the similar book on the Comprehensive/English/Reading portion. While I ended up doing ok on that section, the results of this book really made a huge difference on the math portion. A great investment!
I'm a 55-year old female that has been in the banking business for 30 years. I graduated from college with a degree in math and science a long time ago. I have always wanted to get a graduate degree, but have been afraid that I would have forgotten so much. The Ultimate Math Refresher took that fear away as it was a great, fun and easy refresher for me. The language is clear and concise, and the practice sets are well designed to refresh your math skills and concepts. I couldn't have found a better step-by-step method to get up to speed for the GMAT--even though I had a Math Degree and use to teach math many years ago!
This is an excellent math refresher book! I'm a returning student and I wanted a quick and easy way to not only refresh my long-forgotten math skills but to really learn them this time around. The book covers basic arithmetic, algebra and geometry concepts which are presented in a very readable and understandable format. The sample problems and answers were also very easy to follow. I especially liked the one page summaries of the mathematic symbols and rules (I had forgotten many of these - the summary was a great resource).
I have a degree in Applied Math so I know a thing or two about the subject in this book. After flipping throught the pages, I can say that this book is a good refresher for those who took math in high school and at the beginning of their college years and now is trying to take GMAT or GRE. This book will not give you an extensive knowledge of math, but it does keep its focus on relevat topics needed to score high on GRE/GMAT.
I hated the idea of tackling math again until I found this little Math Refresher book. The book gave me the easy-to-understand brush-up that I needed and it didn't hurt a bit! For anyone who hasn't tackled math in awhile and who needs an inexpensive jump-start, this book is a great buy.
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Information for New Engineering Students
All new CU-Boulder engineering students should take the ALEKS math assessment, which assesses your pre-calculus algebraic and trigonometric preparedness. A minimum ALEKS score is required to be eligible to enroll in selected mathematics, science, economics, and business courses. For more information about the ALEKS math assessment and to take the test, please visit the ALEKS website.
If you are a new engineering first-year freshman student, you will likely be pre-enrolled in an Applied Math (APPM) course based on your ALEKS score (see courses and minimum scores for enrollment at the ALEKS website). If you have any questions about your course enrollment, you should consult with your academic advisor.
APPM 1235 Pre-Calculus for Engineers - A four-credit course developing techniques and concepts prerequisite to engineering calculus through the study of trigonometric, exponential, logarithmic, polynomial, and other functions. Requires an ALEKS score of 61% or higher.
APPM 1340/1345 - Calculus 1A with Algebra (APPM 1340) and Calculus 1B with Algebra (APPM 1345) are each three-credit courses that together cover the material of Calculus 1 for engineering-bound students, spread over two semesters to allow for a slower-paced development of Calculus 1 foundation, knowledge and skills. The APPM 1340/1345 two-course sequence includes a four-week review of algebra and trigonometry, although not as much as in APPM 1235 (Pre-Calculus for Engineers). Thus, the APPM 1340/1345 sequence is appropriate for students with high proficiency in trigonometry who want to take calculus at a slower pace.
NOTE: The APPM 1340/1345 course sequence is equivalent to APPM 1350 (Calculus 1) plus 2 credits of algebra/trig review. Students must complete both APPM 1340 and APPM 1345 to receive credit for Calculus 1. Enrollment in APPM 1340 requires an ALEKS score of 61% or higher.
NOTE FOR EITHER OPTION ABOVE: A strong math foundation is crucial for your success in engineering; this can be best achieved by solid preparation (i.e., enrolling in either the precalculus course or the two-semester Calculus 1 course). However, this means you may need to enroll in Calculus 2 the summer after your freshman year in order to stay on track for a four-year engineering degree program. Otherwise, your time-to-degree may require more than four years.
APPM 1350 Calculus 1 for Engineers - A four-credit course in analytical geometry and calculus including limits, rates of change of functions, derivatives and integrals of algebraic and transcendental functions, applications of differentiations and integration. Requires an ALEKS score of 76% or higher.
Concurrent Mathematics Work Groups – Students enrolled in APPM 1350 (Calculus 1 for Engineers) may consider concurrently enrolling in the accompanying COEN 1350 (Calculus 1 Work Group), which is a one-credit, pass/fail course which emphasizes collaborative learning techniques. Similarly, students enrolled in APPM 1235 (Pre-Calculus for Engineers) may consider concurrently enrolling in the accompanying COEN 1236 (Pre-Calculus Work Group). At each weekly class period of Work Group, you will work in groups of 3-4 students on problems that are different than your regular class homework. Before moving on to a new problem, all students in the group must understand, and be able to explain, the solution.
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Cryptography: Math and Codes
Course Description
Description
Cryptography: Math and Codes introduces students to the exciting practice of making and breaking secret codes. This popular course is designed for for mathematical enrichment for students in grades 6-9.
Students begin with simple Caesar Ciphers, learning to encrypt and decrypt messages as well as the history behind the cipher. They will move through history and more advanced mathematical concepts to learn substitution ciphers, Vigenère ciphers, and multiplicative and affine ciphers. Students will need to put all their newly acquired knowledge to the test by finishing with public key cryptography and the modern day RSA cryptosystem. This course intersects the disciplines of mathematics, computer science, and electrical engineering. Applications of cryptography include ATM cards, computer passwords, and electronic commerce.
Throughout the course, students will have the opportunity to encrypt and decrypt messages, invent their own ciphers, discuss relevant historical events and literature, and learn some mathematical concepts that are often not seen until college!
Mathematical Topics covered Include:
positive and negative numbers
decimals and percents
data analysis and probability
prime numbers and factorization
modular arithmetic
inverses
exponentiation
Assignments are based on a text that is purchased separately by the student.
Materials Needed
There is a textbook purchase required for this course:
The Cryptoclub: Using Mathematics to Make and Break Secret Codes.Janet Beissinger and Vera Pless (AK Peters, 2006)
Public Key Cryptography
Demo
Demo
In addition to the textbook, there are interactive websites and videos in the course.
In a Caesar Cipher, the alphabet is shifted a certain number of places and each letter of a message is replaced by the corresponding shifted letter. With this tool, you can shift the alphabet using the cipher wheel then encrypt your own message!
This course uses an online classroom for individual or group discussions with the instructor. The classroom works on standard computers with the Adobe Flash plugin, and also tablets or handhelds that support the Adobe Connect Mobile app.
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Single Pack of 1 or more items
Click on the Google Preview image above to read some pages of this book!
Maths Quest 8 for the Australian Curriculum provides students with essential mathematical skills and knowledge through the content strands of Number and Algebra, Measurement and Geometry, and Statistics and Probability. The Curriculum focuses on students becoming proficient in mathematical understanding, fluency, reasoning and problem solving. Maths Quest 8 for the Australian Curriculum is specifically written and designed to meet the requirements and aspirations of the Australian Mathematics Curriculum.
The student textbook contains the following features:
clear and engaging design
judicious use of ICT resources
a numeracy chapter
two chapters on problem solving
Individual pathways activities for every exercise
a Hungry brain class activity for each chapter
two new ProjectsPLUS activities
interactivities
eLessons
references to the content and proficiency strands of the new Australian Mathematics Curriculum What is eBookPLUS?
This title features eBookPLUS: an electronic version of the textbook and a complementary set of targeted digital resources. These flexible and engaging ICT activities are available to you online at the JacarandaPLUS website (
ISBN: 9781742463063 ISBN-10: 1742463061 Series: Maths Quest for Aust Curriculum Series Audience:
Primary / High School
Format:
Single Pack of 1 or more items
Language:
English
Published: 27th June 2013
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Maple includes more than 5,000 mathematical functions and a variety of options for high-performance computing. Maple 16 makes available new techniques and tools in the Clickable Math collection, such as "drag-to-solve" and "smart popups." Drag-to-solve allows users to tackle problems by dragging terms to the desired location and save each step in the calculation. Smart popups let users adjust a single part of a highlighted calculation and preview the answer. It also notifies the user which mathematical identities can be used, if a subexpression can be factored, and shows the plot.
"Teaching Calculus with Maple: A Complete Kit" was developed over five years by Jack Weiner, professor emeritus of mathematics at University of Guelph in Ontario, Canada. The kit has been field- tested in classrooms with 15 to 600 students.
Notes for the teachers with annotation and demonstrations using Maple;
Automatically graded homework assignments for Maple T.A. users, including feedback and hints, as well as assignments for users who don't have access to Maple T.A.; and
Extra homework assignments from Stewart's Calculus.
"Using Maple as the foundation of each lecture has resulted in fun, extremely interactive classes. We're reaching more students now than ever before," said Weiner. "In addition, the use of Maple T.A. to reinforce learning by providing lots of practice and immediate feedback gives students more confidence, which ultimately leads to greater success."
"Teaching Calculus with Maple: A Complete Kit" is free for schools that use Maple or Maple T.A
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Videos Will Help Students "Ace" Math
02/01/97
Ace-Math, an award-winning video tutorial series, is suited for students trying to grasp fundamental mathematical concepts, parents who want to help their child with their homework, or people who need to brush up on math skills for a specialized license or test.
There are nine separate series, each with many individual videos: Basic Mathematical Skills, Pre-Algebra, Algebra I, Algebra II, Advanced Algebra, Trigonometry, Calculus, Geometry, and Probability and Statistics. Each series except Algebra I consists of 30-minute videotapes explaining various concepts. Algebra I has 16 hour-long videos.
For only $29.95, Ace-Math purchasers get a 30-minute tape with the right to make two back-up copies. This lets educators keep the tape in the learning center and let students check out a copy to take home -- with the added security of another back-up copy!
These innovative tapes have been purchased by institutions such as NASA, the U.S. Coast Guard and IBM, and are in use at institutions such as the Los Angeles Public Library and New York Public Library.Video Resources Software, Miami, FL, (888) ACE-MATH
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Abstract Algebra II –
mth402
(3 credits)
This is the second course in a two-part course sequence presenting students with the applications of abstract algebraic theories. Students will investigate rings, fields, and the basic theorems of Galois theory.
Galois Theory: Overview
Detect the relationship between powers of prime numbers and the order of a finite field.
Galois Theory and Geometric Constructions
Determine the correspondence between the set of all subgroups of the Galois group and the set of all subfields of the splitting field.
Identify properties of separable polynomials and normal extensions.
Apply Galois theory in fields and polynomials.
Assess the relationship between solvability of polynomials by radicals and properties of Galois groups
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Trigonometry
9780495108351
ISBN:
0495108359
Edition: 6 Pub Date: 2007 Publisher: Thomson Learning
Summary: Gain a solid understanding of the principles of trigonometry and how these concepts apply to real life with McKeague/Turner's TRIGONOMETRY, Sixth Edition. This books proven approach presents contemporary concepts in brief, manageable sections using current, detailed examples and interesting applications. Captivating illustrations drawn from Lance Armstrongs cycling success, the Ferris wheel, and even the human cannon...ball show trigonometry in action. Unique Historical Vignettes offer a fascinating glimpse at how many of the central ideas in trigonometry began.
McKeague, Charles P. is the author of Trigonometry, published 2007 under ISBN 9780495108351 and 0495108359. Two hundred twenty one Trigonometry textbooks are available for sale on ValoreBooks.com, sixty eight used from the cheapest price of $24.29, or buy new starting at $116Bellevue
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Paperback
Click on the Google Preview image above to read some pages of this book!
Take it step-by-step for algebra success!
The quickest route to learning a subject is through a solid grounding in the basics. So what you won't find in Easy Algebra Step-by-Step is a lot of endless drills. Instead, you get a clear explanation that breaks down complex concepts into easy-to-understand steps, followed by highly focused exercises that are linked to core skills--enabling learners to grasp when and how to apply those techniques.
This book features:
Large step-by-step charts breaking down each step within a process and showing clear connections between topics and annotations to clarify difficulties
Stay-in-step panels show how to cope with variations to the core steps
Sandra Luna McCune, Ph.D. is Regents Professor currently teaching as a mathematics specialist in the Department of Elementary Education at Stephen F. Austin State University. She is also an in-demand statistical/mathematical consultant.
William D. Clark, Ph.D. has been a professor of mathematics at Stephen F. Austin State University for more than 30 years.
just like back in school only actually getting to take it in at a pace that can sink in and find useful. If had to help a child out with their homework this is a fantastic book for a parent. Enabling to remember all the steps so can explain.
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Four Pillars of Geometry - 05 edition
Summary: This new textbook demonstrates that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Euclid-style construction and axiomatics seem the best way to start, but linear algebra smooths the later stages by replacing some tortuous arguments by simple calculations. And how can one avoid projective geometry? It not only explains why objects look the way they do; it also explains why...show more geometry is entangled with algebra. Finally, one needs to know that there is not one geometry, but many, and transformation groups are the best way to distinguish between them. In this book, two chapters are devoted to each approach, the first being concrete and introductory, while the second is more abstract.
Geometry, of all subjects, should be about taking different viewpoints, and geometry is unique among mathematical disciplines in its ability to look different from different angles. Some students prefer to visualize, while others prefer to reason or to calculate. Geometry has something for everyone, and students will find themselves building on their strengths at times, and working to overcome weaknesses at other times. This book will be suitable for a second course in geometry and contains more than 100 figures and a large selection of exercises in each chapter. ...show less
Edition/Copyright:05 Cover: Hardcover Publisher:Springer-Verlag New York Published: 08/09/2005 International: No
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More About
This Textbook
Overview
What does your math course have to do with the latest TV shows or Hollywood movies? Plenty–if you're using the right text. Mathematical Ideas, Twelfth Edition brings the best of Hollywood into the classroom through descriptions of video clips from popular cinema and television. Well-known author John Hornsby's innovative approach is enhanced with great care in this revision, and refined to serve the needs of you and your instructor. Streamlined and updated, it offers a modernized design, new bubble pointers for Example annotations, and much more. It retains the consistent features, friendly writing style, clear examples, and exercise sets for which this text is known.
Editorial Reviews
Booknews
A textbook designed with a variety of students in mind and suited for several types of courses, including mathematics for liberal arts students, survey courses in mathematics, and mathematics for prospective and in-service elementary and middle-school teachers. Some 80% of the exercises are new to this edition, which also sports extensive use of color and changes in format to create a fresh look. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Related Subjects
Meet the Author
Charles Miller has taught at America River College for many years.
Vern Heeren received his bachelor's degree from Occidental College and his master's degree from the University of California, Davis, both in mathematics. He is a retired professor of mathematics from American River College where he was active in all aspects of mathematics education and curriculum development for thirty-eight years. Teaming with Charles D. Miller in 1969 to write Mathematical Ideas, the pair later collaborated on Mathematics: An Everyday Experience; John Hornsby joined as co-author of Mathematical Ideas on the later six editions. Vern enjoys the support of his wife, three sons, three daughters in-law, and eight grandchildren.
John Hornsby: When a young John Hornsby enrolled in Lousiana State University, he was uncertain whether he wanted to study mathematics education or journalism. Ultimately, he decided to become a teacher. After twenty five years in high school and university classrooms, each of his goals has been realized. His passion for teaching and mathematics manifests itself in his dedicated work with students and teachers, while his penchant for writing has, for twenty five years, been exercised in the writing of mathematics textbooks. Devotion to his family (wife Gwen and sons Chris, Jack, and Josh), numismatics (the study of coins) and record collecting keep him busy when he is not involved in teaching or writing. He is also an avid fan of baseball and music of the 1960's. Instructors, students, and the 'general public' are raving about his recent Math Goes to Hollywood presentations
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Linear algebra was one of the Mathematics"Since vectors, as n-tuples, are ordered lists of n components, it is possible to summarize and manipulate data efficiently in this framework. For example, in economics, one can create and use, say, 8-dimensional vectors or 8-tuples to represent the Gross National Product of 8 countries. One can decide to display the GNP of 8 countries for a particular year, where the countries' order is specified, for example, (United States, United Kingdom, France, Germany, Spain, India, Japan, Australia), by using a vector (v1, v2, v3, v4, v5, v6, v7, v8) where each country's GNP is in its respective position."
is misleading and incorrect. There is a big difference between a tuple and a vector. The tuple of GNP values of 8 countries does not behave like a vector. For example, how would it behave under a linear transformation? What are its basis vectors?
Consider the space C([0,1],R) of all continuous real-valued functions over the closed interval [0,1], this is a vector space, since linear combinations of continuous functions are continuous. The vectors in C([0,1],R) do not have "elements" in the same way n-tuples of real or complex numbers do. Also, the fact that polynomials of arbitrarily high degree exist in the space means that it is not finite-dimensional.
However, one way to unify the two ideas is to think about n-tuples as functions from the finite set {1,2,...,n} to R. Now finite-dimensional vector space Rn can be seen as a space of functions whose domain is finite, whereas the infinite-dimensional vector space C([0,1],R) is a space of functions whose domain is infinite.
"However, it has few, if any, applications in the natural sciences and the social sciences, and is rarely used except in esoteric mathematical disciplines."
This is just plain wrong. Linear algebra is used in both the natural and social sciences. Physics and Chemistry are obvious. Biology uses matrices and all that malarkey when looking at coupled ODEs. Social sciences use them in some stats work and in ODEs/PDEs. Anywho, the above statement is misleading and should be removed.--137.205.132.41 10:20, 16 January 2007 (UTC)
In computational number theory you sometimes get people doing linear algebra on matrices made out of integers modulo a prime. Often the prime is 2, but larger ones are also used.
My guess is the elements have to be from a ring or maybe a field. Anyway something with a group operation on the whole set, another group operation on the set except for identity of the first group, distributive law between the two group operations.
You can do linear algebra over any field. If you're working with rings, they're called modules. Modules share many of the properties of vector spaces, but certain important basic facts are no longer true (the term dimension doesn't make much sense anymore, as bases may not have the same cardinality.) --Seb
A vector space, as a purely abstract concept about which we prove theorems, is part of abstract algebra, and well integrated into this field. Some striking examples of this are the group of invertible linear maps or matrices,
This is truly a striking example :-)
Toby Bartels and I are going to correct this and I think we're also going to write about linear algebra over a rig (algebra) (this is not a typo!). -- Miguel
A similar request (this time from a non-mathematician). I plan to introduce a proof from linear algebra into the arbitrage pricing theory article. Firstly, I would like to tighten up the wording such that it is acceptable; secondly, I would like to link the argument to the appropriate linear algebra theorem. Hope that's do-able - and thanks if it is. Basically, this is how the derivation there usually goes (where the generic-vectors below have a financial meaning): "If the fact that (1) a vector is orthogonal to n-1 vectors, implies that (2) it is also orthogonal to an nth vector, then (3) this nth vector can be formed as a linear combination of the other n-1 vectors." Fintor 13:38, 23 October 2006 (UTC)
Quote from the article: "In 1843, William Rowan Hamilton (from whom the term vector stems) discovered the quaternions."
Huh? I didn't find the answer on a quick perusal of the William Rowan Hamilton article either. I didn't see it in quaternions either. It sounds like an interesting story, but what (or where) is the story? Spalding 18:25, Oct 4, 2004 (UTC)
The statement about definite and semi-definite matrices is not correct as stated. Matrices should be assumed to be symmetric. Moreover, this is slightly off-topic: it is rather part of bilinear algebra rather than linear algebra.
The statement ``A non-zero matrix A with n rows and n columns is non-singular if there exists a matrix B that satisfies AB = BA = I where I is the identity matrix is much more a definition than a theorem
In my opinion, the main non-trivial result of linear algebra says that the Dimension of a vector space is well defined: Theorem: If a vector space has two bases, then they have the same cardinality.
A matrix is orthogonal diagonalizable if and only if it is normal (please check and edit!!!) —Preceding unsigned comment added by Niv.sarig (talk • contribs) 23:06, 15 December 2009 (UTC)
Clearly, A matrix with only positive eigenvalues is not necessarily positive definite (also not semi for non-negative). These should be erased. (A counter example is M=[2 0;5 1] with eigenvalues 1,2>0 and vector v=[1;-1] but the inner product is v^tMv=-2<0.) —Preceding unsigned comment added by Niv.sarig (talk • contribs) 20:46, 2 October 2010 (UTC)
I assume the part about "systems of linear equations in finite dimensions" was intended to distinguish the subject of linear algebra from, say functional analysis. However, the distinction lies not in the number of dimensions, but in whether the linear structure is studied as a thing in itself (as opposed to being studied in the context of a topology). In other words, pure vector spaces are the province of linear algebra, while topological vector spaces are the province of functional analysis. Thus, even infinite-dimensional linear phenomena, if studied from a purely algebraic standpoint, are technically part of linear algebra.--Komponisto 21:36, 26 July 2006 (UTC)
I think it would be helpful if someone clarified the meaning of "over a field" from the first sentence of the fourth paragraph in the 'Elementary Introduction' section. The sentence reads as follows: 'A vector space is defined over a field, such as the field of real numbers or the field of complex numbers.' -- —The preceding unsigned comment was added by DrEricH (talk • contribs) .
That's a fair point. I reformulated it so that it does not use the phrase "over a field" anymore. -- Jitse Niesen (talk) 02:12, 16 August 2006 (UTC)
For small systems, ad hoc methods certainly suffice. Larger systems, however, require more systematic methods. The approach generally used today was beautifully explained 2,000 years ago in a Chinese text, the Nine Chapters on the Mathematical Art (Jiuzhang Suanshu, 九章算術).
Which is taken from Linear Algebra with Applications Third Edition by Otto Bretscher.
To me, it sounded a bit too similar to the original text.
Just something I noticed.
Thanks for your message. I agree that the similarity is too much to be coincidence. The first two paragraphs in the "History" section were added in a single edit, so they are both suspect. I thus deleted them. -- Jitse Niesen (talk) 12:05, 23 January 2007 (UTC)
A number of editors of linear algebra/vector space articles are uncomfortable about making statements which rely on the axiom of choice without mentioning it. To some extent, I share their unease (or to put it more light-heartedly, "You say every vector space has a basis? Great! Now give me a well-ordering of the real numbers - it might come in handy..."). On the other hand, these are articles about linear algebra, so it is a pity to constantly distract the reader with digressions into logic and set theory. So we have a choice (fortunately a finite choice!): do we mention choice or not? I know this has been discussed on a few talk pages before, but my recent edit of this article suggests a compromise: footnoting references to choice. I imagine the use of footnotes may polarize opinion, but it might be a sensible way forward in this case, so let me know what you think. And I'll make a few other similar edits to stimulate the discussion :) Geometry guy 21:42, 13 February 2007 (UTC)
(Hmmm... I'm quite proud of that split infinitive.) I've footnoted the choices in Dual space. One obvious question that arises is whether it would be better to have just one choice-related note with all the relevant caveats, or several. I'd be inclined to put them together to avoid repetition. Geometry guy 22:18, 13 February 2007 (UTC)
seems silly to avoid all together making statements that require the axiom of choice in linear algebra articles. however, so long as the topic belongs in linear algebra proper, probably good to discuss the finite dimensional case first if possible. it should be explicitly stated when the axiom of choice is needed. i agree that doing so via footnotes, as you've done in the dual space article, is a good idea. (even better if they are expanded a bit). Mct mht 15:17, 14 February 2007 (UTC)
I actually agree entirely, but wanted to invite opinion. (Also I have a slight preference for avoiding choice when it is not needed. For example, a lot of the claims about dual spaces do hold without choice for duals of vector spaces with bases. For another example, I prefer the statement "does not have a countable basis" to the statement "has an uncountable basis".) Anyway, I'm glad you like the footnotes idea, and would be happy for them to be expanded. Ultimately there might be a place for an article on "choice in linear algebra". Geometry guy 23:22, 14 February 2007 (UTC)
a seperate article that collects relevant linear algebraic results and delineate when the axiom of choice is needed and not needed looks like the best solution. let's hope someone will take that up. :-) Mct mht 02:52, 15 February 2007 (UTC)
I have to (regretfully) state that for a "top importance" article, this one is remarkably incoherent. Problems are manifold, but just for starters, there is the issue of consistency within the article itself and in wikipedia in general.
Moreover, since according to the article on abstract algebra, linear algebra is its proper part, it would seem circular to state that linear algebra is widely used in both abstract algebra and ... On the other hand, applications of linear algebra to differential equations are not even mentioned (and no, it's not covered by a reference to functional analysis).
Why is matrix theory not referenced at all? One would hope it's not because we cannot explain the difference between it and linear algebra!
The History section seems particularly weak. As the article on matrices discusses, they were introduced in ancient times and used throughout the Middle Ages. Of course, Gauss's work in the beginning of 19th century is very relevant for development of linear algebra, but so is, for example, Laplace's work before, and Cauchy's after, neither of which is mentioned. Arthur Cayley only introduced notation for determinants and abstract matrices, it's hardly proper to credit him with invention of linear algebra! In fact, there are [1], [2] two articles on history of linear algebra in MacTutor History of Mathematics Archive, which, while not complete, nonetheless make me think that it's better to scrap the current history section altogether as simplistic and factually wrong, and rewrite it anew.
The section Elementary introduction is a weird mixture, an ad hoc explanation of vector spaces (and as someone has already commented above, a tuple is not at all a representative object for linear algebra as a discipline), with matrices, determinants, and the general idea of linearity interspersed.
All but the very first Useful theorems deal with matrices, would it not be more natural to put them into the article on matrix theory?
And the list goes on, and on, and on. Arcfrk 15:01, 19 March 2007 (UTC)
Although the commentary here is a bit harsh (and some of these issues are easily fixed, for example by referring to applications of linear algebra in other areas of abstract algebra, for instance using representation theory), I do agree with the substance of the criticisms. This really is one of the most fundamental articles in pure mathematics, and I think we have a real opportunity here to expand and enhance it. Geometry guy 22:24, 19 March 2007 (UTC)
How about rewriting it? I mean, really. Readable articles for basic mathematics topics shouldn't be _too_ hard for us, should they? I tried turning the intro into grammatically correct english; as for content, however, I came here to learn and my linear algebra, history thereof, etc. is still weak. Please help! User:x14n 10th-ish Oct. 2007
I'm hopefully gonna spend some time in the next few days reworking some pieces of this article. As is, it really is a complete mess. A couple of thoughts that spring to mend:
-There should be a definition of vector space, some substantial mention of module theory and a couple of comments about why vector spaces and modules are different and why they are the same.
-The example given about the GNP of 8 countries is misleading in its triviality. Vectors are much more than just lists of numbers. Towards the end of explaining what linear algebra is and what a vector space really is, this article should have some well developed heuristic explanations of the concept of "linearity".
-I think the section that just lists important theorems should be trashed. It is completely unenlightening to just list off a bunch of results that all involve technical concepts, none of which have been defined.
Its late for me and these comments might be a little bit vague but please respond. Ill try to realize some of this stuff when ive had some sleep.Jrdodge 08:31, 11 November 2007 (UTC)
I was taught this exact statement in school, and it cost me time and effort when I started trying to work with matrices over rings other than the integers or the reals. I'd rather not see anyone else misled by this implicit assumption. 91.84.221.238 (talk) 02:24, 15 January 2008 (UTC)
From the structure of the article, I interpreted the assumption of the previous section (that scalars come from a field) as carrying over to the subsequent section ("Some useful theorems"). The next section ("Generalisations and related topics") discusses matrices over other algebraic objects. Myasuda (talk) 02:52, 15 January 2008 (UTC)
I tried to fix this in a way that would not reduce the accessibility of the page, by replacing "non-zero" with "has an inverse", and then pointing out what this implies for real/complex or for integer entries, without mentioning that this holds over any commutative ring. MvH (talk) 18:19, 9 April 2014 (UTC)MvH
PS. There are many students taking abstract algebra that make the mistake non-zero det --> invertible, long after the point where they should know better! MvH (talk) 18:22, 9 April 2014 (UTC)MvH
I was reading my linear Algebra book for class (Otto Bretsher's Linear Algebra with Applications. 3 edition. Upper Saddle River, New Jersey: Pearson Education, 2005) when I came across something interesting on page 8: "When mathematicans in ancient China had to solve a system of simultaneous linear equations such as, they took all the numbers involved in this system and arranged them in a rectangular pattern (Fang Cheng in Chinese as follows:
All the information about this system is conveniently stored in this array of numbers. The entries were respresented by counting rods; [...] the equations were then solved in a hands-on fashion, by manipulating the rods" I did some googling and found out that how Fang Chang is Chapter 8 in a book called Nine Chapters on the Mathematical Art which shows how almost 2000 years ago Chinese had a method similar to Guassian Elimination for solving linear equations even though they didn't call it Linear Alegbra. I thought it would be an interesting side note to add to the history section. the link about the book is here Nine Chapters on wiki and here Nine Chapters on google books—Preceding unsigned comment added by 128.61.43.160 (talk) 18:05, 12 June 2008 (UTC)
I agree with people elsewhere on this page who feel that this entry is in need of major rewriting and changing. I have just made a dent in this. I began by tweaking the "history" section slightly (to treat the history of linear algebra as synonymous with the history of "abstract", ie, post-1850s linear algebra, is inaccurate) and removing a good chunk of stuff about quaternions. (Although it is certainly related to linear algebra, so are vectors--- which predate quaternions by centuries--- and it seems unusual to give so much attention to quaternions here.)
The statement that the use of Cramer's rule (which dates to the 1700s, mind you, not the 1850s) to solve partial differential equations led to the introduction of linear algebra to the math curriculum is, to this reader, laughable, and as it was not sourced in a way that makes sense to me I took it out. Copson's quote seems to have much more bearing on the difference in education between two universities, one English and one Scottish, in the beginning of the 19th century, than it does on anything specific to linear algebra. (In any case, the quoted portion of Copson does not say anything about the role of Cramer's rule in partial differential equations.)
Frankly, the article in its current form reads like a mish-mash of submissions by beginning students of linear algebra, well-intentioned people relating the observations of third parties (e.g. footnotes in general science works, or introductory paragraphs in introductory textbooks), and people too inclined to include abstract technical detail that is probably better left to more specific entries than "linear algebra". In my very humble opinion. 75.167.204.90 (talk) 05:08, 7 July 2009 (UTC)
I've started a revamping of the article, which was long overdue. In the first installment, I've replaced a rambling and incoherent section Elementary introduction with a synopsis of the first few chapters in a standard linear algebra text. The section on history needs to go, too: there is no excuse to having such poor quality material, especially since a very good historical account is contained in the article on vector spaces (and, perhaps, elsewhere on wikipedia). It seems unwise to fork the content, especially from the maintenance point of view. I also feel that the list of results doesn't add much, but if anyone has ideas about how to incorporate some of them into the narrative, please, share them here or implement them yourself. Arcfrk (talk) 05:07, 9 February 2010 (UTC)
This article must speak to the non-mathematical reader and to students. Mathematicians already know linear algebra and do not need to read about it on Wikipedia. Every line of the article should pass two tests. It should be mathematically accurate. And it should be readable by someone who is not a mathematician. The later parts of this article can address students who already know the material in the earlier parts of the article, but there should not be anything in the article that you need a Ph.D. in mathematics to read.
I've watched this article over the years swing back and forth between extremes. Sometimes it is oversimplified, sometimes too technical. I would like to see it at least move out of the start class, and I think there are currently some editors here working toward the same goal.
Turning now to the picture, I find both the picture and the caption confusing. Were I beginning reader, I might think that all subspaces were lines through the origin. And the three dimensional effect is not clear: I'm not sure where the colored planes intersect. Also, vector subspaces are fundamental to the study of vector spaces, but more important in linear algebra, I think, is the use of a matrix to transform one vector to another. Does anyone have a picture showing this? Rick Norwood (talk) 17:06, 9 February 2010 (UTC)
A "Matrix" isn't listed in the section on main structures, yet references to matricies are all over this page especially in the "most useful theorems" section. —Preceding unsigned comment added by 65.50.39.118 (talk) 05:39, 7 September 2010 (UTC)
The statement "any claim that the concepts of linear algebra were known to mathematicians prior to the end of the nineteenth century is inaccurate, an instance of the historical error of anachronism." seems strange. Herman Grassmann's The theory of Linear Extension (1834) seems to deal with linear algebra. 128.240.229.7 (talk) 07:35, 21 January 2011 (UTC) Niko
If we have a source that confirms that this work indeed deals with linear algebra, then we can add the sentence "... althouh Herman Grassmann's The theory of Linear Extension (1834) deals with linear algebra." But perhaps we need to be wp:BOLD and just remove that sentence, since (1) it sounds like a declaration—i.e. unencyclopedic—, (2) it is not sourced, and (3) I think it is very unlikely that a source will ever be found. So I have removed the assertion. If someone has a good source for it, it can of course gladly be reinserted. DVdm (talk) 19:18, 21 January 2011 (UTC)
Hi, I tried to re-write the introduction but someone reversed my changes. I think defining linear algebra as a branch of mathematics that studies vectors is not quite correct and also kind of circular. I'll re-write again if some people have suggestions where what I wrote wasn't clear, but I don't feel like writing everything out again only to have it deleted. Loadedsalt (talk) 22:35, 18 February 2011 (UTC)
Clearly, I agree that linear algebra does not study vectors, and I have just changed it. Your re-write, however, was a bit too radical for a single change and, perhaps, hard for non-experts. Let's see if my revision survives longer :) 2andrewknyazev (talk) 00:55, 19 February 2011 (UTC)
I don't have a problem with the recent edits. A larger concern is that the article needs a substantial expansion. Someone needs to write a section on solving linear systems and a section on applications l, for starters. As the article improves, generally the lead needs to be rewritten anyway. Sławomir Biały (talk) 13:06, 19 February 2011 (UTC)
I have been editing this page lately, and here are some of my ideas on how to improve the page. Please feel free to act on these ideas, comment on them, reject them, and add your own thoughts!
1) In the scopes of study section, elaborate on determinants and inner product spaces, since they are important concepts. I am not sure if we should, but we could add some information on Hermitian and normal operators and the fact that they are diagonalizable and have orthonormal basis of eigenvectors.
2) In the applications section, elaborate on the solution of linear equations. I do not know if it is best to introduce this application through an example or through theory. I also don't know if we should use the augmented matrix notation or the equation notation that we are currently using; my inclination is that the augmented matrix notation might be cleaner and might connect better with the rest of the article.
3) Add the section on best-fit lines. Personally, I am not too familiar with this subject, so it might be better if someone else writes that section, but if no one will do it, I could relearn that material and write it up.
4) Add more applications? There are so many, so it is debatable how many we should include.
5) Expand the history section. The history of mathematics is really interesting, and when I read Wikipedia articles on math concepts, I like to read the history section.
6) Flesh out the generalization section. Maybe we should elaborate on what linear algebra theorems remains true in module theory and what becomes false, or put to symbols the concept of multilinear algebra. We should not write a whole expose of the subject, but link the subject to linear algebra.
7) Maybe mention its role in a mathematical education? That it is often used as a bridge to abstract math?
I thought that was strange, too. Maybe it should redirect to Nonlinear system for the time being? I don't want to change it myself, as I know nothing about the topic and am not sure which page is more closely related. 138.16.18.24 (talk) 16:10, 17 April 2014 (UTC)
The following comment was removed from the eigenvectors and eigenvalues section today:
but diagonalizable matrices form a dense subset of all matrices.
Perhaps true, but without reference, and inserted into introductory material, the comment is out of place. The topic requires a topology of matrices and an indication of density, beyond the scope of this article.Rgdboer (talk) 22:16, 7 October 2012 (UTC)
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