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Elementary Algebra - 3rd edition
Summary: Elementary Algebra is a book for the student. The authors' goal is to help build students' confidence, their understanding and appreciation of math, and their basic skills by presenting an extremely user-friendly text that models a framework in which students can succeed. Unfortunately, students who place into developmental math courses often struggle with math anxiety due to bad experiences in past math courses. Developmental students often have never developed nor ...show moreapplied a study system in mathematics. To address these needs, the authors have framed three goals for Elementary Algebra: 1) reduce math anxiety, 2) teach for understanding, and 3) foster critical thinking and enthusiasm.The authors' writing style is extremely student-friendly. They talk to students in their own language and walk them through the concepts, explaining not only how to do the math, but also why it works and where it comes from, rather than using the "monkey-see, monkey-do" approach that some books take1577299 used book - free tracking number with every order. book may have some writing or highlighting, or used book stickers on front or back77290-5-0VeryGood
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Mathematics Education resources
At the Times Higher Awards ceremony
on 24th November 2011, it was
announced that Loughborough and
Coventry Universities had won the
award for Outstanding Support for
Students, in recognition of the work
of sigma, Centre for Excellence in
University-wide mathematics and
statistics support.
Whilst sigma at Coventry and Loughborough Universities received the award, the real winner was mathematics and statistics support across the country. In this booklet,
we outline how sigma's work has contributed to the growing recognition of the importance of mathematics and statistics support and to the development of a national and international community of practitioners. Authors : Ciaran Mac an Bhaird and Duncan Lawson
This guide edited by Michael Grove and Tina Overton has been developed for those looking to begin pedagogic research within the science, technology, engineering and mathematics (STEM) disciplines. Its purpose is to provide an accessible introduction to pedagogic research along with a practical
guide containing hints and tips on how to get started. The guide was produced following a series of national workshops and conferences that were started in
2011 by the National HE STEM Programme and continued in 2012 with the support of the Higher Education Academy.
Recruitment to post-graduate mathematics programmes and to lecturer positions in mathematics departments in UK universities has become dominated by international students and staff. Although mathematics is generally regarded as 'the universal language', the reality is that different countries have very different cultures when it comes to the teaching and learning of mathematics. There are significant variations in the pre-university mathematical experience, in terms of the curriculum content, learning styles, levels of abstraction, and assessment methods. Even within the UK, a considerable number of pre-higher education mathematics qualifications are available and, it is not always clear what mathematics can be expected when students commence their degree programmes. With increasing numbers of international students and academic staff in UK HE, the scene is becoming more complicated. Students enter degree courses with a wide range of backgrounds and bring with them very different experiences. At the same time, academic staff, having experienced different education systems, may have some unrealistic expectations from their students.
With an HEA Teaching Development Grant (Individual Scheme 2012 -2013), this research by Aiping Xu, Coventry University has investigated the mathematical cultures of a range of the main international suppliers (of students and staff) to UK HE mathematics departments. Using semi-structured interviews and online questionnaires, personal experiences of academic staff who have studied or taught more than two educational systems have been drawn upon. Some examinations have also been studied in detail.
This report is based on a presentation given by the author, Josh
Hillman, on 17 March at the first Q-Step conference, Counting them
in: quantitative social science and the links between secondary and higher
education. Other presentations from the day are available at
Josh Hillman is Director of Education at the Nuffield Foundation.
Josh Hillman, Mathematics after 16: the state of play, challenges and
ways ahead, (London: Nuffield Foundation, 2014)
A report containing the Royal Society's Vision for science and mathematics education over the next 20 years. This includes a proposal for a broad and balanced curriculum, where young people study science and mathematics until 18 alongside arts, humanities and social sciences. The Royal Society Policy Centre report 01/14 issued June 2014 DES3090.
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Mathematics
The mission of the KoC Mathematics Department is to provide a friendly atmosphere where students can learn and become masters of mathematical application. Moreover, the department promises to contribute to the development of students as logical thinkers, enabling them to become life-long learners, to continue to grow in their future professions, and to function as p
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Easy Algebra Step-By-Step
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 ...Show synopsisTake 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 Step-it-up exercises link practice to the core steps already presented Missteps and stumbles highlight common errors to avoid You can master algebra as long as you take it Step-by-Step!. Take it step-by-step for algebra success! The quickest...New. 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...
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Product Description
Integrate TI graphing calculator technology into mathematics instruction using these resource books. Move students from the concrete to the abstract in three steps: explain the concept, use the graphing calculator, and apply the concept. Improve students' use of math language with an extensive glossary. Supports both students and teachers with step-by-step instructions, including keystrokes and screen shots. Increase student achievement with lessons and strategies that have been classroom tested. Helps you prepare students for testing situations that permit the use of graphing calculators. Correlated to NCTM Standards, as well as standards from all 50 states. 240-page book includes a CD-ROM.
CCSS Product Alignment Math Grade 8 8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. 8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p, is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. 8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. 8.EE.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 8.SP.2
Prices listed are U.S. Domestic prices only and apply to orders shipped within the United States.
Orders from outside the
United States may be charged additional distributor, customs, and shipping charges.
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Algebra 2 covers several methods for solving quadratic equations, such as factoring, completing the square, and graphing. The text also introduces trigonometry and exponential functions—vital concepts for real world applications. Filled with full-color illustrations and examples throughout, Algebra 2 will motivate your child to learn. Overall, this high-interest text makes it easy for you to engage students in Algebra.
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An essential tool for standardized tests, the Spectrum Math series offers grade-appropriate coverage of basic arithmetic and math skills. Each book features drill and skill practice in math fundamentals, as well as applications of mathematics in everyday settings. Chapter Pre-Tests, Chapter Tests, Mid-Book Tests and Final Tests all contribute to an extended familiarity with developmental, problem-solving and analytical exercises. An assignment record sheet, record of test scores sheet and answer key are included.
Both of my sons used an earlier edition of this book for "afterschooling" when they came home from the public school each day. I loved how concise this series was compared to all the other math textbooks at this grade level, so I could be sure I was tutoring them on all the typical Gr. 3 math topics, without the "twaddle."
Share this review:
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1of1voted this as helpful.
Review 2 for Spectrum Math, 2007 Edition, Grade 3
Overall Rating:
5out of5
Date:May 7, 2010
Lisa
My daughter used this workbook this year for 3rd grade. She did wonderfull with math! She does very well with her CAT tests too. When she needs extra practice on something we just make up quick worksheets on the computer to go along with what she is doing in this book. It's a great guide for what they need to learn in thier grades at a good price with great explainations so they can do it on thier own and with the key in the back so no need for a teacher manual.
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Find a Gainesville, GA StatisticsCan help you understand the basics of the Microsoft Word toolbar. Frequently used Microsoft PowerPoint during MBA coursework. Can help you use templates, create appealing transitions, and link or embed various items.
...In addition, I have tutored students in different concepts pertaining to discrete math. I have taught a Quantitative Reasoning course which contains some of the concepts covered in discrete math. Finite math is a compilation of various mathematical topics.
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Introduction to Mathematica
Posted by: Nicholas O'Brien on November 18, 2010
Mathematica is a mathematical package that is quite intuitive and easy to use. On the plus side it is very good for calculus, graph theory, and 3D graphic rotations. It also allows easy in-line commenting. On the down side Mathematica requires one use strict syntax, and that you load external packages to perform many functions. Additionally, it undergoes big changes version to version so reverse compatibility is a problem.
At Bates, Mathematica is used some in math, and also in the physics department.
For more information about using Mathematica, thes resources are available online for you and yous students:
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Math "Head Start" Workshops
The Tunxis Community College Academic Support Center offers Tunxis students who are planning to enroll in Elementary or Intermediate Algebra the opportunity to get a free head start in math before the stress of the semester begins.
You will refresh your memory on the foundations of the course, and you will learn key math study skills. This is your chance to ask the questions that have always plagued you about math.
Elementary Algebra
The session is designed to prepare you for MAT*095 Elementary Algebra Foundations by refreshing your memory on key concepts like:
Integers/ Fractions
Combining Like Terms
Word problems
And more
When?
Tuesday, January 21st at 9:30 – 11:30am OR Wednesday, January 22nd at 2:00 – 4:00pm OR Thursday, January 23rd at 5:00 – 7:00pm
Sign Up for an Elementary Algebra Workshop
If you have registered for Elementary Algebra Foundations
Intermediate Algebra
The session is designed to prepare you for MAT*137 Intermediate Algebra by refreshing your memory on key concepts like:
Fun with polynomials
Laws of exponents
Linear Equations
Word problems
And more
When?
Tuesday January 21st at 5:00 – 7:00pm OR Wednesday, January 22nd at 9:30 – 11:30am OR Thursday, January 23rd at 2:00 – 4:00pm
Sign Up for an Intermediate Algebra Workshop
If you have registered for Intermediate Algebra
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Featured Events
ACE Pre-Conference
Tip Of The Week
Did You Know... Students often have difficulty with algebra because of misconceptions in various areas. One common misconception is that students think that algebraic variables stand for things, such as 6d = 12 , where d stands for donuts, not the cost of a donut.
How can the problem be solved?
Make sure students understand the difference between abbreviations and variable, such as: "6 meters" instead of "6m."
Have students write out the literal translation of an equation, such as: 6P=S, where 6 times the number of professors is equivalent to the number of students. It's all in how we teach the basics!
About IPDAE
The Institute for the Professional Development of Adult Educators is an initiative supported by Florida Department of Education to offer information, resources and professional development for adult education career pathways programs. Learn more
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Welcome to Beyond Calculus
Developed by Jake Chipps
Why Use This Site?
Beyond Calculus is an online video-based textbook that provides support to students taking AP Calculus AB. Are you sitting at home wracking your brains over your homework, and need help? Try watching the videos in whatever section you are stumped. Is your math class going too fast? The instructional videos on this site may just provide the understanding you need to excel. Cramming for the AP exam or a final? Why not review by going through this e-book? You will find instructional videos here rather than academic text.
How to Use This Site
Each section of Calculus A and B is organized in the table of contents. Click on any topic to view that particular lesson. Videos are organized to enhance understanding. The first video provides a big-picture view of the topic, and each subsequent video provides the user with more difficult examples. For the best experience, It is suggested that you press pause when you need to think about a concept. Rewatch videos if necessary, and you will be amazed at your level of understanding by the end.
Why Is This E-Book Free?
Information should be free and available to all. Not only should information be free, but it should be available anywhere at any time. This e-book can be viewed on all devices, including desktops, iPads and other mid-size devices, and phones with internet access. After creating online videos to supplement and enhance my classroom instruction, I decided to compile my work into an online textbook. Rather than charge for these services, I am providing my work to the world free of charge, as it should be.
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Euclidean and Non-Euclidean Geometry - 4th edition
Summary: This is the definitive presentation of the history, development and philosophical significance of non-Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Appropriate for liberal arts students, prospective high school teachers, math. majors, and even bright high school students. The first eight chapters are mostly accessible to any educated reader; the last two chapters and the two a...show moreppendices contain more advanced material, such as the classification of motions, hyperbolic trigonometry, hyperbolic constructions, classification of Hilbert planes and an introduction to Riemannian geometryback NEAR FINE Hardback, 500
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More About
This Textbook
Overview
MODERN GEOMETRY was written to provide undergraduate and graduate level mathematics education students with an introduction to both Euclidean and non-Euclidean geometries, appropriate to their needs as future junior and senior high school mathematics teachers. MODERN GEOMETRY provides a systematic survey of Euclidean, hyperbolic, transformation, fractal, and projective geometries. This approach is consistent with the recommendations of the National Council of Teachers of Mathematics (NCTM), the International Society for Technology in Education (ISTE), and other professional organizations active in the preparation and continuing professional development of K-12 mathematics
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Tucson SAT Math the student must learn to see it from the perspective of functions: polynomial, rational, radical, exponential and trigonometric functions. Calculus, as Algebra, is an art; the artist needs a pallet. In this case the pallet is the coordinate plane and the student must learn to draw the faces (graphs) of all the functions and recognize them from their faces am very familiar with methods for teaching these concepts. I
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MATH 210 Introduction to Proofs
Introduces the central idea of proof in mathematics and some standard proof formats that are used throughout the math major. The course includes propositional logic, an introduction to predicate logic, direct proof, proof by contradiction, and mathematical induction. Liberal Arts.
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Lesson 8.1ExploringExponentialModels Mrs. Snow, Instructor Exponential functions are similar in looks to our other functions involving exponents, but there is a big difference. The variable is now the power, rather than the base.
Title: ExploringExponential Growth and Decay Functions Brief Overview: ... 1Answers will vary, but to insure accurate ... Which models of cars are more/less likely to become antiques? When is that likely to happen?
Understand and use basic exponential functions as models of real phenomena. f ... reviewed and students practice working with rational exponents. ... where a is a non-zero real number and b is a positive real number other than 1. An exponential growth function has a value of b that is greater ...
exploringexponentialmodels more accessible. It is important to emphasize the patterns in the tables, graphs, and algebraic rules for ... 1. Answers will vary. Number of M&M's Remaining Trial Number Number of M&M's remaining 0 140 1 76 2 39 3 22 4 12 5 8 6 3
when a > 0 and b > 1 when a > 0 and 0 < b < 1 This is called an exponential regression model. You will need to plot data points using the STAT menu. 1) Plot Data Points as described on previous worksheet. 2) Press [STAT], press [ ] ( Calc). Press ...
Describe the scenario from page 1 of the Exploring "Geometric sequences and series." ... What is the constant multiplier of the exponential function that models this situation? What is its domain? ... More practice, pages 6-8 Student Activity Sheet 3, question 6.
[SAS 3, question 1] Exploring "Geometric sequences and series" ... What is the constant multiplier of the exponential function that models this situation? ... Have them practice writing the sum in sigma notation. [SAS 3, question 4]
linear and exponentialmodels. ... Common Core State Standards for Mathematical Practice1. Make sense of problems and persevere in solving them. ... When making statistical models, technology is valuable for varying assumptions, exploring
... quadratic, and exponential through graph models and algebraic models. ... answers by using "benchmarks" to estimate measures and other strategies to approximate a ... Thinking With Mathematical Models (Inv. 1, 2) 8.SP.4
The Mathematical Practice Standards apply throughout each course and, ... (1) Exploring Data: observing patterns and departures from patterns (2) ... be able to explain their answers using arguments, graphs, and statistical skills that they will learn in the
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Math Courses of Study: Grade by Grade
See Below For Grade by Grade Goals
Although mathematics curricula will vary from state to state and country to country, you'll find that this list provides the basic concepts that are addressed and required for each grade. The concepts have been divided by topic and grade for easy navigation. Mastery of the concepts at the previous grade is assumed. Students preparing for each grade will find the listings to be extremely helpful. When you understand the topics and concepts that are required, you will find tutorials to help you prepare under the perspective subjects on the home page. Calculators and computer applications are also required as early as kindgarten. Most curriculum documents request that you are also able to use the corresponding technologies such as software applications, regular calculators, and graphing calculators.
For more specific details regarding the math requirements for each grade, you may want to do a search for the curriculum in your state, province or country. Most boards of education will provide you with the details to access the documents.
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College Algebra : Graphs and Models -Text Only - 4th edition
Summary: The approach of this text is more interactive than most precalculus texts and the goal of the author team is to enhance the learning process through the use of technology and to provide as much support and help for students as possible. In Connecting the Concepts, comprehension is streamlined and retention is maximized when the student views a concept in visual, rather than paragraph, form. Zeros, Solutions, and X-Intercepts Theme Carried Throughout helps students vi...show moresualize and connect the following three concepts when they are solving problems: the real zeros of the function, the solutions of the associated equation, and the x-coordinates of the x-intercept of the graph of the function. Each chapter begins with a relevant application highlighting how concepts presented in the chapter can be put to use in the real world. These applications are accompanied by numerical tables, equations, and grapher windows to show students the many different ways in which problems can be examined. End-of-Chapter material includes a summary and review of properties and formulas along with a complete set of review exercises. Review exercises also include synthesis, critical thinking, and writing exercises. The answers to all of the review exercises appear in the back of the text and have text section references to further aid students. For anyone interested in learning algebra2.00 +$3.99 s/h
Good
Campus_Bookstore Fayetteville, AR
Used - Good Hardcover. Textbook only
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According to Leonard M. Keenedy and Steve Tipps (2000): "Mathematics is alanguage for describing common events in everyday life and complex events in business,science and technology"Below are some examples of application of Mathematics in traveling, astronomy,engineering and manufacturing.When a family plans a vacation, they use mathematics to estimate distances, timesfor departure and return, fuel needed, food and other supplies required and costs of maintaining the family vehicle. As astronomers and engineers plan space travel,mathematics is used to calculate distances, times for departure and return, fuel needed,food and other supplies and costs of maintaining the space vehicle. When manufacturers plan for distribution across the Brunei Darussalam, they employ mathematics to calculatedistances, time for departure and return, fuel needed, food and other supplies required,and costs of maintaining the vehicle.Three different problems, three levels of mathematics and three needs for precision, and all these three require the same thinking process. As a result of this,mathematics is a tool and a language for solving problems great and small.
Page
2
PS 4305 CT Mathematics I
2008
This report will focus on the use of mathematics in other subject includingdifferent aspect of mathematics in different area or field. New mathematics courses have been developed that focus on "nontraditional" topics and applications in real world. Astoday's student learn mathematical concepts and thinking, they must apply, adapt andextend old concepts to new tasks and existing ideas into real world. In the 21
st
century,children will need mathematics for complex and common applications.The report of the Cockcroft Committee Mathematics Count (HMSO, 1982)comments that although there are some books linking mathematics to other subjects suchas art and science, more are needed. Paragraph 292 of the Report states:
Almost all children find pleasure in working with shapes, and work of this canencourage the development of positive attitudes towards mathematics in those who are finding difficulty with number work.
In later chapter, we will discuss about mathematics in different fields, namelymathematics through art and design, mathematics in everyday life, mathematics inmedicine, mathematics in sports, mathematics in cooking, mathematics in psychology,mathematics in architecture and mathematics in nature.
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Math and Computer Science Courses
Mathematics can be described as a language, a tool, a science, and an art. Computer Science is an area of study that continues to gain importance for its rich theory and wide applications to physical and social sciences. In CTY's math and computer science courses, students move beyond basic skills to gain greater understanding of both the underlying principles and the intriguing ways these concepts can be applied and extended to a range of contexts.
Students have the opportunity to explore challenging material and strengthen their problem-solving skills. They investigate advanced concepts through a process of discovery and engagement that promotes a lifelong interest in the discipline. Through hands-on, thought-provoking exercises, students learn to make connections between abstract ideas and their uses in a range of fields, including science, engineering, economics, and advanced mathematics.
Math Course Descriptions and Syllabi
Paradoxes and Infinities
The second sentence is true. The first sentence is false. Are these sentences true or false? How is it that observing an orange pumpkin is seemingly evidence for the claim that all ravens are black? Students in this course explore conundrums like these as they analyze a range of mathematical and philosophical paradoxes.
Students begin by considering Zeno's paradoxes of space and time, such as The Racecourse in which Achilles continually travels half of the remaining distance and so seemingly can never reach the finish line. To address this class of paradoxes, students are introduced to the concepts of infinite series and limits. Students also explore paradoxes of set theory, self-reference, and truth, such as Russell's Paradox, which asks who shaves a barber who shaves all and only those who do not shave themselves. Students analyze the Paradox of the Ravens as they study paradoxes of probability and inductive reasoning. Finally, they examine the concept of infinity and its paradoxes and demonstrate that some infinities are bigger than others.
Through their investigations, students acquire skills and concepts that are foundational for higher-level mathematics. Students learn and apply the basics of set theory, logic, and mathematical proof. They leave the course with more nuanced problem-solving skills, an enriched mathematical vocabulary, and an appreciation for and insight into some of the most perplexing questions ever posed.
Geometry through Art
"Geometry is the right foundation of all painting." In this way, the German artist Albrecht Dürer described a connection between mathematics and art that can be found in every culture. In this introductory geometry course, students learn about geometric figures, properties, and constructions, and use this knowledge to analyze works of art ranging from ancient Greek statues to the modern art of Salvador Dalí.
Beginning with the foundations of Euclidean geometry, including lines, angles, triangles, and other polygons, students examine tessellations and two-dimensional symmetry. Using what they learn about points, lines, and planes, students investigate the development of perspective in Renaissance art. Next they venture into three dimensions, analyzing the geometry of polyhedra and considering their place in ancient art. Finally, students explore non-Euclidean geometry and its links to twentieth-century art, including the drawings of M. C. Escher.
Through lectures, discussions, hands-on modeling, and small group work, students gain a strong foundation for the further study of geometry, as well as an appreciation of the mathematical aspects of art.
Note: Students who have taken CTY's Geometry and Its Applications class, or a high school geometry class, should not take this course.
Note: This course exposes students to geometric properties and concepts but should not be used to replace a year-long high school geometry course.
Sample text:Squaring the Circle: Geometry in Art and Architecture, Calter.
Mathematical Modeling
Mathematics is more than just numbers and symbols on a page. Applications of mathematics are indispensable in the modern world. Math can be used to determine whether a meteor will impact Earth, predict the spread of an infectious disease, or analyze a remarkably close presidential election. In this course, students learn how to create mathematical models to represent and solve problems across a broad range of disciplines, including political science, economics, biology, and physics.
Students in this class investigate voting systems by constructing mathematical models of how groups make decisions and how elections are conducted. They consider how goods, property, and even political power can be fairly divided and apportioned. Students learn how to use Euler and Hamilton circuits to find the optimal solutions in a variety of real-world situations, such as determining the most efficient way to schedule airline travel. While investigating growth and symmetry, students develop linear and exponential growth models and explore fractals and the Fibonacci numbers. Students leave this course with the ability to use the seemingly abstract language of mathematics to gain a greater understanding of the world around them.
The Mathematics of Money
From managing one's personal investments to examining the profitability of a multibillion-dollar global corporation, the mathematics of money is at the heart of successful financial endeavors. Why are round-trip fares from Orlando to Kansas City higher than those from Kansas City to Orlando? How do interest rate adjustments made by the Federal Reserve affect the real estate market? How does one calculate the price-earnings ratio of a stock and use that result to help predict that stock's future performance? Mathematics plays an indispensable part in answering each of these questions.
This course provides students with a mathematical grounding in central concepts of business and finance. Students investigate the mathematics of buying and selling, and apply these principles to real- world situations. They gain fluency with the concepts of simple and compound interest and learn how these affect the present and future value of loans, mortgages, and interest-bearing accounts. Students investigate various forms of taxes, considering their impact on personal and governmental budgets. In their examination of these topics, students manipulate and solve algebraic expressions, and also learn to apply a range of mathematical concepts including direct and indirect variation and arithmetic and exponential growth. Through simulations, entrepreneurial projects, and classroom investigations, this course provides students with the foundation required to be more secure in their financial management and enhances their understanding of the broader economic conditions that shape investments in the public and private sector.
Game Theory and Economics
Thomas J. Watson, the founder of IBM, once said, "Business is a game—the greatest game in the world if you know how to play it." In today's global marketplace, understanding game theory, the branch of mathematics which focuses on the application of strategic reasoning to competitive behavior, is crucial to understanding business and economics.
In this course, students use game theory as a framework from which to analyze a variety of real-world economic situations. Students begin the course by analyzing simple games, such as two-person, zero-sum games, and learn how these games can be used to model actual situations encountered by entrepreneurs and economists. For instance, students may apply the concept of Nash equilibria to find the optimum strategy for the pricing of pizza in the competition between Domino's and Pizza Hut.
As they acquire an understanding of more complex games, students apply these methods to analyze a variety of economic situations, which may include auctions and bidding behavior, fair division and profit sharing, monopolies and oligopolies, and bankruptcy. Through class discussions, activities, research, and mathematical analysis, students learn to predict and understand human behavior in a variety of real-world contexts in business and economics.
Discrete Math
Can any given map be colored with just four different colors such that no two regions sharing a common edge are the same color? Mathematicians took more than 100 years to answer this question in the affirmative, establishing the result known as the Four-Color Theorem. Discrete math introduces students to questions such as this as they learn math from a range of disciplines including set theory, combinatorics and graph theory, and number theory. This leads them to important real-world applications such as determining the number of ways to create a password of a given length or finding the shortest path for a taxi between multiple locations.
Students in this course begin by building a foundation in set theory and proof. They then explore combinatorics, examining the number of possible configurations of different sets of objects. Students move on to investigate graph theory, an area that introduces them to both historic problems such as the Bridges of Königsberg and the Traveling Salesman, as well as more modern applications such as the analysis of social networks and traffic patterns.
Students leave the course not only with a familiarity with a flourishing branch of mathematics, but also with an enriched mathematical vocabulary and an improved ability to understand and create mathematical arguments.
Computer Science Course Descriptions and Syllabi
Foundations of Programming
Students in this course gain insight into methods of computer programming and explore the algorithmic aspects of computer science. They learn the theoretical constructs common to all high-level programming languages by studying the syntax and basic commands of a particular programming language such as Java, C, C++, or Python*. Building on this knowledge, students move on to study additional concepts of programming, such as object-oriented programming or graphical user interfaces. By solving a variety of challenging problems, students learn to start with a concept and work through the steps of writing a program: defining the problem and its desired solution, outlining an approach, encoding the algorithm, and debugging the code.
Through a combination of individual and group work, students complete supplemental problems, lab exercises, and various programming projects in order to reinforce concepts learned in class. By the end of the course, students can develop more complex programs and are familiar with some of the standards of software development practiced in the professional world. Students leave with an understanding of how to apply the techniques learned to other high-level programming languages.
*Note: The programming language learned may change based on the instructor's preference.
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Black's Academy
Online electronic mathematics textbooks where all problems also have solutions worked out. Covers mathematics for AQA, OCR, and Edexcel. Suitable for K12+.
Symmetry
Visual journey through the worlds of symmetry, with many photos and drawings.
Precalculus Concepts
A different approach to algebra and trig that is designed to prepare students for calculus. Graphics calculators play an important role.
Computational Beauty of Nature, The
Companion site for the book by Gary William Flake. Contains applets and source code for simulations of fractals, chaos, complex systems, and adaptation. mitpress.mit.edu/books/FLAOH/cbnhtml
Advanced Math SAT Workbook
Workbook with strategies for students striving for a perfect score on common standardized admissions tests.
Handbook of Analysis and Its Foundations
Mathematics mini-encyclopedia for advanced undergraduates and beginning graduate students. Excerpts about the Axiom of Choice, orderings, norms, etc. math.vanderbilt.edu/~schectex/ccc
Ecological Numeracy
By Robert A. Herendeen, published by John Wiley & Sons. Relatively simple mathematics to understand quantitative aspects of environmental issues.
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Linear Algebra with and accessible book from one of the leading figures in the field of linear algebra provides readers with both a challenging and broad understanding of linear algebra. The author infuses key concepts with their modern practical applications to offer readers examples of how mathematics is used in the real world. Topics such as linear systems theory, matrix theory, and vector space theory are integrated with real world applications to give a clear understanding of the material and the application of the concepts to solve real world problems. Each chapter contains integrated worked examples and chapter tests. The book stresses the important role geometry and visualization play in understanding linear algebra.For anyone interested in the application of linear algebra theories to solve real world problems.
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
| 677.169 | 1 |
Summary: Viewing stained glass from different angles or in various lights is necessary to discover its many qualities. Likewise, viewing solutions of differential equations from several points of view is essential to fully understand their behavior. Lomen and Lovelock provide an active environment for students to explore differential equations by using analytical, numerical, graphical, and descriptive techniques, and for students to use ODEs as a natural tool for modeling man...show morey interesting processes in science and engineering. ...show less
Basic Concepts. Autonomous Differential Equations. First Order Differential Equations - Qualitative and Quantitative Aspects. Models and Applications Leading to New Techniques. First Order Linear Differential Equations and Models. Interplay Between First Order Systems and Second Order Equations. Second Order Linear Differential Equations with Forcing Functions. Second Order Linear Differential Equations - Qualitative and Quantitative Aspects. Linear Autonomous Systems. Nonlinear Autonomous Systems. Using Laplace Transforms. Using Power Series46 +$3.99 s/h
Good
Big Planet Books Burbank, CA
1998-11-09 Paperback Good Expedited shipping is available for this item!
$53.46 +$3.99 s/h
Good
Big Planet Books Burbank, CA
1998-11-09 Paperback Good Expedited shipping is available for this item!
$78
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Saxon Math Homeschool Curriculum for Grades 7-12
If you have a math student who likes to work a variety of problems and would be bored with twenty-five of the same type of problems, then your student will like Saxon Math. If your student needs to work twenty-five problems of the same type to learn a concept, this might not be the best program for you unless you adapt the problem sections. Our daughters who chose Saxon for their math did quite well in math courses at college.
$5 Flat Rate Standard Shipping On Any Pre-Paid Order to Continental US
We think the Solution Manuals are a must for Algebra ½ and up! Not just the answers are included; the problems are worked for you! They come with the homeschool kits for Saxon Math 54 to Saxon Math 87.
Instruction CDs
If you need help with Saxon Math or want offer additional instruction, then we can highly recommend that you purchase an instruction CD. Two instruction CD options are available.
The Saxon Homeschool Teacher Lesson and Test CDs offer instruction for each lesson. The instructor shows how to work every problem in the book. In the Saxon Teacher CDs, a teacher with extensive Saxon teaching experience goes over a lesson and works every practice problem problem set, and test problem. The Saxon Teacher instruction is secular, and there are several different instructors.
Math 76
Saxon Math 76 Homeschool Kit 4th Edition
Publisher: Saxon Homeschool.
ISBN-13:9781591413493
The Math 76 Homeschool Kit 4th Edition includes a softcover textbook, a softcover Tests and Worksheets, and a softcover Solutions Manual for advanced grade 6 and regular grade 7.
List $103.60
Price $83.99
Saxon Math 76 Student Textbook 4th Edition
ISBN-13:9781591413196
List $48.65
Price $39.99
Saxon Math 76 Tests and Worksheets 4th Edition
ISBN-13:9781591413233
If you have an additional student and want to have worksheets and tests for each one, then you need this.
Most of the pages in this book are not reproducible; however, there are 5 recording forms in the back of the book that are reproducible and include:
Mixed Practice Solutions – Grid layout for student to show work and answers to Mixed Practice problems)
Scorecard – Helps track scores on daily assignments and tests.
Test Solutions – Grid layout for student to show work and answers to tests.
List $29.90
Price $24.99
Saxon Math 87 Solutions Manual 3rd Edition
ISBN-13: 9781591413288
Note: This is compatible only with the homeschool version.
This is the solutions manual for Math 87 3rd Edition. It has the solutions [the work to solve the problems is shown] for the problems in the book.
List $35.80
Price $28.99
Algebra 1/2
Saxon Algebra 1/2 3rd Edition Homeschool Kit
Publisher: Saxon Homeschool.
ISBN-13:9781565774995
The Algebra ½ Kit 3rd Edition covers all topics normally taught in pre-algebra, as well as additional topics from geometry and discrete mathematics. It is recommended for seventh-graders who plan to take first-year algebra in the eighth grade, or for eighth-graders who plan to take first-year algebra in the ninth grade.
The Saxon Algebra ½ Homeschool Kit 3rd Edition contains one of each of the following items.
Saxon Algebra 1/2 3rd Edition Homeschool Kit with Solutions Manual
The Algebra ½ Kit 3rd Edition with Solutions Manual includes one of each of the following items.
Algebra ½ Student Textbook
Algebra ½ Answer Key and Test forms 9781591411727
Algebra ½ Solutions Manual 9781565771314
Grade 8.
List $115.95
Price $93.99
Saxon Algebra ½ Solutions Manual, 3rd Edition*
ISBN-13: 9781565771314
This is the Saxon Algebra ½ Solutions Manual, 3rd Edition. It has the solutions [the work to solve the problems is shown for the problems in the book.
List $43.55
Price $34.99
Saxon Algebra ½ Answer Key and Tests 3rd Edition
ISBN-13: 9781591411727
List $22.90
Price $18.99
Algebra 1
Saxon Algebra 1 3rd Ed. Homeschool Kit
Publisher: Saxon Homeschool.
ISBN-13: 9781565771239
The Saxon Algebra 1, Third Edition is made up of five instructional components: Introduction of the New Increment, Examples with Complete Solutions, Practice of the Increment, Daily Problem Set, and Cumulative Tests.
The Saxon Algebra 1, Third Edition Homeschool Kit includes one of each of the following:
Algebra 1 Student text
Algebra 1 Answer Key and Test Forms 9781565771383
A solutions manual is available separately.
Grade 9.
List $80.85
Price $65.99
Saxon Algebra 1 3rd Edition Homeschool Kit with Solutions Manual
Publisher: Saxon Homeschool.
ISBN-13:9781600329715
The Algebra 1 Kit 3rd Edition with Solutions Manual includes one of each of the following items.
Algebra 1 Student Textbook
Algebra 1 Answer Key and Test forms 9781565771383
Algebra 1 Solutions Manual 9781565771376
Grade 8.
List $124.20
Price $97.99
Saxon Algebra 1 3rd Edition Solutions Manual*
ISBN-13: 9781565771376
The Algebra 1 3rd Edition Solutions Manual has the solutions [the work to solve the problems is shown] for the problems in the book.
Grade 9.
List $44.85
Price $35.99
Saxon Algebra 1 3rd Edition Answer Key and Tests
ISBN-13: 9781565771383
Grade 9.
List $22.90
Price $18.99
Saxon Geometry First Edition
Geometry In Saxon Algebra 1 and 2
The new Saxon Geometry Kit is available now. For those of you who want a separate geometry from Saxon, this should work. In the meantime, Saxon Algebra 1 Third Edition and Algebra 2 Third Edition still include geometry. If you are required by law to keep a log, please log the lessons on geometry separately as geometry.
Saxon Geometry First Edition Homeschool Kit With Solutions Manual
Publisher: Saxon Homeschool.
ISBN-13: 9781600329760
This kit includes one of each of the following:
Geometry textbook (9781602773059)
Homeschool Testing Book with cumulative tests, answer forms, and answer key (9781600329777), and
Saxon Geometry Solutions Manual (978160275619).
List $130.55
Price $104.99
Saxon Geometry, 1st Edition Homeschool Testing Book
Publisher: Saxon Homeschool.
ISBN-13: 9781600329777
Unlike the Homeschool Packets in other Saxon Math curriculum, the Saxon Geometry Homeschool Testing book has the test forms and the answers for those tests. There are no answers to problems in the book.
It makes sense. There are complete solutions in the Solutions Manual which is included in the kit and is not sold separately.
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Math 1351 Foundations of Mathematics II Information
LSC-CyFair Math Department
Catalog Description This is designed specifically for students who seek elementary and middle school teacher certification. Topics include concepts of geometry, probability, and statistics, as well as applications of the algebraic properties of real numbers to concepts of measurement with an emphasis on problem solving and critical thinking.
Course Learning Outcomes The student will: • Explore the geometric attributes of physical objects in order to classify and to form definitions. • Analyze spatial characteristics such as direction, orientation, and perspective. • Connect geometric ideas to numbers and measurement. • Use geometric models to solve problems. • Explore and understand measurement and estimation. • Analyze data and statistics. • Use probability with simple and complex experiments. • Understand surface area and volume through discovery.
Billstein, Libeskind, Lott; A Problem Solving Approach to Math for Elementary Teachers, 11th ed.; Pearson Required: Students must buy an access code to MyMathLab, an online course management system which includes a complete eBook; students will first need a Course ID provided by the instructor in order to register; online purchase of MyMathLab access at hard copies of access codes available with ISBN: 9780558357603 Hardbound text (optional), ISBN: 0321756665 Hardbound text + free MyMathLab access, ISBN: 032182802X
Calculator: Graphing Calculator required. TI 83, TI 84 or TI 86 series calculators recommended. Calculators capable of symbolic manipulation will not be allowed on tests. Examples include, but are not limited to, TI 89, TI 92, and Nspire CAS models and HP 48 models. Neither cell phones nor PDA's can be used as calculators. Calculators may be cleared before tests.
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College Algebra and Trigonometry - 5th edition
Summary: This text provides a supportive environment to help students successfully learn the content of a standard algebra and trigonometry course. By incorporating interactive learning techniques, the Aufmann team helps students to better understand concepts, focus their studying habits, and obtain greater mathematical success.
Many new components added to this edition of College Algebra and Trigonometry have been designed to help students diagnose and review weak ...show morealgebra skills. Prerequisite review is include in the textbook (and supporting materials) so that instructors can spend less time covering review material and students can still fill in the gaps in their mathematical knowledge. ...show less
Section 7.1 The Law of Sines Section 7.2 The Law of Cosines and Area Section 7.3 Vectors Section 7.4 Trigonometric Form of Complex Numbers Section 7.5 De Moivre's Theorem Exploring Concepts with Technology: Optimal Branching of ArteriesCD MissingGood
Susies Books Garner, NC
2004 Hardcover COVER, CORNER WEAR This book looks good. It is like any used book you would expect to find in a used book shop.
$9.95 +$3.99 s/h
Good
invisibledog Salt Lake City, UT
0618386807 Some marking.
$9.99 +$3.99 s/h
Good
harambee Kansas city, MO
2004 Hardcover Good condition, jacket is slightly torn and taped on the corners
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Therefore, in Precalculus, students will be introduced to the important and basic mathematical concepts inquired before in algebra with deeper and higher details. They comprise, but not limited in, inequalities, equations, absolute values, and graphs of lines and circles. Students also focus on functions and their graphs.
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In this paper, we explore the use of dynamic geometry software (DGS) as a medium for
changing student and teacher interactions (and attitudes) with functions. We o er three
examples of sketches that may be used to encourage students to build their own functions.
Moreover, we share a strategy for developing additional sketches, namely our three-step
MTA process (Measure - Trace - Algebratize). Note that these steps roughly correspond
to concrete, iconic, and symbolic levels of representation proposed by Bruner (1960; 1966).
As our examples illustrate, the MTA approach provides students with opportunities to
explore and construct remarkably non-standard functions - often beautiful, unexpected,
and thoroughly original.
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A series of reports produced by the Higher Education Academy STEM project: Skills in Mathematics and Statistics in the disciplines and tackling transition are published today.
The five discipline-specific reports accompany an overarching report, Mathematical transitions, and present findings of a major project funded and run by the HEA which looked at the mathematical and statistical needs of undergraduate students in disciplines including Business and Management, Chemistry, Economics, Geography, and Sociology. Broadly speaking these are disciplines where there is a clear need for Mathematics and Statistics, but where an A-level in Mathematics is not usually a pre-requisite for acceptance at university.
The main findings of the project suggest that many undergraduate students are surprised at the amount of mathematical content in their degree programmes and some struggle to cope with this content. For some time, there has been concern in the HE sector about the mathematical and statistical skills of students entering undergraduate degree courses in these disciplines. In particular there are many questions regarding the extent to which students' skills match the actual requirements of university degree courses.
The reports consider the mathematical demands of the subjects, what the HE departments are doing to meet the students' needs, staff and student expectations and the signalling HE provides about the need for mathematical and statistical skills. Throughout a particular emphasis is placed on the transition into university study.
Dr Mary McAlinden, Discipline Lead for Mathematics, Statistics and Operational Research at the HEA says: "Mathematical and statistical skills are embedded within the university curricula of many subjects both in STEM and more widely. They are fundamental tools which students need to acquire in order to be able to understand and appreciate academic literature and research findings within their subject domains. There is, therefore, a clear need for a greater understanding between the HE and pre-university sectors about the need for students to be able to apply their mathematical and statistical skills within their subject domains."
The Higher Education Academy (HEA) invites expressions of interest from HEA subscribing higher education providers from across the UK to participate in three strategic enhancement programmes commencing in October 2014.
The three programmes – Embedding employability into the curriculum, Internationalising the curriculum, and Engaged student learning – form key elements of the HEA's four enhancement workstreams for 2014-15. A call for participation in two further strategic enhancement programmes relating to flexible learning, and retention and attainment, will be issued in October 2014.
Participating institutions will receive HEA-facilitated support for enhancement initiatives that they are taking forward during 2014-15 that align to the programme. Support will take the form of a series of network meetings tailored for each of the three programmes, plus bespoke support for each participating institution. Specific projects designed to strengthen the evidence base in the areas covered by the programmes will be commissioned by the HEA.
The deadline for submission of applications is: noon on Monday 15 September 2014.
A report presenting the findings of a study commissioned by the HEA and conducted by the National Union of Students (NUS)/NUS Services has found that on the whole students are positive about the use of OERs.
Silicon Valley has a famously rocky relationship with higher education. While many of the brightest minds in the tech industry were educated at prestigious universities such as Stanford and Harvard, many more take pride in having dropped out of their degrees to pursue start-up glory, including Steve Jobs, Mark Zuckerberg and Bill Gates. Peter Thiel, the co-founder of PayPal, is so disdainful of university education that he established a fellowship offering several young people $100,000 (£60,000) per year to drop out and start a company insteadAn independent task force has called on the Government to invest £20 million to help embed the new computing curriculum in schools, warning that the UK could struggle to fill digital roles in the future
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Calculators can perform math functions quickly and easily. The most common functions are addition, subtraction,...
see more
Calculators can perform math functions quickly and easily. The most common functions are addition, subtraction, multiplication, and division. This course will use the Touch Method, which means using the calculator without looking at the keys. Using this method will help develop competency. After building competency, students will be able to use 10-key calculators to enter numeric data and perform calculations efficiently.
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Larkfield, CA ACTThe student gains an appreciation for the power of mathematics to model the real-world. Calculators extend a student?s ability to calculate and visualize mathematics. However, it is important that the student understands the concepts that underlie what the graphing calculator produces.
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Elementary Linear Algebra - 6th edition
ISBN13:978-0618783762 ISBN10: 0618783768 This edition has also been released as: ISBN13: 978-0547004815 ISBN10: 0547004818
Summary: The cornerstone of Elementary Linear Algebra is the authors' clear, careful, and concise presentation of material--written so that students can fully understand how mathematics works. This program balances theory with examples, applications, and geometric intuition for a complete, step-by-step learning system.The Sixth Edition incorporates up-to-date coverage of Computer Algebra Systems (Maple/MATLAB/Mathematica); additional support is provided in a corresponding tec...show morehnology guide. Data and applications also reflect current statistics and examples to engage students and demonstrate the link between theory and practice.38.61 +$3.99 s/h
VeryGood
Bang-for-Bucks sterling, VA
2008-07-03 Hardcover Very Good No markings, No highlighting, clean inside and outside; We ship daily and provide tracking numbersWe Beat the Lowest Amazon Prices All the time! ! !
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SLAE Solver - SLAE SOLVER allows to find on a personal computer high accuracy solutionsSLAE SOLVER allows to find on a personal computer high accuracy solutions of linear algebraic systems with N equations, where N may reach hundreds or thousands
Alphabet Flash Cards - Learning the alphabet can be confusing to toddlers.Learning the alphabet can be confusing to toddlers. Alphabet Flash Cards helps parents teach the alphabet to their children by removing all unnecessary distractions and by focusing on the main...
Quick Guide to English Verbs - Free English4Today studyGuide: Guide to English language verbs and tenses with optional online support materials and exercises. Part of a series of free studyGuides developed by English4Today.com for school, college,university and EFL...
Satellite Antenna Alignment - The program "Satellite Antenna Alignment" is used to calculate the angles necessary for installing satellite dishes. The main difference from similar software is the possibility to calculate the position for all satellites at once.The program...
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Get everything you need for a successful and pain-free year of learning math! This kit includes Saxon's 3rd Edition Math 8/7 textbook, solutions manual, and tests/worksheets book, as well as the DIVE Math 8/7 CD-ROM. A balanced, integrated mathematics program that has proven itself a leader in the math teaching field, Math 8/7 covers concepts such as arithmetic calculation, measurements, geometry and other skills are reviewed, while new concepts such as pre-algebra, ratios, probability and statistics are introduced as preparation for upper level mathematics.
The DIVE software teaches each Saxon lesson concept step-by-step on a digital whiteboard, averaging about 10-15 minutes in length; because each lesson is stored separately, you can easily move about from lesson-to-lesson as well as maneuver within the lesson you're watching. DIVE teaches the same concepts as Saxon, but does not use the problems given in the text; it cannot be used as a solutions guide.
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This radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. Aimed at undergraduate students in mathematics, physics, and engineering, the book's intuitive explanations, lack of advanced prerequisites, and consciously user-friendly prose style will help students to master the subject more readily than was previously possible. The key to this is the book's use of new geometric arguments in place of the standard calculational ones. These geometric arguments are communicated with the aid of hundreds of diagrams of a standard seldom encountered in mathematical works. A new approach to a classical topic, this work will be of interest to students in mathematics, physics, and engineering, as well as to professionals in these fields.
Review:
The book recently won First Prize in the National Jesuit Book Award Contest for the best mathematics or computer science book published in 1994, 1995, or 1996.
Book Description:Fairlawn, New Jersey, U.S.A.: Clarendon Pr, 1997. Soft cover. Book Condition: New. Dust Jacket Condition: New. 1st66020543
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Pre-Calculus Gr. 9-12
Pre-calculus is the bridge between Algebra II and Calculus, and is a great way to get acquainted with ideas like function and rate of change. Analyze angles and geometric shapes to find absolute values. Discover new ways to record solutions with interval notation, and plug trig identities into your equations
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A unique online parsing system that produces partial-credit scoring of students' constructed responses to mathematical...
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A unique online parsing system that produces partial-credit scoring of students' constructed responses to mathematical questions is presented. The parser is the core of a free college readiness website in mathematics. The software generates immediate error analysis for each student response. The response is scored on a continuous scale, based on its overall correctness and the fraction of correct elements. The parser scoring was validated against human scoring of 207 real-world student responses (r = 0.91). Moreover, the software generates more consistent scores than teachers in some cases. The parser analysis of students' errors on 124 additional responses showed that the errors were factored into two groups: structural (possibly conceptual), and computational (could result from typographical errors). The two error groups explained 55% of students' scores variance (structural errors: 36%; computational errors: 19%). In contrast, these groups explained only 33% of the teacher score variance (structural: 18%; computational: 15%). There was a low agreement among teachers on error classification, and their classification was weakly correlated to the parser's error groups. Overall, the parser's total scoring closely matched human scoring, but the machine was found to surpass humans in systematically distinguishing between students' error patternsCalc XT is a full feature scientific calculator for iPad. It turns your iPad into a life-size realistic calculator. In...
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'Calc XT is a full feature scientific calculator for iPad. It turns your iPad into a life-size realistic calculator. In landscape mode, a memo pad is also available that you can jot note easily.''- Most scientific calculator features.- Automatically save states while application quits and restore while application restarts.- Different output mode. Normal, Scientific, Fixed, Engineering Mode.- Memo pad, with pen, highlight and eraser.- Copy to pasteboard or mail the memo.- Click on the digits display area to copy/paste the value.- Two percent modes:* scientific: 200+50%=200.5, 200*50%=100* financial: 200+50%=300, 200*50%=100- memo can set left or right hand side- multiple memos- Label tape, that you can type text on the memo or store numbers from calculator'This app costs $0.99
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Starting at $12 6Number Power Series (Revised) The first choice for those who want to develop and improve their math skills. Every Number Power book targets a particular set of math skills with straightforward explanations, easy-to-follow, step-by-step instruction, real-life examples, and extensive reinforcement exercises. Use these texts across the full scope of the basic math curriculum, from whole numbers to pre-algebra and geometry. NUMBER POWER 2: FRACTIONS, DECIMALS, AND PERCENTS Straightforward calculating and problem solving with fractions, decimals, and percents.
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11,969including precalculus precalculus precalculus
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Intermediate Algebra
Todays Developmental Math students enter college needing more than just the math, and this has directly impacted the instructors role in the ...Show synopsisTodays Developmental Math students enter college needing more than just the math, and this has directly impacted the instructors role in the classroom. Instructors have to teach to different learning styles, within multiple teaching environments, and to a student population that is mostly unfamiliar with how to be a successful college student. Authors Andrea Hendricks and Pauline Chow have noticed this growing trend in their combined 30+ years of teaching at their respective community colleges, both in their face-to-face and online courses. As a result, they set out to create course materials that help todays students not only learn the mathematical concepts but also build life skills for future success. Understanding the time constraints for instructors, these authors have worked to integrate success strategies into both the print and digital materials, so that there is no sacrifice of time spent on the math. Furthermore, Andrea and Pauline have taken the time to write purposeful examples and exercises that are student-centered, relevant to todays students, and guide students to practice critical thinking skills. Intermediate Algebra and its supplemental materials, coupled with ALEKS or Connect Math Hosted by ALEKS, allow for both full-time and part-time instructors to teach more than just the math in any teaching environment without an overwhelming amount of preparation time or even classroom time.Hide synopsis
Description:Hardcover. Instructor Edition: Same as student edition with...Hardcover. Instructor Edition: Same as student edition with additional notes or answers. New Condition. SKU: 978007336097384269.
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The Subject of Engineering Mathematics is being introduced into the Diploma Course to provide mathematical background to the students so that they can be able to grasp the engineering subjects, which they will come across in their higher classes properly. The course will give them the insight to understand and analyse the engineering problems scientifically based on Mathematics.
The subject is divided into two papers, viz. Engineering Mathematics - I and Engineering Mathematics - II. The curriculum of Engineering Mathematics - II consists of the following:
1. Calculus
2. Vector Algebra and Statics
3. Differential Equations
4. Dynamics
The details of the above broad topics have been provided in the curriculum.
Objectives
By covering the course in Engineering Mathematics - II, the students will be able to:
Know the basics of Differential and Integral Calculus, the meaning of limit, continuity and derivative of a single variable and their applications to engineering problems, the various methods of integration, how to solve simple ordinary differential equation of 1st and 2nd order, the concept of Vector Algebra, how to apply concepts of Vector Algebra to Statics, how to apply the concepts of Differential and Integral Calculus in solving the problems of Dynamics.
Understand their engineering application
Solve related simple numerical problems which will help them to understand the subject.
Integration: Integration as inverse process of differentiation, Introduction, Integration by transformation, Integration by Substitution and Integration by parts.
03
01.10
The Definite Integral: Properties of the definite integral. Problem of area by Integration method.
04
02 - Vectors and Statics
Topics
Content
Periods
02.01
Introduction to Vectors: Definition of Scalars and Vectors with example, Representation of a vector, type of vectors (Unit vector, Zero vector, negative of a vector and Equality of vectors), Addition and Substraction of vectors, Multiplication of vectors by a scalar.
03
02.02
Position vector: Position vector of a Point Resolution of vectors (coplanar vectors and space vectors) : Point of Division, Centroid of triangle.
02
02.03
Product of two vectors: Scalar or Dot Product, Vector or Cross Product. Geometrical interpretation and their properties.
04
02.04
Product of three vectors: Scalar Product of three vectors, Vector Product of three vectors and its geometrical meaning.
04
02.05
Physical application: Test of collinearity, coplanarity and linear dependence of vectors, work done as a scalar product.
Projectile: Terminology: Motion of a Projectile velocity at any point, Greatest height, Time of Flight and Horizontal Range, Two directions of projectile, Minimum Speed for a Range, Motion of a given height.
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covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here."
(David Parrott, Australian Mathematical Society)
This book offers a collection of historical essays detailing a large variety of mathematical disciplines and issues; it's accessible to a broad audience. This third edition includes new chapters on simple groups and new sections on alternating groups and the Poincare conjecture. Many more exercises have been added as well as commentary that helps place the exercises in context.
Most helpful customer reviewsMathematics and Its HistoryI wonder what is going on at canada post.This book was at missausaga on the 13 sept and was scheduled to be delivered on the 15th but now it has not moved from there. I wonder when canada post will manage to move it on from there to reach its destination....
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
13 reviews
88 of 91 people found the following review helpful
An intellectually satisfying history of mathematicsFeb. 18 2005
By
Viktor Blasjo
- Published on Amazon.com
Format: Hardcover
This is a brilliant book that conveys a beautiful, unified picture of mathematics. It is not an encyclopedic history, it is history for the sake of understanding mathematics. There is an idea behind every topic, every section makes a mathematical point, showing how the mathematical theories of today has grown inevitably from the natural problems studied by the masters of the past.
Math history textbooks of today are often enslaved by the modern curriculum, which means that they spend lots of time on the question of rigor in analysis and they feel obliged to deal with boring technicalities of the history of matrix theory and so on. This is of course the wrong way to study history. Instead, one of the great virtues of a history such as Stillwell's is that it studies mathematics the way mathematics wants to be studied, which gives a very healthy perspective on the modern customs. Again and again topics which are treated unnaturally in the usual courses are seen here in their proper setting. This makes this book a very valuable companion over the years.
Another flaw of many standard history textbooks is that they spend too much time on trivial things like elementary arithmetic, because they think it is good for aspiring teachers and, I think, because it is fashionable to deal with non-western civilisations. It gives an unsound picture of mathematics if Gauss receives as much attention as abacuses, and it makes these books useless for understanding any of the really interesting mathematics, say after 1800. Here Stillwell saves us again. The chapter on calculus is done by page 170, which is about a third of the book. A comparable point in the more mainstream book of Katz, for instance, is page 596 of my edition, which is more than two thirds into that book.
Petty details aside, the main point is the following: This is the single best book I have ever seen for truly understanding mathematics as a whole.
45 of 50 people found the following review helpful
concise and well written summary of mathematicsOct. 2 2000
By
G W Thielman
- Published on Amazon.com
Format: Hardcover
Stillwell covers a lot of ground in a short undergraduate text intended to unify various mathematical disciplines. Naturally, _Mathematics_and_its_History_ begins with the early Greeks and in particular geometry (which is how mathematics was typically expressed then). The development of algebra and polynomial forms is described followed by perspective geometry. The invention of calculus and the closely related discovery of infinite series provide the backdrop for short biographies of prominent mathematicians (mostly dead white males to multicultural deconstructionists). The development of elliptic integrals (used in solving functions with specified boundary conditions such as a Neumann problem found in fluid mechanics). The treatment then diverges to physical problems including the vibrating string and hydrodynamics, together with a note on the renown Bernoulli family. Then Stillwell returns to the esoteric in complex numbers, topology, group theory and logic with some comments on computation at the end. Some mathematicians may find the overview to lack comprehensiveness, but the book's brevity for each topic and biographical notes present a balanced approach to the more casual reader about this important field of study and how it developed.
28 of 34 people found the following review helpful
Relationship between algebra and geometryNov. 2 2003
By
Ng Chi Chun
- Published on Amazon.com
Format: HardcoverThis review is not negative based on the content of the book. This review is negative because of the poor job by the publisher in printing this book.
The problem is: the "new" copy of the book I received had 15 to 20 page ranges which were totally unprinted. To phrase it another way, all the correct number of pages were in the book but there were places where anywhere from 2 to 12 consecutive pages were not printed - not even the page numbers were printed on such page.
The material on 90%+ of the 600+ seemed fine but several of the sections I had intended to read had large (and important) gaps.
Springer this was an inferior print job - I have never seen a single book so poorly printed.
I would still like a "good" copy of this book but I will never pay for it!!
8 of 10 people found the following review helpful
Simply Outstanding!Oct. 6 2011
By
Mathbuff
- Published on Amazon.com
Format: Hardcover
Every page is filled with fresh insights, genuine scholarships, clarity, connections, and understandings. Leaves all other textbooks on history of math in the dust. Never blindly follows the crowd of other authors to repeat after each other the muddled, and often untrue, interpretations and stories. Makes me want to have a photographic memory to take in everything in the book and use them to motivate and inspire my own teaching. Also makes me want to read many of the original sources Professor Stillwell's vast scholarship has traveled through.
It's a great page-turner and at the same time a fine wine to be sipped and appreciated sentence by sentence.
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Mathematics for Elementary Teachers-Activity Manual - 3rd edition
Summary: An integral part of the text written by Beckmann herself, the Activities Manual contains fully integrated activities getting students engaged in exploring, discussing, and ultimately reaching a true understanding of mathematics. The manual is included with every new copy of the text.
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Used - Acceptable Front cover is torn, but the spine is intact and the pages look fine. 3rd Edition Not perfect, but still usable for class
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Gauss-Kronrod Integration is an adaptation of Gaussian quadrature used on some graphing calculators. This Java applet outlines the mathematical computations involved and visually demonstrates the process the calculator uses to evaluate the integral.
This utility uses the free Flash player plug-in resident in most browsers to allow the user to plot a parametrically defined surface on a customized scale and dynamically rotate the three-dimensional picture.
This collection of resources is designed to supplement a modern algebra course. They are designed to help students visualize many of the important concepts from a first semester undergraduate abstract algebra course.
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In particular, proportions are solved and linear and quadratic equations are solved and graphed. Along the way, factoring polynomials and properties of square roots are introduced. It is customary to include some introductory Geometry topics, such as the Pythagorean theorem.
| 677.169 | 1 |
Intermediate Algebra
Designed for first-year developmental math students who need support in intermediate algebra, the Fourth Edition of Intermediate Algebra owes its ...Show synopsisDesigned for first-year developmental math students who need support in intermediate algebra, the Fourth Edition of Intermediate Algebra owes its success to the hallmark features for which the Larson team is known: learning by example, accessible writing style, emphasis on visualization, and comprehensive exercise sets. These pedagogical features are carefully coordinated to ensure that students are better able to make connections between mathematical concepts and understand the content. The new Student Support Edition continues the Larson tradition of guided learning by incorporating a comprehensive range of student success materials throughout the text. Additionally, instructors and students alike can track progress with HM Assess, a new online diagnostic assessment and remediation tool from Houghton Mifflin
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Ready... Set... Calculus
"Ready.. Set.. Calculus" is a book that has been designed to
help guide incoming Science and Engineering majors in
assessing and practicing their "initial" mathematical skills.
The problems involve arithmetic, algebra, inequalities, trigonometry,
logarithms, exponentials and graph recognition and do not
require the use of a calculator. (Please note that
calculators are NOT allowed on quizzes and exams in Calculus at Rensselaer.)
The book has problems, examples and links to web pages with further help.
| 677.169 | 1 |
How to solve it; a new aspect of mathematical method by George Pólya(
Book
) 97
editions published
between
1945
and
2013
in
English and Undetermined
and held by
3,407 WorldCat member
libraries
worldwide
Outlines a method of solving mathematical problems for teachers and students based upon the four steps of understanding the problem, devising a plan, carrying out the plan, and checking the results
Mathematics and plausible reasoning by George Pólya(
Book
) 62
editions published
between
1954
and
1990
in
4
languages
and held by
1,998 WorldCat member
libraries
worldwide
"Here the author of How to Solve It explains how to become a "good guesser." Marked by G. Polya's simple, energetic prose and use of clever examples from a wide range of human activities, this two-volume work explores techniques of guessing, inductive reasoning, and reasoning by analogy, and the role they play in the most rigorous of deductive disciplines."--Book cover
Problems and theorems in analysis by George Pólya(
Book
) 90
editions published
between
1925
and
1998
in
5
languages
and held by
1,771 WorldCat member
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Mathematical methods in science by George Pólya(
Book
) 22
editions published
between
1963
and
2012
in
English
and held by
1,600 WorldCat member
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Notes on introductory combinatorics by George Pólya(
Book
) 22
editions published
between
1983
and
2010
in
English and German
and held by
1,019 WorldCat member
libraries
worldwide
Inequalities by G. H Hardy(
Book
) 61
editions published
between
1934
and
2004
in
English and Undetermined
and held by
1,010 WorldCat member
libraries
worldwide accessible to a wide audience of mathematicians
Applied combinatorial mathematics by Edwin F Beckenbach(
Book
) 14
editions published
between
1964
and
1981
in
English and Undetermined
and held by
865 WorldCat member
libraries
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Complex variables by George Pólya(
Book
) 13
editions published
between
1974
and
1984
in
English
and held by
701 WorldCat member
libraries
worldwide
The Stanford mathematics problem book: with hints and solutions by George Pólya(
Book
) 19
editions published
between
1974
and
2013
in
English
and held by
537 WorldCat member
libraries
worldwide
This volume features a complete set of problems, hints, and solutions based on Stanford University's well-known competitive examination in mathematics. It offers students at both high school and college levels an excellent mathematics workbook. Filled with rigorous problems, it assists students in developing and cultivating their logic and probability skills. 1974 edition
Problems and theorems in analysis by George Pólya(
Book
) 52
editions published
between
1972
and
2004
in
English and German
and held by
205 WorldCat member
libraries
worldwide
From the reviews: "The work is one of the real classics of this century; it has had much influence on teaching, on research in several branches of hard analysis, particularly complex function theory, and it has been an essential indispensable source book for those seriously interested in mathematical problems. These volumes contain many extraordinary problems and sequences of problems, mostly from some time past, well worth attention today and tomorrow. Written in the early twenties by two young mathematicians of outstanding talent, taste, breadth, perception, perseverence, and pedagogical skill, this work broke new ground in the teaching of mathematics and how to do mathematical research. (Bulletin of the American Mathematical Society)
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Introduction to graph theory
In recent years graph theory has emerged as a subject in its own right, as well as being an important mathematical tool in such diverse subjects as ...Show synopsisIn recent years graph theory has emerged as a subject in its own right, as well as being an important mathematical tool in such diverse subjects as operational research, chemistry, sociology and genetics. Robin Wilsonn++s book has been widely used as a text for undergraduate courses in mathematics, computer science and economics, and as a readable introduction to the subject for non-mathematicians. The opening chapters provide a basic foundation course, containing definitions and examples, connectedness, Eulerian and Hamiltonian paths and cycles, and trees, with a range of applications. This is followed by two chapters on planar graphs and colouring, with special reference to the four-colour theorem. The next chapter deals with transversal theory and connectivity, with applications to network flows. A final chapter on matroid theory ties together material from earlier chapters, and an appendix discusses algorithms and their efficiency 8vo-Between 7 3/4" and 9 3/4" Tall. 166 page...Good. No Jacket. 8vo-Between 7 3/4" and 9 3/4" Tall. 166 page book is in good condition with edgewear, slight scuffing of cover, sun fading on spine, yellowing pages, and a price sticker inside the front cover. The subject of this book has become an important mathematical tool in such diverse subjects as operational research, chemistry, sociology, and genetics. It has been used as a text for both undergraduate and graduate mathematics courses, and a readable introduction to the subject for non-mathematicians
| 677.169 | 1 |
Great idea for those of us who need or want to work at A* level. One big disappointment here is that there are no worked examples so you can't check and review your work, although it does say it was a workbook.
Broken down into manageable sections. Does not have a teach sections so is purely a workbook. Would be useful to individuals who have a tutor, friend or relative that can help them when they get stuck.
| 677.169 | 1 |
the fundamental concepts and techniques of real analysis for students in all of these areas. It helps one develop the ability to think deductively, analyse mathematical situations and extend ideas to a new context. Like the first three editions, this edition maintains the same spirit and user-friendly approach with addition examples and expansion on Logical Operations and Set Theory. There is also content revision in the following areas: introducing point-set topology before discussing continuity, including a more thorough discussion of limsup and limimf, covering series directly following sequences, adding coverage of Lebesgue Integral and the construction of the reals, and drawing student attention to possible applications wherever possible.
| 677.169 | 1 |
This textbook series is designed for grades 9-12.
A textbook for each year consists of five units. The
textbooks are labeled as Year 1, Year 2, Year 3, and Year 4. Only
Years 1, 2, and 3 were evaluated. Year 4 was not complete at the time
of the analysis.
Activities:
A typical unit engages students in a central problem
through class activities and homework. Class activities include examining
new concepts, reviewing daily homework, and solving Problems of the
Week (POWs) in which students describe their work on a problem and
explain their reasoning in write-ups. Each unit contains a section
of supplemental problems.
Assessment:
Assessments include in-class and take-home
assessments at the end of each unit. These assessments do not cover
the concepts/skills of the entire unit. They allow students to demonstrate
some of what they have learned. Some supplemental problems may be
used as assessment items as well.
| 677.169 | 1 |
MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques.a. Represent functions using function notation.b. Graph the basic functions f(x) = x
n
where n = 1 to 3, f(x) =x, f(x) =x, and f(x) = 1/x.c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes.d. Investigate and explain the characteristics of a function
:
domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimumvalues, and end behavior.e. Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior.
Goal
:
Create a scrapbook for six function families. Your scrapbook will include detailed graphs, characteristics, theeffects of transformations, and tables of values.A family photo album tells the visual story of a family.Your album will tell the visual story of 6 families
:
y
L
inear
y
Absolute Value
y
Q
uadratic
y
Cubic
y
Radical
y
RationalBecause each function is the
parent
of a
family
, youwill also investigate the functions
children
. For example, linear functions have 3children, shown below
:
So, there you have it
:
make a family album for all 6
families
, including listing their
children
and everyones
statistics
.Refer to the rubric to learn how to complete the project.[This project is a
metaphor
, which is language that connects unrelated subjects. The metaphor here is that everyfunction is described as a
family
using the language of families to help you understand functions. America is a meltingpot is an example of a metaphor.]
| 677.169 | 1 |
asio's latest and most advanced scientific calculator features new Natural Textbook Display and improved math functionality. FX-115ES PLUS has been designed to be the perfect choice for high school and college students learning General Math, Trigonometry, Statistics, Algebra I and II, Calculus, Engineering, Physics,
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Stopping times, Filtration, Martingales, The system works a lot better if you only ask one question per question. And if you want to get useful answers then it's helpful if you can indicate what you've tried and why it didn't work.
Any good books on Mathematics and Programming? For the benefit of people from different educational systems, what does "still struggling with pre-calculus" mean? Can you solve a quadratic? How about simultaneous equations? What geometry are you comfortable with?
| 677.169 | 1 |
This website is a
complete online study guide for Pre-Calculus Math 40S, which follows the Manitoba math curriculum. With
32 lessons spanning the entire curriculum, the content found
here can serve as a supplement to classroom learning or correspondence modules.
The lessons and practice exams are free for students and teachers to download
and print. Teachers may photocopy this content for use as a student handout.
Math animations and workbook for the new Manitoba math curriculum (2010).
Pre-Calculus Math 40s - Practice Exams:
Based on released questions from previous Manitoba provincial examinations, these
practice exams can be used as homework assignments or resource material for
review classes.
*The newest version of
Adobe Reader is required to view the PDF files, and may be downloaded by
clickinghere.
Linking to this site:
Educational institutions may link to this site from their website without
requesting permission. Please display the link as
Downloading Tips: Most of the files are between 1 - 3
megabytes and take about 30 seconds to download to your computer. Please be
patient! If you get a message saying a link is unavailable, refreshing the page
usually clears that up. You may also wish to save the file on your computer. Make
sure you create a folder to put the files in, as all these lessons will easily
clutter your desktop.
Printing Tips:If you would
like the page to print bigger and fill out the page more, select "none" for page
scaling when the printing window comes up. If your printer is adding color and you wish to print in black and white, the "properties"
button will take you to a screen where you can adjust this setting.
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Over 100 math formulas at high school level. The covered areas include algebra, geometry, calculus, trigonometry, probability and statistics. Most of the formulas come with examples for better understa
Remember the Star Trek computer? It's finally happening--with Wolfram|Alpha. Building on 25 years of development led by Stephen Wolfram, Wolfram|Alpha has rapidly become the world's definitive source f
Math Ref Free is a free version of the award winning education app Math Ref. This app gives you just a sample (over 700) of the over 1,400 helpful formulas, figures, tips, and examples that are include
English (Polish, see below):Mathematical Formulas is the perfect app for you who likes mathematics and easily forgets formulas which you need in certain situation. Without a good app, it's tough to rem
Formula MAX is a universal app with a collection of over 1150+ Physics, Chemistry and Maths formulas, more formulas to be added constantly through updates. Use your Formula MAX app across your iOS devi
"over 100 math formulas at high school level" - "Over 100 math formulas at high school level. The covered areas include algebra, geometry, calculus, trigonometry, probability and statistics. Most of t...
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In this year-long project, students design, "build," and "sell" a house; after which they simulate investment of the profits in the stock market. Along the way, students make scale drawings, compute w... More: lessons, discussions, ratings, reviews,...
Tutorial fee-based software for PCs that must be downloaded to the user's computer. It covers topics from pre-algebra through pre-calculus, including trigonometry and some statistics. The software pos
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Project MOSAIC
Project MOSAIC is a community of educators working to develop a new way to introduce mathematics, statistics, computation and modeling to students in colleges and universities.
Our goal: Provide a broader approach to quantitative studies that provides better support for work in science and technology. The focus of the project is to tie together better diverse aspects of quantitative work that students in science, technology, and engineering will need in their professional lives, but which are today usually taught in isolation, if at all.
Modeling. The ability to create, manipulate and investigate useful and informative mathematical representations of a real-world situations.
Statistics. The analysis of variability that draws on our ability to quantify uncertainty and to draw logical inferences from observations and experiment.
Computation. The capacity to think algorithmically, to manage data on large scales, to visualize and interact with models, and to automate tasks for efficiency, accuracy, and reproducibility.
Calculus. The traditional mathematical entry point for college and university students and a subject that still has the potential to provide important insights to today's students.
The name MOSAIC reflects the first letters — M, S, C, C — of these important components of a quantitative education. Project MOSAIC is motivated by a vision of quantitative education as a mosaic where the basic materials come together to form a complete and compelling picture.
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Contents
CHAPTER 4 The Chain Rule
4.1 Derivatives by the Chain Rule
4.2 Implicit Differentiation and Related Rates
4.3 Inverse Functions and Their Derivatives
4.4 Inverses of Trigonometric Functions
CHAPTER 5 Integrals
5.1 The Idea of the Integral 177
5.2 Antiderivatives 182
5.3 Summation vs. Integration 187
5.4 Indefinite Integrals and Substitutions 195
5.5 The Definite Integral 201
5.6 Properties of the Integral and the Average Value 206
5.7 The Fundamental Theorem and Its Consequences 213
5.8 Numerical Integration 220
CHAPTER 6 Exponentials and Logarithms
6.1 An Overview 228
6.2 The Exponential ex 236
6.3 Growth and Decay in Science and Economics 242
6.4 Logarithms 252
6.5 Separable Equations Including the Logistic Equation 259
6.6 Powers Instead of Exponentials 267
6.7 Hyperbolic Functions 277
CHAPTER 7 Techniques of Integration
7.1 Integration by Parts
7.2 Trigonometric Integrals
7.3 Trigonometric Substitutions
7.4 Partial Fractions
7.5 Improper Integrals
CHAPTER 8 Applications of the Integral
8.1 Areas and Volumes by Slices
8.2 Length of a Plane Curve
8.3 Area of a Surface of Revolution
8.4 Probability and Calculus
8.5 Masses and Moments
8.6 Force, Work, and Energy
CHAPTER 6
Exponentials and Logarithms
This chapter is devoted to exponentials like 2" and 10" and above all ex. The goal is
to understand them, differentiate them, integrate them, solve equations with them,
and invert them (to reach the logarithm). The overwhelming importance of ex makes
this a crucial chapter in pure and applied mathematics.
In the traditional order of calculus books, ex waits until other applications of the .
integral are complete. I would like to explain why it is placed earlier here. I believe
that the equation dyldx = y has to be emphasized above techniques of integration.
The laws of nature are expressed by drflerential equations, and at the center is ex. Its
applications are to life sciences and physical sciences and economics and engineering
(and more-wherever change is influenced by the present state). The model produces
a differential equation and I want to show what calculus can do.
The key is always bm+" (bm)(b3.
= Section 6.1 applies that rule in three ways:
1. to understand the logarithm as the exponent;
2. to draw graphs on ordinary and semilog and log-log paper;
3. to find derivatives. The slope of b" will use bX+*" (bx)(bh").
=
hAn Overview
6.1
There is a good chance you have met logarithms. They turn multiplication into
addition, which is a lot simpler. They are the basis for slide rules (not so important)
and for graphs on log paper (very important). Logarithms are mirror images of
exponentials-and those I know you have met.
Start with exponentials. The numbers 10 and lo2 and lo3 are basic to the decimal
system. For completeness I also include lo0, which is "ten to the zeroth power" or
1. The logarithms of those numbers are the exponents. The logarithms of 1 and 10 and
100 and 1000 are 0 and 1 and 2 and 3. These are logarithms "to base 1 , because
0"
the powers are powers of 10.
Question When the base changes from 10 to b, what is the logarithm of l ?
Answer Since b0 = 1, logJ is always zero. To base b, the logarithm of bn is n.
6.1 An Overview 229
Negative powers are also needed. The number 10x is positive, but its exponent x can
be negative. The first examples are 1/10 and 1/100, which are the same as 10-' and
10- 2 . The logarithms are the exponents -1 and -2:
1000 = 103 and log 1000 = 3
1/1000 = 10- 3
and log 1/1000 = - 3.
Multiplying 1000 times 1/1000 gives 1 = 100. Adding logarithms gives 3 + (- 3) = 0.
m "
Always 10 times 10" equals 10 +" .In particular 103 times 102 produces five tens:
(10)(10)(10) times (10)(10) equals (10)(10)(10)(10)(10) = 105.
The law for b" times b" extends to all exponents, as in 104.6 times 10'. Furthermore
the law applies to all bases (we restrict the base to b > 0 and b - 1). In every case
multiplication of numbers is addition of exponents.
6A bm times b" equals b'",so logarithms (exponents) add
b' divided by b" equals b", so logarithms (exponents) subtract
logb(yZ) = lOgby + lOgbz and logb(Y/Z) = lOgby - lOgbz. (1)
Historical note In the days of slide rules, 1.2 and 1.3 were multiplied by sliding
one edge across to 1.2 and reading the answer under 1.3. A slide rule made in
Germany would give the third digit in 1.56. Its photograph shows the numbers on a
log scale. The distance from 1 to 2 equals the distance from 2 to 4 and from 4 to 8.
By sliding the edges, you add distances and multiply numbers.
Division goes the other way. Notice how 1000/10 = 100 matches 3 - 1 = 2. To divide
1.56 by 1.3, look back along line D for the answer 1.2.
The second figure, though smaller, is the important one. When x increases by 1, 2 x
is multiplied by 2. Adding to x multiplies y. This rule easily gives y = 1, 2, 4, 8, but
look ahead to calculus-which doesn't stay with whole numbers.
Calculus will add Ax. Then y is multiplied by 2ax. This number is near 1. If
ax
Ax = A then 2" 1.07-the tenth root of 2. To find the slope, we have to consider
(2 ax - 1)/Ax. The limit is near (1.07 - 1)/- = .7, but the exact number will take time.
^ ^
2>
1 1+1 1+1+1
Fig. 6.1 An ancient relic (the slide rule). When exponents x add, powers 2x multiply.
6 Exponentials and Logarithms
Base Change Bases other than 10 and exponents other than 1,2,3, ... are needed
for applications. The population of the world x years from now is predicted to grow
by a factor close to 1.02". Certainly x does not need to be a whole number of years.
And certainly the base 1.02 should not be 10 (or we are in real trouble). This prediction
will be refined as we study the differential equations for growth. It can be rewritten
to base 10 if that is preferred (but look at the exponent):
1.02" is the same as 10('Og .02)".
When the base changes from 1.02 to 10, the exponent is multiplied-as we now see.
For practice, start with base b and change to base a. The logarithm to base a will
be written "log." Everything comes from the rule that logarithm = exponent:
base change for numbers: b = d o g b .
Now raise both sides to the power x. You see the change in the exponent:
base change for exponentials: bx = a('0g
,Ix.
Finally set y = bX.Its logarithm to base b is x. Its logarithm to base a is the exponent
on the right hand side: logay = (log,b)x. Now replace x by logby:
base change for logarithms: log, y = (log, b) (log, y ).
We absolutely need this ability to change the base. An example with a = 2 is
b = 8 = Z3 g2 = (z3), = 26 log, 64 = 3 2 = (log28)(log864).
The rule behind base changes is (am)"= am". When the mth power is raised to the
xth power, the exponents multiply. The square of the cube is the sixth power:
(a)(a)(a)times (a)(a)(a) equals (a)(a)(a)(a)(a)(a): (a3),=a6.
Another base will soon be more important than 10-here are the rules for base
changes:
The first is the definition. The second is the xth power of the first. The third is the
logarithm of the second (remember y is bx). An important case is y = a:
log, a = (log, b)(logba) = 1 so log, b = 1/log, a. (3)
EXAMPLE 8 = 23 means 8lI3 = 2. Then (10g28)(l0g82) (3)(1/3) = 1.
=
This completes the algebra of logarithms. The addition rules 6A came from
(bm)(b") bm+".The multiplication rule 68 came from (am)"= am". We still need to
=
deJine b" and ax for all real numbers x. When x is a fraction, the definition is easy.
The square root of a8 is a4 (m = 8 times x = 112). When x is not a fraction, as in 2",
the graph suggests one way to fill in the hole.
.
23141100,.. . As the fractions r approach
We could defne 2" as the limit of 23, 231110,
7t, the powers 2' approach 2". This makes y = 2" into a continuous function, with the
desired properties (2")(2") = 2"'" and (2")" = 2""-whether m and n and x are inte-
gers or not. But the E'S and 6's of continuity are not attractive, and we eventually
choose (in Section 6.4) a smoother approach based on integrals.
GRAPHS O b" AND logby
F
It is time to draw graphs. In principle one graph should do the job for both functions,
because y = bx means the same as x = logby. These are inverse functions. What one
function does, its inverse undoes. The logarithm of g(x) = bXis x:
In the opposite direction, the exponential of the logarithm of y is y:
g(g - = b('08b~)=
Y. (9
This holds for every base b, and it is valuable to see b = 2 and b = 4 on the same
graph. Figure 6.2a shows y = 2" and y = 4". Their mirror images in the 45" line give
the logarithms to base 2 and base 4, which are in the right graph.
When x is negative, y = bx is still positive. If the first graph is extended to the left,
it stays above the x axis. Sketch it in with your pencil. Also extend the second graph
down, to be the mirror image. Don't cross the vertical axis.
Fig. 6.2 Exponentials and mirror images (logarithms). Different scales for x and y.
There are interesting relations within the left figure. All exponentials start at 1,
because b0 is always 1. At the height y = 16, one graph is above x = 2 (because 4' =
16). The other graph is above x = 4 (because 24 = 16). Why does 4" in one graph equal
2," in the other? This is the base change for powers, since 4 = 2,.
The figure on the right shows the mirror image-the logarithm. All logarithms
start from zero at y = 1. The graphs go down to - co at y = 0. (Roughly speaking
2-" is zero.) Again x in one graph corresponds to 2x in the other (base change for
logarithms). Both logarithms climb slowly, since the exponentials climb so fast.
The number log, 10 is between 3 and 4, because 10 is between 23 and 24. The slope
of 2" is proportional to 2"-which never happened for xn. But there are two practical
difficulties with those graphs:
1. 2" and 4" increase too fast. The curves turn virtually straight up.
2. The most important fact about Ab" is the value of 6-and the base
doesn't stand out in the graph.
There is also another point. In many problems we don't know the function y =
f(x). We are looking for it! All we have are measured values of y (with errors mixed
in). When the values are plotted on a graph, we want to discover f(x).
Fortunately there is a solution. Scale the y axis dfferently. On ordinary graphs,
each unit upward adds a fixed amount to y. On a log scale each unit multiplies y by
6 Exponentials and Logarithms
aJixed amount. The step from y = 1 to y = 2 is the same length as the step from 3 to
6 or 10 to 20.
On a log scale, y = 11 is not halfway between 10 and 12. And y = 0 is not there at
all. Each step down divides by a fixed amount-we never reach zero. This is com-
pletely satisfactory for Abx, which also never reaches zero.
Figure 6.3 is on semilog paper (also known as log-linear), with an ordinary x axis.
The graph of y = Abx is a straight line. To see why, take logarithms of that equation:
log y = log A + x log b. (6)
The relation between x and log y is linear. It is really log y that is plotted, so the graph
is straight. The markings on the y axis allow you to enter y without looking up its
logarithm-you get an ordinary graph of log y against x.
Figure 6.3 shows two examples. One graph is an exact plot of y = 2 loX.It goes
upward with slope 1, because a unit across has the same length as multiplication by
10 going up. lox has slope 1 and 10("gb)" (which is bx) will have slope log b. The
crucial number log b can be measured directly as the slope.
Fig. 6.3 2 = 10" and 4 10-"I2 on semilog paper. Fig. 6.4 Graphs of AX^ on log-log paper.
The second graph in Figure 6.3 is more typical of actual practice, in which we start
with measurements and look for f(x). Here are the data points:
We don't know in advance whether these values fit the model y = Abx. The graph is
strong evidence that they do. The points lie close to a line with negative slope-
indicating log b < 0 and b < 1. The slope down is half of the earlier slope up, so the
6.1 An Overview
model is consistent with
y = Ado-X12 or log y = l o g A - f x . (7)
When x reaches 2, y drops by a factor of 10. At x = 0 we see A z 4.
Another model-a power y = Axk instead of an exponential-also stands out with
logarithmic scaling. This time we use log-log paper, with both axes scaled. The
logarithm of y = Axk gives a linear relation between log y and log x:
log y = log A + k log x. (8)
The exponent k becomes the slope on log-log paper. The base b makes no difference.
We just measure the slope, and a straight line is a lot more attractive than a power
curve.
4
The graphs in Figure 6.4 have slopes 3 and and - 1. They represent Ax3 and
A& and Alx. To find the A's, look at one point on the line. At x = 4 the height is
8, so adjust the A's to make this happen: The functions are x3/8 and 4& and 32/x.
On semilog paper those graphs would not be straight!
You can buy log paper or create it with computer graphics.
THE DERIVATIVES OF y = bxAND x= log,y
This is a calculus book. We have to ask about slopes. The algebra of exponents is
done, the rules are set, and on log paper the graphs are straight. Now come limits.
The central question is the derivative. What is dyldx when y = bx? What is dxldy
when x is the logarithm logby? Thpse questions are closely related, because bx and
logby are inverse functions. If one slope can be found, the other is known from
dxldy = l/(dy/dx). The problem is to find one of them, and the exponential comes
first.
You will now see that those questions have quick (and beautiful) answers, except
for a mysterious constant. There is a multiplying factor c which needs more time. I
think it is worth separating out the part that can be done immediately, leaving c in
dyldx and llc in dxldy. Then Section 6.2 discovers c by studying the special number
called e (but c # e).
I 6C The derivative of bX is a multiple ebx. The number c depends on the
base b. I
The product and power and chain rules do not yield this derivative. We are pushed
all the way back to the original definition, the limit of AylAx:
Key idea: Split bx+hinto bXtimes bh. Then the crucial quantity bx factors out. More
than that, bx comes outside the limit because it does not depend on h. The remaining
limit, inside the brackets, is the number c that we don't yet know:
This equation is central to the whole chapter: dyldx equals cbx which equals cy. The
rate of change of y is proportional to y. The slope increases in the same way that bx
increases (except for the factor c). A typical example is money in a bank, where
6 Exponentials and Logarithms
interest is proportional to the principal. The rich get richer, and the poor get slightly
richer. We will come back to compound interest, and identify b and c.
The inverse function is x = logby. Now the unknown factor is l/c:
I 6D The slope of logby is llcy with the same e (depending on b). I
Proof If dy/dx = cbx then dxldy = l/cbx = llcy. (11)
That proof was like a Russian toast, powerful but too quick! We go more carefully:
f(bx) = x (logarithm of exponential)
f '(bx)(cbx)= 1 (x derivative by chain rule)
f '(bx) = l/cbx (divide by cbx)
f '(y) = l/cy (identify bx as y)
The logarithm gives another way to find c. From its slope we can discover l/c. This
is the way that finally works (next section).
-1 0 1
Fig. 6.5 The slope of 2" is about .7 2". The slope of log2y is about 11.7 ~.
Final remark It is extremely satisfying to meet an f(y) whose derivative is llcy.
At last the " - 1 power" has an antiderivative. Remember that j'xndx = xn+'/(n 1) +
is a failure when n = - 1. The derivative of x0 (a constant) does not produce x-'.
'
We had no integral for x - , and the logarithm fills that gap. If y is replaced by x or t
(all dummy variables) then
d 1 d 1
-log,x=- and -log,t=-.
dx cx dt ct
The base b can be chosen so that c = 1. Then the derivative is llx. This final touch
comes from the magic choice b = e-the highlight of Section 6.2.
6.1 EXERCISES
Read-through questions On ordinary paper the graph of y = I is a straight line.
Its slope is m . On semilog paper the graph of y = n
In lo4 = 10,000, the exponent 4 is the a of 10,000. The is a straight line. Its slope is 0 . On log-log paper the
base is b = b . The logarithm of 10" times 10" is c . graph of y = p is a straight line. Its slope is 9 .
The logarithm of 10m/lOn is d . The logarithm of 10,000"
is e . If y = bX then x = f . Here x is any number, The slope of y = b" is dyldx = r , where c depends on
b. The number c is the limit as h - 0 of s . Since x =
,
and y is always s .
k logby is the inverse, (dx/dy)(dy/dx)= t . Knowing
A base change gives b = a -and b" = a - . Then dyldx = cb" yields dxldy = u . Substituting b" for y, the
8' is 2". In other words log2y is i times log8y. When slope of log,?; is v . With a change of letters, the slope of
y = 2 it follows that log28 times log82 equals k . log,x is w .
6.1 An Overview
Problems 1-10 use the rules for logarithms. 14 Draw semilog graphs of y = lo1-' and y = ~fi)".
1 Find these logarithms (or exponents): 15 The Richter scale measures earthquakes by loglo(I/Io)=
(a)log232 (b) logz(1/32) ( 4 log32(1/32) R. What is R for the standard earthquake of intensity I,? If
the 1989 San Francisco earthquake measured R = 7, how did
(d) (e) log, dl0-) (f) log2(l0g216) its intensity I compare to I,? The 1906 San Francisco quake
2 Without a calculator find the values of had R = 8.3. The record quake was four times as intense with
(a)310g35 (b) 3210835 R= .
(c) log, 05 + log1o2 (d) (l0g3~)(logbg) 16 The frequency of A above middle C is 440/second. The
(e) 10510-4103 (f) log256- log27 frequency of the next higher A is . Since 2'/l2 x 1.5,
the note with frequency 660/sec is
3 Sketch y = 2-" and y = g4") from -1 to 1 on the same
graph. Put their mirror images x = - log2y and x = log42y 17 Draw your own semilog paper and plot the data
on a second graph.
4 Following Figure 6.2 sketch the graphs of y = (iy and x =
Estimate A and b in y = Abx.
logl12y.What are loglI22and loglI24?
5 Compute without a computer:
18 Sketch log-log graphs of y = x2 and y = &.
(a)log23 + log2 3 (b) log2(i)10 19 On log-log paper, printed or homemade, plot y = 4, 11,
(c) log,010040 21, 32, 45 at x = 1, 2, 3, 4, 5. Estimate A and k in y = AX^.
( 4 (log1 0 4(loge10)
(e) 223/(22)3 (f logdlle)
Questions 20-29 are about the derivative dyldx = cbx.
6 Solve the following equations for x:
20 g(x) = bx has slope g' = cg. Apply the chain rule to
(a)log10(10")= 7 (b) log 4x - log 4 = log 3
g f (y))= y to prove that dfldy = llcy.
(
(c) logXlO 2
= (d) 10g2(l/x) 2
,=
(e) log x + log x = log 8 (f) logx(xx) 5
=
21 If the slope of log x is llcx, find the slopes of log (2x) and
log (x2)and log (2").
7 The logarithm of y = xn is logby= . 22 What is the equation (including c) for the tangent line to
*8 Prove that (1ogba)(logdc) (logda)(logbc).
= ?
y = 10" at x = O Find also the equation at x = 1.
9 2'' is close to lo3 (1024 versus 1000). If they were equal 23 What is the equation for the tangent line to x = log, ,y at
then log,lO would be . Also logl02 would be y = l? Find also the equation at y = 10.
instead of 0.301.
24 With b = 10, the slope of 10" is c10". Use a calculator for
10 The number 21°00has approximately how many (decimal) small h to estimate c = lim (loh- l)/h.
digits?
25 The unknown constant in the slope of y = (.l)" is
Questions 11-19 are about the graphs of y = bx and x = logby. L =lim (. l h- l)/h. (a) Estimate L by choosing a small h.
(b) Change h to -h to show that L = - c from Problem 24.
11 By hand draw the axes for semilog paper and the graphs
of y = l.lXand y = lq1.1)". 26 Find a base b for which (bh- l)/h x 1. Use h = 114 by hand
or h = 1/10 and 1/100 by calculator.
12 Display a set of axes on which the graph of y = loglox is
a straight line. What other equations give straight lines on 27 Find the second derivative of y = bx and also of x = logby.
those axes? 28 Show that C = lim (lWh- l)/h is twice as large as c =
13 When noise is measured in decibels, amplifying by a factor lim (10" - l)/h. (Replace the last h's by 2h.)
A increases the decibel level by 10 log A. If a whisper is 20db 29 In 28, the limit for b = 100 is twice as large as for b = 10.
and a shout is 70db then 10 log A = 50 and A = . So c probably involves the of b.
236 6 Exponentials and Logarithms
h
6.2 T e Exponential eX
The last section discussed bx and logby. The base b was arbitrary-it could be 2 or 6
or 9.3 or any positive number except 1. But in practice, only a few bases are used.
I have never met a logarithm to base 6 or 9.3. Realistically there are two leading
candidates for b, and 10 is one of them. This section is about the other one, which is
an extremely remarkable number. This number is not seen in arithmetic or algebra
or geometry, where it looks totally clumsy and out of place. In calculus it comes into
its own.
The number is e. That symbol was chosen by Euler (initially in a fit of selfishness,
but he was a wonderful mathematician). It is the base of the natural logarithm.
It also controls the exponential ex, which is much more important than In x.
Euler also chose 7c to stand for perimeter-anyway, our first goal is to find e.
Remember that the derivatives of bx and logby include a constant c that depends
on b. Equations (10) and (1 1) in the previous section were
d d 1
-b" = cb"
dx
and - logby = -. (1)
d~ CY
At x = 0, the graph of bx starts from b0 = 1. The slope is c. At y = 1, the graph of
logby starts from logbl = 0. The logarithm has slope llc. With the right choice of the
base b those slopes will equal 1 (because c will equal 1).
For y = 2" the slope c is near .7. We already tried Ax = .1 and found Ay z -07. The
base has to be larger than 2, for a starting slope of c = 1.
We begin with a direct computation of the slope of logby at y = 1:
1 1
- = slope
C
at 1 = lim - [logb(l
h+O h
+ h) - logbl] = hlim logb[(l + h)'lh].
-0
Always logbl = 0. The fraction in the middle is logb(l + h) times the number l/h. This
number can go up into the exponent, and it did.
The quantity (1 + h)'Ih is unusual, to put it mildly. As h + 0, the number 1 h is +
approaching 1. At the same time, l/h is approaching infinity. In the limit we have
1". But that expression is meaningless (like 010). Everything depends on the
balance bet.ween "nearly 1" and "nearly GO." This balance produces the extraordinary
number e:
DEFINITION The number e is equal to lim (1 +'h)'lh. Equivalently e = lim
h+O n+ c
o
Before computing e, look again at the slope llc. At the end of equation (2) is the
logarithm of e:
When the base is b = e, the slope is logee = 1. That base e has c = 1 as desired
1
The derivative of ex is 1 ex and the derivative of log,y is -
1 my' (4)
This is why the base e is all-important in calculus. It makes c = 1.
To compute the actual number e from (1 + h)'lh, choose h = 1, 1/10, 1/100, ... . Then
the exponents l/h are n = 1, 10, 100, .... (All limits and derivatives will become official
in Section 6.4.) The table shows (1 + h)lih approaching e as h - 0 and n - oo:
, ,
6 2 The Exponential eX
.
The last column is converging to e (not quickly). There is an infinite series that
converges much faster. We know 125,000 digits of e (and a billion digits of n). There
are no definite patterns, although you might think so from the first sixteen digits:
e = 2.7 1828 1828 45 90 45 .-. (and lle z .37).
The powers of e produce y = ex. At x = 2.3 and 5, we are close to y = 10 and 150.
The logarithm is the inversefunction. The logarithms of 150 and 10, to the base e,
are close to x = 5 and x = 2.3. There is a special name for this logarithm--the natural
logarithm. There is also a special notation "ln" to show that the base is e:
In y means the same as log,y. The natural logarithm is the exponent in ex = y.
The notation In y (or In x-it is the function that matters, not the variable) is standard
in calculus courses. After calculus, the base is generally assumed to be e. In most of
science and engineering, the natural logarithm is the automatic choice. The symbol
"exp (x)" means ex, and the truth is that the symbol "log x" generally means In x.
Base e is understood even without the letters In. But in any case of doubt-on a
calculator key for example-the symbol "ln x" emphasizes that the base is e.
THE DERIVATIVES OF ex AND In x
Come back to derivatives and slopes. The derivative of bx is cbx, and the derivative
of log, y is llcy. If b = e then c = 1 . For all bases, equation (3) is llc = logbe.
This gives c-the slope of bx at x = 0:
c = In b is the mysterious constant that was not available earlier. The slope of 2" is
In 2 times 2". The slope of ex is In e times ex (but In e = 1). We have the derivatives
on which this chapter depends:
6F The derivatives of ex and In y are ex and 1fy. For other bases
d d 1
- bx = (In b)bx and - logby= --- (6)
dx d~ (in b ) ~ '
To make clear that those derivatives come from the functions (and not at all from
the dummy variables), we rewrite them using t and x:
d d 1
-e'=ef and -lnx=-.
dt dx x
6 Exponentials and Logarithms
Remark on slopes at x = 0: It would be satisfying to see directly that the slope of 2"
is below 1, and the slope of 4" is above 1. Quick proof: e is between 2 and 4.
But the idea is to see the slopes graphically. This is a small puzzle, which is fun to
solve but can be skipped.
2" rises from 1 at x = 0 to 2 at x = 1. On that interval its average slope is 1. Its
slope at the beginning is smaller than average, so it must be less than 1-as desired.
:
On the other hand 4" rises from at x = - to 1 at x = 0. Again the average slope
,,
is L/L = 1. Since x = 0 comes at the end of this new interval, the slope of 4" at that
point exceeds 1. Somewhere between 2" and 4" is ex, which starts out with slope 1.
This is the graphical approach to e. There is also the infinite series, and a fifth
definition through integrals which is written here for the record:
1. e is the number such that ex has slope 1 at x = 0
2. e is the base for which In y = log,y has slope 1 at y = 1
:r
3. e is the limit of 1 + - as n - co
( ,
5. the area 5; x - l dx equals 1.
The connections between 1, 2, and 3 have been made. The slopes are 1 when e is the
limit of (1 + lln)". Multiplying this out wlll lead to 4, the infinite series in Section 6.6.
The official definition of in x comes from 1dxlx, and then 5 says that in e = 1. This
approach to e (Section 6.4) seems less intuitive than the others.
Figure 6.6b shows the graph of e-". It is the mirror image of ex across the vertical
axis. Their product is eXe-" = 1. Where ex grows exponentially, e-" decays
exponentially-or it grows as x approaches - co. Their growth and decay are faster
than any power of x. Exponential growth is more rapid than polynomial growth, so
that e"/xn goes to infinity (Problem 59). It is the fact that ex has slope ex which keeps
the function climbing so fast.
Fig. 6.6 ex grows between 2" and 4". Decay of e-", faster decay of e-"'I2.
The other curve is y = e-"'I2. This is the famous "bell-shaped curve" of probability
theory. After dividing by fi,
it gives the normal distribution, which applies to so
many averages and so many experiments. The Gallup Poll will be an example in
Section 8.4. The curve is symmetric around its mean value x = 0, since changing x to
- x has no effect on x2.
About two thirds of the area under this curve is between x = - 1 and x = 1. If you
pick points at random below the graph, 213 of all samples are expected in that
interval. The points x = - 2 and x = 2 are "two standard deviations" from the center,
6.2 The Exponential ex 239
enclosing 95% of the area. There is only a 5% chance of landing beyond. The decay
is even faster than an ordinary exponential, because -ix2 has replaced x.
THE DERIVATIVES OF eX AND eu x)
The slope of ex is ex. This opens up a whole world of functions that calculus can deal
with. The chain rule gives the slope of e3 x and esinx and every e"(x):
6G The derivative of euix) is eu(x) times du/dx. (8)
Special case u = cx: The derivative of e" is cecx. (9)
3 3
EXAMPLE 1 The derivative of e x is 3e x (here c = 3). The derivative of esinx is
esin x cos x (here u = sin x). The derivative of f(u(x)) is df/du times du/dx. Here
f= e"so df/du = e". The chain rule demands that secondfactor du/dx.
EXAMPLE 2 e(In 2 is the same as 2x. Its derivative is In times 2x. The chain rule
2)x 2
rediscovers our constant c = In 2. In the slope of bx it rediscovers the factor c = Inb.
Generally ecx is preferred to the original bx. The derivative just brings down the
constant c. It is better to agree on e as the base, and put all complications (like c =
b)
In up in the exponent. The second derivative of ecx is c2ecx.
EXAMPLE 3 The derivative of e-x2/2 is - xe -x 2/ 2 (here u = - x 2/2 so du/dx= - x).
EXAMPLE 4 The second derivative off= e - x2/2, by the chain rule and product rule,
is
2 - 2/2
f" (-1)
= e- 2/2 +
x (
2 2
x) 2 e-x / = ( - l)e x . (10)
Notice how the exponential survives. With every derivative it is multiplied by more
factors, but it is still there to dominate growth or decay. The points of inflection,
where the bell-shaped curve hasf" = 0 in equation (10), are x = 1 and x = - 1.
"n
EXAMPLE 5 (u = n Inx). Since en is x"in disguise, its slope must be nx -1:
slope = e""nx (n In x)= x(n) = nx (11)
This slope is correctfor all n, integer or not. Chapter 2 produced 3x2 and 4x 3 from
the binomial theorem. Now nx"- 1 comes from In and exp and the chain rule.
EXAMPLE 6 An extreme case is xx = (eInx)x. Here u = x In and we need du/dx:
x
d (x) = exnxIn x+ x- = xx(ln x + 1).
dx x)
INTEGRALS OF e" AND e" du/dx
The integral of ex is ex. The integral of ecx is not ecx. The derivative multiplies by c so
the integral divides by c. The integralof ecx is ecx/c (plus a constant).
+
EXAMPLES e2xdx - e2x + C bxdx = C
2 f Inb
6 Exponentiais and Logarithms
The first one has c = 2. The second has c = In b-remember again that bx = e('nb)x.
The integral divides by In b. In the third one, e3("+')is e3" times the number e3 and
that number is carried along. Or more likely we see e3'"+'I as eu. The missing du/dx =
3 is fixed by dividing by 3. The last example fails because duldx is not there. We
cannot integrate without duldx:
Here are three examples with du/dx and one without it:
The first is a pure eudu. So is the second. The third has u = and du/dx = l/2&,
so only the factor 2 had to be fixed. The fourth example does not belong with the
others. It is the integral of du/u2, not the integral of eudu. I don't know any way to
tell you which substitution is best-except that the complicated part is 1 + ex and it
is natural to substitute u. If it works, good.
5
Without an extra ex for duldx, the integral dx/(l + looks bad. But u = 1 + ex
is still worth trying. It has du = exdx = (u - 1)dx:
That last step is "partial fractions.'' The integral splits into simpler pieces (explained
in Section 7.4) and we integrate each piece. Here are three other integrals:
5
The first can change to - eudu/u2, hich is not much better. (It is just as impossible.)
w
The second is actually J u d u , but I prefer a split: 54ex and 5e2" are safer to do
5
separately. The third is (4e-" + l)dx, which also separates. The exercises offer prac-
tice in reaching eudu/dx - ready to be integrated.
Warning about dejinite integrals When the lower limit is x = 0, there is a natural
tendency to expect f(0) = 0-in which case the lower limit contributes nothing. For
a power f = x3 that is true. For an exponential f = e3" it is definitely not true, because
f(0) = 1:
6 2 The Exponential eX
. 241
6.2 EXERCISES
Read-through questions 24 The function that solves dyldx = - y starting from y = 1
The number e is approximately a . It is the limit of (1 + h)
at x = 0 is . Approximate by Y(x h) - Y(x)= +
to the power b . This gives l.O1lOOwhen h = c . An
- hY(x). If h = what is Y(h)after one step and what is Y ( l )
after four steps?
equivalent form is e = lim ( d )".
25 Invent three functions f, g, h such that for x > 10
When the base is b = e, the constant c in Section 6.1 is e .
Therefore the derivative of y = ex is dyldx = f . The deriv-
+
(1 llx)" <f ( x )< e" < g(x)< e2" < h(x)< xx.
ative of x = logey is dxldy = g . The slopes at x = 0 and 26 Graph ex and #
at x = - 2, -1, 0, 1, 2. Another form
y = 1 are both h . The notation for log,y is I , which offiis .
is the I logarithm of y.
Find antiderivatives for the functions in 27-36.
The constant c in the slope of bx is c = k . The function
bx can be rewritten as I . Its derivative is m . The
derivative of eU(") n . The derivative of ednX
is is 0 .
The derivative of ecxbrings down a factor P .
The integral of ex is q . The integral of ecxis r .
The integral of eU(")du/dx s . In general the integral of
is +
33 xeX2 xe-x2 34 (sin x)ecO" + (cos x)e"'""
eU(") itself is t to find.
by +
35 @ (ex)' 36 xe" (trial and error)
37 Compare e-" with e-X2.Which one decreases faster near
Find the derivatives of the functions in 1-18. x = O Where do the graphs meet again? When is the ratio of
?
e-x2 to e-X less than 1/100?
38 Compare ex with xX:Where do the graphs meet? What
are their slopes at that point? Divide xx by ex and show that
the ratio approaches infinity.
39 Find the tangent line to y = ex at x = a. From which point
on the graph does the tangent line pass through the origin?
40 By comparing slopes, prove that if x > 0 then
(a)ex> 1 + x (b)e-"> 1 - x .
41 Find the minimum value of y = xx for x >0.Show from
dZy/dx2that the curve is concave upward.
42 Find the slope of y = x1lXand the point where dy/dx = 0.
17 esinx
+ sin ex 18 x- 'Ix (which is e-) Check d2y/dx2to show that the maximum of xllx is
19 The difference between e and (1 + l/n)" is approximately 43 If dyldx = y find the derivative of e-"y by the product
Celn. Subtract the calculated values for n = 10, 100, 1000 from rule. Deduce that y(x) = Cex for some constant C.
2.7183 to discover the number C.
44 Prove that xe = ex has only one positive solution.
20 By algebra or a calculator find the limits of ( 1 + l/n)2n
and
+
(1 l / n ) 4 Evaluate the integrals in 45-54. With infinite limits, 49-50 are
...
21 The limit of (11/10)1°,(101/100)100, is e. So the limit of "improper."
(1 - l/ny.
...
(10111)1°, (100/101)100, is
(lO/ll)ll , (100/101)101,. is
..
. So the limit of
. The last sequence is 46 Jb" sin x ecoSx
dx
22 Compare the number of correct decimals of e for
(l.OO1)lOOO (l.OOO1)lOOOO if possible (l.OOOO1)lOOOOO.
and and
48 Sl 2-. dx
Which power n would give all the decimals in 2.71828?
23 The function y = ex solves dyldx = y. Approximate this
50 J; xe-.. dx
equation by A Y A x = Y; which is Y(x+ h) - Y(x)= h Y(x).
With h = & find Y(h) after one step starting from Y(0)= 1.
What is Y ( l )after ten steps?
242 6 Exponentials and Logarithms
53 :
1 cos
2sinx x dx 54 1'' (1 -ex)'' ex dx
59 This exercise shows that F(x) = x"/ex - 0 as x + m.
,
(a) Find dF/dx. Notice that F(x) decreases for x > n > 0.
The maximum of xn/e", at x = n, is nn/en.
55 Integrate the integrals that can be integrated: (b) F(2x) = (2x)"/ezx= 2"xn/eXex < 2"n"/en ex.
Deduce that F(2x) + 0 as x + bo. Thus F(x) + 0.
60 With n = 6, graph F(x) = x6/ex on a calculator or com-
puter. Estimate its maximum. Estimate x when you reach
F(x) = 1. Estimate x when you reach F(x) = 4.
61 Stirling's formula says that n! z @JZn. it to esti-
Use
56 Find a function that solves yl(x) = 5y(x) with y(0) = 2. mate 66/e6 to the nearest whole number. Is it correct? How
57 Find a function that solves yl(x) = l/y(x) with y(0) = 2.
many decimal digits in lo!?
62 x6/ex - 0 is also proved by l'H6pital's rule (at x = m):
58 With electronic help graph the function (1 + llx)". What ,
are its asymptotes? Why? lim x6/ex= lim 6xs/ex = fill this in = 0.
6.3 Growth and Decay in Science and Economics
The derivative of y = e" has taken time and effort. The result was y' = cecx, which
means that y' = cy. That computation brought others with it, virtually for free-the
derivatives of bx and x x and eu(x). I want to stay with y' = cy-which is the most
But
important differential equatibn in applied mathematics.
Compare y' = x with y' = y. The first only asks for an antiderivative of x . We quickly
find y = i x 2 + C. The second has dyldx equal to y itself-which we rewrite as dy/y =
d x . The integral is in y = x + C. Then y itself is exec. Notice that the first solution is
$x2 plus a constant, and the second solution is ex times a constant.
There is a way to graph slope x versus slope y. Figure 6.7 shows "tangent arrows,"
which give the slope at each x and y. For parabolas, the arrows grow steeper as x
1 2 1
Fig. 6.7 The slopes are y' =x and y' = y. The solution curves fit those slopes.
6.3 Growth and Decay in Science and Economics 243
grows-because y' = slope = x. For exponentials, the arrows grow steeper as y
grows-the equation is y'= slope = y. Now the arrows are connected by y = Aex.
A differential equation gives afield of arrows (slopes). Its solution is a curve that stays
tangent to the arrows - then the curve has the right slope.
A field of arrows can show many solutions at once (this comes in a differential
equations course). Usually a single Yo is not sacred. To understand the equation we
start from many yo-on the left the parabolas stay parallel, on the right the heights
stay proportional. For y' = - y all solution curves go to zero.
From y' = y it is a short step to y' = cy. To make c appear in the derivative, put c
into the exponent. The derivative of y = ecx is cecx, which is c times y. We have reached
the key equation, which comes with an initial condition-a starting value yo:
dy/dt = cy with y = Yo at t = 0. (1)
A small change: x has switched to t. In most applications time is the natural variable,
rather than space. The factor c becomes the "growth rate" or "decay rate"-and ecx
converts to ect.
The last step is to match the initial condition. The problem requires y = Yo at
t = 0. Our ec' starts from ecO = 1. The constant of integration is needed now-the
solutions are y = Ae". By choosing A = Yo, we match the initial condition and solve
equation (1). The formula to remember is yoec'.
61 The exponential law y = yoec' solves y' = cy starting from yo.
The rate of growth or decay is c. May I call your attention to a basic fact? The
formula yoec' contains three quantities Yo, c, t. If two of them are given, plus one
additional piece of information, the third is determined. Many applications have one
of these three forms: find t, find c, find yo.
1. Find the doubling time T if c = 1/10. At that time yoecT equals 2yo:
In 2 .7
e T = 2 yields cT= In 2 so that T= I --. (2)
c .1
The question asks for an exponent T The answer involves logarithms. If a cell grows
at a continuous rate of c = 10% per day, it takes about .7/.1 = 7 days to double in
size. (Note that .7 is close to In 2.) If a savings account earns 10% continuous interest,
it doubles in 7 years.
In this problem we knew c. In the next problem we know T
2. Find the decay constant c for carbon-14 if y = ½yo in T= 5568 years.
ecr = 4 yields cT= In I so that c (In 5)/5568. (3)
After the half-life T= 5568, the factor e T equals 4. Now c is negative (In = - In 2).
c
Question 1 was about growth. Question 2 was about decay. Both answers found
ecT as the ratio y(T)/y(O). Then cT is its logarithm. Note how c sticks to T.
T has the units of time, c has the units of "1/time."
Main point: The doubling time is (In 2)/c, because cT= In 2. The time to multiply
by e is 1/c. The time to multiply by 10 is (In 10)/c. The time to divide by e is - 1/c,
when a negative c brings decay.
3. Find the initial value Yo if c = 2 and y(l) = 5:
y(t) = yoec' yields Yo = y(t)e - c = 5e-2
6 Exponentials and Logarithms
.
(1.05 13)20
(1 .05l2O
2
simple interest
f
cT=ln2 5 10 15 20 years
Fig. 6.8 Growth (c > 0) and decay (c < 0. Doubling time T = (In 2)lc. Future value at 5%.
)
All we do is run the process backward. Start from 5 and go back to yo. With time
reversed, ect becomes e-". The product of e2 and e-2 is 1-growth forward and
decay backward.
Equally important is T + t. Go forward to time Tand go on to T + t:
which is (yoecT)ect.
y(T+ t) is yoec(T+t) (4)
Every step t, at the start or later, multiplies by the same ect.This uses the fundamental
property of exponentials, that eT+'= eTet.
EXAMPLE 1 Population growth from birth rate b and death rate d (both constant):
dyldt = by - dy = cy (the net rate is c = b - d).
The population in this model is yoect= yoebte-dt.It grows when b > d (which makes
c > 0). One estimate of the growth rate is c = 0.02/year:
In2 .7
The earth's population doubles in about T = -x - = 35 years.
c .02
First comment: We predict the future based on c. We count the past population
to find c. Changes in c are a serious problem for this model.
Second comment: yoectis not a whole number. You may prefer to think of bacteria
instead of people. (This section begins a major application of mathematics to economics
and the life sciences.) Malthus based his theory of human population on this equation
y' = cy-and with large numbers a fraction of a person doesn't matter so much. To
use calculus we go from discrete to continuous. The theory must fail when t is very
large, since populations cannot grow exponentially forever. Section 6.5 introduces the
logistic equation y' = cy - by2, with a competition term - by2 to slow the growth.
Third comment: The dimensions of b, c, d are "l/time." The dictionary gives birth
rate = number of births per person in a unit of time. It is a relative rate-people
divided by people and time. The product ct is dimensionless and ectmakes sense (also
-
dimensionless). Some texts replace c by 1 (lambda). Then 1/A is the growth time or
decay time or drug elimination time or diffusion time.
EXAMPLE 2 Radioactive dating A gram of charcoal from the cave paintings in
France gives 0.97 disintegrations per minute. A gram of living wood gives 6.68 disin-
tegrations per minute. Find the age of those Lascaux paintings.
The charcoal stopped adding radiocarbon when it was burned (at t = 0). The
amount has decayed to yoect.In living wood this amount is still yo, because cosmic
6.3 Growth and ÿ gay in Science and Economics
rays maintain the balance. Their ratio is ect= 0.97/6.68. Knowing the decay rate c
from Question 2 above, we know the present time t:
ct = ln (~3 5568 0.97
yields t = -in -
-.7 (6.68)
= 14,400 years.
f
Here is a related problem-the age o uranium. Right now there is 140 times as much
U-238 as U-235. Nearly equal amounts were created, with half-lives of (4.5)109 and
(0.7)109 years. Question: How long since uranium was created? Answer: Find t by
sybstituting c = (In $)/(4.5)109and C = (ln ;)/(0.7)109:
In 140
ect/ect=140 * -
ct - Ct = In 140 =. t = - 6(109) years.
c-C
EXAMPLE 3 Calculus in Economics: price inflation and the value o money
f
We begin with two inflation rates - a continuous rate and an annual rate. For the
price change Ay over a year, use the annual rate:
Ay = (annual rate) times (y) times (At). (5)
Calculus applies the continuous rate to each instant dt. The price change is dy:
k
dy = (continuous rate) times (y) times (dt). (6)
Dividing by dt, this is a differential equation for the price:
dyldt = (continuous rate) times (y) = .05y.
The solution is yoe.05'.Set t = 1. Then emo5= 1.0513 and the annual rate is 5.13%.
When you ask a bank what interest they pay, they give both rates: 8% and 8.33%.
The higher one they call the "effective rate." It comes from compounding (and depends
how often they do it). If the compounding is continuous, every dt brings an increase
of dy-and eeo8is near 1.0833.
Section 6.6 returns to compound interest. The interval drops from a month to a
day to a second. That leads to (1 + lln)", and in the limit to e. Here we compute the
effect of 5% continuous interest:
Future value A dollar now has the same value as esoST dollars in T years.
Present value A dollar in T years has the same value as e--OSTdollars now.
Doubling time Prices double (emosT= 2) in T= In 21.05 x 14 years.
With no compounding, the doubling time is 20 years. Simple interest adds on 20
times 5% = 100%. With continuous compounding the time is reduced by the factor
In 2 z -7, regardless of the interest rate.
EXAMPLE 4 In 1626 the Indians sold Manhattan for $24. Our calculations indicate
that they knew what they were doing. Assuming 8% compound interest, the original
$24 is multiplied by e.08'. After t = 365 years the multiplier is e29.2and the $24 has
grown to 115 trillion dollars. With that much money they could buy back the land
and pay off the national debt.
This seems farfetched. Possibly there is a big flaw in the model. It is absolutely
true that Ben Franklin left money to Boston and Philadelphia, to be invested for 200
years. In 1990 it yielded millions (not trillions, that takes longer). Our next step is a
new model.
6 Exponentlals and Logarithms
Question How can you estimate e2'm2 with a $24 calculator (log but not In)?
Answer Multiply 29.2 by loglo e = .434 to get 12.7. This is the exponent to base 10.
After that base change, we have or more than a trillion.
GROWTH OR DECAY WlTH A SOURCE TERM
The equation y' = y will be given a new term. Up to now, all growth or decay has
started from yo. No deposit or withdrawal was made later. The investment grew by
itself-a pure exponential. The new term s allows you to add or subtract from the
account. It is a "source"-or a "sink" if s is negative. The source s = 5 adds 5dt,
proportional to dt but not to y:
Constant source: dyldt = y + 5 starting from y = yo.
Notice y on both sides! My first guess y = et+' failed completely. Its derivative is et+'
+
again, which is not y + 5. The class suggested y = et 5t. But its derivative et + 5 is
still not y + 5. We tried other ways to produce 5 in dyldt. This idea is doomed to
failure. Finally we thought o y = Aet - 5. That has y' = Aet = y + 5 as required.
f
Important: A is not yo. Set t = 0 to find yo = A - 5. The source contributes 5et - 5:
+
The solution is (yo+ 5)e' - 5. That is the same as yOef 5(et- 1).
s = 5 multiplies the growth term ef - 1 that starts at zero. yoefgrows as before.
EXAMPLE 5 dyldt = - y + 5 has y = (yo- 5)e-' + 5. This is y0e-' + 5(1 - e-'). 7 ,lOet-5
That final term from the soul-ce is still positive. The other term yoe-' decays to zero.
The limit as t + is y, = 5 . A negative c leads to a steady state y,.
Based on these examples with c = 1 and c = -- 1, we can find y for any c and s.
Oet -5
EQUATION WlTH SOURCE 2 = cy + s starts from y = yo at t = 0.
dt (7)
5e&+5
The source could be a deposit of s = $1000/year, after an initial investment of yo = 5 =Y,
$8000. Or we can withdraw funds at s = - $200/year. The units are "dollars per year"
to match dyldt. The equation feeds in $1000 or removes $200 continuously-not all 0 -5e-'+5
at once. 1
Note again that y = e(c+s)t not a solution. Its derivative is (c + sly. The combina-
is Rgmdm9
tion y = ect+ s is also not a solution (but closer). The analysis of y' = cy + s will be
our main achievement for dzrerential equations (in this section). The equation is not
restricted to finance-far from it-but that produces excellent examples.
I propose to find y in four ways. You may feel that one way is enough.? The first
way is the fastest-only three lines-but please give the others a chance. There is no
point in preparing for real problems if we don't solve them.
Solution by Method 1 (fast way) Substitute the combination y = Aec' + B. The solu-
tion has this form-exponential plus constant. From two facts we find A and B:
the equation y' = cy + s gives cAect= c(Aect+ B) + s
the initial value at t = 0 gives A + B = yo.
tMy class says one way is more than enough. They just want the answer. Sometimes I cave
in and write down the formula: y is y,ect plus s(e" - l)/c from the source term.
6.3 Growth and Decay in Science and Economics
The first line has cAect on both sides. Subtraction leaves cB + s = 0 or B = - SIC.
,
Then the second line becomes A = yo - B = yo + (slc):
y = yoect+ -(ect - 1).
S
KEY FORMULA y = or
C
With s = 0 this is the old solution yoect (no source). The example with c = 1 and
s = 5 produced ( y o + 5)ef - 5. Separating the source term gives yo& + 5(et - 1).
Solution by Method 2 (slow way) The input yo produces the output yo@. After t
years any deposit is multiplied by ea. That also applies to deposits made after the
account is opened. If the deposit enters at time 'IS the growing time is only t - T
-
Therefore the multiplying factor is only ec(t This growth factor applies to the small
deposit (amount s d T ) made between time T and T + dT.
Now add up all outputs at time t. The output from yo is yoea. The small deposit
dT.
s dTnear time T grows to ec('-T)s The total is an integral:
This principle of Duhamel would still apply when the source s varies with time.
Here s is constant, and the integral divides by c:
That agrees with the source term from Method 1, at the end of equation (8). There
we looked for "exponential plus constant," here we added up outputs.
Method 1 was easier. It succeeded because we knew the form A&'+ B-with
"undetermined coefficients." Method 2 is more complete. The form for y is part of
the output, not the input. The source s is a continuous supply of new deposits, all
growing separately. Section 6.5 starts from scratch, by directly integrating y' = cy + s.
Remark Method 2 is often described in terms of an integrating factor. First write
the equation as y' - cy = s. Then multiply by a magic factor that makes integration
possible:
( y r - cy)e-ct = se-c' multiply by the factor e-"
S
ye-"]: = - - e - ~ t $ integrate both sides
C
S
ye - C t - yo = - - (e- C f - 1) substitute 0 and t
C
y = ectyo+ - (ect- 1 )
S
isolate y to reach formula (8)
C
The integrating factor produced a perfect derivative in line 1. I prefer Duhamel's idea,
that all inputs yo and s grow the same way. Either method gives formula (8) for y.
H
T E MATHEMATICS OF FINANCE (AT A CONTINUOUS RATE)
The question from finance is this: What inputs give what outputs? The inputs can
come at the start by yo, or continuously by s. The output can be paid at the end or
continuously. There are six basic questions, two of which are already answered.
The future value is yoect from a deposit of yo. To produce y in the future, deposit
the present value ye-". Questions 3-6 involve the source term s. We fix the continuous
6 Exponentlab and Logarithms
rate at 5% per year (c = .05), and start the account from yo = 0. The answers come
fast from equation (8).
Question 3 With deposits of s = $1000/year, how large is y after 20 years?
One big deposit yields 20,000e z $54,000. The same 20,000 via s yields $34,400.
Notice a small by-product (for mathematicians). When the interest rate is c = 0,
our formula s(ec'- l)/c turns into 010. We are absolutely sure that depositing
$1000/year with no interest produces $20,000 after 20 years. But this is not obvious
from 010. By l'H6pital's rule we take c-derivatives in the fraction:
s(ec'- 1) steC'
lim -= lim - = st. This is (1000)(20)= 20,000.
c+O C c-ro 1 (11)
Question 4 What continuous deposit of s per year yields $20,000 after 20 years?
S
.05
1000
20,000 = -(e(.0"(20) 1) requires s = - 582.
-
e- 1
-
Deposits of $582 over 20 years total $11,640. A single deposit of yo = 20,00O/e =
$7,360 produces the same $20,000 at the end. Better to be rich at t = 0.
Questions 1and 2 had s = 0 (no source). Questions 3 and 4 had yo = 0 (no initial
deposit). Now we come to y = 0. In 5, everything is paid out by an annuity. In 6,
everything is paid up on a loan.
Question 5 What deposit yo provides $1000/year for 20 years? End with y = 0.
y = yoec' + - (ec'- 1) = 0 requires yo = -(1 - e-").
S -S
C C
Substituting s = - 1000, c = .05, t = 20 gives yo x 12,640. If you win $20,000 in a
lottery, and it is paid over 20 years, the lottery only has to put in $12,640. Even less
if the interest rate is above 5%.
Question 6 What payments s will clear a loan of yo = $20,000 in 20 years?
Unfortunately, s exceeds $1000 per year. The bank gives up more than the $20,000
to buy your car (and pay tuition). It also gives up the interest on that money. You pay
that back too, but you don't have to stay even at every moment. Instead you repay
at a constant rate for 20 years. Your payments mostly cover interest at the start and
principal at the end. After t = 20 years you are even and your debt is y = 0.
This is like Question 5 (also y = O), but now we know yo and we want s:
y = yoec'+ - (ec' - 1)= 0 requires s = - cyoec'/(ec'- 1).
S
C
The loan is yo = $20,000, the rate is c = .05/year, the time is t = 20 years. Substituting
in the formula for s, your payments are $1582 per year.
Puzzle How is s = $1582 for loan payments related to s = $582 for deposits?
0 -+ $582 per year + $20,000 and $20,000 + - $1582 per year + 0.
6.3 Growth and Decay in Science and Economics 249
That difference of exactly 1000 cannot be an accident. 1582 and 582 came from
e 1 e-1
1000 • and 1000 with difference 1000 - 1000.
e-1 e-1 e-1
Why? Here is the real reason. Instead of repaying 1582 we can pay only 1000 (to
keep even with the interest on 20,000). The other 582 goes into a separate account.
After 20 years the continuous 582 has built up to 20,000 (including interest as in
Question 4). From that account we pay back the loan.
Section 6.6 deals with daily compounding-which differs from continuous com-
pounding by only a few cents. Yearly compounding differs by a few dollars.
34400
s = 1000 y'= - 3y + 6
+
20000 - 20000 s =-1582 6 2
12640 Yoo - 3 - 1
s= 582 +2
20 s =-1000 20
Fig. 6.10 Questions 3-4 deposit s. Questions 5-6 repay loan or annuity. Steady state -s/c.
TRANSIENTS VS. STEADY STATE
Suppose there is decay instead of growth. The constant c is negative and yoec" dies
out. That is the "transient" term, which disappears as t -+ co. What is left is the
"steady state." We denote that limit by y.
Without a source, y, is zero (total decay). When s is present, y, = - s/c:
6J The solution y = Yo + - e" - - approaches y, =- - when ec -*0.
At this steady state, the source s exactly balances the decay cy. In other words
cy + s = 0. From the left side of the differential equation, this means dy/dt = 0. There
is no change. That is why y, is steady.
Notice that y. depends on the source and on c-but not on yo.
EXAMPLE 6 Suppose Bermuda has a birth rate b = .02 and death rate d = .03. The
net decay rate is c = - .01. There is also immigration from outside, of s = 1200/year.
The initial population might be Yo = 5 thousand or Yo = 5 million, but that number
has no effect on yo. The steady state is independent of yo.
In this case y. = - s/c = 1200/.01 = 120,000. The population grows to 120,000 if
Yo is smaller. It decays to 120,000 if Yo is larger.
EXAMPLE 7 Newton's Law of Cooling: dy/dt = c(y - y.). (12)
This is back to physics. The temperature of a body is y. The temperature around it
is y.. Then y starts at Yo and approaches y,, following Newton's rule: The rate is
proportionalto y - y. The bigger the difference, the faster heat flows.
The equation has - cy. where before we had s. That fits with y. = - s/c. For the
solution, replace s by - cy. in formula (8). Or use this new method:
6 Exponentlab and bgariihms
Solution by Method 3 The new idea is to look at the dzrerence y - y, . Its derivative
is dy/dt, since y, is constant. But dy/dt is c(y - y,)- this is our equation. The differ-
ence starts from yo - y,, and grows or decays as a pure exponential:
d
-(y-y,)=c(y-y,) hasthesolution (y-y,)=(yo-y,)e". (13).
dt
This solves the law of cooling. We repeat Method 3 using the letters s and c:
(y + :) = c(y + :) has the solution (y + f) = (yo + :)ect. (14)
Moving s/c to the right side recovers formula (8). There is a constant term and an
exponential term. In a differential equations course, those are the "particularsolution"
and the "homogeneous solution." In a calculus course, it's time to stop.
EXAMPLE 8 In a 70" room, Newton's corpse is found with a temperature of 90". A
day later the body registers 80". When did he stop integrating (at 98.6")?
Solution Here y, = 70 and yo = 90. Newton's equation (13) is y = 20ec' 70. Then +
y = 80 at t = 1 gives 206 = 10. The rate of cooling is c = In ). Death occurred when
+
2 0 8 70 = 98.6 or ect= 1.43. The time was t = In 1.43/ln ) = half a day earlier.
6.3 EXERCISES
Read-through exercises Solve 5-8 starting from yo = 10. At what time does y increase
to 100 or drop to l?
If y' = cy then At) = a . If dyldt = 7y and yo = 4 then
y(t) = b . This solution reaches 8 at t = c . If the dou-
bling time is Tthen c = d . If y' = 3y and y(1) = 9 then yo
was e . When c is negative, the solution approaches
f astjoo. 9 Draw a field of "tangent arrows" for y' = -y, with the
solution curves y = e-" and y = - e-".
The constant solution to dyldt = y + 6 is y = g . The
general solution is y = Aet - 6. If yo = 4 then A = h . The 10 Draw a direction field of arrows for y' = y - 1, with solu-
solution of dyldt = cy + s starting from yo is y = Ae" + B = tion curves y = eX + 1 and y = 1.
i . The output from the source s is i . An input at
time T grows by the factor k at time t. Problems 11-27 involve yoect. They ask for c or t or yo.
At c = lo%, the interest in time dt is dy = 1 . This 11 If a culture of bacteria doubles in two hours, how many
equation yields At) = m . With a source term instead of hours to multiply by lo? First find c.
yo, a continuous deposit of s = 4000/year yields y = n 12 If bacteria increase by factor of ten in ten hours, how
after 10 years. The deposit required to produce 10,000 in 10 many hours to increase by 100? What is c?
years is s = 0 (exactly or approximately). An income of
4000/year forever (!) comes from yo = P . The deposit to 13 How old is a skull that contains 3 as much radiocarbon
give 4OOOIyear for 20 years is yo = 9 . The payment rate as a modern skull?
s to clear a loan of 10,000 in 10 years is r . 14 If a relic contains 90% as much radiocarbon as new mate-
The solution to y' = - 3y + s approaches y, = s . rial, could it come from the time of Christ?
15 The population of Cairo grew from 5 million to 10 million
Solve 1-4 starting from yo = 1 and from yo = - 1. Draw both in 20 years. From y' = cy find c. When was y = 8 million?
solutions on the same graph.
16 The populations of New York and Los Angeles are grow-
ing at 1% and 1.4% a year. Starting from 8 million (NY) and
6 million (LA), when will they be equal?
6.3 Growth and Decay in Sclenco and Economics 251
17 Suppose the value of $1 in Japanese yen decreases at 2% 30 Solve y' = 8 - y starting from yo and y = Ae-' + B.
per year. Starting from $1 = Y240, when will 1 dollar equal 1
yen?
18 The effect of advertising decays exponentially. If 40% Solve 31-34 with yo = 0 and graph the solution.
remember a new product after three days, find c. How long
will 20% remember it?
19 If y = 1000 at t = 3 and y = 3000 at t = 4 (exponential
growth), what was yo at t = O?
20 If y = 100 at t = 4 and y = 10 at t = 8 (exponential decay)
when will y = l? What was yo?
35 (a) What value y = constant solves dy/dt = - 2y + 12?
(b) Find the solution with an arbitrary constant A.
21 Atmospheric pressure decreases with height according to (c) What solutions start from yo = 0 and yo = lo?
dpldh = cp. The pressures at h = 0 (sea level) and h = 20 km (d) What is the steady state y,?
are 1013 and 50 millibars. Find c. Explain why p =
halfway up at h = 10. 36 Choose + +
signs in dyldt = 3y f 6 to achieve the
following results starting from yo = 1. Draw graphs.
22 For exponential decay show that y(t) is the square root of
y(0) times y(2t). How could you find y(3t) from y(t) and y(2t)? (a) y increases to GO (b) y increases to 2
(c) y decreases to -2 (d) y decreases to - GO
23 Most drugs in the bloodstream decay by y' = cy @st-
order kinetics). (a) The half-life of morphine is 3 hours. Find 37 What value y = constant solves dyldt = 4 - y? Show that
its decay constant c (with units). (b) The half-life of nicotine +
y(t) = Ae-' 4 is also a solution. Find y(1) and y, if yo = 3.
is 2 hours. After a six-hour flight what fraction remains? +
38 Solve y' = y e' from yo = 0 by Method 2, where the
24 How often should a drug be taken if its dose is 3 mg, it is deposit eT at time Tis multiplied by e'-T. The total output
cleared at c =.Ol/hour, and 1 mg is required in the blood- ',
at time t is y(t) = j eTe' - d ~ = . Substitute back to
stream at all times? (The doctor decides this level based on check y' = y + et.
body size.) 39 Rewrite y' = y + et as y' - y = et. Multiplying by e-', the
25 The antiseizure drug dilantin has constant clearance rate left side is the derivative of . Integrate both sides
,
y' = - a until y = yl . Then y' = - ayly . Solve for y(t) in two from yo = 0 to find y(t).
pieces from yo. When does y reach y,? 40 Solve y' = - y + 1 from yo = 0 by rewriting as y' + y = 1,
26 The actual elimination of nicotine is multiexponential:y = multiplying by et, and integrating both sides.
+
Aect ~ e ~The first-order equation (dldt - c)y = 0 changes
' . 41 Solve y' = y + t from yo = 0 by assuming y = Aet + Bt + C.
to the second-order equation (dldt - c)(d/dt - C)y = 0. Write
out this equation starting with y", and show that it is satisfied
by the given y.
Problems 42-57 are about the mathematics of finance.
27 True or false. If false, say what's true.
42 Dollar bills decrease in value at c = - .04 per year because
(a) The time for y = ec' to double is (In 2)/(ln c). of inflation. If you hold $1000, what is the decrease in dt
(b) If y' = cy and z' = cz then (y + 2)' = 2c(y + z). years? At what rate s should you print money to keep even?
(c) If y' = cy and z' = cz then (ylz)' = 0.
43 If a bank offers annual interest of 74% or continuous
(d)If y' = cy and z' = Cz then (yz)' = (c + C)yz. interest of 74%, which is better?
m
28 A rocket has velocity u. Burnt fuel of mass A leaves at 44 What continuous interest rate is equivalent to an annual
velocity v - 7. Total momentum is constant: rate of 9%? Extra credit: Telephone a bank for both rates
and check their calculation.
+
m = (m - Am)(v Av) + Am(u - 7).
u
45 At 100% interest (c = 1)how much is a continuous deposit
What differential equation connects m to v? Solve for v(m) not of s per year worth after one year? What initial deposit yo
v(t), starting from vo = 20 and mo = 4. would have produced the same output?
46 To have $50,000 for college tuition in 20 years, what gift
Problems 29-36 are about solutions of y' = cy + s. yo should a grandparent make now? Assume c = 10%. What
continuous deposit should a parent make during 20 years? If
29 Solve y' = 3y+ 1 with yo = 0 by assuming y = Ae3' + B the parent saves s = $1000 per year, when does he or she reach
and determining A and B. $50,000 arid retire?
252 6 Exponentials and Logarithms
47 Income per person grows 3%, the population grows 2%, Problems 58-65 approach a steady state y, as t -+ m.
the total income grows . Answer if these are (a)
58 If dyldt =- y + 7 what is y,? What is the derivative of
annual rates (b) continuous rates.
y - y,? Then y - y, equals yo - y , times .
48 When dyldt = cy + 4, how much is the deposit of 4dT at
time T worth at the later time t? What is the value at t = 2 of 59 Graph y(t) when y' = 3y - 12 and yo is
deposits 4dTfrom T= 0 to T= I? (a)below 4 (b) equal to 4 (c) above 4
49 Depositing s = $1000 per year leads to $34,400 after 20 60 The solutions to dyldt = c(y - 12) converge to y , =
years (Question 3). To reach the same result, when should you provided c is .
deposit $20,000 all at once?
61 Suppose the time unit in dyldt = cy changes from minutes
50 For how long can you withdraw s = $500/year after to hours. How does the equation change? How does dyldt =
depositing yo = $5000 at 8%, before you run dry? +
- y 5 change? How does y , change?
51 What continuous payment s clears a $1000 loan in 60
days, if a loan shark charges 1% per day continuously?
62 True or false, when y, and y, both satisfy y' = cy + s.
52 You are the loan shark. What is $1 worth after a year of
(a)The sum y = y, + y, also satisfies this equation.
continuous compounding at 1% per day? (b)The average y = $(yl + y2) satisfies the same equation.
53 You can afford payments of s = $100 per month for 48 (c) The derivative y = y; satisfies the same equation.
months. If the dealer charges c = 6%, how much can you 63 If Newton's coffee cools from 80" to 60" in 12 minutes
borrow? (room temperature 20G),find c. When was the coffee at 100G?
54 Your income is Ioe2" per year. Your expenses are Eoect
64 If yo = 100 and y(1) = 90 and y(2) = 84, what is y,?
per year. (a) At what future time are they equal? (b) If you
borrow the difference until then, how much money have you 65 If yo = 100 and y(1) = 90 and y(2) = 81, what is yr?
borrowed?
66 To cool down coffee, should you add milk now or later?
55 If a student loan in your freshman year is repaid plus 20% The coffee is at 70°C, the milk is at lo0, the room is at 20".
four years later, what was the effective interest rate?
(a) Adding 1 part milk to 5 parts coffee makes it 60". With
56 Is a variable rate mortgage with c = .09 + .001t for 20 y, = 20", the white coffee cools to y(t) = .
years better or worse than a fixed rate of lo%?
(b)The black coffee cools to y,(t) = . The milk
57 At 10% instead of 8%, the $24 paid for Manhattan is warms to y,(t) = . Mixing at time t gives
worth after 365 years. (5yc + y J 6 =-
-
6.4 Logarithms
We have given first place to ex and a lower place to In x. In applications that is
absolutely correct. But logarithms have one important theoretical advantage (plus
many applications of their own). The advantage is that the derivative of In x is l/x,
whereas the derivative of ex is ex. We can't define ex as its own integral, without
circular reasoning. But we can and do define In x (the natural logarithm) as the
integral of the " - 1 power" which is llx:
Note the dummy variables, first x then u. Note also the live variables, first x then y.
Especially note the lower limit of integration, which is 1 and not 0. The logarithm is
the area measured from 1. Therefore In 1 = 0 at that starting point-as required.
6.4 Logarithms 253
Earlier chapters integrated all powers except this "-1 power." The logarithm is
that missing integral. The curve in Figure 6.11 has height y = 1/x-it is a hyperbola.
At x = 0 the height goes to infinity and the area becomes infinite: log 0 = - 00.
The minus sign is because the integral goes backward from 1 to 0. The integral
does not extend past zero to negative x. We are defining In x only for x > O.t
1I
1 1
Fig. 6.11
x 1 a ab
Logarithm as area. Neighbors In a + In b = In ab. Equal areas: -In
In2-
1/2 1 2 4
= In 2 = In 4.
With this new approach, In x has a direct definition. It is an integral (or an area).
Its two key properties must follow from this definition. That step is a beautiful
application of the theory behind integrals.
Property 1: In ab = In a + In b. The areas from 1 to a and from a to ab combine into
a single area (1 to ab in the middle figure):
a 1 ab fab
Neighboring areas: dx + - dx - dx. (2)
x x x
The right side is In ab, from definition (1). The first term on the left is In a. The
problem is to show that the second integral (a to ab) is In b:
- d x
du = In b. (3)
We need u = 1 when x = a (the lower limit) and u = b when x = ab (the upper limit).
The choice u = x/a satisfies these requirements. Substituting x = au and dx = a du
yields dx/x = du/u. Equation (3) gives In b, and equation (2) is In a + In b = In ab.
Property2: In b" = n In b. These are the left and right sides of
{b"1 dx (?) n -Jdu. (4)
This comes from the substitution x = u". The lower limit x = 1 corresponds to u = 1,
and x = b" corresponds to u = b. The differential dx is nu"-ldu. Dividing by x = u"
leaves dx/x = n du/u. Then equation (4) becomes In b" = n In b.
Everything comes logically from the definition as an area. Also definite integrals:
3x3x
EXAMPLE I Compute - dt. Solution: In 3x - In x = In - In 3.
EXAMPLE 2 Compute
11
- dx. Solution: In 1 - In .1 = In 10. (Why?)
tThe logarithm of -1 is 7ni (an imaginary number). That is because e"'= -1. The logarithm
of i is also imaginary-it is ½7i. In general, logarithms are complex numbers.
254 6 Exponentials and Logarithms
EXAMPLE 3 Compute ' du. Solution: In e2 = 2. The area from 1 to e2 is 2.
Remark While working on the theory this is a chance to straighten out old debts.
The book has discussed and computed (and even differentiated) the functions ex and
bx and x", without defining them properly. When the exponent is an irrational number
how
like rt, do we multiply e by itself i times? One approach (not taken) is to come
closer and closer to it by rational exponents like 22/7. Another approach (taken now)
is to determine the number e' = 23.1 ... its logarithm.t Start with e itself:
by
e is (by definition) the number whose logarithm is 1
e"is (by definition) the number whose logarithm is 7r.
When the area in Figure 6.12 reaches 1, the basepoint is e. When the area reaches 7E,
the basepoint is e'. We are constructing the inverse function (which is ex). But how
do we know that the area reaches 7t or 1000 or -1000 at exactly one point? (The
area is 1000 far out at e1000 . The area is -1000 very near zero at e-100ooo0.) To define
e we have to know that somewhere the area equals 1!
For a proof in two steps, go back to Figure 6.11c. The area from 1 to 2 is more
than 1 (because 1/x is more than - on that interval of length one). The combined area
from 1 to 4 is more than 1. We come to area = 1 before reaching 4. (Actually at
e = 2.718....) Since 1/x is positive, the area is increasing and never comes back to 1.
To double the area we have to square the distance. The logarithm creeps upwards:
Inx
In x -+ oo but --*0. (5)
x
The logarithm grows slowly because ex grows so fast (and vice versa-they are
inverses). Remember that ex goes past every power x". Therefore In x is passed by
every root x'l". Problems 60 and 61 give two proofs that (In x)/xl"I approaches zero.
We might compare In x with x/. x = 10 they are close (2.3 versus 3.2). But out
At
at x = e'o the comparison is 10 against e5, and In x loses to x.
I
e
e
e 1 ex e
Fig. 6.12 Area is logarithm of basepoint. Fig. 6.13 In x grows more slowly
than x.
tChapter 9 goes on to imaginary exponents, and proves the remarkable formula e"' = - 1.
6.4 Logarithms 255
APPROXIMATION OF LOGARITHMS
The limiting cases In 0 = - co and In oo = + co are important. More important are
1=
- logarithms near the starting point In 1 = 0. Our question is: What is In (1 + x) for x
T
x near zero? The exact answer is an area. The approximate answer is much simpler.
area x If x (positive or negative) is small, then
minus
area x2/2 In (1 +x) x and ex ;1 + x.
1 1+x
The calculator gives In 1.01 = .0099503. This is close to x = .01. Between 1 and 1 + x
S= ex the area under the graph of 1/x is nearly a rectangle. Its base is x and its height is 1.
I areax2/2 So the curved area In (1 + x) is close to the rectangular area x. Figure 6.14 shows
area x how a small triangle is chopped off at the top.
The difference between .0099503 (actual) and .01 (linear approximation) is
Ox
-. 0000497. That is predicted almost exactly by the second derivative: ½ times (Ax)2
Rg. 6.14 times (In x)" is (.01)2( - 1)= - .00005. This is the area of the small triangle!
In(1 + x) . rectangular area minus triangular area = x - Ix 2.
3
The remaining mistake of .0000003 is close to x (Problem 65).
May I switch to ex? Its slope starts at eo = 1, so its linear approximation is 1 + x.
Then In (ex) %In(1 + x) x x. Two wrongs do make a right: In (ex) = x exactly.
0"1
The calculator gives e as 1.0100502 (actual) instead of 1.01 (approximation). The
second-order correction is again a small triangle: ix 2 = .00005. The complete series
for In (1 + x) and ex are in Sections 10.1 and 6.6:
In (1+x)= x- x 2 /2 + x 3 /3- ... ex = 1 + x + x 2/2+ x 3/6 + ....
DERIVATIVES BASED ON LOGARITHMS
Logarithms turn up as antiderivatives very often. To build up a collection of integrals,
we now differentiate In u(x) by the chain rule.
6K The derivative of In x is -.
1 The derivative of In u(x) is
du
x u .:x
The slope of In x was hard work in Section 6.2. With its new definition (the integral
of 1/x) the work is gone. By the Fundamental Theorem, the slope must be 1/x.
For In u(x) the derivative comes from the chain rule. The inside function is u, the
outside function is In. (Keep u > 0 to define In u.) The chain rule gives
d 1 1 ( !) d 3
dIn cx= -c- In X 3 = 3x 2
/x 3 =3
dx cx x dx x
d
d -sin x
d In (x 2 + 1)= 2x/(x 2 + 1) in cos x - tan x
dx dx cos x
d 11
In ex = exlex = 1 In (In x)= I
dx dx In x x
Those are worth another look, especially the first. Any reasonable person would
expect the slope of In 3x to be 3/x. Not so. The 3 cancels, and In 3x has the same
slope as In x. (The real reason is that In 3x = In 3 + In x.) The antiderivative of 3/x is
not In 3x but 3 In x, which is In x 3.
6 Exponentials and Logarithms
Before moving to integrals, here is a new method for derivatives: logarithmic dzreren-
tiation or LD. It applies to products and powers. The product and power rules are
always available, but sometimes there is an easier way.
Main idea: The logarithm of a product p(x) is a sum of logarithms. Switching to
In p, the sum rule just adds up the derivatives. But there is a catch at the end, as you
see in the example.
EXAMPLE 4 Find dpldx if p(x) = xxJx - 1. Here ln p(x) = x in x + f ln(x - 1).
1 1
ld
Take the derivative of In p: --p = x . - + l n x + -
pdx x 2(x - 1)'
Now multiply by p(x):
The catch is that last step. Multiplying by p complicates the answer. This can't be
helped-logarithmic differentiation contains no magic. The derivative of p =fg is the
same as from the product rule: In p = l n f + In g gives
For p = xex sin x, with three factors, the sum has three terms:
In p = l n x + x + l n sin x and p l = p
L
We multiply p times pl/p (the derivative of In p). Do the same for powers:
AE
INTEGRALS B S D ON LOGARITHMS
Now comes an important step. Many integrals produce logarithms. The foremost
example is llx, whose integral is In x. In a certain way that is the only example, but
its range is enormously extended by the chain rule. The derivative of In u(x) is uf/u,
so the integral goes from ul/u back to In u:
dx = ln u(x) or equivalently = In u.
Try to choose u(x) so that the integral contains duldx divided by u.
EXAMPLES
6.4 Logarithms
Final remark When u is negative, In u cannot be the integral of llu. The logarithm
is not defined when u < 0. But the integral can go forward by switching to - u:
jdu? I-du/dx
d x = - = In(- u).
dx
-U
Thus In(- u) succeeds when In u fails.? The forbidden case is u = 0. The integrals In u
and In(- u), on the plus and minus sides of zero, can be combined as lnlul. Every
integral that gives a logarithm allows u < 0 by changing to the absolute value lul:
The areas are -1 and -In 3. The graphs of llx and l/(x - 5) are below the x axis.
We do not have logarithms of negative numbers, and we will not integrate l/(x - 5)
from 2 to 6. That crosses the forbidden point x = 5, with infinite area on both sides.
The ratio dulu leads to important integrals. When u = cos x or u = sin x, we are
integrating the tangent and cotangent. When there is a possibility that u < 0, write
the integral as In lul.
Now we report on the secant and cosecant. The integrals of llcos x and llsin x
also surrender to an attack by logarithms - based on a crazy trick:
1 sec dx = 1 GeC + sec x
+
tan x
tan x) dx = In isec x + tan XI. (9)
1 CSC j
x dx = csc x
csc x - cot x
(CSC - X) dx = ln csc x - cot xi. (10)
+
Here u = sec x + tan x is in the denominator; duldx = sec x tan x sec2 x is above it.
The integral is In lul. Similarly (10) contains duldx over u = csc x - cot x.
In closing we integrate In x itself. The derivative of x In x is In x + 1. To remove
I
the extra 1, subtract x from the integral: ln x dx = x in x -x.
In contrast, the area under l/(ln x) has no elementary formula. Nevertheless it is
the key to the greatest approximation in mathematics-the prime number theorem.
The area J: dxlln x is approximately the number o primes between a and b. Near eloo0,
f
about 1/1000 of the integers are prime.
6.4 EXERCISES
Read-through questions e . As x + GO, In x approaches f . But the ratio
The natural logarithm of x is a . This definition leads
(ln x)/& approaches g . The domain and range of in x
are h .
to In xy = b and In xn = c . Then e is the number
whose logarithm (area under llx curve) is d . Similarly
ex is now defined as the number whose natural logarithm is The derivative of In x is I . The derivative of ln(1 + x)
x
?The integral of llx (odd function) is In 11 (even function). Stay clear of x = 0.
258 6 Exponentials and logarithms
is I . The tangent approximation to ln(1 + x) at x = 0 is
k . The quadratic approximation is I . The quadratic
approximation to ex is m .
The derivative of In u(x) by the chain rule is n . Thus
(ln cos x)' = 0 . An antiderivative of tan x is P . The
Evaluate 37-42 by any method.
product p = x e5" has In p = q . The derivative of this equ-
ation is r . Multiplying by p gives p' = s , which is
LD or logarithmic differentiation.
The integral of ul(x)/u(x) is t . The integral of
2x/(x2+ 4) is u . The integral of llcx is v . The integ-
+
ral of l/(ct s)is w . The integral of l/cos x, after a trick,
is x . We should write In 1 1 for the antiderivative of llx,
x
since this allows Y . Similarly Idu/u should be written
d
41 - ln(sec x + tan x) +
42 lsec2x sec x tan x
dx
2 .
dx sec x + tan x
Find the derivative dyldx in 1-10. Verify the derivatives 43-46, which give useful antiderivatives:
3 y=(ln x)-' 4 y = (ln x)/x
5 y = x ln x - x d x-a 2a
6 y=loglox 44 -In - --
dx (x + a) - (X2 a')
-
Find the indefinite (or definite) integral in 11-24.
Estimate 47-50 to linear accuracy, then quadratic accuracy,
by ex x 1 + x + ix2. Then use a calculator.
In(' ex- 1
51 Compute lim - 52 Compute lim -
+
x+O x x-ro x
bX- 1
53 Compute lim logdl x, 9 Compute lim -
x x
+
x+O x-ro
19 1- cos x dx
sin x
55 Find the area of the "hyperbolic quarter-circle" enclosed
byx=2andy=2abovey=l/x.
56 Estimate the area under y = l/x from 4 to 8 by four upper
21
I tan 3x dx 22
I
cot 3x dx
rectangles and four lower rectangles. Then average the
answers (trapezoidal rule). What is the exact area?
1
57 Why is - + - +
1
--•
1
+ - near In n? Is it above or below?
2 3 n
58 Prove that ln x < 2(& - 1)for x > 1. Compare the integ-
rals of l/t and 1 1 4 , from 1 to x.
25 Graph y = ln (1 x) + 26 Graph y = In (sin x)
59 Dividing by x in Problem 58 gives (In x)/x < 2(&
- l)/x.
Deduce that (In x)/x - 0 as x - co.Where is the maximum
, ,
Compute dyldx by differentiating In y. This is LD:
of (In x)/x?
27 y=,/m 28 Y=,/m
Jn 60 Prove that (In x)/xlln also approaches zero. (Start with
29 y = esinx 30 =x-llx (In xlln)/xlln- 0 )Where is its maximum?
, .
6.5 Separable Equations Including the Logistic Equation 259
61 For any power n, Problem 6.2.59 proved ex > xnfor large 70 The slope of p = xx comes two ways from In p = x In x:
x. Then by logarithms, x > n In x. Since (In x)/x goes below 1 Logarithmic differentiation (LD): Compute (In p)' and
l/n and stays below, it converges to . multiply by p.
62 Prove that y In y approaches zero as y -+ 0, by changing 2 Exponential differentiation (ED): Write xX as eXlnX,
y to llx. Find the limit of yY(take its logarithm as y + 0). take its derivative, and put back xx.
What is .I.' on your calculator? 71 If p = 2" then In p = . LD gives p' = (p)(lnp)' =
63 Find the limit of In x/log,,x as x + co. . ED gives p = e and then p' = .
64 We know the integral th-' dt = [th/h]Z = (xh- l)/h. 72 Compute In 2 by the trapezoidal rule and/or Simpson's
Its limit as h + 0 is . rule, to get five correct decimals.
65 Find linear approximations near x = 0 for e-" and 2". 73 Compute In 10 by either rule with Ax = 1, and compare
with the value on your calculator.
66 The x3 correction to ln(1 + x) yields x - i x 2 + ix3. Check
that In 1.01 x -0099503and find In 1.02. 74 Estimate l/ln 90,000, the fraction of numbers near 90,000
that are prime. (879 of the next 10,000 numbers are actually
67 An ant crawls at 1foot/second along a rubber band whose prime.)
original length is 2 feet. The band is being stretched at 1
footlsecond by pulling the other end. At what time T, ifever, 75 Find a pair of positive integers for which xY= yx. Show
does the ant reach the other end? how to change this equation to (In x)/x = (In y)/y. So look for
One approach: The band's length at time t is t + 2. Let y(t) two points at the same height in Figure 6.13. Prove that you
be the fraction of that length which the ant has covered, and have discovered all the integer solutions.
explain *76 Show that (In x)/x = (In y)/y is satisfied by
(a) y' = 1/(t + 2) (b)y = ln(t + 2) - ln 2 (c) T = 2e - 2.
68 If the rubber band is stretched at 8 feetlsecond, when if
ever does the same ant reach the other end?
+
69 A weaker ant slows down to 2/(t 2) feetlsecond, so y' = with t # 0. Graph those points to show the curve xY= y. It
'
+
2/(t 2)2. Show that the other end is never reached. crosses the line y = x at x = , where t + co.
6.5 Separable Equations Including the Logistic Equation
This section begins with the integrals that solve two basic differential equations:
dy -
- - CY and dy
- - cy + s.
dt dt
We already know the solutions. What we don't know is how to discover those solu-
tions, when a suggestion "try eC"' has not been made. Many important equations,
including these, separate into a y-integral and a t-integral. The answer comes directly
from the two separate integrations. When a differential equation is reduced that far-
to integrals that we know or can look up-it is solved.
One particular equation will be emphasized. The logistic equation describes the
speedup and slowdown of growth. Its solution is an S-curve, which starts slowly,
rises quickly, and levels off. (The 1990's are near the middle of the S, if the
prediction is correct for the world population.) S-curves are solutions to nonlinear
equations, and we will be solving our first nonlinear model. It is highly important
in biology and all life sciences.
6 Exponeniials and Logarithms
SEPARABLE EQUNIONS
The equations dyldt = cy and dyldt = cy + s (with constant source s) can be solved
by a direct method. The idea is to separate y from t:
9= c dt
Y
and - c dt.
dy -
Y + (sld
All y's are on the left side. All t's are on the right side (and c can be on either side).
This separation would not be possible for dyldt = y + t.
Equation (2) contains differentials. They suggest integrals. The t-integrals give ct
and the y-integrals give logarithms:
In y = ct + constant and In (3)
The constant is determined by the initial condition. At t = 0 we require y = yo, and the
right constant will make that happen:
lny=ct+lnyo and
( 3 (
In y + - = c t + l n y o + - .
3
Then the final step isolates y. The goal is a formula for y itself, not its logarithm, so
take the exponential of both sides (elnyis y):
y = yoeC' and y +: + = (yo :)ec'.
It is wise to substitute y back into the differential equation, as a check.
This is our fourth method for y' = cy + s. Method 1 assumed from the start that
+
y = Aect B. Method 2 multiplied all inputs by their growth factors ec('- )' and added
up outputs. Method 3 solved for y - y,. Method 4 is separation of variables (and all
methods give the same answer). This separation method is so useful that we repeat
its main idea, and then explain it by using it.
To solve dyldt = u(y)v(t), separate dy/u(y)from v(t)dt and integrate both sides:
Then substitute the initial condition to determine C, and solve for y(t).
EXAMPLE I dyldt = y2 separates into dyly2 = dt. Integrate to reach - l/y = t + C.
Substitute t = 0 and y = yo to find C = - l/yo. Now solve for y:
1
--=
1 Yo
t-- and y=-.
Y Yo 1 - tYo
This solution blows up (Figure 6.15a) when t reaches lly,. If the bank pays interest
on your deposit squared (y' = y2), you soon have all the money in the world.
EXAMPLE 2 dyldt = ty separates into dy/y = t dt. Then by integration in y = f t2 + C.
Substitute t = 0 and y = yo to find C = In yo. The exponential of *t2 + In yo gives
y = yoe'2'2. When the interest rate is c = t, the exponent is t2/2.
EXAMPLE 3 dyldt = y + t is not separable. Method 1 survives by assuming y =
6.5 Separable Equations Including the Logistic Equation
I I
I
blowup times r =
I
l
Yo
0 1 2 0 1
dy dy dy
Fig. 6.15 The solutions to separable equations - = y2 and - = n- or -= n-.dt
Y
dt d t t y t
+
Ae' B + Dt-with an extra coefficient D in Problem 23. Method 2 also succeeds-
but not the separation method.
E A P E 4 Separate dyldt = nylt into dyly = n dtlt. By integration In y = n In t + C.
XML
Substituting t = 0 produces In 0 and disaster. This equation cannot start from time
zero (it divides by t). However y can start from y, at t = 1, which gives C = In y, . The
,
solution is a power function y = y t ".
This was the first differential equation in the book (Section 2.2). The ratio of dyly
to dtlt is the "elasticity" in economics. These relative changes have units like
dollars/dollars-they are dimensionless, and y = tn has constant elasticity n.
On log-log paper the graph of In y = n In t + C is a straight line with slope n.
THE LOGISTIC EQUATION
The simplest model of population growth is dyldt = cy. The growth rate c is the birth
rate minus the death rate. If c is constant the growth goes on forever-beyond the
point where the model is reasonable. A population can't grow all the way to infinity!
Eventually there is competition for food and space, and y = ectmust slow down.
The true rate c depends on the population size y. It is a function c(y) not a constant.
The choice of the model is at least half the problem:
Problem in biology or ecology: Discover c(y).
Problem in mathematics: Solve dyldt = c(y)y.
Every model looks linear over a small range of y's-but not forever. When the rate
drops off, two models are of the greatest importance. The Michaelis-Menten equation
has c(y) = c/(y + K). The logistic equation has c(y) = c - by. It comes first.
The nonlinear effect is from "interaction." For two populations of size y and z, the
number of interactions is proportional to y times z. The Law of Mass Action produces
a quadratic term byz. It is the basic model for interactions and competition. Here we
have one population competing within itself, so z is the same as y. This competition
slows down the growth, because - by2 goes into the equation.
The basic model of growth versus competition is known as the logistic equation:
Normally b is very small compared to c. The growth begins as usual (close to ect).
The competition term by2 is much smaller than cy, until y itselfgets large. Then by2
6 Exponentlals and Logarithms
(with its minus sign) slows the growth down. The solution follows an S-curve that
we can compute exactly.
What are the numbers b and c for human population? Ecologists estimate the
natural growth rate as c = .029/year. That is not the actual rate, because of b. About
1930, the world population was 3 billion. The cy term predicts a yearly increase of
(.029)(3billion) = 87 million. The actual growth was more like dyldt = 60 millionlyear.
That difference of 27 millionlyear was by2:
27 millionlyear = b(3 b i l l i ~ n leads to b = 3 10- 12/year.
)~
Certainly b is a small number (three trillionths) but its effect is not small. It reduces
87 to 60. What is fascinating is to calculate the steady state, when the new term by2
equals the old term cy. When these terms cancel each other, dyldt = cy - by2 is zero.
The loss from competition balances the gain from new growth: cy = by2 and y = c/b.
The growth stops at this equilibrium point-the top of the S-curve:
c .029
Y,=T;= -1012= 10 billion people.
3
According to Verhulst's logistic equation, the world population is converging to 10
billion. That is from the model. From present indications we are growing much faster.
We will very probably go beyond 10 billion. The United Nations report in Section 3.3
predicts 11 billion to 14 billion.
Notice a special point halfway to y, = clb. (In the model this point is at 5 billion.)
It is the inflection point where the S-curve begins to bend down. The second derivative
d2y/dt2is zero. The slope dyldt is a maximum. It is easier to find this point from the
differential equation (which gives dyldt) than from y. Take one more derivative:
y" = (cy - by2)' = cy' - 2byy' = (c - 2by)y'. (8)
The factor c - 2by is zero at the inflection point y = c/2b, halfway up the S-curve.
THE S-CURVE
The logistic equation is solved by separating variables y and t:
J
dyldt = cy - by2 becomes dy/(cy - by2)= dt. )
The first question is whether we recognize this y-integral. No. The second question
is whether it is listed in the cover of the book. No. The nearest is Idx/(a2 - x2),which
can be reached with considerable manipulation (Problem 21). The third question is
whether a general method is available. Yes. "Partial fractions" is perfectly suited to
l/(cy - by2), and Section 7.4 gives the following integral of equation (9):
Y
In-=ct+C andthen Yo
In-=C.
c - by (10)
c - YO
That constant C makes the solution correct at t = 0. The logistic equation is integ-
rated, but the solution can be improved. Take exponentials of both sides to remove
the logarithms:
-- - ect Yo
y
c-by c-byo'
This contains the same growth factor ec' as in linear equations. But the logistic
6.5 Separable Equations Including the Logistic Equation 263
equation is not linear-it is not y that increases so fast. According to (ll), it is
y/(c - by) that grows to infinity. This happens when c - by approaches zero.
The growth stops at y = clb. That is the final population of the world (10 billion?).
We still need a formula for y. The perfect S-curve is the graph of y = 1/(1 + e-'). It
equals 1 when t = oo, it equals 4 when t = 0, it equals 0 when t = - co. It satisfies
y' = y - y2, with c = b = 1. The general formula cannot be so beautiful, because it
allows any c, b, and yo. To find the S-curve, multiply equation (11) by c - by and
solve for y:
When t approaches infinity, e-" approaches zero. The complicated part of the for-
mula disappears. Then y approaches its steady state clb, the asymptote in Figure 6.16.
The S-shape comes from the inflection point halfway up.
1 2 3 4 1988
Fig. 6.16 The standard S-curve y = 1/(1 + e - ' ) . The population S-curve (with prediction).
Surprising observation: z = l/y satisjes a linear equation. By calculus z' = - y'/y2. So
This equation z' = - cz + b is solved by an exponential e-" plus a constant:
Year US Model
Population
1790 3.9 = 3.9
1800 5.3 5.3 Turned upside down, y = l/z is the S-curve (12). As z approaches blc, the S-curve
1810 7.2 7.2 approaches clb. Notice that z starts at l/yo.
1820 9.6 9.8
1830 12.9 13.1 EXAMPLE 1 (United States population) The table shows the actual population and
1840 17.1 17.5
the model. Pearl and Reed used census figures for 1790, 1850, and 1910 to compute
1850 23.2 = 23.2
c and b. In between, the fit is good but not fantastic. One reason is war-another is
1860 31.4 30.4
1870 38.6 39.4 depression. Probably more important is immigration."fn fact the Pearl-Reed steady
1880 50.2 50.2 state c/b is below 200 million, which the US has already passed. Certainly their model
1890 62.9 62.8 can be and has been improved. The 1990 census predicted a stop before 300 million.
1900 76.0 76.9 For constant immigration s we could still solve y' = cy - by2 + s by partial fractions-
1910 92.0 = 92.0 but in practice the computer has taken over. The table comes from Braun's book
1920 105.7 107.6 DifSerentiaE Equations (Springer 1975).
1930 122.8 123.1
1940 131.7 # 136.7
1950 150.7 149.1 ?Immigration does not enter for the world population model (at least not yet).
6 Exponentials and Logarithms
Remark For good science the y2 term should be explained and justified. It gave a
nonlinear model that could be completely solved, but simplicity is not necessarily
truth. The basic justification is this: In a population of size y, the number of encounters
is proportional to y2. If those encounters are fights, the term is - by2. If those
encounters increase the population, as some like to think, the sign is changed. There
is a cooperation term + by2, and the population increases very fast.
EXAMPLE 5 y' = cy + by2: y goes to infinity in afinite time.
EXAMPLE 6 y' = - dy + by2: y dies to zero if yo < dlb.
In Example 6 death wins. A small population dies out before the cooperation by2
can save it. A population below dlb is an endangered species.
The logistic equation can't predict oscillations-those go beyond dyldt =f(y).
The y line Here is a way to understand every nonlinear equation y' =f(y). Draw a
" y line." Add arrows to show the sign of f(y). When y' =f ( y ) is positive, y is increasing
(itfollows the arrow to the right). When f is negative, y goes to the left. When f is zero,
the equation is y' = 0 and y is stationary:
y' = cy - by2 (this is f(y)) y' = - dy + by2 (this is f(y))
The arrows take you left or right, to the steady state or to infinity. Arrows go toward
stable steady states. The arrows go away, when the stationary point is unstable. The
y line shows which way y moves and where it stops.
The terminal velocity of a falling body is v, = &
in Problem 6.7.54. For f ( y ) =
sin y there are several steady states:
falling body: dvldt = g - v2 dyldt = sin y
EXAMPLE 7 Kinetics of a chemical reaction mA + nB -+ pC.
The reaction combines m molecules of A with n molecules of B to produce p
molecules of C. The numbers m, n, p are 1, 1,2 for hydrogen chloride: H, + C1, =
2 HCl. The Law of Mass Action says that the reaction rate is proportional to the
product of the concentrations [ A ] and [ B ] . Then [ A ] decays as [ C ] grows:
d[A]/dt= - r [ A ][ B ] and d [Clldt = + k [ A ][ B ] .
(15)
Chemistry measures r and k. Mathematics solves for [ A ] and [ C ] . Write y for the
concentration [ C ] , the number of molecules in a unit volume. Forming those y
molecules drops the concentration [ A ] from a, to a, - (m/p)y. Similarly [B] drops
from b, to b, - (n/p)y.The mass action law (15)contains y2:
6.5 Separable Equations Including the laglttlc Equation
This fits our nonlinear model (Problem 33-34). We now find this same mass action
in biology. You recognize it whenever there is a product of two concentrations.
THE MM EQUATION wdt=- cy/(y+ K)
Biochemical reactions are the keys to life. They take place continually in every living
organism. Their mathematical description is not easy! Engineering and physics go
far with linear models, while biology is quickly nonlinear. It is true that y' = cy is
extremely effective in first-order kinetics (Section 6.3), but nature builds in a nonlinear
regulator.
It is enzymes that speed up a reaction. Without them, your life would be in slow
motion. Blood would take years to clot. Steaks would take decades to digest. Calculus
would take centuries to learn. The whole system is awesomely beautiful-DNA tells
amino acids how to combine into useful proteins, and we get enzymes and elephants
and Isaac Newton.
Briefly, the enzyme enters the reaction and comes out again. It is the catalyst. Its
combination with the substrate is an unstable intermediate, which breaks up into a
new product and the enzyme (which is ready to start over).
Here are examples of catalysts, some good and some bad.
The platinum in a catalytic converter reacts with pollutants from the car engine.
(But platinum also reacts with lead-ten gallons of leaded gasoline and you
can forget the platinum.)
Spray propellants (CFC's) catalyze the change from ozone (03) into ordinary
oxygen (0J. This wipes out the ozone layer-our shield in the atmosphere.
Milk becomes yoghurt and grape juice becomes wine.
Blood clotting needs a whole cascade of enzymes, amplifying the reaction at
every step. In hemophilia-the "Czar's diseasew-the enzyme called Factor VIII
is missing. A small accident is disaster; the bleeding won't stop.
Adolph's Meat Tenderizer is a protein from papayas. It predigests the steak.
The same enzyme (chymopapain) is injected to soften herniated disks.
Yeast makes bread rise. Enzymes put the sour in sourdough.
Of course, it takes enzymes to make enzymes. The maternal egg contains the material
for a cell, and also half of the DNA. The fertilized egg contains the full instructions.
We now look at the Michaelis-Menten (MM) equation, to describe these reactions.
It is based on the Law o Mass Action. An enzyme in concentration z converts a
f
substrate in concentration y by dyldt = - byz. The rate constant is 6, and you see
the product of "enzyme times substrate." A similar law governs the other reactions
(some go backwards). The equations are nonlinear, with no exact solution. It is
typical of applied mathematics (and nature) that a pattern can still be found.
What happens is that the enzyme concentration z(t) quickly drops to z, K/(y + K).
The Michaelis constant K depends on the rates (like 6) in the mass action laws.
Later the enzyme reappears (z, = 2,). But by then the first reaction is over. Its law
of mass action is effectively
with c =.bz,K. This is the Michaelis-Menten equation-basic to biochemistry.
The rate dyldt is all-important in biology. Look at the function cy/(y + K):
when y is large, dyldt x -c when y is small, dyldt x - cylK.
6 Exponentials and Logarithms
The start and the finish operate at different rates, depending whether y dominates K
or K dominates y. The fastest rate is c.
A biochemist solves the MM equation by separating variables:
S y d y =
Set t = 0 as usual. Then C = yo
-
S+
c dt gives y + K In y = - ct + C.
K In yo. The exponentials of the two sides are
We don't have a simple formula for y. We are lucky to get this close. A computer
can quickly graph y(t)-and we see the dynamics of enzymes.
Problems 27-32 follow up the Michaelis-Menten theory. In science, concentrations
and rate constants come with units. In mathematics, variables can be made dimen-
sionless and constants become 1. We solve d v d T = Y/(Y + 1) and then $witch back
to y, t, c, K. This idea applies to other equations too.
Essential point: Most applications of calculus come through dzrerential equations.
That is the language of mathematics-with populations and chemicals and epidemics
obeying the same equation. Running parallel to dyldt = cy are the difference equations
that come next.
6.5 EXERCISES
Read-through questions
The equations dy/dt = cy and dyldt = cy + s and dyldt =
u(y)v(t) are called a because we can separate y from t.
1
Integration of idyly = c dt gives b . Integration of
6 dy/dx=tan ycos x, y o = 1
1 +
dy/(y sjc) = i c dt gives c . The equation dyldx = 7 dyldt = y sin t, yo = 1
- xly leads to d . Then y2 + x2 = e and the solution
stays on a circle. 8 dyldt = et-Y, yo = e
9 Suppose the rate of rowth is proportional to & instead
The logistic equation is dyldt = f . The new term - by2 of y. Solve dyldt = c&starting from yo.
represents g when cy represents growth. Separation gives
i dy/(cy - by2)= [ dt, and the y-integral is l/c times In h . 10 The equation dyjdx = nylx for constant elasticity is the
Substituting yo at t = 0 and taking exponentials produces same as d(ln y)/d(ln x) = . The solution is In y =
y/(c - by) = ect( i ). As t + co,y approaches i . That
is the steady state where cy - by2 = k . The graph of y 11 When c = 0 in the logistic equation, the only term is y' =
looks like an I , because it has an inflection point at - by2. What is the steady state y,? How long until y drops
y= m . from yo to iyo?
In biology and chemistry, concentrations y and z react at 12 Reversing signs in Problem 11, suppose y' = + by2. At
a rate proportional to y times n . This is the Law of
what time does the population explode to y = co, starting
o . In a model equation dyldt = c(y)y, the rate c depends from yo = 2 (Adam + Eve)?
on P . The M M equation is dyldt = q . Separating
variables yields j r dy = s = - ct + C. Problems 13-26 deal with logistic equations y' = cy - by2.
13 Show that y = 1/(1+ e-') solves the equation y' = y - y2.
Draw the graph of y from starting values 3 and 3.
Separate, integrate, and solve equations 1-8.
14 (a) What logistic equation is solved by y = 2/(1 + e-')?
(b) Find c and b in the equation solved by y = 1/(1 + e-3t).
15 Solve z' = - z + 1 with zo = 2. Turned upside down as in
3 dyjdx = xly2, yo = 1 (1 3), what is y = l/z?
6.6 Powers Instead of Exponential6 267
16 By algebra find the S-curve (12) from y = l/z in (14). aspirin follows the MM equation. With c = K = yo = 1, does
17 How many years to grow from yo = $c/b to y = #c/b? Use
aspirin decay faster?
equation (10) for the time t since the inflection point in 1988. 28 If you take aspirin at a constant rate d (the maintenance
When does y reach 9 billion = .9c/b? dose), find the steady state level where d = cy/(y + K). Then
18 Show by differentiating u = y/(c - by) that if y' = cy - by2 y' = 0.
then u' = cu. This explains the logistic solution (11) - it is 29 Show that the rate R = cy/(y + K) in the MM equation
u = uoect. increases as y increases, and find the maximum as y -* a.
19 Suppose Pittsburgh grows from yo = 1 million people in 30 Graph the rate R as a function of y for K = 1 and K =
1900 to y = 3 million in the year 2000. If the growth rate is 10. (Take c = 1.) As the Michaelis constant increases, the rate
y' = 12,00O/year in 1900 and y' = 30,00O/year in 2000, substi- . At what value of y is R = *c?
tute in the logistic equation to find c and b. What is the steady
31 With y = KY and ct = KT, find the "nondimensional"
state? Extra credit: When does y = y, /2 = c/2b?
MM equation for dY/dT. From the solution erY=
20 Suppose c = 1 but b = - 1, giving cooperation y' = y + y2. e-= eroYorecover the y, t solution (19).
Solve for fit) if yo = 1. When does y become infinite?
32 Graph fit) in (19) for different c and K (by computer).
21 Draw an S-curve through (0,O) with horizontal asymp-
33 The Law of Mass Action for A + B + C is y' =
+
totes y = - 1 and y = 1. Show that y = (et- e-')/(et e-') has
k(ao- y)(bo- y). Suppose yo = 0, a. = bo = 3, k = 1. Solve for
those three properties. The graph of y2 is shaped like
y and find the time when y = 2.
34 In addition to the equation for d[C]/dt, the mass action
22 To solve y' = cy - by3 change to u = l/y2. Substitute for
law gives d[A]/dt =
y' in u' = - 2y'/y3 to find a linear equation for u. Solve it as
in (14) but with uo = ljy;. Then y = I/&. 35 Solve y' = y + t from yo = 0 by assuming y = Aet + B + Dt.
Find A, B, D.
23 With y = rY and t = ST, the equation dyldt = cy - by2
changes to d Y/d T = Y- Y '. Find r and s. 36 Rewrite cy - by2 as a2 - x2, with x = Gy - c/2$ and
24 In a change to y = rY and t = ST,how are the initial values
a= . Substitute for a and x in the integral taken
from tables, to obtain the y-integral in the text:
yo and yb related to Yo and G?
25 A rumor spreads according to y' = y(N - y). If y people --In- {A=-ln-
1 Y
know, then N - y don't know. The product y(N - y) measures -a2-x2 2a x
a '-x cy-by2 c c-by
the number of meetings (to pass on the rumor). 37 (Important) Draw the y-lines (with arrows as in the text)
(a) Solve dyldt = y(N - y) starting from yo = 1. for y' = y/(l - y) and y' = y - y3. Which steady states are
(b) At what time T have N/2 people heard the rumor? approached from which initial values yo?
(c) This model is terrible because T goes to as 38 Explain in your own words how the y-line works.
N + GO. A better model is y' = by(N - y).
39 (a) Solve yl= tan y starting from yo = n / 6 to find
26 Suppose b and c are bcth multiplied by 10. Does the sin y = $et.
middle of the S-curve get steeper or flatter? (b)Explain why t = 1 is never reached.
(c) Draw arrows on the y-line to show that y approaches
71 - when does it get there?
12
Problems 27-34 deal with mass action and the MM equation
40 Write the logistic equation as y' = cy(1 - y/K). As y'
+
y' = - cy/(y K).
approaches zero, y approaches . Find y, y', y" at the
27 Most drugs are eliminated acording to y' = - cy but inflection point.
6.6 Powers lnstead of Exponentials
You may remember our first look at e. It is the special base for which ex has slope 1
at x = 0 That led to the great equation of exponential growth: The derivative of
.
ex equals ex. But our look at the actual number e = 2.71828 ... was very short.
6 Exponentlals and Logarithms
It appeared as the limit of (1 + lln)". This seems an unnatural way to write down
such an important number.
+
I want to show how (1 lln)" and (1 + xln)" arise naturally. They give discrete
growth infinite steps-with applications to compound interest. Loans and life insur-
ance and money market funds use the discrete form of yf = cy + s. (We include extra
information about bank rates, hoping this may be useful some day.) The applications
in science and engineering are equally important. Scientific computing, like account-
ing, has diflerence equations in parallel with differential equations.
Knowing that this section will be full of formulas, I would like to jump ahead and
tell you the best one. It is an infinite series for ex. What makes the series beautiful is
that its derivative is itself:
Start with y = 1 + x. This has y = 1 and yt = 1 at x = 0. But y" is zero, not one.
Such a simple function doesn't stand a chance! No polynomial can be its own deriva-
tive, because the highest power xn drops down to nxn-l. The only way is to have no
highest power. We are forced to consider infinitely many terms-a power series-to
achieve "derivative equals function.''
+
To produce the derivative 1 + x, we need 1 x + ix2. Then i x 2 is the derivative
of Ax3, which is the derivative of &x4. The best way is to write the whole series at
once:
+ + i x 2 + 4x3 + &x4 + -.
Infinite series ex = 1 x (1)
This must be the greatest power series ever discovered. Its derivative is itself:
The derivative of each term is the term before it. The integral of each term is the one
after it (so jexdx = ex + C). The approximation ex = 1 + x appears in the first two
are
terms. Other properties like (ex)(ex) eZX not so obvious. (Multiplying series is
=
hard but interesting.) It is not even clear why the sum is 2.718 ... when x = 1.
Somehow 1 + 1 + f + & + equals e. That is where (1 + lln)" will come in.
Notice that xn is divided by the product 1 2 3 * - . - n. This is "n factorial." Thus
-
x4 is divided by 1 2 3 4 = 4! = 24, and xS is divided by 5! = 120. The derivative of .
x5/120 is x4/24, because 5 from the derivative cancels 5 from the factorial. In general
xn/n! has derivative xn- '/(n - l)! Surprisingly O! is 1.
Chapter 10 emphasizes that xn/n! becomes extremely small as n increases. The
infinite series adds up to a finite number-which is ex. We turn now to discrete
growth, which produces the same series in the limit.
This headline was on page one of the New York Times for May 27, 1990.
213 Years After Loan, Uncle Sam is Dunned
San Antonio, May 26-More than 200 years ago, a wealthy Pennsylvania
merchant named Jacob DeHaven lent $450,000 to the Continental Congress to
rescue the troops at Valley Forge. That loan was apparently never repaid.
So Mr. DeHaven's descendants are taking the United States Government to
court to collect what they believe they are owed. The total: $141 billion if the
interest is compounded daily at 6 percent, the going rate at the time. If com-
pounded yearly, the bill is only $98 billion.
The thousands of family members scattered around the country say they are
not being greedy. "It's not the money-it's the principle of the thing," said
Carolyn Cokerham, a DeHaven on her father's side who lives in San Antonio.
6.6 Powen Instead of Exponentlals
"You have to wonder whether there would even be a United States if this man
had not made the sacrifice that he did. He gave everything he had."
The descendants say that they are willing to be flexible about the amount of
settlement. But they also note that interest is accumulating at $190 a second.
"None of these people have any intention of bankrupting the Government,"
said Jo Beth Kloecker, a lawyer from Stafford, Texas. Fresh out of law school,
Ms. Kloecker accepted the case for less than the customary 30 percent
contingency.
It is unclear how many descendants there are. Ms. Kloecker estimates that
based on 10 generations with four children in each generation, there could be as
many as half a million.
The initial suit was dismissed on the ground that the statute of limitations is
six years for a suit against the Federal Government. The family's appeal asserts
that this violates Article 6 of the Constitution, which declares as valid all debts
owed by the Government before the Constitution was adopted.
Mr. DeHaven died penniless in 1812. He had no children.
C O M P O U N D INTEREST
The idea of compound interest can be applied right away. Suppose you invest $1000
at a rate of 100% (hard to do). If this is the annual rate, the interest after a year is
another $1000. You receive $2000 in all. But if the interest is compounded you receive
more:
after six months: Interest of $500 is reinvested to give $1500
end of year: New interest of $750 (50% of 1500) gives $2250 total.
The bank multiplied twice by 1.5 (1000 to 1500 to 2250). Compounding quarterly
multiplies four times by 1.25 (1 for principal, .25 for interest):
after one quarter the total is 1000 + (.25)(1000) = 1250
after two quarters the total is 1250 + (.25)(1250)= 1562.50
after nine months the total is 1562.50 + (.25)(1562.50)= 1953.12
after a full year the total is 1953.12 + (.25)(@53.12) = 2441.41
Each step multiplies by 1 + (l/n), to add one nth of a year's interest-still at 100%:
quarterly conversion: (1 + 1/4)4x low = 2441.41
monthly conversion: (1 + 1/12)" x 1 Q h 2613.04
=
daily conversion: (1 + 1/365)36% 1000 = 2714.57.
Many banks use 360 days in a year, although computers have made that obsolete.
Very few banks use minutes (525,600 per year). Nobody compounds every second
(n = 31,536,000). But some banks offer continuous compounding. This is the limiting
case (n -+ GO) that produces e:
x 1000 approaches e x 1000 = 2718.28.
(1+
1
1. Quick method for (1 + lln)": Take its logarithm. Use ln(1 + x) x x with x = -:
n
6 Exponentlals and Logartthms
As l/n gets smaller, this approximation gets better. The limit is 1. Conclusion:
+
(1 l/n)" approaches the number whose logarithm is 1. Sections 6.2 and 6.4 define
the same number (which is e).
2. Slow method for (1 + l/n)": Multiply out all the terms. Then let n + a.
This is a brutal use of the binomial theorem. It involves nothing smart like logarithms,
but the result is a fantastic new formula for e.
Practice for n = 3:
Binomial theorem for any positive integer n:
Each term in equation (4) approaches a limit as n + a.Typical terms are
Next comes 111 2 3 4. The sum of all those limits in (4) is our new formula for e:
In summation notation this is Z,"=, l/k! = e. The factorials give fast convergence:
Those nine terms give an accuracy that was not reached by n = 365 compoundings.
A limit is still involved (to add up the whole series). You never see e without a limit!
It can be defined by derivatives or integrals or powers (1 + l/n)" or by an infinite
series. Something goes to zero or infinity, and care is required.
All terms in equation (4) are below (or equal to) the corresponding terms in (5).
The power (1 + l/n)" approaches efrom below. There is a steady increase with n. Faster
compounding yields more interest. Continuous compounding at 100% yields e, as
each term in (4) moves up to its limit in (5).
Remark Change (1 + lln)" to (1 + xln)". Now the binomial theorem produces ex:
Please recognize ex on the right side! It is the infinite power series in equation (1).
The next term is x3/6 (x can be positive or negative). This is a final formula for ex:
The logarithm of that power is n In(1 + x/n) x n(x/n) = x. The power approaches ex.
To summarize: The quick method proves (1 + lln)" + e by logarithms. The slow
method (multiplying out every term) led to the infinite series. Together they show the
agreement of all our definitions of e.
DIFFERENCE EQUATIONS VS. DIFFERENTIAL EQUATIONS
We have the chance to see an important part of applied mathematics. This is not a
course on differential equations, and it cannot become a course on difference equ-
ations. But it is a course with a purpose-we aim to use what we know. Our main
application of e was to solve y' = cy and y' = cy + s. Now we solve the corresponding
difference equations.
f
Above all, the goal is to see the connections. The purpose o mathematics is to
understand and explain patterns. The path from "discrete to continuous" is beautifully
illustrated by these equations. Not every class will pursue them to the end, but I
cannot fail to show the pattern in a difference equation:
Each step multiplies by the same number a. The starting value yo is followed by ay,,
a2yo,and a3y0. The solution at discrete times t = 0, 1,2, ... is y(t) = atyo.
f
This formula atyo replaces the continuous solution ectyoo the differential equation.
decaying
Fig. 6.17 Growth for la1 > 1, decay for la1 < 1. Growth factor a compares to ec.
A source or sink (birth or death, deposit or withdrawal) is like y' = cy + s:
y(t + 1)= ay(t) + s.
Each step multiplies by a and adds s. The first outputs are
We saw this pattern for differential equations-every input s becomes a new starting
point. It is multiplied by powers of a. Since s enters later than yo, the powers stop at
t - 1. Algebra turns the sum into a clean formula by adding the geometric series:
y(t)= atyo+ s[at-' +at-' + +
+ a + 1]= atyo s(at- l)/(a- 1). (9)
EXAMPLE 1 Interest at 8% from annual IRA deposits of s = $2000 (here yo = 0).
The first deposit is at year t = 1. In a year it is multiplied by a = 1.08, because 8% is
added. At the same time a new s = 2000 goes in. At t = 3 the first deposit has been
multiplied by (1.08)2,the second by 1.08, and there is another s = 2000. After year t,
y(t) = 2000(1.08' - 1)/(1.08 - 1). (10)
+
With t = 1 this is 2000. With t = 2 it is 2000 (1.08 1)-two deposits. Notice how
a - 1 (the interest rate .08) appears in the denominator.
EXAMPLE 2 Approach to steady state when la1 < 1. Compare with c < 0.
With a > 1, everything has been increasing. That corresponds to c > 0 in the
differential equation (which is growth). But things die, and money is spent, so a can
be smaller than one. In that case atyo approaches zero-the starting balance disap-
f
pears. What happens if there is also a source? Every year half o the balance y(t) is
6 Exponentials and Logartthms
spent and a new $2000 is deposited. Now a = +:
y(t + 1) = $y(t) + 2000 yields y(t) = (f)ty, + 2000[((+)' - I)/(+- I)].
The limit as t - co is an equilibrium point. As (fy goes to zero, y(t) stabilizes to
,
y, = 200qO - I)/($ - 1) = 4000 = steady state. (11)
Why is 4000 steady? Because half is lost and the new 2000 makes it up again. The
, +
iteration is y,,, = fy,, 2000. Ztsfied point is where y, = fy, + 2000.
In general the steady equation is y, = ay, +
s. Solving for y, gives s/(l - a).
Compare with the steady differential equation y' = cy + s = 0:
S S
y, = - - (differential equation) us. y, =-(difference equation). (12)
c 1-a
EXAMPLE 3 Demand equals supply when the price is right.
Difference equations are basic to economics. Decisions are made every year (by a
farmer) or every day (by a bank) or every minute (by the stock market). There are
three assumptions:
1. Supply next time depends on price this time: S(t + 1) = cP(t).
+
2. Demand next time depends on price next time: D(t 1) = - dP(t + 1) + b.
3. Demand next time equals supply next time: D(t + 1) = S(t + 1).
Comment on 3: the price sets itself to make demand = supply. The demand slope - d
is negative. The supply slope c is positive. Those lines intersect at the competitive
price, where supply equals demand. To find the difference equation, substitute 1 and
2 into 3:
+ +
Difference equation: - dP(t 1) b = cP(t)
Steady state price: - dP, + b = cP,. Thus P, = b/(c + d).
If the price starts above P,, the difference equation brings it down. If below, the
price goes up. When the price is P,, it stays there. This is not news-economic
theory depends on approach to a steady state. But convergence only occurs if c < d.
f
I supply is less sensitive than demand, the economy is stable.
+
Blow-up example: c = 2, b = d = 1. The difference equation is - P(t 1) + 1 = 2P(t).
From P(0) = 1 the price oscillates as it grows: P = - 1, 3, - 5, 11, ... .
Stable example: c = 112, b = d = 1. The price moves from P(0) = 1 to P(m) = 213:
1 1 3 5 2
- P(t + 1) + 1 = - P(t) yields
2
P = 1' - - - "" approaching - .
2' 4' 8'
3
Increasing d gives greater stability. That is the effect of price supports. For d = 0
(fixed demand regardless of price) the economy is out of control.
H
T E MATHEMATICS OF FINANCE
It would be a pleasure to make this supply-demand model more realistic-with
curves, not straight lines. Stability depends on the slope-calculus enters. But we
also have to be realistic about class time. I believe the most practical application is
to solve the fundamentalproblems offinance. Section 6.3 answered six questions about
continuous interest. We now answer the same six questions when the annual rate is
x = .05 = 5% and interest is compounded n times a year.
6.6 Powers Instead of Exponentials
First we compute eflective rates, higher than .05 because of compounding:
( T:.
compounded quarterly 1 + - = 1.0509 [effective rate .0509 = 5.09%]
compounded continuously eno5= 1.O513 [effective rate 5.13%]
Now come the six questions. Next to the new answer (discrete) we write the old
answer (continuous). One is algebra, the other is calculus. The time period is 20 years,
so simple interest on yo would produce (.05)(20)(yo).That equals yo -money doubles
in 20 years at 5% simple interest.
Questions 1and 2 ask for the future value y and present value yo with compound
interest n times a year:
1. y growing from yo: y = (1 + yonyo y = e(~OS,(20)yo
yo = e-(-05)(20)y
2. deposit yo to reach y: yo = (1 + :F20ny
Each step multiplies by a = (1 + .05/n). There are 20n steps in 20 years. Time goes
backward in Question 2. We divide by the growth factor instead of multiplying. The
future value is greater than the present value (unless the interest rate is negative!). As
n + GO the discrete y on the left approaches the continuous y on the right.
Questions 3 and 4 connect y to s (with yo = 0 at the start). As soon as each s is
deposited, it starts growing. Then y = s + as + a2s + --.
(1 + .05/n)20n I]
- y = s [e(.05)(20) I]
-
3. y growing from deposits s: y = s[ .05/n .05
4. deposits s to reach y:
Questions 5 and 6 connect yo to s. This time y is zero-there is nothing left at the
end. Everything is paid. The deposit yo is just enough to allow payments of s. This
is an annuity, where the bank earns interest on your yo while it pays you s (n times
a year for 20 years). So your deposit in Question 5 is less than 20ns.
Question 6 is the opposite-a loan. At the start you borrow yo (instead of giving
the bank yo). You can earn interest on it as you pay it back. Therefore your payments
have to total more than yo. This is the calculation for car loans and mortgages.
5. Annuity: Deposit yo to receive 20n payments of s:
6. Loan:. Repay yo with 20n payments of s:
Questions 2 , 4 , 6 are the inverses of 1,3,5. Notice the pattern: There are three num-
f
bers y, yo, and s. One o them-is zero each time. If all three are present, go back to
equation (9).
The algebra for these lines is in the exercises. I t is not calculus because At is not dt.
All factors in brackets [ 1 are listed in tables, and the banks keep copies. It might
6 Exponenlials and Logartthms
also be helpful to know their symbols. If a bank has interest rate i per period over
N periods, then in our notation a = 1 + i = 1 + .05/n and t = N = 20n:
future value of yo = $1 (line 1):y(N) = (1 + i)N
present value of y = $1 (line 2): yo = (1 + i)-N
future value of s = $1 (line 3): y(N) = s~~= [(I + i)N- l]/i
present value of s = $1 (line 5): yo = a~~= [ l - (1 + i)-']/i
To tell the truth, I never knew the last two formulas until writing this book. The
mortgage on my home has N = (12)(25) monthly payments with interest rate i =
.07/12. In 1972 the present value was $42,000 = amount borrowed. I am now going
to see if the bank is honest.?
Remark In many loans, the bank computes interest on the amount paid back
instead of the amount received. This is called discounting. A loan of $1000 at 5%
for one year costs $50 interest. Normally you receive $1000 and pay back $1050.
With discounting you receive $950 (called the proceeds) and you pay back $1000.
The true interest rate is higher than 5%-because the $50 interest is paid on the
smaller amount $950. In this case the "discount rate" is 501950 = 5.26%.
IFRNI L IFRN E
SCIENTIFIC COMPUTING: DF E E TA EQUATIONS BY DF E E C EQUATIONS
In biology and business, most events are discrete. In engineering and physics, time
and space are continuous. Maybe at some quantum level it's all the same, but the
equations of physics (starting with Newton's law F = ma) are differential equations.
The great contribution of calculus is to model the rates of change we see in nature.
But to solve that model with a computer, it needs to be made digital and discrete.
These paragraphs work with dyldt = cy. It is the test equation that all analysts use,
as soon as a new computing method is proposed. Its solution is y = ect,starting from
yo = 1. Here we test Euler's method (nearly ancient, and not well thought of). He
replaced dyldt by AylAt:
The left side is dyldt, in the limit At + 0. We stop earlier, when At > 0.
The problem is to solve (13). Multiplying by At, the equation is
y(t + At) = (1 + cAt)y(t) (with y(0) = 1).
Each step multiplies by a = 1 + cAt, so n steps multiply by an:
y = an= (1 + cAt)" at time nAt. (14)
This is growth or decay, depending on a. The correct ectis growth or decay, depending
on c. The question is whether an and eczstay close. Can one of them grow while the
other decays? We expect the difference equation to copy y' = cy, but we might be
wrong.
A good example is y' = - y. Then c = - 1 and y = e-'-the true solution decays.
?It's not. s is too big. I knew it.
The calculator gives the following answers an for n = 2, 10,20:
The big step At = 3 shows total instability (top row). The numbers blow up when
they should decay. The row with At = 1 is equally useless (all zeros). In practice the
magnitude of cAt must come down to .10 or .05. For accurate calculations it would
have to be even smaller, unless we change to a better difference equation. That is the
right thing to do.
Notice the two reasonable numbers. They are .35 and .36, approaching e- = .37. '
They come from n = 10 (with At = 1/10) and n = 20 (with At = 1/20). Those have the
same clock time nAt = 1:
The main diagonal of the table is executing (1 + xln)" - e" in the case x = - 1.
,
Final question: How quickly are .35 and .36 converging to e-' = .37? With At = .10
the error is .02. With At = .05 the error is .01. Cutting the time step in half cuts the
error in half. We are not keeping enough digits to be sure, but the error seems close
to *At. To test that, apply the "quick method" and estimate an= (1 - Atr from its
logarithm:
ln(1- Atr = n ln(1- At) z n[- At - + ( ~ t ) = ] 1 - f At.
~-
The clock time is nAt = 1. Now take exponentials of the far left and right:
' '.
The differencebetween an and e- is the last term *Ate- Everything comes down
to one question: Is that error the same as *At? The answer is yes, because e-'12 is
115. If we keep only one digit, the prediction is perfect!
That took an hour to work out, and I hope it takes longer than At to read. I wanted
you to see in use the properties of In x and e". The exact property In an= n In a came
first. In the middle of (15) was the key approximation ln(1 + x) z x - f x2, with x =
- At. That x2 term uses the second derivative (Section 6.4). At the very end came
e"xl+x.
+
A linear approximation shows convergence: (1 x/n)" - ex. A quadratic shows the
,
error: proportional to At = l/n. It is like using rectangles for areas, with error propor-
tional to Ax. This minimal accuracy was enough to define the integral, and here it is
enough to define e. It is completely unacceptable for scientific computing.
The trapezoidal rule, for integrals or for y' = cy, has errors of order (Ax)2and (At)2.
All good software goes further than that. Euler's first-order method could not predict
the weather before it happens.
dy
Euler's Method for - = F(y, t): Y(' + At) - y(t) = ~ ( ~ ( t). ,
t)
dt At
276 6 Exponentials and Logarithms
6.6 EXERCISES
Read-through questions limit of (1 - l/n)". What is the sum of this infinite series -
the exact sum and the sum after five terms?
The infinite series for e" is a . Its derivative is b . The
denominator n! is called " c " and it equals d . At x = 9 Knowing that (1 + l/n)" -+ e, explain (1 + l/n)2n e2 and
-+
1 the series for e is e . (1 + 2/N)N-+e2.
To match the original definition of e, multiply out 10 What are the limits of (1 + l/n2)" and (1 + l/n)"*?
(1 + l/n)" = f (first three terms). As n + co those terms OK to use a calculator to guess these limits.
approach Q in agreement with e. The first three terms of
11 (a) The power (1 + l/n)" (decreases) (increases) with n, as
(1 + xln)" are h . As n + co they approach 1 in
we compound more often. (b) The derivative of f(x)=
agreement with ex. Thus (1 + xln)" approaches I .A
x ln(1 + llx), which is , should be (<0)(> 0). This is
+
quicker method computes ln(1 xln)" x k (first term
confirmed by Problem 12.
only) and takes the exponential.
Compound interest (n times in one year at annual rate x)
12 Show that ln(1 + l/x) > l/(x + 1) by drawing the graph of
llt. The area from t = 1 to 1 + l/x is . The rectangle
multiplies by ( I )". As n -+ co, continuous compounding
inside it has area .
multiplies by m . At x = 10% with continuous compound-
ing, $1 grows to n in a year. 13 Take three steps of y(t + 1) = 2y(t) from yo = 1.
The difference equation y(t + 1) = ay(t) yields fit) = o 14 Take three steps of y(t + 1) = 2y(t) + 1 from yo = 0.
times yo. The equation y(t + 1) = ay(t) + s is solved by y =
+ +
atyo+ $1 a + -.- at-']. The sum in brackets is P .
Solve the difference equations 15-22.
When a = 1.08 and yo = 0, annual deposits of s = 1 produce
y = q after t years. If a = 9 and yo = 0, annual deposits
of s = 6 leave r after t years, approaching y, = s .
The steady equation y, = ay, + s gives y, = t .
When i = interest rate per period, the value of yo = $1 after
N periods is y(N) = u . The deposit to produce y(N) = 1
is yo = v . The value of s = $1 deposited after each period
grows to y(N) = w . The deposit to reach y(N) = 1 is s =
In 23-26, which initial value produces y, = yo (steady state)?
x .
Euler's method replaces y' = cy by Ay = cyAt. Each step 23 y(t + 1) = 2y(t) - 6 24 y(t + 1) = iy(t) - 6
multiplies y by Y . Therefore y at t = 1 is (1 + cAt)ll'yo, 25 y(t + 1)= - y(t) + 6 26 y(t + 1)= - $y(t) + 6
which converges to as At -+ 0. The error is proportional
27 In Problems 23 and 24, start from yo = 2 and take three
to A , which is too B for scientific computing.
steps to reach y,. Is this approaching a steady state?
1 Write down a power series y = 1 - x + .-.whose derivative 28 For which numbers a does (1 - at)/(l - a) approach a limit
is -y. as t -+ oo and what is the limit?
2 Write down a power series y = 1 + 2x + .--whose deriva- 29 The price P is determined by supply =demand or
tive is 2y. -dP(t + +
1) b = cP(t). Which price P is not changed from
one year to the next?
3 Find two series that are equal to their second derivatives.
30 Find P(t) from the supply-demand equation with c = 1,
4 By comparing e = 1 + 1 + 9 + 4 + + -.. with a larger
d = 2, b = 8, P(0) = 0. What is the steady state as t -+ co?
series (whose sum is easier) show that e < 3.
5 At 5% interest compute the output from $1000 in a year
Assume 10% interest (so a = 1 + i = 1.1) in Problems 31-38.
with 6-month and 3-month and weekly compounding.
31 At 10% interest compounded quarterly, what is the effec-
6 With the quick method ln(1 + x) z x, estimate ln(1- lln)"
tive rate?
and ln(1 + 2/n)". Then take exponentials to find the two limits.
32 At 10% interest compounded daily, what is the effective
7 With the slow method multiply out the three terms of
rate?
(1 - $)2 and the five terms of (1 - $I4.
What are the first three
terms of (1 - l/n)", and what are their limits as n -+ oo? 33 Find the future value in 20 years of $100 deposited now.
8 The slow method leads to 1 - 1 + 1/2! - 1/3! + -.-for the 34 Find the present value of $1000 promised in twenty years.
6.7 Hyperbolic Functions 277
35 For a mortgage of $100,000 over 20 years, what is the do you still owe after one month (and after a year)?
monthly payment?
41 Euler charges c = 100% interest on his $1 fee for discover-
36 For a car loan of $10,000 over 6 years, what is the monthly ing e. What do you owe (including the $1) after a year with
payment? (a) no compounding; (b) compounding every week; (c) con-
tinuous compounding?
37 With annual compounding of deposits s = $1000, what is
the balance in 20 years? 42 Approximate (1 + 1/n)" as in (15) and (16) to show that
you owe Euler about e - e/2n. Compare Problem 6.2.5.
38 If you repay s = $1000 annually on a loan of $8000, when
are you paid up? (Remember interest.) 43 My Visa statement says monthly rate = 1.42% and yearly
rate = 17%. What is the true yearly rate, since Visa com-
39 Every year two thirds of the available houses are sold, and
pounds the interest? Give a formula or a number.
1000 new houses are built. What is the steady state of the
housing market - how many are available? 44 You borrow yo = $80,000 at 9% to buy a house.
40 If a loan shark charges 5% interest a month on the $1000 (a) What are your monthly payments s over 30 years?
you need for blackmail, and you pay $60 a month, how much (b) How much do you pay altogether?
I 6.7 Hyperbolic Functions
This section combines ex with e - x. Up to now those functions have gone separate
ways-one increasing, the other decreasing. But two particular combinations have
earned names of their own (cosh x and sinh x):
ex + e - x ex - e-x
hyperbolic cosine cosh x-= -
hyperbolic sine sinh x =
2 2
The first name rhymes with "gosh". The second is usually pronounced "cinch".
The graphs in Figure 6.18 show that cosh x > sinh x. For large x both hyperbolic
functions come extremely close to ½ex. When x is large and negative, it is e- x that
dominates. Cosh x still goes up to + 00 while sinh x goes down to - co (because
sinh x has a minus sign in front of e-x).
1 1 1 1
cosh x = eX+ e-x sinh x = -ex e
2 2 2 2
\ /I
1 1
e-X 1 ex
2 2
-1 1
Fig. 6.18 Cosh x and sinh x. The hyperbolic Fig. 6.19 Gateway Arch courtesy of the St.
functions combine 'ex and ½e- x . Louis Visitors Commission.
The following facts come directly from ((ex + e - x) and ½(ex - e-X):
cosh(- x) = cosh x and cosh 0 = 1 (cosh is even like the cosine)
sinh(- x) = - sinh x and sinh 0 = 0 (sinh is odd like the sine)
6 Exponentials and Logarithms
The graph of cosh x corresponds to a hanging cable (hanging under its weight).
Turned upside down, it has the shape of the Gateway Arch in St. Louis. That must
be the largest upside-down cosh function ever built. A cable is easier to construct
than an arch, because gravity does the work. With the right axes in Problem 55, the
height of the cable is a stretched-out cosh function called a catenary:
y = a cosh (x/a) (cable tension/cable density = a).
Busch Stadium in St. Louis has 96 catenary curves, to match the Arch.
The properties of the hyperbolic functions come directly from the definitions. There
are too many properties to memorize-and no reason to do it! One rule is the most
important. Every fact about sines and cosines is reflected in a correspondingfact about
sinh x and cosh x. Often the only difference is a minus sign. Here are four properties:
1. (cosh x)2 - (sinh x)2 = 1 instead of (cos x) 2 + (sin x)2 = 1]
- 2
e 2 x+2+e-2x e2 x+2 -e x
ex e-x 2 x e- 2 =
Check:
2. d (cosh x) = sinh x instead of d (cos x) - sin x
dx dx
3. d (sinh x) = cosh x like d sin x = cos x
4. f sinh x dx = cosh x + C and f cosh x dx = sinh x + C
t, sinh t)
t)
Fig. 6.20 The unit circle cos 2 t + sin 2 t = 1 and the unit hyperbola cosh 2 t - sinh 2 t = 1.
Property 1 is the connection to hyperbolas. It is responsible for the "h" in cosh and
sinh. Remember that (cos x)2 + (sin x)2 = 1 puts the point (cos x, sin x) onto a unit
circle. As x varies, the point goes around the circle. The ordinary sine and cosine are
"circular functions." Now look at (cosh x, sinh x). Property 1 is (cosh x) 2 - (sinh x) 2 =
1, so this point travels on the unit hyperbola in Figure 6.20.
You will guess the definitions of the other four hyperbolic functions:
sinh x ex - e-x cosh x ex + e-x
tanh x - - coth x - - -
cosh x ex + e - x sinh x ex - e - x
1 2 1 2
sech x cosh x ex + e-x
csch x sinh x ex - e-x
I think "tanh" is pronounceable, and "sech" is easy. The others are harder. Their
6.7 Hyperbolic Functions
properties come directly from cosh2x- sinh2x = 1. Divide by cosh2x and sinh2x:
1 - tanh 2x = sech2x and coth2x - 1 = csch2x
(tanh x)' = sech2x and (sech x)' = -sech x tanh x
1 sinh x
tanh x dx = S=dx = ln(cosh x) + C.
N E S Y E B LC
I V R E H P R O I FUNCTIONS
You remember the angles sin-'x and tan-'x and sec-'x. In Section 4.4 we
differentiated those inverse functions by the chain rule. The main application was to
integrals. If we happen to meet jdx/(l+ x2), it is tan-'x + C. The situation for
sinh- 'x and tanh- 'x and sech- 'x is the same except for sign changes - which are
expected for hyperbolic functions. We write down the three new derivatives:
y = sinh-'x (meaning x = sinh y) has 9= J 21T i
dx
1
y = tanh-'x (meaning x = tanh y) has 9= -
dx 1 - x2
-1
'
y = sech - x (meaning x = sech y) has dy =
dx X J i 7
Problems 44-46 compute dyldx from l/(dx/dy). The alternative is to use logarithms.
Since In x is the inverse of ex, we can express sinh-'x and tanh-'x and sech-'x as
logarithms. Here is y = tanh- 'x:
The last step is an ordinary derivative of 4 ln(1 + x) - ln(1 - x). Nothing is new
except the answer. But where did the logarithms come from? In the middle of the
following identity, multiply above and below by cosh y:
-- x - 1 + tanh y - cosh y + sinh y - - - - e2y.
1+ - eY
1 - x 1- tanh y cosh y - sinh y e-y
Then 2y is the logarithm of the left side. This is the first equation in (4), and it is the
third formula in the following list:
Remark 1 Those are listed onlyfor reference. If possible do not memorize them. The
derivatives in equations (I), (2), (3) offer a choice of antiderivatives - either inverse
functions or logarithms (most tables prefer logarithms). The inside cover of the book
has
1% = fln[E] +C (in place of tanh- 'x + C).
Remark 2 Logarithms were not seen for sin- 'x and tan- 'x and sec - 'x. You might
6 Exponentials and Logarithms
wonder why. How does it happen that tanh-'x is expressed by logarithms, when the
parallel formula for tan-lx was missing? Answer: There must be a parallel formula.
To display it I have to reveal a secret that has been hidden throughout this section.
The secret is one of the great equations of mathematics. What formulas for cos x
and sin x correspond to &ex + e-x) and &ex- e-x)? With so many analogies
(circular vs. hyperbolic) you would expect to find something. The formulas do exist,
but they involve imaginary numbers. Fortunately they are very simple and there is
no reason to withhold the truth any longer:
1 1 .
cosx=-(eix+eix) and sin~=-(e'~--e-'~). (5)
2 2i
It is the imaginary exponents that kept those identities hidden. Multiplying sin x by
i and adding to cos x gives Euler's unbelievably beautiful equation
cos x + i sin x = eiX. (6)
That is parallel to the non-beautiful hyperbolic equation cosh x + sinh x = ex.
I have to say that (6) is infinitely more important than anything hyperbolic will
ever be. The sine and cosine are far more useful than the sinh and cosh. So we end
our record of the main properties, with exercises to bring out their applications.
Read-through questions Find the derivatives of the functions 9-18:
Cosh x = a and sinh x = b and cosh2x - sinh2x = 9 cosh(3x + 1) 10 sinh x2
c . Their derivatives are d and e and f .
The point (x, y) = (cosh t , sinh t ) travels on the hyperbola 11 l/cosh x 12 sinh(1n x)
- -
g . A cable hangs in the shape of a catenary y = h . 13 cosh2x + sinh2x 14 cosh2x - sinh2x
The inverse functions sinh-'x and t a n h l x are equal to 15 tanh , =
, / 16 (1 + tanh x)/(l - tanh x)
ln[x + ,/x2 + 11 and 4ln I . Their derivatives are i
17 sinh6x 18 ln(sech x + tanh x)
and k . So we have two ways to write the anti I . The
parallel to cosh x + sinh x = ex is Euler's formula m . 19 Find the minimum value of cosh(1n x) for x > 0.
The formula cos x = $(eix+ ePix)involves n exponents.
20 From tanh x = +find sech x, cosh x, sinh x, coth x, csch x.
The parallel formula for sin x is o .
21 Do the same if tanh x = - 12/13.
1 Find cosh x + sinh x, cosh x - sinh x, and cosh x sinh x.
22 Find the other five values if sinh x = 2.
2 From the definitions of cosh x and sinh x, find their deriv-
atives. 23 Find the other five values if cosh x = 1.
3 Show that both functions satisfy y" = y. 24 Compute sinh(1n 5) and tanh(2 In 4).
4 By the quotient rule, verify (tanh x)' = sech2x.
5 Derive cosh2x + sinh2x = cosh 2x, from the definitions. Find antiderivatives for the functions in 25-32:
6 From the derivative of Problem 5 find sinh 2x. 25 cosh(2x + 1) 26 x cosh(x2)
7 The parallel to (cos x + i sin x r = cos nx + i sin nx is a 27 cosh2x sinh
hyperbolic formula (cosh x + sinh x)" = cosh nx + .
sinh x ex + e P x
8 Prove sinh(x + y) = sinh x cosh y + cosh x sinh y by 30 ~ 0 t h = ex ---
x -
- e-"
changing to exponentials. Then the x-derivative gives 29 1 +cosh x
cosh(x + y) = 31 sinh x + cosh x 32 (sinh x + cosh x)"
6.7 Hyperbolic Functions 281
33 The triangle in Figure 6.20 has area 3 cosh t sinh t.
(a) Integrate to find the shaded area below the hyperbola
(b)For the area A in red verify that dA/dt = 4
(c) Conclude that A = it + C and show C = 0.
Sketch graphs of the functions in 34-40.
34 y = tanh x (with inflection point)
35 y = coth x (in the limit as x 4 GO) 54 A falling body with friction equal to velocity squared
obeys dvldt = g - v2.
36 y = sech x
(a) Show that v(t) = & tanh &t satisfies the equation.
(b)Derive this v yourself, by integrating dv/(g - v2)= dt.
38 y=cosh-lx for x 3 1 (c) Integrate v(t) to find the distance f(t).
39 y = sech- 'x for 0 c x d 1 55 A cable hanging under its own weight has slope S = dyldx
40 : (i':)
= tanh-'x = - In - for lxlc 1
that satisfiesdS/dx = c d m . The constant c is the ratio of
cable density to tension.
(a) Show that S = sinh cx satisfies the equation.
41 (a) Multiplying x = sinh y = b(ey - e - Y by 2Y gives
) e (b)Integrate dyldx = sinh cx to find the cable height y(x),
(eq2- 2 4 8 ) - 1 = 0. Solve as a quadratic equation for eY. if y 0 = llc.
()
(b)Take logarithms to find y = sinh - 'x and compare with (c) Sketch the cable hanging between x = - L and x = L
the text. and find how far it sags down at x = 0.
42 (a) Multiplying x = cosh y = i ( 8 + ebY) by 2ey gives 56 The simplest nonlinear wave equation (Burgers' equation)
( e ~- 2x(e") + 1 = 0. Solve for eY.
)~ yields a waveform W(x) that satisfies W" = WW' - W'. One
(b)Take logarithms to find y = cosh- 'x and compare with integration gives W' = 3w2- W .
the text. (a) Separate variables and integrate:
43 Turn (4) upside down to prove y' = - l/(l - x2), if y = dx=dw/(3w2- W)=-dW/(2- W)-dW/W.
coth- 'x. (b) Check W' = 3W2- W.
44 Compute dy/dx = I/,/= by differentiating x = sinh y 57 A solitary water wave has a shape satisfying the KdV
and using cosh2y - sinh2y= 1. equation y" = y' - 6yy'.
(a) Integrate once to find y". Multiply the answer by y'.
45 Compute dy/dx = l/(l - x2) if y = tanh- 'x by differen-
tiating x = tanh y and using sech2y+ tanh2y = 1. (b) Integrate again to find y' (all constants of integration
are zero).
46 Compute dyldx = -l / x J E ? for y = sech- 'x, by (c) Show that y = 4 sech2(x/2) gives the shape of the
differentiating x = sech y. "soliton."
From formulas (I), (2), (3) or otherwise, find antiderivatives in 58 Derive cos ix = cosh x from equation (5). What is the
47-52: cosine of the imaginary angle i =
59 Derive sin ix = i sinh x from (5). What is sin i?
60 The derivative of eix= cos x + i sin x is
MIT OpenCourseWare
Resource: Calculus Online Textbook
Gilbert Strang
The following may not correspond to a particular course on MIT OpenCourseWare, but has been
provided by the author as an individual learning resource.
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Latest Posts
After our TMC14 Algebra 2 session, Brooke and I decided to reorder our pacing guide together in a way that was similar to Glenn's.Here is a poster of the functions in the order we plan to teach them.The idea is to start the course with asking what they notice and wonder about the equations. Then as we study that equation, we have a giant You Are Here arrow.Hopefully the students will see more connections and realize that only the pictures are changing.Here is the powerpoint if you want to […]
For the first time in 15 years both PD1 & PD2 will join us on vacation. Space is crammed, so drastic decisions are required… ThePartner and I travel down south 4 or 5 times a year and we usually take along a huge pile of books. Not that we intend to read them all, but ...
Try as I might to focus this post on other things, I think I first need to address the fact that up until TMC14 I considered myself a math outsider.
Here's a Moss Egg drawing Using GeoGebra. The idea came from a book by Dixon, Robert, called Mathographics. There's an online source by Freyja Hreinsdóttir called Euclidean Eggs, University of Iceland, Lifelong Learning Programme. Here's a link to her version which was very helpful when creating this drawing. She also has other styles of eggs for [...]
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Web Site Dave's Short Trig Course Check out the short trigonometry course and learn the new way of learning trig. This short course breaks into sections and allows user to learn at his/her o... Curriculum: Mathematics Grades: 9, 10, 11, 12
33.
Web Site S.O.S. Mathematics - Calculus Check out a good list of calculus problems with solutions. This is a free resource for math review material from Algebra to Differential Equations!
Web Site Order of Operations When a numerical expression involves two or more operations, there is a specific order in which these operations must be performed. The phrase PEMDA (Parenth... Curriculum: Mathematics Grades: 5, 6, 7, 8
38.
Web Site Intermediate Algebra Tutorials 42 Tutorials that math teachers can use with student or students can work on their own to reinforce skills, as homework, or review during class. Tutorials in... Curriculum: Mathematics Grades: 6, 7, 8, 9, 10, 11, 12, Junior/Community College, University
39.
Web Site Variables This site covers symbol variables and substitution of symbols to discover unknown values. In simple terms it shows you how a box is waiting for a value. (Key... Curriculum: Mathematics Grades: 6, 7, 8
40.
Web Site Introduction to Algebra Think Algebra is hard? Think again - this site explains the history along with simple equations. Each paragraph scaffolds skills until you get it. Than at th... Curriculum: Mathematics Grades: 3, 4, 5, 6
By Resource Type:
Web Site Document or Handout Image Template Book Video
"I used My eCoach for a few years as my classroom website and for surveys and quizzes.
Two years ago, the district asked us to learn to use moodle because it is linked to student's grades, etc... At first, the moodle class page was separate, but needed to be accessed by student username and password. Many students forgot passwords or changed them or couldn't get on because of other reasons. This year there is a link within the students portal site to a moodle classroom page. (They still need their username and password.) This link was not to the webpage I had made earlier, but to a new one. I switched all the information over to the new because I was told the parents would be able to access it from their portal site, also. This was not true. I then switched back to My eCoach and found it so easy to use!
The parents and students both can keep up with assignments because they do not need to remember passwords! This past week, I put together an online test in portal-moodle because I was told it would grade the test and link the grade to my gradebook. What a disaster! The transferred grade was not correct and I am dealing with many e-mails from students who could not submit their finished tests. I believe I will try the on-line quiz in My eCoach next time and just copy over the grade.
I am still only using a small portion of what is available in ecoach, but am looking forward to trying all the new things that Barbara and her team have developed over the past year or so."
Susan Brown
High School Science Teacher
Osceola High School - Pinellas, Florida
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Semester Projects
Document Actions
Project Goals
The goal of these projects is to give you the opportunity to make your own connections between mathematics and modern society by considering a wide variety of problems ranging from economic and environmental issues to social and political situations that can be modeled and solved by mathematical means. They will help you establish connections between Math 10260 and your other courses, and they will allow you to make contributions in areas of your interest and expertise. In addition, they will provide you with an opportunity to interact and collaborate with your classmates.
Rules for the Project
You can work in groups of size 1-4 students (from any section of Math 10260).
Each group submits one (typed) paper (and an electronic copy, if possible).
Each member of the group receives the same score - a number between 0 and 10 - which will count toward your 20 participation points.
Final version due by December 7. If you choose option d above, a first draft for approval is due by November 9. You must include: Project title, the names of your team members and the class section each member belongs to.
Alternative Project Topics
The Social Security. Some experts project that the Social Security shortfall over the next 75 years will be about four trillion dollars. Is that true? How do they know? Make your contribution in the national debate about saving Social Security using ideas and techniques you learned in Math 10260 (for example, income streams).
The Deficit. Visit the Webpage of the Congressional Budget Office (CBO) at and try to make sense of the numbers you will find in "Current Budget Projections." Note that income streams are useful in making projections.
Sub-prime Loans. What are sub-prime loans and what they have to do with the current housing and banking crisis?
Ponzi Scheme. Currently the Securities and Exchange Commission (SEC) is investigating an alleged $50 billion fraud, a Ponzi scheme, perpetrated by Bernard Madoff and the asset management company that he ran. Explain what is a Ponzi scheme and why in the Madoff's case it went unnoticed by the SEC for so long that it became so massive.
Arctic National Wildlife Refuge: To Drill or not to drill? A question for public debate these days is whether the Arctic National Wildlife Refuge (ANWR) contains enough oil to make its extraction worth both the economic cost and the environmental risk. Make your contribution by doing the numbers.
Energy Conservation. Some claim that there are ways for saving about 20% of the energy we consume today. Examine this claim by doing some quantitative analysis on energy consumption and energy wasted.
Oil Price. Is the current oil price the result of world demand & supply or/and market manipulation? Draw your own conclusions by collecting data from reliable sources and analyzing them using the mathematics you learned in Math 10260.
Wind Energy. Collect data about wind energy production in the U.S. since 2000 and draw a curve that fits these data. Also, draw the oil-price curve using data from reliable sources. Furthermore, compare the shape of these curves and make sense of the current projections of wind energy productions for the next 10-20 years. Finally, find out for which country in the world the percentage of the energy it uses from wind is maximum.
Solar Energy. Collect data about solar energy production in the U.S. since 2000 and draw a curve that fits these data. Also, draw the oil-price curve using data from reliable sources. Furthermore, compare the shape of these curves and make sense of the current projections of solar energy production for the next 10-20 years. Finally, find out for which country in the world the percentage of the energy it uses from the sun is maximum.
How much renewable energy do we use? Visit the Energy Information Administration to find U.S energy statistics (e.g. see and provide an answer to this question. Use the quantitative skills you acquired in Math 10260 to make your answer clear and informative.
Sustainable Development. How would you define the concept of sustainable development? Is the current socioeconomic system consistent with such a concept? If not, then provide some explanation.
The nearly $800 billion stimulus package. The following is from a speech given by President-elect Barack Obama on January 11, 2009, at George Mason University, to defend his stimulus package (text is taken from New York Times.): "I don't believe it's too late to change course, but it will be if we don't take dramatic action as soon as possible. If nothing is done, this recession could linger for years. The unemployment rate could reach double digits. Our economy could fall $1 trillion short of its full capacity, which translates into more than $12,000 in lost income for a family of four. We could lose a generation of potential and promise, as more young Americans are forced to forgo dreams of college or the chance to train for the jobs of the future. And our nation could lose the competitive edge that has served as a foundation for our strength and standing in the world...
I understand that some might be skeptical of this plan. Our government has already spent a good deal of money, but we haven't yet seen that translate into more jobs or higher incomes or renewed confidence in our economy. That's why the American Recovery and Reinvestment Plan won't just throw money at our problems - we'll invest in what works. The true test of the policies we'll pursue won't be whether they're Democratic or Republican ideas, but whether they create jobs, grow our economy, and put the American Dream within reach of the American people."
Using the materials we learned in Math10260 and your business knowledge, make a quantitative analysis of the proposed economic stimulus package. You may wish to explore its impact to the Deficit problem the country faces.
Mountains Beyond Mountains (preview). In this inspiring book. Tracey Kidder describes the quest of Dr. Paul Farmer, a man who would cure the world. Curing infectious diseases and bringing the lifesaving tools of modern medicine to those who need them most is his life calling. Read this book and use the mathematics you have learned in Math10260 to try to understand, analyze and propose possible solutions to the global health problem.
Universal Health Care. What are the benefits and problems of a universal health care system? Examine and compare the health care system of the U.S. and one or two from other developed countries like the U.K., Germany, France, Japan, etc.
The End of Poverty (preview). In the preface of this book its author Dr. Jeffrey Sachs (Quetelet Professor of Sustainable Development at Columbia University, Direct of the Earth Institute, and Director of the United Nations Millennium Project) writes: "When the end of poverty arrives, as it can and should in our generation, it will be citizens in a million communities in rich and poor countries alike, rather than a handful of political leaders, who will have turned the tide. The fight for the end of poverty is a fight that all of us must join in our own way." Read this very interesting book and use the mathematics you have learned in Math 10260 to try to understand (quantify, analyze) poverty as a world problem, and propose possible solutions that our generation can realize.
Top Ten. What are the top 10 major challenges for your generation? Provide some numbers to justify your choices.
The Paradox of Choice (preview). In this book, Barry Schwartz, among many other things, claims that freedom of choice can turn into a tyranny of choice. He even uses some math to make his point. For example, in pages 67-73 he uses familiar curves to give a general explanation of how we go about evaluating options and making decisions. Write a report on this very interesting book and try to relate it to ideas you learned in Math 10260.
Demand and Supply. Read carefully section 6.1 on consumer and producer surplus, compare it with writings in economics' literature, and explain how demand and supply are curves determined.
Flatland (read online). Imagine that you live in a plane (a 2D-space) and that you are not able to see 3D shapes. Then, think of ways for visualizing such shapes. A good source of ideas is the book "Flatland" by Edwin Abbott. Read this book and extend its ideas to describe how inhabitants of 3D-space (i.e., humans) could visualize 4D shapes.
A) Income distribution and Lorentz curves. The way that income is distributed throughout a given society is an important object of study for economists. The U.S. Census Bureau collects and analyzes income data, which it makes available at its website, In 2001, for instance, the poorest 20% of the U.S. population received 3.5% of the money income, while the richest 20% received 50.1% The cumulative proportions of population and income are shown in the following table:
Proportion of Population
Proportion of Income
0
0
0.20
0.035
0.40
0.123
0.60
0.268
0.80
0.499
1.00
1.00
For instance, the table shows that the lowest 40% of the population received 12.3% of the total income. We can think of the data in this table as being given by a functional equation y = f(x), where x is the cumulative proportion of the population and y is the cumulative proportion of income. For instance, f(0.60) = 0.268 and f(0.80) = 0.499. Such a function (or, more properly speaking, its graph) is called a Lorentz curve.
Show that f(x) = 0.1x + 0.9x2 is a possible Lorentz curve. Also, compute the income received by the lowest 0%, 50%, and 100% of the population.
Show that f(x) = 0.3x + 0.9x2 is not a Lorentz curve.
For the Lorentz curve in (i) show the following properties:
f(0) = 0, f(1) = 1, and 0 ≤ f(x) ≤ 1 for all 0 ≤ x ≤ 1,
f(x) is an increasing function,
f(x) ≤ x for all x, 0 ≤ x ≤ 1
Explain why properties (a) - (c) hold for every Lorentz curve.
Write many other different formulas for Lorentz curves.
Using real data, produce Lorentz curves for the U.S. in 2006
Sketch the graph of a Lorentz curve and compare it with the line y = z.
B) Coefficient of Inequality. If the Lorentz curve of a country is given by f(x) = x, then its total income is distributed equally. Otherwise, there are inequalities present in the distribution of income which are measured by the following number:
coefficient of inequality =
which is also called the Gini Index.
Compute the coefficient of inequality when f(x) = 0.1x + 0.9x2.
Show that the Gini Index is the ratio of the area of the region between y = f(x) and y = x to the area of the region under y = x, and provide an economic interpretation of this ratio.
Using real data estimate the Gini Index of the U.S in 2006.
A) The Cobb-Douglas Production Function. Show that the production function Q(K,L) having the properties:
(Marginal Product of Capital) * (Capital) = α * (output)
(Marginal Product of Labor) * (Labor) = (1-α) * (output)
for some constant α, 0 < α < 1, must be of the form Q(K,L)=KαL1-α, for some constant A. B) Read and understand the Solow Growth Model (Section 9.3) and do exercise 1, or 2, or 3 on page 607.
A) You are 35 years old and your company offers you the following three retirement plans:
At the beginning it deposits $50,000 into an account A and nothing more during the next 30 years.
For the next 30 years it deposits money continuously into an account B at a rate of $10,000 per year
At the age of 65, you will receive $1,2000,000 and nothing more during the next 30 years you will be working there
If the accounts A and B yield 8% interest, compounded continuously, which option will you choose? Explain your answer. B) Do part A again with interest rate at 10% compounded monthly. For Plan 2, assume that money will be deposited monthly into account B. To complete this part, you will have to set up a geometric series that gives the value of your retirement account. Go to your notes for continuous compounding and modify the setup for discrete compounding. Explain what each of the terms in the geometric series means. You should state clearly the first term, common ratio, and the formula you use to obtain the value of your retirement account from the geometric series. How would this change your decision in part A?
A) A homeowner takes out a 20-year mortgage with an interest rate of 5% compounded continuously. The homeowner plans to make payments totaling $1,500 per month. Let M(t) be the account owed after t years. Write an initial value problem modeling this situation. Then find the maximum amount of mortgage that the homeowner can afford. B) Do part A again with interest rate at 5% compounded quarterly. To complete this part, you will have to set up a geometric series that gives the value of the mortgage. Go to your notes for continuous compounding and modify the set up for discrete compounding. Explain what each of the terms in the geometric series means. You should state clearly the first term, common ratio, and the formula you use to obtain the mortgage value from the geometric series. (Hint: You should prorate the interest because you are paying monthly.)
Read carefully section 6.4 on population models and then do exercise 27 and 28 on page 445.
What does calculus have to do with change? The two central concepts in calculus are the derivative (instantaneous rate of change) and the integral (total change). Both are based on the fundamental calculus idea of "using elementary concepts (like slope of a line and area of a rectangle) to approximate advanced concepts (like slope of a curve and area enclosed by a curve)." Write in your own words the way you understand these concepts. Give examples from mathematics and its applications to demonstrate them.
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Calcula = THE CALCULATOR ... but not limited to the calculator. Calcula is not a scientific calculator. Calcula is a tool 'all-in-one': instead of having 1, 2, 5, 10, 20 applications that serve as 'technical means' we have only one: Calcula, indeed!So, what is and what makes Calcula?. calculator with the 4 operations, percentage, square root, exponentiation of x, a fraction of 1, form, and factor accumulation and subtraction in memory ... what they do all the calculators, some (not quite all, actually!). storing the list of all the transactions like a roll of paper with its zoom. button to cancel last input CI, C key to cancel the entire operation and key 'tearing paper' to delete all memorized transactions. selection of the number of decimal places, from 0 to 5, with which to develop. currency conversion online, in real time and then, leaning on a free service of common good (the result can be integrated in the operation in progress). conversion between many units of measurement: length, weight, volume, area, etc.. (The result can be integrated in the operation in progress). conversion between different number systems: decimal, octal, hexadecimal and binary (the result, of course decimal) can be integrated in the operation in progress). calculating perimeter, area and volume of many geometric shapes with a list of requests for images and input context to the figure (within the perimeter of the circle or to calculate the volume of the cylinder, or more or less according to your traps, etc..)(The result can be integrated in the operation in progress). expression processing up to 26 variables and many functions available, such as cos, sin, tan, etc.. (The result can be integrated in the operation in progress). development of algebraic proportions of the type: b = x: c-fit of the 3 known values and the processing of the result in 4 combinations (the result can be integrated in the operation in progress). generation of random numbers indicating the amount of numbers to be generated and the minimum and maximum limits (ability to select whether the numbers generated should all be different or with repetitions). elaborations of summations, differences between dates with even numbers add or subtract days. elaboration of summations, differences between zones with even add or subtract a preset time. stopwatch with lap times list the possibility of. flashlight (beam) with a selection of different colors. in cm and inch ruler, and color-changing background and calibration lines for even better viewing of the backlit. compass needle or rotary dial with digital indication of the degree. level graphics and digital indication of the degree of vertical tilt and horizontal. selection if it beeps when you press any buttons or voice with repetition of numbers typed and conducting operations in. ability to change the background color. appropriate option for the configuration settings. Detailed help on all aspects. Calcula the program is released with 2 screens, others are making and will be issued free of charge even after the purchase) to have more or fewer buttons then more or less the same size buttons. In version 1.1.00 there are 2 screens: the no. 0 with all the buttons available, some with 2 or 3 functions enabled via special button shift, the no. 1 instead of the calculator and all transactions with a button that serves as a menu to call up all the other functions.. all the screens are operated by the minimum resolution is 320x480 portrait or landscape (480x320). ON / OFF switch!The program is released in Italian,
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Course Description: Algebraic geometry is the study of solutions to systems of polynomial equations. The algebra/geometry dictionary establishes a bijection between such solution sets and certain ideals in a polynomial ring. Hence, techniques from commutative algebra can be used to analyze these geometric objects. The course will provide an introduction to algebraic geometry. We will emphasize not only the theory, but also practical aspects of how to compute with polynomial ideals using Groebner bases. We will follow the book closely and discuss the following topics: Groebner bases, affine varieties, morphisms and rational maps, elimination theory, the Nullstellensatz, primary decomposition, projective varieties, Grassmannians, and Hilbert Functions.
Homework: Homework will be assigned regularly at the beginning of the semester and is due in class on Fridays. Students must write up their own solutions. Please indicate on your homework any sources that you used in preparing solutions (e.g. if another student helped with a solution, or you found the solution in a book). Students are encouraged to prepare homework solutions in LateX. A file explaining how to prepare your homework can be found here. Homework assignments can be found here.
Final Project: Students will be expected to prepare a final project. This will consist of critically reading a paper on a topic related to the material from this course or working on a research problem in algebraic geometry, preparing a paper describing your project, and giving a twenty minute presentation to the class on your project during the final weeks of class. Project topics will be chosen halfway through the course, based on some suggested topics on the course website and in discussion with the instructor. Suggested topics can be found here.
Exams: There will be no exams. However, students will be expected to attend the final exam period as this will be used for student final presentations (Friday, 12/10 8–11 AM).
Grades: Grades will be based on Homework (50%) and a Final Project and Presentation (50%).
Attendance: Students are expected to arrive on time, to contribute to group work and class discussions, and to stay until the class ends. Attendance at all meetings of the class is expected. Occasional absences will be approved if they meet University policies.
Adverse Weather: Announcements regarding scheduled delays or the closing of the University due to adverse weather conditions will be broadcast on local radio and television stations and posted on the University homepage.
H1N1 Policy:
If you are ill with symptoms of H1N1 influenza (i.e. fever over 100, sore throat, cough, stuffy or runny nose, fatigue, headache, body aches, vomiting and diarrhea) please do not come to class. Instead, immediately contact your medical provider or Student Health Services (515-7107) for advice or to arrange an appointment. If you are diagnosed with H1N1, please inform me immediately. You will be required to be isolated away from class for up to 7 days or 24 hours after symptoms subside, whichever is longest.
Cell Phones: Pagers, cellular phones and other types of telecommunication equipment are prohibited from use during class. Make sure that any pagers, phones or other equipment are turned off during the class period. If you have a special need to have your pager or phone on during class, please let me know.
Academic Integrity Statement: Students are required to follow the NCSU policy . "Academic dishonesty is the giving, taking, or presenting of information or material by a student that unethically or fraudulently aids oneself or another on any work which is to be considered in the determination of a grade or the completion of academic requirements or the enhancement of that student's record or academic career.'' (NCSU Code of Student Conduct). The Student Affairs website has more information.
Students with Disabilities: Reasonable accommodations will be made for students with verifiable disabilities. In order to take advantage of available accommodations, students must register with Disabilities Services for Students.
Class Evaluations: Online class evaluations will be available for students to complete during the last two weeks of class. Students will receive an email message directing them to a website where they can login using their Unity ID and complete evaluations. All evaluations are confidential; instructors will never know how any one student responded to any question, and students will never know the ratings for any particular instructors.
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Introduction to Topology We will study the basic theory and the topological properties of the Möbius bands, the torus and the Klein bottles to explain why a topologist cannot distinguish between a doughnut and a tea cup.
You Can Count on It - Maths in Finance In this brief course we shall look at how mathematics contributes to finance and business. Our course is suitable for people with previous experience of mathematics at the sixth-form level and aims to provide an elementary introduction to the mathematics.
Puzzles and Pastimes Puzzles and pastimes can be both amazing and amusing. Mathematical puzzles often seem like magic. Mechanical examples include Rubik's cube. Other topics include board and card games, game theory (hawks and doves) and Sudoku..
Alternatively you can perform a keyword search on all our courses using the 'Find courses' box on this page.
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Synopses & Reviews
Publisher Comments:
Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.
Book News Annotation:
Not a textbook, and not just for the classroom, this book which comes in two parts, is a collection of largely independent chapters on the major concepts related to the science and mathematics of fractals, chaos, and dynamics, presented at the mathematical level of an advanced secondary school student. A wide range of illustrations (including 14 color plates) are integrated with an enjoyable text, bringing the concepts alive and making them understandable to a large audience. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Synopsis:
"Synopsis"
by Ingram,
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Matlab: An Introduction WithMATLAB: An Introduction with Applications 5th Edition walks readers through the ins and outs of this powerful software for technical computing. The text describes basic features of the program and shows how to use it in simple arithmetic operations with scalars. The topic of arrays (the basis of MATLAB) is examined, along with a wide range of other applications. MATLAB: An Introduction with Applications 5th Edition is presented gradually and in great detail, generously illustrated through computer screen shots and step-by-step tutorials, and applied in problems in mathematics, science, and engineering
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Geometry : a high school course by Serge Lang(
Book
) 40
editions published
between
1983
and
2011
in
3
languages
and held by
601 WorldCat member
libraries
worldwide
From the reviews: "A prominent research mathematician and a high school teacher have combined their efforts in order to produce a high school geometry course. The result is a challenging, vividly written volume which offers a broader treatment than the traditional Euclidean one, but which preserves its pedagogical virtues. The material included has been judiciously selected: some traditional items have been omitted, while emphasis has been laid on topics which relate the geometry course to the mathematics that precedes and follows. The exposition is clear and precise, while avoiding pedantry. There are many exercises, quite a number of them not routine. The exposition falls into twelve chapters: 1. Distance and Angles.- 2. Coordinates.- 3. Area and the Pythagoras Theorem.- 4. The Distance Formula.- 5. Some Applications of Right Triangles.- 6. Polygons.- 7. Congruent Triangles.- 8. Dilatations and Similarities.- 9. Volumes.- 10. Vectors and Dot Product.- 11. Transformations.- 12. Isometries. This excellent text, presenting elementary geometry in a manner fully corresponding to the requirements of modern mathematics, will certainly obtain well-merited popularity. Publicationes Mathematicae Debrecen#1
Geometry by Serge Lang(
Book
) 3
editions published
between
1997
and
2010
in
English and German
and held by
10 WorldCat member
libraries
worldwide
| 677.169 | 1 |
Algebra Antics
Presenting algebra exercises in which FUN is the "unknown quantity!" Each page features 12 to 24 skill-building algebra problems. After students have simplified expressions or solved equations, the answers provide clues for drawing lines to reveal a secret picture in the coordinate grid. Algebra Antics is a unique and fresh approach to algebra practice; great for students who enjoy visual challenges and direct feedback.
for Algebra Antics
Author: Barbara Smead Date: 01/18/2014 Demographic: Parent Rating: 4 out of 5 stars My daughter has had a great time working the book. She has almost finished. Not bad for a non math freak!
Author: Sheryl Reed Date: 01/15/2014 Demographic: Parent Rating: 5 out of 5 stars She has done several pages already and really loves it. Great car activist when there is no homework.
Author: LS Date: 11/07/2013 Demographic: Teacher Rating: 5 out of 5 stars My middle-school students love this book's way of combining hard problems with a picture. I love how the books make my students practice difficult pre-algebra problems and graphing coordinates, so their brains have to switch gears and thereby make stronger memories.
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07682025Pre-Algebra Made Simple, Middle School (Teaching Resource)
Now it's simple to make Pre-Algebra fun, relevant, interesting, and exciting. This book is designed to help students develop a basic understanding of algebraic concepts using everyday applications. Includes activities on whole numbers and integers, solving equations, geometry, logic, problem solving and patterning, and statistics and probability. Background information, extension activities, group learning, and school-home connections are provided along with an answer key
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More About
This Textbook
Overview
Kaufmann and Schwitters have built this text's reputation on clear and concise exposition, numerous examples, and plentiful problem sets. This traditional text consistently reinforces the following common thread: learn a skill; practice the skill to help solve equations; and then apply what you have learned to solve application problems. This simple, straightforward approach has helped many students grasp and apply fundamental problem-solving skills necessary for future mathematics courses. Algebraic ideas are developed in a logical sequence, and in an easy-to-read manner, without excessive vocabulary and formalism. The open and uncluttered design helps keep students focused on the concepts while minimizing distractions. Problems and examples reference a broad range of topics, as well as career areas such as electronics, mechanics, and health, showing students that mathematics is part of everyday life. The text's resource package—anchored by Enhanced WebAssign, an online homework management tool—saves instructors time while providing additional help and skill-building practice for students outside of class.
Editorial Reviews
From the Publisher
"This text is a very straightforward approach to learning skills needed in college algebra. There are plenty of examples and problems to help students prepare for college algebra."- Joseph Eyles, Morehouse College
"The one thing about Kaufmann books is that they are very clear in their intent. This is one of the strongest assets of the book. It's not all fancy; it just says this is what we're going to do and then it does it . . ."- Patrick Webster, El Camino College
Booknews
A text for college students who need an algebra course that bridges the gap between elementary algebra and the more advanced courses in precalculus mathematics, covering intermediate algebra topics. Algebraic ideas are developed in logical sequence in an easy-to-read manner without excessive formalism. Concepts are developed through examples and problem solving. Includes chapter summaries, problems, and chapter and cumulative tests, with answers. This sixth edition contains word problems, coverage of all real numbers, incorporation of a graphing approach, and earlier introduction of the graphing calculator. The authors are affiliated with Seminole Community College. Annotation c. Book News, Inc., Portland, OR booknews.com
Product Details
Meet the Author
Jerome E. Kaufmann received his Ed.D. in Mathematics Education from the University of Virginia. Now a retired Professor of Mathematics from Western Illinois University, he has more than 30 years of teaching experience at the high school, two-year, and four-year college levels. He is the author of 45 college mathematics textbooks.
Karen L. Schwitters graduated from the University of Wisconsin with a B.S. in Mathematics. She earned an M.S. Ed. in Professional Secondary Education from Northern Illinois University. Schwitters is currently teaching at Seminole Community College in Sanford, Florida, where she is very active in multimedia instruction and is involved in planning distance learning courses with multimedia materials. She is an advocate for Enhanced WebAssign and uses it in her classroom. In 1998, she received the Innovative Excellence in Teaching, Learning, and Technology
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This course will provide an introduction to the use of computers to solve problems arising in the physical, biological, and engineering sciences. Various computational approaches commonly used to solve mathematical problems (including systems of linear equations, curve fitting, integration, and differential equations) will be presented. Both the theory and application of each numerical method will be demonstrated. The student will gain mathematical judgment in selecting tools to solve scientific problems through in-class examples and programming homework assignments. MATLAB will be used as the primary environment for numerical computation. An overview of MATLAB's syntax, code structure, and algorithms will be given. Although the subject matter of Scientific Computing has many aspects that can be made rather difficult, the material in this course is an introduction to the field and will be presented in a simple as possible way. Theoretical aspects will be mentioned throught the course, but more complicated issues such as proofs of relevant theorems/schemes will not be presented. Applications will be emphasized.
Objectives
To make MATLAB superstars!
Syllabus
Week 1 - Basics of MATLAB and Introduction
We will begin with a brief review of MATLAB and its basic functionality, including plotting, implementing IF and FOR logicals and the construction of matrices and vectors.
(a) constructing matrices and vectors in MATLAB (text 1.1)
(b) FOR and IF statements for program logic (text 1.2)
(c) inputing/exporting data and plotting (text 1.5)
Week 2
Linear Algebra and Ax=b
We discuss direct solutions to matrix systems of equations Ax=b. Computational complexity, i.e. how methods scale with size of system, is considered.
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often referred to as industrial mathematics is becoming a more important focus of applied mathematics. An increased interest in undergraduate control theory courses for mathematics students is part of this trend. This is due to the fact that control theory is both quite mathematical and very important in applications. Introduction to Feedback Control provides a rigorous introduction to input/output, controller design for linear systems to junior/senior level engineering and mathematics students. All explanations and most examples are single-input, single-output for ease of exposition. The student is assumed to have knowledge of linear ordinary differential equations and complex variables. Written specifically for the undergraduate mathematics student Concise and clear examples that illustrate theoryAuthor is faculty member at University of Waterloo; the largest mathematics department in the worldContains exercisesIncludes MATLAB
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proven system of teaching math for classroom teachers, and for parents who are homeschooling. The easy-to-follow program requires only 20 minutes a day. Short, concise, and self-contained lessons help students to master, maintain and reinforce math skil
Customer Reviews
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
61 reviews
36 of 37 people found the following review helpful
Our girls no longer complained!Feb. 24 2009
By
K. Webster
- Published on Amazon.com
Format: Paperback
Last year in our homeschool, we switched to "Mastering Essential Math Skills: Middle Grade/High School, and it was the best switch we could have made. Both girls were to the point of dreading math, and after changing to this program, they now looked forward to math class and could make sense of it because of the step-by-step instructions. Great for 7th-8th grades. Our 2nd daughter actually moved from 5th grade to 7th grade because this program was available.
65 of 75 people found the following review helpful
Not as advertisedFeb. 17 2007
By
Parent
- Published on Amazon.com
Format: Paperback
Verified Purchase
I loved the grade 6-8 edition of this book for reviewing math with my daughter. It is an excellent step by step approach to middle school math. This edition is the same book with very little added. It is not "high school" math. I wouldn't call it "Book Two". I was expecting something very different based on the title. Highly recommended for 6-7 grade math practice and review.
17 of 17 people found the following review helpful
Good math book, but has typo's.July 17 2007
By
The Way
- Published on Amazon.com
Format: Paperback
This book is helping me getting freshed up in math but it contains a few typo's. For example, on page 7 it says there are two #5 problems. There should be obviously one problem #5 and the other is problem #6. Nevertheless, it is a good math book. I just wished it gave more examples on how to actually DO some of the work.
9 of 9 people found the following review helpful
Miscategorized.Aug. 20 2011
By
Cautious Shopper
- Published on Amazon.com
Format: Paperback
Verified Purchase
This book seemed more geared toward elementary/middle school than middle school/high school. Anyone going into 9th grade might find this way too simple.
10 of 11 people found the following review helpful
Exactly what I was looking for!June 16 2009
By
L. Halling
- Published on Amazon.com
Format: Paperback
There's nothing like having a personal tour guide to lead you through the jungle of math skills. Richard Fisher's 30 years of experience teaching math shines through in his Mastering Essential Math Skills Books 1 and 2, shedding light on the dark and daunting paths to the realm of math comprehension. Book One for Grades 4 and 5, takes us all the way from basic adding, subtracting, multiplying and dividing to number theory and Algebra: effectively, concisely, and simply. I'm not a math expert but I would say both books will not only bring your student up to grade level (if they weren't there already) but, I dare say, will accelerate them. In the companion DVDs included with both books, Mr. Fisher's easy-going, encouraging manner walks us, at a comfortable pace, through the maze of math. A feature I consider ingenious is the built in speed drills on each one page lesson. It is so cleverly designed, it leaves one thinking, "why didn't I think of that?" Book Two is designated for Middle Grades and High school. It contains the same sequence of development but goes a little further into each skill giving students a more comprehensive understanding of the necessary skills to tackle Algebra I (coming soon). I plan to use Book One with my 14 year old as a review to begin with this fall before taking on Algebra second semester. After going through Mr. Fisher's Pre-Algebra Concepts book with DVD this year, my son has actually requested, nay, begged, to use Mr. Fisher's math curriculum for the remainder of his schooling! Need I say more?
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Course Descriptions
This course, in the calculus of a single variable, concerns
recognizing, analyzing, and calculating problems in the
following topic areas: the calculus of inverse trigonometric
functions, integration techniques, application of
integration, L'Hopital's Rule, improper integrals, infinite
sequences and series, plane curves, parametric equations,
polar coordinates, and polar curves.
PR: MAT 180 or consent of the department
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Product Description
The Algebra 1: The Complete Course DVD Series will help students build confidence in their ability to understand and solve algebraic problems.
In this episode, students will learn to use concrete examples and practical applications to understand algebraic concepts. Students will learn about sets of natural and whole numbers, sets of integers, sets of rational numbers and sets of real numbers. Grades 5-9. 30 minutes on DVD.
DVD Playable in Bermuda, Canada, United States and U.S. territories. Please check if your equipment can play DVDs coded for this region. Learn more about DVDs and Videos
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Product Details
See What's Inside
Product Description
By Matthew LarsonFocusing on similarities between these new standards and those outlined in NCTM's influential Principles and Standards for School Mathematics, this handbook efficiently highlights reasoning processes that are essential in any high-quality mathematics program. Students who know these processes are set to "do math" in any context, real or abstract.
Research indicates that many individuals within a school can and do contribute to the work of leading and managing a school. This guide can support anyone who is working to improve mathematics education—either alone or with others.
Make the most of the rare opportunity that the Common Core State Standards offer for rethinking school mathematics and creating exciting new pathways from high school to college and beyond.
Includes answers to "Frequently Asked Questions" and detailed lists of resources to support school mathematics programsCustomers Who Bought This Also Bought... proof and the process of proving. It is organized around five big ideas, supported by multiple smaller, interconnected ideas—essential understandings"Powerfully affirmative to teachers who may feel less comfortable teaching math than reading . . . equally appealing to teachers who are math gurus and those who are less confident in their math skills." —Sally Moomaw, Ed.D., Assistant Professor of Early Childhood Education, University of Cincinnati
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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Location
Hours of operation
Calculator Policy
Students must demonstrate the ability to perform basic operations at least once without the use of a
four-function calculator. To this end calculators will not be allowed in the following:
Math 27 (self-paced basic skills mathematics)
The college's math placement test
The first chapter of Math 34 (pre-algebra)
Following this, students will be allowed to use a four-function calculator during class and during tests in
any math course offered by the college, provided they supply their instructors with a letter explaining their
accommodations at least two days prior.
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Applied Combinatorics - 2nd edition
Summary: For courses in undergraduate Combinatorics for juniors or seniors.
This carefully crafted text emphasizes applications and problem solving. It is divided into 4 parts. Part I introduces basic tools of combinatorics, Part II discusses advanced tools, Part III covers the existence problem, and Part IV deals with combinatorial optimization.
Book has very light external/internal wear. It may have creases on the cover and some folded pages.This is a USED book.
$12.4612
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Mathematics for Physicists
9780534379971
ISBN:
0534379974
Pub Date: 2003 Publisher: Thomson Learning
Summary: This essential new text by Dr. Susan Lea will help physics undergraduate and graduate student hone their mathematical skills. Ideal for the one-semester course, MATHEMATICS FOR PHYSICISTS has been extensively class-tested at San Francisco State University--and the response has been enthusiastic from students and instructors alike. Because physics students are often uncomfortable using the mathematical tools that they... learned in their undergraduate courses, MATHEMATICS FOR PHYSICISTS provides students with the necessary tools to hone those skills. Lea designed the text specifically for physics students by using physics problems to teach mathematical concepts.
Lea, Susan M. is the author of Mathematics for Physicists, published 2003 under ISBN 9780534379971 and 0534379974. Five hundred eighty five Mathematics for Physicists textbooks are available for sale on ValoreBooks.com, one hundred eighteen used from the cheapest price of $32.97, or buy new starting at $69
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Books on Mathematics > Algebra > Linear
6 new & used from sellers starting at 1,582 In Stock.Ships Free to India in
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Renowned professor and author Gilbert Strang demonstrates that linear algebra is a fascinating subject by showing both its beauty and value. While the mathematics is there, the effort is not all concentrated on proofs. Strang's emphasis is on understanding. He explains concepts, rather than...... more
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This clear, concise and highly readable text is designed for a first course in linear algebra and is intended for undergraduate courses in mathematics. It focusses throughout on geometric explanations to make the student perceive that linear algebra is nothing but analytic geometry of n dimensions. From the very start,... more
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An effort has been made to present the various topics in the theory of graphs in a logical order, to indicate the historical background, and to clarify the exposition by including figures to illustrate concepts and results. In addition, there are three appendices which provide diagrams of graphs, directed graphs,... more
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This text offers a comprehensive and coherent introduction to the fundamentals of graph theory. Written in a reader-friendly style and with features that enhance students- comprehension, the book focuses on the structure of graphs and techniques used to analyze problems. Greatly expanded and reorganized, this edition is integrated with key... more
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In this international version of the first edition, Principles of Signal Processing and Linear Systems, the author emphasizes the physical appreciation of concepts rather than the mere mathematical manipulation of symbols Avoiding the tendency to treat engineering as a branch of applied mathematics, the text uses mathematics not so much... more
Available.
The state space method developed in the last decades allows us to study the theory of linear systems by using tools from the theory of linear operators; conversely, it had a strong influence on operator theory introducing new questions and topics. The present volume contains a collection of essays representing... more
Available.
Written For The Undergraduate Linear Algebra Student, Linear Algebra: Theory And Applications, Serves As The Ideal Text For Science And Engineering Students, Who Are Interested Principally In Applications, As Well As For Mathematics Students, Who Wish To Acquire A Mastery Of Theoretical Linear Algebra. This Flexible Blend Serves Diverse Groups... more
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This translation of the 1987 German edition is an introduction into the classical parts of algebra with a focus on fields and Galois theory. It discusses nonstandard topics, such as the transcendence of pi, and new concepts are defined in the framework of the development of carefully selected problems. It... more
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This volume presents a fairly self-contained theory of certain singular coverings of toposes, including branched coverings. This is a field that should be of interest to topologists working in knot theory, as well as also to certain categorists. An unusual feature which distinguishes this book from classical treatments of the... more
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The present book deals with factorization problems for matrix and operator functions. The problems originate from, or are motivated by, the theory of non-selfadjoint operators, the theory of matrix polynomials, mathematical systems and control theory, the theory of Riccati equations, inversion of convolution operators, theory of job scheduling in operations... more
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Matrix-Based Multigrid introduces and analyzes the multigrid approach for the numerical solution of large sparse linear systems arising from the discretization of elliptic partial differential equations. Special attention is given to the powerful matrix-based-multigrid approach, which is particularly useful for problems with variable coefficients and nonsymmetric and indefinite problems. This... more
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Written by the recipient of the 1997 MAA Chauvenet Prize for mathematical exposition, this book tells how the theory of Lie groups emerged from a fascinating cross fertilization of many strains of 19th and early 20th century geometry, analysis, mathematical physics, algebra and topology. The reader will meet a host... more
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The Structural Theory of Probability addresses the interpretation of probability, often debated in the scientific community. This problem has been examined for centuries; perhaps no other mathematical calculationsuffuses mankind's efforts at survival as amply as probability. In the dawn of the 20th century David Hilbert included the foundations of the... more
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This volume aims to provide the fundamental knowledge to appreciate the advantages of the J-matrix method and to encourage its use and further development. The J-matrix method is an algebraic method of quantum scattering with substantial success in atomic and nuclear physics. The accuracy and convergence property of the method... more
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Linear algebra permeates mathematics, perhaps more so than any other single subject. It plays an essential role in pure and applied mathematics, statistics, computer science, and many aspects of physics and engineering. This book conveys in a user-friendly way the basic and advanced techniques of linear algebra from the point... more
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Examines linear structures, the topology of metric spaces, and continuity in infinite dimensions, with detailed coverage at the graduate level Includes applications to geometry and differential equations, numerous beautiful illustrations, examples, exercises, historical notes, and comprehensive index May be used in graduate seminars and courses or as a reference text... more
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Table of Contents Graphs Euler and Hamiltonian Graphs Tree Cut Sets and Network Flow Planar Graphs Vector Spaces of a Graph Matrix Representation of Graphs Colouring of Graphs Enumeration of Graphs Directed Graphs Previous Years Question Papers Index... more
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Canadian Senior and Intermediate Mathematics Contests
The Canadian Senior and Intermediate Mathematics Contests (CSMC and CIMC) are two contests designed to give students the opportunity to have fun and to develop their mathematical problem solving ability.
Audience
Students in Grades 10 or 9 or below (CIMC)
and senior secondary school and CÉGEP students (CSMC);
motivated students in lower grades are also encouraged to write these contests.
Date
Thursday, November 20, 2014
Format
9 questions; 6 are answer only and 3 are full solution
marks for full solution questions assigned for form and style of presentation
2 hours
60 total marks
non-programmable calculators permitted provided they are without graphical displays
Mathematical Content
Most of the CIMC problems are based on the mathematical curriculum up to and including Grade 10. Most of the CSMC problems are based on the mathematical curriculum up to and including the final year of secondary school.
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Permutations, combinations, and basic probability, 12. Matrices. Often coming between Algebra I and Algebra II, Geometry is the study of the properties and uses of geometric figures in two and three dimensions.Organic Chemistry is one of the most difficult subjects for students to understand. It
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97803064614odes on Algebraic Curves
This book provides a self-contained introduction to the theory of error-correcting codes and related topics in number theory, Algebraic Geometry and the theory of Sphere Packings. The material is presented in an easily understandable form. This book is devoted to geometric Goppa codes; the recently discovered areas which combines Coding Theory, Algebraic Geometry, Number Theory, and Theory of Sphere Packings. It has an interdisciplinary nature and demonstrates the close interconnection of Coding Theory with various classical areas of mathematics. There are four main themes in the book. The first is a brief exposition of the basic concepts and facts of error-correcting code theory. The second is a complete presentation of the theory of algebraic curves; especially the curves defined over finite fields. The third is a detailed description of the theory of elliptic and modular codes, and their reductions modulo a prime number. The fourth is a construction of geometric Gappa codes producing rather long linear codes with very good parameters coming from algebraic curves, and with a lot of rational points. The aim of the book is to present these themes in a simple, easily understandable manner, and explain their close interconnection. At the same time the book introduces the reader to topics which are at the forefront of current
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Prealgebra
This book's clear, well-constructed and straightforward writing style makes it accessible to even the most apprehensive math students.The primary ...Show synopsisThis book's clear, well-constructed and straightforward writing style makes it accessible to even the most apprehensive math students.The primary focus of the pedagogy, presentation and other elements is to ease the transition into algebra; for example, emphasis is placed on basic arithmetic operations within algebraic contexts. The Second Edition includes a greater integration of NCTM and AMATYC standards, including more emphasis on visualization, problem solving and data analysis.Hide synopsis
Description:Fine. Paperback. Almost new condition. SKU: 9780321955043-2-0-3...Fine. Paperback. Almost new condition. SKU: 978032195504319550432886Description:Good. Paperback. May include moderately worn cover, writing,...Good. Paperback. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780321628862
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Book Summary of S.Chand's Mathematics For Class Ix Term ...
(Paperback)
A Complete Textbook of Mathematics for Term -II absolutely based on new pattern of examination CCE for both Formative and Summative Assessments with following key features and worksheets at the end of each chapter for practising the problems based on : True / False | Fill in the blanks | Match the Columns | MCQ's | Asserstion Reasoning | Comprehension | Riddles | Special Worksheets | Activities for Lab Manual | Important Facts | Ten CBSE Model Papers
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Chases & Escapes The Mathematics of Pursuit & Evasion
9780691125145
ISBN:
0691125147
Pub Date: 2007 Publisher: Princeton University Press
Summary: Ideal both for self-study and as supplemental readings by students and/or professors in any of the mathematical and physical sciences, this text presents the historical development of the differential equations of pursuit theory.
Nahin, Paul J. is the author of Chases & Escapes The Mathematics of Pursuit & Evasion, published 2007 under ISBN 9780691125145 and 0691125147. Two hundred seventy six Chases & Escap...es The Mathematics of Pursuit & Evasion textbooks are available for sale on ValoreBooks.com, one hundred six used from the cheapest price of $5.76, or buy new starting at $22Princeton. 2007. Princeton University Press. 1st American Edition. Very Good In Dustjacket. 270 pages. hardcover. 9780691125145. keywords: Mathematics. inventory # 36711. FROM THE PUBLISHER-We all played tag when we were kids. The rules couldn't be easier?one player is designated ?it' and must try to tag out one of the others. What most of us don't realize is that this simple chase game is in fact an application of pursuit theory, and that the same principles of games like tag, dodgeball, and hide-and-seek are at play in military strategy, high-seas chases by the Coast Guard, even romantic pursuits. In Chases and Escapes, Paul Nahin gives us the first complete history of this fascinating area of mathematics. Writing in an accessible style that has been enjoyed by popular-math enthusiasts everywhere, Nahin traces the development of modern pursuit theory from its classical analytical beginnings to the present day. Along the way, he informs his mathematical discussions with fun facts and captivating stories. Nahin invites readers to explore the different approaches to solving various chase-and-escape problems. He draws upon game theory, geometry, linear algebra, target-tracking algorithms?and much more. Nahin offers an array of challenging puzzles for beginners on up, providing historical background for each problem and explaining how each one can be applied more broadly. Chases and Escapes includes solutions to all problems and provides computer programs that readers can use for their own cutting-edge analysis. This informative and entertaining book is the first comprehensive treatment of the subject, one that is sure to appeal to anyone interested in the mathematics that underlie the all-too-human endeavor of pursuit and evasion. Paul J. Nahin is Professor Emeritus of Electrical Engineering at the University of New Hampshire. His books include Dr. Euler's Fabulous Formula, When Least Is Best, and Duelling Idiots and Other Probability Puzzlers (all Princeton).[less]
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Summary: These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short exercises that focus on developing a particular skill, mostly requiring students to draw or interpret sketches and graphs.
New Jersey 2007 Paperback 2nd Revised edition. Revised. Good. Go green, recycle! Book may have wear from reading, may contain some library markings. 480 p. Intended for professional and scholarly au...show moredience0321513576
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Find a Humble can almost guarantee that you will have "Aha! So that's how it works!" moments as algebra becomes more familiar and understandable. Algebra 2 builds on the foundation of algebra 1, especially in the ongoing application of the basic concepts of variables, solving equations, and manipulations such as factoring.
| 677.169 | 1 |
College Algebra and Trigonometry: A Unit Circle Approach
With an emphasis on problem solving and critical thinking, Mark Dugopolski's College Algebra and Trigonometry: A Unit Circle Approach, Sixth Edition ...Show synopsisWith an emphasis on problem solving and critical thinking, Mark Dugopolski's College Algebra and Trigonometry: A Unit Circle Approach, Sixth Edition gives students the essential strategies to help them develop the comprehension and confidence they need to be successful in this course. Students will find carefully placed learning aids and review tools to help them do the math1916492 / 9780321916495 College Algebra and Trigonometry: A Unit Approach Plus NEW MyMathLab with Pearson eText -- Access Card Package Package consists of: 0321431308 / 9780321431301 MyMathLab -- Glue-in Access Card 0321654064 / 9780321654069 MyMathLab Inside Star Sticker 0321916522 / 9780321916525 College Algebra and Trigonometry: A Unit Circle ApproachHide synopsis
Description:New. 0321644778 New book with very minor shelf wear. STUDENT US...New. 0321644778
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College Algebra : Concepts and Models - 5th edition
Summary: College Algebra: Concepts and Models provides a solid understanding of algebra, using modeling techniques and real-world data applications. The text is effective for students who will continue on in mathematics, as well as for those who will end their mathematics education with college algebra. Instructors may also take advantage of optional discovery and exploration activities that use technology and are integrated throughout the text.
The Fifth Edition en...show morehances problem solving coverage through Make a Decision features. These features are threaded throughout each chapter, beginning with the Chapter Opener application, followed by examples and exercises, and ending with the end-of-chapter project. This edition also features Eduspace, Houghton Mifflin's online learning tool, which allows instructors to teach all or part of a course online, and provides students with additional practice, review, and homework problems.
A brief version of this text, College Algebra: A Concise Course, provides a shorter version of the text without the introductory review.
New! Make a Decision features thread through each chapter beginning with the Chapter Opener application, followed by examples and exercises, and ending with the end-of-chapter project. Students are asked to choose which answer fits within the context of a problem, to interpret answers in the context of a problem, to choose an appropriate model for a data set, or to decide whether a current model will continue to be accurate in future years. The student must examine all data and decide upon a final answer.
Chapter Projects extend applications designed to enhance students understanding of mathematical concepts. Real data is previewed at the beginning of the chapter and then analyzed in detail in the Project at the end of the chapter. Here the student is guided through a set of multi-part exercises using modeling, graphing, and critical thinking skills to analyze the data.
A variety of exercise types are included in each exercise set. Questions involving skills, modeling, writing, critical thinking, problem-solving, applications, and real data sets are included throughout the text. Exercises are presented in a variety of question formats, including free response, true/false, and fill-in the blank.
New! "In the News" Articles from current media sources (magazines, newspapers, web sites, etc.) have been added to every chapter. Students answer questions that connect the article and the algebra learned in that section. This feature allows students to see the relevancy of what they are learning, and the importance of everyday mathematics.
Discussing the Concept activities end most sections and encourage students to think, reason, and write about algebra. These exercises help synthesize the concepts and methods presented in the section. Instructors can use these problems for individual student work, for collaborative work or for class discussion. In many sections, problems in the exercise sets have been marked with a special icon in the instructor's edition as alternative discussion/collaborative problem.
Discovery activities provide opportunities for the exploration of selected mathematical concepts. Students are encouraged to use techniques such as visualization and modeling to develop their intuitive understanding of theoretical concepts. These optional activities can be omitted at the instructor's discretion without affecting the flow of the material.49 +$3.99 s/h
Good
Silver Arch Books St Louis, MO
Book has a small amount of wear visible on the binding, cover, pages. Selection as wide as the Mississippi.
$12.49 +$3.99 s/h
Good
AllAmericanTextbooks Ypsilanti, MI
061849281X Multiple available! Some wear to cover. Text is in great shape. ISBN|061849281X, College Algebra: Concepts and Models (C.)2006 (MAD) WQ
Ships today or the next business day. Cover and binding show moderate wear. Text may contain moderate writing/highlightings but is easily readable. 28-12.
$2727Good
booksmostly Pasadena, CA
2005 Hardcover Good Few if any marks, if so large majority of pages clean, boards/binding with light wear/stickers.
$68
| 677.169 | 1 |
Often, trigonometry students leave class believing that they understand a concept but are unable to apply that understanding when they get home and attempt their homework problems. This mainstream yet innovative text is written by an experienced professor who has identified this gap as one of the biggest challenges that trigonometry professors face. She uses a clear voice that speaks directly to students- similar to how instructors communicate to them in class. Students learning from this text will overcome common barriers to learning trigonometry and will build confidence in their ability to do mathematics.
| 677.169 | 1 |
OxtonHouse Publishers,LLC
Pathways from the Past II
Pathways from the Pastis a pair of teacher's manuals and two sets of activity sheets
for helping students master various topics in arithmetic, pre-algebra, and algebra.
Building on ideas from their award-winning book, Math through the Ages, Drs. Berlinghoff
and Gouvêa have crafted teacher-friendly tools for helping students understand basic
topics that often cause trouble. Their approach implements the spirit of the NCTM
document, Reasoning and Sense Making, as well as the "Standards for Mathematical
Practice" of the Common Core State Standards for Mathematics (CCSSM). The second
set is
Pathways from the Past
II: Using History to Teach Algebra
This is a 72-page manual and a set of reproducible masters for 18 two-page activity
sheets that you can use with your students to strengthen their ability to think clearly
and reason effectively as they learn algebra.
Contents
First Thoughts for Teachers
1. Writing Algebra: Using Algebraic Symbols
Sheet 1-1: Symbols of Arithmetic
Sheet 1-2: Algebra in Italy, 1200 - 1550
Sheet 1-3: Germany and France, 1450 - 1600
Sheet 1-4: Letters for Numbers
2. Linear Thinking: Ratio, Proportion, and Slope
Sheet 2-1: The Rule of Three Direct Sheet 2-2: The Rule of Three Inverse
A section for each Activity Sheet starts with its main mathematical ideas and its
pedagogical purpose, followed by a detailed solution for each question.
The Content
The four chapters in this set progress from deciphering basic algebraic symbols to
tracing how the pursuit of solutions to cubic equations led to major changes in the
way we think about numbers. There's something for every level of algebra instruction,
from the basic ideas of proportion and linearity to the equation-solving that led
to the complex number system. Connections to European and early American history
provide a fertile ground for interdisciplinary student activities.
| 677.169 | 1 |
Do the Math: Secrets, Lies, and Algebra
About the Book
In the eighth grade, 1 math whiz < 1 popular boy, according to Tess's calculations. That is, until she has to factor in a few more variables, like: 1 stolen test (x), 3 cheaters (y), and 2 best friends (z) who can't keep a secret. Oh, and she can't forget the winter dance (d)!
Then there's the suspicious guy Tess's parents know, but that's a whole different problem.
| 677.169 | 1 |
kerja kursus additional mathematics 2011Before, shop-fitting primarily contains installing surfaces, shelves and additional essential accessories that have been needed to keep and show the retailer's products and was regarded significantly less significant to companies with
| 677.169 | 1 |
From Newton's Law of Gravity to the Black-Scholes model used by bankers to predict the markets, equations, are everywhere - and they are fundamental to everyday life. In this book, the author sets out seventeen equations that have altered the course of human history. It is also an exploration - and explanation - of life on earth.
In its updated second edition, this book explores computational methods for problems arising in the areas of classical analysis, approximation theory, and ordinary differential equations, among others. Includes new exercises, and a complete solutions manual.
Helping Children Learn to Love Their Most Hated Subject--And Why It's Important for America
by: Jo Boaler
$26.86
$29.85 inc GST
$24.42
$27.14 ex GST
ISBN: 9780670019526
The U.S. is rapidly falling behind the rest of the developed world in terms of math education. In this straightforward and inspiring book, Boaler presents concrete solutions to help reverse this trend, including classroom approaches, essential strategies for students, and advice for parents.
Making good decisions under conditions of uncertainty requires an appreciation of the way random chance works. In this Very Short Introduction, John Haigh provides a brief account of probability theory; explaining the philosophical approaches, discussing probability distributions, and looking its applications in science and economics.
In this Very Short Introduction, Jacqueline Stedall explores the rich historical and cultural diversity of mathematical endeavour from the distant past to the present day, using illustrative case studies drawn from a range of times and places; including early imperial China, the medieval Islamic world, and nineteenth-century Britain.
Leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the notions of hyperspace. This title reveals how simple questions anyone might ask about space in the living room or in some other galaxy have been the hidden engine of science's highest achievements.
Provides plain-English explanations of the most challenging aspects of trig, plus numerous practice problems, and their easy-to-follow solutions. This helpful guide is the next best thing to a personal trigonometry tutor!--
| 677.169 | 1 |
the fundamentals of discrete mathematics with DISCRETE MATHEMATICS FOR COMPUTER SCIENCE with Student Solutions Manual CD-ROM! An increasing number of computer scientists from diverse areas are using discrete mathematical structures to explain concepts and problems and this mathematics text shows you how to express precise ideas in clear mathematical language. Through a wealth of exercises and examples, you will learn how mastering discrete mathematics will help you develop important reasoning skills that will continue to be useful throughout your career.
Customer Reviews
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
3 reviews
1 of 1 people found the following review helpful
A text for the masochistic learnerJan. 14 2014
By
Jwheezy
- Published on Amazon.com
Format: Hardcover
This was a required textbook for a course at my university. My professor pulled all the homework from the ends of each chapter. This part of the book is one of my biggest gripes. The reading sections of this book pack a large amount of material in a brief page or two for each section followed by homework exercises. The exercise sections have are about as long as the actual information sections, meaning they are packed with questions. This would be a positive for this book except the questions aren't similar, so the included CD with the odd problems solved will often be of little help because question 3 will be a completely different sort of problem than question 4. Since each problem is so unique, you'll often be left dealing with problems that are considerably more complex than anything found in the reading sections of the text. If you are using the questions of this book for homework, be prepared to use google extensively. As an example, the book may explain how to perform an operation on 2 sets of numbers. Then in the homework, it will ask you to perform the same operation on 5 sets abstract sets without ever explaining how to go about doing that.
I ended up receiving an A in the course, but that was after spending ~8 hours for each 10-14 question homework. Most of that time was spent on the internet trying to learn the material from whatever sites I could find. The reading sections of this text are an excersize in frustration. In one of the explanations for a concept in the book, the author literally uses the phrase "from [problem], it is obvious that the answer is [answer]." That was the entire explanation on the topic. A textbook should never say the phrase "from X, it is obvious that Y" if the whole section is supposed to be telling you how to find Y from X in the first place. This is an introductory text into formal logic, proofs, and set mathematics. Yet, you'll often find that the author skips steps in his solutions which may be obvious to someone familiar with the material but that is obviously not the target of this text. There is an occasional table for reference which doesn't explain what the relationship between anything on the table is (I'm looking at you, Table of Commonly Used Tautologies....). This book covers a great number of topics in a fairly small book, for a textbook that is. However, this book suffers from a lack of depth necessary to reach its potential.
If you have a choice, skip this text. If, like me, you are required to use this text.... Google everything and god help you.
Extremely poor organization.Jan. 13 2014
By
Dan G.
- Published on Amazon.com
Format: Hardcover
Verified Purchase
This book has an extremely poor organization of information. It's like the authors just threw a bunch of information at the book without thinking about how a student has to go through learning the mathematical concepts. The only reason I have to use this book is because a professor from my university was one of the authors. Get another book on discrete mathematics if you want to really learn the material.
1 of 8 people found the following review helpful
Great TextbookSept. 7 2011
By
mfox
- Published on Amazon.com
Format: Hardcover
Verified Purchase
This textbook was the exact same one I needed for class and was MUCH cheaper than buying from the school store. It was even in better condition than what was advertised! I would definitely recommend this book.
| 677.169 | 1 |
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