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The Most Common Errors in Undergraduate Mathematics
The Most Common Errors in Undergraduate Mathematics
This web page describes the errors that I have seen most frequently in undergraduate mathematics, the likely causes of those errors, and their remedies. I am tired of seeing these same old errors over and over again. (I would rather see new, original errors!) I caution my undergraduate students about these errors at the beginning of each semester. Outline of this web page:ERRORS IN COMMUNICATION, including teacher hostility or arrogance, student shyness, unclear wording, bad handwriting, not...
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<p>This web page describes the errors that I have seen most frequently in undergraduate mathematics, the likely causes of those errors, and their remedies. I am tired of seeing these same old errors over and over again. (I would rather see <em>new, original</em> errors!) I caution my undergraduate students about these errors at the beginning of each semester. <strong>Outline of this web page:</strong></p><p><strong>ERRORS IN COMMUNICATION</strong>, including <a href=" hostility or arrogance</a>, <a href=" shyness</a>, <a href=" wording</a>, <a href=" handwriting</a>, <a href=" reading directions</a>, <a href=" of invisible parentheses</a>, <a href=" lost inside an ellipsis</a> <strong>ALGEBRA ERRORS</strong>, including <a href=" errors</a>, <a href=" is additive</a>, <a href=" is commutative</a>, <a href=" cancellations</a>, <a href=" errors</a> <strong>CONFUSION ABOUT NOTATION</strong>, including <a href=" inverses</a>, <a href=" roots</a>, <a href=" of operations</a>, <a href=" written fractions</a>, <a href=" notations</a>. <strong>ERRORS IN REASONING</strong>, including <a href=" over your work</a>, <a href=" irreversibility</a>, <a href=" checking for extraneous roots</a>, <a href=" a statement with its converse</a>, <a href=" backward</a>, <a href=" with quantifiers</a>, <a href=" methods that work</a>, <a href=" faith in calculators</a>. <strong>UNWARRANTED GENERALIZATIONS</strong>, including <a href=" square root error</a>, <a href=" <strong>OTHER COMMON CALCULUS ERRORS</strong>, including <a href=" to conclusions about infinity</a>, <a href=" or misuse of constants of integration</a>, <a href=" of differentials</a> Common Errors in Undergraduate Mathematics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material The Most Common Errors in Undergraduate Mathematics
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Discussion
Discussion for The Most Common Errors in Undergraduate Mathematics
Mary Martin
(Faculty)
I did not use this in class because I thought it would infuriate my students. I may go back and test this idea...the material is thorough. Pedagogically, it is best not to embed wrong ideas by focusing on them.
Time spent reviewing site:
10
12 years ago
dixon
(Student)
In reading " The Most Common Errors in Undergraduate Mathematics" is some what very boring because the auther is telling the reader that there are common trends the students make in math. On the other hand the site has many example of the mistakes and corections. That earned three ***. If you are told by a teacher that you mess up because of common errors then this site will help you.
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This new edition is intended for a one semester course in optics for juniors and seniors in science and engineering. It uses scripts from Maple, MathCad, Mathematica, and MATLAB to provide a simulated laboratory where students can learn by exploration and discovery. The text covers all the standard topics of a traditional optics course. It contains step by step derivations of all basic formulas in geometrical, wave and Fourier optics. The basic text is supplemented by over 170 files, each suggesting programs to solve a particular problem, and each linked to a topic in or application of optics. The computer files are dynamic, allowing the reader to see instantly the effects of changing parameters in the equations. The book is written for the study of particular projects but can easily be adapted to other situations. The three fold arrangement of text, applications and files makes the book suitable for "self-learning".
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Quotable: Why Study Algebra?
One reason to study algebra: because it's a building block. And just as it was really hard at first to get those blocks to do what you wanted them to do, so also it can be really hard at first to get algebra to work. But if you persevere, who knows what you might build someday?
Algebra is the beginning of a journey that gives you the skills to solve more complex problems.
…
So, try not to think of Algebra as a boring list of rules and procedures to memorize. Consider algebra as a gateway to exploring the world around us all
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A Course in Mathematical Biology: Quantitative Modeling with Mathematical and Computational Methods
Publisher:
SIAM
Number of Pages:
xii + 309
Price:
48.30
ISBN:
0898716128
The goal of A Course in Mathematical Biology is to introduce students problem solving in the context of biology, and to do so by integrating analytical and computational tools for modeling biological processes. According to the introduction, this book is directed principally toward undergraduates in mathematics, physics, biology and other quantitative sciences.
The contents fall neatly into three parts. The first part addresses basic techniques of analytical modeling: differential and difference equations, probability and cellular automata, parameter estimation and model comparison.
The second part concentrates on computational tools for modeling biological processes. The authors use Maple as their standard software, but examples and exercises in this part could be easily adapted to use MATLAB or Mathematica instead. The last part of the book provides open-ended problems from physiology, ecology and epidemiology. A concluding section presents "solved projects", a detailed review of modeling efforts in cell competition and chemotaxis developed by students during a workshop.
The mathematical threshold is set pretty high in this text. It is accessible to upper level mathematics and physics majors, but I would expect most biology majors, for example, to find this pretty tough sledding. It is not so much that the prerequisites are extensive; no more than basic calculus and some familiarity with linear algebra and differential equations are assumed. It is more that a relatively high level of mathematical sophistication and comfort with mathematical thinking are required. There is simply a lot of new mathematics introduced as the book progresses. By the time we have reached page 100, we see, for example, linear stability analysis for difference equations, Jury conditions for the eigenvalues of the Jacobian, and bifurcation analysis. Shortly thereafter, the authors plunge into partial differential equations and the reaction-diffusion equation. Too much, too fast for many potential readers!
A small thing, but one which shows up more and more frequently in textbooks at a similar level, is the treatment of existence and uniqueness for solutions of differential equations. Here the authors define Lipschitz continuity, immediately state the Picard-Lindelöf theorem, and then stop. This, I think, has vanishingly small value to the student. Why not either spend some time discussing the issues of existence and uniqueness with examples and pictures, or state that you're going to assume that all equations under discussion have unique solutions and leave it at that?
Complaints aside, this is a fine text for those students with an appropriate background. There is a nice mix of biological applications, ranging from the more common examples in epidemiology and population genetics to forest ecology, the polymerase chain reaction, movement of flagellated bacteria, and ocular dominance patterns. The chapter on parameter estimation and model comparison is well-done, but at a rather sophisticated level — it begins with a discussion of the likelihood function and accelerates from there. The authors do not give much attention to the verification and validation of models. This is an important topic — particularly for biological models — and deserves a more extensive treatment.
The authors have provided an annotated section of "Further Reading" as well as an extensive bibliography. Both of these are valuable for a field that has grown enormously over the last ten years.
Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.
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Is maths REALLY required for programming?
This is a discussion on Is maths REALLY required for programming? within the A Brief History of Cprogramming.com forums, part of the Community Boards category; There are 10 kinds of people in the world - those that can count in binary and those that can't
...
I'm starting college in the fall, and the booklet says to get a computer science degree you need calculus 1, 2, 3, vector calculus, and 3 physics classes. I only hope I can do better in college math than I did in high school.
Detailed understanding of language features - even of all features of a language - cannot compensate for lack of an overall view of the language and the fundamental techniques for using it. - Bjarne Stroustrup
Originally posted by ZakkWylde969 What type of math do you really need? Like I want to program games with C++ in general. Should I just have a firm grasp of math in general or what?
You had better buy a Linear Algebra book and read it cover to cover.
Well I can only speak for game programming because that is what I do. I'd say a good fundamental understanding of mathematics and physics is a must for any game programmer. I mean, I guess you can write scripts and stuff like that without knowing too much math but those jobs are pretty low on the priority list. If you want to do anything graphics related you need a FIRM understanding of Linear Algebra. If you want to write a physics simulator you need general calculus knowledge, etc.
I really enjoy math. There are so many applications for it in programming it's sweet. However, I'm sure you can get away with looking stuff up as it is needed.
At many game companies ( i.e. Nintendo, M$ XBox Division ) the interviews will ask a lot of math questions. Everything from deriving the matrix for rotation of a point around an arbitrary vector to the explicit equation of a line in 3-space. Also basic things like Plane - Line intersection tests. I know this because I have friends at each of those companies who told me about their interviews. When I interviewed at the company I work at I wasn't asked to derive anything. They did ask how many and which math classes I had taken. They were very pleased that I had a decent number under my belt.
You don't need to memorize fancy formulas when you can derive them yourself. I'd suggest to strive to learn the under workings of any of the math functions you use. In Direct3D and OpenGL it's really easy to just call the function to make your projection matrix for you but how many of you actually understand how it is computed and why? If you want to invent faster functions and develop new algorithms you really need to know whats going on behind the scenes.
For graphics, you definitely need linear algebra. Though, if your not terribly familiar with the subject of vectors, I'd start with them first, and then move onto linear algebra (as a course, linear algebra is usually taken after calculus II or III, just for reference).
Also, regardless of what your programming, a good background in discrete math is a good idea - understanding the concepts involved with the development and use of algorithms.
Graph theory is also very important, especially for algorithmic work. Many data structures you run into (most trees for examples) are examples of directed-acyclic-graphs.
If your going with cryptography, then your gonna need a lot of abstract algebra - group theory, field theory, etc.
Its possible to program without all the math, but your usefullness is severely limited. As for how much time you spend, it really depends on how quickly you pick things up, and what you are trying to learn. Just stay with it until it makes sense.
i didnt really think you needed math, but the project im working on now at work requires me knowing matrices and finding determinant and all that stuff...its audio processing. but for audio and gfx you need math def
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Course Content and Outcome Guide for MTH 63
Course Description
Introduces algebraic concepts and processes with a focus on functions, polynomials, and quadratic equations. Emphasizes applications, graphs, functions, formulas, and proper mathematical notation. A scientific calculator is required. The TI-30X II is recommended. Completion of MTH 62 and MTH 63 is equivalent to MTH 65. The PCC math department recommends that students take MTH courses in consecutive terms. Prerequisites: (MTH 62) AND (RD 80 or ESOL 250). Audit available.
Addendum to Course Description
Students will be evaluated not only on their ability to get correct answers and perform correct steps, but also on the accuracy of the presentation itself.
Application problems must be answered in complete sentences.
Intended Outcomes for the course
1. Recognize and differentiate between linear and quadratic patterns in ordered paired data, graphs, and equations. 2. Use variables to represent unknowns in quadratic problems, create a quadratic equation that represents the situation, and find the solution to the problem using algebra. 3. Be prepared for future coursework that requires the use of basic algebraic concepts and an understanding of functions.
Outcome Assessment Strategies
Assessment shall include:
The following must be assessed in a proctored, closed-book, no-note, and no-calculator setting: simplifying expressions; binomial and trinomial factoring; extracting roots; solving quadratic equations by factoring and using the quadratic formula; and graphing quadratic functions.
At least two proctored closed-book, closed-note examinations (one of which is the comprehensive final). These exams must consist primarily of free response questions although a limited number of multiple choice and/or fill in the blank questions may be used where appropriate.
Assessment must include evaluation of the student€™s ability to arrive at correct and appropriate conclusions using proper mathematical procedures and proper mathematical notation. Additionally, each student must be assessed on their ability to use appropriate organizational strategies and their ability to write conclusions appropriate to the problem.
At least one of the following additional measures:
Take-home examinations
Graded homework
Quizzes
Projects
In-class activities
Portfolios
Course Content (Themes, Concepts, Issues and Skills)
THEMES:
Functions
Graphical understanding
Algebraic manipulation
Number sense
Problem solving
Applications, formulas, and modeling
Critical thinking
Effective communication
SKILLS:
WORKING WITH ALGEBRAIC EXPRESSIONS
Understand nonvariable square roots
Simplify using the product rule of square roots
Recognize like radical terms
Rationalize denominators
Estimate square roots
FACTORING POLYNOMIALS
Factor the greatest common factor from a polynomial
Factor a polynomial of four terms with the grouping method
Factor trinomials that have leading coefficients of 1
Factor trinomials that have leading coefficients other than 1
Factor the difference of squares
Recognize and factor the sum and difference of cubes
QUADRATIC EQUATIONS IN ONE VARIABLE
Solve quadratic equations by using the zero product principle (factoring)
Evaluate the function at a particular input value and interpret its meaning
Given a functional value (output), find and interpret the input
Interpret the vertex using proper units
Interpret the vertical intercept using proper units
Interpret the horizontal intercept(s) using proper units
RELATIONS AND FUNCTIONS
Use the definition of a function to determine whether a given relation represents a function
Determine the domain and range of a function given as a graph or as a table
Apply function notation in graphical, algebraic, and tabular settings
Understand the difference between the input and output
Identify ordered pairs from function notation
Given an input, find an output
Given an output, find input(s)
Interpret function notation in real world applications
Evaluate the function at a particular input value and interpret its meaning
Given a functional value (output), find and interpret the input
ADDENDUM:
MTH 63 is the third term of a three-term sequence in beginning algebra. One major problem experienced by beginning algebra students is difficulty conducting operations with fractions and negative numbers. It would be beneficial to incorporate these topics throughout the course, whenever possible, so that students have ample exposure. Encourage them throughout the course to get better at performing operations with fractions and negative numbers, as it will make a difference in this and future math courses.
Vocabulary is an important part of algebra. Instructors should make a point of using proper vocabulary throughout the course. Some of this vocabulary should include, but not be limited to, inverses, identities, the commutative property, the associative property, the distributive property, equations, expressions and equivalent equations.
The difference between expressions, equations, and inequalities needs to be emphasized throughout the course. A focus must be placed on helping students understand that evaluating an expression, simplifying an expression, and solving an equation or inequality are distinct mathematical processes and that each has its own set of rules, procedures, and outcomes.
Proper usage of equal signs must be stressed at all times. Students need to be taught that equal signs are used to communicate multiple ideas and they need to be taught the manner in which equal signs are used to communicate these ideas.
Equivalence of expressions is always communicated using equal signs. Students need to be taught that when they simplify or evaluate an expression they are not solving an equation despite the presence of equal signs. Instructors should also stress that it is not acceptable to write equal signs between nonequivalent expressions.
Instructors should demonstrate that both sides of an equation need to be written on each line when solving an equation. An emphasis should be placed on the fact that two equations are not equal to one another but they can be equivalent to one another.
Instructors should demonstrate and emphasize the importance of performing operations in a vertical format. Equal signs must be used when changing the form of an expression. Examples of a vertical format are as follows:
The distinction between an equal sign and an approximately equal sign should be noted and students should be taught when it is appropriate to use one sign or the other.
The manner in which one presents the steps to a problem is very important. We want all of our students to recognize this fact; thus the instructor needs to emphasize the importance of writing mathematics properly and students need to be held accountable to the standard. When presenting their work", all students in a Math 65 course should consistently show appropriate steps using correct mathematical notation and appropriate forms of organization. All axes on graphs should include scales and labels. A portion of the grade for any free response problem should be based on mathematical syntax.
The concept of functions should be covered before quadratic equations", and continually revisited. Use quadratic equations as an example of a function to reinforce the use of function notation, and the concepts of domain and range throughout the course.
Instructors should remind students that the topics discussed in MTH 60 and MTH 65 will be revisited in MTH 95 and beyond, but at a much faster pace while being integrated with new topics.
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Product Overview
Geared specifically toward the homeschool classroom, Saxon Algebra 2 3rd edition is a college-prep course designed to build the mathematical foundation necessary for students to transition successfully into higher-level math courses. Students completing Algebra 2 will have studied the equivalent of one semester of informal geometry.
This Teacher CD contains over 110 hours of content, including instruction for every part of every lesson and complete solutions for every example problem, practice problem, problem set and test problem. Offers students helpful navigation tools within a customized player and is compatible with both Windows and Mac systems.
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Contest Preparation
Computer Programming
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Group Theory
Group theory is the study of symmetry. Objects in nature (math, physics, chemistry, etc.) have beautiful symmetries and group theory is the algebraic language we use to unlock that beauty. Group theory is the gateway to abstract algebra which is what tells us (among many other things) that you can't trisect an angle, that there are finitely many regular polyhedra, and that there is no closed form for solving a quintic. In this class we will get a glimpse of the mathematics underlying these famous questions. This course will focus specifically on building groups from other groups, exploring groups as symmetries of other objects, and using the tools of group theory to construct fields.
Diagnostics
Schedule
AoPS Holidays
There are no classes on May 25, July 3, September 7, or November 23-27.
Who Should Take?
This class is aimed primarily at students who have mastered the standard high school curriculum and do not have access to a strong post-secondary curriculum. We assume fluency with modular arithmetic, the complex numbers, and basic combinatorics, and also a good background in forming mathematical arguments and writing proofs. The class will be on the level of the most difficult Art of Problem Solving courses. We will not assume any calculus, but we will rely on precalculus, number theory, and counting extensively.
Lessons
Lesson 1
Symmetry
Lesson 2
Examples of Groups
Lesson 3
Cyclic Groups
Lesson 4
Abelian Groups
Lesson 5
Group Actions I
Lesson 6
Group Actions II
Lesson 7
Quotients
Lesson 8
Functions from Groups to Groups
Lesson 9
Group Presentations
Lesson 10
Symmetric and Alternating Groups
Lesson 11
Group Classification and Sylow's Theorems
Lesson 12
Examples of Fields
Lesson 13
Building Finite Fields
Lesson 14
Polynomials
Lesson 15
Vector Spaces and Dimension
Lesson 16
Number Fields and Constructions
Lesson 17
Galois Groups of Finite Fields
Lesson 18
Automorphisms of Number Fields and Solving Quintics
I loved this class! The understanding of abstract algebra that I received will definitely help me in future classes. From this class I have also gained a deeper understanding of linear algebra. I enjoyed the low student/teacher ratio and receiving quick answers to my questions. Thanks a lot for this class.
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Complex Variables with Applications, 3e
Written for advanced undergraduate and graduate courses in engineering, applied mathematics, and physics, this text presents the theory and practical application of complex variables. Topics covered include complex arithmetic, derivatives, and analyticity; complex functions and series; integration in the complex plane; Laplace transforms; and conformal mapping. MATLAB is used to solve end-of-chapter example problems throughout the text. In addition, a solutions manual that includes MATLAB solutions is available to instructors who adopt the text.
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It covers intermediate calculus topics in plain English, featuring in-depth coverage of integration, including substitution, integration techniques and when to use them, approximate integration, and improper integrals. This hands-on guide also covers sequences and series, with introductions to multivariable calculus, differential equations, and numerical analysis. Best of all, it includes practical exercises designed to simplify and enhance understanding of this complex subject.
Introduction to integration
Indefinite integrals
Intermediate Integration topics
Infinite series
Advanced topics
Practice exercises
Confounded by curves? Perplexed by polynomials? This plain-English guide to Calculus II will set you straight!
This item is Non-Returnable.
Details
ISBN: 9781118204269
Publisher: Wiley
Imprint: For Dummies
Date: Jan
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Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
Mathematical expeditions : exploring word problems across the ages
"A collection of over 500 culturally and historically diverse mathematical problems carefully chosen to enrich mathematics teaching from middle school through the college level."--Provided by publisher.
Abstract:
An introduction to mathematics from around the world. It collects over 500 culturally and historically diverse mathematical problems chosen to enrich mathematics teaching from middle school through the college level. It tells us what various cultures knew about math and how they came to learn it.Read more...
Reviews
Editorial reviews
Publisher Synopsis
Swetz has collected word problems, or story problems, used to teach mathematics around the world and throughout history, so mathematics teachers in middle and secondary schools can use them today. University students of mathematics and its history might also find them useful as well as entertaining. Reference and Research Book News Mathematical Expeditions is a wonderful resource for any teacher who would like to use old problems in a course to help students understand the context of mathematical ideas. -- Victor J. Katz Mathematical Reviews The book is well thought-out and is recommended to readers interested in the history of mathematics. -- E. Keith Lloyd London Mathematical Society Newsletter One of my graduate students, who is majoring in mathematics, was excited when I showed her a sample of problems in the book. A month later, she asked whether I had finished my review-she wanted to borrow the book! -- Winifred A. Mallam Mathematics TeacherRead more...
""A collection of over 500 culturally and historically diverse mathematical problems carefully chosen to enrich mathematics teaching from middle school through the college level."--Provided by publisher."@en
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of Mathematics
The primary emphasis of this textbook is to convey some of the great ideas of mathematics and, in addition, to teach effective thinking. A diverse number of topics are covered such as numbers, infinity, geometry, contortions of space, chaos and fractals, uncertainty, and decision making. Numerous illustrations are used to explain key ideas. The writing style is very informal and allows the reader to more easily grasp complex mathematical concepts. Text would be ideal for a liberal arts mathematics course, an enrichment course for high school or middle school teachers, or as a course for advanced high school students. Text comes with 3D glasses and a interactive explorations CD. In summary, the authors have produced a reader/learner friendly text which does a nice job of conveying the beauty of mathematics and effective thinking strategies.
1 out of 1 people found this review helpful.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
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The book is a collection of solved problems in linear algebra, this first volume covers linear equations, matrices and determinants. All examples are solved, and the solutions usually consist of step-by-step instructions, and are designed to assist students in methodically solving problems.
With more than two million inhabitants is Manchester one of the largest urban areas outside of London. The city is located in the north west of England. In this free travel guide you can read more about Manchesterís history. As well you find several daytrips which will guide you along the sights you must see. Thanks to the maps you are sure you wonít get lost.
Do you have a career path? Do you know exactly what you want in terms of your career? Do you know what youíll be best suited to? All of these questions need answering if you want a happy and fruitful career. You spend a great proportion of your life at work so it pays to choose and manage your career wisely! In this textbook you will work through a series of exercises and content so that you will be able to plan and map out a rewarding career for yourself.
Madrid is the beautiful capital of Spain. Here you find many great and historical buildings, with the royal palace as an absolute highlight. In this free travel guide you find several inspiring daytrips. Stroll down in the many small backstreets and enjoy the poetic and southern atmosphere of this amazing Spanish city. In this guide you as well find an historical outline and convenient maps.
Mathematics is an exceptionally useful subject, having numerous applications in business, computing, engineering and medicine to name but a few. `Applied mathematics' refers to the study of the physical world using mathematics. This book approaches the subject from an oft-neglected historical perspective. A particular aim is to make accessible to students Newton's vision of a single system of law governing the falling of an apple and the orbital motion of the moon. The book and its associated volume of practice problems give an excellent introduction to applied mathematics.
If you have never worked with Word before, you will probably soon find yourself in the wilderness of possibilities the program offers. In this book the author will attempt to guide you through that wilderness, so you can learn the things that are necessary for you to use the program effectively.
Barcelona is the capital of Catalonia and it combines many historical sites with the warm Mediterranean coast. In this free travel guide you find seven tours which will guide you through the city. Visit the impressive La Sagrada Familia made by Antonio Gaudi or relax in Barcelonaís oldest park; Parc del Laberint dí Horta. In this guide you as well find an historical outline and convenient maps.
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Algebra: Concepts and Applications, Volume 2 ...
Read More-8 of "Algebra: Concepts & Applications" plus an initial section called Chapter A. Chapter A includes a pretest, lessons on prerequisite concepts, and a posttest. Designed for students who are challenged by high school mathematics, the 2007 edition has many new features and support components
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Video tutorial & screenshots
Editorial review
The following review written by one of the Software Informer contributors applies to version 2.0
Graphmatica is a small program designed for students and teachers of Mathematics. With it, you can graph your own equation just typing it over a text field and pressing enter.
Graphmatica is very easy-to-use and have a lot of interesting options.
You can easily switch the graph paper (ex. to show polar coordinates instead of the standard x-y graph). Moreover you can change any graph color: the background, grids, and function line colors can be altered to easily match your needs.
Looking at the math features, after you draw a line over the graph you can easily (I mean 1-click) find derivatives for such equation. The same occurs with numerical integrations: just select the area and the program makes the rest.
Personally, I found this program very interesting. The only drawback is that you could use it just for 30 days, after that you have to pay for it. Another issue to mention is that you have no help at all when typing the equations. If you misspelled something, then you get just an error message. It will be crucial to read carefully the help file before to start using it seriously.
Pros
Very fast drawing.
Complete set of graphs.
Cons
The errors on equations are not very descriptive.
What's new in version 2.4
New/improved features:
- Curves are now highlighted as you mouse over them or select them in the equation list.
- Added Pick Line Style to the context menu for equations, allowing you to choose from solid, dashed, dotted, and dash-dot line styles.
- Fixed-increment Cartesian, polar, and parametric graphs (i.e. with the step rate specified as a third parameter in the domain) now display as a series of points; point tables are aligned to display the same values.
- The parser now accepts functions run together with a single-variable parameter, e.g. y=logx.
- Added support for specifying powers of functions before the parameter e.g. y=sin^2x or y=cos²x
- Any equation/inequality that is explicitly specified as x=f(y) is now graphed as a function of y, even if it can be solved for y instead.
Bug fixes:
- Fixed bug which could cause graphs to be drawn slightly past the end of their specified domain.
- Find All Graphs now adjusts the range based only on the requested domain of the equations (rather than the maximum possible values that would be visible on the screen with an unrestricted domain).
- Selection highlighting now uses a dark transparent overlay (instead of a hard to see white overlay) for bright colors other than pure white.
- Fixed a number of issues with graphing inequalities like x < 1/y, where it is important to evaluate asymptotes as a function of y instead of x.
- Fixed a problem which caused ODEs (and implicit Cartesian functions) to fail to register (and later disappear) when evaluation failed due to a non-fatal error (overflow, out of domain, etc.). They now just stop drawing at that point.
- Eliminated extra vertical asymptote drawn as part of y=acot x.
- Fixed cubert(x) function to be defined for x < 0.
What's new in version 2.2 beta
Corrected domain specifications in one more demo file and a help page.
XP-style common controls are now loaded on 64-bit versions of Windows.
The installer is now digitally signed to verify that you have downloaded an authentic version of the program.
What's new in version 2.0
1. Fixed infinite loop parsing comma operator in domains and 2-variable functions when decimal separator is also set to ",".
2. Fixed crash graphing equations with free variables that have no on-screen solutions.
3. Added independently-settable font for bottom labels.
4. Graphs are now recalculated automatically upon changing the Theta Range.
5. The fraction characters ¼, ½, and ¾ are now accepted in place of normal decimals (before they could cause an infinite loop in the equation parser).
6. The custom increment for point table spacing now works for polar graphs as well as cartesian and parametric.
7. Switched from WinHelp to HTML help format, which is supported on Vista.
Publisher's description
Graphmatica is a powerful, easy-to-use, equation plotter with numerical and calculus features. It provides support for Graph Cartesian functions, relations, and inequalities, plus polar, parametric, and ordinary differential equations. You can print your graphs or copy them to clipboard in bitmap or vector format in black-and-white or color
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Hands-on Start to Mathematica (Spanish)
This tutorial screencast encourages users to work along in Mathematica 7 as they learn the basics to create their first notebook, calculations, visualizations, and interactive examples
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Concepts of Modern MathematicsNew:
BRAND NEW 0486284247 352 p. Contains: Illustrations. Dover Books on Mathematics.
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$23.08
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New BRAND NEW BOOK! Shipped within 24-48 hours. Normal delivery time is 5-12 days.
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About the Book
In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts underlying "new math"--groups, sets, subsets, topology, Boolean algebra, and more. By the time readers finish this book, they shall have a much clearer grasp of how modern mathematicians look at figures, function, and formulas, leading them to a better comprehension of the nature of the mathematics itself.
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Computational Number Theory and Modern Cryptography
The only book to provide a unified view of the interplay between computational number theory and cryptography
Computational number theory and modern cryptography are two of the most important and fundamental research fields in information security. In this book, Song Y. Yang combines knowledge of these two critical fields, providing a unified view of the relationships between computational number theory and cryptography. The author takes an innovative approach, presenting mathematical ideas first, thereupon treating cryptography as an immediate application of the mathematical concepts. The book also presents topics from number theory, which are relevant for applications in public-key cryptography, as well as modern topics, such as coding and lattice based cryptography for post-quantum cryptography. The author further covers the current research and applications for common cryptographic algorithms, describing the mathematical problems behind these applications in a manner accessible to computer scientists and engineers.
- Makes mathematical problems accessible to computer scientists and engineers by showing their immediate application - Presents topics from number
theory relevant for public-key cryptography applications - Covers modern topics such as coding and lattice based cryptography for post-quantum cryptography - Starts with the basics, then goes into applications and areas of active research - Geared at a global audience; classroom tested in North America, Europe, and Asia - Incudes exercises in every chapter - Instructor resources available on the book's Companion Website
Computational Number Theory and Modern Cryptography is ideal for graduate and advanced undergraduate students in computer science, communications engineering, cryptography and mathematics. Computer scientists, practicing cryptographers, and other professionals involved in various security schemes will also find this book to be a helpful reference
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More About
This Textbook
Overview
Become fluent in "the language of algebra" with INTERMEDIATE ALGEBRA. This textbook will improve your ability to read, write, and communicate mathematical ideas. Tussy and Gustafson make learning easy with their five-step problem-solving approach: analyze the problem, form an equation, solve the equation, state the result, and check the solution. This edition also includes iLrn Testing and Tutorial; Personal Tutor with SMARTHINKING live online tutoring; the Interactive Video Skillbuilder CD-ROM with MathCue; a Book Companion Web Site featuring online graphing calculator resources; and The Learning Equation, powered by iLrn.
Editorial Reviews
From the Publisher
You have done a great job explaining to the students how to read and write mathematically. The writing component helps the student to retain the information and [it will] be helpful to them in a subsequent math course.
I really like these explanatory [author's] notes. They are similar to ones that I like to write on the chalkboard when I am lecturing. This is perhaps my favorite feature of the text. Martin-Gay does have similar explanatory notes, but Tussy/Gustafson has more detail. The authors have done a good job with this.
Meet the Author
Alan Tussy teaches all levels of developmental mathematics at Citrus College in Glendora, CA. He has written nine math books — a paperback series and a hard-cover series. An extraordinary author, he is dedicated to his students' success, relentlessly meticulous, creative, and a visionary who maintains a keen focus on his students' greatest challenges. Alan received his Bachelor of Science degree in Mathematics from the University of Redlands and his Master of Science degree in Applied Mathematics from California State University, Los Angeles. He has taught up and down the curriculum from prealgebra to differential equations. He is currently focusing on the developmental math courses. Professor Tussy is a member of the American Mathematical Association of Two-Year Colleges
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Product Description
All the fantastic features that have made other Teaching Textbooks are so popular are included in the new Math 5. Designed for independent students, the Teaching Textbooks learner will discover a wealth of instruction written directly to them, clear examples, fun hand-drawn illustrations, highlighted important concepts, and of course, step-by-step solutions to every problem. Plus, a 5-10 minute interactive lecture is included for every lesson, and includes a print summary that reinforces the key concepts to remember. Automated grading, non-required additional practice problems, lectures and step-by-step audiovisual solutions to every homework and quiz problem make for a thorough and easy to use curriculum for both parent and student. Consistent review and real-world illustrations help reinforce both concepts taught, as well as the relevance of what they're learning.
I pulled my son from ps in 3rd grade because he was struggling with the math he was bringing home (common core) We would sit at the table for 3 hours and he would cry and get so upset... That year we taught him thru a Cyber school... Same thing, but better... I was able to explain it right away instead of having to wait till he was doing homework, but that following year the Cyber school turn over to Common core.. So I decided to teach him traditionally... So I ordered this... He is Excelling! He is finally "getting it" and maintaining an A average! I highly recommend Teaching Textbooks! Wish they made K-2nd for my younger two!
I love Teaching Textbooks! My child was struggling in Public School because of Common Core. This math helped him gain the understanding and confidence that he needed. We look forward to continuing Teaching Textbooks in the future!
My son has always done well in math, but has struggled to keep up with all the extra work. Consequently in school, he grew to hate math. Since we started Teaching Textbooks, math is his favorite subject. The Teaching Textbook program is clear and concise. He is learning and is thrilled to be able to get it done in a reasonable amount of time. What amazes me the most is that he is now doing complicated math in his head because he really understands the concepts.
This is our first time using this product, but I will definitely be using it again!
This is the best teaching method for math for homeschoolers. I've used teaching textbook since pre-algebra and was thrilled to see that teaching textbook was available starting in the 3rd grade. My youngest son loves it...he's been using since 3rd grade and now is on the 5th grade....Thank you!
My Son loves this program so much! He can work independently. The program would be a good fit for several types of learners. He used to complete one math lesson a week, now he completes up to three in one day. We will continue to buy this curriculum.
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I'm not sure what the gist of your response was, and upon reading my message to
which you responded, I see that I was too vague!
Let me say, first, that I don't think we should teach (or expect) the students
to compete with computers. We need to teach low-level skills, mid-level
skills, and higher order skills (though at higher order, the word "skills"
sounds too limiting). What we can't afford to do, as the examples you listed
illustrate, is to use mastery of low-level skills as a pre-requisite for
exposure to higher order mathematical thinking skills. That only serves to
limit the access of otherwise active thinkers.
An example I remember from my undergraduate days (over 20 years ago!) is the
Economics course I took--it required Calculus as a pre-requisite, and there
were several graduate students in the course who really struggled with the
mechanics of taking derivatives (calculating and using the chain rule are fairly
low-level skills in Calculus), but their understanding of economics was still
possible because their exposure to the concepts of rate of change (or marginal
whatever in economics) was at a higher-order level
| 677.169 | 1 |
Mathematics
Mathematics Courses
20-804-210-00 Introduction to Computers and Their Use Introduces hardware, software, and information processing methodologies as problem-solving
tools in liberal arts, sciences, and education. Addresses the history and social impact
of computers. Students will use productivity tools such as word processing, spreadsheet,
and database software. Lecture/Lab. 3 credits.
20-804-220-00 Intermediate Algebra Studies the construction and resulting properties of the real number system. Students
simplify and factor algebraic expressions using fundamental laws and order of operations;
solve first and second degree equations and inequalities in one variable, systems
of equations, and exponential and logarithmic equations; graph first degree and second
degree equations and inequalities in two variables; and solve equations involving
rational expressions, fractional exponents and radicals. Prerequisite: 10-834-110
with a grade of "C" or better, one year of high school algebra with a grade of "C"
or better, or suitable UW Math Placement Test score. Lecture. 4 credits.
20-804-224-00 Algebra for Calculus Covers properties of the real number system, algebraic expressions, equations and
inequalities, functions and graphs, polynomial and rational functions, exponential
and logarithmic functions, analytic geometry, matrices, determinants, and systems
of linear equations, sequences and series227-00 Elementary Math Education I Covers mathematics content necessary for prospective early childhood and elementary
teachers. Topics include foundational and historical concepts from arithmetic and
algebra. Prerequisite: 20-804-220 with a grade of "C" or better, two years of high
school algebra with a grade of "C" or better, or suitable score on the UW Math Placement
Test. Lecture. 4 credits.
20-804-228-00 Plane Trigonometry Covers trigonometric functions and their inverse functions, graphing trigonometric
functions, trigonometric identities, solving triangles, solving equations and inequalities,
complex numbers in trigonometric form, and polar curves. Prerequisite: 20-804-220
with a grade of "C" or better, two years of high school algebra with a grade of "C"
or better, or suitable score on the UW Math Placement Test. Lecture. 3 credits.
20-804-230-00 Statistics Studies statistical techniques for the systematic collection, presentation, analysis
and interpretation of data. Studies statistical inference, including confidence intervals,
Types I and II errors, hypothesis testing. Also includes descriptive statistics, basic
probability theory, the Central Limit Theorem, distributions, linear regression, and
correlation. May require use of a graphing calculator or computer software. Prerequisite:
10-834-110 with a grade of "C" or better, one year of high school algebra with a grade
of "C" or better, or suitable UW Math Placement Test score. Lecture. 3 credits.
20-804-236-00 Calculus and Analytic Geometry I Covers limits and continuity of functions, the derivative, and its applications. Prerequisites:
20-804-224 and 20-804-228 with a grade of "C" or better, two years of high school
algebra with a grade of "C" or better and one year of trigonometry with a grade of
"C" or better, or suitable score on the UW Math Placement Test. Lecture. 5 credits.
20-804-237-00 Elementary Math Education II Includes concepts of proportionality, statistics and probability, plane geometry,
the geometry of solids, and measurement241-00 Calculus and Analytic Geometry III Topics covered include differentiation of vectors, space curves and curvature, functions
of more than one variable, level curves and level surfaces, limits and continuity,
partial derivatives, total differential, tangent planes, the gradient operator, the
directional derivative, multivariable forms of the chain rule, locating maxima, minima,
saddle points, the method of Lagrange multipliers, multiple integrals in rectangular,
polar, cylindrical and spherical coordinates, transformations of multiple integrals
and the Jacobian, surface area, applications of multiple integrals to geometry and
mechanics, line integrals in two and three dimensions, vector fields, circulation
and flux in two dimensions, and Green's Theorem. Prerequisite: 20-804-240 with a grade
of "C" or better or suitable score on the UW Math Placement Test. Lecture. 5 credits.
20-804-250-00 Quantitative Reasoning Intended to develop analytic reasoning and the ability to solve quantitative problems.
Topics to be covered include construction and interpretation of graphs, functional
relationships and mathematical modeling, descriptive statistics, basic probability,
geometry, and spatial visualizations. This is a suitable final mathematics course
for students who do not intend to take Calculus. Prerequisite: 20-804-220 with a grade
of "C" or better, two years of high school algebra with a grade of "C" or better,
or suitable score on the UW Math Placement Test or instructor consent. Lecture. 4
credits.
20-804-290-00 Topics in Mathematics Pursues advanced or specialized mathematics topics in a traditionally structured,
independent study, or service learning format. Depending on the structure, requirements
and topics are developed in advance by the instructor or by the student in consultation
with the instructor. Lecture. 3 credits.
20-804-290-01 Differential Equations Linear Algebra Differential equations are the fundamental tools that modern science and engineering
use to model physical reality. Linear algebra is a part of mathematics concerned with
the structure inherent in mathematical systems. Students will see that solutions of
certain differential equations in fact form a vector space, and techniques from linear
algebra will allow us to solve systems of linear differential equations. Topics covered
will include first order differential equations, differential models, linear systems
and matrices including solving systems of equations by Gaussian elimination, matrix
operations, determinants, vector spaces, higher order linear differential equations,
exponential methods with matrices, and nonlinear systems. Prerequisite: 20-804-240.
Lecture. 3 credits.
20-804-290-02 Topics in Advanced Calculus Designed for students who can work independently, studying higher-level mathematical
principles in the field of calculus. Students will learn to interpret three-dimensional
coordinates, general level curves and level surfaces, compute limits of multivariate
functions, compute partial derivatives of multivariate functions, and evaluate double
and triple integrals. Lecture. 1 credit.
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MATHalino.com - Engineering Math Review
enQuiz 1 - Basic AlgebraThere are 10 problems in this quiz taken randomly from a pool of 100 problems. If you are able to finish this quiz and will take it again, chances are, you will encounter different problems. Unfinished quiz is internally saved, it will resume where you leave with the same set of problems the next time you run the quiz. The system however will delete any incomplete quiz after 10 days.</p>
<p>This quiz is designed for 30 minutes, you are given three minutes per problem although most of the problems can be answered in less than a minute. The total time allotted for this quiz is 40 minutes, the ten minutes allowance is for the loading of each problem. We assume each problem will load in one minute, we understand that the number is too conservative.</p>
<p>The topics in this quiz are as follows: logarithm, inequality, absolute value, radicals, factoring, determinants, worded problems, polynomials, variations, complex numbers, partial fractions, complex fractions, quadratic equations, progressions, sequence, and other topics in Algebra. You may not encounter all the topics as you are given only 10 items per quiz.</p>
<p>You can take this quiz as many times as you want, and hopefully you will encounter all the 100 problems from the pool where the 10 problems are randomly generated. All of your finished quizzes are stored under your account and you can access it any time you want.</p>
<p>You need to login in order to <a href="/node/2417/take">take</a> this quiz. Good luck and enjoy!</p>
</div></div></div></div><table id="quiz-view-table">
<tr><td class="quiz-view-table-title"><strong>Questions:</strong></td><td class="quiz-view-table-data"><em>10</em></td></tr>
<tr><td class="quiz-view-table-title"><strong>Attempts allowed:</strong></td><td class="quiz-view-table-data"><em>UnlimitedTime limit:</strong></td><td class="quiz-view-table-data"><em>0:40:00</em></td></tr>
<tr><td class="quiz-view-table-title"><strong>Backwards navigation:</strong></td><td class="quiz-view-table-data"><em>Allowed</em></td></tr>
</table>
Tue, 19 May 2015 12:43:36 +0000Romel Verterra2417 at 86464</strong><br />
An 18-ft beam, simply supported at 4 ft from each end, carries a uniformly distributed load of 200 lb/ft over its entire length. Compute the value of EI<span class="mathsymbol">δ</span> at the middle and at the ends.<br />
</p>
<div align="center"><a href="/reviewer/strength-materials/problem-864-deflection-three-moment-equation"><img src="/sites/default/files/reviewer-strength/08-continuous-beams/864-overhanging-simple-bem.gif" width="456" height="137" alt="864-overhanging-simple-bem.gif" /></a></div>
<p> </p>
</div></div></div></div>Fri, 08 May 2015 22:02:51 +0000Romel Verterra2407 at 86363</strong><br />
For the beam shown in Fig. P-863, determine the value of EI<span class="mathsymbol">δ</span> midway between the supports and at the left end.<br />
</p>
<div align="center"><a href="/reviewer/strength-materials/problem-863-deflection-three-moment-equation"><img src="/sites/default/files/reviewer-strength/08-continuous-beams/863-verhang-beam-given.gif" width="400" height="163" alt="863-verhang-beam-given.gif" /></a></div>
<p> </p>
<p><strong>Answer</strong></p>
<dd>At midway between the supports<br />
$\delta = \dfrac{1066.67}{EI} ~ \text{ upward}$<br />
<p>At the left end<br />
$\delta = \dfrac{16,000}{EI} ~ \text{ downward}$</p></dd>
<p> </p>
<p>For the complete solution using the three moment equation, see it here: <a href=" />
</p>
</div></div></div></div>Tue, 05 May 2015 19:00:36 +0000Romel Verterra2405 at 86262</strong><br />
Determine the value of EI<span class="mathsymbol">δ</span> at B for the beam shown in Fig. P-862.<br />
</p>
<div align="center"><a href="/reviewer/strength-materials/problem-862-deflection-three-moment-equation"><img src="/sites/default/files/reviewer-strength/08-continuous-beams/862-simple-beam-given.gif" width="234" height="130" alt="862-simple-beam-given.gif" /></a></div>
<p> </p>
</div></div></div></div>Tue, 28 Apr 2015 01:58:34 +0000Romel Verterra2403 at 86161</strong><br />
For the beam shown in Fig. P-861, determine the value of EI<span class="mathsymbol">δ</span> at 2 m and 4 m from the left support.<br />
</p>
<div align="center"><a href="/reviewer/strength-materials/problem-861-deflection-three-moment-equation"><img src="/sites/default/files/reviewer-strength/08-continuous-beams/861-simple-beam-given.gif" width="274" height="155" alt="861-simple-beam-given.gif" /></a></div>
<p> </p>
</div></div></div></div>Mon, 27 Apr 2015 03:21:49 +0000Romel Verterra2402 at 86060</strong><br />
Determine the value of EI<span class="mathsymbol">δ</span> at the end of the overhang and midway between the supports for the beam shown in Fig. P-860.<br />
</p>
<div align="center"><a href="/reviewer/strength-materials/problem-860-deflection-three-moment-equation"><img src="/sites/default/files/reviewer-strength/08-continuous-beams/860-overhang-beam-given.gif" width="232" height="122" alt="860-overhang-beam-given.gif" /></a></div>
<p> </p>
</div></div></div></div>Sat, 25 Apr 2015 03:01:27 +0000Romel Verterra2401 at Polar Coordinate System
<a hrefIn Polar Coordinate System, the references are a fixed point and a fixed line. The fixed point is called the <strong>pole</strong> and the fixed line is called the <strong>polar axis</strong>. The location of a point is expressed according to its distance from the pole and its angle from the polar axis. The distance is denoted by r and the angle by <span class="mathsymbol">θ</span>.</p>
<div align="center"><a href="/reviewer/polar-coordinate-system"><img src="/sites/default/files/reviewer-analytic/polar-ccordinates.gif" width="572" height="548" alt="polar-ccordinates.gif" /></a></div>
<p> </p>
</div></div></div></div>Mon, 30 Mar 2015 05:23:42 +0000Romel Verterra2397 at Location of the third point on the parabola for largest triangleThe line y = 2x + 8 intersects the parabola y = x<sup>2</sup> at points A and B. Point C is on the parabolic arc AOB where O is the origin. Locate C to maximize the area of the triangle ABC.<br />
</p>
<div align="center"><a href=" src="/sites/default/files/reviewer-diffcalc/02-largest-triangle-line-parabola.jpg" width="266" height="390" alt="02-largest-triangle-line-parabola.jpg" /></a></div>
<p> </p>
</div></div></div></div>Wed, 07 Jan 2015 22:28:37 +0000Romel Verterra2366 at for grazing by the goat tied to a siloA goat is tied outside a silo of radius 10 m by a rope just long enough for the goat to reach the opposite side of the silo. Find the area available for grazing by the goat. Note that the goat may not enter the silo.<br />
</p>
<div align="center"><a href=" src="/sites/default/files/reviewer-integral/grazing-area-goat-figure-1.jpg" width="550" height="600" alt="grazing-area-goat-figure-1.jpg" /></a></div>
<p> </p>
</div></div></div></div>Mon, 05 Jan 2015 03:58:02 +0000Romel Verterra2365 at of the curve r = 4(1 + sin theta) by integrationWhat is the perimeter of the curve r = 4(1 + sin θ)?<br />
</p>
<div align="center"><a href=" src="/sites/default/files/reviewer-integral/cardioid-pointing-upward.jpg" width="431" height="413" alt="cardioid-pointing-upward.jpg" /></a></div>
<p> </p>
<p>The answer is 32 units. For detailed solution, follow the link by clicking on the figure.<br />
</p>
</div></div></div></div>Sun, 04 Jan 2015 01:45:02 +0000Romel Verterra2364 at
| 677.169 | 1 |
Tag Info
There are several
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Well I'm not a programmer, but I'm pretty sure that you need math to program like the air you breathe.
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Perhaps you can use colours and simple objects like eggs, which students in grade 6 can easily acquaint with.
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I can't imagine that A graduate program in economy would include a course in graph theory as an obligatory class. It may be available as an optional course though.
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| 677.169 | 1 |
If I consider the math courses I took in college undergrad (I was a C-S major) there are some that are useful in everyday day life like pre-Calc (aka. algebra + trig) when I build a new deck or figure out how much grass seed to buy. Others courses such as Calc I and II are useful in understanding the issues and trends around me, both business and political. But the vast majority are valuable to me every day as I work on systems software: Mathematical Logic and Discrete Math, Graph Theory, Linear Algebra with Differential Eq & Intro to the Theory of computation.
| 677.169 | 1 |
This recording is for memory improvement, examination success, effective studying, speedy learning, and improved mathematics understanding. Also, you will have better concentration, meaningful participation, and contribution in class discussions.
Publisher's Summary
This recording is for memory improvement, examination success, effective studying, speedy learning, and improved mathematics understanding. Also, you will have better concentration, meaningful participation, and contribution in class discussions. You will gain creative thinking and improved writing abilities and greater desire to know. This recording will help every student to improve their grade and general academic abilities, by at least 25 percent.
A lot of students hate mathematics, but love language. Therefore, using Neuro-Linguistic Programing (NLP), this Excellent Academic Performance hypnotherapy session is programmed to make every student love mathematics and find it as easy to study and understand as the English language or their native
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GeoGebra portable is free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus in one easy-to-use package. Features includes Graphics, ...
GeoGebra is free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus in one easy-to-use package. Features includes Graphics, algebra ...
... and Regression are fundamental and important calculations in mathematics. Mr. Newton and Mr. Gauss were engaged in-depth ... point-and-click. So this program is not only for mathematics and engineers. The applications are many and reach ... ...
CalculPro will help your elementary school students practice mental arithmetic or to do fractions. Choose addition, subtraction, multiplication, division or fractions; then set the range of numbers and the number of ...
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CalculationLaboratory - mathematical program that can handle scripts (your own programming language), also can process scripts extreme numbers (both decimal binary(256-bit number) and a 16-bit format). There is also a Grapher. ...
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... a nice experience dealing with this chapter of mathematics.? Moreover, the interface language can be selected from English, French, Spanish and Romanian. Several examples to test are delivered along with ...
Calcul will help your elementary school students practice mental arithmetic or to do fractions. Choose addition, subtraction, multiplication, division or fractions; then set the range of numbers and the number of ...
| 677.169 | 1 |
AlgebratoAlgorithm
By junp.
Price: Free.
Category: Nonfiction / Children's Books / Mathematics / Algebra.
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Nonfiction / Children's Books / Mathematics / AlgebraTue, 26 Nov 2013 10:52:28 -0800Coordinate Geometry 10:19:57 -0800PatternsSun, 24 Nov 2013 23:57:22 -0800
| 677.169 | 1 |
t
he phrases the phrasesFour Ways to Represent a Function. Mathematical Models. New Functions from Old Functions. Graphing Calculators and Computers. Exponential Functions. Inverse Functions and Logarithms. Parametric Curves. Review. Principles of Problem Solving.
2. LIMITS AND DERIVATIVES.
The Tangent and Velocity Problems. The Limit of a Function. Calculating Limits Using the Limit Laws. Continuity. Limits Involving Infinity. Tangents, Velocities, and Other Rates of Change. Derivatives. The Derivative as a Function. Linear Approximations. What does f' say about f? Review. Focus on Problem Solving.
3. DIFFERENTIATION RULES.
Derivatives of Polynomials and Exponential Functions. The Product and Quotient Rules. Rates of Change in the Natural and Social Sciences. Derivatives of Trigonometric Functions. The Chain Rule. Implicit Differentiation. Derivatives of Logarithmic Functions. Linear Approximations and Differentials. Review. Focus on Problem SolvingWonder Book Frederick, MD
Cengage Learning, 11/05/2004, Hardcover, Good condition. With CDSusies Books Garner, NC
2004 Hardcover COVER SHOWS WEAR This book looks good. It is like any used book you would expect to find in a used book shop.
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Used - Good Hardcover. includes bind in access code With CD! 3rd Edition May contain highlighting/underlining/notes/etc. May have used stickers on cover. Ships same or next day. Expedited shipping tak...show morees 2-3 business days;
| 677.169 | 1 |
Precalculus A Right Triangle Approachatti and McWaters have combined years of lecture notes and firsthand experience with students to bring readers a book series that teaches at the same level and in the style as the best math instructors. An extensive array of exercises and learning aids further complements the instruction readers would receive in class and during office hours. Basic Concepts of Algebra, Equations and Inequalities, The Coordinate Plane, Polynomial and Rational Functions, Exponential and Logarithmic Functions, Trigonometric Functions Angles and Their Measure, Trigonometric Identities, Applications of Trigonometric Functions, Systems of Equations and Inequalities, Matrices and Determinants, Conic Sections, Further Topics in Algebra For readers interested in precalculus.
J.S. Ratti has been teaching mathematics at all levels for over 35 years. He is currently a full professor of mathematics and director of the "Center for Mathematical Services" at the University of South Florida. Professor Ratti is the author of numerous research papers in analysis, graph theory, and probability. He has won several awards for excellence in undergraduate teaching at University of South Florida and known as the coauthor of a successful finite mathematics textbook.
Marcus McWaters is currently the chair of the Mathematics Department at the University of South Florida, a position he has held for the last eleven years. Since receiving his PhD in mathematics from the University of Florida, he has taught all levels of undergraduate and graduate courses, with class sizes ranging from 3 to 250. As chair, he has worked intensively to structure a course delivery system for lower level courses that would improve the low retention rate these courses experience across the country. When not involved with mathematics or administrative activity, he enjoys playing racquetball, spending time with his two daughters, and traveling the world with his wife.
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Numerical Methods, Software, and Analysis, Second Edition introduces science and engineering students to the methods, tools, and ideas of numerical computation. Introductory courses in numerical methods face a fundamental problem-there is too little time to learn too much. This text solves that problem by using high-quality mathematical software.... more...
Numerical Analysis is an elementary introduction to numerical analysis, its applications, limitations, and pitfalls. Methods suitable for digital computers are emphasized, but some desk computations are also described. Topics covered range from the use of digital computers in numerical work to errors in computations using desk machines, finite difference... more...
Basic Numerical Mathematics, Volume 1: Numerical Analysis focuses on numerical analysis, with emphasis on the ideas of "controlled computational experiments" and "bad examples". The concepts of convergence and continuity are discussed, along with the rate of convergence, acceleration, and asymptotic series. The more traditional topics of interpolation,... more...
Analytic Computational Complexity contains the proceedings of the Symposium on Analytic Computational Complexity held by the Computer Science Department, Carnegie-Mellon University, Pittsburgh, Pennsylvania, on April 7-8, 1975. The symposium provided a forum for assessing progress made in analytic computational complexity and covered topics ranging... more...
Computer Science and Applied Mathematics: Introduction to Numerical Computations, Second Edition introduces numerical algorithms as they are used in practice. This edition covers the usual topics contained in introductory numerical analysis textbooks that include all of the well-known and most frequently used algorithms for interpolation and approximation,... more...
Applications of Number Theory to Numerical Analysis contains the proceedings of the Symposium on Applications of Number Theory to Numerical Analysis, held in Quebec, Canada, on September 9-14, 1971, under the sponsorship of the University of Montreal's Center for Research in Mathematics. The symposium provided a forum for discussing number theory and... more...
Functional Analysis and Numerical Mathematics focuses on the structural changes which numerical analysis has undergone, including iterative methods, vectors, integral equations, matrices, and boundary value problems. The publication first examines the foundations of functional analysis and applications, including various types of spaces, convergence... more...
Computer Science and Applied Mathematics: Numerical Analysis: A Second Course presents some of the basic theoretical results pertaining to the three major problem areas of numerical analysis-rounding error, discretization error, and convergence error. This book is organized into four main topics: mathematical stability and ill conditioning, discretization... more...
Exponential Fitting is a procedure for an efficient numerical approach of functions consisting of weighted sums of exponential, trigonometric or hyperbolic functions with slowly varying weight functions. This book is the first one devoted to this subject. Operations on the functions described above like numerical differentiation, quadrature, interpolation... more...
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Math for the Automotive Trade, 5th Edition
MATH FOR THE AUTOMOTIVE TRADE, 5E is an up-to-date, highly practical book that helps you develop a real-world understanding of math concepts and applications in the modern automotive repair trade. Written at a beginner's level, this book is a comprehensive instructional workbook that shows you how to solve the types of math problems faced regularly by automotive technicians. Unique to MATH FOR THE AUTOMOTIVE TRADE, 5E are realistic practice exercises that allow you to determine if your answers fall within manufacturers' specifications and repair orders that are completed by finding the appropriate information in the professional literature and reference material, included in the book's valuable appendices168.95
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Course content:
This course concerns itself with the concepts
of limit and continuity, which are the basis of mathematical analysis
and the calculus from which it evolved. We focus on limits of
sequences and of functions, and continuity of functions. While
the main context is the real numbers, i.e. limits of sequences of real
numbers, continuity of functions from the real numbers to the real
numbers, and so on, we will introduce a more general context, that of metric spaces,
in which to study the same concepts. This will allow us to extend
our results (often with no added effort) to functions on n, sequences in , functions on spaces of functions, etc. A highlight of the course is the interaction of continuity with the concepts of completeness, compactness, and connectedness,
which give us rigorous proofs of the seemingly obvious (but definitely
not obvious) existence of global extrema and Intermediate Value
Theorem. As time permits, we may begin the concept of
differentiability before semester break.
Note: In Real Analysis II, we will see that
the "proofs" about derivatives that we saw in Calculus I were mostly
quite rigorous, if we have a rigorous understanding of limits.
Therefore, we will quickly review what we already know about
derivatives in a rigorous setting. Integrals, however, are far
more complicated than a Calculus I/II course would lead us to believe,
especially if they involve discontinous functions. Thus much of
Real Analysis II will be concerned with properties of the Riemann
integral. Other material will include some introductory material
on measure theory and/or sequences of functions, e.g. uniform convergence.
Course Goals: At the completion of this course, the successful student will have demonstrated
these abilities:
The ability to understand both abstract and concrete
mathematical reasoning.
The ability to differentiate between sound mathematical
reasoning, flawed reasoning, and non-rigorous reasoning.
The
ability to use the basic tools and methods of proof seen in analysis,
in particular set theory and epsilon-delta and epsilon-n arguments.
The ability to formulate and prove theorems that arise from the
definitions and concepts of the course, and the ability to
apply those theorems to specific examples.
The ability to write up, and occasionally present orally, one's mathematical proofs and arguments in
a
clear and compelling manner.
Grading
Three to five quizzes
75-125 points
Two or three "midterm" exams:
200-300 points
Two or three in class presentations
80-120 points
Final exam
200 points
Around 7 homework sets at ~20 pts each
~140 points
Total:
695-985 points
For each graded piece of work, I will post cutoff scores for grades
of A-, B-, C, C-, and D. At the end of the term, if your point total is
more than the total of the A- cutoffs, your grade will be an A- or better,
and so on. Cutoffs will never be higher than: A-: 90%
B-: 80% C: 70% C-: 67%
D: 60% ... but they are often lower.
Exams: We will have a mix of
in-class and take-home exams. Exams will be announced several
days in advance. The "rules" for take-home exams will be provided
with the exam. Final exam date: Thursday, Dec. 11, 2014 from 2-5 p.m.. For borderline grades, I tend
to pay more attention to the final exam score. The final may
be take-home, in which case the above is the due date.
.
Exam makeup policy: Midterm make-ups or early midterms are given
only for verifiable illness or for university-sanctioned intercollegiate
activities. For collegiate activities, you must see me before you
leave to arrange a makeup time. In any case, you must contact me in
advance
except in emergencies.If the final is in-class, it will definitely not be given
early; we will try to finalize this by mid-semester, but please plan
accordingly.
Quizzes are 10
to 25 minutes long. Some will be primarily content quizzes, e.g.
stating definitions and theorems from class and assigned reading; these
may be given without notice. Other quizzes will be short
reasoning and proof exercises, closely related to previously assigned
homework. Presentations: I will
periodically assign individual presentations, typically proofs of
moderate difficulty, to be given during class meetings. For the
first presentation, I will ask students to give a preview in my office
so we can be confident that things are going well. Homework is assigned daily but
collected
sporadically. Approximately every other week, I will ask you to
turn in a specific subset of the assigned work. If you have
additional problems on the same page, it's fine if you turn them in as
well, but the problems specified for collection need to be thoroughly
legible and well written. If it is necessary to achieve
legibility and coherence, students may choose to (re)write up a clean
copy of their work to turn in (by the original deadline given). Participation: In advanced mathematics, the step from things that can be easily
understood to things that may as well be spoken in the Martian language
is a short step. To stay ahead of this "step," one needs to keep
up with the concepts and definitions already covered. It is vital
that students do so. See also the "Workload" section below. The occasional "pop" quiz on definitions and theorems
(see Quizzes above) and class discussion will
help motivate students to keep up and help me assess how well students
are doing. Please come prepared to discuss the
homework and assigned reading from the previous class.
As
with all 400-level mathematics courses at Willamette, students of this
course are expected to attend the department colloquium at least four
times this semester. The purpose of this requirement is to
instill an appreciation of the breadth and variety of mathematical
topics, techniques, people and (academic) programs. If scheduling
conflicts make colloquium attendance impossible, alternative short
written assignments will be made. If a student anticipates
needing this alternative for more than one colloquium session, they
should contact the instructor well in advance so that suitable
assignments can be prepared. Failure to complete this requirement
will result in a modest grade penalty for the course.
Accommodations for students with disabilities: Accomodations
required by students with disabilities will be provided upon reasonable
advance notice and verification of requirement/eligibility from the Office of Disability Services (Bishop
Wellness Center). If you forsee needing an accomodation,
it is probably best to inquire at the Office of Disability Services at
the start of the semester. If Disability Services prescribes
extra time for exams, you must remind your instructor of your needs at
least one week before an exam and send an email reminder at least three
days before the exam to ensure appropriate accommodations have been
made.
Policy on in-class distractions and cell phones: It
is important to respect the concentration and attention of each student
in the class. Class time is limited, precious, and the tuition is
quite expensive per minute.
Arriving late for class is not acceptable barring genuine
emergencies.
Electronic devices must be turned off during class time, including laptop computers
and smartphones.
Exceptions include special-needs aids such as
lecture-recording devices. If your cell phone rings during
regular class time, you will be required to bring pastries or cookies for the entire
class at the next class meeting. During exams and quizzes, the
penalty is more specific: two points for each ring, or one point per
second of audible sound, whichever is greater. Please help me
hold distractions for your fellow students to an absolute minimum.
Workload and time committment expectations
It is the policy of the College of Liberal Arts that for every hour of
class time there is an expectation of "two to three hours" of work
outside of class. This specific class, however, is widely
recognized as the most challenging course we offer in Math, so you
should probably plan on three hours of study per hour of class.
Thus, for our class you should anticipate spending nine hours
weekly outside of class engaged in study time, reading and homework
assignments, and preparing presentations.
Academic Honesty Expectations All exams and quizzes are to be taken with books and notes closed (except
as noted on the exam paper), completely on your own. When we have take-home exams, specific expectations
will be distributed with those exams.
While you are allowed (and even encouraged) to discuss homework
problems in general terms with your peers, and to discuss methods and
the like, you should not see a peer's written homework (or a
transcription of it on a whiteboard, e.g.) until after your own graded
homework is returned to you. With that in mind, plagiarism is
work copied or paraphrased from another source without
proper written acknowledgement. All students involved in
plagiarized work will recieve a zero grade on the assignment. I
will not try to tell who was "copied from" and who "did the copying."
In keeping with college policy, plagiarism will be reported to the
dean
for possible referral to the Honor Council per the Willamette Ethic. Systematic or organized plagiarism on exams or
quizzes will result in course failure. If you are uncertain about some
aspect of the academic honesty policy, it is your responsibility to get
clarification from the instructor.
| 677.169 | 1 |
...
Show More it provides custom classes for parsing and graphing mathematical expressions. The tiered learning approach starts from scratch and builds each example upon what has come before, eventually producing dozens of classic applets. While many examples have been chosen to reflect the needs and interests of mathematics and science specialists, the book has a lot to offer general Flash developers who have found our focused examples of functionality and user interface extremely useful. This book offers a step-by-step path to learning essential ActionScript programming that is based on the popular tutorials at FlashandMath.com
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Learning/NBML/Calculus - Revision history2015-05-29T23:13:14ZRevision history for this page on the wikiMediaWiki 1.19.1 Created page with '=== Introduction === The most relevant Calculus for machine learning involves the [ Derivatives], and the multi-dimensional version of the …'2011-01-06T16:59:38Z<p>Created page with '=== Introduction === The most relevant Calculus for machine learning involves the [ Derivatives], and the multi-dimensional version of the …'</p>
<p><b>New page</b></p><div>=== Introduction ===<br />
The most relevant Calculus for machine learning involves the [ Derivatives], and the multi-dimensional version of the derivative, known as the [ Gradient]. If you can compute gradients, you can do [ Gradient Descent], the most basic form of [ Optimization].<br />
<br />
=== Resources ===<br />
*Set of [ video lectures].</div>Mschachter
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Synopses & Reviews
Please note that used books may not include additional media (study guides, CDs, DVDs, solutions manuals, etc.) as described in the publisher comments.
Publisher Comments:
This is a serious - but not solemn - textbook that attempts to make a clear, conceptual understanding of calculus accessible to all liberal arts students. It presents mathematics as growing out of the classical liberal arts to form a natural bridge between the humanities and the sciences, integrating the history and pedagogy of mathematics in a way that may be of interest to prospective teachers as well. Instead of a pre-calculus review, this book offers an historical development of much of the geometry and algebra needed, emphasizing the fundamental need for students to develop a clear style of writing. Calculus is here largely restricted to the study of algebraic functions, but all the usual aspects of the interplay between functions and derivatives are covered: optimization, instantaneous rates, Newton's method, freely falling bodies, antiderivatives, integrals, areas, volumes, etc. The fundamental theorem is prominently featured and carefully treated. A brief final chapter about the intellectual climate surrounding the development of calculus offers students further insight into the place of mathematics as an element in the history of thought.
Synopsis:
"Synopsis"
by Springer,
| 677.169 | 1 |
Professional development is often determined by black and white thinking. Either issues are considered as being good or bad, or statements like teachers should or teachers must are transported. However, it is easily forgotten from which perspective the judgment is taken, surely it is not the teacher's one. Profoundly respecting and cherishing the... more...
Mathematicians is a remarkable collection of ninety-two photographic portraits, featuring some of the most amazing mathematicians of our time. Acclaimed photographer Mariana Cook captures the exuberant and colorful personalities of these brilliant thinkers and the superb images are accompanied by brief autobiographical texts written by each mathematician.... more...
This book serves as a reference to help prepare and support effective math content coaches. It provides insight into the leadership skills necessary to mentor other teachers, establish collaborative teacher teams, influence school culture positively, and improve student achievement. more...
Designed to support both teachers and university-based tutors in mentoring pre-service and newly qualified mathematics teachers at both primary and secondary levels, Mentoring Mathematics Teachers offers straightforward practical advice that is based on practice, underpinned by research, and geared specifically towards this challenging subject area.... more...
Tips for simplifying tricky basic math and pre-algebra operations Whether you're a student preparing to take algebra or a parent who wants or needs to brush up on basic math, this fun, friendly guide has the tools you need to get in gear. From positive, negative, and whole numbers to fractions, decimals, and percents, you'll build necessary math... more...
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In this new text, Steven Givant?the author of several acclaimed books, including works co-authored with Paul Halmos and Alfred Tarski?develops three theories of duality for Boolean algebras with operators. Givant addresses the two most recognized dualities (one algebraic and the other topological) and introduces a third duality, best understood as... more...
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Actually since you are in 8th grade, try focusing on Algebra I.
Then do geometry, Algebra II, Precal, then Calculus if you are able to in high school.
Once you get into college do a major in math and/or computer science
Computer science majors take so much math in college they just do a double major.
2 months is not a correct amount of time to learn this if you have not already mastered Algebra I
| 677.169 | 1 |
Personally, buying a cheap textbook at a second hard store is probably your best bet.
I bought mine for like $9 or so. I learned a lot. Although it wasn't as thorough as others, I thought it was perfect for a first timer. Perfect for an undergraduate anyways. When I went on to take Topology as a course, we had Munkres. I was able to answer questions without too much difficulty and so on, so I knew the cheap textbook taught me well. It felt good knowing I didn't waste my time reading it.
The book I got, in case you're wondering, is by Theral H. Moore. I saw it at the University library as well, so I would imagine there exists enough copies around to get a hold of it. The questions weren't too hard. I say there were just right for a beginner. Honestly, I think they should teach out of this textbook as a Topology I course for like 2nd year students.
there is a high school level intro to topology by chinn and steenrod, with almost "no" prerequisites. it is excellent, chinn is a decorated high school teacher and steenrod is one of the most outstanding topologists of the 20th century, from princeton.
If you were asking whetehr chinn and steenrod has akll one needs to elarn diff geom right, the answer is NO.
but it is a start. the right question is not does this book have everything i need, but does this book have something i need.
to lesarn differential geometry one needs very little topology. maybe you should search the web for free topology books. there are not many but there are several with the sort of boring basic point set topology, out there which may be what you want.
the more interesting and fun stuff involves some algebra, homology, homotopy, intersection theiory. after reading my posts in who wants to be a mathematician thread referred to above, here is more free introduction to ideas of topology: (but some msymbols like little curly d's, and maybe infinity symbols will not reproduce right)
4220, lecture 1,
First steps, some problems and approaches to them.
The goal of this course is to use calculus (i.e. the concepts of continuity and differentiability) to prove statements such as: a complex polynomial of positive degree always has roots, a smooth self mapping of the disc always has fixed points, or a smooth
vector field on a sphere always has zeroes, and higher dimensional generalizations of them.
These are called existence statements. They are called such because usually we do not produce the solutions whose existence is claimed, rather we deduce some contradiction from the assumption that no solution exists. Thus it is entirely another matter to obtain specific information about these solutions. I want you to give some thought in each case to the problem of actually finding, or at least approximating, these solutions.
On the other hand we will often prove results about the "number" of such solutions. We use quotation marks because again there is no guarantee the actual number of solutions will obey our prediction. We will define at times a "weighting" for each solution, and will prove that either there are infinitely many solutions, or if the number of solutions is finite, the sum of the weights equals our number.
Since a non solution has weight zero, it follows that if our predicted number is non zero, then there is at least one solution. Moreover if there is only one solution, then it must have weight equal to our predicted number. This is a big improvement, since in some cases we can actually find some of the solutions and their weights, and if their sum is deficient from our prediction, we then conclude there must be more solutions. This is a useful tool in plane algebraic geometry called the strong Bezout theorem.
To take advantage of this, poses the challenge of actually computing these weights. In each case we encounter, please give some thought to how to calculate the weights, or the actual number of solutions.
Let's recall one of the earliest cases of an existence theorem of this type, the so called "intermediate value theorem".
Theorem (IVT):
If f:[a,b]-->R is continuous, and f(a) < 0 while f(b) > 0, then there is a number c with a < c < b and f(c) = 0.
This proof is based on the completeness axiom for the real numbers: every non empty set of reals which is bounded above has a real least upper bound.
proof of theorem: Consider the set S = {x in [a,b] such that f(x) <= 0}. S contains a, so S is non empty, and b is an upper bound for S, so S has a least upper bound L with a <= L <= b. If f(L) < 0 then L < b so f is also negative at some number between L and b, so L is not even an upper bound for S. If f(L) > 0, then L > a, so all members of S are less than L but there is an interval of numbers less than L where f is also positive, i.e. which are not elements of S. Hence every element of that interval is an upper bound of S, so L is not the least upper bound of S. This contradiction proves that f(L) is neither positive nor negative, hence must be zero. QED.
abstract version of this proof: If f is continuous on D, and D is connected, then f(D) is also connected. In R the only connected sets are intervals, thus f([a,b]) is an interval. QED.
Let's extend the argument a bit.
Corollary: If f is a polynomial of odd degree, with real coefficients, then f has a real root.
proof: Assume f is monic. Then the limit of f(x) is ? as x-->?, and is -? as x --> -?. Hence there exist a,b, with a < b and f(a) < 0 and f(b) > 0. QED.
Corollary: If f is a differentiable function with no critical points in [a,b] and with f(a) < 0 < f(b), then f has exactly one root in [a,b].
proof: Uniqueness follows from the MVT. QED.
Cor: If f is a polynomial of even (odd) degree with real coefficients, and no root is a critical point, then f has a finite even (odd) number of roots.
proof: (even case) By the hypothesis that f'(x) != 0 when f(x) = 0, the graph of f crosses from one side to the other of the x axis at each root, but the limit of f(x) is ? both as x--> ? and as x--> - ?. So it crosses the x axis an even number of times, provided the number of crossings, i.e. roots, is finite.
To show the number of roots is finite, note that if no root is a critical point, i.e. if f'(x) != 0 when f(x) = 0, then each root is isolated, i.e. each belongs to an open interval in which there are no other roots. Now since the limit of f(x) is ? both as x--> ? and as x--> - ?, we can choose a bounded closed interval [a,b] containing all the roots. Then by compactness of this interval an infinite set of roots would have an accumulation point. But such an accumulation point would be a non isolated root, contrary to hypothesis. QED.
Note: This last corollary introduces and exploits the concepts of transversality and regular values. Notice too that it includes a version of the idea of the weight of a solution, since where the graph crosses the x axis the weight is +1 if f crosses in an increasing direction and -1 if f crosses in a decreasing direction. Then the theorem says the sum of the weights is one for odd degree monic polynomials and zero for even degree ones.
We could also define the weight of a root where the graph stays on the same side of the x axis (and f' = 0) to be zero, provided we relax the assumption that roots are not critical points. (in this case however there can be infinitely many roots, so adding up their weights may not be possible.) Note that a root of weight zero can disappear if the graph is wiggled arbitrarily little, but not so for a root of non zero weight. This introduces the concept of "stability" of a root of non zero weight.
Here as an easy corollary is a "fixed point" theorem.
Cor: If f:[a,b]-->[a,b] is continuous, then f has a fixed point, i.e. there is some point x in [a,b] with f(x) = x.
proof: We try to translate the conclusion into that of the IVT, i.e. try to replace the existence of a fixed point by the existence of a zero. Consider g(x) = f(x) - x. Then f has a fixed point at x if and only if g(x) = 0. Now g:[a,b}--> R is continuous and g(a) = f(a)-a >= 0 since f(a) is in [a,b]. Also g(b) = f(b)-b <= 0. Thus either a or b is a fixed point, or if neither is, then f has one between them, by the IVT. QED.
Note: These corollaries are really easy and fun, as compared to the IVT which is hard and boring. Why then do we teach only the IVT in calculus courses? Lucky us, this course is about deducing the easy fun stuff from the hard boring stuff. (I admit I like the hard boring stuff too, so remind me to ease up if I begin to stress it too much.)
Limitations on the method:
I am tempted to call these "chicken little" arguments, i.e. we prove "either there is a solution or else the sky would fall, and anyone can see the sky is not falling." In this case the "sky falling" is the statement the reals are not complete.
I.e. how close are we to actually finding a solution of f(x) = 0 by the IVT argument? We know there is a solution between a and b, but where? Well we could subdivide the interval further, into subintervals of length 1/10n, say. If we evaluate f at every endpoint of these subintervals, we can reapply the argument and deduce that there is a solution, not necessarily the ¬ in the proof, in one of these subintervals, assuming it is actually practical to evaluate f at these points. Thus in a finite amount of time, maybe large, we can find a number that is within any desired distance of a solution. I.e. we can find for each n, a point that agrees with a solution up to the first n decimal places.
How close is such an approximation to being a solution? There are two ways to measure how close µ is to being a solution of f(x) = 0. We could ask that |L-µ| be small, where L is an actual solution, or we could ask that f(µ) be close to zero. These are somewhat different. I.e. given any N, and any L in [a,b], it is possible to construct a continuous function f on [a,b] which equals -1,000 at every point x <= L -1/10N, and equals 1,000 at every point x >= L + 1/10N. Thus if N is very large, and if in the finite amount of time we allot to the problem, we do not choose x close enough to L, we might always have |f(x)| >= 1,000, for all "approximate solutions" of f(x) = 0.
Improvements on the method
What we really want in an approximate solution of f(x) = 0, is a point x which is near a solution, and also with f(x) close to zero. For this it would help to know that f is differentiable and to have a bound on the derivative of f. I.e. if we knew the derivative of f was never larger than a certain bound, then by the MVT we could bound how fast f grows, and thus know how fine to make our subdivision to get a good approximate solution, i.e. one with f(x) small. E.g. if we know the derivative of f is never greater than 10M, then on a subinterval interval of length 1/10N+M it cannot grow more than 1/10N. Thus for any subinterval with f negative at one end and positive at the other, all points x of this subinterval will be close to an actual solution, and will also satisfy |f(x)| <= 1/10N. Thus differential calculus is a big help in studying these questions.
Moral:
Are these computational limitations so serious as to make our use of topological and differential tools pointless? I think not. It is true we are limited in the precision of the results we obtain, but without these existential tools we would be very hard pressed to say anything at all about such difficult questions as those we will attack. In fact it is exactly their lack of precision that gives these tools their increased flexibility and hence their power. If we are not happy with these imprecise results, fine, try and do better. If we cannot, accept what they give us in the meantime.
Here is an example from my own research interests in algebraic geometry. It was proved over 20 years ago, using a variation of the inverse function theorem, that most "Prym varieties" arise from a unique double cover of curves, but to this day almost no examples of Prym varieties which do in fact arise uniquely are known. This is the question of the number of solutions of the equation p(X) = P, where p is the "Prym map".
To this day the much harder problem of making this result precise is still unsolved. One step toward this result, uses deformation theory (i.e. abstract differential calculus).
In this problem there is a good guess as to what the answer is, i.e. what the right collection of unique solutions X is. In that situation one can try to use differential topological methods to verify the guess. In this way one can sometimes use these tools to get precise results too. For example suppose one wishes to show that a map from the circle to the plane is everywhere injective. If one can show the map has injective derivative everywhere, that the range is contained in the circle, and that there is at least one point of the range with only one preimage, then I claim that all points of the circle have exactly one preimage. Can you see intuitively why?
Generalizations
Now if we really understand the IVT argument above, shouldn't we be able to generalize it? Suppose we consider a complex polynomial
f:C-->C and ask whether it has a zero. This time restricting to an interval is not so useful since we cannot conclude that the image curve in C passes through zero just from knowing its endpoints are on opposite sides of zero. Try to understand the structure of the previous argument, i.e. the topology. We had an interval [a,b], and we deduced the existence of a solution in the interval just by looking at the function restricted to the endpoints {a,b}. What would be an analog of that in two dimensions?
We could look at our polynomial f on a square in C, or perhaps a disc. In order to deduce the existence of a root of f(x) = 0 lying in the disc, what would we look at? As an analog of the endpoints of the interval perhaps we should look at the restriction of f to the boundary of the disc. What behavior of f on the boundary would lead us to conclude there is a solution inside the disc? Let us take a very simple case. What if f were the identity on the boundary of the unit disc? Would it follow that f(x) = 0 has a solution inside the disc?
This is a direct analog of the IVT as follows. Consider this question: Is there a continuous map g:[a,b]--> {a,b}, from the interval [a,b] to its set of endpoints {a,b}, which maps a to a, and b to b? If there were a continuous map f:[a,b]-->R with f(a) < 0 and f(b) > 0, but with no point mapping to zero, then we could find a map g as above in the following way. Since the map R- {0}--> {a,b} sending negative numbers to a and positive numbers to b is continuous, if we compose f with this map we would get a map g:[a,b]--> {a,b}, with g(a) = a and g(b) = b.
Thus the question whether there is a continuous map f:[a,b]-->R-{0} with f(a) < 0, and f(b) > 0, is equivalent to the question of whether there is a continuous map g:[-1,1]--> {-1,1} with f(-1) = -1, and f(1) = 1. This form of the question can be generalized to higher dimensions.
Conjecture: suppose f:C-->C is continuous and that the restriction of f to the unit circle {z: |z|*= 1} is the identity map. Then we claim f(z) = 0 has at least one solution for |z| < 1.
How would we prove this? Remember the philosophy is to deduce a catastrophe from the falsity of the result, so assume it is false. I.e. that f maps the whole unit disc into C-{0}. Then what could one do? By analogy with the final form above of the IVT, after composing f with a retraction onto the circle, via z --> z/|z|, one could deduce there is a continuous map of the disc to its boundary circle which restricts to the identity on the boundary. Is this possible? Why or why not?
Retraction problem: There is no continuous map D-->?D of the disc onto its boundary, which restricts to the identity map on the boundary.
This is the hard result, analogous to the IVT above. It already implies the conjecture above that a polynomial f:C-->C which restricts to the identity on the unit circle has a root inside that circle. We can also use it to easily prove the Brouwer fixed point theorem.
Fixed point theorem: Any continuous map f:D-->D from the disc to itself has a fixed point.
proof: Assume not. Then for every x in D, f(x) and x are different points, hence determine a vector. Running along this vector from f(x) to x and then on until we reach the boundary gives a continuous map from D-->?D, which is the identity on the boundary, contradicting the retraction principle. QED.
Counting roots
What about results on the number of roots? I.e. if a complex polynomial restricts to the identity on the unit circle, how many roots do you think it should have inside that circle? How does the derivative of a complex polynomial behave? We will need this to generalize the methods above. I.e. recall the derivative of a complex polynomial at the point a, is a complex linear transformation L:C-->C.
If L is non zero, it is a complex linear isomorphism, in particular it is an orientation preserving injection. Thus if x is a root of f which is not a critical point, then near x, by the inverse function theorem f itself is also an orientation preserving injection. In particular f maps a small oriented disc centered at a, onto a small similarly oriented (deformed) disc centered at 0. If f restricts to the identity on the unit circle then how many roots do you think f can have inside the circle? Why?
Do you know how f behaves near a point where the derivative is zero? Does this suggest to you whether the derivative of an f can be zero inside the unit circle, if f itself is the identity on the unit circle? What is the key concept involved here, and how can we measure it? (Winding number. Sum of interior winding numbers at all roots, equals winding number of boundary.)
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Get a good grade in your precalculus course with Precalculus, Seventh Edition. Written in a clear, student-friendly style, the book also provides a graphical perspective so you can develop a visual understanding of college algebra and trigonometry. With great examples, exercises, applications, and real-life data--and a range of online study resources--this book provides you with the tools you need to be successful in your course.
In this DVD, Ryan Church introduces the viewer to Corel Painter. Ryan teaches the way he uses the program on a daily basis to create production artwork for the film and video game industries. From the basic first steps of setting up the Painter workspace and menus, to using brushes, layers, papers and textures, this DVD is perfect for anyone interested in learning traditional illustration techniques using Painter's powerful features and tools. As the lecture progresses, Ryan walks you through his process for creating a finished production illustration in real time, from sketching and designing, to composing and adding final details.
James Stewart, author of the worldwide, best-selling Calculus texts, along with two of his former Ph.D. students, Lothar Redlin and Saleem Watson, collaborated in writing this text to address a problem they frequently saw in their calculus courses: many students were not prepared to think mathematically but attempted instead to memorize facts and mimic examples. College Algebra was written specifically to help students learn to think mathematically and to develop true problem-solving skills. This comprehensive, evenly paced book highlights the authors' commitment to encouraging conceptual understanding. To implement this goal, Stewart, Redlin, and Watson incorporate technology, the rule of four, real-world applications, and extended projects and writing exercises to enhance a central core of fundamental skills.
Objects created using 3D computer graphics have a tendency to appear too perfect and therefore slightly unrealistic. To create more natural textures, it is necessary to be able to observe and understand how real world objects become worn and dirty with age. This DVD focuses on how the understanding and use of aging, and "wear and tear" on an object can help add realism to your textures. Paul demonstrates how different surface materials can be simulated using simple color, bump, specular, reflection and transparency maps. These principles are illustrated by the texturing of a rustic lantern and weathered house using Adobe Photoshop®.
The Eighth Edition features a new design, enhancing the Aufmann Interactive Method and the organization of the text around objectives, making the pages easier for both students and instructors to follow.
Eight-DVD set (approx. ten hours total) with David Hobby from the website Strobist.com. It is aimed at advanced amateur photographers who would like to gain a better understanding of how use small flashes off-camera to improve the quality of light in their photos. In English with English Subtitles.
Introduction to Mechatronics and Measurement Systems provides comprehensive and accessible coverage of the evolving field of mechatronics for mechanical, electrical and aerospace engineering majors. The author presents a concise review of electrical circuits, solid-state devices, digital circuits, and motors- all of which are fundamental to understanding mechatronic systems.
An engaging introduction to how people use IS to solve business problems. Using MIS explains why MIS is the most important course in the business school by showing readers how businesses use information systems and technology to accomplish their goals, objectives, and competitive strategy. With a new edition now publishing each year, Using MIS, 4e, contains fresh, new, and current material to help keep you up to date.
. Electric Blues – At a Glance features these lessons: Blues Progressions, B.B. King Style, Stevie Ray Vaughan Style, and Progressive Licks.
THE ECONOMIST is a global weekly magazine written for those who share an uncommon interest in being well and broadly informed. Each issue explores the close links between domestic and international issues, business, politics, finance, current affairs, science, technology and the arts. In addition to regular weekly content, Special Reports are published approximately 20 times a year, spotlighting a specific country, industry, or hot-button topic. The Technology Quarterly, published 4 times a year, highlights and analyzes new technologies that will change the world we live in.
Bloomberg Markets is a US based magazine for financial professionals throughout the world. Globally the largest publication of its kind, Bloomberg Markets magazine can keep you up to date on commodities, rates and bonds, futures and currencies. Bloomberg Markets includes detailed explorations of the current stock market and financial markets, outlines financial strategies, and features articles on some of the most successful businesses and affluent personalities in the world. The magazine has earned numerous accolades, scooping awards for its journalism and editorial and design distinction.
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Algebra 2, Student Edition
9780078738302
ISBN:
007873830X
Edition: 1 Pub Date: 2006 Publisher: Glencoe/McGraw-Hill
Summary: THE PROGRAM STUDENTS NEED; THE FOCUS TEACHERS WANT! "Glencoe Algebra 2" is a key program in our vertically aligned high school mathematics series developed to help all students achieve a better understanding of mathematics and improve their mathematics scores on today's high-stakes assessments.
McGraw-Hill Education is the author of Algebra 2, Student Edition, published 2006 under ISBN 9780078738302 and 0078...73830X. Four hundred forty one Algebra 2, Student Edition textbooks are available for sale on ValoreBooks.com, two hundred seventy seven used from the cheapest price of $8.95, or buy new starting at $145.00
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Cabri™ Jr. Geometry App
Now you can add a new dimension to your students' learning experience interactive geometry for your calculator. Create excitement in the classroom as you construct, analyze, and transform mathematical models and geometric diagrams.
Explore the cutting-edge technology developed for TI by CabriLog® with renowned French mathematician Jean-Marie Laborde. Just take a look at what Cabri™ Jr. geometry app can do:
Perform analytic, transformational and Euclidean geometric functions
Build geometric constructions interactively with points, a set of points for locus, lines, polygons, circles, and other basic geometric objects
Alter geometric objects on the fly to see patterns, make conjectures, and draw conclusions -- and, you'll get a more intuitive and highly interactive interface
The Cabri Jr. Geometry App is also available for the TI-83 Plus, TI-84 Plus, TI-84 Plus Silver Edition Graphing Calculators.
Language Localization Apps
Periodic Table
Students can utilize this app to access a graphical, electronic Periodic Table the moment they need it! This app is not just your basic periodic table - students can also study and review trends in the periodic table easily with their TI graphing calculator.
Additional benefits include:
15 properties and facts about 109 elements
Graphs of the periodic nature of the elements
Easy navigation between the elements
The Periodic Table App is also available for the TI-83 Plus, TI-84 Plus, and TI-84 Plus Silver Edition Graphing Calculators
In addition, students can export data for further exploration.
The Probability Simulation App is also available for the TI-73 Explorer, TI-83 Plus, TI-84 Plus, TI-84 Plus Silver Edition Graphing Calculators.
The TestGuard App is also available for the TI-83 Plus, TI-84 Plus, and TI-84 Plus Silver Edition Graphing Calculators.
* Disable feature works only on student TI-84 Plus calculators with OS version 2.40 or higher. If TI TestGuard is set up to disable programs and/or applications and the student has a TI-83 Plus or a TI-84 Plus with an earlier OS, the programs and/or applications will be deleted instead, with a configuration comparable to what was specified.
Transformation Graphing App
Students can visually draw conclusions about functions and improve graphing comprehension with this App. To take advantage of this App's capabilities, simply input functions, and view changes in the function as the parameters are changed.
Students can:
Instantly see how changing the value of a coefficient transforms a graph
Visually fit equations to data plots by manipulating coefficients
The Transformation Graphing App is also available for the TI-83 Plus, TI-84 Plus, and TI-84 Plus Silver Edition Graphing Calculators.
Vernier EasyData™ App
The EasyData™ App from Vernier Software & Technology is simple data collection software for the TI-84 Plus family of graphing calculators. Explore the world around you using this App. The EasyData App auto-launches data collection when used with Vernier EasyTemp™ sensor, and loads built-in experiments for every supported Vernier sensor. EasyData™ supports data collection with Vernier USB sensors (TI-84 Plus family only), CBR™, CBR™ 2, CBL™ 2 and LabPro® devices using the TI-84 Plus family and TI-83 Plus family.
The EasyData App is also available for the TI-83 Plus, TI-84 Plus, and TI-84 Plus Silver Edition Graphing Calculators.
App4Math
This App, by I.Q. Joe, the creators of Zoom Math, provides an easy-to-use interface so you can enter the math problem the same way you see it in your textbook, thereby reinforcing math notation and students' understanding. Visualize and work with correct math notation for fractions, radicals, exponents and math symbols like π. Students can also grab and edit previous inputs, saving time and typos!
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A parent writes:I'm trying to find other options for my son in computing math problems. He is a sophomore this year and is taking applied math. He has visual impairment due to saccadic eye movements. He also has involuntary muscle movements. Fine motor skills are very difficult, if not impossible. Can anyone suggest a calculator that would work for him? For instance, are there portables with enlarged keys that repeat entries with speech synthesis...AND are easy to use? Any suggestions would be appreciated
Susan replies: Since your son is taking applied math, I would recommend a five function (+, -, *, /, %) talking calculator. There are several models and prices and various vendors. I will give you a list to choose from, but there may be other sources closer to home where you can find bargains. The calculators described in depth below are the ones I have successfully used with those of my applied math students who had great difficulty with talking calculators having small keys and compact keypads.
They have a talking desktop calculator with a large 8 digit LCD readout, a repeat key, and enlarged keys for $18.95 (with earphone $28.95). They have quite a variety of other talking and/or large number calculators as well.
Call for a catalog, but be sure to check out your local resources. It's always better to try things out in person if possible. See Vendors of Math Materials for more detailed information on the above vendors.
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Elementary and Intermediate Algebra: A Practical Approach provides concise, manageable treatment of the mathematics required for the combined algebra course. While emphasizing problem solving and the real-world applications of algebra, the text provides solid coverage of core mathematical concepts and essential symbol manipulation skills. Furthermore, the text encourages students to use graphing technology while still requiring them to master pencil-and-paper techniques for certain tasks. Authors Craine, McGowan, and Ruben combine their experience and expertise as a math educator and author, a math researcher, and an authority on math anxiety to deliver a balanced, targeted text that enables instructors to cover all the material for the combined course in one text
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DIFFERENCE EQUATIONS
Simple Linear Forms
Amortization
An Iterative Geometric Construct: The Koch Curve
Solution of Linear Constant Coefficients Difference Equations
Convolution-Summation of a First-Order System with Constant Coefficients
General First-Order Linear Difference Equations
Nonlinear Difference Equations
Fractals and Computer Art
Generation of Special Functions from Their Recursion Relations
ELEMENTARY FUNCTIONS AND SOME OF THEIR USES
Function Files
Examples with Affine Functions
Examples with Quadratic Functions
Examples with Polynomial Functions
Examples with the Trigonometric Functions
Examples with the Logarithmic Function
Examples with the Exponential Function
Examples with the Hyperbolic Functions and Their Inverses
Commonly Used Signal Processing Functions
Animation of a Moving Rectangular Pulse
Use of the Function Handle
MATLAB Commands Review
ROOT SOLVING AND OPTIMIZATION METHODS
Finding the Real Roots of a Function of One Variable
Roots of a Polynomial
Optimization Methods for Functions of One Variable
The Zeros and the Minima of Functions in Two Variables
Finding the Minima of Functions with Constraints Present
MATLAB Commands Review
COMPLEX NUMBERS
Introduction
The Basics
Complex Conjugation and Division
Polar Form of Complex Numbers
Analytical Solutions of Constant Coefficients ODE
Phasors
Interference and Diffraction of Electromagnetic Waves
Solving ac Circuits with Phasors: The Impedance Method
Transfer Function for a Difference Equation with Constant Coefficients
MATLAB Commands Review
MATRICES
Setting up Matrices
Adding Matrices
Multiplying a Matrix by a Scalar
Multiplying Matrices
Inverse of a Matrix
Solving a System of Linear Equations
Application of Matrix Methods
Eigenvalues and Eigenvectors
The Cayley-Hamilton and Other Analytical Techniques
Special Classes of Matrices
Transfer Matrices
Covariance Matrices
MATLAB Commands
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Article excerpt
The teaching of algebra in Victorian secondary schools has changed substantially in the last decade. Here we present implications for curriculum policy arising from research into students' algebra learning in Years 7 to 11. Data were collected from approximately 3000 students in 34 schools. Information about programs offered was obtained from teachers, by textbook analysis and by some lesson observation and teaching interventions. Performance varied considerably between schools and classes. Some differences are attributable to teaching methods, the content taught, and the arrangement of the curriculum. Subtle reductions in goals and the isolation of topics in the curriculum were disturbing trends. We discuss findings that have important implications for mathematics education policy Australia-wide.
Over the six years 1991-96, we have studied the learning of school algebra, with particular focus on the causes of common misunderstandings and low attainment. In the course of this research, we tested and talked to students, observed teaching, discussed approaches with teachers, and analysed textbooks and curriculum documents. Readers will find details of the data collection and specific findings in our research reports and articles (see Stacey & MacGregor, 1997c). In this article, we discuss the `broad brush' issues and the implications for curriculum policy that arise from them. We intend that our discussion and analysis will stimulate and inform debate on issues such as whether algebra as currently being taught is achieving its purpose for various groups of students, the levels of achievement that we expect, and the quantity and quality of instruction needed to reach such levels.
The first section of this article places algebra in context in the secondary school mathematics curriculum and summarises our research procedure and findings. The subsequent sections highlight aspects of the findings relating to the issues above.
School algebra and students' achievement
Place of algebra in the mathematics curriculum
Algebra is one of the five strands of content in A national statement on mathematics for Australian schools (Australian Education Council, 1991) and the Curriculum and standards framework: Mathematics (Board of Studies, 1995). As highlighted in the National statement, ideas essential for learning algebra have a place in the primary curriculum, but only in secondary school do students begin formal algebra, which for us is signified by the use of letters to denote unknown or variable quantities. This late introduction reflects the special role of algebra as a gateway to higher mathematics. Algebra is the language of higher mathematics and is also a set of methods to solve problems encountered in professional, rather than everyday, life.
Some algebra is taught to all junior secondary students in normal Victorian (and Australian) schools. This is done for two reasons. First, some familiarity with algebra is considered to be important for informed participation in a democratic society, and therefore all students should learn about its key concepts. Secondly, since algebra is important for further mathematics, on grounds of equity no student should be denied access to it. The inclusion of algebra in a common curriculum for all secondary school students is not universal, however. In the United States, for example, a different curriculum structure means that many students never take the one-year algebra course that is a prerequisite for subsequent mathematics courses. There is at present a campaign for all students to take this course, so that decisions about future access to mathematics are not made so early. In Australia, since all students have the opportunity to learn some algebra in every year level, the questions that must be asked about `algebra for all' are `What type of algebra?' and `How much for whom?'. These are important questions.
In recent years, there have been substantial changes in the way algebra is taught, in Australia and in other countries, at least as indicated in curriculum documents and textbooks. …
Further Pure Mathematics with Technology: Developing a New ?-Level Mathematics Unit That Uses Technology in the Teaching, the Learning, and the Assessment Button, Tom.
Mathematics Teaching, No. 235, July 2013
Schools and Businesses Must Unite for Future; AFTER Taking over as Head of Skills Specialist Semta, Ann Watson Explains How Employers and Educators Are Joining Forces to Improve the Quality of Teaching and Training in Science Technology, Engineering and Mathematics (STEM) Subjects - Essential to Engineering and the Future of the UK Economy Watson, Ann.
The Journal (Newcastle, England), October 20, 2014
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About the book:
Based on lectures given at Claremont McKenna College, this text constitutes a substantial, abstract introduction to linear algebra. The presentation emphasizes the structural elements over the computational - for example by connecting matrices to linear transformations from the outset - and prepares the student for further study of abstract mathematics. Uniquely among algebra texts at this level, it introduces group theory early in the discussion, as an example of the rigorous development of informal axiomatic systems.
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Hardcover, ISBN 0387940995 Publisher: Springer, 1999 0387940995 Publisher: Springer Science and Med.,387940995 Publisher: Springer-Verlag New York Inc.,
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This series of videos, created by Salman Khan of the Khan Academy, features topics covered from very basic algebra all the way through algebra II. This is the best algebra playlist to start at if you've never seen...
This lesson from Illuminations asks students to measure the diameter and circumference of various circular objects, plot the measurements on a graph, and relate the slope of the line to ?, the ratio of circumference to ...
This series of videos contains 180 Worked Algebra I examples (problems written by the Monterey Institute of Technology and Education). You should look at the "Algebra" playlist if you've never seen algebra before or if...
This lesson helps students further their understanding of linear functions by applying the material to a real-world example. The class will use data on an airline flight including travel time, ground speed, time...
With this lesson, students will use tables of fees from a few different cell phone providers to create an algebraic expression that reflects billing for services. The example helps students apply algebraic functions to...
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summer.
About the Series Math in a Cultural Context
This series is a supplemental math curriculum based on the traditional wisdom and practices of the Yup'ik people of southwest Alaska. The result of more than a decade of collaboration between math educators and Yup'ik elders, these modules connect cultural knowledge to school mathematics. Students are challenged to communicate and think mathematically as they solve inquiry-oriented problems, which require creative, practical and analytical thinking. Classroom-based research strongly suggests that students engaged in this curriculum can develop deeper mathematical understandings than students who engage only with a procedure-oriented, paper-and-pencil
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2001 Trade paperback Revised ed. Illustrated. New. Trade paperback (US). Glued binding. 409 p. Contains: Illustrations. Audience: General/trade. New and unmarked copy. This book ...revolutionizes the prevailing understanding and teaching of math. The addition of this book is a must for all upper-level Christian school curricula and for college students and adults interested in math or related fields of science and religion. It will serve as a solid refutation for the claim, often made in court, that mathematics is one subject, which cannot be taught from a distinctively Biblical perspective. Helps support Christian Homeschooling family.Read moreShow Less
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This book revolutionizes the prevailing understanding and teaching of math. This book is a must for all upper-level christian school curricula and for college students and adults interested in math or related fields of science and religion. It will serve as a solid refutation for the claim, often made in court, that mathematics is one subject which cannot be taught from a distinctively biblical
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Tips and personal email to me if you need help or want more of a specific problem!!!
This app has many different problems with a total of 160+ problems making sure you know every area of Calculus!!!
Only 99 cents!!!! Limited time only- A program features tasks of different complexity and complete solutions with answers to every task. - For: high school students of final years of study. - Develops analytical skills and non-standard approach to math. - Adaptive solutions. The most complex issues are marked as optional commentaries, so that the students who don't want it too complex can read the necessary and the most easy part while still getting an understanding. - Topics: Numbers, Calculus, Geometry, Algebra, Probability theory. - Checks answers which are entered by the user and gives final number of points collected. - The task bank is updated regularly. - The program is available in English and Russian. - Private tutorship and consultancy over all topics are possible via skype with the author of the appplication.
This app takes education to a whole new level by reinterpreting known theories with artistic depictions. Scattered with engineered interactivity and reinforced with fully illustrated practice questions, the Calculus App presents a new educational paradigm that will enrich the learning experience of our knowledge-seeking audience.
Now you can learn about differentiation and integration minus the boring bits of simply going through equations and formulas
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- 6 Complete Chapters to cover the fundamentals of Calculus including: Introduction to Differentiation, Applications of Differentiation, Transcendental Functions, Introduction to Integration, Applications of Integration, and Integration Techniques. - 3 Feature Videos to introduce the topic to you - Fully narrated tutorials guide you on the basic concepts of Calculus - Fun-to-watch videos introduce you to the world of Calculus; learn about fun facts and not just the technical concepts - Animated and illustrated concepts makes learning from scratch or refreshing easy - Interactive sections reinforce important concepts - Move away from the usual style of learning in the classroom - Learn at your own pace: pause or replay the tutorials at your command - Track your progress from the Navigation Menu - Try the sample math questions and check your answers against model solutions
Unsure about the Calculus Interactive App Full Edition? Try out the FREE Lite Edition ⇒
Calculus made Easy is an app for students wanting to master Calculus the easy way. Using the knowledge of a teacher who has taught for over 38 years with a Pure and Applied Mathematics Degree, you will find learning calculus a breeze.
Calculus made Easy has t utored lessons which shows step by step worked examples. Calculus Made Easy is designed to help you understand Calculus, having your own Maths Tutor when and where you need it.
This app covers the following topics: * Definition of Function * First Principles for Differentiation * The Gradient of a Tangent * Rules for Differentiation * Function of a function rule * Product Rule * Quotient Rule * The derivatives of the function of Trigonometry.
"iTeachers" aim to provide easy to understand lessons using new age technology, taught the old fashioned way.
With our International content, easy to understand, fast-paced lessons. You will soon be top of the class.
Calculus has been known to bring students to tears. Now you have an expert in your corner.
This application contains a rich collection of examples, tutorials and solvers, crafted by a professional math tutor with over 20 years of applied mathematics and teaching experience. "Calculus Pro" covers:
Calculus Reference Tool Lite is the perfect tool for any student taking a calculus course, or those looking for a brush up! It contains detailed descriptions with clear and helpful examples on over 15 calculus topics, ranging from limits and the definition of a derivative, up to partial derivatives and triple integrals using multiple coordinate systems. A must have to accompany any student!
Another plus, it's ad free!
Upgrade to the Pro version for a very helpful list of cheat sheets and a vector calculator!
CalculusGENIUS (formerly CalculusGenie) is the most engaging Calculus App yet. Calculus is actually very interesting and fun, if taught properly! We work with teachers who have decades of college teaching experience. They have seen it all ... So you don't have to!
Combine this with ZeGenie's Interactive Learning Engine, and you have the best Calculus App yet!
This App covers all the basis of Calculus I (Differential Calculus), with more advanced topics coming soon!
This free app provides access to a large sample of Magoosh's popular Calculus lessons. Sign in with a Magoosh Premium account to get access to the full library, along with all the online lessons on the Magoosh Calculus website. See for more information.
Who makes the lessons? === Magoosh's expert tutor, Mike McGarry. He has over twenty years of experience teaching in a variety of subjects, specializing in math, science and standardized exams.
What is Magoosh? === Magoosh is an online education company with a focus on teaching through videos and personalized customer support Calculus Lessons today! very important and if you can learn to follow them you will have a successful relationship!
History facts is a simple fact generator of 100+ facts so far. They have many great world facts for most countries. I will add more when people download it more and rate it more. Tell your friends and family about it for more facts. These facts took time to research and find and that's why I assure you will learn a new fact in each fact. This app is updated around every week to every month with about 4-10 new facts each update. Leave feedback about them! They are great facts to read. Download History facts today to learn all about history!
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THE NEW AND IMPROVED WEIRD NOISES!!! Download now! Weird Noises will soon have more sounds, currently has an animation feature, and will plan to expand soon! I plan to get better sound quality and better funnier sounds! Thank you!
Do not forget to check out my brother's cool youtube videos. :) is based off Campbells Biology Book 9th edition. It covers most of the book. There are exactly 300 slides of information in app over Biology I!!! Very useful summarization of Biolgy in general. Tired of using your book? Use this app!
Now includes 2nd half of General Biology in depth in a flashcard edition of it. 282 flashcards over biology terms and questions.
In general, there is nearly 600 slides of information over Biology.
SPECIAL THANKS TO Mrs. Killion!!!(Biology I) AND Mrs. Karels!!!(Biology II).Living healthy, is living good. Download this app now and understand how to live a healthier life. So many of us today are going to fast food chains nearly every day for food. Not only are they deteriorating to your health, but you're losing money to it too. Several of these tips were summarized by health.gov, a website created by the government to advise us on our health. Download the app and learn to live a healthier life.
Think about it, you're not paying very much for access to the only app on the market that includes so many features for relationships. If you've been following the app, you know it's changed a lot and many features are a part of the app now. More are to come for this app. It's not much to invest in that will earn you huge rewards in the long run having this app. Relationships today are degrading and this is designed to help you in modern relationships!
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You probably know at least a handful of people going through problems in their relationships and wonder if you'll experience the same. This can all stop if you stick to reading a few of these tips a day and following them.
Relationship Tips Pro is the app that will give you experienced advice for your relationship. It is simple, easy and to the point. The tips are very important and if you can learn to follow them you will have a successful relationship!
The revolutionary magazine that brings you the latest trends in fashion, couture, fashion weeks and integrates it with an online catalog of latest brands so that readers could shop for their favorite brands whilst still reading.
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Show More algebra. This nearly 900-page material combines trigonometry and algebra in a straightforward manual designed to provide reviews of equations and functions, as well as complex numbers, quadratics, conics, factoring, and multivariable polynomials. Precalculus: Mathematics for Calculus (ISBN 9780840068071) simplifies difficult topics such as mathematical modeling and problem solving in the first chapters, helping students develop a more reliable foundation for advanced math. The latter chapters discuss irrational roots, sequences and series, identities and induction, binomial coefficients, exponents, logarithms, and functional equations. Each section in Precalculus: Mathematics for Calculus caps off with a cumulative review test and a focus on modeling, giving readers a closer look at various topics including vector fields, fitting exponential and power curves to data, and conics in architecture. By the end of this book, students are expected to have developed better mathematical skills for required to progress to calculus, where they need to solve mathematical problems dealing properties of integrals and derivatives using methods based on infinitesimal differences and summation. James Stewart and fellow best-selling authors, Lothar Redlin and Saleem Watson use practical examples and exercises in varying levels of difficulty to cater to both advanced high school learners as well as university students.
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Tough Test Questions? Missed Lectures? Not Enough Time? Fortunately for you, there's Schaum's. More than 40 million students have trusted Schaum's Outlines:
Master discrete mathematics and ace your exams with this easy-to-use guide that reinforces problem-solving skills and reduces your study time! Students of discrete mathematics love Schaum's--the first edition of this book was a major bestseller--and this edition will show you why! Schaum's Outline of Discrete Mathematics lets you focus on the problems that are at the heart of the subject. It cuts your study time by eliminating the extraneous material that clutters up so many textbooks
Confusing Textbooks? Missed Lectures? Not Enough TimeThis Student Solution Manual provides complete solutions to all the odd-numbered problems in Foundation Mathematics for the Physical Sciences. It takes students through each problem step-by-step, so they can clearly see how the solution is reached, and understand any mistakes in their own working. Students will learn by example how to arrive at the correct answer and improve their problem-solving skills.
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Math 126: Calculus With Analytic Geometry III
Section A: Summer 2009
Instructor: Mark Hubenthal
Email: hubenjm@math.washington.edu
Class Website:
126 Materials Website:
Office: Padelford C-20
Office Hours: Mon 11:30am-12:30pm, Wed 11:00am-12:00pm, or by appointment
Teaching Assistant: Joel Barnes, jbarnes2@math.washington.edu
Schedule:
Lecture: MWF 1:10 - 2:10 in SIG (Sieg) 227
Quiz section: TuTh 1:10-2:10 in BLM (Balmer) 413
The Course: This course covers a few miscellaneous topics from calculus. Most of our time will be spent
on calculus in three dimensions. This part of the course is interesting because we actually live in three
dimensional space. We first cover the basics of analytic geometry in three-space. We then discuss parametric
equations and introduce the differential calculus of vector valued functions. This part of the class finishes
with an introduction to multivariable integration. The last two weeks are devoted to a study of sequences
and series. This culminates in Taylor's beautiful theorem. This material is useful for solving differential
equations and for making approximations. We will use locally produced notes for this part of the class
instead of the textbook. These notes are available online at
Grading:
Worksheets 5%
Homework 15%
Quizzes 20%
Midterm 20%
Final 40%
Homework:
There is weekly homework, assigned daily. The problems assigned during the week will be collected in
lecture on the following Wednesday. Since the answers to many of the exercises are available to you, it is
important that you write out complete solutions to all assigned problems. No credit will be given for simply
writing the correct answer. You are encouraged to talk to your classmates and discuss both the homework
and the material you are learning. However, please make sure you write up solutions on your own. It is
essential to fully understand how to solve the homework problems and to acquire enough practice to be able
to do problems relatively quickly.
Unfortunately, the amount of homework that can be graded is limited. Each week two of the problems
will be chosen at random to grade. They will be worth 3 points each. This makes a total of 6 points.
In addition to this, you will receive a score out of 4 points reflecting the percentage of the homework you
completed. (For example, if you completed about 75% of the assignment, you would receive an additional 3
1
points.) Thus the total possible score for each assignment is 10 points. The lowest weekly homework score
will be dropped. No late homework will be accepted.
The homework problems assigned will be posted on the course webpage.
The homework makes up 15% of your course grade.
Quizzes: There will be a 20 minute quiz on most Thursdays that there is not a worksheet, see the course
webpage: These will usually cover the homework from the
preceeding Monday or Friday. They will be very similar to the homework problems. The TA's will grade
them and return them to you the following Tuesday. The quizzes are closed book/closed notes and you
cannot use a graphing calculator. Makeup quizzes can be arranged only if you have a valid reason and come
talk to me or your TA in advance. I will drop your lowest quiz score.
The quizzes make up 20% of your course grade.
Worksheets:
There are worksheets to be completed on Thursdays in the quiz section. You will do these problems
in small groups and your TA will help you work through them. Worksheets give you enough supervised
practice to go off and do the homework. They may also be used to introduce new ideas and methods that
have not been covered in lecture. Treat the worksheets seriously as they help you learn how to think and
write mathematics with your TA present to help you if you make a mistake. Your TA will will keep a record
of your participation and performance in these worksheet sessions.
The worksheets are posted on the course website You
should bring a copy of the worksheet to the quiz section.
The worksheets make up 5% of your course grade.
Exams:
There will be one midterm exam and one final exam. The midterm exam will be held in the Tuesday
quiz section on July 21 (week 5). The final exam will be given during the last two days of class on August
20 and 21 (Thurs/Fri), in two parts.
You must bring a photo ID to each exam. You may bring one 8.5x11 handwritten sheet of notes (writing
allowed on both sides). You may use a scientific (but *not* graphing) calculator.
The exams count for 60% of your course grade – 20% for the midterm and 40% for the final.
Make-Ups:
Late homework assignments and worksheets will not be accepted for any reason. However, the lowest
weekly homework score will be dropped.
There are no make-up exams. If you have a compelling and well-documented reason for missing a test,
speak to the instructor about it.
Calculators:
Graphing calculators are *not* allowed on the exams and quizzes. You may use a scientific calculator
with trigonometric functions, logarithms and exponential functions.
Other Resources:
1. The math study center:
2. CLUE:
3. My office hours (see above).
Textbook:
The course text is Multivariable Calculus by James Stewart (The Sixth Edition).
2
Note: We are using the new 6th edition this year and the homework problems are different from the 5th
edition.
3
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. Used books do not include components, such as CDs, DVDs, Study Guides/Readers, or access cards and one time use codes. Quality text from a reliable seller. Speedy service! ...Choose EXPEDITED for fastest shippingTried and true, Gustafson and Frisk's ALGEBRA FOR COLLEGE STUDENTS teaches solid mathematical skills while supporting the student with careful pedagogy. Each book in this series maintains the authors' proven style through clear, no-nonsense explanations, as well as the mathematical accuracy and rigor that only Gustafson and Frisk can deliver. The text's clearly useful applications emphasize problem solving to effectively develop the skills students need for future mathematics courses, such as college algebra, and for real life. The Seventh Edition of ALGEBRA FOR COLLEGE STUDENTS also features a robust suite of online course management, testing, and tutorial resources for instructors and students. This includes BCA/iLrn Testing and Tutorial, vMentor live online tutoring, the Interactive Video Skillbuilder CD-ROM with MathCue, a Book Companion Web Site featuring online graphing calculator resources, and The Learning Equation (TLE), powered by BCA/iLrn. TLE provides a complete courseware package, featuring a diagnostic tool that gives instructors the capability to create individualized study plans. With TLE, a cohesive, focused study plan can be put together to help each student succeed in math.
Related Subjects
Meet the Author
R. David Gustafson is Professor Emeritus of Mathematics at Rock Valley College in Illinois and has also taught extensively at Rockford College and Beloit College. He is coauthor of several best-selling mathematics textbooks, including Gustafson/Frisk/Hughes' COLLEGE ALGEBRA, Gustafson/Karr/Massey's BEGINNING ALGEBRA, INTERMEDIATE ALGEBRA, BEGINNING AND INTERMEDIATE ALGEBRA, BEGINNING AND INTERMEDIATE ALGEBRA: A COMBINED APPROACH, and the Tussy/Gustafson and Tussy/Gustafson/Koenig developmental mathematics series. His numerous professional honors include Rock Valley Teacher of the Year and Rockford's Outstanding Educator of the Year. He has been very active in AMATYC as a Midwest Vice-president and has been President of IMACC, AMATYC's Illinois affiliate. He earned a Master of Arts from Rockford College in Illinois, as well as a Master of Science from Northern Illinois University.
Peter D. Frisk, Professor Emeritus of Mathematics at Rock Valley College, earned both a Bachelor of Arts and a Master of Arts in mathematics from University of Illinois. He is the coauthor of several best-selling math texts including BEGINNING ALGEBRA, INTERMEDIATE ALGEBRA, BEGINNING AND INTERMEDIATE ALGEBRA: A COMBINED APPROACH, ALGEBRA FOR COLLEGE STUDENTS, and COLLEGE ALGEBRA. He has been the recipient of the Rock Valley Teacher
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This Textbook
Overview
This text is written for the standard, one-semester, undergraduate course in elementary partial differential equations. The topics include derivations of some of the standard equations of mathematical physics (including the heat equation, the wave equation, and Laplace's equation) and methods for solving those equations on bounded and unbounded domains. Methods include eigenfunction expansions, or separation of variables, and methods based on Fourier and Laplace transforms.
Editorial Reviews
From the Publisher
From the reviews of the second edition:
"This second edition of the short undergraduate text provides a fist course in PDE aimed at students in mathematics, engineering and the sciences. The material is standard … . Strong emphasis is put on modeling and applications throughout; the main text is supplied with many examples and exercises." (R. Steinbauer, Monatshefte für Mathematik, Vol. 150 (4), 2007)
"This book contains an elementary introduction of partial differential equations to undergraduate students in mathematics, engineering, and physical sciences. … This is a unique book in the sense that it provides a coverage of the main topics of the subject in a concise style which is accessible to science and engineering students. … Reading this book and solving the problems, the students will have a solid base for a course in partial differential equations … ." (Tibor Krisztin, Acta Scientiarum Mathematicarum, Vol. 74, 2008)
Related Subjects
Meet the Author
J. David Logan is Willa Cather Professor of Mathematics at the University of Nebraska Lincoln. He received his PhD from The Ohio State University and has served on the faculties at the University of Arizona, Kansas State University, and Rensselaer Polytechnic Institute. For many years he served as a visiting scientist at Los Alamos and Lawrence Livermore National Laboratories. He has published widely in differential equations, mathematical physics, fluid and gas dynamics, hydrogeology, and mathematical biology. Dr. Logan has authored 7 books, among them A First Course in Differential Equations, 2nd ed.,
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Microsoft Releases Math 4.0 Free
Microsoft has released a new version of its math education software Mathematics 4.0, making it available as a free download for the first time.
By Dian Schaffhauser
03/10/11
Microsoft said the new version of its math program has been downloaded 250,000 times since its quiet January 2011 release.
Microsoft Mathematics 4.0, designed for students in middle school, high school, and early college, is intended to teach users how to solve equations while bolstering their understanding of fundamental math and science concepts. Although the company charged for its last version, this latest edition is free.
The new program works on computers running Windows XP, Vista, and 7, as well as Windows Server 2003 and 2008. The software includes a graphing calculator capable of plotting in 2D and 3D, a formulas and equations library, a triangle solver, a unit conversion tool, and ink handwriting support for tablet or ultra-mobile PC use. One new feature enables a user to create a custom movie where a 3D graphed image shifts among multiple shapes as variables change.
An 18-page step-by-step guide provides basic documentation to use the program's functions.
Microsoft Mathematics 4.0 is available now. Further information can be found here
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More About
This Textbook
Overview
A. H. Studenmund's practical introduction to econometrics combines single-equation linear regression analysis with real-world examples and exercises. Using Econometrics: A Practical Guide provides a thorough introduction to econometrics that avoids complex matrix algebra and calculus, making it the ideal text for the beginning econometrics student, the regression user looking for a refresher or the experienced practitioner seeking a convenient 7, 2006
Ideal for practical econometrics
This book is very reader-friendly and hardly requires any prior studies in statistics and calculus it has a whole chapter on essential statistics for beginners. You wont find much elaborate theoretical proofs and such in this book, instead the focus is on giving the reader practical skills to use econometrics immediately. This book gives a solid foundation in econometrics, especially for those who hate theory and elaborate manipulation of equations where calculus is required. If you mainly seek the practical techniques of econometrics to use in everyday business analysis, and leave theory to the academics, then this book is ideal. It gave me a good foundation before doing intermediate and more theoretical econometrics.
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Monday, April 15, 2013
Introduction to Real Analysis 4th Edition, Bartle and Sherbert
Introduction to Real Analysis 4th Edition PDF Download Ebook. Robert G. Bartle and Donald R. Sherbert describe 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.
Several new examples have been added to this edition to make the text more up-to-date and relevant. New exercises have been added throughout to give students more material to practice and solidify their understanding of the material. Coverage of the Darboux integral has been added in Section 7.4. This text offers detailed and rigorous treatment of the basic theory of functions of one real variable.
This book is accessible and user-friendly because concepts are developed in a reasonably paced manner with many examples to illustrate the theory. Every concept is illustrated by examples and special cases. There is also a wide range of exercises, and hints for some of them are provided in the back of the text. Approximation Methods and Numerical Calculation are emphasized whenever appropriate, making this text suitable for computer science students.
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Mathematics Learning Centre
Introduction to Differential Calculus
Christopher Thomas
c 1997 University of Sydney
Acknowledgements
Some parts of this booklet appeared in a similar form in the booklet Review of Differen-
tiation Techniques published by the Mathematics Learning Centre.
I should like to thank Mary Barnes, Jackie Nicholas and Collin Phillips for their helpful
comments.
Christopher Thomas
December 1996
Contents
1 Introduction 1
1.1 An example of a rate of change: velocity . . . . . . . . . . . . . . . . . . . 1
1.1.1 Constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Non-constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Other rates of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 What is the derivative? 6
2.1 Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 The derivative: the slope of a tangent to a graph . . . . . . . . . . . . . . 7
3 How do we find derivatives (in practice)? 9
3.1 Derivatives of constant functions and powers . . . . . . . . . . . . . . . . . 9
3.2 Adding, subtracting, and multiplying by a constant . . . . . . . . . . . . . 12
3.3 The product rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 The Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 The composite function rule (also known as the chain rule) . . . . . . . . . 15
3.6 Derivatives of exponential and logarithmic functions . . . . . . . . . . . . . 18
3.7 Derivatives of trigonometric functions . . . . . . . . . . . . . . . . . . . . . 21
4 What is differential calculus used for? 24
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Optimisation problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2.1 Stationary points - the idea behind optimisation . . . . . . . . . . . 24
4.2.2 Types of stationary points . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.3 Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 The clever idea behind differential calculus (also known as differentiation
from first principles) 31
6 Solutions to exercises 35
Mathematics Learning Centre, University of Sydney 1
1 Introduction
In day to day life we are often interested in the extent to which a change in one quantity
affects a change in another related quantity. This is called a rate of change. For example,
if you own a motor car you might be interested in how much a change in the amount of
fuel used affects how far you have travelled. This rate of change is called fuel consumption.
If your car has high fuel consumption then a large change in the amount of fuel in your
tank is accompanied by a small change in the distance you have travelled. Sprinters are
interested in how a change in time is related to a change in their position. This rate
of change is called velocity. Other rates of change may not have special names like fuel
consumption or velocity, but are nonetheless important. For example, an agronomist
might be interested in the extent to which a change in the amount of fertiliser used on a
particular crop affects the yield of the crop. Economists want to know how a change in
the price of a product affects the demand for that product.
Differential calculus is about describing in a precise fashion the ways in which related
quantities change.
To proceed with this booklet you will need to be familiar with the concept of the slope
(also called the gradient) of a straight line. You may need to revise this concept before
continuing.
1.1 An example of a rate of change: velocity
1.1.1 Constant velocity
Figure 1 shows the graph of part of a motorist's journey along a straight road. The
vertical axis represents the distance of the motorist from some fixed reference point on
the road, which could for example be the motorist's home. Time is represented along the
horizontal axis and is measured from some convenient instant (for example the instant an
observer starts a stopwatch).
Di stance
(metres)
300
200
100
2.00 4.00 6.00 8.00
t ime (seconds)
Figure 1: Distance versus time graph for a motorist's journey.
Mathematics Learning Centre, University of Sydney 2
Exercise 1.1
How far is the motorist in Figure 1 away from home at time t = 0 and at time t = 6?
Exercise 1.2
How far does the motorist travel in the first two seconds (ie from time t = 0 to time t = 2)?
How far does the motorist travel in the two second interval from time t = 3 to t = 5? How far
do you think the motorist would travel in any two second interval of time?
The shape of the graph in Figure 1 tells us something special about the type of motion
that the motorist is undergoing. The fact that the graph is a straight line tells us that the
motorist is travelling at a constant velocity.
• At a constant velocity equal increments in time result in equal changes in distance.
• For a straight line graph equal increments in the horizontal direction result in the
same change in the vertical direction.
In Exercise 1.2 for example, you should have found that in the first two seconds the
motorist travels 50 metres and that the motorist also travels 50 metres in the two seconds
between time t = 3 and t = 5.
Because the graph is a straight line we know that the motorist is travelling at a constant
velocity. What is this velocity? How can we calculate it from the graph? Well, in this
situation, velocity is calculated by dividing distance travelled by the time taken to travel
that distance. At time t = 6 the motorist was 250 metres from home and at time t = 2
the motorist was 150 metres away from home. The distance travelled over the four second
interval from time t = 2 to t = 6 was
distance travelled = 250 − 150 = 100
and the time taken was
time taken = 6 − 2 = 4
and so the velocity of the motorist is
distance travelled 250 − 150 100
velocity = = = = 25 metres per second.
time taken 6−2 4
But this is exactly how we would calculate the slope of the line in Figure 1. Take a look
at Figure 2 where the above calculation of velocity is shown diagramatically.
The slope of a line is calculated by vertical rise divided by horizontal run and if we were
to use the two points (2, 150) and (6, 250) to calculate the slope we would get
rise 250 − 150
slope = = = 25.
run 6−2
To summarise:
The fact that the car is travelling at a constant velocity is reflected in the fact that the
distance-time graph is a straight line. The velocity of the car is given by the slope of this
line.
Mathematics Learning Centre, University of Sydney 3
Di stance
(metres)
300
200 250 - 150 = 100
6 - 2= 4
100
2.00 4.00 6.00 8.00
t ime (seconds)
Figure 2: Calculation of the velocity of the motorist is the same as the calculation of the slope
of the distance - time graph.
1.1.2 Non-constant velocity
Figure 3 shows the graph of a different motorist's journey along a straight road. This
graph is not a straight line. The motorist is not travelling at a constant velocity.
Exercise 1.3
How far does the motorist travel in the two seconds from time t = 60 to time t = 62?
How far does the motorist travel in the two second interval from time t = 62 to t = 64?
Since the motorist travels at different velocities at different times, when we talk about
the velocity of the motorist in Figure 3 we need to specify the particular time that we
mean. Nevertheless we would still like somehow to interpret the velocity of the motorist
as the slope of the graph, even though the graph is curved and not a straight line.
Distance in metres
1080
1060
1040
1020
59 60 61 62 63 64 65
Time in seconds
Figure 3: Position versus time graph for a motorist's journey.
Mathematics Learning Centre, University of Sydney 4
What do we mean by the slope of a curve? Suppose for example that we are interested
in the velocity of the motorist in Figure 3 at time t = 62. In Figure 3 we have drawn in
a dashed line. Notice that this line just grazes the curve at the point on the curve where
t = 62. The dashed line is in fact the tangent to the curve at that point. We will talk
more about tangents to curves in Section 2. For now you can think of the dashed line
like this: if you were going to draw a straight line through this point on the curve, and if
you wanted that straight line to look as much like the curve near that point as it possibly
could, this is the line that you would draw. This solves our problem about interpreting
the slope of the curve at this point on the curve.
The slope of the curve at the point on the curve where t = 62 is the slope of the tangent
to the curve at that point: that is the slope of the dashed line in Figure 3.
The velocity of the motorist at time t = 62 is the slope of the dashed line in that figure.
Of course if we were interested in the velocity of the motorist at time t = 64 then we
would draw the tangent to the curve at the point on the curve where t = 64 and we would
get a different slope. At different points on the curve we get different tangents having
different slopes. At different times the motorist is travelling at different velocities.
1.2 Other rates of change
The situation above described a car moving in one direction along a straight road away
from a fixed point. Here, the word velocity describes how the distance changes with time.
Velocity is a rate of change. For these type of problems, the velocity corresponds to the
rate of change of distance with respect to time. Motion in general may not always be in
one direction or in a straight line. In this case we need to use more complex techniques.
Velocity is by no means the only rate of change that we might be interested in. Figure 4
shows a graph representing the yield a farmer gets from a crop depending on the amount
of fertiliser that the farmer uses.
The shape of this graph makes good sense. If no fertiliser is used then there is still some
crop yield (50 tonnes to be precise). As more fertiliser is used the crop yield increases,
Crop Yield (Tonnes )
200
slope = 50
s lope = 2 5
150
100
50
1 2 3 4
Fer ti l i ser Useage (Tonnes )
Figure 4: Crop yield versus fertiliser useage for a hypothetical crop.
Mathematics Learning Centre, University of Sydney 5
as you would expect. Note though that at a certain point putting on more fertiliser does
not improve the yield of the crop, but in fact decreases it. The soil is becoming poisoned
by too much fertiliser. Eventually the use of too much fertiliser causes the crop to die
altogether and no yield is obtained.
On the graph the tangents to the curve corresponding to fertiliser usage of 1 tonne (the
dotted line) and of 1.5 tonnes (the dashed line) are drawn. The slope of these tangents
give the rate of change of crop yield with respect to fertiliser usage.
The slope of the dotted tangent is 50. This means that if fertiliser usage is increased from
1 tonne by a very small amount then the crop yield will increase by 50 times that small
change. For example an increase in fertiliser usage from 1 tonne (1000 kg) to 1005 kg
will increase the crop yield by approximately 50 × 5 = 250 kg. If we are using 1 tonne
of fertiliser then the rate of change of crop yield with respect to fertiliser useage is quite
high. On the other hand the slope of the dashed tangent is 25. The same increase (by 5
kg) in fertiliser useage from 1500 kg (1.5 tonnes) to 1505 kg will increase the crop yield
by about 25 × 5 = 125 kg.
Mathematics Learning Centre, University of Sydney 6
2 What is the derivative?
If you are not completely comfortable with the concept of a function and its graph then
you need to familiarise yourself with it before continuing. The booklet Functions published
by the Mathematics Learning Centre may help you.
In Section 1 we learnt that differential calculus is about finding the rates of change of
related quantities. We also found that a rate of change can be thought of as the slope of
a tangent to a graph of a function. Therefore we can also say that:
Differential calculus is about finding the slope of a tangent to the graph of a function, or
equivalently, differential calculus is about finding the rate of change of one quantity with
respect to another quantity.
If we are going to go to all this trouble to find out about the slope of a tangent to a graph,
we had better have a good idea of just what a tangent is.
2.1 Tangents
Look at the curve and straight line in Figure 5.
A
B
Figure 5: The line is tangent to the curve at point A but not at point B.
Imagine taking a very powerful magnifying glass and looking very closely at this figure near
the point A. Figure 6 shows two views of this curve at successively greater magnifications.
The closer we look at the curve near the point A the straighter the curve appears to be.
The more we zoom in the more the curve begins to look like the straight line. This straight
line is called the tangent to the curve at the point A. If we want to draw a straight line
that most resembles the curve near the point A, the tangent line is the one that we would
draw. It is pretty clear from Figure 5 that no matter how closely we look at the curve
near the point B the curve is never going to look like the straight line we have drawn in
here. That line is tangent to the curve at A but not at B. The curve does have a tangent
at B, but it is not shown on Figure 5.
Note that it is not necessarily true that the tangent line only cuts the curve at one point
or that curve lies entirely on one side of the line. These properties hold for some special
curves like circles, but not for all curves, and certainly not for the one in Figure 5.
Mathematics Learning Centre, University of Sydney 7
A
A
Figure 6: Two close up views of the curve in Figure 5 near the point A. The closer we look
near the point A the more the curve looks like the tangent.
2.2 The derivative: the slope of a tangent to a graph
Terminology The slope of the tangent at the point (x, f (x)) on the graph of f is called
the derivative of f at x and is written f (x).
Look at the graph of the function y = f (x) in Figure 7. Three different tangent lines have
B (0.5, f (0.5))
A (−0.5, f (−0.5))
C (1.5, f (1.5))
−0.5 0.5 1.0 1.5 x
Figure 7: Tangent lines to the graph of f (x) drawn at three different points on the graph.
been drawn on the graph, at A, B and C, corresponding to three different values of the
independent variable, x = −0.5, x = 0.5 and x = 1.5.
If we were to make careful measurements of the slopes of the three tangents shown we
would find that f (−0.5) ≈ 2.75, f (0.5) ≈ −1.25 and f (1.5) ≈ 0.75. Here the symbol
≈ means 'is approximately'. We can only say approximately here because there is no
way that we can make completely accurate measurements from a graph, and no way
even to draw a completely accurate graph. However this graphical approach to finding
the approximate derivative is often very useful, and in some situations may be the only
technique that we have.
At different points on the graph we get different tangents having different slopes. The
slope of the tangent to the graph depends on where on the graph we draw the tangent.
Because we can specify a point on the graph by just giving its x coordinate (the other
Mathematics Learning Centre, University of Sydney 8
coordinate is then f (x)), we can say that the slope of the tangent to the graph of a function
depends on the value of the independent variable x, or the value of f (x) depends on x.
In other words, f is a function of x.
Terminology The function f is called the derivative of f .
Terminology The process of finding the derivative is called differentiation.
The derivative of a a function f is another function, called f , which tells us about the
slopes of tangents to the graph of f . Because there are several different ways of writing
functions, there are several different ways of writing the derivative of a function. Most of
the ways that are commonly used are expressed in the following table.
Function Derivative
df (x)
f (x) f (x) or dx
df
f f or dx
dy
y y or dx
dy(x)
y(x) y (x) or dx
Exercise 2.1 (You will find this exercise easier to do if you use graph paper.)
Draw a careful graph of the function f (x) = x2 . Draw the tangents at the points x = 1,
x = 0 and x = −0.5. Find the slopes of these lines by picking two points on them and
using the formula
y2 − y1
slope = .
x2 − x1
These slopes are the (approximate) values of f (1), f (0) and f (−0.5) respectively.
Exercise 2.2
Repeat Exercise 2.1 with the function f (x) = x3 .
Mathematics Learning Centre, University of Sydney 9
3 How do we find derivatives (in practice)?
Differential calculus is a procedure for finding the exact derivative directly from the for-
mula of the function, without having to use graphical methods. In practise we use a few
rules that tell us how to find the derivative of almost any function that we are likely to
encounter. In this section we will introduce these rules to you, show you what they mean
and how to use them.
Warning! To follow the rest of these notes you will need feel comfortable manipulating
expressions containing indices. If you find that you need to revise this topic you may find
the Mathematics Learning Centre publication Exponents and Logarithms helpful.
3.1 Derivatives of constant functions and powers
Perhaps the simplest functions in mathematics are the constant functions and the func-
tions of the form xn .
d
Rule 1 If k is a constant then k = 0.
dx
d n
Rule 2 If n is any number then x = nxn−1 .
dx
Rule 1 at least makes sense. The graph of a constant function is a horizontal line and a
horizontal line has slope zero. The derivative measures the slope of the tangent, and so
the derivative is zero.
How you approach Rule 2 is up to you. You certainly need to know it and be able to use
it. However we have given no justification for why Rule 2 works! In fact in these notes we
will give little justification for any of the rules of differentiation that are presented. We
will show you how to apply these rules and what you can do with them, but we will not
make any attempt to prove any of them.
Examples If f (x) = x7 then f (x) = 7x6 .
If y = x−0.5 then dy
dx
= −0.5x−1.5 .
d −3
x = −3x−4 .
dx
If g(x) = 3.2 then g (x) = 0.
If f (t) = t 2 then f (t) = 1 t− 2 .
1 1
2
If h(u) = −13.29 then h (u) = 0.
Mathematics Learning Centre, University of Sydney 10
In the examples above we have used Rules 1 and 2 to calculate the derivatives of many
simple functions. However we must not lose sight of what it is that we are calculating
here. The derivative gives the slope of the tangent to the graph of the function.
For example, if f (x) = x2 then f (x) = 2x. To find the slope of the tangent to the graph
of x2 at x = 1 we substitute x = 1 into the derivative. The slope is f (1) = 2 × 1 = 2.
Similarly the slope of the tangent to the graph of x2 at x = −0.5 is found by substituting
x = −0.5 into the derivative. The slope is f (−0.5) = 2 × −0.5 = −1. This is illustrated
in Figure 8.
3.00
2.00
1.00 ( 1, 1 )
( -0.5, 0.2 5 ) slope = f ' (1) = 2
slope = f ' (-0.5) = - 1
-1.00 -0.50 0.50 1.00 1.50
Figure 8: Slopes of tangents to the graph of y = x2 .
Example
Find the slope of the tangent to the graph of the function g(t) = t4 at the point on the
graph where t = −2.
Solution
The derivative is g (t) = 4t3 , and so the slope of the tangent line at t = −2 is g (−2) =
4 × (−2)3 = −32.
Example
1
Find the equation of the line tangent to the graph of y = f (x) = x 2 at the point x = 4.
Solution
1 √
f (4) = 4 2 = 4 = 2, so the coordinates of the point on the graph are (4, 2). The
derivative is
x− 2
1
1
f (x) = = √
2 2 x
and so the slope of the tangent line at x = 4 is f (4) = 1 . We therefore know the slope of
4
the line and we know one point through which the line passes.
Any non vertical line has equation of the form y = mx + b where m is the slope and b the
vertical intercept.
Mathematics Learning Centre, University of Sydney 11
In this case the slope is 1 , so m = 1 , and the equation is y =
4 4
x
4
+ b. Because the line
passes through the point (4, 2) we know that y = 2 when x = 4.
4 x
Substituting we get 2 = 4
+ b, so that b = 1. The equation is therefore y = 4
+ 1.
Notice that in the examples above the independent variable is not always called x. We
have also used u and t, and in fact we can and will use many different letters for the
independent variable. Notice also that we might not stick to the symbol f to stand
for function. Many other symbols are used. Some of the common ones are g and h.
Throughout this booklet we will use a variety of symbols for functions and variables to
get you used to the fact that our choice of symbols makes no difference to the ideas that
we are introducing. On the other hand, we can make life easier for ourselves if we make
sensible choices of symbols. For example if we were discussing the revenue obtained by
a manufacturer who sells articles for a certain price it might be sensible for us to choose
the symbol p to mean price, and r to mean revenue, and to write r(p) to express the
fact that the revenue is a function of the price. In this way the symbols we have chosen
remind us of their meaning, much more than if we had chosen x to represent price and f
to represent revenue and written f (x). On the other hand, because the symbol d has a
(x)
special use in calculus, to express the derivative dfdx , we almost never use d for any other
purpose. For this reason you will often see the letter s used to represent diSplacement.
We now know how to differentiate any function that is a power of the variable. Examples
are functions like x3 and t−1.3 . You will come across functions that do not at first appear
to be a power of the variable, but can be rewritten in this form. One of the simplest
examples is the function √
f (t) = t,
which can also be written in the form
1
f (t) = t 2 .
The derivative is then
t− 2
1
1
f (t) = = √ .
2 2 t
Similarly, if
1
h(s) = = s−1
s
then
1
h (s) = −s−2 = − .
s2
Examples
1 1 4
If f (x) = √ = x− 3 then f (x) = − x− 3 .
1
3
x 3
1 dy 3 5
If y = √ = x− 2 then = − x− 2 .
3
x x dx 2
Mathematics Learning Centre, University of Sydney 12
Exercises 3.1
Differentiate the following functions:
a. f (x) = x4 b. y = x−7 c. f (u) = u2.3
d. f (t) = t− 3 g(z) = z − 2
1 22 3
e. f (t) = t 7 f.
3
g. y = t−3.8 h. z = x 7
Exercise 3.2
Express the following as powers and then differentiate:
1 √ √
a. b. t t c. 3 x
x 2
√
1 1 s3 s
d. √ e. √ f. √
x2 x x4x 3
s
√
1 t 1 x
g. 3
h. 2 √ i. x 2
u t t x
Exercise 3.3
√
Find the equation of the line tangent to the graph of y = 3 x when x = 8.
3.2 Adding, subtracting, and multiplying by a constant
So far we know how to differentiate powers of the independent variable. Many of the
functions that you will encounter are made up in simple ways from powers. For example,
a function like 3x2 is just a constant multiple of x2 . However neither Rule 1 nor Rule 2
tell us how to differentiate 3x2 . Nor do they tell us how to differentiate something like
x2 + x3 or x2 − x3 .
Rules 3 and 4 specify how to differentiate combinations of functions that are formed by
multiplying by constants, or by adding or subtracting functions.
Rule 3 If f (x) = cg(x), where c is a constant, then f (x) = cg (x).
Rule 4 If f (x) = g(x) ± h(x) then f (x) = g (x) ± h (x).
d 2
Examples If f (x) = 3x2 then f (x) = 3 × x = 6x.
dx
d 2 d
If g(t) = 3t2 + 2t−2 then g (t) = 3t + 2t−2 = 6t − 4t−3 .
dt dt
√
− 2x 3 x = 3x− 2 − 2x 3 then = − 3 x− 2 − 8 x 3 .
3 1 4 dy 3 1
If y = √
x dx 2 3
If y = −0.3x−0.4 then dy
dx
= 0.12x−1.4 .
d
dx
2x0.3 = 0.6x−0.7 .
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Warning! Although Rule 4 tells us that dx (f (x) ± g(x)) = f (x) ± g (x), the same is not
d
true for multiplication or division. To differentiate f (x) × g(x) or f (x) ÷ g(x) we cannot
simply find f (x) and g (x) and multiply or divide them. Be careful of this! The methods
of differentiating products of functions or quotients of functions are discussed in Sections
3.3 and 3.4.
Exercise 3.4
Differentiate the following functions:
√
a. f (x) = 5x2 − 2 x b. y = 2x−7 + 3
x2
c. f (t) = 2.5t2.3 + √t
t
d. h(z) = z − 3 + 5z
1 5
e. f (u) = u 3 − 3u−7 f. g(z) = 8z −2 − 5
z
h. z = 4x 7 + 2x− 2
1 1
g. y = 5t−8 + √t
t
3.3 The product rule
Another way of combining functions to make new functions is by multiplying them to-
gether, or in other words by forming products. The product rule tells us how to differen-
tiate functions like this.
Rule 5 (The product rule) If f (x) = u(x)v(x) then
f (x) = u(x)v (x) + u (x)v(x).
Examples
If y = (x + 2)(x2 + 3) then y = (x + 2)2x + 1(x2 + 3).
√ √
x(3x2 − 6x) + 1 x− 2 (x3 − 3x2 + 7).
1
If f (x) = x(x3 − 3x2 + 7) then f (x) = 2
√ √
= (t2 + 3)( 1 t− 2 + 3t2 ) + 2t( t + t3 ).
dz 1
If z = (t2 + 3)( t + t3 ) then dt 2
Mathematics Learning Centre, University of Sydney 14
Exercise 3.5
Use the product rule to differentiate the functions below:
a. f (x) = (4x3 + 2)(1 − 3x)
b. g(x) = (x2 + x + 2)(x2 + 1)
c. h(x) = (3x3 − 2x2 + 8x − 5)(x2 − 2x + 4)
1 2
d. f (s) = (1 − s )(3s + 5)
2
√ 1
e. g(t) = ( t + )(2t − 1)
t
√
f. h(y) = (2 − y + y 2 )(1 − 3y 2 )
Exercise 3.6
1
If r = (t + )(t2 − 2t + 1), find the rate of change of r with respect to t when t = 2.
t
Exercise 3.7
Find the slope of the tangent to the curve y = (x2 − 2x + 1)(3x3 − 5x2 + 2) at x = 2.
3.4 The Quotient Rule
This rule allows us to differentiate functions which are formed by dividing one function
by another, ie by forming quotients of functions. An example is such as
2x + 3
f (x) = .
3x − 5
Rule 6 (The quotient rule)
u(x)
f (x) =
v(x)
v(x)u (x) − u(x)v (x)
f (x) =
[v(x)]2
vu − uv
= .
v2
Warning! Because of the minus sign in the numerator (ie in the top line) it is important
to get the terms in the numerator in the correct order. This is often a source of mistakes,
so be careful. Decide on your own way of remembering the correct order of the terms.
Mathematics Learning Centre, University of Sydney 15
Examples
2x2 + 3x dy (x3 + 1)(4x + 3) − (2x2 + 3x)3x2
If y = , then = .
x3 + 1 dx (x3 + 1)2
√
( t + 1)(2t + 3) − (t2 + 3t + 1)( 1 t− 2 )
1
t2 + 3t + 1
If g(t) = √ then g (t) = √ 2
.
t+1 ( t + 1)2
Exercise 3.8
Use the Quotient Rule to find derivatives for the following functions:
x−1 2x + 3
a. f (x) = b. g(x) =
x+1 3x − 2
x2 + 2 2t
c. h(x) = d. f (t) =
x2 + 5 1 + 2t2
√
1+ s x2 − 1
e. f (s) = √ f. h(x) =
1− s x3 + 4
u3 + u − 4 t(t + 6)
g. f (u) = h. g(t) =
3u4 + 5 t2 + 3t + 1
3.5 The composite function rule (also known as the chain rule)
Have a look at the function f (x) = (x2 + 1)17 . We can think of this function as being
the result of combining two functions. If g(x) = x2 + 1 and h(t) = t17 then the result of
substituting g(x) into the function h is
h(g(x)) = (g(x))17 = (x2 + 1)17 .
Another way of representing this would be with a diagram like
g h
x −→ x2 + 1 −→ (x2 + 1)17 .
We start off with x. The function g takes x to x2 + 1, and the function h then takes
x2 + 1 to (x2 + 1)17 . Combining two (or more) functions like this is called composing the
functions, and the resulting function is called a composite function. For a more detailed
discussion of composite functions you might wish to refer to the Mathematics Learning
Centre booklet Functions.
Using the rules that we have introduced so far, the only way to differentiate the function
f (x) = (x2 + 1)17 would involve expanding the expression and then differentiating. If the
function was (x2 + 1)2 = (x2 + 1)(x2 + 1) then it would not take too long to expand these
two sets of brackets. But to expand the seventeen sets of brackets involved in the function
f (x) = (x2 + 1)17 (or even to expand using the binomial theorem) would take a long time.
The composite function rule shows us a quicker way.
Rule 7 (The composite function rule (also known as the chain rule))
If f (x) = h(g(x)) then f (x) = h (g(x)) × g (x).
Mathematics Learning Centre, University of Sydney 16
In words: differentiate the 'outside' function, and then multiply by the derivative of the
'inside' function.
To apply this to f (x) = (x2 + 1)17 , the outside function is h(·) = (·)17 and its derivative
is 17(·)16 . The inside function is g(x) = x2 + 1 which has derivative 2x. The composite
function rule tells us that f (x) = 17(x2 + 1)16 × 2x.
As another example let us differentiate the function 1/(z 3 + 4z 2 − 3z − 3)6 . This can be
rewritten as (z 3 + 4z 2 − 3z − 3)−6 . The outside function is (·)−6 which has derivative
−6(·)−7 . The inside function is z 3 + 4z 2 − 3z − 3 with derivative 3z 2 + 8z − 3. The chain
rule says that
d 3
(z + 4z 2 − 3z − 3)−6 = −6(z 3 + 4z 2 − 3z − 3)−7 × (3z 2 + 8z − 3).
dz
There is another way of writing down, and hence remembering, the composite function
rule.
Rule 7 (The composite function rule (alternative formulation))
If y is a function of u and u is a function of x then
dy dy du
= × .
dx du dx
dy
This makes the rule very easy to remember. The expressions du and du are not really
dx
fractions but rather they stand for the derivative of a function with respect to a variable.
However for the purposes of remembering the chain rule we can think of them as fractions,
dy
so that the du cancels from the top and the bottom, leaving just dx .
To use this formulation of the rule in the examples above, to differentiate y = (x2 + 1)17
put u = x2 + 1, so that y = u17 . The alternative formulation of the chain rules says that
dy dy du
= ×
dx du dx
= 17u16 × 2x
= 17(x2 + 1)16 × 2x.
which is the same result as before. Again, if y = (z 3 + 4z 2 − 3z − 3)−6
then set u = z 3 + 4z 2 − 3z − 3 so that y = u−6 and
dy dy du
= ×
dx du dx
= −6u−7 × (3z 2 + 8z − 3).
You select the formulation of the chain rule that you find easiest to use. They are equiv-
alent.
Mathematics Learning Centre, University of Sydney 17
Example
Differentiate (3x2 − 5)3 .
Solution
The first step is always to recognise that we are dealing with a composite function and
then to split up the composite function into its components. In this case the outside
function is (·)3 which has derivative 3(·)2 , and the inside function is 3x2 − 5 which has
derivative 6x, and so by the composite function rule,
d(3x2 − 5)3
= 3(3x2 − 5)2 × 6x = 18x(3x2 − 5)2 .
dx
Alternatively we could first let u = 3x2 − 5 and then y = u3 . So
dy dy du
= × = 3u2 × 6x = 18x(3x2 − 5)2 .
dx du dx
Example
dy
√
Find dx
if y = x2 + 1.
Solution
√
· = (·) 2 which has derivative 1 (·)− 2 , and the inside function is
1 1
The outside function is 2
x2 + 1 so that
1
y = (x2 + 1)− 2 × 2x.
1
2
√ 1
Alternatively, if u = x2 + 1, we have y = u = u 2 . So
dy 1 1
= u− 2 × 2x = (x2 + 1)− 2 × 2x.
1 1
dx 2 2
Exercise 3.9
Differentiate the following functions using the composite function rule.
a. (2x + 3)2 b. (x2 + 2x + 1)12 c. (3 − x)21
√
d. (x3 − 1)5 e. f (t) = t2 − 5t + 7 f. g(z) = √ 1
2−z 4
√ 3
g. y = (t3 − t)−3.8 h. 1
z = (x + x ) 7
Mathematics Learning Centre, University of Sydney 18
Exercise 3.10
Differentiate the functions below. You will need to use both the composite function rule
and the product or quotient rule.
a. (x + 2)(x + 3)2 b. (2x − 1)2 (x + 3)3 c. x (1 − x)
x
√
1 2
d. x 3 (1 − x) 3 e.
1 − x2
3.6 Derivatives of exponential and logarithmic functions
If you are not familiar with exponential and logarithmic functions you may wish to consult
the booklet Exponents and Logarithms which is available from the Mathematics Learning
Centre.
You may have seen that there are two notations popularly used for natural logarithms,
loge and ln. These are just two different ways of writing exactly the same thing, so that
loge x ≡ ln x. In this booklet we will use both these notations.
The basic results are:
d x
e = ex
dx
d 1
(loge x) = .
dx x
We can use these results and the rules that we have learnt already to differentiate functions
which involve exponentials or logarithms.
Example
Differentiate loge (x2 + 3x + 1).
Solution
We solve this by using the chain rule and our knowledge of the derivative of loge x.
d d
loge (x2 + 3x + 1) = (loge u) (where u = x2 + 3x + 1)
dx dx
d du
= (loge u) × (by the chain rule)
du dx
1 du
= ×
u dx
1 d
= 2 × (x2 + 3x + 1)
x + 3x + 1 dx
1
= 2 × (2x + 3)
x + 3x + 1
2x + 3
= 2 .
x + 3x + 1
Mathematics Learning Centre, University of Sydney 19
Example
d 2
Find dx (e3x ).
Solution
This is an application of the chain rule together with our knowledge of the derivative of
ex .
d 3x2 deu
(e ) = where u = 3x2
dx dx
deu du
= × by the chain rule
du dx
du
= eu ×
dx
2 d
= e3x × (3x2 )
dx
2
= 6xe3x .
Example
d 3
Find dx (ex +2x ).
Solution
Again, we use our knowledge of the derivative of ex together with the chain rule.
d x3 +2x deu
(e ) = (where u = x3 + 2x)
dx dx
du
= eu × (by the chain rule)
dx
3 +2x d 3
= ex ×(x + 2x)
dx
3
= (3x2 + 2) × ex +2x .
Example
Differentiate ln (2x3 + 5x2 − 3).
Solution
We solve this by using the chain rule and our knowledge of the derivative of ln x.
d d ln u
ln (2x3 + 5x2 − 3) = (where u = (2x3 + 5x2 − 3)
dx dx
d ln u du
= × (by the chain rule)
du dx
1 du
= ×
u dx
1 d
= 3 + 5x2 − 3
× (2x3 + 5x2 − 3)
2x dx
1
= × (6x2 + 10x)
2x3 + 5x2 − 3
6x2 + 10x
= .
2x3 + 5x2 − 3
Mathematics Learning Centre, University of Sydney 20
There are two shortcuts to differentiating functions involving exponents and logarithms.
The four examples above gave
d 2x + 3
(loge (x2 + 3x + 1)) = 2
dx x + 3x + 1
d 3x2 2
(e ) = 6xe3x
dx
d x3 +2x 2
(e ) = (3x2 + 2)e3x
dx
d 6x2 + 10x
(loge (2x3 + 5x2 − 3)) = .
dx 2x3 + 5x2 − 3
These examples suggest the general rules
d f (x)
(e ) = f (x)ef (x)
dx
d f (x)
(ln f (x)) = .
dx f (x)
x ln 1
These rules arise from the chain rule and the fact that de = ex and d dxx = x . They can
dx
speed up the process of differentiation but it is not necessary that you remember them.
If you forget, just use the chain rule as in the examples above.
Exercise 3.11
Differentiate the following functions.
7
a. f (x) = ln(2x3 ) b. f (x) = ex c. f (x) = ln(11x7 )
f (x) = loge (7x−2 ) f. f (x) = e−x
2 +x3
d. f (x) = ex e.
x2 + 1
g. f (x) = ln(ex + x3 ) h. f (x) = ln(ex x3 ) i. f (x) = ln
x3 − x
Mathematics Learning Centre, University of Sydney 21
3.7 Derivatives of trigonometric functions
To understand this section properly you will need to know about trigonometric functions.
The Mathematics Learning Centre booklet Introduction to Trigonometric Functions may
be of use to you.
There are only two basic rules for differentiating trigonometric functions:
d
sin x = cos x
dx
d
cos x = − sin x.
dx
For differentiating all trigonometric functions these are the only two things that we need
to remember.
Of course all the rules that we have already learnt still work with the trigonometric
functions. Thus we can use the product, quotient and chain rules to differentiate functions
that are combinations of the trigonometric functions.
sin x
For example, tan x = cos x
and so we can use the quotient rule to calculate the derivative.
sin x
f (x) = tan x = ,
cos x
cos x.(cos x) − sin x.(− sin x)
f (x) =
(cos x)2
cos2 x + sin2 x 1
= = (since cos2 x + sin2 x = 1)
cos x cos2 x
= sec2 x
Note also that
cos2 x + sin2 x cos2 x sin2 x
2x
= 2x
+ 2x
= 1 + tan2 x
cos cos cos
so it is also true that
d
tan x = sec2 x = 1 + tan2 x.
dx
Mathematics Learning Centre, University of Sydney 22
Example
Differentiate f (x) = sin2 x.
Solution
f (x) = sin2 x is just another way of writing f (x) = (sin x)2 . This is a composite function,
with the outside function being (·)2 and the inside function being sin x.
By the chain rule, f (x) = 2(sin x)1 × cos x = 2 sin x cos x. Alternatively using the other
method and setting u = sin x we get f (x) = u2 and
df (x) df (x) du du
= × = 2u × = 2 sin x cos x.
dx du dx dx
Example
Differentiate g(z) = cos(3z 2 + 2z + 1).
Solution
Again you should recognise this as a composite function, with the outside function being
cos(·) and the inside function being 3z 2 + 2z + 1. By the chain rule g (z) = − sin(3z 2 +
2z + 1) × (6z + 2) = −(6z + 2) sin(3z 2 + 2z + 1).
Example
et
Differentiate f (t) = sin t
.
Solution
By the quotient rule
et sin t − et cos t et (sin t − cos t)
f (t) = = .
sin2 t sin2 t
Example
Use the quotient rule or the composite function rule to find the derivatives of cot x, sec x,
and cosec x.
Solution
These functions are defined as follows:
cos x
cot x =
sin x
1
sec x =
cos x
1
csc x = .
sin x
Mathematics Learning Centre, University of Sydney 23
By the quotient rule
d cot x − sin2 x − cos2 x −1
= 2 = .
dx sin x sin2 x
Using the composite function rule
d sec x d(cos x)−1 sin x
= = −(cos x)−2 × (− sin x) = .
dx dx cos2 x
d csc x d(sin x)−1 cos x
= = −(sin x)−2 × cos x = − 2 .
dx dx sin x
Exercise 3.12
Differentiate the following:
a. cos 3x b. sin(4x + 5) c. sin3 x d. sin x cos x e. x2 sin x
sin x 1 √ 1 1
f. cos(x2 + 1) g. h. sin i. tan( x) j. sin
x x x x
Mathematics Learning Centre, University of Sydney 24
4 What is differential calculus used for?
4.1 Introduction
The development of mathematics stands as one of the most important achievements of
humanity, and the development of the calculus, both the differential calculus and integral
calculus is one of most important achievements in mathematics. The practical applications
of differential calculus are so wide ranging that it would be impossible to mention them
all here. Suffice to say that differential calculus is an indispensable tool in every branch
of science and engineering.
In elementary mathematics there are two main applications of differential calculus. One is
to help in sketching curves, and the other is in optimisation problems. For a treatment of
the uses of calculus in curve sketching see the Mathematics Learning Centre publication
Curve Sketching. In this section we will give a brief introduction to how differential
calculus is used in optimisation problems.
4.2 Optimisation problems
There are many practical situations in which we would like to make a quantity as small
as we possibly can or as large as we possibly can. For example, a manufacturer of bicycles
trying to decide how much to charge for a model of bicycle would think that if he charges
too little for the bicycles then he will probably sell a lot of bicycles but that he won't
make much profit because the price is too low, and that if he charges too much for the
bicycle then he won't make much profit because not many people will buy his bicycles.
The manufacturer would like to find just the right price to charge to maximise his profit.
Similarly a farmer might realise that if she uses too little fertiliser on her crops then her
yield will be very low, and if she uses too much fertiliser then she will poison the soil and
her yield will be low. The farmer might like to know just how much fertiliser to use to
maximise the crop yield. A manufacturer of sheet metal cans that are meant to hold one
litre of liquid might like to know just what shape to make the can so that the amount of
sheet metal that is used is a minimum. These are all examples of optimisation problems.
If we were to draw a graph of the profit versus price for the bicycle manufacturer mentioned
above then finding the maximum profit is equivalent to finding the highest point on the
graph. Similarly a minimisation problem may be thought of geometrically as finding the
lowest point on the graph of a funcion.
4.2.1 Stationary points - the idea behind optimisation
As a thought experiment, let us imagine that a person wearing a blindfold is walking
along a road, and that the road has a hill on it. Let us imagine also that the blindfolded
person is searching for the highest point on the road. How would this person be able to
decide when they were at the top of the hill? Well, while they were walking uphill the
person would know that this wasn't the top of the hill - because they are still going up!
And of course while they are walking downhill they would know that they are not at the
top of the hill because they are going down. In other words, while they are on a sloping
bit of the road the blindfolded person would know that this is not the top of the hill.
Mathematics Learning Centre, University of Sydney 25
Right at the top of the hill there would be a little bit of level road. The slope at the
top of the hill would be zero! Without even being able to see the road, the blindfolded
person would know that they could not possibly be at the top of the hill unless they were
standing on level ground. The same idea would apply if the road had a valley in it, and
the person was searching for the lowest piece of road. Right at the lowest point of the
valley the slope of the road would be zero.
So if we are searching for the highest (or lowest) point on a road, of all the possible
places we only have to consider those places where the road has slope zero. This is the
idea behind using calculus for optimisation. If we are searching for the highest or lowest
points on the graph of a function we have to look for those places where the graph has
slope zero. These points are called stationary points.
Definition For a function y = f (x) the points on the graph where the graph has zero
slope are called stationary points. In other words stationary points are where f (x) = 0.
To find the stationary points of a function we differentiate, set the derivative equal to
zero and solve the equation.
Example Find the stationary points of the function f (x) = 2x3 + 3x2 − 12x + 17.
Solution f (x) = 6x2 + 6x − 12. Setting f (x) = 0 and solving we obtain
6x2 + 6x − 12 = 0
x2 + x − 2 = 0
(x − 1)(x + 2) = 0
x = 1, −2.
This gives us the values of x for which the function f is stationary. The corresponding
values of the function are found by substituting 1 and −2 into the function.
They are f (1) = 2 × 13 + 3 × 12 − 12 × 1 + 17 = 10 and
f (−2) = 2 × (−2)3 + 3 × (−2)2 − 12 × (−2) + 17 = 37. The stationary points are therefore
(1, 10) and (−2, 37).
2
Example Find the stationary points of the function g(t) = et .
Solution Differentiating and setting the derivative equal to zero we obtain the equa-
2 2
tion g (t) = 2tet = 0. Since et is never zero, the only solution to this equation is
where 2t = 0, ie t = 0. Substituting into the formula for g we obtain the function value
2
g(0) = e0 = 1. Thus the stationary point is (0, 1).
4.2.2 Types of stationary points
In our thought experiment above we mentioned two types of stationary points: one was
the top of the hill and the other was the bottom of the valley. The top of the hill is called
a local maximum, and the bottom of the valley is called a local minimum. The word
'local' conveys the fact that at the top of the hill the blindfolded person is not necessarily
at the highest point in the world, but merely at the highest point in the local vicinity.
Sometimes you will see local maxima and local minima called relative maxima and relative
Mathematics Learning Centre, University of Sydney 26
local ma x imum
local minimum
Figure 9: Graph of a function showing a local maximum and a local minimum.
minima. Figure 9 shows a function with a local maximum and a local minimum. Note
that at each of these points the slope of the curve is zero.
Local maxima and local minima are not the only types of stationary points. There is a
third kind. Figure 10 shows a stationary point that is neither a local maximum nor a
local minimum. This type of stationary point is called a stationary point of inflection.
Don't worry about why it is given this name. That is beyond the scope of this booklet.
You just need to be aware of the fact that stationary points exist that are neither local
maxima nor local minima.
s t ationary point of inflection
Figure 10: Graph of a function showing a stationary point of inflection.
Let us now return to the first of the examples in the previous section. We found that
the function f (x) = 2x3 + 3x2 − 12x + 17 had stationary points at (1, 10) and (−2, 37).
What type of stationary points are they? At the moment you probably have no idea just
what the graph of f (x) = 2x3 + 3x2 − 12x + 17 looks like. How can you tell what type
of stationary points these are? If you could see the graph you would be able to tell what
types of stationary points they were, but it takes a lot of work to draw the graph of a
function. What we need is a way of testing a stationary point that will tell us whether we
have found a local maximium, a local minimum or neither (in other words a stationary
point of inflection) without drawing the graph. There are several ways of doing this, but
in this booklet we will look at only one of them. This is called the first derivative test.
Mathematics Learning Centre, University of Sydney 27
Really we are pretty much in the shoes of the blindfolded person now. We can't see the
whole graph, so how can we tell what type of stationary point we have got? Imagine the
blindfolded person standing on a piece of level ground, and wanting to know whether this
was the top of a hill (a local maximum), the bottom of a valley (a local minimum) or
neither (a stationary point of inflection). One thing the person could do is take a step
backwards from the level spot. Which way does the ground slope here? And then take a
step forwards from the level spot. Which way does the ground slope here? If the person
took a step backward and found that the ground in front of them sloped up, then returned
to the original position and took a step forward and the ground sloped down, then the
level spot must have been the top of the hill. On the other hand if the person took a step
backward and the ground sloped down, and a step forward and the ground sloped up,
then the level spot must have been the bottom of a valley. You should be able to figure
out what the blindfolded person would find for a stationary point of inflection. This idea
is the basis of the first derivative test.
The first derivative test
If x0 is a stationary point of the function f , so that f (x0 ) = 0 then to find out the nature
of the stationary point check the sign (ie positive or negative) of f just either side of x0 .
If f < 0 to the left of x0 (ie for x < x0 ) and f > 0 to the right of x0 (ie for x > x0 ) then
x0 is a local minimum. If f > 0 to the left of x0 and f < 0 to the right of x0 then x0 is a
local maximum. Otherwise x0 is a stationary point of inflection. Have a look at Figures
11 and 12.
local ma x imum f' = 0
f'>0 f'<0
f' <0 f' > 0
f' =0
local m inimum
Figure 11: First derivative test for local minima and local maxima.
f' >0
f' = 0
f' >0
s t ationary point of inflection
Figure 12: First derivative test for point of inflection.
Mathematics Learning Centre, University of Sydney 28
4.2.3 Optimisation
Okay, now we are in a position to be able to do some optimisation problems. To maximise
a function f (x) in a certain region of the x values we are looking for the greatest value
that f (x) can possibly take for x in the region that we are interested in. This may or
may not be at a stationary point. Figure 13 illustrates this. In this figure, we are looking
for the maximum and minimum of the function in the region 2 ≤ x ≤ 7. In this region
there are two stationary points, one a local maximum and one a local minimum. However
notice that the maximum value of the function does not occur at the local maximum, but
at the endpoint of the region, ie where x = 7. This point is not at the top of the hill, so it
is not a stationary point, but it is still the maximum value of the function for 2 ≤ x ≤ 7
because we are ignoring any x which is bigger than 7. On the other hand, in this case the
minimum value of this function for 2 ≤ x ≤ 7 is found at a stationary point. Now we are
in a position to tell you exactly how to find the maximum or minimum of a function.
35.0
30.0 t h e maximu m for the region under conside ra t ion
25.0
20.0
15.0
local maximum
10.0
5.0
t h e minimum f o r the r egion unde r cons ider at i on
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00
Figure 13: The maximum is found at the endpoint of the region under consideration, and not at
a stationary point. The minimum is found inside the region under consideration at a stationary
point
The location of maxima and minima
A function f (x) may or may not have a maximum or minimum value in a particular region
of x values. However, if they do exist the maximum and the minimum values must occur
at one of three places:
1. At the endpoints (if they exist) of the region under consideration.
2. Inside the region at a stationary point.
3. Inside the region at a point where the derivative does not exist.
Notes
1. It is easy to find an example of a function which has no maximum or minimum in
a particular region. For example the function f (x) = x has neither a maximum nor a
Mathematics Learning Centre, University of Sydney 29
minimum value for −∞ < x < ∞. Its graph simply keeps increasing as the values of x
increase. Referring to Point 1 above, if for example the region under consideration was
−∞ < x < ∞ then this region has no endpoints. As another example, the region x ≥ 1
has only one endpoint, x = 1.
2. A note about Point 3 above: in this booklet we will not treat points where the derivative
does not exist. However you should be aware that there may be such points, and that
the maximum or minimum may be found at one. For more information consult a more
comprehensive calculus text.
Now that we know exactly where the maxima or minima can occur, we can give a proce-
dure for finding them.
Procedure for finding the maximum or minimum values of a function.
1. Find the endpoints of the region under consideration (if there are any).
2. Find all the stationary points in the region.
3. Find all points in the region where the derivative does not exist.
4. Substitute each of these into the function and see which gives the greatest (or
smallest) function value.
Example Find the minimum value and the maximum value of the function f (x) = x2 ex
for −4 ≤ x ≤ 1.
Solution We will follow the procedure outlined above. The endpoints are −4 and 1.
Differentiating we obtain f (x) = x2 ex + 2xex = x(x + 2)ex . Setting f (x) = 0 and solving
we get stationary points at x = 0 and x = −2. There are no points where the derivative
does not exist. Therefore the maximum and minimum values will be found at one of the
points x = −4, −2, 0, 1. Substituting we obtain f (−4) ≈ 0.29, f (−2) ≈ 0.54, f (0) = 0
and f (1) = e ≈ 2.7. therefore the maximum value occurs at x = 1 and is equal to e, and
the minimum value occurs at x = 0 and is 0.
Example Find the maximum and minimum values of the function g(t) = 1 t3 − t + 2 for
3
0 ≤ t ≤ 3.
Solution The endpoints are t = 0 and t = 3. Differentiating and equating to zero
we get g (t) = t2 − 1 = (t − 1)(t + 1) = 0 so the stationary points are at t = −1, 1. Since
−1 is not in the region, the possible locations of the maximum and the minimum are
t = 0, 1, 3. Substituting into g we obtain g(0) = 2, g(1) = 4 and g(3) = 8. The maximum
3
is therefore g(3) = 8 and the minimum is g(1) = 4 .3
Example A farmer is to make a rectangular paddock. The farmer has 100 metres of fenc-
ing and wants to make the rectangle that will enclose the greatest area. What dimensions
should the rectangle be?
Solution There are many rectangular paddocks that can be made with 100 metres of
fencing. If we call one side of the rectangle x, then because the perimeter is 100, the other
side of the rectangle is 50 − x. The area of the paddock is then A(x) = x(50 − x). We
must maximise the function A(x) for 0 ≤ x ≤ 50 (since the sides of the rectangle cannot
Mathematics Learning Centre, University of Sydney 30
have negative length). Now dA = 50 − 2x which is zero when x = 25. Thus x = 25 is
dx
the only stationary point and the maximum is found at one of the points x = 0, 25, 50.
Substituting these values into A(x) we find that the maximum occurs when x = 25. The
rectangular paddock with the maximum area is a square.
Exercise 4.1 Find the maximum and the minimum of the function f (x) = x4 − 2x2 for
−1 ≤ x ≤ 2
Exercise 4.2 Maximise the function g(t) = te−t for −2 < t < 2.
2
Exercise 4.3 Find the minimum value of h(u) = 2u3 + 3u2 − 12u + 5 in the region
−3 ≤ u ≤ 2.
Exercise 4.4 A farmer wishes to make a rectangular chicken run using an existing wall
as one side. He has 16 metres of wire nettting. Find the dimensions of the run which will
give the maximum area. What is this area?
Mathematics Learning Centre, University of Sydney 31
5 The clever idea behind differential calculus (also
known as differentiation from first principles)
In this section we will have a look at the idea behind differential calculus. While it is
important that you at least see this idea once, in practice you calculate the derivative of
a function using the procedures explained in Section 3. These procedures work because
of the clever idea that we are going to describe now, but in practice we just use them
without keeping in mind the whole time where they came from.
Figure 14 shows a portion of the graph of the function f (x) = x2 .
y
(0.8, 0.64)
0.600
0.400
(0.5, 0.25)
0.200
0.500 1.000
x
Figure 148, 0.64)
which both lie on the graph of the function.
The tangent to the graph at the point (x, y) = (0.5, (0.5)2 ) is represented by the solid
line. We are going to find the exact slope of this tangent.
To work out the slope of a line we need to know two points on the line. If we know the
points (x1 , y1 ) and (x2 , y2 ) on the line then the rise between these two points is y2 − y1 ,
and the run between them is x2 − x1 , and so the slope of the line is given by
rise y2 − y1
slope = =
run x2 − x1
We cannot use this formula directly to work out the slope of the tangent, because we only
know the exact location of one point on the tangent line, the point (0.5, 0.25). If we were
to pick another point from the diagram that looks like it is on the line, then we would
be back to using the approximate graphical methods from Section 2. If we want to get
the exact answer, we must use another way. This is the clever idea behind differential
calculus.
We look at another line which which has slope nearly equal to the slope of the tangent,
and on which we do know two points. In Figure 14 we have drawn such a line (the dotted
Mathematics Learning Centre, University of Sydney 32
line) going through the points (0.5, (0.5)2 ), which is the actual point on the curve where
we are trying to find the tangent, and (0.8, (0.8)2 ) which is another point on the curve.
This second point is not far from the point (0.5, (0.5)2 ), and so the slope of the line joining
them is not too different from the slope of the tangent at (0.5, (0.5)2 ). Because we know
two points on the dotted line, we can work out its slope. It is
(0.8)2 − (0.5)2
slope =
0.8 − 0.5
= 1.3
The slope of the tangent is therefore about (but certainly not exactly) 1.3.
To get a better approximation we might try taking the second point closer to the point
(0.5, (0.5)2 ). In Figure 15 we have done this.
(0.6, 0.36)
(0.5, 0.25)
Figure 156, 0.36)
which both lie on the graph of the function.
Here the second point is just 0.1 units to the right of (0.5, (0.5)2 ). The slope of the dotted
line in this figure is
(0.6)2 − (0.5)2
slope =
0.6 − 0.5
= 1.1
The exact slope of the tangent is closer to 1.1 than it is to 1.3, though we still don't know
its precise value.
If we wanted an even better approximation then we could choose the second point to be
even closer to (0.5, (0.5)2 ). For example we could try the second point to be (0.49, (0.49)2 ).
Notice that this point is to the left of (0.5, (0.5)2 ), whereas previously we had chosen points
to the right of the point. This is not important. What is important is that it is just 0.01
Mathematics Learning Centre, University of Sydney 33
units to the left of this point. The line joining these two points is very close to the actual
tangent, and the slope of this line is
(0.49)2 − (0.5)2
slope =
0.49 − 0.5
= 0.99
It seems that the closer the second point gets to the point (0.5, (0.5)2 ) the closer the slope
of the line joining the two points gets to 1. We might guess that the slope of the tangent
to the curve at the point (0.5, (0.5)2 ) must be 1. We can be sure of this with the following
calculation.
Suppose that the second point is just h units to the right (or left if h < 0) of x = 1.
We can think of h as being a very small number. For example we have used h = 0.3,
h = 0.1, and h = −0.01 in our examples above. The coordinates of the second point
will be (0.5 + h, (0.5 + h)2 ). We can work out the slope of the line joining the points
(0.5, (0.5)2 ) and (0.5 + h, (0.5 + h)2 ) in the same way that we did above. It is
(0.5 + h)2 − 0.52
Slope =
(0.5 + h) − 0.5
(0.5 + h)2 − 0.52
=
h
(.25 + h + h2 ) − .25
=
h
h + h2
=
h
= 1+h
Moving the second point closer and closer to the first is the same as making h closer and
closer to zero. But the slope is 1 + h so the closer that h gets to zero the closer the slope
gets to 1. The slope of the tangent is therefore exactly 1. This puts the matter beyond
doubt. We are no longer relying on approximations or guesses. We have shown that the
slope of the tangent to the graph of y = x2 at the point (0.5, 0.25) is exactly 1. In symbols,
d 2
x = 1.
dx x=0.5
With the same method we could have found the slope of the tangent to the curve when
x = 1, or x = −0.37, or indeed at any value of the independent variable x.
Exercise 5.1 (a) Using the same technique as above, find the slope of the tangent
to the graph of x2 at x = 2. Check that this agrees with the
answer that you would have obtained using the results of Section
3.
(b) Using the idea introduced above, find the slope of the tangent
to the graph of x3 at x = 1. Check your answer by using the
techniques from Section 3.
Mathematics Learning Centre, University of Sydney 34
We don't really have to specify any particular value of x, but can leave it as unknown.
(x + h)2 − x2
Slope of line through (x, x2 ) and (x + h, (x + h)2 ) =
x+h−x
x2 + 2xh + h2 − x2
=
h
= 2x + h
and as h → 0 the line through (x, x2 ) and (x + h, (x + h)2 ) gets closer to the tangent
at x and the slope of this line gets closer and closer to 2x. The slope of the tangent is
therefore exactly 2x. This works no matter what the value of x is. For example the slope
of the tangent at x = −0.37 is 2 × (−0.37) = −0.74. We can say that the derivative of
the function x2 is 2x. In symbols,
d 2
x = 2x.
dx
We have chosen the function f (x) = x2 for this example, because it is perhaps the simplest
function that gets across the idea. The same method works for any function, though the
resulting algebra will often be more difficult.
Exercise 5.2 (a) Using the ideas of this section, find the derivative of x3 .
(b) Use the same ideas to find the derivative of x3 + 2x .
By now you probably have little doubt that the derivative of x2 is 2x, and that the
derivative of x3 is 3x2 . Hopefully you are now willing to believe that the derivative of xn
is nxn−1 , no matter what value n has. It would not be too difficult for us to prove this
fact, but the proof is beyond the scope of this booklet. However what you have seen in
this section is the basic idea that underlies all of differential calculus, and all of the rules
and techniques of Section 3 come from it.
Mathematics Learning Centre, University of Sydney 35
6 Solutions to exercises
Exercise 1.1
From the graph, at time t = 0 the motorist is 100 metres from home, and at time t = 6
the motorist is 250 metres from home.
Exercise 1.2
At time t = 0 the motorist is 100 metres from home and at time t = 2 the motorist is
150 metres from home, so in the first 2 seconds the motorist has travelled 150 − 100 = 50
metres. At time t = 3 the motorist is 175 metres from home and at time t = 5 the
motorist is 225 metres from home so in the time from t = 3 to t = 5 the motorist has
travelled 225 − 175 = 50 metres.
Exercise 1.3
A time t = 60 the motorist is 1008 metres from home and at time t = 62 the motorist is
1032 metres from home so in the 2 second interval from time t = 60 to time t = 62 the
motorist travelled 1032 − 1008 = 24 metres. A time t = 64 the motorist is 1072 metres
from home so in the 2 second interval from time t = 62 to time t = 64 the motorist has
travelled 1072 − 1032 = 40 metres.
Exercise 2.1
Refer to Figure 16. We have used the indicated points on the lines to calculate the slopes.
You may have chosed different points, but your answers should be close to those here.
Remember this is only an approximate way of finding the slopes, so you shouldn't consider
yourself wrong if you don't get exactly the same answers as here.
4.00
3.00
2.00
1.00
3.00 -2.00 -1.00 1.00 2.00 3.0
Figure 16: Tangents to graph of f (x) = x2 .
Slope of tangent to f (x) = x2 at x = 1 is
3−1
f (1) ≈ = 2.
2−1
Mathematics Learning Centre, University of Sydney 36
The tangent at x = 0 is the x-axis, which has slope 0, so f (0) = 0.
Slope of tangent to f (x) = x2 at x = −0.5 is
2.75 − 0.75
f (−0.5) ≈ = −1.
−3 − (−1)
Exercise 2.2
Refer to Figure 17.
1.50
1.00
0.50
-1.00 -0.50 0.50 1.00 1.50
-0.50
Figure 17: Tangents to graph of f (x) = x3 .
Slope of tangent to f (x) at x = 1 is
1 − (−0.5)
f (1) ≈ = 3.
1 − 0.5
As in Exercise 2.1, the tangent at x = 0 is the x-axis which has slope 0, so f (0) = 0.
Slope of tangent to f (x) at x = −0.5 is
0.25 − (−0.5)
f (−0.5) ≈ = 0.75.
0 − (−1)
Exercise 3.1
dy 1 4
(a) f (x) = 4x3 (b) = −7x−8 (c) f (u) = 2.3u1.3 (d) f (t) = − t− 3
dx 3
22 15 3 5 dy dz 3 4
(e) f (t) = t7 (f ) g (z) = − z − 2 (g) = −3.8t−4.8 (h) = x− 7
7 2 dt dx 7
Mathematics Learning Centre, University of Sydney 37
Exercise 3.2
1 d 1 √ d √ 3 1
= x−2 so = −2x−3
3
(a) 2
(b) t t = t 2 so t t = t2
x dx x2 dt 2
d √ d 1 1 2 d 1 d −5 5 7
(c) 3
x= x 3 = x− 3 (d) √ = x 2 = − x− 2
dx dx 3 dx x2 x dx 2
√
d 1 d 5/4 −5 − 9 d s3 s d 19 19 13
(e) √ =− x = x 4 (f ) √ = s6 = s6
dx x x
4
dx 4 dx 3
s ds 6
d 1 du−3 d t d −3 3 5
(g) = = −3u−4 (h) √ = t 2 = − t− 2
du u3 du dt t2 t dt 2
√
d 1 x d
(i) x2 = 1=0
dx x dx
Exercise 3.3
√
= 1 x− 3 and
dy 2
When x = 8 we have y = 3 8 = 2 so the point (8, 2) is on the line. Now dx 3
dy 1
so dx = 12 when x = 8. The tangent therefore has equation
1
y= x + b.
12
Substituting x = 8 and y = 2 into this equation we obtain
1
2= 8+b
12
so that b = 4 . The equation is therefore y =
3
x
12
+ 4.
3
Exercise 3.4
−1
(a) f (x) = 10x − x− 2
1
(b) dy
dx
= −14x−8 − 6x−3 (c) f (t) = 5.75t1.3 + 1 t
2
2
(d) h (z) = − 1 z − 3 + 5
4 2
3
(e) f (u) = 5 u 3 + 21u−8
3
(f ) g (z) = −16z −3 + 5z −2
= −40t−9 + 1 t− 2 = 4 x− 7 − x− 2
dy 1 dz 6 3
(g) dt 2
(h) dx 7
Mathematics Learning Centre, University of Sydney 38
Exercise 3.5
(a) f (x) = 12x2 (1 − 3x) − 3(4x3 + 2)
(b) g (x) = (2x + 1)(x2 + 1) + (x2 + x + 2)2x
(c) h (x) = (9x2 − 4x + 8)(x2 − 2x + 4) + (3x3 − 2x2 + 8x − 5)(2x − 2)
s2
(d) f (s) = −s(3s + 5) + 3(1 − 2
)
−1 √
(e) g (t) = ( t 22 − t−2 )(2t − 1) + 2( t + 1 )
t
−1 √
(f ) h (y) = (− y 2 + 2y)(1 − 3y 2 ) − 6y(2 −
2
y + y2)
Exercise 3.6
The rate of change of r with respect to t is
dr 1
= (1 − t−2 )(t2 − 2t + 1) + (t + )(2t − 2).
dt t
Substituting t = 2 we obtain (1 − 1 )(4 − 4 + 1) + (2 + 1 )(4 − 2) =
4 2
23
4
.
Exercise 3.7
The gradient of the tangent is given by
dy
= (2x − 2)(3x3 − 5x2 + 2) + (x2 − 2x + 1)(9x2 − 10x).
dx
Substituting x = 2 we obtain 28.
Exercise 3.8
(x + 1) − (x − 1) 2
(a) f (x) = 2
=
(x + 1) (x + 1)2
(3x − 2)2 − (2x + 3)3 −13
(b) g (x) = =
(3x − 2)2 (3x − 2)2
(x2 + 5)2x − (x2 + 2)2x 6x
(c) h (x) = = 2
(x2 + 5)2 (x + 5)2
(1 + 2t2 )2 − 8t2 2 − 4t2
(d) f (t) = =
(1 + 2t2 )2 (1 + 2t2 )2
√ √
(1 − s) 1 s− 2 + (1 + s) 1 s−1/2
1
s− 2
1
(e) f (s) = 2
√ 2
= √
(1 − s)2 (1 − s)2
(x3 + 4)2x − (x2 − 1)3x2 −x4 + 3x2 + 8x
(f ) h (x) = =
(x3 + 4)2 (x3 + 4)2
(3u4 + 5)(3u2 + 1) − (u3 + u − 4)12u3 −3u6 − 9u4 + 48u3 + 15u2 + 5
(g) f (u) = =
(3u4 + 5)2 (3u4 + 5)2
(t2 + 3t + 1)(2t + 6) − (t2 + 6t)(2t + 3) −3t2 + 2t + 6
(h) g (t) = = 2
(t2 + 3t + 1)2 (t + 3t + 1)2
Mathematics Learning Centre, University of Sydney 39
Exercise 3.9
d
(a) (2x + 3)2 = 8x + 12
dx
d
(b) (x2 + 2x + 1)12 = 12(x2 + 2x + 1)11 (2x + 2)
dx
d
(c) (3 − x)21 = −21(3 − x)20
dx
d
(d) (x3 − 1)5 = 5(x3 − 1)4 3x2 = 15x2 (x3 − 1)4
dx
d√2 d 1
t − 5t + 7 = (t2 − 5t + 7) 2 = (t2 − 5t + 7)− 2 (2t − 5)
1 1
(e)
dt dt 2
d 1 d
(2 − z 4 )− 2 = 2z 3 (2 − z 4 )− 2
1 3
(f ) √ =
dz 2−z 4 dz
d √ √ 1
(g) (t3 − t)−3.8 = −3.8(t3 − t)−4.8 (3t2 − √ )
dt 2 t
d 1 3 3 1 4 1
(h) (x + ) 7 = (x + )− 7 (1 − 2 )
dx x 7 x x
Exercise 3.10
d
(a) (x + 2)(x + 3)2 = (x + 3)2 + 2(x + 2)(x + 3)
dx
d
(b) (2x − 1)2 (x + 3)3 = 4(2x − 1)(x + 3)3 + 3(2x − 1)2 (x + 3)2
dx
d √ √ x
(c) x 1−x = 1−x− √
dx 2 1−x
d 1 2 2 1
x 3 (1 − x) 3 = x− 3 (1 − x) 3 − x 3 (1 − x)− 3
1 2 2 1
(d)
dx 3 3
√
1 − x2 + x2 (1 − x2 )− 2
1
d x
(e) √ =
dx 1 − x2 1 − x2
Exercise 3.11
6x2 3
(a) f (x) = 3
=
2x x
1
Alternatively write f (x) = ln 2 + 3 ln x so that f (x) = 3 .
x
7
(b) f (x) = 7x6 ex
7
(c) f (x) = x
2 +x3
(d) f (x) = (2x + 3x2 )ex
(e) Write f (x) = loge 7 − 2 loge x so that f (x) = − x .
2
(f ) f (x) = −e−x
ex + 3x2
(g) f (x) =
ex + x3
Mathematics Learning Centre, University of Sydney 40
3
(h) Write f (x) = ln ex + 3 ln x so that f (x) = 1 + .
x
2x 3x2 − 1
(i) Write f (x) = ln(x2 + 1) − ln(x3 − x) so that f (x) = − 3 .
x2 + 1 x −x
Exercise 3.12
d
(a) cos 3x = −3 sin 3x
dx
d
(b) sin(4x + 5) = 4 cos(4x + 5)
dx
d
(c) sin3 x = 3 sin2 x cos x
dx
d
(d) sin x cos x = cos2 x − sin2 x
dx
d 2
(e) x sin x = 2x sin x + x2 cos x
dx
d
(f ) cos(x2 + 1) = −2x sin(x2 + 1)
dx
d sin x x cos x − sin x
(g) =
dx x x2
d 1 1 1
(h) sin = − 2 cos
dx x x x
d √ 1 √
(i) tan x = √ sec2 x
dx 2 x
d 1 1 1 1 1 1
(j) sin = − 2 sin − 3 cos
dx x x x x x x
Exercise 4.1
f (x) = 4x3 − 4x so f (x) = 0 at x = 0, ±1 and the maxima and minima must occur at
the points x = −1, 0, 1, 2. Substituting these values into f (x) we find that the maximum
occurs at x = 2 and the minimum occurs at x = −1 and at x = 1.
Exercise 4.2
g (t) = (1 − 2t2 )e−t . Setting this equal to zero and solving we find that the stationary
2
points are at t = ± √2 and the maximum must occur at one of the points t = −2, ± √2 , 2.
1 1
1
Substituting into g(t) we find that the maximum value occurs at t = √2 .
Exercise 4.3
h (u) = 6u2 + 6u − 12 = 6(u2 + u − 2). The stationary points are at u = −2, 1 and the
minimum value occurs at one of the points u = −3, −2, 1, 2. Substituting into h(u) we
find that the minimum occurs at u = 1.
Mathematics Learning Centre, University of Sydney 41
Exercise 4.4
If we let the side of the run that is opposite the existing wall have length x, then the other
side of the run has length 8 − x .
2
The area of the run is A(x) = x(8 − x ) and we must maximise this function in the region
2
0 ≤ x ≤ 16. Differentiating gives A (x) = 8 − x so the only stationary point is at x = 8.
The maximum occurs at one of x = 0, 8, 16. Substituting, we see that the maximum
occurs when x = 8, giving an area of 32 square metres.
8 - x /2
x
Figure 18: A chicken run built against the side of an existing wall, with 16 metres of netting.
Exercise 5.1
b.
(1 + h)3 − 13
Slope =
(1 + h) − 1
1 + 3h + 3h2 + h3 − 1
=
h
3h + 3h2 + h3
=
h
= 3 + 3h + h2 .
So, the slope of the tangent to the graph of x3 at x = 1 is 3.
Exercise 5.2
b. Slope of the line through (x, x3 + 2x) and ((x + h), (x + h)3 + 2(x + h)) is
((x + h)3 + 2(x + h)) − (x3 + 2x)
Slope =
(x + h) − x
x + 3x h + 3xh2 + h3 + 2x + 2h − x3 − 2x
3 2
=
h
3x2 h + 3xh2 + h3 + 2h
=
h
2
= 3x + 3xh + h2 + 2.
As h → 0 the slope of this line → 3x2 + 2.
So, the slope of the tangent to the curve is exactly 3x2 + 2.
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Good. Names on inside cover and numbers on bookedge; no other internal marking/highlighting. Unless specifically stated as present, assume no CD, DVD, access code or other support materials is available.
All Editions of Math at Hand: A Mathematics Handbook
Customer Reviews
This is a wonderful tool for my daughter to use for everyday math. It provides a great reference point for homework or to get an understanding of what she will be working on next.
nursepolly92
Oct 4, 2007
Great Reference Tool
This series of books are great resources for parents to use at home (the reason I bought them!) to help with homework. I was finding so many times my kids would bring home papers to turn back in the next day and there would be no explanation on how to do the work. He would get frustrated and I couldn't remember from 25 years ago how to do some problems step by step. They speak in common everyday language and take you through processes step by step in easy to understand format. These books are ALL a great reference set to have for elementary through middle school. I would highly recommend purchasing them
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This Book
multiple perspectives on mathematical thinking, this volume presents elaborations on principles reflecting the progress made in the field over the past 20 years and represents starting points for understanding mathematical learning today. This volume will be of importance to educational researchers, math educators, graduate students of mathematical learning, and anyone interested in the enterprise of improving mathematical learning
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Classes
MTH 067: Foundations of Mathematics
This is the first of two courses in the developmental math sequence. The focus of this course is to develop students' problem-solving and basic algebra skills. Topics for this course include applications involving integers, decimals and fractions, as well as applications of percents, proportions and consumer credit, algebraic expressions, algebraic properties, algebraic operations and multi-step equation-solving. The Cartesian Coordinate system and applications of algebra are also introduced. Students who complete this course, and demonstrate competency on an exit test, are eligible to enroll in the second course.
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Excellent Book. Belongs on Your Bookshelf.Aug 03, 2003
By Michael Wischmeyer Courant's 500-page text is not entirely suitable for the layman. Its target audience includes those who enjoy reading and studying mathematics and have a good background through precalculus or higher. "What is Mathematics?" is a mathematics book, not a book about mathematics.
"What is Mathematics?" is not a new book. It was first published by Oxford University Press in 1941 with later editions in 1943, 1945, and 1947. Good quality soft cover copies are still in print as Oxford Paperbacks.
The authors indicate that it is no means necessary to "plow through it page by page, chapter by chapter". I fully agree. I have skipped around, jumping to chapters of particular interest, but I have now read nearly every chapter.
I initially skipped to page 165 and delved directly into projective geometry (chapter IV), proceeded to topology (chapter V), and then jumped backwards to the beginning to explore the theory of numbers. After moving to geometry, I finally returned to the later chapters on functions and limits, maxima and minima, and the calculus.
Courant engages the reader in discussions on mathematical concepts rather than focusing on applications and problem solving. "What is Mathematics?" is a great textbook for students that have completed a year or more of calculus and wish to pull all of their mathematical learning together before moving on to more advanced studies. I suspect that it would even be welcomed by students that have completed an undergraduate degree in mathematics.
I cannot resist quoting Albert Einstein's comment on What is Mathematics? - "A lucid representation of the fundamental concepts and methods of the whole field of mathematics...Easily understandable."
Richard Courant was a highly respected mathematician. He taught in Germany and in Cambridge and was director of the Institute of Mathematical Sciences at New York University (now renamed the Courant Institute of Mathematical Sciences). Courant has authored other widely acclaimed mathematical texts including Methods of Mathematical Physics (co-authored with David Hilbert) and his popular Differential and Integral Calculus.
99 of 100 found the following review helpful:
the best bargain in introductory math books in existenceApr 20, 2005
By twit This book genuinely has more mathematical content, for around $15-$25, than most, maybe all, "bridge" texts for college math majors, costing 5 or 10 times as much.
This book was written by a master, for an intelligent person knowing only 1950's style high school mathematics (some trig, algebra, and geometry).
When I fiorst tried to read it as a youngster however I was not used to books that required actually thinking about each statement, before proceeding to the next. Hence I could not read it at the pace I thought normal.
So this is not a breezy read, but is an outstanding one. It has literally no competitor to my knowledge at the present time, in quantity of material, quality of material, and quality of exposition.
Even experts may learn something here about the most familiar topics. E.g. in presenting the proof of the well known fact that all integers greater than one have unique prime factorizations, the authors show how a clever use of induction avoids developing the characterization of a gcd, which usually precedes this theorem. I had never seen that before.
If you are looking for a miracle book that treats the reader like a baby, and still covers calculus, this is not it. But if you have the prerequisites of a good high school course of elementary math, and are willing to spend time on the arguments, there is no better book for beginners and intelligent laypersons.
70 of 73 found the following review helpful:
InspiringOct 24, 1999
Although I was always good in math in high school, I never really appreciated it. One summer I found this book in a dusty little corner of a bookshelf and I started reading it. I still remember how for the first time, I was inspired by the subject while reading this book. I couldn't stop reading it, until I finished it. At the time, I didn't really know Calculus or any advanced subject and I had never read any math books other than the high school textbooks. This book literally changed my life. I might have forgotten who my first love was, but I remember very well this book after 25 years!
38 of 38 found the following review helpful:
A MasterpieceJun 24, 2000
By Ary Armando Perez Jr. If you start to read "What is Mathematics?" in order to find a direct answer to the title's issue, forget it! I would like to adapt a piece of "My Brain is Open", by Bruce Schechter, in the following way: "Asking a mathematician to explain exactly what is mathematics is a little like asking a poet what a poem is, or a musician what jazz is. Asked this last question, Louis Armstrong replied, `Man, if you gotta ask, you'll never know.'" On the other hand, if you start to read just to go deeper and deeper in the beautiful, and sometimes magic, structure of Math than I say: Go ahead! Because this book is a perennial source of pleasure. Of course it demands a lot of work to solve some of its problems (at least for me!), but as Courant says, you cannot learn music only by listening! I have reproduced almost all the calculations of this book and I know that it demands a lot of effort, but it is one of the few books I know where each small piece of calculation has its own reward! This book is my definition of perfect guide to Math style! Try it!
60 of 64 found the following review helpful:
Timeless.Apr 16, 2003
By Palle E T Jorgensen
"Palle Jorgensen"
Einstein writes..."Easily understandable." And Herman Weyl,..."It is a work of high perfection." It is both for beginners and for scholars. The first edition by Courant and Robbins, has been revised, with love and care, by Ian Stewart. Of the sciences, math stands out in the way some central ideas and tools are timeless. Key math ideas from our first mathematical experiences, perhaps early in life, often have more permanence this way. While the fads do change in math, there are some landmarks that remain, and which inspire generations. And they are as useful now as they were at their inception, the fundamentals of numbers, of geometry, of calculus and differential equations, and more. Much of it is presented with an eye to applications. The book is a classic and a masterpiece. The co-authors are ambitious (and remarkably sucessful)in trying to cover the essetials within the span of 500 plus pages. You find the facts, presented in clear and engaging prose, and with lots of illustrations. The book has been used by generations of readers, and it still points to the future.
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book containing over 200 problems spanning over 70 specific topic areas covered in a typical Algebra II course. Learners can encounter a selection of application problems featuring astronomy, earth science and space exploration, often with...(View More) more than one example in a specific category. Learners will use mathematics to explore science topics related to a wide variety of NASA science and space exploration endeavors. Each problem or problem set is introduced with a brief paragraph about the underlying science, written in a simplified, non-technical jargon where possible. Problems are often presented as a multi-step or multi-part activities. This book can be found on the Space Math@NASA website.(View Less)
In this problem set, learners will consider the temperature in Kelvin of various places in the universe and use equations to convert measures from the three temperature scales to answer a series of questions. Answer key is provided. This is part of...(View More) Earth Math: A Brief Mathematical Guide to Earth Science and Climate Change.(View Less)
This is a booklet containing 37 space science mathematical problems, several of which use authentic science data. The problems involve math skills such as unit conversions, geometry, trigonometry, algebra, graph analysis, vectors, scientific...(View More) notation, and many others. Learners will use mathematics to explore science topics related to Earth's magnetic field, space weather, the Sun, and other related concepts. This booklet can be found on the Space Math@NASA website.(View Less)
In this problem set, learners will refer to the tabulated data used to create the Keeling Curve of atmospheric carbon dioxide to create a mathematical function that accounts for both periodic and long-term changes. They will use this function to...(View More) answer a series of questions, including predictions of atmospheric concentration in the future. A link to the data, which is in an Excel file, as well as the answer key are provided. This is part of Earth Math: A Brief Mathematical Guide to Earth Science and Climate Change.(View Less)
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Generalized Polygons is the first book to cover, in a coherent manner, the theory of polygons from scratch. In particular, it fills elementary gaps in the literature and gives an up-to-date account of current research in this area, including most proofs, which are often unified and streamlined in comparison to the versions generally known. Generalized... more...
The aim of this book is to throw light on various facets of geometry through development of four geometrical themes.The first theme is about the ellipse, the shape of the shadow cast by a circle. The next, a natural continuation of the first, is a study of all three types of conic sections, the ellipse, the parabola and the hyperbola.The third theme... more...
The abstract homotopy theory is based on the observation that analogues of much of the topological homotopy theory and simple homotopy theory exist in many other categories (e.g. spaces over a fixed base, groupoids, chain complexes, module categories). Studying categorical versions of homotopy structure, such as cylinders and path space constructions,... more...
Riemannian geometry has today become a vast and important subject. This new book of Marcel Berger sets out to introduce readers to most of the living topics of the field and convey them quickly to the main results known to date. These results are stated without detailed proofs but the main ideas involved are described and motivated. This enables the... more...
An ingenious problem-solving solution for befuddled math students.
A bestselling math book author takes what appears to be a typical geometry workbook, full of solved problems, and makes notes in the margins adding missing steps and simplifying concepts so that otherwise baffling solutions are made perfectly clear. By learning how to interpret... more...
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Doing Physics with Quaternions
by Douglas B. Sweetser , 2005 Quaternions, like the real numbers, can be added, subtracted, multiplied, and divided. They are composed of four numbers that work together as one. This book contains a brief summary of important laws in physics written as quaternions. (8948 views)
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This course is designed to help you work through critical mathematical concepts that are needed for college level mathematics classes. We will start the class with a written pre-assessment to find out what material you personally will need to focus on, then you will start on an individualized program of self-study. Your instructor is here to help you work through any difficulties you encounter, and the on-line learning supplement will also act as your personal tutor. This course is designed for YOUR SUCCESS. Please read through the information below to learn more about how the course is structured and what the expectations will be.
Topics:
na
Submitted by:
Kris Johnson
Date Submitted:
12/10/14
Math 103
Title:
Algebra Methods and Introduction to Functions
Prerequisites:
Math 100 or 40% on ALEKS exam as of 1/2015, subject to change.
Comments:
Course renewal Fall 2014
Text:
Intermediate Algebra with POWER Learning, Messersmith, Perez and Feldman by McGraw Hill
ALEKS Student Access Code required. See instructor for your course code for this class.
By the end of the course you are expected to be able to solve expressions, then solving the equations to determine a specific result.
Use properties of real numbers and properties of exponents to manipulate and simplify exponential expressions and solve simple exponential equations.
Use properties of real numbers, properties of radicals and properties of exponents to simplify racial expressions and solve simple radical equations.
Add, subtract and multiply polynomial expressions.
Use properties of real numbers and properties of exponents to factor polynomial expressions.
Solve simple polynomial equations and simple absolute value equations.
Set up equations to represent data given an application problem and use it to solve for a specific outcomes.
Topics:
Use properties of real numbers and properties of exponents to add, subtract, multiply, divide and simplify.
Recognize the difference between an algebraic expression and an algebraic equations and use this information to construct expressions from the context of real-life situation.
Analyze a real-life situation and convert it into an appropriate mathematical statement.
Solve linear inequalities; linear, quadratic, rational, and radical equations.
Use properties of real numbers and properties of exponents to manipulate and simplify rational expressions and solve simple rational equations.
Determine slopes of lines, parallel lines, and perpendicular lines.
Use properties of real numbers and properties of exponents to manipulate and simplify rational expressions and solve simple rational equations.
Use properties of real numbers and properties of exponents to manipulate and simplify rational expressions and solve simple rational equations.
Math 106
Title:
College Algebra
Prerequisites:
Math 103 with a grade of C or better, or 70% on ALEKS exam as of 1/2015, subject to change..
Accelerated Version: Math 103 with a grade of B or better, 75% on ALEKS exam as of 1/2015, subject to change.
Precalculus by Eric Schulz, Julianne Connell Sachs, William Briggs and Lyle Cochran. This is an eBook published by Pearson. MyMathLab, Explorations and Notes.
Text Sections Covered:
Class will work with polynomial, rational, exponential, and logarithmic functions. For each class of functions, you will study the domains, ranges, graphs, special properties and application. By the end of the course you should understand their properties, the shapes of their graphs and be able to solve problems and equations involving the. These are the functions that will be used in later classes as calculus, physics, biology and engineering.
Comments:
Credit not granted for both Math 106 and Math 107.
Topics:
Understand and apply quantitative principles and methods to define, analyze and solve problems.
Integrate and synthesize knowledge and different techniques to solve problems.
Draw conclusions from computational and symbolic representations in order to check the logic and validity of statement and models.
Clearly communicate your reasoning and findings.
Use properties of real numbers and properties of exponents to maniulate and simplify mathematical expressions.
Solve: linear equations and inequalities; absolute value equations and inequalities, rational, radical and polynomial equations; polynomial and rational inequalities; exponential and logarithmic equations.
Read and create representations of data using tables and graphs, interpret this information in the context of a real-life situation and determine whether your answer makes sense in the context of the problems.
Math 108
Trigonometry
Math 106 with a grade of C or better. ALEKS assessment is NOT use to place into Math 108.
Comments:
Graphs, properties and applications of trigonometric functions.
Text:
Precalculus by Eric Schulz, Julianne Connell Sachs, William Briggs and Lyle Cochran. This is an eBook published by Pearson.
MyMathLab
Explorations and Notes by Eric Schulz and Julianne Connell Sachs.
Text Sections Covered:
You will learn the trigonometric functions as derived from the unity circle and from right triangles and the trigonometric identities. For each class of functions, you will study the domains, ranges, transformations, graphs, special properties and applications. By the end of the course you should understand all these concepts and be able to solve problems and equations involving all six trigonometric functions. These functions that will be used in classes such as calculus, physics, biology and engineering.
Comments:
Develop learning skills that are important for your success in this and future courses.
Understand and apply quantitative principles and methods to define, analyze and solve problems.
Integrate and synthesize knowledge and different techniques to solve problems.
Draw conclusions from computational and symbolic representations in order to check the logic and validity of statements and models.
Clearly communicate your reasoning and findings.
Topics:
Concepts of radian and degree and convert from one to the other.
Circle definitions and right triangle definitions of the six trigonometric functions.
Use the six inverse trigonometric functions to find angles.
Know the properties of the trigonometric functions
Identify periodic functions and their periods.
Submitted by:
Kris Johnson
Date Submitted:
12/12/14
Math 110
Title:
Mathematics Tutorial for Math 107
Prerequisites:
Must be currently enrolled in 107.
Comments:
Support course for Math 107.
Text:
-
Text Sections Covered:
-
Comments:
-
Topics:
As the tutorial for Precalculus, Math 110 emphasizes and reviews key concepts and skills required for Math 107. To this end, this course supplies facilities and time for individual work on ALEKS, an online tutoring program required for Math 107 (see website: The rest of the class time is geared towards general review and Q&A, where students work individually or in small groups.
Submitted by:
Corby Harwood
Date Submitted:
12/07/09
Math 111
Title:
Mathematics Tutorial for Math 201
Prerequisites:
Must be currently enrolled in Math 201
Comments:
Support course for Math 201.
Text:
-
Text Sections Covered:
-
Comments:
-
Topics:
Content in this course is aligned with specific content of Math 201. Specifically, students will focus on correctly and efficiently performing algebraic manipulations on a variety of function types including, but not limited to, linear, quadratic, exponential, and logarithmic.
Submitted by:
Christy Jacobs
Date Submitted:
6/18/07
Math 140
Title:
Calculus for Life Science
Prerequisites:
Math 106 & 108 with a grade of C or better, or 80% on ALEKS exam as of 1/2015, subject to change.
Comments:
Credit not normally granted for more than one of Math 140, 171, 202, 206.
Text:
Calculus for Life Sciences by Schreiber, published by Wiley, ISBN: 9781119074113.
Recognition and solution of linear, fractional, radical, quadratic equations; solution of linear and absolute-value inequalities; definition of a function; recognition and graphing of linear and quadratic functions; recognition and solution of linear and non-linear systems of equations using substitution; matrix algebra; recognition and solution of linear systems of equations using elimination, matrix reduction, matrix inversion; solution of linear programming problems by graphing (using the corner-point principle) and by using the Simplex method; permutations, combinations, and other basic counting principles.
Submitted by:
Jessica Cross
Date Submitted:
4/28/15
Math 202
Title:
Calculus for Business and Economics
Prerequisites:
Math 106 or 201 with a grade of C or better, 80% on ALEKS exam as of 1/2015, subject to change.
Comments:
Differential and integral calculus of the polynomial, exponential, and logarithmic functions.
1.1 The Slope of a Straight Line; 1.2 The Slope of a Curve at a Point; 1.3 The Derivative; 1.4 Limits and the Derivative; 1.5 Differentiability and Continuity; 1.6 Some Rules for Differentiation
1.7 More About Derivatives; 1.8 The Derivative as a Rate of Change; 2.1 Describing Graphs of Functions; 2.2 The First and Second Derivative Rules; 2.3 The First and Second Derivative Tests and Curve Sketching; 2.4 Curve Sketching (Conclusion); 2.5 Optimization Problems; 2.6 Further Optimization Problems; 3.1 The Product and Quotient Rules; 3.2 The Chain Rule and the General Power Rule; 3.3 Implicit Differentiation and Related Rates; 4.1 Exponential Functions
4.2 The Exponential Function e^x; 4.3 Differentiation of Exponential Functions; 4.4 The Natural Logarithm Function; 4.5 The Derivative of ln x; 4.6 Properties of the Natural Logarithm Function; 5.1 Exponential Growth and Decay; 5.2 Compound Interest; 5.4 Further Exponential Models; 6.1 Antidifferentiation; 6.2 Areas and Riemann Sums; 6.3 Definite Integrals and the Fundamental Theorem; 6.4 Areas in the xy-Plane; 6.5 Applications of the Definite Integral; 6.6 Techniques of Integration; 6.7 Improper Integrals; 8.1 Radian Measure of Angles; 8.2 The Sine and the Cosine; 8.3 Differentiation and Integration of sin t and cos t; 8.4 The Tangent and Other Trigonometric Functions
Comments:
ALEKS: This is a web-based assessment and learning system that uses adaptive questioning to help a student determine which background areas, if any, need strengthening. Since success in calculus depends on mastery of algebra and precalculus, you will be required to demonstrate mastery of these topics via ALEKS in the first five weeks of the course.
PROJECT: Many ideas in calculus can be explored using software packages such as Maple. This project will give you the opportunity to explore the concepts we are learning without having to spend time on lengthy calculations. This assignment will be worked individually and will involve the use of the internet
Logical and historical development of present-day number systems and associated algorithms, methods of problem solving. An introduction to problem solving, Polyas 4 Problem-Solving Principles, problem-solving strategies, sets and operations on sets, counting, and the whole numbers, addition and subtraction of whole numbers, multiplication and division of whole numbers, non-decimal positional systems, algorithms for adding and subtracting whole numbers, algorithms for multiplication and division of whole numbers, mental arithmetic and estimation, divisibility of natural numbers, tests for divisibility, greatest common divisors and least common multiples, representation of integers, addition and subtraction on integers, multiplication and division of integers, basic concepts of fractions and rational numbers, the arithmetic of rational numbers, the rational numbers system, decimals, computations with decimals, ratio and proportion, percent.
Classification of differential equations First order differential equations General theory of higher order linear differential equations Equations with constant coefficients Methods of undetermined coefficients and variation of parameters Series solutions of second order linear equations Laplace Transform Systems of first order linear differential equations
Submitted by:
Eric Remaley
Date Submitted:
4/23/09
Math 320
Title:
Elementary Modern Algebra
Prerequisites:
Math 220
Comments:
-
Text:
Instructor's Lecture Notes
Text Sections Covered:
-
Comments:
-
Topics:
We will cover the rudiments of modern algebra and their application to solving polynomial equations, particularly the determination of the solvability of the quintic polynomial. Furthermore, time permitting, the classical Greek problems will be briefly discussed. Close attention will be paid to the historical development of the algebraic concepts with the intent of demonstrating how modern algebraic concepts arose from the problem of solving polynomial equations. The Application of these algebraic concepts to the above-mentioned problems will be examined, and other uses will be discussed.
Math 330
Methods of Teaching Secondary Mathematics
No text book required, however, you are required to purchase a bundle of software.
Software Bundle includes: Geometers Sketchpad (version 4), Fathom and Tinker Tots. Key College Publishing Co.
Text Sections Covered:
Various articles will be supplied or put on reserve in Brain Education Library in Cleveland Hall, throughout the semester.
Comments:
-
Topics:
Each of you will be able: to create and implement effective pedagogical strategies incorporate collaborative learning and appropriate technology design a variety of assessment tools connect mathematics to the real world incorporate inclusive teaching strategies acquire knowledge of the state learning goals and Essential Academic Learning Requirements by (a) demonstrating knowledge of the goals (b) demonstrating skill in developing curriculum, instruction, and assessment of students in grades 4-12 Washington Math Standards, and (c) demonstrating the ability to have a positive impact on 4-12 students learning in the Washington Math Standard
Submitted by:
Kim Vincent
Date Submitted:
7/6/07
Math 340
Title:
Introduction to Mathematical Biology
Prerequisites:
Math 140, 172, and 3 hours of biology.
Comments:
Mathematical biology and development of mathematical modeling for solutions to problems in the life sciences.
MATH 398
Mathematical Snapshots
Character, life work, and historical importance of mathematicians from various eras and branches of mathematics
Text:
A Concise History of Mathematics, 4th Edition by Struik. Published by Dover
Text Sections Covered:
----
Comments:
Each student will write an independent term paper on a mathematical topic not discussed in class such as: a biographical sketch of a famous mathematician; the development of an important mathematical concept; the evolution of mathematical notation; a famous controversy over priority of discovery; an exposition of mathematics in an ancient culture, etc. The paper should be written carefully with respect to content, format, and language.
Topics:
Egyptian and Babylonian Mathematics; Arithmetic of Central and South America; Ancient Chinese Mathematics; Early Greek Mathematics; Apollonius, Archimedes, and Euclid; Ancient Indian Mathematics; Reawaking of European Mathematics; Irish Mathematicians; The Story of Two Greek Mathematicians of Modern Times; Sophia Kobalevskaya and Mathematics in 19th Century Russia; The Life of Alan Turing; Women in Mathematics.
Sequences and series of functions; Infinite series of constants; Continuous real-valued functions of n variables; Partial derivatives and the differential; The Chain Rule and Taylors Theorem; Linear Transformations and Matrices
Submitted by:
Alexander Panchenko
Date Submitted:
4/24/09
Math 403
Title:
Higher Geometry
Prerequisites:
Math 220 or permission of the instructor
Comments:
Students are expected to have had a high school geometry course and are somewhat familiar with the basics of analytic geometry. The course has natural historical and philosophical aspects, and it reviews some of the more intellectual challenging problems solved over the centuries. Although Intended for potential secondary teachers, the course should be rewarding to many liberal arts students.
Text:
Geometry: A Historical Perspective, by M. Kallaher
Text Sections Covered:
-
Comments:
Format will be a mixture of group discussion, small group interaction, lecture.
Grading will be based on group problem sets, tests and one project.
Topics:
The theme will be historical with emphasis on the development of geometry from the time of Euclid to modern times.
During the semester various types of geometries (including non-euclidian, projective, finite) will be discussed.
Math 421
Algebraic Structures
A First Course in Abstract Algebra, 7th Edition by J. Fraleigh; Published by Addison Wesley
Text Sections Covered:
Chapters: 1-18
Comments:
-
Topics:
We shall cover basic results about groups, rings and fields, together with some of their applications.
Submitted by:
Judi McDonald
Date Submitted:
10/11/07
Math 423/523
Title:
Statistical Methods for Engineers and Scientists
Prerequisites:
Prerequisite for Math 423: Math 220, 360 or other statistics course. Math 523 prerequisite is graduate standing. Credit not normally granted for both 423 and 523
Comments:
This class is a continuation of the material presented in Math/Stat 360 and is the compliment of those topics that are virtually indispensable for engineers and scientists in an age of global competition for manufacturing quality items. The principle focus is design of experiments with applications to quality control where analysis of data from industry will be an integral part of the course.
Text:
Probability and Statistics for Engineering and the Sciences, 7th Edition by Jay L. Devore; Published by Duxbury Press.
Text Sections Covered:
Chapters 7 - 13.
Comments:
All methods will be illustrated with actual problems originating from industry during the laboratory sessions. These sessions will provide instruction in, and implementation of, commonly used statistical software such as Minitab or SAS.
An analysis of intersection of culture, gender & math. Including, but not be limited to: eurocentrism & androcentrism in math, the role of culture in the development, learning of math, a study of gender and race/ethncity differences in math, their social consequences, factors influencing these differences, historic roles of women people of color.
Comments:
-
Topics:
Critically evaluated eurocentrism & androcentrism in math. Explore the ways culture affects the development & learning of math. Investigate gender and race differences in math and their sociological consequences. Examine factors influencing gender & and race differences in mathematics and learning styles. Critically evaluate research on the intersections of gender, race, mathematics, and mathematics education. Understand culturally responsive teaching. Create projects for high schools and/or middle schools that are suitable for a culturally responsive classroom.
Submitted by:
Sandy Cooper
Date Submitted:
5/20/08
Math 432/532
Title:
Mathematics for College and Secondary Teachers
Prerequisites:
Teaching experience or intention; Calculus and linear algebra.
Comments:
Credit not granted for both Math 432 and 532
Text:
Mathematics for High School Teachers: From an Advanced Perspective, First Edition, by Usiskin, Peressini, Marchisotto, Stanley; published by Prentice Hall, 2003,
Text Sections Covered:
to be decided
Comments:
--This course is intended for students of senior status or beyond. However, any juniors who will not be here in two years should take this course now for it will be offered in the spring of even years only. --This course will look at the mathematical content in courses taken prior to Calculus from an "advanced perspective", meaning we will use mathematics from your college career to develop a deeper understanding of the mathematics content in the high school curriculum. --For graduate students enrolled in Math 532, some of their homework will be different than the undergraduates in Math 432; some of their work on exams, homework, and projects will have a higher level of mathematics embedded and expected.
Topics:
Pre-algebra, algebra functions and geometry examined from an advanced perspective
series solutions of ordinary differential equations; the method of Frobenius; development of Bessel functions of the first and second kinds;Legendre polynomials; Sturm-Liouville problems;Laplace transforms and inverse transforms, and their use in solving differential equations; the Wave Equation and solution; the Heat Equation and solution.
MATH 448/548
Numerical Analysis
The course is designed to teach science and engineering students how to derive and use standard numerical methods for mathematically posed problems from science and engineering. The course is cross-listed with Cpt S 430/530. It is normally offered Fall and Spring semesters.
Most of Chapters 1-6 will be covered. The Matlab Primer is supplementary.
Comments:
Computing:
--Computing is an essential part of the course and some of the assignments will require computer programming work. Completion of these computing assignments if necessary for receiving a good course grade. The course textbook authors provide FORTRAN, C and MATLAB software (see text page vii for access information) for algorithms discussed in the text. I strongly recommend using the MATLAB software. The supplementary text provides detailed MATLAB information and there are on-line MATLAB information links at the course website, where there is also information about on-line access to MATLAB for all students in the course.
Math 456/556
Introduction to Statistical Theory
Introduction to Probability and Mathematical Statistics, 2nd Edition by Bain & Engelhandt; Published by Duxbury
Text Sections Covered:
We aim to cover Chapters 8-12 from the text, emphasizing certain sections more than the others. Time permitting, we would cover either Chapter 13 or Chapter 14.
Comments:
This is a course on Mathematical Statistics. The aim is to have an in-depth understanding of the theory behind Inferential Statistics. Substantial importance will be given to proofs of fundamental results in Mathematical Statistics.
Math 464
Title:
Linear Optimization
Prerequisites:
Math 220 and Math 273, Math 364 is recommended.
Comments:
Familiarity with elementary concepts of linear algebra including matrices and vectors is expected along with experience in doing proofs. Cooperative course taught by WSU, open to U of I students (Math 464).
Text:
Introduction to Linear Optimization by D. Bertsimas & J. Tsitsiklis; Published by Athena Scientific (1997); A recommended book: Linear Programming and Network Flows, by Bazaraa, Jarvis and Sherali, published by Wiley.
Text Sections Covered:
The first part of the course will be based on the first 7 chapters of the text. Most of the second part of the course would be devoted to topic (9). References to topics (7), (8) and (9) will be provided as they are treated in class.
Sections from Chapters 1,2,3,4 and 6.
Comments:
-
Topics:
The course will consist of two parts. The first part will consist the following topics. (1) Introduction. (2) A brief review of some results from linear algebra and convex analysis including convex sets and functions. (3) The simplex method. (4) Starting solutions and convergence. (5) Duality, sensitivity and the dual simplex algorithm. (6) The decomposition principle. In the second part, we shall consider problems that arise in the following areas. (7) Allocation and scheduling. (8) Approximating data by linear functions. (9) Integer programming. Formulating linear programs.
Network flow problems form an important class of linear optimization problems with applications to several areas including chemistry, computer networking, engineering, public policy, scheduling, telecommunications, transportation, and many others. This course will provide an integrated view of the theory, algorithms, and the applications of key network optimization problems. Emphasis will be on powerful algorithm strategies, rigorous analysis of the algorithms, and data structures for their implementation.
Complex functions and continuous functions Complex differential calculus Holomorphy and conformality Modes of convergence, power series Transcendental functions Complex integral calculus Integral theorems and power series development Consequences of the integral theorems Meromorphic functions Laurent series Residue calculus and its application to definite integrals
Math 505
Abstract Algebra
Basic group theory will be reviewed. The Sylow Theorems will be covered in detail. Modules and Fields will also be studied in some depth.
Submitted by:
Jessica Cross
Date Submitted:
4/28/15
MATH 507
Title:
Advanced Theory of Numbers
Prerequisites:
General mathematical maturity
Comments:
------
Text:
Cryptanalysis of Number Theoretic Ciphers, 1st Edition by Samuel Wagstaff; Published by Chapman & Hall/CRC Press
Text Sections Covered:
Chapters 1-14, 17, 18, 23-25
Comments:
We will review the major topics from elementary number theory, then look at how number theory is used in cryptology. Special emphasis will be on factoring of large numbers and solving the discrete logarithm problem. These concerns will lead to some more advanced topics.
The course basically deals with two topics: 1. The variety of ways in which a problem Lu = f can be solved where L is a linear matrix, differential, or integral operator. 2. Various asymptotic representation of integrals I(k) as k goes to infinity, where k is a large positive parameter. -- The methods of solution for (1) include eigenvector expansion and direct approaches, the former introducing the concepts of adjoint linear operators and eigenvalue problems while the latter develop inverse operators, Greens functions, and distributions. Then these two approaches are related by means of spectral representations of operators. -- For (2) appropriated asymptotic expansions of integrals of particular forms are deduced by Watsons Lemma, LaPlaces Method and the Method of Stationary Phase, respectively. Finally these general expressions are applied to deduce asymptotic representations for the gamma, error and Bessels functions as well as Legendre polynomials.
Submitted by:
David Wollkind
Date Submitted:
3/15/06
Math 510
Title:
Topics in Probabilities and Statistics
Prerequisites:
One 3 hour statistics course.
Comments:
Graduate-level counterpart of Math 410; Credit not granted for both Math 410 and 510.
Text:
-
Text Sections Covered:
-
Comments:
For more information, see Stat 510 (taught by Department of Statistics).
We will focus on spectral theory of matrices, including unitary equivalence, similarity, normal matrices, Jordan canonical form, as well as material on inner product spaces and matrix norms. More details will be given during the first lecture.
Math 512
Ordinary Differential Equations
Prerequisites:
Math 402
Comments:
-
Text:
Differential Dynamical Systems, James Meiss (SIAB, 2007)
Text Sections Covered:
Chapters 1-5 and selected sections of latter chapters
Comments:
Computer: We will learn something about XPPAUT, a software package for making a range of computations related to the initial value problem for ODEs. This has been developed by Bard Ermentrout, of the University of Pittsburgh.
Combinatorics has become a catch-all discipline of mathematics encompassing such things as graph theory, enumeration, analysis of algorithms, recursion, et al. At the heart of it all is the desire to count things in an elegant manner.
Math 570
Mathematical Foundations of Continuum Mechanics I
Mathematics Applied to Deterministic Problems in the Natural Sciences, by C.C.Lin & L.A.Segel; Mathematics Applied to Continuum Mechanics, by L.A.Segel, both Published by SIAM
Text Sections Covered:
This basically is an introduction to the foundations and techniques of modeling natural phenomena from a deterministic continuous viewpoint. It includes the following topics to be presented in a two semester sequence: Cartesian tensors, eigenvalue problems, the continuum hypothesis, Eulerian and Lagrangian coordinates, the Reynolds transport theorem, the DuBois-Reymond lemma, conservation of mass, balance of linear and angular momentum, the principle of local stress equilibrium, conservation of energy, the Clausius-Duhem inequality, equations of state and constitutive relations, boundary conditions and surfaces of discontinuity, asymptotic expansions, regular and singular perturbation theory, and linear and nonlinear stability analyses. These topics are developed in the context of various problems in the continuum with an emphasis on fluid mechanics but with an inclusion of metallurgical solidification and chemical Turing pattern formation as well.
Comments:
There are ten required problem sets and a take home final containing case studies closely related to those presented in class.
There are ten problem sets each worth 40 points covering the topics enumerated above for a total of 400 points. No examinations. All three lessons in week 15 will be slide shows highlighting research areas. These deal with weakly nonlinear stability
Math 574
Topics in Optimization
Decent level of analytical ability and the interest to learn problems from biology required. Course will be adapted to accommodate students from non-mathematical backgrounds.
Text:
An Introduction to Bioinformatics Algorithms, by Neil C. Jones and Pavel A. Pevzner; Published by MIT Press; ISBN: 0262101068
Text Sections Covered:
Portions from Chapters 2-6, 8, 9, 11
Comments:
Students will be graded through homework assignments and projects -- there will be no exams. Main emphasis will be on learning how to apply optimization techniques rather than on the theory behind them.
Check course web page:
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Find an Ashburn, VA MathAlgebra is the abstraction of basic arithmetic, using letters to stand in for specific known or unknown numbers. The abstract notation of algebra often gives new students difficulty, but the concept, when properly explained, is not difficult. A good working knowledge of algebra is essential to fields like science, engineering, math, economics, and finance
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Mathematics has been described as an "invisible culture," shunned by
many educated adults, yet exercising profound influence on all aspects
of society--from engineering to economics, from strategic planning to
political polls. The avoidance of mathematics extends also to campuses,
where students, faculty, and administrators expend valuable educational
energies arguing about multiculturalism and politically correct
curricula, meanwhile ignoring less glamorous yet equally important
educational policy concerning the nature of the university's mathematics
curriculum.
Make no mistake about it: even as mathematics conveys the power of
ideas, it also entails profound socio-political consequences. Success or
failure in mathematics determines access to courses and curricula which
lead to positions of influence in society. The increasing role of
technology in the world of work amplifies the already strong signal sent
by the scientific revolution that the language of mathematics is an
essential component of literacy for our age.
At its best, education can be the equalizer of socio-economic
differences. Yet more than any other subject, mathematics serves as a
filter, enhancing or blocking access to professional careers in a manner
that has disproportionately negative consequences for women and
minorities. When students drop out of college for academic reasons, the
culprit is often mathematics--not just because of one poor grade, but
because failure in mathematics prevents further progress in so many
other subjects.
For years, educators assumed that this was just in the nature of
things--that mathematics learning was the result of genetic and cultural
factors that predisposed certain people to success and others to
failure. This belief endured--and still persists--despite significant
evidence that it simply is not true. The records of many small colleges
(especially women's colleges and historically black colleges), the
success of special intervention programs at various universities, and
evidence from educational research show that the traditional lecture
style is effective only for students who arrive with uncommon levels of
motivation and persistence; that most students learn better with more
active, varied modes of instruction; and that virtually all students can
succeed in mathematics provided they are placed in and supported by an
appropriate community of learning.
There is no longer any excuse for excessive failure rates in college
mathematics. Examples abound of ways to improve success rates for all
students, even for those with poor mathematics preparation. Although
effective programs may appear to cost more than ineffective ones, the
benefits of success--in reduced repetition of courses, in improved
retention and graduation rates, in increased opportunities for
students--far outweigh the visible costs of these programs. In some
cases, effective programs may actually be less expensive overall.
Deans and provosts who wish to improve mathematics education on their
campuses need to recognize two realities:
Mathematics canbe learned by most students.
The cost of failure is often higher than the cost of success.
Armed with these convictions, they can begin the process of self-renewal
by posing to their mathematics departments a series of critical yet
frequently unasked questions:
About Students
Who are your students? More fundamentally, do you know who your
students are and why they are in your courses? Students who enroll in
college mathematics courses arrive with amazing mixtures of aspirations
and anxieties, often exaggerated, always intensely personal. Since
student attitudes towards mathematics frequently have more influence on
performance than do remembered skills or school-based learning, the
first step to improved success is better understanding of the
motivations of your students.
Do departmental priorities match the institutional mission? More
pointedly, is your mathematics faculty committed to teaching the
students you have? It is all too easy for faculty to covet students who
fit an imagined mold of young scholars created in the faculty's image,
or to view every first year student as a potential mathematics major.
Instructional practice based on false assumptions yields disillusionment
for both students and faculty. Effective instruction must harmonize the
goals of the institution with the expectations of its students.
Do you believe that your department should educate all students?
More concretely, do you believe that all students can learn mathematics?
Do you offer appropriate and appealing courses that meet the needs of
all students who enroll in your institution? Do you apply as much
creative energy to improving the most elementary courses (those often
termed "developmental" or, more derisively, "remedial") as to those that
are more advanced? Most mathematics used in the world, after all, is
just simple school mathematics applied in unusual contexts.
Do you have explicit goals for increasing the number of students from
under-represented groups who succeed with mathematics courses?
Vague intentions without explicit goals are too easily ignored. There
are precious few departments of mathematics in the country whose records
of success with Black, Hispanic, and other under-represented groups
could not be significantly improved. Specific goals must relate to
specific institutions, but surely one aspiration must be that
mathematics becomes a pump rather than a filter for students who have
been traditionally under-represented in the professional fields that
build upon college-level mathematics.
What do your students achieve? More specifically, how do you
know what your students have really accomplished? End-of-term exams
generally reveal only short-lived mastery of procedural skills. Do you
have any evidence concerning long-term goals or broad objectives? Do
you ever ask students to solve authentic, open-ended problems, or to
write, read, or speak about mathematics? Do courses provide
opportunities for students to learn anything other than textbook-based
template exercises? To what extent are course grades based on an
examination of these broader goals of mathematics education?
Do you know what happens to students after they leave your
courses? Do students who take mathematics courses go on to use
their mathematics in subsequent courses? What about students who drop
out or fail: do they give up on mathematics, or do they return and
succeed in a subsequent course? Do those who receive good grades find
that what they learned serves them well in subsequent courses? Do you
have a mechanism for adjusting curricular emphases based on feedback
from students who have taken your courses?
About Curriculum
Do departmental objectives support institutional goals? Despite
variations in rhetoric, widely shared goals for mathematics education
are entirely consistent with broad goals of higher education: to
develop students' capabilities for critical thinking, for creative
problem solving, for analytic reasoning, and for communicating
effectively about quantitative ideas. Yet the implicit objectives of
many mathematics departments, as inferred from curricula, exams, and
student performance, are often focussed on mastering relatively
sophisticated yet intellectually limited procedural skills. Departments
must express for themselves--and even more so, for their students--how
their course objectives advance their institution's educational
goals.
Do your courses reflect current mathematics? Since mathematics is
such an old subject, it is all too easy for its curriculum to become
ossified. Strong departments find that half of their courses are
replaced or changed significantly approximately once a decade. As new
mathematics is continually created, so mathematics courses must be
continually renewed. Does the mathematics curriculum reveal to students
a level of innovation and attractiveness that reflects the excitement of
contemporary mathematical practice?
Are your faculty aware of the new NCTM Standards for school
mathematics? More importantly, are you making plans to provide an
appropriate curriculum that builds on the foundation of these
Standards , following their spirit as well as their content?
Colleges must be prepared for students arriving with increasingly
disparate backgrounds--many from traditional authoritarian,
exercise-based courses while an increasing number of others will enter
college fresh from an active, project-centered approach that typifies
the new school Standards. It would be ironic, indeed tragic, if
intransigent college mathematics departments were to hold back reform of
school mathematics by refusing to adapt to the new reality of a more
diverse and powerful secondary school curriculum.
Are calculators and computers used extensively and effectively?
Beginning with placement exams and continuing all the way through senior
courses, calculators and computers should be used in every appropriate
context. Since the mathematics used in the scientific and business world
is a mathematics fully integrated with calculators and computers, the
mathematics taught in college must reflect this reality. Anything less
short-changes students, parents, and taxpayers.
Do you know what your majors do after graduation? How many take
jobs in which they use their mathematics training? How many enter
secondary or elementary teaching? What about graduate school--in
mathematical sciences, in other sciences--or professional school? How
well suited is your curriculum to the actual experience of your
graduates?
Does your program help students see how mathematics connects to broad
issues of human concern? Specifically, do your faculty and courses
connect mathematics to student aspirations, to liberal education, to
other disciplines? Do students who study mathematics emerge empowered to
think and act mathematically in broad contexts beyond the classroom?
Unless this happens, students feel cheated by lack of reward
commensurate with effort required in typical mathematics courses.
About Faculty
How does the scholarship of your faculty relate to the teaching
mission of your department? Does your faculty subscribe to a
narrow view of research or to a broad perspective on scholarship? Does
the department both expect and support professional development in its
varied forms? Is there a departmental commitment to offer all majors
suitable professional, scholarly, research, or internship opportunities?
Traditional standards of mathematical research make direct connections
to undergraduate teaching rather difficult, whereas a "reconsidered"
view of scholarship eases constructive engagement in which faculty can
thrive professionally and students can become junior colleagues.
What steps has your department taken to be sure that your faculty are
well informed about curriculum studies and research on how students
learn? Part of the professional responsibility of faculty is to
know the scholarship that undergirds college teaching. Everyone has
opinions about curriculum and pedagogy, but professionals need to
support their opinions with evidence. Since graduate education in
mathematics rarely provides any introduction to this arena of
scholarship, departments must accept it as part of their
responsibilities. Regular faculty seminars on issues of curriculum,
teaching, and educational research help focus faculty attention on these
important issues at the same time as they help create an environment for
learning that is crucial to student success.
What are your priorities for teaching assignments? In
particular, do you assign your best teachers to beginning courses? Are
courses for non-majors given the same priority as those for majors? Do
your faculty prefer students who learn without being taught or those who
challenge teachers to teach effectively? How do faculty rewards reflect
the teaching challenges they undertake? Is the quality of faculty
teaching measured by the good students they attract to their courses or
by the improvement of students in their courses?
Are your faculty fulfilling their responsibility for the preparation
and continuing professional education of teachers? The new
Standards for school mathematics include clear expectations for
both content and pedagogical style in the mathematical preparation of
school teachers at all grade levels. How many members of your
mathematics department are familiar with these expectations? To what
extent do your courses conform to these standards? What steps are you
taking to ensure that all mathematics courses taken by prospective
teachers meet appropriate professional expectations?
How are faculty resources allocated between courses that serve the
major and those that serve general education? Typically 80% of the
students in a mathematics department are enrolled either in service
courses or in general education courses. Often these 80% of students
command only 20% of faculty time and energy. Yet it is these students
who will go on to be future policy leaders of society--members of boards
of education and city councils, editors of local newspapers and leaders
of Chambers of Commerce.
About Costs
Have you calculated the true cost of the status quo? Courses
staffed on the cheap (the "cash cow" approach to funding mathematics
departments) result in students repeating courses, or failing related
science courses, or dropping out of college altogether. Students who
succeed in their first college mathematics course are far more likely to
succeed in college than those who who do not. Cheap courses are not
necessarily as cost-effective as they appear.
Are you aware that mathematics departments exercise disproportionate
influence over an institution's graduation rate? A small increase
in the percentage of students who complete mathematics courses with a
well-earned sense of accomplishment can translate into increased
graduation rates in many disciplines that depend on mathematics.
Conversely, any decline in the success rate in elementary mathematics
courses cascades into even larger drop-out rates by students who find
themselves lacking prerequisites for key courses in their majors.
What resources are required to achieve your objectives? This is
the most important question of all. Mathematics cannot be taught
successfully without resources adequate to the task. Many mathematics
departments suffer not only from insufficient resources, but also from
inefficient distribution of existing resources. In return for a prudent
self-study by a department of mathematics, the institution must be
prepared to focus resources on promising new approaches in which the
cost of success compares favorably with the cost of failure.
The spotlight of national attention that has been aimed at mathematics
and science education has revealed not only weaknesses in the present
system, but also outstanding examples of success. Mathematics need not
remain a barrier to higher education. Investment in programs that make
possible increased success in mathematics provides great leverage for
any institution that wishes to improve the overall education of its
students.
| 677.169 | 1 |
Steven G. Krantz is Chairman of the Mathematics Department at Washington University in St. Louis. An award-winning teacher and author, Dr. Krantz has written more than 30 books on mathematics, including Calculus DeMYSTiFieD , and Differential Equations DeMYSTiFieD . He is the former Deputy Director at the American Institute of Mathematics. more...
LEARNING CALCULUS JUST GOT A LOT EASIER! Here?s an innovative shortcut to gaining a more intuitive understanding of both differential and integral calculus. In Calculus Demystified an experienced teacher and author of more than 30 books puts all the math background you need inside and uses practical examples, real data, and a totally different... more...
Here's the perfect self-teaching guide to help anyone master differential equations--a common stumbling block for students looking to progress to advanced topics in both science and math. Covers First Order Equations, Second Order Equations and Higher, Properties, Solutions, Series Solutions, Fourier Series and Orthogonal Systems, Partial Differential... more...
Krantz takes the reader on a journey around the globe and through centuries of history, exploring the many transformations that mathematical proof has undergone from its inception at the time of Euclid and Pythagoras to its versatile, present-day use. The author elaborates on the beauty, challenges and metamorphisms of thought that have accompanied... more...,... more...
This concise, well-written handbook provides a distillation of real variable theory with a particular focus on the subject's significant applications to differential equations and Fourier analysis. Ample examples and brief explanations---with very few proofs and little axiomatic machinery---are used to highlight all the major results of real... more...
This text provides a masterful and systematic treatment of all the basic analytic and geometric aspects of Bergman's classic theory of the kernel and its invariance properties. These include calculation, invariance properties, boundary asymptotics, and asymptotic expansion of the Bergman kernel and metric. Moreover, it presents a unique compendium... more...
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Most Helpful Customer Reviews
I have no hesitation in recommending this book, it is one of the best books on the interface of pure mathematics and Applied mathematics. Often Applied maths or mathematical methods courses for physicists and engineers are simply an introduction to a series of techniques how to solve differential equations in various co-ordinate systems, a bewildering range of special functions, Bessel, Hermite, Legendre and the ridicuously named Confluent Hypergeometric function. There is no attempt to explain how all these functions have similar properties when considered as orthonormal vectors in a vector space. Neither is there any real justification given for why Fourier series actually converge to a given solution. On the other hand most pure maths will just concentrate on the underlying structure of the mathematical object they are concentrating on with no real hint as to why the stuff they are studying is useful. This book is one of the few I know of that gives an introduction to the underlying mathematical structure behind all the various solutions to linear differential equations as well as including plenty of concrete examples and introducing the reader to a wide range of techniques. Thus it has one foot on the ground in the fact that the reader is shown numerous techniques for solving differential equations in various coordinate systems and one foot (for want of a better phrase) in the heavens as the underlying mathematical structure is also given. The analogy between geometric vectors and functions in function space is one of the profoundest in all of mathematics and its applications to physics both classical and quantum. This book provides the best introduction to this subject I am aware of.
| 677.169 | 1 |
Graph plotter programEquation graph plotter gives engineers and researchers the power to graphically review equations, by putting a large number of equations at their fingertips. The program is also indispensable for students and teachersFeatures:
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The program allows you to solve algebraic equations in the automatic mode. You just enter an equation in any form without any preparatory operations. Step by step Equation Wizard reduces it to a canonical form performing all necessary operations. After that it determines the order of the equation, which can be any - linear, square, cubic or, for instance, of the 7-th power. The program finds the roots of the equation - both real and imaginary.
Statistical Analysis and Inference Software for Windows covering everything from Average, Mode and Variance through to Hypothesis Analysis, Time Series and Linear Regression. Includes Online Help, Tutorials, Graphs, Summaries, Import/Export, Customisable Interface, Calculator, Live Spell Check, Install/Uninstall and much more.
The Eye4Software Coordinate Calculator is an easy to use tool to perform map or GPS coordinate conversions. The software can be used to transform a single coordinate, or a batch of coordinates read from a comma separated,database or ESRI shapefiles. Supported map projections include: Transverse Mercator, Oblique Mercator, Mercator, Oblique Stereographic, Polar Stereographic, Albers Equal Area Conic, Krovak, Lambert Conformal Conic and more.
Need a quick and easy way to visually discover the difference between Fahrenheit and Celcius? Fahrenheit Converter is the way to go! Simply watch conversions appear before your eyes as you slide up and down the temperature scale! Its that easy. You will soon discover general relations between Metric and US measuring systems visually. Fahrenheit Converter is a great tool for both teacher and student alike! absolute minimum; i.e., a mathematical model, constraints, and the objective (function) definition. Minimizing the amount of code allows the user to concentrate on the science or engineering problem at hand.
absolute minimum; i.e., a mathematical model, constraints, and the objective (function) definition. Minimizing the
amount of code allows the user to concentrate on the science or engineering problem at hand. 60+ examplFindGraph is a curve fitting and digitizing tool for engineers, scientists and businesss. Convert plots back to X,Y data from any published report for later processing and manipulations. The program offers 12 generic fits, including linear and nonlinear regression, Logistic functions, Fourier approximation, Neural Networks, B-splines curves, cycles extraction, SSA forecasting, and plus a library of over 300 industry-specific formulas.
Fraction Calculator name speaks for itself. This piece of software will help you to add, subtract, multiply and divide fractions. Enter fraction #1 and fraction #2. Make sure you use / between numerator and denominator (e.g. 1/3 for one third). Select what operation you want to perform and click Calculate Fraction button to see result.
Two programs in one. One that can Add, Subtract, Divide and Multiply fractions and another that can convert a decimal to a fraction or a fraction to a decimal. It's a breeze with "Fractions". Quick and easy interface. No confusing menus. Many fraction programs only convert fractions to a decimal. This unique program converts decimals to fractions. Great for school or work. Handy for STOCK quote conversions. Get the Pocket PC edition for Windows CE absolutely FREE when you register the desktop version.
Free Random Number Generator is an advanced tool developed to generate list of true random numbers. Free Random Number Generator is very fast in executing and highly optimized for generating truly random numbers. Software also provides you the feature to create list of sequential numbers and constant value number. Random Number Generating tool is designed with easy to understand Interface.
Free Random Number Generator allows you to create list of thousands random numbers with only single mouse click. Random Number list creator is generates 5000 random or sequential numbers at a time and create much longer numbers, as longer as 18 digits. Random Number Generator Software also provides you option to add suffix and prefix with numbers. Random Number list generating software is developed in Windows environment so it is compatible with all Windows versions.
Features:
? Developed with Powerful algorithm to generate True Random Number List.
? Generates passwords of any length till 18 digits.
? Generate Li ...
Free Square Footage Calculator can operate in your browser. No additiional software required. Intended for the Commercial Cleaning Industry, but available here to anyone with a requirement for square footage calculations.
Unlike traditional calculators this easy to use software calculator displays all input data ,making it easy to double check on data entered ,data can also be randomly altered without having to re-input the entire calculation . It also has full data saving , back-up and protection features, is capable of doing long and multiple calculations and has Vat capabilities.Has many other features not available in traditional calculators including text input, data print-out, function key assignment, template download and file transferring abilities.Will generate up to 25 pages or approx 5000 entries.A very handy little calculator to have on your desktop.
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One of my favorite ways to teach about functions is to use the machine analogy.
There are several applets out there that use this. I just searched the web and
found the one cited below which is part of a comprehensive unit on functions. It
is worth a look.
Also, I have found the machine analogy to be very useful for explaining
composition of functions. Also, several years ago some memebers of Bolt, Beranek
and Newman's educational group developed a piece of software called function
machines that was very well done. I haven't seen it in years. If anyone reading
this knows of its' current state please contact me or post its' where abouts
here
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Forgotten Algebra 2ND Edition a Self Teaching Re
Synopses & Reviews
Publisher Comments:
This updated book is a self-teaching brush-up course for students who need more math background before taking calculus, or who are preparing for a standardized exam such as the GRE or GMAT. Set up as a workbook, Forgotten Algebra is divided into 31 units, starting with signed numbers, symbols, and first-degree equations, and progressing to include logarithms and right triangles. Each unit provides explanations and includes numerous examples, problems, and exercises with detailed solutions to facilitate self-study. Optional sections introduce the use of graphing calculators. Units conclude with exercises, their answers given at the back of the book. Systematic presentation of subject matter is easy to follow, but contains all the algebraic information learners need for mastery of this
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The Advantage Series
These multidisc programs—for elementary, middle, and high school kids, as well as math for grades 6 through 12—are not new, but that doesn't diminish their quality.
Elementary Advantage covers ten subjects with best-of-breed children's software. Its mid-1990s writing program, Storybook Weaver Deluxe, is still one of the most effective programs for engaging kids in writing. Middle School Advantage covers ten subjects, including algebra, typing, and Spanish, and it is one of the few offerings that teaches grammar.
Math Advantage is very comprehensive including topics such as business math, precalculus, calculus, and statistics; in fact, it may be too diverse. High School Advantage, the weakest of the series, has solid math help in algebra, geometry, and trigonometry. But writing is treated as a rote task, and only the 20th century is covered in the world history section
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Basic Requirements for Math HTML Projects
Since presentations dates are Tue. Dec. 3—Thu. Dec. 5,
presentation order is not considered critical, but will normally
proceed in the order listed.
For Geometry the topics are biographies of famous Geometers.
Creating a web page as specified below is part and parcel
of this project.
All partners and topics must be selected by Friday,
November 1, 2002 or will be assigned.
Students will create an HTML file.
The file must be formatted as discussed below.
It should be about 2 or 4 pages in a format
(font size, text density, and style) similar to the 15
numbers and 8
statistics lectures already available.
Students will present their topic to the class in a 6 to 8 minute presentation.
This is a presentation.
Reading of your web page is not allowed.
Powerpoint and such may not be available.
Most biographies are for people not done before.
Those which were (Pythagoras, Gauss, Dedekind, Riemann, Hilbert)
might check with last year's presentors for their paper and critique.
Many were worse than the web page the year before which may be available online.
Plans have been to make copies of each
appropriate presentation for each student.
Important project material will be tested on semester exams.
Your presentation should focus on 3 to 5 specific questions
concerning your topic. E-mail these to
calkins@andrews.edu
before Nov. 15 for approval. Be sure to answer these questions.
All necessary additions, corrections, and final alterations
shall be made to your web page by Friday, Dec. 13, 2001 and stored in
the PC in SH100 in the directory \calkins\www\math\biograph.
All information collected should be "common knowledge", that is
widely available in most common sources. Thus references are not
required and are actually to be avoided in the final product.
However, good research principles dictates that you retain such
information at least until the project is completed, accepted, and graded.
File changes, if from a prior year, should not be frivolous but adhere to
the file format requirements listed below. Even more so than reading,
this was the biggest problem last year.
Roughly 50% of the web page should focus on their mathematics,
25% on how their work related to others, leaving
no more than 25% to cover personal aspects.
File Formatting Requirements for these Projects
HTML files are "flat ASCII" files, otherwise known as text files.
They thus differ from many word processor files which may have
"control" or other "unprintable" characters. The following are
requirements and not guidelines—failure to follow them will
severely impact your final grade assignment. Some may seem
capricious and arbitrary, but are present to ensure long-term
preservation and availability over a wide range of operating environments
and presentation formats.
Your file will be named either "bioxxxxx.htm", where
xxxxx is up to a 5 letter abbreviation for the person's last name or the topic.
Only lower case letters may appear in the file name.
Do not change the name of the file if one is given you.
The file name will thus not exceed the former DOS "8.3" file name format.
If any additional files are used,
they must adhere to this naming convention, except the last "x"
in the file name above may be replaced with a digit.
No JAVA, JAVAScript, TCL, CGI, or other similar HTML
language extensions are allowed. No FRAMES are allowed.
You should focus on the content, not endlessly on such distractions.
Focus on content and structure, especially emphasizing facts to remember.
Double-check your spelling and your facts---we intend to!
Only GIF/JPG files for special characters already in
may be used. Any
other graphics files must be original work with all rights transferred,
OR a link established to a reliable source. Reliable means
likely to be there for the next 50 years as well as correct/accurate.
An ALT="[graphic description]" entry must be included for each graphic.
No background colors are allowed.
Students tend to pick dark colors which do not
show up well on the data video projector.
Also, no specification for text/link color is allowed (no white text!).
Several links to relevant web resources should be provided.
"Records" should not exceed 99 characters, thus ensuring
editting to be a one dimensional process.
HTML tags shall all be in upper case so they can be easily
ignored while editting the text.
File heading, footing, and table tags should be specified
exactly as displayed below.
The files should thus not take on a C program nature, where
white space detracts from the content.
Other specifications may be added as necessary.
File Preamble
<HTML><HEAD><TITLE>Biography of xxxxx</TITLE></HEAD><BODY>
<A HREF="biotoc.htm">Back to the Table of Contents</A>
<H1 ALIGN=CENTER>Biographies of Mathematicians - xxxxx</H1>
Tables for important facts
The most important facts to remember should be blue-boxed exactly as below.
<TABLE WIDTH="100%" BGCOLOR=LIGHTBLUE CELLPADDING=5><TD ALIGN=CENTER>
The most important facts to remember should be blue-boxed exactly as below.
</TD></TABLE><P>
File Postamble
The file should conclude with the following information.
You may enclose project, yyyyy, and zzzzz with links to
your version of the paper and your corresponding home pages.
xxxxx is replaced by part of the mathematician's last name.
This should appear below last year's entry–do not replace.
Use one line per year, with breaks not paragraphs.
If prior students no longer have an AU account, remove their link.
This project was prepared and presented by students yyyyy and zzzzz in 2000.
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Meyer's Geometry and Its Applications, Second Edition , combines traditional geometry with current ideas to present a modern approach that is grounded in real-world applications. It balances the deductive approach with discovery learning, and introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The text... more...
Deformable objects are ubiquitous in the world, on various levels from micro to macro. The need to study such shapes and model their behavior arises in a wide spectrum of applications, ranging from medicine to security. This book provides an overview of the state of science in analysis and synthesis of non-rigid shapes. more...
Like other areas of mathematics, geometry is a continually growing and evolving field. Computers, technology, and the sciences drive many new discoveries in mathematics. For geometry, the areas of quantum computers, computer graphics, nanotechnology, crystallography, and theoretical physics have been particularly relevant in the past few years. There... more...
The family in this book is moving to a new neighborhood. They have a lot of work to do! They need to unload the moving truck, unpack boxes, and put everything away. The kids make new friends and discover all the fun they can have with the empty boxes. While building forts from the empty packing boxes, the kids discover many new shapes and their dimensions.... more...... more...
This lively book explains many of the key concepts in the field, starting with the Golden Number and taking the reader on a geometrical journey via Shapes and Solids, through the Fourth Dimension, finishing up with Einstein's Theories of Relativity. more...
This book mainly deals with the Bochner–Riesz means of multiple Fourier integral and series on Euclidean spaces. It aims to give a systematical introduction to the fundamental theories of the Bochner–Riesz means and important achievements attained in the last 50 years. For the Bochner–Riesz means of multiple Fourier integral, it... more...
The first edition of Connections was chosen by the National Association of Publishers (USA) as the best book in "Mathematics, Chemistry, and Astronomy — Professional and Reference" in 1991. It has been a comprehensive reference in design science, bringing together in a single volume material from the areas of proportion in architecture... more...
Features the classical themes of geometry with plentiful applications in mathematics, education, engineering, and science Accessible and reader-friendly, Classical Geometry: Euclidean, Transformational, Inversive, and Projective introduces readers to a valuable discipline that is crucial to understanding bothspatial relationships and logical... more...
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Sample Lesson
From the book: Introduction to the Seventh Grade Year
The Live Education Books for Seventh Grade
The Live Education Books for
Seventh Grade
Perspective Drawing
Perspective drawing is a technique used by Renaissance
artists to create the illusion of three-dimensional space on a two-dimensional
surface. In the same way that all the lines of light stream to a single point
entering the pupil of the eye so too do the appearance of all three-dimensional
lines defining the forms of the outer worldseem to coalesce at single points out on the horizon. These are called the vanishing points.
Perspective drawing is a kind of projected geometry to these vanishing points
on the horizon. It builds upon the basic Euclidean geometry principles studied
in the fifth grade and the sacred geometry of nature studied in the sixth
grade. Most students are very enthusiastic to discover that they are able to
draw accurate three-dimensional figures such as one would find in a cityscape
or in the representation of simple architecture. Your Live Education
perspective geometry book leads the student and teacher through a simple
step-by-step procedure beginning with the simplest forms such as a row of
columns and progressing to complex architecture and cityscapes. The entry into
exact geometrical construction is prefaced by easy methods of achieving three
dimensional representations through the techniques of overlapping, atmospheric
perspective and color perspective. Since geometrical perspective drawing was
developed during the Renaissance, the book also includes historical and
cultural themes from that era to be used in a main lesson presentation. There is enough material in the book to cover
three to four weeks of the main lesson block, but one can also continue with
the techniques learned for other drawing projects.
Introduction to Algebra
The Introduction to Algebra book shows the teacher how to
lead the student into that domain of mathematic variables where letters
substitute for numbers. There are 16 lessons in the book; some will take two
main lessons to complete, so 3 to 4 weeks of main lessons are required. Many
teachers have used this book successfully as imaginative presentations in the "after
main lesson practice times." Several practice periods per week are often needed
in the upper grades as supplemental work to the materials presented in main
lesson, and during this time, the student might have three lessons of algebra
per week for several months, and then the practice time is used for English and
grammar studies three times per week for several months.
The student is led gently into the realm of algebra
through imaginative stories and artistic work. Simple calculations for area,
perimeters, and circumferences are where the study begins, leading into the
concept of negative numbers, simple linear equations, and Cartesian coordinates.
It is always a story, an artistic project, or an imagination that opens the
door to the algebraic concepts; facility with the conceptual mathematical abstraction
being the final result of the process.
European Geography
The study of European geography is a natural enhancement
to the history of the Middle Ages studied in sixth grade as well as the Renaissance
and Age of Exploration of the seventh grade. The approach is to gain an
overview of all the modern countries on the European continent along with the
major watersheds, mountain ranges, characteristic flora and fauna of each
region, and an understanding of the cultural differences that exist from
country to country. Student activities include map making, drawing and watercolor
painting, modeling with clay or beeswax, narrative writing, cooking, and drama.
The Live Education book is designed to cover a 3 to 4 weeks long main lesson
block.
Physics II
Physics II is a four week study in the seventh grade that
builds upon the simple phenomena and concepts of sound, light and heat learned
in the sixth grade year. [See the Live
Education book Introduction to Physics]
For example, in the introductory book,
simple acoustic phenomena such as the quality of sound—pitch, timbre,
dissonance, etc.—are built upon with their mathematical correlates: pitch is now
seen as frequency measured in vibrations per minute or hertz; consonance and
dissonance are seen as mathematical proportions, sound intensity as decibel
levels, etc. The study of acoustics in the seventh grade leads into other
phenomena such as sympathetic vibrations, amplitude, and the mathematical
ratios of musical intervals and finds its summation in the study of the human
ear. Similarly, color and optics leads us through the study of refraction,
reflection and magnification and find their summation in the study of the human
eye.
The survey of simple phenomena in magnetism and
electricity leads to a survey of electromagnetism, the basis for so many
devices that surround us in our everyday lives: electric motors and alternators
in our cars, telephones, speakers and the stereo, a variety of electronic
components, transformers and the transmission of power over high voltagelines.
Finally, the study of physics in the seventh grade leads
into "hands on" applications with simple machines such as the lever and pulley,
and then introduces the ideas and calculations involved with work, force, and
energy.
Wish Wonder Surprise
Wish Wonder and Surprise is a study in the modes of
expression and genres of writing in English literature that capture the essence
of human expressions of awe, aspiration, and astonishment. In this 4 to 6 week
study, the student will create and experiment with various literary genres such
as rhetoric, narrative, scientific description, oratory, poetry, drama, and
story. The student will read from some of world literature's finest examples
written by authors such as Shakespeare, Robert Frost, Thoreau, Emerson,
Coleridge, etc.
The wish, the sense of wonder, and the feeling of surprise
are three fundamental modes of human experience by which we perceive and think
about the world. Wonder and awe motivate not only the poet but also the
scientist bent on discovery. The wish
partakes of a very real idealism, a vision of the future of "what is to be" or
"what is possible" and in this way embodies every human hope and aspiration
from an intimate prayer to an architect's designs for a high-rise skyscraper. The surprise always shocks us out of our
ordinary perceiving and thinking. It is often the doorway into the
extraordinary.The literature and
student activities in this block follow from these three modes of expression.
Renaissance Biographies
Renaissance Biographies constitutes a three to four week
block of history studies roughly covering the period of 1300 to 1550 CE. The
major cultural developments in Europe during this period are seen through the
lives of four major historical figures: Michelangelo, Leonardo da Vinci,
Raphael, and the alchemist Paracelsus. Through the lives of these four important
figures, one meets the Medici's, Verrochio, Marsilio Ficino, Savonarola, Pope
Leo, and other key actors on the European Renaissance stage of history. The
book is filled with many activities for the students including beginning art
studies in portraiture, and the history of commerce and money.
The Age of Exploration and Discovery
The Age of Exploration and Discovery is a another three to
four week main lesson block in history that surveys the experiences and
adventures of the explorers during the centuries beginning with the Renaissance
and leading into the late 1600's. The study begins with a look back to the
early Viking explorers of the American continent as well as Marco Polo's
exploration into China. These precursors are followed by the earliest European
explorers who ventured southward along the west coast of the African continent
leading into the opening of the spice trade to the east after successfully
rounding the continent and the eventual circumnavigation of the globe.
Conquistadores such as Cortez, Pizarro, and Balboa venturing into Central and
South America and their encounters with the indigenous peoples concludes this
block.
Recommended:
The Sentence Sounds a Melody: English Grammar for the
Upper Grades
This Live Education book which is included in the sixth
grade curriculum is designed as a source book on English grammar for the
teacher throughout the sixth, seventh, and eighth grades. The focus is upon
grammar studies including such topics as the perfect tense irregular verbs,
split infinitives, subjunctive mood, advanced punctuation such as the correct
use of colons and semicolons, where to use commas, etc.But it is also useful for developing the
student's vocabulary through a survey of Latin and Greek roots, with
corresponding suffixes and prefixes. The important topics that teach strategies
and skill development exercises in composition, business letters, essays, reports,
and effective note-taking are also covered.
If you do not have this
book it is recommended for English and grammar studies supplementing Wish Wonder and Surprise in the seventh
grade, and The Art of the Short Story in
eighth grade.
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The CME Project is a four-year, NSF-funded, comprehensive high school mathematics program that is problem-based, student-centered, and organized around the familiar themes of Algebra 1, Geometry, Algebra 2, and Precalculus. The CME Project sets as its goal robust mathematical proficiency for all students by emphasizing the interplay between mathematical thinking and essential technical skills. It provides a coherent curriculum with mathematical ideas, skills, and themes introduced early and deepened throughout the program. See workshop information, book tables of contents, events and presentations, Frequently Asked Questions (FAQ), and sign up for newsletters.
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Algebra II Introduction
Cite This Source
Applications in Science and Engineering
Algebra II has tons of real-world applications. Power functions are often used to model things in nature, like how populations grow. If we wanted to model the trends that make a newly uploaded video go viral, we could use what we learn in Algebra II to do just that. Future biologists, get your natural number on.
From supply and demand to compound interest, we need the basics of Algebra II to master the math we'll encounter in Economics. Not only that, but Algebra II will help us to generally understand the graphs we see on a day-to-day basis. These might include graphs of stock market trends or graphs of the cuteness of kittens. (The latter tends to be exponential.)
Matrices are also important in science. In physics, we can learn a little thing call the cross product that will make electricity and magnetism (slightly) easier. Speaking of physics, it's possible to get through a lot of basic calculations once you've mastered this topic.
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Product Description
Ensure your students know the real-life value of math! Integrating a problem-solving focus into your math classes, this guide to data analysis covers high-school topics in the context of everyday scenarios. Topics covered include basics of statistics, the ways statistics are used, how to determine when data changes are statistically significant, and even how statistics are misused. 86 reproducible pages, softcover. Grades 9-12. Revised edition.
Product Information
ISBN: 0825163227 ISBN-13: 9780825163227 Availability: Expected to ship on or about 06/25/15. Series:Real Life Math
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G-C.5A.APR.3
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
A.SSE.2
Use the structure of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
A.CED.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
F.IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F.IF.7C
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
F-LE.5
Interpret the parameters in a linear or exponential function in terms of a context.
S-ID.6
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
S-ID.6a
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
S-ID.6b
Informally assess the fit of a function by plotting and analyzing residuals.
S-ID.6c
Fit a linear function for a scatter plot that suggests a linear association.
S-ID.7
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
G-SRT.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G-SRT.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G-GPE.1
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
F-TF.8
Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
A-APR.4
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples.
8.G.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
7.EE.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.4
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
6.EE.6
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.2c6.EE.1
Write and evaluate numerical expressions involving whole-number exponents.
6.EE.2
Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.2a
Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation "Subtract y from 5" as 5 – y.
5.OA.1
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols4.NF.5
Use for the unknown number to represent the problem.1
1.OA.1
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books.google.com - INTER.... Algebra
Intermediate Algebra: Connecting Concepts Through Applications
INTER. It modifies the rule of four, integrating algebraic techniques, graphing, the use of data in tables, and writing sentences to communicate solutions to application problems. The authors have developed several key ideas to make concepts real and vivid for students.First, the authors integrate applications, drawing on real-world data to show students why they need to know and how to apply math. The applications help students develop the skills needed to explain the meaning of answers in the context of the application.Second, they emphasize strong algebra skills. These skills support the applications and enhance student comprehension. Third, the authors use an eyeball best-fit approach to modeling. Doing models by hand helps students focus on the characteristics of each function type. Fourth, the text underscores the importance of graphs and graphing. Students learn graphing by hand, while the graphing calculator is used to display real-life data problems. In short, INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS, International Edition takes an application-driven approach to algebra, using appropriate calculator technology as students master algebraic concepts and skills.
About the author (2011)
Mark Clark graduated from California State University, Long Beach, with a Bachelor's and Master's in Mathematics. He is a full-time associate professor at Palomar College and has taught there for the past thirteen years. He is committed to teaching his students through applications and using technology to help them both understand the mathematics in context and communicate their results clearly. Intermediate algebra is one of his favorite courses to teach, and he continues to teach several sections of this course each year.
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More About
This Textbook
Overview
"This book would be a great tool for helping [today's future elementary teachers] acquire a 'gut level' understanding of mathematics concepts."
- Hester Lewellen, Baldwin-Wallace College, OH
"The writing in this text is very clear and would easily be understood by the intended audience. The real-world examples put the various math concepts into a context that is easily understood. The vignettes at the beginning of each chapter are interesting and they get the reader to begin thinking about the math concepts that will follow. Each of the chapters seem to build on one another and the author often refers back to activities and concepts from previous chapters which is meaningful to the reader because it lets the reader know that the information they are learning builds their conceptual understanding of other mathematical concepts. "
- Melany L. Rish, University of South Carolina, Aiken
Organized around five key concepts or "powerful ideas" in mathematics, this book presents elementary mathematics content in a concise and nonthreatening manner for teachers.
Designed to sharpen teachers' mathematics pedagogical content knowledge, the friendly writing style and vignettes relate math concepts to "real life" situations so that they may better present the content to their students. The five "powerful ideas" (composition, decomposition, relationships, representation, and context) provide an organizing framework and highlight the interconnections between mathematics topics. In addition, the book thoroughly integrates discussion of the five NCTM process strands.
Features:
Icons highlighting the NCTM process standards appear throughout the book to indicate where the text relates to each of these.
Practice exercises and activities and their explanations reinforce math concepts presented in the book and provide an opportunity for reflection and practice.
Related Subjects
Meet the Author
James Schwartz is currently an Associate Professor of Education at Ralph C. Wilson School of Education, St. John Fisher College. He has a rich experience as both an elementary teacher and teacher-educator. James has contributed to various texts including Essentials of Educational Technology 1/e, (Allyn and Bacon, 1999); Essentials of Elementary Mathematics 2/e, (Allyn and Bacon, 1999); and Teaching Elementary School Mathematics 6/e, (Allyn and Bacon
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Eligible Content
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
Write and identify 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).
Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Big Ideas
•
Mathematical functions are relationships that assign each member of one set (domain) to a unique member of another set (range), and the relationship is recognizable across representations.
•
Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms.
•
Numerical measures describe the center and spread of numerical data.
•
Patterns exhibit relationships that can be extended, described, and generalized.
•
Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations.
•
Some questions can be answered by collecting, representing, and analyzing data, and the question to be answered determines the data to be collected, how best to collect it, and how best to represent it.
•
The set of real numbers has infinite subsets including the sets of whole numbers, integers, rational, and irrational numbers.
•
There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities.
Concepts
•
Distance, Pythagorean Theorem
•
Linear equations and inequalities
•
Linear functions
•
Polygons and Polyhedra
•
Rate of change
•
Representations
Competencies
•
Understand and apply the Pythagorean Theorem to find distances between points in a coordinate plane and to analyze polygons and polyhedra.
•
Use fundamental facts about distances and angles to describe and analyze figures and situations in 2- and 3-dimensional spaces and to solve problems including those with multiple steps.
•
Use linear functions, linear equations, and linear inequalities to represent, analyze, and solve a variety of problems.
•
Use the appropriate graphical data representation and extend understanding of the influence of scale in data interpretation.
Objectives
In this unit, students will learn to use equations and inequalities to model real-world situations and to solve basic linear equations and inequalities, including the following forms/techniques. Students will:
isolate a variable using inverse operations.
combine like terms.
balance variables on both sides of an equation or inequality.
use the distributive property.
write an equation in slope-intercept form.
Essential Questions
How do we recognize when it is appropriate to use a linear model to represent a real-world situation and what are the benefits of using a linear model to answer questions about the situation?
Divide both sides by −15 or division property of inequality (Multiplying by −1/15 also acceptable, with explanation of Multiply both sides by −1/15 or multiplication property of inequality)
(Also note: The student must reverse the inequality sign to get full credit.)
x < 8 Simplify
Points
Description
2
Correctly and completely solves the inequality
Demonstrates thorough understanding of finding the LCD, the distributive property, the addition/subtraction property of inequalities, and the multiplication/division property of inequalities
Supports each step with an explanation or identification of the correct property.
1
Correctly solves the inequality, but the answer may be incomplete and does not show all the steps or shows all the correct procedures but includes one calculation error
Demonstrates partial understanding of finding the LCD, the distributive property, the addition/subtraction property of inequalities, or the multiplication/division property of inequalities
Attempts to support each step with an explanation or identification of the correct property
0
Makes no attempt at solving or incorrectly attempts to solve the inequality
Demonstrates no understanding of finding the LCD, the distributive property, the addition/subtraction property of inequalities, or the multiplication/division property of inequalities
Does not give any explanation or property identification
Performance Assessment:
Eighth Grade Talent Show/Fundraiser:
The Eighth Grade Student Council in your school has been given permission to use the gymnasium to hold a Talent Show to raise funds for the local food shelf. As part of the planning group, you have the chance to help.
Set a goal of how much money you would like to raise. Think of different ways to make this money at the event, including ticket sales and refreshments. Do you want to charge the same for all ages or have separate prices? If you are using different prices, estimate the fraction of the total that will be in the different age brackets.
After deciding upon the ticket pricing and other items to be sold, list these different amounts. Use p for the number of people attending, and if some amounts involve only some of the attendees (for example, a portion of the attendees will purchase a soda, a portion of the attendees will be under 10 years old, etc.), estimate these amounts and represent them as fractions of the total p.
Research and determine any expenses involved with the event. These could include making programs, purchasing refreshments to sell, and producing posters and flyers to advertise the event.
Using the amounts from problem 2, write an equation that can be used to determine the number of people attending (p) needed to make your goal amount of money. Be sure to include expenses in this equation.
Does your answer in problem 4 sound reasonable? If it does not seem possible to get that many attendees, what other solution(s) could be used? Show how a new solution would change your equation.
Write a plan that your group will submit to the principal. Include the estimates and equation that you wrote.
Performance Assessment Scoring Rubric:
Points
Description
4
Responds completely with detailed explanation
Contains no math/calculation errors
Demonstrates complete understanding of how to model a real-world situation in mathematical terms
Shows complete understanding of the questions, mathematical ideas, and processes
Goes beyond what is required by the problem, shows creativity
3
Responds completely with clear explanation
Contains no major math errors or conceptual/procedural errors
Demonstrates understanding of how to model a real-world situation in mathematical terms
Shows substantial understanding of the questions, mathematical ideas, and processes
Meets the problem requirements
2
Responds unclearly or has some parts missing.
Contains several minor errors or one or more serious math errors or conceptual/procedural errors
Demonstrates some understanding of how to model real-world situations
Shows some understanding of the problem
Partially meets the problem requirements
1
Misses key points and/or sections
Contains major math errors or serious conceptual/procedural errors
Demonstrates lack of understanding of how to model real-world situations
Shows lack of understanding of the problem
Does not meet the problem requirements
0
Fails to complete or incorrectly completes most sections
Contains major math errors or serious conceptual/procedural errors
Demonstrates complete lack of understanding of how to model real-world situations
Shows complete lack of understanding of the problem
Does not meet the problem requirements
Final 04/12/13
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Calculators. The 2-line display scientific calculator combines statistics and advanced scientific functions and is a durable and affordable calculator for the classroom. The 2-Line display helps students explore math and science concepts in the classroom.
Calculators. From the kitchen table to the playground, children are intrigued by their world. The TI-15 is a pedagogically sound tool that helps students make connections between classroom learning and real-world situations.The TI-15 combines the fraction ...
Calculators. Scientific calculator performs 469 functions and features Direct Algebraic Logic (DAL) to simplify the entry of equations. DAL allows students to enter the elements of an expression in the exact order they appear in the textbook. The Multi-Line ...
Calculators. The TI-34 MultiView scientific calculator comes with the same features that made the TI-34 II Explorer Plus so helpful at exploring fraction simplification, integer division and constant operators.
Calculators. Scientific calculator performs 556 Scientific calculator performs 335 The two-line display scientific calculator combines statistics and advanced scientific functions and is a durable and affordable calculator for the classroom. The two-line display helps students explore math and science concepts in the classroom.
Calculators. The two-line display scientific calculator combines statistics and advanced scientific functions and is a durable and affordable calculator for the classroom. The two-line display helps students explore math and science concepts in the classroom.
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are absolutely right. Algebra is something that every subject that uses math is built on. It's the key to a lot of really cool stuff, like programming, physics etc.
I think the school should use more time to try out what they learn practically. For math a good example would be physics where you really get to see why stuff like algebra and differentiation is useful.
When I was doing trig in HS, the teacher brought in one of his friends from the Navy and taught us about trajectory and a bunch of other math stuffs (it was a long time ago & I don't use math - forgive me) used for targeting and such. It was rather interesting at the time.
Exactly, if you wanted to make a good math class instead of boring questions make a huge game of battleships and get each student to map a trajectory/speed/distance etc to shoot each of their other classmates battleship.
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Summary:Signals can be represented by discrete quantities instead of as a function of a
continuous variable. These discrete time signals do not
necessarily have to take real number values. Many properties of
continuous valued signals transfer almost directly to the discrete
domain.
Summary:This module defines eigenvalues and eigenvectors and explains a method of finding them given a matrix. These ideas are presented, along with many examples, in hopes of leading up to an understanding of the Fourier Series.
Summary:This module will introduce the Fourier Series and its Fourier coefficients using the concepts of eigenfunctions and basis. We will show several examples of how to decompose a signal and find the Fourier coefficients.
Summary: ... of discontinuities.[Expand Summary]. It does not seem
possible to exactly reconstruct a discontinuous function from a
set of continuous ones. In fact, it is not. However, it can be
if we relax the condition of exactly and replace it with the idea
of almost everywhere. This is to say that the reconstruction is
exactly the same as the original signal except at a finite number
of points. These points, not necessarily suprisingly, occur at
the points of discontinuities.[Collapse Summary]
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In algebra 2, students build upon the foundation of algebra 1, and become ready to tackle higher level of mathematical problems. topics as systems of equations, inequalities, and functions, exponential and logarithmic functions, logarithms, imaginary and complex numbers, conic sections, and matri...
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Find a BedfordI have also used math in my industrial profession as a research chemist. Prealgebra suggests that students are laying down the basics of numbers, areas, etc., before tackling how to deal with "unknowns" and "equations". I can guide the students in mastering the multiplication tables and how to proceed up and down the number line and getting comfortable in dealing with numbers.
...For quick look-ups, use the simple one. For in-depth look-ups, use the great one -- but any unfamiliar words you encounter in the great book's definition, you look up in the simple dictionary. That way, you don't ever stack more than one or two words deep
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Groups - Modular Mathematics Series
This text provides an introduction to group theory with an emphasis on clear examples. The authors present groups as naturally occurring structures arising from symmetry in geometrical figures and other mathematical objects. Written in a 'user-friendly' style, where new ideas are always motivated before being fully introduced, the text will help readers to gain confidence and skill in handling group theory notation before progressing on to applying it in complex situations. An ideal companion to any first or second year course on the topic
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More About
This Textbook
Overview
clear exposition and the consistency of presentation make learning arithmetic accessible for all. Key concepts are presented in section objectives and further defined within the context of How and Why; providing a strong foundation for learning. The predominant emphasis of the book focuses on problem-solving, skills, concepts, and applications based on "real world" data, with some introductory algebra integrated throughout. The authors feel strongly about making the connection between mathematics and the modern, day-to-day activities of students. This textbook is suitable for individual study or for a variety of course formats: lab, self-paced, lecture, group or combined formats. Though the mathematical content of FUNDAMENTALS OF MATHEMATICS is elementary, students using this textbook are often mature adults, bringing with them adult attitudes and experiences and a broad range of abilities. Teaching elementary content to these students, therefore, is effective when it accounts for their distinct and diverse adult needs. Using Fundamentals of Math meets three needs of students which are: students must establish good study habits and overcome math anxiety; students must see connections between mathematics and the modern day-to-day world of adult activities; and students must be paced and challenged according to their individual level of understanding.
Related Subjects
Meet the Author
James Van Dyke has been an instructor of mathematics for over 30 years, teaching courses at both the high school and college levels. He is the co-author of eight different mathematics textbooks, including Fundamentals of Mathematics, published by Cengage Learning.
James Rogers has been teaching high school and college-level mathematics courses for more than 30 years. Though retired, he currently teaches part-time at Portland Community College. He is the co-author of nine mathematics textbooks, including Fundamentals of Mathematics, published by Cengage Learning.
Hollis Adams teaches mathematics at Portland Community College, a position she has been in for nearly 20 years. She is the co-author of two mathematics textbooks, including Fundamentals of Mathematics, published by Cengage Learning
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All students meeting general education or introductory requirements
in the mathematical sciences should be enrolled in courses designed to
• Engage students in a meaningful and positive intellectual
experience;
• Increase quantitative and logical reasoning abilities needed for
informed citizenship and in the workplace;
• Strengthen quantitative and mathematical abilities that will be
useful to students in other disciplines;
• Improve every student's ability to communicate quantitative
ideas orally and in writing;
• Encourage students to take at least one additional course in the
mathematical sciences.
One of the characteristics that distinguishes this CUPM Curriculum Guide from
previous sets of CUPM recommendations is the attention paid to the students
we teach who do not intend to major in mathematics. One of our most important
audiences consists of the math-averse, those who feel that they are constitutionally
incapable of doing mathematics, consider it irrelevant to their interests, or
have had a bad experience with mathematics. According to the CBMS [1],
more than 75% of students taking mathematics in college are taking courses classified
as "remedial" or "introductory". Many if not most of
these students qualify as math-averse. They would rather not be there.
This is the first of three recommendations that the CUPM Curriculum Guide makes
for the courses aimed at these students. It makes the case that we need to be
clear about what mathematics these students actually need. For many of them,
a course in quantitative literacy (aka quantitative reasoning, quantitative
thinking, numeracy) will serve far better than a repeat of algebra.
Quantitative literacy is "the power and habit of mind to search out quantitative
information, critique it, reflect on it, and apply it in one's public,
personal, and professional life" (National Numeracy Network [2]).
The mathematics can be very simple. It is the ability to work in context that
makes this a demanding discipline, and for quantitative literacy, context is
everything. The goal is to empower students to reason with the complex quantitative
information that is omnipresent in today's world.
Many colleges and universities are wrestling with how to teach this. It is
not the unique domain of mathematicians. The American Sociological Association
in now encouraging its members to promote quantitative literacy. The National
Numeracy Network includes geographers and geologists, physicists, economists,
biologists, and statisticians as well as sociologists and mathematicians. The
MAA's Special Interest Group on Quantitative Literacy [3]
was established as a forum for the exchange of information on innovative ways
to develop and implement QL curricula. The MAA has made available two of the
most useful publications on quantitative literacy, Mathematics and Democracy:
The Case for Quantitative Literacy [4] and Achieving Quantitative
Literacy: An Urgent Challenge for Higher Education [5].
Macalester College is developing its own program in quantitative thinking [6],
assisted by funding from the Department of Education's Fund for the Improvement
for Post-Secondary Education and from NSF Department of Undergraduate Education.
We have identified several key components of quantitative thinking and created
lessons to teach them. These include:
• Trade-offs Identifying and working through the complexities
of conflicting goals. One of the weaknesses we observe in many of our students
is a tendency to latch onto a single worthwhile goal and to ignore the effect
that maximizing that single good has on other desirable ends. Recognizing
the difficulty of balancing competing goals motivates the need for quantitative
thinking.
• Rates and comparisons How the quantities composing
a rate can inform or mislead a discussion. Understanding the distinction between
aggregate and differential rates, total and partial rates, and knowing when
each is appropriate.
• Change over time Exponential versus linear growth
and decay. Compounding. Restricted growth. Limits to prediction and extrapolation.
• Variability and bias Distinguishing among what is
normal, what is average, what is typical. Knowing how to assess the reliability
of measurements.
• Causation and association Understanding the difference
and why an insubstantial association might still pass the test of "significance".
Recognizing the hallmarks of reliable research.
• Uncertainty and risk Assessing, comparing, and balancing
risks. Ability to understand conditional statements and probabilities and
draw on them to assess risk.
• Estimation, modeling, and scale Ability to do "back
of the envelope" calculations. Recognizing the usefulness and limitations
of models. Knowing that big and small are not absolutes but always relative.
What I have described is not a traditional mathematics class. In fact, it is
not clear where such a class belongs or who is best trained to teach it. But
quantitative literacy is a set of capabilities that every educated person should
possess, and it is a first step in demonstrating to the math-averse the importance
of mathematics. If mathematicians do not promote quantitative literacy, who
will?
[1] Includes 2-year and 4-year colleges and universities.
At 4-year institutions, it is over 60%. Tables E.2 and TYR.4 in Statistical
Abstract of Undergraduate Programs in the Mathematical Sciences in the United
States: Fall 2000 CBMS Survey, David J. Lutzer, James W. Maxwell, Stephen B.
Rodi, editors, American Mathematical Society, Providence, RI, 2002.
We would appreciate more examples that document experiences with the use of
technology as well as examples of interdisciplinary cooperation.
David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester
College in St. Paul, Minnesota, he was one of the writers for the Curriculum
Guide, and he currently serves as Chair of the CUPM. He wrote this column
with help from his colleagues in CUPM, but it does not reflect an official
position of the committee. You can reach him at bressoud@macalester.edu.
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Samples in periodicals archive:
Now in an updated and expanded third edition, "A Concise Introduction To Pure Mathematics" by Martin Liebeck provides an informed and informative presentation into a representative selection of fundamental ideas in mathematics including the theory of solving cubic equations, the use of Euler's formula to study the five Platonic solids, the use of prime numbers to encode and decode secret information, the theory of how to compare the sizes of two infinite sets, the limits of sequences and continuous functions, the use of the intermediate value theorem to prove the existence of nth roots, and so much more.
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Most mathematicians, engineers, and many other scientists are well-acquainted with theory and application of ordinary differential equations. This book seeks to present Volterra integral and functional differential equations in that same framwork, allowing the readers to parlay their knowledge of ordinary differential equations into theory and application of the more general problems. Thus, the presentation starts slowly with very familiar concepts and shows how these are generalized in a natural way to problems involving a memory. Liapunov's direct method is gently introduced and applied to many particular examples in ordinary differential equations, Volterra integro-differential equations, and functional differential equations.
By Chapter 7 the momentum has built until we are looking at problems on the frontier. Chapter 7 is entirely new, dealing with fundamental problems of the resolvent, Floquet theory, and total stability. Chapter 8 presents a solid foundation for the theory of functional differential equations. Many recent results on stability and periodic solutions of functional differential equations are given and unsolved problems are stated.
Key Features:
Smooth
transition from ordinary differential equations to integral and functional differential equations. Unification of the theories, methods, and applications of ordinary and functional differential equations. Large collection of examples of Liapunov functions. Description of the history of stability theory leading up to unsolved problems. Applications of the resolvent to stability and periodic problems.
1. Smooth transition from ordinary differential equations to integral and functional differential equations. 2. Unification of the theories, methods, and applications of ordinary and functional differential equations. 3. Large collection of examples of Liapunov functions. 4. Description of the history of stability theory leading up to unsolved problems. 5. Applications of the resolvent to stability and periodic problems
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Algebra Demystified 1st edition
0071389938
9780071389938
Details about Algebra Demystified:
Whether you want to learn more about algebra, refresh your skills, or improve your classroom performance,Algebra Demystifiedis the perfect shortcut.Knowing algebra gives you a better choice of jobs, helps you perform better in science, computing, and math courses, ups your score on competitive exams, and improves your ability to do daily computations. And there's no faster or more painless way to master the subject thanAlgebra Demystified! Entertaining author and experienced teacher Rhonda Huettenmueller provides all the math background you need and uses practical examples, real data, and a totally different approach to life the "myst" from algebra.WithAlgebra Demystified, you master algebra one simple step at a time--at your own speed. Unlike most books on the subject, general concepts are presented first--and the details follow. In order to make the process as clear and simple as possible, long computations are presented in a logical, layered progression with just one execution per step.THIS ONE-OF-A-KIND SELF-TEACHING TEXT OFFERS:Questions at the end of every chapter and section to reinforce learning and pinpoint weaknessesA 100-questions final exam for self-assessmentAn intensive focus on word problems and fractions--help where it's most often neededDetailedexamples and solutions
Back to top
Rent Algebra Demystified 1st edition today, or search our site for Rhonda textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by McGraw-Hill Professional Publishing.
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Math modeling handbook now available
Apr 23, 2014
This is the cover of the free math modeling handbook published by the Society for Industrial and Applied Mathematics this month. Credit: SIAM
Math comes in handy for answering questions about a variety of topics, from calculating the cost-effectiveness of fuel sources and determining the best regions to build high-speed rail to predicting the spread of disease and assessing roller coasters on the basis of their "thrill" factor. How does math do all that?
That is the topic of a free handbook published by the Society for Industrial and Applied Mathematics (SIAM) this month: "Math Modeling: Getting Started and Getting Solutions."
Finding a solution to any of the aforementioned problems—or the multitude of other unanswered questions in the real world—will likely involve the creation, application, and refinement of a mathematical model. A math model is a mathematical representation of a real-world situation intended to gain a qualitative or quantitative understanding in order to predict future behavior. Such predictions allow us to come up with novel findings, enable scientific advances, and make informed decisions.
The handbook provides instructions and a process for building mathematical models using a variety of examples to answer wide-ranging questions.
The inspiration for the handbook came from Moody's Mega Math (M3) Challenge, a high school applied math contest organized by SIAM. Despite the tremendous success of the nine-year-old Challenge, which is currently available to 45 U.S. states and Washington, D.C., organizers found that many participating students—high school juniors and seniors—were having trouble coming up with approaches and solutions to the open-ended realistic problems posed by the contest. Participants expressed their frustration in post-contest surveys and emails.
"We have been enthusiastic about the high level of insight and analysis demonstrated by participants in the Challenge, especially the winning teams," says M3 Challenge Project Director Michelle Montgomery. "However, it became clear to us that, given the lack of modeling courses in most high school curricula, many of the participants did not have access to basic resources necessary to create a successful model. We came up with the handbook to give every participant these tools."
This type of thinking created an "aha" moment, so to speak, for handbook authors Karen Bliss, Katie Fowler, and Ben Galluzzo, long-time Challenge judges who have been part of the contest's problem development team for the past two years.
"All students, especially those interested in STEM disciplines, need as much practice in solving open-ended problems as possible, but they often do not get many chances to do that in school,"says Fowler, who is an associate professor of mathematics at Clarkson University. "Math modeling skills allow students to approach problems they initially may feel are outside of their comfort zone, and we want to give them the confidence to tackle them."
Further motivated by a series of SIAM-National Science Foundation (NSF) workshops on the topic of math modeling across the curriculum, the trio began work on a modeling guide. What started as a pamphlet with step-by-step guidance about the modeling process grew into a 70-page, full color handbook, with a companion document that makes connections to the Common Core State Standards as well as easy-to-use reference cards for those who want to get straight to the crux of modeling. The guide is suitable for teachers as well as high school and undergraduate students interested in learning how to model.
"Math modeling is challenging, but it's also surprisingly accessible. The guidebook is designed to remove perceived roadblocks by presenting modeling as a highly-creative iterative process in which multiple approaches—to the same problem—can lead to meaningful results," says Galluzzo, an assistant professor of mathematics at Shippensburg University.
The handbook, as well as the Challenge itself, has another, more pressing goal: motivating our younger generation to pursue higher education and careers in science and math. "SIAM does a big service to the math community at large by giving high school students the opportunity to see how math is more than just a series of formulas and rote memorization," says Bliss, an assistant professor of mathematics at Quinnipiac University. "Students at all levels have the means to produce highly creative solutions to interesting problems. Seeing that math can be a powerful tool for solving truly important problems through M3 Challenge participation might be just enough to encourage a student to study math or another STEM discipline in college."
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Precalculus
You may think that precalculus is simply the course you take before calculus. You would be right, of course, but that definition doesn't mean anything unless you have some knowledge of what calculus is. Let's keep it simple, shall we? Calculus is a conceptual framework which provides systematic techniques for solving problems. These problems are appropriately applicable to analytic geometry and algebra. Therefore....precalculus gives you the background for the mathematical concepts, problems, issues and techniques that appear in calculus, including trigonometry, functions, complex numbers, vectors, matrices, and others. There you have it ladies and gentlemen....an introduction to precalculus!
What is the probability of getting a diamond or an ace from a deck of cards? Well I could get a diamond that is not an ace, an ace that is not a diamond, or the ace of diamonds. This tutorial helps us think these types of situations through a bit better (especially with the help of our good friend, the Venn diagram).
What is the probability of making three free throws in a row (LeBron literally asks this in this tutorial).
In this tutorial, we'll explore compound events happening where the probability of one event is not dependent on the outcome of another (compound, independent, events).
What's the probability of picking two "e" from the bag in scrabble (assuming that I don't replace the tiles). Well, the probability of picking an 'e' on your second try depends on what happened in the first (if you picked an 'e' the first time around, then there is one less 'e' in the bag). This is just one of many, many type of scenarios involving dependent probability.
You want to display your Chuck Norris dolls on your desk at school and there is only room for five of them. Unfortunately, you own 50. How many ways can you pick the dolls and arrange them on your desk?
You are already familiar with calculating permutation ("How many ways could 7 different people sit in 4 different seats?"). But what if you didn't care about which seat they sat in? What if you just cared about which 4 people were in the car? Or put another way, you want to know how many combinations of 4 people can you stick in the car from a pool of 7 candidates. Or how many ways are there to choose 4 things from a pool of 7? Look no further than this tutorial to answer your questions.
This tutorial will apply the permutation and combination tools you learned in the last tutorial to problems of probability. You'll finally learn that there may be better "investments" than poring all your money into the Powerball Lottery.
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This lesson from Illuminations looks at inverse variation. Students are asked to use estimates to measure the height of an object, visualize two-dimensional objects and calculate their area, and sketch graphs to...
This lesson from Illuminations explores conic shapes. The shapes circle, ellipse, hyperbola, and parabola are illustrated. Students will discover how to cut a double-napped cone to create the various conic sections. The...
This lesson from Illuminations allows students to learn about linear equations in a real-world setting. The material applies linear equations to the concept of supply and demand. Students will be able to translate...
This lesson from Illuminations introduces modular arithmetic. The common barcode system, as used for UPCs and ISBNs, is discussed as an application of this type of arithmetic. The lesson also explains how modular...
This lesson from Illuminations asks students to look at different classes of polynomial functions by exploring the graphs of the functions. Students should already have a grasp of linear functions, quadratic functions,...
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