The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guides) - Softcover

Banner, Adrian

Softcover

ISBN 10: 0691130884 ISBN 13: 9780691130880

Verlag: Princeton University Press, 2007

Inhaltsangabe

For many students, calculus can be the most mystifying and frustrating course they will ever take. The Calculus Lifesaver provides students with the essential tools they need not only to learn calculus, but to excel at it.

All of the material in this user-friendly study guide has been proven to get results. The book arose from Adrian Banner's popular calculus review course at Princeton University, which he developed especially for students who are motivated to earn A's but get only average grades on exams. The complete course will be available for free on the Web in a series of videotaped lectures. This study guide works as a supplement to any single-variable calculus course or textbook. Coupled with a selection of exercises, the book can also be used as a textbook in its own right. The style is informal, non-intimidating, and even entertaining, without sacrificing comprehensiveness. The author elaborates standard course material with scores of detailed examples that treat the reader to an "inner monologue"--the train of thought students should be following in order to solve the problem--providing the necessary reasoning as well as the solution. The book's emphasis is on building problem-solving skills. Examples range from easy to difficult and illustrate the in-depth presentation of theory.

The Calculus Lifesaver combines ease of use and readability with the depth of content and mathematical rigor of the best calculus textbooks. It is an indispensable volume for any student seeking to master calculus.

• Serves as a companion to any single-variable calculus textbook


• Informal, entertaining, and not intimidating


• Informative videos that follow the book--a full forty-eight hours of Banner's Princeton calculus-review course--is available at Adrian Banner lectures


• More than 475 examples (ranging from easy to hard) provide step-by-step reasoning


• Theorems and methods justified and connections made to actual practice


• Difficult topics such as improper integrals and infinite series covered in detail


• Tried and tested by students taking freshman calculus

Über diesen Titel

Über die Autorin bzw. den Autor

Adrian Banner is Lecturer in Mathematics at Princeton University and Director of Research at INTECH.

Von der hinteren Coverseite

"I used Adrian Banner's The Calculus Lifesaver as the sole textbook for an intensive, three-week summer Calculus I course for high-school students. I chose this book for several reasons, among them its conversational expository style, its wealth of worked-out examples, and its price. This book is designed to supplement any standard calculus textbook, thus my students will be able to use it again when they take later calculus courses. The students in my class came from diverse backgrounds, ranging from those who had already seen much of the material to others who were struggling with basic algebra. They all uniformly praised the book for being one of the clearest mathematics texts they have ever read, and because it reviews the required prerequisite material. The numerous worked-out examples are an ideal supplement to the lectures. The only difficulty in using this book as a primary text is the lack of additional exercises in the text. However, there are so many sites and sources for calculus problems that this was not a problem. I would definitely use this book again."--Steven J. Miller, Brown University

"Banner's book is a chatty, user-friendly guide to calculus that will be a useful addition to the resources available to students. Banner does an exceptionally thorough job while maintaining an engaging style."--Gerald B. Folland, author of Advanced Calculus

"This is an engaging read. Each page engenders at least one smile, often a chuckle, occasionally a belly laugh."--Charles R. MacCluer, author of Honors Calculus

"This book is significant. The author's attempt to give an 'inner monologue' into the thought process that is needed to solve calculus problems rather than just providing worked examples is novel and is in line with his purpose of helping the reader get a deeper understanding of calculus. The book is well written and the author's examples are clear and complete."--Thomas Seidenberg, Phillips Exeter Academy

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The Calculus Lifesaver

All the tools you need o excel at calculusBy ADRIAN BANNER

PRINCETON UNIVERSITY PRESS

Copyright © 2007 Princeton University Press
All right reserved.
ISBN: 978-0-691-13088-0

Contents

Welcome....................................................................xviiiHow to Use This Book to Study for an Exam..................................xixTwo all-purpose study tips.................................................xxKey sections for exam review (by topic)....................................xxAcknowledgments............................................................xxiii1 Functions, Graphs, and Lines.............................................12 Review of Trigonometry...................................................253 Introduction to Limits...................................................414 How to Solve Limit Problems Involving Polynomials........................575 Continuity and Differentiability.........................................756 How to Solve Differentiation Problems....................................997 Trig Limits and Derivatives..............................................1278 Implicit Differentiation and Related Rates...............................1499 Exponentials and Logarithms..............................................16710 Inverse Functions and Inverse Trig Functions............................20111 The Derivative and Graphs...............................................22512 Sketching Graphs........................................................24513 Optimization and Linearization..........................................26714 L'Hôpital's Rule and Overview of Limits............................29315 Introduction to Integration.............................................30716 Definite Integrals......................................................32517 The Fundamental Theorems of Calculus....................................35518 Techniques of Integration, Part One.....................................38319 Techniques of Integration, Part Two.....................................40920 Improper Integrals: Basic Concepts......................................43121 Improper Integrals: How to Solve Problems...............................45122 Sequences and Series: Basic Concepts....................................47723 How to Solve Series Problems............................................50124 Taylor Polynomials, Taylor Series, and Power Series.....................51925 How to Solve Estimation Problems........................................53526 Taylor and Power Series: How to Solve Problems..........................55127 Parametric Equations and Polar Coordinates..............................57528 Complex Numbers.........................................................59529 Volumes, Arc Lengths, and Surface Areas.................................61730 Differential Equations..................................................645Appendix A Limits and Proofs...............................................669A.1 Formal Definition of a Limit...........................................669A.2 Making New Limits from Old Ones........................................674A.3 Other Varieties of Limits..............................................678A.4 Continuity and Limits..................................................684A.5 Exponentials and Logarithms Revisited..................................689A.6 Differentiation and Limits.............................................691A.7 Proof of the Taylor Approximation Theorem..............................700Appendix B Estimating Integrals............................................703B.1 Estimating Integrals Using Strips......................................703B.2 The Trapezoidal Rule...................................................706B.3 Simpson's Rule.........................................................709B.4 The Error in Our Approximations........................................711List of Symbols............................................................717Index......................................................................719

Chapter One

Functions, Graphs, and Lines

Trying to do calculus without using functions would be one of the most pointless things you could do. If calculus had an ingredients list, functions would be first on it, and by some margin too. So, the first two chapters of this book are designed to jog your memory about the main features of functions. This chapter contains a review of the following topics:

? functions: their domain, codomain, and range, and the vertical line test;

? inverse functions and the horizontal line test;

? composition of functions; ? odd and even functions;

? graphs of linear functions and polynomials in general, as well as a brief survey of graphs of rational functions, exponentials, and logarithms; and

? how to deal with absolute values.

Trigonometric functions, or trig functions for short, are dealt with in the next chapter. So, let's kick off with a review of what a function actually is.

1.1 Functions

A function is a rule for transforming an object into another object. The object you start with is called the input, and comes from some set called the domain. What you get back is called the output; it comes from some set called the codomain.

Here are some examples of functions:

? Suppose you write f(x) = x². You have just defined a function f which transforms any number into its square. Since you didn't say what the domain or codomain are, it's assumed that they are both R, the set of all real numbers. So you can square any real number, and get a real number back. For example, f transforms 2 into 4; it transforms -1/2 into 1/4; and it transforms 1 into 1. This last one isn't much of a change at all, but that's no problem: the transformed object doesn't have to be different from the original one. When you write f(2) = 4, what you really mean is that f transforms 2 into 4. By the way, f is the transformation rule, while f(x) is the result of applying the transformation rule to the variable x. So it's technically not correct to say "f(x) is a function"; it should be "f is a function."

? Now, let g(x) = x² with domain consisting only of numbers greater than or equal to 0. (Such numbers are called nonnegative.) This seems like the same function as f, but it's not: the domains are different. For example, f(-1/2) = 1/4, but g(-1/2) isn't defined. The function g just chokes on anything not in the domain, refusing even to touch it. Since g and f have the same rule, but the domain of g is smaller than the domain of f, we say that g is formed by restricting the domain of f.

? Still letting f(x) = x², what do you make of f(horse)? Obviously this is undefined, since you can't square a horse. On the other hand, let's set

h(x) = number of legs x has,

where the domain of h is the set of all animals. So h(horse) = 4, while h(ant) = 6 and h(salmon) = 0. The codomain could be the set of all nonnegative integers, since animals don't have negative or fractional numbers of legs. By the way, what is h(2)? This isn't defined, of course, since 2 isn't in the domain. How many legs does a...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2007
• Sprache: Englisch
• ISBN 10: 0691130884
• ISBN 13: 9780691130880
• Einband: Taschenbuch
• Anzahl der Seiten: 728
• Kontakt zum Hersteller: Princeton University Press; https://press.princeton.edu/books; ; 7th Floor, New South Building; PRINCETON; NJ; 8544; USA
• Verantwortliche Person: Nicht verfügbar