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

Banner, Adrian

Softcover

ISBN 10: 0691130884 ISBN 13: 9780691130880

Publisher: Princeton University Press, 2007

Synopsis

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

About this title

About the Author

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

From the Back Cover

"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

Excerpt. © Reprinted by permission. All rights reserved.

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 "2" have, after all? The question doesn't really make sense. You might also think that h(chair) = 4, since most chairs have four legs, but that doesn't work either, since a chair isn't an animal, and so "chair" is not in the domain of h. That is, h(chair) is undefined.

? Suppose you have a dog called Junkster. Unfortunately, poor Junkster has indigestion. He eats something, then chews on it for a while and tries to digest it, fails, and hurls. Junkster has transformed the food into ... something else altogether. We could let

j(x) = color of barf when Junkster eats x,

where the domain of j is the set of foods that Junkster will eat. The codomain is the set of all colors. For this to work, we have to be confident that whenever Junkster eats a taco, his barf is always the same color (say, red). If it's sometimes red and sometimes green, that's no good: a function must assign a unique output for each valid input.

Now we have to look at the concept of the range of a function. The range is the set of all outputs that could possibly occur. You can think of the function working on transforming everything in the domain, one object at a time; the collection of transformed objects is the range. You might get duplicates, but that's OK.

So why isn't the range the same thing as the codomain? Well, the range is actually a subset of the codomain. The codomain is a set of possible outputs, while the range is the set of actual outputs. Here are the ranges of the functions we looked at above:

? If f(x) = x² with domain R and codomain R, the range is the set of nonnegative numbers. After all, when you square a number, the result cannot be negative. How do you know the range is all the nonnegative numbers? Well, if you square every number, you definitely cover all nonnegative numbers. For example, you get 2 by squaring v2 (or v2).

? If g(x) = x², where the domain of g is only the nonnegative numbers but the codomain is still all of R, the range will again be the set of nonnegative numbers. When you square every nonnegative number, you still cover all the nonnegative numbers.

? If h(x) is the number of legs the animal x has, then the range is all the possible numbers of legs that any animal can have. I can think of animals that have 0, 2, 4, 6, and 8 legs, as well as some creepy-crawlies with more legs. If you include individual animals which have lost one or more legs, you can also include 1, 3, 5, and 7 in the mix, as well as other possibilities. In any case, the range of this function isn't so clear-cut; you probably have to be a biologist to know the real answer.

? Finally, if j(x) is the color of Junkster's barf when he eats x, then the range consists of all possible barf-colors. I dread to think what these are, but probably bright blue isn't among them.

1.1.1 Interval notation

In the rest of this book, our functions will always have codomain R, and the domain will always be as much of R as possible (unless stated otherwise). So we'll often be dealing with subsets of the real line, especially connected intervals such as {x : 2 = x < 5}. It's a bit of a pain to write out the full set notation like this, but it sure beats having to say "all the numbers between 2 and 5, including 2 but not 5." We can do even better using interval notation.

We'll write [a, b] to mean the set of all numbers between a and b, including a and b themselves. So [a, b] means the set of all x such that a = x = b. For example, [2, 5] is the set of all real numbers between 2 and 5, including 2 and 5. (It's not just the set consisting of 2, 3, 4, and 5: don't forget that there are loads of fractions and irrational numbers between 2 and 5, such as 5/2, v7, and p.) An interval such as [a, b] is called closed.

If you don't want the endpoints, change the square brackets to parentheses. In particular, (a, b) is the set of all numbers between a and b, not including a or b. So if x is in the interval (a, b), we know that a < x < b. The set (2, 5) includes all real numbers between 2 and 5, but not 2 or 5. An interval of the form (a, b) is called open.

You can mix and match: [a, b) consists of all numbers between a and b, including a but not b. And (a, b] includes b but not a. These intervals are closed at one end and open at the other. Sometimes such intervals are called half-open. An example is the set {x : 2 = x < 5} from above, which can also be written as [2, 5).

There's also the useful notation (a, 8) for all the numbers greater than a not including a; [a, 8) is the same thing but with a included. There are three other possibilities which involve -8; all in all, the situation looks like this:

1.1.2 Finding the domain

Sometimes the definition of a function will include the domain. (This was the case, for example, with our function g from Section 1.1 above.) Most of the time, however, the domain is not provided. The basic convention is that the domain consists of as much of the set of real numbers as possible. For example, if k(x) = vx, the domain can't be all of R, since you can't take the square root of a negative number. The domain must be [0, 8), which is just the set of all numbers greater than or equal to 0.

OK, so square roots of negative numbers are bad. What else can cause a screw-up? Here's a list of the three most common possibilities:

1. The denominator of a fraction can't be zero.

2. You can't take the square root (or fourth root, sixth root, and so on) of a negative number.

3. You can't take the logarithm of a negative number or of 0. (Remember logs? If not, see Chapter 9!)

You might recall that tan(90°) is also a problem, but this is really a special case of the first item above. You see,

tan(90°) = sin(90°)/cos(90°) = 1/0.

so the reason tan(90°) is undefined is really that a hidden denominator is zero. Here's another example: if we try to define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then what is the domain of f? Well, for f(x) to make sense, here's what needs to happen:

? We need to take the square root of (26-2x), so this quantity had better be nonnegative. That is, 26 - 2x = 0. This can be rewritten as x = 13.

? We also need to take the logarithm of (x + 8), so this quantity needs to be positive. (Notice the difference between logs and square roots: you can take the square root of 0, but you can't take the log of 0.) Anyway, we need x + 8 > 0, so x > -8. So far, we know that -8 < x = 13, so the domain is at most (-8, 13].

? The denominator can't be 0; this means that (x-2) [not equal to] 0 and (x+19) [not equal to] 0. In other words, x [not equal to] 2 and x [not equal to] -19. This last one isn't a problem, since we already know that x lies in (-8, 13], so x can't possibly be -19. We do have to exclude 2, though.

So we have found that the domain is the set (-8, 13] except for the number 2. This set could be written as (-8, 13]\{2}. Here the backslash means "not including."

1.1.3 Finding the range using the graph

Let's define a new function F by specifying that its domain is [-2, 1] and that F(x) = x² on this domain. (Remember, the codomain of any function we look at will always be the set of all real numbers.) Is F the same function as f, where f(x) = x² for all real numbers x? The answer is no, since the two functions have different domains (even though they have the same rule). As in the case of the function g from Section 1.1 above, the function F is formed by restricting the domain of f.

Now, what is the range of F? Well, what happens if you square every number between -2 and 1 inclusive? You should be able to work this out directly, but this is a good opportunity to see how to use a graph to find the range of a function. The idea is to sketch the graph of the function, then imagine two rows of lights shining from the far left and far right of the graph horizontally toward the y-axis. The curve will cast two shadows, one on the left side and one on the right side of the y-axis. The range is the union of both shadows: that is, if any point on the y-axis lies in either the left-hand or the right-hand shadow, it is in the range of the function. Let's see how this works with our function F:

The left-hand shadow covers all the points on the y-axis between 0 and 4 inclusive, which is [0, 4]; on the other hand, the right-hand shadow covers the points between 0 and 1 inclusive, which is [0, 1]. The right-hand shadow doesn't contribute anything extra: the total coverage is still [0, 4]. This is the range of F.

1.1.4 The vertical line test

In the last section, we used the graph of a function to find its range. The graph of a function is very important: it really shows you what the function "looks like." We'll be looking at techniques for sketching graphs in Chapter 12, but for now I'd like to remind you about the vertical line test.

You can draw any figure you like on a coordinate plane, but the result may not be the graph of a function. So what's special about the graph of a function? What is the graph of a function f, anyway? Well, it's the collection of all points with coordinates (x, f(x)), where x is in the domain of f. Here's another way of looking at this: start with some number x. If x is in the domain, you plot the point (x, f(x)), which of course is at a height of f(x) units above the point x on the x-axis. If x isn't in the domain, you don't plot anything. Now repeat for every real number x to build up the graph.

(Continues...)

Excerpted from The Calculus Lifesaverby ADRIAN BANNER Copyright © 2007 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Edition details

• Publisher: Princeton University Press
• Publication date: 2007
• ISBN 10: 0691130884
• ISBN 13: 9780691130880
• Binding: Paperback
• Language: English
• Number of pages: 752