Quick Calculus: A Self-Teaching Guide 3rd Edition

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Book Preface

Quick Calculus is designed for you to learn the basic techniques of differential and integral calculus with a minimum of wasted effort, studying by yourself. It was created on a premise that is now widely accepted: in technical subjects such as calculus, students learn by doing rather than by listening. The book consists of a sequence of relatively short discussions, each followed by a problem whose solution is immediately available. One’s path through the book is directed by the responses. The text is aimed at newcomers to calculus, but additional topics are discussed in the final chapter for those who wish to go further.
The initial audience for Quick Calculus was composed of students entering college who did not wish to postpone physics for a semester in order to take a prerequisite in calculus. In reality, the level of calculus needed to start out in physics is not high and could readily be mastered by self-study.
The readership for Quick Calculus has grown far beyond novice physics students, encom-passing people at every stage of their career. The fundamental reason is that calculus is empow-ering, providing the language for every physical science and for engineering, as well as tools that are crucial for economics, the social sciences, medicine, genetics, and public health, to name a few. Anyone who learns the basics of calculus will think about how things change and influence each other with a new perspective. We hope that Quick Calculus will provide a firm launching point for helping the reader to achieve this perspective.

Daniel Kleppner Peter Dourmashkin Cambridge, Massachusetts

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Contents
Preface iii
Chapter One Starting Out 1
1.1 A Few Preliminaries 1
1.2 Functions 2
1.3 Graphs 5
1.4 Linear and Quadratic Functions 11
1.5 Angles and Their Measurements 19
1.6 Trigonometry 28
1.7 Exponentials and Logarithms 42
Summary of Chapter 1 51
Chapter Two Differential Calculus 57
2.1 The Limit of a Function 57
2.2 Velocity 71
2.3 Derivatives 83
2.4 Graphs of Functions and Their Derivatives 87
2.5 Differentiation 97
2.6 Some Rules for Differentiation 103
2.7 Differentiating Trigonometric Functions 114
2.8 Differentiating Logarithms and Exponentials 121
2.9 Higher-Order Derivatives 130
2.10 Maxima and Minima 134
2.11 Differentials 143
2.12 A Short Review and Some Problems 147
Conclusion to Chapter 2 164
Summary of Chapter 2 165
Chapter Three Integral Calculus 169
3.1 Antiderivative, Integration, and the Indefinite Integral 170
3.2 Some Techniques of Integration 174

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Contents
3.3 Area Under a Curve and the Definite Integral 182
3.4 Some Applications of Integration 201
3.5 Multiple Integrals 211
Conclusion to Chapter 3 219
Summary of Chapter 3 219
Chapter Four Advanced Topics: Taylor Series, Numerical Integration, and Differential Equations 223
4.1 Taylor Series 223
4.2 Numerical Integration 232
4.3 Differential Equations 235
4.4 Additional Problems for Chapter 4 244
Summary of Chapter 4 248
Conclusion (frame 449) 250
Appendix A Derivations 251
A.1 Trigonometric Functions of Sums of Angles 251
A.2 Some Theorems on Limits 252
A.3 Exponential Function 254
A.4 Proof That dy
A.5 Differentiating x n 256
A.6 Differentiating Trigonometric Functions 258
A.7 Differentiating the Product of Two Functions 258
A.8 Chain Rule for Differentiating 259
A.9 Differentiating ln x 259
A.10 Differentials When Both Variables Depend on a Third Variable 260
A.11 Proof That if Two Functions Have the Same Derivative They Differ Only by a Constant 261
A.12 Limits Involving Trigonometric Functions 261
Appendix B Additional Topics in Differential Calculus 263
B.1 Implicit Differentiation 263
B.2 Differentiating the Inverse Trigonometric Functions 264
B.3 Partial Derivatives 267
B.4 Radial Acceleration in Circular Motion 269
B.5 Resources for Further Study 270
Answers to Selected Problems from the Text 273
Review Problems 277
Chapter 1 277
Chapter 2 278
Chapter 3 282
Tables 287
Table 1: Derivatives 287
Table 2: Integrals 288
Indexes 291
Index 291
Index of Symbols 295