Quick Calculus: A Self–Teaching Guide

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

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/dx = 1 dx/dy 255

A.5 Differentiating xⁿ 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

Frame Problems Answers 273

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

Daniel Kleppner (Belmont, MA) is the Lester Wolfe Professor Emeritus of Physics at MIT and Director Emeritus of the MIT-Harvard Center for Ultracold Atoms. He is the winner of the 2005 Wolf Prize in Physics, the National Medal of Science in 2006, the 2007 Frederic Ives Medal, the 2014 Benjamin Franklin Medal, and the 2017 American Physical Society Career Achievement Award.   Together with Norman Ramsey, he authored Quick Calculus. Together with Robert J. Kolenkow, he authored a popular introductory mechanics textbook for advanced students. 

Peter Andrew Dourmashkin (Cambridge, MA) is a Senior Lecturer at MIT that has taught in the Physics Department, and at a variety of specialized programs at MIT including the Experimental Study Group, Integrated Studies Program, Seminar XL and Project Interphase both of which are sponsored by the Office of Minority Education. He has been awarded the Irwin Sizer Award for Most Significant Improvement to MIT Education  and the Buechner Faculty Teaching Prize. In addition to his work developing physics courses at MIT, he assisted Professor Eric Mazur (Harvard University) in writing a first-year introductory physics textbook published in 2014.

The late Norman RAMSEY was the Higgins Professor of Physics at Harvard University and the recipient of the 1989 Nobel Prize in Physics.