Quick Calculus

Daniel KLEPPNER is the Lester Wolfe Professor of Physics at MIT. He was awarded the National Medal of Science and the Oersted Medal of the American Association of Physics Teachers. peter DOURMASHKIN is Senior Lecturer at MIT. The late Norman RAMSEY was the Higgins Professor of Physics at Harvard University and the recipient of the 1989 Nobel Prize in Physics.

• Preface iiiChapter One Starting Out 11.1 A Few Preliminaries 11.2 Functions 21.3 Graphs 51.4 Linear and Quadratic Functions 111.5 Angles and Their Measurements 191.6 Trigonometry 281.7 Exponentials and Logarithms 42Summary of Chapter 1 51Chapter Two Differential Calculus 572.1 The Limit of a Function 572.2 Velocity 712.3 Derivatives 832.4 Graphs of Functions and Their Derivatives 872.5 Differentiation 972.6 Some Rules for Differentiation 1032.7 Differentiating Trigonometric Functions 1142.8 Differentiating Logarithms and Exponentials 1212.9 Higher-Order Derivatives 1302.10 Maxima and Minima 1342.11 Differentials 1432.12 A Short Review and Some Problems 147Conclusion to Chapter 2 164Summary of Chapter 2 165Chapter Three Integral Calculus 1693.1 Antiderivative, Integration, and the Indefinite Integral 1703.2 Some Techniques of Integration 1743.3 Area Under a Curve and the Definite Integral 1823.4 Some Applications of Integration 2013.5 Multiple Integrals 211Conclusion to Chapter 3 219Summary of Chapter 3 219Chapter Four Advanced Topics: Taylor Series, Numerical Integration, and Differential Equations 2234.1 Taylor Series 2234.2 Numerical Integration 2324.3 Differential Equations 2354.4 Additional Problems for Chapter 4 244Summary of Chapter 4 248Conclusion (frame 449) 250Appendix A Derivations 251A.1 Trigonometric Functions of Sums of Angles 251A.2 Some Theorems on Limits 252A.3 Exponential Function 254A.4 Proof That dy/dx = 1/dx∕dy 255A.5 Differentiating Xn 256A.6 Differentiating Trigonometric Functions 258A.7 Differentiating the Product of Two Functions 258A.8 Chain Rule for Differentiating 259A.9 Differentiating Ln X 259A.10 Differentials When Both Variables Depend on a Third Variable 260A.11 Proof That if Two Functions Have the Same Derivative They Differ Only by a Constant 261A.12 Limits Involving Trigonometric Functions 261Appendix B Additional Topics in Differential Calculus 263B.1 Implicit Differentiation 263B.2 Differentiating the Inverse Trigonometric Functions 264B.3 Partial Derivatives 267B.4 Radial Acceleration in Circular Motion 269B.5 Resources for Further Study 270Frame Problems Answers 273Answers to Selected Problems from the Text 273Review Problems 277Chapter 1 277Chapter 2 278Chapter 3 282Tables 287Table 1: Derivatives 287Table 2: Integrals 288Indexes 291Index 291Index of Symbols 295