Table of Contents
How to Use This Book vi
Chapter 1 Limits and Continuity 1
Definition of a Limit 1
Graphic Approach to Limits 2
Graphic Approach to Limits at Infinity 5
Properties of Limits 9
Algebraic Approach to Limits 13
Algebraic Approach to Limits at Infinity 20
Continuity 28
Intermediate Value Theorem 32
Brain Ticklers-The Answers 34
Chapter 2 Differentiation 35
Average Rate of Change 35
Slope of a Tangent Line 38
Definition of a Derivative Function 43
Differentiability 46
Continuity and Differentiability 51
Brain Ticklers-The Answers 52
Chapter 3 Derivatives 53
Notation 53
Properties of Derivatives 55
Power Rule 56
Product Rule 59
Quotient Rule 60
Chain Rule 63
Trigonometric Functions 64
Exponential Functions 66
Logarithmic Functions 68
Sneaky Derivatives 70
Implicit Differentiation 72
Logarithmic Differentiation 76
Inverse Functions 77
Brain Ticklers-The Answers 81
Chapter 4 Applications of Derivatives 83
Equation of a Tangent Line 83
Equation of a Normal Line 86
Local Linear Approximation 88
L'Hopital's Rule 92
Curve Sketching 99
Graphical Representation of the Derivative 118
Second Derivative Test 121
Absolute Maxima and Minima 124
Extreme Value Theorem 126
Optimization 127
Rectilinear Motion 131
Mean Value Theorem 140
Related Rates 142
Brain Ticklers-The Answers 153
Chapter 5 Antidifferentiation 155
Indefinite Integrals 156
Properties of Indefinite Integrals 158
Integral Formulas 160
Integration by U-Substitution 162
Integration by Parts 176
Integration by Partial Fraction Decomposition 185
Integration by Completing the Square 194
Integration with Powers of Trigonometric Functions 197
Integration by Trigonometric Substitutions 201
Brain Ticklers-The Answers 207
Chapter 6 Definite Integrals 209
Notation 209
Properties of Definite Integrals 213
Riemann Sum 218
Trapezoidal Sum 229
Fundamental Theorem of Calculus 237
Mean Value Theorem for Integrals 244
Average Function Value 246
Brain Ticklers-The Answers 250
Chapter 7 Applications of Definite Integrals 251
Area Between Curves 251
Volume of a Solid of Revolution 267
Volume of a Solid with a Known Cross Section 279
Arc Length 289
Rectilinear Motion 292
Brain Ticklers-The Answers 298
Chapter 8 Differential Equations 299
Understanding Differential Equations 299
Separation of Variables 299
Exponential Growth and Decay 304
Logistic Growth 307
Slope Fields 312
Euler's Method 322
Brain Ticklers-The Answers 330
Appendix: Important Derivatives, Formulas, Integrals, Limits, and Theorems 333
Important Derivatives 333
Important Formulas 335
Important Integrals 335
Important Limits 336
Important Theorems 336
Index 339
