On the Shoulders of Giants: A Course in Single Variable Calculus

On the Shoulders of Giants - A Course in Single Variable Calculus 1 CONTENTS 2 CHAPTER 1 TERROR, TRAGEDY AND BAD VIBRATIONS 8 1.1 INTRODUCTION 8 1.2 THE TOWER OF TERROR 8 1.3 INTO THIN AIR 11 1.4 MUSIC AND THE BRIDGE 14 1.5 DISCUSSION 15 1.6 RULES OF CALCULATION 16 CHAPTER 2 FUNCTIONS 18 2.1 RULES OF CALCULATION 18 2.2 INTERVALS ON THE REAL LINE 22 2.3 GRAPHS OF FUNCTIONS 24 2.4 EXAMPLES OF FUNCTIONS 27 CHAPTER 3 CONTINUITY AND SMOOTHNESS 34 3.1 SMOOTH FUNCTIONS 34 3.2 CONTINUITY 37 CHAPTER 4 DIFFERENTIATION 48 4.1 THE DERIVATIVE 48 4.2 RULES FOR DIFFERENTIATION 55 4.3 VELOCITY, ACCELERATION AND RATES OF CHANGE 60 CHAPTER 5 FALLING BODIES 64 5.1 THE TOWER OF TERROR 64 5.2 SOLVING DIFFERENTIAL EQUATIONS 69 5.3 GENERAL REMARKS 72 5.4 INCREASING AND DECREASING FUNCTIONS 75 5.5 EXTREME VALUES 77 CHAPTER 6 SERIES AND THE EXPONENTIAL FUNCTION 82 6.1 THE AIR PRESSURE PROBLEM 82 6.2 INFINITE SERIES 88 6.3 CONVERGENCE OF SERIES 91 6.4 RADIUS OF CONVERGENCE 97 6.5 DIFFERENTIATION OF POWER SERIES 100 6.6 THE CHAIN RULE 103 6.7 PROPERTIES OF THE EXPONENTIAL FUNCTION 106 6.8 SOLUTION OF THE AIR PRESSURE PROBLEM 109 CHAPTER 7 TRIGONOMETRIC FUNCTIONS 116 7.1 VIBRATING STRINGS AND CABLES 116 7.2 TRIGONOMETRIC FUNCTIONS 118 7.3 MORE ON THE SINE AND COSINE FUNCTIONS 121 7.4 TRIANGLES, CIRCLES AND THE NUMBER 126 7.5 EXACT VALUES OF THE SINE AND COSINE FUNCTIONS 129 7.6 OTHER TRIGONOMETRIC FUNCTIONS 132 CHAPTER 8 OSCILLATION PROBLEMS 134 8.1 SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS 134 8.2 COMPLEX NUMBERS 141 8.3 COMPLEX SERIES 147 8.4 COMPLEX ROOTS OF THE AUXILIARY EQUATION 150 8.5 SIMPLE HARMONIC MOTION AND DAMPING 152 8.6 FORCED OSCILLATIONS 160 CHAPTER 9 INTEGRATION 174 9.1 ANOTHER PROBLEM ON THE TOWER OF TERROR 174 9.2 MORE ON AIR PRESSURE 175 9.3 INTEGRALS AND PRIMITIVE FUNCTIONS 177 9.4 AREAS UNDER CURVES 178 9.5 AREA FUNCTIONS 181 9.6 INTEGRATION 183 9.7 EVALUATION OF INTEGRALS 189 9.8 THE FUNDAMENTAL THEOREM OF THE CALCULUS 194 9.9 THE LOGARITHM FUNCTION 195 CHAPTER 10 INVERSE FUNCTIONS 204 10.1 THE EXISTENCE OF INVERSES 207 10.2 CALCULATING FUNCTION VALUES FOR INVERSES 212 10.3 THE OSCILLATION PROBLEM AGAIN 221 10.4 INVERSE TRIGONOMETRIC FUNCTIONS 225 10.5 OTHER INVERSE TRIGONOMETRIC FUNCTIONS 228 CHAPTER 11 HYPERBOLIC FUNCTIONS 232 11.1 HYPERBOLIC FUNCTIONS 232 11.2 PROPERTIES OF THE HYPERBOLIC FUNCTIONS 234 11.3 INVERSE HYPERBOLIC FUNCTIONS 237 CHAPTER 12 METHODS OF INTEGRATION 242 12.1 INTRODUCTION 242 12.2 CALCULATION OF DEFINITE INTEGRALS 244 12.3 INTEGRATION BY SUBSTITUTION 246 12.4 INTEGRATION BY PARTS 248 12.5 THE METHOD OF PARTIAL FRACTIONS 250 12.6 INTEGRALS WITH A QUADRATIC DENOMINATOR 254 12.7 CONCLUDING REMARKS 256 CHAPTER 13 A NONLINEAR DIFFERENTIAL EQUATION 258 13.1 THE ENERGY EQUATION 259 13.2 CONCLUSION 266 ANSWERS 268 Index 288