On the shoulders of giants: A course in single variable calculus

CONTENTS......Page 5
Preface......Page 9
1.2 The Tower of Terror......Page 11
1.3 Into thin air......Page 14
1.4 Music and the bridge......Page 17
1.5 Discussion......Page 18
1.6 Rules of calculation......Page 19
2.1 Rules of calculation......Page 21
2.2 Intervals on the real line......Page 25
2.3 Graphs of functions......Page 27
2.4 Examples of functions......Page 30
3.1 Smooth functions......Page 37
3.2 Continuity......Page 40
4.1 The derivative......Page 51
4.2 Rules for differentiation......Page 58
4.3 Velocity, acceleration and rates of change......Page 63
5.1 The Tower of Terror......Page 67
5.2 Solving differential equations......Page 72
5.3 General remarks......Page 75
5.4 Increasing and decreasing functions......Page 78
5.5 Extreme values......Page 80
6.1 The air pressure problem......Page 85
6.2 Infinite series......Page 91
6.3 Convergence of series......Page 94
6.4 Radius of convergence......Page 100
6.5 Differentiation of power series......Page 103
6.6 The chain rule......Page 106
6.7 Properties of the exponential function......Page 109
6.8 Solution of the air pressure problem......Page 112
7.1 Vibrating strings and cables......Page 119
7.2 Trigonometric functions......Page 121
7.3 More on the sine and cosine functions......Page 124
7.4 Triangles, circles and the number π......Page 129
7.5 Exact values of the sine and cosine functions......Page 132
7.6 Other trigonometric functions......Page 135
8.1 Second order linear differential equations......Page 137
8.2 Complex numbers......Page 144
8.3 Complex series......Page 150
8.4 Complex roots of the auxiliary equation......Page 153
8.5 Simple harmonic motion and damping......Page 155
8.6 Forced oscillations......Page 163
9.1 Another problem on the Tower of Terror......Page 177
9.2 More on air pressure......Page 178
9.3 Integrals and primitive functions......Page 180
9.4 Areas under curves......Page 181
9.5 Area functions......Page 184
9.6 Integration......Page 186
9.7 Evaluation of integrals......Page 192
9.8 The fundamental theorem of the calculus......Page 197
9.9 The logarithm function......Page 198
10 Inverse functions......Page 207
10.1 The existence of inverses......Page 210
10.2 Calculating function values for inverses......Page 215
10.3 The oscillation problem again......Page 224
10.4 Inverse trigonometric functions......Page 228
10.5 Other inverse trigonometric functions......Page 231
11.1 Hyperbolic functions......Page 235
11.2 Properties of the hyperbolic functions......Page 237
11.3 Inverse hyperbolic functions......Page 240
12.1 Introduction......Page 245
12.2 Calculation of definite integrals......Page 247
12.3 Integration by substitution......Page 249
12.4 Integration by parts......Page 251
12.5 The method of partial fractions......Page 253
12.6 Integrals with a quadratic denominator......Page 257
12.7 Concluding remarks......Page 259
13 A nonlinear differential equation......Page 261
13.1 The energy equation......Page 262
13.2 Conclusion......Page 269
Answers......Page 271
D......Page 291
L......Page 292
S......Page 293
W......Page 294