1 The Real Numbers ............................................ 1
1.1 Intervals and Absolute Value ........................... 1
1.2 Rational and Irrational Numbers ........................ 3
1.3 Sequences .............................................. 5
1.4 Increasing Sequences .................................. 11
1.5 The Increasing Bounded Sequence Property .............. 13
1.6 The Nested Interval Property .......................... 17
Exercises .................................................. 18
References ................................................. 24
2 Famous Inequalities ........................................ 25
2.1 Bernoulli's Inequality and Euler's Number e ........... 25
2.2 The AGM Inequality .................................... 28
2.3 The Cauchy-Schwarz Inequality ......................... 34
Exercises .................................................. 37
References ................................................. 50
3 Continuous Functions ....................................... 53
3.1 Basic Properties ...................................... 53
3.2 Bolzano's Theorem ..................................... 56
3.3 The Universal Chord Theorem ........................... 59
3.4 The Intermediate Value Theorem ........................ 61
3.5 The Extreme Value Theorem ............................. 65
Exercises .................................................. 66
References ................................................. 71
4 Differentiable Functions ................................... 73
4.1 Basic Properties ...................................... 73
4.2 Differentiation Rules ................................. 77
4.3 Derivatives of Transcendental Functions ............... 80
4.4 Fermat's Theorem and Applications ..................... 83
Exercises .................................................. 87
References ................................................. 94
5 The Mean Value Theorem ..................................... 97
5.1 The Mean Value Theorem ................................ 97
5.2 Applications ......................................... 100
5.3 Cauchy's Mean Value Theorem .......................... 104
Exercises ................................................. 107
References ................................................ 116
6 The Exponential Function .................................. 119
6.1 The Exponential Function, Quickly .................... 119
6.2 The Exponential Function, Carefully .................. 122
6.3 The Natural Logarithmic Function ..................... 126
6.4 Real Exponents ....................................... 130
6.5 The AGM Inequality Again ............................. 132
6.6 The Logarithmic Mean ................................. 135
6.7 The Harmonic Series and Some Relatives ............... 137
Exercises ................................................. 142
References ................................................ 156
7 Other Mean Value Theorems ................................. 159
7.1 Darboux's Theorem .................................... 159
7.2 Flett's Mean Value Theorem ........................... 162
7.3 Pompeiu's Mean Value Theorem ......................... 163
7.4 A Related Result ..................................... 165
Exercises ................................................. 165
References ................................................ 169
8 Convex Functions and Taylor's Theorem ..................... 171
8.1 Higher Derivatives ................................... 171
8.2 Convex Functions ..................................... 176
8.3 Jensen's Inequality .................................. 179
8.4 Taylor's Theorem: e Is Irrational .................... 183
8.5 Taylor Series ........................................ 187
Exercises ................................................. 189
References ................................................ 207
9 Integration of Continuous Functions ....................... 209
9.1 The Average Value of a Continuous Function ........... 209
9.2 The Definite Integral ................................ 215
9.3 The Definite Integral as Area ........................ 220
9.4 Some Applications .................................... 223
9.5 Famous Inequalities for the Definite Integral ........ 228
9.6 Epilogue ............................................. 233
Exercises ................................................. 234
References ................................................ 247
10 The Fundamental Theorem of Calculus ....................... 249
10.1 The Fundamental Theorem .............................. 249
10.2 The Natural Logarithmic and Exponential Functions
Again ................................................ 260
Exercises ................................................. 264
References ................................................ 279
11 Techniques of Integration ................................. 283
11.1 Integration by u-Substitution ........................ 283
11.2 Integration by Parts ................................. 287
11.3 Two Consequences ..................................... 290
11.4 Taylor's Theorem Again ............................... 294
Exercises ................................................. 296
References ................................................ 309
12 Classic Examples .......................................... 311
12.1 Wallis's Product ..................................... 311
12.2 π Is Irrational ...................................... 314
12.3 More Irrational Numbers .............................. 315
12.4 Euler's Sum Σ l/n2 = π2/6 ............................ 318
12.5 The Sum Σ l/p of the Reciprocals of the Primes
Diverges ............................................. 321
Exercises ................................................. 323
References ................................................ 328
13 Simple Quadrature Rules ................................... 331
13.1 The Rectangle Rules .................................. 331
13.2 The Trapezoid and Midpoint Rules ..................... 333
13.3 Stirling's Formula ................................... 337
13.4 Trapezoid Rule or Midpoint Rule: Which Is Better? .... 341
Exercises ................................................. 342
References ................................................ 354
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