Inside Calculus

Cover......Page 1
Undergraduate Texts in Mathematics......Page 2
Inside Calculus......Page 4
Copyright - ISBN: 0387989323......Page 5
Contents......Page 14
Introduction......Page 8
Acknowledgments......Page 13
1 Limits......Page 20
1.1.1 Absolute Value......Page 21
1.2 First Limit......Page 23
1.2.1 The Graphical Method......Page 24
1.2.2 Exercises......Page 25
1.3 Second Limit......Page 26
1.3.1 Exercises......Page 28
1.4 Anxiety, and Some Limits that Don\'t Work......Page 30
1.4.1 Exercises......Page 33
1.5 More Limits that Don\'t Exist......Page 34
1.5.1 Exercises......Page 36
2.1 Continuity at a Point......Page 38
2.2 Naive Continuity vs. Continuity......Page 40
2.3 Continuity on a Set......Page 43
2.3.1 Exercises......Page 44
2.3.2 Continuous Functions......Page 46
2.3.3 Exercises......Page 48
3 The Language of Theorems......Page 50
3.1 Implication......Page 51
3.1.1 Exercises......Page 53
3.2 For Every......Page 54
3.2.1 Exercises......Page 56
3.3.1 Exercises......Page 57
4 Theorems about Continuous Functions......Page 58
4.1 The Intermediate Value Theorem......Page 59
4.1.1 Exercise......Page 61
4.1.2 Why These Hypotheses?......Page 62
4.1.3 Generalizations......Page 63
4.1.5 Who Cares?......Page 64
4.1.6 Exercises......Page 65
4.2.1 Preliminaries......Page 66
4.2.2 Exercises......Page 67
4.2.3 Guarantees of Maxima......Page 68
4.2.5 Applications......Page 71
4.3 Digression: Compare and Contrast......Page 72
4.3.1 The Payoff......Page 73
5 Limit Proofs......Page 74
5.1.1 Existence Template I......Page 75
5.1.3 Existence Template II......Page 76
5.1.4 Exercises......Page 78
5.1.5 Universal Template......Page 79
5.1.6 Exercises......Page 82
5.2 Limits, Finally......Page 83
5.2.2 Nasty Technical Details......Page 86
5.2.3 Exercises......Page 87
6 Limit Theorems......Page 90
6.1 A Single δ......Page 92
6.1.2 A Single δ, but Worse......Page 94
6.1.3 Exercises......Page 96
6.2 Two δ’s: The Sum Theorem......Page 98
6.3 Two δ’s (?): The Product Theorem......Page 100
6.4 Two δ’s: The Quotient Theorem......Page 102
6.5.1 Composition of Functions......Page 103
6.5.3 Limits of Composite Functions......Page 106
6.6 Three δ’s: The Squeeze Theorem......Page 109
6.6.1 Exercises......Page 113
6.7 Change of Limiting Variable......Page 114
6.7.1 Exercise......Page 115
7.1 Preview: Direct from the Limit Theorems......Page 116
7.2 Small Integer Powers of x......Page 117
7.2.1 Exercises......Page 118
7.3 Mathematical Induction......Page 119
7.3.2 Use of the Induction Theorem......Page 120
7.4 Better Use of the Induction Theorem: Polynomials Plus......Page 122
7.4.2 Rational Functions......Page 125
7.5 Trigonometric Functions (Trouble)......Page 126
7.5.1 Continuity of the Sine at 0......Page 127
7.5.2 What\'s the Problem?......Page 128
7.5.3 A Lemma: Continuity of the Square Root......Page 129
7.5.5 Continuity of the Cosine at Zero......Page 130
7.6 Logs and Exponential Functions (Worse Trouble)......Page 131
7.6.1 Continuity of ln......Page 132
7.6.2 Exercises......Page 134
8.1 General Derivative Theorems......Page 136
8.2 Continuity Meets Differentiability......Page 139
8.3 The Monster, Weakened......Page 141
8.4 Polynomial and Rational Derivatives......Page 143
8.4.1 Exercises......Page 144
8.5.1 Useful Trigonometric Limits......Page 145
8.6 The Derivative Function......Page 147
8.7 Derivatives of Rational Powers......Page 149
8.8 Derivatives of Exponential and Logarithmic Functions......Page 151
8.8.1 Exercises......Page 153
9.1.1 Preliminaries......Page 154
9.1.2 The Derivative and Local Extrema......Page 156
9.1.3 Exercises......Page 159
9.2 The Mean Value Theorem......Page 160
9.2.2 Rolle\'s Theorem......Page 161
9.2.3 Exercises......Page 162
9.2.4 The Theorem Itself......Page 163
9.2.5 Exercises......Page 164
9.3 Consequences of the Mean Value Theorem......Page 165
9.3.1 Exercises......Page 167
10.1 Limits Involving Infinity......Page 170
10.1.1 Exercises......Page 171
10.2.1 The Definition of Sequence......Page 172
10.2.2 Sequence Limits......Page 173
10.2.3 Exercises......Page 175
10.3 Functions of Several Variables......Page 176
10.3.1 Exercises......Page 178
10.3.2 Limits for Functions from R to R[sup(2)]......Page 182
10.3.3 Exercises......Page 183
10.3.4 Limits for Functions from R[sup(2)] to R......Page 184
10.3.5 Exercises......Page 186
10.4 The Integral......Page 187
10.4.1 Difficulties with the Integral Definition......Page 188
10.4.2 Exercises......Page 190
10.4.3 Preparation: l.u.b. and g.l.b.......Page 191
10.4.5 Upper and Lower Sums......Page 192
10.4.6 Exercises......Page 198
A Hints for Selected Exercises......Page 200
References......Page 224
C......Page 226
I......Page 227
L......Page 228
Q......Page 229
V......Page 230