Calculus, once again

4. MODIFICATIONS......Page 3
10. FUTURE REVISIONS OF THIS LICENSE......Page 4
Preface......Page 8
Sets......Page 9
Homework......Page 11
Numerical Functions......Page 12
Injective and Surjective Functions......Page 13
Inverse Image......Page 14
Inverse Function......Page 15
Homework......Page 16
Countability......Page 17
Groups and Fields......Page 18
Homework......Page 20
Addition and Multiplication in R......Page 21
Homework......Page 26
Order Axioms......Page 27
Absolute Value......Page 28
Triangle Inequality......Page 29
Rearrangement Inequality......Page 31
Arithmetic Mean-Geometric Mean Inequality......Page 32
Cauchy-Bunyakovsky-Schwarz Inequality......Page 34
Minkowski's Inequality......Page 35
Homework......Page 36
Completeness Axiom......Page 37
Homework......Page 40
Intervals......Page 42
Dense Sets......Page 44
Open and Closed Sets......Page 45
Interior, Boundary, and Closure of a Set......Page 47
Compact Sets......Page 49
Homework......Page 51
R......Page 52
The Cantor Set......Page 53
Convergence of Sequences......Page 55
Homework......Page 59
Classical Limits of Sequences......Page 61
Homework......Page 66
Averages of Sequences......Page 67
Orders of Infinity......Page 70
Cauchy Sequences......Page 74
Topology of sequences. Limit Superior and Limit Inferior......Page 75
Homework......Page 76
Convergence and Divergence of Series......Page 78
Homework......Page 80
Convergence and Divergence of Series of Positive Terms......Page 81
Homework......Page 87
Summation by Parts......Page 88
Homework......Page 91
Absolute Convergence......Page 92
Limits of Functions......Page 93
Homework......Page 97
Continuity......Page 98
Algebraic Operations with Continuous Functions......Page 101
Monotonicity and Inverse Image......Page 102
Graphs of Functions......Page 103
Trigonometric Functions......Page 104
Continuity of the Exponential and Logarithmic Functions......Page 105
Homework......Page 106
Inequalities Obtained by Continuity Arguments......Page 107
Intermediate Value Property......Page 111
Variation of a Function and Uniform Continuity......Page 113
Classical Limits......Page 117
Homework......Page 121
Differentiation Rules......Page 122
Homework......Page 127
Rolle's Theorem and the Mean Value Theorem......Page 128
Homework......Page 129
Extrema......Page 130
Convex Functions......Page 131
Homework......Page 134
Inequalities Obtained Through Differentiation......Page 135
Homework......Page 137
Asymptotic Preponderance......Page 138
Asymptotic Equivalence......Page 142
Homework......Page 147
Asymptotic Expansions......Page 148
Homework......Page 150
The Area Problem......Page 151
Integration......Page 160
Homework......Page 165
Euler's Summation Formula......Page 167
Uniform Convergence......Page 168
Maclaurin Expansions to know by inspection......Page 169
Homework......Page 170
Answers and Hints......Page 171