Contents
Chapter lPreliruinaries 1
I.I Some Set Theory Notation for the Study of Calculus 1
1.1.1 Definition of Set 1
1.1.2 Descriptions of set 1
1.1.3 Set Operations 2
1.1.4 Interval 3
1.1.5 Neighbourhood 3
1.2 The Rectangular Coordinate System 3
1.2.1 Cartesian Coordinates 3
1.2.2 Distance Formula 4
1.2.3 The Equation of a Circle 4
1.3 The Straight Line 6
1.3.1 The Slope of' a Line 6
1.3.2 The Equation of a Line 6
1.4 Graphs of Equations 7
1.4.1 The Graphing Procedure 7
1.4.2 Symmetry of a Graph 7
1.4.3 Intercepts 9
1.4.4 Problems for Chapter 10
Chapter 2Functions and Limits 12
2.1 Functions 12
2.1.1 Definition of Function 12
2.1.2 Properties of Functions 14
2.1.3 0perations on Functions 16
2.1.4 Elementary Functions 18
2.1.5 Problems for Section 2.1 19
2.2 Limits 20
2.2.1 Introduction to Limits 20
2.2.2 Definition of Limit 99
2.2.3 0perations on Limits 25
2.2.4 Limits at Infinity and Infinite Limits 29
2.2.5 Infinitely Small Quantity (or Infinitesimal) 33
2.2.6 Problems for Section 2.2 35
2.3 Continuity of Functions 36
2.3.1 Definition of Continuity 36
2.3.2 Continuity under Function Operations 38
2.3.3 Continuity of Elementary Functions 38
2.3.4 Intermediate Value Theorem 39
2.3.5 Problems for Section 2.3 39
2.4 Chapter Review 40
2.4.1 Drills 40
2.4.2 Sample Test Problems 41
Chapter 3 Differentiation 43
3.1 Derivatives 43
3.1.1 Two Problems with One Theme 43
3.1.2 Definition 45
3.1.3 Rules for Finding Derivatives 45
3.1.4 Problems for Section 3.1 52
3.2 Higher-Order Derivatives 53
3.2.1 Definition 53
3.2.2 Sum, Difference and Product Rules 55
3.2.3 Problems for Section 3.2 56
3.3 Implicit Differentiation 57
3.3.1 Guidelines for implicit Differentiation 57
3.3.2 Related Rates 59
3.3.3 Problems for Section 3.3 61
3.4 Differentials and Approximations 62
3.4.1 Definition of Differential 62
3.4.2 Differential Rules63
3.4.3 Approximations 64
3.4.4 Problems for Section 3.4 67
3.5 Chapter Review 68
3.5.1 Drills 68
3.5.2 Sample Test Problems 69
Chapter 4 Applications of Differentiation 70
4.1 Maxima and Minima 70
4.1.1 Extrema on an Interval 70
4.1.2 Problems for Section 4.1 74
4.2 Monotonicity and Concavity 75
4.2.1 The First Derivative and Monotonicity 75
4.2.2 The Second Derivative and Concavity 77
4.2.3 Problems for Section 4.2 81
4.3 Local Maxima and Minima 82
4.3.1 Definition 82
4.3.2 Tests for Local Maxima and Minima 82
4.3.3 More Maxima and Minima Problems 84
4.3.4 Problems for Section 4.3 87
4.4 Sophisticated Graphing 88
4.4.1 Asymptote 88
4.4.2 Sophisticated Graphing 90
4.4.3 Problems for Section 4.4 94
4.5 The Mean Value Theorem 94
4.5.1 Rolle's Theorem 94
4.5.2 Lagrange's Mean Value Theorem 96
4.5.3 Cauchy's Mean Value Theorem 99
4.5.4 Problems for Section 4.5 99
4.6 L'Hopital's Rule 100
4.6.1 Indeterminate Forms of Type 100
4.6.2 Indeterminate Forms of Type里 101
4.6.3 0ther Indeterminate Forms 102
4.6.4 Problems for Section 4.6 103
4.7 Chapter Review 104
4.7.1 Drills 104
4.7.2 Sample Test Problems 106
Chapter5 Indefinite Integrals 108
5.1 Definition and Properties of Indefinite Integrals 108
5.1.1 Definition of Indefinite Integrals 108
5.1.2 Standard Integral Forms 110
5.1.3 Properties of' Indefinite Integrals 111
5.1.4 Problems for Section 5.1 113
5.2 Integration by Substitution 114
5.2.1 The Mehod of Substitution 114
5.2.2 Rationalizing Substitutions 121
5.2.3 0ther Substitutions (Inverse) 125
5.2.4 Problems for Section 5.2 126
5.3 Integration by Parts 127
5.3.1 Problems for Section 5.3 132
5.4 Integration of Rational Functions 132
5.4.1 Partial Fraction Decompositions 133
5.4.2 Integration of Rational Functions Using Partial Fractions 135
5.4.3 Problems for Section 5.4 137
5.5 Chapter Review 137
5.5.1 Drills 137
5.5.2 Sample Test Problems 138
Chapter6 Definite Integrals 140
6.1 Definition and Properties of Definite Integrals 140
6.1.1 Two Problems 140
6.1.2 Definition of Definite Integrals 143
6.1.3 Existence of' Definite Integrals 145
6.1.4 Geometric Interpretation 145
6.1.5 Properties of Definite Integrals 149
6.1.6 Problems for Section 6.1 152
6.2 Fundamental Theorem of Calculus 153
6.2.1 Problems for Section 6.2 157
6.3 Evaluation of Definite Integrals 158
6.3.1 Integration by Substitution 158
6.3.2 Integration by Parts 161
6.3.3 Problems for Section 6.3 163
6.4 hnproper Integrals 164
6.4.1 Infinite Limits of Integration 164
6.4.2 Infinite Integrands 166
6.4.3 Problems for Section 6.4 168
6.5 Chapter Review 169
6.5.1 Drills 169
6.5.2 Sample Test Problems 169
Chapter7 Applications of Integration 172
7.1 The Area of a Plane Region 172
7.1.1 Problems for Section 7.1 175
7.2 Volumes of Solids: Slabs, Disks and Washers 175
7.2.1 Volume of a Cylinder and a Solid 175
7.2.2 Solids of' Revolution: Disk Method 177
7.2.3 0ther Examples 180
7.2.4 Problems for Section 7.2 181
7.3 Volumes of Solids of Revolution: Shells 182
7.3.1 Problems for Section 7.3 185
7.4 Length of a Plane Curve 185
7.4.1 Problems for Section 7.4 189