Chapter 1 Functions and Limits 1 1.1 Sets and Functions 1 1.1.1 Sets 1 1.1.2 Functions 3 1.1.3 Properties of Functions 5 1.1.4 Combining Functions and Inverse Functions 7 1.1.5 Elementary Functions 11 1.2 The Limits of Functions 15 1.2.1 Limits of Sequence 15 1.2.2 Limits of Functions at Infinities and a Point 17 1.2.3 Calculating Limits 20 1.3 Two Important Limits 23 1.4 Infinitesimal and Infinite 26 1.4.1 Infinitesimal 26 1.4.2 Infinite 28 1.5 Continuous Functions 30 1.5.1 Continuity of Functions 30 1.5.2 Discontinuity 32 1.5.3 Properties of Continuous Functions on a Closed Interval 34 Chapter Review 37 Chapter 2 Differentiation and Applications of Differentiation 39 2.1 The Definition of Derivatives 39 2.1.1 The Tangent Line Problem 39 2.1.2 The Derivative of a Function 40 2.1.3 One��Side Derivatives 42 2.1.4 Differentiability and Continuity 42 2.2 Basic Differentiation Rules and Derivatives of Inverse Functions 43 2.2.1 Basic Differentiation Rules 43 2.2.2 Derivatives of Inverse Functions 44 2.3 The Chain Rule and Higher��Order Derivatives 45 2.3.1 The Chain Rule 45 2.3.2 Higher��Order Derivatives 47 2.4 Derivatives of Implicit and Parametric Functions 48 2.4.1 Implicit Differentiation 48 2.4.2 Derivatives of Parametric Functions 50 2.5 Differentials 52 2.5.1 Tangent Line Approximations 52 2.5.2 Differentials Expression 52 2.6 The Mean Value Theorem 54 2.7 L�� Hospital��s Rule 56 2.8 Monotonicity of Functions and Convexity of Curves 60 2.8.1 Monotonicity of Functions 60 2.8.2 Convexity of Curves 62 2.9 Maximum and Minimum Values 65 2.9.1 Extreme and Local Values 65 2.9.2 The Closed Interval Method 67 2.9.3 Optimization Problems 67 Chapter Review 69 Chapter 3 InDefinite Integrals 71 3.1 The Definition and Properties of InDefinite Integrals 71 3.1.1 Antiderivatives and InDefinite Integrals 71 3.1.2 The Properties of InDefinite Integrals 73 3.2 The Substitution Rule for InDefinite Integrals 74 3.2.1 Integration by Substitution of the First Type 74 3.2.2 Integration by Substitution of the Second Type 76 3.3 Integration by Parts 77 3.4 Integration of Rational Functions by Partial Fractions 80 Chapter Review 83 Chapter 4 Definite Integrals 85 4.1 Introduction of the Definite Integrals 85 4.1.1 Areas and Distances 85 4.1.2 The Definition of Definite Integrals 88 4.1.3 Geometric Meaning of Definite Integrals 89 4.1.4 Properties of Definite Integrals 90 4.2 The Fundamental Theorems of Calculus 92 4.2.1 Variable Upper Bound Integral Function 92 4.2.2 Fundamental Theorems 95 4.3 The Calculation of Definite Integrals 98 4.3.1 The Methods of Substitution and Partial Integration for Definite Integrals 98 4.3.2 Integration by Parts 101 4.4 Improper Integrals on an Infinite Interval 103 4.5 An Application of Integration: Areas Between Curves 105 Chapter Review 111 Chapter 5 Differential Equations 113 5.1 General Differential Equations and Solutions 113 5.2 Separable Equations 115 5.3 First��Order Linear Equations 119 5.4 Homogeneous Second��Order Linear Equations with Constant Coefficients 124 5.4.1 Second��Order Linear Equations 124 5.4.2 Homogeneous Second��Order Linear Equations with Constant Coefficients 125 5.4.3 Summary 130 Chapter Review 130 Chapter 6 Infinite Series 132 6.1 Sequences 132 6.1.1 Introduction to Sequences 132 6.1.2 Convergence and Divergence 133 6.1.3 Calculating Limits of Sequences 135 6.1.4 Bounded Monotonic Sequences 137 6.2 Infinite Series 138 6.2.1 Partial Sums 138 6.2.2 Geometric Series 139 6.2.3 Telescoping Series 141 6.2.4 Harmonic Series 141 6.2.5 The n��th Term Test for a Divergent Series 142 6.2.6 Combining Series 143 6.2.7 Adding or Deleting Terms 144 6.2.8 Reindexing 144 6.3 The Comparison Tests 145 6.3.1 The Direct Comparison Test 145 6.3.2 The Limit Comparison Test 147 6.4 Alternating Series 148 6.5 Absolute Convergence and the Ratio and Root Tests 150 6.5.1 Absolute Convergence 150 6.5.2 The Ratio Test 152 6.5.3 The Root Test 153 6.5.4 Rearranging Series 155 6.5.5 Strategy for Testing Series 156 Chapter Review 157 Bibliography 160