CONTENTS
Chapter 1 Functions, Limits, and Continuity 1
1.1 Functions 1
1.1.1 Linear and Quadratic Functions 1
1.1.2 Concept of Function 3
1.1.3 Polynomial and Rational Functions 5
1.1.4 Exponential and Logarithmic Functions 6
1.1.5 Trigonometric Functions and Functional Properties 8
1.2 Limits of Function 10
1.2.1 The Concept of Limit 10
1.2.2 Computation of Limits 15
1.3 Continuity of Function 18
1.3.1 The Continuity of Function 18
1.3.2* Continuous Compounding 21
Chapter Summary 22
Review Exercises 23
Chapter 2 Differentiation of One Variable 25
2.1 The Concept of Derivative 25
2.1.1 Instantaneous Velocity and Derivative 25
2.1.2 Slope of Tangent Line on Geometric Interpretation of Derivative 26
2.1.3 Definition of Derivative and Rates of Change 27
2.2 Computations of Derivatives 28
2.2.1 Techniques of the Differentiation 28
2.2.2 Calculation Rules of Derivative 30
2.3 Compound Function and Its Chain Rule 31
2.3.1 Compound Function and Its Chain Rule 31
2.3.2 Implicit Differentiation 33
2.4 Second-Order Derivative and Differential 34
2.4.1 Second-Order Derivative 34
2.4.2 The Concept and Computation of Differential 35
2.5 Application of the Derivative 36
2.5.1 Increasing and Decreasing Functions in the Derivative 37
2.5.2 Concavity and Points of Inflection of Functions 38
2.5.3 Relative Maximum and Relative Minimum of Functions 41
Chapter Summary 44
Review Exercises 45
Chapter 3 Integration of One Variable 46
3.1 Indefinite Integration 46
3.1.1 The Concept of Indefinite Integration 46
3.1.2 The Computing Rules and Formulas of Indefinite Integration 48
3.1.3 Integration by Substitution 50
3.1.4 Integration by Parts 52
3.2 Definite Integration 55
3.2.1 Definite Integral and the Fundamental Theorem of Calculus 55
3.2.2 The Computation of Definite Integral 59
3.2.3 Applications of Integration 62
3.2.4 Improper Integrals 67
Chapter Summary 70
Review Exercises 71
Chapter 4 Calculus of Several Variables 73
4.1 Functions of Several Variables 73
4.1.1 Functions of Two or More Variables 73
4.1.2 Graphs of Functions of Two Variables 74
4.2 Partial Derivatives 78
4.2.1 Compute and Interpret Partial Derivatives 78
4.2.2 Geometric Interpretation of Partial Derivatives 79
4.2.3 Second-order Partial Derivatives 80
4.2.4 The Chain Rule for Partial Derivatives 80
4.3 Optimizing Functions of Two Variables 82
4.3.1 The Extreme Value Property for a Function of Two Variables 82
4.3.2 Apply the Extreme Value Property to the Functions of Two Variables 84
4.3.3* The Method of Least-Squares 86
4.3.4* The Least-Squares Line 88
4.4 Double Integrals 90
4.4.1 The Double Integral over a Rectangular Region 90
4.4.2 Double Integrals over Nonrectangular Regions 91
4.4.3 The Applications of Double Integrals 93
Chapter Summary 97
Review Exercises 98
APPENDIXES 100
APPENDIX A 100
APPENDIX B 100
APPENDIX C English-Chinese Vocabulary 101