国外高校优秀教材审定委员会 主任委员：杨叔子 委员（按姓氏笔画为序）： 王先逵、王大康、白峰杉、史荣昌、朱孝禄、陆启韶、张润琦、张策、张三慧、张福润、张延华、吴宗泽、吴麒、宋心琦、李俊峰、佘远斌、陈文楷、陈立周、俞正光、赵汝嘉、翁海珊、龚光鲁、章栋恩、黄永畅、谭泽光

1 Preliminaries
1.1 The Real Number System
1.2 Decimals, Calculators, Estimation
1.3 Inequalties
1.4 Absolute Values, Square Roots, Squares
1.5 The Rectangular Coordinate System
1.6 The Straight Line
1.7 Graphs of Equations
1.8 Chapter Review
Technology Project1.1 Graphing
Technology Project1.2 Solving Equations by Zooming
2 Functions and Limits
2.1 Functions and Their Graphs
2.2 Operations on Functions
2.3 The Trigonometric Functions
2.4 Introduction to Limits
2.5 Rigorous Study of Limits
2.6 Limit Theorems
2.7 Limits Involving Trigonometric Functions
2.8 Limits at Infinity, Infinite Limits
2.9 Continuity of Functions
2.10 Chapter Reciew
2.11 Additional Problems
Technology Project2.1 Shifting and Scaling the Graph of a Function
Technology Project2.2 Limits
3 The Derivative
3.1 Two Problems with One Theme
3.2 The Derivative
3.3 Rules for Finding Derivatives
3.4 Derivatives of Trigonometric Functions
3.5 The Chain Rule
3.6 Leibniz Notation
3.7 Higher-Order Derivatives
3.8 Implicit Differentiation
3.9 Related Rates
3.10 Differentials and Approximations
3.11 Chapter Review
3.12 Additional Problems
Technology Project3.1 Secant and Tangent Lines
Technology Project3.2 Linear Approximation to a Function
4 Applications of the Derivative
4.1 Maxima and Minima
4.2 Monotonicity and Concavity
4.3 Local Maxima and Minima
4.4 More Max-Min Problems
4.5 Econimic Applications
4.6 Sophisticated Graphing
4.7 The Mean Value Theorem
4.8 Chapter Review
4.9 Additional Problems
Technology Project4.1 Reflection and Refraction of Light
Technology Project4.2 An Optimization Problem
5 The Integral
5.1 Antiderivatives (Indefinite Integrals)
5.2 Introduction to Differential Equations
5.3 Sums and Sigma Notion
5.4 Introduction to Area
5.5 The Definite Integral
5.6 The First Fundamental Theorem of Calculus
5.7 The First Fundamental Theorem of Calulus
Theorem for Integrals
5.8 Evaluating Definite Integrals
5.9 Chapter Review
5.10 Additional Problems
Technology Project5.1 Riemann Sums
Technology Project5.2 Accumulation Functions
6 Applications of the Integral
6.1 The Area of a Plane Region
6.2 Volumes of Solids: Slabs, Disks, Washers
6.3 Volumes of Solids of Revolution: Shells
6.4 Length of a Plane Curve
6.5 Work
6.6 Moments, Center of Mass
6.7 Chapter Review
6.8 Additional Problems
Technology Project6.1 Volume in an Elliptical Cylinder
Technology Project6.2 Arc Length
7 Transcendetal Functions
7.1 The Natural Logarithm Function
7.2 Inverse Functions and Their Derivatives
7.3 The Natural Exponetial Function
7.4 General Exponential and Logarithmic Functions
7.5 Exponential Growth and Decay
7.6 First-Order Linear Differential Equations
7.7 The Inverse Trigonometric Functions and Their Derivatives
7.8 The Hyperbolic Functions and Their Inverses
7.9 Chapter Reciew
7.10 Additional Problems
Technology Project7.1 Special Functions
Technology Project7.2 Population Growth and Least Squares
8 Techniques of Integration
8.1 Integration by Substitution
8.2 Some Trigonometric Integrals
8.3 Rationalizing Substitutions
8.4 Integration by Parts
8.5 Integration of Rational Functions
8.6 Chapter Review
Technology Project8.1 Integration Using a Computer Algebra System
Technology Project8.2 The Logistic Differential Equation
9 Indeterminate Forms and Improper Integrals
9.1 Indeterminate Forms of Type 0/0
9.2 Other Indeterminate Forms
9.3 Improper Integrals: Infinite Limits of Integration
9.4 Improper Integrals: Infinite Integrands
9.5 Chapter Review
9.6 Additional Problems
Technology Project9.1 Probability Density Functions
Technology Project9.2 The Normal Distribution
10 Infinite Series
10.1 Infinite Sequences
10.2 Infinite Series
10.3 Positive Series: The Integral Test
10.4 Positive Series: Other Tests
10.5 Alternating Series, Absolute Convergence, and Conditional Convergence
10.6 Power Series
10.7 Operations on Power Series
10.8 Taylor and Maclaurin Series
10.9 Chapter Review
Technology Project10.1 Using Infinite Series to Approximate π
Technology Project10.2 Euler#s Derivation of
11 Numerical Methods, Approximations
11.1 The Taylor Approximation to a Function
11.2 Numerical Integration
11.3 Solving Equations Numerically
11.4 The Fixed-Point Algorithm
11.5 Approximations for Differential Equations
11.6 Chapter Review
Technology Project11.1 Maclaurin Polynomials
Technology Project11.2 Numerical Integration
Technology Project11.3 Fixed-Point Methods
12 Conics and Polar Coordinates
12.1 The Parabola
12.2 Ellipses and Hyperbolas
12.3 More on Ellipses and Hyperbolas
12.4 Translation of Axes
12.5 Rotation of Axes
12.6 The Polar Coordinate System
12.7 Graphs of Polar Coordinates
12.8 Calulus in Polar Coordinates
12.9 Chapter Review
Technology Project12.1 Rotations in the Plane
Technology Project12.2 Another Kind of Rose
13 Geometry in the Plane, Vectors
13.1 Plane Curves: Parametric Representation
13.2 Vectors in the Plane: Geometric Approach
13.3 Vectors in the Plane: Algebraic Approach
13.4 Vector-Valued Functions and Curvilinear Motion
13.5 Curvature and Acceleration
13.6 Chapter Review
Technology Project13.1 Hypocycloids
Technology Project13.2 Measuring Home Run Distance
14 Geometry in Space, Vectors
14.1 Cartesian Coordinates in Three-Space
14.2 Vectors in Tree-Space
14.3 The Cross Product
14.4 Line and Curves in Three-Space
14.5 Velocity, Acceleration, and Curvature
14.6 Surfaces in Three-Space
14.7 Cylindrical and Spherical Coordinates
14.8 Chapter Review
Technology Project14.1 Curves in Three-Space
Technology Project14.2 The Ferris Wheel and the Corkscrew Roller Coaster
15 The Derivative in n-Space
15.1 Functions of Two or More Variables
15.2 Partial Derivatives
15.3 Limits and Continuity
15.4 Differentiability
15.5 Directional Derivatives and Gradients
15.6 The Chain Rule
15.7 Tangent Planes, Approximations
15.8 Maxima and Minima
15.9 Lagrange#s Method
15.10 Chapter Review
Technology Project15.1 Newton#s Method for Two Equations in Two Unknows
Technology Project15.2 Visualizing the Directional Derivative
16 The Integral in n-Space
16.1 Double Integrals over Rectangles
16.2 Interated Integrals
16.3 Double Integrals over Nonrectangular Regions
16.4 Double Integrals in Polar Coordinates
16.5 Applications of Double Integrals
16.6 Surface Area
16.7 Triple Integrals(Cartesian Coordinates)
16.8 Triple Integrals(Cylindrical and Spherical Coordinates)
16.9 Chapter Review
Technology Project16.1 Newton#s Law of Gravitation
Technology Project16.2 Monte Carlo Integration
17 Vector Calculus
17.1 Vector Fields
17.2 Line Integrals
17.3 Independence of Path
17.4 Green#s Theorem in the Plane
17.5 Surface Integrals
17.6 Gauss#s Divergence Theorem
17.7 Stokes#s Theorem
17.8 Chapter Review
Technology Project17.1 Line Integrals and Work
Technology Project17.2 Parametrized Surfaces
18 Differential Equations
18.1 Linear Homogeneous Equations
18.2 Nonhomogeneous Equations
18.3 Applications of Second-Order Equations
18.4 Chapter Review
Technology Project18.1 Vibrating Spring
Technology Project18.2 Phase Portraits
Appendix
A.1 Mathematical Induction
A.2 Proofs of Several Theorems
A.3 A Backward Look
Answers to Odd-Numbered Problems