1 Functions

1．1 Sets

1．1．1 Definition of Set

1．1．2 Operations upon Sets

1．1．3 The Set of Real Numbers

1．2 Functions

1．2．1 Definition

1．2．2 Some Properties of Functions

1．3 Composite Functions and Inverse Functions

1．3．1 Composite Functions

1．3．2 Inverse Functions

1．4 Elementary Functions

1．4．1 Constant functions

1．4．2 Power functions

1．4．3 Exponential functions

1．4．4 Logarithmic functions

1．4．5 Trigonometric functions

1．4．6 Anti-trigonometric functions

1．5 Exercises

2 Limits and Continuity

2．1 Limits of Sequences

2．1．1 Definition

2．2 Limits of Functions

2．2．1 A Limit of a Function f(x)as x Tends to a Real Number x0

2．2．2 Limits Involving Infinity

2．3 Techniques for Finding Limits

2．4 Continuous Functions

2．5 Exercises

3 The Derivative

3．1 Tangent lines and Rates of Change

3．2 Definition of Derivative

3．3 Differentiation Formulas

3．4 Derivatives of Logarithmic Functions

3．5 Derivatives of Trigonometric Functions

3．6 The Chain Rule

3．7 Derivatives of Inverse Functions and Implicit Differentiation

3．8 Higher Derivatives

3．9 Differentials and Linear Approximations

3．9．1 Diffcrentials

3．9．2 Linear Approximations

3．10 Exercises

4 Applications of Derivative

4．1 The Mean Value Theorem

4．2 Indeterminate Forms and L HOSPITAL S Rule

4．2．1 The Forms 0/0 and ∞/∞

4．2．2 The Forms 0·∞,00, ∞0,1∞ and ∞-∞

4．3 Monotonic Functions

4．4 Concavity and Points of Inflection

4．5 Extrema of Functions

4．6 Applications to Economics

4．7 Exercises

5 Indefinite Integrals

5．1 Antiderivatives and the Indefinite Integral

5．2 Substitution Rules

5．3 Integration by Parts

5．4 Exercises

6 Definite Integrals

6．1 Area and the Definite Integral

6．2 Properties of the Definite Integral

6．3 The Fundamental Theorem of Calculus

6．4 Techniques of Integration

6．4．1 Formula for integration by substitution

6．4．2 Formula for integration by parts

6．5 Improper Integrals

6．5．1 Type 1:Infinite Intervals

6．5．2 Type 2:Discontinuous Integrand

6．6 Exercises

7 Applications of Definite Integrals

7．1 Area between Curves

7．2 Volume

7．3 Are Length

7．4 Area of a Surface of Revolution

7．5 Work

7．6 Applications in Business and Economics

7．6．1 Continuous Income Stream

7．6．2 Consumers and Producers Surplus

7．7 Exercises

8 Series

8．1 Numerical Series

8．1．1 Fundamental Concepts

8．1．2 Elementary Properties

8．1．3 Infinite Series of Nonnegative Terms

8．1．4 Alternating Series

8．1．5 Absolute and Conditional Convergence

8．2 Functional Series

8．2．1 Power Series

8．2．2 Properties of Power Series

8．3 Taylor Series

8．4 Exercises

9 Vector Algebra and Space Ana？ytic Geometry

9．1 Rectangular Coordinates in Space

9．2 Vector Algebra

9．2．1 Operations of Vectors

9．2．2 The Coordinates of a Vector

9．2．3 The Scalar Product

9．2．4 The Vector Product

9．3 The Planes and Lines in Space

9．3．1 The Point-Normal Form Equations of a Plane

9．3．2 Distance from a Point to a Plane

9．3．3 The Angle between Two Planes

9．3．4 The General Equation of a Line in Space

9．3．5 Equations of a Line

9．3．6 The Angle between Two Lines

9．4 Equations for a Surface or a Curve

9．4．1 The Equation for a Sphere

9．4．2 The Equation of a Cylindrical Surface with Generators Paralleling to a Coordinate Axis

9．4．3 Equation for the Intersection of Two Curved Surfaces

9．4．4 The Parametric Equation of a Space Curve

9．4．5 Equation for the Projecting Curve on a Coordinate Plane of a Space Curve

9．5 Surfaces of Revolution

9．6 Quadratic Surfaces

9．6．1 Ellipsoids

9．6．2 Hyperboloids of One Sheet

9．6．3 Hyperboloids of Two Sheets

9．6．4 Quadratic Cones

9．6．5 Paraboloids

9．6．6 Quadratic Cylinders

9．7 Exercises

10 Functions of Several Variables

10．1 Fundamental Concepts

10．2 Limits and Continuity

10．3 Partial Derivatives

10．4 The Chain Rule

10．5 Approximation and Total Differential

10．6 Applications of Partial Derivatives

10．6．1 Geometric App？ication

10．6．2 Extreme Values of Functions of Two Variables

10．7 Exercises

11 Multiple Integrals

11．1 Double Integrals

11．2 Properties of Double Integral

11．3 Evaluation of Double Integrals

11．4 Triple Integrals

11．4．1 The Mass of an Object of Nonhomogeneous Density

11．4．2 The Definition of Triple Integral

11．4．3 Evaluation of Triple Integrals in Rectangular Coordinates

11．5 Exercises