全新高等数学北京邮电大学高等数学双语教学组 编9787563552726

Chapter 1 Fundamental Knowledge of Calculus
1.1 Mappings and Functions
1.1.1 Sets and Their Oraios
1.1.2 Mappings and Functions
1.1.3 Elementary Properties of Functions
1.1.4 Coite Functions and Inverse Functions
1.1.5 Basic Elementary Functions and Elementary Functions
Exercises 1.1 A
Exercises 1.1 B
1.2 Limits of Sequences
1.2.1 The Definition of Limit of a Sequence
1.2.2 Properties of Limits of Sequences
1.. Oraios of Limits of Sequences
1.2.4 Some Criteria for Existence of the Limit of a Sequence
Exercises 1.2 A
Exercises 1.2 B
1.3 The Limit of a Function
1.3.1 Concept of the Limit of a Function
1.3.2 Properties and Oraios of Limits for Functions
1.3.3 Two Important Limits of Functions
Exercises 1.3 A
Exercises 1.3 B
1.4 Infinitesimal and Infinite ntities
1.4.1 Infinitesimal ntities
1.4.2 Infinite ntities
1.4.3 The Order of Infinitesimals and Infinite ntities
Exercises 1.4 A
Exercises 1.4 B
1.5 Continuous Functions
1.5.1 Continuity of Functions
1.5.2 Properties and Oraios of Continuous Functions
1.5.3 Continuity of Elementary Functions
1.5.4 Discontinuous Points and Their Classification
1.5.5 Properties of Continuous Functions on a Closed Interva
Exercises 1.5 A
Exercises 1.5 B
Chapter 2 Derivative and Differentia
2.1 Concept of Derivatives
2.1.1 Introductory Examples
2.1.2 Definition of Derivatives
2.1.3 Geometric Meaning of the Derivative
2.1.4 Relationship between DerivabiliyndCntinuity
Exercises 2.1 A
Exercises 2.1 B
2.2 Rules of Finding Derivatives
2.2.1 Derivation Rules of Rational Oraios
2.2.2 Derivation Rules of Coite Functions
2.. Derivative of Inverse Functions
2.2.4 Derivation Formulas of Fundamental Elementary Functions
Exercises 2.2 A
Exercises 2.2 B
. Higher Order Derivatives
Exercises . A
Exercises . B
2.4 Derivation of Implicit Functions and Parametric Equations，
Related Rates
2.4.1 Derivation of Implicit Functions
2.4.2 Derivation of Parametric Equations
2.4.3 Related Rates
Exercises 2.4 A
Exercises 2.4 B
2.5 Differential of the Function
2.5.1 Concept of the Differential
2.5.2 Geometric Meaning of the Differential
2.5.3 Differential Rules of Elementary Functions
2.5.4 Differential in Linear Approximate Computation
Exercises 2.5
Chapter 3 The Mean Value Theorem and Applications of Derivatives
3.1 The Mean Value Theorem
3.1.1 Rolles Theorem
3.1.2 Lagranges Theorem
3.1.3 Cauchys Theorem
Exercises 3.1 A
Exercises 3.1 B
3.2 LHospitals Rule
Exercises 3.2 A
Exercises 3.2 B
3.3 Taylors Theorem
3.3.1 Taylors Theorem
3.3.2 Applications of Taylors Theorem
Exercises 3.3 A
Exercises 3.3 B
3.4 Monotonicity， Extreme Values， Global Maxima and Minima of Functions
3.4.1 Monotonicity of Functions
3.4.2 Extreme Values
3.4.3 Global Maxima and Minima and Its Application
Exercises 3.4 A
Exercises 3.4 B
3.5 Convexity of Functions， Inflections
Exercises 3.5 A
Exercises 3.5 B
3.6 Asytte and Graphing Functions
Exercises 3.6
Chapter 4 Indefinite Integrals
4.1 ConcepsndPrperties of Indefinite Integrals
4.1.1 Antiderivatives and Indefinite Integrals
4.1.2 Formulas for Indefinite Integrals
4.1.3 Oraio Rules of Indefinite Integrals
Exercises 4.1 A
Exercises 4.1 B
4.2 Integration by Substitution
4.2.1 Integration by the First Substitution
4.2.2 Integration by the Second Substitution
Exercises 4.2 A
Exercises 4.2 B
4.3 Integration by Parts
Exercises 4.3 A
Exercises 4.3 B
4.4 Integration of Rational Functions
4.4.1 Rational Functions and Partial Fractions
4.4.2 Integration of Rational Fractions
4.4.3 Antiderivatives Not Expressed by Elementary Functions
Exercises 4.4
Chapter 5 Definite Integrals
5.1 ConcepsndPrperties of Definite Integrals
5.1.1 Instances of Definite Integral Problems
5.1.2 The Definition of the Definite Integral
5.1.3 Properties of Definite Integrals
Exercises 5.1 A
Exercises 5.1 B
5.2 The Fundamental Theorems of Calculus
5.2.1 Fundamental Theorems of Calculus
5.2.2 Newton  Leibniz Formula for Evaluation of Definite Integrals
Exercises 5.2 A
Exercises 5.2 B
5.3 Integration by Substitution and by Parts in Definite Integrals
5.3.1 Substitution in Definite Integrals
5.3.2 Integration by Parts in Definite Integrals
Exercises 5.3 A
Exercises 5.3 B
5.4 Improper Integral
5.4.1 Integration on an Infinite Interval
5.4.2 Improper Integrals with Infinite Discontinuities
Exercises 5.4 A
Exercises 5.4 B
5.5 Applications of Definite Integrals
5.5.1 Method of Setting up Elements of Integration
5.5.2 The Area of a Plane Region
5.5.3 The Arc Length of Plane Curves
5.5.4 The Volume of a Solid by Slicing and Rotation about an Axis
5.5.5 Applications of Definite Integral in Physics
Exercises 5.5 A
Exercises 5.5 B
Chapter 6 Differential Equations
6.1 Basic Concepts of Differential Equations
6.1.1 Examples of Differential Equations
6.1.2 Basic Concepts
Exercises 6.1
6.2 First  Order Differential Equations
6.2.1 First  Order Separable Differential Equation
6.2.2 Equations can be Reduced to Equations with Variables Separable
6.. First  Order Linear Equations
6.2.4 Bernoullis Equation
6.2.5 Some Examples that can be Reduced to First  Order Linear Equations
Exercises 6.2
6.3 Reducible Second  Order Differential Equations
Exercises 6.3
6.4 Higher  Order Linear Differential Equations
6.4.1 Some Examples of Linear Differential Equation of Higher  Order
6.4.2 Structure of Solutions of Linear Differential Equations
Exercises 6.4
6.5 Linear Equations with Constant Coefficients
6.5.1 Higher  Order Linear Homogeneous Equations with Constant Coefficients
6.5.2 Higher  Order Linear Nonhomogeneous Equations with Constant Coefficients
Exercises 6.5
6.6 *Eulers Differential Equation
Exercises 6.6
6.7 Applications of Differential Equations
Exercises 6.7
Bibliography