Advanced Mathematics (Second Edition, Volume 1)

Author: Hubei Provincial Department of Education
Publisher: Wuhan University Press
Book Description:
The textbook "Advanced Mathematics" for 21st-century vocational colleges and higher vocational colleges includes content such as functions, limits and continuity, derivatives and differentials, the application of derivatives, integral calculus of single-variable functions, differential equations, vector algebra and spatial analytical geometry, multivariable calculus, and infinite series. The book is divided into two volumes, with a total of 10 chapters, each accompanied by a historical review and commentary at the end, primarily introducing the history of mathematics and relevant mathematical masters. The entire book is approximately 400,000 words.
This textbook mainly reflects the following features:
First, it pays attention to the connection with high school mathematics knowledge, strengthens the teaching of power functions, exponential functions, logarithmic functions, and trigonometric functions, and includes a brief table of elementary mathematics formulas at the end for students to review and self-study. For theorems, corollaries, propositions, etc. mentioned in the book, it neither pursues detailed proofs nor compromises on the rigor of mathematical theory.
Second, it emphasizes the integration of mathematical modeling concepts into teaching, strengthening the connection with practice, enabling students to flexibly apply mathematical knowledge to solve real-world problems, thereby enhancing their innovative abilities and better serving the construction of the socialist motherland in the future.
Third, it emphasizes the integration of new teaching methods and new educational philosophies into teaching practice. It uses mathematical software to calculate integrals and lists the commands and usage of Mathematica software in the appendix, enabling students to master modern computer technology. Teachers reform teaching methods and use modern multimedia technology for instruction, thereby improving teaching quality.
Table of Contents:
Chapter 1: Functions, Limits, and Continuity
§1.1 Functions
§1.2 Basic Elementary Functions and Elementary Functions
§1.3 Common Functions in Economics
§1.4 Limits of Sequences
§1.5 Limits of Functions
§1.6 Infinitesimals and Infinites
§1.7 Limit Laws and Two Important Limits
§1.8 Continuity of Functions
Historical Review and Commentary
Chapter 2: Derivatives and Differentials
§2.1 Concept of Derivatives
§2.2 Derivative Rules
§2.3 Basic Derivative Formulas
§2.4 Derivative Rules for Implicit Functions and Functions Determined by Parametric Equations
§2.5 Higher-Order Derivatives
§2.6 Differentials
Historical Review and Commentary
Chapter 3: Applications of Derivatives
§3.1 Mean Value Theorem and L'H?pital's Rule
§3.2 Monotonicity and Extrema of Functions
§3.3 Maximum and Minimum Values and Economic Applications
§3.4 Economic Analysis Models—Marginal and Elasticity Analysis
§3.5 Concavity and Inflection Points of Curves, Graphing Functions
Historical Review and Commentary
Chapter 4: Indefinite Integrals
§4.1 Concept of Indefinite Integrals
§4.2 Substitution Integration
§4.3 Integration by Parts
§4.4 Using Integral Tables and Mathematica to Solve Indefinite Integrals
Historical Review and Commentary
Chapter 5: Definite Integrals and Models
§5.1 Concept of Definite Integrals
§5.2 Fundamental Theorem of Calculus
§5.3 Substitution and Integration by Parts for Definite Integrals
§5.4 Improper Integrals
§5.5 Mathematical Models of Definite Integral Applications—the "Elementary Method"
Historical Review and Commentary
Chapter 6: Differential Equations
§6.1 Basic Concepts of Differential Equations
§6.2 Separable Differential Equations
§6.3 First-Order Linear Differential Equations
§6.4 Second-Order Linear Differential Equations with Constant Coefficients
§6.5 Second-Order Non-Homogeneous Linear Differential Equations with Constant Coefficients
Historical Review and Commentary
Appendix 1: Brief Commands of Mathematica 4.1
Appendix 2: Derivative and Differential Formulas
Appendix 3: Indefinite Integral Formulas
Appendix 4: Simple Integration Table
Appendix 5: Common Elementary Mathematics Formulas
Appendix 6: Reference Answers to Exercises
References