微积分是高等数学中研究函数的微分、积分以及有关概念和应用的数学分支，它是数学的一个基础学科，是理工科院校一门重要的基础理论课。它推动了其他学科的发展，推动了人类文明与科学技术的发展，它的作用是举足轻重的。微积分（I）是本科生的一门必修课，内容主要包括函数、极限、函数连续性、导数及其应用、积分及其应用、不定型的极限及广义积分。极限是微积分的基本概念，微分和积分是特定过程特定形式的极限。通过全英教学，学生在学会用英语获取数学知识的同时又通过学习数学掌握和运用英语，达到双赢的目的。从而培养具有国际竞争力并适应国家和社会需要的国际化人才。
Calculus is the branch of mathematics that studies differentiation, integration and related concepts and applications in advanced mathematics. It is a basic subject of mathematics. It is an important basic theory course in universities of science and engineering. It has promoted the development of other disciplines and human civilization and science and technology, and its function is of great importance. Calculus (I) is a compulsory course for undergraduates. The basic requirements of the course include functions, limits, continuity of function, derivatives and their applications, integrals and their applications, the limits of indefinite forms and generalized integrals. The limit is the basic concept of calculus. Differential and integral are the limits of particular forms of a particular process.Through the teaching of English, students learn to acquire mathematical knowledge in English while mastering and using English in the process of learning mathematics to achieve a win-win goal. Therefore, we can cultivate international talents with international competitiveness and meet the needs of the state and society.
本课程的目的是使学生掌握一元微积分的基本概念，理论及其应用。通过本课程的学习，在理论上，使学生获得一元函数微积分的基本概念、基本理论和基本运算技能；在具体传授知识的过程中，在教学中注意培养学生抽象思维能力、逻辑推理能力、空间想象能力和自学能力，特别是综合运用所学知识去分析问题和解决问题的能力。
Teaching Objective:
The purpose of this course is to enable students to master the basic concepts, theories and operations of one variable calculus. By the study of this course, in theory, students can master the basic definition, basic theory and basic operation skills of one variable calculus. At the same time, we should pay attention to the cultivation of students’ abstract thinking ability, logical reasoning ability, spatial imaginary ability and self-learning ability in the process of imparting knowledge and teaching. In particular, the ability of analyzing and solving problems is trained by using the learned knowledge.
Course Introduction
Course Introduction
Chapter 1 Limits
1.1 Introduction to Limits
1.2 Rigorous Study of Limits
1.3 Limit Theorems
1.4 Limits Involving Trigonometric Functions
1.5 Limits at Infinity, Infinite Limits
1.6 Continuity of Functions
1.7 Chapter Review
Supplementary Material for Chapter One
Homework for Chapter One
Answer to Chapter One
Discussion Topics of Chapter 1
Chapter 1 Limits
Assignment 1 for Chapter 1
Assignment 2 for Chapter 1
Chapter 2 The Derivative
2.1 Two Problems with One Theme
2.2 The Derivative
2.3 Rules for Finding Derivatives
2.4 Derivate of Trigonometric Functions
2.5 The Chain Rule
2.6 Higher-Order Derivative
2.7 Implicit Differentiation
2.8 Related Rates
2.9 Differentials and Approximations
2.10 Chapter Review
Homework for Chapter Two
Answer to Chapter Two
Discussion Topics of Chapter 2
Chapter 2 The Derivative
Assignment 1 for Chapter 2
Assignment 2 for Chapter 2
Chapter 3 Applications of the Derivative
3.1 Maxima and Minima
3.2 Monotonicity and Concavity
3.3 Local Extrema and Extrema on Open Intervals
3.4 Practical Problems
3.5 Graphing Functions Using Calculus
3.6 The Mean Value Theorem for Derivatives
3.7 Solving Equations Numerically
3.8 Anti-derivatives
3.9 Introduction to Differential Equations
3.10 Chapter Review
Supplementary Material for Chapter Three
Homework for Chapter Three
Answer to Chapter Three
Discussion Topics of Chapter 3
Assignment 1 for Chapter 3
Assignment 2 for Chapter 3
Test 1
Chapter 4 The Definite Integral
4.1 Introduction to Area
4.2 The Definite Integral
4.3 The First Fundamental Theorem of Calculus
4.4 The Second Fundamental Theorem of Calculus and the Method of Substitution
4.5 The Mean Value Theorem for Integrals and the Use of Symmetry
4.6 Numerical Integration
4.7 Chapter Review
Homework for Chapter Four
Answer to Chapter Four
Discussion Topics of Chapter 4
Chapter 4 The Definite Integral
Assignment 1 for Chapter 4
Assignment 2 for Chapter 4
Chapter 5 Applications of the Integral
5.1 The Area of a plane region
5.2 Volumes of Solids: Slabs, Disks
5.3 Volumes of Solids of Revolution: Shells
5.4 Length of a plane curve
5.5 Work and Fluid Force
5.6 Moments and Center of Mass
5.7 Probability and Random Variables
5.8 Chapter Review
Homework for Chapter Five
Answer to Chapter Five
Discussion Topics of Chapter 5
Chapter 5 Applications of the Integral
Assignment 1 for Chapter 5
Assignment 2 for Chapter 5
Chapter 6 Transcendental and Functions
6.1 The Natural Logarithm Function
6.2 Inverse Functions
6.3 The Natural Exponential Function
6.4 General Exponential and Logarithm Function
6.5 Exponential Growth and Decay
6.6 First-Order Linear Differential Equations
6.7 Approximations for Differential Equations
6.8 The Inverse Trigonometric Functions and Their Derivatives
6.9 The Hyperbolic Functions and Their Derivatives
6.10 Chapter Review
Chapter 7 Techniques of Integration
7.1 Basic Integration Rules
7.2 Integration by parts
7.3 Some Trigonometric Integrals
7.4 Rationalizing Substitutions
7.5 Integration of Rational Functions Using Partial Fraction
7.6 Strategies for Integration
7.7 Chapter Review
Homework for Chapter Seven
Answer to Chapter Seven
Discussion Topics of Chapter 7
Chapter 7 Techniques of Integration
Assignment 1 for Chapter 7
Assignment 2 for Chapter 7
Chapter 8 Indeterminate Forms and Improper Integrals
8.1 Indeterminate Forms of Type
8.2 Other Indeterminate Forms
8.3 Improper Integrals: Infinite Limits of Integration
8.4 Improper Integrals: Infinite Integrands
8.5 Chapter Review
Supplementary Material for Chapter Eight
Homework for Chapter Eight
Answer to Chapter Eight
Discussion Topics of Chapter 8
Assignment 1 for Chapter 8
Assignment 2 for Chapter 8
Test 2
Exercises
Limit of sequence and limit of function
Continuity, differentiation and derivative
Derivative
Indefinite integral Ι
Indefinite integral II
Indefinite integral III
Definite integral
Supplement
较扎实的高中数学基础；较好的英语听、说、读、写的能力。
A solid foundation of high school mathematics; a good command of English in listening, speaking, reading and writing.
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配套课本 ：微积分=calculus,（美/第九版）沃伯格（Varberg, D），柏塞尔（Purcell, E.J.），里格登（Rigdon,S.E.）著, 北京：机械工业出版社，2009.8.
参考资料：高等数学（上下册），同济大学数学系编.上海：同济大学出版社，2006.7.
Matching textbook：Calculus, Ninth Edition (Varberg, D., Purcell, E.J., Rigdon, S.E.). Beijing: Mechanical Industry Publisher, 2009.8.
Reference materials ：Advanced Mathematics (I and II), Department of Mathematics, Tongji University. Shanghai: Tongji University Publisher, 2006.7.
Q :
A : 需要教材： 微积分=calculus,（美/第九版）沃伯格（Varberg, D），柏塞尔（Purcell, E.J.），里格登（Rigdon,S.E.）著, 北京：机械工业出版社，2009.8.
Q :
A : 自学内容大部分为高中基础知识，需要复习和理解基本概念。
Q :
A : 了解基本概念，基本定理并掌握基本数学计算，同时反馈知识难点。
Q :
Is there any matching textbook?
A : Textbook, Calculus, Ninth Edition (Varberg, D., Purcell, E.J., Rigdon, S.E.). Beijing: Mechanical Industry Publisher, 2009.8.
Q :
What are the requirements for the self-study content?
A : Most of the self-study contents are the basic knowledge of high school. Students need to review and understand the basic concepts.
Q :
What tasks do we need to finish in advance?
A : Understand the basic concepts, basic theorems. Master the basic mathematical calculations and skills. Give the feedback of knowledge difficulties.
教授
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