微积分 I（双语版）

目录
Chapter 1Functions
第1章函数
1.1Functions and Their Graphs
1.1函数及其图像
1. The Domain and the Range of
a Function
1. 函数的定义域和值域
2. The Graph of a Function
2. 函数的图像
3. The Vertical Line Test for a Function
3. 函数的垂直线测试
4. Examples of Functions
4. 函数的例子
1.2The Special Properties of Functions
1.2函数的特性
1. The Boundness of a Function
1. 函数的有界性
2. The Monotonicity of a Function
2. 函数的单调性
3. The Symmetry of a Function
3. 函数的对称性
4. The Periodicity of a Function
4. 函数的周期性
1.3The Operations of Functions
1.3函数的运算
1. The Arithmetic of Functions
1. 函数的四则运算
2. The Composition of Functions
2. 函数的复合
3. The Transformations of Functions
3. 函数的变换
1.4Elementary Functions
1.4初等函数
1. Basic Elementary Functions
1. 基本初等函数
2. Elementary Functions
2. 初等函数
Exercises 1
习题1
Chapter 2Limits
第2章极限
2.1The Limit of a Sequence
2.1数列的极限
1. The Definition of the Convergent
Sequence
1. 收敛数列的定义
2. The Properties of a Convergent
Sequence
2. 收敛数列的性质
2.2The Limit of a Function
2.2函数的极限
1. The Limit of a Function as x→x0
1. 函数在x→x0时的极限
2. Onesided Limits
2. 单侧极限
3. The Limit of a Function as x→∞
3. 函数在x→∞ 时的极限
4. Infinite Limits
4. 无穷极限
5. The Properties of Limits
5. 极限的性质
2.3Limit Laws
2.3极限运算法则
2.4Limit Existence Rules and
Two Important Limits
2.4极限存在准则和两个重要极限
2.5The Continuity of Functions
2.5函数的连续性
1. Continuity at a Point
1. 在一点处的连续性
2. Several Common Types of
Discontinuities
2. 间断点的几种常见类型
3. Continuity on an Interval
3. 区间上的连续性
4. The Operations of Continuous Functions
4. 连续函数的运算
5. The Properties of Continuous
Functions on a Closed Interval
5. 闭区间上连续函数的性质
2.6Infinitesimals and Infinitys
2.6无穷小量和无穷大量
1. Infinitesimals
1. 无穷小量
2. Infinitys
2. 无穷大量
3. Compare of Infinitesimals
3. 无穷小量的比较
Exercises 2
习题2
Chapter 3The Derivative and the Differential
第3章导数和微分
3.1The Concept of the Derivative
3.1导数的概念
1. Introducing Examples
1. 引例
2. The Derivative of Function at a Point
2. 函数在一点处的导数
3. Onesided Derivatives
3. 单侧导数
4. The Derivative of a Function
4. 函数的导数
5. Relationship Between Differentiability
and Continuity
5. 可导与连续的关系
3.2The Rules for Finding Derivatives
3.2求导法则
1. The Constant Multiple Rule
1. 常数乘法法则
2. The Sum Rule
2. 和法则
3. The Difference Rule
3. 差法则
4. The Product Rule
4. 乘积法则
5. The Quotient Rule
5. 商法则
6. The Rule for the Derivative of
an Inverse Function
6. 反函数求导法则
7. The Derivative Formulas of Basic
Elementary Functions
7. 基本初等函数的导数公式
8. The Chain Rule
8. 链式法则
3.3Higherorder Derivatives
3.3高阶导数
3.4The Derivatives of Implicit Functions
and Functions Determined by Parameter
Equations
3.4隐函数及由参数方程确定的函数
的导数
1. The Derivative of an Implicit
Function
1. 隐函数的导数
2. The Derivative of a Function Determined by a Parameter Equation
2. 由参数方程确定的函数的导数
3.5The Differential and the Approximation
3.5微分和近似
1. The Definition of the Differential
1. 微分的定义
2. The Rules of the Differential
2. 微分法则
3. The Differential Formulas of Basic
Elementary Functions
3. 基本初等函数的微分公式
4. The Linear Approximation of
a Function
4. 函数的线性近似
Exercises 3
习题3
Chapter 4Applications of the Derivative
第4章导数的应用
4.1The Mean Value Theorem
4.1微分中值定理
4.2The L’Hospital Rule
4.2洛必达法则
4.3The Criterion of the Monotonicity of
Functions
4.3函数的单调性判别法
1. The First Derivative and Monotonicity
1. 函数的一阶导数与单调性
2. The Second Derivative and Concavity
2. 二阶导数和凹性
4.4Maxima and Minima
4.4最大值和最小值
1. The Existence Question
1. 存在性问题
2. Where Do Extreme Values Occur?
2. 最值在哪里出现？
3. How to Find Extreme Values?
3. 如何求最值?
4.5Local Extrema and Local Extrema
on Open Intervals
4.5局部极值与开区间上的局部
极值
1. Where Do Local Extreme Values Occur?
1. 局部极值存在于何处?
2. Extrema on an Open Interval
2. 开区间上的最值
4.6Graphing Functions
4.6作函数的图像
Exercises 4
习题4
Chapter 5The Indefinite Integral
第5章不定积分
5.1The Concept and the Properties of
the Indefinite Integral
5.1不定积分的概念与性质
1. The Concepts of the Primitive Function
and the Indefinite Integral
1. 原函数与不定积分的概念
2. Basic Formulas of Integrals
2. 基本积分公式
3. The Properties of the Indefinite Integral
3. 不定积分的性质
5.2Integration by Substitution
5.2换元积分法
1. The Substitution Rule 1
1. 第一换元法
2. The Substitution Rule 2
2. 第二换元法
5.3Integration by Parts
5.3分部积分法
5.4The Indefinite Integral of the Rational
Function
5.4有理函数的不定积分
1. The Indefinite Integral of
the Rational Function
1. 有理函数的不定积分
2. The Indefinite Integral of
the Rational Function with
Trigonometric Function
2. 三角函数有理式的不定积分
3. The Indefinite Integral of the Simple
Irrational Function
3. 简单无理函数的不定积分
Exercises 5
习题5
Chapter 6The Definite Integral
第6章定积分
6.1The Concept and the Properties of
the Definite Integral
6.1定积分的概念与性质
1. Examples of the Definite Integral
1. 定积分问题举例
2. The Definition of the Definite Integral
2. 定积分的定义
3. The Geometric Significance of
the Definite Integral
3. 定积分的几何意义
4. The Properties of the Definite Integral
4. 定积分的性质
6.2The Fundamental Formula of Calculus
6.2微积分基本公式
1. The Function of Integral Upper Limit
and Its Derivative
1. 积分上限函数及其导数
2. The NewtonLeibniz Formula
2. 牛顿莱布尼茨公式
6.3Definite Integration by Substitution
and Parts
6.3定积分的换元法和分部积分法
1. Definite Integration by Substitution
1. 定积分的换元法
2. Definite Integration by Parts
2. 定积分的分部积分法
6.4The Improper Integral
6.4反常积分
1. The Improper Integral of Infinite Limit
1. 无穷限的反常积分
2. The Improper Integral of
the Unbounded Function
2. 无界函数的反常积分
6.5Applications of the Definite Integral
6.5定积分的应用
1. The Infinitesimal Method
1. 微元法
2. Applications in Geometry
2. 在几何中的应用
3. Applications in Economics
3. 在经济中的应用
4. Application in Physics
4. 在物理中的应用
Exercises 6
习题6