微積分 I（雙語(yǔ)版）

目錄
Chapter 1Functions
第1章函數(shù)
1.1Functions and Their Graphs
1.1函數(shù)及其圖像
1. The Domain and the Range of
a Function
1. 函數(shù)的定義域和值域
2. The Graph of a Function
2. 函數(shù)的圖像
3. The Vertical Line Test for a Function
3. 函數(shù)的垂直線測(cè)試
4. Examples of Functions
4. 函數(shù)的例子
1.2The Special Properties of Functions
1.2函數(shù)的特性
1. The Boundness of a Function
1. 函數(shù)的有界性
2. The Monotonicity of a Function
2. 函數(shù)的單調(diào)性
3. The Symmetry of a Function
3. 函數(shù)的對(duì)稱性
4. The Periodicity of a Function
4. 函數(shù)的周期性
1.3The Operations of Functions
1.3函數(shù)的運(yùn)算
1. The Arithmetic of Functions
1. 函數(shù)的四則運(yùn)算
2. The Composition of Functions
2. 函數(shù)的復(fù)合
3. The Transformations of Functions
3. 函數(shù)的變換
1.4Elementary Functions
1.4初等函數(shù)
1. Basic Elementary Functions
1. 基本初等函數(shù)
2. Elementary Functions
2. 初等函數(shù)
Exercises 1
習(xí)題1
Chapter 2Limits
第2章極限
2.1The Limit of a Sequence
2.1數(shù)列的極限
1. The Definition of the Convergent
Sequence
1. 收斂數(shù)列的定義
2. The Properties of a Convergent
Sequence
2. 收斂數(shù)列的性質(zhì)
2.2The Limit of a Function
2.2函數(shù)的極限
1. The Limit of a Function as x→x0
1. 函數(shù)在x→x0時(shí)的極限
2. Onesided Limits
2. 單側(cè)極限
3. The Limit of a Function as x→∞
3. 函數(shù)在x→∞ 時(shí)的極限
4. Infinite Limits
4. 無(wú)窮極限
5. The Properties of Limits
5. 極限的性質(zhì)
2.3Limit Laws
2.3極限運(yùn)算法則
2.4Limit Existence Rules and
Two Important Limits
2.4極限存在準(zhǔn)則和兩個(gè)重要極限
2.5The Continuity of Functions
2.5函數(shù)的連續(xù)性
1. Continuity at a Point
1. 在一點(diǎn)處的連續(xù)性
2. Several Common Types of
Discontinuities
2. 間斷點(diǎn)的幾種常見(jiàn)類型
3. Continuity on an Interval
3. 區(qū)間上的連續(xù)性
4. The Operations of Continuous Functions
4. 連續(xù)函數(shù)的運(yùn)算
5. The Properties of Continuous
Functions on a Closed Interval
5. 閉區(qū)間上連續(xù)函數(shù)的性質(zhì)
2.6Infinitesimals and Infinitys
2.6無(wú)窮小量和無(wú)窮大量
1. Infinitesimals
1. 無(wú)窮小量
2. Infinitys
2. 無(wú)窮大量
3. Compare of Infinitesimals
3. 無(wú)窮小量的比較
Exercises 2
習(xí)題2
Chapter 3The Derivative and the Differential
第3章導(dǎo)數(shù)和微分
3.1The Concept of the Derivative
3.1導(dǎo)數(shù)的概念
1. Introducing Examples
1. 引例
2. The Derivative of Function at a Point
2. 函數(shù)在一點(diǎn)處的導(dǎo)數(shù)
3. Onesided Derivatives
3. 單側(cè)導(dǎo)數(shù)
4. The Derivative of a Function
4. 函數(shù)的導(dǎo)數(shù)
5. Relationship Between Differentiability
and Continuity
5. 可導(dǎo)與連續(xù)的關(guān)系
3.2The Rules for Finding Derivatives
3.2求導(dǎo)法則
1. The Constant Multiple Rule
1. 常數(shù)乘法法則
2. The Sum Rule
2. 和法則
3. The Difference Rule
3. 差法則
4. The Product Rule
4. 乘積法則
5. The Quotient Rule
5. 商法則
6. The Rule for the Derivative of
an Inverse Function
6. 反函數(shù)求導(dǎo)法則
7. The Derivative Formulas of Basic
Elementary Functions
7. 基本初等函數(shù)的導(dǎo)數(shù)公式
8. The Chain Rule
8. 鏈?zhǔn)椒▌t
3.3Higherorder Derivatives
3.3高階導(dǎo)數(shù)
3.4The Derivatives of Implicit Functions
and Functions Determined by Parameter
Equations
3.4隱函數(shù)及由參數(shù)方程確定的函數(shù)
的導(dǎo)數(shù)
1. The Derivative of an Implicit
Function
1. 隱函數(shù)的導(dǎo)數(shù)
2. The Derivative of a Function Determined by a Parameter Equation
2. 由參數(shù)方程確定的函數(shù)的導(dǎo)數(shù)
3.5The Differential and the Approximation
3.5微分和近似
1. The Definition of the Differential
1. 微分的定義
2. The Rules of the Differential
2. 微分法則
3. The Differential Formulas of Basic
Elementary Functions
3. 基本初等函數(shù)的微分公式
4. The Linear Approximation of
a Function
4. 函數(shù)的線性近似
Exercises 3
習(xí)題3
Chapter 4Applications of the Derivative
第4章導(dǎo)數(shù)的應(yīng)用
4.1The Mean Value Theorem
4.1微分中值定理
4.2The L’Hospital Rule
4.2洛必達(dá)法則
4.3The Criterion of the Monotonicity of
Functions
4.3函數(shù)的單調(diào)性判別法
1. The First Derivative and Monotonicity
1. 函數(shù)的一階導(dǎo)數(shù)與單調(diào)性
2. The Second Derivative and Concavity
2. 二階導(dǎo)數(shù)和凹性
4.4Maxima and Minima
4.4最大值和最小值
1. The Existence Question
1. 存在性問(wèn)題
2. Where Do Extreme Values Occur?
2. 最值在哪里出現(xiàn)？
3. How to Find Extreme Values?
3. 如何求最值?
4.5Local Extrema and Local Extrema
on Open Intervals
4.5局部極值與開(kāi)區(qū)間上的局部
極值
1. Where Do Local Extreme Values Occur?
1. 局部極值存在于何處?
2. Extrema on an Open Interval
2. 開(kāi)區(qū)間上的最值
4.6Graphing Functions
4.6作函數(shù)的圖像
Exercises 4
習(xí)題4
Chapter 5The Indefinite Integral
第5章不定積分
5.1The Concept and the Properties of
the Indefinite Integral
5.1不定積分的概念與性質(zhì)
1. The Concepts of the Primitive Function
and the Indefinite Integral
1. 原函數(shù)與不定積分的概念
2. Basic Formulas of Integrals
2. 基本積分公式
3. The Properties of the Indefinite Integral
3. 不定積分的性質(zhì)
5.2Integration by Substitution
5.2換元積分法
1. The Substitution Rule 1
1. 第一換元法
2. The Substitution Rule 2
2. 第二換元法
5.3Integration by Parts
5.3分部積分法
5.4The Indefinite Integral of the Rational
Function
5.4有理函數(shù)的不定積分
1. The Indefinite Integral of
the Rational Function
1. 有理函數(shù)的不定積分
2. The Indefinite Integral of
the Rational Function with
Trigonometric Function
2. 三角函數(shù)有理式的不定積分
3. The Indefinite Integral of the Simple
Irrational Function
3. 簡(jiǎn)單無(wú)理函數(shù)的不定積分
Exercises 5
習(xí)題5
Chapter 6The Definite Integral
第6章定積分
6.1The Concept and the Properties of
the Definite Integral
6.1定積分的概念與性質(zhì)
1. Examples of the Definite Integral
1. 定積分問(wèn)題舉例
2. The Definition of the Definite Integral
2. 定積分的定義
3. The Geometric Significance of
the Definite Integral
3. 定積分的幾何意義
4. The Properties of the Definite Integral
4. 定積分的性質(zhì)
6.2The Fundamental Formula of Calculus
6.2微積分基本公式
1. The Function of Integral Upper Limit
and Its Derivative
1. 積分上限函數(shù)及其導(dǎo)數(shù)
2. The NewtonLeibniz Formula
2. 牛頓萊布尼茨公式
6.3Definite Integration by Substitution
and Parts
6.3定積分的換元法和分部積分法
1. Definite Integration by Substitution
1. 定積分的換元法
2. Definite Integration by Parts
2. 定積分的分部積分法
6.4The Improper Integral
6.4反常積分
1. The Improper Integral of Infinite Limit
1. 無(wú)窮限的反常積分
2. The Improper Integral of
the Unbounded Function
2. 無(wú)界函數(shù)的反常積分
6.5Applications of the Definite Integral
6.5定積分的應(yīng)用
1. The Infinitesimal Method
1. 微元法
2. Applications in Geometry
2. 在幾何中的應(yīng)用
3. Applications in Economics
3. 在經(jīng)濟(jì)中的應(yīng)用
4. Application in Physics
4. 在物理中的應(yīng)用
Exercises 6
習(xí)題6