目錄
Chapter 1Functions
第1章函數
1.1Functions and Their Graphs
1.1函數及其圖像
1. The Domain and the Range of
a Function
1. 函數的定義域和值域
2. The Graph of a Function
2. 函數的圖像
3. The Vertical Line Test for a Function
3. 函數的垂直線測試
4. Examples of Functions
4. 函數的例子
1.2The Special Properties of Functions
1.2函數的特性
1. The Boundness of a Function
1. 函數的有界性
2. The Monotonicity of a Function
2. 函數的單調性
3. The Symmetry of a Function
3. 函數的對稱性
4. The Periodicity of a Function
4. 函數的周期性
1.3The Operations of Functions
1.3函數的運算
1. The Arithmetic of Functions
1. 函數的四則運算
2. The Composition of Functions
2. 函數的復合
3. The Transformations of Functions
3. 函數的變換
1.4Elementary Functions
1.4初等函數
1. Basic Elementary Functions
1. 基本初等函數
2. Elementary Functions
2. 初等函數
Exercises 1
習題1
Chapter 2Limits
第2章極限
2.1The Limit of a Sequence
2.1數列的極限
1. The Definition of the Convergent
Sequence
1. 收斂數列的定義
2. The Properties of a Convergent
Sequence
2. 收斂數列的性質
2.2The Limit of a Function
2.2函數的極限
1. The Limit of a Function as x→x0
1. 函數在x→x0時的極限
2. Onesided Limits
2. 單側極限
3. The Limit of a Function as x→∞
3. 函數在x→∞ 時的極限
4. Infinite Limits
4. 無窮極限
5. The Properties of Limits
5. 極限的性質
2.3Limit Laws
2.3極限運算法則
2.4Limit Existence Rules and
Two Important Limits
2.4極限存在準則和兩個重要極限
2.5The Continuity of Functions
2.5函數的連續性
1. Continuity at a Point
1. 在一點處的連續性
2. Several Common Types of
Discontinuities
2. 間斷點的幾種常見類型
3. Continuity on an Interval
3. 區間上的連續性
4. The Operations of Continuous Functions
4. 連續函數的運算
5. The Properties of Continuous
Functions on a Closed Interval
5. 閉區間上連續函數的性質
2.6Infinitesimals and Infinitys
2.6無窮小量和無窮大量
1. Infinitesimals
1. 無窮小量
2. Infinitys
2. 無窮大量
3. Compare of Infinitesimals
3. 無窮小量的比較
Exercises 2
習題2
Chapter 3The Derivative and the Differential
第3章導數和微分
3.1The Concept of the Derivative
3.1導數的概念
1. Introducing Examples
1. 引例
2. The Derivative of Function at a Point
2. 函數在一點處的導數
3. Onesided Derivatives
3. 單側導數
4. The Derivative of a Function
4. 函數的導數
5. Relationship Between Differentiability
and Continuity
5. 可導與連續的關繫
3.2The Rules for Finding Derivatives
3.2求導法則
1. The Constant Multiple Rule
1. 常數乘法法則
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. 反函數求導法則
7. The Derivative Formulas of Basic
Elementary Functions
7. 基本初等函數的導數公式
8. The Chain Rule
8. 鏈式法則
3.3Higherorder Derivatives
3.3高階導數
3.4The Derivatives of Implicit Functions
and Functions Determined by Parameter
Equations
3.4隱函數及由參數方程確定的函數
的導數
1. The Derivative of an Implicit
Function
1. 隱函數的導數
2. The Derivative of a Function Determined by a Parameter Equation
2. 由參數方程確定的函數的導數
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. 基本初等函數的微分公式
4. The Linear Approximation of
a Function
4. 函數的線性近似
Exercises 3
習題3
Chapter 4Applications of the Derivative
第4章導數的應用
4.1The Mean Value Theorem
4.1微分中值定理
4.2The L’Hospital Rule
4.2洛必達法則
4.3The Criterion of the Monotonicity of
Functions
4.3函數的單調性判別法
1. The First Derivative and Monotonicity
1. 函數的一階導數與單調性
2. The Second Derivative and Concavity
2. 二階導數和凹性
4.4Maxima and Minima
4.4值和小值
1. The Existence Question
1. 存在性問題
2. Where Do Extreme Values Occur?
2. 值在哪裡出現？
3. How to Find Extreme Values?
3. 如何求值?
4.5Local Extrema and Local Extrema
on Open Intervals
4.5局部極值與開區間上的局部
極值
1. Where Do Local Extreme Values Occur?
1. 局部極值存在於何處?
2. Extrema on an Open Interval
2. 開區間上的值
4.6Graphing Functions
4.6作函數的圖像
Exercises 4
習題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
1. The Concepts of the Primitive Function
and the Indefinite Integral
1. 原函數與不定積分的概念
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
5.2Integration by Substitution
5積分法
1. The Substitution Rule 1
1法
2. The Substitution Rule 2
2. 法
5.3Integration by Parts
5.3分部積分法
5.4The Indefinite Integral of the Rational
Function
5.4有理函數的不定積分
1. The Indefinite Integral of
the Rational Function
1. 有理函數的不定積分
2. The Indefinite Integral of
the Rational Function with
Trigonometric Function
2. 三角函數有理式的不定積分
3. The Indefinite Integral of the Simple
Irrational Function
3. 簡單無理函數的不定積分
Exercises 5
習題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
1. Examples of the Definite Integral
1. 定積分問題舉例
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
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. 積分上限函數及其導數
2. The NewtonLeibniz Formula
2. 牛頓萊布尼茨公式
6.3Definite Integration by Substitution
and Parts
6.3定積法和分部積分法
1. Definite Integration by Substitution
1. 定積法
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. 無窮限的反常積分
2. The Improper Integral of
the Unbounded Function
2. 無界函數的反常積分
6.5Applications of the Definite Integral
6.5定積分的應用
1. The Infinitesimal Method
1法
2. Applications in Geometry
2. 在幾何中的應用
3. Applications in Economics
3. 在經濟中的應用
4. Application in Physics
4. 在物理中的應用
Exercises 6
習題6