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)的運算 
1. The Arithmetic of Functions 
1. 函數(shù)的四則運算 
2. The Composition of Functions 
2. 函數(shù)的復合 
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時的極限 
2. One�瞫ided Limits 
2. 單側極限 
3. The Limit of a Function as x→∞ 
3. 函數(shù)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(zhì) 
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(shù)的連續(xù)性 
1. Continuity at a Point 
1. 在一點處的連續(xù)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(qū)間上的連續(xù)性 
4. The Operations of Continuous Functions 
4. 連續(xù)函數(shù)的運算 
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(shù)和微分 
3.1The Concept of the Derivative 
3.1導數(shù)的概念 
1. Introducing Examples 
1. 引例 
2. The Derivative of Function at a Point 
2. 函數(shù)在一點處的導數(shù) 
3. One�瞫ided Derivatives 
3. 單側導數(shù) 
4. The Derivative of a Function 
4. 函數(shù)的導數(shù) 
5. Relationship Between Differentiability 
and Continuity 
5. 可導與連續(xù)的關系 
3.2The Rules for Finding Derivatives 
3.2求導法則 
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)求導法則 
7. The Derivative Formulas of Basic 
Elementary Functions 
7. 基本初等函數(shù)的導數(shù)公式 
8. The Chain Rule 
8. 鏈式法則 
3.3Higher�瞣rder Derivatives 
3.3高階導數(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) 
的導數(shù) 
1. The Derivative of an Implicit 
Function 
1. 隱函數(shù)的導數(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 
習題3 
Chapter 4Applications of the Derivative 
第4章導數(shù)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(shù)的單調(diào)性判別法 
1. The First Derivative and Monotonicity 
1. 函數(shù)的一階導數(shù)與單調(diào)性 
2. The Second Derivative and Concavity 
2. 二階導數(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. 開區(qū)間上的最值 
4.6Graphing Functions 
4.6作函數(shù)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(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. 簡單無理函數(shù)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(zhì) 
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(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)及其導數(shù) 
2. The Newton�睱eibniz Formula 
2. 牛頓�踩R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. 無窮限的反常積分 
2. The Improper Integral of 
the Unbounded Function 
2. 無界函數(shù)的反常積分 
6.5Applications of the Definite Integral 
6.5定積分的應用 
1. The Infinitesimal Method 
1. 微元法 
2. Applications in Geometry 
2. 在幾何中的應用 
3. Applications in Economics 
3. 在經(jīng)濟中的應用 
4. Application in Physics 
4. 在物理中的應用 
Exercises 6 
習題6