一生的夢與愛在此奔騰

eCalculus@CSU

CHAPTER 1  LIMITS OF FUNCTIONS

Section 1-1     Limits

Section 1-2     One-Sided Limit

Section 1-3     Continuity

Section 1-4     A Limit at Infinity and Infinite Limit

CHAPTER 2  DERIVATIVE

Section 2-1    Definition of Derivative

Section 2-2    The Rule of Differentiation

Section 2-3    Chain Rule and Implicit Differentiation

Section 2-4    Derivatives of Exponential and Logarithmic F

Section 2-5    Numerical Approximate –Differentials

Section 2-6    Derivatives of Trigonometric Functions

Section 2-7    Derivatives of Inverse Trigonometric F

CHAPTER 3  APPLICATIONS OF DERIVATIVES

Section 3-1    The Mean Value Theorem and its Applications

Section 3-2    Increasing and Decreasing Functions

Section 3-3    Maximum and Minimum Values

Section 3-4    The Max -Min Problems

Section 3-5    Concavity and Points of Inflection

Section 3-6    Asymptotes

Section 3-7    Sketching curve

Section 3-8    L' Hopital's Rule

Section 3-9    Taylor Series

Section 3-10  Applications In Marginal Analysis

Section 3-11  Elasticity

CHAPTER 4    THE INDEFINITE INTEGRALS

Section 4-1     Antiderivative and The Indefinite Integrals

Section 4-2     Integration by Changing Variables

Section 4-3     Integration by Parts

Section 4-4     The Trigonometric Integrals

Section 4-5     The Integration by Partial Fractions

Section 4-6     Trigonometric and Half-Angle Substitution

CHAPTER 5    THE DEFINITE INTEGRALS

Section 5-1    Areas and the Definition of Definite Integral

Section 5-2    The Fundamental Theorem of Calculus

Section 5-3    The Approximate Integration

Section 5-4    The Improper Integrals

CHAPTER 6    APPLICATIONS OF INTEGRATION

Section 6-1    Areas between Curves

Section 6-2    Areas in Polar Coordinates

Section 6-3    Arc Length

Section 6-4    Volumes and The Volumes of Revolution

Section 6-5    Area of a Surface of Revolution

Section 6-6    Centroid of A Plane Region

Section 6-7    Work and The Problems of The Engineering

CHAPTER 7     PARTIAL DERIVATIVES

Section 7-1     Limits and Continuity

Section 7-2     Partial Derivatives

Section 7-3     The Differentials and Chain Rules

Section 7-4     Extrema of Functions of Two Variables

Section 7-5     Directional Derivatives, Gradient and Tangent Plane

CHAPTER 8      MULTIPLE INTEGRALS

Section 8-1     Integrals over a Rectangle

Section 8-2     Integrals over a Region

Section 8-3     Three-Dimensional Iterated Integrals

Section 8-4     Multiple Integration in Polar, Cylindrical and  Spherical Coordinates

Section 8-5     Applications of Multiple Integrals

微積分

一）極限
1-1  極限的概念
1-2  單邊極限
1-3  連續性
1-4  無窮極限


二）導函數
2-1  導數的意義與求法
2-2  微分的方法
2-3  連鎖律與隱微分法
2-4  三角函數的導函數
2-5  反三角函數的導函數
2-6  對數、指數函數的導函數
2-7  微分與近似值


三）微分的應用
3-1  函數的遞增與遞減
3-2  曲線的凹性
3-3  曲線作圖
3-4  函數的極值

3-5  極值的應用
3-6  商業及經濟上的應用(商用)需求彈性和收益彈性(商用)牛頓求根法
3-8 
3-7 

3-9  均值定理

3-10 泰勒展開式
3-11 羅必達法則



四）不定積分與反導數
4-1  反導數與不定積分
4-2  變數變換法
4-3  分部積分法
4-4  三角函數的積分(工用)         
4-5  三角代換及半角代換(工用)
4-6  部分分式的積分法     

五）定積分
5-1  定積分的定義與性質
5-2  微積分的基本定理
5-3  定積分的近似值
5-4  廣義積分




六）定積分的應用

6-1  曲線間的面積
6-2  極座標曲線所圍區域的面積
6-3  曲線長
6-4  旋轉體的體積
6-5  旋轉體之側表面積
6-6  形心
6-7  功與工程問題

6-8 消費者剩餘與生產者剩餘 (商用)

6-9 機率(商用)

6-10 複利與折現(商用)

七）三維空間的基本概念
7-1  平面向量及其運算 (高二上5-1)

7-2  空間向量及其運算 (高二上5-2)

7-3  向量的內積與外積 (高二上5-3)
7-4  直線與平面的向量式
7-5  柱面、球面；柱面座標以及球面座標



八）偏導函數
8-1  極限與連續性
8-2  偏導函數

8-3  連鎖規則 、全微分與近似估計
8-4  梯度、方向導數與切平面

8-5  偏導函數的應用–求極值

九）重積分及其應用
9-1  二重積分的定義
9-2  二重積分的運算
9-3  三重積分

9-4 重積分的轉換
9-5 重積分的應用