AP考试系列教程 AP微积分2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Chapter 1 Function函数1

1.1 Polynomial Functions多项式函数1

1.1.1 Definition多项式函数的定义1

1.1.2 Division of Polynomial Functions带余除法与综合除法1

1.1.3 Remainder Theorem and Factor Theorem余数定理与因式定理4

1.1.4 Partial Fraction待定系数法与部分分式5

1.2 Inverse Function反函数6

1.2.1 Definition反函数的概念6

1.2.2 The Graphs of Inverse Functions互为反函数的函数图像间的关系7

1.2.3 Inverses of Trigonometric Functions反三角函数8

1.3 Parametrically Defined Functions参变量函数14

1.3.1 Definition参数方程的意义14

1.3.2 Parameter Equation and Ordinary Equation参数方程与普通方程的互化15

1.3.3 Line and Conic Curve直线与圆锥曲线的参数方程17

1.4 Polar Functions极坐标函数19

1.4.1 Polar System极坐标系20

1.4.2 Polar System and Cartesian System极坐标与直角坐标的互化21

1.4.3 Polar Curves曲线的极坐标方程的意义22

1.4.4 Some Curves常见曲线的极坐标方程23

1.5 Composition Functions复合函数25

1.6 Graphs of Functions函数的图像变换27

1.6.1 Translation Transform平移变换27

1.6.2 Symmetry Transform对称变换29

1.6.3 Stretching Transform伸缩变换30

Practice Exercises32

Chapter 2 Limits极限33

2.1 Limits of Sequence数列的极限33

2.1.1 Definition数列极限的定义33

2.1.2 Rules of Limits数列极限的运算法则35

2.1.3 Sum of the Infinite Geometric Series无穷等比数列各项的和37

2.2 Limits of Functions函数的极限39

2.2.1 Definition函数极限的定义39

2.2.2 Infinitesimal and Infinity无穷小量与无穷大量42

2.2.3 Rules of Operation函数极限的运算法则44

2.2.4 Two Important Limits两个重要极限46

2.3 Asymptote曲线的渐近线48

2.4 Continuity函数的连续性49

2.4.1 Definition函数连续性定义49

2.4.2 Properties of Continuous Functions连续性函数的基本性质51

Practice Exercises52

Chapter 3 Differentiation微分54

3.1 Derivative导数54

3.1.1 Definition导数的概念54

3.2 Derivatives导数的运算56

3.2.1 Formulas常见函数的导数56

3.2.2 Some Basic Rules函数的和、差、积、商的导数58

3.2.3 Special Rules几种特殊形式函数的求导法则59

3.3 Applications of Differentiation微分的应用62

3.3.1 The Physical and Geometrical Meaning of Differentiation微分的物理意义和几何意义62

3.3.2 Monotonicity and Extreme Value of the Functions函数的单调性与极值64

3.3.3 Graphs of the Functions函数图像的描绘67

3.3.4 The Mean Value Theorem中值定理68

3.3.5 L'Hopital's Rule洛必达法则69

3.4 真题集锦72

Practice Exercises80

Chapter 4 Antiderivative不定积分81

4.1 Definition定义81

4.2 Basic Formulas基本公式82

4.3 Techniques of Antiderivative积分方法83

4.3.1 U-substitution第一积分换元法83

4.3.2 Integration by Parts分部积分88

4.3.3 Integration by Simple Partial Fractions简单部分分数积分92

4.3.4 Tips for Antiderivative积分方法归纳94

Practice Exercises96

Chapter 5 Definite Integrals定积分97

5.1 Definition：the limit of Riemann Sum基本定义：黎曼和的极限形式97

5.1.1 Definition of Riemann Sum黎曼和97

5.1.2 More about Riemann Sums特殊黎曼和98

5.1.3 Definition of Definite Integrals定积分的定义104

5.2 Fundamental Theorems of Calculus微积分基本定理106

5.2.1 The First FTC微积分基本定理106

5.2.2 The Second FTC微积分基本定理二108

5.2.3 Properties of Definite Integrals定积分性质109

5.3 Improper Integrals广义积分110

Practice Exercises112

Chapter 6 Applications of Integrals积分应用113

6.1 Computing Areas Bounded by Curves面积计算113

6.1.1 Region Below the X-axis函数取负值的情形113

6.1.2 Region Between two Curves曲线间面积114

6.1.3 Region Bounded by Polar Curve极曲线图形118

6.2 Volume体积121

6.2.1 Solids with Known Cross Sections截面已知的体积121

6.2.2 Solids of Revolution旋转体122

6.3 Arc Length弧长125

6.4 More Applications of Integrals积分应用补充130

6.4.1 Motion运动130

6.5真题集锦131

Practice Exercises137

Chapter 7 Differential Equations微分方程139

7.1 Definition of Differential Equation定义139

7.2 Graphical Method：Slope Fields图像法：斜率场141

7.3 Numerical Method：Euler's Method数值法：欧拉法146

7.4 Analytical Method：Separating Variables解析解：分离变量法149

7.5 Application of Differential Equation微分方程应用152

7.5.1 Exponential Growth指数增长152

7.5.2 Restricted Growth约束增长153

7.5.3 Logistic Growth逻辑斯蒂增长155

7.6真题集锦156

Practice Exercises162

Chapter 8 Series级数163

8.1 Concepts of Series级数的概念163

8.1.1 Definition of Series级数的定义163

8.1.2 Convergence and Divergence收敛性和发散性164

8.1.3 Theorems about Convergence or Divergence of Infinite Series级数收敛性定理165

8.2 Series of Constants常数级数166

8.2.1 Tests for Convergence of Infinite Series无穷级数收敛性判别166

8.3 Power Series幂级数170

8.3.1 Definition of Power Series幂级数定义170

8.3.2 Functions Defined by Power Series幂级数定义的函数173

8.4 Taylor Series泰勒级数173

8.4.1 Taylor Series and Maclaurin Series泰勒级数以及迈克劳林级数173

8.4.2 Some Basic Maclaurin Series基本函数的迈克劳林展开174

8.5 Application of Series级数应用175

8.5.1 Computations with Power Series级数计算175

8.5.2 Lagrange Error Bound拉格朗日误差界176

8.6真题集锦176

Practice Exercises184