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 Exercises 32
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 Exercises 52
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 Exercises 80
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 Exercises 96
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 Exercises 112
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 Exercises 137
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 Exercises 162
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 Exercises 184