第1 章 一元函数极限及连续 ·········································································· 1
1.1 函数的概念 ····················································································· 1
1.1.1 集合与邻域 ······························································································· 1
1.1.2 函数的概念 ······························································································· 2
1.1.3 函数的性质 ······························································································· 4
1.1.4 初等函数的概念 ························································································· 6
习题1.1 ································································································ 6
1.2 数列极限 ························································································ 7
1.2.1 数列的概念 ······························································································· 7
1.2.2 数列的极限 ······························································································· 8
1.2.3 数列极限的性质及收敛准则 ·········································································· 9
1.2.4 数列极限的四则运算法则 ··········································································· 14
习题1.2 ······························································································· 14
1.3 函数的极限 ···················································································· 15
1.3.1 函数极限的定义 ······················································································· 15
1.3.2 函数极限的性质 ······················································································· 19
习题1.3 ······························································································· 20
1.4 无穷小量、无穷大量 ········································································ 20
1.4.1 无穷小量 ································································································ 20
1.4.2 无穷大量 ································································································ 23
1.4.3 无穷大与无穷小的关系 ·············································································· 25
习题1.4 ······························································································· 25
1.5 函数极限的运算法则 ········································································ 26
1.5.1 函数极限的运算法则 ················································································· 26
1.5.2 复合函数的极限运算法则 ··········································································· 29
习题1.5 ······························································································· 30
1.6 两个重要极限 ················································································· 30
1.6.1 重要极限 ················································································· 31
1.6.2 重要极限············································································· 32
习题1.6 ······························································································· 35
1.7 无穷小的比较 ················································································· 36
1.7.1 无穷小的比较 ·························································································· 36
1.7.2 关于等价无穷小量的性质和定理 ·································································· 38
习题1.7 ······························································································· 40
1.8 函数的连续性 ················································································· 40
1.8.1 函数连续性的概念 ···················································································· 41
1.8.2 函数的间断点 ·························································································· 44
1.8.3 连续函数的基本性质 ················································································· 45
1.8.4 初等函数的连续性 ···················································································· 46
习题1.8 ······························································································· 48
1.9 闭区间上连续函数的性质 ·································································· 49
1.9.1 最大值最小值定理 ···················································································· 49
1.9.2 零点定理 ································································································ 50
1.9.3 介值定理 ································································································ 51
习题1.9 ······························································································· 52
总习题一 ······························································································ 52
第2 章 导数与微分 ····················································································· 55
2.1 导数的概念 ···················································································· 55
2.1.1 导数的定义 ····························································································· 55
2.1.2 导数的几何意义 ······················································································· 59
2.1.3 可导与连续的关系 ···················································································· 60
习题2.1 ······························································································· 61
2.2 函数的求导法则和基本求导公式 ························································· 62
2.2.1 函数求导的四则运算法则 ··········································································· 62
2.2.2 反函数求导法则 ······················································································· 63
2.2.3 复合函数求导法则 ···················································································· 64
习题2.2 ······························································································· 67
2.3 高阶导数 ······················································································· 69
习题2.3 ······························································································· 71
2.4 隐函数及参数方程确定的函数的导数 ··················································· 72
2.4.1 隐函数的导数 ·························································································· 72
2.4.2 对数求导法 ····························································································· 73
2.4.3 参数方程求导法则 ···················································································· 74
习题2.4 ······························································································· 77
2.5 函数的微分 ···················································································· 78
2.5.1 微分的概念 ····························································································· 78
2.5.2 可微与可导的关系 ···················································································· 79
2.5.3 微分运算法则和基本微分公式 ····································································· 79
2.5.4 微分的几何意义 ······················································································· 81
2.5.5 微分在近似计算中的应用 ··········································································· 81
习题2.5 ······························································································· 83
总习题二 ······························································································ 84
第3 章 微分中值定理与导数的应用 ································································ 88
3.1 微分中值定理 ················································································· 88
习题3.1 ······························································································· 91
3.2 洛必达法则 ···················································································· 92
习题3.2 ······························································································· 96
3.3 函数的单调性、极值、最值 ······························································· 96
3.3.1 函数的单调性 ·························································································· 96
3.3.2 函数的极值 ····························································································· 99
3.3.3 函数的最大值与最小值 ············································································· 101
习题3.3 ····························································································· 102
3.4 曲线的凸性、拐点与渐近线 ····························································· 103
3.4.1 曲线的凸性和拐点 ··················································································· 103
3.4.2 曲线的渐近线 ························································································· 105
3.4.3 函数图形的描绘 ······················································································ 106
习题3.4 ····························································································· 107
3.5 导数在经济中的应用 ······································································ 108
3.5.1 边际与边际分析 ······················································································ 108
3.5.2 弹性与弹性分析 ······················································································ 109
习题3.5 ····························································································· 111
3.6 泰勒公式 ····················································································· 112
习题3.6 ····························································································· 117
总习题三 ···························································································· 117
第4 章 不定积分 ······················································································ 122
4.1 不定积分的概念与性质 ··································································· 122
4.1.1 原函数与不定积分 ··················································································· 122
4.1.2 基本积分表 ···························································································· 123
4.1.3 不定积分的性质 ······················································································ 124
4.1.4 不定积分的几何意义 ················································································ 126
习题4.1 ····························································································· 127
4.2 换元积分法 ·················································································· 128
4.2.1 第一类换元积分法 ··················································································· 128
4.2.2 第二类换元积分法 ··················································································· 131
习题4.2 ····························································································· 135
4.3 分部积分法 ·················································································· 137
习题4.3 ····························································································· 140
4.4 有理函数的积分 ············································································ 141
4.4.1 有理函数及其积分 ··················································································· 141
4.4.2 可化为有理函数的积分举例 ······································································· 146
习题4.4 ····························································································· 148
总习题四 ···························································································· 149
第5 章 定积分及其应用 ············································································· 153
5.1 定积分的概念与性质 ····································································· 153
5.1.1 定积分的定义 ························································································· 153
5.1.2 定积分的几何意义 ··················································································· 157
5.1.3 定积分的基本性质 ··················································································· 157
习题5.1 ····························································································· 160
5.2 微积分基本定理 ············································································ 160
5.2.1 积分上限函数 ························································································· 161
5.2.2 牛顿-莱布尼茨公式 ·················································································· 162
习题5.2 ····························································································· 164
5.3 定积分的积分法 ············································································ 164
5.3.1 定积分的换元积分法 ················································································ 165
5.3.2 定积分的分部积分法 ················································································ 167
习题5.3 ····························································································· 169
5.4 广义积分 ····················································································· 170
5.4.1 无穷区间上的广义积分 ············································································· 170
5.4.2 无界函数的广义积分 ················································································ 172
习题5.4 ····························································································· 174
5.5 定积分的应用 ··············································································· 175
5.5.1 平面图形的面积 ······················································································ 175
5.5.2 旋转体的体积 ························································································· 179
5.5.3 定积分在经济学的应用 ············································································· 182
习题5.5 ····························································································· 185
总习题五 ···························································································· 186
第6 章 多元函数微积分 ············································································· 189
6.1 空间解析几何 ··············································································· 189
6.1.1 空间直角坐标系 ······················································································ 189
6.1.2 空间两点间的距离 ··················································································· 191
6.1.3 曲面及其方程 ························································································· 191
习题6.1 ····························································································· 195
6.2 多元函数的基本概念 ······································································ 196
6.2.1 平面点集 ······························································································· 196
6.2.2 平面点集的基本概念 ················································································ 197
6.2.3 二元函数的概念 ······················································································ 198
6.2.4 二元函数的极限 ······················································································ 199
6.2.5 二元函数的连续性 ··················································································· 200
习题6.2 ····························································································· 200
6.3 偏导数 ························································································ 201
6.3.1 偏导数的概念 ························································································· 201
6.3.2 偏导数的计算 ························································································· 203
6.3.3 偏导数和连续的关系 ················································································ 203
习题6.3 ····························································································· 204
6.4 全微分 ························································································ 205
6.4.1 全微分的定义 ························································································· 205
6.4.2 可微性的应用 ························································································· 208
习题6.4 ····························································································· 208
6.5 复合函数微分法与隐函数微分法 ······················································· 208
6.5.1 复合函数的求导法则 ················································································ 208
6.5.2 一阶微分形式不变性 ················································································ 211
6.5.3 隐函数可微性定理 ··················································································· 211
习题6.5 ····························································································· 214
6.6 多元函数的极值及其求法 ································································ 215
6.6.1 二元函数的极值问题 ················································································ 215
6.6.2 条件极值与拉格朗日乘数法 ······································································· 216
习题6.6 ····························································································· 220
6.7 二重积分的概念与性质 ··································································· 221
6.7.1 二重积分的概念 ······················································································ 221
6.7.2 二重积分的性质 ······················································································ 223
习题6.7 ····························································································· 224
6.8 在直角坐标系下的二重积分的计算 ···················································· 224
6.8.1 在直角坐标系下二重积分的计算 ································································· 224
6.8.2 交换二次积分次序 ··················································································· 230
6.8.3 利用对称性和奇偶性化简二重积分的计算 ····················································· 232
习题6.8 ····························································································· 233
6.9 在极坐标系下的二重积分的计算 ······················································· 234
习题6.9 ····························································································· 237
总习题六 ···························································································· 238
第7 章 常微分方程 ··················································································· 242
7.1 基本概念和例题 ············································································ 242
习题7.1 ····························································································· 244
7.2 一阶可分离变量的微分方程 ····························································· 244
习题7.2 ····························································································· 245
7.3 一阶线性微分方程 ········································································· 246
习题7.3 ····························································································· 248
7.4 一阶微分方程的变量代换法 ····························································· 249
习题7.4 ····························································································· 251
7.5 可降阶的微分方程 ········································································· 251
7.5.1 形如 y(n) ? f (x)的微分方程 ········································································ 251
7.5.2 形如 y? ? f (x, y?)的微分方程 ······································································ 252
7.5.3 形如 y? ? f (y, y?)的微分方程 ······································································ 252
习题7.5 ····························································································· 253
7.6 二阶常系数线性微分方程 ································································ 253
7.6.1 二阶常系数齐次线性微分方程的求解 ··························································· 253
7.6.2 二阶常系数非齐次线性微分方程的求解 ························································ 256
习题7.6 ····························································································· 258
7.7 一般的二阶线性微分方程 ································································ 259
习题7.7 ····························································································· 260
7.8 差分方程* ···················································································· 261
7.8.1 差分的概念与性质 ··················································································· 261
7.8.2 差分方程的概念 ······················································································ 262
7.8.3 一阶常系数线性差分方程 ·········································································· 262
习题7.8 ····························································································· 264
总习题七 ···························································································· 264
第8 章 无穷级数 ······················································································ 267
8.1 无穷级数的概念与性质 ··································································· 267
8.1.1 无穷级数的定义 ······················································································ 267
8.1.2 收敛级数的基本性质 ················································································ 270
习题8.1 ····························································································· 272
8.2 常数项级数的审敛法 ······································································ 273
8.2.1 正项级数及其审敛法 ················································································ 273
8.2.2 交错级数及其审敛法 ················································································ 279
8.2.3 绝对收敛与条件收敛 ················································································ 281
习题8.2 ····························································································· 282
8.3 幂级数 ························································································ 284
8.3.1 函数项级数的概念 ··················································································· 284
8.3.2 幂级数及其收敛性 ··················································································· 284
8.3.3 幂级数的运算 ························································································· 288
习题8.3 ····························································································· 290
8.4 函数展开成幂级数 ········································································· 291
8.4.1 泰勒级数 ······························································································· 291
8.4.2 函数展开成幂级数 ··················································································· 293
8.4.3 函数的幂级数展开式的应用 ······································································· 295
习题8.4 ····························································································· 298
总习题八 ···························································································· 298
主要参考文献 ···························································································· 302
习题参考答案 ···························································································· 303
