第1 章 函数、极限与连续 ············································································· 1
1.1 映射与函数 ························································································· 1
1.1.1 集合的概念 ··················································································· 1
1.1.2 区间与邻域 ··················································································· 2
1.1.3 映射与函数的概念 ·········································································· 3
1.1.4 函数的几种特性 ············································································· 6
1.1.5 反函数与复合函数、函数的运算 ························································ 9
1.1.6 基本初等函数、初等函数 ································································ 13
1.1.7 函数的参数表示和极坐标表示 ·························································· 18
习题1.1 ··································································································· 22
1.2 数列的极限 ························································································ 24
1.2.1 数列极限的定义 ············································································ 24
1.2.2 数列极限的性质 ············································································ 27
习题1.2 ··································································································· 29
1.3 函数的极限 ························································································ 30
1.3.1 自变量趋向无穷大时函数的极限 ······················································· 30
1.3.2 自变量趋于有限值时函数的极限 ······················································· 31
1.3.3 函数极限的性质 ············································································ 34
习题1.3 ··································································································· 37
1.4 无穷小量与无穷大量 ············································································ 38
1.4.1 无穷小量的概念 ············································································ 38
1.4.2 无穷小量的性质 ············································································ 39
1.4.3 无穷大量 ····················································································· 39
习题1.4 ··································································································· 42
1.5 极限的运算法则 ·················································································· 42
1.5.1 极限的四则运算法则 ······································································ 42
1.5.2 复合函数的极限运算法则 ································································ 45
习题1.5 ··································································································· 46
1.6 极限存在准则与两个重要极限 ································································ 48
1.6.1 极限存在准则 ··············································································· 48
1.6.2 两个重要极限 ··············································································· 50
习题1.6 ··································································································· 55
1.7 无穷小量的比较 ················································································· 56
1.7.1 无穷小量的比较 ············································································ 57
1.7.2 利用等价无穷小求极限 ··································································· 58
习题1.7 ··································································································· 60
1.8 函数的连续与间断 ·············································································· 61
1.8.1 函数连续性的概念 ········································································· 61
1.8.2 连续函数的运算法则与初等函数的连续性 ··········································· 62
1.8.3 函数的间断点 ··············································································· 63
1.8.4 闭区间上连续函数的性质 ································································ 66
习题1.8 ··································································································· 69
拓展阅读 ································································································· 70
总习题一 ································································································· 71
第2 章 导数与微分 ····················································································· 76
2.1 导数的概念 ······················································································· 76
2.1.1 引例 ··························································································· 76
2.1.2 导数的定义 ·················································································· 77
2.1.3 导数的几何意义 ············································································ 80
2.1.4 函数可导性与连续性的关系 ····························································· 81
习题2.1 ··································································································· 82
2.2 导数的求导法则 ················································································· 83
2.2.1 导数的四则运算法则 ······································································ 83
2.2.2 反函数的求导法则 ········································································· 85
2.2.3 复合函数的求导法则 ······································································ 86
2.2.4 导数表(常数和基本初等函数的导数公式) ··········································· 87
习题2.2 ··································································································· 88
2.3 高阶导数 ·························································································· 89
2.3.1 高阶导数的概念 ············································································ 89
2.3.2 高阶导数的计算 ············································································ 89
习题2.3 ··································································································· 92
2.4 隐函数和参数方程所确定的函数的导数及其相关变化率 ······························ 93
2.4.1 隐函数的导数 ··············································································· 93
2.4.2 参数方程所确定的函数的导数 ·························································· 95
2.4.3 相关变化率 ·················································································· 97
习题2.4 ··································································································· 98
2.5 函数的微分 ······················································································ 100
2.5.1 微分的概念 ················································································ 100
2.5.2 基本初等函数的微分公式和微分法则 ··············································· 101
2.5.3 微分的几何意义及在近似计算中的应用 ············································ 103
习题2.5 ································································································· 105
拓展阅读 ······························································································· 107
总习题二 ······························································································· 108
第3 章 一元函数微分学的应用 ···································································· 113
3.1 微分中值定理 ··················································································· 113
3.1.1 罗尔中值定理 ············································································· 113
3.1.2 拉格朗日中值定理 ······································································· 115
3.1.3 柯西中值定理 ············································································· 117
习题3.1 ································································································· 119
3.2 洛必达(L' Hospital)法则 ····································································· 119
3.2.1 “
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”型未定式 ·········································································· 120
3.2.2 “
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”型未定式 ·········································································· 121
3.2.3 其他型的未定式 ·········································································· 123
习题3.2 ································································································· 124
3.3 泰勒公式 ························································································· 125
3.3.1 函数逼近 ··················································································· 125
3.3.2 泰勒中值定理 ············································································· 125
3.3.3 泰勒公式的应用 ·········································································· 128
习题3.3 ································································································· 129
3.4 函数的单调性、极值和最值 ································································· 130
3.4.1 函数的单调性 ············································································· 130
3.4.2 函数的极值 ················································································ 132
3.4.3 最大值与最小值问题 ···································································· 135
习题3.4 ································································································· 136
3.5 曲线的凹凸性与拐点 ·········································································· 137
3.5.1 曲线的凹凸性 ············································································· 138
3.5.2 曲线的拐点 ················································································ 140
习题3.5 ································································································· 142
3.6 曲率 ······························································································· 142
3.6.1 弧微分 ······················································································· 142
3.6.2 曲率及其计算公式 ········································································ 143
习题3.6 ·································································································· 146
3.7 函数的性态和图形 ············································································· 146
3.7.1 渐近线 ······················································································· 147
3.7.2 函数图形描绘 ·············································································· 148
习题3.7 ·································································································· 150
拓展阅读 ································································································ 150
总习题三 ································································································ 151
第4 章 不定积分 ······················································································· 154
4.1 不定积分的概念与性质 ······································································· 154
4.1.1 原函数与不定积分的概念 ······························································· 154
4.1.2 基本积分表 ················································································· 155
4.1.3 不定积分的性质 ··········································································· 156
习题4.1 ·································································································· 158
4.2 不定积分的换元积分法 ······································································· 159
4.2.1 第一类换元法 ·············································································· 159
4.2.2 第二类换元法 ·············································································· 162
习题4.2 ·································································································· 166
4.3 不定积分的分部积分法 ······································································· 167
习题4.3 ·································································································· 170
4.4 有理函数的积分 ················································································ 170
4.4.1 有理函数的积分 ··········································································· 170
4.4.2 三角有理式的积分 ········································································ 173
4.4.3 简单无理式的积分 ········································································ 174
习题4.4 ·································································································· 176
拓展阅读 ································································································ 176
总习题四 ································································································ 177
第5 章 定积分 ·························································································· 179
5.1 定积分的概念和性质 ·········································································· 179
5.1.1 引例 ·························································································· 179
5.1.2 定积分的定义 ·············································································· 181
5.1.3 定积分的近似计算 ········································································ 183
5.1.4 定积分的性质 ·············································································· 185
习题5.1 ································································································· 188
5.2 微积分基本公式 ················································································ 189
5.2.1 引例 ························································································· 190
5.2.2 积分上限的函数及其导数 ······························································ 190
5.2.3 牛顿-莱布尼茨公式 ······································································ 193
习题5.2 ································································································· 194
5.3 定积分的换元法和分部积分法 ······························································ 196
5.3.1 定积分的换元法 ·········································································· 196
5.3.2 定积分的分部积分法 ···································································· 200
习题5.3 ································································································· 201
5.4 反常积分 ························································································· 202
5.4.1 无穷区间上的反常积分 ································································· 203
5.4.2 无界函数的反常积分 ···································································· 205
习题5.4 ································································································· 208
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5.5 反常积分的审敛法和Γ函数 ······························································ 208
5.5.1 无穷区间上反常积分的判别法 ························································ 208
5.5.2 无界函数的反常积分的审敛法 ························································ 211
5.5.3 ? 函数 ······················································································ 213
*习题5.5 ································································································ 214
拓展阅读 ······························································································· 215
总习题五 ······························································································· 216
第6 章 定积分的应用 ················································································ 220
6.1 定积分的元素法 ················································································ 220
6.2 定积分在几何学上的应用 ···································································· 222
6.2.1 平面图形的面积 ·········································································· 222
6.2.2 体积 ························································································· 225
6.2.3 平面曲线的弧长 ·········································································· 228
习题6.2 ································································································· 231
6.3 定积分在物理学上的应用 ···································································· 232
6.3.1 变力沿直线所做的功 ···································································· 232
6.3.2 水压力 ······················································································ 233
6.3.3 引力 ························································································· 235
习题6.3 ································································································· 236
拓展阅读 ······························································································· 237
总习题六 ······························································································· 239
第7 章 微分方程 ······················································································· 241
7.1 微分方程的基本概念 ·········································································· 241
习题7.1 ·································································································· 244
7.2 可分离变量的微分方程 ······································································· 244
习题7.2 ·································································································· 246
7.3 齐次方程 ························································································· 247
7.3.1 齐次方程及其解法 ········································································ 247
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7.3.2 可化为齐次方程的微分方程 ··························································· 250
习题7.3 ·································································································· 252
7.4 一阶线性微分方程 ············································································· 253
7.4.1 一阶线性微分方程及常数变易法 ······················································ 253
7.4.2 伯努利方程 ················································································· 256
习题7.4 ·································································································· 257
7.5 可降阶的二阶微分方程 ······································································· 258
7.5.1 不含未知函数 y和 y?的方程 y?? ? f (x) ··············································· 258
7.5.2 不含未知函数 y的方程 y?? ? f (x, y?) ·················································· 259
7.5.3 不含自变量x的方程 y?? ? f ( y, y?) ····················································· 262
习题7.5 ·································································································· 263
7.6 二阶线性微分方程解的结构 ································································· 264
7.6.1 二阶线性微分方程 ········································································ 264
7.6.2 线性微分方程解的结构 ·································································· 265
习题7.6 ·································································································· 267
7.7 二阶常系数齐次线性微分方程 ······························································ 268
习题7.7 ·································································································· 272
7.8 二阶常系数非齐次线性微分方程 ··························································· 272
7.8.1 ( ) e ( ) x
m fx P x ?
? 型 ········································································· 273
7.8.2 1( ) e ( )cos x
m f x P x x ? ? ? 或 2( ) e ( )sin x
m f x P x x ? ? ? 型 ································ 276
习题7.8 ·································································································· 279
7.9 欧拉方程 ························································································· 280
习题7.9 ·································································································· 281
7.10 常系数线性微分方程组 ······································································ 281
习题7.10 ································································································ 283
拓展阅读 ································································································ 283
总习题七 ································································································ 285
数学实验(上) ···························································································· 287
Python 简介及基本操作 ············································································· 287
实验1 割圆术、生长模型 ········································································ 323
实验2 陈酒出售的最佳时机问题 ······························································· 327
实验3 泰勒展开与e 的计算 ····································································· 331
实验4 氢燃料电池堆优化设计 ·································································· 335
实验5 智慧农场中根系层含水量估计与智能灌溉决策 ···································· 343
习题答案··································································································· 351
第1 章 函数、极限与连续 ······································································· 351
第2 章 导数与微分 ················································································ 357
第3 章 一元函数微分学的应用 ································································· 364
第4 章 不定积分 ··················································································· 369
第5 章 定积分 ······················································································ 374
第6 章 定积分的应用 ············································································· 378
第7 章 微分方程 ··················································································· 380
主要参考文献 ···························································································· 387
附录 积分表 ···························································································· 389
