目录
第1 章 实数集与初等函数··················1
1．1 实数集····································1
1．1．1 集合及其运算························1
1．1．2 映射···································3
1．1．3 可数集································3
1．1．4 实数集的性质························5
1．1．5 戴德金原理···························8
1．1．6 确界原理·····························8
习题1．1····································．10
1．2 初等函数······························．11
1．2．1 函数的概念························．11
1．2．2 函数的一些特性··················．12
1．2．3 函数的运算························．13
1．2．4 基本初等函数·····················．14
1．2．5 反函数及其存在条件·············．18
1．2．6 反三角函数························．19
*1．2．7 双曲函数和反双曲函数··········．22
*1．2．8 双曲函数与三角函数之间
的联系·····························．24
习题1．2····································．24
第2 章 数列极限····························．27
2．1 数列极限的概念·····················．27
习题2．1····································．30
2．2 数列极限的性质·····················．31
习题2．2····································．35
2．3 几类特殊的数列·····················．36
2．3．1 无穷大数列与无穷小数列·······．36
2．3．2 无穷大数列与无界数列··········．36
2．3．3 Stolz 定理·························．38
习题2．3 ···································．40
2．4 实数连续性定理····················．41
2．4．1 单调有界定理·····················．41
2．4．2 闭区间套定理·····················．43
2．4．3 Bolzano-Weierstrass 定理·········．44
2．4．4 柯西收敛准则·····················．45
*2．4．5 有限覆盖定理·····················．47
*2．4．6 聚点定理··························．48
习题2．4 ···································．48
*2．5 上极限与下极限···················．50
习题2．5 ···································．54
第3 章 函数极限与连续··················．55
3．1 函数极限的概念····················．55
3．1．1 函数在一点的极限···············．55
3．1．2 函数在无穷远处的极限··········．58
习题3．1 ···································．58
3．2 函数极限的性质及运算···········．59
3．2．1 函数极限的性质··················．59
3．2．2 函数极限的四则运算·············．60
3．2．3 复合函数的极限··················．62
习题3．2 ···································．62
3．3 函数极限的存在条件··············．63
3．3．1 函数极限与数列极限的关系·····．63
3．3．2 两个重要极限·····················．65
3．3．3 无穷大量与无穷小量·············．67
3．3．4 等价无穷小量代换求极限········．69
习题3．3····································．71
3．4 函数的连续···························．73
3．4．1 函数连续的概念··················．73
3．4．2 间断点及其分类··················．74
3．4．3 连续函数的局部性质·············．78
习题3．4····································．78
3．5 闭区间上连续函数的性质········．79
3．5．1 闭区间上连续函数的基本性质··．79
3．5．2 反函数的连续性··················．82
3．5．3 一致连续性························．83
习题3．5····································．86
第4 章 导数与微分·························．89
4．1 导数的概念···························．89
4．1．1 导数概念的引出··················．89
4．1．2 函数可导的条件与性质··········．91
习题4．1····································．92
4．2 求导法则······························．94
4．2．1 导数的四则运算法则·············．94
4．2．2 反函数求导法则··················．96
4．2．3 复合函数的导数――链式法则···．97
4．2．4 隐函数求导法则··················．98
4．2．5 参数方程求导法则················．99
习题4．2····································102
4．3 函数的微分···························103
4．3．1 可微的概念························103
4．3．2 可微与可导的关系················104
4．3．3 微分在函数近似计算中的应用··105
4．3．4 微分的运算法则··················106
习题4．3····································106
4．4 高阶导数与高阶微分··············106
4．4．1 高阶导数·························．107
4．4．2 高阶微分·························．109
4．4．3 复合函数的微分·················．109
习题4．4 ··································．110
第5 章 微分学基本定理及应用········．112
5．1 微分中值定理······················．112
5．1．1 极值的概念与费马定理·········．112
5．1．2 微分中值定理····················．113
习题5．1 ··································．118
5．2 洛必达法则··························．120
5．2．1 0
0
型不定式极限·················．120
5．2．2 ∞
∞
型不定式极限················．123
5．2．3 其他类型不定式极限············．125
习题5．2 ··································．126
5．3 泰勒公式及应用···················．127
5．3．1 泰勒公式·························．128
5．3．2 基本初等函数的展开式·········．130
5．3．3 泰勒公式的应用·················．134
习题5．3 ··································．138
5．4 单调性与极值······················．140
5．4．1 函数的单调性····················．140
5．4．2 函数取极值的条件··············．142
习题5．4 ··································．145
5．5 函数的凸性与函数作图··········．147
5．5．1 函数的凸性······················．147
5．5．2 曲线的渐近性····················．152
5．5．3 函数作图·························．153
习题5．5 ··································．155
*5．6 方程求根的牛顿迭代公式·····．155
第6 章 不定积分···························．160
6．1 原函数与不定积分················．160
6．1．1 原函数与不定积分的概念······．160
6．1．2 不定积分的线性运算·············162
6．1．3 常用的不定积分公式·············162
习题6．1····································163
6．2 不定积分计算························164
6．2．1 分部积分法························165
6．2．2 积分换元法························166
习题6．2····································174
6．3 有理函数的不定积分··············175
习题6．3····································178
6．4 可化为有理函数的不定积分·····179
6．4．1 三角有理函数的不定积分········179
6．4．2 某些无理函数的不定积分········182
习题6．4····································185
第7 章 定积分·······························187
7．1 定积分的概念及可积条件········187
7．1．1 引例································187
7．1．2 定积分的概念·····················188
7．1．3 定积分的几何意义················189
7．1．4 可积的必要条件··················190
7．1．5 可积准则··························191
习题7．1····································195
7．2 可积函数类及定积分的性质·····195
7．2．1 闭区间上的可积函数类··········195
*7．2．2 再论可积的充要条件·············196
7．2．3 定积分的性质·····················200
习题7．2····································203
7．3 定积分的计算························204
7．3．1 变上限积分························205
7．3．2 微积分基本定理··················208
7．3．3 积分换元法和分部积分法········210
习题7．3····································213
7．4 积分中值定理························216
习题7．4····································221
7．5 定积分的应用······················．221
*7．5．1 分析学应用······················．221
7．5．2 定积分的几何应用··············．224
7．5．3 定积分的物理应用··············．232
习题7．5 ··································．236
第8 章 广义积分···························．239
8．1 无穷积分·····························．239
8．1．1 无穷积分的概念·················．239
8．1．2 无穷积分求值····················．240
8．1．3 无穷积分敛散性判别法·········．241
习题8．1 ··································．246
8．2 瑕积分································．248
8．2．1 瑕积分收敛的概念··············．248
8．2．2 无穷积分与瑕积分的关系······．249
8．2．3 瑕积分敛散性判别法············．250
习题8．2 ··································．254
第9 章 常微分方程························．255
9．1 常微分方程的概念················．255
9．1．1 引例······························．255
9．1．2 常微分方程的概念··············．257
9．1．3 常微分方程的解·················．257
习题9．1 ··································．258
9．2 一阶常微分方程的初等解法····．259
9．2．1 可分离变量的微分方程·········．259
9．2．2 齐次方程·························．261
9．2．3 可化为齐次方程类型的方程····．262
9．2．4 常数变易法······················．263
9．2．5 伯努利方程······················．265
习题9．2 ··································．267
9．3 一阶微分方程初值问题的解····．268
9．3．1 初值问题解的存在专享性
定理······························．268
*9．3．2 奇解······························．268
9．4 高阶线性常微分方程··············269
9．4．1 可降阶的高阶微分方程··········269
9．4．2 高阶线性常微分方程解
的结构·····························273
9．4．3 高阶非齐次方程的常数
变易法·····························278
习题9．4····································280
9．5 常系数高阶线性常微分方程·····281
9．5．1 常系数齐次线性常微分方程的
特征值法··························281
9．5．2 常系数非齐次线性常微分方程
的待定系数法·····················285
*9．5．3 常系数线性常微分方程的
应用――质点的振动···············289
习题9．5····································291
9．6 欧拉方程······························292
习题9．6····································294
9．7 一阶线性常微分方程组··········．294
9．7．1 解的叠加原理及解的存在
专享性····························．294
9．7．2 一阶线性常微分方程组解的
结构······························．295
9．7．3 一阶非齐次线性常微分方程组的
常数变易法······················．298
9．7．4 从方程组的观点看高阶微分
方程······························．299
9．8 常系数线性常微分方程组·······．301
9．8．1 矩阵A 可对角化的情形·········．301
9．8．2 矩阵A 不可对角化的情形······．302
9．8．3 矩阵A 有复特征根的情形······．305
*9．8．4 方程组初值问题解的
一般形式·························．307
*9．8．5 非齐次方程的通解··············．309
习题9．8 ··································．310