目 录 
第1 章函数 ··································································································································· 1 
1.1  知识要点 ··························································································································· 1 
1.1.1 函数、邻域的概念 ····································································································· 1 
1.1.2 函数的基本特性 ········································································································· 1 
1.1.3 反函数与复合函数 ····································································································· 2 
1.1.4 基本初等函数与初等函数 ··························································································· 3 
1.1.5 一些常用公式 ············································································································ 3 
1.2  典型例题分析 ··················································································································· 4 
1.2.1 题型一：函数定义域的求解 ························································································ 4 
1.2.2 题型二：函数表达式的求解 ························································································ 4 
1.2.3 题型三：反函数的求解 ······························································································· 5 
1.2.4 题型四：复合函数的求解 ··························································································· 6 
1.2.5 题型五：函数的四种基本特性 ····················································································· 7 
1.3  深化训练 ··························································································································· 8 
1.4  深化训练详解 ··················································································································· 9 
1.5  综合提高训练 ················································································································· 10 
第2 章极限与连续 ····················································································································· 12 
2.1  知识要点 ························································································································· 12 
2.1.1 极限的概念 ············································································································· 12 
2.1.2 无穷小量与无穷大量 ································································································ 12 
2.1.3 极限的性质与运算法则 ····························································································· 14 
2.1.4 极限存在准则与两个重要极限 ··················································································· 14 
2.1.5 函数的连续性 ·········································································································· 15 
2.1.6 函数的间断点 ·········································································································· 15 
2.1.7 连续函数的性质 ······································································································· 16 
2.1.8 闭区间上的连续函数的性质 ······················································································ 16 
2.1.9 一些重要的结论 ······································································································· 17 
2.2  典型例题分析 ················································································································· 17 
2.2.1 题型一：极限的概念与性质问题 ··············································································· 17 
2.2.2 题型二：利用极限的四则运算法则求极限 ·································································· 18 
2.2.3 题型三：利用单侧极限的性质求极限 ········································································· 19 
2.2.4 题型四：利用两个重要极限求极限 ············································································ 20 
2.2.5 题型五：利用等价无穷小量替换求极限 ····································································· 21 
2.2.6 题型六：利用极限存在准则求极限 ············································································ 22 
2.2.7 题型七：函数的连续性问题 ······················································································ 23 
2.2.8 题型八：连续函数的等式证明问题 ············································································ 24 
2.3  深化训练 ························································································································· 25 
2.4  深化训练详解 ················································································································· 27 
2.5  综合提高训练 ················································································································· 31 
第3 章导数与微分 ····················································································································· 35 
3.1  知识要点 ························································································································· 35 
3.1.1 导数的概念 ············································································································· 35 
3.1.2 导数的几何意义 ······································································································· 35 
3.1.3 基本导数公式 ·········································································································· 35 
3.1.4 导数的四则运算法则 ································································································ 36 
3.1.5 常用求导法则 ·········································································································· 36 
3.1.6 高阶导数 ················································································································· 37 
3.1.7 微分的概念与性质 ··································································································· 38 
3.1.8 导数在经济学中的应用 ····························································································· 39 
3.2  典型例题分析 ················································································································· 40 
3.2.1 题型一：导数与微分的定义问题 ··············································································· 40 
3.2.2 题型二：分段函数的求导问题 ··················································································· 42 
3.2.3 题型三：导数的几何意义 ························································································· 43 
3.2.4 题型四：导函数的几何特性问题 ··············································································· 44 
3.2.5 题型五：利用可导性求参数值（域） ········································································· 44 
3.2.6 题型六：高阶导数问题 ····························································································· 45 
3.2.7 题型七：反函数、复合函数的求导问题 ····································································· 46 
3.2.8 题型八：隐函数的求导问题 ······················································································ 47 
3.2.9 题型九：导函数的连续性问题 ··················································································· 48 
3.2.10 题型十：导数在经济学中的应用 ·············································································· 49 
3.3  深化训练 ························································································································· 49 
3.4  深化训练详解 ················································································································· 52 
3.5  综合提高训练 ················································································································· 56 
第4 章中值定理与导数的应用 ·································································································· 57 
4.1  知识要点 ························································································································· 57 
4.1.1 中值定理 ················································································································· 57 
4.1.2 洛必达法则 ············································································································· 57 
4.1.3 函数的单调区间 ······································································································· 58 
4.1.4 函数的极值与最值 ··································································································· 58 
4.1.5 函数的凹凸区间与拐点 ····························································································· 58 
4.1.6 曲线的渐近线 ·········································································································· 59 
4.1.7 函数作图 ················································································································· 59 
4.1.8 一些常用的麦克劳林公式 ························································································· 59 
4.2  典型例题分析 ················································································································· 60 
4.2.1 题型一：利用中值定理证明等式问题 ········································································· 60 
4.2.2 题型二：利用中值定理证明不等式问题 ····································································· 62 
4.2.3 题型三：利用洛必达法则求极限 ··············································································· 63 
4.2.4 题型四：关于函数的单调性与极值问题 ····································································· 64 
4.2.5 题型五：函数的凹凸性与拐点问题 ············································································ 64 
4.2.6 题型六：显式不等式的证明问题 ··············································································· 66 
4.2.7 题型七：函数的零点（方程的根）问题 ····································································· 68 
4.2.8 题型八：渐近线问题 ································································································ 68 
4.2.9 题型九：泰勒公式的应用 ························································································· 69 
4.2.10 题型十：应用题 ····································································································· 70 
4.3  深化训练 ························································································································· 71 
4.4  深化训练详解 ················································································································· 73 
4.5  综合提高训练 ················································································································· 80 
第5 章不定积分 ························································································································· 83 
5.1  知识要点 ························································································································· 83 
5.1.1 不定积分的概念与几何意义 ······················································································ 83 
5.1.2 不定积分的性质 ······································································································· 83 
5.1.3 换元积分法 ············································································································· 83 
5.1.4 分部积分法 ············································································································· 85 
5.1.5 有理函数的积分法 ··································································································· 85 
5.1.6 三角函数有理式的积分法 ························································································· 86 
5.1.7 简单无理函数的积分法 ····························································································· 86 
5.1.8 常用积分公式表 ······································································································· 86 
5.2  典型例题分析 ················································································································· 87 
5.2.1 题型一：不定积分的定义与性质问题 ········································································· 87 
5.2.2 题型二：求解分段函数的不定积分 ············································································ 88 
5.2.3 题型三：直接积分法求解不定积分 ············································································ 89 
5.2.4 题型四：利用换元积分法求解不定积分 ····································································· 90 
5.2.5 题型五：利用分部积分法求解不定积分 ····································································· 91 
5.2.6 题型六：求解三角函数有理式的不定积分 ·································································· 93 
5.2.7 题型七：求解有理函数的不定积分 ············································································ 95 
5.2.8 题型八：求解简单无理函数的不定积分 ····································································· 97 
5.3  深化训练 ························································································································· 98 
5.4  深化训练详解 ················································································································· 99 
5.5  综合提高训练 ··············································································································· 107 
第6 章定积分 ···························································································································· 115 
6.1  知识要点 ························································································································ 115 
6.1.1定积分的定义 ········································································································· 115 ·VII·
6.1.2 定积分的几何意义与物理意义 ·················································································· 115 
6.1.3 定积分的基本性质 ·································································································· 116 
6.1.4 变上限积分函数 ······································································································ 117 
6.1.5 定积分的计算 ········································································································· 117 
6.1.6 广义积分 ················································································································ 117 
6.1.7 定积分的几何应用 ·································································································· 118 
6.1.8 定积分的经济应用 ·································································································· 119 
6.1.9 几个重要的结论 ····································································································· 120 
6.2  典型例题分析 ··············································································································· 120 
6.2.1 题型一：利用几何意义计算定积分 ·········································································· 120 
6.2.2 题型二：有关定积分的性质问题 ············································································· 121 
6.2.3 题型三：利用定积分的定义求解极限 ······································································· 122 
6.2.4 题型四：变限积分问题 ··························································································· 123 
6.2.5 题型五：利用换元法、分部积分法求解定积分 ························································· 126 
6.2.6 题型六：利用奇偶性、周期性计算定积分 ································································ 127 
6.2.7 题型七：分段函数的积分问题 ················································································· 128 
6.2.8 题型八：某些不易求出原函数的积分计算问题 ························································· 129 
6.2.9 题型九：定积分相关的证明问题 ············································································· 130 
6.2.10 题型十：广义积分问题 ························································································· 131 
6.2.11 题型十一：积分的应用问题··················································································· 132 
6.3  深化训练 ······················································································································· 134 
6.4  深化训练详解 ··············································································································· 137 
6.5  综合提高训练 ··············································································································· 147 
第7 章多元函数微积分学 ······································································································· 152 
7.1  知识要点 ······················································································································· 152 
7.1.1 二元函数的定义 ····································································································· 152 
7.1.2 二元函数的极限与连续 ··························································································· 152 
7.1.3 偏导数 ·················································································································· 152 
7.1.4 全微分 ·················································································································· 153 
7.1.5 高阶偏导数 ··········································································································· 154 
7.1.6 多元函数的求导法则 ······························································································ 155 
7.1.7 二元函数的极值 ····································································································· 155 
7.1.8 二重积分的概念与性质 ··························································································· 156 
7.1.9 利用直角坐标系计算二重积分 ················································································· 158 
7.1.10 利用极坐标计算二重积分 ······················································································ 158 
7.1.11 利用对称性求解二重积分 ······················································································ 159 
7.2  典型例题分析 ··············································································································· 160 
7.2.1 题型一：多元函数的概念问题 ················································································· 160 
7.2.2 题型二：求解多元函数的极限 ················································································· 161 
7.2.3 题型三：求解多元函数的偏导数 ············································································· 162 
7.2.4 题型四：计算多元函数的全微分 ············································································· 164 
7.2.5 题型五：抽象复合函数的偏导数的求解 ··································································· 166 
7.2.6 题型六：隐函数的求导问题 ···················································································· 167 
7.2.7 题型七：求多元函数的极值和最值 ·········································································· 169 
7.2.8 题型八：二重积分的计算 ······················································································· 171 
7.2.9 题型九：实际应用题 ······························································································ 175 
7.3  深化训练 ······················································································································· 176 
7.4  深化训练详解 ··············································································································· 179 
7.5  综合提高训练 ··············································································································· 190 
第8 章无穷级数 ······················································································································· 196 
8.1  知识要点 ······················································································································· 196 
8.1.1 无穷级数的概念 ····································································································· 196 
8.1.2 无穷级数的性质 ····································································································· 196 
8.1.3 常见级数的敛散性 ································································································· 197 
8.1.4 正项级数敛散性的判别法 ······················································································· 197 
8.1.5 任意项级数的敛散性 ······························································································ 198 
8.1.6 函数项级数的概念 ································································································· 198 
8.1.7 幂级数的概念 ········································································································ 199 
8.1.8 幂级数的和函数的性质 ··························································································· 199 
8.1.9 函数的幂级数展开 ································································································· 200 
8.1.10 常见的麦克劳林公式 ···························································································· 200 
8.2  典型例题分析 ··············································································································· 200 
8.2.1 题型一：利用定义与性质判断级数的敛散性 ····························································· 200 
8.2.2 题型二：判断正项级数的敛散性 ············································································· 202 
8.2.3 题型三：判断任意项级数的敛散性 ·········································································· 203 
8.2.4 题型四：函数项级数收敛域的求解 ·········································································· 205 
8.2.5 题型五：讨论幂级数的收敛半径及收敛域 ································································ 205 
8.2.6 题型六：求幂级数的和函数 ···················································································· 208 
8.2.7 题型七：函数展开成幂级数问题 ·············································································· 211 
8.2.8 题型八：无穷级数的应用问题 ················································································· 213 
8.3  深化训练 ······················································································································· 213 
8.4  深化训练详解 ··············································································································· 215 
8.5  综合提高训练 ··············································································································· 224 
第9 章常微分方程 ··················································································································· 229 
9.1  知识要点 ······················································································································· 229 
9.1.1 微分方程的概念 ····································································································· 229 
9.1.2 一阶微分方程及解法 ······························································································ 229 
9.1.3二阶线性微分方程 ································································································· 231 ·IX·
9.2  典型例题分析 ··············································································································· 232 
9.2.1 题型一：分离变量法求解微分方程 ·········································································· 232 
9.2.2 题型二：求解齐次微分方程 ···················································································· 233 
9.2.3 题型三：求解一阶线性微分方程 ············································································· 235 
9.2.4 题型四：求解伯努利方程 ······················································································· 236 
9.2.5 题型五：求解二阶线性微分方程 ············································································· 237 
9.2.6 题型六：应用题 ····································································································· 238 
9.3  深化训练 ······················································································································· 240 
9.4  深化训练详解 ··············································································································· 242 
9.5  综合提高训练 ··············································································································· 247 
2013 年考研数学三高等数学考题 ······························································································ 251 
2014 年考研数学三高等数学考题 ······························································································ 256 
2015 年考研数学三高等数学考题 ······························································································ 262 
2016 年考研数学三高等数学考题 ······························································································ 268