本书依据全国高校数学基础课程授课的内容组织编写，按微积分（经管类）课程等知识点分专题进行讲授，对所涉及的知识点和考点进行分类整合，精选了典型例题和拓展习题进行讲解或解答，化解难点。本书编写特色在于知识的高度综合性和交叉性，在一定高度上进行数学思想的糅合。知识点可以前后穿插，以训练学生的数学思维能力，锻炼学生的独立思考的能力，提高学生的解题水平。

前 言 为了更好地帮助普通高等院校工科类、经管类本科生学好大学数学，同时为了满足众多考生考研的需要，我们结合多年的考研辅导经验，编写了“普通高等学校工科类、经管类数学深化训练与考研辅导丛书”，该丛书包括微积分、高等数学、线性代数、概率论与数理统计以及大学生数学竞赛等数学课程的训练辅导用书，由首都经济贸易大学的刘强教授担任丛书主编。 本书为微积分分册，内容涵盖了考研“数学三”中关于微积分内容的全部考点。本书编写的主要目的有两个：一是帮助学有余力的在校学生更好地学习“微积分”课程，以开阔学习视野，拓展解题思路；二是满足学生报考研究生的需要，因此本书编写紧扣“数学三”考研大纲，贴近考试实际，做到分门别类、详略得当，帮助考研学生在短时间内迅速掌握各种解题方法和技巧，提升综合分析问题、解决问题的能力，以达到融会贯通、举一反三的学习效果。 全书共分为9 章，每章包括5 个模块，即知识要点、典型例题分析、深化训练、深化训练详解及综合提高训练，具体模块内容为： 一、知识要点：本模块对基本概念、基本理论、基本公式等内容进行系统梳理，方便读者查阅相关内容。 二、典型例题分析：本模块是作者在多年来考研辅导经验的基础上，创新性地构思了大量有代表性的例题，并选编了部分国内外优秀教材、辅导资料的经典题目，汇集了一些有代表性的考研真题，按照知识结构、解题思路、解题方法等对典型例题进行了系统归类，通过专题讲解，详细阐述了相关问题的解题方法与技巧。 三、深化训练：本模块精心选编了部分具有代表性的习题以及历年的考研真题，帮助读者巩固强化所学知识，提升读者学习效果，做到举一反三、融会贯通。 四、深化训练详解：本部分对深化训练给出了详细的解答过程，部分习题给出多种解法，以开拓读者的解题思路，培养读者的分析能力和发散思维。 五、综合提高训练：本部分的例题综合性较强，有较高的难度和较强的灵活性，通过本模块的学习，有效提升读者的综合能力和应变能力。 为了便于读者阅读本书，书中有一定难度的结论、例题和综合练习题等将用“**”标出。另外为了方便读者查阅，本书在考研真题后面加上了标志，例如【2010（3）】表示该题是2010 年硕士研究生入学考试“数学三”考题，【2010（1，3）】表示该题是2010 年“数学一”和“数学三”考题，其余类推。 本丛书在编写过程中，得到了北京工业大学李高荣教授，首都经济贸易大学张宝学教授、马立平教授、吴启富教授，昆明理工大学吴刘仓教授，北京化工大学李志强副教授，以及同事们的大力支持，电子工业出版社的徐颢编辑和高教分社的谭海平社长也为丛书的出版付出了大量努力，在此表示诚挚的感谢。 本书既可以作为普通高等学校工科类、经管类本科生学习“微积分”课程的深化训练用书，也可以作为全国硕士研究生统一入学考试的辅导用书。由于作者水平有限，书中仍可能存在不妥甚至错漏之处，恳请读者和同行们不吝指正。邮件地址为：cuebliuqiang@163.com 。 编 者 2016 年1 月

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

刘强，理学博士，教授，博士生导师，现任首都经济贸易大学统计学院副院长，兼任全国工业统计教学研究会常务理事兼常务副秘书长，北京应用统计学会常务理事，北京大数据协会理事等．先后入选北京市中青年骨干人才，北京市优秀人才和北京市中青年拔尖人才．主讲本科生课程：微积分，线性代数，概率论与数理统计，高等数学，多元统计分析，数学竞赛等；主讲研究生课程：高等数理统计，应用数理统计，数据分析与R语言等；主讲博士生课程：非参与半参数回归等．主要研究方向：经济数据分析，非参数计量经济和复杂数据分析等。__eol__贾尚晖，理学博士，中央财经大学统计与数学学院教授，硕士生导师。主讲本科生课程：高等数学，微积分，概率论与数理统计，线性代数，数学竞赛等；主讲研究生课程：最优化理论，数理经济学等。主要研究方向：计算数学，数量经济学。主持2项国家自然科学基金课题，1项北京市社科基金课题，参与多项国家自然科学基金课题。