本书可作为同济大学数学系编写的《高等数学》（上册）的同步考研辅导教材，专为大一学生量身打造，首创国内高等数学“课程学习+考研启蒙”双轨并行的辅导体系。全书以课程标准为根基、考研要求为导向，通过五大模块设计——知识结构、知识点讲解、考研大纲要求、双轨习题（典型习题与考研典型例题）解析、考研自测习题，系统构建“基础巩固—思维进阶—考研衔接”的立体化学习路径。书中精选考研典型例题及自测习题，科学划分难度梯度，配套浙江省线上线下混合式一流课程视频资源（扫码即学），助力学生同步夯实课程基础，前瞻性培养考研实战能力。

王玲娜，女，副教授，是一位在教育领域尤其是高等数学教学中卓有成就的杰出教育者。自其学术生涯以来，王玲娜副教授始终致力于教学创新与质量提升，不断探索适应新时代需求的教学模式。2019年，她成功主持了校级教学改革项目《翻转课堂在高等数学中的应用》，该项目有效推动了数学教学方式的革新，增强了学生的学习兴趣与自主性。次年，她再接再厉，主持了校级课堂教学改革项目《多元微积分试题库建设》，为学生提供了更加丰富、系统的学习资源。

第一章 函数与极限 ·······················································································.1
第一节 映射与函数··················································································.1
一、考研典型例题解析··································································.2
二、考研自测习题········································································.2
第二节 数列的极限··················································································.3
一、典型习题解析········································································.4
二、考研典型例题解析··································································.4
三、考研自测习题········································································.5
第三节 函数的极限··················································································.7
一、典型习题解析········································································.7
二、考研典型例题解析··································································.8
三、考研自测习题········································································.9
第四节 无穷小与无穷大 ··········································································.10
一、典型习题解析·······································································.11
二、考研典型例题解析·································································.11
三、考研自测习题·······································································.13
第五节 极限运算法则··············································································.13
一、典型习题解析·······································································.14
二、考研典型例题解析·································································.14
三、考研自测习题·······································································.16
第六节 极限存在准则及两个重要极限 ························································.17
一、典型习题解析·······································································.18
二、考研典型例题解析·································································.18
三、考研自测习题·······································································.21
第七节 无穷小的比较··············································································.23
一、典型习题解析·······································································.23
二、考研典型例题解析·································································.25
三、考研自测习题·······································································.28
第八节 函数的连续性与间断点 ·································································.30
一、典型习题解析·······································································.31
二、考研典型例题解析·································································.31
三、考研自测习题·······································································.33
第九节 连续函数的运算与初等函数的连续性 ···············································.35
一、典型习题解析·······································································.35
高等数学习题解答及考研辅导（上册） VI
二、考研典型例题解析·································································.36
三、考研自测习题·······································································.37
第十节 闭区间上连续函数的性质 ······························································.38
一、典型习题解析·······································································.38
二、考研典型例题解析·································································.39
三、考研自测习题·······································································.40
第二章 导数与微分 ······················································································.42
第一节 导数概念····················································································.42
一、典型习题解析·······································································.42
二、考研典型例题解析·································································.45
三、考研自测习题·······································································.49
第二节 函数的求导法则 ··········································································.51
一、典型习题解析·······································································.52
二、考研典型例题解析·································································.53
三、考研自测习题·······································································.55
第三节 高阶导数····················································································.56
一、典型习题解析·······································································.56
二、考研典型例题解析·································································.57
三、考研自测习题·······································································.59
第四节 隐函数及参数方程所确定的函数的导数相关变化率 ·····························.60
一、典型习题解析·······································································.60
二、考研典型例题解析·································································.62
三、考研自测习题·······································································.65
第五节 函数的微分·················································································.67
一、考研典型例题解析·································································.67
二、考研自测习题·······································································.68
第三章 微分中值定理与导数应用····································································.70
第一节 微分中值定理··············································································.70
一、典型习题解析·······································································.70
二、考研典型例题解析·································································.71
三、考研自测习题·······································································.74
第二节 洛必达法则·················································································.77
一、典型习题解析·······································································.77
二、考研典型例题解析·································································.79
三、考研自测习题·······································································.81
第三节 泰勒公式····················································································.83
一、典型习题解析·······································································.83
二、考研典型例题解析·································································.84
目 录 VII
三、考研自测习题·······································································.87
第四节 函数的单调性与曲线的凹凸性 ························································.89
一、典型习题解析·······································································.90
二、考研典型例题解析·································································.90
三、考研自测习题·······································································.94
第五节 函数的极值与最大值最小值 ···························································.96
一、典型习题解析·······································································.97
二、考研典型例题解析·································································.97
三、考研自测习题·····································································.101
第四章 不定积分 ·······················································································.103
第一节 不定积分的概念与性质 ·······························································.103
一、典型习题解析·····································································.103
二、考研典型例题解析·······························································.104
三、考研自测习题·····································································.105
第二节 换元积分法···············································································.107
一、典型习题解析·····································································.107
二、考研典型例题解析·································································110
三、考研自测习题·······································································112
第三节 分部积分法·················································································114
一、典型习题解析·······································································114
二、考研典型例题解析·································································115
三、考研自测习题·······································································117
第四节 有理函数的积分 ········································································.120
一、典型习题解析·····································································.120
二、考研典型例题解析·······························································.122
三、考研自测习题·····································································.125
第五章 定积分 ··························································································.128
第一节 定积分的概念和性质 ··································································.128
一、考研典型例题解析·······························································.128
二、考研自测习题·····································································.130
第二节 微积分基本公式 ········································································.132
一、典型习题解析·····································································.132
二、考研典型例题解析·······························································.133
三、考研自测习题·····································································.136
第三节 定积分换元法和分部积分法 ·························································.137
一、典型习题解析·····································································.138
二、考研典型例题解析·······························································.138
三、考研自测习题·····································································.141
高等数学习题解答及考研辅导（上册） VIII
第四节 反常积分··················································································.143
一、典型习题解析·····································································.144
二、考研典型例题解析·······························································.144
三、考研自测习题·····································································.147
第六章 定积分的应用 ·················································································.150
第一节 定积分在几何学上的应用 ····························································.150
一、典型习题解析·····································································.150
二、考研典型例题解析·······························································.152
三、考研自测习题·····································································.154
第二节 定积分在物理学上的应用 ····························································.155
一、考研典型例题解析·······························································.156
二、考研自测习题·····································································.158
第七章 微分方程 ·······················································································.159
第一节 微分方程的基本概念 ··································································.159
一、考研典型例题解析·······························································.159
二、考研自测习题·····································································.159
第二节 可分离变量的微分方程 ·······························································.160
一、考研典型例题解析·······························································.160
二、考研自测习题·····································································.161
第三节 齐次方程··················································································.162
一、考研典型例题解析·······························································.162
二、考研自测习题·····································································.163
第四节 一阶线性微分方程 ·····································································.164
一、考研典型例题解析·······························································.164
二、考研自测习题·····································································.165
第五节 可降阶的高阶微分方程（仅数学一、数学二要求） ···························.166
一、考研典型例题解析·······························································.166
二、考研自测习题·····································································.167
第六节 高阶线性微分方程 ·····································································.168
一、考研典型例题解析·······························································.168
二、考研自测习题·····································································.169
第七节 常系数齐次线性微分方程 ····························································.169
一、考研典型例题解析·······························································.170
二、考研自测习题·····································································.170
第八节 常系数非齐次线性微分方程 ·························································.171
一、考研典型例题解析·······························································.171
二、考研自测习题·····································································.172