本书以全国硕士研究生招生考试数学考试大纲为依据，精讲了高等数学中的重要知识点，同时配备了相应的例题和习题。全书共11 章，分上、下两册，每章都由知识点提要、例题与方法、习题三部分组成。 其中，知识点提要部分精讲了每章的重要知识点；例题与方法、习题部分包含历年考研真题和相似类型的练习题，便于学生练习和巩固。

前言 高等数学是本科高校所有理工科大一新生入学后都要学习的一门公共基础课，其涉及的内容非常多，难度大，一直以来都令大一学生感到“头疼”. 同时，数学也是全国硕士研究生招生考试中十分重要的一门课程，所以高等数学是学生在大学数学学习中的重中之重. 本书是针对非数学专业本科生学习高等数学的辅导资料，旨在帮助学生能更好地理解和掌握高等数学的定义、定理、公式、解题方法. 本书共分11章，分上、下两册，每章都由知识点提要、例题与方法、习题三部分构成. 其中，知识点提要部分精讲了高等数学中的重要知识点，帮助学生理解基本概念，掌握基本理论，熟悉基本公式；例题与方法部分详细讲解了解题的方法和技巧，帮助学生梳理解题思路，澄清困惑；习题部分由历年考研真题和相似类型的练习题构成，使学生感受考研真题的难易，体会考研数学的命题特点，从而提高学生学习高等数学的能力和参加全国硕士研究生招生考试的应试能力. 本书作者都是常年从事高等数学教学工作的一线教师，对高等数学中知识点的重点、难点的把握及易错点的分析有着丰富的经验，在选择例题和习题时特别注重题目的针对性. 本书特色如下： ? 按照高等数学的教学进度安排章节内容，而且内容全面、精炼，知识点总结细致，典型例题讲解透彻，易错点分析清晰. 本书是对主教材的一个很好补充，适合大一、大二学生初学和考研学生进行系统复习时使用. ? 习题难度由浅入深、循序渐进，包含历年考研真题和一些更难的题目，使学生在提前感受考研的同时，还能进行考前强化训练，以开阔眼界. ? 书后配有习题解析，学生查阅十分方便. 本书上册共7章，内容包括函数、极限与连续、导数与微分、中值定理及导数应用、不定积分、定积分及反常积分、微分方程；下册共4章，内容包括空间解析几何、多元函数微分学、多元函数积分学、无穷级数.

第1 章 函数································································································································1 1.1 知识点提要·····················································································································1 1.1.1 函数的概念·············································································································1 1.1.2 具有特殊性质的函数································································································1 1.2 例题与方法·····················································································································2 1.2.1 函数定义域的确定及函数关系符号的使用··································································2 1.2.2 函数性质的鉴别······································································································3 1.3 习题································································································································4 第2 章 极限与连续·····················································································································8 2.1 知识点提要·····················································································································8 2.1.1 极限、连续、间断点的概念及其等价概念··································································8 2.1.2 无穷小····················································································································9 2.1.3 极限和连续的性质···································································································9 2.1.4 闭区间上连续函数的三个原理···················································································9 2.2 例题与方法·····················································································································9 2.2.1 极限的计算·············································································································9 2.2.2 函数连续性与间断点的鉴别····················································································19 2.2.3 闭区间上连续函数的三个原理的应用·······································································20 2.3 习题······························································································································21 第3 章 导数与微分···················································································································28 3.1 知识点提要···················································································································28 3.1.1 导数与微分的概念、关系及实际意义·······································································28 3.1.2 导数公式··············································································································29 3.2 例题与方法···················································································································30 3.2.1 导数与微分的计算·································································································30 3.2.2 导数意义的应用····································································································37 3.2.3 微分与近似计算····································································································39 3.3 习题······························································································································40 第4 章 中值定理及导数应用···································································································45 4.1 知识点提要···················································································································45 4.1.1 微分中值定理及推论······························································································45 4.1.2 应用定理··············································································································46 4.1.3 应用公式··············································································································47 4.2 例题与方法···················································································································48 4.2.1 函数性质··············································································································48 4.2.2 利用中值定理求极限和高阶导数·············································································52 4.2.3 方程根的存在性及个数···························································································53 4.2.4 不等式的证明········································································································54 4.2.5 微分中值定理的证明······························································································55 4.2.6 恒等式的证明········································································································59 4.2.7 中值定理的中间值极限问题····················································································59 4.3 习题······························································································································60 第5 章 不定积分······················································································································66 5.1 知识点提要···················································································································66 5.1.1 原函数与不定积分的概念·······················································································66 5.1.2 不定积分的性质····································································································66 5.1.3 不定积分基本公式·································································································66 5.1.4 原函数的存在性····································································································67 5.1.5 不定积分的计算····································································································67 5.2 例题与方法···················································································································68 5.2.1 第一换元积分法（凑微分法）·················································································68 5.2.2 分部积分法···········································································································70 5.2.3 第二换元积分法····································································································72 5.2.4 有理真分式的积分·································································································74 5.2.5 三角有理式的积分·································································································75 5.2.6 无理函数的积分····································································································75 5.3 习题······························································································································76 第6 章 定积分及反常积分·······································································································79 6.1 知识点提要···················································································································79 6.1.1 定积分的定义和几何意义·······················································································79 6.1.2 定积分的性质········································································································80 6.1.3 定积分存在的条件·································································································81 6.1.4 微积分基本定理····································································································81 6.1.5 定积分的计算········································································································82 6.1.6 反常积分（或广义积分） ·······················································································83 6.1.7 定积分的应用········································································································84 6.2 例题与方法···················································································································86 6.2.1 定积分的概念、性质及几何意义·············································································86 6.2.2 定积分的计算········································································································88 6.2.3 变限积分函数及其应用···························································································93 6.2.4 定积分的证明········································································································96 6.2.5 反常积分··············································································································98 6.2.6 定积分的应用········································································································99 6.3 习题····························································································································104 第7 章 微分方程····················································································································109 7.1 知识点提要·················································································································109 7.1.1 微分方程的基本概念····························································································109 7.1.2 一阶微分方程······································································································109 7.1.3 高阶微分方程的几种特殊类型··············································································.112 7.1.4 n 阶线性微分方程·······························································································.112 7.2 例题与方法················································································································.114 7.2.1 一阶微分方程·····································································································.114 7.2.2 高阶微分方程的几种特殊类型···············································································121 7.2.3 n 阶线性微分方程································································································123 7.2.4 微分方程的应用··································································································132 7.3 习题····························································································································137 习题解析····································································································································142 第1 章 函数习题解析······································································································142 第2 章 极限与连续习题解析···························································································145 第3 章 导数与微分习题解析···························································································158 第4 章 中值定理及导数应用习题解析···········································································164 第5 章 不定积分习题解析·······························································································183 第6 章 定积分及反常积分习题解析···············································································190 第7 章 微分方程习题解析·······························································································204