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高等数学考点分析与题解（广东版）

作译者：官金兰,吴锡明,付德刚，蒋芬

出版时间：2025-11 千字数：608 版次：01-01 页数：212

开本：16开装帧： I S B N ：9787121516542

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纸质书定价：￥71.8

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内容简介

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作者简介

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音视频专区

第 1 章 函数、极限及连续················································································1
1.1 函数·······························································································.1
1.1.1 函数的定义 ············································································.1
1.1.2 函数的性质 ············································································.3
1.1.3 复合函数 ···············································································.4
1.1.4 基本初等函数 ·········································································.5
1.1.5 初等函数 ···············································································.5
1.2 极限的概念······················································································.5
1.2.1 数列的极限 ············································································.5
1.2.2 函数的极限 ············································································.7
1.3 极限的计算······················································································.9
1.3.1 极限的四则运算······································································.9
1.3.2 无穷大与无穷小·····································································.12
1.3.3 两个重要极限 ········································································.16
1.4 函数连续的定义···············································································.18
1.4.1 函数的连续性 ········································································.18
1.4.2 函数的间断点 ········································································.20
1.4.3 初等函数的连续性··································································.22
1.4.4 闭区间上连续函数的性质·························································.23
1.5 知识点汇总·····················································································.24
1.6 本章真题归类··················································································.25
1.7 本章检测························································································.28
第 2 章 一元函数微分学················································································.30
2.1 导数的概念·····················································································.31
2.1.1 导数的定义 ···········································································.31
2.1.2 导数的几何意义·····································································.33
2.1.3 可导与连续的关系··································································.34
2.2 导数的计算·····················································································.35
2.2.1 导数的四则运算·····································································.35
2.2.2 复合函数和反函数的导数·························································.37
2.2.3 隐函数的导数 ········································································.38
2.2.4 对数求导法 ···········································································.39
2.2.5 高阶求导 ··············································································.41
2.2.6 参数方程确定的函数的导数······················································.42
2.3 函数的微分·····················································································.44
2.3.1 微分的定义及运算··································································.44
2.3.2 微分在近似计算中的应用·························································.46
2.4 微分中值定理及其应用······································································.47
2.4.1 微分中值定理 ········································································.47
2.4.2 洛必达法则 ···········································································.49
2.5 导数的应用·····················································································.52
2.5.1 函数的单调性 ········································································.52
2.5.2 函数的极值及其求法·······························································.55
2.5.3 函数的最值及其求法·······························································.57
2.5.4 函数图象的凹凸性与渐近线······················································.58
2.5.5 曲线的渐近线 ········································································.59
2.6 知识点汇总·····················································································.62
2.7 本章真题归类··················································································.63
2.8 本章检测························································································.71
第 3 章 一元函数积分学················································································.72
3.1 不定积分基础知识············································································.72
3.1.1 不定积分的概念·····································································.72
3.1.2 不定积分的几何意义·······························································.73
3.1.3 不定积分的性质·····································································.73
3.1.4 基本积分公式 ········································································.74
3.2 不定积分的计算方法·········································································.77
3.2.1 第一类换元法（凑微分法）······················································.77
3.2.2 第二类换元法 ········································································.78
3.2.3 分部积分法 ···········································································.81
3.3 定积分的基本概念············································································.82
3.3.1 定积分的概念 ········································································.82
3.3.2 定积分的几何意义··································································.83
3.3.3 定积分的性质 ········································································.84
3.4 微积分基本定理和定积分的计算··························································.86
3.4.1 变上限积分函数·····································································.86
3.4.2 微积分基本定理·····································································.87
3.4.3 定积分的换元积分法·······························································.89
3.4.4 分部积分法 ···········································································.90
3.4.5 无穷区间上的反常积分····························································.92
3.5 定积分的应用··················································································.93
3.5.1 直角坐标系下平面图形面积的计算·············································.93
3.5.2 旋转体体积的计算方法····························································.95
3.5.3 曲线弧长的计算方法·······························································.96
3.6 知识点汇总·····················································································.98
3.7 本章真题归类··················································································.99
3.8 本章检测······················································································.107
第 4 章 微分方程·························································································109
4.1 微分方程的概念·············································································.109
4.2 一阶微分方程··················································································110
4.2.1 可分离变量的微分方程····························································111
4.2.2 齐次微分方程 ·······································································112
4.2.3 一阶线性微分方程··································································114
4.2.4 可降阶的高阶微分方程···························································117
4.3 二阶常系数线性微分方程···································································119
4.3.1 二阶常系数齐次线性微分方程···················································119
4.3.2 二阶常系数非齐次线性微分方程·············································.121
4.4 知识点汇总···················································································.123
4.5 本章真题归类················································································.124
4.6 本章检测······················································································.127
第 5 章 多元函数微积分················································································128
5.1 多元函数的基本概念·······································································.129
5.2 多元函数的偏导数与全微分······························································.131
5.2.1 多元函数的一阶偏导数··························································.131
5.2.2 多元函数的高阶偏导数··························································.133
5.2.3 多元函数全微分···································································.135
5.2.4 多元复合函数求导································································.136
5.2.5 隐函数的偏导数···································································.139
5.3 二重积分······················································································.141
5.3.1 二重积分的概念和性质··························································.141
5.3.2 二重积分的计算···································································.145
5.4 知识点汇总···················································································.152
5.5 本章真题归类················································································.153
5.6 本章检测······················································································.159
第 6 章 常数项级数······················································································161
6.1 常数项级数的概念与性质·································································.161
6.1.1 数项级数的定义与敛散性·······················································.161
6.1.2 常数项级数的基本性质··························································.165
6.2 正项级数的审敛法··········································································.166
6.2.1 比较审敛法 ·········································································.166
6.2.2 比值审敛法 ·········································································.170
6.2.3 根值审敛法 ·········································································.171
6.3 任意项级数的审敛法·······································································.172
6.3.1 交错级数与莱布尼茨判别法····················································.172
6.3.2 绝对收敛与条件收敛·····························································.173
6.4 知识点汇总···················································································.174
6.5 本章真题归类················································································.175
6.6 本章检测······················································································.180
第 7 章 新增考点预测···················································································182
7.1 曲率····························································································.182
7.1.1 弧微分 ···············································································.182
7.1.2 曲率及其计算公式································································.183
7.1.3 曲率圆与曲率半径································································.184
7.2 泰勒公式······················································································.184
7.2.1 泰勒公式 ············································································.184
7.2.1 麦克劳林公式 ······································································.186
7.3 函数图形的绘制·············································································.187
7.4 定积分的物理学应用·······································································.189
7.4.1 变力沿直线所做的功·····························································.189
7.4.2 水压力 ···············································································.190
7.4.3 引力 ··················································································.191
7.5 定积分的几何应用··········································································.192
7.5.1 平面图形的面积···································································.192
7.5.2 体积 ··················································································.193
7.5 三重积分······················································································.197
7.5.1 三重积分的定义···································································.197
7.5.2 直角坐标系下三重积分的计算·················································.198

广东省普通高等学校专科升本科招生考试（以下简称“广东省专升本考试”）是专科生升入全日制本科院校的主要途径，被誉为“第二次高考”。近年来随着报考人数持续攀升，考试竞争愈发激烈，考生对高质量备考资料的需求也日益增长。
为帮助广大考生高效备考，精准把握考试方向，我们组织广东省内长期从事专升本高等数学教学与命题研究的专家团队，精心编写了《高等数学考点分析与题解（广东版）》，以及配套的《高等数学真题与模拟试卷》。
本书严格依据最新考试大纲，融合 2015～2025 年共 11 年真题的命题规律，结合未来考试趋势，为考生提供全方位、系统化的备考资料。无论是高职院校高本衔接班的本科生、计划参加专升本考试的专科生，还是需要教学参考的高职院校教师，本书都能成为您不可或缺的权威辅导资料。
一、本书核心特色
1. 真题导向，科学归类，精准预测
11 年真题大数据分析：本书对 2015～2025 年广东省专升本高等数学考试真题进行系统整理，按照考点、题型、难度进行科学归类，帮助考生快速掌握高频考点和命题趋势。倒序真题归类（2025～2021 年）：每章末尾特别设置“本章真题归类”版块，从最新年份（2025 年）回溯至 2021 年，让考生直观感受近年命题变化，精准锁定复习重点。
2. 科学编排，打造高效学习体系
本书采用“知识点梳理归纳+真题归类+章节检测”三维备考模式，确保考生从基础到冲刺全程无忧。知识点精讲：每章开篇系统梳理核心概念、公式及定理，并标注考频（高频/中频/低频），让考生复习时有的放矢。
经典例题解析：精选典型真题，分步骤详细讲解，并总结解题技巧和易错点，帮助考生举一反三。
章节强化训练：每节后配备预测题，每章末设置“章节检测”（含选择题、填空题、计算题、证明题等全题型），帮助考生巩固所学知识。
思维导图总结：每章最后提供高清思维导图，直观呈现知识框架，方便考生快速回顾核心考点。
3. 超值附赠资料，助力高效提分
《高等数学公式大全+高频考点》：整理高等数学核心公式、定理及推导过程，方便考生随身携带，随时查阅；归纳近 5 年高频考点，标注命题规律，帮助考生在冲刺阶段快速抓重点，提升复习效率。
5 套历年真题及详解（2021～2025 年）：单独成册，附完整解析，帮助考生熟悉真实考试难度，掌握标准答题格式。
2026 年 5 套权威模拟题：特邀资深命题专家结合最新考纲和命题趋势，精心命制 5 套高仿真模拟题，并附详细解析，帮助考生提前适应考试节奏，查漏补缺。
4. 名师团队编写，权威可靠
本书由广东省多所高职和本科院校的数学教学专家联合编写，编委团队均具有多年专升本辅导经验，部分成员曾参与命题或阅卷工作，对考试趋势和评分标准有深刻理解。
本书由广东农工商职业技术学院官金兰老师、广州体育职业技术学院付德刚老师、广州软件学院蒋芬老师、广东食品药品职业学院吴锡明老师、广东外语艺术职业学院黄玮欣老师、广东培正学院郭慧敏老师、清远工贸职业技术学校薛妙珠老师合作进行编写。其中，黄玮欣老师负责第 1 章和全书的思维导图的编写，吴锡明老师负责第 2 章的编写，蒋芬老师负责第 3 章和第 7 章的编写，付德刚和官金兰老师负责第 4 章到第 6 章和全书常考公式的编写。郭慧敏老师负责全书的章节检测和真题与模拟试卷的编写，薛妙珠老师协助编辑排版和校对。
此外，广东农工商职业技术学院计算机学院的优秀学生团队（24 级计算机学院的林俊豪、罗斌、杨春玲、萧李桐、邢烘嘉、李涛、梁施婷、纪淼珣、李思宜、杨佳涵、梁汶琪、莫镇源、贺晓勤、林雅仪、陈子乔、黄锦俊、吴润怡、张紫彤、白岂丞、吴文怡等同学）参与了公式校对和排版优化工作，确保本书内容准确、易读。
三、适用人群
高职院校高本衔接班学生：系统复习专升本数学考点，提前适应考试难度。
专升本考生：高效备考，精准提分，冲刺理想本科院校。
普通专科生：夯实数学基础，为未来升本或考研做准备。
高职院校教师：可作为教学参考书，或用于专升本辅导班授课的教材。

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