高等数学考点分析与题解（广东版）

定　　价：71.8 元

• 作者：官金兰
• 出版时间：2025/11/1
• ISBN：9787121516542
• 出 版 社：电子工业出版社

• 中图法分类：O13-44 
• 页码：212
• 版次：01
• 开本：16开

内容简介

本选题属于专升本考试用书系列（广东版）图书之一，适合广东省高职院校与本科合作的高本协同班的学生以及广东省专升本考试学生使用，覆盖范围较广。本选题包含高等数学六章内容，分别是第一章函数、极限与连续，第二章一元函数微分学，第三章一元函数积分学，第四章常微分方程，第五章多元函数微积分，第六章数项级数。每个章节内容通俗易懂，针对广东省专升本高等数学考试大纲设计例题、预测题和章节检测题，并在题目详解中融合AI大模型技术，提供基于DeepSeek的详细解答，帮助学生提高复习与应试的效率。

作者介绍

官金兰，女，华南农业大学生物数学专业研究生毕业，广东农工商职业技术学院数学教研室主任，主持多项省部级教学改革研究课题，指导学生获得多项全国数学建模一等奖，具有丰富的实践和教学经验。

目 录

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