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书名:高等数学习题与解析(上册)
定价:52.0
ISBN:9787121426438
作者:无
版次:第1版
出版时间:2022-02
内容提要:
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。
作者简介:
任雪昆,哈尔滨工业大学数学学院副教授,硕导,博士毕业于哈尔滨工业大学基础数学专业,研究领域为模糊分析学,获得多项国家自然科学基金项目。
目录:
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 **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
定价:52.0
ISBN:9787121426438
作者:无
版次:第1版
出版时间:2022-02
内容提要:
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。
作者简介:
任雪昆,哈尔滨工业大学数学学院副教授,硕导,博士毕业于哈尔滨工业大学基础数学专业,研究领域为模糊分析学,获得多项国家自然科学基金项目。
目录:
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 **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
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