高等数学同济第四版笔记
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第一章 函数与极限.......................................................................................................................1 一、函数...................................................................................................................................1 1.1、注意以前所学习的一些概念 ..................................................................................1 1.2、函数概念 ..................................................................................................................1 1.3、函数的几种特性 ......................................................................................................1 1.4、反函数......................................................................................................................1 1.5、几种特殊的函数 ......................................................................................................1 二、初等函数...........................................................................................................................2 2.1、基本初等函数 ..........................................................................................................2 2.2、双曲函数 ..................................................................................................................2 三、数列的极限.......................................................................................................................3 3.1、数列极限的定义 ......................................................................................................3 3.2、收敛数列的特性 ......................................................................................................3 四、函数的极限.......................................................................................................................4 4.1、定义..........................................................................................................................4 4.2、函数极限的特性 ......................................................................................................5 五、无穷小与无穷大 ...............................................................................................................8 5.1、无穷小的定义 ..........................................................................................................8 5.2、无穷大的定义 ..........................................................................................................8 5.3、相关定理 ..................................................................................................................8 六、极限的运算法则 ...............................................................................................................9 极限的叠代求法 .............................................................................................................10 七、极限存在准则：两个重要极限 .....................................................................................12 7.1、准则I：夹逼准则 ..................................................................................................12 7.2、准则II：单调有界数列必有极限 .........................................................................12 7.3、两个重要极限 ........................................................................................................12 八、无穷小的比较 .................................................................................................................15 8.1、无穷小的阶 ............................................................................................................15 8.2、等阶无穷小的性质 ................................................................................................15 8.3、一些常用的等阶无穷小 ........................................................................................15 九、函数的连续性与间断点 .................................................................................................17 9.1、函数的连续性 ........................................................................................................17 9.2、函数的间断点 ........................................................................................................17 十、连续函数的运算与初等函数的连续性 .........................................................................18 10.1、连续函数的和、积及商的连续性 ......................................................................18 10.2、反函数与复合函数的连续性 ..............................................................................18 10.3、初等函数的连续性 ..............................................................................................20 十一、闭区间上的连续函数的性质 .....................................................................................21 11.1、最大值和最小值定理 ..........................................................................................21 11.2、介值定理 ..............................................................................................................21 11.3、一致连续性 ..........................................................................................................21 第二章 导数与微分.....................................................................................................................23
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第一章 函数与极限.......................................................................................................................1 一、函数...................................................................................................................................1 1.1、注意以前所学习的一些概念 ..................................................................................1 1.2、函数概念 ..................................................................................................................1 1.3、函数的几种特性 ......................................................................................................1 1.4、反函数......................................................................................................................1 1.5、几种特殊的函数 ......................................................................................................1 二、初等函数...........................................................................................................................2 2.1、基本初等函数 ..........................................................................................................2 2.2、双曲函数 ..................................................................................................................2 三、数列的极限.......................................................................................................................3 3.1、数列极限的定义 ......................................................................................................3 3.2、收敛数列的特性 ......................................................................................................3 四、函数的极限.......................................................................................................................4 4.1、定义..........................................................................................................................4 4.2、函数极限的特性 ......................................................................................................5 五、无穷小与无穷大 ...............................................................................................................8 5.1、无穷小的定义 ..........................................................................................................8 5.2、无穷大的定义 ..........................................................................................................8 5.3、相关定理 ..................................................................................................................8 六、极限的运算法则 ...............................................................................................................9 极限的叠代求法 .............................................................................................................10 七、极限存在准则：两个重要极限 .....................................................................................12 7.1、准则I：夹逼准则 ..................................................................................................12 7.2、准则II：单调有界数列必有极限 .........................................................................12 7.3、两个重要极限 ........................................................................................................12 八、无穷小的比较 .................................................................................................................15 8.1、无穷小的阶 ............................................................................................................15 8.2、等阶无穷小的性质 ................................................................................................15 8.3、一些常用的等阶无穷小 ........................................................................................15 九、函数的连续性与间断点 .................................................................................................17 9.1、函数的连续性 ........................................................................................................17 9.2、函数的间断点 ........................................................................................................17 十、连续函数的运算与初等函数的连续性 .........................................................................18 10.1、连续函数的和、积及商的连续性 ......................................................................18 10.2、反函数与复合函数的连续性 ..............................................................................18 10.3、初等函数的连续性 ..............................................................................................20 十一、闭区间上的连续函数的性质 .....................................................................................21 11.1、最大值和最小值定理 ..........................................................................................21 11.2、介值定理 ..............................................................................................................21 11.3、一致连续性 ..........................................................................................................21 第二章 导数与微分.....................................................................................................................23