考研数学之高等数学讲义考点知识点,概念定理总结)

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1、考研数学之高等数学讲义考点知识点,概念定理总结)总结)高等数学讲义目录第一章第二章第三章第四章第五章第六章第七章第八章函数、极限、连续·······························································1一元函数微分学···································································24一元函数积分学······························································

2、·····49常微分方程··········································································70向量代数与空间解析几何···················································82多元函数微分学···································································92多元函数积分学·········································

3、··························107无穷级数（数一和数三）···················································129第一章函数、极限、连续1.1函数(甲)内容要点一、函数的概念1．函数的定义2．分段函数二、基本初等函数的概念、性质和图象三、复合函数与初等函数四、考研数学中常出现的非初等函数1．用极限表示的函数(1)y?limfn(x)n??3．反函数4．隐函数(2)y?limf(t,x)t?x2．用变上、下限积分表示的函数(1)y?(2)y?则?xaf(t)dt其中f(t)连

4、续，则dy?f(x)dx???2(x)1(x)f(t)dt其中?1(x),?2(x)可导，f(t)连续，dy?(x)?f[?1(x)]?1?(x)?f[?2(x)]?2dx五、函数的几种性质1．有界性：设函数y?f(x)在X内有定义，若存在正数M，使x?X都有f(x)?M，则称f(x)在X上是有界的。2．奇偶性：设区间X关于原点对称，若对x?X，都有f(?x)??f(x)，则称f(x)在X上是奇函数。若对x?X，都有f(?x)?f(x)，则称f(x)在X上是偶函数，奇函数的图象关于原点对称；偶函数图象关于y轴对称。3．单调性：设f(x)在X上有定

5、义，若对任意x1?X,x2?X，x1?x2都有f(x1)?f(x2)[f(x1)?f(x2)]则称f(x)在X上是单调增加的[单调减少的]；若对任意x1?X，x2?X,x1?x2都有f(x1)?f(x2)[f(x1)?f(x2)]，则称f(x)在X上是单调不减[单调不增]（注意：有些书上把这里单调增加称为严格单调增加；把这里单调不减称为单调增加。）4．周期性：设f(x)在X上有定义，如果存在常数T?0，使得任意x?X，x?T?X，都有f(x?T)?f(x)，则称f(x)是周期函数，称T为f(x)的周期。由此可见，周期函数有无穷多个周期，一般我们把

6、其中最小正周期称为周期。1.2极限(甲)内容要点一、极限的概念与基本性质1．极限的概念(1)数列的极限limxn?An??(2)函数的极限limf(x)?A；limf(x)?A；limf(x)?Ax???x???x??f(x)?A；limf(x)?Alimf(x)?A；lim??x?x0x?x0x?x02．极限的基本性质定理1（极限的唯一性）设limf(x)?A，limf(x)?B，则A=B定理2（极限的不等式性质）设limf(x)?A，limg(x)?B若x变化一定以后，总有f(x)?g(x)，则A?B反之，A?B，则x变化一定以后，有f(x)

7、?g(x)（注：当g(x)?0，B?0情形也称为极限的保号性）定理3（极限的局部有界性）设limf(x)?A则当x变化一定以后，f(x)是有界的。定理4设limf(x)?A，limg(x)?B则（1）lim[f(x)?g(x)]?A?B（2）lim[f(x)?g(x)]?A?B（3）lim[f(x)?g(x)]?A?B（4）limf(x)A?(B?0)g(x)B（5）lim[f(x)]g(x)?AB(A?0)二、无穷小lim1．无穷小定义：若limf(x)?0，则称f(x)为无穷小（注：无穷小与x的变化过程有关，当x??时1?0，x??x11为无

8、穷小，而x?x0或其它时，不是无穷小）xx2．无穷大定义：任给M>0，当x变化一定以后，总有f(x)?M，则称f(x)为无穷大，记