2021年高等数学(同济第七版)上册-知识点总结.pdf

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1、高等数学(同济第七版)上册-知识点总结第一章函数与极限一.函数的概念1.两个无穷小的比较设0)(lim,0)(limxgxf且lxgxf)()(lim（1）l=0，称 f(x)是比 g(x)高阶的无穷小，记以 f(x)=0)(xg，称g(x)是比f(x)低阶的无穷小。（2）l 0，称f(x)与g(x)是同阶无穷小。（3）l=1，称f(x)与g(x)是等价无穷小，记以 f(x)g(x)2.常见的等价无穷小当x 0时sin x x，tan x x，xarcsin x，xarccos x，1-cos x 2/2x，xe-1 x，)1ln(x x，1)1(x x二 求极限的方法1两个准则准则 1.单调

2、有界数列极限一定存在准则 2.（夹逼定理）设 g(x)f(x)h(x)若AxhAxg)(lim,)(lim，则Axf)(lim2两个重要公式公式 11sinlim0 xxx公式 2exxx/10)1(lim|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 1 页，共 21 页3用无穷小重要性质和等价无穷小代换4用泰勒公式当 x0时，有以下公式，可当做等价无穷小更深层次)()!12()1(.!5!3sin)(!.!3!2112125332nnnnnxxonxxxxxxonxxxxe)(!2)1(.!4!21cos2242nnnxonxxxx)()1(

3、.32)1ln(132nnnxonxxxxx)(!)1().(1(.!2)1(1)1(2nnxoxnnxxx)(12)1(.53arctan1212153nnnxonxxxxx5洛必达法则定理 1 设函数)(xf、)(xF满足下列条件：（1）0)(lim0 xfxx，0)(lim0 xFxx；（2）)(xf与)(xF在0 x的某一去心邻域内可导，且0)(xF；（3）)()(lim0 xFxfxx存在（或为无穷大），则这个定理说明：当)()(lim0 xFxfxx存在时，)()(lim0 xFxfxx也存在且等于)()(lim0 xFxfxx；当)()(lim0 xFxfxx为无穷大时，)()(

4、lim0 xFxfxx也是无穷大这种在一定条件下通过分子分母分别求导再求极限来确定未定式的极限值的方法称为洛必达（HLospital）法则.型未定式定理 2 设函数)(xf、)(xF满足下列条件：)()(lim)()(lim00 xFxfxFxfxxxx|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 2 页，共 21 页文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9

5、C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:

6、CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 H

7、M6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 Z

8、Z9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编

9、码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9

10、 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9

11、 ZZ9C5S4F7Z5文档编码:CL1K9K3O2B9 HM6N10Q8H6F9 ZZ9C5S4F7Z5（1）)(lim0 xfxx，)(lim0 xFxx；（2）)(xf与)(xF在0 x的某一去心邻域内可导，且0)(xF；（3）)()(lim0 xFxfxx存在（或为无穷大），则注：上述关于0 xx时未定式型的洛必达法则，对于x时未定式型同样适用使用洛必达法则时必须注意以下几点：（1）洛必达法则只能适用于“00”和“”型的未定式，其它的未定式须先化简变形成“00”或“”型才能运用该法则；（2）只要条件具备，可以连续应用洛必达法则；（3）洛必达法则的条件是充分的，但不必要因此，在该法则失效

12、时并不能断定原极限不存在6利用导数定义求极限基本公式)()()(lim0000 xfxxfxxfx(如果存在）7.利用定积分定义求极限基本格式101)()(1limdxxfnkfnnkn（如果存在）三函数的间断点的分类函数的间断点分为两类：（1）第一类间断点设0 x是函数 y=f(x)的间断点。如果f(x)在间断点0 x处的左、右极限都存在，则称0 x是 f(x)的第一类间断点。左右极限存在且相同但不等于该点的函数值为可去间断点。左右极限不存在为跳跃间断点。第一类间断点包括可去间断点和跳跃间断点。（2）第二类间断点第一类间断点以外的其他间断点统称为第二类间断点。常见的第二类间断点有无)()(l

13、im)()(lim00 xFxfxFxfxxxx|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 3 页，共 21 页文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8

14、H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6

15、J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8

16、E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V1

17、0B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC

18、4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10

19、E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6穷间

20、断点和振荡间断点。四闭区间上连续函数的性质在闭区间 a,b 上连续的函数 f(x)，有以下几个基本性质。这些性质以后都要用到。定理1（有界定理）如果函数f(x)在闭区间 a,b 上连续，则 f(x)必在 a,b上有界。定理2（最大值和最小值定理）如果函数f(x)在闭区间 a,b 上连续，则在这个区间上一定存在最大值 M 和最小值 m。定理3（介值定理）如果函数 f(x)在闭区间 a,b 上连续，且其最大值和最小值分别为 M 和m，则对于介于 m 和M 之间的任何实数 c，在 a,b 上至少存在一个，使得f()=c推论：如果函数 f(x)在闭区间 a,b 上连续，且f(a)与f(b)异号，则在(

21、a,b)内至少存在一个点，使得f()=0这个推论也称为零点定理|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 4 页，共 21 页文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文

22、档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF

23、8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L

24、6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM

25、8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V

26、10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 Z

27、C4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D1

28、0E8I6第二章导数与微分一基本概念1可微和可导等价，都可以推出连续，但是连续不能推出可微和可导。二求导公式三常见求导|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 5 页，共 21 页文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J

29、7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E

30、2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10

31、B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4

32、P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E

33、8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编

34、码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H

35、1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I61.复合函数运算法则2.由参数方程确定函数的运算法则设x=(t)，y=)(t确定函数y=y(x)，其中)(),(tt存在，且)(t 0，则)()(ttdxdy3.反函数求导法则设y=f(x)的反函数 x=g(y)，两者皆可导，且 f(x)0则)0)()(1)(1)(xfygfxfyg4.隐函数运算法则设y=y(x)是由方程 F(x,y)=0 所确定，求 y的方法如下：把F(x,y)=0两边的各项对 x求导，把 y 看作中间变量，用复合函数求导公式计算，然后再解出 y 的表达式（允许出现 y 变量）5.对数求导法则（指数类型如xx

36、ysin）先两边取对数，然后再用隐函数求导方法得出导数y。对数求导法主要用于：幂指函数求导数多个函数连乘除或开方求导数（注意定义域。关于幂指函数 y=f(x)g(x)常用的一种方法,y=)(ln)(xfxge这样就可以直接用复合函数运算法则进行。6.求n阶导数（n 2，正整数）先求出 y,y,总结出规律性，然后写出 y(n)，最后用归纳法证明。有一些常用的初等函数的n 阶导数公式（1）xnxeyey)(,（2）nxnxaayay)(ln,)(（3）xysin,)2sin()(nxyn（4）xycos,)2cos()(nxyn|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|

37、欢.|迎.|下.|载.第 6 页，共 21 页文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V

38、10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 Z

39、C4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D1

40、0E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文

41、档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF

42、8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L

43、6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6(5)xyln,nnnxny)!1()1(1)(第三章微分中值定理与导数应用一.罗尔定理设函数 f(x)满足（1）在闭区间

44、 a,b 上连续；（2）在开区间(a,b)内可导；（3）f(a)=f(b)则存在(a,b)，使得 f()=0二 拉格朗日中值定理设函数 f(x)满足（1）在闭区间 a,b 上连续；（2）在开区间(a,b)内可导；则存在(a,b)，使得)()()(fabafbf推论1若f(x)在(a,b)内可导，且 f(x)0，则f(x)在(a,b)内为常数。推论2若f(x),g(x)在(a,b)内皆可导，且 f(x)g(x)，则在(a,b)内f(x)=g(x)+c，其中 c为一个常数。三.柯西中值定理设函数 f(x)和g(x)满足：（1）在闭区间 a,b 上皆连续；（2）在开区间(a,b)内皆可导；且g(x)

45、0则存在(a,b)使得)()()()()()(gfagbgafbf)(ba（注：柯西中值定理为拉格朗日中值定理的推广，特殊情形g(x)=x 时，柯西中值定理就是拉格朗日中值定理。）|精.|品.|可.|编.|辑.|学.|习.|资.|料.*|*|*|*|欢.|迎.|下.|载.第 7 页，共 21 页文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V1

46、0B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC

47、4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10

48、E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档

49、编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8

50、H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6J7 HM8E2F3V10B2 ZC4P5D10E8I6文档编码:CF8H1V8L6