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sinx~x，tanx~x，xarcsin~x，xarccos~x，1-cosx~2/2^x，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11sinlim0xxx公式2exxx/10)1(lim3．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2)()!12()1(...!5!3sin)(!...!3!2112125332nnnnnxxonxxxxxxonxxxxe)(!2)1(...!4!21cos2242nnnxonxxxx)()1(...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)(lim0xfxx，0)(lim0xFxx；（2）)(xf与)(xF在0x的某一去心邻域内可导，且0)(xF；（3）)()(lim0xFxfxx存在（或为无穷大），则这个定理说明：当)()(lim0xFxfxx存在时，)()(lim0xFxfxx也存在且等于)()(lim0xFxfxx；当)()(lim0xFxfxx为无穷大时，)()(lim0xFxfxx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：（1）)(lim0xfxx，)(lim0xFxx；（2）)(xf与)(xF在0x的某一去心邻域内可导，且0)(xF；（3）)()(lim0xFxfxx存在（或为无穷大），则注：上述关于0xx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；)()(lim)()(lim00xFxfxFxfxxxx)()(lim)()(lim00xFxfxFxfxxxx3236)()()(lim0'000xfxxfxxfx(7.101)()(1limdxxfnkfnnkn10xy=f(x)f(x)0x0xf(x)2[a,b]f(x)1f(x)[a,b]f(x)[a,b]2f(x)[a,b]Mm3f(x)[a,b]MmmMc[a,b]f()=cf(x)[a,b]f(a)f(b)(a,b)f()=04151.2.x=(t)y=)(ty=y(x))('),('tt)('t0)(')('ttdxdy3.y=f(x)x=g(y)f(x)0)0)('())(('1)('1)('xfygfxfyg4.y=y(x)F(x,y)=0yF(x,y)=0xyyy5.xxysinyy=[f(x)]g(x),y=)(ln)(xfxge6.nn2y,y,,,,y(n)n1xnxeyey)(,2nxnxaayay)(ln,)(3xysin,)2sin()(nxyn4xycos,)2cos()(nxyn(5)xyln,nnnxny)!1()1(1)(6.f(x)1[a,b]2(a,b)3f(a)=f(b)(a,b)f()=0f(x)1[a,b]2(a,b)(a,b))(')()(fabafbf1f(x)(a,b)f(x)0f(x)(a,b)2f(x),g(x)(a,b)f(x)g(x)(a,b)f(x)=g(x)+cc.f(x)g(x)1[a,b]2(a,b)g(x)0(a,b))(')(')()()()(gfagbgafbf)(bag(x)=x.1nf(x)0xn2nf(x)0x(a,b)n+1[a,b]nx[a,b],0(x)n0x=0n7(8)8f(x)0x0xf(x)0)('0xfx0)('0xf0x)(xf1.)(xf0x0)(0xf0xx,0)(xf0xx0)(xf0x0xx0)(xf0xx0)(xf0x0x)(xf0x.2.)(xf0x0)(0xf0)(0xf0)(0xf0x0)(0xf0x.3..1f(x)I12x,xf(x)Iy=f(x)y=f(x)y=f(x)y=f(x)23f(x)(a,b))(''xf(a,b)x)(''xf>0y=f(x)(a,b)9(a,b)x)(''xf<0y=f(x)(a,b)y=f(x))(''xfkxxx,...2,110CaxxaxdxCshxchxdxCchxshxdxCaadxaCxctgxdxxCxdxtgxxCctgxxdxxdxCtgxxdxxdxxx)ln(lncsccscsecseccscsinseccos22222222CaxxadxCxaxaaxadxCaxaxaaxdxCaxarctgaxadxCctgxxxdxCtgxxxdxCxctgxdxCxtgxdxarcsinln21ln211csclncscseclnsecsinlncosln22222222CaxaxaxdxxaCaxxaaxxdxaxCaxxaaxxdxaxInnxdxxdxInnnnarcsin22ln22)ln(221cossin222222222222222222222020111)()(d)()]([xuduufxxxf2)(1d)()]([)(xttttfdxxfvduuvudv)(xu)('xv)(xu)(xuxdxexarcsinxarcsin)(xu)()()(xQxPxf)()(xQxP21)()(,1)()(xxPxfxxPxf))(()()(bxaxxPxfbaxxPxf2)()()(12.121niiibaxfdxxf10)(lim)(210(3)133.xadttfx)()()()(xfx)()]([)()]([)()()(xxfxxfdttfdxdxxNL)(xF)(xf)()()(aFbFdxxfba4.14151.badxxfxfA)]()([122.dA)]()([2121221.a)xbxaxxfy,,),(xbaxdxxfV)(2b)xbxaxxfy,,),(y16baydxxxfV)(2.1.badxxfs2)(12.dttts22)()(ds22)()(171...2....(1).dxxfdyyg)()(dxxfdyyg)()((2).)(xydxdyxyudxduxudxdy)(yxdydxyxvdydvyvdydx(3).)()(xQyxPdxdyCdxexQeydxxPdxxP)()()((4).1)()(xfynn2),(yxfyypypy3),(yyfyxpydydppy121,yy2211yCyC221,yy2211yCyC3*2211yyCyCy21,yy18*y.0qyypy02qprr21,rrxrxreCeCy2121221prrxrexCCy1)(21ir,21)sincos(21xCxCeyx)(xfqyypy1)()(xPexfmx)(*xQexymxk,,,k2102xxPxxPexfnlxsin)(cos)()(xxRxxRexymmxksin)(cos)()2()1(*},max{nlmiik,1,0
