高等数学(上)复习题

第一章函数与极限(－)5x)x）1x2+）））(x)x5x42f（）x11x()2B.-31xln）x2)(-∪)）x32e1x)24e1x）aaxx）2a1当0x）1x1x1x2x1当x0是x）x()在yfxxyf(xh)f(x)h0000是h是hyf(x)yh是hf(x)eg(x)x2x2x0设.f）(x)g(x)f(x)g(x)是是..(x)g(x)是f(x)g(x)与.f()11x1x1；xxxx0x2x1；1；xxxx0x1x2lim1（）x112e-2e221）2nn.0122)（）xxxe-2e-1213x)2xx32230（）x032∞sinxlim(xx1∞2x2x12lim（）x3xx11012xx(1)x1202sinmxlimm)()xx011mm(xh)2x3)。hh0h0sin2xlim（）x(x2)x0∞1mx)e设2（）xx01122sin4xlim（）2x)x0a)bxcx）x.1.be.e.cex(x22））x1xx1设则为3x,x1x1x1x1f(x)）3sinxx））x3x22xx12x3f(x)在()在limf(x)xlimf(x)limf(x)xx11x),x0x设在（）a,x01e1xsin,x0设在（）xk,x0x00xx(x)设f在（）xa1x1x设f(x)在xk，x00x，x01；；；；2ax,x0设在)x),x01012ln(1-2x)设22)在x14|x|x则)。limx011在x在x）00(x)设f在x）0f(x)f(x)xx0xx0()()fxfxxx．xxsinxlim.x2)2＝x.xxx121.xxax32lim()a．设1x1x2x11(x)limf(x)，limf(x)．设f1x0x01exx2,x0则.2xx021则1xax),x0(x)0a．设f在xx2,xasinxx0设在ex1x01sinx,x0x(x),x0x0f在。1xsinx0x51xf(x)()(0)fxf=设。x622x,1xx(x)1a，b．设f在x1x,bln(2x)11xarcsin(y；yx(x22x2(x)22()ln(x2；ff；fxxxxx(ex)x(x)f(x)(x0)；；e1xxxf(x)x(x0)．3xn2lim)nnnx221n21nn4x1limlimx213x)x2x6x2x1xx1xxx22x0x0xxxlimx2x1x2xx0x23x2x1xlim()lim()3x21x31x1x1x2sin2xlim)3xsinxxx0xxe11xxlim.lim()xsinx1xx2x01.limxxsin).limx3lnx.2xx0xsinx1xlim。)]xexxxcosxxx0x1111()lim[]xe1x)xxx0x032xxxx3lim.lim.xx1x0x6xbxc2lim3b、c1xx1xaxb2lim3。1xx11xx,x0f(x)xx2x0第二章导数与微分:点xε）0x-,xεxε]0000εε）xεxε）00f(xh)f(x)yf(x)xx在f(x)22.且，则00h00h0（）112-22h1f(x)xlim，fx）,则,且fx2hfx）400h000()242f(xh)f(x)yf(x)xx在f(x)24.且，则00h00h0（）11-222f2h)f1f'f(x)x1()h2h011121244f(x)f(xh)若lim()hh0f(x)且flim()00xxxx00f)00711设在xxv(x)=（）v(x)v(x)11v(x)-v(x)2v(x)2v(x)2v(x)2设（）a1111xxxaxa设yarcsinx2则（）112x2xdxdxdx1x41x41x41x4设=（）2x2y()xxxxx设则=（）x2x-12x设y的ny=（）(n)ππ)·)22π·)2e,y(0)设y.(则n()（）x(n1ny=（）1xx22secxx2tdydx设（）y4t21xsintdy,则设（）ycos2tdx11.2t4t8设则()dx设则dy（）x1y1xyxyx1y1x在）(0)f(x)2x1xf（）12dy设则（）22dx4y3x4y3x4y--4y3x设则（）22x2222eax,x0(x)设f,,(bxx02b1ab1ab0ab1axx）000处1在xyx1,0xx(x)0f在x()x0,0xMM）241111（,））（,））2442x2xyMM）））））91xx1x）0020011(,ln)22(-x(3-372,）224）f(x)(x)x0在f(0)0则设f,且.xx0在limf(x).x0f(0)设ΔΔΔ则f2xf(2x)xlim则limf(x)x0x0f(x2h)f(x)()00设fx.0hh0xxy3x1.2y1sine设y.xxdy则t设yetdxdyxdx设2102dxdxd1x21,00x2xx在x0.yx0,12x)x0(x)()0.fx在x设f2x10e:x1yarcsin(2x21ty；；1x2yx)lny2sec；x1tsinxecos2xln12yyxxxxxdydxe.ydy设x)2dx21xdydxt设ytt3x1xarcsint,设，求。y1t2,2设x1y.ee0yy是,求xy．xyxx02x3y1y1.22第三章中值定理与导数应用:4xy）x(,2),(2,)(2)11(,(2,)(2)1）x2-,）-）-))x)y(,)(0,)）(,0)）()34334)),))4）2)((,)(2,)ex2(.-)xbf(x)f(x)0yf(x)(a,b)当a）沿x沿x沿x沿x）321xy（x1x）)xx3((x)xx)f223；；；；）x22x12()122/3x若f(x)0）c）11ee1－1223earctanxy上()x若xyf(x)()0()0()0fxfx00()0()()fxfx或fx000）2是是0若fx(00()1fx(x)xx在limf(x)f(x)是的fx20x0xx00）3）））(x)(a,b)在f(x)0f()0fx()fx(x)(a,b)在f(x)f()0fx()fx1y1x2x2f(x).1x13.(x)a,b(a,b)(a,b),使f在f.(x)xf(x)xf(x)f在0在0．0ax2xc2a，c．x1y2()fx(xg(x)f(0)g(0)0x0g(x)0时limA设f()gxx0f(x)lim．g(x)x0若limf(x)cxxey．x(x)e在区间[0,fxe1exxxlim；lim；e1xxsinxxxx0x01；）lim1xe；xxx3x0x011xsin1ex2xx）limxlim）xxx0x03x12x1eexx）lim）limsinxx1x0ln(x1x2)y(x)2x3x12x14f3232axbxcx2x1a、b、cy32在Vr.141(x)fxfx0()1,()f在ex使f)ln.x1x0x)x．lnx1x1x10.x22,xex第四章不定积分）11x2x||2lnx与x2与2141xxx2x与2与2224221）x12x1xx1x225）e5x1ee5x51）111212(3x23若f(x)dxF(x)C,则f(2x（）11F(2xCF(x)C(22)df(x)dxf(xdxdx(x)f(x)dfddxdf(x)f(xf(x)dxf(x)c15()()设Fxfx）(x)f(x)Cf(x)dxF(x)CdFdF(x)f(x)CF(x)dxf(x)C设是xf(x)dx(x)F(x)cef(e若f()xx(e)cF(e)cF(e)cF(e)cFxxxxf(lnx)dx（x）-x11CCxx1（）x311114xCx.4－C－442x2x22x(x)C()设ffx（lnlnx）xx1121xx)2x2x2设f(xcosxC则设是（）f2x)dx112x)C-2x)C22x若f(x)dx2eC2x2x21x2eee2设a0(axb)9dx（）1a1(b)C9()axb(b)C(b)C8C8910a)816111997992（）11-)cos3xCcos3xC33xsin2(21111xxCxC21212211xxCxC22223=（）2xxe11eeC.eex3x3x3x333）(4)xdxxCsinxdxcosxC343dx3CxCxdx23xxxxdx(0,）252x1(x)12x(x)C5552d(x)f(x)dx．设ftanxdx2．xx3edx.11设则f()dxx2x1(x)dx_________x2tgx2e______________.________dx2)xdx．1exd．x2171x12x2））；）dx；xx2xx)x22dx；）；xx22x(2)23dxdx）；）；xx244xx)dxx2x23dx））；）；x2e1xxsinxdx2；）2；2x1x2)）ex；）；x1）dx；）dx；24x23/2costgx)x3tanxdx；）x2；（）1x2sinxxf(x)dx.x第五章定积分及其应用(x)[a,b]()上连续，xx()()(x)f(x)是的设f在区间）ftdtaxb，则a（在[a,b]sintdtx2lim（0）x3x01112D.14323tF（）2x72272dbf(t)dt()x18在=yfb)f(a)1(x)()bbfxdxfbabaaa1x2y5x2）4221112(x25x2)dx1(5x2x2)dx4421111(5x2x2)dx(x25x2)dx442x()设ftdt（）0xte2φ=tφ（）01122xee2xex2xex22x2x212设(x)则(x)（）1t2sinx1cosx2cosx21221sinx1sinx1sinx1sinxtdtaaf(x)f(x)x设f22，x12a2x2xaa22x1aa2x41x4f(t)dtx则f(x)dx(x002xcos3xdx（）224203334x（）sinx3（）12cosx0aaxdxa22（）0a2a2a4192设(x22x)dx,则I）02I00I21I1D.1I4222设IlnlnI与I）121211与I121212122(3x（）1972212dx()x)224323431设(2xa)dxa（）012(edx11sinx0dxcotx2dx1dxxlnxxx4x2204x（）111）xp01111）111xdxxx2x1111()1111lnxdx1dxdxxx1x211()1cosxdxsinxdxlnx