2000年全国硕士研究生入学统一考试数学二试题

53)xx.2x)3x0yy(x)2xydy.xyx0.(xx221yxe.x10002300AB(EA)(EA)则设，E为4104500067(EB).153xf(x)(,)limf(x)0,a,b在()aebxxa0,ba0,ba0,ba0,bf(x)f(x)[f(x)]xf0()2ff(x)是.ff(x)是.点fyf(x).ff(x)fyf(x).f(xg(x)f'(x)g(x)f(x)g'(x)0,axb)设有()f(x)gb)fb)g(x)f(x)g(x)fb)gb)f(x)g(a)f(a)g(x)f(x)g(x)f(a)g(a)16x(x)6f(x)0lim若为()x3x2x0x0.e,y2xe,yey的3()xxx123yyyyyyyy6622yyyyyyyy5分)x)(lnx)f(x)dx.设fx5分)(x,y)0x1,0yl:xytt.若设DSt)DlSt)x.x05分).f(0)(n3)(x)xln(1x)0nf在x2n6分)(x)|txS，0(1)xn2nS(x)2(n；当nnS(x).lim求xx7分)VV，每年排入湖泊内含污染物AA的6VV5mA630mA0VAmA)06分)fx(x)0,()(0)f在fx00,1)f)0.f2127分)2(x)x0f5f(1x)3f(1x)8x(x)f(x)x1(x)x0yf(x)x在f.8分)ax(a0,x0)y1x2AOAy与2ax线y2问ax8分)(x)[0,)f1在f1()()fxfxftdt()0,xx10()fx；0()1exfxx6分)11012,,,B0,A设是TTT21802BAxAxBx22447分)0ab13911,2,12,0,623123071131,,,ab.312331611arctanxxarctanxxx16ln2323lim1x2洛2limlimlim22ln12x32x3x2x1xx0x0x0x0y(x)2xydyxy.yx022xyxy2ln2(xdyydx)dxdy.xy0y1时x0y12，.x.将(ln21).x0xy2ln2(xyy)1y.xy0y1时y21(ln21).当xx032.x.2t,x2tdxtdt,xdxt1t22arctan.dt(x7)x2tt3332322002x1yy为ykxbyf(x)k,b[f(x)kx]xxxxxxxy11)ekxxxxx412e21ublim(y2x)lim[(2x1)e2x]令ulim(e)uxxuu0xxeue)e1ue)211uuuuuuuu21xyx21xyx.当.1xex.21yyx1000120002300034(EB)B(EA)(EA)11(EB)E(E)(E)11(E)(E)(E)(E)1112(E)(E)122000100012001240020460023000340068D(,)f(x)内f(x)在0aebxxx处ax00()limf(x),b0a0a0xlimf(x)0(D).(B)C).和设xCf(x)()0fx在x，00f(x)在x()0()0当fxfx0005(x)当()0ffx在x0020()[()]200fxfxx中fxf.：()(()12()()fxxfx2fxfx0f1以x()(0)()fxfxf(0)limlim1.fx0xx0x00f(x)x与x0yf(x)x0yf(x)xyf(x)f点A.f(x)f'(x)g(x)f(x)g'(x).g(x)f(x)f'(x)g(x)f(x)g'(x)g设F(x)F(x)(x)()gx(x)axb在axbF(a)F(x)Fb)，则Ff(a)f(x)fb)g(a)g(x)gb)(x)gb)fb)g(xaxb()，.得f()C(x)f(x)f.：66f(x)6xxf(x)6xsin6xsin6xxf(x)x2x3x312(6x)26xsin6x66cos6x6(1cos6x)2洛limlimlimlim36而x33x23x2x2x0x0x0x011由于1cosxx1cos(6x)2(6x))2226f(x)6xsin6xsin6xxf(x)limlimlim36036x2x3x3x0x0x0sin6xxf(x)alima0，x3x06x3sin6xxf(x)ax3f(x)xax3sin6x66f(x)ax6xsin6x3xx2x2x36f(x)6x6x6x6x3limax2x3x3x0x0x0x012(6x)266cos6x3x220limlim36x2x0x0sin6xx0在3x：将(6x)36x6x(x)6xx(x3336x(x)6x(x)x(x)6f(x)(x)333,x3x3x2x36f(x)6x(x)(x)300x2x3x3x0x0x0Be,y2xeyxx12r1yer11x23(rr1)rrr10,2327yyyyf(x)lnxtxetx)e)tft)fx)xetf(x)dxee)dxe)dexxxxexee)exxx1ex1eexxee)xx1exexee))dxxx1exexee)dxdxxx1ex1ee)dxdexxx1ex1ee)dxd(exxx1exee)xe)Cxxxlntx，1t)1f(x)ft)t)dtt2tt)1[]ttt)t)11()dttt1tt)11t)dtdt)lntt)Ctt1tteln(1e)xln(1e)C.xxxt)S1)8Ot1当0t11St)t；22当1t2yty与1t1x1(1)2tt111St)1(2t)1(tt4)tt1；222222当2S(t)1，t则1t,0t1,221St)1(2t),1t2,221,2.11txx33;60x1()xStx当t22023001111tdt[1t2)dt当1x2StdtStdtStdtxxx222200101111x133(x(xxx326666Stx()112()2()StxStxx当x00221x30x11x2611xStdtxxx326301x2x(uv)uvCuvCuvuv.(n)(n)1(n1)k(nk)(k)(n)nnx)的nyx)，1y；1x911y；1x1x221122yy；1x1x3312123(4)31x1x44((nn1yn()x)n((nn1x)(n)即x)n设ln(1x)vx2x2，u.((n((n((nn1n2n3f()(x)xn22n(n.x)n(1x)n1(1x)n20x(n!n1f()n(nnn3(n,n3,4.n2f(x)：y(0)(0)n()fff(x)f(0)f求fn(0)(n3)(0)xxxnx()n22!n!yf(x)xn!yf(x)x0nf(0).(n)nx2x3xn2x)x((xn1n223n2x4x5xn2xx)x((x23n1n23n2n()ffff(x)fxxx(x2nnn!10f((n)n1n!n2(n!n1得fn(),n3,4.n2x0(1)nxn，xxn(x.x000xaxx，a0coscosxdxxdx2cosxdxcosxdx00(nnxn(xdx)2002nxx)n2n202(nn02n2(nnSxn2()x即02nS(x)2(n(1)xnn(nx命n1)2n222(n1)2nlimlimlimlim，1(n1)nnnnnS(x)2lim.xxmVtm0，tA11mVtdtdt)mt,tdt]Adt00V66mVmdtdtA.V33mmAdm()dt到tt.063dmdtmm(0)63mmd(0)63mm3tC)tC0mm)63mm)636311(0(0Cmmmmtmt13mteC)ee10100336336336mmeCetmCet,CeC)3m1103303229mtm(0)5mCm.m9e)032020m6ln3.6ln3Ammt00Am.0(x)ft)xF()0xFF.0(x0f00f(x(x)00F(xxF(xF(x000(0,)使0F(xF0(0,)(0,F.)0sin0[0,]，,]F(x)(0,),)与上对用罗尔定理，推知存在，使1212))0f))0FFf1212(x)0(0,)()0.ff(0,)110(0,)内仅有一个零点1与内.内f(x)(0,),)f(x)111()(0f(x)0(,)()0fx.fx内fx1000f(x)cosxdxf(x)cosdxf(x)(cosxcos)dx11000()(coscos)f(x)(cosxcosdxfx)1xdx1101cosxcosf(x)(cosxcos)0当0时，，；当时，xx1111cosxcosf(x)(cosxcos)000f.(x)11(0,)(0,)在内.2f(x)f(x)fyf(x)在yx61x6.5xx1.(1x)3f(1x)8x(x)x0将fff(1)3f(1)lim(8x(x))02f(1)0x0(1)0f(x)x1在x故f，fsinx)f(1)sinxf(1sinx)f(1)8x(x)sinxlim3limlimlimsinxsinxx0x0x0x08xx(x)x(1)3(1)limlim84f(1)8ffxsinxxsinxx0x0(1)2(52fffff(yffxf(1)f2y2(x2xy以f即11xax0x，x221a131aax(x,y)(,).y.1a11aab()Vf2xa21ax11ax222()4Vdxaxdxaxdx22a1a1a11a1a0001a2ax2ax235a1a)55a)20为了求Va55322aa)aa)22aa2a)5222a)55252a)53(1a)[2a(1a)a][2a(1a)a]22222(1a)57a)251[2a2aa][2aa]222[4aa]222150a151572772a)a)2a)dVda50,a4a4VV命.a41()()()ftdt()0(xxfxfxfxx10(xf(x)(xf(x)f(t)x0x()(()()(()()0fxfxfxfxfxxx即(x1)f(x)(xf(x)0.f(x)(x1)u(x2)u0uu(x(x14(x1u(xx1u(xln(x1))C即11uexx(1)1)(ee)1xCx1CexCexef(x)u令CCu1.x1x110f(0)f(0)0f(t)dtf(0)f(0)0f1x10ex1C()fx.fx1.edt.tf(x)fft()1xxt100et0tee1e0牛xxx而xttt1000(1)et10xex，t10etf(x)11x即ext102.f(x)0f(x)x0f(x)1时.因f(x)ef(x)e0(x)f(x)e，令要证f，可转化为证明，则xxxex110()()xfxefx()0x0()xx10(x)0时()fxe.xx0()1exfxx..1512101112121022A210210，B.TT111102284BA2AAAB2，4；，2TTT2BAxAxBx224416816x82AExx,x,xTxE1231xx02122xx01212x21x1x232xx0121xx2x121231k,x2k,xkx2123k10x1x2kk20()k2111xk322,,,12313913913920606012,,12331701000022知,,.故,,,,，,,123123123123161300ab112101,,1231100ab3023ab,,123,,又,,2,,,31231233123将,,,12331392