2003年考研數學三真題及全面解析

64.1若x若x,x(x)〔1〕設fx2.當x0.當111若x若xsin,x1x2()fxxx2limf(x)0f(0).x0xaxb與xbb4a6.〔y2a322y0b2與a.330a2.yx2a2x20y0x3axb0302,0故b2x(3ax)a4a4a.202202246,0ax(x)g(x)〔3〕設，f而D表示全平面，那么其他，If(x)g(yx)dxdy=a.2D0x0yx1.f(x)g(yx)dxdyI=adxdy2D0x1,0yx1101x1axa].=a22x20x〔n維向量(a,0,,0,a),a0TE為n.1AE，BE，TTaA.2aE這里為nTT2.1AB(E)(E)TTa11TT=ETTTTTaa11a1TT=E=E=Ea2aTTa1(12a)E,Ta1112a02aa10a,a1.故2a2X0.4〔X和YZY與Z..Y,Z)Y,X0.4)Y(X0.4)]EY)E(X0.4)=E()0.4EY)EY)E(X)0.4EY)DX.且DZY,Z)X,Y)0.9.=XY,X,,X〔X2XX12n1n1nYX2n.2nii1【分析】X,X,,Xn121n1nnpXin(nii1i1.,X,,XX21222n111EX2DX(EX)2=()2422iii1n1n1.2nnYnX2i2ii1i164.f(x)〔f(0)g(x)x在.在.[D],.為f(x)f(x)f(0)limg(x)limlimf(0).xx0x0x0x0(x,y)〔00(x,y)yyf(x,y)yy在.f在.〕.00000(x,y)yyf(x,y)yy在.f在000[A].(x,y)【詳解】f(x,y)00(,)0fxyy(,)fxy在yy,0000aaaaqn，n〔p，nnnn22nap與q.nnnn1n1n1ap與q.nnnn1n1n1ap與q.nnnn1n1n1.假設a絕對收斂，那么np與q斂散性都不定.nnn1n1n1[B].aa【詳解】annnn1n1n1aaaapnqp與qnnnn22nnnn1n1abbbab〔AAbba或或ab且ab且[C]AA.Aabbbab(ab)(ab)20ab0或bba2,ab且,,,〔n12s,k,,kkkk0，k12s1122ss,,,.12s,,,,,,kkk112s2s1k0.kk122ss,,,12s,,,.[B]12s..,k,,k【詳解】假設對于任意一組不全為零的數k,都有12s1k0,,,,,,kk122ss12s12s.,k,,kkkk0.k.12s1122ss,,,12sk,k,kk,1kk0..12s122ss,,,,,,12s12s,,,.12s,,,線性無關12s.〔A，A12，A，A34,A,AA,A,AA...A,A,A.123234,A,AA[C]123234.12121214P(A)(A)P(A)P(A)，P，，，12341111(AA)P(AA)P(AA)P(AA)P(AAA)0，，且P，，444412132324123P(AA)P(A)P(A)，P(AA)P(A)P(A)，P(AA)P(A)P(A)，121213132323P(AAA)P(A)P(A)P(A)，P(AA)P(A)P(A).1231232424,A,AA,A,A故A123234三8設1111f(x),x[.)x2xx1[在.2.limf(x)x.111limf(x)lim[]=)xxxxx111x)x)==xxx11xxxxx11sinx2lim==coscossinxxx2xx11.1[在21f，1[使在.2四8ff1fxyx221(,)[,()]設gxy2y2,uv222gg22.求xy221fu,v)uxy,v(xy)g，22，2ff22.uvugxffyx，uvgyffxy.uvgffff2222y2x故22，xuuvvv222.gffff2222x2xyy.vv22yuu222ggff2222(xy)(xy)2222xyuv2222y.=x22五8Iesin(xy).(x2y2)22D{(x,y)xy}.22.rcos,yrsinxIeesin(xy)dxdy(x2y2)22D2sinr2dr.=edrer00r令t2.Ieet0記Aet0Aett0[esintecostdt]=tt00==t0[ecostetdtsin]tt00=e1.1Ae)，2eIe)e229.x2n1(n(x.2nn1.x()(2n1.fxxn1x2n10到xt1f(x)f(0)dtx).x21t220由得1f(x)1x),(x22()0令fx1x2()fx,x)22(0)10f，在9設在(,)()()fxgx，gx()()fxfxgx()()2.ex求..由()()()()()Fxfxgxfxgx(x)f(x)=g22[f(x)g(x)]2f(x)g(x)=2e)2x.()2()4.FxFxe2x2F(x)e[4eeC]22x[edxC]=e2x4x.=e2x2x將F(x)e2e2.xx8f)0.cf(c)1ff(0)ff(2).131.值Mmf(0)M，mfM，mf(2)M.故f(0)ff(2)mM.3c[0,2]f(0)ff(2)f(c)3且f(x)在[c,3](c,3)(cf)0..(ab)xaxaxax112233nnax(ab)xaxax112233nnax(ab)xaxax112233nnaxaxax11()abx2233nnana,a,,a和bi12ni1....abaaaaaa1233nnnaab21Aaaab123a1aaab23nnba=bn1ii1n0ba0.當bii1當axaxax0.1122nn由an0a(i,n)0a.ii1i1aaa(,0)(,0),(,0,0,.T，TT，23n1a2ana111na0當bbii1.naaaaaaaa1i233nnni1aana12ii1naaaa123ii1naaaaa123nii1112nnaii1naaaaa1i23ni1110010100011aa211〔n行2n11001010.10010000xx，xx，,xx.2131n1.Tf(x,x,x)XAXax22x22x22bxx(b0)，T12312313A求f.AAA..f0baA020.02b(設Ai2(2)1，ia123a0b0204ab12.2123b02A102EA020，(2)(22023.得A123(2E)x012(，.TT123(3)0EAx32).T3,,,,得1231232112()(,0,).，，TTT5555123201550201Q，1231055Q..200QT020，00f2y22y23y2.123X1,若,xf(x)3x32;是X..,Y的(0F(X)y.當]x1F(x)x1.x33t1320時，y1時，設.y[yG(y)Y}{F(X)