2018年高等数学二试题及完全解析(Word版)

2018年全国硕士研究生入学统一考试数学二考研真题与全面解析（Word版）84.1limeaxbx12若x2（）xx01111,b1a,b1a,b1a,b1a2222B)111limeaxbxlim1(eaxbx1)x2x22x2xx0x0•，exax211eax21xex0lim1(eaxbx1)x2x2exax21x2x01()bxxxxaxx2222eaxbx1x22lim0lim0x2x0x01(a)xb)x(x)2212lim0a0,1b0x22x0e1e2be2a12ax2x(xb00，x22x22x0x0x01故a,b12x0）(x)xsinxf(x)xxff(x)cosxf(x)xD)1第页共15页xxxxxf(x)f0xxx0x0x0xsinxxxf(x)f(0)limlimlim0,xxxx0x0x01x2cosx1f(x)f(0)2limlimlim0xxxx0x0x0112xxcosx122Dlimlimlimxxxx0x0x02ax,x11,x0R(x)(),1x0()()f，gxxfxgx在.1,x0x,x03,b1a3,b2a3,b1a3,b2aD）1ax,x1F(x)f(x)g(x)x1,1x0令，xb1,x0F(1)1a,F(0)1b,F(10)2,F(00)1,则1a2,1b1a3,b2，.f(x)[0,1]在1(xdx0f（）011()0fx()0()0fx()0fff2211()0fx()0()0fx()0f22D）12x0。1f(x)x在202第页共15页111f()112f(x)f()f()x()xx2。2222111)111f10f(xdxf()f()(x)dx(x)dx22222!20001)1f()21(x)dxf2，2!20()11f()0fx1dx()0f()0x2.2!220121f(x)10f()0f(x)x1fx()0，。2011(x)x()0fx，()20fxf()0f21。32011f(x)xf(x)10f()01fx()0，.。220x)21x21xdx，Kxdxcos)2dx设M，N）22ex222MNKMKNKMNKNM）））D）C）x)21x21x2x2x2Mdxdx)dx，2221x21x2222Kcosx)dx1dx，2222f(x)e1x,x(,)()1fxexx(,0)f(x)0x令当，222x(0,)fx()0(,)x()(0)0fxf对2223第页共15页xdxdx12C2ex22x2(1)01dx（）1x0x5）））6xyx2x2(1)xydydxxydy01x0x，1)20DD110011）001））））0010010010014110H011r(H)3tr(H)31001,B,C,D四个选项，它们与矩阵Hr(EH)2r(E)2r(EB)1,，r(EC)1r(ED)1AXkEAkEX与,kr(kEA)r(kEX),,B是nr(X)X(X,Y)设）r(,AB)r()r(,BA)r()））D）r(,B)max{r(),r(B)}r(,B)r(A,B)TT）A）A(,,),AB(,,),AB12n12n,,,,,,组与12n12n12n,,,,r(,AB)r(A)A，12n12n4.limx[arctan(x1)arctanx]2若。x1arctan(1)arctanx,xx1,x02，111x2x21，1x)11x21x)21x222xxx2x[arctan(xx]1。021x2x5第页共15页arctan(x1)arctanxlimx[arctan(x1)arctanx]lim2x2xx111x)1x1xx)x223222x32(1x(1x)]22xx12xx431。2(1x(1x)]22xyx22lnx。；y4x3.224(0,)y2xy2y，，。xx2x3y01y(1)0(1,1)x令y(1)443yx。dx；x4x3251ln2。21111x32x1122。x4x32x3x12555xt3t。4yt32。3dy3sintcostdy2(tant)sect22tant，，dx3costsintdx3costsint3costsint22226第页共15页sect2y3cossin12tt2KtK。33costsint433t(1y)tant)42222zzz(x,y)lnzezxy1。x1(2,)21。41zzxz12ez1yx2,yxxzx14z1。1(2,)2,,AA设为3，1231123AAA，。2233232.200(,,),,),,)111，123123123121200P(,,),C11令,123121PACAC，于则,200EC111(2)(23)021212。7第页共15页1PC)1142PC),114PC)22.earctanedx2xx.1eedxe2xxx2x211earctane1edarctane12xx2xx2121de1xearctane1e2xx2x21211(e1)x2earctane1ede12xxxx21211earctane1(e1)de1de12xxxxx212121earctane1(e1)e1C2xxx3x262f(x)ftdttf(xtdtaxxx200f(x)f(x)在区间[0,1]1auxt(xtdt(xu)fuduxfuduufuduxxxxtf，0000xxxftdtxfuduufuduax2x000xf(x)fudu2axf(0)0，0f(x)fuduf(x)x0()()2fxfxa8第页共15页xx，将有。1e1f(xdxf(x)2a(1e)a。20xy1cost。Dyx2，000Dxtsint,y1cost,(0t2)20D223，00202(cos2cossinsincossin2)tttttttt222，02(1cos)2(1cos3cos3cos)tttdt332，00(x2yd2故。Dkln21(xxlnx2klnx1)0。29(x1)(x1)0x1lnx（x0xlnx2klnx10；22当令x1xlnx2klnx10。lnxkx2lnxk()12F(x)xlnx2klnx1Fx2，xxx2G(x)x2lnx2kG(x)1令。x0x1G(x)0,G(x)G(x)G(1)2ln210，()0()F(x)F0FxFx，221x2()102()10xGxGx。xx1,G(x)x2lnx2kx2G(2)0故在，G(x)x)G(x)0F(x)02xx()0Fxx)()Fx()0FxF有，xlnx2klnx102。x(0,)(xxlnx2klnx1)02。2m1S。433min2x4y3z2xyz，3S(x,y,z)xyz222，42x4y3z20,y0,z0x，第页共15页3S(x,y,z)xyz22243222令410Lxxy2y3223LxyzzS.4L:yx2(x0)O(0,0)(0,1)PL点9OAAPLP(3,4)Stx2tx42xx3x1xuS22，292024x102。9x3x3xen0,xen1(1,2,3,)。nx1xnnx。xnnne1x0e21。x1x1e1xee(0,x)1ex21x10xx0xx21n1ne10xxxen2e(0x)n1n1，n2xn1n1xnxnxAxeenAeeAxxA设解n1nnnA0x0。nnxn0x：ne1x11x0x1x2ln10；x2ln1e1n当1x1nkx0k；e1x1k1lnexxkln10xk1kxkn。xk1kx0nnN，。xne1e1e1xnxnxxxlnxlnlnelnnnx，n1nxnnxnxexnnf(x)e1xe(x0)xex0f(x)0xxx设，f(x)f(x)f0e1xx。第页共15页e1xxxln10xxnx1nxex1nnnnxxnnxaxeen1nnaeexxaa设nnna0x0。nnf(x,x,x)(xxx)(xx)(xax)222a1231232313f(x,x,x)0f(x,x,x)123123f(x,x,x)0123xxx0123x0x23xax0131111111111A0110110100a210a01a1f(x,x,x)0x(0,0,0)T当a2。1231020112A，当a000f(x,x,x)0xk(2,1,1)kT，123x,x,xf(x,x,x)a2123123f(x,x,x)fyyyyy(,,)y212223。123123a2当第页共15页f(x,x,x)(xxx)(xx)(xax)22212312323132x2x6x2xx6xx2122231213213B120306213EB120(1018)0230657,57,0123f(z,z,z)zz2122。12312aa130A27a1a2aP？B011APB111