2018年研究生入學考試數(shù)學一真題及標準答案

2018年研究生入學考試數(shù)學一試題及標準答案1.y|x|sin|x|.y|x|sin|x|.ycos|x|.ycos|x|1|x|cos|x|1122limlim，xxx0x01|x|cos|x|1122limlimxxx0x0xy2與z22.zxyz1.zC.yxyz1.y2x2yz2xyB與z222x(X1)2Z0(2x,2y,xyxB11)2n3n3）((2n1)!n0..C..2n3(1)x2n1則S(x)n(2n1)!n02n311S'(x)(1)x(1)x3(1)x2nn2nn2nn(2n)!(2n1)!(2n)!n0n1n0xsinx3cosx2n3xxx,(1)S(1)cos12sin1S(x)n(2n1)!n0x)1xdxK，2dxN，cosx)dxM,N,K4）M2221xex2222.MNKB.MKNC.KMND.KNMCx)2x)dx2Mdxdx2221x1x222221xe01，故因為f(x)在（）點的切線方程是y1x，所以xex1xNKdxdx22ex22dx2cosx)dxcosxdx2220221105011001111101111101.011.011C.010.010001001001001A110111Q011A011，r(EQ)2,A001001A)2為，r(E6,B為nr(X)(XY)X.r(AAB)r(A).r(ABA)r(A)C.r(AB)max{r(A),r(B)}.r(AB)r(AB)TTn,r(AAB)r(A(E,B))r(A)r(E,B)7f(x)fx)fx),f(x)dx0.6{X20..C..N)Af(x)X20}{XN(,)x,x,,x8X，2212n,H:令H:00100.05H0.01H000.05H0.01H0.01H00若顯著性水平0.05H00若顯著性水平0.05H0.01H000.05時，拒絕域為|z|z|z|z）20.01時，接受域為|z|z|z|z當2）22D11tanxsinkx9）limek1tanxx02k11tanxxx111sinlimelimelimx01x1xsinsin1tanxx0x02x12limee2xkx0f(x)(0,0)ya(1,2)1x)）yx0222111x)'(x)'(x)10f'(xfff22000）F(x,y,z)ijkyizjxkikrotFxyzxyyzzxyz1與xyz0xydSS由x222xyz11222yzxz得xyL:xyz0213161dS2(xyyzzx)dS故xydS63LLL）二階矩陣A有兩個不同的特征值，,是A的兩個線性無關的特征向量，12A()則|A|21212)AA222A(2212121122122(0,10,10(22211221211211|A|1或1212114,P()P(B)P(AC|ABC)）,B獨立，,CBCP(C)214[(CP()P(AB)PC)P()P(|C)P(C)1PC)P()PC)142PC)P()P(B)PC)1PC)4dxearctane2xx1111earctanedx=arctanede=earctane1edx)2xxx2x2xx2x222e1x121dxue1u1)dxuuC1e2xx23e1x311（earctane1(e1)e1C)=2xxx3x23yz2x,y,zx，x2yz2322S(x,y,z)234xyz21322(xyz2)L(x,y,z,)22234xL'(x)L'(y)0(1)(2)3y0123418z3L'(z)08L'()xyz204168,x,y,z3343343343341S33413y3zxdydz(yz)dxdzzdxdy）xx223313y3zx3y3z1，Px,Qyz,Rz22222333y3z122x01xPQR)dxdydz3y3z)dxdydzxdydz(yz)dxdzzdxdy(3322xyz12941445dxdydz6ydxdydz24514xdydz(yz)dxdzzdxdy03345111yyf(x)xf(x)f(x)e(xexdxdxC)e((x1)eC)x1Ce1f(x)xxxT)f(x)Tf(x，e(f(x)exdxdxC)exf(x)edxCe令Y(x)xxT)exT(f(x)edxC)e(f(x)eeC)xT方程有通解，則Y(x。xTxTee(ef(x)edx)()eef(x)edx)eY(x)xTTxTxxxTxT0,xee1{x}limx）數(shù)列{x},xxxnn1n1nnnn1e1e1e1xxxexln2由xen1n得en1nxn1lnn1，xnxxxnn1e1x,f'(x)e10,(x0)設f(x)f(0)=0f(x)xxe1e1xx0e1x,(x0)1,xln0，且f(x)11xxx2110由數(shù)學歸納法可知，對任意n,有xne1e1nxnxxxlnxlng(x)e1xe,g'(x)xex0xxxn1nxnnxexnne1xg'(x)0以g(x)g(0)0x0g(x)001，xe1e1xnxxxlnxln0，即數(shù)列{x}nn1nxnnxexnnnAAee1,{x}limxnAAnne1xex0A0。g(x)xx(xxx)(xx)(xx)f(x,x,x)22212312323130f(x,x,x)312f(x,x,x)123xxx012301f(x,x,x)得xx012323xx0131111111111010110112100110011111012當時，0101100101方程組由唯一解，002001xxx0123111102x21k1,kR。2當011011x2000000x13yxxx11232yxxyyy222222313yxx313f(x,x,x)(xxx)(xx)(x2x)2當2221231232313x3x3(xx)2f(x,x,x)2x2x6x2xx6xxx)222222323221231323131y2y21212a1a2011aA130B27a111aBP12a12a12a130，R(B)R()2R()，13013039000027a1a21a21a2B011011011a211101a30a20AXBBP）即1221221063441300110121112721110000006k36k46k41232k12k12k1PkkP1k2k3k231236k36k46k41232k12k12k1，k,k,kkk。P1k2k3k12323123y,{xy，Y22X,Y{X12ZX,Z)Zcov(X,XY)E(XY)E(X)E(XY)1）cov(X,Z)2E(X)EY)E(X)EY)D(X)EY)221,2,，Z0}{XY0}{XY0}e）0,Z11Z{XYe，Z{XYe221e|k|Zk},k1,2,,Z0}e2|k|!1|x|e,(x)x,x,,x23Xf(x)，的X12n0。E(),D().n|x|i1ei1,