2012年中国科学院数学分析试题及答案

¥I‰ÆïÄ)2012cçÂôÖa¬Æ ïÄ)\ÆÁêÆ©Ûë)‰15¿:1.ò
8‡ŒK,11,2K©O¹küK,÷©150©,ÜÁžmoO180©¨;2.ÁK=k˜?†©ØÓ,31.(2)¥,©¥én4,ùw,ØÎÜ¿,ÆUéx4;3.)‰=Šë,†Ø3¤J,•ž1µ.1.(1502)OŽ4:(1)limn!1n32sin1nsin2n;(2)limx!1rcos1x2!x4:)(1)limn!1n32sin1nsin2n=limn!1n32sin1n2sin1ncos1n=limn!1n3
2sin1n
1cos1n=limn!12n3
1n
121n2=1;(2)limx!1rcos1x2!x4=limx!1cos1x2x42=limx!1ex42
logcos1x2=limx!1ex42
log(1+cos1x21)=limx!1ex42
(1+cos1x2)=limx!1ex42
12(1x2)2=14pe:2.(1502)OŽÈ©:(1)I=Z20dx1+tan3x;(2)J=ZZSx(1+yf(x2+y2))dxdy;Ù¥Sd­‚y=x3;y=1;x=1¤Œ¤«,f(x)¢ŠëY¼ê.E-mail:[e-mail]¥I‰ÆïÄ)2012cçÂôÖa¬Æ ïÄ)\ÆÁêÆ©Ûë)‰2)(1)I=Z20dx1+tan3x=Z20cos3xsin3x+cos3xdx=Z201sin3xsin3x+cos3xdx=2Z20dx1+cot3x=2Z02dt1+cot32t
(ŠC†x=2t)=2Z20dt1+tan3t=2I;I=4;(2)·F(x)=xZ1x3yf(x2+y2)dy;y²F(x)[1;1]þÛ¼ê.é?¿x2[1;1],F(x)+F(x)=xZ1x3yf(x2+y2)dyxZ1x3yf(x2+y2)dy=xZ1x3yf(x2+y2)dyZ1x3yf(x2+y2)dy=xZx3x3yf(x2+y2)dy(ȼê3é¡«mþ'uyÛ¼ê)=0;(Û¼ê3é¡«mþÈ©0)¤±F(x)[1;1]þÛ¼ê.(ܱþ,dK¿Œ:J=ZZSx1+yf(x2+y2)
dxdy=Z11dxZ1x3x1+yf(x2+y2)dy=Z11x(1x3)+Z1x3yf(x2+y2)dydx=Z11x(1x3)dx+Z11xZ1x3yf(x2+y2)dydx=25:3.(150)¦e˜?êÂñ:1Xn=1xn1+12+


+1n:(1)E-mail:[e-mail]¥I‰ÆïÄ)2012cçÂôÖa¬Æ ïÄ)\ÆÁêÆ©Ûë)‰3)·Hn=nXk=11k;n2N+;K˜?ê(1)ÂñŒ»R=1limsupn!1npHn=1:x=1ž,(1)=1Xn=11Hn;Hnlogn(n!1),…?ê1Xn=11lognuÑ,¤±x=1ž,(1)ØÂñ;x=1ž,(1)=1Xn=1(1)nHn;(2)Šâ†?êLeibnizO{=(2)Âñ.nܱþŒ,¼ê‘?ê(1)Âñ[1;1).4.(150)y²:¼êsn(x)=x1+n2x2(n1)3«m(1;+1)þ˜—Âñ;¼êtn(x)=nx1+n2x2(n1)3«m(0;1)þؘ—Âñ.yϏlimn!1sn(x)=0é?¿¢êx¤á,…jsn(x)j6x2nx=12n;dN=Œysn(x)3Rþ˜—Âñ;betn(x)3(0;1)þ˜—Âñ,K7½˜—Âñu0,é?¿¯k‰½ƒê,3N2N+,é¤knN,تjtn(x)j(3)éx2(0;1)˜—/¤á.0=13,Ϗtn1n
=12,ùw,´ØU÷vت(3),bØý,y..5.(150)3«m[a;b]þ,f(x)ëY,g(x)ŒÈ,¿…f(x)0;g(x)0.y²:limn!1Zbafn(x)g(x)dx!1n=maxa6x6bf(x):(4)E-mail:[e-mail]¥I‰ÆïÄ)2012cçÂôÖa¬Æ ïÄ)\ÆÁêÆ©Ûë)‰4yϏg(x)ŒÈ,Œb½jg(x)j6L(L0),qPmaxa6x6bf(x)=M:kyت(4)†à؇Lmà.dÈ©'5ŸkZbafn(x)g(x)dx!1n6ZbaMn
Ldx!1n=MnpL(ba)!M;n!1;2y²(4)†àØumà.é¯k?¿‰½ê,ϏfëY¼ê,34«m[c;d][a;b],¦f(x)3[c;d]þ÷vf(x)M0:2dÈ©'5ŸknsZbafn(x)g(x)dxnsZdcfn(x)g(x)dxnsZdc(M)ng(x)dx=(M)nsZdcg(x)dx:(5)dŒÈ7‡¿©^‡Œ[c;d]þgmä:Nÿ8,g3[c;d]þ7½3ëY:,qdug(x)0u[c;d]??¤á,dëYÛÜ5ŸÚÈ©'5ŸŒZdcg(x)dx0:3ت(5)¥4n!1,¤±é?¿ê,~klimn!1Zbafn(x)g(x)!1nM:Šâ?¿5Œ(4)†àØumà.y..6.(150)3«m[0;a]þ,f(x)gŒ,¿…jf(x)j61;jf00(x)j61,Kx2[0;a]ž,jf0(x)j62a+a2.y?x2(0;a),Ϗf3[0;a]þgŒ‡,dTaylor½nkf(0)=f(x)f0(x)x+f00()2x2;(6)f(a)=f(x)+f0(x)(ax)+f00()2(ax)2;(7)E-mail:[e-mail]¥I‰ÆïÄ)2012cçÂôÖa¬Æ ïÄ)\ÆÁêÆ©Ûë)‰5Ù¥0xa.ò(6),(7)ƒ~,=af0(x)=f00()2x2f00()2(ax)2+(f(a)f(0)):Ïdéz‡x2(0;a),~kjf0(x)j612a
x2+(ax)2+2a6supx2(0;a)12a
x2+(ax)2+2a=a2+2a;dfëY5ŒØª3x=0;x=aü:Ѥá,²¤–y.7.(150)n´˜‡ê.y²:§xn+nx1=0k˜¢Šxn,¿…1ž,?ê1Pn=1xnÂñ.y·f(x)=xn+nx1;x2R,w,f(x)RþëY¼ê.Ϗf(0)=10;f1n
=1n
n0,d0Š½n,3n20;1n
,¦f(n)=0,¢Š35y.Ϗ¼êg(x):=xn3(0;+1)þî‚üN4O,¼êh(x):=1nx3Rþî‚üN4~,…g(n)=h(n),¤±f(x)=g(x)h(x):8:0;x2(0;n);=0;x=n;0;x2(n;+1);f(x)=0k˜‡¢Šn=xn,d±þ35ƒy²Œxn20;1n
.Ϗ1ž,?ê1Xn=11nÂñ,â'O{=1Xn=1xnÂñ.8.(150)(x;y;z)´:Oý¥¡x22+y22+z2=1þŒÜ©(=÷vz0Ü©)?˜:(x;y;z)?ƒ²¡ål,¦È©ZZz(x;y;z)dS:)éuþ?¿˜:(x0;y0;z0),­¡3(x0;y0;z0)?ƒ²¡§x0x2+y0y2+z0z=1;E-mail:[e-mail]¥I‰ÆïÄ)2012cçÂôÖa¬Æ ïÄ)\ÆÁêÆ©Ûë)‰6:ƒ²¡ål(x0;y0;z0)=1qx02
2+y02
2+z20;ÏdZZz(x;y;z)dS=ZZzr14(x2+y2)+112(x2+y2)dS=ZZs114(x2+y2)
112(x2+y2)dS:·z=(x;y)=r112(x2+y2);K@@x=12
xq112(x2+y2)@@y=12
yq112(x2+y2):u´–¦ƒÈ©==zI=ZZ(x;y)(x;y;(x;y))s1+@@x2+@@y2dxdy=ZZs114(x2+y2)
112(x2+y2)
1+14
x2+y2112(x2+y2)dxdy;Ù¥=f(x;y)2R2:x2+y262g.·(x=rcos;y=rsin;Ù¥(r;)2
=f(r;):r2[0;p2];2[0;2]g,I=ZZ
r
vuut114r2
112r2
1+14
r21r22!drd=ZZ
r114r2drd=32:E-mail:[e-mail]