2004年考研数学二真题

2004.6424..12(1)()lim1nnxfxnx→∞−=+,()fxx=____.00x=()0fx=0x≠2221(1)(1)1()limlim11nnxnxxnfxnxxxxn→∞→∞−−====++()fx0,01,0xxx=⎧⎪=⎨≠⎪⎩,001lim()lim(0)xxfxfx→→==∞≠0x=()fx.2()yx333131xttytt⎧=++⎪⎨=−+⎪⎩,()yyx=x_______.1−∞∞22222331213311dydyttdtdxdxtttdt−−====−+++,222223214113(1)3(1)dyddydttdtdxdxdxttt′⎛⎞⎛⎞==−⋅=⎜⎟⎜⎟+++⎝⎠⎝⎠,220dydx⇒0t.331xtt=++,0t,(,1)x∈−∞(∵0t=1x=⇒x∈(,1]−∞.)3121dxxx+∞=−∫______.2π221002sectansecsectan21dxttxtdtdtttxxπππ+∞⋅===⋅−∫∫∫.01120110222111()arcsin21111dxtxdtdttttxxttπ+∞=−===−−−∫∫∫4(,)zzxy=232xzzey−=+,3zzxy∂∂+=∂∂______.2232xzzey−=+xyz,xy23(23)xzzzexx−∂∂=−∂∂23(3)2xzzzeyy−∂∂=−+∂∂2323213xzxzzexe−−∂=∂+23213xzzye−∂=∂[phone]xzxzzzexye−−∂∂++=⋅=∂∂+23(,,)20xzFxyzeyz−=+−=232xzFex−∂=⋅∂2Fy∂=∂23(3)1xzFez−∂=−−∂2323232322(13)13xzxzxzxzFzeexFxeez−−−−∂∂⋅∂∴=−=−=∂∂−++∂232322(13)13xzxzFzyFyeez−−∂∂∂=−=−=∂∂−++∂232323313221313xzxzxzzzexyee−−−⎛⎞∂∂+=+=⎜⎟∂∂++⎝⎠23(23)2xzdzedxdzdy−=−+2323223xzxzedxdyedz−−=+−2323(13)22xzxzedzedxdy−−+=[phone]xzxzxzedzdxdyee−−−∴=+++2323213xzxzzexe−−∂=∂+23213xzzye−∂=∂+32zzxy∂∂+=∂∂53()20yxdxxdy+−=165xy==______.315yxx=+21122dyyxdxx−=102dyydxx−=12dydxyx=1lnlnln2yxc=+ycx⇒=()ycxx=2111()()()222cxxcxcxxxxx′+−=321()2cxx′=352211()25cxxdxCxC=+=+∫53211()55yxxCCxx=+=+1615xyC==⇒=315yxx=+21122dyyxdxx−=1122212dxdxxxyexedxC−⎡⎤∫∫=+⎢⎥⎣⎦∫11lnln22212xxexedxC−⎡⎤=+⎢⎥⎣⎦∫35221125xxdxCxxC⎡⎤⎡⎤=+=+⎢⎥⎢⎥⎣⎦⎣⎦∫6(1)15yC=⇒=315yxx=+.6210120001A⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠,B2ABABAE∗∗=+,A∗A,E,B=_______.192ABABAE∗∗=+2ABABAE∗∗⇔−=,(2)AEBAE∗⇔−=,21AEBAE∗∴−==,221111010(1)(1)392100001BAEAA∗====−⋅−−−.1AAA∗−=11122ABABAEABAABAAAA∗∗−−−=+⇒=+2AABABA⇒=+(2)AAEBA⇒−=32AAEBA⇒−=21192BAAE∴==−.8432.,.70x+→20cosxtdtα=∫,20tanxtdtβ=∫,30sinxtdtγ=∫,,A,,.αβγB,,.αγβC,,.βαγD,,.βγαB302000sinlimlimcosxxxxtdttdtγα++→→=∫∫∵32201sin2limcosxxxx+→⋅=3200limlim022xxxxx++→→===o()γα=200030tanlimlimsinxxxxtdttdtβγ++→→=∫∫23002tan22limlim011sin22xxxxxxxx++→→⋅===⋅o()βγ=αγβ,B.8()(1)fxxx=−,A0x=()fx,(0,0)()yfx=.B0x=()fx,(0,0)()yfx=.C0x=()fx,(0,0)()yfx=.D0x=()fx,(0,0)()yfx=.C()fx=(1),10(1),01xxxxxx−−−≤⎧⎨−⎩()fx′=12,1012,01xxxx−+−⎧⎨−⎩()fx′′=2,102,01xx−⎧⎨−⎩10x−,()fx,10x,()fx,(0,0).(0)0f=,01x≠,()0fx,0x=.,0x=,(0,0)()yfx=,C.922212limln(1)(1)(1)nnnnnn→∞+++A221lnxdx∫.B212lnxdx∫.C212ln(1)xdx+∫.D221ln(1)xdx+∫B22212limln(1)(1)(1)nnnnnn→∞+++212limln(1)(1)(1)nnnnnn→∞⎡⎤=+++⎢⎥⎣⎦212limln(1)ln(1)(1)nnnnnn→∞⎡⎤=++++++⎢⎥⎣⎦11lim2ln(1)nniinn→∞==+∑102ln(1)xdx=+∫2112lnxttdt+=∫212lnxdx=∫B.10()fx,(0)0f′,0δ,A()fx(0,)δ.B()fx(,0)δ−.C(0,)xδ∈()(0)fxf.D(,0)xδ∈−()(0)fxf.C0()(0)(0)lim00xfxffx→−′=−,,0δ∃,xδ,()(0)0fxfx−0xδ,()(0)fxf0xδ−,()(0)fxf,C.1121sinyyxx′′+=++A2(sincos)yaxbxcxAxBx∗=++++.B2(sincos)yxaxbxcAxBx∗=++++.C2sinyaxbxcAx∗=+++.D2cosyaxbxcAx∗=+++A0yy′′+=210λ+=,iλ=±,2021(1)yyxex′′+=+=+,0,21yaxbxc∗=++sin()ixmyyxIe′′+==,i,2(sincos)yxAxBx∗=+21sinyyxx′′+=++2(sincos)yaxbxcxAxBx∗=++++12()fu,{}22(,)2Dxyxyy=+≤,()Dfxydxdy∫∫A221111()xxdxfxydy−−−−∫∫.B222002()yydyfxydx−∫∫.C2sin200(sincos)dfrdrπθθθθ∫∫.D2sin200(sincos)dfrrdrπθθθθ∫∫D2221(1)01(1)()()yyDfxydxdydyfxydx−−−−−=∫∫∫∫22111111()xxdxfxydy+−−−−=∫∫AB.cossinxryrθθ=⎧⎨=⎩xyo⋅121−12sin200()(sincos)Dfxydxdydfrrdrπθθθθ=∫∫∫∫,D.13A3,A12B,B23C,AQC=QA010100101⎛⎞⎜⎟⎜⎟⎜⎟⎝⎠.B010101001⎛⎞⎜⎟⎜⎟⎜⎟⎝⎠.C010100011⎛⎞⎜⎟⎜⎟⎜⎟⎝⎠.D011100001⎛⎞⎜⎟⎜⎟⎜⎟⎝⎠.D010100001BA⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠100011001CB⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠010100100011001001CA⎛⎞⎛⎞⎜⎟⎜⎟∴=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠011100001AAQ⎛⎞⎜⎟==⎜⎟⎜⎟⎝⎠011100001Q⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠D.14A,B0AB=,AA,B.BA,B.CA,B.DA,B.A(),ijlmAa×=()ijmnBb×=()12mAAAA=0AB=⇒()11121212221212nnmmmmnbbbbbbAAAbbb⎛⎞⎜⎟⎜⎟⎜⎟⋅⋅⋅⎜⎟⎜⎟⎝⎠()1111110mmnmnmbAbAbAbA=++++=10B≠,0ijb≠1,1imjn≤≤≤≤1,112210jjijimmbAbAbAbA+++++=,12,,,mAAA12mBBBB⎛⎞⎜⎟⎜⎟=⎜⎟⎜⎟⎜⎟⎝⎠,0AB=⇒11121121222212mmlllmmaaaBaaaBaaaB⎛⎞⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⋅⋅⋅⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠1111221211222211220mmmmlllmmaBaBaBaBaBaBaBaBaB+++⎛⎞⎜⎟+++⎜⎟==⎜⎟⎜⎟⎜⎟+++⎝⎠0A≠0ija≠1,1iljm≤≤≤≤11220iiijjimmaBaBaBaB++++=12,,,mBBBA.AmnBnsAB0r(A)+r(B)n.ABr(A)0,r(B)0,r(A)n,r(B)n,ABA..994..15103012coslim13xxxx→⎡⎤+⎛⎞−⎢⎥⎜⎟⎝⎠⎢⎥⎣⎦.3012coslim13xxxx→⎡⎤+⎛⎞−⎢⎥⎜⎟⎝⎠⎢⎥⎣⎦2cosln3301limxxxex+⎛⎞⎜⎟⎝⎠→−=202cosln3limxxx→+⎛⎞⎜⎟⎝⎠=20ln2cosln3limxxx→+−=01sin2coslim2xxxx→⋅−+=011sin1lim22cos6xxxx→=−⋅=−+3012coslim13xxxx→⎡⎤+⎛⎞−⎢⎥⎜⎟⎝⎠⎢⎥⎣⎦2cosln3301limxxxex+⎛⎞⎜⎟⎝⎠→−=202cosln3limxxx→+⎛⎞⎜⎟⎝⎠=20cos1ln3limxxx→−+=20cos11lim36xxx→−==−1610()fx,−∞+∞,[0,2],2()(4)fxxx=−,x()(2)fxkfx=+,k.()fx[2,0]−;k,()fx0x=.20x−≤022x≤+()(2)fxkfx=+2(2)[(2)4](2)(4)kxxkxxx=++−=++.(0)0f=.200()(0)(4)(0)limlim40xxfxfxxfxx+++→→−−′===−−00()(0)(2)(4)(0)limlim80xxfxfkxxxfkxx−−−→→−++′===−.(0)(0)ff−+′′=,12k=−.12k=−,()fx0x=.17112()sinxxfxtdtπ+=∫,()fxπ;()fx.32()sinxxfxtdtπππ+++=∫,tuπ=+,22()sin()sin()xxxxfxuduudufxππππ+++=+==∫∫,()fxπ.sinx(,)−∞+∞π,[0,]π.()sin()sincossin2fxxxxxπ′=+−=−,()0fx′=,14xπ=,234xπ=,344()sin24ftdtπππ==∫,554433443()sinsinsin224ftdttdttdtπππππππ==−=−∫∫∫,20(0)sin1ftdtπ==∫,32()(sin)1ftdtπππ=−=∫,∴()fx22−,2,()fx[22,2]−.18122xxeey−+=0,(0)xxtt==0y=.x,()Vt,()St,xt=()Ft.()()StVt()lim()tStFt→+∞20()21tStyydxπ′=+∫22022124xxxxteeeedxπ−−⎛⎞+−+=+⎜⎟⎝⎠∫2022xxteedxπ−⎛⎞+=⎜⎟⎝⎠∫,2200()2xxtteeVtydxdxππ−⎛⎞+==⎜⎟⎝⎠∫∫,()2()StVt∴=.22()2ttxteeFtyππ−=⎛⎞+==⎜⎟⎝⎠20222()limlim()2xxttttteedxStFteeππ−→+∞→+∞−⎛⎞+⎜⎟⎝⎠=⎛⎞+⎜⎟⎝⎠∫222lim222ttttttteeeeee−−−→+∞⎛⎞+⎜⎟⎝⎠=⎛⎞⎛⎞+−⎜⎟⎜⎟⎝⎠⎝⎠lim1ttttteeee