2000年考研数学二真题及答案详解

20001()30arctanlimln12xxxx→−=+.16−.()232300011arctanarctan1limlimlim26ln12xxxxxxxxxxx→→→−−−+==+()2220lim6116xxxx→−=+=−2()yyx=2xyxy=+0|xdy==.()ln21dx−()2ln2xyydxxdydxdy+=+0x=1y=ln2,dxdxdy−=()0ln21,|xdydx==−2xyxy=+x2ln21xydydyyxdxdx⎛⎞⋅+=+⎜⎟⎝⎠0x=1y=0x=1y=()ln2101dydx+=+ln21dydx=−()0ln21,|xdydx==−3()272dxxx+∞=+−∫.3π2,xt−=22,2,xtdxtdt=+=()()22200022lim99722limarctan333|bbbbdxtdtdttttxxtπ+∞+∞→+∞→+∞==+++−⎛⎞=⎜⎟⎝⎠=∫∫∫4()121xyxe=−.21yx=+()111111limlim22lim2lim2121lim11xxxxxxxxxxyaexxbyxxeeeex→∞→∞→∞→∞→∞⎛⎞==−=⎜⎟⎝⎠⎡⎤⎛⎞=−=−−⎢⎥⎜⎟⎢⎥⎝⎠⎣⎦⎡⎤⎛⎞−⎢⎥⎜⎟⎝⎠⎢⎥=−=⎢⎥⎢⎥⎢⎥⎣⎦21yx=+510002300,04500067AE⎡⎤⎢⎥−⎢⎥=⎢⎥−⎢⎥−⎣⎦4()()1BEAEA−=+−,()1BE−+.1000120002300034⎡⎤⎢⎥−⎢⎥⎢⎥−⎢⎥−⎣⎦()()1BEAEA−=+−()EABEA+=−()()2,2,ABABEEEAEBE+++=++=()()1,2EAEBE++=()()1100012001023020034BEEA−⎡⎤⎢⎥−⎢⎥+=+=⎢⎥−⎢⎥−⎣⎦1()bxxfxae=+(),−∞+∞()lim0xfx→−∞=,abA0,0abB0,0abC0,0ab≤D0,0ab≥D()fx(),−∞+∞(),x∈−∞+∞0a≥.()lim0xfx→−∞=()limbxxae→−∞+=∞0bD2()fx()()2''',fxfxx⎡⎤+=⎣⎦()'00f=A()0f()fxB()0f()fxC()()0,0f()yfx=D()0f()fx()()0,0f()yfx=C()'00f=()()2''',fxfxx⎡⎤+=⎣⎦()''00,f=()()0,0f.()()2''',fxfxx⎡⎤=−=⎣⎦()fx()()()''''''21fxfxfx=−+()'''01f=0x=()''0fx0x=()''0fx.()()0,0f()yfx=3()(),fxgx()()()()''0,fxgxfxgx−axbA()()()()fxgbfbgxB()()()()fxgafagxC()()()()fxgxfbgbC()()()()fxgxfagaA.()()()()()()()'''20fxfxgxfxgxgxgx⎡⎤−=⎢⎥⎣⎦axb()()()(),fxfbgxgb()()()()fxgbfbgxA.4()30sin6lim0,xxxfxx→+=()206limxfxx→+A0B6C36D∞C()()331sin6663!xxxox=−+()()()()33330020636sin6limlim6lim360xxxxxoxxfxxxfxxxfxx→→→−+++=+⎡⎤=−⎢⎥⎣⎦=()206limxfxx→+36()()()3300320sin6sin666limlim6sin66lim0xxxxxfxxxxxyxxxfxxxxx→→→+−++=+⎡⎤−=+⎢⎥⎣⎦=()23200006sin666cos66limlimlim312sin6lim362xxxxfxxxxxxxxx→→→→+−−=−=−−=−=5123,2,3xxxyeyxeye−−===3A''''''0yyyy−−+=B''''''0yyyy+−−=C''''''61160yyyy−+−=D''''''220yyyy−−+=B1231,1λλλ==−=()()2321110λλλλλ+−=+−−=''''''0yyyy+−−=B()()ln1ln,xfxx+=().fxdx∫ln,xt=txe=()()ln1,ttefte+=()()()()()()()()()ln1ln11ln1ln1111ln1ln11ln1xxxxxxxxxxxxxxxxefxdxdxedeeeeedxeedxeeeexeCxeeC−−−−+==−+⎛⎞=−++=−++−⎜⎟++⎝⎠=−++−++=−+++∫∫∫∫∫xOy(){},|01,01Dxyxy=≤≤≤≤():0lxytt+=≥()StDl()()00.xStdtx≥∫()221,012121,1221,2ttSttttt⎧≤≤⎪⎪⎪=−+−≤⎨⎪⎪⎪⎩01x≤≤()230011;26xxStdttdtx==∫∫12x≤()22001321121221163xxxStdttdtttdtxxx⎛⎞=+−+−⎜⎟⎝⎠=−+−+∫∫∫2x()()()20021xxStdtStdtStdtx=+=−∫∫∫()33201,01,611,12631,2xxxStdtxxxxxx⎧≤≤⎪⎪⎪=−+−+≤⎨⎪−⎪⎪⎩∫()()2ln1fxxx=+0x=n()()()03nfn≥()()()()()()'''200002!!nnfffxffxxxn=+++++()()()2321224213ln11232142nnnnxxxxxxxnxxxn−−−−⎡⎤+=−+−+−+⎢⎥−⎣⎦=−++−+−nx()()()()101!2nnfnn−−=−()()()()11!02nnnfn−−=−()()()()()()()()01201'2''nnnnnnnuvuvCuvCuvuv−−=++++()()()()()111!ln11kkkkxx−−−+=⎡⎤⎣⎦+k()()()()()()()()()()()()12321211!12!13!21111nnnnnnnnnnfxxnxnnxxx−−−−−−−−−−−=++−+++()()()()()()131!0113!2nnnnfnnn−−−=−−−=−()0cos,xSxtdt=∫1n()1nxnππ≤+()()221;nSxn≤+2()limxSxx→+∞1()1nxnππ≤+()()100coscosnnxdxSxxdxππ+≤∫∫cosxπ00coscos2nxdxnxdxnππ==∫∫()()()100cos1cos21nnxdxnxdxnππ+=+=+∫∫()1nxnππ≤+()()221;nSxn≤+21()1nxnππ≤+()()()2121Sxnnnxnππ++x→+∞n→∞()2limxSxxπ→+∞=,VA6VA6V3V1999A05m2000A0.mVA0mA20000t=A()mt,mV[],ttdt+A00.,66mmVdtdtV=.,33mVmdtdtV=[],ttdt+()mt063mmdmdt⎛⎞=−⎜⎟⎝⎠032tmmCe−=−()005mm=092Cm=−03192tmme−⎛⎞=+⎜⎟⎝⎠0mm=6ln3t=6ln3A0m.()fx[]0,π()()000,cos0,fxdxfxxdxππ==∫∫()0,π12,ξξ()()120.ffξξ==()()0,Fxftdtπ=∫()()00,0FFπ==()()()()()000000coscoscossinsin|ftxdxxdFxFxxFxxdxFxxdxπππππ===+=∫∫∫∫()()0sinGxFttdtπ=∫()()00,GGπ==()0,ξπ∈()sin0.Fξξ=()0,ξπ∈sin0ξ≠()0Fξ=()()()00FFFξπ===()Fx[]0,ξ[],ξπ()()120,,,ξξξξπ∈∈()()''120,FFξξ==()()120,ffξξ==()()0,Fxftdtπ=∫()()00,0FFπ==()10,ξπ∈()()'110,Ffξξ==()0,π()0fx=1xξ=()00fxdxπ=∫()fx()10,ξ()1,ξπ.()10,ξ()0fx()()000,cos0,fxdxfxxdxππ==∫∫cosx[]0,π()()()()()()11101100coscoscoscoscoscos0fxxdxfxxdxfxxdxπξπξξξξ=−=−+−∫∫∫()0,π1ξ()0fx=2ξ12,ξξ()()120.ffξξ==()fx50x=()()()1sin31sin8fxfxxax+−−=+()ax0x→x()fx1x=()yfx=()()6,6f.()()()1sin31sin8fxfxxax+−−=+()()()00lim1sin31sinlim8xxfxfxxax→→+−−=+⎡⎤⎡⎤⎣⎦⎣⎦()()1310,ff−=()10,f=()()()001sin31sin8limlim8sinsinsinxxfxfxaxxxxxxx→→+−−⎡⎤=+⋅=⎢⎥⎣⎦()()()()()()()()()00'''1sin31sinlimsin1sin11sin1lim3sinsin131418xxfxfxxfxffxfxxfff→→+−−+−−−=+−=+==()'12f=()()5,fxfx+=()()610ff==()'1f()'6f()()''612ff==()26yx=−2120xy−−=()20,0yaxax=≥21yx=−,AOA2yax=ax0x≥221yaxyx⎧=⎨=−⎩1,11axyaa==++OA1axya=+()()1222410122235150212315151|aaaxVaxdxaaaaxxaaπππ++⎛⎞=−⎜⎟+⎝⎠⎡⎤=−=⋅⎢⎥+⎣⎦+∫()()()()()()53222527252112215140151aaaadVdaaaaaaππ+−⋅+=⋅+−=+0,dVda=0a4a=4a=522163251518755Vππ=⋅=()fx[]0,+∞()01f=()()()'010.1xfxfxftdtx+−=+∫1()'fx20x≥()1xefx−≤≤.1()()()()()'0110.xxfxxfxftdt+++−=∫x()()()()'''12.xfxxfx+=−+()()''21dfxxdxfxx+=−+()()'lnln1lnfxxxC=−+++0x=()()'000ff+=()01f=()'01f=−()'fx1,C=()'1xefxx−=−+20x≥()'0,fx()fx()01f=()()01.fxf≤=()(),xxfxeϕ−=−()()()''00,,1xxxxfxeexϕϕ−−==+=+0x≥()'0xϕ≥()xϕ()()00,xϕϕ≥=(),xfxe−≥0x≥()1.xefx−≤≤()()()()'001xftdtfxffx=−=−∫()'fx()01.1xxefxdtt−=−+∫0x≥0001.1txxtxedtedtet−−−≤≤=−+∫∫()1xefx−≤≤.11012,,0,,2180TTABBαβγαβα⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥=====⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎢⎥⎣⎦,Tββ22442.BAxAxBxγ=++11012121,,0210211102TAαβ⎡⎤⎢⎥⎡⎤⎢⎥⎛⎞⎢⎥==⎜⎟⎢⎥⎢⎥⎝⎠⎢⎥⎢⎥⎣⎦⎢⎥⎣⎦111,,022,21B⎡⎤⎛⎞⎢⎥==⎜⎟⎢⎥⎝⎠⎢⎥⎣⎦()2428TTTTAAAAαβαβαβαβ====16816AxAxxγ=++()82AExγ−=()123,,,Txxxx=121212310,220,1212xxxxxxx⎧−+=⎪⎪−=⎨⎪⎪+−=⎩12,1kξ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦k*10,0,2Tη⎛⎞=−⎜⎟⎝⎠*,xξη=+1020112xk⎡⎤⎢⎥⎡⎤⎢