微积分全英微积分全英 (78).pdf

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1、问题问题 1：(0,)Let arctansin(),|=+=zxyxydz A.(1)B.(1)C.(1)D.(1)+dxdydxdydxdydxdy 22(0,)Anscos()cos(),1 sin()1 sin()(1)|：+=+=zyxyzxxyxxyxyyxyxydzdxdy 答案：A 问题问题 2：()(1,1)(1),A.(2ln2 1)B.(2ln2 1)()C.(2ln2 1)D.2ln2(2ln2 1)|xyxzdzydxdydxdydxdydxdy=+=+2(1,1)Ans:lnln(1)11ln(1)1ln(1)let 1,12ln2 1,(2ln2 1)(2ln2

2、1)()|=+=+=+=+=+=+xxzyyzxxz xyyxyzxxxz yyyxyzzxyxydzdxdy 答案：C 问题问题 3：(1,1)(1,1)31(,)is differentiable at point(1,1),and(1,1)1,2,3,(),(,).()A.45B.48C.51D.54|xffzf x yfxydxf x f x xxdx=3211212121Ans:(1)1,(1,1)(1,1)1,()3()()3(),(,),(,)(,)(,)=3 1 2+3(2+3)=51|=+xxxfffddxxxdxdxxf x f x xf x f x xf x xfx x

3、答案：C 问题问题 4：212222122221222122(,)has a second order continuous partial derivative,(,),A.B.C.D.zf u vzf x xyx yfffxffxyffxyfxff=+12211122212212222Ans:0(0)=+=+zfyfxzfxffy fxfx yxffxyf 答案：B 问题问题 5：222Find the total differential of function(,)determined by equation 2 at point(1,0,1)A.2B.2()C.2D.2=+=+zz

4、x yxyzxyzdxdydxdydxdydxdy 222222222222222(1,0,1)(1,0,1)Ans:21,22.|=+=+=+=+=zzdzdxdyxyyz xyzxzxxy xyzzxyzxyzxz xyzyzyxy xyzzzzdzdxdyxy 答案：A 问题问题 6：2The function(,)determined by the equation(,)0 F is a differentiable function,and 0,A.B.C.D.0y zzz x yFx xzzFxyxyyxz=+=,12221222Ans:1()0(,)0110()+=+=+=+=y

5、zzFFy zxx xxFzx xFFxy xzzzzxyFzFxyzxyxy 答案：C 问题问题 7：2222222Let transformation simplify the equation 60 to 0find the value of A.9B.6C.3D.2uxyzzzzvxayxx yyu va=+=+2222222222222222222222222222Ans22(2)24460(105)(6：=+=+=+=+=+=+zzzzxuu vvzzzzzzzxuvaazzzx yuu vvayuvzzzzaayuu vvzzzzaaaxx yyu v222)06031050=+

6、=+zvaaaa 答案：C 问题问题 8：22It is known that the tangent plane of point on surface 4 is parallel to plane 10.Find A.(1,1,2)B.(2,1,2)C.(2,2,1)D.(1,1,1)=+=PzxyxyzP 0000000000Ans:Let the coordinates of is(,),then the normal vector of the surface at P is(2,2,1)tangent plant parallel to 22102211,12.221=+=Pxy

7、znxyxyzxyxyz 答案：A 问题问题 9：Find the tangent plane equation of surface 22 at points(1,2,0).A.220B.240C.240D.40zzexyxyxyxyxy+=+=+=+=+=Ans:(,)232,2,1The normal vector of surface 23 at points(1,2,0)is(4,2,0)Tangent plane equation:4(1)2(2)0240=+=+=+=+=zzxyzzF x y zzexyFy Fx Fezexynxyxy 答案：B 问题问题 10：22Find

8、the tangent plane equation of surface parallel to plane 220.2A.20B.230C.230D.2230 xzyxyzxyzxyzxyzxyz=+=+=+=+=+=00022002200000000Ans:Let the tangent point is(,),the normal vector of surface at points is(,2,1)2The normal vector of the given plane is(2,2,1)21,22122,1,3.The t=+=+=P xy zxzyPxyxyxzyxyzang

9、ent plane is 2(2)2(1)(3)02230+=+=xyzxyz 答案 D 问题问题 11：222Find the normal equation of surface 2321 at points(1,2,2)122A.146122B.123122C.362122D.521+=+=+=+=+=xyzxyzxyzxyzxyz 222Ans:Let(,)2321(1,2,2)2,(1,2,2)8,(1,2,2)12122normal equation:146=+=+=xyzF x y zxyzFFFxyz 答案：A 问题问题 12：22Find the tangent plane

10、equation of surface parallel to plane 240.A.235B.225C.245D.25zxyxyzxyzxyzxyzxyz=+=+=+=+=+=000220000000Ans:Let the tangent point is(,),The normal vector of surface at points is(2,2,1)2The normal vector of the given plane is(2,4,1)2212411,2,5The tangent p=+=P xy zxzyPxyxyxyzlane is 2(1)4(2)(5)0245+=+=

11、xyzxyz 答案：C 问题问题 13：2222(,),(1,0)A.1B.3C.2D.4xxyxyf x yfexy xy+=+222221Ans:(,)(,0)(1,0)22|=+=+=xyxxxyf x yf xxexy xyfx 答案：C 问题问题 14：23223(),when 0,A.()B.3()C.D.3()xxx yxx yxx yxx yzzeyf xyyzxxexyeexyeeeexye+=+=+23232Ans:()()()3()+=+=+=+=+xxxx yxx yzeyf xyxef xzeyxyezexyex 答案：B 问题问题 15：210(2),A.2B.1C

12、.3D.5|=yxxyzzxyx 222 ln2110Ans:(2)0,()(2ln2)22|时=yxxxxxxxxxyzxyyzxzexxzzx 答案：A 问题问题 16：2(1,)2222(,)ln|sin(),1A.11B.1(1)1C.1(2)1D.12(1)|=+=+ff x yxyxyx y 2222(1,)Ans:1cos()1cos()sin()()11(1)|=+=+=+fyxyxxyfxyxyxyx yxyfx y 答案：B 问题问题 17：23(0,0)(,)is determined by the equation1,A.21B.()31C.(2)31D.3|：+=+=

13、+xyzzz x yexyzdzdxdydxdydxdydxdy 2332323(0,0)Ans:0,01101(23)02301(2)3|代入+=+=+=+=+=+xyzzxyzxyzxyexyzezexyzedxdydzyzdzxzdyxydzdxdydzdzdxdy 答案：C 问题问题 18：22(0,1)The function(,)is differentiable,(,)is determined by the equation(1)(,),A.2B.C.2D.2|：=+=+f u vzz x yxzyx f xz y dzdxdydxdydxdydxdy 22222(0,1)(0

14、,1)Ans:(1)(,)(1)2(,)(,)20,1(1)(,)1202|时+=+=+=+=+=+xzyx f xz yxdzzdxydyx df xz yf xz yxdxxyxzyx f xz yzdzdxdydzdxdy 答案：A 问题问题 19：21112222111222211122(,),(,)has a second order continuous partial derivativeA.()B.()C.2()D.(),=+=+xyxyxyxyxyxyxyzef xy xyf u vzx yexyefxy fxyffefxy fxyffexyefxy ffexy f12222

15、+xyff 1221112222Ans:(,)()=+=+=+xyxyxyxyzef xy xyzyefyfxzexyefxy fxyffx y 答案：A 问题问题 20：22The plane region is surrounded by 3(1),3 and y-axiscalculate3A.(1)1623B.(1)823C.(1)223D.(1)42=DDyxyxx dxdy 2113(1)2222220301122322001222242004013202Ans:3(1)3131sincos (sin)1 1 cos4832116=xxDx dxdydxx dyxxx dxxx d

16、xx dxxx dxttdtxttdtx dxx dxdy3(1)162=D 答案：A 问题问题 21：221111=(,)|1,11,calculate 1A.(1 2)61B.(1 2)21C.(1 2)31D.(1 2)9yDDx yxyxx edxdyeeee 2221222221222001132001210011101Ans:=2=221 331 (|2)311 (|)(1 2)33 is=+=yyyyDDyyyyyx edxdyx edxdyedyx dxey dyy dey eyedyeeeDthe part of where 0Dx 答案：C 问题问题 22：222(,)|2

17、,calculate()8A.53B.52C.452D.25=+DDx yxyxyx xy dxdy 22122201222011224001222200Ans:is symmetric about y-axis0()2 2(2)22224sincos=+=DxxDDDxydxdyx xy dxdyx dxdydxx dyxxx dxxx dxx dxxx dxttdt44014011 cos428152()45=+=Dtdtx dxx xy dxdy 答案：C 问题问题 23：2222(,)|14,0,0 sin()calculate 3A.53B.43C.2D.1=+DDx yxyxyxx

18、ydxdyxy 2222222201Ans:rotational?symmetrysin()sin()=1 sin()21 sin()2+=+=DDDxxyyxydxdydxdyxyxyxydxdydrr dr3 4=答案：B 问题问题 24：22110calculate()1A.(1)2B.11C.(1)31D.(1)4xyyedyedxxeeee 22222222211111100011000111000Ans:()=(1)=+xxyyyyyxxyxyyeedyedxdydxdye dxxxedxdyey dyxe dxe dyye dy1 =(1)2e 答案：A 问题问题 25：1 is

19、 an unbounded region bounded by,and y-axiscalculate.1.2.31.2=xDDyx yxe xydxdyABCD 110120112001100Ans:=1 (1)21 (1)21 21 2|=+=+=xxxxDxxxxxe xydxdydxe xydyex dxexxe dxxee dx 答案：D 问题问题 26：22The plane is bounded by,circle 2 and axis,find 7A.21B.37C.121D.4=+=DDyxxyyyxyd 2sin320444246624Ans:cos sin2 sin =c

20、os sin42227 =sin1()33212|=DIxyddrdrd 答案：C 问题问题 27：22(,)has a second order continuous partial derivative,(1,)(,1)0,(,),(,)|01,01,calculate(,).2.2=DxyDf x yfyf xf x y dxdyaDx yxyIxy fx y dxdyAaB aC aD a1100111000110011100010Ans:(1,)(,1)0(1,)0,(,1)0(,)(,)(,)(,)(,)(,)(,)(,|=yxxyxyDyxxyxxxfyf xfyfxIxy fx

21、 y dxdyxdxyfx y dyx yfx yfx y dy dxdyxfx y dyxf x yf x y dx dydyf x10)=y dxa 答案：A 问题问题 28：5The plane is bounded by sin,and 1(1)2A.B.2C.3D.4，=DDyx xyxydxdy 1552sin26226622522Ans:(1)(1)11 (sin1 sin)6611,sin,sin are odd function.6611(sinsin)066(1)(1)=+=xDDxydxdydxxydyxxxx dxxxxxxxxx dxxydxdydx 答案：A 问题问

22、题 29：the plane region is bounded by a curve 1 cos(0)and a polar axiscalculate 15A.167B.813C.163D.4，=+DDrxyd 31 cos300404056Ans:sincos sincos1 (1 cos)sincos41 (1 cos)1(1 cos)(1 cos)412216 ()45615+=+=+=+=DDxydrdrddrdrdd 答案：A 问题问题 30：2222 is a bounded region bounded by lines 1,and,A.12B.22C.32D.2=+Dxxy

23、yDyyxyxdxdyxy 2222222222312224sin2043244Ans:about-axis symmetry0(cossin)1 (csc2)2=+=+=DDDxyDydxdyxyxxyyxydxdydxdyxyxyrdrdrrd3441 (cot2)2 12|=答案：A 问题问题 31：222(,)|2,calculate(1)5A.43B.41C.2D.=+DDx yxyyxdxdy 222sin23200420202200Ans:(1)(21)=cos =4sincos1 cos2 =sin 2211 =sin 2cos2 sin 2221 =(14 +=+DDDxdx

24、dyxxdxdyx dxdydrdrdddd02cos4)45(21)0(1)4=+=+=+=DDdxdxdyxdxdy 答案：A 问题问题 32：113012.(2 21)91.(2 21)92 2.94 2.9=+=yIdyx dxABCD 2111133230000Ans:2111(2 21)9=+=+=+=xyIdyx dxdxx dyxx dx 答案：A 问题问题 33：2215101111(),calculate()1.(21)61.(21)121.31.6=xyf xedyx f x dxAeBeCeDe 222222115500115001301220120Ans:()1 61

25、 121 (1)121 (212|=xyyyyyyx f x dxx dxedyedyx dxy edyy edyye11)e 答案：B 问题问题 34：10sinA.1 sin1B.2sin1C.2cos1D.1 cos1=xxyIdxdyy 2101010101100Ans:sinsin (1)sin(1)cos(1)coscos 1 sin1|=xxyyyIdxdyyydydxyyydyydyyyydy 答案：A 问题问题 35：1121112241212121231.8231.423.833.82=+=xxxxyyxIdxe dydxe dyAeeBeeCeeDee 211211122

26、4112112121212Ans:()()231 82|=+=+=xxxxyyxxyyyyyyIdxe dydxe dydye dxy eedyeyyeeee 答案：A 问题问题 36：1121330112A.4B.3C.2D.1+=+=xxxxIdxydydxydydxydy 31131321Ans:1 42|=yyIdyydxydyy 答案：A 问题问题 37：Calculate,is the region bounded by the curve sin2(0)21A.152B.151C.54D.15=DxydDr 3sin2320042055200520Ans:cos sin cos

27、sin1 cos sinsin 2411 sin 2sin(2)81611 4 21 sin 2188 5 315 =DDxydrddr drddtdt td 答案：A 问题问题 38：222123221calculate(0),is the area enclosed by a short arc tangent 2to the coordinate axis and the circle()()8.(2 2)38.(2 2)38.(2 2)38.(2 2)3+=DdaDaxxayaaAaBaCaDa 22200200320Ans:(,)|0,0211221 (2)2 22 (22)3|=a

28、aax xDaaaDx yxayaaxxddxdyaxaxaaxxdxaxaxdaxaaxx328 (2 2)3=a 答案：B 问题问题 39：11Conditional convergence of series,what kind of point is 3 and 3 of series(1)?Convergence pointConvergence pointConvergence pointDivergence pointDivergence pointConvergA.、B.、C.、=nnnnnaxxnaxence pointDivergence pointDivergence p

29、ointD.、11111111Ans:Conditional convergence of series Convergence radius of is 1if 1 is divergenceif 1 is absolute convergence.Convergence radius|;=nnnnnnnntnnnnntnnaa tRRa taRa ta111111 of()is 1when 3(31)is absolute convergence;when 3(3 1)is divergence.，=nnnnnnnnnnnnnnnna ttna tta tRxnaxna 答案：B 问题问题

30、 40：11112211 is monotonically increasing and bounded,What the convergence of the following series isA.1B.(1)C.(1)D.()=+=nnnnnnnnnnnnuunuuuuu 221122222211111Ans:is monotonically increasing and boundedlim exists,mark it.The sum of the first n terms of sequence()is,limlim()+=+=nnnnnnnnnnnnnuuauuSSuuSuu

31、au 答案：D 问题问题 41：121111(1)2,5,find.9 .3 .7 .8nnnnnnnaaaABCD=1212122111111Ans:5,(1)=23538=+=nnnnnnnnnnnnnaaaaaa 答案：D 问题问题 42：21Constant0 judging convergence and divergence(1).divergence .absolute convergence .conditional convergence .Convergence and ,=+nnknknABCDdivergence are independent of valuek222

32、21112222111Ans:1(1)(1)(1)1(1)and(1)are convergence,so(1)is convergence.1|(1)|1 is convergence.(1)is divergence.is div=+=+=+nnnnnnnnnnnnnnknknnnkknnnnknknnnkknnnn21ergence.(1)is conditional convergence.=+nnknn 答案：C 问题问题 43：21sin1 is constant judging convergence and divergence().divergence .absolute c

33、onvergence .conditional convergence .Convergence a,=nnaannABCDnd divergence are independent of valuea222211211Ans:sin11sin|,is convergence.is convergence,1sin1 is divergence,()is divergence=nnnnnanannnnnannn 答案：A 问题问题 44：2211|Constant and is convergence judging convergence and divergence(1).divergen

34、ce .absolute convergence .conditional convergenc0,=+nnnnnaanABCe .Convergence and divergence are independent of valueD22222222111Ans:|111|()|(1)|()22|1 and are convergence(1)is absolute convergence.=+nnnnnnnnnaababannaann 答案：B 问题问题 45：1210(1,2,3,.)is convergence Constant 2judging convergence and div

35、ergence(1)(tan).divergence .absolute convergence .c,.(0,)=nnnnnnanananABC onditional convergence .Convergence and divergence are independent of valueD22211Ans:(1)(tan)lim|limtan(0)2 is convergence.(1)(tan)is absolute convergence.=nnnnnnnnnnnannananan 答案：B 问题问题 46：1211212111 is convergence which of t

36、he following series is convergent?.(1).().(),=+=+nnnnnnnnnnnnnuuABunCuuDuu 111211212111Ans:(1)1(1)is divergenceln(1)ln(1)(1)1 is divergence112 ()()is divergence212.=+=+nnnnnnnnnnnnnnnnuunnnnuunnuunnn 答案：D 问题问题 47：1111111 is convergence which of the following series is convergent?.|.(1).2,=+=+nnnnnnn

37、nnnnnnaAaBaaaCa aD 111111111Ans:(1)1|is divergence.1 (1)is divergence.11 is divergence.1(1)=+=+nnnnnnnnnnnnnnaannana ann n 答案：D 问题问题 48：111122111Serieslim0,.is convergence.is convergence.is divergence.is divergence.|is convergence.is convergence.|is diverg：,=nnnnnnnnnnnnnnnnnnnnnabaAba bBba bCba bDb221ence.is divergence.=nnna b 11122211Ans:(1)1 is divergence.11 is convergence.=nnnnnnnnnnnnnaba bnnaba bnn 答案：C