南京郵電大學(xué)

1、南京郵電大學(xué)高等數(shù)學(xué)（下冊(cè)）習(xí)題參考答案2.(1)D(xy)2d(xy)3d;D(2)D(xy)3d(xy)2d;D(3)1xyzdvxyzdv;(4),22(xyz2)2dv222(x2y2z2)dv3.(1)0I2;(2)36I100;(3)323233I第七章3342x1.(1)odxof(x,y)dy或f(x,y)dx;22rJrxrdxof(x,y)dy或22；dy二f(x，y)dx；y2xdxf(x,y)dy或x1222idy1f(x,y)dx1dyyf(x,y)dx;i,廠x2(4)idxL(x,y)dy1.1-x2idxL(x,y)dy1.I4x22dxkf(x,y)dy2n2

2、1dxlf(x,y)dy或4y24y221dy4y2f(x,y)dx12dy4y24y2f(x,y)dx11dy1y4y2f(x,y)dxr214y1dy.12f(x,y)dx.12.01dxxf(x,y)dy;(2)4dx0xx2f(x,y)dy;(3)11dx10x2f(x,y)dy;1(4)1Fy12y21f(x,y)dx;2-(5)1edy0eyf(x,y)dx;01dy2arcsiny1f(x,y)dx0arcsinydyf(x,y)dxarcsiny203.(1)3;(2)3;26;55(4)e1e;(5)94;(6)21.4.-.35.71216.17.629.(1)0bdaf(

3、cos,sin)d;(2)d2cos0f(cos,sin)d;2(cossin)(3)2d0f(cos,0sin)d.seccsc10.(1)4d0f(cos,sin0)d2d0f(cos,sin)d42sec0f(cossin1f(cossin)d;04dsecsectanf(cos,sin11.(1)3a421;e41);326412.(1)g(2);14a4;2 .3(b3a).213.a.414.(1)6;715.(1)ln2;32如e二2;1(3)ab.216.(1)提示:作變換(2)提示：作變換11x211.(1)1dxdyx2y2f(x,y,z)dz;11x22x21dx1x2d

4、yx22y2f(x,y,Z)dZ;11x2y21dx好dy0f(x,y,z)dz;11xxy(4)0dx0dy0f(x,y,z)dz.11(2)-(ln2258);222.(1)“;(3)0;(4)hR;364417164.(1)-;84(2)127;4(3).3455、5.(1)一;5611x2a;2(3)(Aa).1522xy6.直角坐標(biāo)系柱面坐標(biāo)系dx1一21 1x2 1dd002一dyx,T22,2f(x,y,z)dz;y2f(cos,sin,z)dz;球面坐標(biāo)系d4d000f(rsincos,rsinsin,rcos327.(1);33a3;2(3)2匚；-(554).632)r2s

5、in8.4t2f(t2).9.kR4.dr.15/26)a2.2.223.16R2.5.(1)x0,y4b(2)xb2ab2a3;2(ab)0,0,3(A48(A3a4).a3);(4)0,0,3.46.Ix72,I596y亍7.(1)84一a;3(2)x0,y0,z7152a;8.FxFy0,Fz2G.(ha)2R22.aR2,yh.0;1126a45總習(xí)題71.(1)(C);(A);(3)(B);(4)(D);(5)(B),(D).2t2.(1);3(2)0;2;(4)4;(5)1m423.(1)R49R;(2).f250168.234.(1)a.334R4.3hf(0).第八章1.(1)

6、Ix2Ly(x,y)ds,Iy(x,y)ds;x(x,y)ds(x,y)dsLy(x,y)dsL(x,y)ds2.(1)2a2n1(5v5126一21)；(27a)2；31(1);(5)9；2563a.15asin4.6k36.(1)-a;絲15.33k2(4ra(5)13;1-.323347.j;(2)11;(3)14;310.(1)P(x,y)Q(x,y)ds；(2)P(x,y)2xQ(x,y)14x2ds;11.L2xx2P(x,y)(1x)Q(x,y)ds.P(x,y,z)L2xQ(x,y,z)3yR(x,y,z)22ds,14x29y21.(1)8;1302.12;(2)0;(3)-

7、a4;2(4)tsin2(1)(1)52;(2)236;o122xy-y;235;(4)-2(2)y2cosxx2cosy;x3y224xy12ey12yey.1.Ix(yz2)(x,y,z)dS.3 3.(1)4.(1)13;61;1493027411110a(a2h2);644(4)一2a.155. 2一152_7(1)R;1051).37.(1)(一P55q3223R)dS;2xP2yQR4x2dS.4y28.8.1.(1)125812.(1)0;a3(22a);1083.(1)2x2y2z;yexyxsin(xy)2、2xzsin(xz);(3)2x.291. 3a;(2)2a(ab)

8、;(3)20;(4)-.2. 2(1)2i4j6k;(2)ij;3. (3)xsin(cosz)xy2cos(xz)iysin(cosz)jy2zcos(xz)x2cosyk0;(2)4.2;(2)12;6.0.總習(xí)題81.12a;(2)4a;(3)4;(4)6;_3.222、_3-(5)2R();(6)2R;(7)(C);(8)(B).2.(1).2ln3(1-)ln242.22arctan.2;18(3)0;(4)2(5)一163.(1)24.8.+Harctan;R125.6.h4；2.7.WmaxabC98.3.2習(xí)題9.11.Un(n1)ln(n1)n1x(n1)!；(2)Un(4)

9、Un1)2.3.4.5.收斂；(4)發(fā)散；(4)1)n1sinnxn發(fā)散.收斂；(5)(1)收斂；(2)發(fā)散;(3)(1)發(fā)散；(2)發(fā)散;(3)提示：利用數(shù)列收斂與其子列收斂之間的關(guān)系提示:s2n1收斂;(6)發(fā)散.$2U2n1.習(xí)題9.21. (1)(1)(7)發(fā)散；發(fā)散；收斂；收斂；發(fā)散；(4)收斂；(5)收斂；發(fā)散；(4)收斂；(5)(8)ba時(shí)收斂，ba時(shí)發(fā)散，收斂.收斂;收斂；(6)收斂；(6)ba不能確定.3.4.(1)收斂;(2)(1)絕對(duì)收斂；收斂；收斂；(4)發(fā)散；(5)收斂；(6)收斂.條件收斂；(3)條件收斂；(4)發(fā)散;(5)條件收斂；(6)條件收斂.6.提示:色bn

10、a1b.7.提示：四n1(un21).n8.提示:0cnanbnan.9.提示:a習(xí)題9.31.(1)1時(shí)絕對(duì)收斂,當(dāng)|a1時(shí)發(fā)散,時(shí)條件收斂，a1時(shí)發(fā)散(4)2.(1)1,1,1；1R一，2,L11井;R1,1,1；)；3,0,6);(6)R!1,0).1 .1x.ln(11xx1)；(2)一(14x34x1)；x1);ln(1x)(1).3.s(x)arctanx,1,1;2arctan皇.24.(1)-；(2)4;4ln2;(4)2.1)習(xí)題9.41.cosx2.(1)(n02nx2nnx1)(2n)!).0(2n1)!(lna)n);(2)ln2nx,n!);(4)1(5)(1)n1n

11、xn1n(n1);3.(1)(上(n1021)n,x(3,1);n1),x);(4)2n1(1)n口(x(2n)!-)2n(x(2n1)!4.(1)(1)nnTx2(2n)!2n2n1(2n1)(n!)1)nnxnna2,2;1)n22n1(2n)!arctan21,1);1,1,2nx(1)n22n12n12nx-,-.221)n1(x1)nn(0,2;1)n(1f(n)(0)12不)(x2)n,(13).0,2(2k)!22k(k!)2,n2k,n2k1.6.f(x)n1nxn1(n1)!).7.(1)0.9848;(2)0.9461.習(xí)題9.52.(1)f(x)1rcos(4n3)xco

12、s(4n1)x,x(2k1),k0,1,2,m4n34n1f(x)J1(1)n(ab)(1)n1(ab).2cosnxsinnx1nnx(2k1),k0,1,2,；(3)f(x)-42cosnx,x3n1n33n(4)f(x)x(2k1),k0,1,2,.ee1(1)-(3cosnxnsinnx),6n19n、541,、r,3.f(x)cos(2n1)x,x1,1,2n0(2n1)224(1)n13.(1)f(x)2cosnx,x,;n14n1f(x)e11(1)nen1cosnxn(1)nne(1)nnsinnx,4.f(x)1-n(1)n1-_-(3n1)631n(1)nsinnx,x(0

13、,5.f(x)Ucosnx,n0,.6.f(x)1cosnhsinnx,xn(0,h)(h,);f(x)sinnhcosnx,nx0,h)(h,).習(xí)題9.61.(1)f(x)(1)n1cosl2nx);(2)f(x)1-sinnf(x)11122nx亍xn12cos2nnkT,k0,1,2,一;(1)x,x1 .x2n.(n1).nx2.(1)f(x)-sinsinsinln2n12lx。身（；;,1f(x)1xcosl12n21n1(ncos2八nxII1)cos,x0,；)e,l;l2241(1)f(x)-3sinnx,x0,1;n1n121(1)nf(x)二2cosnx,x0,1.6n

14、1n4.(1)f(x)5.6.7.(4)f(x)f(x)f(x)f(t)總習(xí)題91.2.(1)C;8;223.4.5.sinnxx(0,2),n)(1)n1-sinnx,xnsin2nx,x(0,)(1(0,);x,x(12n11n(1)n1.sinnn1n(1)n1o.nnsinnx,(1)n|inxen02E1.n-sin1n(3,5)x(3,5)ntcos1,1).2kl,k0,1,2,.C;B;1,0p(4)A;1,PA.0;R2；2,2),s(x)2x-ln(1-),x2,0)(0,2),x21x0;(2)發(fā)散；發(fā)散；(4)發(fā)散；(5)a1時(shí)收斂，a1時(shí)發(fā)散，a1且k1絕對(duì)收斂；收斂

15、；0a(1)(6)(1)1k一時(shí)收斂，k2絕對(duì)收斂；條件收斂；-時(shí)發(fā)散.2(5)1時(shí)收斂，0a時(shí)收斂，a1且0條件收斂.1時(shí)發(fā)散；k1時(shí)發(fā)散.116.捐11(-,-);e(2,0);(4)(1,1).1(11arctanx21);(2)s(x)1-)ln(1x),x0,1,(1,0)(0,1,x0,x1;x1(0(3)(2x)28.1ln3;49.(1)8n(xxn010.f(x)81n1習(xí)題10.11.(1)1;:22.1不是;x2);8n1)(12227sin(2n1)x(2n1)32(4)(22土(2x.2)x)1);En2n11).(0);322;2不是;313不是;42,4是,4.1

16、y2(1y2xy2xy2y0,2xy22y；2y2xxe,yy0,22)4;5.16.2x習(xí)題10.21.2.(x1)2y2C；2sinycosxC；4yC(xa)(1ay)；6y212ln(ex1)2ln(e1)；2(1x)y1；4353Inx1(1x2)(1y2)Cx210x10yCy(x一x2一1)Cyarctanxe4；lny.xtan-23.f(x)4.cxyxe；2(與3Ceytan(xC)5.6.2xsin()x3xCeTsinx2(xC)ex(lnlnxC)xaeabesinx2esinx2;3y22x2Cexx8.1y2ex1；39.1是，x_2xyy33是，(1e2)C7.

17、c；y5(5x3Cx5)12，51y2(1Ce)；1cosx6x(xeC)3、1x24334xy是,x213x2(12exC(1x82)92lnx)ycosxxcosy10.1x44xy2y2C；.x.一arctan(-)xCy3.4秒,13.1xarctany42yx2xIn2C1x3C2x2C4G(xx)C2；C1xC244(Gy1)C12(xC2)21.(1)相關(guān);2無(wú)關(guān);3無(wú)關(guān);4相關(guān),2.y(C12C2x)e,3.5.6.C1xC2e2x；2yC1e2xC2(2x1)x)C2(x21)1,G(1(1)k(2k1)!2k、x/C2(1)kLk1(2k)!k0(2k1)!2ky1yyG(

18、x22k1、x)1kx,1(2k1)!7.22x3xC1eC2e；2C1C2e4x2C1e(12)xC2e(1xe(G3cos一x2C2sinW)2(5)當(dāng)a0時(shí)，yG*%C2eE0時(shí)，yC1C2xyC1cosOxC2sinOx；(6當(dāng)1時(shí)，yC1e(21)xC2e(21)x1時(shí),yC1exC2xex；1時(shí)，yex(C1cos、12xC2sin12x)；7yC1exC2exC3cosxC4sinx；cos.12x8yC1cosxC2sinxC3；9y(C1C2x)cosx(C3C4x)sinx；10x(C1C2x)e(C3C4x)e2x；11eax(C1C2xC3x2）；12y(CiC2x)e

19、xC3cosxC4sinx；8.1x3x4e2e；(2x)eex(cos3xsin3x)5xcost3y1.-tsint2(4x22x)e；9.ycos3x1.csin3x,310.C1exC2e3x22(C1C2x)e2x1)；11.561C1C2ex-x33x6x7CeC2ecosx742xC1cosxC2sinx1xcosx;25一sinx:74x3e2(C1cosx2C242Aex(BCx)cosx(D、2.-3-x)sin2xcos2x2626Ex)sinx；3yexx2(BCx)(Dcos2xEsin2x)；4ye3xx(Ax2BxC)(Dcos2xEsin2x)；5yx(BCx)

20、cosx(DEx)sinx;6yA2x,1x12x11c1ye-ex_;2y一cos3x-cosx;222483yex(xsinx)；4y2xexsinx,1y1_(C1InxC2):2yC1C2Inxax；x3yx(C1InxC2)xln2x;4yx(C1InxC2)C3x2(DEx)sin2x；y12.13.xacosft1.9秒,4x_xe(BCx)cos2x14.1.32x3(233lnx)C；42.4.6.5f(x)x(cx)cxCarctan.C1；63.(x)cosxsinxxCyn或y5.Cx1y(CiC2x)e2xxGexC?e1一e161cos2x，102yC1cosxx.

21、cos3xC2sinxsinx:4164yC1cos3lnx)C2sin(龍Inx)烏sin(lnx)，7.(x)C1exC2e2x8x.yeJsinx2習(xí)題11.11.(1)Rez邑,Imz13Rez3,Imz2Rez7-,Imz2,(4)Rez1,Imz2.x1,y11.3.(1)i1x(222x1)e,9.2-宿,z1-2,z13,z3,z1cosisin22ie2;(3)sin6.(1)8i;7.1.9.(1)約2.8秒.32.1Ai,z,Argz1313131.10-i,z,Argz5293i,zarctanZ31arctan-32k2k(k(kZ);Z)；13i,z,Argz10,

22、Argzarctan31cosisin26arctan72k(kZ).2k(kZ)；2i16316i;(3)62e仍i,62eT2i,62e3i；(4)22iicoscos()isin()e6;(4)3667.5為中心，半徑為2的圓周；直線x3；2(cos41；2，isin)2e4431.一i.22(4)中心在2i，半徑為1的圓周及其外部區(qū)域；10.(1)直線yx;雙曲線xy1；(3)雙曲線xy(4)不包含實(shí)軸的上半平面1在第一象限中的一支；(4)拋物線yx21.習(xí)題11.21.(1)w1i,w222i,w38i；(2)0argw1221221(3) (1)圓周(u-)v日；(2)圓周uv-;

23、1直線vu；(4)直線u分.2. (1)不存在；(2)0;(3)不存在.3. (1)處處連續(xù)；(2)除zi外處處連續(xù).習(xí)題11.31.(3) (1)在直線y；上可導(dǎo)，在復(fù)平面上處處不解析；(2)在直線屈思y0上可導(dǎo)，在復(fù)平面上處處不解析;在z0點(diǎn)可導(dǎo)，在復(fù)平面上處處不解析;在復(fù)平面上處處可導(dǎo)、處處解析.3.(1)除zi外在復(fù)平面上處處解析，f(z)2z(-1)2(2)當(dāng)c0時(shí)除z外在復(fù)平面上處處解析c(z)adbc(czd)24.In323, m1,f(z)iz,f(z)3iz.習(xí)題11.41.ei;避4，e(122.1-In2i(-242ke3.；(2)k4.i；(2)專i)；ch1;(4)

24、sin1ch2icos1sh2.kZ;(2)In54iarctan-3(2k1)i,kZ(2k1)e4,In2(cos2.In2isin),k2Z.;(2k1)i;(4)k,4這里k0,1i;(3)(2k2)i,這里k0,1,2,.2(lnz)2k),(4)1,2,.5.lnz與Lnz在除原點(diǎn)與負(fù)實(shí)軸外處處解析，且(Lnz)6.eLnz對(duì)每個(gè)單值分支在除原點(diǎn)與負(fù)實(shí)軸外處處解析，且(z)總習(xí)題113,2,Imz(2)充分，必要；1.(1)Rez32C;3333.-2,argz,z-i;2422(4)a2,b3,c2；(4)siniish1一2ke2(kr、1Z),ln(1i)1ln2-i.2k2

25、.(1)2(13i);(2)cos4-2kisin,k0,1,2,3.43.(1).2(22)i;(2)itane,0,.455556.f(z)處處不可導(dǎo)、處處不解析._28.(1)ln2g2k),kZ;(2)e.3習(xí)題12.12.1333i)；13133(33(3i).15153.(1)-i；i.4.(1)i;2i.66665.4i;8i.6.(1)0;(2)0.習(xí)題12.21.(1)0；0；(3)0；(4)0.2.相等；不能利用閉路變形原理.3.0.4.(1)0；.5.i.6.0;(1-sh2)i;(3)sin1，cos1；(4),/1(tan12212tan1-th1)ith12習(xí)題12

26、.32.2e2i；J；(3)一；(4)0;(5)0；(6)0;0；(8)-.ae123.(1)0；0,蘭NI1時(shí)；ie，當(dāng)11時(shí).4.當(dāng)與都不在C的內(nèi)部時(shí)，積分值為0;當(dāng)與中有一個(gè)在C的內(nèi)部時(shí)，積分值為i;當(dāng)與都在C的內(nèi)部時(shí)，積分值為2i.習(xí)題12.44. vx22xyy2C,f(z)(1i)z2iC(C為實(shí)數(shù)).5. (1)i(z1)2；(1i)z2iC(C為實(shí)數(shù))；iz21；(4)lnzC.6.當(dāng)P1時(shí)，v為調(diào)和函數(shù)；當(dāng)p1時(shí)，f(z)ezC(CR);當(dāng)P1時(shí)，f(z)ezC(CR).總習(xí)題121. (1)D;(2)D;(3)C;(4)D;(5)B.2. (1)0;(2);(3)i;(4)

27、2i；(5)12i；(6)643. (1)0;(2)2i;(3)ei;(4)(2e)i.5.2i.9.uC1(axby)C2.習(xí)題13.14. (1)收斂，極限為1;(2)收斂，極限為0;5. (1)發(fā)散；(2)發(fā)散；(3)絕對(duì)收斂；(4)條件收斂i.收斂,.極限為0;(4)發(fā)散6.(1)2；(2)12；(3)1；(4)1.習(xí)題13.22.3.n(z1)nnz11)o2n1z2n,R(2n)!；(5)n1(1(z1)n,R2；2n2n1,R0(2n1)!(EpU4ncz1)n,R;(2n)!(6)MRn0n!)(z2)n,R3；3nzo(13i)n1z(1i)103(4)(1)n11-(z1)n,R1習(xí)題13.3(n0習(xí)題(z1)n,(1)n1.(z-、n2，n1n02)1,21(，5zz211z2zzz2482.(1)nn3z16(n1)；2)zn,(1)n(z1)n；2n(n1)n-(zi)13.4(zi)n0(2i)n1)in13,1_J2i)0(zn2，i)(5)(6)(112!z1)n