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Baixe (SOLUTIONS MANUAL) Calculo com Geometria Analítica - Earl Swokowski vol.2 e outras Manuais, Projetos, Pesquisas em PDF para Engenharia Mecânica, somente na Docsity!
ery A. Cole -K. Rockswold Instructor's Solutions Manual Volume 2 to Accompany Swokowski's Calculus: The Classic Edition Jeffery A. Cole Anoka-Ramsey Community College Gary K. Rockswold Minnesota State University, Mankato Brooks/Cole Thomson Learning. Australia + Canada + Mexico + Singapore + Spain + United Kingdom + United States PREFACE This manual contains answers to all exercises in Chapters 9 through 19 of the text, Calculus: The Classic Edition, by Earl W. Swokowski. For most problems, a reasonably detailed solution is included. We have tried to correlate the length of the solutions with their difficulty. It is our hope that by merely browsing through the solutions, the instructor will save time in determining appropriate assignments for their particular class. AM figures are new for this edition. Most function values have been plotted using computer software, and we are very happy with the high precision provided by this method. We would appreciate any feedback concerning errors, solution correctness, solution style, or manual style. These and any other comments may be sent directly to us or in care of the publisher. We would like to thank: Editor Dave Geggis, for entrusting us with this project and continued support; Earl Swokowski, for his assistance; Sally Lifland and Gail Magin of Lifland, et al. Bookmakers, for assembling the final manuscript; and George and Bryan Morris, for preparing the new figures. We dedicate this book to our wives, Joan and Wendy, and thank them for their support and understanding. Jeffery A. Cole Gary K. Rockswold Anoka-Ramsey Community College Minnesota State University, Mankato 11200 Mississippi Blvd. NW P.O. Box 41 Coon Rapids, MN 55433 Mankato, MN 56002 The following notations are defined in the manual, and also listed here for convenience. DNE LIS TS 2 (8) (6) GD AC, CC DERIV VI (Does Not Exist) (the original limit, integral, or series ) (the result is obtained from using the trapezoidal rule or Simpson's rule ) (integration by parts has been applied— the parts are defined following the solution ) (LºHôpital's rule is applied when this symbol appears ) (converges or convergent, diverges or divergent ) (absolutely convergent, conditionally convergent ) (see notes in $11.8 and $11.9 for this notation ) (vertex, focus, and directrix of a parabola ) C, V, F, M (center, vertices, foci, and end points of the minor axis of an ellipse ) C, V, F, W (center, vertices, foci, and end points of the conjugate axis of a hyperbola ) D (discriminant value (B? — 44C) in $12.4) vT, HT (vertical tangent, horizontal tangent ) 14 (increasing, decreasing ) cx (critical number(s) ) PI (point(s) of inflection ) CU, CD | (concave up, concave down) MAX, MIN (absolute maximum or minimum ) sp [saddle point) En] (equation number n) s (surface area) J (Jacobian) A (the value of F-<—fi, —f,1>) IF (integrating factor ) LMAX, LMIN (local maximum or minimum ) VA, HA, OA (vertical, horizontal, or oblique asymptote ) QL QI, QUI, QIV [quadrants 1, II, HI, IV) NTH, INT, BCT, LCT, RAT, ROT, AST (various series tests: nth-term, integral, basic comparison, limit comparison, ratio, root, the alternating series ) INSTRUCTOR'S SOLUTIONS MANUAL VOLUME 2 13 14 15 16 vi PLANE CURVES AND POLAR COORDINATES 13.1 13.2 13.3 134 13.5 13.6 VECTORS AND SURFACES...... 141 14.2 14.3 14.4 14.5 14.6 14.7 VECTOR-VALUED FUNCTIONS .. 15.1 15.2 15.3 15.4 15.5 15.7 PARTIAL DIFFERENTIATION.. 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 Plane Curves e 159 Tangent Lines and Arc Length e 170 Polar Coordinates e 176 Integrals in Polar Coordinates e 188 Polar Equations of Conies e 194 Review Exercises e 199 Vectors in Two Dimensions e 209 Vectors in Three Dimensions e 213 The Dot Product e 216 “The Vector Product e 220 Lines and Planes e 223 Surfaces e 230 Review Exercises e 237 243 Vector-Valued Functions and Space Curves e 243 Limits, Derivatives, and Integrals e 248 Motion e 254 Curvature e 258 Tangential and Normal Components of Acceleration e 266 Review Exercises e 269 Functions of Several Variables e 273 Limits and Continuity e 278 Partial Derivatives e 281 Increments and Differentials e 288 Chain Rules e 294 Directional Derivatives e 300 Tangent Planes and Normal Lines e 305 Extrema of Functions of Several Variables e 310 Lagrange Multipliers e 319 16.10 Review Exercises e 323 17 MULTIPLE INTEGRALS .. 18 19 171 17.2 17.3 174 17.5 17.6 17.7 17.8 17.9 Double Integrals e 331 Area and Volume e 339 Double Integrals in Polar Coordinates e 343 Surface Area e 348 Triple Integrals e 350 Moments and Center of Mass e 356 Cylindrical Coordinates e 362 Spherical Coordinates e 366 Change of Variables and Jacobians e 370 17.10 Review Exercises e 379 VECTOR CALCULUS ... 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 DIFFERENTIAL EQUATIONS... 19.1 19.2 19.3 19.4 19.5 19.6 19.7 APPENDICES..... I Vector Fields 389 Line Integrals e 394 Independence of Path e 399 Green?s Theorem e 402 Surface Integrals e 406 The Divergence Theorem e 411 Stokes Theorem e 415 Review Exercises e 419 Separable Differential Equations e 425 First-Order Linear Differential Equations e 429 Second-Order Linear Differential Equations e 433 Nonhomogeneous Linear Differential Equations e 435 Vibrations e 438 Series Solutions e 440 Review Exercises e 445 Mathematical Induction e 451 2 EXERCISES 9.1 [DI É stato [E ado = tanto qnt +3+0 A. = tuo de= plgdodo=dov=s MD! 8 conte | Emdo=asntiz4 (l-2+0 1l—r A. v=siêto de= lodo do= do 052 Hg 2 geme Bjo a = 3 (gma 2) + C A v=ad=tld dd v= 3 0912 je mz-lfzdr=30(3nz-1)+C A u=had=Ld,do=rd,v=i 031 À zcotz + Jcotzdz = —zcotz + Inlsinal + C A u=z du = dz, do = esclzdr, v=—ctz Aja? tania = [Edo = de tanto (1 ge= 32 tanclz — dá + Jtan” lz+C A. u= tanta do = Lodo, do= sdo,v= bo? "2 sinz + Je" coszde É — sin + [—e* cosz— [e*sinzdr] > 2= A. v=sing, du = coszde, do = € td v= —E* “*sinz — e*cosz=I = —le*(sinz + cosa) + C B. u=cosz du= —sinzdz, do= e dev= eºº cos2z + 2e'* sindz — 4º? cosdade => 181 = Jeºº cos2z + 2e%º sindz BEI À qe cos2a + 3[e%º sindede E 26% cosdo + [e sindz — 3j e% cos2z dz | =1 - >I= e" (3cos2z + 2sin2a) + C A. u=cos2z, du = —2sin2rda, do = e de, v =)” B. u= inda, du = 2cos2zdr, do = e de, v = Je* [BLet y = cosz. Then dy = —sinzde. By Example 3,1 = —[Inydy=—ylny+y+C=Yl-ny)+O A st] ER, si] ==] Bm 8 [pec + [setas = [peer] + [er] = = -0)+(el-)=1-cl=013 A u=2?du=2ede,dv= cedo v= es? EXERCISES 9.1 3 À ccscrcotz — Jesczcotirde = —escacotz — Jesce(esciz — 1) dz = —esezcotz — [escêzdz + fesczdz => 21 = —csez cotz + Inlescz — cota] == -lesczcotz + ilnlescz — cotz] + O A. u=csca, du = —cscr cotzda, dy = cscizdz, v = —cotz EMI É tanzsec?z — 3/ secêztan?rdo = tanz seca — 3/sec%a(sectz — 1) de = tanz sec?z — 3[secêzda + 3/secêzdz. By Example 6, 41 = tan z secºz + 3tanzsecz + Slnjsecz + tanz). Thus, I = jtanz sec?z + E(tanz secz + Insecz + tanzl) + C. A. u = sec?z, du = 3secir tanzdz, du = sec?:dz, v = tanz era [2[2 +) oferta [224] [8 + 0] = - (2-)-HNB-D=32-5)= 020 A u=2du=2rde, v= ei dê + [291 É ssinlnz — fcoslnade 8 esinlno — [reoslnz + [sinlnede] > 2 =zsinlnz-—zcoslnz>I= dz(sinlnz — coslnz) + € A. u=sinlnz, du = CSl0Z gr dy = day B. u=cosnz du = —SBhZgz dy=d,v=s papça Je cos2e];7? + 177 contado = [-Iecos2a]5/? + [Asinzo];? = =X-5-0)+30-0)=7 = 0.79 A. u=2,du = da, do = sindzda, v = —Icos2z [BJI.2 jo tanõe — [Hanõade = io tanõe + dlnlosõe] + O A u=2du= dz,do= sec?5zdz, v = itanõz EMI É siga(2o + 3100 — hoj (2o + 3) de = aoor(2z + 3) — ral +34 o = auhvo(2z + 3)!0[202z — (22 + 3)] + C = qoimo(2z + 3)10(2007 — 3) + O A u=a du = de, do = (20 + 39 de, v= sla(2 + 3100 Bar 2h +o2fi-sa=-[1— =-Hh-a[s o midi en) sa E) ga +o A u=7,du= 37d, do = EXERCISES 9.1 5 dr + + DL df(o+ Ma > BI He+ +! ala+1)2+C talz + Dº[IXc +29) (2+1)]+C=3iz+0Mz+2)+C A u=2+2du=dodo=(r+1)Cdo=i(r+ EI 2 cm — mfar-te do A u=2",du=ma"-ld dv= Edrv=e 40) 1 É —2” cosz + mf2"! coszdz A u=2”,du= ma”-lde, do = sinzdo, v= —cosz 2 e(lno)” — mi (Ina)"-tdr m(Ina)moA Auv=(n”, das qdo=d,v=a [1 É sec"-?rtanz — (m — 2)fsec"-2z tanizdz = sec"-2rtanz — (m — 2)J sec"-2s(sectz — 1) de = sec"-2r tanz — (m — 2)J sec” dz + (m — 2) [sec"-2zdr > (m— 1)1 = sec”-?r tanz + (m — 2)[sec"-2zdr > ma 1= tec rtanz | mM psemadoform fl. A. u=sec"-2z, du = (m — 2)sec"-2z tanzdz, du = sec?zdz, v = tanz 1 = —5fie do = ate — s[ate — af de) = — sete + 20[2 — 3[22e*do] = E — bote? + 202%eº — 60[22eº — af ne" dr ] = ate? — Gate? + 202% — 6022eº + 120[2eº — [e dz] = e(aº — 52! + 207º — 607º + 1207 — 120) + O 91 = z(ina)t — 4J (Ina)? do = a(lna)* — a[z(ina)º — 3] (n2)? dz] = (Ino)! — do(n o)? + 12[2(Ina)? — 2fInzdz] z(Ina)* — 4z(ln0)?.+ 12z(In20)? — 24(2Inz — 2) + C (by Example 3) a[(no)! — ano)? + 12(no)? - 24m= +24] + O H = [2a [e sinz, A "2 , (T2cosyz =[ sinqedo = [" RES E 8 [agreste]; +]; “E = [52 cos ve + 2sine [O = (2040) (0-0) =24 1 de, do = SUNZ go, o = 2085 6 EXEROISES 9.1 n/2 x/2 6) Using disks, V = 1| (alsmz)?do = 1] "Poesia o o > r, n/2 [52 cosa]? + 2x] zcoszdz o lo , n/2 «[-2 cosz + 2esinz [7 — 2m[ sinzde o = -[-* cosz + 2rsinz + 2cosz 2 =a(r-2) 8 359. A u=2?du=22de, do = sinzdov= —cosz B. u=2,du= dz dv=coszdz,v=sinz [EM Using shells, V = dm 2lnsés & om[je? n=]| — 2m[d Jjede= 2m[h na — 35º) = 3[2(2me — v]= A u=lnsd=ld,do=rdov=i W= O do= Ih Era A pes +] — 3], A + 1) do =D +” a e nt = [00 at” + [2-2 + 4]=40B+0=02 2 du= 37 de, do= 2 [P+Iidov= AS +41) [EB] Let f(2) = e*, 9(2) = 0, and p = lin (6.35). m= [Pe de=. o (e? + 1) = 13.18. A u= Ins Ins <=3 SP dr> = cd Are E - Me = df; (de= 2. my = [] ze ds 8 [ze nd =3ln3— 2. 2=4 = 3-2 06andy= UE =2=1. A u= du = de do= de v=e ED sy) = [ugdi= [1a A Met pipe ta = Me! 14 O ()=0>-1+C=0=C=lands(t)= ie? le 41. A u=tdu=d,do=eld,v= le? [6] Substituting (v + C) for vin (9.1) yields Judo = u(v+ C)— J(v+ C)du = uv + uC— fudu— Cu = uv fudu, which is (9.1). [52] Since f! is only positive or only negative, q = f" exists on [a, 5]. Using Figure 52a and disks, V = 7 ) “Ee dz £ r[a[H0)]) — 2), zH(2)f'(a) de eiliagad = em, =f(o) flo) do. (cont) 8 EXERCISES 9.2 I= [(t=ges2e) ar =4J(1 — 3cos2z + 3cos?2z — cos? 22) de =4J[1 — 3cos22 + 3(1 + cos4s) — (1 — sin?22)cos2z | dz = 4I(g — 4cos2z + Scos4z + sin? 22 cos 27) dr = &(iz — 2sin2z + gsin4z + |sin?22) + C I= [(t=ges2e (1 + gosze) go = 3541 — cos2z — cos22z + cos? 27) dz = A(O — cos2z — X(1 + cos4z) + (1 — sin?22)cos2z ]dz = 4J(4 — jcos4z — sin? 27 cos22) dz = A(iz — Asin4z — sin? 27) + € I= ftanºz(1 + tan?z)sec?rdz u = tanz, du = secizdz > I=[(+u)du=iut ++ o S(1 + tanZa)2secêzdz, u = tanz, du = sec?rdz > I=[0+0)du=](1+2+u)du=uriu ++ J(sec?z — 1)sec?z secz tanzdz; u = secz, du = secztanzdz > I=J(ut-uW)du=iê + C = J(sec?z — 1)?secz tanzdz; u = secz, du = secztanzdz > I=f(u-1du=](ut-+idu=iê-i +u+C [EB = Jtanfz(secêz — 1) de = J[ tantz sec?z — tan?e(sec?z — 1) Jd = J(tanzsecêz — tan?z sec?z + sec?z — 1) dz =itanfr — ltanfz+tanz— 2+C = Jcote(escz — 1) dz = J[ cot?zcsctz — (escta — 1)]Jdo = —jctês+cotr + z+O =J(sino)'2( — sin?o)coszds u = sinz, du = cosede > 1= (2 8 qu = Qu? — J(sin 2 — sin?ocoszdz u = sin, du = coszde > 2 qu = 2”? 2 4 O = I(tan?z + 2 + cot?) do = J[ (tanta + 1) + (1 + cot?s)Jdo = q!/2 +C J(sec?z + csc?z) dz = tanz — cotz + O [BI = f(csc?z — 1)csc?z cscz cotzdz; u = esc, —du = cscrcotzdr > I="[(t-W)d=i-bS+C n/4 . EI = F: (1 — costz)sinzdz = [-cos= + cost [! =3— o EmI= [ea Sm a * 0.08 -iJé = [Etan (53) a | =$-1x0.27 EXERCISES 9.2 9 Note: Exercises 21-24 use the trigonometric product-to-sum formulas. BD 1 = 1J(cos2z — cos8z) de = Másin2z — Jsin8a) + O ,/4 if; [cos6z + cos(—42)] de = A[sin6z + Asindz ||! = —3 ia ” af; [ne + sins]de = 3[cosõe — cos= [7 =3[0-(-]=3 Bal = If[sin7z + sinz]dz = —IicosTz + cosz) + C B5I = Jcotiz(1 + cot?z)cscirdz; u = cotz, —du = cscizdz > 1=—S(ut+u)du= du + O EB1= [1 + 2cos2)!/? + cosz ]sinzds, u = cosz, —du = sinzde > I=-[(142 + udu= 2-sinz, —du=coszdz51=-[Ãdu=-he+Cu>0 EI = J(sin?z — cos?) dz = | —cos2zdz = —isin2z + C Eu=1+tanz du=sectrdr>1=[uldu=-1+0 [B01 = Jtanfzseczdz = J(sec?z — 1)?secztanzdz; u = secz, du = secr tanzdr > I=[(2-1du=[(u!-2+Ndu=is-I pu+C sing syrametry and disks, V = af “atadiflio= am[;! “[ia-+ cos2a)]º do = =[” [1 + 200822 +41 + cos4z) |dz = n[3z + sin2z + gsindz]? = 222 a 7.40. /4 [BB] Using disks, V = pe m(tan2a)? do = «f; tanZo(secto — 1)de = =[º [tan2z sec? — (secêz — 1) Jde = m[Btan?a — tono + 2]! = =(5-3) = 037. BB] Since (1) = cos? xt > 0, the distance traveled in any 5-second interval is given by d2+5)-d)= pe agdi= [E costmias = [Ea + cos2ri) dt = u+ desinzmi | = [M0+5) + &sin[2n(2 + 5)]] - [32 + &sin(205)] = 5 + &[sin(2rz + 107) — sin(272)] = 3, since sin(272 + 107) = sin(272). a(1) = sin?t cost > (1) = Jsin?tcostdt = Isint + O. (0) =10> C= s(1) = J(gsin?t + 10) dt = 3J(1 — cos?ysintdt + [10d = -jest+jcs*t + 10t+D. d(0)=0=-]+]+D=0>D=2 
