; TeX output 1997.03.13:1656y?*K`y cmr1018.01UUSolutions,ProblemSet4AM !", cmsy10Wx[q˲4.5UU#2WA՝largesphericalsnowballismeltingattherateof2 b> cmmi101ٲft^ٓRcmr73|s=h.Atthe WmomentUUwhenitis30inchesindiameter,determine」_ra)mhowUUfasttheradiusischanging,#m"V cmbx10Solution:*': cmti10TheFvolumeisV= K4K&fes3 )[rG^3ֺwher}'er6cistheradius.bSo<$"ydV"ywfe B (֍8dt_o=m4[rG2<$õdrßwfe (֍dt(|.iWhenYthediameteris30in,!ther}'adiusis15in= K5K&fes4 .Jft.Thismgives2"=<$K25Kwfe (֍4<$oѵdroџwfe (֍dt%5qwher}'e<$drwfe (֍dtisinft/h.So<$drwfe (֍dt=<$zL8Kwfe  (֍25gft/h.?^9b)mandUUhowfastthesurfaceareaischanging.mSolution:*The%surfac}'eareaisA==4[rG^2Ð,!uso<$8XdA8Xwfe s (֍dt;=8r<$zPdrzPwfe (֍dt .PWhenmr5= K5K&fes4 غft,<$drwfe (֍dt=<$zL8Kwfe  (֍25gft/h,and<$dAwfe s (֍dtuز=<$K16Kwfe (֍5Ѕft^2|s=h.PMWx[q˲4.5UU#3WSandisbGeingpouredontoaconicalpileattheconstantrateof50ft^3|s=min. WF*rictionalQBforcesinthesandaresuchthattheheightofthepileisalwaysWequalltotheradiusofitsbase.BHowfastistheheightofthepileincreasingWwhenUUthesandis5ftdeep?WSolution:*The5Bvolumeofac}'oneisV= K1K&fes3 )[rG^2Ðhwherer|_istheradiusand(nWh{istheheight.QIfrGi=Lh,`thenV90= 313&fes3 c%[h^3/and<$段dV殟wfe B (֍8dt\ز=h2<$dhwfe i (֍CdtB.QIf<$段dV殟wfe B (֍8dt\ز=50Wft^3|s=minandh=5ft,then<$dhwfe i (֍Cdtβ=2=ft/min.MWx[q˲4.5UU#6WAlightatthetopofapGoleis80fthigh.]AballisdroppGedfromthesameWheight6cfromapGoint20ftawayfromthelight.gv(AssumethattheballfallsWs=16t^2ȲftUUintseconds.)o箍>5PSfile=4.5.6.eps llx=0 lly=0 urx=272 ury=164 rwi=1791WFindUUhowfasttheshadowoftheballismovingalongtheground*y?_ra)m1UUsecondlater;^mSolution:*Similartriangles(dr}'awapicture)showthatx=80=20=s mwher}'e^xisthedistancebetweentheshadowandthepole. Thisgivesjmx=1600s^ O!cmsy71oCand<$dxwfe 뎟 (֍ Udtײ=1600s2<$ 淋ds 履wfe r (֍dtL.MSinc}'eϵs=16t^2/Band<$dswfe r (֍dt=32t,mthis?gives<$rֵdxr֟wfe 뎟 (֍ Udtʲ=3200=t3(wher}'ealldistancesareinfeet,jandtime\ mis7inse}'conds).Therefore7whent_=1,!the7shadowismovingat200mft/se}'c.^9b)m2UUsecondslater.mSolution:*Whent=2,theshadowismovingat25ft/se}'c.^MWx[q˲4.5UU#11/WAZAsphericalZmeteoriteenterstheearth'satmosphereandburnsupataWratepropGortionaltoitssurfacearea.9^ShowthatitsradiusdecreasesataWconstantUUrate.MWSolution:*Sinc}'eݙV= K4K&fes3 )[rG^3Ð,wehave<$̵dV̟wfe B (֍8dtM²=4rG2<$õdrßwfe (֍dt(|.\Sinc}'e<$̵dV̟wfe B (֍8dtdCispropor-Wtional>tothesurfac}'earea,O/whichis4[rG^2Ð,wehave<$q4dVq4wfe B (֍8dt*=4[rG2A~#CforsomeW(ne}'gative)S1constantC.Therefore<$ddrdwfe (֍dt5=C,`#soS1ther}'adiusisdecreasingataWc}'onstantrate.썍MWx[q˲4.6UU#2_ra)mShow`thatx^3+a3x^26,=0`hasonlyonerealroGot,andcalculateit mtoUUsixdecimalplacesofaccuracy*.mSolution:*TheXfunctionf(x)=x^34+@3x^26Xhasderivativef^0Ȳ(x)=m3x^2Q+6x,!Xwhichiszer}'owhenx=2orx=0. We ndthatf(x)misincr}'easingon(1;2), decreasingon(2;0)andincreasingonm(0;1).0Atpx˯=2,ther}'eisarelativemaximumwithvaluef(2)˯=m2,L and:atx=0,ther}'e:isarelativeminimumwithvaluef(0)=6.m(SketchH>thegr}'aph.)Thegraphthereforecrossesthex-axisatmostmonc}'e,Ήandsincef(1)=2andf(2)=14,Ήther}'eisexactlyoneroot,msomewher}'ebintheinterval(1;2).Wesetx1C=1andrepeatedlyapplymNewtonP'smetho}'d,whichinthiscasegivestheformulaqhx 0ercmmi7n+1X=xn^<$lx^3፴n+83x^2፴n6lwfe8 (֍ O3xr2S+86x=b8:;mThisgivesx2 ͷQD1:222222,Tx31:196215,Tx41:195823,Tx5m1:195823,:::6.^9b)mShow5_thatx^3uh+3x=85_hasonlyonerealroGot,;andcalculateittosixmdecimalUUplacesofaccuracy*. y?mSolution:*TheCfunctionf(x)=x^3dz+K@3x8Chasderivativef^0Ȳ(x)= m3x^2Ū+I73,whichtisalwaysp}'ositive,sof(x)isalwaysincr}'easing,sotitmhasatmostonezer}'o.Sincef(1)=4andf(2)=6,ther}'eisazeromsomewher}'eintheinterval(1;2).ApplyingNewtonP'sMethod:Hxn+1X=xn^<$lx^3፴n+83xn8lwfe8 (֍ +B3xr2S+83=b8:븍mwith'x1O=I1:5givesx2I1:512821,Lx31:512745,Lx41:512745,m:::zUQ.^MWx[q˲4.6UU#6tꍑWConsiderasphericalshell1ftthickwhosevolumeequalsthevolumeofWthehollowspaceinsideit.UseNewton'smethoGdtocalculatetheshell'sWouterUUradiustosixdecimalplacesofaccuracy*.HWSolution:*L}'etרrźdenotetheouterradiusinfeet.dTheoutersphereisWsupp}'osedtohavetwic}'ethevolumeoftheinnersphere,?ZwhichhasradiusWr81.Ther}'efore,wehavetosolvethee}'quationo<$̅4̅wfe (֍3[rG3=28<$l4lwfe (֍3 G(r1)3|s;HuWore}'quivalentlyrG^3߾.2(rcK1)^3C=0.0Sowearelookingforazerooff(x)=Wx^3=Hղ2(x1)^2|s..A\go}'od\startingp}'ointisx1C=5,gsince5^3"isalittlesmallerWthan284^3|s.NewtonP'sMetho}'dgivestheformula:"fLxn+1X=xn^<$x^3፴n82(xn1)^3lwfeJ (֍3xr2\nn86(xn1)r2Nʵ;cWso1x2ht4:857143,x24:847367,x34:847322,x34:847322,:::Ӻ.W(It'snothar}'dtosolvetheaboveequationalgebraically;thesolutionisWr5=0ZtZcmr53g Pp wPfeE2x=(Zt3Pp_PfeE2.@81).)MWx[q˲4.6UU#13tꍑWFindthesmallestpGositivesolutionofeachofthefollowingequations,8cor-WrectUUtosixdecimalplaces.ԍ_ra)m4(x81)=sin޵x;ԍmSolution:*L}'etڵf(x)=4(xݸ1)sinx.nThenf^0Ȳ(x)=4ݸcosxismalwaysp}'ositve,!sof(x)isalwaysincreasing,!sothereisatmostonemsolutionSoff(x)"=0.Sinc}'eSf(1)=sinn1Sisne}'gativeandf(2)"=m4 sin 2޺isp}'ositive,thereisasolutionsomewher}'eintheintervalm(1;2).XWe~Gstartwithx1!=o1:5andapplyNewtonP'sMetho}'d,whichmgives5+xn+1X=xn^<$l4(xn81)sin*xnlwfeN{֟ (֍48cosGxnS:mSoaMx2;01:244862,x31:236140,x41:236130,x51:236130,m:::zUQ.y?^9b)mx^2C=sin޵x;mSolution:*L}'etf(x)2 =x^2csinUnx.JF;romthediscussioninclass,ݳwe mar}'ezlookingforazerooff(x)intheinterval(0:5;1),ŞsowemightmstartJwithx1C=0:75forexample.>NewtonP'smetho}'dgivestheformula|ixn+1X=xn^<$zLx^2፴n8sin*xnlwfe6J (֍2xn8cosGxn;7;ύmsoux2ܗ`$0:905066,'x30:877663,'x30:876727,'x40:876726,mx5C0:876726,:::6.MWx[q˲12.2UUFindthefollowinglimits.jZ#2nMXlim-mx!1<$lnx\wfe (֍x81G.1^mSolution:*L}'ettingf(x)=lnxandg[ٲ(x)=x1,thehyp}'othesesmforĀl'H^opital'sRulear}'esatis ed(f(x)andg[ٲ(x)aredi erentiableatmx=1andf(1)=g[ٲ(1)=0.)Ther}'efore܍1lim-Nx!1<$ղlnԵx@ewfe (֍x81*=nlim-x!1<$#/1=x#/wfe (֍[1)Ԙ=1:#kZ#4nMXlim-mx!0<$\e^xAĸ8e^x\wfe%ک (֍sin5xi.ImSolution:*ThesChyp}'othesesforl'H^opital'sRuleareagainsatis ed,mso3wTlim-)x!0<$Ȇe^xAĸ8e^xȆwfe%ک (֍sin5x[ =nlim-x!0<$#/e^xAIJ+8e^x#/wfe%ک (֍V5cos5x?#=<$K2Kwfe (֍5 -:퍍Z#9nMXlim-mx!0<$\e^xAĸ81x\wfe,پ (֍Sxr2i .mSolution:*Applyingl'H^opital'sRuleagaingivesZ2lim-x!0<$صe^xAĸ81x؟wfe,پ (֍Sxr2=nlim-x!0<$#/e^xAĸ81#/wfeğ (֍T2x2@&;Imtowhichwec}'anagainapplytheruletogetlim-ȷSx!0<$je^xAĸ81jwfeğ (֍T2xy=nlim-x!0<$#/e^x#/wfe (֍Xr2#_=<$K1Kwfe (֍2 -:#7U#11nMXlim-mx!0<$x^3\wfe%ܟ (֍sin Ƶx8x((.%mSolution:*L'H^opital'sRuleappliesagaintogive|Qlim-x!0<$ॹx^3wfe%ܟ (֍sin Ƶx8x9=nlim-x!0<$!3x^2#/wfe%2 (֍cos7x81=T;*y?mandagaintogivep#klim-x!0<$}3x^22,wfe%2 (֍cos7x81*=nlim-x!0<$T6x#/wfe (֍sinnx4q ;mandagaintogive)^lim-kx!0<$9H6xwfe (֍sinnxu=nlim-x!0<$"6#/wfe7 (֍cosߵx8T=6:ΞU#22nMXlim-mx!0<$\sinMߵx^3\wfe%U ݍsin G*3n9x.mSolution:*Applyingl'H^opital'sRulegives|Ӳlim-z}x!0<$%sinZx^3%wfe%U ݍsin G*3n9xE4=nlim-x!0<$-3x^2'cos5Rx^3#/wfe5? ݍ3sin Ɵ*2xcosߵxO]s=`u cmex10 Rlim-x!0<$#xwfe (֍sin Ƶx0(`6 ݱ2>e}lim-='x!0<$Pt>cos_ux^3Pt>wfeAƟ (֍>:cosLqxmO=1;msinc}'ecos0^3C=cosO0=1and=lim-x!0<$sin$ĵxwfe (֍x.+=1.MWx[q˲12.3UUEvqaluatethefollowinglimitsbyanymethoGd.iZ#2pKlimmx!1<$Xln(ln x)XwfeӐ (֍;ln:x_N.yBmSolution:*L'H^opital'sRuleappliestothe\indeterminateform"<$͸1͟wfe  (֍1,imgiving5Ųlim@6x!1<$ln(ln x)wfeӐ (֍;ln:xf=limx!1<$1=(xln x)wfe( (֍ q1=xF=limx!1<${.1wfe (֍ln x+ ;薍mwhichis0sinc}'ethedenominator!1asx!1.g.U#10nMXlim-mx!0(xcotx.wmSolution:*Substituting~xoa=0,we ndthishastheindeterminateۍmformc'01.VWewriteitinste}'adas}lim-x!0<$>xcosߵx>wfe' (֍?sin0x7|andapplyL'H^opital'sImRule,whichgiveseOlim-Jx!0<$̦xcosߵx̦wfe' (֍?sin0x=nlim-x!0<$#/cos%1fx8xsinnx#/wfe<ן (֍²cos"xS_9;!=mwhichSis1sinc}'ethenumeratoranddenominatorboth!1asx!0.5U#20nMXlim-mx!0(^<$1wfe (֍xۼ<$ 1lwfe (֍sin ƵxH(^Gv.mSolution:*ThisEhastheindeterminateform11.WeEwriteitasc[nMXlim-mx!0<$\sinMߵx8x\wfe%ܟ (֍GxsinnxandapplyL'H^opital'sRule,whichgives lim-x!0<$sin߅x8xwfe%ܟ (֍Gxsinnx=nlim-x!0<$!(cos06x81#/wfe<ן (֍sin Ƶx8+xcosߵxS_9:mL'H^opital'sRuleappliesagain,giving*lim-{Դx!0<$>cosux81wfe<ן (֍sin Ƶx8+xcosߵx =nlim-x!0<$(sinnx#/wfeB (֍2cosߵx8xsinnx\=<$K0Kwfe (֍2 =0:7fy?U#27q"limmx!0+!Ҳ(ex0r'1)x.The\expressionhastheindeterminateform0^0 7ϲatmx=0,*soweconsideritslogarithminstead:ln h(e^xYu1)^x=xln (e^x1).܍mThiskhastheindeterminateform0nD1,qsokwewriteitas<$؞ln -(e^xAĸ81)؞wfe+9 (֍ 1=xnminsteadD(indeterminateform<$w1wwfe  (֍1))andapplyl'H^opital'sRule.lAW*eget .limx!0+<$lnF(e^xAĸ81)wfe+9 (֍ 1=x==8limx!0+<$õe^x=(e^xAĸ81)ßwfe1aȟ (֍ 1=xr2Q"ֲ=8limx!0+<$øx^2|se^xßwfe (֍`erxAĸ818: umThishastheindeterminateform<$K0Kwfe (֍0 ;yapplyingl'H^opital'sRuleagainmgivesElim{x!0+<$ʚx^2|se^xʚwfe (֍`erxAĸ81pw=8limx!0+<$ø(x^2|se^xAIJ+82xe^x)ßwfeD (֍1VerxcԞ=8limx!0+(x2S+82x)=0:ɍmThereforeUUthelogarithmoftulimx!0+w%(exAĸ81)x^9is0,so㢍f?limGx!0+h(exAĸ81)x=e0C=1:'MWFindUUthelinearizationofmu3]$p J$fe18+x&atx=0.?ɍWSolution:*withSf(x)=ru3K$p $fe18+x) cmmi10 0ercmmi7K`y cmr10ٓRcmr7Zcmr5u cmex10Vy