f 3 3.(0pts)Find dy dy dx = f x hastwoinflectionpointsatx=candx=dwith C D 5.(0 pts) Evaluate the indefinite integral. f x

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1 UofUMath0-3Olie Youg-Seo Lee. WeBWorK assigmet umber Practice. due/3/04at:0pm. ThemaipurposeofthisfirstWeBWorKsetisto review some prerequisites for this class ad to help you familiarize yourself with WeBWorK. HerearesomehitsohowtouseWeBWorKeffectively: After first loggig ito WeBWorK chage your password. Fidouthowtopritahardcopyothecomputersystemthatyouaregoigtouse.Cotactmeifyouhaveayproblems.Pritahard copy of this assigmet. Note, however, that the olie versios of these problems may havelikstothewebpagesofthiscoursethat areabsetothehardcopy. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Not olywillthisgiveyouthechacetofigure outwhat swrogifaaswerisotaccepted, youalsowillavoidthelikelyrushadcogestio prior to the deadlie. The primary purpose of the WeBWorK assigmetsithisclassistogiveyoutheopportuity to lear by havig istat feedback o your active solutio of relevat problems. Makethebestofit! Procrastiatio is hazardous!.(0pts)iff 6l 4 4l,fidf. Fidf 3..(0 pts) Let f Use logarithmic differetiatio to determie the derivative. f f 3 3.(0pts)Fid dy d foreachofthefollowigfuctios y 6 0 l dy d = 7 y cos dy d = 4.(0pts)Cosiderthefuctiof e 8. f hastwoiflectiopoitsat=cad=dwith C D wherecis addis Fially for each of the followig itervals, tell whetherf iscocaveup(typeicu)orcocave dow(typeicd). C: CD: D l 9 5.(0 pts) Evaluate the idefiite itegral. d 6.(0 pts) Evaluate the idefiite itegral. e 3 d 7.(0 pts) Evaluate the idefiite itegral. 5e 5 si e 5 d 8.(0 pts) Evaluate the idefiite itegral. d e 5 9.(0 pts) Evaluate the defiite itegral. d l C 0.(0pts)Note: Youcagetfullcreditforthis problembyjusteterigtheaswertothelastquestio correctly. The iitial questios are meat as hits

2 towardsthefialasweradalsoallowyoutheopportuity to get partial credit. Cosider the idefiite itegral 4 4 e d The most appropriate substitutio to simplify this itegralisu f where f = Wethehave d g u du where g u = Hit:youeedtobacksubstituteforitermsof uforthispart. After substitutig ito the origial itegral we obtai h u duwhere h u = To evaluate this itegral rewrite the umerator as 4 u u 4 simplify, the itegrate, thus obtaiig h u du H u where H u = +C Aftersubstitutigbackforuweobtaiourfial aswer 4 4 ed= +C Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur

3 Math 0-3 Youg-Seo Lee. WeBWorK Assigmet Number. Due//04at:59PM. ThemaipurposeofthisfirstWeBWorKsetisto review some prerequisites for this class ad to help you familiarize yourself with WeBWorK. HerearesomehitsohowtouseWeBWorKeffectively: After first loggig ito WeBWorK chage your password. Fidouthowtopritahardcopyothecomputersystemthatyouaregoigtouse.Cotactmeifyouhaveayproblems.Pritahard copy of this assigmet. Note, however, that the olie versios of these problems may havelikstothewebpagesofthiscoursethat areabsetothehardcopy. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Not olywillthisgiveyouthechacetofigure outwhat swrogifaaswerisotaccepted, youalsowillavoidthelikelyrushadcogestio prior to the deadlie. The primary purpose of the WeBWorK assigmetsithisclassistogiveyoutheopportuity to lear by havig istat feedback o your active solutio of relevat problems. Makethebestofit! Procrastiatio is hazardous!.( pt) Evaluate the followig epressios. (a)log 3 8 (b)log 4 (c)log 4 64 (d)3 log 3 6.( pt) Evaluate the followig epressios. (a)le 3 (b)e l l7 (c)e (d)l e 4 3.(pt)Solvethegiveequatiofor (pt) Ifl 6 4 3,the. 5.(pt)Ife 6,the 5. 6.(pt) l r 5 s 5 4 r 6 s 4 isequalto wherea Alr Bls adwhereb 7.(pt)Theequatioe 7e 0hastwo solutios. The smaller oe is: adthelargeroeis:. 8.(pt)Letf l cos Assumethatisrestrictedsothatlisdefied.Fid f π Aswer: 9.( pt) Fid the itegral 0 t t 4t 3 dt Aswer: 0.( pt) Suppose y 3 Fid dy d bylogarithmicdifferetiatio.seeeample7isectio7.ofyourtet. Aswer:.(pt)Therateoftrasmissioiatelegraphcableisobservedtobeproportioalto l,where istheratiooftheradiusofthecoretothethickess oftheisulatio(0!! ).Whatvalueofgives the maimum rate of trasmissio? Aswer:.( pt) Evaluate Aswer: π"3 0 tad

4 3.(pt)Letf 3.Thethederivative is D f 3 Thus,f ismootoeo # because A.f %$ 0 B.f %! 0 Hecef hasaiversefuctio.sicef & as&,adf '& as&,itfollows thattheiversefuctiof isdefiedforallumbers. Fid the derivative: D f 40 4.(pt)Supposey e " e.fidd y. D y ( 5.(pt)Supposee y y.fidd y. D y 6.( pt) Fid the itegral e e d Aswer: 7.( pt) Fid the itegral e 3" d C Aswer: 8.(pt)Theregioboudedbyy e,y 0, 0,ad isrevolvedaboutthey-ais. Fid the volume of the resultig solid. Aswer: 9.(pt)Let f ) l l fori 0. Fid a)lim * 0+ f, b)lim * f 0.( pt) Evaluate the followig limit: " lim * 0 Aswer:.(pt)Supposey si si.fiddyd. Aswer:.(pt)Theloudessofsoudismeasuredi decibels i hoor of Aleader Graham Bell(847-9), ivetor of the telephoe. If the variatio i pressureisppoudspersquareich,thetheloudesslidecibelsis L 0log 0 3P - Fidthevariatioipressurecausedbyarock bad at 5 decibels. Aswer: pouds per square ich. Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur

5 Math 0-3 Youg-Seo Lee. WeBWorK Problem Set. Due/30/04at:59PM. HerearesomehitsohowtouseWeBWorKeffectively: Fidouthowtopritahardcopyothecomputersystemthatyouaregoigtouse.Cotactmeifyouhaveayproblems.Pritahard copy of this assigmet. Note, however, that the olie versios of these problems may havelikstothewebpagesofthiscoursethat areabsetothehardcopy. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Not olywillthisgiveyouthechacetofigure outwhat swrogifaaswerisotaccepted, youalsowillavoidthelikelyrushadcogestio prior to the deadlie. The primary purpose of the WeBWorK assigmetsithisclassistogiveyoutheopportuity to lear by havig istat feedback o your active solutio of relevat problems. Makethebestofit! Procrastiatio is hazardous!.(pt)let f 5 log f.(pt)fidthelimit *. lim / e 4" Aswer:. 3.( pt) I 380 days, a ukow radioactive substacedecayto38percetofitssize. t (a)whatisthehalflifeofthissubstace? (days) (b)howlogwillittakeforasampleof00mgto decayto7mg? T 4.(pt)Acellofsomebacteriadividesitotwo cells every 0 miutes.the iitial populatio is bacteria. (a) t 0 Fid the populatio after t hours y (fuctio of t) (b) 7 Fid the populatio after 7 hours. y (c) Whe will the populatio reach 4? T hour(s). 5.( pt) A bacteria culture starts with 460 bacteria adgrowsataratepropotioaltoitssize. After5 hours there are 300 bacteria. (a) Fid the populatio after t hours t 0 y 9 y (fuctiooft) (b) Fid the populatio after 9 hours. (c)whewillthepopulatioreach80? T 6.(pt)Cosiderthecircuit(seefigure)withL 3 herys,r 5ohmsadE t 0e 8t. FidthecurretIasafuctiooftime. Assume thatthereisocurreti 0 0whetheswitchis closedattimet 0. (Clickoimageforalargerview) The curret is I t. 7.( pt) Solve the differetial equatio y y 5 wherey 5whe 0. y. 8.(pt)Let, ( f.

6 (a)evaluatef 3 (b)evaluate f (pt)Let f 54 l t dt (a)evaluatef 0 (b)evaluate f ( pt) Solve the differetial equatio with give iitial coditio dy dt 05y;y 00 y.( pt) How may years would it take your moey to double: (a) At 0% iterest compouded yearly. years. (b) At 0% iterest compouded weekly. years ad weaks. (c) At 0% iterest compouded cotiuously. years..(pt)someboesareehumedfromaaciet burialgroud.iftheratioofcarbo4itheboes is65%oftheratioitheboesofalivighuma, thehowoldaretheboes?(thehalf-lifeofcarbo 4is5730yearsaditbegistodecayimmediately after a perso dies). years. Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur

7 Math 0-3 θ = θismaimizedwhe 0 = Youg-Seo Lee. WeBWorK Problem Set 3. Due/4/04at:59PM. HerearesomehitsohowtouseWeBWorKeffectively: Fidouthowtopritahardcopyothecomputersystemthatyouaregoigtouse.Cotactmeifyouhaveayproblems.Pritahard copy of this assigmet. Note, however, that the olie versios of these problems may havelikstothewebpagesofthiscoursethat areabsetothehardcopy. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Not olywillthisgiveyouthechacetofigure outwhat swrogifaaswerisotaccepted, youalsowillavoidthelikelyrushadcogestio prior to the deadlie. The primary purpose of the WeBWorK assigmetsithisclassistogiveyoutheopportuity to lear by havig istat feedback o your active solutio of relevat problems. Makethebestofit! Procrastiatio is hazardous!.( pt) Evaluate the defiite itegral. 4 0 d.(pt)iff 3arcta 4si 4 6,fidf. 3.( pt) The Nearsighted Cow Problem: A Calculus Classic. Arectagularbillboard9feetiheightstadsia fieldsothatitsbottomisfeetabovethegroud. Aearsightedcowwitheyelevelat4feetabovethe groud stads feet from the billboard. Epress θ, the vertical agle subteded by the billboard at her eye,itermsof.thefidthedistace 0 thecow must stad from the billboard to maimize θ. (Clickoimageforalargerview) 4.(pt)a)Fid ta si 5 7 cos 5 0 =. (Make sure your aswer is a algebraic epressio with square roots but without trigoometric or iverse trigometric fuctios.) b)epressitermsof: si ta =. 5.(pt)a)Let 889! πady cos 7.The the derivative D y=. b)let 88:! π adz l sec ta. Thethe derivative D z=. 6.(pt)a)Letf 8 ta 8. b)letg 64 3si 8. Fidf =. Fidg =. 7.(pt)a)Let0!! ady sech 9.The the derivative (Note: Look at the page for hyperbolic futctios available i WeBWorK.) D y=. b)let $ 0adz l sech tah.thethe derivative D z=.

8 8.( pt) Fid cosh 5 z 5 z Aswer: +C. 9.(pt)Fidtheareaoftheregioboudedby y tah,y 0ad 5. Area =. 3 cosh 0.( pt) Fid 3 sih d. Aswer:. dz.(pt)lety ; l cosh 7 =< cosh >. ThederivativeD y=..(pt)thecurvey sih,0 7isrevolved aroudthe-ais.fidtheareaoftheresultigsurface. [Hit:ItegralissimilartoProb.7.7.7orseeItegral Table No. 44 iside back cover.] (Note: Look at the page for hyperbolic futctios available i WeBWorK.) Area =. Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur

9 Math 0-3 Youg-Seo Lee. WeBWorK Problem Set 4. Due//04at:59PM. HerearesomehitsohowtouseWeBWorKeffectively: Here s the list of the fuctios which WeB- WorK uderstads. Give4or5sigificatdigitsfor(floatig poit) umerical aswers. For most problems whe eterig umerical aswers, you ca if you wish eter elemetary epressios such as 3 isteadof8si 3pi isteadof el isteadof ta 4 si isteadof760343etc Gettoworkothissetrightawayadaswerthese questios well before the deadlie. Not oly will this giveyouthechacetofigureoutwhat swrogifa aswerisotaccepted,youalsowillavoidthelikely rush ad cogestio prior to the deadlie. Procrastiatio is hazardous!.(pt)letabeafiedpositiveumber.fidthe idefiite itegrals: (a) 4 si cos a d (b) 4 cos si a d.( pt) Perform the followig itegratio: cos 3 d +C. +C. Aswer: C. 3.( pt) Evaluate the followig itegral: L L cos mπ L cosπ adm,areitegers. L d wherem A Aswer:. 4.( pt) Evaluate the followig itegral: t 0 t dt Aswer:. 5.(pt)Fidthevalueof π" π"5 6.( pt) Evaluate the defiite itegral. 0 0 si 5 cos 4 d 7.( pt) Evaluate the idefiite itegral. 6cos 3 75 d 8.( pt) Evaluate the idefiite itegral. si 7 si 8 d si si d. C C [NOTE:Remembertoeterallecessary*,(,ad)!! Eterarcta()forta,si()forsi...] 7si π 4 9.( pt) Evaluate the defiite itegral d [NOTE:Remembertoeterallecessary*,(,ad)!! Eterarcta()forta,si()forsi...] 0.( pt) Evaluate the defiite itegral d [NOTE:Remembertoeterallecessary*,(,ad)!! Eterarcta()forta,si()forsi...].( pt) Evaluate the idefiite itegral d [NOTE:Remembertoeterallecessary*,(,ad)!! Eterarcta()forta,si()forsi...].(pt) 0 d= +C 89 WeBWorKotatioforsi isarcsi()or asi(),forta it sarcta()orata().

10 3.( pt) Fid Aswer: dz +C. 4.(pt)Fid Aswer: d C Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur

11 Math 0-3 Youg-Seo Lee. WeBWorK Problem Set 5. Due/8/04at:59PM. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Notolywillthisgiveyouthechacetofigureoutwhat swrogifaaswerisotaccepted,youalsowillavoidthelikelyrushad cogestio prior to the deadlie. Procrastiatio is hazardous!.( pt) Iverse Fuctios vs. Reciprocals Letf Fidf Fid f.(pt)modelligthespreadofnews[seeprob ] Supposethatewsspreadsthroughacityoffied sizeof800000peopleatatimerateproportioalto theumberofpeoplewhohaveotheardtheews. (a.) Formulate a differetial equatio ad iitial coditiofory t,theumberofpeoplewhohave heardtheewstdaysafterithashappeed. Nooehasheardtheewsatfirst,soy The timerateoficreaseitheumberofpeoplewho haveheardtheewsisproportioaltotheumberof people who have ot heard the ews traslates ito the differetial equatio dy dt k, where k is the proportioality costat. (b.) 5daysafterascadaliCityHallwasreported, a poll showed that people have heard the ews. Usig this iformatio ad the differetial equatio, solve for the umber of people who have heardtheewsaftertdays. y t B 3.( pt) Fid the idefiite itegrals: (a) 4 ( 3 d +C. (b) 4 3 +C. ( d 4.( pt) Fid the defiite itegrals: (a) ( 4( 8 ( ( d (b) ( 5 ( ( d 5.( pt) Fid the idefiite itegrals: (a) 4 ( 5 d ( ( 5 d 4 (b) +C. +C. 6.( pt) Evaluate the followig defiite itegral: 4 4 d. Sketch for yourself the regio whose area is computed by this itegral. 7.( pt) Fid the followig idefiite itegrals: (a) 4 3 d +C. (b) 4 3 d +C. (c) 4 3 d +C. 8.( pt) Evaluate the followig defiite itegrals: (a) 4 0 e d. (b) 4 0 d. (c) 4 l 6 d. π (d) 4 0 d cos6. 9.( pt) Evaluate the followig idefiite itegrals: (a) 4 +C. ( C( d 4 d ( ( d (b) 4 +C. (c) 4 +C. (Note: Look at the page for hyperbolic fuctios available i WeBWorK.) 0.( pt) Evaluate the idefiite itegral. 5 = C [NOTE:Remembertoeterallecessary*,(,ad)!! Eterarcta()forta,si()forsi...] (Note: Look at the page for hyperbolic fuctios available i WeBWorK.) d.( pt) Evaluate the idefiite itegral.

12 d 8 4 = C. [NOTE:Remebertoeterallecessary*,(,ad)!! Eterarcta()forta,si()forsi...].( pt) Fid the coefficiets of the partial fractios epasio D E 6 A B C 3 6 A. B. C. D. E. 3.( pt) Evaluate the idefiite itegral. 7 9 d = C [NOTE:Remebertoeterallecessary*,(,ad)!! Eterarcta()forta,si()forsi...] 4.( pt) Evaluate the idefiite itegral d = +C [NOTE:Remebertoeterallecessary*,(,ad)!! Eterarcta()forta,si()forsi...] Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur

13 Math 0-3 If the itegral diverges, eter diverge as aswer. 6.( pt) Evaluate the followig improper itegral: Youg-Seo Lee. WeBWorK Problem Set 6. Due/5/04at:59PM. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Notolywillthisgiveyouthechacetofigureoutwhat swrogifaaswerisotaccepted,youalsowillavoidthelikelyrushad cogestio prior to the deadlie. Procrastiatio is hazardous!.( pt) Fid the followig limit usig l Hopital s Rule: * l lim Aswer:..( pt) Fid the followig limit usig l Hopital s Rule: lim * 0D 3si Aswer:. 3.( pt) Fid the followig limit usig l Hopital s Rule: tcostdt * 0+ lim 4 0 Aswer:. Etertheword ifiity iftheasweris. 4.( pt) Evaluate the followig improper itegral: e 4 d Aswer:. If the itegral diverges, eter diverge as aswer. 5.( pt) Evaluate the followig improper itegral: 0 d Aswer:. l e d Aswer:. If the itegral diverges, eter diverge as aswer. 7.( pt) Evaluate the followig improper itegral: e d Aswer:. If the itegral diverges, eter diverge as aswer. 8.(pt)Fidtheareaoftheregiouderthecurve y totherightof. Aswer:. 9.( pt) Evaluate the limit lim * ( pt) Fid the followig limits, usig l Hôpital s rule if appropriate lim * arcta 8 = 8 lim l * 0+ =.(pt) Evaluate the limit usig L Hospital s rule if ecessary 5 0 lim *.( pt) Determie whether the itegral is diverget or coverget. If it is coverget, evaluate it. If ot, state your aswer as diverget " d 3.(pt)Ialltheseproblems,writeIifthelimit iseither or. Fid the followig limits: (a)lim * 4 3 3( 3 ( ( (b)lim * 0 cos..

14 (c)lim * 0 e 4.(pt)Ialltheseproblems,writeIifthelimit iseither or. Fid the followig limits: (a)lim * 0 l. (b)lim * 0+ l 7.. (c)lim * 0 e. 5.(pt)Ialltheseproblems,writeIifthelimit iseither or. Fid the followig limits: (a)lim * D (b)lim * si 0.. Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur

15 Math 0-3 Youg-Seo Lee. WeBWorK Problem Set 7. Due3/4/04at:59PM. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Notolywillthisgiveyouthechacetofigureoutwhat swrogifaaswerisotaccepted,youalsowillavoidthelikelyrushad cogestio prior to the deadlie. Procrastiatio is hazardous!.( pt) Evaluate the followig improper itegral. If the itegral is diverget, eter diverget as aswer "3 d Aswer:..( pt) Evaluate the followig improper itegral. If the itegral is diverget, eter diverget as aswer. 3 0 d Aswer:. 3.( pt) Determie whether the itegral is diverget orcoverget.ifitiscoverget,evaluateit.ifitdiverges to ifiity, state your aswer as INF (without the quotatio marks). If it diverges to egative ifiity, state your aswer as MINF. If it diverges without beig ifiity or egative ifiity, state your asweras DIV. 9E5 0E5 3 5 d 4.( pt) Cosider the followig itegrals. Label eachas P, C, D,accordigastheitegralis proper, improper but coverget, or improper ad diverget.. 5π si ta d 5π π" π"5 5 d l 5 d se 7s ds si 7 d t 49 dt ta 7 d 5 d 5.( pt) Determie whether the sequece is diverget or coverget. If it is coverget, evaluate its limit. Ifitdivergestoifiity,stateyourasweras INF (without the quotatio marks). If it diverges to egative ifiity, state your aswer as MINF. If it diverges without beig ifiity or egative ifiity, state your aswer as DIV. lim * 6 7 si (pt)Fidthelimitofthesequecewhoseterms aregiveby a cos E8-7.( pt) Match each sequece below to statemet thatbestfitsit. STATEMENTS Z. The sequece coverges to zero; I. The sequece diverges to ifiity; F. The sequece has a fiite o-zero limit; D. The sequece diverges. SEQUENCES.l l lf E0 5.! si 7.arcta 8.si

16 8.( pt) Determie whether the sequeces are icreasig, decreasig, or ot mootoic. If icreasig, eterasyouraswer.ifdecreasig,eter asyour aswer. If ot mootoic, eter 0 as your aswer..a ( 7 7(.a ( cos 3.a 4.a ( 9.( pt) Cosider the sequece a cos π Writethefirstfivetermsofa,adfidlim * a. If the sequece diverges, eter diverget i the aswerboforitslimit. a)firstfiveterms:,,,,. b)lim * a. 0.( pt) Cosider the sequece a l Writethefirstfivetermsofa,adfidlim * a. If the sequece diverges, eter diverget i the aswerboforitslimit. a)firstfiveterms:,,,,. b)lim * a..( pt) Suppose a a a a a a)fidaeplicitformulafora :. b) Determie whether the sequece is coverget or diverget:. (Eter coverget or diverget as appropriate.) c)ifitcoverges,fidlim * a..( pt) Suppose a a ( a a Fidlim * a. Hit: Let a lim *. The, sice a ( a a,wehavea fora. a a. Nowsolve 3.(pt)Cosidertheseries G 8 the-thpartialsum;thatis, s Fids 4 ads 8 s 4 = s 8 = 6 4.(pt)Letr 4 8 ig i 4 ( 4. Lets be For both of the followig aswer blaks, decide whether the give sequece or series is coverget or diverget. If coverget, eter the limit(for a sequece)orthesum(foraseries).ifdiverget,eter INFifitdivergestoifiity,MINFifitdivergesto mius ifiity, or DIV otherwise. I A.Cosiderthesequece Hr. lim * r B.Takemywordforitthatitcabeshowthat ir ig ( i r ( r r r Nowcosidertheseries G r. G r 5.(pt)Matcheachofthefollowigwiththecorrect statemet. C stads for Coverget, D stads for Diverget l G 9 8 G 0 G 7 6 G 5 G 9 36 G 6.(pt)Determiethesumofthefollowigseries. 7.(pt)Determiethesumofthefollowigseries. 6

17 4 5 G 0 8.( pt) Epress JK as a ratioal umber,itheform p q wherepadqhaveocommofactors. p= ad q= 9.(pt)Aballdropsfromaheightoffeet. Eachtimeithitsthegroud,itboucesup40percetsoftheheightitfall.Assumeitgoesoforever, fid the total distace it travels. 0.(pt)Computethevalueofthefollowigimproper itegral if it coverges. If it diverges, eter INFifitdivergestoifiity,MINFifitdivergesto mius ifiity, or DIV otherwise(hit: itegrate by parts). 4l 4 d 4l.(pt)Fidthevalueof d 7 8 Determie whether 7 8 G Determie whether G 4 isacovergetseries.eterciftheseriesiscoverget,ordifitisdiverget. EterAifseriesiscoverget,Bifseriesisdiverget..(pt)Testeachofthefollowigseriesforcovergece by the Itegral Test. If the Itegral Test ca beappliedtotheseries,eterconvifitcoverges ordivifitdiverges.iftheitegraltestcaotbeappliedtotheseries,eterna.(note:thismeasthat eveifyoukowagiveseriescovergesbysome othertest,buttheitegraltestcaotbeappliedto it,theyoumusteternaratherthaconv.) G G G G G e 3 e 3 9 l 4 l 5 9 l 4 Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur 3

18 Math ( pt) Determie the covergece or divergece of the followig series. Youg-Seo Lee. WeBWorK Problem Set 8. Due3//04at:59PM. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Notolywillthisgiveyouthechacetofigureoutwhat swrogifaaswerisotaccepted,youalsowillavoidthelikelyrushad cogestio prior to the deadlie. Procrastiatio is hazardous!.( pt) Cosider the series: 3 3 kg 5 k k a) Determie whether the series is coverget or diverget:. (Eter coverget or diverget as appropriate.) b)ifitcoverges,fiditssum:. If the series diverges, eter here diverget agai..(pt) HowlargemustNbeiorderfor S N N kg to eceed 4? Note: Computer calculatios show thatfors N toeceed0,n adfor S N toeceed00,n L 5M Aswer:N=. 3.( pt) Determie the covergece or divergece of the followig series. A. coverget B. diverget G k A. coverget B. diverget G 3 k k k! 5.( pt) Determie the covergece or divergece of the followig series. G A. coverget B. diverget N cos PO 6.( pt) Determie whether the followig series is G ( 5 E A. coditioally coverget B. absolutely coverget C. diverget 7.( pt) Determie whether the followig series is G A. absolutely coverget B. coditioally coverget C. diverget 8.(pt) a Computes 3 (the3rdpartialsum)of 9 s G 4 5 b Estimatetheerroriusigs 3 asaapproimatio ofthesumoftheseries.(i.e.use c Use=3ad s f d s s tofidabetterestimateofthesum. 3 f dq R 3 ) ( f d s 9.(pt)Testeachofthefollowigseriesforcovergece by either the Compariso Test or the Limit ComparisoTest.Ifeithertestcabeappliedtothe

19 series,eterconvifitcovergesordivifitdiverges. Ifeithertestcabeappliedtotheseries, eterna.(note: thismeasthateveifyoukow agiveseriescovergesbysomeothertest,butthe compariso tests caot be applied to it, the you must eter NA rather tha CONV.) cos G G 5 6 G G G 0.(pt)Testeachofthefollowigseriesforcovergece by either the Compariso Test or the Limit ComparisoTest.Ifeithertestcabeappliedtothe series,eterconvifitcovergesordivifitdiverges. Ifeithertestcabeappliedtotheseries, eterna.(note: thismeasthateveifyoukow agiveseriescovergesbysomeothertest,butthe compariso tests caot be applied to it, the you must eter NA rather tha CONV.) G cos. G 5 cos 3. G G G (pt)Eachofthefollowigstatemetsisaattempttoshowthatagiveseriesiscovergetorot usig the Compariso Test(NOT the Limit Compariso Test.) For each statemet, eter C(for correct ) iftheargumetisvalid,oreteri(for icorrect ) ifaypartoftheargumetisflawed. (Note:ifthe coclusioistruebuttheargumetthatledtoitwas wrog, you must eter I.).Forall $, 3 3!,adtheseries coverges, so by the Compariso Test, the series 3 3 coverges..forall $, l!,adtheseries diverges, so by the Compariso Test, the series l diverges. 3.Forall$, arcta π!,adtheseries 3 3 π coverges,sobythecomparisotest, 3 theseries arcta coverges. 3 4.Forall$,!,adtheseries coverges, so by the Compariso Test, the series coverges. 5.Forall$, 5!,adtheseries 3 coverges, so by the Compariso Test, the series 5 coverges. 3 6.Forall$, l $,adtheseries coverges, so by the Compariso Test, the series l coverges..(pt)thethreeseries A, B,ad C have terms A 9 B 5 C Use the Limit Compariso Test to compare the followigseriestoayoftheaboveseries.foreachof theseriesbelow,youmustetertwoletters.thefirst istheletter(a,b,orc)oftheseriesabovethatitca be legally compared to with the Limit Compariso Test. ThesecodisCifthegiveseriescoverges, ordifitdiverges. Soforistace,ifyoubelieve the series coverges ad ca be compared with series Cabove,youwouldeterCC;orifyoubelieveitdivergesadcabecomparedwithseriesA,youwould eter AD G G G

20 3.( pt) Select the FIRST correct reaso why the give series coverges. A. Coverget geometric series B. Coverget p series C. Compariso(or Limit Compariso) with a geometric or p series D.Caotapplyaytestdoesofariclass G G G G G G 5 cos π l l e 7 cos π 4.(pt)Foreachsequecea fidaumberrsuch that a r has a fiite o-zero limit. (Thisisofuse,becausebythelimitcomparisotest the series a ad G 5 R ( C.a ( ( 3 5 ( 6 ( 5 r= D.a ( 3( 0S6 5 + ( 6( 5 r= 5.( pt) Select the FIRST correct reaso why the give series diverges. A. Diverges because the terms do t have limit zero B. Diverget geometric series C. Diverget p series D. Itegral test E. Compariso with a diverget p series F. Diverges by limit compariso test 3 G.Caotapplyaytestdoesofariclass 5. G 5 cos π G l 6 G l G G G 4 4!! G G G G G 3 si r G bothcovergeorbothdi- verge.) A.a r= B.a r= 6.(pt)Matcheachofthefollowigwiththecorrect statemet. A. The series is absolutely coverget. C. The series coverges, but is ot absolutely coverget. D. The series diverges. 7.(pt)Matcheachofthefollowigwiththecorrect statemet. A. The series is absolutely coverget. C. The series coverges, but is ot absolutely coverget. D. The series diverges G G G 3!! 6! 5 5 ( 4

21 4. 5. G G! G 5 7 G 7 G l 4 G ( G 7 G ( pt) Cosider the series G e a 6 3 3! a where IthisproblemyoumustattempttousetheRatioTest to decide whether the series coverges. Compute a L lim ( * TT TT a TT TT 8.(pt)Matcheachofthefollowigwiththecorrect statemet. A. The series is absolutely coverget. C. The series coverges, but is ot absolutely coverget. D. The series diverges. EtertheumericalvalueofthelimitLifitcoverges,INFifitdivergestoifiity,MINFifitdivergestoegativeifiity,orDIVifitdivergesbut ot to ifiity or egative ifiity. L Which of the followig statemets is true? A.TheRatioTestsaysthattheseriescovergesabsolutely. B.TheRatioTestsaysthattheseriesdiverges. C.TheRatioTestsaysthattheseriescovergescoditioally. D.TheRatioTestisicoclusive,buttheseriescoverges absolutely by aother test or tests. E.TheRatioTestisicoclusive,buttheseriesdiverges by aother test or tests. F.TheRatioTestisicoclusive,buttheseriescoverges coditioally by aother test or tests. Eter the letter for your choice here: 0.( pt) Cosider the series a G where a 5 IthisproblemyoumustattempttousetheRatioTest to decide whether the series coverges. Compute L lim a * T ( T a T T T T T T EtertheumericalvalueofthelimitLifitcoverges,INFifitdivergestoifiity,MINFifitdivergestoegativeifiity,orDIVifitdivergesbut ot to ifiity or egative ifiity. L Which of the followig statemets is true? A.TheRatioTestsaysthattheseriescovergesabsolutely. B.TheRatioTestsaysthattheseriesdiverges. C.TheRatioTestsaysthattheseriescovergescoditioally. D.TheRatioTestisicoclusive,buttheseriescoverges absolutely by aother test or tests. E.TheRatioTestisicoclusive,buttheseriesdiverges by aother test or tests. F.TheRatioTestisicoclusive,buttheseriescoverges coditioally by aother test or tests. Eter the letter for your choice here: Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur 4

22 Math 0-3 Youg-Seo Lee. WeBWorK Problem Set 9. Due3/4/04at:59PM. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Notolywillthisgiveyouthechacetofigureoutwhat swrogifaaswerisotaccepted,youalsowillavoidthelikelyrushad cogestio prior to the deadlie. Procrastiatio is hazardous!.( pt) Fid thecovergece set of the give power series:! G The above series coverges for!!. Eter ifiity for ad -ifiity for..( pt) Fid thecovergece set of the give power series: G The above series coverges for. Eter ifiity for ad -ifiity for. 3.( pt) A famous sequece f, called the Fiboacci Sequece after Leoardo Fiboacci, who itroduceditarouda.d.00,isdefiedbytherecursio formula f f f ( f ( f Fid the radius of covergece of f G Radius of covergece:. 4.( pt) Fid the iterval of covergece for the give power series. 6 3 G The series is coverget: from=,leftedicluded(y,n): to=,rightedicluded(y,n): 5.(pt)Matcheachofthepowerserieswithitsiterval of covergece G 4. G G! 8! G 8 A. B. 06 C. 4 4 D. H84 I 6.( pt) Fid the power series represetatio for f, ad specify the radius of covergece. f e a p G wheree A.- B. C.0 wherea, adp. Radius of covergece:. 7.( pt) Fid the power series represetatio for f f e a! p G 0 wherea adp. 8.( pt) Fid the power series represetatio for

23 f U 0 ta t t f e a p G wheree A. B.- C.0 ada, adp. 9.(pt)Fidthesumof G for!!. 0.(pt)FidtheTaylorseriesi a through a 3 for f ta a dt f π π 4 4 π 3 4 O π 4 4. π 4.(pt)FidtheTaylorseriesi a 3 for a through f 3 3 a V f 3..( pt) Calculate the followig itegral, accurate to five decimal places: 0 0E5 si d Aswer: 3.(pt)Supposethatf adg aregiveby the power series f XW W W ad g XW W WY By multiplyig power series, fid the first few termsoftheseriesfortheproduct h f ;Wg c 0 c c c 3 3 XW W6W: c 0 = c = c = c 3 = 4.(pt)Supposethatf adg aregiveby the power series f, XW W6W ad g XW W W: Bydividigpowerseries,fidthefirstfewterms oftheseriesforthequotiet h c 0 = c = c = c 3 = g f c 0 c c c 3 3 XW W W: 0 5.( pt) Suppose that 5 c G 0 Fid the first few coefficiets. c 0 c c c 3 c 4 FidtheradiusofcovergeceRofthepowerseries. R. 6.(pt)Thefuctio f 7 Z isrepresetedasapowerseries 7 f, c G 0 Fid the first few coefficiets i the power series. c 0 c c c 3 c 4 FidtheradiusofcovergeceRoftheseries. R. 9 arcta 4 isrep- 7.(pt)Thefuctiof 0 reseted as a power series f, c G 0 Whatisthedegreeofthelowesttermwithaozero coefficiet. FidtheradiusofcovergeceRoftheseries.

24 R. 8.(pt)TheTaylorseriesforf [ 3 at-is c G 0 Fid the first few coefficiets. c 0 c c c 3 c 4 Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur 3

25 Math 0-3 Youg-Seo Lee. WeBWorK Problem Set 0. Due3/3/04at:59PM. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Notolywillthisgiveyouthechacetofigureoutwhat swrogifaaswerisotaccepted,youalsowillavoidthelikelyrushad cogestio prior to the deadlie. Procrastiatio is hazardous! c G 9 0.(pt)Represetthefuctio 0E4 asapowerseries c 0 c c c 3 Fid the left edpoit of the iterval of covergece. lefted=. Fid the right edpoit of the iterval of covergece. righted=..( pt) Compute the 0th derivative of f arcta 7 at 0. f0 0 Hit:UsetheMacLauriseriesforf. 3.( pt) Compute the 6th derivative of at 0. f6 0, f cos6 Hit:UsetheMacLauriseriesforf. 4.(pt)TheTaylorseriesforf U l sec at c G 0 a 0is Fid the first few coefficiets. c 0 c c c 3 c 4 Fidtheeacterroriapproimatigl sec 0 by itsfourthdegreetaylorpolyomialata 0 Theerroris 5.(pt)LetT 5 bethefifthdegreetaylorpolyomialofthefuctiof cos 06 ata 0. A.FidT 5 -(Eterafuctio.) T 5 B.Fidthelargestitegerksuchthatforallfor which 88\! thetaylorpolyomialt 5 approimatesf with error k lesstha 0 k 6.(pt)LetF 0 si 8t dt. FidtheMacLauripolyomialofdegree7forF. 0E6 Use this polyomial to estimate the value of si 8 d. 0 7.(pt)FidTaylorseriesoffuctiof l ata 8. f c G 8 0 c 0 c c c 3 c 4 Fid the iterval of covergece. The series is coverget: from= to= 8.( pt) Evaluate,leftedicluded(Y,N):,rightedicluded(Y,N): lim * 0 l 7 3

26 Hit: Use power series. 9.( pt) Evaluate Hit: Use power series. e lim * (pt)Assumethatsi equalsitsmaclauri seriesforall. UsetheMaclauriseriesforsi 7 toevaluatethe itegral 0E7 0 si 7 d. Youraswerwillbeaifiiteseries.Usethefirst two terms to estimate its value..(pt)fidt 5 :Taylorpolyomialofdegree 5ofthefuctiof cos ata 0. (You eed to eter fuctio.) T 5 Fidallvaluesofforwhichthisapproimatiois withi of the right aswer. Assume for simplicitythatwelimitourselvesto 88\. 88\.(pt)LetT 4 : bethetaylorpolyomialof degree4ofthefuctiof cos ata 0. Supposeyouapproimate f byt 4,adif 88:,whatistheboudforyourerrorofyourestimate?(Hit: use the alteratig series approimatio.) 3.(pt)LetT k : bethetaylorpolyomialof degreekofthefuctiof si ata 0. Supposeyouapproimate f byt k,adif 88Y,howmaytermsdoyoueed(thatis,what isk)foryoutohaveyourerrortobelesstha 0? (Hit: use the alteratig series approimatio.) 4.(pt)LetT 4 : bethetaylorpolyomialof degree4ofthefuctiof l ata 0. Supposeyouapproimatef byt 4,fidall positive values of for which this approimatio is withi 0.00 of the right aswer.(hit: use the alteratig series approimatio.) 0! 5.(pt)LetCbeasemicircleofradiusr $ 0cetered at the origi. LetPbeapoitothe-aiswhosecoordiatesare P r rt0 wheret $ 0. LetLbealiethroughPwhichistagettothesemicircle. Let A deote the triagular regio betwee the circle adthelieadabovethe-ais(seefigure.) (Clickoimageforalargerview) FidtheeactareaofAitermsofradt. Area A 0. UseaMaclauriPolyomialtogetasimpleapproimatiofortheareaofAforsmallt.: Area A 0L. Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur

27 Math 0-3 Youg-Seo Lee. WeBWorK Problem Set. Due4/8/04at:59PM. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Notolywillthisgiveyouthechacetofigureoutwhat swrogifaaswerisotaccepted,youalsowillavoidthelikelyrushad cogestio prior to the deadlie. Procrastiatio is hazardous!.(pt)supposethatweusethebisectioalgorithmtoapproimater 4 which the greatest zeroofthefuctio f P 4 Webegiby fidigtwoumbers,say,a 6adb 7which bracketthezero. Thisisbecause f a Z! 0ad f b '$ 0Thewefidm a b 6 65ad h b a 05 We proceed with the bisectio algorithm. Suppose thata adb bracketthezero. Thewecompute m a b adh ]8b a 8^ Iff m _ 0westopbecauser m isthedesiredzero. If f m [$ 0them becomestheewrightedpoit, soweseta ( : a adb ( : m Iff m `! 0 them becomestheewleftedpoit,soweset a ( : m adb ( : b Them isaapproimatiotorwithaerrorofh Complete the followig table: a b h m The isaapproimatiosofartorwitha error of..(pt) Suppose that we use Newto s Method to approimater 0whichthezeroofthefuctiof B Webegiwithagoodguess,say 6 The Newto s Method proceeds by the recursio f ( fab Computethefirstfewtermsofthesequece obtaied from Newto s method ( pt) FIXED POINT ALGORITHM. Ifgisacotiuousfuctiotakigtheiterval ab toitself,theithasafiedpoitrc ab sothat r g r Ifiadditio,gisdifferetiableadsatisfies 8g \8Y Mforalla bwherem! isa costat,thetherecursio ( g - c ab yieldsasequecethatcoverges & ras& Cosider the equatio 30 Usig the FiedPoitAlgorithmstartigwith 5,fid to Solveforthe(positive)i 30 Evaluate < XW W6Wd 4.(pt)Givethefollowigitegraladvalueof, approimate the followig itegral usig the methods idicated(roud your aswers to si decimal places): (a) Trapezoidal Rule (b) Midpoit Rule (c) Simpso s Rule 0 e 5 d 4

28 5.(pt)UseSimpso sruleadallthedataithe followig table to estimate the value of the itegral 3 5 yd y (pt) Determieasothatthetrapezoidalrulewillapproimate the itegral 7d 6 7 withaerrore satisfyig 8E 8e The theoretical error boud for the Trapezoid rule isgiveby b a 3 E f f ^ c wherecissomepoitbetweeaadb.itpredicts thatthedesiredaccuracywillbeachievediftheumberoftermsisatleast. 7.(pt)SupposethatweuseEuler smethodtoapproimate the solutio to the differetial equatio dy d 3 y ; y 0 Letf y 3 y Welet 0 0ady 0 6adpickastepsize h 0 Euler s method is the the followig algorithm. From ady ourapproimatiostothesolutio ofthedifferetialequatioatthethstage,wefid the et stage by computig ( h y ( y h Wf y Complete the followig table: y The eact solutio ca also be foud usig separatioofvariables.itis 6 y Thustheactualvalueofthefuctioatthepoit y. 8.( pt) Suppose that we use the Improved Euler s method to approimate the solutio to the differetial equatio dy d 05y; y 05 7 Letf y, 05y Welet 0 05ady 0 7adpickastepsize h 05 The improved Euler method is the the followig algorithm. From y - ourapproimatiotothe solutio of the differetial equatio at the -th stage, wefidtheetstagebycomputigthe-step ( h adthek theslopeat y - Thepredictedewvalueofthesolutioisz ( y h Wk Thewefidtheslopeatthepredictedewpoit k f ( z ( adgetthecorrectedpoitbyaveragig slopes y ( y h k k - Complete the followig table: y k z ( Theeactsolutiocaalsobefoudfortheliear equatio.writetheaswerasafuctioof. y Thustheactualvalueofthefuctioatthepoit 5is y 5. 9.(pt)Theparabolay 5 6hasitsfocusatthe poit b0 wherebis 0.(pt)Theparabolay 5hasitsfocusat thepoit bc where b c

29 .(pt)theellipse0 y hasitsceteratthepoit bc where b c Thelegthofthemajordiameterofthisellipseis.( pt) Determie the distace D betwee the verticesof 9 8 4y 4y 9 0. D 3.(pt)Fidthepoit y of 4y 49y 00thatisclosesttotheorigiadliesithefirst quadrat. y 4.(pt)Matcheachequatiobelowtothecurve itrepresets.eachaswershouldbea,b,c,d,e, F,orG. CURVES A. Circle, B. Ellipse, C. Poit, D. Parabola, E. Empty Set, F. Itersectig lies, G. Hyperbola, EQUATIONS.9 4y 7 6y 4 0. y y 0 3. y y y 7 6y y y 8 y y 5 4y y 6 y (pt)Theequatioofaellipsewithceter 3 thatpassesthroughthepoits 63 ad 5 hastheformf y.fidf 0. 6.(pt)Abridgeuderpassitheshapeofaellipticalarch,thatis,halfofaellipse,is50feetwide ad3feethigh.aeightfootwiderectagulartruck istodrive(safely)udereath.howhighcaitbe? h Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur 3

30 Math 0-3 Youg-Seo Lee. WeBWorK Problem Set. Due4/5/04at:59PM. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Notolywillthisgiveyouthechacetofigureoutwhat swrogifaaswerisotaccepted,youalsowillavoidthelikelyrushad cogestio prior to the deadlie. Procrastiatio is hazardous!.(pt) Fidtheslopeofthetagetlie: (a)totheparabolay 4atthepoit m. (b)totheellipse y atthepoit m. (c) To the hyperbola y at the poit 3 m..(pt) The stadard equatio of a elliptic curve is: y a b fora $ b $ 0 Fidtheslopeoftheellipticcurvey 3 atthepoit < 3. m. 3.(pt) Covert the followig rectagular coordiates ito polar coordiates. Always choose 0 θ! π. (a) 05 r, θ. (b) 3 r, θ. (c) 3 r, θ. (d) 4 3 r, θ. 4.(pt) Covert the followig polar coordiates ito rectagular coordiates. π (a) 4,y. π (b) 4 3,y. 3π (c) 6,y. 5π (d) 6 3,y. 5.(pt) Covert the followig rectagular equatios ito polar equatios. (a) y r acos θ ; a. (b) y r cosaθ cosbθ; a,b. (c) 4 y 4 y r ata bθ ; a,b. 6.(pt)Acurvewithpolarequatio r 8 6siθ 3cosθ represets a lie. This lie has a Cartesia equatio oftheform y m b,wheremadbarecostats.givethe formula foryitermsof.foreample,iftheliehadequatio y 3thetheaswerwouldbe? 3. 7.( pt) Match each polar equatio below to the best descriptio. Possible aswers are C,E,H,L,P,R,S,V,ad Z. DESCRIPTIONS C. Circle cetered at origi, E. Ellipse, H. Hyperbola, L. Lie either vertical or horizotal, P. Parabola, R. Circle ot cetered at origi, S. Spiral, V. Vertical Lie, Z. Horizotal Lie POLAR EQUATIONS 8.r 6siθ( 7cos θ.r V 6 3.r 8 Hit:siθ siθcosθ siθ 4.r 7cos θ 5.r 7siθ

31 8.( pt) Match each polar equatio below to the best descriptio. Each aswer should be C,F,I,L,M,O,or T. DESCRIPTIONS C. Cardioid, F. Rose with four petals, I. Iwardly spiralig spiral, L. Lemaco, M. Lemiscate, O. Outwardly spiralig spiral, T. Rose with three petals POLAR EQUATIONS.r 4cosθ.r 7 4cosθ 3.r 8siθ 4.r 7cos3θ 5.r 7θr$ 0 6.r 7 θ r$ 0 7.r 7 7siθ 9.( pt) Make the chage of variables ucosθ vsiθ y usiθ vcosθ wheretheagle0 θ! πischoseiorderto elimiate the cross product term i y y 6 Thefidthestadardformofequatioithe uv variables.(eterafuctioof uv.) 0.(pt)Ifyoumakethechageofvariables ucosθ vsiθ y usiθ vcosθ wheretheagle0 θ! πischoseiorderto elimiate the cross product term i 5 4y 8y 90 Whatistheagleyouwoulduse? θ..( pt) Make the chage of variables ucosθ vsiθ y usiθ vcosθ wheretheagle0 θ! πischoseiorderto elimiate the cross product term i 97 9y 53y y g 5 Thefidthestadardformofequatioithe uv variables.(eterafuctioof uv.) Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur

32 Math 0-3 Youg-Seo Lee. WeBWorK Problem Set 3. Due4//04at:59PM. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Notolywillthisgiveyouthechacetofigureoutwhat swrogifaaswerisotaccepted,youalsowillavoidthelikelyrushad cogestio prior to the deadlie. Procrastiatio is hazardous!.(pt)fidtheareaoftheregioiside: r 9siθbutoutside:r.(pt)Fidtheareaoftheregioboudedby: r 7 siθ 3.(pt)AcircleChasceterattheorigiadradius4.AothercircleKhasadiameterwithoeed attheorigiadtheotheredatthepoit 0 ThecirclesCadKitersectitwopoits.LetPbe thepoitofitersectioofcadkwhichliesithe firstquadrat.let r θ bethepolarcoordiatesofp, chosesothatrispositivead0 θ Fidrad θ. r θ 4.(pt)Fidthelegthofthecurver θ from θ 0to θ 3. 5.(pt)Fidtheslopeofthetagettothecurve r V 8 7cosθatthevalue θ π 6.(pt)Fidtheeactlegthofthepolarcurvedescribed by: θ r 3e otheiterval 4 5π θ 6π. 7.(pt)Fidtheareaoftheregioboudedby: r 8cosθ 8.(pt)Fidtheareaoftheregiooutsider 9 9siθ,butisider 7siθ. 9.( pt) Solve the followig differetial equatio: y ^ 3y 0y 0; y y 0at 0 Aswer:y. 0.( pt) Solve the followig differetial equatio: y ^ 0y 5y 0 Aswer:y C C. NOTE:Theorderofyouraswersisimportati this problem. For eample, webwork may epect the aswer A+B buttheasweryougiveis B+A. Both aswers are correct but webwork will oly accept the former..( pt) Solve the followig differetial equatio: y ^ 9y 0; y 3 y 3at π3 Aswer:y..( pt) Solve the followig differetial equatio: y ^ y y 0 Aswer:y C C. NOTE:Theorderofyouraswersisimportati this problem. For eample, webwork may epect the aswer A+B buttheasweryougiveis B+A. Both aswers are correct but webwork will oly accept the former. 3.( pt) Solve the followig differetial equatio: y ^ y y 0 adepressyourasweritheform ce α si β γ Aswer: α, β. Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur

33 Math 0-3 Youg-Seo Lee. Both aswers are correct but webwork will oly accept the former. 4.( pt) Use the method of udetermied coefficiets to solve the followig differetial equatio: WeBWorK Problem Set 4. Due4/9/04at:59PM. Gettoworkothissetrightawayadaswer these questios well before the deadlie. Notolywillthisgiveyouthechacetofigureoutwhat swrogifaaswerisotaccepted,youalsowillavoidthelikelyrushad cogestio prior to the deadlie. Procrastiatio is hazardous!.( pt) Use the method of udetermied coefficiets to solve the followig differetial equatio: y h y 4 Aswer:y C C. NOTE:Theorderofyouraswersisimportati this problem. For eample, webwork may epect the aswer A+B buttheasweryougiveis B+A. Both aswers are correct but webwork will oly accept the former..( pt) Use the method of udetermied coefficiets to solve the followig differetial equatio: y ^ 6y 9y e Aswer:y C C. NOTE:Theorderofyouraswersisimportati this problem. For eample, webwork may epect the aswer A+B buttheasweryougiveis B+A. Both aswers are correct but webwork will oly accept the former. 3.( pt) Use the method of udetermied coefficiets to solve the followig differetial equatio: y ^ 4y 4 Aswer:y C C. NOTE:Theorderofyouraswersisimportati this problem. For eample, webwork may epect the aswer A+B buttheasweryougiveis B+A. y ^ 6y 9y si Aswer:y C C. NOTE:Theorderofyouraswersisimportati this problem. For eample, webwork may epect the aswer A+B buttheasweryougiveis B+A. Both aswers are correct but webwork will oly accept the former. 5.(pt)Letybethesolutiooftheiitialvalue problem y ^ y y 0y 0 0y 0 7 Themaimumvalueofyover0! is. 6.(pt)Asprigwithasprigcostatkof00 pouds per foot is loaded with -poud weight ad brought to equilibrium. It is the stretched a additioalichadreleased.fidtheequatioofmotio, the amplitude, ad the period. Neglect frictio. y t 0, wheretistimeady t isdisplacemetitime. Amplitude: ich(es) Period: secod(s). 7.(pt)Asprigwithasprigcostatkof0 poudsperfootisloadedwitha0-poudweight ad allowed to reach equilibrium. It is the displaced foot dowward ad released. If the weight eperiecesaretardigforceipoudsequaltofourtimes thevelocityateverypoit,fidtheequatioofmotio. y t 0, wheretistimeady t isdisplacemetitime. 8.(pt)Solvetheequatioy ^ y ep wherey 0 y 0 0 y 9.(pt)Letybethesolutiooftheiitialvalue problem y h y V si 7y 0 0y 0 0 Themaimumvalueofyis,

34 0.( pt) A alteratig curret E t i 0si t hasbeeruigthroughasimplecircuitforalogtime. Thecircuithasaiductace ofl 048herys,aresistorofR 0ohmsada capacitor of capcitace C 0 09 farads. WhatistheamplitudeofthecurretI?.( pt) Solve the followig differetial equatio: y ^ 4y si 3 Aswer:y C C. Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, cur