5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t

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1 AP CALCULUS FINAL UNIT WORKSHEETS ACCELERATION, VELOCTIY AND POSITION In problms -, drmin h posiion funcion, (), from h givn informaion.. v (), () = 5. v ()5, () = b g. a (), v() =, () = -. a (), v() = 5, () = 5. An objc moving along an -coordina ais wih is scal masurd in mrs has a vlociy of 6 m/sc in h posiiv dircion a im. A = h objc is a =. Whr is h objc a = 8? 6. A sor aks in mony a h ra of dollars/hour hours afr i has opnd. How much dos i ak in during is n-hour day? 7. War flows ino a ank a h ra of f /day a im (days) and laks ou a h consan ra of 6 f /day. A = h ank conains 5 f of war. How much dos h ank conain on day lar? 8. An objc s vlociy in h posiiv dircion on h -ais is v () yards/sc a im (sconds). A = i is a = yards. a) Whn is h vlociy zro and whr is i a hos ims? b) A wha im(s) is i again a = yards? 9. An objc s acclraion on h -ais is m/sc a im (sconds). A = i is a = mrs and is vlociy is m/sc. How long dos i ak o rach = mrs and wha is is vlociy a ha im? ANSWERS:. () 5. ()5 6. (). () m 6. $ f 8. a) = and = ; = and = 59/6 b) =.5 9. sc and m/sc NET DISTANCE/TOTAL DISTANCE Th funcion, v() is h vlociy in mrs pr scond of a body moving along a coordina lin. Find : a) h oal disanc ravld by h body during h givn im inrval b) h shif in h body s posiion. v() 5 cos. v() 6sin /. v() v () v () / 6 Th following graphs ar vlociy graphs of four bodis moving on coordina lins. Find h disanc ravld by ach body and h posiion shif for h givn im inrval. NOTE: h firs hr graphs ar no coninuous / ANSWERS:. a) m b) m. a) 6 m b) m. a) 5 m b) m. a) m b) 9m 5. a) 5m b) m 6. d=, s= 7. d=, s= 8. d=, s= 9. d=, s=

2 SEPARABLE DIFFERENTIAL EQUATIONS Solv h diffrnial quaions and iniial valu problms in problms -7.. dy ysin, y()=. y jdy. d y dy dy. cos y 5. dy y, y()= 6. y y dy 7. y y dy b g b g j j 8. Th hicknss () (in inchs) of ic forming on a lak saisfis h diffrnial quaion. A im d = days h ic is on inch hick. Whn is h ic wo inchs hick? 9. A nwly crad lak is sockd wih fish. Thrafr, h populaion incrass a h ra of P fish pr monh. Wri an prssion P() o modl h populaion of h fish a any im. ANSWERS:. y cos. y. y sin( C). y an bcg k an C 5. y ln L HOPITAL S RULE 6. ln yln y C 7. an ln y Find h following limis. Suppor all answrs. jc 8. day 9. P (). lim. lim. lim an. lim sin cos F 5. lim ln 6. lim HG I K J F I 7. lim HG K J F I 8. lim HG K J 9. lim b sin g. lim / b g. lim b g. lim sin sin F HG I K J ANSWERS:. ln SOLUTIONS BY EULER S METHOD. L dy and suppos y =.5 whn =. a) Us Eulr s mhod o sima h soluion of h diffrnial quaion whn =.5. Us 5 sps, i.., l n = 5. Calcula your answr by hand and lis all inrmdia answrs. b) Us sps and calcula h sam answr. c) Did you ovrsima or undrsima h ac answr?. L dy y and suppos y = whn =. a) Us Eulr s mhod o sima h soluion whn = by using 5 sps. Show all inrmdia answrs. b) Us a calculaor program and sps o sima h soluion whn =. c) Hav you ovrsimad or undrsimad? Eplain your rasoning. d) Consruc a slop fild for dy. Do you hink your rasoning for par c) was corrc? coninud

3 . If P dollars is invsd a 8% pr yar compoundd coninuously and A dollars is h amoun of h da invsmn afr yars, hn.8a d. Suppos $5 is invsd iniially. Us Eulr s mhod o sima h amoun of h invsmn afr 5 yars using sps of: a) on yar. b) si monhs. c) on monh.. Th vlociy of a paricl moving along a horizonal lin a any im is givn by v (), whr v is masurd in f/sc and is masurd in sconds. If h paricl is on h on foo poin whn =, sima h posiion of h paricl using Eulr s mhod a h nd of sconds. Us 6 sps. dp 5. Suppos h ra of growh of a prdaor populaion is givn by. P( P), whr P is h d populaion of h prdaor in a confind ara a im yars. If h populaion of h prdaor numbrs 5 in, how many will b in h prdaor populaion a h sam im in? ANSWERS:. a).7 b).75 c) undrsimad. a).88 b).7889 c) undrsimad slop fild is concav up. a) $7.66 b) $7. c) $ ~66 (66.75) INTEGRAL AS AN ACCUMULATOR. Suppos a car is moving wih non-dcrasing spd according o h abl blow. TIME (sc) 6 8 SPEED (f/sc) a) Wha is an uppr sima for h disanc ravld in h firs wo sconds? b) Drmin uppr and lowr simas for h chang in posiion for h firs sconds.. Suppos h dnsiy of cars, in cars/mil, for h firs mils along h Mass Pik from Boson during crain hours of h day can b modld by: p ( ).. whr rprsns h numbr of mils from Boson. a) Wri a funcion ha givs h numbr of cars on h Mass Pik from Boson o a poin mils from Boson. b) Us his funcion o drmin h numbr of cars on his mil srch of road. b g. If y' ln sin and y() =, wha is y()?.. Oil is laking from a ankr a h ra of R () gallons/hour, whr is masurd in hours. How much oil has lakd ou of h ankr afr hours? 5. Th ra a which war flows ou of a pip, in gallons, is givn by a diffrniabl (hours funcion R of im. Th abl shows h ra as masurd vry hours for a -hour priod. a) Us a midpoin Rimann sum wih subdivisions of qual lngh o approima R() d. Using corrc unis, plain h maning of your answr in rms of war flow. b) Th ra of war flow R() can b approimad by Q ( ) Us Q ( ) o approima h avrag ra of flow during h -hour priod. Indica unis of masur. j R() (gallons/hour) coninud

4 6. Th mpraur ousid a hous during a -hour priod is givn by F ( ) 8 cos,, whr F ( ) is masurd in dgrs Fahrnhi and is masurd in hours. a) Find h avrag mpraur, o h nars dgr Fahrnhi, bwn 6 and. b) An air condiionr coold h hous whnvr h ousid mpraur was a or abov 78 dgrs Fahrnhi. For wha valus of was h air condiionr cooling h hous? c) Th cos of cooling h hous accumulas a h ra of $.5 pr hour for ach dgr h ousid mpraur cds 78 dgrs Fahrnhi. Wha was h oal cos, o h nars cn, o cool h hous for his -hour priod? 7. Th vlociy of a cork bobbing up and down on h wavs in h sa is givn by h graph blow. Upward is considrd posiiv. Dscrib h moion of h cork a ach of h labld poins. A which poin(s), if any, is h acclraion zro. Skch a graph of h high of h cork abov h sa floor as a funcion of im. vlociy A B im D C 8. Suppos h vlociy, v, of a car ravling along a sraigh road is approimad by h picwis graph blow. Wri a picwis funcion for h disanc of h car from is saring poin as a funcion of im in hours. Draw a graph of his funcion. vlociy (mph) im (hours) ANSWERS:. a) 7 f b) lowr: 6 f uppr: 76 f. a) cars = n i p ( ) b) 55 cars gallons 5. a) 58.6 gallons b).785 gal/hr 6. a) 87 F b) c) 5.96 $5. 7. rising a A and ma high a B, falling a C and minimum high a D, acclraion = a A and C 8. s () R S T 5 5,, 5, i i or C ( ) pd ( ) z

5 AP CALCULUS FINAL UNIT REVIEW Evalua h ingrals. z.. z. Compu h ara boundd by y = and y Evalua h limis. z lim 6. lim ln 7. lim 8. ln 9. Using h following informaion lim b g / a() = 7 m/sc, v() = m/sc and () = 5 m find: a) () b) h oal disanc and h n disanc ravld on h inrval.. L f ( ). a) Wha is h valu of b) Wha is h valu of f '() f ( ) whn =? Solv h following diffrnial quaions and iniial valu problms. Solv ach for y. j j. dy 5 y ; y() =. y y y dy ; y() =. Us Nwon s mhod o find all of h soluions o h nars housandh for sin 5.. A abl of valus for a coninuous funcion, f() is givn blow. Us ha abl o approima h ingral. Sa which mhod you usd. z f ( ).5.5 f() dy 5. L y and suppos y =.5 whn =. a) Using sps us Eulr s mhod o sima h soluion of h diffrnial quaion whn =. Calcula your answr by hand and lis all inrmdia answrs. ANSWERS:.. divrgs wih no limi a) () 5 b) n disanc: 7 m oal disanc: m. a) f () b) f '(). y an 5 5. y.,. 96. LRAM = 8, RRAM = or Trapzoid Rul = a) y() 9.5