# AP Calculus AB 2008 Scoring Guidelines

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1 AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity. Foundd in 9, th association is composd of mor than 5,4 schools, collgs, univrsitis, and othr ducational organizations. Each yar, th Collg Board srvs svn million studnts and thir parnts,, high schools, and,5 collgs through major programs and srvics in collg admissions, guidanc, assssmnt, financial aid, nrollmnt, and taching and larning. Among its bst-known programs ar th SAT, th PSAT/NMSQT, and th Advancd Placmnt Program (AP ). Th Collg Board is committd to th principls of cllnc and quity, and that commitmnt is mbodid in all of its programs, srvics, activitis, and concrns. 8 Th Collg Board. All rights rsrvd. Collg Board, AP Cntral, Advancd Placmnt Program, AP, SAT, and th acorn logo ar rgistrd tradmarks of th Collg Board. PSAT/NMSQT is a rgistrd tradmark of th Collg Board and National Mrit Scholarship Corporation. All othr products and srvics may b tradmarks of thir rspctiv ownrs. Prmission to us copyrightd Collg Board matrials may b rqustd onlin at: AP Cntral is th onlin hom for AP tachrs: apcntral.collgboard.com.

2 8 SCORING GUIDELINES Qustion Lt R b th rgion boundd by th graphs of y = sin( π ) and y = 4, as shown in th figur abov. (a) Find th ara of R. (b) Th horizontal lin y = splits th rgion R into two parts. Writ, but do not valuat, an intgral prssion for th ara of th part of R that is blow this horizontal lin. (c) Th rgion R is th bas of a solid. For this solid, ach cross sction prpndicular to th -ais is a squar. Find th volum of this solid. (d) Th rgion R modls th surfac of a small pond. At all points in R at a distanc from th y-ais, th dpth of th watr is givn by h ( ) =. Find th volum of watr in th pond. (a) sin ( π ) = 4 at = and = Ara ( ( ) ( )) = sin π 4 d = 4 : : limits : intgrand : answr (b) 4 = at r = and s =.6759 s Th ara of th statd rgion is ( ( 4 )) r d : { : limits : intgrand (c) Volum = ( sin ( π ) ( 4 )) d = : { : intgrand : answr (d) Volum = ( )( sin ( π ) ( 4 )) d = 8.69 or 8.7 : { : intgrand : answr 8 Th Collg Board. All rights rsrvd.

3 8 SCORING GUIDELINES Qustion t (hours) Lt ()(popl) Concrt tickts wnt on sal at noon ( t = ) and wr sold out within 9 hours. Th numbr of popl waiting in lin to purchas tickts at tim t is modld by a twic-diffrntiabl function L for t 9. Valus of Lt () at various tims t ar shown in th tabl abov. (a) Us th data in th tabl to stimat th rat at which th numbr of popl waiting in lin was changing at 5: P.M. ( t = 5.5 ). Show th computations that lad to your answr. Indicat units of masur. (b) Us a trapzoidal sum with thr subintrvals to stimat th avrag numbr of popl waiting in lin during th first 4 hours that tickts wr on sal. (c) For t 9, what is th fwst numbr of tims at which L () t must qual? Giv a rason for your answr. (d) Th rat at which tickts wr sold for t 9 is modld by rt () = 55t t tickts pr hour. Basd on th modl, how many tickts wr sold by P.M. ( t =, ) to th narst whol numbr? L( 7) L( 4) 5 6 (a) L ( 5.5) = = 8 popl pr hour 7 4 (b) Th avrag numbr of popl waiting in lin during th first 4 hours is approimatly L( ) + L( ) L() ( ) ( ) ( ) ( ) + L L ( ) + L 4 ( 4 ) = 55.5 popl (c) L is diffrntiabl on [, 9 ] so th Man Valu Thorm implis L () t > for som t in (, ) and som t in ( 4, 7 ). Similarly, L () t < for som t in (, 4 ) and som t in ( 7, 8 ). Thn, sinc L is continuous on [, 9 ], th Intrmdiat Valu Thorm implis that L () t = for at last thr valus of t in [, 9 ]. OR Th continuity of L on [, 4 ] implis that L attains a maimum valu thr. Sinc L( ) > L( ) and L( ) > L( 4 ), this maimum occurs on (, 4 ). Similarly, L attains a minimum on (, 7 ) and a maimum on ( 4, 8 ). L is diffrntiabl, so L () t = at ach rlativ trm point on (, 9 ). Thrfor L () t = for at last thr valus of t in [, 9 ]. [Not: Thr is a function L that satisfis th givn conditions with L () t = for actly thr valus of t.] (d) rt () = Thr wr approimatly 97 tickts sold by P.M. : { : stimat : units : trapzoidal sum : { : answr : : : considrs chang in sign of L : analysis : conclusion OR : considrs rlativ trma of L on (, 9) : analysis : conclusion : { : intgrand : limits and answr 8 Th Collg Board. All rights rsrvd.

4 8 SCORING GUIDELINES Qustion Oil is laking from a piplin on th surfac of a lak and forms an oil slick whos volum incrass at a constant rat of cubic cntimtrs pr minut. Th oil slick taks th form of a right circular cylindr with both its radius and hight changing with tim. (Not: Th volum V of a right circular cylindr with radius r and hight h is givn by V = π r h. ) (a) At th instant whn th radius of th oil slick is cntimtrs and th hight is.5 cntimtr, th radius is incrasing at th rat of.5 cntimtrs pr minut. At this instant, what is th rat of chang of th hight of th oil slick with rspct to tim, in cntimtrs pr minut? (b) A rcovry dvic arrivs on th scn and bgins rmoving oil. Th rat at which oil is rmovd is R() t = 4 t cubic cntimtrs pr minut, whr t is th tim in minuts sinc th dvic bgan working. Oil continus to lak at th rat of cubic cntimtrs pr minut. Find th tim t whn th oil slick rachs its maimum volum. Justify your answr. (c) By th tim th rcovry dvic bgan rmoving oil, 6, cubic cntimtrs of oil had alrady lakd. Writ, but do not valuat, an prssion involving an intgral that givs th volum of oil at th tim found in part (b). (a) Whn r = cm and h =.5 cm, and dr =.5 cm min. dr dh = πr h + πr = π( )(.5)(.5) + π( ) dh =.8 or.9 cm min dh = cm min dr : = and =.5 4 : : prssion for : answr (b) = R() t, so = whn Rt () =. This occurs whn t = 5 minuts. Sinc > for < t < 5 and < for t > 5, th oil slick rachs its maimum volum 5 minuts aftr th dvic bgins working. : : Rt () = : answr : justification (c) Th volum of oil, in cm, in th slick at tim t = 5 minuts 5 is givn by 6, + ( R() t ). : limits and initial condition : { : intgrand 8 Th Collg Board. All rights rsrvd.

5 8 SCORING GUIDELINES Qustion 4 A particl movs along th -ais so that its vlocity at tim t, for t 6, is givn by a diffrntiabl function v whos graph is shown abov. Th vlocity is at t =, t =, and t = 5, and th graph has horizontal tangnts at t = and t = 4. Th aras of th rgions boundd by th t-ais and th graph of v on 5, 6 ar 8,, and, rspctivly. At tim t =, th particl is at =. th intrvals [, ], [, 5 ], and [ ] (a) For t 6, find both th tim and th position of th particl whn th particl is farthst to th lft. Justify your answr. (b) For how many valus of t, whr t 6, is th particl at = 8? Eplain your rasoning. (c) On th intrval < t <, is th spd of th particl incrasing or dcrasing? Giv a rason for your answr. (d) During what tim intrvals, if any, is th acclration of th particl ngativ? Justify your answr. (a) Sinc vt () < for < t < and 5 < t < 6, and vt () > for < t < 5, w considr t = and t = 6. ( ) = + v( t) = 8 = 6 ( 6) = + v( t) = 8 + = 9 Thrfor, th particl is farthst lft at tim t = whn its position is ( ) =. (b) Th particl movs continuously and monotonically from ( ) = to ( ) =. Similarly, th particl movs continuously and monotonically from ( ) = to ( 5) = 7 and also from ( 5) = 7 to ( 6) = 9. By th Intrmdiat Valu Thorm, thr ar thr valus of t for which th particl is at t () = 8. (c) Th spd is dcrasing on th intrval < t < sinc on this intrval v < and v is incrasing. (d) Th acclration is ngativ on th intrvals < t < and 4 < t < 6 sinc vlocity is dcrasing on ths intrvals. : idntifis t = as a candidat 6 : : considrs vt () : conclusion : : positions at t =, t = 5, and t = 6 : dscription of motion : conclusion : answr with rason : { : answr : justification 8 Th Collg Board. All rights rsrvd.

6 8 SCORING GUIDELINES Qustion 5 dy y Considr th diffrntial quation =, whr. d (a) On th as providd, sktch a slop fild for th givn diffrntial quation at th nin points indicatd. (Not: Us th as providd in th am booklt.) (b) Find th particular solution y = f( ) to th diffrntial quation with th initial condition f ( ) =. (c) For th particular solution y = f( ) dscribd in part (b), find lim f ( ). (a) : zro slops : { : all othr slops (b) dy = d y ln y = + C y = + C y = y = k, whr k = ± = k k = C ( ) ( ) f =, > C 6 : : sparats variabls : antidiffrntiats : includs constant of intgration : uss initial condition : solvs for y Not: ma 6 [----] if no constant of intgration Not: 6 if no sparation of variabls ( ) (c) lim = : limit 8 Th Collg Board. All rights rsrvd.

7 8 SCORING GUIDELINES Qustion 6 ln Lt f b th function givn by f( ) = for all >. Th drivativ of f is givn by ln f ( ) =. (a) Writ an quation for th lin tangnt to th graph of f at =. (b) Find th -coordinat of th critical point of f. Dtrmin whthr this point is a rlativ minimum, a rlativ maimum, or nithr for th function f. Justify your answr. (c) Th graph of th function f has actly on point of inflction. Find th -coordinat of this point. (d) Find lim f ( ). + ln (a) f( ), ln = = f ( ) = = 4 ( ) An quation for th tangnt lin is y = ( ) 4. ( ) ( ) : : f and f : answr (b) f ( ) = whn =. Th function f has a rlativ maimum at = bcaus f ( ) changs from positiv to ngativ at =. : : = : rlativ maimum : justification ( ln ) ln (c) f ( ) + = = for all > 4 f ( ) = whn + ln = : f ( ) : : answr = Th graph of f has a point of inflction at f ( ) changs sign at =. = bcaus (d) ln lim + = or Dos Not Eist : answr 8 Th Collg Board. All rights rsrvd.

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