AP Calculus Multiple-Choice Question Collection connect to college success

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1 AP Calculus Multipl-Choic Qustion Collction connct to collg succss

2 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,7 schools, collgs, univrsitis, and othr ducational organizations. Each yar, th Collg Board srvs ovr thr and a half 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. Copyright 5 by Collg Board. All rights rsrvd. Collg Board, AP Cntral, APCD, Advancd Placmnt Program, AP, AP Vrtical Tams, Pr-AP, SAT, and th acorn logo ar rgistrd tradmarks of th Collg Entranc Eamination Board. Admittd Class Evaluation Srvic, CollgEd, Connct to collg succss, MyRoad, SAT Profssional Dvlopmnt, SAT Radinss Program, and Stting th Cornrstons ar tradmarks ownd by th Collg Entranc Eamination Board. PSAT/NMSQT is a tradmark of th Collg Entranc Eamination Board and National Mrit Scholarship Corporation. Othr products and srvics may b tradmarks of thir rspctiv ownrs. Prmission to us copyrightd Collg Board matrials may b rqustd onlin at: Visit th Collg Board on th Wb: AP Cntral is th official onlin hom for th AP Program and Pr-AP: apcntral.collgboard.com. AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com. ii

3 K- Accss and Equity Initiativs Equity Policy Statmnt Th Collg Board blivs that all studnts should b prpard for and hav an opportunity to participat succssfully in collg, and that quitabl accss to highr ducation must b a guiding principl for tachrs, counslors, administrators, and policymakrs. As part of this, all studnts should b givn appropriat guidanc about collg admissions, and providd th full support ncssary to nsur collg admission and succss. All studnts should b ncouragd to accpt th challng of a rigorous acadmic curriculum through nrollmnt in collg prparatory programs and AP courss. Schools should mak vry ffort to nsur that AP and othr collg-lvl classs rflct th divrsity of th studnt population. Th Collg Board ncourags th limination of barrirs that limit accss to dmanding courss for all studnts, particularly thos from traditionally undrrprsntd thnic, racial, and socioconomic groups. For mor information about quity and accss in principl and practic, plas snd an mail to [email protected]. AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com. iii

4 Tabl of Contnts About This Collction vi Qustions 969 AP Calculus AB Eam, Sction 969 AP Calculus BC Eam, Sction 97 AP Calculus AB Eam, Sction 97 AP Calculus BC Eam, Sction AP Calculus AB Eam, Sction AP Calculus BC Eam, Sction AP Calculus AB Eam, Sction AP Calculus BC Eam, Sction AP Calculus AB Eam, Sction AP Calculus BC Eam, Sction AP Calculus AB Eam, Sction Part A Part B AP Calculus BC Eam, Sction Part A Part B 998 AP Calculus AB Eam, Sction 5 Part A 5 Part B 998 AP Calculus BC Eam, Sction 8 Part A 8 Part B 7 AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com. iv

5 Tabl of Contnts Answr Ky 5 Solutions Calculus AB Calculus BC Calculus AB 7 97 Calculus BC Calculus AB Calculus BC Calculus AB Calculus BC 99 Calculus AB 6 99 Calculus BC 997 Calculus AB 7 Part A 7 Part B 997 Calculus BC Part A Part B Calculus AB 8 Part A 8 Part B 998 Calculus BC Part A Part B 6 AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com. v

6 About This Collction About This Collction Multipl-choic qustions from past AP Calculus Eams provid a rich rsourc for taching topics in th cours and rviwing for th am ach yar. Ovr th yars, som topics hav bn addd or rmovd, but almost all of th old qustions still offr intrsting opportunitis to invstigat concpts and assss studnt undrstanding. Always consult th most rcnt Cours Dscription on AP Cntral for th currnt topic outlins for Calculus AB and Calculus BC. Plas not th following: Th solution to ach multipl-choic qustion suggsts on possibl way to solv that qustion. Thr ar oftn altrnativ approachs that produc th sam choic of answr, and for som qustions such multipl approachs ar providd. Tachrs ar also ncouragd to invstigat how th incorrct options for ach qustion could b obtaind to hlp studnts undrstand (and avoid) common typs of mistaks. Scintific (nongraphing) calculators wr rquird on th AP Calculus Eams in 99. Graphing calculators hav bn rquird on th AP Calculus Eams sinc 995. In 997 and 998, Sction I, Part A did not allow th us of a calculator; Sction I, Part B rquird th us of a graphing calculator. Matrials includd in this rsourc may not rflct th currnt AP Cours Dscription and am in this subjct, and tachrs ar advisd to tak this into account as thy us ths matrials to support thir instruction of studnts. For up-to-dat information about this AP cours and am, plas download th official AP Cours Dscription from th AP Cntral Wb sit at apcntral.collgboard.com. AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com. vi

7 969 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. Which of th following dfins a function f for which f ( ) = f( )? f ( ) = f ( ) = sin f ( ) = cos f ( ) = log f ( ) =. ln ( ) < if and only if < < < < < > >. If f( ) =, for, f() = k and if f is continuous at =, thn k = d + = 6 5. If + y+ y =, thn th valu of dy d at = is not dfind AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

8 969 AP Calculus AB: Sction I 6. What is h h 8 lim? h Th limit dos not ist. It cannot b dtrmind from th information givn. 7. For what valu of k will k + hav a rlativ maimum at =? Non of ths = + + and if th rmaindr is whn p( ) is dividd by, thn k = 8. If p( ) ( )( k) 6 9. Whn th ara in squar units of an panding circl is incrasing twic as fast as its radius in linar units, th radius is t. Th st of all points (, t ), whr t is a ral numbr, is th graph of y = ln ln. Th point on th curv + y = that is narst th point, occurs whr y is non of th abov AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

9 969 AP Calculus AB: Sction I. If f( ) = and g ( ), = thn th solution st of f ( g ( )) g( f( ) ) = is { } { } {,},. Th rgion boundd by th -ais and th part of th graph of y = cos btwn = and = is sparatd into two rgions by th lin = k. If th ara of th rgion for k is thr tims th ara of th rgion for k, thn k = arcsin arcsin 6. If th function f is dfind by f ( ) = 5 f ( ) =, thn f, th invrs function of f, is dfind by If f ( ) and g ( ) ist and f ( ) > g ( ) for all ral, thn th graph of y = f( ) and th graph of y = g( ) intrsct actly onc. intrsct no mor than onc. do not intrsct. could intrsct mor than onc. hav a common tangnt at ach point of intrsction. AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

10 969 AP Calculus AB: Sction I 6. If y is a function of such that y > for all and y < for all, which of th following could b part of th graph of y = f( )? 7. Th graph of 5 y = 5 has a point of inflction at (,) only (,6) only (,56) only (,) and (,6) (,) and (,56) 8. If f( ) = + for all, thn th valu of th drivativ f ( ) at = is nonistnt 9. A point movs on th -ais in such a way that its vlocity at tim t ( t > ) is givn by At what valu of t dos v attain its maimum? v = ln t. t Thr is no maimum valu for v. AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

11 . An quation for a tangnt to th graph of arcsin y = at th origin is 969 AP Calculus AB: Sction I y = y = = y = y =. At =, which of th following is tru of th function f dfind by f ( ) = +? f is incrasing. f is dcrasing. f is discontinuous. f has a rlativ minimum. f has a rlativ maimum. d. ( ln ) d =. Th ara of th rgion boundd by th curv qual to y =, th -ais, th y-ais, and th lin = is y. If sin =, < <, what is dy d in trms of? tan cot cot tan csc AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

12 969 AP Calculus AB: Sction I 5. A rgion in th plan is boundd by th graph of = m, m >. Th ara of this rgion y =, th -ais, th lin = m, and th lin is indpndnt of m. incrass as m incrass. dcrass as m incrass. dcrass as m incrass whn incrass as m incrass whn m < ; incrass as m incrass whn m >. m < ; dcrass as m incrass whn m > d is non of th abov dy 7. If tan, d = thn y = tan + C sc + C ln sc + C ln cos + C sc tan + C 8. Th function dfind by f ( ) = cos+ sin has an amplitud of + AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

13 969 AP Calculus AB: Sction I 9. cos d sin = ln ln ln ln ln. If a function f is continuous for all and if f has a rlativ maimum at (,) and a rlativ minimum at (, ), which of th following statmnts must b tru? Th graph of f has a point of inflction somwhr btwn = and =. f ( ) = Th graph of f has a horizontal asymptot. Th graph of f has a horizontal tangnt lin at =. Th graph of f intrscts both as.. If f ( ) = f( ) and f () =, thn f ( ) = +. If abcd,,,, and ar ral numbrs and a, thn th polynomial quation 7 5 a + b + c + d + = has only on ral root. at last on ral root. an odd numbr of nonral roots. no ral roots. no positiv ral roots.. What is th avrag (man) valu of t t ovr th intrval t? AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

14 969 AP Calculus AB: Sction I. Which of th following is an quation of a curv that intrscts at right angls vry curv of th family y = + k (whr k taks all ral valus)? y = y = y = y = y = ln 5. At t = a particl starts at rst and movs along a lin in such a way that at tim t its acclration is t ft pr scond pr scond. Through how many ft dos th particl mov during th first sconds? Th approimat valu of y = + sin at =., obtaind from th tangnt to th graph at =, is Which is th bst of th following polynomial approimations to cos nar =? d = ln + C ln + C + C + C + C 9. If y = tan u, u = v, and v= ln, what is th valu of dy v d at =? sc AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

15 n n. If n is a non-ngativ intgr, thn = ( ) 969 AP Calculus AB: Sction I d d for no n n vn, only n odd, only nonzro n, only all n. If f = ( ) 8 for, f( ) = lswhr, thn f ( d ) is a numbr btwn and 8 8 and 6 6 and and and. What ar all valus of k for which th graph of -intrcpts? y = + k will hav thr distinct All k > All k < k =, < k < All k. sin ( ) + d = cos ( + ) + C cos( + ) + C cos( ) cos cos + + C 5 ( + ) + C ( ) + + C. Th fundamntal priod of th function dfind by f( ) = cos is 5 6 d d d g ( ) f ( ), d 5. If ( f ( ) ) = g( ) and ( ) d = thn ( f( )) 6 f ( ) g( ) g( ) 9 f ( 6 ) + 6g( ) f ( 6 ) + g( ) d = AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

16 969 AP Calculus BC: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. t Th asymptots of th graph of th paramtric quations, y t t =, y = = only =, y = = only =, y =. What ar th coordinats of th inflction point on th graph of y = ( + )arctan? (,) (, ) ( ),,,. Th Man Valu Thorm guarants th istnc of a spcial point on th graph of y =,. What ar th coordinats of this point? btwn (, ) and ( ) (, ) (, ) (, ), Non of th abov. 8 d + = 6 5. If + y+ y =, thn th valu of dy d at = is not dfind AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

17 6. What is h h 8 lim? h 969 AP Calculus BC: Sction I Th limit dos not ist. It cannot b dtrmind from th information givn. 7. For what valu of k will k + hav a rlativ maimum at =? Non of ths 8. If h ( ) = f ( ) g ( ), f ( ) = g ( ), and g ( ) = f( ), thn h ( ) = f ( g ) ( ) ( g ( )) ( f( ) ) ( g ( ) + f( ) ) 9. Th ara of th closd rgion boundd by th polar graph of r = + cosθ is givn by th intgral + cosθdθ + cosθdθ ( + cosθ) dθ + cosθ dθ ( ) + cosθdθ. + d = ln ln + AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

18 969 AP Calculus BC: Sction I. Th point on th curv non of th abov + y = that is narst th point, occurs whr y is. If t F( ) = dt, thn F ( ) = + +. Th rgion boundd by th -ais and th part of th graph of y = cos btwn = and = is sparatd into two rgions by th lin = k. If th ara of th rgion for k is thr tims th ara of th rgion for k, thn k = arcsin arcsin 6. If = + and =, thn dy y u du = + ( ) 6 + AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

19 969 AP Calculus BC: Sction I 5. If f ( ) and g ( ) ist and f ( ) > g ( ) for all ral, thn th graph of y = f( ) and th graph of y = g( ) intrsct actly onc. intrsct no mor than onc. do not intrsct. could intrsct mor than onc. hav a common tangnt at ach point of intrsction. 6. If y is a function such that y > for all and y < for all, which of th following could b part of th graph of y = f( )? 7. Th graph of 5 y = 5 has a point of inflction at (, ) only (,6 ) only ( ) (, ) and (,6 ) (, ) and (, 56 ), 56 only 8. If f( ) = + for all, thn th valu of th drivativ f ( ) at = is nonistnt AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

20 969 AP Calculus BC: Sction I 9. A point movs on th -ais in such a way that its vlocity at tim t ( t > ) is givn by At what valu of t dos v attain its maimum? ln t v =. t Thr is no maimum valu for v.. An quation for a tangnt to th graph of arcsin y = at th origin is y = y = = y = y =. At =, which of th following is tru of th function f dfind by f ( ) = +? f is incrasing. f is dcrasing. f is discontinuous. f has a rlativ minimum. f has a rlativ maimum.. If f ( ) = dt, which of th following is FALSE? t + f () = f is continuous at for all. f () > f () = f ( ) > AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

21 . If th graph of y f( ) + = contains th point ( ) 969 AP Calculus BC: Sction I dy,, = and f( ) > for all, thn f ( ) = d y y. If sin =, < <, what is dy in trms of? d tan cot cot tan csc 5. A rgion in th plan is boundd by th graph of = m, m >. Th ara of this rgion y =, th -ais, th lin = m, and th lin is indpndnt of m. incrass as m incrass. dcrass as m incrass. dcrass as m incrass whn m < ; incrass as m incrass whn m >. incrass as m incrass whn m < ; dcrass as m incrass whn m > d is non of th abov AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

22 969 AP Calculus BC: Sction I dy 7. If tan, d = thn y = tan + C sc + C ln sc + C ln cos + C sc tan + C 8. What is lim tan? Th limit dos not ist. 9. ( ) d =. ( ) is th Taylor sris about zro for which of th following functions? n! n= n n sin cos ln( + ). If f ( ) = f( ) and f () =, thn f ( ) = +. For what valus of dos th sris n + convrg? No valus of < > All valus of. What is th avrag (man) valu of t t ovr th intrval t? AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

23 969 AP Calculus BC: Sction I. Which of th following is an quation of a curv that intrscts at right angls vry curv of th family y = + k (whr k taks all ral valus)? y = y = y = y = y = ln 5. At t = a particl starts at rst and movs along a lin in such a way that at tim t its acclration is t ft pr scond pr scond. Through how many ft dos th particl mov during th first sconds? Th approimat valu of y = + sin at =., obtaind from th tangnt to th graph at =, is Of th following choics of δ, which is th largst that could b usd succssfully with an arbitrary ε in an psilon-dlta proof of ( ) lim 5? = δ= ε δ=ε ε δ = ε δ = ε δ= 5 8. If ( ) ( f ) ( ) = +, thn f () = ln(8 ) ln(8 ) ln() 8 9. If y = tan u, u = v, and v= ln, what is th valu of dy v d at =? sc AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

24 n n. If n is a non-ngativ intgr, thn = ( ) 969 AP Calculus BC: Sction I d d for no n n vn, only n odd, only nonzro n, only all n. If f = ( ) 8 for, f( ) = lswhr, thn f ( d ) is a numbr btwn and 8 8 and 6 6 and and and. If cos d = f ( ) sin d, thn f ( ) = sin + cos+ C sin + C cos sin+ C cos sin+ C ( ) cos sin+ C. Which of th following intgrals givs th lngth of th graph of y = tan btwn = a and = b, whr < a< b<? b a b a b a b a b a + tan + tan d + sc + tan + sc d d d d AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

25 . If f ( ) f ( ) f( ) =, f () =, and f () =, thn f () = 969 AP Calculus BC: Sction I + C) 5. Th complt intrval of convrgnc of th sris k= ( + ) k is k < < < < AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

26 97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = and ( ) 5, g= thn ( ( )) g f = Th slop of th lin tangnt to th graph of y ln ( ) = at = is. If f ( ) = + sin, thn f ( ) = + cos cos cos sin cos sin + cos 5. If f ( ) =, which of th following lins is an asymptot to th graph of f? y = = y = y = y = 6. If f( ) = for all, thn f () = + AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

27 97 AP Calculus AB: Sction I 7. Which of th following quations has a graph that is symmtric with rspct to th origin? + y = = + y = ( + ) y ( ) 5 y = + y = A particl movs in a straight lin with vlocity tims t = and t =? vt () = t. How far dos th particl mov btwn If y = cos, thn dy d = 6sin cos cos cos 6cos sin cos 5. Th drivativ of f( ) = attains its maimum valu at = 5 5. If th lin y = is tangnt in th first quadrant to th curv y = + k, thn k is 8. If f( ) = + A + B 5 and if f() = and f( ) = 7, what is th valu of A+ B? 6 It cannot b dtrmind from th information givn. AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

28 97 AP Calculus AB: Sction I. Th acclration α of a body moving in a straight lin is givn in trms of tim t by α= 8 6t. If th vlocity of th body is 5 at t = and if st ( ) is th distanc of th body from th origin at tim t, what is s() s()? 8. If f( ) ( ) = for all, thn th domain of f is { } { > } { } { and } { is a ral numbr} 5. Th ara of th rgion boundd by th lins =, =, and y = and th curv y = is ( ) 6. Th numbr of bactria in a cultur is growing at a rat of 5 pr unit of tim t. At t =, th numbr of bactria prsnt was 7,5. Find th numbr prsnt at t = 5. t,, 7,5 5 7,5 5, What is th ara of th rgion compltly boundd by th curv y =? y = and th lin d d 8. ( arcsin ) = AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

29 97 AP Calculus AB: Sction I 9. Suppos that f is a function that is dfind for all ral numbrs. Which of th following conditions assurs that f has an invrs function? Th function f is priodic. Th graph of f is symmtric with rspct to th y-ais. Th graph of f is concav up. Th function f is a strictly incrasing function. Th function f is continuous.. If F and f ar continuous functions such that F ( ) = f( ) for all, thn f ( d ) is a F ( a) F ( b) F ( b) F ( a) Fa ( ) Fb ( ) Fb ( ) Fa ( ) non of th abov b. + ( + ) d=. Givn th function dfind by concav up. 5 f ( ) =, find all valus of for which th graph of f is > < < or > < < or > > < < AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

30 97 AP Calculus AB: Sction I. + h lim ln h is h nonistnt. Lt f ( ) cos( arctan) =. What is th rang of f? < < { < < } { } { < } { } 5. tan d = + 6. Th radius r of a sphr is incrasing at th uniform rat of. inchs pr scond. At th instant whn th surfac ara S bcoms squar inchs, what is th rat of incras, in cubic inchs pr scond, in th volum V? S = r and V = r d = ln A point movs in a straight lin so that its distanc at tim t from a fid point of th lin is 8t t. What is th total distanc covrd by th point btwn t = and t =? 5 5 AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

31 97 AP Calculus AB: Sction I 9. Lt f( ) = sin. Th maimum valu attaind by f is. d = ln ln ln + a log =, thn a = a. If a ( ) d = ( + ) + C 5arctan C 5 ln ( ) + ( ) + + C 5ln + + C C. Suppos that f is an odd function; i.., f ( ) f( ) Which of th following must ncssarily b qual to f ( )? f ( ) ( ) f = for all. Suppos that ( ) f ists. f ( ) f ( ) Non of th abov AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

32 . Th avrag valu of ovr th intrval is 97 AP Calculus AB: Sction I 5. Th rgion in th first quadrant boundd by th graph of y = sc, =, and th as is rotatd about th -ais. What is th volum of th solid gnratd? 8 n 6. If y =, thn n d y d n = n n n! n n n n n n n! dy 7. If = y and if y = whn =, thn y = d c 8. If f ( c) d= 5 whr c is a constant, thn ( ) f d = c 5 + c 5 5 c c Th point on th curv y = narst to ( ), is (, ) (, ) (, ) (, ) (,8 ). If tan( y) =, thn dy d = ytan( y)sc( y) tan( y)sc( y) sc ( y) y cos ( y ) cos ( y) cos ( y) y AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

33 . Givn + for <, f( ) = cos for, f ( d ) 97 AP Calculus AB: Sction I = + +. Calculat th approimat ara of th shadd rgion in th figur by th trapzoidal rul, using 5 divisions at = and = If th solutions of f( ) = ar and, thn th solutions of f = ar and 5 and and and and. For small valus of h, th function 6 + h is bst approimatd by which of th following? h + h + h h h AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

34 97 AP Calculus AB: Sction I 5. If f is a continuous function on [ ab, ], which of th following is ncssarily tru? f ists on ( ab., ) If f ( ) is a maimum of f, thn ( ) lim f ( ) = f lim for ( a, b) f ( ) = for som a, b [ ] f =. Th graph of f is a straight lin. AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

35 97 AP Calculus BC: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. If f ( ) =, thn f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, thn th st of valus for which f incrass is (, ] [, ) [,] (, ) (, ) (,) (, ). For what non-ngativ valu of b is th lin givn by y = + b normal to th curv y =? 5. d is nonistnt 6. If f( ) = for all, thn f () = + AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

36 7. If y ( y ) = ln +, thn th valu of dy at th point (,) is d 97 AP Calculus BC: Sction I undfind 8. If y = sin and intgr n for which ( n) y mans th nth drivativ of y with rspct to, thn th smallst positiv ( n) y = y is If y = cos, thn dy d = 6sin cos cos cos 6cos sin cos. Th lngth of th curv y = ln sc from = to = b, whr < b <, may b prssd by which of th following intgrals? b sc d b sc b d (sc tan d ) b ( ) + lnsc d b + ( sc tan ) d. Lt y = +. Whn = and d =, th valu of dy is AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

37 97 AP Calculus BC: Sction I. If n is a known positiv intgr, for what valu of k is n k n d=? n n n n n n n. Th acclration α of a body moving in a straight lin is givn in trms of tim t by α= 8 6t. If th vlocity of th body is 5 at t = and if st ( ) is th distanc of th body from th origin at tim t, what is s() s()? 8. If t = t and y =, thn dy d = t t t t t t t t t 5. Th ara of th rgion boundd by th lins =, =, and y = and th curv y = is ( ) 6. A sris pansion of sin t t is 6 t t t + +! 5! 7! 5 t t t t! +! 6! + 6 t t t ! 5! 7! 5 t t t t +! +! + 6! t t t t + +! 5! 7! AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

38 97 AP Calculus BC: Sction I 5 7. Th numbr of bactria in a cultur is growing at a rat of, t pr unit of tim t. At t =, th numbr of bactria prsnt was 7,5. Find th numbr prsnt at t = 5.,, 7,5 5 7,5 5, Lt g b a continuous function on th closd intrval [, ]. Lt g () = and g () =. Which of th following is NOT ncssarily tru? Thr ists a numbr h in [ ] For all a and b in [ ], such that gh ( ) g ( ),, if a= b, thn ga ( ) = gb ( ). Thr ists a numbr h in [, ] such that Thr ists a numbr h in [, ] such that For all h in th opn intrval ( ) 9. Which of th following sris convrg? gh ( ) =. gh ( ) =.,, lim g ( ) = gh ( ). h for all in [ ],. I. II. n n= III. n n= n= ( ) n n I only III only I and II only I and III only I, II, and III. d= ( ) + C ( ) + C ( ) + C ( ) + C ( ) + C. + ( + ) d= AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

39 97 AP Calculus BC: Sction I. A particl movs on th curv y = ln so that th -componnt has vlocity ( t) = t+ for t. At tim,. At tim t =, th particl is at th point ( ) t =, th particl is at th point ( ),ln (,) (,ln ),ln 5 5,ln. + h lim ln h is h nonistnt. Lt f ( ) = + for all ral and lt ε >. For which of th following choics of δ is f( ) 7 <ε whnvr <δ? ε ε ε ε + ε + ε ε 5. tan d = + 6. Which of th following is tru about th graph of y = ln in th intrval (,)? It is incrasing. It attains a rlativ minimum at (, ). It has a rang of all ral numbrs. It is concav down. It has an asymptot of =. 7. If f ( ) = + 5 and th domain is th st of all such that 9, thn th absolut maimum valu of th function f occurs whn is 6 9 AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

40 97 AP Calculus BC: Sction I 8. If th substitution = sin y is mad in th intgrand of d, th rsulting intgral is sin ydy sin y dy cos y sin ydy sin ydy 6 sin ydy 9. If y = y and if y = y = whn =, thn whn =, y = + ( ) ( ). d ln ln ln + ( ) = ln ln, thn f ( ) =. If f ( ) ln ln ln. If ln, y = thn y is ln ln ln ln ln ln ln Non of th abov AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

41 . Suppos that f is an odd function; i.., f ( ) f( ) Which of th following must ncssarily b qual to f ( )? f ( ) ( ) f f ( ) f ( ) Non of th abov. Th avrag (man) valu of ovr th intrval is 97 AP Calculus BC: Sction I = for all. Suppos that ( ) f ists. 5. Th rgion in th first quadrant boundd by th graph of y = sc, =, and th as is rotatd about th -ais. What is th volum of th solid gnratd? d is + ln ln ln ln divrgnt 7. cos ( ) lim = c 8. If f ( c) d= 5 whr c is a constant, thn ( ) f d = c 5 + c 5 5 c c 5 5 AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

42 9. Lt f and g b diffrntiabl functions such that f () =, f () =, f () =, g () =, g () =, g () = 5. ( ) = ( ), thn h () = If h f( g) 97 AP Calculus BC: Sction I 9 5. Th ara of th rgion nclosd by th polar curv r = cosθ is. Givn + for <, f( ) = cos for, f ( d ) = + +. Calculat th approimat ara of th shadd rgion in th figur by th trapzoidal rul, using 5 divisions at = and = AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

43 97 AP Calculus BC: Sction I. arcsin d = sin d ( arcsin ) arcsin + + C d arccos arcsin d d. If f is th solution of f ( ) f( ) = such that ( ), f = thn ( ) f = C) 5. Suppos g ( ) < for all and statmnts is FALSE? F( ) = tg ( t) dt for all. Which of th following F taks on ngativ valus. F is continuous for all >. F( ) g( ) g( t) dt = F ( ) ists for all >. F is an incrasing function. AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

44 985 AP Calculus AB: Sction I 9 Minuts No Calculator Nots: () In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ). () Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. d = If f( ) ( ) = +, thn th th drivativ of f ( ) at = is 8 8. If dy y =, thn + d = 6 ( + ) ( + ) 6 ( + ) ( + ) dy, thn y d = =. If cos( ) cos cos sin ( ) + C ( ) + C ( ) sin sin + C ( ) + C ( ) + C 5. n lim is n, n n +,5 nonistnt AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

45 985 AP Calculus AB: Sction I 6. If f ( ) =, thn f (5) = Which of th following is qual to ln? ln + ln ln 8 ln t dt ln d dt t 8. Th slop of th lin tangnt to th graph of y = ln at = is 8 9. If d= k, thn d= k k k k k. If ( ) y =, thn dy d = ( ) ( ) ln ( ) ( ) ( ) ( ) ln ( ) ( ) ln ( ) ( ). Th position of a particl moving along a straight lin at any tim t is givn by st () = t + t+. What is th acclration of th particl whn t =? 8. If f ( g) ( ) f ( ) ( ) = ln +, ( ) = ln, and g> ( ) for all ral, thn g () = AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

46 985 AP Calculus AB: Sction I. If + y+ y =, thn, in trms of and y, dy d = + y + y + y + y + y + y + y + y. Th vlocity of a particl moving on a lin at tim t is mtrs did th particl travl from t = to t =? v= t + 5t mtrs pr scond. How many Th domain of th function dfind by f( ) ln( ) = is th st of all ral numbrs such that < > is a ral numbr 6. Th function dfind by f ( ) = for all ral numbrs has a rlativ maimum at = 7. d= 8. If y = cos sin, thn y = sin( ) cos ( + sin) cos ( sin) 9. If f ( ) f ( ) f ( ) dfin f? + = + for all ral numbrs and, which of th following could f ( ) = + f ( ) = f( ) = f ( ) = f ( ) = AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

47 985 AP Calculus AB: Sction I. If y ( ) = arctan cos, thn dy d = sin + cos ( arccos ) + ( ) ( ) arcsc( cos ) sin arcsc( cos ) + cos. If th domain of th function f givn by f( ) = is { } : >, what is th rang of f? { : < < } { : < < } { : < < } { : < < } { :< < }. d = + 5 ln. d at is d + = 6 6. If ( 7 ) + k d= 6, thn k = 5. If f ( ) =, which of th following is qual to f ()? lim h h + h lim h + h h lim h + h h lim h + h h lim h + h h AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

48 985 AP Calculus AB: Sction I 6. Th graph of y = + 9 is symmtric to which of th following? I. Th -ais II. Th y-ais III. Th origin I only II only III only I and II only I, II, and III 7. d = If th position of a particl on th -ais at tim t is for t is 5t, thn th avrag vlocity of th particl Which of th following functions ar continuous for all ral numbrs? I. II. III. y = y = y = tan Non I only II only I and II I and III. tan ( ) d = ln cos( ) + C ln cos( ) + C ln cos( ) + C ln cos( ) + C sc( )tan( ) + C AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

49 985 AP Calculus AB: Sction I. Th volum of a con of radius r and hight h is givn by V = r h. If th radius and th hight both incras at a constant rat of cntimtr pr scond, at what rat, in cubic cntimtrs pr scond, is th volum incrasing whn th hight is 9 cntimtrs and th radius is 6 cntimtrs? 5 8. ( ) sin d =. Th graph of th drivativ of f is shown in th figur abov. Which of th following could b th graph of f? AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

50 985 AP Calculus AB: Sction I. Th ara of th rgion in th first quadrant that is nclosd by th graphs of y = + 8 is y = + 8 and Th figur abov shows th graph of a sin function for on complt priod. Which of th following is an quation for th graph? y = sin y = sin( ) y = sin ( ) y = sin ( ) y = sin( ) 6. If f is a continuous function dfind for all ral numbrs and if th maimum valu of f ( ) is 5 and th minimum valu of f ( ) is 7, thn which of th following must b tru? I. Th maimum valu of f ( ) is 5. II. Th maimum valu of f ( ) is 7. III. Th minimum valu of f ( ) is. I only II only I and II only II and III only I, II, and III 7. lim ( csc ) is AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

51 985 AP Calculus AB: Sction I 8. Lt f and g hav continuous first and scond drivativs vrywhr. If f ( ) g( ) for all ral, which of th following must b tru? I. f ( ) g ( ) for all ral II. f ( ) g ( ) for all ral III. f ( d ) gd ( ) Non I only III only I and II only I, II, and III 9. If ln f( ) =, for all >, which of th following is tru? f is incrasing for all gratr than. f is incrasing for all gratr than. f is dcrasing for all btwn and. f is dcrasing for all btwn and. f is dcrasing for all gratr than.. Lt f b a continuous function on th closd intrval [ ] possibl valu of f ( d ) is,. If f( ), thn th gratst 8 6. If lim f ( ) = L, whr L is a ral numbr, which of th following must b tru? a f ( a) ists. f ( ) is continuous at = a. f ( ) is dfind at = a. f ( a) = L Non of th abov AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

52 985 AP Calculus AB: Sction I. d d + t dt = An quation of th lin tangnt to y = + + at its point of inflction is y = 6 6 y = + y = + y = y = +. Th avrag valu of = + on th closd intrval [ ] f( ), is Th rgion nclosd by th graph of y =, th lin =, and th -ais is rvolvd about th y -ais. Th volum of th solid gnratd is AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

53 985 AP Calculus BC: Sction I 9 Minuts No Calculator Nots: () In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ). () Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. Th ara of th rgion btwn th graph of y = + and th -ais from = to = is At what valus of dos 5 f( ) = hav a rlativ maimum? only only only and only, and. + d = + ln 8 ln ln 8 ln ln 8 ln ln+. A particl movs in th y-plan so that at any tim t its coordinats ar At t =, its acclration vctor is = t and y = t t. (, ) (, ) (, ) (, ) (,8 ) AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

54 985 AP Calculus BC: Sction I 5. Th curvs y f( ) and y g( ) ara of th shadd rgion nclosd by ths curvs and th lin = is givn by = = shown in th figur abov intrsct at th point (, ) a ( ( ) ( )) + ( ( ) + ( )) f g d f g d ab. Th b g ( ) d + f ( ) d b c c ( ( ) ( )) a f g d ( ( ) ( )) f g d a ( ( ) ( ) ) f g d 6. If f( ) =, thn tan f = + AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

55 985 AP Calculus BC: Sction I 7. Which of th following is qual to arcsin 5 5 d? + C arcsin + C arcsin + C C 5 + C 8. If f is a function such that f( ) f() lim =, which of th following must b tru? Th limit of f ( ) as approachs dos not ist. f is not dfind at =. Th drivativ of f at = is. f is continuous at =. f () = 9. If y + y = 8, thn, at th point (, ), y is 5. For n= n+ n ( ) < < if f( ) =, thn f ( ) = n n= n= + ( ) n n ( ) n n ( ) n n n= ( ) n n n= ( ) n + n n= AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

56 985 AP Calculus BC: Sction I. d ln = d ( ). d = ( )( + ) ln + C + ( ln )( ln ) + + C + ln + C ln ( )( + ) + C ln ( )( + ) + C. Lt f b th function givn by f ( ) =. What ar all valus of c that satisfy th conclusion of th Man Valu Thorm of diffrntial calculus on th closd intrval [, ]? only only only and and. Which of th following sris ar convrgnt? I. II. III n n n+ ( ) n I only III only I and III only II and III only I, II, and III 5. If th vlocity of a particl moving along th -ais is vt ( ) = t and if at t = its position is, thn at any tim t its position ( t ) is t t t t t t+ t t t t+ AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

57 985 AP Calculus BC: Sction I 6. Which of th following functions shows that th statmnt If a function is continuous at =, thn it is diffrntiabl at = is fals? f ( ) = f ( ) = f ( ) = f ( ) = f ( ) = 7. If f ( ) ( ) = ln, thn f ( ) = ln ( ) + ln ( ) + ln ( ) + 8. sin ( ) + d = cos + + C cos( + ) + C cos( + ) + C ( ) cos ( + ) + C cos( ) + + C 9. If f and g ar twic diffrntiabl functions such that thn h ( ) = f ( ) g ( ) = and ( ) g ( ) = h( ) f, f ( ) f ( ) ( ) + f ( ) + ( f ( ) ) ( f ( ) + f ( ) ) f ( ) + f ( ) f ( ) + f ( ). Th graph of y f( ) = on th closd intrval [ ],7 is shown abov. How many points of inflction dos this graph hav on this intrval? On Two Thr Four Fiv AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

58 985 AP Calculus BC: Sction I. If f ( )sin d = f ( )cos + cos d, thn ( ) f could b sin cos. Th ara of a circular rgion is incrasing at a rat of 96 squar mtrs pr scond. Whn th ara of th rgion is 6 squar mtrs, how fast, in mtrs pr scond, is th radius of th rgion incrasing? lim h + h 5 h + 8 d is nonistnt. Th ara of th rgion nclosd by th polar curv sin ( ) r = θ for θ is 8 5. A particl movs along th -ais so that at any tim t its position is givn by valus of t is th particl at rst? t () t = t. For what No valus only only only and 6. For, < < if y ( ) = sin, thn dy d is ln ( sin ) ( sin ) cot ( sin ) ( cos ) ( sin ) ( cos + sin ) ( sin ) ( cot + ln ( sin ) ) AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

59 985 AP Calculus BC: Sction I 7. If f is th continuous, strictly incrasing function on th intrval a b as shown abov, which of th following must b tru? b a b a b a I. f ( d ) < fb ( )( b a) II. f ( d ) > fa ( )( b a) III. f ( d ) = fc ( )( b a) for som numbr c such that a< c< b I only II only III only I and III only I, II, and III 8. An antidrivativ of f( ) = + is + + ( ) sin lim is nonistnt. If = t t and y = t+, thn dy d at t = is What ar all valus of for which th sris n= ( ) n convrgs? n < < < < AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

60 985 AP Calculus BC: Sction I. An quation of th lin normal to th graph of y = at th point whr = is + y = y = y = + y = 5 + y = 5 dy. If y dt = and if y = whn t =, what is th valu of t for which y =? ln ln ln. Which of th following givs th ara of th surfac gnratd by rvolving about th y-ais th arc of = y from y = to y =? y + 9y dy 6 y + y dy y + y dy y + 9y dy 6 y + y dy 5. Th rgion in th first quadrant btwn th -ais and th graph of y = 6 is rotatd around th y-ais. Th volum of th rsulting solid of rvolution is givn by 6 ( ) 6 d 6 ( ) 6 d 6 ( ) 6 6 d ( + ) 9 9 y dy ( + ) 9 y dy AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

61 985 AP Calculus BC: Sction I 6. d is 6 6 nonistnt 7. Th gnral solution for th quation dy + y = is d y = + C y = + + C C y = + + y = + C y = C + C 8. ( ) lim 5 + is 5 nonistnt 9. Th bas of a solid is th rgion nclosd by th graph of y =, th coordinat as, and th lin =. If all plan cross sctions prpndicular to th -ais ar squars, thn its volum is ( 6 ) 6 6. If th substitution u = is mad, th intgral d = u du u u du u u du u u du u u du u AP Calculus Multipl-Choic Qustion Collction 55 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

62 985 AP Calculus BC: Sction I. What is th lngth of th arc of y = from = to =? Th cofficint of in th Taylor sris for about = is 6 9. Lt f b a function that is continuous on th closd intrval [,] such that f () dos not ist, f () =, and f ( ) < for all cpt =. Which of th following could b th graph of f?. At ach point (, y ) on a crtain curv, th slop of th curv is point (,8 ), thn its quation is y. If th curv contains th y = 8 y = + 8 y = + 7 y ( ) = ln y = If n is a positiv intgr, thn lim n n can b prssd as n n n n d d d d d AP Calculus Multipl-Choic Qustion Collction 56 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

63 988 AP Calculus AB: Sction I 9 Minuts No Calculator Nots: () In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ). () Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. = thn dy If y, d = ( ) + ( + ) + +. What is th domain of th function f givn by { : } f( ) =? { : } { : } { : and } { : and }. A particl with vlocity at any tim t givn by vt () = movs in a straight lin. How far dos th particl mov from t = to t =? t. Th graph of 5 y = is concav downward for all valus of such that < < < 5 > > 5. sc d = tan + C csc + C cos + C sc + C sc tan + C AP Calculus Multipl-Choic Qustion Collction 57 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

64 988 AP Calculus AB: Sction I 6. If ln y =, thn dy d = ln ln + ln 7. d + 5 = ( ) C ( ) C ( ) 5 9 ( ) C ( ) C + + C 8. Th graph of y = f( ) is shown in th figur abov. On which of th following intrvals ar dy d > and d y d <? I. a< < b II. b< < c III. c< < d I only II only III only I and II II and III AP Calculus Multipl-Choic Qustion Collction 58 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

65 988 AP Calculus AB: Sction I 9. If + =, thn at th point ( ) y y,, dy d is nonistnt k. If ( ) k d = 8, thn k = An quation of th lin tangnt to th graph of = at th point (, ) f ( ) ( ) is y = 7+ 6 y = 6+ 5 y = + y = y = 7 8. If f ( ) = sin, thn f =. If th function f has a continuous drivativ on [,c ], thn c f ( d ) = f ( c) f() f() c f() f ( c ) f ( ) + c f ( c) f (). cos θ sin d θ= + θ ( ) ( ) ( + ) AP Calculus Multipl-Choic Qustion Collction 59 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

66 988 AP Calculus AB: Sction I 5. If f ( ) =, thn f () = 6. A particl movs along th -ais so that at any tim t its position is givn by () t = t t 9t+. For what valus of t is th particl at rst? No valus only only 5 only and d = 7. ( ) If y = cos, thn d y d = 8cos cos sin cos cos 9. + d = ln ln ln ln ln 5. Lt f b a polynomial function with dgr gratr than. If a b and f( a) = f( b) =, which of th following must b tru for at last on valu of btwn a and b? I. f( ) = II. f ( ) = III. f ( ) = Non I only II only I and II only I, II, and III AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

67 . Th ara of th rgion nclosd by th graphs of y = and 988 AP Calculus AB: Sction I y = + is. If ln ln =, thn =. If f ( ) = cos and g ( ) = for all, and if f() = g() =, thn f ( ) lim g ( ) is nonistnt d. ( ln ) d = ln ( ln ) ( ln ln )( ) ln ( ln )( ) ln ln ( )( ) 5. For all >, if f ( ) = dt, thn f ( ) = t ln ln 6. cos d= AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

68 988 AP Calculus AB: Sction I 7. At =, th function givn by f ( ) <, = 6 9, is undfind. continuous but not diffrntiabl. diffrntiabl but not continuous. nithr continuous nor diffrntiabl. both continuous and diffrntiabl. 8. d = Th h ( ) tan + h tan lim h is sc ( ) sc ( ) cot( ) nonistnt. A rgion in th first quadrant is nclosd by th graphs of y = =, and th coordinat as. If th rgion is rotatd about th y -ais, th volum of th solid that is gnratd is rprsntd by which of th following intgrals?, y ln d d d ln ydy ydy AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

69 988 AP Calculus AB: Sction I. If f( ) =, thn th invrs function, + f, is givn by f ( ) = + +. Which of th following dos NOT hav a priod of? f ( ) = sin f ( ) = sin f ( ) = tan f ( ) = tan f ( ) = sin. Th absolut maimum valu of = + on th closd intrval [, ] f( ) occurs at =. Th ara of th shadd rgion in th figur abov is rprsntd by which of th following intgrals? c ( ( ) ( ) ) a c b f g d c a f ( d ) gd ( ) c a ( ( ) ( )) c a g f d ( ( ) ( )) b a f g d ( ( ) ( )) + ( ( ) ( )) c b g f d f g d AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

70 988 AP Calculus AB: Sction I 5. cos + = cos sin cos sin cos + sin cos + sin cos + 6. What is th avrag valu of y for th part of th curv y = which is in th first quadrant? If f ( ) sin =, thn th numbr of zros of f on th closd intrval [ ], is 8. For du >, d = u C C ln ( ln ) + C ln ( ) + C ( ln ) + C 9. If f( ) d= and f( ) d= 7, thn f ( d= ) AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

71 988 AP Calculus AB: Sction I y z dz. Th sids of th rctangl abov incras in such a way that dt = and d dy =. At th instant dt dt whn = and y =, what is th valu of d dt? 5 5. If lim f( ) = 7, which of th following must b tru? I. f is continuous at =. II. f is diffrntiabl at =. III. f () = 7 Non II only III only I and III only I, II, and III. Th graph of which of th following quations has y = as an asymptot? y = ln y = sin y = + y = y =. Th volum of th solid obtaind by rvolving th rgion nclosd by th llips about th -ais is + 9y = AP Calculus Multipl-Choic Qustion Collction 65 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

72 988 AP Calculus AB: Sction I. Lt f and g b odd functions. If p, r, and s ar nonzro functions dfind as follows, which must b odd? I. p( ) = f ( g( ) ) II. r ( ) = f( ) + g ( ) III. s ( ) = f( g ) ( ) I only II only I and II only II and III only I, II, and III 5. Th volum of a cylindrical tin can with a top and a bottom is to b 6 cubic inchs. If a minimum amount of tin is to b usd to construct th can, what must b th hight, in inchs, of th can? 8 AP Calculus Multipl-Choic Qustion Collction 66 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

73 988 AP Calculus BC: Sction I 9 Minuts No Calculator Nots: () In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ). () Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. Th ara of th rgion in th first quadrant nclosd by th graph of y ( ) = and th -ais is ( ) + d= ( ) = ln, thn f ( ) =. If f ( ). If u, v, and w ar nonzro diffrntiabl functions, thn th drivativ of uv w is uv + u v w uvw uvw w uvw uv w u vw w uvw + uvw + uvw w uv w + u vw uvw w AP Calculus Multipl-Choic Qustion Collction 67 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

74 988 AP Calculus BC: Sction I 5. Lt f b th function dfind by th following. sin, <, < f( ) =, <, For what valus of is f NOT continuous? only only only and only,, and 6. If y y = 6, thn dy d = y y y y y y y y y 7. d + is ln nonistnt 8. If f ( ), = thn ln ( ()) f = AP Calculus Multipl-Choic Qustion Collction 68 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

75 988 AP Calculus BC: Sction I 9. Which of th following pairs of graphs could rprsnt th graph of a function and th graph of its drivativ? I only II only III only I and III II and III. sin lim h ( ) + h sin h is sin cos nonistnt. If 7 y 9 + = is an quation of th lin normal to th graph of f at th point ( ),, thn f () = A particl travls in a straight lin with a constant acclration of mtrs pr scond pr scond. If th vlocity of th particl is mtrs pr scond at tim sconds, how far dos th particl travl during th tim intrval whn its vlocity incrass from mtrs pr scond to mtrs pr scond? m m 7 m 6 m m AP Calculus Multipl-Choic Qustion Collction 69 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

76 988 AP Calculus BC: Sction I. sin ( ) = 5 n n ( ) + + +! 5!! ( n ) 5 n n ( ) ( ) ( ) ( ) + + +! 5!! ( n) ( n ) n n ( ) ( ) ( ) ( ) + + +!!! 6 n !! 6!! ( n) 5 n ( ) ( ) ( ) ! 5!! ( n ). If t dt thn F ( ) F( ) = +, = t + t dt 5. For any tim t, if th position of a particl in th y-plan is givn by y = ln t+, thn th acclration vctor is ( ) = t + and, t ( t + ), t ( t + ), (t + ), (t + ), (t + ) AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

77 988 AP Calculus BC: Sction I 6. d= + C + C + + C + + C + C 7. ( )( ) d = ln 8 ln 5 ln 5 8. If thr qual subdivisions of [, ] ar usd, what is th trapzoidal approimation of d? ( ) ( ) A polynomial ( ) p has a rlativ maimum at (, ), a rlativ minimum at ( ) maimum at ( 5,7 ) and no othr critical points. How many zros dos p( ) hav?,, a rlativ On Two Thr Four Fiv. Th statmnt lim f ( ) = L mans that for ach ε >, thr ists a δ > such that a if < a <ε, thn f( ) L <δ if < f( ) L <ε, thn a <δ if f( ) L <δ, thn < a <ε < a <δ and f( ) L <ε if < a <δ, thn f( ) L <ε AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

78 . Th avrag valu of on th closd intrval [, ] is 988 AP Calculus BC: Sction I ln ln ln. If f ( ) ( ) = +, thn f ( ) = ( + ) ( + ) ln ( + ) ln ( ) ( ) ln( ) Which of th following givs th ara of th rgion nclosd by th loop of th graph of th polar curv r = cos( θ ) shown in th figur abov? 6 cos( θ) dθ cos( θ)dθ 8 cos ( θ)dθ 6 θ θ 6 6 cos ( ) d cos ( θ)dθ AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

79 988 AP Calculus BC: Sction I. If c is th numbr that satisfis th conclusion of th Man Valu Thorm for th intrval, thn c = f ( ) = on 5. Th bas of a solid is th rgion in th first quadrant nclosd by th parabola y =, th lin =, and th -ais. Each plan sction of th solid prpndicular to th -ais is a squar. Th volum of th solid is If f is a function such that f ( ) ists for all and f( ) > for all, which of th following is NOT ncssarily tru? f( ) d> f ( d ) = f( d ) f ( d ) = f( d ) f ( d ) = f( d ) f ( d ) = f( d ) + f( d ) 7. If th graph of = + + has a point of inflction at (, 6) y a b, what is th valu of b? It cannot b dtrmind from th information givn. AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

80 988 AP Calculus BC: Sction I d 8. ln cos d is cos tan cos tan tan 9. Th rgion R in th first quadrant is nclosd by th lins = and y = 5 and th graph of y = +. Th volum of th solid gnratd whn R is rvolvd about th y -ais is i= n i = n n n n n+. d = 8 6. Th gnral solution of th diffrntial quation y = y+ is y = C C + + C + C C AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

81 988 AP Calculus BC: Sction I. Th lngth of th curv y = from = to = is givn by 6 + d + d + 9 d + 9 d + 9 d. A curv in th plan is dfind paramtrically by th quations An quation of th lin tangnt to th curv at t = is = t + t and y = t + t. y = y = 8 y = y = 5 y = If k is a positiv intgr, thn lim + k is k! nonistnt 6. Lt R b th rgion btwn th graphs of y = and th solid obtaind by rvolving R about th -ais is givn by y = sin from = to =. Th volum of sind cosd ( ) sin d sin d ( ) sin d 7. A prson mtrs tall walks dirctly away from a strtlight that is 8 mtrs abov th ground. If th prson is walking at a constant rat and th prson s shadow is lngthning at th rat of 9 mtr pr scond, at what rat, in mtrs pr scond, is th prson walking? AP Calculus Multipl-Choic Qustion Collction 75 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

82 988 AP Calculus BC: Sction I 8. What ar all valus of for which th sris n= n convrgs? n < < < < All ral 9. If dy = ysc and y = 5 whn =, thn y = d tan + tan tan tan + 5 tan + 5. Lt f and g b functions that ar diffrntiabl vrywhr. If g is th invrs function of f and if g( ) = 5 and f (5) =, thn g ( ) = 5 5. lim n n = n n n n d d d d d. If f( ) d= 6, what is th valu of f (5 d )? 6 6 AP Calculus Multipl-Choic Qustion Collction 76 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

83 988 AP Calculus BC: Sction I. Bactria in a crtain cultur incras at a rat proportional to th numbr prsnt. If th numbr of bactria doubls in thr hours, in how many hours will th numbr of bactria tripl? ln ln ln ln ln ln ln 7 9 ln. Which of th following sris convrg? I. n= ( ) n+ n + II. n= n n III. ln n= n n I only II only III only I and III only I, II, and III 5. What is th ara of th largst rctangl that can b inscribd in th llips + 9y = 6? 6 6 AP Calculus Multipl-Choic Qustion Collction 77 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

84 99 AP Calculus AB: Sction I 9 Minuts Scintific Calculator Nots: () Th act numrical valu of th corrct answr dos not always appar among th choics givn. Whn this happns, slct from among th choics th numbr that bst approimats th act numrical valu. () Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. If f( ) =, thn f () = Which of th following rprsnts th ara of th shadd rgion in th figur abov? d ( ( )) f ( ydy ) c ( b a) [ f( b) f( a) ] ( d c) [ f( b) f( a) ] b d f d f ( b) f ( a) a. n 5n lim n n n + is 5 nonistnt AP Calculus Multipl-Choic Qustion Collction 78 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

85 99 AP Calculus AB: Sction I. If + y+ y = 7, thn in trms of and y, dy d = + y + y + y + y + y + y + y y + y 5. If th function f is continuous for all ral numbrs and if thn f ( ) = f( ) = + whn, 6. Th ara of th rgion nclosd by th curv y =, th -ais, and th lins = and = is 5 6 ln ln ln ln 6 7. An quation of th lin tangnt to th graph of + y = at th point (, 5 ) is y = 8 + y = 8 y = 6 + y = 66 + y = AP Calculus Multipl-Choic Qustion Collction 79 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

86 99 AP Calculus AB: Sction I 8. If y = tan cot, thn dy d = sc csc sc csc sc + csc sc csc sc + csc 9. If h is th function givn by h ( ) = f( g ( )), whr f( ) = and g ( ) =, thn h ( ) =. If f ( ) = ( ) sin, thn f () =. Th acclration of a particl moving along th -ais at tim t is givn by at () = 6t. If th vlocity is 5 whn t = and th position is whn t =, thn th position () t = 9t + t t+ t t + t+ 6 t t + 9t 6t t 77t+ 55. If f and g ar continuous functions, and if f( ) for all ral numbrs, which of th following must b tru? b b b I. f ( gd ) ( ) f( d ) gd ( ) = a a a b b b a a a II. ( ) f ( ) + g ( ) d= f( d ) + gd ( ) b III. f ( ) d= f( ) d a b a I only II only III only II and III only I, II, and III AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

87 99 AP Calculus AB: Sction I. Th fundamntal priod of cos( ) is 6. d = C + + C + + C ln + + C ln( + ) + C 5. For what valu of dos th function f( ) = ( )( ) hav a rlativ maimum? Th slop of th lin normal to th graph of y = ln(sc ) at = is nonistnt AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

88 99 AP Calculus AB: Sction I 7. ( + ) d = ( + ) ( + ) 6 + C + C + + C ( + ) 5 + C C 5 8. If f( ) = sin, thn thr ists a numbr c in th intrval < < that satisfis th conclusion of th Man Valu Thorm. Which of th following could b c? Lt f b th function dfind by about f is tru? f( ) = for, for >. Which of th following statmnts f is an odd function. f is discontinuous at =. f has a rlativ maimum. f () = f ( ) > for AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

89 99 AP Calculus AB: Sction I. Lt R b th rgion in th first quadrant nclosd by th graph of y = ( + ), th lin = 7, th -ais, and th y-ais. Th volum of th solid gnratd whn R is rvolvd about th y -ais is givn by 7 ( + ) d 7 ( + ) d ( + ) d ( + ) d 7 ( y ) dy. At what valu of dos th graph of y = hav a point of inflction? At no valu of. An antidrivativ for + is ( + ) ln( + ) ln + arcsc( ) arctan( ). How many critical points dos th function 5 f( ) = ( + ) ( ) hav? On Two Thr Fiv Nin. If f( ) = ( ), thn f () is AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

90 99 AP Calculus AB: Sction I d 5. ( ) d = ( ) ( )ln ( )ln ln 6. A particl movs along a lin so that at tim t, whr t, its position is givn by t st ( ) = cost +. What is th vlocity of th particl whn its acclration is zro? Th function f givn by f( ) = + is incrasing for <, dcrasing for < <, incrasing for > dcrasing for <, incrasing for > incrasing for all dcrasing for all dcrasing for <, incrasing for < <, dcrasing for > 8. 5 ( ) 5 d ( ) + d = cos lim θ sin θ θ is 8 nonistnt. Th rgion nclosd by th -ais, th lin =, and th curv y = is rotatd about th -ais. What is th volum of th solid gnratd? AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

91 99 AP Calculus AB: Sction I. If ln( ) f( ) =, thn f ( ) = ln( ) ln( ) ln( ) 6(ln ) d = 6 ln ln. If dy = y and if y = whn =, thn whn =, y = d. Th top of a 5-foot laddr is sliding down a vrtical wall at a constant rat of ft pr minut. Whn th top of th laddr is 7 ft from th ground, what is th rat of chang of th distanc btwn th bottom of th laddr and th wall? 7 ft pr minut 8 7 ft pr minut ft pr minut ft pr minut ft pr minut 5. If th graph of thn a+ c= a + b y = + c has a horizontal asymptot y = and a vrtical asymptot =, 5 5 AP Calculus Multipl-Choic Qustion Collction 85 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

92 6. If th dfinit intgral 99 AP Calculus AB: Sction I d is first approimatd by using two inscribd rctangls of qual width and thn approimatd by using th trapzoidal rul with n =, th diffrnc btwn th two approimations is If f is a diffrntiabl function, thn f ( a) is givn by which of th following? I. II. III. lim h lim a lim a f ( a+ h) f( a) h f ( ) f( a) a f ( + h) f( ) h I only II only I and II only I and III only I, II, and III 8. If th scond drivativ of f is givn by f ( ) = cos, which of th following could b f ( )? + cos + cos + + cos + sin + + sin + 9. Th radius of a circl is incrasing at a nonzro rat, and at a crtain instant, th rat of incras in th ara of th circl is numrically qual to th rat of incras in its circumfrnc. At this instant, th radius of th circl is AP Calculus Multipl-Choic Qustion Collction 86 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

93 99 AP Calculus AB: Sction I. Th graph of y = f( ) is shown in th figur abov. Which of th following could b th graph of y = f ( )?. d cos( udu d ) is sin cos( ) cos( ) cos( ). A puppy wighs. pounds at birth and.5 pounds two months latr. If th wight of th puppy during its first 6 months is incrasing at a rat proportional to its wight, thn how much will th puppy wigh whn it is months old?. pounds.6 pounds.8 pounds 5.6 pounds 6.5 pounds AP Calculus Multipl-Choic Qustion Collction 87 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

94 99 AP Calculus AB: Sction I. f( ) d= f( ) f ( ) d f ( ) ( ) f d f( ) f( ) + C f( ) f ( ) d f ( d ). What is th minimum valu of f ( ) = ln? f ( ) has no minimum valu. 5. If Nwton s mthod is usd to approimat th ral root of + =, thn a first approimation = would lad to a third approimation of = AP Calculus Multipl-Choic Qustion Collction 88 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

95 99 AP Calculus BC: Sction I 9 Minuts Scintific Calculator Nots: () Th act numrical valu of th corrct answr dos not always appar among th choics givn. Whn this happns, slct from among th choics th numbr that bst approimats th act numrical valu. () Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. Th ara of th rgion nclosd by th graphs of y = and y = is If f( ) = +, thn lim f ( ) f() is nonistnt. If p is a polynomial of dgr n, n >, what is th dgr of th polynomial Q ( ) = ptdt ( )? n n n + dy d. A particl movs along th curv y =. If = and =, what is th valu of dt dt? AP Calculus Multipl-Choic Qustion Collction 89 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

96 99 AP Calculus BC: Sction I 5. Which of th following rprsnts th graph of th polar curv r = scθ? 6. If = t + and y = t, thn d y d = t t t 6t 7. d= ( ) ( ) 8. If f( ) = ln( ), thn f ( ) = AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

97 99 AP Calculus BC: Sction I 9. If f ( ) = +, which of th following is NOT tru? f is continuous for all ral numbrs. f has a minimum at =. f is incrasing for >. f ( ) ists for all. f ( ) is ngativ for >.. Which of th following functions ar continuous at =? I. ln II. III. ln( ) I only II only I and II only II and III only I, II, and III. d is nonistnt. Th position of a particl moving along th -ais is ( t) = sin( t) cos( t) for tim t. Whn t =, th acclration of th particl is If dy y d =, thn y could b ln AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

98 . Th drivativ of f is maimum? 99 AP Calculus BC: Sction I ( )( + ). At how many points will th graph of f hav a rlativ Non On Two Thr Four 5. If tan f( ) =, thn f ( ) = tan sc tan tan tan tan tansc tan tan 6. Which of th following sris divrg? I. II. k= k + k= 6 7 k III. k= ( ) k k Non II only III only I and III II and III 7. Th slop of th lin tangnt to th graph of ln( y) = at th point whr = is 8. If f ( ) = +, thn f ( ) = ( ) + ( ) ln( + ) AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

99 99 AP Calculus BC: Sction I 9. Th shadd rgion R, shown in th figur abov, is rotatd about th y -ais to form a solid whos volum is cubic units. Of th following, which bst approimats k? A particl movs along th -ais so that at any tim t th acclration of th particl is t at () =. If at t = th vlocity of th particl is 5 and its position is 7, thn its position at any tim t > is () t = t + t + t t 9 + t+ 5 + t + t + t +. Th valu of th drivativ of y = at = is AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

100 99 AP Calculus BC: Sction I. If f ( ) =, thn th graph of f is dcrasing for all such that < < < > < >. Th lngth of th curv dtrmind by th quations = t and y = t from t = to t = is t+ dt t + dt t + dt t + dt t + dt. Lt f and g b functions that ar diffrntiabl for all ral numbrs, with g ( ) for. If lim f( ) = lim g( ) = and lim f ( ) f ( ) ists, thn lim is g ( ) g ( ) f ( ) g ( ) f ( ) lim g ( ) f ( g ) ( ) f( g ) ( ) ( f( ) ) nonistnt 5. Considr th curv in th y-plan rprsntd by = and lin tangnt to th curv at th point whr = is t y = t t for t. Th slop of th AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

101 99 AP Calculus BC: Sction I 6. If y = arctan( ), thn dy d = Th intrval of convrgnc of ( ) is n n= n < < < < 8. If a particl movs in th y-plan so that at tim at tim t =, its vlocity vctor is t > its position vctor is ( ln( t t), t ) +, thn,8,,8 8, 8 5, 6 9. sc d= tan + C tan + C sc + sc tan + C tan ln cos + C tan + ln cos + C. What is th volum of th solid gnratd by rotating about th -ais th rgion nclosd by th curv y = sc and th lins =, y =, and =? 8 ln + AP Calculus Multipl-Choic Qustion Collction 95 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

102 . If s n n (5 + n) 5 n+ = 5 ( + n) 99 AP Calculus BC: Sction I, to what numbr dos th squnc { s n } convrg? Th squnc dos not convrg. b. If f ( ) d= b 5 and ( ) a gd=, which of th following must b tru? a I. f ( ) > g( ) for a b b a b a f( ) + g( ) d= II. ( ) III. ( f g ) ( ) ( ) d= 5 I only II only III only II and III only I, II, and III. Which of th following is qual to sin d? cos d sin d cos d sin d sin d AP Calculus Multipl-Choic Qustion Collction 96 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

103 99 AP Calculus BC: Sction I. In th figur abov, PQ rprsnts a -foot laddr with nd P against a vrtical wall and nd Q on lvl ground. If th laddr is slipping down th wall, what is th distanc RQ at th instant whn Q is moving along th ground as fast as P is moving down th wall? If F and f ar diffrntiabl functions such that Fb ( ) = whr a < b, which of th following must b tru? f( ) = for som such that a< < b. f( ) > for all such that a< < b. f( ) < for all such that a< < b. F( ) for all such that a< < b. F( ) = for som such that a< < b. F( ) = f( t) dt, and if Fa ( ) = and 6. Considr all right circular cylindrs for which th sum of th hight and circumfrnc is cntimtrs. What is th radius of th on with maimum volum? cm cm cm cm cm AP Calculus Multipl-Choic Qustion Collction 97 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

104 99 AP Calculus BC: Sction I 7. If for f( ) = for >, thn f ( d ) = + 8. During a crtain pidmic, th numbr of popl that ar infctd at any tim incrass at a rat proportional to th numbr of popl that ar infctd at that tim. If, popl ar infctd whn th pidmic is first discovrd, and, ar infctd 7 days latr, how many popl ar infctd days aftr th pidmic is first discovrd?,,67,,57 9. If dy d =, thn th avrag rat of chang of y with rspct to on th closd intrval [ ], is ln ln 5. Lt R b th rgion in th first quadrant nclosd by th -ais and th graph of y = ln( + ). If Simpson s Rul with subintrvals is usd to approimat th ara of R, th approimation is Lt t f ( ) = dt. At what valu of is f ( ) a minimum? For no valu of. csc lim( + ) = AP Calculus Multipl-Choic Qustion Collction 98 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

105 . Th cofficint of 6 in th Taylor sris pansion about 99 AP Calculus BC: Sction I = for f ( ) sin( ) = is 6 6. If f is continuous on th intrval [ ab, ], thn thr ists c such that a c b < < and f ( d= ) a b f () c b a f ( b) f( a) b a f ( b) f( a) f ()( c b a) f ()( c b a) 5. If f ( ) ( ) sin, k= k f is = thn () AP Calculus Multipl-Choic Qustion Collction 99 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

106 997 AP Calculus AB: Sction I, Part A 5 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. ( 6 ) d= 6 6. If f( ) =, thn f ( ) = b. If f ( ) d= a+ b, thn ( ( ) 5) a f + d= a b a+ b+ 5 5b 5a 7b a 7b 5a 7b 6a. If f( ) = + +, thn f ( ) = 5 AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

107 997 AP Calculus AB: Sction I, Part A 5. Th graph of y = is concav down for < > < or > < or > < < 6. dt = t t t + C t + C + C t + C t + C 7. d cos ( ) = d 6 sin( )cos( ) 6 cos( ) sin ( ) 6 sin( )cos( ) sin( )cos( ) AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

108 997 AP Calculus AB: Sction I, Part A Qustions 8-9 rfr to th following situation. A bug bgins to crawl up a vrtical wir at tim t =. Th vlocity v of th bug at tim t, t 8, is givn by th function whos graph is shown abov. 8. At what valu of t dos th bug chang dirction? What is th total distanc th bug travld from t = to t = 8? 8 6. An quation of th lin tangnt to th graph of y = cos( ) at = is y = y = y = y = y = AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

109 997 AP Calculus AB: Sction I, Part A. Th graph of th drivativ of f is shown in th figur abov. Which of th following could b th graph of f?. At what point on th graph of y = is th tangnt lin paralll to th lin y =?,, 8,, (, ) AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

110 997 AP Calculus AB: Sction I, Part A. Lt f b a function dfind for all ral numbrs. If intrval f ( ) =, thn f is dcrasing on th (,) (, ) (, ) (, ) (, ). Lt f b a diffrntiabl function such that f () = and f () = 5. If th tangnt lin to th graph of f at = is usd to find an approimation to a zro of f, that approimation is Th graph of th function f is shown in th figur abov. Which of th following statmnts about f is tru? lim f ( ) = lim f( ) a b lim f ( ) = a lim f ( ) = b lim f ( ) = b lim f ( ) dos not ist. a AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

111 997 AP Calculus AB: Sction I, Part A 6. Th ara of th rgion nclosd by th graph of y = + and th lin y = 5 is If + y = 5, what is th valu of d y d at th point (, )? tan cos d is + 9. If f( ) = ln, thn f ( ) = AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

112 997 AP Calculus AB: Sction I, Part A. Th avrag valu of cos on th intrval [,5] is sin 5 sin 8 sin 5 sin sin sin 5 sin + sin 5 sin + sin 5 8. lim ln is nonistnt. What ar all valus of for which th function f dfind by f ( ) = ( ) is incrasing? Thr ar no such valus of. < and > < < < < All valus of. If th rgion nclosd by th y-ais, th lin y =, and th curv y = is rvolvd about th y-ais, th volum of th solid gnratd is AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

113 997 AP Calculus AB: Sction I, Part A. Th prssion is a Rimann sum approimation for d 5 d 5 5 d d d 5. sin( ) d= cos( ) + sin( ) + C cos( ) sin( ) + C cos( ) sin( ) + C cos( ) + sin( ) + C cos( ) sin( ) + C AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

114 997 AP Calculus AB: Sction I, Part B Minuts Graphing Calculator Rquird Nots: () Th act numrical valu of th corrct answr dos not always appar among th choics givn. Whn this happns, slct from among th choics th numbr that bst approimats th act numrical valu. () Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr. 76. If f( ) =, thn f ( ) = ( ) (+ ) ( ) 77. Th graph of th function y = cos changs concavity at = Th graph of f is shown in th figur abov. If F() F() = f( ) d=. and F ( ) f( ), = thn AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

115 997 AP Calculus AB: Sction I, Part B 79. Lt f b a function such that f( + h) f() lim = 5. Which of th following must b tru? h h I. f is continuous at =. II. f is diffrntiabl at =. III. Th drivativ of f is continuous at =. I only II only I and II only I and III only II and III only 8. Lt f b th function givn by ( ) f th graph of f at (, f( )) qual to? =. For what valu of is th slop of th lin tangnt to A railroad track and a road cross at right angls. An obsrvr stands on th road 7 mtrs south of th crossing and watchs an astbound train travling at 6 mtrs pr scond. At how many mtrs pr scond is th train moving away from th obsrvr sconds aftr it passs through th intrsction? If y = 8, what is th minimum valu of th product y? What is th ara of th rgion in th first quadrant nclosd by th graphs of y = cos, y =, and th y-ais? Th bas of a solid S is th rgion nclosd by th graph of y = ln, th lin =, and th -ais. If th cross sctions of S prpndicular to th -ais ar squars, thn th volum of S is ( ) AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

116 997 AP Calculus AB: Sction I, Part B 85. If th drivativ of f is givn by hav a rlativ maimum valu? f ( ) =, at which of th following valus of dos f Lt f ( ) =. If th rat of chang of f at = c is twic its rat of chang at =, thn c = 87. At tim t, th acclration of a particl moving on th -ais is at ( ) = t+ sin t. At t =, th vlocity of th particl is. For what valu t will th vlocity of th particl b zro? AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

117 997 AP Calculus AB: Sction I, Part B 88. Lt f ( ) = h( t) dt, whr h has th graph shown abov. Which of th following could b th graph of f? a AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

118 997 AP Calculus AB: Sction I, Part B f( ) A tabl of valus for a continuous function f is shown abov. If four qual subintrvals of [, ] ar usd, which of th following is th trapzoidal approimation of f ( d )? Which of th following ar antidrivativs of f ( ) = sincos? I. II. III. sin F( ) = cos F( ) = cos( ) F( ) = I only II only III only I and III only II and III only AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

119 997 AP Calculus BC: Sction I, Part A 5 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. ( ) + d= If t = and y = sin( t), thn dy d = t cos(t) t cos(t) sin(t) t cos(t) t cos(t) t. Th function f givn by 5 f ( ) = has a rlativ maimum at = d. ( ln d ) = If f( ) = ( ) +, thn f () = Th lin normal to th curv y = 6 at th point (, ) has slop AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

120 997 AP Calculus BC: Sction I, Part A Qustions 7-9 rfr to th graph and th information blow. Th function f is dfind on th closd intrval [,8 ]. Th graph of its drivativ f is shown abov. 7. Th point ( ) (,5 ) is y = y = 5 y 5= ( ) y+ 5= ( ) y+ 5= ( + ),5 is on th graph of y = f( ). An quation of th lin tangnt to th graph of f at 8. How many points of inflction dos th graph of f hav? Two Thr Four Fiv Si AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

121 997 AP Calculus BC: Sction I, Part A 9. At what valu of dos th absolut minimum of f occur? 6 8. If y = y+ +, thn whn =, dy d is nonistnt. d is ( + ) divrgnt. Th graph of f, th drivativ of f, is shown in th figur abov. Which of th following dscribs all rlativ trma of f on th opn intrval ( ab, )? On rlativ maimum and two rlativ minima Two rlativ maima and on rlativ minimum Thr rlativ maima and on rlativ minimum On rlativ maimum and thr rlativ minima Thr rlativ maima and two rlativ minima AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

122 997 AP Calculus BC: Sction I, Part A. A particl movs along th -ais so that its acclration at any tim t is at () = t 7. If th initial vlocity of th particl is 6, at what tim t during th intrval t is th particl farthst to th right?. Th sum of th infinit gomtric sris is 6 8, Th lngth of th path dscribd by th paramtric quations t, is givn by = cos t and y = sin t, for cos t+ sin t dt cos t sin t+ sin t cos t dt 9cos t+ 9sin t dt 9cos t sin t+ 9sin t cos t dt 6 6 cos t+ sin t dt 6. lim h h h is nonistnt AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

123 7. Lt f b th function givn by f ( ) ln( ) = is 997 AP Calculus BC: Sction I, Part A =. Th third-dgr Taylor polynomial for f about ( ) ( ) ( ) + ( ) ( ) ( ) ( ) + ( ) + ( ) ( ) ( ) ( ) + + ( ) ( ) ( ) + 8. For what valus of t dos th curv givn by th paramtric quations y = t + t 8t hav a vrtical tangnt? = t t and only only and only,, and No valu 9. Th graph of y = f( ) is shown in th figur abov. If A and A ar positiv numbrs that rprsnt th aras of th shadd rgions, thn in trms of A and A, f d ( ) f( d ) = A A A A A A+ A A+ A AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

124 997 AP Calculus BC: Sction I, Part A. What ar all valus of for which th sris ( ) convrgs? n n n= n < < < 5 5 < 5. Which of th following is qual to th ara of th rgion insid th polar curv r = cosθ and outsid th polar curv r = cos θ? θ θ cos d θ θ cos d cos θ dθ cos θdθ cos θ d θ. Th graph of f is shown in th figur abov. If g ( ) = ftdt ( ), for what valu of dos g ( ) hav a maimum? a b c d It cannot b dtrmind from th information givn. a AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

125 997 AP Calculus BC: Sction I, Part A. In th triangl shown abov, if θ incrass at a constant rat of radians pr minut, at what rat is incrasing in units pr minut whn quals units? 5 9. Th Taylor sris for sin about = is ( ) f ( ) = sin, thn th cofficint of 5 +. If f is a function such that! 5! 7 in th Taylor sris for f ( ) about = is 7! 7 7! 5. Th closd intrval [ ab, ] is partitiond into n qual subintrvals, ach of width numbrs,,..., n whr a= < < < < n < n = b. What is ( ) b a b a ( ) b a b a ( b ) a n i =, by th n lim i? AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

126 997 AP Calculus BC: Sction I, Part B Minuts Graphing Calculator Rquird Nots: () Th act numrical valu of th corrct answr dos not always appar among th choics givn. Whn this happns, slct from among th choics th numbr that bst approimats th act numrical valu. () Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr. 76. Which of th following squncs convrg? I. II. III. 5n n n n n + n I only II only I and II only I and III only I, II, and III 77. Whn th rgion nclosd by th graphs of y = and volum of th solid gnratd is givn by ( ) d y = is rvolvd about th y-ais, th ( ) d ( ) d ( ) d ( ) d AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

127 997 AP Calculus BC: Sction I, Part B 78. h ( h) ln + lim is h f ( ), whr f( ) = ln ln f (), whr f() = f (), whr f( ) = ln f (), whr f( ) = ln ( + ) f (), whr f( ) = ln 79. Th position of an objct attachd to a spring is givn by yt ( ) = cos(5 t) sin(5 t), whr t is 6 tim in sconds. In th first sconds, how many tims is th vlocity of th objct qual to? Zro Thr Fiv Si Svn 8. Lt f b th function givn by f ( ) = cos( ) + ln( ). What is th last valu of at which th graph of f changs concavity? Lt f b a continuous function on th closd intrval [, 6]. If f ( ) = and ( ) th Intrmdiat Valu Thorm guarants that f () = f () c = for at last on c btwn and 6 9 f( ) for all btwn and 6 f() c = for at last on c btwn and 6 f( c ) = for at last on c btwn and f 6 =, thn AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

128 8. If, of th following, which is th gratst valu of such that 997 AP Calculus BC: Sction I, Part B ( ) t t dt tdt? dy = + y and if y = whn =, thn y = d 8. If ( ln ) + ln ln ln + ln 8. sin d= cos sin cos + C cos + sin cos + C cos + sin + cos + C cos + C cos+ C 85. Lt f b a twic diffrntiabl function such that f () = and f () = 7. Which of th following must b tru for th function f on th intrval? I. Th avrag rat of chang of f is 5. II. Th avrag valu of f is 9. III. Th avrag valu of f is 5. Non I only III only I and III only II and III only AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

129 997 AP Calculus BC: Sction I, Part B 86. d = ( )( + ) ln + + ln + C + C ln + + C ( )( ) + ln ( )( + ) ln ( )( ) + + C + C 87. Th bas of a solid is th rgion in th first quadrant nclosd by th graph of y = and th coordinat as. If vry cross sction of th solid prpndicular to th y-ais is a squar, th volum of th solid is givn by ( ) y dy ( ) y dy ( ) d ( ) d ( ) d AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

130 997 AP Calculus BC: Sction I, Part B 88. Lt f ( ) = sint dt. At how many points in th closd intrval, rat of chang of f qual th avrag rat of chang of f on that intrval? dos th instantanous Zro On Two Thr Four 89. If f is th antidrivativ of + 5 such that f ( ) =, thn ( ) f = A forc of pounds is rquird to strtch a spring inchs byond its natural lngth. Assuming Hook s law applis, how much work is don in strtching th spring from its natural lngth to 6 inchs byond its natural lngth? 6. inch-pounds 5. inch-pounds. inch-pounds 5. inch-pounds 7. inch-pounds AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

131 998 AP Calculus AB: Sction I, Part A 55 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. What is th -coordinat of th point of inflction on th graph of y = + 5 +? 5 5. Th graph of a picwis-linar function f, for, is shown abov. What is th valu of f ( ) d? d = 7 ln AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

132 998 AP Calculus AB: Sction I, Part A. If f is continuous for a b and diffrntiabl for a< < b, which of th following could b fals? f ( b) f( a) f () c = for som c such that a< c< b. b a f ( c) = for som c such that a< c< b. f has a minimum valu on a b. f has a maimum valu on a b. b f ( d ) ists. a 5. sin tdt= sin cos cos cos cos 6. If + y =, thn whn =, dy = d d = + 8. Lt f and g b diffrntiabl functions with th following proprtis: (i) g> ( ) for all (ii) f () = If h ( ) = f( g ) ( ) and h ( ) = f( ) g ( ), thn f ( ) = f ( ) g ( ) AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

133 998 AP Calculus AB: Sction I, Part A 9. Th flow of oil, in barrls pr hour, through a piplin on July 9 is givn by th graph shown abov. Of th following, which bst approimats th total numbr of barrls of oil that passd through th piplin that day? 5 6,,,8. What is th instantanous rat of chang at = of th function f givn by f( ) =? 6 6 b f a. If f is a linar function and < a< b, thn ( d ) = ab b a b a. If ln for < f( ) = ln for <, thn lim f ( ) is ln ln 8 ln6 nonistnt AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

134 998 AP Calculus AB: Sction I, Part A. Th graph of th function f shown in th figur abov has a vrtical tangnt at th point (, ) and horizontal tangnts at th points (, ) and (, ). For what valus of, < <, is f not diffrntiabl? only and only and only,, and only,,, and. A particl movs along th -ais so that its position at tim t is givn by what valu of t is th vlocity of th particl zro? () t = t 6t+ 5. For 5 5. If thn () F( ) = t + dt, F = 8 ( ) = sin, thn f ( ) = 6. If f ( ) cos( ) cos( ) + cos( ) cos( ) cos( ) AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

135 998 AP Calculus AB: Sction I, Part A 7. Th graph of a twic-diffrntiabl function f is shown in th figur abov. Which of th following is tru? f () < f () < f () f () < f () < f () f () < f () < f () f () < f () < f () f () < f () < f () 8. An quation of th lin tangnt to th graph of y = + cos at th point (, ) is y = + y = + y = y = y = = +, thn th graph of f has inflction points whn = 9. If f ( ) ( )( ) only only and only and only,, and only. What ar all valus of k for which k d =? and,, and. If dy = ky and k is a nonzro constant, thn y could b dt kty kt kt + kty + 5 ky + AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

136 998 AP Calculus AB: Sction I, Part A. Th function f is givn by incrasing? f( ) = +. On which of th following intrvals is f, (, ) (,),, AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

137 998 AP Calculus AB: Sction I, Part A. Th graph of f is shown in th figur abov. Which of th following could b th graph of th drivativ of f? AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

138 998 AP Calculus AB: Sction I, Part A. Th maimum acclration attaind on th intrval t by th particl whos vlocity is givn by vt () = t t + t+ is 9 5. What is th ara of th rgion btwn th graphs of y = and y = from = to =? 8 6 f ( ) k 6. Th function f is continuous on th closd intrval [, ] and has valus that ar givn in th tabl abov. Th quation f( ) = must hav at last two solutions in th intrval [, ] if k = 7. What is th avrag valu of = + on th intrval [ ] y,? If f ( ) = tan( ), thn f = 6 8 AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

139 998 AP Calculus AB: Sction I, Part B 5 Minuts Graphing Calculator Rquird Nots: () Th act numrical valu of th corrct answr dos not always appar among th choics givn. Whn this happns, slct from among th choics th numbr that bst approimats th act numrical valu. () Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr. 76. Th graph of a function f is shown abov. Which of th following statmnts about f is fals? f is continuous at = a. f has a rlativ maimum at = a. = a is in th domain of f. lim f ( ) a + is qual to lim f ( ). a lim f ( ) ists. a 77. Lt f b th function givn by f ( ) = and lt g b th function givn by valu of do th graphs of f and g hav paralll tangnt lins? g ( ) = 6. At what AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

140 998 AP Calculus AB: Sction I, Part B 78. Th radius of a circl is dcrasing at a constant rat of. cntimtr pr scond. In trms of th circumfrnc C, what is th rat of chang of th ara of th circl, in squar cntimtrs pr scond? (.) C (.) C (.) C ( ). C (.) C 79. Th graphs of th drivativs of th functions f, g, and h ar shown abov. Which of th functions f, g, or h hav a rlativ maimum on th opn intrval a< < b? f only g only h only f and g only f, g, and h 8. Th first drivativ of th function f is givn by dos f hav on th opn intrval (, )? cos f ( ) =. How many critical valus 5 On Thr Four Fiv Svn AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

141 8. Lt f b th function givn by f ( ) I. f is continuous at =. II. f is diffrntiabl at =. III. f has an absolut minimum at =. 998 AP Calculus AB: Sction I, Part B =. Which of th following statmnts about f ar tru? I only II only III only I and III only II and III only 8. If f is a continuous function and if F ( ) f( ) F() F() F() F() F(6) F() F(6) F() = for all ral numbrs, thn f ( ) d= F(6) F() 8. If a, thn lim a a a is a a 6a nonistnt 8. Population y grows according to th quation dy ky dt =, whr k is a constant and t is masurd in yars. If th population doubls vry yars, thn th valu of k is AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

142 998 AP Calculus AB: Sction I, Part B f ( ) 85. Th function f is continuous on th closd intrval [,8 ] and has valus that ar givn in th tabl abov. Using th subintrvals [,5 ], [ 5,7 ], and [ 7,8 ], what is th trapzoidal approimation of 8 f ( d )? Th bas of a solid is a rgion in th first quadrant boundd by th -ais, th y-ais, and th lin + y = 8, as shown in th figur abov. If cross sctions of th solid prpndicular to th -ais ar smicircls, what is th volum of th solid? Which of th following is an quation of th lin tangnt to th graph of point whr f ( ) =? f ( ) = + at th y = 8 5 y = + 7 y = +.76 y =. y = Lt F( ) b an antidrivativ of ( ln ). If F () =, thn F (9) = ,6.5 AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

143 89. If g is a diffrntiabl function such that g< ( ) for all ral numbrs and if ( ) f ( ) = g( ), which of th following is tru? f has a rlativ maimum at = and a rlativ minimum at =. f has a rlativ minimum at = and a rlativ maimum at =. f has rlativ minima at = and at =. f has rlativ maima at = and at =. It cannot b dtrmind if f has any rlativ trma. 998 AP Calculus AB: Sction I, Part B 9. If th bas b of a triangl is incrasing at a rat of inchs pr minut whil its hight h is dcrasing at a rat of inchs pr minut, which of th following must b tru about th ara A of th triangl? A is always incrasing. A is always dcrasing. A is dcrasing only whn b A is dcrasing only whn b A rmains constant. < h. > h. 9. Lt f b a function that is diffrntiabl on th opn intrval ( ) f (9) = 5, which of th following must b tru?,. If f() = 5, f(5) = 5, and I. f has at last zros. II. Th graph of f has at last on horizontal tangnt. III. For som c, < c< 5, f( c) =. Non I only I and II only I and III only I, II, and III 9. If k < and th ara undr th curv y = cos from = k to = is., thn k = AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

144 998 AP Calculus BC: Sction I, Part A 55 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. What ar all valus of for which th function f dfind by incrasing? f( ) = is < < < < < or > < or > All ral numbrs. In th y-plan, th graph of th paramtric quations = 5t + and y =, t for t, is a lin sgmnt with slop Th slop of th lin tangnt to th curv y + ( y+ ) = at (, ) is. 6 8 d = + ln ln + C + C ln + C ( )( ) ln + + C ( )( ) ln ( )( ) + C AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

145 ( ) = ( ), thn h ( ) = 5. If f and g ar twic diffrntiabl and if h f( g) ( )[ ] + ( ) f g ( ) g ( ) f g ( ) g ( ) f g ( ) g ( ) f g ( ) g ( ) ( ) + ( ) f ( g ( ))[ g ( ) ] f g ( ) g ( ) ( ) f ( g ( )) 998 AP Calculus BC: Sction I, Part A AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

146 998 AP Calculus BC: Sction I, Part A 6. Th graph of y = h( ) is shown abov. Which of th following could b th graph of y = h ( )? AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

147 998 AP Calculus BC: Sction I, Part A 7. d = + 8. If dy sin cos d = and if y = whn =, what is th valu of y whn =? 9. Th flow of oil, in barrls pr hour, through a piplin on July 9 is givn by th graph shown abov. Of th following, which bst approimats th total numbr of barrls of oil that passd through th piplin that day? 5 6,,,8. A particl movs on a plan curv so that at any tim t > its -coordinat is t t and its y-coordinat is ( t ). Th acclration vctor of th particl at t = is (, ) (, ) (,6 ) ( 6, ) ( 6, ) b f a. If f is a linar function and < a< b, thn ( d ) = ab b a b a AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

148 998 AP Calculus BC: Sction I, Part A. If f( ) = ln for < ln for <, thn lim f ( ) is ln ln 8 ln6 nonistnt. Th graph of th function f shown in th figur abov has a vrtical tangnt at th point (, ) and horizontal tangnts at th points (, ) and (, ). For what valus of, < <, is f not diffrntiabl? only and only and only,, and only,,, and. What is th approimation of th valu of sin obtaind by using th fifth-dgr Taylor polynomial about = for sin? AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

149 5. cos d= 998 AP Calculus BC: Sction I, Part A sin cos + C sin + cos + C sin + cos + C sin + C sin + C 6. If f is th function dfind by inflction for th graph of f? 5 f ( ) = 5, what ar all th -coordinats of points of and,, and 7. Th graph of a twic-diffrntiabl function f is shown in th figur abov. Which of th following is tru? f () < f () < f () f () < f () < f () f () < f () < f () f () < f () < f () f () < f () < f () AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

150 8. Which of th following sris convrg? 998 AP Calculus BC: Sction I, Part A I. n II. n n= + cos( n) III. n n= n= n Non II only III only I and II only I and III only 9. Th ara of th rgion insid th polar curv r = sinθ and outsid th polar curv r = is givn by ( sinθ ) dθ ( sinθ ) dθ 6 ( sinθ ) 5 6 ( 6sin θ ) dθ ( 6sin θ ) 6 dθ 5 6 dθ. Whn = 8, th rat at which is incrasing is k tims th rat at which is incrasing. What is th valu of k? 6 8. Th lngth of th path dscribd by th paramtric quations t, is givn by = t and y = t, whr 6 t + t + dt + t dt t t dt + t dt t t + 9dt AP Calculus Multipl-Choic Qustion Collction Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

151 . If d lim b is finit, thn which of th following must b tru? b p 998 AP Calculus BC: Sction I, Part A convrgs p n n= divrgs p n n= convrgs p n n= convrgs p n n= divrgs p+ n n=. Lt f b a function dfind and continuous on th closd intrval [ ab, ]. If f has a rlativ maimum at c and a< c< b, which of th following statmnts must b tru? I. f () c ists. II. If f () c ists, thn f () c =. III. If f () c ists, thn f () c. II only III only I and II only I and III only II and III only AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

152 998 AP Calculus BC: Sction I, Part A. Shown abov is a slop fild for which of th following diffrntial quations? dy d = + dy d = dy y d = + dy d y dy = ln d = y 5. d is divrgnt dp P 6. Th population Pt () of a spcis satisfis th logistic diffrntial quation = P dt 5, whr th initial population P () =, and t is th tim in yars. What is lim Pt ( )? t,5,, 5,, 7. If n a n is a Taylor sris that convrgs to ( ) n= a f for all ral, thn f () = an n= nan n= n= na n n 8. lim t dt is nonistnt AP Calculus Multipl-Choic Qustion Collction 6 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

153 998 AP Calculus BC: Sction I, Part B 5 Minuts Graphing Calculator Rquird Nots: () Th act numrical valu of th corrct answr dos not always appar among th choics givn. Whn this happns, slct from among th choics th numbr that bst approimats th act numrical valu. () Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr. 76. For what intgr k, k >, will both n= ( ) kn and n n= n k convrg? If f is a vctor-valud function dfind by f () t ( t,cost) t + sin t cost ( t,cost) =, thn f ( t) = t t (, sint) ( t, cost) 78. Th radius of a circl is dcrasing at a constant rat of. cntimtr pr scond. In trms of th circumfrnc C, what is th rat of chang of th ara of th circl, in squar cntimtrs pr scond? (.) C (.) C (.) C ( ). C (.) C AP Calculus Multipl-Choic Qustion Collction 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

154 998 AP Calculus BC: Sction I, Part B ( )( ) 79. Lt f b th function givn by f( ) =. For what positiv valus of a is f a continuous for all ral numbrs? Non only only only and only 8. Lt R b th rgion nclosd by th graph of y ln( cos ) and =. Th closst intgr approimation of th ara of R is = +, th -ais, and th lins = 8. If dy y, d = thn d y d = y y y y y 8. If f( ) = g( ) + 7 for 5, thn [ ( ) ( )] 5 f + g d= 5 gd+ ( ) 7 5 gd+ ( ) 5 gd+ ( ) gd+ ( ) 7 gd+ ( ) AP Calculus Multipl-Choic Qustion Collction 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

155 =, is ( ) n+ ( ) 998 AP Calculus BC: Sction I, Part B 8. Th Taylor sris for ln, cntrd at. Lt f b th function givn by n= n th sum of th first thr nonzro trms of this sris. Th maimum valu of ln f( ) for..7 is n 8. What ar all valus of for which th sris n= ( + ) n convrgs? n < < < < f ( ) 85. Th function f is continuous on th closd intrval [,8 ] and has valus that ar givn in th tabl abov. Using th subintrvals [,5 ], [ 5,7 ], and [ 7,8 ], what is th trapzoidal approimation of 8 f ( d )? Th bas of a solid is a rgion in th first quadrant boundd by th -ais, th y-ais, and th lin + y = 8, as shown in th figur abov. If cross sctions of th solid prpndicular to th -ais ar smicircls, what is th volum of th solid? AP Calculus Multipl-Choic Qustion Collction 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

156 998 AP Calculus BC: Sction I, Part B 87. Which of th following is an quation of th lin tangnt to th graph of point whr f ( ) =? f ( ) = + at th y = 8 5 y = + 7 y = +.76 y =. y =.6 a 88. Lt g ( ) = ftdt ( ), whr a b. th following could b th graph of f on [ ab, ]? Th figur abov shows th graph of g on [, ] ab. Which of AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

157 89. Th graph of th function rprsntd by th Maclaurin sris ( ) n n intrscts th graph of y = at =!! n! 998 AP Calculus BC: Sction I, Part B A particl starts from rst at th point (, ) and movs along th -ais with a constant positiv acclration for tim t. Which of th following could b th graph of th distanc st ( ) of th particl from th origin as a function of tim t? AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

158 998 AP Calculus BC: Sction I, Part B t (sc) 6 at ()(ft/sc ) Th data for th acclration at ( ) of a car from to 6 sconds ar givn in th tabl abov. If th vlocity at t = is ft pr scond, th approimat valu of th vlocity at t = 6, computd using a lft-hand Rimann sum with thr subintrvals of qual lngth, is 6 ft/sc ft/sc 7 ft/sc 9 ft/sc ft/sc 9. Lt f b th function givn by f ( ) = +. Th tangnt lin to th graph of f at = is usd to approimat valus of f ( ). Which of th following is th gratst valu of for which th rror rsulting from this tangnt lin approimation is lss than.5? AP Calculus Multipl-Choic Qustion Collction 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.