1985 AP Calculus AB: Section I
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1 985 AP Calculus AB: Sction I 9 Minuts No Calculator Nots: () In tis amination, ln dnots t natural logaritm of (tat is, logaritm to t bas ). () Unlss otrwis spcifid, t domain of a function f is assumd to b t st of all ral numbrs for wic f () is a ral numbr.. d = (D) If f( ) ( ) = +, tn t t drivativ of f ( ) at = is 8 (D) 8. If dy y =, tn + d = 6 ( + ) ( + ) 6 ( + ) (D) ( + ) dy, tn y d = =. If cos( ) cos cos sin ( ) ( ) ( ) sin sin (D) ( ) ( ) 5. n lim is n, n n +,5 (D) nonistnt AP Calculus Multipl-Coic Qustion Collction 8 Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
2 985 AP Calculus AB: Sction I 6. If f ( ) =, tn f (5) = 5 (D) Wic of t following is qual to ln? ln + ln ln 8 ln t dt (D) ln d dt t 8. T slop of t lin tangnt to t grap of y = ln at = is 8 (D) 9. If d= k, tn d= k k k (D) k k. If ( ) y =, tn dy d = ( ) ( ) ln ( ) ( ) (D) ( ) ( ) ln ( ) ( ) ln ( ) ( ). T position of a particl moving along a straigt lin at any tim t is givn by st () = t + t+. Wat is t acclration of t particl wn t =? (D) 8. If f ( g) ( ) f ( ) ( ) = ln +, ( ) = ln, and g> ( ) for all ral, tn g () = (D) + + AP Calculus Multipl-Coic Qustion Collction 9 Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
3 985 AP Calculus AB: Sction I. If + y+ y =, tn, in trms of and y, dy d = + y + y + y + y + y (D) + y + y + y. T vlocity of a particl moving on a lin at tim t is mtrs did t particl travl from t = to t =? v= t + 5t mtrs pr scond. How many 6 (D) T domain of t function dfind by f( ) ln( ) = is t st of all ral numbrs suc tat < > (D) is a ral numbr 6. T function dfind by f ( ) = for all ral numbrs as a rlativ maimum at = (D) 7. d= (D) 8. If y = cos sin, tn y = sin( ) (D) cos ( + sin) cos ( sin) 9. If f ( ) f ( ) f ( ) dfin f? + = + for all ral numbrs and, wic of t following could f ( ) = + f ( ) = f( ) = (D) f ( ) = f ( ) = AP Calculus Multipl-Coic Qustion Collction Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
4 985 AP Calculus AB: Sction I. If y ( ) = arctan cos, tn dy d = (D) sin + cos ( arccos ) + ( ) ( ) arcsc( cos ) sin arcsc( cos ) + cos. If t domain of t function f givn by f( ) = is { } : >, wat is t rang of f? { : < < } { : < < } { : < < } (D) { : < < } { :< < }. d = + (D) 5 ln. d at is d + = 6 (D) 6. If ( 7 ) + k d= 6, tn k = (D) 5. If f ( ) =, wic of t following is qual to f ()? lim + lim + lim + (D) lim + lim + AP Calculus Multipl-Coic Qustion Collction Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
5 985 AP Calculus AB: Sction I 6. T grap of y = + 9 is symmtric to wic of t following? I. T -ais II. T y-ais III. T origin I only II only III only (D) I and II only I, II, and III 7. d = (D) If t position of a particl on t -ais at tim t is for t is 5t, tn t avrag vlocity of t particl 5 5 (D) 5 9. Wic of t following functions ar continuous for all ral numbrs? I. II. III. y = y = y = tan Non I only II only (D) I and II I and III. tan ( ) d = ln cos( ) ln cos( ) ln cos( ) (D) ln cos( ) sc( )tan( ) AP Calculus Multipl-Coic Qustion Collction Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
6 985 AP Calculus AB: Sction I. T volum of a con of radius r and igt is givn by V = π r. If t radius and t igt bot incras at a constant rat of cntimtr pr scond, at wat rat, in cubic cntimtrs pr scond, is t volum incrasing wn t igt is 9 cntimtrs and t radius is 6 cntimtrs? π π π (D) 5π 8π. ( ) π sin d = (D). T grap of t drivativ of f is sown in t figur abov. Wic of t following could b t grap of f? AP Calculus Multipl-Coic Qustion Collction Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
7 985 AP Calculus AB: Sction I. T ara of t rgion in t first quadrant tat is nclosd by t graps of y = + 8 is y = + 8 and (D) T figur abov sows t grap of a sin function for on complt priod. Wic of t following is an quation for t grap? π y = sin (D) y = sin( π ) y = sin ( ) y = sin ( π ) y = sin( ) 6. If f is a continuous function dfind for all ral numbrs and if t maimum valu of f ( ) is 5 and t minimum valu of f ( ) is 7, tn wic of t following must b tru? I. T maimum valu of f ( ) is 5. II. T maimum valu of f ( ) is 7. III. T minimum valu of f ( ) is. I only II only I and II only (D) II and III only I, II, and III 7. lim ( csc ) is (D) AP Calculus Multipl-Coic Qustion Collction Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
8 985 AP Calculus AB: Sction I 8. Lt f and g av continuous first and scond drivativs vrywr. If f ( ) g( ) for all ral, wic of t 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 (D) I and II only I, II, and III 9. If ln f( ) =, for all >, wic of t following is tru? f is incrasing for all gratr tan. f is incrasing for all gratr tan. f is dcrasing for all btwn and. (D) f is dcrasing for all btwn and. f is dcrasing for all gratr tan.. Lt f b a continuous function on t closd intrval [ ] possibl valu of f ( d ) is,. If f( ), tn t gratst (D) 8 6. If lim f ( ) = L, wr L is a ral numbr, wic of t following must b tru? a f ( a) ists. f ( ) is continuous at = a. f ( ) is dfind at = a. (D) f ( a) = L Non of t abov AP Calculus Multipl-Coic Qustion Collction 5 Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
9 985 AP Calculus AB: Sction I. d d + t dt = (D) An quation of t lin tangnt to y = + + at its point of inflction is y = 6 6 y = + y = + (D) y = y = +. T avrag valu of = + on t closd intrval [ ] f( ), is (D) 6 5. T rgion nclosd by t grap of y =, t lin =, and t -ais is rvolvd about t y -ais. T volum of t solid gnratd is 8π 5 π 6 π (D) π 8 π AP Calculus Multipl-Coic Qustion Collction 6 Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
10 985 Answr Ky 985 AB 985 BC. D. E. A. C 5. D 6. C 7. E 8. B 9. D. D. B. C. A. D 5. C 6. B 7. C 8. C 9. B. A. B. A. B. D 5. E 6. E 7. D 8. C 9. D. B. C. D. B. A 5. D 6. B 7. D 8. C 9. E. D. E. C. B. A 5. A. D. A. B. D 5. D 6. E 7. A 8. C 9. B. A. A. A. B. C 5. C 6. C 7. B 8. C 9. D. C. B. A. C. D 5. C 6. E 7. E 8. E 9. D. B. D. E. C. A 5. B 6. E 7. A 8. C 9. A. A. C. E. E. A 5. D AP Calculus Multipl-Coic Qustion Collction 55 Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
11 985 Calculus AB Solutions. D. E. A. C 5. D d = = = 8. f ( ) = (+ ), f () = ( + ), f () = (+ ), () f () =! = 8 y = ( + ) so 6 y = ( + ) ( ) = ( + ) Or using t quotint rul dirctly givs cos( ) d = cos( )( d) = sin( ) n lim = lim = n n + n n + n ( + )() ( ) 6 ( + ) ( + ) y = = 6. C f ( ) = f (5) = 7. E 8. B t dt t = ln = ln ln = ln y = ln = ln ln, y =, y () = 9. D Sinc is vn, d= d= k d. D ( ) ( ) ( ) y = ln() ( ) = ln() d. B vt ( ) = t+ at ( ) = a() = f g( ) = ln g( ) = ln + g( ) = +. C ( ) ( ) ( ) AP Calculus Multipl-Coic Qustion Collction 8 Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
12 985 Calculus AB Solutions. A + y + y + y+ y y = y = + y. D 5 Sinc vt ( ), distanc = v() t dt t 5t dt t t = 8 + = + = 5. C 6. B > > f ( ) = 6= ( ) cangs sign from positiv to ngativ only at =. 7. C Us t tcniqu of antidrivativs by parts: u = dv= d du = d v = ( ) + d= = 8. C y = cos sin = cos, y = sin 9. B Quick solution: lins troug t origin av tis proprty. Or, f + f = + = + = f + ( ) ( ) ( ) ( ) dy d sin = = d + cos d + cos. A ( cos ). B. A > > f( ) < for all in t domain. lim f( ) =. option tat is consistnt wit ts statmnts is. + ( )( ) d = d = ( ) d = ( ) = + + lim f( ) =. T only d. B ( ) ( ) d + = + + = + = = = 7 7. D ( ) 6 = ( + k) d = d + k d = + ( ) k = k k = AP Calculus Multipl-Coic Qustion Collction 8 Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
13 985 Calculus AB Solutions 5. E + f ( + ) f( ) f () = lim = lim 6. E I: Rplac y wit ( y) : ( y) = + 9 y = + 9, no cang, so ys. II: Rplac wit ( ) : y = ( ) + 9 y = + 9, no cang, so ys. III: Sinc tr is symmtry wit rspct to bot as tr is origin symmtry. 7. D T grap is a V wit vrt at =. T intgral givs t sum of t aras of t two triangls tat t V forms wit t orizontal ais for from to. Ts triangls av aras of / and rspctivly. 8. C Lt () t = 5t b t position at tim t. Avrag vlocity () () 5 = = = 5 9. D T tangnt function is not dfind at = π so it cannot b continuous for all ral numbrs. Option E is t only on tat includs itm III. In fact, t functions in I and II ar a powr and an ponntial function tat ar known to b continuous for all ral numbrs.. B. C sin( ) tan( ) d = d = ln cos( ) cos( ) dv dr d V = π r, = π r + r = π (6)(9) + 6 = π dt dt dt = = π = π π. D sin() d cos() ( cos cos). B f cangs sign from positiv to ngativ at = and trfor f cangs from incrasing to dcrasing at =.. A ( ) Or f cangs sign from positiv to ngativ at = and from ngativ to positiv at =. Trfor f as a local maimum at = and a local minimum at =. ( ) 8 ( 8) d ( ) d + + = = = AP Calculus Multipl-Coic Qustion Collction 85 Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
14 985 Calculus AB Solutions 5. D T amplitud is and t priod is. π π y = Asin B wr A = amplitud = and B= = =π priod 6. B II is tru sinc 7 = 7 will b t maimum valu of f ( ). To s wy I and III do not 5 if 5 av to b tru, considr t following: f ( ) = if 5 < < 7 7 if 7 For f ( ), t maimum is and t minimum is D lim csc = lim = sin 8. C To s wy I and II do not av to b tru considr f ( ) = sin and g( ) = +. Tn f ( ) g( ) but nitr f ( ) g ( ) nor f ( ) < g ( ) is tru for all ral valus of. III is tru, sinc f( ) g( ) g( ) f( ) ( ) g( ) f ( ) d f ( ) d g( ) d 9. E. D f ( ) = ln = ( ln ) < for >. Hnc f is dcrasing. for >. f ( ) d d = 8. E Considr t function wos grap is t orizontal lin y = wit a ol at = a. For tis function lim f ( ) = and non of t givn statmnts ar tru. a. C Tis is a dirct application of t Fundamntal Torm of Calculus: f ( ) = +. B y = + 6, y = 6+ 6= for =. y ( ) =. Only option B as a slop of. d d 6 + = + = + = 6 9. A ( ) ( ) ( ) ( ) AP Calculus Multipl-Coic Qustion Collction 86 Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
15 5. A Wasrs: ( ) π R r y wr R=, r = ( ) dy y dy y y Volum =π =π ( ) =π = 8π 985 Calculus AB Solutions AP Calculus Multipl-Coic Qustion Collction 87 Copyrigt 5 by Collg Board. All rigts rsrvd. Availabl at apcntral.collgboard.com.
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