1973 AP Calculus BC: Section I

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1 97 AP Calculus BC: Scio I 9 Mius No Calculaor No: I his amiaio, l dos h aural logarihm of (ha is, logarihm o h bas ).. If f ( ) =, h f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, h h s of valus for which f icrass is (, ] [, ) [,] (, ) (, ) (,) (, ). For wha o-gaiv valu of b is h li giv by y = + b ormal o h curv y =? 5. d is ois 6. If f( ) = for all, h f () = + AP Calculus Mulipl-Choic Qusio Collcio 9 Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

2 7. If y ( y ) = l +, h h valu of dy a h poi (,) is d 97 AP Calculus BC: Scio I udfid 8. If y = si ad igr for which ( ) y mas h h drivaiv of y wih rspc o, h h smalls posiiv ( ) y = y is If y = cos, h dy d = 6sicos cos cos 6cos sicos. Th lgh of h curv y = l sc from = o = b, whr < b <, may b prssd by which of h followig igrals? b sc d b sc b d (sc a d ) b ( ) + lsc d b + ( sc a ) d. L y = +. Wh = ad d =, h valu of dy is AP Calculus Mulipl-Choic Qusio Collcio Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

3 97 AP Calculus BC: Scio I. If is a kow posiiv igr, for wha valu of k is k d=?. Th acclraio α of a body movig i a sraigh li is giv i rms of im by α= 8 6. If h vlociy of h body is 5 a = ad if s ( ) is h disac of h body from h origi a im, wha is s() s()? 8. If = ad y =, h dy d = 5. Th ara of h rgio boudd by h lis =, =, ad y = ad h curv y = is ( ) 6. A sris pasio of si is 6 + +! 5! 7! 5! +! 6! ! 5! 7! 5 +! +! + 6! ! 5! 7! AP Calculus Mulipl-Choic Qusio Collcio Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

4 97 AP Calculus BC: Scio I 5 7. Th umbr of bacria i a culur is growig a a ra of, pr ui of im. A =, h umbr of bacria prs was 7,5. Fid h umbr prs a = 5.,, 7,5 5 7,5 5, L g b a coiuous fucio o h closd irval [, ]. L g () = ad g () =. Which of h followig is NOT cssarily ru? Thr iss a umbr h i [ ] For all a ad b i [ ], such ha gh ( ) g ( ),, if a= b, h ga ( ) = gb ( ). Thr iss a umbr h i [, ] such ha Thr iss a umbr h i [, ] such ha For all h i h op irval ( ) 9. Which of h followig sris covrg? gh ( ) =. gh ( ) =.,, lim g ( ) = gh ( ). h for all i [ ],. I. II. = III. = = ( ) I oly III oly I ad II oly I ad III oly I, II, ad III. d= ( ) + C ( ) + C ( ) + C ( ) + C ( ) + C. + ( + ) d= AP Calculus Mulipl-Choic Qusio Collcio Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

5 97 AP Calculus BC: Scio I. A paricl movs o h curv y = l so ha h -compo has vlociy ( ) = + for. A im,. A im =, h paricl is a h poi ( ) =, h paricl is a h poi ( ),l (,) (,l ),l 5 5,l. + h lim l h is h ois. L f ( ) = + for all ral ad l ε >. For which of h followig choics of δ is f( ) 7 <ε whvr <δ? ε ε ε ε + ε + ε ε 5. a d = + 6. Which of h followig is ru abou h graph of y = l i h irval (,)? I is icrasig. I aais a rlaiv miimum a (, ). I has a rag of all ral umbrs. I is cocav dow. I has a asympo of =. 7. If f ( ) = + 5 ad h domai is h s of all such ha 9, h h absolu maimum valu of h fucio f occurs wh is 6 9 AP Calculus Mulipl-Choic Qusio Collcio Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

6 97 AP Calculus BC: Scio I 8. If h subsiuio = si y is mad i h igrad of d, h rsulig igral is si ydy si y dy cos y si ydy si ydy 6 si ydy 9. If y = y ad if y = y = wh =, h wh =, y = + ( ) ( ). d l l l + ( ) = l l, h f ( ) =. If f ( ) l l l. If l, y = h y is l l l l l l l No of h abov AP Calculus Mulipl-Choic Qusio Collcio Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

7 . Suppos ha f is a odd fucio; i.., f ( ) f( ) Which of h followig mus cssarily b qual o f ( )? f ( ) ( ) f f ( ) f ( ) No of h abov. Th avrag (ma) valu of ovr h irval is 97 AP Calculus BC: Scio I = for all. Suppos ha ( ) f iss. 5. Th rgio i h firs quadra boudd by h graph of y = sc, =, ad h as is road abou h -ais. Wha is h volum of h solid grad? d is + l l l l divrg 7. cos ( ) lim = c 8. If f ( c) d= 5 whr c is a cosa, h ( ) f d = c 5 + c 5 5 c c 5 5 AP Calculus Mulipl-Choic Qusio Collcio 5 Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

8 9. L f ad g b diffriabl fucios such ha f () =, f () =, f () =, g () =, g () =, g () = 5. ( ) = ( ), h h () = If h f( g) 97 AP Calculus BC: Scio I 9 5. Th ara of h rgio closd by h polar curv r = cosθ is. Giv + for <, f( ) = cos for, f ( d ) = + +. Calcula h approima ara of h shadd rgio i h figur by h rapzoidal rul, usig 5 divisios a = ad = AP Calculus Mulipl-Choic Qusio Collcio 6 Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

9 97 AP Calculus BC: Scio I. arcsi d = si d ( arcsi ) arcsi + + C d arccos arcsi d d. If f is h soluio of f ( ) f( ) = such ha ( ), f = h ( ) f = C) 5. Suppos g ( ) < for all ad sams is FALSE? F( ) = g ( ) d for all. Which of h followig F aks o gaiv valus. F is coiuous for all >. F( ) g( ) g( ) d = F ( ) iss for all >. F is a icrasig fucio. AP Calculus Mulipl-Choic Qusio Collcio 7 Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

10 97 Aswr Ky 97 AB 97 BC. E. E. B. A 5. A 6. D 7. B 8. B 9. A. C. B. C. D. D 5. C 6. C 7. C 8. D 9. D. D. B. B. C. B 5. B 6. E 7. E 8. C 9. C. B. D. D. A. C 5. C 6. A 7. A 8. B 9. B. E. D. D. E. B 5. C. A. D. A. C 5. B 6. D 7. D 8. B 9. A. A. E. D. D. A 5. C 6. A 7. C 8. D 9. D. E. B. C. C. A 5. B 6. D 7. E 8. C 9. A. B. E. C. A. C 5. C 6. E 7. E 8. B 9. D. C. D. D. E. A 5. E AP Calculus Mulipl-Choic Qusio Collcio 5 Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

11 97 Calculus BC Soluios. A ( ) d f = = d =. D. A ( + ) = ( + ) = = (8 ) = d ( + )( ) f ( ). = = f ( ) for ad for > < >. f is icrasig for ad for.. C Th slops will b gaiv rciprocals a h poi of irscio. = =± ad, hus = ad h y valus mus b h sam a =. + b= b= 5. B 6. D 7. D 8. B d = d + d = + = ()( + ) ( )() f ( ) =, f () = = ( + ) dy + y dy = d a (, ) y = = d + y ( y = si, y = cos, y = si, y = cos, y ) = si 9. A cos d d y = (cos ) = cos ( si ) ( ) = cos ( si ) d d y = 6sicos AP Calculus Mulipl-Choic Qusio Collcio 77 Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

12 97 Calculus BC Soluios b sc a b L = + y d = d + sc. A ( ) b b b ( ) = + a d = sc d = sc d dy = d ; dy = ( + )() =. E ( ) ( ) ( ). D k d k k = = = ; k = k =. D v ( ) = 8 + C ad v() = 5 C= so v () = 8 +. ( ) s() s() = v( ) d = + =. A 5. C dy dy = d = = d d d Ara = d= = ( ) 6. A si si = + + = + +! 5! 7!! 5! 7! 7. C dn d 5 5 =, N = 75 + C ad N() = 75 C = 5 N = 75, N(5) = D D could b fals, cosidr g ( ) = o [,]. A is ru by h Erm Valu Thorm, B is ru bcaus g is a fucio, C is ru by h Irmdia Valu Thorm, ad E is ru bcaus g is coiuous. AP Calculus Mulipl-Choic Qusio Collcio 78 Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

13 97 Calculus BC Soluios 9. D I is a covrg p-sris, p = > II is h Harmoic sris ad is kow o b divrg, III is covrg by h Alraig Sris Ts.. E ( ) ( ) ( ) ( ) d= d = + C = + C. B ( ) d (( ) d) ( ) ( ) = + = = =. C ( ) = + ( ) = ( + ) + C ad () = C = ( ) = ( + ) () =, y() = l ;, l. C l( + h) l lim f () whr f( ) l h h = = = = ; f ( ) f ( ). A This im uss h formal dfiiio of a limi ad is o logr par of h AP Cours ε Dscripio. f ( ) 7 = ( + ) 7 = 6 = <ε whvr <. ε ε ε ε Ay δ< will b suffici ad <, hus h aswr is. a sc a = = = 5. B d ( ) d ( ) 6. D For i h irval (, ), Thrfor g ( ) = = ( ) ad so ( )( ) ( )( ) y y =, = = < ( ) ( ) y = l g( ) = l( ( )). AP Calculus Mulipl-Choic Qusio Collcio 79 Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

14 97 Calculus BC Soluios Alraiv graphical soluio: Cosidr h graphs of g( ) ad l g( ) =. cocav dow g ( ) = l 7. E f ( ) = 8+ = ( )( 6); h cadidas ar: =,,6,9 f ( ) h maimum is a = 9 8. C = si y d = si y cos y dy; wh =, y = ad wh =, y = si y = = d si y cos y dy si y dy si y 9. A L z = y. Th z = wh =. Thus y = y z = z. Solv his diffrial quaio. z = C ; = C C = y = z = +. Solv his diffrial quaio. ( ) + + y = + K; = + K K = ; y = +, y() = + = + Alraiv Soluio: y = y y = C =. Thrfor ad so y () y () = y ( ) d = y ( ) d = y() y() y () y () + y() + y() = =. y () =. AP Calculus Mulipl-Choic Qusio Collcio 8 Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

15 . B d d ( ) ( ). E f ( ) 97 Calculus BC Soluios = = l + = l+ l+ = l d ( l ) = d = = l l l l. C Tak h log of ach sid of h quaio ad diffria. ( ) l y = l = l l = l y d l = l ( l) = l y = l y d. A f ( ) = f( ) f ( ) ( ) = f ( ) f ( ) = f ( ) hus f ( ) = f ( ).. C d= = = 5. C Washrs: r whr r = y = sc. d Volum sc a a a = = = = 6. E L + L + d = lim d = lim l + L L + + L ( L L ) = lim l + l =. Divrg 7. E cos si si si lim = lim = lim = = 8. B L z c c =. 5 = ( ) = ( ) f c d f z dz c 9. D h ( ) = f ( g( ) ) g ( ) ; ( ) h () = f g() g () = f () g () = ( )( ) = AP Calculus Mulipl-Choic Qusio Collcio 8 Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

16 . C Ara ( cos ) d ( cos cos ) = θ θ= θ+ θ dθ 97 Calculus BC Soluios ; cos θ = ( + cos θ ). D Ara = cos ( cos ) d si si θ+ + θ θ= θ θ+ θ = f ( d ) = ( + ) d+ cos( d ) ( ) si( ) ( si si ) = + + = + =. D 5 7 = ; T = = 5. E Us h chiqu of aidrivaivs by par: u = si dv = d d du = v = d d si = si. A Muliply boh sids of = f ( ) f( ) by. Th f ( ) f( ) d f( ) = = d. f( ) Thus w hav = l + C f( ) = ( l + C) = ( l ) sic f ( ) =. Thrfor ( ) f ( ) = l = ( ) = This was mos likly h soluio suds wr pcd o produc whil solvig his problm o h 97 mulipl-choic am. Howvr, h problm islf is o wll-dfid. A soluio o a iiial valu problm should b a fucio ha is diffriabl o a irval coaiig h iiial poi. I his problm ha would b h domai < sic h soluio rquirs h choic of h brach of h logarihm fucio wih <. Thus o cao ask abou h valu of h fucio a =. 5. E ( ) ( ) ( ) ( ) F = g wih ad g < F < F is o icrasig. AP Calculus Mulipl-Choic Qusio Collcio 8 Copyrigh 5 by Collg Board. All righs rsrvd. Availabl a apcral.collgboard.com.

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( A) ( B) ( C) ( D) ( E) d Smsr Fial Exam Worksh x 5x.( NC)If f ( ) d + 7, h 4 f ( ) d is 9x + x 5 6 ( B) ( C) 0 7 ( E) divrg +. (NC) Th ifii sris ak has h parial sum S ( ) for. k Wha is h sum of h sris a? ( B) 0 ( C) ( E) divrgs