1985 AP Calculus BC: Section I

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1 985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b th st of all ral umbrs for which f () is a ral umbr.. Th ara of th rgio btw th graph of y = + ad th -ais from = to = is At what valus of dos 5 f( ) = hav a rlativ maimum? oly oly oly ad oly, ad. + d = + l 8 l l 8 l l 8 l l+. A particl movs i th y-pla so that at ay tim t its coordiats ar At t =, its acclratio vctor is = t ad y = t t. (, ) (, ) (, ) (, ) (,8 ) AP Calculus Multipl-Choic Qustio Collctio 7 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

2 985 AP Calculus BC: Sctio I 5. Th curvs y f( ) ad y g( ) ara of th shadd rgio closd by ths curvs ad th li = is giv by = = show i th figur abov itrsct at th poit (, ) 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( ) =, th ta f = + AP Calculus Multipl-Choic Qustio Collctio 8 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

3 985 AP Calculus BC: Sctio I 7. Which of th followig is qual to arcsi 5 5 d? + C arcsi + C arcsi + C C 5 + C 8. If f is a fuctio such that f( ) f() lim =, which of th followig must b tru? Th limit of f ( ) as approachs dos ot ist. f is ot dfid at =. Th drivativ of f at = is. f is cotiuous at =. f () = 9. If y + y = 8, th, at th poit (, ), y is 5. For = + ( ) < < if f( ) =, th f ( ) = = = + ( ) ( ) ( ) = ( ) = ( ) + = AP Calculus Multipl-Choic Qustio Collctio 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

4 985 AP Calculus BC: Sctio I. d l = d ( ). d = ( )( + ) l + C + ( l )( l ) + + C + l + C l ( )( + ) + C l ( )( + ) + C. Lt f b th fuctio giv by f ( ) =. What ar all valus of c that satisfy th coclusio of th Ma Valu Thorm of diffrtial calculus o th closd itrval [, ]? oly oly oly ad ad. Which of th followig sris ar covrgt? I. II. III ( ) I oly III oly I ad III oly II ad III oly I, II, ad III 5. If th vlocity of a particl movig alog th -ais is vt ( ) = t ad if at t = its positio is, th at ay tim t its positio ( t ) is t t t t t t+ t t t t+ AP Calculus Multipl-Choic Qustio Collctio 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

5 985 AP Calculus BC: Sctio I 6. Which of th followig fuctios shows that th statmt If a fuctio is cotiuous at =, th it is diffrtiabl at = is fals? f ( ) = f ( ) = f ( ) = f ( ) = f ( ) = 7. If f ( ) ( ) = l, th f ( ) = l ( ) + l ( ) + l ( ) + 8. si ( ) + d = cos + + C cos( + ) + C cos( + ) + C ( ) cos ( + ) + C cos( ) + + C 9. If f ad g ar twic diffrtiabl fuctios such that th h ( ) = f ( ) g ( ) = ad ( ) g ( ) = h( ) f, f ( ) f ( ) ( ) + f ( ) + ( f ( ) ) ( f ( ) + f ( ) ) f ( ) + f ( ) f ( ) + f ( ). Th graph of y f( ) = o th closd itrval [ ],7 is show abov. How may poits of iflctio dos this graph hav o this itrval? O Two Thr Four Fiv AP Calculus Multipl-Choic Qustio Collctio 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

6 985 AP Calculus BC: Sctio I. If f ( )si d = f ( )cos + cos d, th ( ) f could b si cos. Th ara of a circular rgio is icrasig at a rat of 96 squar mtrs pr scod. Wh th ara of th rgio is 6 squar mtrs, how fast, i mtrs pr scod, is th radius of th rgio icrasig? lim h + h 5 h + 8 d is oistt. Th ara of th rgio closd by th polar curv si ( ) r = θ for θ is 8 5. A particl movs alog th -ais so that at ay tim t its positio is giv by valus of t is th particl at rst? t () t = t. For what No valus oly oly oly ad 6. For, < < if y ( ) = si, th dy d is l ( si ) ( si ) cot ( si ) ( cos ) ( si ) ( cos + si ) ( si ) ( cot + l ( si ) ) AP Calculus Multipl-Choic Qustio Collctio 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

7 985 AP Calculus BC: Sctio I 7. If f is th cotiuous, strictly icrasig fuctio o th itrval a b as show abov, which of th followig 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 umbr c such that a< c< b I oly II oly III oly I ad III oly I, II, ad III 8. A atidrivativ of f( ) = + is + + ( ) si lim is oistt. If = t t ad y = t+, th dy d at t = is What ar all valus of for which th sris = ( ) covrgs? < < < < AP Calculus Multipl-Choic Qustio Collctio 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

8 985 AP Calculus BC: Sctio I. A quatio of th li ormal to th graph of y = at th poit whr = is + y = y = y = + y = 5 + y = 5 dy. If y dt = ad if y = wh t =, what is th valu of t for which y =? l l l. Which of th followig givs th ara of th surfac gratd by rvolvig 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 rgio i th first quadrat btw th -ais ad th graph of y = 6 is rotatd aroud th y-ais. Th volum of th rsultig solid of rvolutio is giv by 6 ( ) 6 d 6 ( ) 6 d 6 ( ) 6 6 d ( + ) 9 9 y dy ( + ) 9 y dy AP Calculus Multipl-Choic Qustio Collctio 5 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

9 985 AP Calculus BC: Sctio I 6. d is 6 6 oistt 7. Th gral solutio for th quatio dy + y = is d y = + C y = + + C C y = + + y = + C y = C + C 8. ( ) lim 5 + is 5 oistt 9. Th bas of a solid is th rgio closd by th graph of y =, th coordiat as, ad th li =. If all pla cross sctios prpdicular to th -ais ar squars, th its volum is ( 6 ) 6 6. If th substitutio u = is mad, th itgral d = u du u u du u u du u u du u u du u AP Calculus Multipl-Choic Qustio Collctio 55 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

10 985 AP Calculus BC: Sctio I. What is th lgth of th arc of y = from = to =? Th cofficit of i th Taylor sris for about = is 6 9. Lt f b a fuctio that is cotiuous o th closd itrval [,] such that f () dos ot ist, f () =, ad f ( ) < for all cpt =. Which of th followig could b th graph of f?. At ach poit (, y ) o a crtai curv, th slop of th curv is poit (,8 ), th its quatio is y. If th curv cotais th y = 8 y = + 8 y = + 7 y ( ) = l y = If is a positiv itgr, th lim ca b prssd as d d d d d AP Calculus Multipl-Choic Qustio Collctio 56 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

11 985 Aswr 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-Choic Qustio Collctio 55 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

12 985 Calculus BC Solutios. D. A. B. D ( + ) d= ( + ) = (6 + ) ( + ) = 7 f ( ) = 5 5 = 5 ( ) = 5 ( )( + ), chags sig from positiv to gativ oly at =. So f has a rlativ maimum at = oly. ( + ) d = = l + = l 8 l d ( ) d d dy d ( t) = t = t ad = ; y( t) = t t = t 6 t ad = t t dt dt dt dt d d y a() t =, = (, t t) a() = (,) dt dt a 5. D Ara = ( top curv bottom curv ), < ; Ara = ( ( ) ( )) d f g d 6. E 7. A ( ) ta sc f( ), f ( ), f = = ta = = ta du u d du = si d = si + C a a u C f( ) f() lim = f () so th drivativ of f at = is. 9. B Tak th drivativ of ach sid of th quatio with rspct to. yy + y + y + y =, substitut th poit (,) ()() y + + ()() y + ()() = y =. A Tak th drivativ of th gral trm with rspct to : ( ) d d d = = = d. A l ( l( ) ) = + AP Calculus Multipl-Choic Qustio Collctio 88 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

13 . A Us partial fractios to rwrit ( )( ) ( )( ) 985 Calculus BC Solutios as + + d= d = ( l l + ) + C = l + C B f () =, f() =, f ( ) = 6 ; by th Ma Valu Thorm, f() f() f () c = = for c (,). So, c 6c c( c ) = =. Th oly valu i th op itrval is.. C I. covrgt: p-sris with p = > II. divrgt: Harmoic sris which is kow to divrg III. covrgt: Gomtric with r = < 5. C t t () t = + (w ) dw= + ( w w) = + t t = t t+ or, () t = t t+ C, () = C = so, () t = t t+ 6. C For ( ) 7. B ( ) f = w hav cotiuity at =, howvr, ( ) d ( ) d f = () l( ) + = l( ) + = l( ) + f = is ot dfid at =. 8. C 9. D si( + ) d = si( + )( d) = cos( + ) + C f ( ) f ( ) f ( ) f ( ) g( ) =, g ( ) = f ( ), g ( ) = f ( ) + f ( ) f ( ) f ( ( )) ( ) f ( ) ( ) g ( ) = f ( ) + f ( ) = h( ) h( ) = f ( ) + f ( ). C Look for cocavity chags, thr ar. AP Calculus Multipl-Choic Qustio Collctio 89 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

14 985 Calculus BC Solutios. B Us th tchiqu of atidrivativs by parts: u = f dv= d ( ) si ( ) cos du = f d v = f ( )si d = f ( )cos + f ( )cos d ad w ar giv that f ( )si d= f( )cos + cos d f ( ) = f( ) =. A A= r, A= 6 wh r = 8. Tak th drivativ with rspct to t. da = r dr ; 96= (8) dr dr = 6 dt dt dt dt. C + h d F( + h) F() 5 F F h h lim = lim = () whr ( ) = + 8 h h. F ()= Altrat solutio by L Hôpital s Rul: + h d ( + h) + 8 lim = lim = 9= h h h d 8. D d ( ) Ara = si ( θ) θ= cos θ θ= θ si θ = 5. C At rst wh vt () =. t t t vt () = t = ( ) t, vt () = at t = oly. 6. E Apply th log fuctio, simplify, ad diffrtiat. l y = l ( si ) = l ( si ) y cos = l ( si ) + = l ( si ) + cot = si l si + cot y ( ) ( ) ( ( ) ) y y si 7. E Each of th right-had sids rprst th ara of a rctagl with bas lgth ( b a). I. Ara udr th curv is lss tha th ara of th rctagl with hight f ( b ). II. Ara udr th curv is mor tha th ara of th rctagl with hight f ( a ). III. Ara udr th curv is th sam as th ara of th rctagl with hight f () c, a< c< b. Not that this is th Ma Valu Thorm for Itgrals. 8. E d= ( d) + u. This is of th form, du u=, so + d= + C AP Calculus Multipl-Choic Qustio Collctio 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

15 985 Calculus BC Solutios 9. D Lt = t. si si t lim = lim = t t. B At t =, dy dy = dt = t + = = d d t t= 8 dt. D Th ctr is =, so oly C, D, or E ar possibl. Chck th dpoits. ( ) At = : covrgs by altratig sris tst. = At = : which is th harmoic sris ad kow to divrg. =. E y( ) = 6, y ( ) = =, th slop of th ormal is for th ormal is = y+ 6 = ( + ) + y = 5. ad a quatio. C This is th diffrtial quatio for potial growth. t t t y = y() = ; = ; t = l t = l = l. A This topic is o logr part of th AP Cours Dscriptio. ρ s whr ρ= = y d ( ) Surfac Ara = y + dy = y + y dy = y + 9y dy dy 5. B Us shlls (which is o logr part of th AP Cours Dscriptio) rh whr r = ad h = y = 6 ( ) 6 Volum = 6 d AP Calculus Multipl-Choic Qustio Collctio 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

16 985 Calculus BC Solutios 6. E d d lim d lim = = + + = which dos ot ist. L L L L 7. A This topic is o logr part of th AP Cours Dscriptio. y = yh + yp whr dy th solutio to th homogous quatio y solutio to th giv diffrtial quatio. Substitut d + = ad p ( ) p yh = c is y = A + B is a particular y ito th diffrtial quatio to dtrmi th valus of A ad B. Th aswr is A=, B=. 8. C ( ) ( + ) l 5 5 l( 5 ) lim l( 5 ) lim lim lim + 5 = lim = = = = 9. A Squar cross sctios: y whr y =. ( V = d= = 6 ). A u =, du = d; wh =, u = ad wh =, u = u u d = du = du u u = = + = + = + = = =. C y, L ( y ) d d ( ) ( ) ( 8 ). E Sic u = + u+ u + u +, th!! 9 Th cofficit w wat is =! ( ) ( ) = !!. E Graphs A ad B cotradict f <. Graph C cotradicts f () dos ot ist. Graph D cotradicts cotiuity o th itrval [,]. Graph E mts all giv coditios.. A dy dy = y = d l y = + K; y = C ad y() = 8 so, y= 8 d y AP Calculus Multipl-Choic Qustio Collctio 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

17 985 Calculus BC Solutios 5. D Th prssio is a Rima sum with = ad f( ) =. Th valuatio poits ar:,,,, Thus th right Rima sum is for = to =. Th limit is qual to d. AP Calculus Multipl-Choic Qustio Collctio 9 Copyright 5 by Collg Board. All rights rsrvd. Availabl at apctral.collgboard.com.

PURE MATHEMATICS A-LEVEL PAPER 1

-AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio