1997 AP Calculus AB: Section I, Part A

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1 997 AP Calculus AB: Sction I, Part A 50 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.. (4 6 ) d= If f( ) =, thn f ( ) = b. If f ( ) d= a+ b, thn ( ( ) 5) a f + d= a b a+ b+ 5 5b 5a 7b 4a 7b 5a 7b 6a 4. If f( ) = + +, thn f ( ) = 5 AP Calculus Multipl-Choic Qustion Collction 00 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

2 997 AP Calculus AB: Sction I, Part A 5. Th graph of 4 y = is concav down for < 0 > 0 < 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 0 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

3 997 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 = 0. Th vlocity v of th bug at tim t, 0 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 = 0 to t = 8? An quation of th lin tangnt to th graph of y = cos( ) at π = is 4 π y = 4 π y = 4 π y = 4 π y = 4 π y = 4 AP Calculus Multipl-Choic Qustion Collction 0 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

4 997 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 4y =?,, 8, 4, (, ) AP Calculus Multipl-Choic Qustion Collction 0 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

5 997 AP Calculus AB: Sction I, Part A. Lt f b a function dfind for all ral numbrs. If intrval f ( ) = 4, thn f is dcrasing on th (,) (, ) (, 4) (, ) (, ) 4. 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 04 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

6 997 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 π 7. If + y = 5, what is th valu of d y d at th point ( 4, )? π 4 tan 0 cos d is If f( ) = ln, thn f ( ) = AP Calculus Multipl-Choic Qustion Collction 05 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

7 997 AP Calculus AB: Sction I, Part A 0. 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 0 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 π 5 6π 6π 5 8π π AP Calculus Multipl-Choic Qustion Collction 06 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

8 997 AP Calculus AB: Sction I, Part A 4. Th prssion is a Rimann sum approimation for d d d d d 5. sin( ) d= cos( ) + sin( ) + C 4 cos( ) sin( ) + C 4 cos( ) sin( ) + C 4 cos( ) + sin( ) + C cos( ) 4sin( ) + C AP Calculus Multipl-Choic Qustion Collction 07 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

9 997 AP Calculus AB: Sction I, Part B 40 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(0) = f( ) d=. and F ( ) f( ), = thn AP Calculus Multipl-Choic Qustion Collction 08 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

10 997 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 0 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 80. Lt f b th function givn by 4 ( ) 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 70 mtrs south of th crossing and watchs an astbound train travling at 60 mtrs pr scond. At how many mtrs pr scond is th train moving away from th obsrvr 4 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 09 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

11 997 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 = At tim t 0, th acclration of a particl moving on th -ais is at ( ) = t+ sin t. At t = 0, th vlocity of th particl is. For what valu t will th vlocity of th particl b zro? AP Calculus Multipl-Choic Qustion Collction 0 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

12 997 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 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

13 997 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 [ 0, ] ar usd, which of th following is th trapzoidal approimation of 0 f ( d )? Which of th following ar antidrivativs of f ( ) = sincos? I. II. III. sin F( ) = cos F( ) = cos( ) F( ) = 4 I only II only III only I and III only II and III only AP Calculus Multipl-Choic Qustion Collction Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

14 997 Answr Ky 997 AB 997 BC. C. A. C 4. D 5. E 6. C 7. D 8. C 9. B 0. E. E. B. A 4. C 5. B 6. D 7. A 8. C 9. D 0. E. E. D. A 4. B 5. A 76. E 77. D 78. D 79. C 80. A 8. A 8. B 8. C 84. C 85. C 86. A 87. B 88. E 89. B 90. D. C. E. A 4. C 5. C 6. A 7. C 8. E 9. A 0. B. C. A. B 4. C 5. D 6. B 7. B 8. C 9. D 0. E. A. C. E 4. D 5. A 76. D 77. E 78. A 79. D 80. B 8. D 8. B 8. E 84. C 85. D 86. A 87. B 88. C 89. D 90. B AP Calculus Multipl-Choic Qustion Collction 58 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

15 997 Calculus AB Solutions: Part A. C 4 (4 6 ) d= ( ) = (6 ) ( ) = 6. A ( ) f( ) = ( ) ; f ( ) = ( ) + ( ) = ( ) ( ) = b b b. C ( f ( ) + 5) d= f( ) d+ 5 d= a+ b+ 5( b a) = 7b 4a a a a 4. D 5. E f( ) = + + ; f ( ) = + ; f ( ) = ( ) + = + = ( ) 4 y = ; y = ; y = = ( )( ) y < 0 for < <, thrfor th graph is concav down for < < 6. C dt = + C t t 7. D d d d cos ( ) = cos( ) (cos( ) = cos( )( sin( ) ( ) d d d = cos( )( sin( )( ) 8. C Th bug chang dirction whn v changs sign. This happns at t = B Lt A b th ara btwn th graph and t-ais for 0 t 6, and lt A b th ara btwn th graph and th t-ais for 6 t 8 Thn A = and A =. Th total distanc is A A + =. π π π 0. E y = cos( ); y = sin( ); y = and y = 0; y = E Sinc f is positiv for < < and ngativ for < and for >, w ar looking for a graph that is incrasing for < < and dcrasing othrwis. Only option E.. B y = ; y = ; W want y = =. So th point is, 8. AP Calculus Multipl-Choic Qustion Collction 7 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

16 997 Calculus AB Solutions: Part A. A 4 f ( ) = ; f is dcrasing whn f < 0. Sinc th numrator is non-ngativ, this is only whn th dnominator is ngativ. Only whn <. 4. C f( ) L( ) = + 5( ); L( ) = 0 if 0 = 5 =.6 5. B Statmnt B is tru bcaus lim f ( ) = = lim f( ). Also, lim f ( ) dos not ist + a a b bcaus th lft- and right-sidd limits ar not qual, so nithr,, nor ar tru. 6. D Th ara of th rgion is givn by d 0 8 (5 ( + )) = (4 ) = 8 = 7. A 8. C 9. D 0. E 4 + y = 5; + y y = 0; + y y = 0; y (4,) = ; y y = 0 + y y + y y = 0; + () y + = 0; y = 7 π 4 tan 0 cos d is of th form π tan π 4 tan cos d = = = u duwhr u = tan.. d f( ) = ln ; f ( ) = ( ) = d 5 cos (sin 5 sin( )) (sin 5 sin ) 8 d= = + ; Not: Sinc th sin is an odd function, 8 8 sin( ) = sin().. E lim ln is nonistnt sinc lim ln = 0 and lim 0.. D f( ) = ( ) ; f ( ) = ( + + ) = ( )( + ); f ( ) > 0 for < <. A Disks whr r =. 4 π 5 π V =π dy =π y dy y 0 = = AP Calculus Multipl-Choic Qustion Collction 8 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

17 4. B Lt [ ] Using f ( ) 997 Calculus AB Solutions: Part A 0, b dividd into 50 subintrvals. = ; =, =, =,, 50 = =, th right Rimann sum f ( i ) is an approimation for d. 5. A Us th tchniqu of antidrivativs by parts, which was rmovd from th AB Cours Dscription in 998. u = dv= sin d du = d v = cos i= 0 sin( ) d = cos( ) + cos( ) cos( ) sin( ) d= + + C 4 AP Calculus Multipl-Choic Qustion Collction 9 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

18 997 Calculus AB Solutions: Part B 76. E 77. D 78. D ( ) f( ) = ; f ( ) = = 4 y = cos. Look at th graph of y = 6+ + cos in th window [, ] sinc that domain contains all th option valus. y changs sign at =.89. F() F(0) = f( ) d= f( ) d+ f( ) d= +.= ( Count squars for f ( d ) ) C Th stm of th qustions mans f () = 5. Thus f is diffrntiabl at = and thrfor continuous at =. W know nothing of th continuity of f. I and II only. 80. A f( ) = ; f ( ) = 6 ; W want 6 =. Graph th drivativ function and th function y =, thn find th intrsction to gt = A Lt b th distanc of th train from th d crossing. Thn 60 dt =. ds d ds d S = + 70 S = =. dt dt dt S dt Aftr 4 sconds, = 40 and so S = 50. ds 40 Thrfor (60) 57.6 dt = 50 = 8. B 8. C P ( ) = 8 ; P ( ) = 4 8; P changs from ngativ to positiv at =. P() = 8 cos = at = Stor this in A. (cos ) d= A 84. C Cross sctions ar squars with sids of lngth y. Volum = y d= ln d= ( ln ) = ( ln ) (0 ) = 85. C Look at th graph of f and locat whr th y changs from positiv to ngativ. = A f( ) = ; f ( ) = ; = c= c 4 AP Calculus Multipl-Choic Qustion Collction 0 Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

19 997 Calculus AB Solutions: Part B 87. B at ( ) = t+ sin t and v(0) = vt ( ) = t cost ; vt ( ) = 0 at t= E f( ) = h( ) d f( a) = 0, thrfor only or ar possibl. But f ( ) = h( ) and a thrfor f is diffrntiabl at = b. This is tru for th graph in option but not in option whr thr appars to b a cornr in th graph at = b. Also, Sinc h is incrasing at first, th graph of f must start out concav up. This is also tru in but not. 89. B 90. D T = ( ) = F( ) = sin F ( ) = sincos Ys F( ) = cos F ( ) = cossin No F( ) = cos( ) F ( ) = sin( ) = sin cos 4 Ys AP Calculus Multipl-Choic Qustion Collction Copyright 005 by Collg Board. All rights rsrvd. Availabl at apcntral.collgboard.com.

1997 AP Calculus AB: Section I, Part A

1997 AP Calculus AB: Section I, Part A 997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=