2008 AP Calculus BC Multiple Choice Exam

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1 008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ

2 AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl at tim t = 3? y has vlocity vctor givn by vt ( ) t,5 t. = What (A) 45 9, (B) 6,5 (C),0 (D) 306 (E) 6. d = (A) + C (B) + C (C) + C (D) + C (E) + C 3. sin cos lim 0 is π (A) (B) 0 (C) (D) 4 (E) nonistnt

3 AP Calculus 008 BC Multipl Choic n 4. Considr th sris. If th ratio tst is applid to th sris, which of th following n! inqualitis rsults, implying that th sris convrgs? (A) lim < n n! (B) (C) n! lim < n n + lim < n (D) lim < n n + (E) lim < n! ( n + ) 5. Which of th following givs th lngth of th path dscribd by th paramtric quations 3 5t = sin t and y = from t = 0 to t = π? ( ) π (A) sin ( ) 0 t + dt 3 0t π (B) cos ( ) 0 t + dt 3 0t π (C) ( ) cos + 5 t π (D) ( ) 0 t t dt cos + 5 t π (E) cos ( 3 ) 0 t t dt t + dt 0 t

4 AP Calculus 008 BC Multipl Choic 4 if f ( ) = if = 6. Lt f b th function dfind abov. Which of th following statmnts about f ar tru? I. f has a limit at =. II. f is continuous at =. III. f is diffrntiabl at =. (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III dy y = and y, d usd with a stp siz of 0.5, starting at =? 7. Givn that ( ) 3 = + what is th approimation for ( ) y if Eulr s mthod is (A) 5 (B) 4.5 (C) 4 (D) 3.75 (E) 3.5

5 AP Calculus 008 BC Multipl Choic f ( ) Th function f is continuous on th closd intrval [, 3] and has valus as shown in th tabl abov. Using th intrvals [, 3], [3, 5], [5, 8], and [8, 3], what is th approimation of ( ) 3 f d obtaind from a lft Rimann sum? (A) 6 (B) 4 (C) 8 (D) 3 (E) 50 O Graph of f 9. Th graph of th picwis linar function f is shown in th figur abov. If g( ) ( ) which of th following valus is gratst? (A) g ( 3) (B) g ( ) (C) g ( 0) (D) g ( ) (E) g ( ) = f t dt,

6 AP Calculus 008 BC Multipl Choic 0. In th y -plan, what is th slop of th lin tangnt to th graph of (, )? + y+ y = 7 at th point (A) 4 (B) 3 5 (C) 4 (D) 4 (E) Lt R b th rgion btwn th graph of y = and th -ais for 3. Th ara of R is (A) 6 (B) 6 (C) 6 π (D) 6 (E) infinit

7 AP Calculus 008 BC Multipl Choic. Which of th following sris convrgs for all ral numbrs? n (A) n n (B) n n (C) n n n (D) n! (E) n! n n 3. + d = (A) (B) + (C) + (D) (E)

8 AP Calculus 008 BC Multipl Choic 0 3 f "( ) Th polynomial function f has slctd valus of its scond drivativ f " givn in th tabl abov. Which of th following statmnts must b tru? (A) f is incrasing on th intrval ( 0, ). (B) f is dcrasing on th intrval ( 0, ). (C) f has a local maimum at =. (D) Th graph of f has a point of inflction at =. (E) Th graph of f changs concavity in th intrval ( 0, ). 5. If f ( ) ( ) = ln, thn f "( ) = (A) (B) (C) (D) (E)

9 AP Calculus 008 BC Multipl Choic 6. What ar all valus of for which th sris + convrgs? (A) < < (B) > only (C) only (D) < and > only (E) and n 7. Lt h b a diffrntiabl function, and lt f b th function dfind by f ( ) = h( ) Which of th following is qual to f '? ( ) (A) h ' ( ) (B) 4 h '( ) (C) 4 h '( ) (D) h '4 ( ) (E) 4 h '( 4) 3.

10 AP Calculus 008 BC Multipl Choic 8. In th y -plan, th lin + y = k, whr k is a constant, is tangnt to th graph of y = What is th valu of k? (A) 3 (B) (C) (D) 0 (E) 9. 7 ( 3)( + ) d = (A) 3 ln 3 + ln + + C (B) 3ln 3 + ln + + C (C) 3ln 3 ln + + C (D) (E) 6 + C ( 3) ( + ) 3 + C ( 3) ( + )

11 AP Calculus 008 BC Multipl Choic ( ) ( ) ln ln 0. What is th sum of th sris + ln +?! + + n! + (A) ln (B) ln ( + ln ) (C) (D) (E) Th sris divrgs. n ( t) O t. A particl movs along a straight lin. Th graph of th particl s position t ( ) at tim t is shown abov for 0< t < 6. Th graph has horizontal tangnts at t = and t = 5 and a point of inflction at t =. For what valus of t is th vlocity of th particl incrasing? (A) 0< t < (B) < t < 5 (C) < t < 6 (D) 3< t < 5 only (E) < t < and 5< t < 6

12 AP Calculus 008 BC Multipl Choic 0 f ( ) 4 f '( ) 6 3 g( ) 4 3 g'( ). Th tabl abov givs valus of f, f '( ) g( ) d = 5, thn ( ) ( ) 0 ' 0 f ', g and g ' for slctd valus of. If f g d= (A) 4 (B) 3 (C) (D) 7 (E) 5 3. If f ( ) = sin ( ), which of th following is th Taylor sris for f about = 0? (A) (B) (C) (D) (E) ! 4! 6! ! 4! 6! ! 5! 7! ! 5! 7! ! 5! 7!

13 AP Calculus 008 BC Multipl Choic Pt ( ) O t 4. Which of th following diffrntial quations for a population P could modl th logistic growth shown in th figur abov? (A) (B) (C) (D) (E) dp 0.P 0.00P dt = dp 0.P 0.00P dt = dp dt = 0.P 0.00 dp dt = 0.P 0.00 dp dt = + 0.P 0.00 P P P f ( ) = c + d for c for > 5. Lt f b th function dfind abov, whr c and d ar constants. If f is diffrntiabl at =, what is th valu of c+ d? (A) 4 (B) (C) 0 (D) (E) 4

14 AP Calculus 008 BC Multipl Choic y 6. Which of th following prssions givs th total ara nclosd by th polar curv r = sin θ shown in th figur abov? (A) (B) (C) (D) (E) π 0 sin π sin d 0 π 0 sin θdθ θ θ 4 π sin 4 d 0 θdθ θ θ π 4 sin d 0 θ θ

15 AP Calculus 008 BC Multipl Choic 7. Which of th following could b th slop fild for th diffrntial quation dy y d =? y y (A) (B) y y (C) (D) y (E)

16 AP Calculus 008 BC Multipl Choic 8. In th y -plan, a particl movs along th parabola y = with a constant spd of d dy 0 units pr scond. If > 0, what is th valu of whn th particl is at th point dt dt (, )? (A) 3 (B) 0 3 (C) 3 (D) 6 (E) 6 0

17 AP Calculus 008 BC Multipl Choic 008 AP Calculus BC Multipl Choic Eam Sction Calculator Activ

18 AP Calculus 008 BC Multipl Choic y y = f '( ) O a b c d 76. Th graph of f ', th drivativ of a function f, is shown abov. Th domain of f is th opn intrval 0 < < d. Which of th following statmnts is tru? (A) f has a local minimum at = c. (B) f has a local maimum at = b. (C) Th graph of f has a point of inflction at ( a f ( a )),. (D) Th graph of f has a point of inflction at ( b f ( b )),. (E) Th graph of f is concav up on th opn intrval ( c, d ). t 77. Watr is pumpd out of a lak at th rat R() t = cubic mtrs pr minut, whr t is t + masurd in minuts. How much watr is pumpd from tim t = 0 to t = 5? (A) cubic mtrs (B) cubic mtrs (C) cubic mtrs (D) cubic mtrs (E) cubic mtrs

19 AP Calculus 008 BC Multipl Choic y (,) (3,) O 3 4 Graph of f 78. Th graph of a function f is shown abov. For which of th following valus of c dos ( ) lim f =? c (A) 0 only (B) 0 and 3 only (C) and 0 only (D) and 3 only (E), 0, and Lt f b a positiv, continuous, dcrasing function such that a = f ( n). If k, which of th following must b tru? n an convrgs to (A) lim a n = k n n f d= k (B) ( ) (C) f ( ) d divrgs. (D) f ( ) d convrgs. (E) ( ) f d = k

20 AP Calculus 008 BC Multipl Choic 80. Th drivativ of th function f is givn by f ( ) ( ) dos th graph of f hav on th opn intrval (, )? (A) On (B) Two (C) Thr (D) Four (E) Fiv ' = cos. How many points of inflction 8. Lt f and g b continuous functions for a b. If a c b, b < < ( ) =, a f d P c ( ) b f ( ) d = Q b, g ( ) d= R b, c and ( ), a g d= S thn ( ) ( ) c a (A) P Q+ R S (B) P Q R+ S (C) P Q R S (D) P+ Q R S (E) P+ Q R+ S f g d=?

21 AP Calculus 008 BC Multipl Choic 8. If an divrgs and 0 an bn for all n, which of th following statmnts must b tru? n an convrgs. (A) ( ) n bn convrgs. (B) ( ) n bn divrgs. (C) ( ) (D) (E) bn convrgs. bn divrgs. 83. What is th ara nclosd by th curvs = and y = + 5? 3 y (A)0.667 (B).833 (C) (D).333 (E) 3

22 AP Calculus 008 BC Multipl Choic 84. Lt f b a function with f ( 3) =, f '3 ( ) =, f "3 ( ) = 6, and ( ) following is th third-dgr Taylor polynomial for f about = 3? 3 (A) ( 3) + 3( 3) + ( 3) 3 (B) ( 3) + 3( 3) + 4( 3) 3 (C) ( 3) + 6( 3) + ( 3) f ''' 3 =. Which of th (D) (E) A particl movs on th -ais with vlocity givn by ( ) 4 v t = 3t t + 9t for 3 t 3. How many tims dos th particl chang dirction as t incrass from 3 to 3? (A) Zro (B) On (C) Two (D) Thr (E) Four

23 AP Calculus 008 BC Multipl Choic 86. On th graph of y f ( ), what is th valu of f ( 3? ) = th slop at any point (, ) y is twic th valu of. (A) 6 (B) 7 (C) 8 (D) 9 (E) 0 If ( ) f = 3, 87. An objct travling in a straight lin has position t ( ) at tim t. If th initial position is ( 0) 3 = and th vlocity of th objct is vt () = + t, what is th position of th objct at tim t = 3? (A) 0.43 (B).54 (C) 4.5 (D) 6.5 (E) 7.408

24 AP Calculus 008 BC Multipl Choic 88. For all valus of, th continuous function f is positiv and dcrasing. Lt g b th function givn by g( ) ( ) = f t dt. Which of th following could b a tabl of valus for g? (A) (B) (C) (D) (E) g( ) g( ) g( ) g( ) g( ) Th function f is continuous for and f ( ) f ( ) = = 0. If thr is no c, whr < c <, for which f '( c ) = 0, which of th following statmnts must b tru? (A) For k, < < ( ) (B) For k, f ' k > 0. < < ( ) (C) For k, f ' k < 0. < < f '( ) (D) For k, < < '( ) k ists. (E) For som k, whr k, f k ists, but f ' is not continuous. < < f '( ) k dos not ist.

25 AP Calculus 008 BC Multipl Choic f ( ) g( ) f '( ) g' ( ) Th tabl abov givs valus of th diffrntiabl functions f and g and of thir drivativs f ' and g ', at slctd valus of. If h( ) = f ( g( ) ), what is th slop of th graph of h at =? (A) -0 (B) -6 (C) 5 (D) 6 (E) 0 f = dt t valus of dos f attain a rlativ maimum? 9. Lt f b th function givn by ( ) cos for. /3 At which of th following 3 (A) and (B) 0.4 and (C) 0.4 only (D) 0.46 (E) 0.999

26 AP Calculus 008 BC Multipl Choic y y= f ( ) O ( ) y = g 9. Th figur abov shows th graphs of th functions f and g. Th graphs of th lins tangnt to th graph of g at = 3 and = ar also shown. If B( ) = g( f ( ) ), what is B '( 3 )? (A) (B) (C) 6 6 (D) 3 (E)

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 for which f () is a ral numbr.. (4 6 ) d= 4 6 6