Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number.

Transcription

1 997 AP Calculus BC: Section I, Part A 5 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number.. ( ) + d= (A) (C) (E). If t = e and y = sin( t), then dy d = (A) t 4e cos(t) t e cos(t) (C) sin(t) e t cos(t) e t (E) cos(t) t e. The function f given by 5 f ( ) = 4 has a relative maimum at = (A) 5 (C) (E) d 4. ( e ln d ) = (A) + + (C) (E) + 5. If e f( ) = ( ) +, then f () = (A) (C) 7 (E) + e 6. The line normal to the curve y = 6 at the point (, 4 ) has slope (A) 8 4 (C) 8 (E) 8 8 AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

2 997 AP Calculus BC: Section I, Part A Questions 7-9 refer to the graph and the information below. The function f is defined on the closed interval [,8 ]. The graph of its derivative f is shown above. 7. The point ( ) (,5 ) is (A) y = y = 5 (C) y 5= ( ) y+ 5= ( ) (E) y+ 5= ( + ),5 is on the graph of y = f( ). An equation of the line tangent to the graph of f at 8. How many points of inflection does the graph of f have? (A) Two Three (C) Four Five (E) Si AP Calculus Multiple-Choice Question Collection 4 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

3 997 AP Calculus BC: Section I, Part A 9. At what value of does the absolute minimum of f occur? (A) (C) 4 6 (E) 8. If y = y+ +, then when =, dy d is (A) (C) (E) noneistent. d is ( + ) (A) (C) 4 4 (E) divergent. The graph of f, the derivative of f, is shown in the figure above. Which of the following describes all relative etrema of f on the open interval ( ab, )? (A) One relative maimum and two relative minima Two relative maima and one relative minimum (C) Three relative maima and one relative minimum One relative maimum and three relative minima (E) Three relative maima and two relative minima AP Calculus Multiple-Choice Question Collection 5 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

4 997 AP Calculus BC: Section I, Part A. A particle moves along the -ais so that its acceleration at any time t is at () = t 7. If the initial velocity of the particle is 6, at what time t during the interval t 4 is the particle farthest to the right? (A) (C) (E) 4 4. The sum of the infinite geometric series is 6 8, 4 (A).6.5 (C).4.45 (E).5 5. The length of the path described by the parametric equations t, is given by = cos t and y = sin t, for (A) (C) (E) cos t+ sin t dt cos t sin t+ sin t cos t dt 4 4 9cos t+ 9sin t dt 4 4 9cos t sin t+ 9sin t cos t dt 6 6 cos t+ sin t dt 6. e lim h h h is (A) (C) e (E) noneistent AP Calculus Multiple-Choice Question Collection 6 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

5 7. Let f be the function given by f ( ) ln( ) = is 997 AP Calculus BC: Section I, Part A =. The third-degree Taylor polynomial for f about (A) (C) (E) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) + ( ) + ( ) ( ) ( ) ( ) + + ( ) ( ) ( ) + 8. For what values of t does the curve given by the parametric equations 4 y = t + t 8t have a vertical tangent? = t t and (A) only only (C) and only,, and (E) No value 9. The graph of y = f( ) is shown in the figure above. If A and A are positive numbers that represent the areas of the shaded regions, then in terms of A and A, 4 4 f d 4 ( ) f( d ) = (A) A A A (C) A A A+ A (E) A+ A AP Calculus Multiple-Choice Question Collection 7 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

6 997 AP Calculus BC: Section I, Part A. What are all values of for which the series ( ) converges? n n n= n (A) < < (C) < 5 5 (E) < 5. Which of the following is equal to the area of the region inside the polar curve r = cosθ and outside the polar curve r = cos θ? (A) θ θ cos d θ θ (C) cos d cos θ dθ cos θdθ (E) cos θ d θ. The graph of f is shown in the figure above. If g ( ) = ftdt ( ), for what value of does g ( ) have a maimum? (A) a b (C) c d (E) It cannot be determined from the information given. a AP Calculus Multiple-Choice Question Collection 8 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

7 997 AP Calculus BC: Section I, Part A. In the triangle shown above, if θ increases at a constant rate of radians per minute, at what rate is increasing in units per minute when equals units? (A) 5 4 (C) 4 9 (E) 4. The Taylor series for sin about = is ( ) f ( ) = sin, then the coefficient of 5 +. If f is a function such that! 5! 7 in the Taylor series for f ( ) about = is (A) 7! 7 (C) (E) 4 7! 5. The closed interval [ ab, ] is partitioned into n equal subintervals, each of width numbers,,..., n where a= < < < < n < n = b. What is (A) ( ) b a b a (C) ( ) b a b a (E) ( b ) a n i =, by the n lim i? AP Calculus Multiple-Choice Question Collection 9 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

8 997 AP Calculus BC: Section I, Part B 4 Minutes Graphing Calculator Required Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approimates the eact numerical value. () Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number. 76. Which of the following sequences converge? I. II. III. 5n n n e n n e + e n (A) I only II only (C) I and II only I and III only (E) I, II, and III 77. When the region enclosed by the graphs of y = and volume of the solid generated is given by (A) ( ) d y = 4 is revolved about the y-ais, the ( 4 ) d (C) ( ) d ( ) d (E) ( ) d AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

9 997 AP Calculus BC: Section I, Part B 78. h ( e h) ln + lim is h (A) f ( e), where f( ) = ln ln f (), e where f() = (C) f (), where f( ) = ln f (), where f( ) = ln ( + e) (E) f (), where f( ) = ln 79. The position of an object attached to a spring is given by yt ( ) = cos(5 t) sin(5 t), where t is 6 4 time in seconds. In the first 4 seconds, how many times is the velocity of the object equal to? (A) Zero Three (C) Five Si (E) Seven 8. Let f be the function given by f ( ) = cos( ) + ln( ). What is the least value of at which the graph of f changes concavity? (A).56.9 (C).8.8 (E) Let f be a continuous function on the closed interval [, 6]. If f ( ) = and ( ) the Intermediate Value Theorem guarantees that (A) f () = 4 f () c = for at least one c between and 6 9 (C) f( ) for all between and 6 f() c = for at least one c between and 6 (E) f( c ) = for at least one c between and f 6 =, then AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

10 8. If 4, of the following, which is the greatest value of such that 997 AP Calculus BC: Section I, Part B ( ) t t dt tdt? (A).5.8 (C).4.48 (E).59 dy = + y and if y = when =, then y = d 8. If ( ln ) (A) e + ln (C) ln (E) ln e + ln e 84. sin d= (A) (C) (E) cos sin cos + C cos + sin cos + C cos + sin + cos + C cos + C cos+ C 85. Let f be a twice differentiable function such that f () = and f () = 7. Which of the following must be true for the function f on the interval? I. The average rate of change of f is 5. II. The average value of f is 9. III. The average value of f is 5. (A) None I only (C) III only I and III only (E) II and III only AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

11 997 AP Calculus BC: Section I, Part B 86. d = ( )( + ) (A) ln ln 4 + C + C ln + + C (C) ( )( ) + ln ( )( + ) (E) ln ( )( ) + + C + C 87. The base of a solid is the region in the first quadrant enclosed by the graph of y = and the coordinate aes. If every cross section of the solid perpendicular to the y-ais is a square, the volume of the solid is given by (A) ( ) y dy ( ) y dy (C) ( ) d ( ) d (E) ( ) d AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

12 997 AP Calculus BC: Section I, Part B 88. Let f ( ) = sint dt. At how many points in the closed interval, rate of change of f equal the average rate of change of f on that interval? does the instantaneous (A) Zero One (C) Two Three (E) Four 89. If f is the antiderivative of + 5 such that f ( ) =, then ( 4) f = (A). (C).6.76 (E) A force of pounds is required to stretch a spring 4 inches beyond its natural length. Assuming Hooke s law applies, how much work is done in stretching the spring from its natural length to 6 inches beyond its natural length? (A) 6. inch-pounds 45. inch-pounds (C) 4. inch-pounds 5. inch-pounds (E) 7. inch-pounds AP Calculus Multiple-Choice Question Collection 4 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

13 997 Answer Key 997 AB 997 BC. C. A. C 4. D 5. E 6. C 7. D 8. C 9. B. E. E. B. A 4. C 5. B 6. D 7. A 8. C 9. D. E. E. D. A 4. B 5. A 76. E 77. D 78. D 79. C 8. A 8. A 8. B 8. C 84. C 85. C 86. A 87. B 88. E 89. B 9. D. C. E. A 4. C 5. C 6. A 7. C 8. E 9. A. B. C. A. B 4. C 5. D 6. B 7. B 8. C 9. D. E. A. C. E 4. D 5. A 76. D 77. E 78. A 79. D 8. B 8. D 8. B 8. E 84. C 85. D 86. A 87. B 88. C 89. D 9. B AP Calculus Multiple-Choice Question Collection 58 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

14 997 Calculus BC Solutions: Part A. C. E. A 4. C 5. C 6. A 5 6 ( + ) d= + d= + = 5 5 t dy cos( t) cos( t) = e, y = sin( t); = = d t t e e 5 4 f( ) = 4 ; f ( ) = 5 = (5 + )( ) = (5 + )( + )( ) ; f changes from positive to negative only at =. d ln ln e = ; so e = and ( ) = d f( ) = ( ) + e ; f ( ) = ( ) + e ; f () = + = y = (6 ) ; y = (6 ) ; y () = ; The slope of the normal line is C The slope at = is. The equation of the tangent line is y 5 = ( ). 8. E Points of inflection occur where f changes from increasing to decreasing, or from decreasing to increasing. There are si such points. 9. A f increases for 6 and decreases for 6 8. By comparing areas it is clear that f increases more than it decreases, so the absolute minimum must occur at the left endpoint, =.. B. C y = y+ + ; y = y + y+ ; at =, y = ; y = y + y = L d L ( + ) = lim ( + ) = lim = L 4 ( + L ) 4. A f changes from positive to negative once and from negative to positive twice. Thus one relative maimum and two relative minimums.. B at ( ) = t 7 and v() = 6; so vt ( ) = t 7t+ 6 = ( t )( t 6). Movement is right then left with the particle changing direction at t =,, therefore it will be farthest to the right at t =. AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

15 997 Calculus BC Solutions: Part A 4. C Geometric Series. r = < convergence. a= so the sum will be S = = D = cos t, y = sin t for t. d dy L= dt + dt dt 4 4 ( cos sin ) (sin cos ) 9 cos sin 9sin cos L= t t + t t dt = t t+ t tdt 6. B 7. B 8. C 9. D. E h h h e e e e lim lim f (), where f( ) e and f (). lim h h h = = = = = h h h f( ) = ln( ); f ( ) =, f ( ) =, f ( ) = ; ( ) ( ) f() =, f () =, f () =, f () = ; a =, a =, a =, a = ( ) ( ) f( ) ( ) 4 dy 4t + 4t 8 4t + 4t 8 = t t, y = t + t 8 t; = =. Vertical tangents at d t t t(t ) 4 4 f ( d ) f( d ) = ( A A) ( A) = A+ A 4 n t =, ( ). The endpoints of the interval of convergence are when ( ) =± ; =, 5. n n= n Check endpoints: = gives the alternating harmonic series which converges. = 5 gives the harmonic series which diverges. Therefore the interval is < 5.. A Area = ((cos ) cos ) d cos d θ θ θ= θ θ. C g ( ) = f( ). The only critical value of g on ( ad, ) is at = c. Since g changes from positive to negative at = c, the absolute maimum for g occurs at this relative maimum. AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

16 997 Calculus BC Solutions: Part A. E 4. D 4 4 = 5sin θ ; d = 5cos θ dθ ; When =, cos θ= ; d = 5 () = dt dt 5 dt 5 ( ) 6 7 f ( ) = sin( ) = + = + f( ) = + The! coefficieint of is A This is the limit of a right Riemann sum of the function f ( ) n b b lim i = d= = b a n a = i ( ) a = on the interval [ ab,, ] so AP Calculus Multiple-Choice Question Collection 4 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

17 997 Calculus BC Solutions: Part B 76. D Sequence 5 I ; sequence II ; sequence III. Therefore I and III only. 77. E Use shells (which is no longer part of the AP Course Description.) rh where r = and h= 4 Volume = (4 ) d= ( ) d 78. A ln( e+ h) ln( e+ h) ln e lim = lim = f ( e) where f( ) = ln h h h h 79. D Count the number of places where the graph of yt ( ) has a horizontal tangent line. Si places. 8 B Find the first turning point on the graph of y = f ( ). Occurs at = D f assumes every value between and on the interval (,6). Thus f( c ) = at least once. 8. B 8. E ( t t) dt tdt ; the interval [,4] that satisfies this inequality is found to occur at = Using the calculator, the greatest value on dy k ln ln ( ln d ) ; ln y ln k ln k; y ee y Ce y = + = + + = + = =. Since ln y = when =, C =. Hence y = e. AP Calculus Multiple-Choice Question Collection 5 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

18 997 Calculus BC Solutions: Part B 84. C sin d; Use the technique of antiderivatives by parts with u = and dv = sin d. It will take iterations with a different choice of u and dv for the second iteration. sin d = cos + cos d ( ) = cos + sin sin d = cos + sin + cos + C f() f() D I. Average rate of change of f is =. True II. Not enough information to determine the average value of f. False III. Average value of f is the average rate of change of f. True A B 86. A Use partial fractions. = + ; = A ( + ) + B ( ) ( )( + ) + Choose = A= and choose = B=. 4 4 d = d d = ln + C ( )( + ) B Squares with sides of length. Volume = dy = ( y) dy 88. C For the average rate of change of f we need to f ( ) = sin tdt; f ( ) = sin( ); determine f () and f ( ). () f = and f ( ) = sin tdt =. The average rate of change of f on the interval is. See how many points of intersection there are for the graphs of y = sin( ) and y = on the interval,. There are two. AP Calculus Multiple-Choice Question Collection 6 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

19 997 Calculus BC Solutions: Part B 89. D 4 ( ) t 5 ; (4) t f = dt f = dt = 5 + t + t.76 Or, (4) () 4 f = f + d= Both statements follow from the Fundamental Theorem of Calculus. 9. B F( ) = k; = 4 k k = ; Work = F( ) d d 45 inch-lbs = = = 4 6 AP Calculus Multiple-Choice Question Collection 7 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.