1985 AP Calculus AB: Section I

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1 985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () 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) 7 (B) 8 (C) 5 6 (D) 8 (E) 5 6. If f( ) ( ) = +, then the th derivative of f ( ) at = is (A) (B) (C) 8 (D) (E) 8. If dy y =, then + d = (A) 6 ( + ) (B) ( + ) (C) 6 ( + ) (D) ( + ) (E) dy, then y d = =. If cos( ) cos cos sin (A) ( ) + C (B) ( ) + C (C) ( ) sin sin + C (D) ( ) + C (E) ( ) + C 5. n lim is n, n n + (A) (B),5 (C) (D) (E) noneistent AP Calculus Multiple-Choice Question Collection 8 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

2 985 AP Calculus AB: Section I 6. If f ( ) =, then f (5) = (A) (B) 5 (C) (D) 5 (E) 5 7. Which of the following is equal to ln? (A) ln + ln (B) ln 8 ln (C) t edt (D) ln d (E) dt t 8. The slope of the line tangent to the graph of y = ln at = is (A) 8 (B) (C) (D) (E) 9. If e d= k, then e d= (A) k (B) k (C) k (D) k (E) k. If ( ) y =, then dy d = (A) ( ) ( ) ln (B) ( ) ( ) (D) ( ) ( ) ln (E) ( ) ( ) ln (C) ( ) ( ). The position of a particle moving along a straight line at any time t is given by st () = t + t+. What is the acceleration of the particle when t =? (A) (B) (C) (D) 8 (E). If f ( g) ( ) f ( ) ( ) = ln +, ( ) = ln, and g> ( ) for all real, then g () = (A) + (B) + (C) + (D) + (E) + AP Calculus Multiple-Choice Question Collection 9 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

3 985 AP Calculus AB: Section I. If + y+ y =, then, in terms of and y, dy d = (A) + y (B) + y + y + y (C) + y (D) + y (E) + y + y. The velocity of a particle moving on a line at time t is meters did the particle travel from t = to t =? v= t + 5t meters per second. How many (A) (B) (C) 6 (D) 8 (E) 8 5. The domain of the function defined by f( ) ln( ) = is the set of all real numbers such that (A) < (B) (C) > (D) (E) is a real number 6. The function defined by f ( ) = for all real numbers has a relative maimum at = (A) (B) (C) (D) (E) 7. e d= (A) e (B) (C) e (D) (E) e 8. If y = cos sin, then y = (A) (B) (C) sin( ) (D) cos ( + sin) (E) cos ( sin) 9. If f ( ) f ( ) f ( ) define f? + = + for all real numbers and, which of the following could (A) f ( ) = + (B) f ( ) = (C) f( ) = (D) f ( ) = e (E) f ( ) = AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

4 985 AP Calculus AB: Section I. If y ( ) = arctan cos, then dy d = (A) (D) sin + cos ( arccos ) + ( ) ( ) (B) arcsec( cos ) sin (C) arcsec( cos ) (E) + cos. If the domain of the function f given by f( ) = is { } : >, what is the range of f? (A) { : < < } (B) { : < < } (C) { : < < } (D) { : < < } (E) { :< < }. d = + (A) (B) (C) (D) 5 (E) ln. d at is d + = (A) 6 (B) (C) (D) (E) 6. If ( 7 ) + k d= 6, then k = (A) (B) (C) (D) (E) 5. If f ( ) = e, which of the following is equal to f ()? e (A) e lim h h + h (B) e lim h + h e h e (C) e lim h e+ h e h (D) e lim h + h h (E) e lim h e+ h e h e AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

5 985 AP Calculus AB: Section I 6. The graph of y = + 9 is symmetric to which of the following? I. The -ais II. The y-ais III. The origin (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III 7. d = (A) (B) (C) (D) 5 (E) 6 8. If the position of a particle on the -ais at time t is for t is 5t, then the average velocity of the particle (A) 5 (B) (C) 5 (D) (E) 5 9. Which of the following functions are continuous for all real numbers? I. II. III. y = y = e y = tan (A) None (B) I only (C) II only (D) I and II (E) I and III. tan ( ) d = (A) ln cos( ) + C (B) ln cos( ) + C (C) ln cos( ) + C (D) ln cos( ) + C (E) sec( )tan( ) + C AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

6 985 AP Calculus AB: Section I. The volume of a cone of radius r and height h is given by V = π r h. If the radius and the height both increase at a constant rate of centimeter per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters? (A) π (B) π (C) π (D) 5π (E) 8π. ( ) π sin d = (A) (B) (C) (D) (E). The graph of the derivative of f is shown in the figure above. Which of the following could be the graph of f? AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

7 985 AP Calculus AB: Section I. The area of the region in the first quadrant that is enclosed by the graphs of y = + 8 is y = + 8 and (A) (B) (C) (D) (E) The figure above shows the graph of a sine function for one complete period. Which of the following is an equation for the graph? π (A) y = sin (D) y = sin( π ) (E) y = sin ( ) (B) y = sin ( π ) (C) y = sin( ) 6. If f is a continuous function defined for all real numbers and if the maimum value of f ( ) is 5 and the minimum value of f ( ) is 7, then which of the following must be true? I. The maimum value of f ( ) is 5. II. The maimum value of f ( ) is 7. III. The minimum value of f ( ) is. (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III 7. lim ( csc ) is (A) (B) (C) (D) (E) AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

8 985 AP Calculus AB: Section I 8. Let f and g have continuous first and second derivatives everywhere. If f ( ) g( ) for all real, which of the following must be true? I. f ( ) g ( ) for all real II. f ( ) g ( ) for all real III. f ( d ) gd ( ) (A) None (B) I only (C) III only (D) I and II only (E) I, II, and III 9. If ln f( ) =, for all >, which of the following is true? (A) f is increasing for all greater than. (B) f is increasing for all greater than. (C) f is decreasing for all between and. (D) f is decreasing for all between and e. (E) f is decreasing for all greater than e.. Let f be a continuous function on the closed interval [ ] possible value of f ( d ) is,. If f( ), then the greatest (A) (B) (C) (D) 8 (E) 6. If lim f ( ) = L, where L is a real number, which of the following must be true? a (A) f ( a) eists. (B) f ( ) is continuous at = a. (C) f ( ) is defined at = a. (D) f ( a) (E) = L None of the above AP Calculus Multiple-Choice Question Collection 5 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

9 985 AP Calculus AB: Section I. d d + t dt = (A) + (B) + 5 (C) + (D) + 5 (E) + 5. An equation of the line tangent to y = + + at its point of inflection is (A) y = 6 6 (B) y = + (C) y = + (D) y = (E) y = +. The average value of = + on the closed interval [ ] f( ), is (A) 6 9 (B) (C) 6 (D) (E) 6 5. The region enclosed by the graph of y =, the line =, and the -ais is revolved about the y -ais. The volume of the solid generated is (A) 8π (B) 5 π (C) 6 π (D) π (E) 8 π AP Calculus Multiple-Choice Question Collection 6 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

10 985 AB Let f be the function defined by 5 f( ) =. (a) Find the domain of f. (b) Write an equation for each vertical and each horizontal asymptote for the graph of f. (c) Find f ( ). (d) Write an equation for the line tangent to the graph of f at the point (, f ()). Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com

11 985 AB/BC A particle moves along the -ais with acceleration given by at ( ) = cost for t. At t =, the velocity vt ( ) of the particle is, and the position ( t ) is 5. (a) Write an epression for the velocity vt ( ) of the particle. (b) Write an epression for the position ( t ). (c) For what values of t is the particle moving to the right? Justify your answer. (d) Find the total distance traveled by the particle from t = to π t =. Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com

12 985 AB Let R be the region enclosed by the graphs of y = e, y = e, and = ln. (a) (b) (c) Find the area of R by setting up and evaluating a definite integral. Set up, but do not integrate, an integral epression in terms of a single variable for the volume generated when the region R is revolved about the -ais. Set up, but do not integrate, an integral epression in terms of a single variable for the volume generated when the region R is revolved about the y-ais. Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com

13 985 AB/BC Let f ( ) = π and π ( ) k sin g = k for k >. (a) Find the average value of f on [, ]. (b) For what value of k will the average value of g on [, k ] be equal to the average value of f on [, ]? Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com

14 985 AB5/BC The balloon shown is in the shape of a cylinder with hemispherical ends of the same radius as that of the cylinder. The balloon is being inflated at the rate of 6π cubic centimeters per minute. At the instant the radius of the cylinder is centimeters., the volume of the balloon is π cubic centimeters and the radius of the cylinder is increasing at the rate of centimeters per minute. (The volume of a cylinder is the volume of a sphere is π r ). (a) At this instant, what is the height of the cylinder? πr hand (b) At this instant, how fast is the height of the cylinder increasing? Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com

15 985 AB6 Note: This is the graph of the derivative of f, not the graph of f. The figure above shows the graph of f, the derivative of a function f. The domain of the function f is the set of all such that. (a) For what values of, < <, does f have a relative maimum? A relative minimum? Justify your answer. (b) For what values of is the graph of f concave up? Justify your answer. (c) Use the information found in parts (a) and (b) and the fact that f ( ) = to sketch a possible graph of f on the aes provided below. Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com

16 985 Calculus AB Solutions. D. E. A. C 5. D d = = = 8. f ( ) = (+ ), f () = ( + ), f () = (+ ), () f () =! = 8 y = ( + ) so 6 y = ( + ) ( ) = ( + ) Or using the quotient rule directly gives cos( ) d = cos( )( d) = sin( ) + C n lim = lim = n n + n n + n ( + )() ( ) 6 ( + ) ( + ) y = = 6. C f ( ) = f (5) = 7. E 8. B t dt t = ln = ln ln = ln y = ln = ln ln, y =, y () = 9. D Since e is even, e d= e d= k d. D ( ) ( ) ( ) y = ln() ( ) = ln() d. B vt ( ) = t+ at ( ) = a() = f g( ) = ln g( ) = ln + g( ) = +. C ( ) ( ) ( ) AP Calculus Multiple-Choice Question Collection 8 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

17 985 Calculus AB Solutions. A + y + y + y+ y y = y = + y. D 5 Since vt ( ), distance = v() t dt t 5t dt t t = 8 + = + = 5. C 6. B > > f ( ) = 6= ( ) changes sign from positive to negative only at =. 7. C Use the technique of antiderivatives by parts: u = dv= e d du = d v = e ( ) e + e d= e e = e 8. C y = cos sin = cos, y = sin 9. B Quick solution: lines through the origin have this property. Or, f + f = + = + = f + ( ) ( ) ( ) ( ) dy d sin = = d + cos d + cos. A ( cos ). B. A > > f( ) < for all in the domain. lim f( ) =. option that is consistent with these statements is (B). + ( )( ) d = d = ( ) d = ( ) = + + lim f( ) =. The only d. B ( ) ( ) d + = + + = + = = = 7 7. D ( ) 6 = ( + k) d = d + k d = + ( ) k = k k = AP Calculus Multiple-Choice Question Collection 8 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

18 985 Calculus AB Solutions 5. E e+ h f ( e+ h) f( e) e e f () e = lim = lim h h h h e 6. E I: Replace y with ( y) : ( y) = + 9 y = + 9, no change, so yes. II: Replace with ( ) : y = ( ) + 9 y = + 9, no change, so yes. III: Since there is symmetry with respect to both aes there is origin symmetry. 7. D The graph is a V with verte at =. The integral gives the sum of the areas of the two triangles that the V forms with the horizontal ais for from to. These triangles have areas of / and respectively. 8. C Let () t = 5t be the position at time t. Average velocity () () 5 = = = 5 9. D The tangent function is not defined at = π so it cannot be continuous for all real numbers. Option E is the only one that includes item III. In fact, the functions in I and II are a power and an eponential function that are known to be continuous for all real numbers.. B. C sin( ) tan( ) d = d = ln cos( ) + C cos( ) dv dr dh V = π r h, = π rh + r = π (6)(9) + 6 = π dt dt dt = = π = π π. D sin() d cos() ( cos cos). B f changes sign from positive to negative at = and therefore f changes from increasing to decreasing at =.. A ( ) Or f changes sign from positive to negative at = and from negative to positive at =. Therefore f has a local maimum at = and a local minimum at =. ( ) 8 ( 8) d ( ) d + + = = = AP Calculus Multiple-Choice Question Collection 85 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

19 985 Calculus AB Solutions 5. D The amplitude is and the period is. π π y = Asin B where A = amplitude = and B= = =π period 6. B II is true since 7 = 7 will be the maimum value of f ( ). To see why I and III do not 5 if 5 have to be true, consider the following: f ( ) = if 5 < < 7 7 if 7 For f ( ), the maimum is and the minimum is D lim csc = lim = sin 8. C To see why I and II do not have to be true consider f ( ) = sin and g( ) = + e. Then f ( ) g( ) but neither f ( ) g ( ) nor f ( ) < g ( ) is true for all real values of. III is true, since f( ) g( ) g( ) f( ) ( ) g( ) f ( ) d f ( ) d g( ) d 9. E. D f ( ) = ln = ( ln ) < for > e. Hence f is decreasing. for > e. f ( ) d d = 8. E Consider the function whose graph is the horizontal line y = with a hole at = a. For this function lim f ( ) = and none of the given statements are true. a. C This is a direct application of the Fundamental Theorem of Calculus: f ( ) = +. B y = + 6, y = 6+ 6= for =. y ( ) =. Only option B has a slope of. d d 6 + = + = + = 6 9. A ( ) ( ) ( ) ( ) AP Calculus Multiple-Choice Question Collection 86 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

20 5. A Washers: ( ) π R r y where R=, r = ( ) dy y dy y y Volume =π =π ( ) =π = 8π 985 Calculus AB Solutions AP Calculus Multiple-Choice Question Collection 87 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

21 985 AB Solution (a) The domain of f is all real numbers ecept = and =. (b) (c) Asymptotes Vertical: =, = Horizontal: y = ( ) ( 5) + 8 ( )( ) f ( ) = = = ( ) ( ) ( ) (d) Tangent line at = 5 f () = f () = The equation of the line is 5 y = ( ) or 5 y = + or + y = 5 Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com

22 985 AB/BC Solution (a) vt () = sin() t + C = sin() + C C = vt () = sin() t + (b) () t = cos() t + t+ C 5 = cos() + () + C C = 6 t () = cos() t + t+ 6 (c) The particle moves to the right when vt ( ) >,. i.e. when sin( t ) + >. This is true for all t because sin( t) < + sin( t) + + for all t. (d) The particle never changes directions since it moves to the right for all t. () = cos() + () + 6 = 5 π = cos π + π + 6 =π+ 6 Distance = π () =π+ or π π Distance = vt () dt= sin() t + dt = π π t dt t t (sin( ) + ) = ( cos + ) =π+ Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com

23 985 AB Solution (a) Intersection is when =. ln Area = ( ) = ( ) ln e + e e d e 9 = + (+ ) = (b) Disks: Volume = ln ( π ) or e e d Shells: Volume = π y(ln + ln y) dy + π y(ln ln y) dy (c) Disks: Volume = π ((ln ) ( ln ) ) +π ((ln ) (ln ) ) or = π (ln ) dy π (ln y) dy = 5 π (ln ) (ln y ) dy π y dy y dy Shells: ln Volume = π ( ) e e d Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com

24 985 AB/BC Solution (a) Average value = d π π π = = (6 ) 9 = 98π (b) Average value = Therefore Hence = k k k = 98 π. π k π sin d k k k π k k cos = ( ) = π k π π k = 9π and so k = 7π. Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com

25 985 AB5/BC Solution (a) V =π r h+ π r π=π () h + π () h = At this instant, the height is centimeters. (b) dv dh dr dr =π r + π rh + π r dt dt dt dt dh 6 π=π () + π ()()() + π () () dt dh 5 dt = At this instant, the height is increasing at the rate of 5 centimeters per minute. Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com

26 985 AB6 Solution (a) f has a relative maimum at = because: f changes from positive to negative at = or f changes from increasing to decreasing at = or f ( ) = and f ( ) < f has a relative minimum at = because: f changes from negative to positive at =. or f changes from decreasing to increasing at =. or f () = and f () > (b) f is concave up on (,) and (,) because: f is increasing on those intervals or f > on those intervals (c) Copyright by College Entrance Eamination Board. All rights reserved. Available at apcentral.collegeboard.com