Unformatted text preview: AP® Calculus AB
2005 FreeResponse Questions
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AP Central is the official online home for the AP Program and PreAP: apcentral.collegeboard.com. 2005 AP® CALCULUS AB FREERESPONSE QUESTIONS (Form B)
CALCULUS AB
SECTION II, Part A
Time—45 minutes
Number of problems—3
A graphing calculator is required for some problems or parts of problems. 1. Let f and g be the functions given by f x
1 sin 2 x and g x
e x 2 . Let R be the shaded region in the
first quadrant enclosed by the graphs of f and g as shown in the figure above. = )( ) ( += )( (a) Find the area of R.
(b) Find the volume of the solid generated when R is revolved about the xaxis.
(c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the xaxis are
f x to y g x . Find the volume of this solid.
semicircles with diameters extending from y
)( = )( = WRITE ALL WORK IN THE TEST BOOKLET. Copyright © 2005 by College Entrance Examination Board. All rights reserved.
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2 2005 AP® CALCULUS AB FREERESPONSE QUESTIONS (Form B)
t 18 t
gallons per hour.
6
) ( 95 t sin 2 = = Wt 0. During the time interval 0 ££ 2. A water tank at Camp Newton holds 1200 gallons of water at time t
hours, water is pumped into the tank at the rate
)( During the same time interval, water is removed from the tank at the rate
t
gallons per hour.
3 15 ? Why or why not? )( (b) To the nearest whole number, how many gallons of water are in the tank at time t = = (a) Is the amount of water in the tank increasing at time t = ( 275sin 2 ) Rt 18 ? (c) At what time t, for 0 t 18, is the amount of water in the tank at an absolute minimum? Show the work
that leads to your conclusion. ££ (d) For t 18, no water is pumped into the tank, but water continues to be removed at the rate R t until the
tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an
equation involving an integral expression that can be used to find the value of k. > )( WRITE ALL WORK IN THE TEST BOOKLET. Copyright © 2005 by College Entrance Examination Board. All rights reserved.
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3 2005 AP® CALCULUS AB FREERESPONSE QUESTIONS (Form B) 3 . The particle is at position x
)  ( = )( (a) Find the acceleration of the particle at time t = + 3t = ln t 2 8 at time t = vt 0. t ££ 3. A particle moves along the xaxis so that its velocity v at time t, for 0 5, is given by 4. (b) Find all times t in the open interval 0 t 5 at which the particle changes direction. During which time
intervals, for 0 t 5, does the particle travel to the left? << 2. (d) Find the average speed of the particle over the interval 0 t ££ = ££ (c) Find the position of the particle at time t 2. WRITE ALL WORK IN THE TEST BOOKLET. END OF PART A OF SECTION II Copyright © 2005 by College Entrance Examination Board. All rights reserved.
Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 4 2005 AP® CALCULUS AB FREERESPONSE QUESTIONS (Form B)
CALCULUS AB
SECTION II, Part B
Time—45 minutes
Number of problems—3
No calculator is allowed for these problems. 4. The graph of the function f above consists of three line segments.
1 , and g  (¢ f t dt. For each of g 1 , g ) ) ( )( Ú = )( or state that it does not exist. 4 1 , find the value  (¢¢ x ) (a) Let g be the function given by g x (b) For the function g defined in part (a), find the xcoordinate of each point of inflection of the graph of g on
3. Explain your reasoning. = )( 0. Ú = which h x 3
x f t dt. Find all values of x in the closed interval
)( < (c) Let h be the function given by h x 4 £ x x £ 4 < the open interval 3 for )( (d) For the function h defined in part (c), find all intervals on which h is decreasing. Explain your reasoning. WRITE ALL WORK IN THE TEST BOOKLET. Copyright © 2005 by College Entrance Examination Board. All rights reserved.
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5 2005 AP® CALCULUS AB FREERESPONSE QUESTIONS (Form B) y
2y  = dy
dx x = (a) Show that 2 xy. + 5. Consider the curve given by y 2
. (b) Find all points x, y on the curve where the line tangent to the curve has slope 1
.
2 ) ( (c) Show that there are no points x, y on the curve where the line tangent to the curve is horizontal.
) + 2 xy. At time t = = ( (d) Let x and y be functions of time t that are related by the equation y 2
dy
dx
of y is 3 and
6. Find the value of
at time t 5.
dt
dt 5, the value = = = xy 2
. Let y
2
2. f x be the particular solution to this differential
)(  = dy
dx
equation with the initial condition f 1 = ) 6. Consider the differential equation ( (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.
(Note: Use the axes provided in the test booklet.) 1 ) ( f x to the given differential equation with the initial condition f = )( = (c) Find the solution y 1. = (b) Write an equation for the line tangent to the graph of f at x 2. WRITE ALL WORK IN THE TEST BOOKLET.
END OF EXAM Copyright © 2005 by College Entrance Examination Board. All rights reserved.
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 Spring '10
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 Calculus, Derivative, College Entrance Examination Board, college entrance examination, Entrance Examination Board, AP® CALCULUS AB

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