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Unformatted text preview: AP® Calculus AB
2007 FreeResponse Questions
Form B The College Board: Connecting Students to College Success
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AP Central is the official online home for the AP Program: apcentral.collegeboard.com. 2007 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. 2 1. Let R be the region bounded by the graph of y = e2 x  x and the horizontal line y = 2, and let S be the region
2 bounded by the graph of y = e2 x  x and the horizontal lines y = 1 and y = 2, as shown above.
(a) Find the area of R.
(b) Find the area of S.
(c) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is
rotated about the horizontal line y = 1. WRITE ALL WORK IN THE EXAM BOOKLET. © 2007 The College Board. All rights reserved.
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2 2007 AP® CALCULUS AB FREERESPONSE QUESTIONS (Form B) () 2. A particle moves along the xaxis so that its velocity v at time t ≥ 0 is given by v(t ) = sin t 2 . The graph of v
is shown above for 0 £ t £ 5p . The position of the particle at time t is x (t ) and its position at time t = 0 is
x (0 ) = 5.
(a) Find the acceleration of the particle at time t = 3.
(b) Find the total distance traveled by the particle from time t = 0 to t = 3.
(c) Find the position of the particle at time t = 3.
(d) For 0 £ t £ 5p , find the time t at which the particle is farthest to the right. Explain your answer. WRITE ALL WORK IN THE EXAM BOOKLET. © 2007 The College Board. All rights reserved.
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3 2007 AP® CALCULUS AB FREERESPONSE QUESTIONS (Form B)
3. The wind chill is the temperature, in degrees Fahrenheit (∞F ) , a human feels based on the air temperature, in
degrees Fahrenheit, and the wind velocity v, in miles per hour (mph ) . If the air temperature is 32∞F, then the
wind chill is given by W (v ) = 55.6  22.1v 0.16 and is valid for 5 £ v £ 60.
(a) Find W ¢(20 ) . Using correct units, explain the meaning of W ¢(20 ) in terms of the wind chill.
(b) Find the average rate of change of W over the interval 5 £ v £ 60. Find the value of v at which the
instantaneous rate of change of W is equal to the average rate of change of W over the interval 5 £ v £ 60.
(c) Over the time interval 0 £ t £ 4 hours, the air temperature is a constant 32∞F. At time t = 0, the wind
velocity is v = 20 mph. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate
of change of the wind chill with respect to time at t = 3 hours? Indicate units of measure. WRITE ALL WORK IN THE EXAM BOOKLET. END OF PART A OF SECTION II © 2007 The College Board. All rights reserved.
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CALCULUS AB
SECTION II, Part B
Time—45 minutes
Number of problems—3
No calculator is allowed for these problems. 4. Let f be a function defined on the closed interval 5 £ x £ 5 with f (1) = 3. The graph of f ¢, the derivative
of f, consists of two semicircles and two line segments, as shown above.
(a) For 5 < x < 5, find all values x at which f has a relative maximum. Justify your answer.
(b) For 5 < x < 5, find all values x at which the graph of f has a point of inflection. Justify your answer.
(c) Find all intervals on which the graph of f is concave up and also has positive slope. Explain your reasoning.
(d) Find the absolute minimum value of f ( x ) over the closed interval 5 £ x £ 5. Explain your reasoning. WRITE ALL WORK IN THE EXAM BOOKLET. © 2007 The College Board. All rights reserved.
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5 2007 AP® CALCULUS AB FREERESPONSE QUESTIONS (Form B) 5. Consider the differential equation dy
1
= x + y  1.
dx
2 (a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.
(Note: Use the axes provided in the exam booklet.) (b) Find d2 y in terms of x and y. Describe the region in the xyplane in which all solution curves to the
dx 2
differential equation are concave up. (c) Let y = f ( x ) be a particular solution to the differential equation with the initial condition f (0 ) = 1. Does f
have a relative minimum, a relative maximum, or neither at x = 0 ? Justify your answer.
(d) Find the values of the constants m and b, for which y = mx + b is a solution to the differential equation. 6. Let f be a twicedifferentiable function such that f (2 ) = 5 and f (5) = 2. Let g be the function given by
g ( x ) = f ( f ( x )) .
(a) Explain why there must be a value c for 2 < c < 5 such that f ¢(c ) = 1.
(b) Show that g ¢(2 ) = g ¢(5) . Use this result to explain why there must be a value k for 2 < k < 5 such that
g ¢¢( k ) = 0.
(c) Show that if f ¢¢( x ) = 0 for all x, then the graph of g does not have a point of inflection.
(d) Let h( x ) = f ( x )  x. Explain why there must be a value r for 2 < r < 5 such that h(r ) = 0. WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM © 2007 The College Board. All rights reserved.
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This note was uploaded on 12/15/2009 for the course SOCIAL STU 129348437 taught by Professor Phalange during the Spring '09 term at Aberystwyth University.
 Spring '09
 Phalange

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