2 Q - AP® Calculus AB 2009 Free-Response Questions Form B...

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Unformatted text preview: AP® Calculus AB 2009 Free-Response Questions Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 5,600 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools and 3,800 colleges through major programs and services in college readiness, college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT® and the Advanced Placement Program® (AP®). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities and concerns. © 2009 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. Permission to use copyrighted College Board materials may be requested online at: www.collegeboard.com/inquiry/cbpermit.html. Visit the College Board on the Web: www.collegeboard.com. AP Central is the official online home for the AP Program: apcentral.collegeboard.com. 2009 AP® CALCULUS AB FREE-RESPONSE 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. At a certain height, a tree trunk has a circular cross section. The radius R(t ) of that cross section grows at a rate modeled by the function dR 1 = 3 + sin t 2 dt 16 ( ( )) centimeters per year for 0 £ t £ 3, where time t is measured in years. At time t = 0, the radius is 6 centimeters. The area of the cross section at time t is denoted by A(t ) . (a) Write an expression, involving an integral, for the radius R(t ) for 0 £ t £ 3. Use your expression to find R(3) . (b) Find the rate at which the cross-sectional area A(t ) is increasing at time t = 3 years. Indicate units of measure. (c) Evaluate Ú0 A¢(t ) dt. Using appropriate units, interpret the meaning of that integral in terms of cross- 3 sectional area. WRITE ALL WORK IN THE EXAM BOOKLET. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -2- 2009 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) 2. A storm washed away sand from a beach, causing the edge of the water to get closer to a nearby road. The rate at which the distance between the road and the edge of the water was changing during the storm is modeled by f (t ) = t + cos t - 3 meters per hour, t hours after the storm began. The edge of the water was 35 meters from 1 - sin t. the road when the storm began, and the storm lasted 5 hours. The derivative of f (t ) is f ¢(t ) = 2t (a) What was the distance between the road and the edge of the water at the end of the storm? (b) Using correct units, interpret the value f ¢( 4 ) = 1.007 in terms of the distance between the road and the edge of the water. (c) At what time during the 5 hours of the storm was the distance between the road and the edge of the water decreasing most rapidly? Justify your answer. (d) After the storm, a machine pumped sand back onto the beach so that the distance between the road and the edge of the water was growing at a rate of g( p ) meters per day, where p is the number of days since pumping began. Write an equation involving an integral expression whose solution would give the number of days that sand must be pumped to restore the original distance between the road and the edge of the water. WRITE ALL WORK IN THE EXAM BOOKLET. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -3- 2009 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) 3. A continuous function f is defined on the closed interval - 4 £ x £ 6. The graph of f consists of a line segment and a curve that is tangent to the x-axis at x = 3, as shown in the figure above. On the interval 0 < x < 6, the function f is twice differentiable, with f ¢¢( x ) > 0. (a) Is f differentiable at x = 0 ? Use the definition of the derivative with one-sided limits to justify your answer. (b) For how many values of a, - 4 £ a < 6, is the average rate of change of f on the interval [a, 6] equal to 0 ? Give a reason for your answer. (c) Is there a value of a, - 4 £ a < 6, for which the Mean Value Theorem, applied to the interval [a, 6], 1 guarantees a value c, a < c < 6, at which f ¢(c ) = ? Justify your answer. 3 (d) The function g is defined by g( x ) = Ú0 f (t ) dt for - 4 £ x £ 6. On what intervals contained in [- 4, 6] x is the graph of g concave up? Explain your reasoning. WRITE ALL WORK IN THE EXAM BOOKLET. END OF PART A OF SECTION II © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. -4- 2009 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) CALCULUS AB SECTION II, Part B Time— 45 minutes Number of problems— 3 No calculator is allowed for these problems. 4. Let R be the region bounded by the graphs of y = x and y = (a) Find the area of R. x , as shown in the figure above. 2 (b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares. Find the volume of this solid. (c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line y = 2. WRITE ALL WORK IN THE EXAM BOOKLET. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -5- 2009 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) 5. Let f be a twice-differentiable function defined on the interval -1.2 < x < 3.2 with f (1) = 2. The graph of f ¢, the derivative of f, is shown above. The graph of f ¢ crosses the x-axis at x = -1 and x = 3 and has a horizontal tangent at x = 2. Let g be the function given by g( x ) = e f ( x ) . (a) Write an equation for the line tangent to the graph of g at x = 1. (b) For -1.2 < x < 3.2, find all values of x at which g has a local maximum. Justify your answer. 2 (c) The second derivative of g is g ¢¢( x ) = e f ( x ) È( f ¢( x )) + f ¢¢( x )˘ . Is g ¢¢( -1) positive, negative, or zero? Î ˚ Justify your answer. (d) Find the average rate of change of g ¢, the derivative of g, over the interval [1, 3]. WRITE ALL WORK IN THE EXAM BOOKLET. © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -6- 2009 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) t (seconds) 0 3 8 5 20 –10 25 –8 32 –4 40 7 v (t ) (meters per second) 6. The velocity of a particle moving along the x-axis is modeled by a differentiable function v, where the position x is measured in meters, and time t is measured in seconds. Selected values of v(t ) are given in the table above. The particle is at position x = 7 meters when t = 0 seconds. (a) Estimate the acceleration of the particle at t = 36 seconds. Show the computations that lead to your answer. Indicate units of measure. (b) Using correct units, explain the meaning of Ú20 v(t ) dt 40 in the context of this problem. Use a trapezoidal sum with the three subintervals indicated by the data in the table to approximate Ú20 v(t ) dt. 40 (c) For 0 £ t £ 40, must the particle change direction in any of the subintervals indicated by the data in the table? If so, identify the subintervals and explain your reasoning. If not, explain why not. (d) Suppose that the acceleration of the particle is positive for 0 < t < 8 seconds. Explain why the position of the particle at t = 8 seconds must be greater than x = 30 meters. WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM © 2009 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com. -7- ...
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