ap05_frq_calculus_ab_b - AP® Calculus AB 2005...

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Unformatted text preview: AP® Calculus AB 2005 Free-Response Questions Form B The College Board: Connecting Students to College Success 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 4,700 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three and a half million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in 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. Copyright © 2005 by College Board. All rights reserved. College Board, AP Central, APCD, Advanced Placement Program, AP, AP Vertical Teams, Pre-AP, SAT, and the acorn logo are registered trademarks of the College Entrance Examination Board. Admitted Class Evaluation Service, CollegeEd, Connect to college success, MyRoad, SAT Professional Development, SAT Readiness Program, and Setting the Cornerstones are trademarks owned by the College Entrance Examination Board. PSAT/NMSQT is a registered trademark of the College Entrance Examination Board and National Merit Scholarship Corporation. Other products and services may be trademarks of their respective owners. Permission to use copyrighted College Board materials may be requested online at: http://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 and Pre-AP: apcentral.collegeboard.com. 2005 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. 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 x-axis. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis 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. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). GO ON TO THE NEXT PAGE. 2 2005 AP® CALCULUS AB FREE-RESPONSE 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. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). GO ON TO THE NEXT PAGE. 3 2005 AP® CALCULUS AB FREE-RESPONSE 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 x-axis 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 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. 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 x-coordinate 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. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). GO ON TO THE NEXT PAGE. 5 2005 AP® CALCULUS AB FREE-RESPONSE 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. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 6 ...
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