_ap05_sg_calculus_ab__46570

_ap05_sg_calculus_ab__46570 - AP® Calculus AB 2005 Scoring...

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Unformatted text preview: AP® Calculus AB 2005 Scoring Guidelines 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. AP® CALCULUS AB 2005 SCORING GUIDELINES (Form B) Question 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 semicircles with diameters extending from y = f ( x ) to y = g ( x ) . Find the volume of this solid. The graphs of f and g intersect in the first quadrant at ( S , T ) = (1.13569, 1.76446 ) . S ∫0 ( f ( x ) − g ( x ) ) dx S = ∫ (1 + sin ( 2 x ) − e x 2 ) dx 0 (a) Area = 1 : correct limits in an integral in (a), (b), or (c) ⎧ 1 : integrand 2: ⎨ ⎩ 1 : answer = 0.429 (b) Volume = π ∫0 ( ( f ( x ) ) S ⎮ = π⌠ S ⌡0 2 ) − ( g ( x ) )2 dx ((1 + sin ( 2x )) 2 ( − ex ) 22 ⎧ 2 : integrand ⎪ −1 each error ⎪ ⎪ Note: 0 2 if integral not of form 3: ⎨ b ⎪ c ∫ R 2 ( x ) − r 2 ( x ) dx a ⎪ ⎪ 1 : answer ⎩ ) dx ( = 4.266 or 4.267 ) S 2 ⌠ π ⎛ f ( x) − g( x) ⎞ (c) Volume = ⎮ ⎜ ⎟ dx 2 ⎠ ⌡0 2 ⎝ S ⎧ 2 : integrand 3: ⎨ ⎩ 1 : answer 2 ⌠ π ⎛ 1 + sin ( 2 x ) − e x 2 ⎞ =⎮ ⎟ dx 2⎜ 2 ⎠ ⌡0 ⎝ = 0.077 or 0.078 Copyright © 2005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 2 AP® CALCULUS AB 2005 SCORING GUIDELINES (Form B) Question 2 A water tank at Camp Newton holds 1200 gallons of water at time t = 0. During the time interval 0 ≤ t ≤ 18 hours, water is pumped into the tank at the rate W ( t ) = 95 t sin 2 ( 6t ) gallons per hour. During the same time interval, water is removed from the tank at the rate R( t ) = 275sin 2 ( 3t ) gallons per hour. (a) Is the amount of water in the tank increasing at time t = 15 ? Why or why not? (b) To the nearest whole number, how many gallons of water are in the tank at time t = 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. (a) No; the amount of water is not increasing at t = 15 since W (15 ) − R (15 ) = −121.09 < 0. (b) 1200 + 18 ∫0 (W ( t ) − R( t ) ) dt = 1309.788 ⎧ 1 : limits ⎪ 3 : ⎨ 1 : integrand ⎪ 1 : answer ⎩ 1310 gallons (c) W ( t ) − R( t ) = 0 t = 0, 6.4948, 12.9748 ⎧ 1 : interior critical points ⎪ 1 : amount of water is least at ⎪ 3: ⎨ t = 6.494 or 6.495 ⎪ ⎪ ⎩ 1 : analysis for absolute minimum t (hours) gallons of water 0 6.495 12.975 18 1 : answer with reason 1200 525 1697 1310 The values at the endpoints and the critical points show that the absolute minimum occurs when t = 6.494 or 6.495. (d) k ∫18 R( t ) dt = 1310 ⎧ 1 : limits 2: ⎨ ⎩ 1 : equation Copyright © 2005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 3 AP® CALCULUS AB 2005 SCORING GUIDELINES (Form B) Question 3 A particle moves along the x-axis so that its velocity v at time t, for 0 ≤ t ≤ 5, is given by ( ) v( t ) = ln t 2 − 3t + 3 . The particle is at position x = 8 at time t = 0. (a) Find the acceleration of the particle at time t = 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? (c) Find the position of the particle at time t = 2. (d) Find the average speed of the particle over the interval 0 ≤ t ≤ 2. (a) 5 a( 4 ) = v′( 4 ) = 7 (b) v( t ) = 0 1 : answer ⎧ 1 : sets v( t ) = 0 ⎪ 3 : ⎨ 1 : direction change at t = 1, 2 ⎪ 1 : interval with reason ⎩ 2 t − 3t + 3 = 1 t 2 − 3t + 2 = 0 ( t − 2 ) ( t −1) = 0 t = 1, 2 v( t ) > 0 for 0 < t < 1 v( t ) < 0 for 1 < t < 2 v( t ) > 0 for 2 < t < 5 The particle changes direction when t = 1 and t = 2. The particle travels to the left when 1 < t < 2. (c) ( ∫0 ln ( u − 3u + 3) du 2 s ( 2 ) = 8 + ∫ ln ( u 2 − 3u + 3) du 0 s( t ) = s( 0 ) + t = 8.368 or 8.369 (d) ) ⎧ 1 : 2 ln u 2 − 3u + 3 du ∫0 ⎪ ⎪ 3: ⎨ 1 : handles initial condition ⎪ ⎪ ⎩ 1 : answer 2 12 v( t ) dt = 0.370 or 0.371 2 ∫0 ⎧ 1 : integral 2: ⎨ ⎩ 1 : answer Copyright © 2005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 4 AP® CALCULUS AB 2005 SCORING GUIDELINES (Form B) Question 4 The graph of the function f above consists of three line segments. (a) Let g be the function given by g ( x ) = x ∫− 4 f ( t ) dt. For each of g ( −1) , g ′( −1) , and g ′′( −1) , find the value or state that it does not exist. (b) For the function g defined in part (a), find the x-coordinate of each point of inflection of the graph of g on the open interval − 4 < x < 3. Explain your reasoning. (c) Let h be the function given by h( x ) = 3 ∫ x f ( t ) dt. Find all values of x in the closed interval − 4 ≤ x ≤ 3 for which h( x ) = 0. (d) For the function h defined in part (c), find all intervals on which h is decreasing. Explain your reasoning. (a) g ( −1) = −1 1 15 ∫− 4 f ( t ) dt = − 2 ( 3)( 5) = − 2 g ′( −1) = f ( −1) = −2 g ′′( −1) does not exist because f is not differentiable at x = −1. ⎧ 1 : g ( −1) ⎪ 3 : ⎨ 1 : g ′( −1) ⎪ 1 : g ′′( −1) ⎩ (b) x =1 g ′ = f changes from increasing to decreasing at x = 1. ⎧ 1 : x = 1 (only) 2: ⎨ ⎩ 1 : reason (c) x = −1, 1, 3 2 : correct values −1 each missing or extra value (d) h is decreasing on [ 0, 2] h′ = − f < 0 when f > 0 ⎧ 1 : interval 2: ⎨ ⎩ 1 : reason Copyright © 2005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 5 AP® CALCULUS AB 2005 SCORING GUIDELINES (Form B) Question 5 Consider the curve given by y 2 = 2 + xy. (a) Show that dy y = . dx 2 y − x (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. (d) Let x and y be functions of time t that are related by the equation y 2 = 2 + xy. At time t = 5, the dy dx = 6. Find the value of at time t = 5. value of y is 3 and dt dt (a) (b) 2 y y′ = y + x y′ ( 2 y − x ) y′ = y y y′ = 2y − x ⎧ 1 : implicit differentiation 2: ⎨ ⎩ 1 : solves for y ′ y 1 = 2y − x 2 2y = 2y − x x=0 y=± 2 y 1 ⎧1: = ⎪ 2y − x 2 2: ⎨ ⎪ 1 : answer ⎩ y =0 2y − x y=0 The curve has no horizontal tangent since ⎧1: y = 0 2: ⎨ ⎩ 1 : explanation ( 0, (c) 2 ) , ( 0, − 2 ) 02 ≠ 2 + x ⋅ 0 for any x. (d) When y = 3, 32 = 2 + 3 x so x = 7 . 3 ⎧ 1 : solves for x ⎪ 3 : ⎨ 1 : chain rule ⎪ 1 : answer ⎩ dy dy dx y dx = ⋅ = ⋅ 2 y − x dt dt dx dt 3 9 dx dx At t = 5, 6 = ⋅ = ⋅ 7 dt 11 dt 6− 3 dx 22 = dt t = 5 3 Copyright © 2005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 6 AP® CALCULUS AB 2005 SCORING GUIDELINES (Form B) Question 6 dy − xy 2 . Let = dx 2 y = f ( x ) be the particular solution to this differential equation with the initial condition f ( −1) = 2. 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.) (b) Write an equation for the line tangent to the graph of f at x = −1. (c) Find the solution y = f ( x ) to the given differential equation with the initial condition f ( −1) = 2. (a) ⎧ 1 : zero slopes 2: ⎨ ⎩ 1 : nonzero slopes − ( −1) 4 =2 2 y − 2 = 2 ( x + 1) (b) Slope = (c) 1 : equation 1 x dy = − dx 2 2 y ⎧ 1 : separates variables ⎪ 2 : antiderivatives ⎪ 6 : ⎨ 1 : constant of integration ⎪ 1 : uses initial condition ⎪ ⎪ ⎩ 1 : solves for y 1 x2 =− +C 4 y 1 1 1 − = − + C; C = − 2 4 4 1 4 y= 2 =2 1 x x +1 + 4 4 − Note: max 3 6 [1-2-0-0-0] if no constant of integration Note: 0 6 if no separation of variables Copyright © 2005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 7 ...
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This note was uploaded on 10/01/2009 for the course OC 9876 taught by Professor Dq during the Spring '09 term at UC Merced.

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