_ap06_calcab_sg

_ap06_calcab_sg - AP® Calculus AB 2006 Scoring Guidelines...

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Unformatted text preview: AP® Calculus AB 2006 Scoring Guidelines 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 5,000 schools, colleges, universities, and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,500 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. © 2006 The 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 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 Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. 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. AP® CALCULUS AB 2006 SCORING GUIDELINES Question 1 Let R be the shaded region bounded by the graph of y = ln x and the line y = x − 2, as shown above. (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the horizontal line y = −3. (c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis. ln ( x ) = x − 2 when x = 0.15859 and 3.14619. Let S = 0.15859 and T = 3.14619 (a) Area of R = T ∫ S ( ln ( x ) − ( x − 2 ) ) dx = 1.949 (b) Volume = π ∫ T S ⎧ 1 : integrand ⎪ 3 : ⎨ 1 : limits ⎪ 1 : answer ⎩ ( ( ln ( x ) + 3)2 − ( x − 2 + 3)2 ) dx 3: { 2 : integrand 1 : limits, constant, and answer 3: { 2 : integrand 1 : limits and constant = 34.198 or 34.199 ⎮ (c) Volume = π ⌠ T −2 ⌡S − 2 ( ( y + 2) 2 () − ey 2 ) dy © 2006 The 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 2006 SCORING GUIDELINES Question 2 At an intersection in Thomasville, Oregon, cars turn t cars per hour left at the rate L( t ) = 60 t sin 2 3 over the time interval 0 ≤ t ≤ 18 hours. The graph of y = L( t ) is shown above. (a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval 0 ≤ t ≤ 18 hours. (b) Traffic engineers will consider turn restrictions when L( t ) ≥ 150 cars per hour. Find all values of t for which L( t ) ≥ 150 and compute the average value of L over this time interval. Indicate units of measure. (c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion. () 18 (a) ∫0 (b) L ( t ) = 150 when t = 12.42831, 16.12166 Let R = 12.42831 and S = 16.12166 L ( t ) ≥ 150 for t in the interval [ R, S ] S 1 ∫ R L ( t ) dt = 199.426 cars per hour S−R L ( t ) dt ≈ 1658 cars 2: { 1 : setup 1 : answer ⎧ 1 : t -interval when L ( t ) ≥ 150 ⎪ 3 : ⎨ 1 : average value integral ⎪ 1 : answer with units ⎩ (c) For the product to exceed 200,000, the number of cars turning left in a two-hour interval must be greater than 400. 15 ∫13 L ( t ) dt = 431.931 > 400 ⎧ 1 : considers 400 cars ⎪ 1 : valid interval [ h, h + 2] ⎪ h+2 4: ⎨ ⎪ 1 : value of ∫ h L ( t ) dt ⎪ ⎩ 1 : answer and explanation OR OR The number of cars turning left will be greater than 400 on a two-hour interval if L( t ) ≥ 200 on that interval. L( t ) ≥ 200 on any two-hour subinterval of [13.25304, 15.32386]. ⎧ 1 : considers 200 cars per hour ⎪ 1 : solves L( t ) ≥ 200 ⎪ 4: ⎨ ⎪ 1 : discusses 2 hour interval ⎪ 1 : answer and explanation ⎩ Yes, a traffic signal is required. © 2006 The 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 2006 SCORING GUIDELINES Question 3 The graph of the function f shown above consists of six line segments. Let g be the function given by g( x) = x ∫0 f ( t ) dt. (a) Find g ( 4 ) , g ′( 4 ) , and g ′′( 4 ) . (b) Does g have a relative minimum, a relative maximum, or neither at x = 1 ? Justify your answer. (c) Suppose that f is defined for all real numbers x and is periodic with a period of length 5. The graph above shows two periods of f. Given that g ( 5 ) = 2, find g (10 ) and write an equation for the line tangent to the graph of g at x = 108. (a) g ( 4) = 4 ∫0 f ( t ) dt = 3 ⎧ 1 : g ( 4) ⎪ 3 : ⎨ 1 : g ′( 4 ) ⎪ 1 : g ′′( 4 ) ⎩ g ′( 4 ) = f ( 4 ) = 0 g ′′( 4 ) = f ′( 4 ) = −2 (b) g has a relative minimum at x = 1 because g ′ = f changes from negative to positive at x = 1. (c) g ( 0 ) = 0 and the function values of g increase by 2 for every increase of 5 in x. g (10 ) = 2 g ( 5 ) = 4 g (108 ) = 105 ∫0 f ( t ) dt + 108 ∫105 f ( t ) dt 2: { 1 : answer 1 : reason ⎧ 1 : g (10 ) ⎪ ⎪ 4: ⎨ ⎧ 1 : g (108 ) ⎪ 3 : ⎪ 1 : g ′(108 ) ⎨ ⎪ ⎪ ⎩ ⎩ 1 : equation of tangent line = 21g ( 5 ) + g ( 3) = 44 g ′(108 ) = f (108 ) = f ( 3) = 2 An equation for the line tangent to the graph of g at x = 108 is y − 44 = 2 ( x − 108 ) . © 2006 The 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 2006 SCORING GUIDELINES Question 4 t (seconds) 0 10 20 30 40 50 60 70 80 v( t ) (feet per second) 5 14 22 29 35 40 44 47 49 Rocket A has positive velocity v( t ) after being launched upward from an initial height of 0 feet at time t = 0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 ≤ t ≤ 80 seconds, as shown in the table above. (a) Find the average acceleration of rocket A over the time interval 0 ≤ t ≤ 80 seconds. Indicate units of measure. (b) Using correct units, explain the meaning of 70 ∫10 v( t ) dt in terms of the rocket’s flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate 70 ∫10 v( t ) dt. 3 feet per second per second. At time t +1 t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time t = 80 seconds? Explain your answer. (c) Rocket B is launched upward with an acceleration of a( t ) = (a) Average acceleration of rocket A is 1 : answer v( 80 ) − v( 0 ) 49 − 5 11 = = ft sec 2 80 − 0 80 20 (b) Since the velocity is positive, 70 ∫10 v( t ) dt represents the distance, in feet, traveled by rocket A from t = 10 seconds to t = 70 seconds. ⎧ 1 : explanation ⎪ 3 : ⎨ 1 : uses v( 20 ) , v( 40 ) , v( 60 ) ⎪ 1 : value ⎩ A midpoint Riemann sum is 20 [ v( 20 ) + v( 40 ) + v( 60 )] = 20 [ 22 + 35 + 44] = 2020 ft (c) Let vB ( t ) be the velocity of rocket B at time t. ⌠ 3 dt = 6 t + 1 + C vB ( t ) = ⎮ ⌡ t +1 2 = vB ( 0 ) = 6 + C ⎧ 1: 6 t +1 ⎪ 1 : constant of integration ⎪ ⎪ 4 : ⎨ 1 : uses initial condition ⎪ 1 : finds v ( 80 ) , compares to v( 80 ) , B ⎪ and draws a conclusion ⎪ ⎩ vB ( t ) = 6 t + 1 − 4 vB ( 80 ) = 50 > 49 = v( 80 ) Rocket B is traveling faster at time t = 80 seconds. 1 : units in (a) and (b) Units of ft sec2 in (a) and ft in (b) © 2006 The 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 2006 SCORING GUIDELINES Question 5 dy 1 + y = , where x ≠ 0. dx x (a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated. (Note: Use the axes provided in the pink exam booklet.) Consider the differential equation (b) Find the particular solution y = f ( x ) to the differential equation with the initial condition f ( −1) = 1 and state its domain. (a) (b) 2 : sign of slope at each point and relative steepness of slope lines in rows and columns 1 1 dy = dx 1+ y x ⎧ 1 : separates variables ⎧ ⎪ 2 : antiderivatives ⎪ ⎪ ⎪ ⎪ 6 : ⎨ 1 : constant of integration ⎪ 1 : uses initial condition ⎪ ⎪ ⎪ ⎪ ⎩ 1 : solves for y 7: ⎨ Note: max 3 6 [1-2-0-0-0] if no ⎪ ⎪ constant of integration ⎪ ⎪ Note: 0 6 if no separation of variables ⎪ ⎪ 1 : domain ⎩ ln 1 + y = ln x + K 1 + y = eln x + K 1+ y = C x 2=C 1+ y = 2 x y = 2 x − 1 and x < 0 or y = −2 x − 1 and x < 0 © 2006 The 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 2006 SCORING GUIDELINES Question 6 The twice-differentiable function f is defined for all real numbers and satisfies the following conditions: f ( 0) = 2, f ′( 0) = − 4, and f ′′ ( 0) = 3. (a) The function g is given by g ( x ) = eax + f ( x ) for all real numbers, where a is a constant. Find g ′ ( 0) and g ′′ ( 0) in terms of a. Show the work that leads to your answers. (b) The function h is given by h( x ) = cos ( kx ) f ( x ) for all real numbers, where k is a constant. Find h′ ( x ) and write an equation for the line tangent to the graph of h at x = 0. (a) g ′( x ) = aeax + f ′( x ) g ′( 0 ) = a − 4 ⎧ 1 : g ′( x ) ⎪ 1 : g ′( 0 ) ⎪ 4: ⎨ ⎪ 1 : g ′′( x ) ⎪ 1 : g ′′( 0 ) ⎩ g ′′( x ) = a 2 eax + f ′′( x ) g ′′( 0 ) = a 2 + 3 (b) h′( x ) = f ′( x ) cos ( kx ) − k sin ( kx ) f ( x ) h′( 0 ) = f ′( 0 ) cos ( 0 ) − k sin ( 0 ) f ( 0 ) = f ′( 0 ) = − 4 h( 0 ) = cos ( 0 ) f ( 0 ) = 2 The equation of the tangent line is y = − 4 x + 2. ⎧ 2 : h′( x ) ⎪ ⎪ ⎧ 1 : h′( 0 ) 5: ⎨ ⎪ ⎪ 3 : ⎨ 1 : h( 0 ) ⎪ ⎪ ⎩ ⎩ 1 : equation of tangent line © 2006 The 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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