_ap05_sg_calculus_ab__46570

The particle is at position x 8 at time t 0 a find the

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Unformatted text preview: 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...
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