AP CALCULUS AB/CALCULUS BC 2015 SCORING GUIDELINES

Size: px

Start display at page:

Download "AP CALCULUS AB/CALCULUS BC 2015 SCORING GUIDELINES"

Clifton Conley
5 years ago
Views:

1 AP CALCULUS AB/CALCULUS BC 15 SCORING GUIDELINES Question 3 t (minutes) vt ( ) (meters per minute) Johanna jogs along a straight path. For t 4, Johanna s velocity is given by a differentiable function v. Selected values of vt ( ), where t is measured in minutes and vt ( ) is measured in meters per minute, are given in the table above. (a) Use the data in the table to estimate the value of v ( 16 ). (b) Using correct units, explain the meaning of the definite integral v( t) dt in the context of the problem. 4 Approximate the value of v( t) dt using a right Riemann sum with the four subintervals indicated in the table. (c) Bob is riding his bicycle along the same path. For t 1, Bob s velocity is modeled by 3 Bt ( ) = t 6t + 3, where t is measured in minutes and Bt ( ) is measured in meters per minute. Find Bob s acceleration at time t = 5. (d) Based on the model B from part (c), find Bob s average velocity during the interval t 1. 4 (a) v ( 16) = 5 meters/min 1 : approximation (b) v( t) dt is the total distance Johanna jogs, in meters, over the time interval t 4 minutes. 4 v( t) dt 1 v( 1) + 8 v( ) + 4 v( 4) + 16 v( 4) = = = 76 meters 1 : explanation 3 : 1 : right Riemann sum 1 : approximation (c) Bob s acceleration is B ( t) = 3t 1 t. B ( 5) = 3( 5) 1( 5) = 15 meters/min (d) Avg ve ( 6 + 3) 15 The College Board. Visit the College Board on the Web: 1 : uses B ( t) : 1 : answer l = 1 t t dt 1 : integral : 1 : antiderivative 1 t 3 = t + 3t : answer 1 1 = + 3 = 35 meters/ min 1 4

5 AP CALCULUS AB 6 SCORING GUIDELINES Question 4 t (seconds) vt () (feet per second) Rocket A has positive velocity vt () after being launched upward from an initial height of feet at time t = seconds. The velocity of the rocket is recorded for selected values of t over the interval t 8 seconds, as shown in the table above. (a) Find the average acceleration of rocket A over the time interval t 8 seconds. Indicate units of measure. 7 (b) Using correct units, explain the meaning of vt () dtin terms of the rocket s flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate vt () dt. 1 3 (c) Rocket B is launched upward with an acceleration of at () = feet per second per second. At time t + 1 t = seconds, the initial height of the rocket is feet, and the initial velocity is feet per second. Which of the two rockets is traveling faster at time t = 8 seconds? Explain your answer. 7 1 (a) Average acceleration of rocket A is 1 : answer v( 8) v( ) ft sec = = 8 8 (b) Since the velocity is positive, vt () dtrepresents the distance, in feet, traveled by rocket A from t = 1 seconds to t = 7 seconds : explanation 3 : 1 : uses v( ), v( 4 ), v( 6) 1 : value A midpoint Riemann sum is [ v( ) + v( 4) + v( 6) ] = [ ] = ft (c) Let vb () t be the velocity of rocket B at time t. 3 vb () t = dt = 6 t C t + 1 = v ( ) = 6 + C B vb () t = 6 t v ( 8) = 5 > 49 = v( 8) B 4 : 1 : 6 t : constant of integration 1 : uses initial condition 1 : finds vb ( 8 ), compares to v( 8 ), and draws a conclusion Rocket B is traveling faster at time t = 8 seconds. Units of ft sec in (a) and ft in (b) 1 : units in (a) and (b) 6 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 5 49

6 AP CALCULUS AB 6 SCORING GUIDELINES (Form B) Question 6 t (sec) vt () ( ft sec ) at () ( ft sec ) A car travels on a straight track. During the time interval t 6 seconds, the car s velocity v, measured in feet per second, and acceleration a, measured in feet per second per second, are continuous functions. The table above shows selected values of these functions. (a) Using appropriate units, explain the meaning of vt () dtin terms of the car s motion. Approximate 6 vt () dtusing a trapezoidal approximation with the three subintervals determined by the table (b) Using appropriate units, explain the meaning of at () dtin terms of the car s motion. Find the exact value 3 of at () dt. (c) For < t < 6, must there be a time t when vt () = 5? Justify your answer. (d) For < t < 6, must there be a time t when at () =? Justify your answer. 6 (a) vt () dtis the distance in feet that the car travels 3 from t = 3 sec to t = 6 sec. Trapezoidal approximation for vt () 6 3 dt: A = ( ) 5 + ( 1)( 15) + ( 1)( 1) = 185 ft 3 (b) at () dtis the car s change in velocity in ft/sec from t = sec to t = 3 sec. 3 3 at () dt= v () t dt= v( 3) v( ) = 14 ( ) = 6 ft/sec (c) Yes. Since v( 35) = 1 < 5 < = v( 5 ), the IVT guarantees a t in ( 35, 5 ) so that vt () = 5. (d) Yes. Since v( ) = v( 5 ), the MVT guarantees a t in (, 5 ) so that at () = v () t =. Units of ft in (a) and ft/sec in (b) : { 1 : explanation 1 : value : { 1 : explanation 1 : value 1 : v( 35) < 5 < v( 5) : 1 : Yes; refers to IVT or hypotheses 1 : v( ) = v( 5) : 1 : Yes; refers to MVT or hypotheses 1 : units in (a) and (b) 6 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 7 48

7 AP CALCULUS AB 4 SCORING GUIDELINES (Form B) Question 3 A test plane flies in a straight line with t (min) positive velocity vt (), in miles per vt ()(mpm) minute at time t minutes, where v is a differentiable function of t. Selected values of vt () for t 4 are shown in the table above. (a) Use a midpoint Riemann sum with four subintervals of equal length and values from the table to 4 approximate vt () dt. Show the computations that lead to your answer. Using correct units, 4 explain the meaning of vt () dtin terms of the plane s flight. (b) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval < t < 4? Justify your answer. t 7t (c) The function f, defined by f() t = 6 + cos( ) + 3sin ( ), is used to model the velocity of the 1 4 plane, in miles per minute, for t 4. According to this model, what is the acceleration of the plane at t = 3? Indicates units of measure. (d) According to the model f, given in part (c), what is the average velocity of the plane, in miles per minute, over the time interval t 4? (a) Midpoint Riemann sum is 1 [ v( 5) + v( 15) + v( 5) + v( 35) ] = 1 [ ] = 9 The integral gives the total distance in miles that the plane flies during the 4 minutes. 3 : 1 : v( 5) + v( 15) + v( 5) + v( 35) 1 : answer 1 : meaning with units (b) By the Mean Value Theorem, v () t = somewhere in the interval (, 15 ) and somewhere in the interval ( 5, 3 ). Therefore the acceleration will equal for at least two values of t. 1 : two instances : 1 : justification (c) f ( 3) =.47 or.48 miles per minute 1 : answer with units 1 4 (d) Average velocity = () 4 f t dt = miles per minute 3 : 1 : limits 1 : integrand 1 : answer Copyright 4 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 46 4

1998 Calculus AB Scoring Guidelines

41 Velocity (feet per second) v(t) 9 8 7 6 5 4 1 O 1998 Calculus AB Scoring Guidelines 5 1 15 5 5 4 45 5 Time (seconds) t t v(t) (seconds) (feet per second) 5 1 1 15 55 5 7 78 5 81 4 75 45 6 5 7. The graph