# AP Calculus BC 2004 Scoring Guidelines

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2 SCORING GUIDELINES Question 1 Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by t () 8 sin ( ) Ft = + for t 3, where F() t is measured in cars per minute and t is measured in minutes. (a) To the nearest whole number, how many cars pass through the intersection over the 3-minute period? (b) Is the traffic flow increasing or decreasing a = 7? Give a reason for your answer. (c) What is the average value of the traffic flow over the time interval 1 t 15? Indicate units of measure. (d) What is the average rate of change of the traffic flow over the time interval 1 t 15? Indicate units of measure. 3 (a) Ft () = 7 cars 1 : limits 1 : integrand (b) F ( 7) = 1.87 or Since F ( 7) <, the traffic flow is decreasing a = 7. with reason 1 15 (c) () cars min 5 Ft = 1 1 : limits 1 : integrand (d) F( 15) F( 1) 15 1 = or cars min Units of cars min in (c) and cars min in (d) 1 : units in (c) and (d) Copyright by College Entrance Examination Board. All rights reserved.

3 SCORING GUIDELINES Question Let f and g be the functions given by f ( x) = x( 1 x) and g( x) = 3( x 1) x for x 1. The graphs of f and g are shown in the figure above. (a) Find the area of the shaded region enclosed by the graphs of f and g. (b) Find the volume of the solid generated when the shaded region enclosed by the graphs of f and g is revolved abouhe horizontal line y =. (c) Let h be the function given by hx ( ) = kx( 1 x) for x 1. For each k >, the region (not shown) enclosed by the graphs of h and g is the base of a solid with square cross sections perpendicular to the x-axis. There is a value of k for which the volume of this solid is equal to 15. Write, but do not solve, an equation involving an integral expression that could be used to find the value of k. 1 (a) Area = ( f( x) g( x) ) 1 = ( x( 1 x) 3 ( x 1 ) x) = : { 1 : integral ( ) 1 1 (b) Volume = π ( g( x) ) ( f( x) ) = π (( 3( x 1) x) ( x( 1 x) ) ) : = : limits and constant : integrand 1 each error Note: if integral not of form b c ( R ( x) r ( x) ) a 1 (c) Volume = ( hx ( ) gx ( )) 1 ( kx( 1 x) 3( x 1) x) = 15 { : integrand Copyright by College Entrance Examination Board. All rights reserved. 3

4 SCORING GUIDELINES Question 3 An object moving along a curve in the xy-plane has position ( x() t, y() t ) aime t with = 3+ cos ( t ). The derivative dy is not explicitly given. Aime t =, the object is at position ( 1, 8 ). (a) Find the x-coordinate of the position of the object aime t =. (b) Aime t =, the value of dy is 7. Write an equation for the line tangeno the curve ahe point ( x(, ) y ( )). (c) Find the speed of the object aime t =. (d) For t 3, the line tangeno the curve at ( x() t, y() t ) has a slope of t + 1. Find the acceleration vector of the object aime t =. (a) ( ) = ( ) + ( 3+ cos( )) ( ( t )) x x t = cos = 7.13 or ( ( t )) 1 : 3 + cos 1 : handles initial condition (b) dy t = dy 7 = = = cos t = y 8 =.983( x 1) dy 1 : finds : t = 1 : equation (c) The speed of the object aime t = is ( x ( ) ) + ( y ( ) ) = 7.38 or (d) x ( ) =.33 dy dy y t = = = t cos t y ( ) =.813 or.81 The acceleration vector a = is.33,.813 or.33,.81. () ( )( ( )) 1 : x ( ) dy 1 : Copyright by College Entrance Examination Board. All rights reserved.

5 SCORING GUIDELINES Question Consider the curve given by x + y = xy. dy 3y x (a) Show that =. 8y 3x (b) Show thahere is a point P with x-coordinate 3 at which the line tangeno the curve at P is horizontal. Find the y-coordinate of P. d y (c) Find the value of ahe point P found in part (b). Does the curve have a local maximum, a local minimum, or neither ahe point P? Justify your answer. (a) x + 8yy = 3y + 3xy ( 8y 3x) y = 3y x 3y x y = 8y 3x 1 : implicit differentiation : 1 : solves for y (b) 3y x 8y 3x = ; 3y x = When x = 3, 3y = 6 y = 3 + = 5 and = 5 Therefore, P = ( 3, ) is on the curve and the slope is ahis point. dy 1 : = 1 : shows slope is at ( 3, ) 1 : shows ( 3, ) lies on curve (c) d y ( 8y 3x)( 3y ) ( 3y x)( 8y 3) = ( 8y 3x) d y ( 16 9)( ) At P = ( 3, ), = =. ( 16 9) 7 Since y = and y < at P, the curve has a local maximum at P. : d y : d y 1 : value of at ( 3, ) 1 : conclusion with justification Copyright by College Entrance Examination Board. All rights reserved. 5

6 SCORING GUIDELINES Question 5 A population is modeled by a function P that satisfies the logistic differential equation dp = P ( 1 P ). 5 1 (a) If P ( ) = 3, what is lim Pt ()? If P ( ) =, what is lim Pt ()? (b) If P ( ) = 3, for what value of P is the population growing the fastest? (c) A different population is modeled by a function Y that satisfies the separable differential equation dy = Y ( 1 t ). 5 1 Find Y() t if Y ( ) = 3. (d) For the function Y found in part (c), what is lim Y() t? (a) For this logistic differential equation, the carrying capacity is 1. If P ( ) = 3, lim Pt () = 1. If P ( ) =, lim Pt () = 1. : (b) The population is growing the fastest when P is half the carrying capacity. Therefore, P is growing the fastest when P = t 1 t (c) = ( 1 ) = ( ) dy Y ln Y Y() t = K = 3 Y() t = 3 = + C 5 1 Ke 5 1 e5 1 5 : 1 : separates variables 1 : antiderivatives 1 : constant of integration 1 : uses initial condition 1 : solves for Y 1 if Y is not exponential Note: max 5 [1-1---] if no constant of integration Note: 5 if no separation of variables (d) lim Y() t = 1 if Y is not exponential Copyright by College Entrance Examination Board. All rights reserved. 6

7 π Let f be the function given by f( x) ( x ) for f about x =. (a) Find Px ( ). AP CALCULUS BC SCORING GUIDELINES Question 6 = sin 5 +, and let Px ( ) be the third-degree Taylor polynomial (b) Find the coefficient of x in the Taylor series for f about x = (c) Use the Lagrange error bound to show that f( ) P( ) < x (d) Let G be the function given by G( x) = f( t). Write the third-degree Taylor polynomial for G about x =. π (a) f ( ) = ( ) = sin π 5 ( ) ( ) f = 5cos = π 5 ( ) ( ) f = 5sin = π 15 ( ) ( ) f = 15cos = Px ( ) = + x x x (! ) ( 3! ) 3 : Px ( ) 1 each error or missing term deduct only once for sin ( π ) evaluation error deduct only once for cos( π ) evaluation error 1 max for all extra terms, +, misuse of equality (b) 5! ( ) 1 : magnitude : 1 : sign 1 1 ( (c) ( ) ( ) ) 1 1 f P max f ( c) ( )( ) c! ( ) = <! : error bound in an appropriate inequality (d) The third-degree Taylor polynomial for G about x x = is = x + x x 1 : third-degree Taylor polynomial for G about x = 1 each incorrect or missing term 1 max for all extra terms, +, misuse of equality Copyright by College Entrance Examination Board. All rights reserved. 7

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