Calculus AB 2014 Scoring Guidelines

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1 P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official online home for the P Program: apcentral.collegeboard.org.

2 P CLCULUS B/CLCULUS BC 014 SCORING GUIDELINES Question 1 Grass clippings are placed in a bin, where they decompose. For 0 t 0, the amount of grass clippings remaining in the bin is modeled by t ( ) = 6.687( 0.91 ), where t () is measured in pounds and t is measured in days. (a) Find the average rate of change of t ( ) over the interval 0 t 0. Indicate units of measure. (b) Find the value of ( 15 ). Using correct units, interpret the meaning of the value in the context of the problem. (c) Find the time t for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval 0 t 0. (d) For t > 0, Lt ( ), the linear approximation to at t = 0, is a better model for the amount of grass clippings remaining in the bin. Use Lt ( ) to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer. t (a) ( 0) ( 0) 0 0 = (or 0.196) lbs/day 1 : answer with units (b) ( 15 ) = (or 0.16) The amount of grass clippings in the bin is decreasing at a rate of (or 0.16) lbs/day at time t = 15 days. 1 : ( 15) : 1 : interpretation (c) 1 0 t ( ) = t ( ) dt t (or ) 0 = : ( t) dt : 0 0 (d) Lt ( ) = ( 0) + ( 0) ( t 0) ( 0) = ( 0) = : expression for Lt ( ) 4 : 1 : Lt ( ) = 0.5 Lt ( ) = 0.5 t= The College Board.

3 P CLCULUS B 014 SCORING GUIDELINES Question 4 Let R be the region enclosed by the graph of f = x.x + 4 and the horizontal line y = 4, as shown in the figure above. (a) Find the volume of the solid generated when R is rotated about the horizontal line y =. (b) Region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in R. Find the volume of the solid. (c) The vertical line x = k divides R into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value k. (a) f = 4 x = 0,.. Volume = π ( 4 ) ( f ) dx = (or ) : integrand 4 : 1 : limits. 1 (b) Volume = ( 4 f ) dx 0 : { : integrand 1 : answer =.574 (or.57) k. : k { (c) ( 4 f ( x )) dx = ( 4 f ( x )) dx 0 1 : area of one region 1 : equation 014 The College Board.

4 The function f is defined on the closed interval [ 5, 4 ]. P CLCULUS B/CLCULUS BC 014 SCORING GUIDELINES Question The graph of f consists of three line segments and is shown in the figure above. Let g be the function defined by g = f ( t) dt. (a) Find g (. ) (b) On what open intervals contained in 5 < x < 4 is the graph of g both increasing and concave down? Give a reason for your answer. g (c) The function h is defined by hx ( ) =. Find h (. ) 5x x (d) The function p is defined by px ( ) f( x x) =. Find the slope of the line tangent to the graph of p at the point where x = 1. (a) g( ) = f ( t) dt = = 9 1 : answer (b) g = f The graph of g is increasing and concave down on the intervals 5 < x < and 0 < x < because g = f is positive and decreasing on these intervals. : { 1 : answer 1 : reason 5xg g 5 5xg 5g (c) h = = ( 5x) 5x : h : ( 5)( ) g ( ) 5g( ) h ( ) = 5 15( ) 5( 9) 75 1 = = = 5 5 (d) p = f ( x x)( x 1) p ( 1) = f ( )( ) = ( )( ) = 6 : p : 014 The College Board.

5 P CLCULUS B/CLCULUS BC 014 SCORING GUIDELINES Question 4 Train runs back and forth on an east-west section of railroad track. Train s velocity, measured in meters per minute, is given by a differentiable function v( t ), where time t is measured in minutes. Selected values for v( t ) are given in the table above. (a) Find the average acceleration of train over the interval t 8. t ( minutes ) v( t ) ( meters minute ) (b) Do the data in the table support the conclusion that train s velocity is 100 meters per minute at some time t with 5< t < 8? Give a reason for your answer. (c) t time t =, train s position is 00 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train, in meters from the Origin Station, at time t = 1. Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time t = 1. (d) second train, train B, travels north from the Origin Station. t time t the velocity of train B is given by vb ( t) = 5t + 60t + 5, and at time t = the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train and train B is changing at time t =. (a) average accel = v ( 8) v( ) m/min = = : average acceleration (b) v is differentiable v is continuous v ( 8) = 10 < 100 < 40 = v ( 5) Therefore, by the Intermediate Value Theorem, there is a time t, 5 < t < 8, such that v ( ) = 100. t 1 : v( 8) < 100 < v( 5) : 1 : conclusion, using IVT v( t) dt 4 (c) s ( 1) = s ( ) + v ( t) dt = 00 + v ( t) dt = 450 s ( 1) = 150 The position of Train at time t = 1 minutes is approximately 150 meters west of Origin Station. (d) Let x be train s position, y train B s position, and z the distance between train and train B. dz dx z = x + y z = x + y dt dt dt x = 00, y = 400 z = 500 vb ( ) = = 15 dz 500 = ( 00)( 100) + ( 400)( 15) dt dz = = 160 meters per minute dt : position expression : 1 : trapezoidal sum 1 : position at time t = 1 : implicit differentiation of : distance relationship 014 The College Board.

6 P CLCULUS B 014 SCORING GUIDELINES Question 5 x < x < < x < < x < f( x ) 1 Positive 8 Positive Positive 7 f 5 Negative 0 Negative 0 Positive 1 g( x ) 1 Negative 0 Positive Positive 1 g Positive Positive 0 Negative The twice-differentiable functions f and g are defined for all real numbers x. Values of f, f, g, and g for various values of x are given in the table above. (a) Find the x-coordinate of each relative minimum of f on the interval [, ]. Justify your answers. (b) Explain why there must be a value c, for 1 < c < 1, such that f ( c) = 0. (c) The function h is defined by hx ( ) = ln ( f ). Find h (. ) Show the computations that lead to your answer. (d) Evaluate f ( g ) g dx. (a) x = 1 is the only critical point at which f changes sign from negative to positive. Therefore, f has a relative minimum at x = 1. (b) f is differentiable f is continuous on the interval 1 x 1 f ( 1) f ( 1) 0 0 = = 0 1 ( 1) 1 : answer with justification 1 : f ( 1) f ( 1) = 0 : 1 : explanation, using Mean Value Theorem Therefore, by the Mean Value Theorem, there is at least one value c, 1 < c < 1, such that f ( c) = 0. 1 (c) h = f f h ( ) = f ( ) = = f ( ) 7 14 x= (d) f ( g ) g dx = f ( g ) x= = f( g( ) ) f( g( ) ) = f ( 1) f ( 1) = 8 = 6 : h : : { : Fundamental Theorem of Calculus 1 : answer 014 The College Board.

7 P CLCULUS B 014 SCORING GUIDELINES Question 6 Consider the differential equation ( y) cos x. dx = Let y = f be the particular solution to the differential equation with the initial condition f ( 0) = 1. The function f is defined for all real numbers. (a) portion of the slope field of the differential equation is given below. Sketch the solution curve through the point ( 0, 1 ). (b) Write an equation for the line tangent to the solution curve in part (a) at the point ( 0, 1 ). Use the equation to approximate f ( 0. ). (c) Find y = f, the particular solution to the differential equation with the initial condition f ( 0) = 1. (a) 1 : solution curve (b) dx ( x, y) = ( 0, 1) = cos 0 = n equation for the tangent line is y = x + 1. f = 1.4 ( ) ( ) 1 : tangent line equation : { 1 : approximation (c) = ( y) cos x dx = cos x dx y ln y = sin x + C ln = sin 0 + C C = ln ln y = sin x ln Because y ( 0) = 1, y <, so y = y sin x y = e sin x y = e Note: this solution is valid for all real numbers. 014 The College Board. 1 : separation of variables : antiderivatives 6 : 1 : constant of integration 1 : uses initial condition 1 : solves for y Note: max 6 [ ] if no constant of integration Note: 0 6 if no separation of variables

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