AP Calculus BC 2012 Scoring Guidelines


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1 AP Calculus BC Scoring Guidelines The College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in 9, the College Board was created to expand access to higher education. Today, the membership association is made up of more than,9 of the world s leading educational institutions and is dedicated to promoting excellence and equity in education. Each year, the College Board helps more than seven million students prepare for a successful transition to college through programs and services in college readiness and college success including the SAT and the Advanced Placement Program. The organization also serves the education community through research and advocacy on behalf of students, educators, and schools. 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. The College Board. College Board, Advanced Placement Program, AP, SAT and the acorn logo are registered trademarks of the College Board. All other products and services may be trademarks of their respective owners. Permission to use copyrighted College Board materials may be requested online at: AP Central is the official online home for the AP Program: apcentral.collegeboard.org.
2 SCORING GUIDELINES Question t (minutes) 4 9 W() t (degrees Fahrenheit) The temperature of water in a tub at time t is modeled by a strictly increasing, twicedifferentiable function W, where W() t is measured in degrees Fahrenheit and t is measured in minutes. At time t =, the temperature of the water is F. The water is heated for 3 minutes, beginning at time t =. Values of W() t at selected times t for the first minutes are given in the table above. (a) Use the data in the table to estimate W ( ). Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem. (b) Use the data in the table to evaluate W () t. Using correct units, interpret the meaning of () W t in the context of this problem. (c) For t, the average temperature of the water in the tub is (). W t Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate (). W t Does this approximation overestimate or underestimate the average temperature of the water over these minutes? Explain your reasoning. (d) For t, the function W that models the water temperature has first derivative given by W () t =.4 tcos(.6 t). Based on the model, what is the temperature of the water at time t =? W( ) W( 9) (a) W ( ) = 9 6 =.7 (or.6) The water temperature is increasing at a rate of approximately.7 F per minute at time t = minutes. : estimate : { : interpretation with units (b) W () t = W( ) W( ) = 7.. = 6 The water has warmed by 6 F over the interval from t = to t = minutes. (c) () ( 4 ( ) ( 4 ) 6 ( 9 ) ( )) W t W + W + W + W = ( ) =.8 = 6.79 This approximation is an underestimate, because a left Riemann sum is used and the function W is strictly increasing. (d) W ( ) 7. W ( t) = = = + : { : integral : { : value : interpretation with units 3 : : left Riemann sum : approximation : underestimate with reason : answer The College Board.
3 SCORING GUIDELINES Question For t, a particle is moving along a curve so that its position at time t is ( x() t, y() t ). At time t =, the t + dy particle is at position (, ). It is known that = and = sin t. t e (a) Is the horizontal movement of the particle to the left or to the right at time t =? Explain your answer. Find the slope of the path of the particle at time t =. (b) Find the xcoordinate of the particle s position at time t = 4. (c) Find the speed of the particle at time t = 4. Find the acceleration vector of the particle at time t = 4. (d) Find the distance traveled by the particle from time t = to t = 4. (a) t= Because = e t= at time t =. >, the particle is moving to the right : moving to the right with reason dy 3 : : considers : slope at t = dy dy t= = = 3. ( or 3.4) t= t= 4 (b) t + x( 4) = + =.3 ( or.) t e : { : integral : answer (c) Speed = ( x ( 4) ) + ( y ( 4) ) =.7 ( or.74) Acceleration = x ( 4, ) y ( 4) =.4,.989 : { : speed : acceleration 4 (d) Distance = ( x () t ) + ( y () t ) =.6 ( or. 6) : { : integral : answer The College Board.
4 SCORING GUIDELINES Question 3 Let f be the continuous function defined on [ 4, 3] whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let g x be the function given by g( x) = f( t). (a) Find the values of g ( ) and g(. ) (b) For each of g ( 3) and g ( 3, ) find the value or state that it does not exist. (c) Find the xcoordinate of each point at which the graph of g has a horizontal tangent line. For each of these points, determine whether g has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers. (d) For 4 < x < 3, find all values of x for which the graph of g has a point of inflection. Explain your reasoning. (a) g( ) = f( t) = ()( ) = 4 : g( ) : : g( ) g( ) = f () t = f() t 3 π π 3 = ( ) = (b) g ( x) = f( x) g ( 3) = f( 3) = g ( x) = f ( x) g ( 3) = f ( 3) = : : g ( 3) : g ( 3) (c) The graph of g has a horizontal tangent line where g ( x) = f( x) =. This occurs at x = and x =. g ( x) changes sign from positive to negative at x =. Therefore, g has a relative maximum at x =. : considers g ( x) = 3 : : x = and x = : answers with justifications g ( x) does not change sign at x =. Therefore, g has neither a relative maximum nor a relative minimum at x =. (d) The graph of g has a point of inflection at each of x =, x =, and x = because g ( x) = f ( x) changes sign at each of these values. : { : answer : explanation The College Board.
5 SCORING GUIDELINES Question 4 x ( x) f The function f is twice differentiable for x > with f () = and f () =. Values of f, the derivative of f, are given for selected values of x in the table above. (a) Write an equation for the line tangent to the graph of f at x =. Use this line to approximate f (.4 ). (b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to.4.4 approximate f ( x). Use the approximation for ( ) f x to estimate the value of f (.4 ). Show the computations that lead to your answer. (c) Use Euler s method, starting at x = with two steps of equal size, to approximate f (.4 ). Show the computations that lead to your answer. (d) Write the seconddegree Taylor polynomial for f about x =. Use the Taylor polynomial to approximate f (.4 ). (a) f () =, f () = 8 An equation for the tangent line is y = + 8( x ). f (.4) + 8(.4 ) = 8. : tangent line : { : approximation.4 (b) f ( x) (.)( ) + (.)( 3) = f(.4) = f( ) + f ( x) 3 : : midpoint Riemann sum : Fundamental Theorem of Calculus : answer f (.4 ) = 9. 6 (c) f(.) f( ) + (.)( 8) = 6.6 f (.4) (.)( ) = 9. : Euler s method with two steps : { : answer (d) T ( x) = + 8( x ) + ( x )! = + 8( x ) + ( x ) : Taylor polynomial : { : approximation f ( 4. ) + 8(.4 ) + (.4 ) = 9. 8 The College Board.
6 SCORING GUIDELINES Question The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time t =, when the bird is first weighed, its weight is grams. If B() t is the weight of the bird, in grams, at time t days after it is first weighed, then db = ( B). Let y = B() t be the solution to the differential equation above with initial condition B ( ) =. (a) Is the bird gaining weight faster when it weighs 4 grams or when it weighs 7 grams? Explain your reasoning. d B d B (b) Find in terms of B. Use to explain why the graph of B cannot resemble the following graph. (c) Use separation of variables to find y = B(), t the particular solution to the differential equation with initial condition B ( ) =. db (a) = ( 6 ) = B= 4 db B= 7 = ( 3 ) = 6 : db : uses : answer with reason db Because > db, the bird is gaining B= 4 B= 7 weight faster when it weighs 4 grams. d B db (b) ( ) ( ) = = B = B Therefore, the graph of B is concave down for B <. A portion of the given graph is concave up. db (c) = ( B) db = B ln B = t + C Because B <, B = B. ln ( ) = ( ) + C ln ( 8) = C t B = 8e t Bt () = 8 e, t : : d B : in terms of B : explanation : separation of variables : antiderivatives : constant of integration : uses initial condition : solves for B Note: max [] if no constant of integration Note: if no separation of variables The College Board.
7 SCORING GUIDELINES Question 6 The function g has derivatives of all orders, and the Maclaurin series for g is n+ 3 n x x x x ( ) = +. n n= (a) Using the ratio test, determine the interval of convergence of the Maclaurin series for g. (b) The Maclaurin series for g evaluated at value to. The approximation for ( ) this approximation differs from ( ) x = is an alternating series whose terms decrease in absolute g using the first two nonzero terms of this series is 7. Show that g by less than. (c) Write the first three nonzero terms and the general term of the Maclaurin series for g ( x). n+ 3 x (a) n + 3 ( n + = 3 n+ ) n + x n + n + 3 ( ) x = x lim n + n x x < < x < The series converges when < x <. : : sets up ratio : computes limit of ratio : identifies interior of interval of convergence : considers both endpoints : analysis and interval of convergence When x =, the series is This series converges by the Alternating Series Test. When x =, the series is This series converges by the Alternating Series Test. Therefore, the interval of convergence is x. (b) ( ) ( ) 7 : uses the third term as an error bound g < = < : 7 4 { : error bound 3 n (c) g n + ( x) ( ) ( ) { = 4 3 n x x x 7 + n + 3 : : first three terms : general term The College Board.
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