AP Calculus AB 2009 Scoring Guidelines


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1 AP Calculus AB 9 Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded in 19, the association is composed of more than,6 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents,, high schools and,8 colleges through major programs and services in college readiness, college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its bestknown programs are the SAT, the PSAT/NMSQT and the Advanced Placement Program (AP ). 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. 9 The College Board. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. Permission to use copyrighted College Board materials may be requested online at: AP Central is the official online home for AP teachers: apcentral.collegeboard.com.
2 9 SCORING GUIDELINES Question 1 Caren rides her bicycle along a straight road from home to school, starting at home at time t = minutes and arriving at school at time t = 1 minutes. During the time interval t 1 minutes, her velocity vt (), in miles per minute, is modeled by the piecewiselinear function whose graph is shown above. (a) Find the acceleration of Caren s bicycle at time t = 7. minutes. Indicate units of measure. (b) Using correct units, explain the meaning of vt () dt in terms of Caren s trip. Find the value 1 of vt () dt. 1 (c) Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer. (d) Larry also rides his bicycle along a straight road from home to school in 1 minutes. His velocity is modeled by the function w given by wt () = sin ( t), where wt () is in miles per minute for 1 1 t 1 minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer. v( 8) v( 7) (a) a( 7.) = v ( 7.) = =.1 miles minute (b) vt () dt is the total distance, in miles, that Caren rode during the 1 minutes from t = to t = v() t dt = v() t dt v() t dt + v() t dt = = 1.8 miles (c) Caren turns around to go back home at time t = minutes. This is the time at which her velocity changes from positive to negative. 1 (d) wt () dt= 1.6; Larry lives 1.6 miles from school. 1 vt () dt= 1.; Caren lives 1. miles from school. Therefore, Caren lives closer to school. : { 1 : units 1 : meaning of integral : { 1 : value of integral : { 1 : reason : : Larry s distance from school 1 : integral 1 : value 1 : Caren s distance from school and conclusion 9 The College Board. All rights reserved.
3 9 SCORING GUIDELINES Question The rate at which people enter an auditorium for a rock concert is modeled by the function R given by R() t = 18t 67t for t hours; R() t is measured in people per hour. No one is in the auditorium at time t =, when the doors open. The doors close and the concert begins at time t =. (a) How many people are in the auditorium when the concert begins? (b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer. (c) The total wait time for all the people in the auditorium is found by adding the time each person waits, starting at the time the person enters the auditorium and ending when the concert begins. The function w models the total wait time for all the people who enter the auditorium before time t. The derivative of w is given by w () t = ( t) R() t. Find w( ) w( 1 ), the total wait time for those who enter the auditorium after time t = 1. (d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part (c). people : { 1 : integral (a) Rt () dt= 98 (b) R () t = when t = and t = The maximum rate may occur at, a = 1.696, or. R ( ) = Ra ( ) = 8.7 R ( ) = 1 : 1 : considers R () t = 1 : interior critical point and justification The maximum rate occurs when t = 1.6 or 1.6. (c) w( ) w() 1 = w () t dt = ( t) R() t dt = The total wait time for those who enter the auditorium after time t = 1 is 87. hours. : { 1 : integral 1 1 (d) w( ) = ( t) R( t) dt = On average, a person waits.77 or.776 hour. : { 1 : integral 9 The College Board. All rights reserved.
4 9 SCORING GUIDELINES Question Mighty Cable Company manufactures cable that sells for $1 per meter. For a cable of fixed length, the cost of producing a portion of the cable varies with its distance from the beginning of the cable. Mighty reports that the cost to produce a portion of a cable that is x meters from the beginning of the cable is 6 x dollars per meter. (Note: Profit is defined to be the difference between the amount of money received by the company for selling the cable and the company s cost of producing the cable.) (a) Find Mighty s profit on the sale of a meter cable. (b) Using correct units, explain the meaning of 6 x dx in the context of this problem. (c) Write an expression, involving an integral, that represents Mighty s profit on the sale of a cable that is k meters long. (d) Find the maximum profit that Mighty could earn on the sale of one cable. Justify your answer. (a) Profit = = dollars : { 1 : integral 1 6 xdx (b) 6 x dx is the difference in cost, in dollars, of producing a cable of length meters and a cable of length meters. with units (c) Profit 1 k k 6 x dx = dollars : { 1 : integral 1 : expression (d) Let P( k ) be the profit for a cable of length k. P ( k) = 1 6 k = when k =. This is the only critical point for P, and P changes from positive to negative at k =. Therefore, the maximum profit is P ( ) = 16, dollars. : 1 : P ( k) = 1 : k = 1 : justification 9 The College Board. All rights reserved.
5 9 SCORING GUIDELINES Question Let R be the region in the first quadrant enclosed by the graphs of y = x and y = x, as shown in the figure above. (a) Find the area of R. (b) The region R is the base of a solid. For this solid, at each x the cross section perpendicular to the xaxis has area Ax ( ) = sin ( x). Find the volume of the solid. (c) Another solid has the same base R. For this solid, the cross sections perpendicular to the yaxis are squares. Write, but do not evaluate, an integral expression for the volume of the solid. (a) Area = ( ) 1 x = x = x x dx : 1 : integrand 1 : antiderivative (b) Volume = sin ( ) = ( x) = cos x dx : 1 : integrand 1 : antiderivative y (c) Volume = ( y ) dy : { : integrand 1 : limits 9 The College Board. All rights reserved.
6 9 SCORING GUIDELINES Question x 8 1 f ( x ) 1 6 Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in the closed interval x 1. (a) Estimate f (. ) Show the work that leads to your answer. 1 (b) Evaluate ( f ( x) ) dx. Show the work that leads to your answer. (c) Use a left Riemann sum with subintervals indicated by the data in the table to approximate f ( x) dx. Show the work that leads to your answer. (d) Suppose f ( ) = and f ( x) < for all x in the closed interval x 8. Use the line tangent to the graph of f at x = to show that f ( 7). Use the secant line for the graph of f on x 8 to show that f ( 7 ). f( ) f( ) (a) f ( ) = (b) ( f ( x) ) dx = dx f ( x) dx = ( 1 ) ( f (1) f ()) = 8 1 : { (c) f( x) dx f( )( ) + f( )( ) + f( )( 8 ) + f( 8)( 1 8) = 18 : 1 1 : uses Fundamental Theorem of Calculus 1 : left Riemann sum (d) An equation for the tangent line is y = + ( x ). Since f ( x) < for all x in the interval x 8, the line tangent to the graph of y = f( x) at x = lies above the graph for all x in the interval < x 8. Therefore, f ( 7) + =. An equation for the secant line is y = + ( x ). Since f ( x) < for all x in the interval x 8, the secant line connecting (, f ( ) ) and ( 8, f ( 8) ) lies below the graph of y = f( x) for all x in the interval < x < 8. Therefore, f ( 7) + =. 1 : tangent line 1 : shows f ( 7) : 1 : secant line 1 : shows f ( 7) 9 The College Board. All rights reserved.
7 9 SCORING GUIDELINES Question 6 The derivative of a function f is defined by g( x) for x f ( x) = x. e for < x The graph of the continuous function f, shown in the figure above, has xintercepts at x = and ( ) is a semicircle, and f ( ) =. x = ln. The graph of g on x (a) For < x <, find all values of x at which the graph of f has a point of inflection. Justify your answer. (b) Find f ( ) and f (. ) (c) For x, find the value of x at which f has an absolute maximum. Justify your answer. (a) f changes from decreasing to increasing at x = and from increasing to decreasing at x =. Therefore, the graph of f has points of inflection at x = and x =. 1 : identifies x = or x = : { with justification (b) f( ) = + g( x) dx = ( 8 ) = x ( ) = + ( ) x = + ( 1e x) f e dx = 8 1e : : f ( ) 1 : integral 1 : value : f ( ) 1 : integral 1 : antiderivative 1 : value (c) Since f ( x) > on the intervals < x < and ( ) ( ) < x < ln, f is increasing on the interval x ln. < on the interval ln( ) < x <, f is decreasing on the interval ln( ) x. Therefore, f has an absolute maximum at x = ( ) Since f ( x) ln. : { 1 : justification 9 The College Board. All rights reserved.
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