AP Calculus AB 2003 Scoring Guidelines


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1 AP Calculus AB Scoring Guidelines The materials included in these files are intended for use y AP teachers for course and exam preparation; permission for any other use must e sought from the Advanced Placement Program. Teachers may reproduce them, in whole or in part, in limited quantities for noncommercial, facetoface teaching purposes. This permission does not apply to any thirdparty copyrights contained herein. This material may not e mass distriuted, electronically or otherwise. These materials and any copies made of them may not e resold, and the copyright notices must e retained as they appear here. These materials were produced y Educational Testing Service (ETS, which develops and administers the examinations of the Advanced Placement Program for the College Board. The College Board and Educational Testing Service (ETS are dedicated to the principle of equal opportunity, and their programs, services, and employment policies are guided y that principle. The College Board is a national nonprofit memership association whose mission is to prepare, inspire, and connect students to college and opportunity. Founded in 9, the association is composed of more than, schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents,, high schools, and, colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its estknown programs are the SAT, the PSAT/NMSQT, and the Advanced Placement Program (AP. The College Board is committed to the principles of equity and excellence, and that commitment is emodied in all of its programs, services, activities, and concerns. For further information, visit Copyright College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Vertical Teams, APCD, Pacesetter, PreAP, SAT, Student Search Service, and the acorn logo are registered trademarks of the College Entrance Examination Board. AP Central is a trademark owned y the College Entrance Examination Board. PSAT/NMSQT is a registered trademark jointly owned y the College Entrance Examination Board and the National Merit Scholarship Corporation. Educational Testing Service and ETS are registered trademarks of Educational Testing Service. Other products and services may e trademarks of their respective owners. For the College Board s online home for AP professionals, visit AP Central at apcentral.collegeoard.com.
2 SCORING GUIDELINES Question Let R e the shaded region ounded y the graphs of y the vertical line x =, as shown in the figure aove. (a Find the area of R. = x and y = e x and ( Find the volume of the solid generated when R is revolved aout the horizontal (c line y =. The region R is the ase of a solid. For this solid, each cross section perpendicular to the xaxis is a rectangle whose height is times the length of its ase in region R. Find the volume of this solid. Point of intersection e x = x at (T, S = (.87,.886 : Correct limits in an integral in (a, (, or (c x (a Area = ( T x e dx =. or. : : integrand : answer ( x ( Volume = ( ( T e x dx =. or. or. : : integrand < > reversal < > error with constant < > omits in one radius < > other errors : answer (c Length = x x e x Height = ( x e x e dx =. x Volume = ( T : : integrand < > incorrect ut has x e as a factor : answer x Copyright y College Entrance Examination Board. All rights reserved.
3 SCORING GUIDELINES Question A particle moves along the xaxis so that its velocity at time t is given y t v( t = ( t + sin. At time t =, the particle is at position x =. (a Find the acceleration of the particle at time t =. Is the speed of the particle increasing at t =? Why or why not? ( Find all times t in the open interval < t < when the particle changes direction. Justify your answer. (c Find the total distance traveled y the particle from time t = until time t =. (d During the time interval t, what is the greatest distance etween the particle and the origin? Show the work that leads to your answer. (a a( = v( =.87 or.88 v ( = sin( < Speed is decreasing since a ( > and v ( <. : : a( : speed decreasing with reason t ( vt ( = when = t = or.6 or.7 Since vt ( < for < t < and vt ( > for < t <, the particle changes directions at t =. : : t = only : justification (c Distance = vt ( dt =. or. : : limits : integrand : answer (d vt ( dt=.6 x( = x( + v( t dt =.6 Since the total distance from t = to t = is., the particle is still to the left of the origin at t =. Hence the greatest distance from the origin is.6. : ± (distance particle travels : while velocity is negative : answer Copyright y College Entrance Examination Board. All rights reserved.
4 SCORING GUIDELINES Question The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given y a twicedifferentiale and strictly increasing function R of time t. The graph of R and a tale of selected values of R( t, for the time interval t 9 minutes, are shown aove. (a Use data from the tale to find an approximation for R (. Show the computations that lead to your answer. Indicate units of measure. ( The rate of fuel consumption is increasing fastest at time t = minutes. What is the value of R (? Explain your reasoning. (c Approximate the value of 9 Rt ( dt using a left Riemann sum with the five suintervals indicated y the data in the tale. Is this numerical approximation less than the value of 9 Rt ( dt? Explain your reasoning. (d For < 9 minutes, explain the meaning of ( R t dt in terms of fuel consumption for the plane. Explain the meaning of R ( t dt in terms of fuel consumption for the plane. Indicate units of measure in oth answers. (a R( R( R( = =. gal/min : ( R ( = since R ( t has a maximum at (c t =. 9 Rt ( dt (( + (( + (( + (( + ((6 = 7 Yes, this approximation is less ecause the graph of R is increasing on the interval. : : : a difference quotient using numers from tale and interval that contains :. gal/min : R( = : reason : value of left Riemann sum : less with reason (d Rt ( dt is the total amount of fuel in gallons consumed for the first minutes. Rt ( dt is the average value of the rate of fuel consumption in gallons/min during the first minutes. Copyright y College Entrance Examination Board. All rights reserved. : : meanings : meaning of Rt ( dt : meaning of Rt ( dt < > if no reference to time : units in oth answers
5 SCORING GUIDELINES Question Let f e a function defined on the closed interval x with f ( =. The graph of f, the derivative of f, consists of one line segment and a semicircle, as shown aove. (a On what intervals, if any, is f increasing? Justify your answer. ( Find the xcoordinate of each point of inflection of the graph of f on the open interval < x <. Justify your answer. (c Find an equation for the line tangent to the graph of f at the point (,. (d Find f ( and f (. Show the work that leads to your answers. (a The function f is increasing on [, ] since f > for x <. : : interval : reason ( x = and x = f changes from decreasing to increasing at x = and from increasing to decreasing at x = : : x = and x = only : justification (c f ( = Tangent line is y = x +. (d f( f( 9 f( = f( + = f( f( = f( t dt = (( (( = = f( t dt = ( 8 ( = 8 + f( = f( 8 + = + : equation ( : ± (difference of areas of triangles : answer for f ( using FTC : : ± ( 8 ( (area of rectangle area of semicircle : answer for f ( using FTC Copyright y College Entrance Examination Board. All rights reserved.
6 AP CALCULUS AB SCORING GUIDELINES Question A coffeepot has the shape of a cylinder with radius inches, as shown in the figure aove. Let h e the depth of the coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume V of coffee in the pot is changing at the rate of h cuic inches per second. (The volume V of a cylinder with radius r and height h is V = r h. dh h (a Show that. dt = dh h ( Given that h = 7 at time t =, solve the differential equation dt = for h as a function of t. (c At what time t is the coffeepot empty? (a V = h dv dh = = h dt dt dh h h = = dt : dv : = h dt dv : computes dt : shows result ( dh dt = h dh = dt h h = t + C 7 = + C ( t 7 h = + : separates variales : antiderivatives : constant of integration : : uses initial condition h = 7 when t = : solves for h Note: max / [] if no constant of integration Note: / if no separation of variales (c ( 7 t + = : answer t = 7 Copyright y College Entrance Examination Board. All rights reserved. 6
7 SCORING GUIDELINES Question 6 Let f e the function defined y x + for x fx ( = x for < x. (a Is f continuous at x =? Explain why or why not. ( Find the average value of f ( x on the closed interval x. (c Suppose the function g is defined y k x + for x gx ( = mx + for < x, where k and m are constants. If g is differentiale at x =, what are the values of k and m? (a f is continuous at x = ecause lim f ( x = lim f( x =. x x + Therefore, lim f ( x = = f(. x : : answers yes and equates the values of the left and righthand limits : explanation involving limits ( f ( x dx = f ( x dx + f ( x dx Average value: ( x ( x x ( ( = = + = ( fxdx= : : k f( x dx + k ( f x dx (where k : antiderivative of x + : antiderivative of x : evaluation and answer (c Since g is continuous at x =, k = m +. k for < x < g ( x = x + m for < x < k lim g ( x = and lim g ( x = m x x + Since these two limits exist and g is differentiale at x =, the two limits are k equal. Thus = m. : k = m + : k : = m : values for k and m 8 8m = m + ; m = and k = Copyright y College Entrance Examination Board. All rights reserved. 7
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