2003 AP Calculus AB Scoring Guidelines - College Board

1 AP Calculus AB 2003 Scoring Guidelines These materials were produced by 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 by that principle. The College Board is a national nonprofit membership association whose mission is to prepare, inspire, and connect students to College and opportunity. Founded in 1900, the association is composed of more than 4,300 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 22,000 high schools, and 3,500 colleges through major programs and services in College admissions, guidance, assessment, financial aid, enrollment, and teaching and learning.

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3 Other products and services may be trademarks of their respective owners. For the College Board s online home for AP professionals, visit AP Central at The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced Placement Program . Teachers may reproduce them, in whole or in part, in limited quantities for noncommercial, face-to-face teaching purposes. This permission does not apply to any third-party copyrights contained herein. This material may not be mass distributed, electronically or otherwise. These materials and any copies made of them may not be resold, and the copyright notices must be retained as they appear here. AP Calculus AB 2003 Scoring Guidelines Copyright 2003 by College Entrance Examination Board . All rights reserved. Available at 2 Question 1 Let R be the shaded region bounded by the graphs of yx= and 3xye = and the vertical line 1,x= as shown in the figure above.

4 (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the horizontal line (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid. Point of intersection 3xex = at (T, S) = ( , ) 1: Correct limits in an integral in (a), (b), or (c) (a) Area = ()13xTxedx = or 2 : 1 : integrand 1 : answer (b) Volume = ()()()122311xTexdxQ = or or 3 : 2 : integrand 1 reversal 1 error with constant 1 omits 1 in one radius < 2 other errors 1 : answer < > < > < > > (c) Length = 3xxe Height = ()35xxe Volume = ()1235xTxedx = 3 : 32 : integrand < 1 > incorrect but has as a factor1 : answerxxe AP Calculus AB 2003 Scoring Guidelines Copyright 2003 by College Entrance Examination Board .

5 All rights reserved. Available at 3 Question 2 A particle moves along the x-axis so that its velocity at time t is given by ()() = + At time 0,t= the particle is at position (a) Find the acceleration of the particle at time Is the speed of the particle increasing at 2?t= Why or why not? (b) Find all times t in the open interval 03t<< when the particle changes direction. Justify your answer. (c) Find the total distance traveled by the particle from time 0t= until time (d) During the time interval 03,tbb what is the greatest distance between the particle and the origin? Show the work that leads to your answer. (a) (2)(2)ava= = or (2)3 sin(2)0v= < Speed is decreasing since (2)0a> and (2)0v<. 2 : 1: (2)1 : speed decreasing with reasona (b) () 0vt= when 22tQ= 2tQ= or or Since () 0vt< for 02tQ<< and () 0vt> for 23tQ<<, the particle changes directions at 2tQ=.

6 2 : 1: 2 only1 : justificationtQ = (c) Distance = 30()vtdt = or 3 : 1: limits 1: integrand1: answer (d) 20( ) ()202(0)( ) += Since the total distance from 0t= to 3t= is , the particle is still to the left of the origin at Hence the greatest distance from the origin is 2 : 1 : (distance particle travels while velocity is negative) 1: answer AP Calculus AB 2003 Scoring Guidelines Copyright 2003 by College Entrance Examination Board . All rights reserved. Available at 4 Question 3 The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function R of time t. The graph of R and a table of selected values of (),Rt for the time interval 090tbb minutes, are shown above. (a) Use data from the table to find an approximation for ()45.

7 Ra Show the computations that lead to your answer. Indicate units of measure. (b) The rate of fuel consumption is increasing fastest at time 45t= minutes. What is the value of ()45 ?Raa Explain your reasoning. (c) Approximate the value of 900()Rtdt using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation less than the value of 900() ?Rtdt Explain your reasoning. (d) For 090b<b minutes, explain the meaning of ()0bRtdt in terms of fuel consumption for the plane. Explain the meaning of ()01bRtdtb in terms of fuel consumption for the plane. Indicate units of measure in both answers. (a) (50) (40) 55 40(45)50 4010 RRR ax= = gal/min2 2 : 21 : a difference quotient using numbers from table and interval that contains 45 1 : gal/min (b) (45) 0 Raa= since ()Rta has a maximum at 45t=.

8 2 : 1 : (45) 0 1 : reasonRaa = (c) 900( )(30)(20) (10)(30) (10)(40)Rtdtx++ (20)(55) (20)(65) 3700++= Yes, this approximation is less because the graph of R is increasing on the interval. 2 : 1 : value of left Riemann sum 1 : less with reason (d) 0()bRtdt is the total amount of fuel in gallons consumed for the first b minutes. 01()bRtdtb is the average value of the rate of fuel consumption in gallons/min during the first b minutes. 3 : 002 : meanings 1 : meaning of ( )1 1 : meaning of ( ) < 1 > if no reference to time 1 : units in both answersbbRtdtRtdtbb AP Calculus AB 2003 Scoring Guidelines Copyright 2003 by College Entrance Examination Board . All rights reserved. Available at 5 Question 4 Let f be a function defined on the closed interval 34x b b with () The graph of ,fa the derivative of f, consists of one line segment and a semicircle, as shown above.

9 (a) On what intervals, if any, is f increasing? Justify your answer. (b) Find the x-coordinate of each point of inflection of the graph of f on the open interval << Justify your answer. (c) Find an equation for the line tangent to the graph of f at the point ()0, 3 . (d) Find ()3f and() Show the work that leads to your answers. (a) The function f is increasing on [3,2] since 0fa> for 32x b< . 2 : 1 : interval1 : reason (b) 0x= and 2x= fa changes from decreasing to increasing at 0x= and from increasing to decreasing at 2x= 2 : 1 : 0 and 2 only 1 : justificationxx == (c) (0)2fa= Tangent line is + 1 : equation (d) (0) ( 3)ff 03()ftdt a= 11 3(1)(1) (2)(2)22 2= = 39(3) (0)22ff =+= (4) (0)ff 40()ftdta= ()218(2) 822QQ= = + (4) (0) 8 25 2ffQQ= += + 4 : ()()21 1 : 22 (difference of areas of triangles)1 : answer for ( 3) using FTC1 1 : 8 (2)2 (area of rectangle area of semicircle)1 : answer for (4) using FTCffQ AP Calculus AB 2003 Scoring Guidelines Copyright 2003 by College Entrance Examination Board .

10 All rights reserved. Available at 6 Question 5 A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let h be 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 5hQ cubic inches per second. (The volume V of a cylinder with radius r and height h is ) (a) Show that .5dhhdt= (b) Given that 17h= at time 0,t= solve the differential equation 5dhhdt= for h as a function of t. (c) At what time t is the coffeepot empty? (a) 25 VhQ= 255dVdhhdtdtQQ== 5255dhhhdtQQ == 3 : 1 : 5 1 : computes 1 : shows resultdVhdtdVdtQ = (b) 5dhhdt= 115dhdth= 125htC= + 217 0C=+ ()211710ht= + 5 : 1 : separates variables 1 : antiderivatives 1 : constant of integration1 : uses initial condition 17 when 0 1 : solves for hth = = Note: max 2/5 [1-1-0-0-0] if no constant of integration Note: 0/5 if no separation of variables (c) ()2117010t += 10 17t= 1 : answer AP Calculus AB 2003 Scoring Guidelines Copyright 2003 by College Entrance Examination Board .