2001 AP Calculus AB Scoring Guidelines - College Board

1 AP Calculus AB 2001 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 dedicated to preparing, inspiring, and connecting students to College and opportunity. Founded in 1900, the association is composed of more than 3,900 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 admission, guidance, assessment, financial aid, enrollment, and teaching and learning.

2 Among its best-known programs are the SAT , the PSAT/NMSQT , the advanced Placement Program (AP ), and Pacesetter . The College Board is committed to the principles of equity and excellence, and that commitment is embodied in all of its programs, services, activities, and concerns. Copyright 2001 by College Entrance Examination Board . All rights reserved. College Board , advanced Placement Program, AP, and the acorn logo are registered trademarks of the College Entrance Examination Board . The materials included in these files are intended for non-commercial 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 face-to-face teaching purposes but may not mass distribute the materials, 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.

3 This permission does not apply to any third-party copyrights contained herein. AP Calculus AB 2001 Scoring Guidelines Copyright 2001 by College Entrance Examination Board . All rights reserved. advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board . 2 Question 1 Let R and S be the regions in the first quadrant shown in the figure above. The region R is bounded by the x-axis and the graphs of 32yx and tanyx . The region S is bounded by the y-axis and the graphs of 32yx and tanyx . (a) Find the area of R. (b) Find the area of S. (c) Find the volume of the solid generated when S is revolved about the x-axis. Point of intersection 32tanxx at ( , )( , )AB (a) Area R = 3230tan2 AAxdxx dx = or Area R = 1/310(2)tanByydy = or Area R = 32330022tanAxdxxxdx = 3 : 1 : limits 1 : integrand 1 : answer (b) Area S = 302tanAxxdx = or or Area S = 211/30tan(2)BBydyydy = or or Area S = 21/31/3100(2)(2)tanBydyyydy = or 3 : 1 : limits 1 : integrand 1 : answer (c) Volume = 23202tanAxxdxQ = or or 3 : 1 : limits and constant 1 : integrand 1 : answer AP Calculus AB 2001 Scoring Guidelines Copyright 2001 by College Entrance Examination Board .

4 All rights reserved. advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board . 3 Question 2 The temperature, in degrees Celsius ( C), of the water in a pond is a differentiable function W of time t. The table above shows the water temperature as recorded every 3 days over a 15-day period. (a) Use data from the table to find an approximation for (12)Wa. Show the computations that lead to your answer. Indicate units of measure. (b) Approximate the average temperature, in degrees Celsius, of the water over the time interval 015tbb days by using a trapezoidal approximation with subintervals of length 3t% days. (c) A student proposes the function P, given by (/3)( )20 10tPtte , as a model for the temperature of the water in the pond at time t, where t is measured in days and ()Pt is measured in degrees Celsius.

5 Find (12)Pa. Using appropriate units, explain the meaning of your answer in terms of water temperature. (d) Use the function P defined in part (c) to find the average value, in degrees Celsius, of ()Pt over the time interval 015tbb days. (a) Difference quotient; (15)(12)1(12)15 123 WWW ax C/day or (12)(9)2(12)12 93 WWW ax C/day or (15)(9)1(12)15 92 WWW ax C/day 2 : 1 : difference quotient1 : answer (with units) (b) 320 2(31) 2(28) 2(24) 2(22) Average temperature 1( ) C 2 : 1 : trapezoidal method 1 : answer (c) /3/31210(12)103tttPete a C/day This means that the temperature is decreasing at the rate of C/day when t = 12 days. 2 : 1 : (12) (with or without units) 1 : interpretationPa (d) 15/30120 C 3 : 1 : integrand 1 : limits and average value constant 1 : answer t (days) ()Wt( C) 0 3 6 9 12 15 20 31 28 24 22 21 AP Calculus AB 2001 Scoring Guidelines Copyright 2001 by College Entrance Examination Board .

6 All rights reserved. advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board . 4 Question 3 A car is traveling on a straight road with velocity 55 ft/sec at time t = 0. For 018tbb seconds, the car s acceleration ()at, in ft/sec2, is the piecewise linear function defined by the graph above. (a) Is the velocity of the car increasing at t = 2 seconds? Why or why not? (b) At what time in the interval 018tbb, other than t = 0, is the velocity of the car 55 ft/sec? Why? (c) On the time interval 018tbb, what is the car s absolute maximum velocity, in ft/sec, and at what time does it occur? Justify your answer. (d) At what times in the interval 018tbb, if any, is the car s velocity equal to zero? Justify your answer. (a) Since (2)(2)vaa and (2)150a , the velocity is increasing at t = 2.

7 1 : answer and reason (b) At time t = 12 because 120(12)(0)( )0vvatdt . 2 : 1 : 121 : reasont (c) The absolute maximum velocity is 115 ft/sec at t = 6. The absolute maximum must occur at t = 6 or at an endpoint. 60(6)55( )155 2(15)(4)(15)115(0)2vatdtv 186()0at dt so (18)(6)vv 4 : 1 : 61 : absolute maximum velocity 1 : identifies 6 and 18 as candidates or indicates that increases, decreases, then increases 1 : eliminates 18tttvt (d) The car s velocity is never equal to 0. The absolute minimum occurs at t = 16 where 166(16)115( )115 105100vatdt . 2 : 1 : answer1 : reason AP Calculus AB 2001 Scoring Guidelines Copyright 2001 by College Entrance Examination Board . All rights reserved. advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board . 5 Question 4 Let h be a function defined for all 0xv such that (4)3h and the derivative of h is given by 22()xhxx a for all 0xv.

8 (a) Find all values of x for which the graph of h has a horizontal tangent, and determine whether h has a local maximum, a local minimum, or neither at each of these values. Justify your answers. (b) On what intervals, if any, is the graph of h concave up? Justify your answer. (c) Write an equation for the line tangent to the graph of h at x = 4. (d) Does the line tangent to the graph of h at x = 4 lie above or below the graph of h for 4x ? Why? (a) () 0hxa at 2x o Local minima at 2x and at 2x 4 : 1 : 2 1 : analysis 2 : conclusions 1 > not dealing with discontinuity at 0x o (b) 22() 10hxxaa for all 0xv. Therefore, the graph of h is concave up for all 0xv. 3 : 1 : ( )1 : ( )01 : answer hxhx aa aa (c) 16 27(4)42h a 73(4)2yx 1 : tangent line equation (d) The tangent line is below the graph because the graph of h is concave up for 4x.

9 1 : answer with reason + 2 0 2x ()hxa 0 0und +AP Calculus AB 2001 Scoring Guidelines Copyright 2001 by College Entrance Examination Board . All rights reserved. advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board . 6 Question 5 A cubic polynomial function f is defined by 32() 4fxxaxbx k where a, b, and k are constants. The function f has a local minimum at 1x , and the graph of f has a point of inflection at 2x . (a) Find the values of a and b. (b) If 10()fxdx = 32, what is the value of k ? (a) 2() 122fxxax ba () 242fxx aaa (1) 12 20faba (2)48 2 0faaa 24a 12 236ba 5 : 1 : ( ) 1 : ( )1 : ( 1)01 : ( 2)0 1 : , fxfxffab a aa a aa (b) 132042436xxxkdx 143 2081827xxxx xkxk 27325kk 4 : 2 : antidifferentiation < 1 > each error 1 : expression in 1 : kk AP Calculus AB 2001 Scoring Guidelines Copyright 2001 by College Entrance Examination Board .

10 All rights reserved. advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board . 7 Question 6 The function f is differentiable for all real numbers. The point 13,4 is on the graph of ()yfx , and the slope at each point ,xy on the graph is given by 262dyyxdx . (a) Find 22dydx and evaluate it at the point 13,4. (b) Find ()yfx by solving the differential equation 262dyyxdx with the initial condition 1(3)4f . (a) 2222(62)2dydyyxydxdx = 3222(6 2) 2yxy 22213,4110248dydx 3 : 22 2 : < 2 > product rule or chain rule error 1 1 : value at 3,4dydx (b) 21(6 2 )dyx dxy 216xx Cy 41899CC 13C 21613yxx 6 : 1 : separates variables 1 : antiderivative of term 1 : antiderivative of term 1 : constant of integration1 1 : uses initial condition (3)4 1 : solves for dydxfy Note: max 3/6 [1-1-1-0-0-0] if no constant of integration Note: 0/6 if no separation of variables