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. Among its best-known programs are the SATÆ, the PSAT/NMSQTô, the Advanced Placement ProgramÆ (APÆ), and PacesetterÆ. 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. This permission does not apply to any third-party copyrights contained herein. www.mymathscloud.com
APÆ CALCULUS AB 2001 SCORING GUIDELINESCopyright © 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 ( , )(0.902155,1.265751)AB (a) Area R = 3230tan2AAxdxx dx ̈ ̈ = 0.729 or Area R = 1/310(2)tanByydy ̈ = 0.729 or Area R =32330022tanAxdxxxdx ̈ ̈ = 0.729 3 : 1 : limits 1 : integrand 1 : answer£¦¦¦¦¦¤¦¦¦¦¦¥(b) Area S = 302tanAxxdx ̈ = 1.160 or 1.161 or Area S = 211/30tan(2)BBydyydy ̈ ̈ = 1.160 or 1.161 or Area S = 21/31/3100(2)(2)tanBydyyydy ̈ ̈ = 1.160 or 1.161 3 : 1 : limits 1 : integrand 1 : answer£¦¦¦¦¦¤¦¦¦¦¦¥(c) Volume = 23202tanAxxdxQ ̈ = 2.652Q or 8.331 or 8.332 3 : 1 : limits and constant 1 : integrand 1 : answer£¦¦¦¦¦¤¦¦¦¦¦¥www.mymathscloud.com
APÆ CALCULUS AB 2001 SCORING GUIDELINESCopyright © 2001 by College Entrance Examination Board. 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. 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 ()Ptover the time interval 015tbb days. (a) Difference quotient; e.g. (15)(12)1(12)15 123WWWax °C/day or (12)(9)2(12)12 93WWWax °C/day or (15)(9)1(12)15 92WWWax °C/day 2 : 1 : difference quotient1 : answer (with units)£¦¦¤¦¦¥(b) 320 2(31) 2(28) 2(24) 2(22) 21376.52 Average temperature 1(376.5)25.115x °C 2 : 1 : trapezoidal method 1 : answer£¦¦¤¦¦¥(c) /3/31210(12)103tttPete a 4300.549e °C/day This means that the temperature is decreasing at the rate of 0.549 °C/day when t = 12 days. 2 : 1 : (12) (with or without units) 1 : interpretationPa£¦¦¦¤¦¦¦¥(d) 15/30120 1025.75715ttedt ̈°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 www.mymathscloud.com
APÆ CALCULUS AB 2001 SCORING GUIDELINESCopyright © 2001 by College Entrance Examination Board. 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 carís 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 carís 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 carís velocity equal to zero? Justify your answer. (a) Since (2)(2)vaa and (2)150a , the velocity is increasing at t = 2. 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 carís velocity is never equal to 0. The absolute minimum occurs at t = 16 where 166(16)115( )115 105100vatdt ̈. 2 : 1 : answer1 : reason£¦¦¤¦¦¥www.mymathscloud.com
APÆ CALCULUS AB 2001 SCORING GUIDELINESCopyright © 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()xhxxa for all 0xv. (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 oLocal minima at 2x and at 2x 4 : 1 : 2 1 : analysis 2 : conclusions1 > 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)42ha 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 . 1 : answer with reason + 20 2x ()hxa0 0und +www.mymathscloud.com
APÆ CALCULUS AB 2001 SCORING GUIDELINESCopyright © 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£¦¦¦¦¦¦¦¤¦¦¦¦¦¦¦¥www.mymathscloud.com
