•••2022 AP® Calculus AB Sample Student Responses and Scoring Commentary Inside: FreeResponse Question 1 •Scoring Guidelines •Student Samples •Scoring Commentary © 2022 College Board. College Board, Advanced Placement, AP, AP Central, and the acorn logo are registered trademarks of College Board. Visit College Board on the web: collegeboard.org. $3&HQWUDOLVWKHRƱFLDORQOLQHKRPHIRUWKH$33URJUDP DSFHQWUDOFROOHJHERDUGRUJ
AP® Calculus AB/BC 2022 Scoring Guidelines Part A (AB or BC): Graphing calculator required Question 1 9 points General Scoring Notes The model solution is presented using standard mathematical notation. Answers (numeric or algebraic) need not be simplified. Answers given as a decimal approximation should be correct to three places after the decimal point. Within each individual free-response question, at most one point is not earned for inappropriate rounding. From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by At450 sin 0.62t, where tis the number of hours after 5 A.M. and Atis measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line. Model Solution Scoring (a)Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at thetoll plaza from 6 A.M. ( t1) to 10 A.M. ( t5 ).The total number of vehicles that arrive at the toll plaza from 6 A.M. to 10 A.M. is given by ³5 Atdt1. Answer 1 point Scoring notes: xThe response mustbe a definite integral with correct lower and upper limits to earn this point.xBecauseAtAtfor 1 ddt5, a response of 5 ³1 450 sin 0.62tdtor 5 ³1 Atdtearns the point.xA response missing dtor using dxis eligible to earn the point.xA response with a copy error in the expression for Atwill earn the point only in the presence of³15 Atdt. Total for part (a) 1 point © 2022 College Board
AP® Calculus AB/BC 2022 Scoring Guidelines (b)Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from6 A.M. ( t1) to 10 A.M. ( t5 ).1 ³5 Average Atdt375.536966 5 11 The average rate at which vehicles arrive at the toll plaza from 6 A.M. to 10 A.M. is 375.537 (or 375.536 ) vehicles per hour. Uses average value formula: 1 bAtdtba³a1 point Answer 1 point Scoring notes: xThe use of the average value formula, indicating that a1 and b5, can be presented in single ormultiple steps to earn the first point. For example, the following response earns both points:³15 Atdt1502.147865, so the average value is 375.536966. xA response that presents a correct integral along with the correct average value, but providesincorrect or incomplete communication, earns 1 out of 2 points. For example, the following responseearns 1 out of 2 points: ³15 Atdt1502.147865 375.536966. xThe answer must be correct to three decimal places. For example,1 5 Atdt375.536966 |37651 ³1 earns only the first point. xDegree mode: A response that presents answers obtained by using a calculator in degree modedoes not earn the first point it would have otherwise earned. The response is generally eligiblefor all subsequent points (unless no answer is possible in degree mode or the question is madesimpler by using degree mode). In degree mode, Atdt79.416068. 14 ³15xSpecial case: 5 15 ³1 Atdt300.429573 earns 1 out of 2 points.Total for part (b) 2 points © 2022 College Board
AP® Calculus AB/BC 2022 Scoring Guidelines (c) Is the rate at which vehicles arrive at the toll plaza at 6 A.M. ( t1) increasing or decreasing? Give a reason for your answer. Ac1148.947272 Because Ac1!0, the rate at which the vehicles arrive at the toll plaza is increasing. Considers Ac11 point Answer with reason 1 point Scoring notes: xThe response need not present the value of Ac1.The second line of the model solution earns both points. xAn incorrect value assigned to Ac1earns the first point (but will not earn the second point). xWithout a reference to t1, the first point is earned by any of the following: o148.947 accurate to the number of decimals presented, with zero up to three decimal places (i.e., 149, 148, 148.9, 148.95, or 148.94 ) oAtc148.947 by itself xTo be eligible for the second point, the first point must be earned. xTo earn the second point, there must be a reference to t1. xDegree mode: Ac123.404311 Total for part (c) 2 points (d) A line forms whenever Att400. The number of vehicles in line at time t, for at4, ddis given by ³atNtAx400dx, where ais the time when a line first begins to form. To the nearest whole number, find the greatest number of vehicles in line at the toll plaza in the time interval t4. addJustify your answer. NctAt400 0 At400 t1.469372, t3.597713 a1.469372 b3.597713 Considers Ntc0 1 point taand tb1 point t³tNtAx400 dxaa0 b71.254129 4 62.338346 The greatest number of vehicles in line is 71. Answer 1 point Justification 1 point © 2022 College Board
AP® Calculus AB/BC 2022 Scoring Guidelines Scoring notes: xIt is not necessary to indicate that At400 to earn the first point, although this statement alone would earn the first point. xA response of “ Att400 when dd1.469372 t3.597713 ” will earn the first 2 points. A response of “ Att400 ” along with the presence of exactly one of the two numbers above will earn the first point, but not the second. A response of “ Att400 ” by itself will not earn either of the first 2 points. xTo earn the second point the values for aand bmust be accurate to the number of decimals presented, with at least one and up to three decimal places. These may appear only in a candidates table, as limits of integration, or on a number line. xA response with incorrect notation involving tor xis eligible to earn all 4 points. xA response that does not earn the first point is still eligible for the remaining 3 points. xTo earn the third point, a response must present the greatest number of vehicles. This point is earned for answers of either 71 or 71.254*** only. xA correct justification earns the fourth point, even if the third point is not earned because of a decimal presentation error. xWhen using a Candidates Test, the response must include the values for Na, Nb, and N4 to earn the fourth point. These values must be correct to the number of decimals presented, with up to three decimal places. (Correctly rounded integer values are acceptable.) xAlternate solution for the third and fourth points: For addtb, Att400. For bddt4, Atd400. Thus, Nt³tAx400dxais greatest at tb. Nb71.254129, and the greatest number of vehicles in line is 71. xDegree mode: The response is only eligible to earn the first point because in degree mode At 400. Total for part (d) 4 points Total for question 1 9 points © 2022 College Board
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