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AP® Calculus AB/BC 2023 Scoring Guidelines© 2023 College BoardPart A (AB or BC):Graphing calculator requiredQuestion 1 9 pointsGeneral Scoring NotesThe 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.( )06090120 135 150(seconds)0 0.1 0.150.1 0.050(gallons per second)tftA customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function ,f where ( )ft is measured in gallons per second and t is measured in seconds since pumping began. Selected values of ()ft are given in the table.Model SolutionScoring(a) Using correct units, interpret the meaning of ( )13560ftdt∫ in the context of the problem. Use a right Riemann sum with the three subintervals []60, 90 ,[]90, 120 , and []120, 135 to approximate the value of ( )13560.ftdt∫( )13560ftdt∫ represents the total number of gallons of gasoline pumped into the gas tank from time 60t= seconds to time 135t= seconds.Interpretation with units1 point( )( )() ( )() ( )()()( ) ( )( ) ()( )1356090 9060120 12090135 135 1200.15 300.1 300.05 158.25ftdtfff≈−+−+−= ++ =∫Form of Riemann sum1 pointAnswer1 pointScoring notes:•To earn the first point the response must reference gallons of gasoline added/pumped and the time interval 60t=to135.t=•To earn the second point at least five of the six factors in the Riemann sum must be correct.•If there is any error in the Riemann sum, the response does not earn the third point.•A response of ()( ) ( )( ) ()( )0.15 300.1 300.05 15++ earns both the second and third points, unless there is a subsequent error in simplification, in which case the response would earn only the second point.
AP® Calculus AB/BC 2023 Scoring Guidelines© 2023 College Board•A response that presents a correct value with accompanying work that shows the threeproducts in the Riemann sum but does not show all six of the factors and/or the sum process, does not earn the second point but does earn the third point. For example, responses of either 4.53.00.75++ or ()( )()( )0.15 30 , 0.1 30 , 0.05 158.25→ earn the third point but not the second.•A response of ( )() ( )() ( )()90 9060120 12090135 135 1208.25fff−+−+− = earns both the second and the third points. •A response that presents an answer of only 8.25 does not earn either the second or third point.•A response that provides a completely correct left Riemann sum with accompanying work, ( )() ( )( ) ( )( )60 3090 30120 159,fff++ =or( )( ) ()( ) ( )( )0.1 300.15 300.1 15++ earns 1 of the last 2 points. A response with any errors or missing factors in a left Riemann sum earns neither of the last 2 points.Total for part (a)3 points(b) Must there exist a value of ,c for 60120,c<< such that ( )0?fc′= Justify your answer.f is differentiable. f⇒ is continuous on []60, 120 .( ) ()120600.10.101206060ff−−==−By the Mean Value Theorem, there must exist a ,c for 60120,c<< such that ( )0.fc′=( ) ( )120600ff−=1 pointAnswer with justification1 pointScoring notes:•To earn the first point a response must present either ( ) ( )120600,ff−=0.10.10−= (perhapsas the numerator of a quotient), or ( )( )60120 .ff=•To earn the second point a response must: ohave earned the first point, ostate that f is continuous because f is differentiable (or equivalent), andoanswer “yes” in some way.•A response may reference either the Mean Value Theorem or Rolle’s Theorem.•A response that references the Intermediate Value Theorem cannot earn the second point.Total for part (b)2 points(c)The rate of flow of gasoline, in gallons per second, can also be modeled by ( )( )( )2cos500120ttgt= for 0150.t≤≤ Using this model, find the average rate of flow of gasoline over the time interval 0150.t≤≤ Show the setup for your calculations.( )150011500gtdt−∫Average value formula1 point0.0959967=Answer1 point
AP® Calculus AB/BC 2023 Scoring Guidelines© 2023 College BoardThe average rate of flow of gasoline, in gallons per second, is 0.096 (or 0.095). Scoring notes:•The exact value of ( )15001150gtdt∫ is ( )1225sin.12516•A response may present the average value formula in single or multiple steps. For example, the following response earns both points: ( )150014.399504gtdt=∫ so the average rate is 0.0959967.•A response that presents the average value formula in multiple steps but provides incorrect or incomplete communication (e.g., ( )150014.3995040.0959967150gtdt==∫)earns 1 out of 2 points.•A response of 15000.0959967()dtgt=∫ does not earn either point.•Degree mode: A response that presents answers obtained by using a calculator in degree mode does not earn the first point it would have otherwise earned. The response is generally eligible for all subsequent points (unless no answer is possible in degree mode or the question is made simpler by using degree mode). In degree mode, ( )150010.149981150gtdt=∫or0.002618.Total for part (c)2 points(d) Using the model g defined in part (c), find the value of ( )140 .g′ Interpret the meaning of your answer in the context of the problem.( )1400.004908g′≈−( )1400.005g′= − (or 0.004−) ( )140g′1 pointThe rate at which gasoline is flowing into the tank is decreasing at a rate of 0.005 (or 0.004) gallon per second per second at time 140t= seconds.Interpretation1 pointScoring notes:•The exact value of ( )140g′ is ( )( )1494949cossin.50036900036−•The value of()140g′ may appear only in the interpretation.•To be eligible for the second point a response must present some numerical value for ( )140 .g′•To earn the second point the interpretation must include “the rate of flow of gasoline is changing at a rate of [the declared value of ( )140g′]” and “at 140t=” (or equivalent).•An interpretation of “decreasing at a rate of –0.005” or “increasing at a rate of0.005” does not earn the second point.•Degree mode: In degree mode, ()1400.001997g′= or 0.00187.Total for part (d)2 pointsTotal for question 19 points
AP® Calculus AB/BC 2023 Scoring Guidelines© 2023 College BoardPart A (AB):Graphing calculator requiredQuestion 2 9 pointsGeneral Scoring NotesThe 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.Stephen swims back and forth along a straight path in a 50-meter-long pool for 90 seconds. Stephen’s velocity is modeled by ()( )0.022.38sin,56tvtetπ−= where t is measured in seconds and ()vt is measured in meters per second.Model SolutionScoring(a) Find all times t in the interval 090t<< at which Stephen changes direction. Give a reason for your answer.For 090,t<<( )056.vtt=⇒=Stephen changes direction when his velocity changes sign. This occurs at 56t= seconds.Considers sign of ()vt1 pointAnswer with reason1 pointScoring notes:•A response that considers ( )0vt= earns the first point.•A response of “Stephen changes direction when his velocity changes sign” earns the first point for considering the sign of( )vt but must include the answer of56t= in order to earn the second point.•A response of 56t= with no supporting work does not earn either point.•Any presented values of t outside the interval 090t<< will not affect scoring.Total for part (a)2 points(b) Find Stephen’s acceleration at time 60t= seconds. Show the setup for your calculations, and indicate units of measure. Is Stephen speeding up or slowing down at time 60t= seconds? Give a reason for your answer.( ) ( )60600.0360162va′== −Stephen’s acceleration at time 60t= seconds is 0.036− meter per second per second.( )60a with setup1 pointAcceleration units 1 point
