2023 AP Calculus AB Free-Response Questions
Advanced PlacementAP Calculus ABFree ResponseMathematics
6 questions54 pts
6 gradable54 rubric criteria0 dependencies
q1Question 1
q1
Question 1
t (seconds) 0 60 90 120 135 150 f(t) (gallons per second) 0 0.1 0.15 0.1 0.05 0 1. A 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 f(t) is measured in gallons per second and t is measured in seconds since pumping began. Selected values of f(t) are given in the table. (a) Using correct units, interpret the meaning of ∫_{60}^{135} f(t) dt 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 ∫_{60}^{135} f(t) dt. (b) Must there exist a value of c, for 60 < c < 120, such that f'(c) = 0 ? Justify your answer. (c) The rate of flow of gasoline, in gallons per second, can also be modeled by g(t) = (t/500) cos((t/120)^2) for 0 ≤ t ≤ 150. Using this model, find the average rate of flow of gasoline over the time interval 0 ≤ t ≤ 150. Show the setup for your calculations. (d) Using the model g defined in part (c), find the value of g'(140). Interpret the meaning of your answer in the context of the problem. Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.
Rubric Criteria (9)
C1The response interprets the definite integral from 60 to 135 of f(t) dt as the total number of gallons of gasoline pumped into the gas tank from time t = 60 seconds to time t = 135 seconds, explicitly referencing gallons and the time interval.
1.0000pts
C2The response presents a right Riemann sum using the three subintervals [60, 90], [90, 120], and [120, 135] with at least five of the six factors correct (the three function values at the right endpoints and the three corresponding widths). A completely correct left Riemann sum with accompanying work earns this point.
1.0000pts
C3The response provides the correct answer 8.25 for the approximation. This point is earned only if the underlying Riemann sum is completely correct or the work shows the three correct products (0.15)(30), (0.1)(30), and (0.05)(15); it is not earned if there is any error in the Riemann sum factors.
1.0000pts
C4The response presents f(120) - f(60) = 0, or 0.1 - 0.1 = 0 (perhaps as the numerator of a quotient), or states f(60) = f(120).
1.0000pts
C5The response states that f is continuous because f is differentiable (or equivalent), references the Mean Value Theorem or Rolle's Theorem, and concludes that there exists a c with 60 < c < 120 such that f'(c) = 0 (or answers yes). This point requires earning the first point of part (b).
1.0000pts
C6The response presents the average value formula (1/(150-0)) integral from 0 to 150 of g(t) dt or equivalent.
1.0000pts
C7The response provides the correct average rate of flow as 0.096 (or 0.0959967, or 0.095, or exact (12/125)sin(25/16)) gallons per second.
1.0000pts
C8The response presents the value of g'(140) as approximately -0.004908, or -0.005, or -0.004, or exact (1/500)cos(49/36) - (49/9000)sin(49/36).
1.0000pts
C9The response interprets g'(140) in context, stating that the rate at which gasoline is flowing into the tank is decreasing at a rate of [the declared value of g'(140)] gallon per second per second at time t = 140 seconds, where the wording is consistent with a negative value (not 'decreasing at a rate of -0.005').
1.0000pts
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