2024 AP Calculus AB Free-Response Questions
Advanced PlacementAP Calculus ABFree ResponseMathematics
6 questions54 pts
6 gradable54 rubric criteria0 dependencies
q1Question 1
q1
Question 1
1. The temperature of coffee in a cup at time t minutes is modeled by a decreasing differentiable function C, where C(t) is measured in degrees Celsius. For 0 ≤ t ≤ 12, selected values of C(t) are given in the table shown. t (minutes) 0 3 7 12 C(t) (degrees Celsius) 100 85 69 55 (a) Approximate C'(5) using the average rate of change of C over the interval 3 ≤ t ≤ 7. Show the work that leads to your answer and include units of measure. (b) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the value of ∫₀¹² C(t) dt. Interpret the meaning of (1/12)∫₀¹² C(t) dt in the context of the problem. (c) For 12 ≤ t ≤ 20, the rate of change of the temperature of the coffee is modeled by C'(t) = (-24.55e^(0.01t))/t, where C'(t) is measured in degrees Celsius per minute. Find the temperature of the coffee at time t = 20. Show the setup for your calculations. (d) For the model defined in part (c), it can be shown that C''(t) = (0.2455e^(0.01t)(100-t))/t². For 12 < t < 20, determine whether the temperature of the coffee is changing at a decreasing rate or at an increasing rate. Give a reason for your answer. 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)
C1Response approximates C'(5) using a difference quotient over the interval [3,7]—such as (C(7)-C(3))/(7-3), (69-85)/4, or an equivalent quotient with a difference and a quotient shown—and presents a numerical approximation.
1.0000pts
C2Response includes correct units of degrees Celsius per minute (or degrees per minute) attached to a numerical approximation of C'(5).
1.0000pts
C3Response presents a left Riemann sum approximation for the integral of C(t) from t=0 to t=12 using the correct form with interval widths and function values from the table, with at least five of the six factors correct (e.g., (3-0)C(0)+(7-3)C(3)+(12-7)C(7)); a completely correct right Riemann sum also earns this point.
1.0000pts
C4Response evaluates the Riemann sum to produce the estimate 985 (or equivalent correct arithmetic using the table values, such as 3·100+4·85+5·69).
1.0000pts
C5Response interprets (1/12) times the integral of C(t) from 0 to 12 as the average temperature of the coffee over the interval from t=0 to t=12, including both the phrase “average temperature” and the time interval.
1.0000pts
C6Response presents a definite integral with integrand C'(t) and correct limits from 12 to 20 in an equation for C(20), with an unambiguous presentation that includes the differential dt or equivalent notation.
1.0000pts
C7Response adds C(12) or 55 to the definite integral with a lower limit of 12, either symbolically or numerically, in an equation for C(20).
1.0000pts
C8Response provides a correct answer of 55 - 14.671, -14.671 + 55, 40.329, or equivalent, with supporting work for the values used, and does not contain a linkage error that omits the initial condition or mislinks the integral to the final answer.
1.0000pts
C9Response states that the temperature is changing at an increasing rate because C''(t) > 0 on the interval 12 < t < 20 (or equivalent correct reason referencing the positive sign of the second derivative of C).
1.0000pts
Dependencies (0)
Page Mapping (1)
Visual Preview
7 boxes
Draw, drag, resize. Advanced coordinates are available if needed.
1
2
3
4
5
6
7
Select a box to edit its role and metadata.
Attribution
This material is the intellectual property of College Board, is not owned by GradeThis, and is shown here for ease of use and demonstration purposes.