2024 AP Calculus AB FRQ #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 t is between 0 and 12, selected values of C(t) are given in the table shown.

(a) Approximate C'(5) using the average rate of change of C over the interval 3 to 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 the integral from 0 to 12 of C(t) dt. Interpret the meaning in the context of the problem.

(c) For t is between 12 and 20, the rate of change of the temperature of the coffee is modeled by C'(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) is equal to . For t is between 12 and 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.

Problem A: 00:00
Problem B: 04:01
Problem C: 10:06
Problem D: 14:10