A drone equipped with a thermocouple flies horizontally away from a campfire
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A drone equipped with a thermocouple flies horizontally away from a campfire. The air temperature (in °C) measured by the thermocouple is modeled by T(x), where x (in meters) is the drone’s horizontal distance from the fire. The drone’s position is modeled by a differentiable function x(t), where t is time in seconds.
The pilot wishes to compare how quickly the temperature reading is changing over time during the interval 10≤t≤25 seconds with how quickly the temperature is changing at the single instant t=10 seconds.
(a) Describe, without performing any calculations, the conceptual difference between the average rate of change of the temperature reading with respect to time over the interval 10≤t≤25 and the instantaneous rate of change at t=10. Your answer should explicitly reference the functions T and x.
(b) Explain why evaluating the instantaneous rate of change of the temperature reading with respect to time at t=10 seconds requires consideration of both how temperature varies with distance and how distance varies with time.
(c) Suppose the pilot computes 25−10T(x(25))−T(x(10))=−0.12 °C/s. Interpret this value in the context of the flight and explain one limitation of using it to predict what happens immediately after t=10 seconds.