A water-quality sensor records the temperature of a stream as a differentiable…
MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives
Problem
A water-quality sensor records the temperature of a stream as a differentiable function T (in °C) of time t (in minutes). At time t=12, the temperature is 18.6°C. Let T(12+h) denote the temperature a short time h minutes later.
(a) Write an expression for the average rate of change of the temperature over the interval [12,12+h].
(b) Explain why the expression from part (a) is not defined when h=0.
(c) Describe how you can use the expression from part (a) to define the instantaneous rate of change of the temperature at t=12 even though h=0 makes the quotient undefined. Your answer should reference appropriate limit ideas without performing any computation.
(d) In complete sentences, interpret the meaning of this instantaneous rate of change in the context of water temperature monitoring.