Activity 1.5.2.
A potato is placed in an oven, and the potato’s temperature \(F\) (in degrees Fahrenheit) at various points in time is taken and recorded in the following table. Time \(t\) is measured in minutes.
| \(t\) | \(0\) | \(15\) | \(30\) | \(45\) | \(60\) | \(75\) | \(90\) |
| \(F(t)\) | \(70\) | \(180.5\) | \(251\) | \(296\) | \(324.5\) | \(342.8\) | \(354.5\) |
(a)
Use a central difference to estimate the instantaneous rate of change of the temperature of the potato at \(t= 30\text{.}\) Include units on your answer.
(b)
Use a central difference to estimate the instantaneous rate of change of the temperature of the potato at \(t= 60\text{.}\) Include units on your answer.
(c)
Without doing any calculation, which do you expect to be greater: \(F'(75)\) or \(F'(90)\text{?}\) Why?
(d)
Suppose it is given that \(F(64) = 330.28\) and \(F'(64) = 1.341\text{.}\) What are the units on these two quantities? What do you expect the temperature of the potato to be when \(t = 65\text{?}\) when \(t = 66\text{?}\) Why?
(e)
Write a couple of careful sentences that describe the behavior of the temperature of the potato on the time interval \([0,90]\text{,}\) as well as the behavior of the instantaneous rate of change of the temperature of the potato on the same time interval.
