Activity 1.6.3.
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. In Activity 1.5.2, we computed approximations to \(F'(30)\) and \(F'(60)\) using central differences. Those values are provided in the second table below, along with several others computed in the same way.
| \(t\) | \(F(t)\) |
| \(0\) | \(70\) |
| \(15\) | \(180.5\) |
| \(30\) | \(251\) |
| \(45\) | \(296\) |
| \(60\) | \(324.5\) |
| \(75\) | \(342.8\) |
| \(90\) | \(354.5\) |
| \(t\) | \(F'(t)\) |
| \(0\) | NA |
| \(15\) | \(6.03\) |
| \(30\) | \(3.85\) |
| \(45\) | \(2.45\) |
| \(60\) | \(1.56\) |
| \(75\) | \(1.00\) |
| \(90\) | NA |
(a)
What are the units on the values of \(F'(t)\text{?}\)
(b)
Use a central difference to estimate the value of \(F''(30)\text{.}\)
(c)
What is the meaning of the value of \(F''(30)\) that you have computed in (b) in terms of the potato’s temperature? Write several careful sentences that discuss, with appropriate units, the values of \(F(30)\text{,}\) \(F'(30)\text{,}\) and \(F''(30)\text{,}\) and explain the overall behavior of the potato’s temperature at this point in time.
(d)
Overall, is the potato’s temperature increasing at an increasing rate, increasing at a constant rate, or increasing at a decreasing rate? Why?
