Activity 1.6.3.
A potato is placed in an oven whose temperature is 350 degrees Fahrenheit, and the potato’s temperature \(F\) (in degrees Fahrenheit) is recorded in Table 1.6.9. Time \(t\) is measured in minutes.
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Thanks to Nick Owad of Hood College for conducting experiments with actual potatoes in his oven in order to generate the data for this activity.
| \(t\) | \(0\) | \(10\) | \(20\) | \(30\) | \(40\) | \(50\) | \(60\) |
| \(F(t)\) | \(65\) | \(94.7\) | \(141.7\) | \(167.6\) | \(182.4\) | \(197.9\) | \(209.3\) |
| \(t\) | \(0\) | \(10\) | \(20\) | \(30\) | \(40\) | \(50\) | \(60\) |
| \(F'(t)\) | NA | \(3.805\) | \(3.645\) | \(2.065\) | \(1.515\) | \(1.35\) | NA |
In Activity 1.5.3, we used this data to compute approximations to \(F'(20)\) and \(F'(40)\) using central differences. Those values are provided in Table 1.6.10, along with several others computed in the same way.
(a)
What are the units on \(F'(t)\text{?}\) What is the precise meaning of the value \(F'(20) = 3.645\text{?}\)
(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 describe the overall behavior of the potato’s temperature at this point in time. In particular, you should cite the values of \(F(30)\text{,}\) \(F'(30)\text{,}\) and \(F''(30)\text{,}\) each with appropriate units. Be sure to explicitly discuss what you expect to happen in the minute that transpires from \(t = 30\) to \(t = 31\text{.}\)
(d)
On the interval from \(t = 10\) to \(t = 40\text{,}\) is the potato’s temperature increasing at an increasing rate, increasing at a constant rate, or increasing at a decreasing rate? Why?
