The table below shows the temperature (in °F) t hours after midnight in Phoenix on March 15. The table shows values of this function recorded every two hours.

Question

1. Estimate the value of T'(6). Give units in your answer.
2. What is the meaning of T'(6)?

t	0	2	4	6	8	10	12	14
T	73	73	70	68	73	80	86	89

Answer 1

There are a number of ways of doing this problem. The derivative is the slope of the tangent line at the point in question. To find the slope at t = 6, you could:

1) Use the difference between t = 4 and t = 6 like this: (f(6) - f(4))/(6 - 4) = (68 - 70)/2 = -1 °F/hr

2) Use the difference between t = 6 and t = 8 like this: (f(8) - f(6))/(8 - 6) = (73 - 68)/2 = 2.5 °F/hr

3) Use the difference between t = 4 and t = 8 like this: (f(8) - f(4))/(8 - 6) = (73 - 70)/2 = 1.5 °F/hr

4) Curve fit the data using a polynomial and then take the derivative. A quartic curve fit gives an r² value of 0.99 (good fit) and the equation is f(x) = -0.00675x⁴ + 0.177x³ - 1.181x² +1.494x + 73.129. The derivative would be f'(x) = -0.027x³ + 0.531x² - 2.362x + 1.494 so f'(6) = 0.61 °F/h

5) Curve fit the data using a sine function and then take the derivative. A sine curve fit gives the following equation: f(x) = 11.324sin(0.259x - 2.669) + 80.084. The derivative would be f'(x) = 2.933cos(0.259x - 2.669) and f'(6) = 1.29 °F/hr

I would probably use 4) or 5) as my best guess (temperature usually fits a sine function really well so perhaps 5) is the best) but I bet your book gives either 1), 2), or 3) as their answer. Based on what I got from 4) and 5), I'd go with the answer from 3).

The meaning is just the change in temperature per hour occurring at 6:00 am in Phoenix on March 15