Question

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.
Estimate the value of T′(8). Give units in your answer.
What is the meaning of T′(8)?


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

Answer 1

We need to estimate the value of T'(8) and then we need to give the meaning of this value. Since we have a table and not an expression of a function our goal is to find the Average Rate of Change between two points.

For a nonlinear graph whose slope changes at each point, the average rate of change between any two points $(x_{1},f(x_{1}) \ and \ (x_{2},f(x_{2})$ is the slope of the line through the two points. So:

$ARC=\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}} =\frac{Change \ in \ y}{Change \ in \ x}=m_{sec}$

1. Estimated value of T′(8).

T'(8) means the derivative of the function T(t) when t = 8 hours. So, we can estimate this value using the average rate of change:

$ARC=\frac{T(8)-T(6)}{8-6} =\frac{73-68}{8-6}=\frac{5}{2} \ degrees \ per \ hours$

2. The meaning of T′(8)

This means the average temperature after midnight in Phoenix on March 15, from t = 6 to t = 8 hours, that is, for each hour the temperature rises 5/2 degrees on the Fahrenheit scale. The points have been plotted in the Figure bellow.

Answer 2

I assume T'(8) is referring to the rate of change in Temperature when t = 8.

To find this use the slope formula with T(8) and nearest point T(6).

$T'(8) = \frac{T(8) - T(6)}{8-6} = \frac{73-68}{8-6} = \frac{5}{2}$

The units will be degrees per hour