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. (10 points)
a. Estimate the value of T'(6). Give units in your answer.
b. 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
To estimate the value of T'(6), we need to find the average rate of change of temperature with respect to time between t = 4 and t = 8.
To do this, we can use the formula for average rate of change:
T'(a) = (T(b) - T(a)) / (b - a)
In this case, a = 4 and b = 8, so we have:
T'(6) = (T(8) - T(4)) / (8 - 4)
Plugging in the values from the table:
T(8) = 73°F
T(4) = 68°F
T'(6) = (73 - 68) / (8 - 4)
T'(6) = 5 / 4
T'(6) = 1.25°F per hour
Therefore, the estimated value of T'(6) is 1.25°F per hour.
The meaning of T'(6) is the rate of change of temperature at t = 6 hours after midnight in Phoenix on March 15. It represents how much the temperature is changing per hour at that specific time.
To estimate the value of T'(6), we need to find the average rate of change of temperature with respect to time between the value of t = 4 and t = 8.
a. To estimate T'(6), we can use the formula:
T'(6) = (T(8) - T(4))/(8 - 4)
Substituting the values from the table:
T'(6) = (73 - 68)/(8 - 4)
= 5/4
Therefore, the estimated value of T'(6) is 5/4 degrees Fahrenheit per hour.
b. The meaning of T'(6) is the rate at which the temperature is changing with respect to time at t = 6. In other words, it represents the average rate of change of the temperature in Phoenix at 6 hours after midnight on March 15. The units for T'(6) are degrees Fahrenheit per hour.
T'(6) could be approximated by the value
(T(8)-T(6))/2 = 5/2
T'(6) means how fast T is changing at t=6, in °F/hr.