he 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'(8). Give units in your answer.
b. 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
t 00 02 04 06 08 10 12 14
T 73 73 70 68 73 80 86 89
There are a million ways :)
One way:
T(10)= 80
T(6)= 68
change in T = 12
change in t = 4
delta T/delta t = 3 degrees/hour
average rate of change between 6 am and 10
am is used as slope at 8 am
To estimate the value of T'(8) and understand its meaning, we need to calculate the derivative of the temperature function, T(t), at t = 8.
a. Estimate the value of T'(8):
To estimate the value of T'(8), we can use the concept of average rate of change over an interval. Since the temperature values are recorded every two hours, we can calculate the average rate of change between the two closest recorded temperatures to t = 8.
Average rate of change = (T(10) - T(6)) / (10 - 6)
Plugging in the values from the table, we get:
Average rate of change = (73 - 68) / (10 - 6) = 5 / 4 = 1.25
Therefore, the estimated value of T'(8) is 1.25 °F per hour.
b. The meaning of T'(8):
T'(8) represents the instantaneous rate of change of temperature with respect to time at t = 8. In simpler terms, it represents how fast the temperature is changing at the specific time of 8 hours after midnight.
In this case, T'(8) = 1.25 °F per hour means that the temperature in Phoenix on March 15 is estimated to be increasing by 1.25 °F per hour at the time of 8 hours after midnight.
To estimate the value of T'(8) and find its meaning, we need to calculate the average rate of change of temperature over a small interval around t=8.
First, we can find the slope between the points (6, 68) and (10, 80), which will give us the average rate of change of temperature between hour 6 and hour 10:
Slope = (Change in T) / (Change in t) = (80 - 68) / (10 - 6) = 12 / 4 = 3 °F per hour
The value of T'(8) is the estimated rate of change of temperature at t=8, which can be approximated by the above calculated slope:
T'(8) = 3 °F per hour
The meaning of T'(8) is the estimated rate at which the temperature is changing in Phoenix at 8 hours after midnight on March 15th. This represents the average increase or decrease in temperature per hour at that specific time.