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′(10). Give units in your answer.
What is the meaning of T′(10)?
t 0 2 4 6 8 10 12 14
T 73 73 70 68 73 80 86 89
My first estimate of T'(10) would be the average change from t=8 to t=12:
(T(12)-T(8))/(12-8) = (86-73)/4 = 3.25
Using a forward estimate from t=10, it might be
(T(12)-T(10))/(12-10) = (86-80)/2 = 3.0
To estimate the value of T′(10), we need to find the average rate of change of T with respect to t between the hours 8 and 10.
Step 1: Calculate the change in T between the hours 8 and 10.
T(10) - T(8) = 80 - 73 = 7
Step 2: Calculate the change in t between the hours 8 and 10.
10 - 8 = 2 hours
Step 3: Calculate the average rate of change.
T′(10) = (T(10) - T(8)) / (10 - 8) = 7 / 2 = 3.5
Therefore, the estimate for T′(10) is 3.5 units per hour.
The meaning of T′(10) is the rate of change of the temperature in Phoenix at 10 hours after midnight on March 15th. It represents how fast the temperature is changing at that specific time.
To estimate the value of T′(10), we need to find the derivative of the temperature function T(t) and evaluate it at t = 10.
To find the derivative, we can use the concept of average rate of change. The average rate of change of T(t) between two time points can be approximated by the difference in temperature divided by the difference in time.
For example, between t = 8 and t = 10, the temperature changes from 73°F to 80°F. The difference in time is 2 hours. Therefore, the average rate of change between t = 8 and t = 10 is (80 - 73) / 2 = 3.5°F per hour.
Following the same approach, we can calculate the average rate of change between t = 10 and t = 12, and between t = 12 and t = 14:
Between t = 10 and t = 12: (86 - 80) / 2 = 3°F per hour
Between t = 12 and t = 14: (89 - 86) / 2 = 1.5°F per hour
Since T(t) is not a linear function and the average rate of change can vary over time intervals, we will approximate T′(10) as the average of these three average rates of change:
T′(10) ≈ (3.5 + 3 + 1.5) / 3 = 2.67°F per hour
Therefore, the estimated value of T′(10) is approximately 2.67°F per hour.
The meaning of T′(10) is the rate at which the temperature in Phoenix on March 15 is changing around 10 hours after midnight. In this case, it indicates how fast the temperature is increasing or decreasing per hour at that specific time.