Final answer:
The average rate of change of the function f(t) = 4t² - 5 on the interval [3, 3.1] is found using the difference in function values over the time interval. Instantaneous rates of change are found by evaluating the derivative, which is 8t, at t = 3 and t = 3.1.
Step-by-step explanation:
The question asks to find the average rate of change of the function f(t) = 4t² − 5 over the interval [3, 3.1] and compare this with the instantaneous rates of change at the endpoints of the interval.
To calculate the average rate of change, we use the formula:
∆f / ∆t = (f(t2) - f(t1)) / (t2 - t1)
In the given interval [3, 3.1]:
∆f = f(3.1) − f(3) = (4(3.1)² − 5) − (4(3)² − 5)
∆t = 3.1 - 3 = 0.1
Substitute these values into the formula to get the average rate of change.
For the instantaneous rate of change at t = 3 and t = 3.1, we take the derivative of the function, which is f'(t) = 8t. Then we evaluate f'(3) and f'(3.1) to get the instantaneous rates at the endpoints.