Question

find the average rate of change of the function over the given interval. f(t) = 6t^2 - 3, 3, 3.1 compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. left endpoint right endpoint

Explanation:

Step1: Compute f(3)

$f(3) = 6(3)^2 - 3 = 51$

Step2: Compute f(3.1)

$f(3.1) = 6(3.1)^2 - 3 = 54.66$

Step3: Average rate of change

$\frac{f(3.1)-f(3)}{3.1-3} = \frac{54.66-51}{0.1} = 36.6$

Step4: Find derivative f'(t)

$f'(t) = 12t$

Step5: Left endpoint rate (t=3)

$f'(3) = 12(3) = 36$

Step6: Right endpoint rate (t=3.1)

$f'(3.1) = 12(3.1) = 37.2$

Answer:

Average rate of change: 36.6
Left endpoint instantaneous rate: 36
Right endpoint instantaneous rate: 37.2
The average rate (36.6) is between the left (36) and right (37.2) instantaneous rates.