find the average rate of change of the function over the given interval. f(t) = 6t^2 - 3, 3, 3.1 compare…

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

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

Answer

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.