Consider the following function.

f(t) = 3t2 − 3

(a) Find the average rate of change of the function below over the interval
[3, 3.1].


(b) Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval.

(at t = 3)

(at t = 3.1)

To find the average rate of change of a function over an interval, we need to calculate the difference in function values between the endpoints of the interval and divide it by the difference in the values of the independent variable (t) between those endpoints.

(a) To find the average rate of change of the function f(t) = 3t^2 - 3 over the interval [3, 3.1], we need to calculate the difference in function values between t = 3 and t = 3.1, and divide it by the difference in t values.

First, calculate f(3):
f(3) = 3(3)^2 - 3
= 27 - 3
= 24

Next, calculate f(3.1):
f(3.1) = 3(3.1)^2 - 3
= 27.63 - 3
= 24.63

The difference in function values is f(3.1) - f(3) = 24.63 - 24 = 0.63.

The difference in t values is 3.1 - 3 = 0.1.

Now, we can calculate the average rate of change:

Average rate of change = (change in function values) / (change in t values)
= 0.63 / 0.1
= 6.3

Therefore, the average rate of change of the function over the interval [3, 3.1] is 6.3.

(b) To compare the average rate of change with the instantaneous rates of change at the endpoints of the interval, we can calculate the instantaneous rates of change at t = 3 and t = 3.1.

The instantaneous rate of change at any point on the function can be calculated by finding the derivative of the function with respect to t and evaluating it at that specific point.

Taking the derivative of the given function, f(t) = 3t^2 - 3, with respect to t gives:

f'(t) = 6t

Now, let's evaluate the derivative at t = 3:

f'(3) = 6(3)
= 18

The instantaneous rate of change at t = 3 is 18.

Similarly, let's evaluate the derivative at t = 3.1:

f'(3.1) = 6(3.1)
= 18.6

The instantaneous rate of change at t = 3.1 is 18.6.

Comparing the average rate of change (6.3) with the instantaneous rates of change at the endpoints:

At t = 3, the instantaneous rate of change is 18, which is significantly different from the average rate of change.

At t = 3.1, the instantaneous rate of change is 18.6, which is also significantly different from the average rate of change.

Therefore, we can conclude that the average rate of change over the interval [3, 3.1] is not representative of the instantaneous rates of change at the endpoints.