Question

Find the average rate of change for f(t) = 2t² + 9 over the interval [6, 6+h].

Answer 1

The average rate of change of the function f(t) = 2t² + 9 over the interval [6, 6+h] is computed by evaluating the function at both endpoints, finding the difference, and dividing it by h. This gives us the formula ([2(6+h)² + 9] - 81) / h for the average rate of change.

Step-by-step explanation:

The average rate of change of a function f(t) over an interval can be found by calculating the change in the function values over that interval divided by the change in t. For the function f(t) = 2t² + 9 and the interval [6, 6+h], where h represents a small change in t, we'll evaluate the function at both endpoints to find the average rate of change.

1. First, find the value of the function at the beginning of the interval: f(6) = 2(6)² + 9 = 2(36) + 9 = 72 + 9 = 81.
2. Next, find the value of the function at the end of the interval: f(6+h) = 2(6+h)² + 9.
3. Then, calculate the difference: f(6+h) - f(6) = [2(6+h)² + 9] - 81.
4. Divide this difference by h, the change in t, to find the average rate of change: ([2(6+h)² + 9] - 81) / h.

This expression simplifies to the average rate of change of f(t) over the interval [6, 6+h].