Final answer:
The average rate of change of the function f(t) = t² + t - 2 over the interval from 3 to 4 is 8. The instantaneous rate of change of the function at t = 3 is 7.
Step-by-step explanation:
The student's question is about finding the average rate of change and instantaneous rate of change of a function over a specific interval and at a specific point, respectively. Given the function f(t) = t² + t - 2, we can find the average rate of change over the interval from 3 to 4 by calculating the difference in the function's values at the endpoints of the interval divided by the interval's width.
To find the average rate of change of f(t) from 3 to 4, evaluate the function at the endpoints:
f(4) = 4² + 4 - 2 = 18, f(3) = 3² + 3 - 2 = 10
Then, the average rate of change is (f(4) - f(3)) / (4 - 3) = (18 - 10) / 1 = 8.
The instantaneous rate of change of the function when t = 3 corresponds to the derivative of the function evaluated at that point. The derivative f'(t) is 2t + 1, and f'(3) is 2(3) + 1 = 7.