Final answer:
The average rate of change of f(t) over the interval 3 to 4 is 10, and the instantaneous rate of change of f(t) when t = 3 is 9.
Step-by-step explanation:
a. To find the average rate of change of f(t) over the interval 3 to 4, we need to find the difference between f(4) and f(3) and divide it by the difference in t values.
Substituting t = 4 into the function f(t) = t^2 + 3t -1, we get:
f(4) = (4)^2 + 3(4) - 1 = 16 + 12 - 1 = 27
Substituting t = 3 into the function f(t), we get:
f(3) = (3)^2 + 3(3) - 1 = 9 + 9 - 1 = 17
The average rate of change is calculated as:
(f(4) - f(3))/(4 - 3) = (27 - 17)/(4 - 3) = 10
b. To find the instantaneous rate of change of f(t) when t = 3, we need to find the derivative of f(t) and evaluate it at t = 3. Differentiating the function f(t) = t^2 + 3t - 1, we get f'(t) = 2t + 3.
Evaluating f'(t) at t = 3, we get:
f'(3) = 2(3) + 3 = 9
Therefore, the average rate of change of f(t) over the interval 3 to 4 is 10, and the instantaneous rate of change of f(t) when t = 3 is 9.