Final answer:
To find the average value of a function over an interval, evaluate the definite integral of the function over that interval and divide by the width of the interval. In this case, the average value of f(x) = 7x⁵ over [4, 9] is 2904.
Step-by-step explanation:
To find the average value of a function over a given interval, you need to evaluate the definite integral of the function over that interval and divide it by the width of the interval. In this case, the function is f(x) = 7x⁵ and the interval is [4, 9].
The definite integral of f(x) = 7x⁵ over [4, 9] is [ (7/6)x⁶] evaluated from 4 to 9. Evaluating the definite integral, we get [(7/6)(9²⁶ - 4²⁶)] = 7(9²⁶ - 4²⁶)/6.
Finally, we divide this value by the width of the interval which is 9 - 4 = 5. So, the average value of f(x) = 7x⁵ over [4, 9] is (7(9²⁶ - 4²⁶)/6) / 5 = 2904.