Question

Find the average value of f(x)=−3x² −1 on [0,1]

Answer 1

The average value of the function f(x) = -3x² - 1 over the interval [0,1] is computed using integral calculus. The final value turns out to be -1.

Step-by-step explanation:

The student is asking for the average value of the function f(x)=-3x² -1 on the interval [0,1]. To find the average value of a continuous function on a closed interval [a,b], we use the formula:

Average value of f(x) = (1/(b-a)) ∫ f(x) dx

Let's follow the steps to calculate it:

1. Given the function f(x) = -3x² - 1 and the interval [0,1], we find that 'a' is 0 and 'b' is 1.
2. We will compute the integral of f(x) from 0 to 1.
3. The integral of -3x² from 0 to 1 is -x³.
4. After integrating, we substitute the limits to get (-1³) - (0³) = -1 - 0 = -1.
5. The average value is (1/(1-0)) × (-1) = -1.

Therefore, the average value of the function on the interval [0,1] is -1.