Question

The average value of the function on the interval [a,b] is equal to 1. Find the value of...

a) Clearly define the problem involving the average value of a function over a specified interval.
b) Instruct the respondent to calculate the value being sought given that the average value is 1.
c) Encourage a step-by-step explanation or solution process, demonstrating an understanding of finding the value of a function based on its average value over an interval.

Ensure clarity in the question to prompt a thoughtful calculation and explanation of the value sought for the given function and interval.

Answer 1

Given the average value of a function on the interval [a, b] is equal to 1, the sought value can be found by integrating the function over the interval [a, b] and dividing the result by the length of the interval (b - a).

Step-by-step explanation:

To calculate the value of the function over the interval [a, b], knowing that its average value is 1, the integral mean value theorem is applied. Denoted as f(x), the function's average value on the interval [a, b] can be represented by the formula 1 = (1 / (b - a)) * ∫[a to b] f(x) dx, where ∫[a to b] f(x) dx denotes the definite integral of the function over the interval [a, b].

To find the sought value, rearrange the equation to solve for the integral of the function ∫[a to b] f(x) dx. Multiply both sides of the equation by (b - a) to obtain ∫[a to b] f(x) dx = (b - a). This implies that the integral of the function over the interval [a, b] is equal to (b - a).

Therefore, the sought value of the function over the interval [a, b] is (b - a), derived from the property that the integral of the function equals the length of the interval when the average value of the function over that interval is known to be 1.