Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval.


ƒ(𝓍) = 1 ― |𝓍| on [―1, 1]

Verified step by step guidance

1. Step 1: Recall the formula for the average value of a function on an interval [a, b]. The average value is given by: $\frac{1}{(b-a)}{\int }_{a,b}ƒ\left(𝓍\right)d𝓍$. For this problem, the interval is [−1, 1] and ƒ(𝓍) = 1 − |𝓍|.
2. Step 2: Compute the definite integral of ƒ(𝓍) over [−1, 1]. Break the integral into two parts because the absolute value function |𝓍| behaves differently for 𝓍 < 0 and 𝓍 ≥ 0. Specifically: ${\int }_{-1,1}(1-|𝓍\left|\right)d𝓍$ = ${\int }_{-1,0}(1-(-𝓍\left)\right)d𝓍$ + ${\int }_{0,1}(1-𝓍)d𝓍$.
3. Step 3: Simplify and evaluate each integral. For the first integral, ${\int }_{-1,0}(1-(-𝓍\left)\right)d𝓍$, rewrite the integrand as $1+𝓍$. For the second integral, ${\int }_{0,1}(1-𝓍)d𝓍$, the integrand remains $1-𝓍$. Compute these integrals separately.
4. Step 4: Divide the result of the definite integral by the length of the interval (b − a = 1 − (−1) = 2) to find the average value of the function on [−1, 1]. This gives the average value of ƒ(𝓍).
5. Step 5: Solve the equation ƒ(𝓍) = average value to find the points where the function equals its average value. Substitute ƒ(𝓍) = 1 − |𝓍| into the equation and solve for 𝓍. Consider both cases for |𝓍|: when 𝓍 ≥ 0 and when 𝓍 < 0.

Key Concepts

Mean Value Theorem for Integrals

The Mean Value Theorem for Integrals states that if a function is continuous on a closed interval [a, b], then there exists at least one point c in (a, b) such that the value of the function at c equals the average value of the function over that interval. This theorem is essential for finding points where a function meets its average value.

Average Value of a Function

The average value of a continuous function f(x) over the interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx. This concept is crucial for determining the specific value that the function must equal at some point within the interval to satisfy the Mean Value Theorem for Integrals.

Continuous Functions

A function is continuous on an interval if there are no breaks, jumps, or holes in its graph over that interval. Continuity is a key requirement for applying the Mean Value Theorem for Integrals, as it ensures that the function behaves predictably and that the average value can be accurately determined.