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) = 8 ― 2𝓍 on [0, 4]

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: $\frac{1}{(b-a)}\int a\to bƒ\left(𝓍\right)d𝓍$. Here, a = 0 and b = 4.
2. Step 2: Compute the definite integral of ƒ(𝓍) = 8 - 2𝓍 over the interval [0, 4]. Set up the integral: $\int 0\to 4(8-2𝓍)d𝓍$. Evaluate this integral step by step.
3. Step 3: Divide the result of the integral by the length of the interval (b - a = 4 - 0 = 4) to find the average value of the function on [0, 4]. This gives the average value of ƒ(𝓍).
4. Step 4: Set ƒ(𝓍) equal to the average value found in Step 3. Solve the equation $8-2𝓍=\mathrm{average}\_\mathrm{value}$ for 𝓍.
5. Step 5: Verify that the solution(s) for 𝓍 lie within the interval [0, 4]. These are the points at which the function equals its average value on the given interval.

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 function's value at c equals the average value of the function over that interval. This average value is calculated as (1/(b-a)) * ∫[a to b] f(x) dx.

Average Value of a Function

The average value of a function f(x) over the interval [a, b] is defined as (1/(b-a)) * ∫[a to b] f(x) dx. This concept allows us to determine a single representative value of the function across the interval, which can then be compared to the function's actual values at specific points within that interval.

Finding Points of Intersection

To find points where a function equals its average value, we set the function f(x) equal to the average value calculated from the previous concepts. This involves solving the equation f(x) = average value, which may require algebraic manipulation or numerical methods to identify the specific x-values where this equality holds.