We speak of averages almost every day. What’s the average temperature? average velocity? average cost? average time? etc.
So wouldn’t it be cool if we could find the average of a function too?
Mean Value Theorem for Integrals
Well with the Average Value or the Mean Value Theorem for Integrals we can.
We begin our lesson with a quick reminder of how the Mean Value Theorem for differentiation allowed us to determine that there was at least one place in the interval where the slope of the secant line equals the slope of the tangent line, given our function was continuous and differentiable.
Next, we will expand upon this idea by looking at the area, and notice that the area under a curve over an interval is equal to the area of a rectangle with the same width.
So this means that the Mean Value Theorem for Integrals guarantees that a continuous function has at least one point in the closed interval that equals the average value of the function, as Math Words nicely states.
Confused?
Formulas for Mean Value Theorem for Integrals
Let’s break it down even further.
First, we are going to use the Mean Value Theorem that we learned with derivatives and transform it into an integral expression so we can calculate the area over a specified region.
Then we are going to use this average value formula to determine the hight of a rectangle that will produce an equal area to that of the area under the curve.
That’s it.
Together we will walk through several examples in detail to ensure mastery and understanding of this great theorem.
Video Tutorial w/ Full Lesson & Detailed Examples
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