# (a) Find the average value of f (x) = x^3 on [0, 4]. Then find c in [0, 4] such that f (c) is...

## Question:

(a) Find the average value of {eq}f (x) = x^3 {/eq} on {eq}[0,\ 4] {/eq}. Then find {eq}c {/eq} in {eq}[0,\ 4] {/eq} such that {eq}f (c) {/eq} is equal to the average value.

(b) For the same function on the same interval as in (a), find the {eq}c {/eq} guaranteed by the differential form of the mean value theorem.

## Mean Value Theorem

There are two statements of the Mean Value Theorem, one for derivatives and one for integrals. The Mean Value Theorem for derivatives relates the average and instantaneous rates of change, and the Mean Value Theorem for integrals relates the average value of a function to the function itself.

{eq}f'(c) = \frac{f(b) - f(a) }{b -a}\\ f(c) = \frac{1}{b-a}\int_a^b f(x) dx {/eq}

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a) Let's begin by applying the Mean Value Theorem for integrals. This theorem states that a continuous function must, at some point in an interval,...

What is the Mean Value Theorem?

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Chapter 7 / Lesson 3
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Three people set off on a car trip. They all start at the same time and end at the same time. Learn what calculus says about how fast they traveled along the way as you study the Mean Value Theorem in this lesson.