Average value of a function

Average value of a function

The average value of a function is just the mean value theorem for integrals. In this lesson, we learn that we can find an area of a rectangle that is exactly the same as the area under the curve. Equating them together and algebraically manipulating the equation will give us the formula for the average value. We will be taking a look at some examples of using this formula, as well as using the formula to find the c value that is between a and b.


The Mean Value Theorem for Integrals:

If f is continuous on [a, b], then there exists a number c in [a, b] such that

f(c)=favc=1baabf(x)dxf(c) = f_{avc}= \frac{1}{b-a}\int_{a}^{b} f(x)dx

In other words.
abf(x)dx=f(c)(ba)\int_{a}^{b} f(x)dx=f(c)(b-a)

  • Introduction

  • 1.
    Find the average value of the function f(x)=4+xx3f(x)=4+x-x^3 on the interval [-2,3].

  • 2.
    Given that f(x)=4x2f(x)=4-x^2, [-1,3]. Use the Mean Value Theorem for Integrals to find cc in [-2,3] such that faverage=f(c)f_{average}= f(c)