Chapter 4: Integration
Section 4.6: Average Value and the Mean Value Theorem
The average value of a function over an interval is made precise by Definition 4.6.1. Theorem 4.6.1 then says that a continuous function attains its average value at least once on a closed, bounded interval.
Definition 4.6.1: Average Value of a Function
If f is a continuous function on the bounded interval a,b, its average value on a,b is
favg=1b−a ∫abfx ⅆx
A formal statement of the Mean Value theorem for integrals is given in Theorem 4.6.1.
Theorem 4.6.1: Mean Value Theorem
f is continuous on the bounded interval a,b
f attains its average value for at least one c in a,b
For a geometric interpretation of the Mean Value theorem, let Fx be an antiderivative for fx in
For the function Fx, the fraction Fb−Fab−a is the slope of the (secant) line connecting the endpoints a,Fa and b,Fb. The number F′c is the slope of the tangent line. Hence, the Mean Value theorem implies that under suitable conditions, there is at least one point on the graph of Fx where the tangent line is parallel to the secant through the endpoints.
Incidentally, this geometric interpretation of the Mean Value theorem is consistent with the linear approximation afforded by Theorem 3.4.1.
From the definition of favg and from the existence of x=c in the Mean Value theorem, it follows that
for some c in a,b. This is the essence of the Integral Mean Value theorem, which gives the value of a definite integral in terms of the integrand evaluated at a single point.
Maple has both a
tutor and a FunctionAverage command, which are used in Examples 4.6.(1-2).
Maple also has both a
tutor and a MeanValueTheorem command, used in Example 4.6.4.
Find the average value of fx=x2 on the interval 0,1.
Obtain the average value of fx=sinx on the interval 0,π.
In the interval 0,π, find all values of c for which fc=favg.
Use fx=x3−x,x∈0,3, to illustrate the connection between the average value of f and the Mean Value theorem.
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