MATH 163, Spring 2001 Due Date: Name(s):
Extra Project 6.5: The Average Value of a Function
Objective
To investigate the average value of a function.
Narrative
By definition, the average value of the function f = f(x) over the interval [a, b] is
1
b - a
b
x=a
f(x) dx.
Task
a) Type the command lines below into Maple in the order in which they are listed. These command lines
compute the average value of f(x) = 18x - x3
over the interval [1, 3].
> # Project 6.5: The Average Value of a Function
> restart;
> with(plots): with(plottools):
> f := x -> 18*x-x^3;
> a := 1.0; b := 3.0;
> plot0 := plot(f(x),x=a..b):
> display(plot0);
> Av f := (1/(b-a))*evalf(int(f(x),x=a..b));
> plot1 := rectangle([a,0],[b,Av f],color=green):
> display({plot0,plot1});
At this point, make a hard-copy of your typed input and Maple's responses. Then, ...
b) There is a number c  [a, b] for which f(c) = Av f; what is c? (Estimate c from the graphic you produced
in part (a).)
c) Does such a c always exist? (That is, if f is any function defined on a closed interval [a, b], is there always
a number c  [a, b] for which f(c) = Av f?) Justify your answer.