Lab 02: Average Values

Objective

We will investigate the average values of a function.

Recall

By definition, the average value of the function f over the interval [a,b] is

(Subscript[Avg, f])([a,b])=(∫_a^b)f(x)dx.

The following Mathematica commands graph and then compute the average value of f(x)=18x-(x^3) over the interval [1,3].

f[x_]=18*x-x^3
a=1.0
b=3.0
plot0=Plot[f[x],{x,a,b}]
avgf=NIntegrate[f[x],{x,a,b}]/(b-a)
plot1=Graphics[{EdgeForm[Dashed],Opacity[0.2],Rectangle[{1,0},{3,avgf}]},AspectRatio->Full,Axes->True]
Show[{plot1,plot0}]

Tasks

There is a number c ∈ [a,b] for which f(c) = avgf ; what is c?  (Estimate the value from the graphic you produced above.  Then use Mathematica to verify your estimate.

Does such a c always exist?  Justify your answer.  (Consider the function g(x)=(x^2)+((|x|)/x)+1 on the interval [-1,1].)

Pick two problems out of section 6.5 and use Mathematica to find the solution.  If possible, pick problems with “CAS” or the calculater symbol next to the problem.