SOLUTION: Please help me solve this equation: Consider the function {{{ f(x)= root(3,x) }}}, {{{ 1<=x<=8 }}}. a) Find the average value of f on the interval [1, 8]. b) Find c such tha

Question 350476: Please help me solve this equation: Consider the function , .
a) Find the average value of f on the interval [1, 8].
b) Find c such that fave. = f (c).
c) Sketch the graph of f and a rectangle whose area is the same as the area under
the graph of f .
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Please help me solve this equation: Consider the function , .
a) Find the average value of f on the interval [1, 8].
For the average value on this interval you will use:

Writing the radcial in fractional exponent form will probably help with finding the integral:

The indefinite integral of is . The definite integral, from 1 to 8, will be:

. So now we have

Now the average value is:

b) Find c such that fave. = f (c).
So we solve:

Cube both sides:

c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f.
The graph of f (without the limited domain) looks like:

Note:

Truncate this graph so that only the part between x = 1 and x = 8 shows.
I think the intention is for the graph of the rectangle to be on the same graph. Specifically, the bottom of the rectangle should be the on the x axis between 1 and 8 and the sides will be 45/28 in height extending up from the x axis (at x = 1 and x = 8). If done well, the top of the rectangle should intersect the graph of f at x = c = 91125/21952 (which is approximately 4.15)

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