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(a) Find the average value of $ f $ on the given interval.

(b) Find $ c $ such that $ f_{ave} = f(c) $.

(c) Sketch the graph of $ f $ and a rectangle whose area is the same as the area under the graph of $ f $.

$ f(x) = \dfrac{1}{x} $ , $ [1, 3] $

(A). $f_{a v g}=\frac{1}{2} \ln 3$

(B). $\frac{2}{\ln (3)}$

(C). SEE SOLUTION

Applications of Integration

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

and this problem, we are going to be using a specific application of the integral, which is finding the average value of our function. So let's just review how we would find that. Remember, on the closed interval, a B theatric value of our function is equivalent to one over B minus a times the integral from A to B of f of x indie X. And in this problem, we're told that f of X equals one over acts and our interval is 123 So we can simply plug these into this general format. So the average value of our function would be equivalent to 1/3 minus one, and we'll multiply that value by the integral from 1 to 3 of one over X in D X. So then we get one half will take the anti derivative, which is the natural log of X, and we'll evaluate that for 123 And you might be asking, why isn't this the natural? The natural log of the absolute value of X and that is you are correct. Um, the only thing that we have to consider here is that our intervals from 1 to 3 were in the positive number, so we can't have a negative value in the natural log. So then, when we evaluate this will do B minus a will take one half times the natural log of three minus the natural log of one. But remember, the actual log of one is zero. So we'll find that the average value of our function is one half times the natural log of three. So now we were asked to find some value. See, and remember, by the mean value the're, um we're told that there exists number C in our average value of our function. FFC So we'll set one oversee equal to one half times the natural log of three will get see moved on to the other side. So I'll get two equals. A natural log of three times see, and then to find C by itself will have two divided by the natural log of three. How and now Finally, we're asked, Let's put everything together that we've solved and really analyze it in a visual format in the graph of our function. So this is this portion is that portion of our function that we just analyzed this is one over X and we are all the wayto one, which is basically the height of our function. And we're evaluating it from 1 to 3 because remember, that's our interval. And this is the portion we just analyzing using the the average value of the function. So I hope that this helps you understand how we can find the average value of a function using integration and then use an application of the mean value for him to find the number see in our in our average of the function. And then finally just thinking about this visually by graphing our function and the portion of our interval that we were looking at.