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Problem 3 Medium Difficulty

The graph of a function $ g $ is shown.

(a) Verify that $ g $ satisfies the hypotheses of the Mean Value Theorem on the interval $ [0, 8] $.
(b) Estimate the value(s) of $ c $ that satisfy the conclusion of the Mean Value Theorem on the interval $ [0, 8] $.
(c) Estimate the value(s) of $ c $ that satisfy the conclusion of the Mean Value Theorem on the interval $ [2, 6] $.

Answer

(a) (1)$g$ is continuous on the closed interval [0,8]
(2) $g$ is differentiable on the open interval (0,8)
(b) $g^{\prime}(c)=\frac{g(8)-g(0)}{8-0}=\frac{4-1}{8}=\frac{3}{8}$
It appears that $g^{\prime}(c)=\frac{3}{8}$ when $c \approx 2.2$ and 6.4
(c) $g^{\prime}(c)=\frac{g(6)-g(2)}{6-2}=\frac{1-3}{4}=-\frac{1}{2}$
It appears that $g^{\prime}(c)=-\frac{1}{2}$ when $c \approx 3.7$ and 5.5

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Video Transcript

Okay, um, the graph of function G is shown which I've gone, and we were being asked to verify if this function satisfies to help partisans of mean value serum. I estimate the values of sea that satisfied the conclusion of the mean, very therm and estimate the value of C that can't satisfy of the Minbari term in a different animal still from two to six. Ah, so let's begin with the high parts of the conditions that are required for to mean very firm. So they're too big conditions that required for the mean value Don't miss that f has. So this function G has to be continuous so continuous on the closed general on whatever in a void is in this case, it is your eight and differential and the second condition of differential ability. So it means it is defensible. You could take the driven of any point on the open interval from zero to a. So if you look at this function, we knows continuous because we can draw this function without picking up our pencil, and you can see that there's no holes or any sort of, you know, weird things that are happening And so that means that this satisfy the hypothesis for me. So what? What is the actual conclusion for the mean value doing? Before we answer this question, we need to know what is the actual conclusion. And so the conclusion actually is is that there would be a number. See, such that primacy, it is equal to effort be which in this case is after eight after eight minus F zero all over, eh? Minus here. So essentially what this means is that the average slope from zero to a said the slope literally from zero to ace. If I draw this line right here trying to show of straight lines just slow grey here from zero to eight, there's some number. There's some number on this interval such that the tangent. So there's a number cease of some number along this egg sacs of such that the tangent on this too derivative of the tangent line On this curve, we'Ll be the exact same slope down at the exact same lines They'LL be parallel to the two this low. So how do we find this value so we can do this Well, first we need to find the slope at what is the slope of this line? Some of this line is so if you look at this value after bait is for and f zeros one. So this will be, ah will be for a four minus one. So for minus one over eight minus zero, which is a and it has become three eight. So that is the slope of this line. So what is this? So one of the values of sea that satisfies this conclusion? Well, since we're being asked to estimate the values, we can simply just look, you know, visually where this is happening. And if we look at this point right here is this one right here looks really similar to this line. The slope, at least. And this is a little less tin. That is a very, very, very bad line. But if you look down this way, it about two point eight. So we can say this horn is two point eight. But we're not done yet. Actually, this this conclusion tells us that there is for sure at least one. Well, you'LL see that exists. And if you look closely, um, there's actually another one three four, five if you look at it, if you look at this point right here, it's a little hard to tell in this crap because it was in Jordan as neatly. But you could see that this is also parallel to this flow. So there's another value in my graph. It looks like it is exactly at six, three, four, five, six maybe a little bit around six in the actual gruff, but it should be nearer the value of sick. But you could see that this is essentially what you're looking for. How parallel floor. So this tells us the answer to death. Right now, I'm gonna go back to clean up the scrap a little. So the second question is asking us to find the to satisfy the conclusion in a different from from two to six. So from two to six is right here at this point. And three, four, five in this point And this slope right here. So we're looking for a slope like Okay, that looking for this slope, they hear this average flow and we can calculate the value. So at this point after I mercy, the effort to FF six is approximately one. It will be one minus and then effort. Coup is approximately three, only one minus story. And then it'Ll be all over six minus two. Because that's how you do it and then just gives us negative, too. Fourth, which is really just negative One hat that is the slope of define negative one half. So now we're looking for a C that kind of looks like the slopes. And for being asked to estimate if we don't have to find the exact you see, so here you could see that this value a distension looks very similar to this. It was parallel to the slopes. This values around a little greater than three. Well, it's a three point five, and then our next value is you could see another one right here, right at this tangents. If we take a line and go through right here. This awful looks very similar to our average slope. And that is around five point five. There you go. Those air are two values that satisfy the mean value. Durham. Thank you.