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Find the number $ c $ that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at $ (c, f(c)) $. Are the secant line and the tangent line parallel?

$ f(x) = e^{-x} $, $ [0, 2] $

$$

c=\ln \frac{2}{1-e^{-2}}

$$

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Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Boston College

All right. So here we're being asked to find the number. See, that satisfies the conclusion of the being very serum on the given interval. And we're being done asked to graph the function Ah, and the secret line through the end point and a tangent line. And the question is, are the seeking run and attention on hair around? So in order to first proceed with this problem, we have to figure out whether this whether this function after avec ses equals e to the negative act, satisfies the condition for the mean value through And since e to the negative X is an exponent function, we know that there are always continuous and defensible on domain, which is all real number. So you know that these two are definitely satisfied. So we now know that this we can go ahead and proceed with the conclusion of the mean Barry there. Um and so the conclusion of the mean value, as you recall that is a numbers see. So standing, the derivative of prime back, it is equal to the average shroh. But in its interval, so in this case is after two minus after zero all over to myself. So in order to So we're just gonna go ahead. And first of all, is to find the average slope just to make this easier. This's much longer outbreak problems. So we're just gonna find the average the average smokers half of two, minus ab zero. But we're going to go ahead, and I'm gonna value so e to the negative, too. That is F to an A minus E to the negative zero. What? All over two months now, Eater. Negativity. Right now, Just any number raised to the power of zero as one. So this simplifies down too. E to the minus two minus one over Q. So this is this right? Here is the average slope from on dearer to two. So what this number is is this city average for So this is how we get our first seek it online. So this this is our secret line equation. So this is going to Hawaii. Calls e to the minus two minus one over two X. This is our first seek it line E equation. This is an important thing to remember. So now the question is we want to find where when the Qianjin line is equal to the same slopes as if they were looking for a line left parallel. So the way we do this is way. Take the derivative of So it's going after the next period. Clean page. So I called that the function was e to the negative X eat a negative x. And so, by taking the derivative of this prime, we'LL have to apply to change rule. So we will take the druid of negative acts, which is negative one and in a delivery of into the act it just it reacts and we keep the same upper bounds. So, Teresa, negative each of the ex And now we're going to plant just we're gonna rewrite this in terms of cease to apply to storm in various rooms. So we're gonna go. Is he to the negative with negative horns? It's the negative C equals thie average slope. But remember, we already calculated our average fruit, which was a to the minus two minus one. Oh, over two. So now I'm actually now we just have to isolate too see? And at this point, it's just there's a lot of algebraic manipulation, but we're going to bring the negative one over here, and we're going to switch these two values to as you're quiet. So we might see the vehicle twenty nine the to the miners to all over, too. And then we're going to take the national log on both sides of our national log. But that's how you get rid of the excellent function we're gonna by that National are going to move. I'm up here so new pride of national Long This goes off this explanation goes away and are left with his negative c equals power. Yeah, of one minus e to the minus two. All of them, too. And then we can plug it. See? It's just a negative. Ah, Born minus B to the martyrs, too. Over one sec. All over, too. Yeah. So this is our value of see that one? So now, in order to find the actual tangent line, remember that this is where this is just a sea. So we want to plunge this in. You want to find the actual tangent line, And so the way we do that is where you do this with the point slope formula. So basically what we're going to write this or trying to find is this value why? And we do this by taking the point f c, which is FC, and then this will equal to the slope. But we already calculated the average slope, so e to the minus two minus one two climes. X minus C on, then we're going to bring why you course you rewrite it. I thought that we're just gonna re just doing some algebraic manipulation. We're going to bring the FC to the other side. And now we're just going to plug a or values tto win our sea from here. And we know our ffc because we complaint this record that I would be to the negative act that was our original function. That was our original function. So now, instead of putting an X, we're going to put in our value sea here. So when we do that, we get why you call it e to the minus two minus one. Oh, you too x minus and then our sea Negative Lock those commie positive log now Positive log of one minus e to the minus two over to clothes too Negative. And then now we're going to put in our value since there's a negative island of one minus e to the negatives too, or two? Yeah, this creates a positive on SNC. This is our slope right here. And if you just do some al Jamaica manipulation, you're going to find out that this function is the exact same slope which we can see right here just have the exact same slope as our original function that we highlight this. This is a secret. And this is the tangent line equation. This equation right here is the tangent line. And so, by just looking at this, we can see that Ah, the values are just simply being moved up and out this e studio and we're also cancel. And then we're just going to be adding there's value so essentially dysfunction right here. It's just the same azi this except shifted up by one minus e to the negative to over too. So the question of whether our the Secret Line and Tanja line Hello? Yes, it isthe. And we know that because you could simply look at Slope