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University of North Texas

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Problem 18 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = \frac{x}{x^3 - 1} $

Answer

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Video Transcript

we want to sketch the curve. Why isn't X over acute minus one? So in this truck that they give us it's laundry bust, that's that's so it's gonna go have all those one by one. First, we need to figure out what our domain is going to be. Whoa. This is just gonna be weird. Denominator is not equal to zero. So something excuse minus one could deserve we just get executed with one or what? So our domain should be negative. 30 two one. Union one, too. The next thing we want to figure out is our interests. So let's go ahead and find our ex intercepts first X intercept. Well, this is going to be where the function is equal. 00 is equal to X over X cubed minus one. Well, we know that our denominator could never make this 10 So we just let the new Marie go to zero. So we get Gerald, is he gonna X? And this implies that this is also our why intercept has well, since it's that exit. So we don't even need to solve for our Why intercept. Next. We want to look for symmetry and function so national functions aren't really known for being periodic. So we can check to see if this isn't even or odd function. So in the new mayor would get negative x on dhe. Mom, they're negative. X cubed minus one. So we could go ahead and rewrite this as a negative X over negative X Q minus one and distributing hated from the numerous to dominate. We just get exploder execute plus one. Now, this here is not equal to FX, nor is even negative FX. So what, This tells us we have no symmetry released, no symmetry about the origin or be why access the next thing we'd all still look for his passenger routes. So from chapter 2.6, when we talk about in behavior, we know that sense, our denominator as a larger power than our numerator, the in behavior or as X goes to positive or negative of FX show just be zero. So we have a horizontal aspecto at why is zero and then we can go ahead and look for vertical assam totes which are going to occur. Where are denominators? Zero. Which is that What? So we need to check till left and right in this. So it's gonna be one from the right over one from the right, cubed minus one. Now one from the right is so going to be a positive number. So are numerous gonna be positive and one from the right cube will be positive and then subtracting one. Well, since we're cubing, something larger than one after we cubit is gonna be larger than one. So our denomination also be possible. So this year should go towards positive infinity. And we can use a nice little trick that since Cube excuse minus one is odd. That means our vertical ascent owt should have similar behavior as an odd function with its in behavior. So since it's positive infinity on one side, that means it should be negative. And the on the other side aren't next. We want to find intervals of where the function is increasing and decreasing, and any local max or men's that may come up. So we're gonna need to find what why Prime is coming into that on another page. So why is equal to expert X cube months one and refined to need to use questionable to take this derivatives of the way I remember, questionable is low behind minus Hi. Deal. Oh, all over the square of what is below executed finest one squid. Now we know the derivative of X is just going to be one. And I actually wrote the same thing twice was going a little too fast in scenes. So I need to switch those places. So it should be X times the derivative of X cubed minus one. What? Now the durability execute minus one is going to be three x squared, and the derivative of one is just zero. So very luck with that. And actually, go ahead and scoot all the silver. Tighten it now, if you were to go through and do all that algebra this year should simplify down to negative two x cubed plus one all over X Q minus one squared. Now, to find our critical points, you're gonna want to set this zero and solving for that we would just get to execute. Plus one is equal to zero or X is going to equal to the cube root of four over too. So now we could go ahead and use that information to find our intervals of increasing, decreasing and are Max Mittens. So this should be negative. Two x cubed plus one over X cubed minus one square. Now this function is going to be increasing when Why Prime is strictly larger than zero. And I went ahead already solved this just for the sake of brevity of the video and it should be from negative infinity two negative cube root before over. And we know this function will be decreasing or where Why Prime is strictly less than zero on Negative Cube grew up for over two, 21 Union 12 in unity and we can go ahead and look at that value that we found to the a possible point of inflection. So that's going to be negative, Hubert of war over too. Now to the left of this value, we know the function is increasing and to the right of this value, we know the function is decreasing. So that tells us this here is going to be a max. What now? The last thing they tell us we need to do is to look for a cavity as well as any inflection points that may arise. So we're gonna need to know what? Why double promise to do this. So we come over here back to this page, and so why Double prime is going to be We're going need to take the derivative of the first room. So I'm first gonna factor that negative out front and again. Hopefully, I don't go too fast, Actually. Write everything in place, so it's gonna be low. D Hi, Minus Hi. Well, and then all over the square of what is below So x cubed minus one toe. That's where in force now should be to thee for power and this Negative shit. Now to take the derivative of X cubed plus one, we're going to use our rules. We're gonna handle it. Six x square and think group of X cubed minus one square. But I need to power and changeable. So first for the outside function would get two x cubed minus one. I shall go and screw this down a little bit. So to speed minus one and then the derivative of the inside function is Excuse my swan, which is going to be three squares now. If you were to go through and do all this algebra, this should simplify down to six x squared x cubed plus two all over X cubed minus one. Q. So we can go ahead and now set this equal to zero. And doing that tells us X is equal to zero and X is equal to the cube root of the negative Cooper. Negative, you brute of the negative Cooper to me. And so now we have our possible points of infection, so you'll use those in the next part to see if they actually are. So now, our second ribbon was six X squared X cubes. Us too. Over. Execute minus one. You. So we want to figure out where the assumptions can keep up a calm down so the function will be con cave up when y double prime is strictly argument zero And again I want to solve this. It would be negative. 32 Negatives. Hubert up to Union 123 and the function is going to be. Khan came down when wider, well primed to strictly less than zero or negative. You grew up too. Zero Union 01 And now let's go ahead in Put those two points that we had for possible points of inflection. So yeah, Ecstasy, which is negative Cube root too. And X is equal to zero. So to the left of the negative Cuba too. The function is calm. Keep up. And after X is equal to zero and between, you know, So we already have the And then the rest of these intervals will be conch aid down since between Native cube root too. It is concrete down and after nearly this honky down. So we actually only have that one of these points is a point of inflection. So X is equal to negative too. It will be conch aid or it will be a change cavity. But at X is equal to zero. It is not so. This was the last thing they told us in the chapter we should do before we start roughing. So let's go ahead and dark graphic. So the only intercept we had was executed. We have no symmetry. We know that at why is it zero or along the x axis? We're going to have a horizontal into and at X is equal to one. We're going to have a vertical acid too. All right, Um, we have a maximum at being a bit more negative, Que group of four two. And we have a change in Kong cavity at the cube groups are the negative cube root of to So that number is going to be to the left. Oh, about so we should about here. So negative, you group of two. So I should say this is an inflection point. And this here is a max and so we have all our important points down. So let's go ahead and start by putting down our behavior around where's ah, vertical acid. So to the right Oh, exceeded one function should be possible. And to the left to puncture should be next. Now to the right of ecstasy with the one we have no important points. So we know this is just going to come down and eventually approach our access. Let me do that in green. Instead, I want to look like this. And now to the left of excessive one, we're going to go up and first hit exiting with zero our origin. Now we know we need a maximum to occur at our next point, and there's going to be no, um, other ex intercepts. So we know that it should peek around here, so is goingto flatten out, and then it's gonna come down until it hits bouts the negative puberty two and then change con cavity. So from concave down to calm, keep up there. So this here, I think, is a sufficient sketch of the graph. Something you may want to do is go back and maybe say what this wide value is for inflection point as well as for maximum. But since we're just trying to sketch it and get a look to see what it kind of looks like, I think that's a good place to stop.