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Use the guidelines of this section to sketch the …

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Problem 9 Medium Difficulty

Use the guidelines of this section to sketch the curve.

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

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Video Transcript

Alright, so here we want to sketch the function Y equals X over x minus one. And so the first thing we'll do is we'll look for basically a horizontal as some toads, vertical ascent. Toads and so forth. So we can get a good picture of the line. Um books I wrote line because that's a said line, horizontal assim toe. Okay, so horizontal acetone is in the limit as X goes to plus or minus infinity of our function and we want to see what we get. So let's take a look at for our function and if it's polynomial you'll get the same value plus or minus. If you have simple polynomial fractions, you don't have to worry about going to the infinity equaling anything different than going to the minus infinity. Alright, so if I look at this, I'm going to infinity. And so what happens is that when I go to infinity then just the high growth rates dominate top and bottom because um X is getting really big. So it certainly dominates the numerator and the bottom is denominated by the X. As well. And you can see that fracture reduces down to one. So I get one. So we have a horizontal assume toda at Y equals one. And let's go ahead and um draw that on there. It's not going to be dotted because I can't easily do dotted lines but there it is. So we will have a horizontal Aston toad at Y equals one. Okay, so next let's try, how about vertical ascent toe, vertical ascent. Toad. Okay, so vertical ascent toad is basically whatever makes the denominator go to zero. So I can see right away that we're not. Uh If I set the denominator equal to zero, X equals one. And that is my vertical ascent total. I think I'll show you the calculus in a sec. But let's just go do that because that one's pretty easy to do. So let's do um Let's do our X equals one. Okay, so that's our vertical ascent toad. And officially vertical ascent toads would be limited As X approaches, say one um of F of X. It does not exist. So our function does not exist As we go to one. and we can actually kind of see what's going to happen. Plus or minus. You can see the bottom, we'll go to zero. But the question is, are we approaching? Uh and when we get a positive or minus infinity, So let's go figure this out. Okay, so let's say approach from the minus approach one from the left. So I am going to look at this and approach one from the left. So let's plug in. Um just do this on the side. Let's imagine that we're plugging in negative, like 1.1. If I go negative 1.1 over -1.1 -1. And I get negative 1.1 over. Um ah Oops, I did the wrong thing. Let's fix that real quick. I want to be. Really? I did it for -1. So let's fix it and do it for one. So let's time that .9. That's just a little bit to the left of one. I accidentally did negative one before. So if I'm a little bit to the left of one That I'm at say .9 Then I get .9 -1. So I get .9 over negative .1 you can see I'm already up at minus nine so you can see I'm definitely going to be going way down and that's gonna be the one boy towards minus infinity. So I can say this is minus infinity and let's take a look at the opposite side. If I do limit X approaches one from the right side, then let's just plug in. All right, Like 1.1 now, 1.1 Over 1.1 -1. I get 1.1 over .1 or 11. So you can see that's going to be going to plus infinity. Okay, so that's kind of showing the calculus limit version, vertical ascent toe. Okay, so just kind of note that we're going to minus infinity on this side and plus infinity on that side. Okay, so next thing, let's see if there are any zeros. So um or intercepts let's see what's going on. So let's say if X equals zero then why equals zero. So we have we definitely know our final result is going to go through the point um 00 and um we can maybe we'll just go ahead and check for max and then usually we can even already solve this As is in fact, I would know that here I am bound by my essence totes and here as well, this would be your typical um answer. But let's go ahead and prove it. We want to prove that there are no maximum values and we can prove Arkan cavity. So let's do that real quick. I'm going to need to clear the screen so I'll be right back after clearing half the screen. Alright, so let's go ahead and find the derivative. We'll do quotient roles. So derivative of the top times the bottom minus the derivative of the bottom times the top all over the bottom squared. And let's see that gives us x minus one minus x. Oh, that's nice. The ex is canceled. So that's helpful. So I get um I get -1 over X -1 quantity squared. So we know that Critical points or when that equals zero does not exist. So we have a DNA case. Um so basically we have a critical point um when y prime does not exist and that occurs on the bottom goes to zero or x equals one and notice that that fits with our vertical isotopes. That's cool. Um Okay, so let's look at the second derivative. Well, so therefore if I were to do a sign chart already, let's just do that, let's do this all the way. So if I would do a sign chart with X and Y. Prime effects, we only have one critical critical point at one. This is a DNA case. And if I were to make X really big then my derivative is negative. And if I were to make X really small, I get a square so I'm still negative. So you can see that I do not have a max for men, I don't have Y prime going from plus to minus or minus two plus. So um that's kind of a proof that I see no max's arm in here. Um because we're approaching as some toads, we don't get a maximum limit. All right, so we're almost there. Let's check on cavity real quick and then we'll have solved everything. So if I take the next derivative, first derivative, by the way, I can rewrite as minus X -1, 2 -2. So I can do power role. So I'll get to X -1. 2 -3. General is one. So I get to over X -1 Cubed same idea. This will not equal zero. And I do have a DNA case at X equals one. Again, if I were to do a sign chart this time of Of the 2nd derivative then if I plug in I noticed I always get positive because I have squared on the bottom, there's nothing I can plug in that will give me a negative. So notice that gives me um oh my bad, it's cubed on the bottom, I wrote it. Sloppy. And so I just have to correct what I just said because this is indeed cubed. So if I plug in a big value for X like to I get a positive value, but if I plug in a smaller than one value for X like zero, I will get a negative. So this is negative. So basically we have on the left, concave down which you can see is true here On my graph and on the right past one I do have a concave up so very cool that we can analyze it all the way through. So anyway, hopefully that helped have an amazing day. See you next time.

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