Problem 65 Hard Difficulty

Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.

$ y = \dfrac{x^2}{x - 1} $

Answer

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Video Transcript

So you're the Dominion cannot be won. So X can't be one. Because it's a rational function and the denominator cannot equals your intercepts. We have 00 symmetry. There is none for us and toes. We have one X equals one. Like we prove this previously said, the denominator cannot equal zero. So X equals one is our vertical and we also have a slant as in tow. Let's find out pulling a mule division of the way to go so exponentially. Okay, How many times can x go into X squared? We left with an X and now multiply X times x x squared. We're gonna switch the sign to cancel the first term X times minus one would normally be negative. Expert, We're gonna make it a positive X. Okay, so cancels. And we're left with X. Okay, X and X plus one one time sex. What it's saying Cancel the first term. One time was one sort of sign cause more. Okay, we could no longer continue polynomial division because this power is, um and we don't have a pact. We don't have any exes here. And this power is hot. You're so we cannot continue So our oceans, it tells us that we have this land hasn't do it. Why you called Exports one. And now we. And now we can do the graphing for the s and toes. But first, let's do, um Why prime? So we know what's going on with local Min and Max. So why prime is X squared minus two X over X minus one squared kosher Earl. Great. That equals said it tickles zero and get X equals zero and actually close to. So if this is our first derivative test, we're testing points around her critical points to see that it's increasing, decreasing and then increasing on these intervals, making this one a local Max and this one a local men. Right? And just so we know we already know that 00 zero's and intercept. You don't see every two. That's four. Just so you know the why value okay. And when we try to sell for our why, double prime will see that it's inconclusive. So just go ahead and graft because we can't use the con cavity in this case. All right, so let's say this is one. This is not going to be drawn into skill. What? We're gonna do a rough sketch. Say this is one. Um, say this is one. Let's say that's too. Okay. And we have us a total X equals one. Well, first of all, the inter sub 00 It's easy. Any of it intercepts X equals one. It was right here. And we have a slant. One at y equals X plus one. And she was gonna be approximately here. So does the refuge. Okay. And now we have a local maximum at 00 which is here. So that means it's going to be increasing and decreasing so increasing and then decreasing. All right. And now we can look at our second half the grafts. We know that it's gonna be We have, ah, minimum value at 24 So let's say, because you see, this is for Okay, So let's just say this is approximately 24 This is a minimum value, and it's gonna be decreasing in an increasing, and that's her graph