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University of North Texas

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Problem 17 Medium Difficulty

Use the guidelines of this section to sketch the curve.

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

Answer

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Video Transcript

We want to use the guidelines in this section to sketch seeker of why is he to expires one over x squared. So they give us this laundry list of steps First bowl. So let's just go through them one by one. So the first thing they tell us is we should find our domain of dysfunction to remember the domain and just tells us where the function is to find it. And so, for rational functions, we just need to make sure that our denominator is not equal to zero. And so we'll x where Dottie go Observe would just mean ex can't be zero. So that was pretty straightforward for us. We're going right underneath. It's our domain was going to be all remembers Excluding zero negative Infinity to zero union, zero to infinity. The next thing they suggest that we find is our intercepts. And so to find the why interest up that is when X is going to be equal zero. But that is excluded from our domain. So we know right away that we have No Why intercept? No. Why intercept? And then to find our ex intercepts, we set the numerator equal to zero and so well, that would just be exploits women secret zero at the one over. So we end up giving that are ex intercept is going to be the 0.10 Next, we want to determine if our function has any kind of symmetry so pure it are rational. Functions are normally not known for being periodic, so we can just go ahead and skip checking that. But we can check to see if it is even or odd. So what we're gonna do, just go ahead and plug in negative X and or in tow? Why, in this case, and see what we get out for our input. So I was gonna be negative X minus one over negative X squared, satisfactory that negative out of the numerator of negative times X plus one all over and then negative X squared should just be X squared. Now, this is not equal to effort, Becks or negative FX. So that means this is going to be neither even nor odd. So neither. So there's gonna be no symmetry or this one. Now, the next thing we need to Chuck is for any kind of acid trips. So ask me toots. So we're already told that over here this would be a vertical awesome total zero. Because, remember, that's just where our function diverges to positive or negative infinity. And when we divide by zero, that's essentially what we're doing. So execute zero would be our boat across in tow and now to bind our horizontal. Remember, we would take the limit as X approaches, plus or minus infinity of why. And you might recall from algebra when the numerator is smaller than the denominator. This just goes to zero. So we'd end up with our or is on tool. Ask himto being the line. Why is equal to zero? All right, so next we want to figure out where the functions increasing or decreasing, and if it has any local men's or Max is so. This is to separate steps, but I think we should go ahead and combine them into one because we need the increasing decreasing to determine local Ben's anyways. So local man slash Mex. So for this, we're going to need to know what why Prime is. So let's go ahead and go to another page to do this really quickly, so I don't wanna have to use question rule to do this. What I'm going to do is divide X word into each. So we'd have one over X minus one over X squared. Now, when we take the derivative, this is just using power roll twice. So we have, like, prime is equal to. So it would be negative. One over X squared, plus two over x cute. And then if we want, we can go ahead and combine that in tow. One fraction by getting a common denominator. So it be negative X plus two all over X squared. So let's go ahead and set the secret zero so we can find our critical numbers. So we get X is equal to zero, so we have negative X plus two equals zero or X is equal to two. So this might be our local men are Max. And then remember, we also want any of the values where the functions undefined and well, that would just be when X is equal to zero. So these were the two points we need to check or increasing decrease. Or I should say how we're gonna break our intervals. So this here waas to minus X over X cute. So let's go ahead and write the numbers that we had. So he had X is equal to zero X is equal to. And so remember, our domain is not even defined at X is equal to zero. So we don't even need to really worry about that being a maximum. But we will need this just to help us for our intervals. All right, so the first thing we're going to do, actually, I should say this first, So we know that a function is going to be decreasing when F prime of X is less than zero. So this is decreasing, and we know a function is going to be increasing when this derivative is positive. So we need to just figure out where this is broken up. And so, by breaking it up into its critical values like this, it will help us. So let's pick a number smaller than zero and plug it into here. So, like, let's do negative one. So to minus minus, one would be three and the negative one cute would be negative one so they could do three. Divided by negative one is negative three. So that means it's going to be negative to the left of X is equal to zero. Now we pick a number between zero and two. So let's do one. So plugging in one we'd get to minus one, which is one and then one. Cubed is also one, so it would be positive there and then we plug it a number larger than two. And doing that would give us um so it's 23 so two minus three is negative one and then three. Huge is 27 so negative one over 27 is negative would be decreasing like that. So since X is equal to zero is undefined. By applying the first derivative test here, we would know that X is equal to two since it's being increased into it. And the decreasing doctor is a local max and now or where it's decreasing. So it's just where the down arrows are, so it should be decreasing from negative infinity to zero union to to infinity. And then after that, we will have the functions increasing just from 0 to 2. All right, so we have all of that now, and the last thing we need to do before we actually start crafting is find our intervals for con cavity as well as any points of inflection that we might have. So for this, remember, we need to know what our second derivative it's. So let's go ahead and find that. So just like before, I'm going to use this to take the derivative because I don't like using questionable. So again we'll just use power rule for each of those. So the 1st 1 would give us two over X cubed than minus six over X to the fourth. And then we can go ahead and combine this into one fraction, and that would give us two X minus six over X to the fourth. And now to find our possible points of Kong cavity, remember, we're gonna go ahead, set this equal to zero, and so that would give us two X minus six. Physical zero. Let's divide each side by too. So we get X is equal to X minus three is equal to zero at the three over. So we get X is equal to three. So this is one of our possible points of inflection. And then remember, we also want to make sure to look at points where this is not equal to Earth, where the denominators equal to zero which would just be a zero. So began just gonna be zero and so are derivative was two X minus six over X to the fourth and our possible points of inflection We're going to be X is equal to zero X is equal to three. Now X is equal to zero again is undefined So we don't need to worry about this. It'll so undefined for our original functions Domain Now X equals three weaken. Try to figure out if that is a point of inflection, So it's gonna be the same thing as before. Points of inflection just means we have a change in con cavity. Oh, we know that if the second derivative is less than zero, this is going to be con cave down. And if our second derivative is larger than zero, this will be con cave. So just like what we did or the derivative, we're gonna pick numbers on either side of how this special function is partitioned. So I must do negative one. So noticed that anything I plug into X to the fourth is always going to be positive. Zero. So we only really need to figure out if the numerator is gonna be positive. Negative. So plugging negative one into their We get two times negative one. So we get negative to minus six. Negative eight. So it's gonna be negative over here now when we pick a number larger than zero Well, this is going to give us. So let's do one. So plugging in one, we get two times 12 minus six, which would be negative for So it's still gonna be calm, keep down, and then we pick a number of larger than three. So let's just before so two times four is eight minus six is too. So we'll be calm cave up on the other side. So notice how we have a changing from cavity around three. So this would be a point of inflection. And then for Conkey about calm, keep down. So we said that it will be calm. Keep down to the left of zeros the negative 30 to 0 union and then zero up to three and then you'll be con cave up from three to infinity. All right? No, that was the last thing we needed to do, so we can go ahead and start sketching. That's so let's go ahead and put a little grid down right here. So the first thing we should probably do is pot our intercepts. So we only had 10 so that should be about here. We had no symmetry. We know that we have a horizontal and burgle awesome toe at zero for each. So So why is equal to zero? We have one, and then also X is equal to zero. We have now some troops. We know that we'll have a max. Our local max set X is equal to 12 So there's a local max somewhere around there. And then we also have a point of con cavity at X is equal to Marie. All right, so now let's think about how we can graft this. Let me go ahead and make soap it darker. So let's start from our X intercept. So on this interval here, this would be in this portion here, so it's saying the function is increasing up till two. So that means we're going to keep on increasing. Until we had our local Max So we hit that. And then it's as we start decreasing after that. So we keep on decreasing like this. And then we have this point of inflection at three. So around here is going to start to change Con cavity. So it looks like from three on word, it is calm. Keep up. And before, that's Colin kept out. All right, so now to the left of our X intercept. So it's still supposed to be increasing, so it just keeps on going like this until we hit our intercept here and now to decide on what side we need to start on over here, then that means we're going to do the following. So we look at X squared because so are denominator is one over X squared, and so remember this has the same in behavior on either side of it. So that means we should start on the same side like this, and we're gonna just keep on coming up. It doesn't have any kind of inflection. Mentor Max on this side, it's the only thing we need to now do is get as close to our horizontal Assam tote as we can as we go out to infinity, so I'll just go ahead and put a little thought here to say, That's our point of inflection. This here is our local, and it's actually are just Max all around. And that would describe just a quick little sketch of this. So if you wanted, you could go in and say what the local Max is and where the point of inflection is. But if we're just sketching it, I think this would be a good place to stop.