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University of North Texas

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Problem 12 Medium Difficulty

Use the guidelines of this section to sketch the curve.

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

Answer

see solution

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Video Transcript

we want to sketched a row of Why is he the one plus one explosive Rex? Where now? I went ahead and just added these up because this was really irrational function in disguise. And we might need to I realize this is a rational function so we can start looking at horizontal and vertical ascent tips. Now, normally, the first thing I would do is try to factor the top and bottom. But we know that X squared plus X plus one does not factor. So our numerator will never even to zero, so we won't be able to factor. And we will build to simplify this at all. All right, so first, um, in Destructor, it gives us a list of steps we should follow. Do you sketch the graph of function? The personal is upon the domain. Now the only thing we will need to worry about is our denominator not be zero. So that would just be where X cannot be. So be negative. 30 20 Union zero depending. Next we want to find her intercepts so and triceps. Now, we already said there's gonna be no ex intercepts just due to the fact our new Marie cannot factor and to find our why intercept. You're gonna go head let X equal to zero. Why intercept? Be zero squared, plus zero plus one over zero. But that's undefined. Oh, undefined. So there's no why intercept either. So we're gonna have no intercepts for our function. Next, we want to look for whether there is any symmetry. So rational functions are really known for being periodic. So we will instead to check to see if this is even her on function. So we can determine if this is a mature about the why access or the origin. So just plugging at Meg new X, we have negative X squared minus X plus one over negative X squared. Well, that's going to be X word mind sex host one over X squared. And this here is not equal to negative FX nor F of X. So that tells us we're going to have no symmetry, barely snow symmetry about E why access and origin. The next thing they tell us to look for our ass in tips so we know when we have the same power are horizontal. Ask him to will be just dividing the largest powers coefficients. So the limit as ex purchase both positive and negative. Infinity, Oh, of X is going to just be full 1,000,000,000. Kokoshin at the top is one. And the bottom is one. So no, just one. Now, limit has x a purchase from the right, and we want to look from the right because we'll have a vertical ascent Toe at X is equal to zero. Since when we plugged in zero, we could be dividing by zero. So oh, uh, ex. So let's just go ahead and plug this end. Zero from the right squared, plus zero from the right, plus one over zero on the right squared. Now, if I square something that is slightly larger than zero is going to be positive, something slightly larger than zero will be positive. And one this positive. So adding, all those up in the new mayor Wow. Something that's positive, are you Marie? And when I square something slightly larger than zero, I'll also be positive. So this year will approach positive that now we can look at the limit as X approaches zero from the left of the Lexx. And so zero from the left squared plus syrah from the left, plus one over zero from the west squared. All right, so just like before, we could go ahead and look at what happens here. So zero from the left will be its slightly small negative number squaring that would make it positive. Zero from the left is gonna be negative, and one is gonna be negative. So now you know, adding to posit numbers will be positive. And then we have to decide if adding something slightly smaller than zero 21 will make it negative. Well, if we make that term small enough than it will get bored by one So we can go ahead and say that our new mayor is going to be positive. And if we square something slightly to the left of zero, even though it's negative, it will become possible. So this will also go to positive infinity. So we have our acidosis done. The next thing they suggest we find is our intervals of increasing and decreasing. And any local max is or men's we may have. So we're going to need to find why front. So let's go ahead and do that on the next page. So why is it to one? So for this, I'm going to use the original equation. Since I don't wanna have to use questionable to take this derivative, you could use question wrong. We'll get the same answer. I just want to be lazy when I do this. So taking the derivative of this derivative of one is going to be zero the derivative of one of her ex. So let's go ahead and rewrite this. So this is extra negative First power and extra negative second power. So to take the derivative of each of those reeds powerful. So the negative one, thanks to the negative second power and then plus make it two hex Judy Negative Third power. And then we could go ahead and rewrite this. Using a bit of algebra was negative one. And actually one factor that negative out. So negative one over X plus two over X cubed. And then we can add those to get X plus two over X. You should've been expired there. All right, so now let's go ahead and figure out where this function is equal to zero at, so the denominator doesn't matter. So it's only matters when the numerator is equal to zero. So you do. X Plus two is equal to zero. R X equals negative too. So this is one of our critical points and also we get a critical point. When are the denominator is equal to zero since it would be undefined. So ex physical zero. But we know our original function ex cannot be equal to zero. So we can just go ahead and ignore this. But keep in mind normally when we have undefined points in our the river, that bills are still critical. All right, so let's go ahead and used that. So get wide prime is able to negative X plus two over. Exe Cute. Now we want to determine where dysfunction is strictly larger than zero intricately, less than zero. So why prime will be strictly largeness or or this will be increasing win. So I want to have an already solved this beforehand. This was taken brevity and it's really negative. 2 to 0 for when dysfunctions Christian. Next. We want to find where the function is decreasing or when y prime is strictly less than zero. And again I went ahead and solved. This would be negative. Indeed too negative too. Union 02 infinity. All right. And so we only found one critical value before being X is equal to negative too. So to be left of negative, too, we would be on negative infinity to negative too. And the function is decreasing there and then on negative to zero. The function is increasing. So this tells us exes at X is equal to negative too. So at exited with a negative to this, there will be a local minimum by using the first real test, right? And then the next thing they said. So this was supposed to be five antics, not just by and then seven. They tell us we should find our cavity as well as any inflection points. So we're gonna need to know what? Why? Double promise. So let's go back over here and we have wide of the point. So that's going to be the derivative of first ribbit. And again, I'm going to take the derivative of That's where we distribute the X squared. But just because once again, I don't wanna have to do the questionable so weaken factor that negative out and then take the derivative of each of those. So one over X words derivative is going to be negative. Two over X. You remember just using the power rule again, just like you did before and then plus negative six over X to the fore. Now we can go ahead and distribute that negative, which would cancel those outside plus plus plus and then combining those into one fraction. We get to expose six over X. Now, let's go ahead. And something secret deserve to find our possible point of inflection. So again, the denominator could never make this equal to zero. So be two X plus six is equal to zero or X is equal to negative three. And again, we would probably need to check our denominator. But in this case, since we know executed cannot meet with zero, and it's also not our domain, we don't need to worry about that. All right, so let's go ahead. All right. Our second revenue, which should be few X plus six over X to the poor. Now, this is going to be con que Well, when by double promise sugar larger than zero. And again, I'm gonna head and solve this. And this should give us Negative 3 to 0. Union zero to infinity and it's going to be Kong Cave down when? Why? Don't cry, Miss Trick. Listen zero which is going to be negative entity to negative three. Now we found that we have a possible point of inflection at X is you do negative three. Let's go ahead and check. So to the left of negative three, this function is Khan Cave down and to the right of negative three. This will be Khan caged up. So this is indeed a point of inflection. Now that we have all that, the next thing they tell us we should do is go ahead and ground. So we are told we have no intercepts. We have no symmetry. So let's go ahead and plot are asking toots. So we found in behavior. Suppose goto one. So we have a vertical. The board's on blast. I'm sorry. Why is it you want? So why is it that one? And we have a vertical lassitude at X is equal to and to the right of zero. We know we're approaching positive energy and also to left of there. We know we're approaching positivity. Next we know we're going to have a minimum at exited with a negative, too. So negative. And we're going to have a point of inflection at X equals negative three. So those points are gonna be important. So let's go ahead and start to the right over ground. So to the right of zero, we have no faxes. Men's the inflection points. So we know we just need to approach our horizontal awesome, too. Approach it from above. Next, we know at exited with a negative to we're going to have some kind of minimum here. I'm going to read. Actually, we'll have some minimum here, so it's going to calm like this and hit exited with a negative, too. And then by the time we hit negative three, it should be going from pocketbook to con cave down. So it should look something kind of like this here, and we'll keep getting closer and closer to why is equal to what? And actually, let's go ahead and find what? Why is he going to negative too? Should, because maybe we actually do go blow are whores on housing. So let's go ahead and find better over here, so of negative too is going to equal to or minus two plus one over or so it is going to be three force. So actually we do get below. So let me go ahead, action. Fix this. So here we have our minimum and we go where we didn't below all horizontal acid trip and then we're going to come back up and they get very close to a horizontal stones from the bottom. So if you have Max's and men's, it is a good idea to go and check it, because otherwise you might do what I want to do it first, a kind of not noticed that we should be going below the horizontal access here, and we could maybe put some values for some of the points. But since we're just sketching, I think that's all that really matters for us.