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University of North Texas



Problem 25 Medium Difficulty

Use the guidelines of this section to sketch the curve.

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


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Video Transcript

we want to sketch the curve. Why is equal to X over the square root of X squared, plus one, the destructor. They give us this laundry list off steps that we should follow. So let's go ahead and just fall. So the first thing they tell us to look for is our domain. Well, since this is a rational function, we need to make sure denominator, never zero well expert plus one will never equal to zero. You'll always be greater than equal to one and or the square root. We need to make sure the inside is going to be strictly greater than that. So we could really just say that we need to ensure that X squared plus one is strictly larger than zero, which in this case it is. So our domain is just going to be all real numbers or negative infinity to infinity. The next thing they tell us look for is our intercepts. So let's find our Exeter sentence first. Remember, X intercept is where the function is equal to. 00 is equal to X over expert plus one square brooded, and this would be only where the murders of reserve so zero X would be where our X intercept occurred. And it just so happens that when exiting zero, it also tells us our wires. So our origin will be our only interest up to this case. Next, we could go ahead and look at the symmetry of our function. So functions like this normally aren't known for being periodic. But we can at least check it F of negative X will give us, um, it being even or odd. And we could use the Cimatu about you buy access or the origin, so plug it and negative X will give us negative X in the new mayor and negative X squared, which would just be X squared plus one. And this here is equal to negative effects. So we know we have an odd function so odd, and that means symmetry about origin. So whatever happens on the right of the function, we will know will happen on the left. We just need to make all the values negative and on the right, I mean to the right of X equals zero. All right, so let's go ahead and look at this symmetry or the isotopes now of our function so we're not going to have any vertical assam tokes due to the fact that are denominator will never equal to zero. But we will have in behavior, um, for a going to impending it. Negative 30. So this might take a little bit of space. Let's do this on a new page. So we want to look at the limit as X approaches Infinity Oh, X over the square root of expert plus one. Now, since this is essentially a rational function, remember to find the in behavior of this. What we would do is identify a large fire in the new mayor and denominator and then just divide the top bottom by that. So in the new merits X and the denominator, it's X squared. But we're square routing it so it would also be accepted on there. So overall, the large power is actually want to divide the top bottom by or most by the top one by one of Rex. So doing that, we get one in our numerator and in the denominator. When I bring in that one of her ex into the square root, I'm going need to square it. So that's going to give that. This here is one plus one over X squared. And now we can go ahead and fly the limit as excess affinity. And so this term here goes to zero. So we would just have one over the square root of one, which is what Now we could go ahead and do the exact same thing that we just did or negative Infinity. I'm just gonna say f bucks, But if we go ahead and use the fact that this is an odd function, well, that means we can rewrite this as the limit as ex purchased infinity of negative FX. And this this since odd function ml of spectral that negative when we get negative limit as expert purchase infinity of f of X, which will be just negative one. So you could go through the same steps as what we did before. But you have to remember, when we pull in, this bottom one will have to multiply by a negative sign early would just switch the signs like we did before. But I think doing that this way would be a little bit. So we now know are in behavior will be one and negative one. All right, so let's go ahead and write that over here. So we have the limit. As X approaches, infinity of F of X is going to be one and the limit. As X approaches negative infinity of Ethel X is going to be equal to negative one now the next steps 56 or to look for intervals with a function increase and decrease. And we also want to use that information to find out if we have any local maxes or men's. So we'll need to find what why Prime is, too so lister that are another page again. So again, why is into X over expert plus one? So take the derivative of this. We're gonna use the questionable. So why prime is going to equal to So it's low d high minus hide below all over square root's below. So the ex of what we have in the numerator minus them the opposite order and x squared plus one square brooded all over the denominator school. Where'd so that the X squared plus one and we would only have absolute value on the outside here. But that's what plus one is always going to be larger than one, so we don't need to worry about that right? And so now the derivative of X is just going to be one the derivative of square root of expert plus one. Well, we'll need to use powerful and changeable. So 1st 1 powerful will get 1/2 X squared, plus one to the negative 1/2 power. And then the derivative on the inside is going to be two X now just save some time. I did this algebra earlier, and that should give us one over X squared plus one to the three house power. Now you might notice that this here is strictly larger than zero for all values, since we'll never equal zero since our numerator is just one. And our denominator will also always be larger than zero since X squared plus one is always larger than one. So we have no critical values for this, and we know the function is always increasing. So let's go ahead and write that down. So why prime waas one over X squared plus one to the 3/2 power and we know that function is increasing when y prime is strictly larger than zero, which happens to be just all real numbers, and then that tells us our function is decreasing with my partners. Listen, zero never and again we'll have no critical value since that function never is equal to zero and why prime is not undefined anywhere. The next thing we want to do is to check for absent of the acid trips, but calm cavity of the function and any inflection points we lay, I have. So we're going to need to know what? Why? Double prime minutes. So let's go ahead and do that. So this time I'm going to first rewrite our function here as X squared, plus one to the negative three house power. That way, we don't need to use questionable, and we could just use power and change. So why Double prime is going to equal to negative free, huh? X squared plus one Now to the negative five pass power. And then we take the dribble on the inside, which is going to be two x. So let's just read like this really quickly and doing that will give negative one are negative. Three x Oh, actually, I was looking at the wrong thing, so this year should be negative. Three x over X squared plus one 35 House. Now, this function here can equal to zero and no equal zero when X is equal to zero since three bits of the numerous able to zero. So we'll go ahead and use this as a possible point of inflection. So we have by double prime waas negative three x over X squared plus one to the 5/2 hour and we know that our function is going to be calm. Kate up. So why double prime stripper Large zero is calm. Keep up on the interval, zero to infinity and the function is going to be Kong cave down. Or why double prime is less than zero on negative infinity to zero. So we said our possible point of inflection is secret zero. So to the left of X is equal to zero, the function is going to be con cave down and to the right of zero, the function is going to be calm. Keep up. So, actually, um, these intervals here should actually be opposite. I wrote them down incorrectly. It sees just because the denominator is always going to be bigger than zero. So the only thing that really matters is when negative X is bigger than zero and less than zero. So this should actually be I'm glad I caught that that the function is concave up on negative infinity to zero and conk eight down from zero to infinity. Yeah, so is calling Kate down to the right of zero and Khan caged up to the left of zero. So it's so going to the inflection point, but we'll need to know this con cavity or the con cavity won't really make much sense when we actually go to sketch it. All right, so we know our intercept was at 00 and we know we have acid totes at Why is a good one and why is it too negative? One bye, little one? Why is it a negative one? And we have no local Max's term ends, But we do know we have an inflection point at X is equal to zero. So let's start on the right side, and then we can use the symmetry of our function. You just draw it pretty much in the opposite manner, going towards negativity. So we know the function used to be conch aid down to the right of zero and it should always be increasing. So it'll look something kind of like this, But we have a horse on cross to it. Why is it the one? So we need to get so Lee closer. And since we know this is it odd function, we have symmetry about the origin. So we'll just draw it in the exact opposite manner going into the other quadrant like so. And we see its Khan gave up still and it would still be increasing from negative zero. So that works up. So you could maybe go on and draw one or two different values here. But as long as we show that we have the change in Kong cavity at X is equal zero. Since we see toe left, its can't give up into the right company down, and it's always increasing. I think that's all we really need to show in this one. So I would stop here