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Use l'Hospital's Rule to help find the asymptotes of $f$ . Then use them, together with information from $f^{\prime}$ and $f^{\prime \prime},$ to sketch the graph of $f .$ Check your work with a graphing device.$f(x)=x e^{-x^{2}}$

$(0,0),\left(\sqrt{\frac{3}{2}}, \sqrt{\frac{3}{2}} e^{-\frac{3}{2}}\right),\left(-\sqrt{\frac{3}{2}},-\sqrt{\frac{3}{2}} e^{-\frac{3}{2}}\right)$

Calculus 1 / AB

Chapter 4

Applications of Derivatives

Section 3

L'Hospital's Rule: Comparing Rates of Growth

Differentiation

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So you want, do you want to grab the function x times e with the minus x square of x, in equal to that? And so after seeing pore, analysis shows us that this function is because, if we pugin minus x, without minus x times to the minus minus x, squared which all this part over here, that is just minus x square, so that this whole expression is equal to minus X, minus x, it seems like i did nothing, but all these analysis is to conclude that you can factor a miners there and or this is the value of the function of x. So this simple analysis shows us that minus x is minus f r x. So this function is so so starting off by that, let's compute! Well, what is the horizontal asymptotes? So what happens when you pontelima 6 cost infinity with this function? So all would have here infinity times e to the minus infinity point: that's going to go to 0 to infinity times 0. So what is that? Well, you don't know, but you can write this expression, as the limit x goes to infinity of x, divided by e to the end, since it is dividing. The sign of the exponents is minus so divided by e to the power of x squared. So in this form we apply. The limit as x goes to infinity we're going to have infinity over an infinity which is the appropriate form for lobido, so we can apply hopital sidoine. This limit is going to be equal to the linear cost infinity of differentiating that over differentiating that so doing that or this limit is going to be equal to it while they, the lining has goes to infinity of other ative of x is 1. The relative of e to the x square is 2 x coming from the chain from the chain rule e to the x squared. So it's gonna be of the form 1 and i say exposition. Baby 3 x squared goes to infinity so 1 over infinite infinity over osso. This limit that limit will be equal to 0, so y goles 0. That'S a horizontal syntaone 100 on well. We can go ahead and have a atlanta and then, from this analysis, what is going to happen. 6 goes to infinity, so the lie 6 goes to minus in 6 to 2 minus infinity, or this is going to be equal to changing. This sign here here goes to war, is infinity, is the same as minus x? To so you got equal to the limit. Is minus x goes to infinity. Well, since we complete the minus x in the argument of this function, that is equal to the limit as x goes to or some other variable x, prime goes to infinity were minus x, prime sort of all that is going to be equal to that. To say that this limit should be equal to minus. The limit of 6 goes to infinity of faithful facts because the function is sobur. You already computed that 1. That is right there it's going to be minus 0, which is equal to 0. So we have the 2 horizontal asymptotes. Both of them are 0 and let's compute, what is prime, so prime. We differentiate this equation or you have the gotiate product so differentiate the first 1, so that is just 1, so e to be minus x, squared plus the first 1 times the relative of the second plus 6 times to the minus x minus x, squared times. Zat of the minus x, squared so times minus 2 x, so putting that together factoring factoring e to the minus 6 square is going to be equal to e to then minus x, squared and then 1 minus 2 x, squared to x, squared the soyo. Have that so orsteinsson first relative is this so that the second deity will be differentiating. Dotata would be double prime, so we differentiate this or differentiate in the first 1. So you get a minus 4 x minus 4 x times e to the minus x are now guys times. The first 1 minus x squared is rative of the second 1, which is going to be in the exponential t times minus 2 x. So putting together that you can, from this expression we can factor the e to the minus x, squared and then well. This part is going to be equal to the times that to minus 2 x and that time plus 2 x, cube x, cube so that all these can be factored as or we have minus minus 4 minus 2 point. So it would be minus 4 minus 2. So it would be minus 6 x, plus 2 x, cube and well again from this. We can further an our x. So it's gonna be e to the minus x squared times x in se, minus 6 plus 2 x squared. So you see here well, since they have there, we have the the fact that the function is old. We only need to do the analysis for the rilatives for say we do it for positive x and then the other side is going to be just reflecting that there origin so reflection. So let's just draw what happens: let's restrict our analysis first for x, positive. So from looking at what is prime, we see that prime is going to be positive for vol 4 x betfabsolute value smaller than 1, so you'll have here x, square 1, half because this comes out from what is the zeros 1 minus 2 makes squared is equal To 0, so then you see that 1 half is equal to x, squared and so so that these will be positive for the apter for 4 x, between minus 1 over square root, 2 and 1. Over square root of 2 point so were restricted to was dix. We have our boxes on. Let me have 01 over square root of 2. You know that the function is increasing there, then so she ate a tiger and then decreasing and now for the cavity. We see that for if double prime you see that here now for positive x, this will be positive, always e to the minus x squared times x. That'S always positive, and now here you will have positive ugality for x, 4 days 6 houston, because this is 0 x equal to x, squared equals to 6 hops so for x square that will be concave up, so we'll get c x is equal to 3 times. Twont, so you can solve that and then you eso, you can get a ma forher so for x, bourdon square root of 3. It'S going to be concave up so for the second liberty. Ye'Ll get 4 square of 3. Like some point around. There is going to be concave up converted there on this conclave bound before and so ll. You know that this function for positive x x to the minus x square for x, positive- this is always positive and so is 00 is just a maximum at 1 over square root of 2 because they relative there 0 and then these stars start decreasing, and it Has a changing gravity at square root of 3, so it's gonna, look something like this for positive, so athesis, concave down here, okay down and then from there on its decreasing, but with positive on the limit of sex, goes to infinity 0. So from this point on it becomes concave on the net work. The analysis there with the the initial all the function is sot sotafor, our graph, you can just you can just be through the origin and we're going to get our 4 goes. So it should look something like that and they have their vertical deorizontal synods going to see goes through, plus or minus infinity.

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