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$37-44$(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the int...

Question

$37-44$(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection points.(e) Use the information from parts (a)-(d) to sketch the graphof $f .$$$f(x)= rac{e^{x}}{1+e^{x}}$$

$37-44$ (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)-(d) to sketch the graph of $f .$ $$f(x)=\frac{e^{x}}{1+e^{x}}$$



Answers

$37-44$
(a) Find the vertical and horizontal asymptotes.
(b) Find the intervals of increase or decrease.
(c) Find the local maximum and minimum values.
(d) Find the intervals of concavity and the inflection points.
(e) Use the information from parts (a)-(d) to sketch the graph
of $f .$
$$f(x)=\frac{e^{x}}{1+e^{x}}$$

All right. So first we're being asked to find the vertical and horizontal asking told of affect, So ah, the vertical and jerkers when the parliament equals zero. So in this case, X square plus four equals zero over a few Attempt to solve that You'll quickly realize that you'LL get an imaginary entrance. So there's actually no WeII this the economic chemicals, you know? So there is no class in tow, so no vertical tinto to find the horizontal axis. So we take the limit of the function to infinity and negative infinity. So we'll just tio we can do infinity and so ex women for over four plus four. So there's a couple of ways to convince us you can just look at the leading coefficient and realize that it's just a ratio of one to one. So be one, for if you want to be a bit more rigorous about it, you can actually pull out the factor of X where, so I'm not going to write the limit. I'm just gonna rewrite it, Write it, um, the better idea so I can show you that I'm actually applying limit loss so I pull out X square this will be one minded or over X squared and this will be all over X squared and I'll do one plus for Rex squared and then what will happen and I'll cancel the deck squares And if you actually applied the infinite if you plan to infinity you realized that this function we'll go to one minus four over infinity, which is zero and then four over infinity again one plus zero and I give you one of the one was just one. And if you do too negative, infinity doesn't make any difference because it's squared So you can also do plus or minus infinity I'll still give you one so we have a horizontal no at y equals one. So don't ask youto at Weyco sworn and that all we have for Athens yourself to find where the function is increasing the decreasing we have to apply the first derivative care So the prime of act and this is a question little problem, so you won't have to buy the question rules. So your final product should be sixteen x over X squared plus four and in the square the whole thing that the secret geo sixteen tactical gear only ex clinical zero and there is no ah, no undefined number for this causes squared on the bottom. So don't worry about that for interval evaluations are now we create a sign chart and we look at around zero for the sign of a crime. So if if it's lessened your your values that are negative, you get values are great and you get a crowd of number. And this tells us that it is decreasing from negative infinity to zero and it's increasing from zero to infinity. And there is also local men occurring because it is decreasing and then increasing. So it looks like this. So there is a loophole Men at X ical Tio Now Ah, we're going to look at the con cavity of this functions. So look at the second derivative cast. So we take the second do everyday again will kind of questionable. It's a bit maciver. It should come out to be sixty four minus forty eight x squared all over X square clothes for and and this is cute. They said the sequel is yours. So you do sixty four months excoriate square forty eight expert equal Tio, you were the X We're good. You Then you'LL get X square is equal to sixty four over forty eight and that's the same thing for third. So then this will give us access Decoy. So you take the square root of personal minus to over three because discredited forced to Now we're going to evaluate it around those two number So the sign chart. So this will be negative to over three and it will be a positive to over three. And then you're going to look at this time after Hubble Prime. The people in power is less than negative. Two or three. You get negative numbers between between these two get positive and greater than to get negative. This is Concord. Down, up, down. Oh, you have Kong kept down occurring from negative infinity to two negative to over three and from to over three to infinity And you have Khan caged up crying betweennegative two or three two positive two, three You're inflected Point occurs when the sign changes. So you have inflexion point at both negative two plus and minus two of the three. So you have class of money two over to me. Now you have enough information to draw the graph. Let's go ahead and draw the graph. So we have a horizontal ass enter at like a wild one. Something I didn't draw this red line, so reminder that it is decreasing zero. And it is. Khan gave down. So it's going to be kind of going down like this. So it has come down and then we have a local men and then it starts increasing at zero. It goes up and we have We have a slight you right here. And then it's going to go back to being a cop. So it's going to come give down and then climb up. And then back to be Kong came down. That's gonna look like this and that the graph of f of X, I believe that the bank Yes, exactly.

The problem is part of a find the vertical and horizontal Azem totes we'LL put a so we have half works is equal to x squared plus one square root of X squared plus one bus Thanks Hams scored a tiff Ex Squire Paswan minus X over square root of X squared plus Juan US max which is equal to one over square root of X squared plus one us max. So we're half the limit. Ex gustatory infinity for box is equal to you Cyril No one acts goes to negative infinity from the function With half after wax goes through infinity over half Why come to zero is horizontal Azem tote And there is no work ical Azem told off this function had to be find the intervals of increase or decrease first of computers Derivative This is Echo two Act's over square root of X squared plus one minus one Not his side Axe is last time Explore your past Come on So it's a generative is smaller than Siro We'Ll all relax So we have I've is in decreasing home Negative infinity to infinity How to see find it a local maximum and minimum values from hot be We know there's no local maximum on minimal values Hot tea Find the intervals off Comm cavity onda the inflection points So we're half second. Derivative is the control square root of X squared plus one minus X times one half times Truax times x squared plus one The power ofthe negative behalf over X squared plus one which is equal to one over X square plus one. It was a pop off three over too, which is always grazes and Cyril. So we're half the function after Lex is Can Kev upward negative infinity to infinity. So seriously, no inflection point how to eat Using the information from parts A to B to scratch the graph of us so we can sketch the graph as follows This is a graph of life.

Okay, So you're being asked to find the very corners on asking Tom affect, so Ah, your coin two first, try to find that there is a bit of class and so so critical acid. It occurs when the denominator is equal to deal. So one minute you too, dear. Because you're also added together. Actually get a boner when you put it back and you plug in zero, or you could just Okay, so there's two ways to do it. You can take the natural laws on both sides and then desperately I want this eliminates toe e and national level one zero. So this will be zero is equal to X. So we have a vertical. Yeah, and X equals zero. Now to find the horizontal ass until we take the limit. As X goes to infinity s o, we first find it an infinity. So that would be it. The X over one, minus eighty react you, Khun, do direct substitution. I'm not your ex amputation. So, um, we're gonna pull out a fact. Evita Deac, toe limit of X goes to infinity. You pull out ttx that this will be one and then you do it director here again. So this will be one Overeating, Jack minus one. And then you cancel your Zodiac. And then if you take the Limited's exclusive infinity, this will become Ah, So this will be one over and then one over infinity and zero zero minus ones. This will be negative form. Now, let's take the limit as X goes to negative in Trinity because we'LL be eating the egg over one minute into the uh, for this one, you have to apply. Ah, let's see. You can apply a sort of same principle, but you can think of the graft to remember that the graph of Italy it looks like this, so it it rises. Looks something like this. So it goes off to infinity. But as it goes to negative, infinity approaches zero. So whenever you see limited expression and infinity of ttx, you could assume that easy except zero. So this Khun b AA zero over one minus zero and this will equal zero. So this was really important to have an idea of the shape of some of the important functions the shape of the graph, and it really helps. You were finding that limit when it comes to things like this to make it more intuit intuitive. So we have two horizontal asking me have won at wife with negative point, you know, and waggles Knight of one. No, the fine where the function of increasing the solution flight thie first river after So we find the first derivative You popped the question rule and you get to the back over one minded. You too, Jack squared and way said, this equals zero Andi supposed to find the critical numbers of this function. But since into the extra nautical zero, we have to look for places where this function is undefined. So when the bottom decoders also one minus into the act squared equals here. So you get out of the school by twenty per sciences have become one of us because it acts of desecration, X equals zero and then you do a sign tried evaluation. So add zero. The sign of prime the worth less and zero. It is actually a positive, and if it's great into, it was positive as well. So this function is always increasing so increasing from negative infinity to infinity. No local men or max or no local men or Mac Ah, To find where the function and find the functions Con cavity would take the thing and do every day. This turned out to be e to the X Class e Teo over one minus into the cube. So the secret hero, the talk cannot equals, you know, it is impossible. So we look at where the denominator Because you're also one minutes, Q Yeah. The Cuba taking cube roots. So the warning calls each X again and now we actually calls, you know. So again, we evaluate around zero We were assigned charge. Sorry, that is not I'm straight. Ah, Teo. So keep looking. Value less angio into F double crime. Okay, you get positive numbers and you put greater than doing it. Negative number. So it's Kong gave up and down and we have inflection point at X equals you. So now we have enough information to draw the graph of our function. So we have the function first of all, as it goes to negative infinity approaches zero. So it will come from the road. No. And then it is Khan cave up, So it's going to increase and then I'll go off to infinity because X equals, you know, the vertical Assam toe. And then he said, it is conclave down, but it is also increasing. So the only way to be conquered down an increasing It has to look like this. And we have a horizontal ass in tow at Michael. Negative one. And that makes sense. So it would come up like this. And that is the graph. What I think.

All right. So we are first being asked to find the vertical and horizontal ask until so, um, dysfunction can be rewritten as one over e to the X Square. And now, if you haven't already noticed, um, this bottom function, I'm dysfunction denominator cannot equal zero because that's what defines vertical. I think that's one of the ways. And since exponential function never equal zero, there is no vertical Jacinto. So no vertical ascent toe. And so now we gotta find the horizontal axis until we find that I find the limit. So the limit is X goes to infinity, and this will be one over e to the X squared. And as you can see as, uh, this number guests, uh, the denominator gets exponentially higher. You get one of the infinity, which is a one of a really big numbers, just zero. And this is a case also for negative numbers because negative numbers squared. It's the same thing as positive number squared. So this was also a plus or minus infinity. So we have a horizontal attitude at y equals zero. Yeah, and then now, to find the intervals in which function increases or decreases, we apply the first derivative cath that I've been taking the first derivative. In this case, you will have to apply the chain rule, so I'll come out to be negative. Two X equals negative. Two x 10 e to the negative x squared. Uh, sorry about that. Just bring this down. I was just negative two x or eat the minus X squared and this can be rewritten at negative two X over E to the X squared. Then you set this equal to zero. The denominator cannot equal zero, so that's not one of the critical number. Negative two X equals zero when X is equal to zero. So we have a signed chart evaluation here, so it's going to be X, and you put it at zero, and then you bring it down and they were looking at the sign of a crime. But people are numbers less than zero. You get positive numbers and you fucking greater than zero. You get negative numbers, so we noticed increasing and decreasing. So we have a local max occurring at X equals zero. Uh, and we know that it is increasing from negative infinity to zero, and it is decreasing from zero to infinity. Now, to find where the funk says, uh, find the functions can cavity would take the second derivative tests that they have been trying in the second derivative. So again, this is a little bit more chain rule and, uh, combination of some protocol, and this will come out to be negative one plus two x squared statistical zero. Um, this part of the function cannot obviously equals zero. So we have to set this part of the pumpernickel zero. So this will be negative one plus two x squared because you have to find a critical number. You add one divide by Tuesday. You got X square called 1/2. This will be X equals. The square root of one is one. So we one over root two plus or minus because we took a square root. Now we can do a sign chart evaluation. This will be negative. 1/2. It would be positive one over. Richer, not a line, but not a line. Looking at the sign of a double prime your fucking values less than negative one or two. You get positive numbers between these two negative and positive. Still can't give up. Down, up. So we have a con cave up interval occurring from negative infinity. Two negative 1/2. And from one over to to infinity. We have concrete down occurring between negative one over to positive 1/2. And we have inflection points occurring when sign change occurs. So that's occurred that both plus and minus route to I mean plus or minus, uh, 1/2. So inflection point point occurred that plus or minus 1/2. Now we have enough information to draw a graph. So it looks something like this. Uh, you know that there is a local max at zero, and we have a concave up shape from here from the negative. So we're kind of coming up like I use so this is like, like up you, and then I'll turn into a concave down shape. So now we start decreasing after zero, you come down and and then I'll go off to zero. And this this max occurring a gray here. Um, the peak is supposed we had zero Mhm. Sorry, if that wasn't clear enough, but this is the graph of F WebEx and it's symmetrical. That's it.


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