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For f (x) = x + 2sin € , I > 0 :Find all - and y- intercepts.b) Find all asymptotes (if any). You must use limits to justify your answer_Find all critical ...

Question

For f (x) = x + 2sin € , I > 0 :Find all - and y- intercepts.b) Find all asymptotes (if any). You must use limits to justify your answer_Find all critical numbers_d) Find all intervals of increase and decrease_Find and classify all local extrema (if any) _f) Find the intervals where f (z) is concave up Or concave down.Find all points of inflection (if any)h) Sketch a graph of f (x_ and label all the points found in parts a-g:

For f (x) = x + 2sin € , I > 0 : Find all - and y- intercepts. b) Find all asymptotes (if any). You must use limits to justify your answer_ Find all critical numbers_ d) Find all intervals of increase and decrease_ Find and classify all local extrema (if any) _ f) Find the intervals where f (z) is concave up Or concave down. Find all points of inflection (if any) h) Sketch a graph of f (x_ and label all the points found in parts a-g:



Answers

(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 $f$ $$f(x)=e^{2 x+4 x}$$

For this program. Um, first part, this about the house methodic. So for the for today, the horizontal symptomatic. We need to evaluate this to limits. So we send X two positive infinity off. If, um, so the result off this will be zero. Because, um, this is because when we evaluates this kind of limits we meet you converts this function to a fraction toe a kind the difference of the square formula. So we have, um, one the white divided If I something are really, really large. So the result is there and the on only on a hand, we also need to evaluates, um, when x approaches to negative infinitely. What's it for? FEC's? So, in this case, um, the result will be just knew finished. So that means very so. 100 Santos in the totemic. Which is why, because zero as x goes toe positive. Infinity. As for the verticals, sympathetic, there's neng because, um, this function is defined it on the whole, um, interval from minus infinity toe quality of the affinity. This is product. And for probably we need to find, um, the purity of first said your activities this, um So it's just about the chin role so we can see. Um, the denominator is always positive because it's a squared off, some quality of thing and the for the numerator. It's also positive because eggs is always sorry. You should be negative because X is always less than Rudolph X Square Pass 12 cities. We just square boats that we have X square less than next passport, which is always true. Okay, so there's no quit your point. That means ah, if prime is always negative. So if is one anatomically decreasing on minus infinity joke infinity. And for the inflection point and the concave Biti's we need to take the second curative. So we have one over one process X square to, ah, 3/2 which is also positive. Full cool, thanks in the interval. Next. Injecting infinity to apart it here for you? Definitely. So if is, um Com cave up for it and the least known infection point. Based on this information, we can sketch the graph off this function, which is a mountain Nikolay decrease in function with with the horizontal sympathetic XY Y causes zero. So it looks like this if I want to be decorated. We can finally y intercept. So the Y intercept way Just project X equals zero wherever one year. Okay, so this is the graph. The sketch off graph off this function.

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.

For this program. First, we need to figure out the s Antarctic. Um so because little man often this function a yes for O minus infinity to infinity. So there is no vertical A sympathetic, but for the for is not so sympathetic. We need to evaluate the following two limits. X goes to infinity and X goes to negative infinity off so I can see that act engine goes toe pie half when x goes to infinity name Is this the mediators to eat the pie have and the act engine goes to negative. I have as x goes to connective infant. So the second image because to you to the ah minus pi half that means we have 213 with politics. Um so the first nice white close to you to the pie half has X goes to positive infinity the second ways. Why cause to e to the minus by half as X goes to minus the affinity. As for the increasing and decreasing cover with tech literate tive And there we see that this the first of the narrative it's known active because both components are asked tricky, positive them use if you see Increasing. Oh, on minus Infinity to infinity. No way! I want to find a can carry t So we take the steak on the narrative than we have you to be. Car engine Hicks. Um Times one minus two x over one plus x square in Madison Square here. So he said the second of the directive because of your we have x equals to 1/2. So from from negative infinity to 1/2 second, the narrative is, um what is positive? The muse. Ah, if his concrete fuck only the interval one have to infinity a safe on the purity of these Negative. So you can keep down now we are ready to schedule graph off f. So, um first, let's put to hurry down toe. I've seen Tom Dick here. Um, so there's another inflection point at 1/2. So the function the great for function Well, looks like this. So first this increasing Montagny clean crazy bats can keep up. Um, when you passed this one, have it becomes conclave thumb. So there is the inflection point of this. At this point, X equals Do I have? And then there are two horizontal sympathetic. So this point is e to the minus. I have another point is to the Italy pie half

Okay, so we're being asked to find the vertical and horizontal as hotel of after Beckers. So in order to do that, we look for where there is such a point where his function will have an under finding point and in this case is a little bit more obvious. So at the bottom, you could see that if you plug in zero, we'll get one plus one over zero plus one over one over deal squared, which is still zero, and this is an undefined ah value. So it is un undefined and it will go off to some sort of infinity. We don't know. We got off the positive or negative acquired further evaluation Should I mean ex surgical zoo with the vertical acto, And then to find out whether there is a horizontal jacinto, All we do is we apply limit. So we look at the limit as it goes to positive infinity and goes negative, infinite limited except twenty of one plus one over X class one of X squared. What happened is as Oko do One of her ex will go to zero because as the bottom gets infinitely big against closer and closer to zero. And so this will be zero. And also there is a just noticed There's a slight mistake. I believe, Um, this should be a minus. So let's just fix that. So then this will be minus zero and this is still one. And if you do negative infinity, you also get the same answers. So this is the limit as X goes to positive. Negative. So you have a warzone passenger at quiet Call one for now to find where the increases or decreases we got to find the first derivative cast. I mean, taking the first derivative. So this will come out to be negative one over X squared, plus two over execute. Oh, also, yeah. And that's where we set this equal to zero toe. Bring over one of X square shall be too over execute equal one over x squared and and less cute squared. So multiply both sides by execute and then we'LL have excuse over X squared, which is just X So that means too is equal to X and we also also we have to take in account vertical ascent. Oh, so we have very glass into X equals zero. So we're going to evaluate it. Also at X equals zero when we're doing a signed chart evaluation. So we'll do it at zero and two. We're looking at the sign for a crime. So if you played about Weston zero you got negative numbers you're talking about between gentry. Positive number. You're talking about greater than two. You get negative number, there's going to decrease, increase and decrease. Also remember that zero is a vertical. Ask himto on our original graph because if you look at it when we did the re evaluated earlier, we said that X ical do the vertical ass in tow. So you would think there's a local men here, but there's not s. So there's no local men. We do have a local max occurring to you. The local Max on tactical No, to find the interval of con cavity, we apply the second derivative cast. So we find a double crime and that comes out to be too over. Execute minus six. Rex extra forth without this secrecy. Well, you bring up nice. That's a terrible zero. Sorry about that. And then we add six over extend fourth, both eyes. We got to x cubed equals sticks over actually fourth, Multiply both sides back and forth. So you have extra phones over X cube, which is just again access would be to X equals six and then this will be X equals three and way Do a sign Chart evaluation. Ah, around three. So this will be with three looking at sign of double crime. But he broke It varies Lesson three You get negative number of putting value greater than three A positive number. So you know that it is Khan cave down when his negative infinity to positive three and con came up from three to infinity also, I forgot to do it on the increasing decreasing So we know it is decreasing from negative infinity to zero and from to to infinity And we knows increasing from between Tonto first the left his page. Now we know the con cavity. We can also identify the inflection point to the sign change occurring around ex ical stories. So inflexion point technical story. So now what we do is we have all the information to go on a graph. So I'm gonna go ahead and jar our horses down to ass until right here, so so looked something like this. This is why calls one and we have a local max. It too was crash it Well, we'Ll draw that in. So what the graph will do is we had con cave down for all values Less than three basically so And we cannot go above this line or this is that Is that less truth? So we first we'll have a decreasing So this is the horizontal ascent Open There's a river to class in tow And this is concave down Shake I was going to look something like this I'll go down and then it goes down to infinity Get closer and closer to the X axis and then it comes up We haven't increasing between zero and two, so it goes up Uh uh And okay, they'll be coming. Ah, no slightly go above like with one because we have a local max it too, and then it changes con cavity at three. So we get a conclave up shape, so we have to make a slightly you shape and it will go straight off into the horizontal acid up. So right here's to and this is the graph of F Becks President


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