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Find the Taylor polynomial Tn(x) for the function f at the number a. Graph f and T3 on the same paper: f (1) = Ce 6T a =0.n =3Ta(x)...

Question

Find the Taylor polynomial Tn(x) for the function f at the number a. Graph f and T3 on the same paper: f (1) = Ce 6T a =0.n =3Ta(x)

Find the Taylor polynomial Tn(x) for the function f at the number a. Graph f and T3 on the same paper: f (1) = Ce 6T a =0.n =3 Ta(x)



Answers

Find the Taylor polynomial $T_{3}(x)$ for the function $f$ centered at the number $a$. Graph $f$ and $T_{3}$ on the same screen.

All right, I'm worried. Welcome back, Su. What are we doing today? Let's say we're in chapter 11, Section 11, telephone or meals. Page 7 74. Number seven which says let ffx equals Ln We want an expansion of this function with the third degree Taylor polynomial centered at a equals one. So, in other words, we can't do McLaurin expansion because l n and its derivatives air all in defined it to your own thing because f prime, you know, is one of Rex. And if you're right, it is a power rule. Then the rest is just power functions at double prime is negative. X the negative two or feel like to write it in rational terms and have triple crime with then being positive to 10 text Negative three. So we evaluate this a one if of one island of 10 mostly, you know, logs are zero at one. Your base to what power is one, right. So unless it's translated and then f prime at one is gonna be 1/1 is one on the double prime. It one is gonna be negative. 1/1 squared, which is a negative one, and if triple prime and one is gonna be to over one cubes, which is just to So what is our Taylor expansion look like? We're gonna proximate f of X. This dot means approximated with a piece of three of X at third degree telephone only. Such that we have f of zero times X minus a 200 factor, but X minus 120 is one of zero. Picture was one time 00 So let's see. What? That Well, I guess you could read it. Explain this one to the zero of zero editorial plus f prime, which is one times x one is a which is one to the 1/1 factorial right, technically plus F double prime, which is negative one times X minus a, which is X minus one. We're expanding near ones or center convergences one. Or at least it's good. At least that one, hopefully have already covered. Created 0/2 factorial. Mm hmm. And have triple prime, which is two plus two times X minus a or X minus one huge over three factors. All right, so what happens here is the first term goes away and we have X minus one to the 1/1 Victoria minus, uh, X minus one squared over two. Factor over just to plus now 203 factorial is 263. So x minus one cubed over three. So this is a serious that loses the pictorials. All right, we'll get the signal notation. We're gonna find out that it's over and that, in fact, already So that's my third degree extension. Let's graphic. So let's go toe computer out of the system and we're gonna say f is Ln so weaken graphic compared to our estimate. Let's show that make sure there's no syntax errors or anything weird. And let's do peace of three of X is Okay, So what do we have? We had X minus one minus X minus one. Squared over two to Victoria. Plus it's an alternating science. Right, let's x minus 16 Powers of X minus one is what ISS X minus one is typical for a log huge over three on. Let's show that I'm sure everything is working correctly. Ignore the output on the bottom from last question. We didn't I didn't run anything yet here. So then I put it has nothing to do with anybody right now. All right, So what's your food? Here goes. And in this system, log means Island Log Basie. Okay. And that's exactly we wrote. Okay, Didn't expand or anything, I guess. A good expanded if you want to. But why? Well, just for fun, what does the expansion look like? We'll show guys. It's a terrible thing. And there's a problem here. Okay, but nobody else that. So let's not get crazy. Okay, so let's grab this stuff. So let's plot list and directors I landed and we was scion. And then p three, the second graph is gonna be green with a third graf. I think it would give you a magenta and keep on changing colors. All right. And we're expanding there. One, right? So I've always say, from 0 to 2. So the center of the screen, in terms of three x coordinates, is the center of the radios to convert the center of the interval Convergence south week. And let's see how good the green curve matches the blue cross blue curve. Isela. Mm. And Alan is not to find out here. So that's a problem. That's What? All these error messages there. Um, now, wait a minute already too. Oh, sorry. At square bracket goes here. Not here. A graft. Waggles to also that's better. Okay, so at when its exact both functions of zero and near one the green curve in the blue curve for the san curve overlap pretty well. There had here diverges three green. Why? Values, servant overestimate. And also over here. But at least from almost 0.5 to almost 1.5 radiance, we have an interval. Convergence with center at one. Right. OK, so there's your output time. There's your input. Does your work hope that was helpful Memory. Good luck with your Mark. See you next time. Bye bye.

Hello. Over one. We're working on problem three. We have that f of X music with two x two native one effort to issue to 1/2. A primal vex is equal to negative one exterminated too. If crime of two is equal to native one force F double prime of X is he could too. Exterminator, three f double primal, too musical, too. Thanks. And F triple prime off X is equal to native six extra money for F triple crime invalidated a two musical native. 6/16. We know the tiller serious centered at two ISS f of x which is equal to one over X which is equal to the summation. Angles from zero to infinity off negative one to the end times and factorial times two to the native and plus one times X minds to and and then divided by n factorial. Okay, so, um, the red part likes minus two to the end. And the factorial is always in the formal. While this one quite right here it is. What? We came up from the derivatives. So we have negative one to the end. Um, because as you could tell, we go from Ah, positive number. And then we'll tow a native number. And it keeps going like this. And now the in factorial boys. Because off the one right here, So the one would be your zero factorial. And then here, that which is one. And then this one right here would be your one factorial, which is one. And then the national would meet to effect oil, which is to and so on. Now the two to the negative end, plus one to to the his partner here. Um, that one is taking care off the bottom says you can see right here. That too. So that, too, is to to the negative zero plays one which would be to to the native ones, too, to NATO oneness 1/2. Okay. And then you're to simplify this. We're going to get the summation of angles from zero to infinity of negative one to end times two to the native and close one times X minus two to the end. Okay, from here. Um, we need to write the Taylor palling around the 3rd 1 So we're running up. I need a 1 to 0 times two to a negative general. Plus one times X wants to zero plus native one to the one times to tune in and watch. Cost one close negative one Squared times two to native to plus one times x minus two square plus negative one cubed times, two to the negative three plus one times X minus two Cube If you were to simplify this, you'll get uh huh minus 1/4 times X minus two. Close 18 Turner's ex Myers two Squared Might has won over 16 times. Backs Myers to here and now It's located graph. So this red one over here No one is your third Taylor Poland on you and then the black one right there. That is our function that is our one over Works, as you could see, is looking like it's following here, up to the point right. But then, from very kind of Dealey

This problem, whereas to expend x type she's your negative X ray Ron zero using Taylor's polynomial off order three. We know that the expression for tennis polo for off ordered three as given, uh, here and we can see that our function a has to sell functions that are functions off. I said his ex and needs you too negative each of the X And instead of calculating t three effects of proto order approximation directly, were going to say that you threw effects physical to hurt order approximation off age multiplied by third order approximation of G, where this X is our function, age and eater Negative X is our function Jeep. And here we can see that any to calculate first order, second order and third order derivatives. So let's do that separately for both age and G, he had h of eggs as equals. X h crime affects would be one h double primer. Facts would be zero an h triple prime. A fax would also be zero. He also have GI effects as you go to eat negative checks, so G prime effects would be negative to each negative. Two ex g double primer, fax would be four each nation to X and the key triple primal facts would be negative. Eight. Eat a negative to X or it. Now let's first start from third order approximation of function. H. We have to function social f off. Eight Sorry h of X as eggs and since we're trying to let that around zero ff zero would be zero plus one times x 10 plus zero since secondary to Syrian. Third order there with his also zero. So farmers we find a three effects. His eagles act so that is equal to you. Your function itself well, let's find G three effects that is each of the negative. Two times zero plus native to Egypt e zero times X minus zero plus four times each of zero times X minus zero squared over two factorial plus NATO eight each of the zero times X minus zero cubed or t factorial. From this, we find GT effects as one one's two X plus two Expert minus age X cubed over three factorial. So now less confident. E three effects for months playing those two. So we have X times one one's two x plus two Expert on this eight X cubed over three factorial that is equal to X minus two X cried plus two X cubed minus state extra to fort over three factorial. Now the important thing here is that we're interested in third order approximation, so we could have most have third order terms. So we're gonna drop this term and we're going to say that this is the part that we're interested in. So that is our T three of X. Where else asked a graft to function. And that's approximation of the same graph. We have our function. Thanks. As does this is function f effects died Isaac of X times each donated to axe. We also have our church order approx mission where we know the function and that is how it behaves. It fellows, the function pretty good around here and here it overshoots and after this point on the negative cited follows function and now they're shoots. So that would be our third order approximation to that function. TT off X

Hello. Everyone does his problem. Number or function is tension in verse. Oh, thanks. And is going to be centered at one. Okay, so our function of Elena one is a point of tension and worse off horn, which is going to be higher once we take the first derivative X is going to be 1/1 plus x square, which is equal to one plus x squared. It is a negative one and we developed it out. One. We're gonna get one for her. One costs one which is a little on that. Taking the second road, er, we're going to get a native to times one plus x square, time to and a relative. And that one We're going t native to times one and one plus one to the to which is gonna equal donated to whore which is equal Need Hong now taking the third derivative. What is going to be negative too? Times one plus x squared to the to plus eight x weird times one plus x square to the new three. Any validators function one we're gonna get native 1/2 plus four, which is equal to 7/2 and now the third to pull in the room is going to pie for plus one, huh? Over one factorial times backs ones one minus one. And then times two Dorial and then Excellent. So one square plush, huh? 03 factorial, times X minus one, Kim and simple flying. This We're going Teoh pyre for Plus Thanks, oneness. 12 lines, X minus one squared only four lost 7 12 times X minus one. You now looking our function right here. This red one right here. He was going to the older tension and her silverbacks and the screen one is going to be our third Tell your Taylor polynomial and, um


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