use-a-computer-algebra-system-to-find-the-taylor-polynomials-t_n-centered-at-a-for-n-2-3-4-5-then--2

Question

Use a computer algebra system to find the Taylor polynomials $ T_n $ centered at $ a $ for $ n $ = 2, 3, 4, 5. Then Graph these polynomials and $ f $ on the same screen.

$ f (x) = \sqrt [3]{1 + x^2,} $		$ a = 0 $

Doruk Isik · Accepted Answer

indescribable. Where us? To find Taylor&#x27;s polynomial off foot cracks around zero for orders, off to tree for five. Then, whereas to use a computer system, I used Wolfram Alpha. You&#x27;re free to use any so afraid that you&#x27;d like. And, uh, this is what to be got 48 order approximation off this function around zero using Taylor&#x27;s pull No meals here. What we observe is does this first term is our zeroth order perks mission. Um, this is our second order approximation. Since we have the power to this is the fourth order approximation. This is the sixth order, and this whole thing is that the eighth order here? We&#x27;re missing the old terms. If you notice we&#x27;re gonna say that tea for sort of approximation is, um, equal to want, just like t off zero. We&#x27;re going to say that second order approximation teaching FX is excreted or three plus one. We know that Fort order approximation is that we know that third order approximation will be go to second order approximation. Also, we can see that Ford order approximation will be equal to fifth order approximation, and that is equal to x 34th or negative off exit. Afford over nine plus X corridor three plus one. Where else has to graft that again? You could use an alleged function plotter since you know the function itself and the approximations to balance function. Here we see that, Bobby Yeah, t not so zero order approximation as just a straight line. And the idea here is that as we increase the number of terms, as you can see that we&#x27;re capturing the exact behavior that is our original function off effects is behaving as you can see, not off the functions captured a behavior that this Ford order function is capturing. If he one of the capture the behavior more accurately and for a longer period of you&#x27;d need to include more terms and use high order derivatives.

Doruk Isik · Answer

in this problem. Where us? To use a computer system to find Taylor polynomial off. Attention. Expert on pyro for off order. 234 and five are usable from ALPA. You&#x27;re free to use anything or any system or so fair that you want. Let&#x27;s just break it down. Um, this is Taylor&#x27;s expansion off. Attention, ex around given point. Powerful pile before off border five. From this, we see it at this first turn as a zero order approximation. This is the first or first order. This would be the 2nd 1 Then it&#x27;s up up to this ter turn. This would be the third order approximation and up to the fourth turn. That would be the fourth quarter approximation. And this whole thing is, if you do order approximation, where else has to grab all on the same graph as well aside, orginal function. This orange jersey, as you can see, are the orginal functions. Those are pretension off X and B. How approximations here. The idea is that, as you can see here, as we increased the number of terms archers get closer and closer to the real solution and if he were to include more terms, you would have a more secret approximation, So we would need to use the higher order approximation

Doruk Isik · Answer

in this problem. Where is to find church order approximation? Um, off each of the eggs and we&#x27;re going to expend each of the ex using Taylor&#x27;s polynomial. All right, now, Bess is what the third order of prop special looks like. So we always need the 1st 2nd and third order derivatives to function so less right. Our function again FX is you two d x first Ferriter. What else to be judy ex second there were two would be to the X and turn Would be to DX as well now less of a way to function of given points. So, um, we have fo form f primal phone f double primal for three, plus primary one. Since all the functions a recall, we have e and e NAMI have everything that we need. So from this turd order approximation around this given point, all functioning to the eggs are using Tell Taylor&#x27;s Porter meals would be, um e plus e times X minus one plus e times X minus one over to factorial squared plus e times X minus one Cube delighted by three factorial. This is deterred order approximation were also asked to graft. Does the function itself and its approximation. Uh, this is what E J D X looks like. This is the original function. Each of the eggs and the approximation looks like this since the function and austerity is our equal. And since Orel, Attila, this is a pretty easy function to guess. Even church order approximation does a pretty good job on it. Almost, um, captures the behavior of the original function.

Doruk Isik · Answer

this problem, whereas to expend x type she&#x27;s your negative X ray Ron zero using Taylor&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;re interested in third order approximation, so we could have most have third order terms. So we&#x27;re gonna drop this term and we&#x27;re going to say that this is the part that we&#x27;re interested in. So that is our T three of X. Where else asked a graft to function. And that&#x27;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&#x27;re shoots. So that would be our third order approximation to that function. TT off X

Doruk Isik · Answer

it&#x27;s problem, whereas to expend tension in verse effects. Iran one. Using Taylor&#x27;s polynomial off order three and this given expression is detailers expansion off any function? Um off order three. All right now, As we can see, we need the 1st 72nd derivative and the third their trucks function. Let&#x27;s cock let those first we have a perfect sense. Tension, Numerous effects. If prime effects would be one over one plus X grade, there was talk that would be negative. Two eggs divided by one plus X squared squared. And there was talk that so Third river 12 x would be six x cried months two divided by one plus X rayed. Cute. All right, Now we have everything that we need. So let&#x27;s plug goes in. He had done t three of X s tangent universe off one plus one or were one plus one. This is the director Times X minus one plus negative to over one plus one squared times X minus one described or two factorial plus six minus two divided by ah, one plus one cube divided by X minus one. Shoot over three factorial. All right, you know we can. Then, right t three effects as synced Agent Inverse off one is part before we&#x27;re going to start with pyro for plus X minus one over to minus X minus one. Squared over four plus X minus one cube over 12. Well, hard. I get that 12. I have four here in the numerator and 16 the denominator here. Also, I have eight here, So we have four over 48 that ISS one over 12 or hurl sass to graft to function and the approximation on the same graph via our function f if extension universal eggs Acting like this, that says F f ax is equal to tension in verse effects. And our we have our approximation, Turk order approximation. And that is behaving as such. It is following to function pretty get up Their point on in deviates from the function Also it pretty much that was the same behavior here. Although I&#x27;m missing one thing this enough function became FX Behaves as such. All right, Okay, now this wrecker is our third order approximation that girl. As you can see up to a region up plus side this interval, its approximate they function pretty good and it be wanted a better approximation maw record when we would need to add more terms

Doruk Isik · Answer

in this problem, whereas to expend nation Le Goff eggs using tethers full meal off order three on dhe around the given points or one a sequel to one. As you can see, we need to let the function at the given point. And 1st 2nd and third derivative Celeste Chocolate. First word 1st 2nd and third order Derivatives First knowing that two functions then she&#x27;ll look effects first sort of derivative is one over X. From this, we calculate the second order derivatives as negative one over X cried and third order there too. Oh, thanks as to or X Cube. So now let&#x27;s just use this expression and find turd Order for expression of dysfunction. He have a natural log off one plus one over one caps X minus one plus negative one over one times X minus one. Spread over two factorial. Plus we have to over one times X minus. Warm, cute or three Factorial. Do you know that natural ago form as zero. From this, we find court order approximation to dysfunction as X minus one minus X minus one. Scribed over to factorial plus X minus. Phone cubed over feet. How did I get that tree. Well, I know that three factories Eagle 26 on diesel can slot and we&#x27;ll end up with three in the denominator. All right, You know the expression. Now, where s a graft to function and this approximation on the same graph? Um, we know that this is how the national lug bikes behaves. This is one and the approximation church order approximation behaves as such. Well, I didn&#x27;t like this part. That me drove it again. All right? Okay. It&#x27;s better now, is it, Jesse? Since this given function is, um, fairly a simple function, even the third order approximation does a pretty good job. And it captures pretty much the behavior in a way, you can see pretty large interval.

Problem

(a) Approximate $ f $ by a Taylor polynomial with…

Problem 12 Medium Difficulty

Use a computer algebra system to find the Taylor polynomials $ T_n $ centered at $ a $ for $ n $ = 2, 3, 4, 5. Then Graph these polynomials and $ f $ on the same screen.

$ f (x) = \sqrt [3]{1 + x^2,} $ $ a = 0 $

Answer

$\sqrt[3]{1+x^{2}} \approx T_{5}(x)=1+\frac{1}{3} x^{2}-\frac{1}{9} x^{4}$

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$\sqrt[3]{1+x^{2}} \approx T_{5}(x)=1+\frac{1}{3} x^{2}-\frac{1}{9} x^{4}$

Calculus

Grace H.

Caleb E.

Kristen K.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

James StewartCalculus

Grace H.

Caleb E.

Kristen K.

Samuel H.

James Stewart

Calculus