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In Discussion 8, you used the Taylor polynomial of f(z) Vx centered at a = 8 with n = 3 to approximate 10p3(x) = 2 + (2 _ 2) - (z - 2)2 + (x _ 2)3 288 20736Use Tayl...

Question

In Discussion 8, you used the Taylor polynomial of f(z) Vx centered at a = 8 with n = 3 to approximate 10p3(x) = 2 + (2 _ 2) - (z - 2)2 + (x _ 2)3 288 20736Use Taylor s Remainder theorem to find an upper bound on the magnitude of the remainder (the error) of the approximation. This topic was covered in the two videos OHl Taylor's Remainder theorem in module 11.3.b) Assuming that the exact value of V10 is given by calculator the absolute error of the approximation can be computed asV1o P3(10

In Discussion 8, you used the Taylor polynomial of f(z) Vx centered at a = 8 with n = 3 to approximate 10 p3(x) = 2 + (2 _ 2) - (z - 2)2 + (x _ 2)3 288 20736 Use Taylor s Remainder theorem to find an upper bound on the magnitude of the remainder (the error) of the approximation. This topic was covered in the two videos OHl Taylor's Remainder theorem in module 11.3. b) Assuming that the exact value of V10 is given by calculator the absolute error of the approximation can be computed as V1o P3(10)| ~ 2.7x10-4 Compare the absolute error with your bound from part a) Does your bound from seem reasonable?



Answers

Use the remainder term to estimate the absolute error in approximating the following quantities with the nth-order Taylor polynomial centered at $0 .$ Estimates are not unique. $$\sin 0.3 ; n=4$$

Miss Proper and based on the remainder theorem, we know that our necks is defended by implants. Want thought iterative some constancy over and press one factorial times X to the in person since the centuries zero So in this case being passed. So I thought that curative you see either plus or minus strikes or plus or minus costs are off X depending on I mean teacher end. But off this case is we have the upper bound off this and press went over the narrative which is given by one. So are four picks. It's less than it costs to next to the five divided by five factorial and we need to taken up. So the value. So in our case, for sigh off 0.3 the era term just prodding 0.3 into this are are four yes, lesser cost 20.3 to 5/5 factorial which is about two times 10 to the minus five

For this problem with set effects to be 10 Genetics. So the remainder term hard to X is defined by, uh, on the third of the purity for F at some constancy divided by three factorial times x cubed Since we home folks on the center a question zero So the third older curative off if is given by two times um, Seacon square comes on the break It she kind of square cross twice often giant squid. I'm saying this remember formula C his pigeon on the center zero index. So in this case, we are looking for our two off three upon three. Um, so this term, because Teoh the upset the value off the third of the narrative off see divided by three factorial times 2.3 cube. So to evaluates, um, this derogative terms, we can check that. Hey, I felt during takeoff obvious increasing on the interval 0 to 0.3 ne muse, the third Earl Curative off See you spawned by the third of the narrative at 0.3. So with a calculator, we can finally get the pound. The upper bound for the are to your 0.3, which is lesser recourse to the third older purity of about 0.3 derived by three factorial tends to your 30.3 Q which is about, UH, 1.3 times 10 to the minus two.

In this programme we are using further the tailor Parnham Noto. Approximate The function is so right out to the duty by the beginning, but off the mother. Same because it's an exponential function. The center is a close to zero, so we have f zero he caused to F prime zero. He goes to EFTA Blueprint zero because the F trip up around zero because the one so the p three because to one process X plus one have X square plus 1/6 x cubed. We want to use this p three toe a proximity to the 0.12 which is f off their 0.12 um so we can use the third or the tailor for nominal toe. Approximate this value. So we have 1.1 to 7 for 88 Now, if you use a calculator, you 20 0.12 It's about, um, I'm going. 127496 menus. The Minnesota era. It's about eight times 10 to the minus six

For this program. Um, we choose our eighth to be route off. One miners are one process X and we want to use p three toe. Approximate this function centre that zero. So we first we take curative. So first of the narrative Sekondi Generative and the third alliterative then plug in X equals zero. So we have this four results. So by the definition, p three, we choose the submission and from 0 to 3, the ends of immaturity of zero divide by n factorial times x months zero to the power and the Pragyan off this curative. So we have this result. So this is P three up to a cubic term. Okay, 76 16. No, we want to approximate this value this number which is a full 0.6 and then we use p three to approximate Dave. So just project 0.6 into a p three. So we have this result and we can also use calculator to directly obtain this result for the root off 1.6 so we can see the areas around five to the five times 10 to the minus seven


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