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(a) Find the Taylor polynomials up to degree 5 for $ f (x) = sin x $ centered at $ a = 0. $ Graph $ f $ and these polynomials on a common screen.

(b) Evaluate $ f $ and these polynomials at $ x = \pi/4, \pi/2, $ and $ \pi $.

(c) Comment on how the Taylor polynomials converge to $ f(x). $

(a) $\begin{array}{|c|c|c|l|}\hline n & f^{(n)}(x) & f^{(n)}(0) & T_{n}(x) \\\hline 0 & \sin x & 0 & 0 \\1 & \cos x & 1 & x \\2 & -\sin x & 0 & x \\3 & -\cos x & -1 & x-\frac{1}{6} x^{3} \\4 & \sin x & 0 & x-\frac{1}{6} x^{3} \\5 & \cos x & 1 & x-\frac{1}{6} x^{3}+\frac{1}{120} x^{5} \\\hline\end{array}$(b) $\begin{array}{|c|c|c|c|c|c|}\hline x & f & T_{0}(x) & T_{1}(x)=T_{2}(x) & T_{3}(x)=T_{4}(x) & T_{5}(x) \\\hline \frac{\pi}{4} & 0.7071 & 0 & 0.7854 & 0.7047 & 0.7071 \\\frac{\pi}{2} & 1 & 0 & 1.5708 & 0.9248 & 1.0045 \\\pi & 0 & 0 & 3.1416 & -2.0261 & 0.5240 \\\hline\end{array}$(c) As $n$ increases, $T_{n}(x)$ is a good approximation to $f(x)$ on a larger and larger interval.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 11

Applications of Taylor Polynomials

Sequences

Series

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in this problem where us to explain side effects using Taylor's full normal, up to degree off five vino that, uh, Taylor's suspension of the Siri's is off form like this, and we can approximate a given function of effects around the point using De Ritter's and actually high order derivatives. So since we are needing to expend it up to or drove five, we need, um, if prime of X after double prime effects, F three put prime of X A four off eggs as well as F five ex. All right, we know that affects a sign effects or the first tear it off that would be co sign effects. David, of course, and would be negative side effects. There is a negative sign is negative. Go sign effects. There were talking. I get it consigned its side effects, and it is it has a psychic pattern is you can see now the critical 50 Every two would again become side effects. May also less a lead dysfunction, um, Iran's 20.0. So one A is zero. We have f prime of zero as one f double problem. Zero is zero f triple prime of syrup is negative one F four off zero is zero since sign 00 in Cosenza is one F five off, 0/5 orbiter off F What else would be? Well, all right now we have pretty much everything that we need. All we need to do is to take these values and employ it in to this question. Damn, he can estimate approximate sign effects. Pass. Um, sign of zero plus f prime of Syria's wants. So one times X minus zero plus if secondary to zero. So I had zero plus 30. Ritter is negative on so that it's negative. One off X minus zero cube divide by three Factorial 40 with zero Soviet plus zero plus 50 Ritter. Oh yes, one. So we have one times X minus 0 to 25 divided by five sectorial. You know that signs here is also zero. So we can ride this one s x minus X cube over three factorial plus exit effect or five factorial From this, what we see is this We see that t zero so's your order is equal to zero. First and second order approximations are equal and that is equal to this term or disturb so that is equal to X. Also, third and fort order approximations are also equal, and that is equal to X minus execute plus three. Factorial said this is up to this part now. We're also asked to graft dysfunctions. You could use any OLAND function, Porter, if you do so, we see that Miss Blaker is the origin of election of effects, which is side effects, that the first sort of rubber information gives us that straight line. That is the red line. This green car is the third order or second order, since they're equal approximation and the blue one is the future order approximation. The idea is, that s we increase the order or as we had more and more terms, the approximation gets closer and closer to the board, you know, function. So let's write it here as, um en increases be approximation gets better. All right. We're also asked to you will a function and the approximations at few given points And those points are so you wanna have Brooks Mitt or every light effects T bone off x t three of fax 95 for fax at points pi over four, power two and pie So let's make a table. All right? Signed pile before s screwed of two or two. That is a good point. 7071 Signed. Pirate too is equal to want and sign a pi zero. So let's fill the table forward for Starter Perks Mission. Since our first order approximation is here, we can see it won't as X uh, whatever value be, have whatever are going to be had for the angle that will return to someone. So, um, first order approximation of what that pyre for will be equal to fire before. And that is able to 0.79 That will be good to pirate too, which is? Five star 1.57 and first order approximation. What excess pi is equal to apply and week, unless write this one as 3.14 No. For third order of purse mission. Uh, this is the expression and using that, uh, playing in the values, we find out what excess pull over for Turtle wrote Turk. Older folks mission is 0.704 when expire to third Order gives us 0.92 And when exes pie 30 or perks mission gives us later. Two points your 26 Um, the whole thing is thief you further approximation. So let's look at that. Does this t fire for facts using this implanting the X X values we see that when expire over for youth order approximation is 0.7 sierra one for prior to that is almost one and four pi. It is even in point of fact before, as you can see, especially for pyro for and pie over to fifth order Brooks mission is against pretty much the same results as orginal function. However, when X is equal to pi, B should terrifically get 04 to function. However, both heard and Ford are approximations are far away from real answer reason. We actually can see it graphically. Here, Um, we should close the X axis right here. However, fifth order approximation overshoots the answer and we can see that it is positive and third order approximation under shoots The answer. So, in order to get our approximate that function at that point accurately, we would need to add more points

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