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(a) Approximate $ f $ by a Taylor polynomial with degree $ n $ at the number $ a. $

(b) Use Taylor's Inequality to estimate the accuracy of the approximation $ f(x) \approx T_n(x) $ when $ x $ lies in the given interval.

(c) Check you result in part (b) by graphing $ \mid R_n(x) \mid . $

$ f (x) = x^{-1/2}, $ $ a = 4, $ $ n = 2, $ $ 3.5 \le x \le 4.5 $

(a) $T_{2}=\frac{1}{2}-\frac{x-4}{16}+\frac{3(x-4)^{2}}{256}$(b) The accuracy is less than or equal to 0.000486992(c) From the graph, the error is less than 0.000347963 on the 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, whereas to approximate to given function if effects, using Taylor's pull No meal. And we want to expend that function when X equals or one as he could before. And we are gonna include to terms that I'll be second order approximation. All right now. T two effects will be off this given form on the right. Obviously, we need to find a way to function at a given point. First, ever talk to function a secondary function. So let's start by calculating first and second inevitable given function. FX. Now we're having that X is equal to or a perfect physical extra t up our nectar. What health we find first server to pass negative one Health X to the negative, too. Artery found three or two and half double prime would then be three over four extra negative five or two now and feel led to function and the directors at a given point, for we see that to function well, you is equal to one health first ever. Two is equal to the negative one over 16 and eat second. Everything given point is equal. Three over 1 28 If you plug those guys in to this given expression, we don't find the second drug approximation Teacher of X as one health minus X minus four over 16 plus three times X minus four squared over one 2 56 and that is part es importante. Whereas to find the make yourself this approximation using Taylor's inequality Now we're interested in hypocrisy and we truncated the Siri's right here. However, after the second derivative term, he also have deterred. There were two term that through private affair and I will decide Oh, or that will determine the accuracy officer approximation. So it means that you're interested in and plus one term so that is equal to three then. So let's see what's happening at the Turk territory dysfunction was that three people prime effects? Well, knowing the second damages right here, if you take their two off that with suspect X one more time, we fund for derivative to be negative 15 or eight times x times negative seven over to In order to understand the behavior off the Turk Territo, we look at the fourth territory dysfunction, right? If for derivatives positive, it means that our interpreter is increasing. Aziz, you can see, this is alternating Siri's. So the function itself has a positive, um, constant, and that is equal to one. The first narrative has a negative constant secondary is positive. Turk, derivative, as you can see, is negative. And for terrorism, it means that will be positive of sense, for determinative is positive. It means that act triple prom. Aw, thanks. So 32 will be and increasing function. And since it has a negative sign and sense into given interval exes, always positive, it means that it will be increasing. But the narrative 32 will always be negative. Now imagine you have a function that is always negative, but it's increasing. Solis ETA. We're in church quadrant. We have a function out as X increases dysfunction also increases. And since we're interested in the absolute value, it means that absolute reality off triple prime of X will be maximum at the lower left end. All right. No, we know that, um, absolute value of F triple off the lower end. So 3.5 will be lesson or equal to a constant M. From this. If you like a tart irritant at a given point, we see that M will then be greater than or equal to 15 over eight times, 3.5 to the power off. Seven over, too. And from Taylor cynical T we know that our end off X absolute value effect will be Lester equal to m, divided by n plus one factorial mode spiked by actions that have explained stated power. And plus one, if you like everything in the sea that are too thanks. Well, then, be less than or equal to 15 over eight times three point by to the power of seven over two, divided by three factorial multiplied by 3.5 minus four. Part three. That is equal to 30.0 for nine. We haven't and now idea on the crease here for a purpose, a mission. And we're gonna set out the approximation as this much. Eckert now imports. You were asked to Graff function de error specifically. What is the arid? Well, there is the difference between the approximation and the actual function. Now we have absolute sign on dhe. We know that the second approximation second order approximation is this part, and this is given function it be plot that And if you believe it? We checked the air at the lower, um, bound. So left and point we see that the actual air is 0.0.0 is your 34 So it means that the actual error is less than what we calculated in part B. Don't forget the party. So Taylor's Nikola T. Gives us an idea. It is not the exact error.

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