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Find the Taylor polynomials $ T_3(x) $ for the function $ f $ centered at the number $ a $ Graph $ f $ and $ T_3 $ on the same screen.

$ f(x) = \tan^{-1} x, $ $ a = 1 $

$T_{3}(x)=\frac{\pi}{4}+\frac{(x-1)}{2}-\frac{(x-1)^{2}}{4}+\frac{(x-1)^{3}}{12}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 11

Applications of Taylor Polynomials

Sequences

Series

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it's problem, whereas to expend tension in verse effects. Iran one. Using Taylor'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'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'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'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'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

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