Solve T4(x) for Taylor Polynomials of f(x)=arctan(11x)

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SUMMARY

The discussion focuses on finding the fourth-degree Taylor polynomial, T4(x), for the function f(x) = arctan(11x) centered at x = 0. The Taylor polynomial formula Tn(x) = f(x) + (f'(x)(x-a)^n)/n! is applied, but the user encounters issues with derivatives yielding zero when evaluated at x = 0. The correct derivatives are identified, with the first derivative being F'(x) = 11/(121x^2 + 1), and it is noted that only the odd derivatives vanish at x = 0, leading to the conclusion that the even derivatives must be calculated for the polynomial.

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Homework Statement



Find T4(x), the Taylor polynomial of degree 4 of the function f(x)=arctan(11x) about x=0.
(You need to enter a function.)


Homework Equations



The taylor polynomial equation

Tn(x)= f(x)+(fn(x)(x-a)^n)/n!...

The Attempt at a Solution



When I take every derivative of f(x)=arctan(11x) I always end up with an x in the numerator, so when i plug in 0 for x my derivative ends up with 0, so theoretically the answer would just be arctan(11x) which is wrong.

This is what I am getting for my first few derivatives

F'(x)=22x/(11x^2+1)
F''(x)=-484x^2/(11x^2+1)^-2

what am i doing wrong?
 
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Those are all wrong.
F'(x)=11/(121x^2+1)
only the odd derivatives will vanish
 

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