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

In summary, the task is to find the Taylor polynomial of degree 4 for the function f(x)=arctan(11x) about x=0. The Taylor polynomial equation is Tn(x)= f(x)+(fn(x)(x-a)^n)/n! and when taking derivatives of f(x), only odd derivatives will vanish. The first few derivatives given are incorrect.
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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.
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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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  • #2
Those are all wrong.
F'(x)=11/(121x^2+1)
only the odd derivatives will vanish
 

1. What is the general formula for finding the Taylor Polynomials of f(x)=arctan(11x)?

The general formula for finding the Taylor Polynomials of f(x)=arctan(11x) is Tn(x) = f(a) + f'(a)(x-a) + (1/2!)f''(a)(x-a)^2 + ... + (1/n!)f^(n)(a)(x-a)^n, where f^(n) represents the nth derivative of f(x).

2. How do you determine the center and radius of convergence for the Taylor Polynomials of f(x)=arctan(11x)?

The center of convergence for the Taylor Polynomials of f(x)=arctan(11x) is a=0, since the function is centered at x=0. The radius of convergence can be determined by finding the interval of x values for which the series converges, using the ratio test or the root test.

3. Can the Taylor Polynomials of f(x)=arctan(11x) be used to approximate the function for all values of x?

No, the Taylor Polynomials of f(x)=arctan(11x) can only be used to approximate the function for values of x within the radius of convergence. Beyond that, the approximation may not be accurate.

4. What are some common applications of using Taylor Polynomials to approximate functions?

Taylor Polynomials are commonly used in fields such as physics, engineering, and economics to approximate complex functions and make calculations easier. They are also used in numerical analysis to find approximate solutions to equations.

5. How do you determine the degree of the Taylor Polynomials for f(x)=arctan(11x)?

The degree of the Taylor Polynomials for f(x)=arctan(11x) is determined by the number of terms included in the polynomial. The degree increases with each added term and can be adjusted to increase the accuracy of the approximation.

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