Given $\displaystyle F(x)=\frac{1+4x^2}{1+x^4}$,

Plot the error of the approximation on the interval [-1, 1] using the data points $\displaystyle x_0=-1,~ x_1=0, ~x_2=1$.

Now, the formula for the error is given as:

$\displaystyle f(x)-p_{2m+1}=(x-x_0)^2(x-x_1)^2\cdots(x-x_m)^2\frac{f^{(2m+2)}(\xi)}{(2m+2)!}$

where $\displaystyle \xi$ is in an interval containing $\displaystyle x$ and $\displaystyle x_i$ (i = 0, 1, 2, ..., m).

Ok, so, even though I've found the actual approximating polynomial, $\displaystyle p_4$ (not 5 since the highest power has zero coefficient), I don't actually need it in this formula, from what I see.

So I can compute the sixth derivative of F(x) in Maple, which is very long to post. And (2m+2)! = 6! = 720. (m=2).

But what do I do about this $\displaystyle \xi$ thing?! I.e. How do I find the correct expression to plot?