Originally Posted by

**billa** I am supposed to use Taylor's theorem to approximate the solutions to

$\displaystyle 2x^2=x sin(x)+cos(x)$ and give a bound on the error

I computed the 4th degree Taylor polynomial, $\displaystyle T_4$ of the right hand side at x=0. I reasoned that the intersection must occur when $\displaystyle |x|<1$, so I bounded the error between the Taylor polynomial and the function within $\displaystyle \pm \frac{1}{24}$.

Then I computed the intersection between $\displaystyle T_4-\frac{1}{24}$ and $\displaystyle 2x^2$ and between $\displaystyle T_4+\frac{1}{24}$ and $\displaystyle 2x^2$. So the solution to the original problem must be between these values. So I took the average of the values as the approximation and half the difference as the bounds for the error...

How does this look? I just ask because my answer had a square root within a square root, and this feels like it is as difficult to approximate as the answer to the original question.

Thanks.