I require a polynomial $p(x)$ such that $\left|p(x) - \int_0^x \cos{(t^2)} dt\right| < \frac{1}{10!}$ for all $x \in [-1, 1]$. I know that I should probably use the fact that if $m\leq f^{n+1} {(t)} \leq M$ for $t$ in an interval containing the point $a$, then $m \frac{(x-a)^{(n+1)}}{(n+1)!} \leq E_n(x)\leq M \frac{(x-a)^{(n+1)}}{(n+1)!}$ for $x > a$ and $m \frac{(a-x)^{(n+1)}}{(n+1)!} \leq (-1)^{(n+1)}E_n(x)\leq M \frac{(a-x)^{(n+1)}}{(n+1)!}$ for $x < a$, where $E_n(x)$ is the Taylor remainder. These facts combined with the Taylor series for $\cos {w}$ followed by the appropriate substitution should give me the desired polynomial, although I keep getting stuck. Any clarification would be helpful.
Thanks.