Lagrange error bound. What is it?

This is a sample from the AP calculus exam. "If the function e^(-3x)/27 is approximated using the Maclaurin polynomial 1/27-x/9+x^2/6, what is the Lagrange error bound for the maximum error on |x|<0.5?" The textbook I'm using has no definition for Lagrange error bound, so I don't know what is being asked or how to find it.

Re: Lagrange error bound. What is it?

It's also called the Lagrange remainder.

Given a Taylor series with n terms, the Lagrange remainder is $\displaystyle R_n=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}$ for some $\displaystyle \xi$.

What you need to do is to find $\displaystyle f^{(n+1)}(x)$ and pick $\displaystyle |\xi|<0.5$ and $\displaystyle |x|<0.5$ so that $\displaystyle R_n$ is maximized. That tells you the maximum error when using the Taylor series with n terms as the approximation.