The Lagrange Error Bound, or Taylor's Remainder Theorem, provides an upper bound on the error of approximating a function $f(x)$ with its n-th degree Taylor polynomial $P_n(x)$ centered at $a$.

The Formula:

The error, or remainder $R_n(x) = f(x) - P_n(x)$, is bounded as follows:

$$ |R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1} $$

Finding M:

The value $M$ is the most crucial part. You must find the maximum absolute value of the $(n+1)$-th derivative of the function on the interval between $a$ and $x$.

$$ M = \max_{z \in [a, x]} |f^{(n+1)}(z)| $$

You must calculate and provide this value of M. For many common functions like sin(x) or cos(x), the maximum value is simply 1. For others, you may need to find where the derivative is largest on the interval.