Lagrange Error Bound Calculator

The Lagrange error bound relies on Taylor's Theorem with remainder to bound the absolute difference between a function f(x) and its n-th degree Taylor polynomial P_n(x) centered at a. The formula is expressed as:

R_n(x) = |f(x) - P_n(x)| ≤ (M / (n + 1)!) * |x - a|^(n + 1)

Here, M represents the upper bound for the absolute value of the (n + 1)-th derivative of the function on the interval between a and x. The variables n and a denote the degree of the polynomial and the center point, respectively.

Imagine we want to approximate f(x) = sin(x) using a 3rd-degree Maclaurin polynomial (n = 3, a = 0) evaluated at x = 0.2 radians. First, we find the fourth derivative of sin(x), which is cos(x). The maximum absolute value of cos(x) on our interval is M = 1. Next, substitute the values into the formula: (1 / 4!) * |0.2 - 0|^(4). Evaluating this gives (1 / 24) * (0.0016) = 0.0000667. Thus, our polynomial approximation is guaranteed to be within 0.0000667 of the actual value of sin(0.2).

When searching for the maximum derivative value M, always look across the entire closed interval between your center point a and your evaluation point x, rather than just at the endpoints. Additionally, remember that increasing the degree n generally shrinks the error bound drastically as (n+1)! grows much faster than polynomial terms for small values of |x - a|.