extrapolation

Using an interpolating polynomial to estimate f outside the range of the nodes it was built from -- a genuinely different, and much riskier, act than interpolating inside that range.

Extrapolation evaluates an interpolating polynomial $P$ at a point $x$ lying outside the interval $[\min_i x_i, \max_i x_i]$ spanned by its nodes. The interpolation error formula still applies formally, but the node-product term $\prod(x-x_i)$ it depends on is no longer bounded by the nodes surrounding $x$ on both sides -- it grows without any such check as $x$ moves farther from the data, so the same polynomial that fit the known data well can diverge sharply and without warning just beyond it.

Examples

A polynomial fit to a stock's last 10 days of prices can track them closely, then swing wildly if evaluated even a few days past the last known point.