I was poking around on Wikipedia, when I came across the page for Runge’s phenomenon. Runge found an example of a smooth function such that if you interpolate it by a high-degree polynomial at fixed points over a finite interval, the approximation of the function by the polynomial becomes very bad. In fact, in the limit as the degree of the interpolated polynomial goes to infinity, the maximum difference between the function and interpolated polynomial also goes to infinity. Interpolating with polynomials is harder than it looks.