This Demonstration calculates the two-point Taylor expansion (2pTE) of a function, which is a polynomial expansion that is increasingly accurate around two reference points. The example is the expansion on the sine function at the points and as the degree of the polynomial increases. The standard one-point Taylor expansion (1pTE) around is shown for comparison.
The two-point Taylor expansion can be used to provide a better approximation of a function over a finite interval than the usual one-point Taylor expansion. This two-point expansion generates a polynomial of degree that is accurate to the order at the two given reference points. However, the approximating function may Have points in the interval between the points that are inaccurate. See  for more detail.