Two-Point Taylor Expansion
This Demonstration calculates recursively the two-point Taylor expansion (2pTE) of a function, which is a polynomial expansion around two reference points. This expansion is increasingly accurate as the number of steps in the recursion progresses. This block shows two examples of the 2pTE applied to two different functions as the degree of the fitting polynomial increases: One is the expansion of the sine function at the points and , while the standard one-point Taylor expansion around is shown for comparison. The second example is the expansion of the reciprocal function at the points and .
The two-point Taylor expansion can be used to provide an approximation of a function at two different points simultaneously. This two-point expansion generates a polynomial of degree that is accurate to order at each of the two given reference points. The expansion can be computed even when the function being approximated is not analytic everywhere. See  for more details.
 R. Hablützel. "Polynomial Division Revisited: -point Taylor Expansion." (Jun 19, 2020) underthemath.wordpress.com/2020/06/12/polynomial-division-revisited.