# Taylor Polynomial Interpolation

This module illustrates Taylor polynomial interpolation. The Taylor polynomial interpolant of degree n for a smooth function f(x) about a point a is given by the truncated Taylor series pn(x) = f(a) + f ′(a)(xa) + (f ″(a) ⁄ 2) (xa)2 + ⋅ ⋅ ⋅ + (f (n)(a) ⁄ n!) (xa)n. It is the unique polynomial pn(x) of degree n whose value and those of its first n derivatives at a agree with those of f, i.e., pn(k)(a) = f (k)(a) for k = 1,…,n, where superscripts indicate derivatives. The Taylor polynomial interpolants approximate f well in some neighborhood of a whose size depends on the location of any singularities of f in the complex plane.

The user begins by selecting a function f(x) from the list of available functions and using the slider to choose the point a at which the function is to be interpolated. The graph of the function is drawn, and the chosen point a is indicated by a black dot. Clicking Next causes Taylor polynomial interpolants of increasing degree to be added to the graph. The color of the interpolants changes from blue to red as the degree increases, and the polynomial written below the graph is updated to reflect the coefficients of the highest degree interpolant shown.

Reference: Michael T. Heath, Scientific Computing, An Introductory Survey, 2nd edition, McGraw-Hill, New York, 2002. See Section 7.3.5.

Developers: Evan VanderZee and Michael Heath