ABSTRACT
Function approximation seeks to find, among a family of functions, the function that best approximates a given function. This chapter examines a number of ways one can carefully and systematically approximate functions, beginning with Taylor polynomials and Lagrange Interpolation polynomials. As an alternative to a tangent line approximation (a first-degree polynomial that agrees with the function value and its derivative value at a single point), one could consider a secant line approximation (which is constructed so that the polynomial and the function have the same value at two distinct points). As with Taylor polynomials, one should investigate how far off these polynomials are from the function they are approximating. It turns out that the error term is reminiscent of the error in Taylor’s Theorem. While Lagrange Interpolation is relatively simple and elegant, it has the drawback that the polynomial must be rebuilt “from the ground up” each time a new node is included.
