Taylor Polynomial Builder
Build the Taylor polynomial of a polynomial function at any center, term by term.
Example: f(x) = x³ − 2x² + 3x + 4 expanded around a = 1. (Exact for polynomials of degree ≤ n.)
① Answer
② Steps
③ Why it works
A Taylor polynomial rebuilds f near a point a out of its derivatives at that single point: each term ck(x − a)k with ck = f(k)(a)/k! matches one more derivative of f at a, so the approximation hugs the curve tighter as n grows. For a polynomial of degree d, the Taylor polynomial of degree d (or higher) is exactly f — just rewritten around a.