Taylor Polynomials

Calculus 2 · Axiom Academy

Building polynomial approximations that match a function and all its derivatives at a single point. We want a polynomial T(x) that approximates a function f(x) near a chosen point x = a . The simplest attempt is the tangent line T_1(x) — it matches f in value and slope at a . Watch what happens as we move away from the center: the line peels off the curve and the error (the red gap) grows. The n -th degree Taylor polynomial of f centered at x = a is a sum, one term per derivative: Each term is built from one derivative of f at a , divided by a factorial. Watch the polynomial snap closer to e^x as each term is switched on — the curve drawn is computed live from the real coefficients , not faked. Matches the function's value at x = a . Matches the first derivative — the slope. Matches the second derivative — the curvature. Matches the k -th derivative — higher-order bend. Take f(x) = e^x centered at a = 0 . Here every derivative of e^x equals 1 , so f^ (k) (0) = 1 and the coefficients are simply . That gives a clean tower of approximations: T_0(x) = 1 (matches the value only) T_1(x) = 1 + x (value and slope) Now let the degree climb . Each T_n in the animation is sampled point-by-point from those coefficients and laid over the true e^x — notice the window where the polynomial hugs the curve grows wider with every degree. 4. Why It Works: Matching Derivatives

This is the written version of the interactive lesson above. See the full Calculus 2 course.