Taylor Polynomial Calculator

Taylor polynomial of e^x at a = 0, degree 4: All derivatives of e^x equal e^0 = 1 at x = 0. P4(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 At x = 1: P4(1) = 1 + 1 + 0.5 + 0.1667 + 0.0417 = 2.7083 Actual e^1 = 2.7183 (error < 0.004)

A Taylor polynomial is a finite sum of terms that approximates a function's value near a specific point, often called the center point 'a'. Named after mathematician Brook Taylor, these polynomials provide a powerful method to represent complex functions using simpler, more manageable polynomial expressions. The accuracy of the approximation generally improves as more terms are included, meaning a higher degree polynomial offers a closer fit to the original function within an interval around 'a'. Each term in the polynomial is constructed using the function's derivatives evaluated at the chosen center point. This ensures that the polynomial matches the function's value, slope, concavity, and higher-order characteristics precisely at that point. For example, a first-degree Taylor polynomial is simply the tangent line to the function at 'a'. As the degree increases, the polynomial's graph more closely mirrors the function's curve, making this mathematical tool fundamental in calculus, numerical analysis, and various scientific and engineering applications for simplifying calculations and understanding local function behavior.