Taylor & Maclaurin Polynomial Grapher (n = 0 to 40)

① This interactive GeoGebra example demonstrates how Taylor and Maclaurin polynomials of a user-defined function are graphed dynamically, with the degree n adjustable from 0 to 40 and the expansion point a changeable, visually showing how polynomials approximate the original curve near a. ② Core mathematical concepts include Taylor series, Maclaurin series, polynomial approximation, remainder terms, convergence behavior, and the role of differentiability in local approximation. ③ Construction highlights: the TaylorSeries command generates the polynomial of specified order; a Textfield allows input of the function; sliders control the expansion point a and degree n; objects include function f, polynomial g, point A, button1, InputBox1, and text1/text2 for dynamic updates and comparison. ④ Suitable for advanced high school or college calculus courses, this tool helps students grasp the geometric meaning of Taylor expansion, observe how increasing n improves approximation near a, and serves as an effective classroom demonstration or self-study resource.