Polynomial Approximation

Real Analysis · Axiom Academy

Plus, times, powers — that's all a polynomial knows. Yet that humble toolkit can shadow any curve you draw, as closely as you demand. A curve a polynomial should have no business drawing A polynomial is built from nothing but addition, multiplication, and whole-number powers of x . The exponential e^x is a wilder animal — it equals its own slope at every point, it never met a power it couldn't outgrow. Asking a polynomial to copy it sounds hopeless. Watch it happen anyway. Below is the curve y = e^x . We lay polynomials on top of it one at a time, each of higher degree: first a flat line, then a tilted line, then a parabola, and on up. Watch each new polynomial hug the curve over a longer stretch before drifting away — and watch the worst gap between them, the number labelled , shrink with every step. Each polynomial is the exact Taylor polynomial of e^x at 0 — its coefficients are the genuine 1/n! , not a hand-tuned fit. The gap really does fall toward zero. Drag the slider to choose the degree, and the matching polynomial is rebuilt and laid over e^x on the spot. Push it low and the polynomial sags away from the curve at the edges; push it high and the two become almost impossible to tell apart. The readout is the largest vertical distance between curve and polynomial across the whole window — your scorecard for how good the copy is.

This is the written version of the interactive lesson above. See the full Real Analysis course.