Local Polynomial Approximation

Calculus 3 · Axiom Academy

Local Polynomial Approximation Zoom in on any smooth curve and it looks like a line. Add a little bend, then a little more, and a polynomial can hug the whole neighborhood. A curve is hard. A polynomial is easy. Can one stand in for the other? Powers of x are the friendliest thing in mathematics: to evaluate you only ever add and multiply. Functions like e^x , , or are not so kind. The idea behind Taylor approximation is to trade the hard function for an easy polynomial — but only locally , near one base point, where a few well-chosen terms are enough. Watch a straight line try first. The tangent line at x=0 is the degree-1 Taylor polynomial of f(x)=e^x : it matches the height and the slope there. Then a parabola takes over — the quadratic , which also matches the bending. Then a cubic. Each one clings to e^x across a wider stretch, and the worst gap between them shrinks. Degree 1 matches the value and slope; degree 2 also matches the curvature; each higher degree matches one more derivative — and tracks e^x over a wider window. Add one term at a time and watch the fit improve Each new term of the Taylor polynomial is chosen to match the next derivative of e^x at x=0 , so the coefficients come out as : . Drag the degree up from a flat constant to a wiggly quintic and watch the polynomial climb onto the curve — the worst gap over the window dropping with every term.

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