Functions as Infinite Polynomials

Calculus 2 · Axiom Academy

Functions as Infinite Polynomials Watch a curvy function like e x get rebuilt from nothing but powers of x — and see the approximation snap tighter with every term you add. A curvy function, built from straight-line ingredients A calculator has no special button hiding inside it for e x , sine, or a logarithm. All it can really do is add, subtract, and multiply — so all it can really build is a polynomial . The astonishing fact of Calculus 2 is that this is enough: glue together enough powers of x, each with the right coefficient, and you can reproduce almost any smooth function as closely as you like. Watch e x (red) get rebuilt one term at a time. We start with the flat guess P(x) = 1, then add x, then x²/2, then x³/6… Each new term bends the blue polynomial closer to the real curve, and the worst-case error keeps collapsing. Same arithmetic a calculator already has — just more terms. That infinite sum of powers is a power series , and the one for e x equals it exactly. More terms, less error — you drive it Drag the slider (or the handle on the plot) to set how many terms you keep. With degree 0 you only have the constant 1, a hopeless flat line. As the degree climbs, the blue polynomial wraps onto e x over a wider and wider stretch, and the worst-case gap on [−2, 2] plummets toward zero. As n → the error → 0 everywhere: the infinite polynomial is e x . This is a Taylor (power) series. Great near the center, hungry for terms far out

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