NapkinCalc

Calculus 2 — Integrals & Series

Taylor series — polynomials impersonating functions

continues from lesson 3 — values defined earlier in the course stay live here

Near a point, any smooth function can be impersonated by a polynomial built from its derivatives. The first two terms of sin(x) are x − x³/6 — plotted below; it hugs sin(x) until about |x| = 1.5, then the lie unravels:

x - x^3/6
-2-1012-202

Taylor approximation of sin(x) — compare against the real sine in your head

✓ pass abs(0.5−0.536−sin(0.5))<0.001\mathrm{abs}\left(0.5 - \frac{0.5^{3}}{6} - \mathrm{sin}\left(0.5\right)\right) < 0.001 two terms already nail sin(0.5) to 3 decimals

This is how your calculator computes sin — it's polynomials all the way down. Where next: Linear Algebra trades curves for grids of numbers.