Order zero is a constant, order one is a tangent line, order two is a parabola. Each adds a derivative.
From the chapter: Chapter 3: Calculus
Glossary: derivative, taylor series
Transcript
We have a smooth curve, sine of x, and we want to approximate it near the origin using only polynomials.
Start with a constant. The flat line at the curve's value. It is right at the touch point and wrong everywhere else.
Add a linear term, the derivative. Now we have the tangent line. Right value, right slope. It hugs the curve in a small neighbourhood, then drifts away.
Add a quadratic term, scaled by one half the second derivative. A parabola that bends with the curve. The match is perfect for value, slope, and curvature at the origin.
Add a cubic. Add a quartic. Add a fifth-order term. Each new term fixes one more derivative.
The Taylor series is the limit. Infinitely many terms, each scaled by the corresponding derivative divided by the factorial of its order.
For nice functions like sine and exponential, the series converges everywhere. For others it converges only inside a disc.
Computers cannot store infinite functions. They store coefficients. A truncated Taylor series turns calculus into arithmetic, and that turn is what makes numerical analysis possible.