Taylor Approximation

Complicated functions like $\sin x$ or $e^x$ can be approximated near a point by polynomials built from their derivatives. Since polynomials use only basic arithmetic, this is how computers actually evaluate them.

The simplest case is the linear approximation: $$f(x) \approx f(a) + f'(a)(x - a)$$ Here, $f(a)$ is the value, $f'(a)$ is the slope, and $(x-a)$ is the displacement from center $a$.

The general Taylor polynomial improves accuracy by adding higher-derivative terms: $$f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3$$ When the expansion center is $a = 0$, it is specifically called a Maclaurin series.

Terms & Pitfalls

Each term in a Taylor polynomial matches a specific geometric feature of the function at the center point $a$.

More terms generally widen the region of accuracy, but a finite polynomial is always a local approximation.