Taylor Series and Approximation

Computers cannot directly evaluate $\sin x$ or $e^x$; they use Taylor series to approximate smooth functions with polynomials built from derivatives.

The Taylor series of $f$ about point $a$ is:

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$$

Truncating this gives a Taylor polynomial that hugs the function near $a$. Setting $a = 0$ yields a Maclaurin series, such as $e^x = 1 + x + \frac{x^2}{2!} + \dots$.

Accuracy and Pitfalls

To compute efficiently, we replace a hard function by an easy polynomial matching it locally. The linear approximation $f(x) \approx f(a) + f'(a)(x-a)$ is the simplest case and underpins numerical algorithms.

Common Pitfall: Expecting a truncated polynomial to work far from $a$. More terms extend the valid region, but finite polynomials remain strictly local approximations.