Taylor Approximation

Approximating with Polynomials

Complicated functions like $\sin x$, $e^x$, or $\ln x$ can be approximated near a point by polynomials built from their derivatives. Since polynomials use only addition and multiplication, this is how such functions are actually computed, and it is a central tool of analysis.

The simplest case is the linear (tangent-line) approximation: $$f(x) \approx f(a) + f'(a)(x - a),$$ the value plus the slope times the displacement — literally the tangent line at $a$. It is excellent for $x$ near $a$ and is the basis of Newton's method and differentials.

The Taylor polynomial improves accuracy by adding higher-derivative terms: $$f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \dots$$ Each term corrects the fit a little more: the constant matches the value, the linear term the slope, the quadratic the curvature, and so on. The special case $a = 0$ (a Maclaurin series) gives clean expansions like $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$.

The key practical facts:

• More terms generally improve accuracy and widen the region where the approximation is good.
• The approximation is local — most accurate near the expansion point $a$, degrading as $|x - a|$ grows.
• The remainder (error) can be bounded, telling you how many terms you need for a desired accuracy.

Taylor approximation replaces a hard function by an easy polynomial that matches it locally — the workhorse behind numerical computation, physics approximations, and much of applied analysis.

Common pitfall: expecting a truncated Taylor polynomial to be accurate far from the expansion point. The approximation matches $f$ best near the center $a$, and the error typically grows as $|x - a|$ increases — a few terms good near $a$ can be wildly wrong far away. More terms widen the accurate region but a finite polynomial never matches everywhere; the approximation is fundamentally local.

A smooth curve with successive Taylor polynomials (degree 1, 2, 3) overlaid, each accent curve hugging the function over a wider region near the center point as the degree increases.

$$f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2$$