Taylor polynomial

Definition

Let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be $k$ times differentiable at $a$. The Taylor polynomial of degree $k$ of $f$ at $a$ is $$P_k(x) = \sum_{j=0}^{k} \frac{f^{(j)}(a)}{j!} (x - a)^j = f(a) + f'(a)(x - a) + \cdots + \frac{f^{(k)}(a)}{k!} (x - a)^k.$$

Remarks

It is the unique polynomial $P$ of degree at most $k$ with $P^{(j)}(a) = f^{(j)}(a)$ for $j = 0, \dots, k$, using higher derivatives. For $k = 1$, its graph is the tangent line. The difference $R_k = f - P_k$ is the remainder. Taylor's theorem says that it is small near $a$. Mathlib has taylorWithinEval for Taylor polynomials within a set.

Explanations

L1 human

The Taylor polynomial copies the value and the first $k$ derivatives of $f$ at $a$. It is the polynomial that best fits $f$ near $a$.