Approximations and expansions Definition

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.

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Limit point of a set definition
Limit of a function definition
Derivative definition
Differentiable function definition
Derivative function definition
Higher derivative definition
Taylor polynomial definition

Tag	`0HFSFVMWQB`
URI	`m2://0HFSFVMWQB`
Kind	definition
Writing	W0 stub
MSC	41A58
Defines	Taylor polynomial
Uses	Higher derivative

Taylor polynomial

Definition

Remarks

Reading path

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