The Taylor polynomial

The Taylor polynomial is the polynomial that best approximates $f$ near a point $x_0$: it has the same value and the same first derivatives as $f$ at that point.

Definition — Taylor polynomial of order $n$

Let $f$ be differentiable $n$ times in a neighbourhood of $x_0$. The Taylor polynomial of order $n$ of $f$ centred at $x_0$ is: $T_n(x) = \sum_{k=0}^{n}\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2}(x-x_0)^2 + \cdots + \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n.$ When $x_0 = 0$ it is called the Maclaurin polynomial.

Remark — Key property

$T_n$ is the unique polynomial of degree $\le n$ such that $T_n^{(k)}(x_0) = f^{(k)}(x_0)$ for every $k = 0,1,\ldots,n$: it has the same value and the same first $n$ derivatives as $f$ at the point $x_0$. For $n = 1$ one recovers the tangent line; for $n = 2$ the osculating parabola.

The approximations $T_1, T_3, T_5$ of $\sin x$ “embrace” the function better and better: near zero $T_5$ is visually indistinguishable from $\sin x$, and as the degree increases the interval of good approximation widens.

Links

Topics: Derivatives
Concepts: Higher-order derivative · Maclaurin polynomial · Taylor polynomial
Methods: Taylor polynomial
Skills: Sketching a graph · Using formulae
People: Maclaurin · Brook Taylor