Taylor expansion

A Taylor expansion about a point consists of successive derivative terms there, together with either a finite remainder or an infinite Taylor series when that series represents the function.

For a scalar function with continuous third derivatives near $a$, apply the one-dimensional Taylor theorem to $g(t)=f(a+th)$. The chain rule gives $g^{\prime }(0)=Df(a)h$ and $g^{\prime \prime }(0)=h^T{H}_f(a)h$, proving the displayed formula. Here $H_f$ is the Hessian matrix. The finite polynomial part is the Multivariate Taylor polynomial; the remainder controls its accuracy.

For a fixed unit direction $v$, the multivariable Taylor expansion gives the displayed leading error. Equal directional derivatives in two directions do not imply equal approximation errors: the linear changes depend on the gradient, while the quadratic errors depend on the Hessian matrix. For example, $f(x,y)=x+y+2x^2+y^2$ at the origin has equal derivatives in the two coordinate directions but errors $2s^2$ and $s^2$.

A Peano zero through order $r$ means that the function admits the zero Taylor polynomial with remainder $o(|t{|}^r)$ at $x_0$. This does not assert the existence of ordinary iterated derivatives on a neighborhood. For example, multiplying $t^{r+1}$ by a bounded continuous function with dense points of nondifferentiability preserves this estimate while destroying neighborhood differentiability.

A parabolic approximation retains terms through second order in a local Taylor expansion. Near the closest point of a circle of radius $a$ and its tangent line, the separation is $x^2/(2a)+O(x^4/a^3)$.