Multivariate Taylor Approximations¶

Definition¶

Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be sufficiently smooth near $\boldsymbol{x}_0$ and let $\boldsymbol{h}$ be a small displacement. The first-order Taylor approximation is

If $f$ is $C^2$ near $\boldsymbol{x}_0$, the second-order approximation is

The first correction uses local slope, and the second correction uses local curvature.

Interpretation¶

Taylor approximations replace a complicated function near a point by simpler local information. The linear term estimates the immediate change, and the quadratic term corrects for bending.

The figure follows the line $\boldsymbol{x}_0+t\boldsymbol{h}$ for a fixed direction $\boldsymbol{h}$. Its first-order approximation is $f(\boldsymbol{x}_0)+t\boldsymbol{D}f(\boldsymbol{x}_0)\boldsymbol{h}$$t\mathbf{D}f({\mathbf{x}}_0)\mathbf{h}$; the second-order approximation adds $\frac{1}{2}t^2\boldsymbol{h}^{\top}\boldsymbol{H}_f(\boldsymbol{x}_0)\boldsymbol{h}$. As $t\to0$$0$, the errors are $o(|t|)$ under differentiability and $o(t^2)$ under the $C^2$ hypothesis.

The Jacobian of a scalar-valued function is a row matrix, so multiplying it by the displacement gives the first-order scalar change. The Hessian contributes the quadratic correction when the $C^2$ hypothesis holds.

Key Results¶

• For a real-valued function, $\boldsymbol{D}f(\boldsymbol{x}_0)\boldsymbol{h}=\boldsymbol{\nabla} f(\boldsymbol{x}_0)^{\top} \boldsymbol{h}$$\nabla f({\mathbf{x}}_0{)}^{\top }\mathbf{h}$.
• If $f$ is $C^2$, then the Hessian is symmetric and the second-order term is a quadratic form.
• At a critical point, the quadratic term is the first potentially nonzero correction. It may also vanish, in which case higher-order information is needed.

Examples¶

• For $f(x,y)=x^2+xy+y^2$ at $(0,0)$, the second-order approximation is exact.
• For $f(x,y)=\mathrm{e}^{x+y}$ at $(0,0)$, the second-order approximation is $1+x+y+\frac{1}{2}(x+y)^2$$y{)}^2$.

Prerequisites¶

• Jacobian Matrix
• Hessian Matrix

Related Concepts¶

• Quadratic Form
• Higher-order Partial Derivative

Sources¶

• Math IV: Multivariate Calculus · Week 2, Lecture 1

Aliases: multivariate Taylor expansion Keywords: linear approximation in Rn, second-order approximation, quadratic approximation, Taylor approximation with Hessian Symbols: nabla gradient, real numbers

Last updated: 27 October 2026 11:52