First-Order Multivariable Taylor Series

Theorem

• Let $f:U\subset {\mathbb{R}}^n\to \mathbb{R}$ be differentiable at $x_0$. We have the approximations: $f(x_0+h)=f(x_0)+{\sum }_{i=1}^n{h}_i\frac{\partial f}{\partial x_i}+R_1(x_0,h)=f(x_0)+[Df(x_0)]h+R_1(x_0,h)$ Where $\frac{{\lim }_{h\to 0}{R}_1(x_0,h)}{||h||}=0$ The first order taylor approximation is: $f(x_0+h)\sim f(x_0)+[Df(x_0)]h$