> [!NOTE] Lemma (Linear Approximation) > The linear approximation of a [[Real-Valued Function on Real n-Space (Multivariable Function)|multivariable real-valued function]] $f:\mathbb{R}^{n} \to \mathbb{R}$ defined by $f(\underline{x})$ near $\underline{x}_{0}$ is given by $f(\underline{x}) \approx f(\underline{x}_{0}) + \nabla f \mid_{\underline{x}_{0}} \cdot \; (\underline{x}-\underline{x}_{0})$ > **Proof**. By [[Taylor's Theorem for Scalar Fields]], $f(x,y)=c_{0}+c_{1}(x-a)+c_{2}(y-b)$Substituting & differentiating accordingly gives $f(x,y)=f(a,b) + (x-a)\frac{ \partial f }{ \partial x }(a,b) + (y-b) \frac{ \partial f }{ \partial y } $ # Applications In this case we get the ***tangent plane*** which has the form $Ax+By+Cz=D$ hence the vector $(A,B,C)$ is a [[Normal Vector of Surface|normal]] to the plane and also the surface.