Summary of Tangent Planes and Linear Approximations

The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.
Tangent planes can be used to approximate values of functions near known values.
A function is differentiable at a point if it is ”smooth” at that point (i.e., no corners or discontinuities exist at that point).
The total differential can be used to approximate the change in a function $z=f(x_{0},y_{0})$ at the point $(x_{0},y_{0})$ for given values of $\Delta{x}$ and $\Delta{y}$ .

Key Equations

Tangent plane
$z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})$
Linear approximation
$L(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})$
Total differential
$dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})(y-y_{0})dy$
Differentiability (two variables)
$f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)$ , where the error term $E$ satisfies $\underset{(x,y)\to (x_{0},y_{0})}{\lim}\frac{E(x,y)}{\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0$
Differentiability (three variables)
$f(x,y,z)=f(x_{0},y_{0},z_{0})+f_{x}(x_{0},y_{0},z_{0})(x-x_{0})+f_{y}(x_{0},y_{0},z_{0})(y-y_{0})+f_{z}(x_{0},y_{0},z_{0})(z-z_{0})+E(x,y,z)$ , where the error term $E$ satisfies $\underset{(x,y,z)\to (x_{0},y_{0},z_{0})}{\lim}\frac{E(x,y,z)}{\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}}=0$

differentiable: a function $f(x,y,z)$ is differentiable at $(x_{0},y_{0})$ if $f(x,y)$ can be expressed in the form $f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)$ , where the error term $E(x,y)$ satisfies $\underset{(x,y)\to{(x_{0},y_{0})}}{\lim}\frac{E(x,y)}{\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0$

linear approximation: given a function $f(x,y)$ and a tangent plane to the function at a point $(x_{0},y_{0})$ we can approximate $f(x,y)$ for points near $(x_{0},y_{0})$ using the tangent plane formula

tangent plane: given a function $f(x,y)$ that is differentiable at a point $(x_{0},y_{0})$ the equation of the tangent plane to the surface $z=f(x,y)$ is given by $z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})$

total differential: the total differential of the function $f(x,y)$ at $(x_{0},y_{0})$ is given by the formula $dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})dy$