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 $\underset{(x,y,z)\to (x_{0},y_{0},z_{0})}{\lim}\frac{E(x,y,z)}{\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}}=0$

Glossary

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 $dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})dy$