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 [latex]z=f(x_{0},y_{0})[/latex] at the point [latex](x_{0},y_{0})[/latex] for given values of [latex]\Delta{x}[/latex] and [latex]\Delta{y}[/latex].

Key Equations

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

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

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

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

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