14.4 Videos Guide

14.4a

• Equation of a tangent plane to a surface $z=f(x,\ y)$ at the point $P(x_{0},\ y_{0},\ z_{0})$
  $z-z_{0}=f_{x}\left(x_{0},\ y_{0}\right)\left(x-x_{0}\right)+f_{y}(x_{0},\ y_{0})(y-y_{0})$

Exercise:

• Find an equation of the tangent plane to the surface $z=ln(x-2y)$ at the point $(3,\ 1,\ 0)$.

14.4b

• The linearization or linear approximation to a surface $S$ given by $z=f(x,\ y)$ at the point $P(a,\ b)$
  $f(x,\ y)\approx L(x,\ y)=f(a,\ b)+f_{x}(a,\ b)(x-a)+f_{y}(a,\ b)(y-b)$

Exercise:

• Explain why the function $f(x,\ y)=\sqrt{xy}$ is differentiable at the point $(1,\ 4)$. Then find the linearization $L(x,\ y)$ of the function at that point.

14.4c

• The total differential $dz$
  $dz=f_{x}(x,\ y)dx+f_{y}(x,\ y)dy$

14.4d

Exercise:

• If $z=x^{2}-xy+3y^{2}$ and $(x,\ y)$ changes from $(3,\ -1)$ to $(2.96,\ -0.95)$, compare the values of $\mathrm{\Delta}z$ and $dz$.