4.4 Tangent planes and linear approximations  (Page 6/11)

Differentiability of a function of three variables

All of the preceding results for differentiability of functions of two variables can be generalized to functions of three variables. First, the definition:

If a function of three variables is differentiable at a point $\left(x_0,y_0,z_0\right),$ then it is continuous there. Furthermore, continuity of first partial derivatives at that point guarantees differentiability.

Key concepts

• 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 $z=f\left(x_0,y_0\right)$ at the point $\left(x_0,y_0\right)$ for given values of $\text{Δ}x$ and $\text{Δ}y.$

Key equations

• Tangent plane
  $z=f\left(x_0,y_0\right)+f_x\left(x_0,y_0\right)\left(x-x_0\right)+f_y\left(x_0,y_0\right)\left(y-y_0\right)$
• Linear approximation
  $L\left(x,y\right)=f\left(x_0,y_0\right)+f_x\left(x_0,y_0\right)\left(x-x_0\right)+f_y\left(x_0,y_0\right)\left(y-y_0\right)$
• Total differential
  $dz=f_x\left(x_0,y_0\right)dx+f_y\left(x_0,y_0\right)dy.$
• Differentiability (two variables)
  $f\left(x,y\right)=f\left(x_0,y_0\right)+f_x\left(x_0,y_0\right)\left(x-x_0\right)+f_y\left(x_0,y_0\right)\left(y-y_0\right)+E\left(x,y\right),$
  where the error term $E$ satisfies
  $\underset{\left(x,y\right)\to \left(x_0,y_0\right)}{\text{lim}}\frac{E\left(x,y\right)}{\sqrt{{\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2}}=0.$
• Differentiability (three variables)
  $\begin{array}{cc}f\left(x,y\right)\hfill & =f\left(x_0,y_0,z_0\right)+f_x\left(x_0,y_0,z_0\right)\left(x-x_0\right)+f_y\left(x_0,y_0,z_0\right)\left(y-y_0\right)\hfill \\ & +f_z\left(x_0,y_0,z_0\right)\left(z-z_0\right)+E\left(x,y,z\right),\hfill \end{array}$
  where the error term $E$ satisfies
  $\underset{\left(x,y,z\right)\to \left(x_0,y_0,z_0\right)}{\text{lim}}\frac{E\left(x,y,z\right)}{\sqrt{{\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2+{\left(z-z_0\right)}^2}}=0.$

For the following exercises, find a unit normal vector to the surface at the indicated point.

For the following exercises, as a useful review for techniques used in this section, find a normal vector and a tangent vector at point $P.$

For the following exercises, find the equation for the tangent plane to the surface at the indicated point. ( Hint: Solve for $z$ in terms of $x$ and $y.)$

For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, $P_0\left(x_{0,}{y}_0,z_0\right),$ and a vector $n=⟨a,b,c⟩$ that is parallel to the line. Then the equation of the line is $x-x_0=at,y-y_0=bt,z-z_0=ct.)$