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

 Page 6 / 11

Find the differential $dz$ of the function $f\left(x,y\right)=4{y}^{2}+{x}^{2}y-2xy$ and use it to approximate $\text{Δ}z$ at point $\left(1,-1\right).$ Use $\text{Δ}x=0.03$ and $\text{Δ}y=-0.02.$ What is the exact value of $\text{Δ}z?$

$\begin{array}{ccc}\hfill dz& =\hfill & 0.18\hfill \\ \hfill \text{Δ}z& =\hfill & f\left(1.03,-1.02\right)-f\left(1,-1\right)=0.180682\hfill \end{array}$

## 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:

## Definition

A function $f\left(x,y,z\right)$ is differentiable at a point $P\left({x}_{0},{y}_{0},{z}_{0}\right)$ if for all points $\left(x,y,z\right)$ in a $\delta$ disk around $P$ we can write

$\begin{array}{cc}\hfill f\left(x,y\right)& =f\left({x}_{0},{y}_{0},{z}_{0}\right)+{f}_{x}\left({x}_{0},{y}_{0},{z}_{0}\right)\left(x-{x}_{0}\right)+{f}_{y}\left({x}_{0},{y}_{0},{z}_{0}\right)\left(y-{y}_{0}\right)\hfill \\ & \phantom{\rule{0.5em}{0ex}}+{f}_{z}\left({x}_{0},{y}_{0},{z}_{0}\right)\left(z-{z}_{0}\right)+E\left(x,y,z\right),\hfill \end{array}$

where the error term E satisfies

$\underset{\left(x,y,z\right)\to \left({x}_{0},{y}_{0},{z}_{0}\right)}{\text{lim}}\frac{E\left(x,y,z\right)}{\sqrt{{\left(x-{x}_{0}\right)}^{2}+{\left(y-{y}_{0}\right)}^{2}+{\left(z-{z}_{0}\right)}^{2}}}=0.$

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)
$\begin{array}{cc}f\left(x,y\right)\hfill & =f\left({x}_{0},{y}_{0},{z}_{0}\right)+{f}_{x}\left({x}_{0},{y}_{0},{z}_{0}\right)\left(x-{x}_{0}\right)+{f}_{y}\left({x}_{0},{y}_{0},{z}_{0}\right)\left(y-{y}_{0}\right)\hfill \\ & \phantom{\rule{0.5em}{0ex}}+{f}_{z}\left({x}_{0},{y}_{0},{z}_{0}\right)\left(z-{z}_{0}\right)+E\left(x,y,z\right),\hfill \end{array}$
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.

$f\left(x,y\right)={x}^{3},\left(2,-1,8\right)$

$\left(\frac{\sqrt{145}}{145}\right)\left(12i-k\right)$

$\text{ln}\left(\frac{x}{y-z}\right)=0$ when $x=y=1$

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

${x}^{2}+xy+{y}^{2}=3,P\left(-1,-1\right)$

Normal vector: $i+j,$ tangent vector: $i-j$

${\left({x}^{2}+{y}^{2}\right)}^{2}=9\left({x}^{2}-{y}^{2}\right),P\left(\sqrt{2},1\right)$

$x{y}^{2}-2{x}^{2}+y+5x=6,P\left(4,2\right)$

Normal vector: $7i-17j,$ tangent vector: $17i+7j$

$2{x}^{3}-{x}^{2}{y}^{2}=3x-y-7,P\left(1,-2\right)$

$z{e}^{{x}^{2}-{y}^{2}}-3=0,$ $P\left(2,2,3\right)$

$-1.094i-0.18238j$

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 $y.\right)$

$-8x-3y-7z=-19,P\left(1,-1,2\right)$

$z=-9{x}^{2}-3{y}^{2},P\left(2,1,-39\right)$

$-36x-6y-z=-39$

${x}^{2}+10xyz+{y}^{2}+8{z}^{2}=0,P\left(-1,-1,-1\right)$

$z=\text{ln}\left(10{x}^{2}+2{y}^{2}+1\right),P\left(0,0,0\right)$

$z=0$

$z={e}^{7{x}^{2}+4{y}^{2}},$ $P\left(0,0,1\right)$

$xy+yz+zx=11,P\left(1,2,3\right)$

$5x+4y+3z-22=0$

${x}^{2}+4{y}^{2}={z}^{2},P\left(3,2,5\right)$

${x}^{3}+{y}^{3}=3xyz,P\left(1,2,\frac{3}{2}\right)$

$4x-5y+4z=0$

$z=axy,P\left(1,\frac{1}{a},1\right)$

$z=\text{sin}\phantom{\rule{0.2em}{0ex}}x+\text{sin}\phantom{\rule{0.2em}{0ex}}y+\text{sin}\left(x+y\right),P\left(0,0,0\right)$

$2x+2y-z=0$

$h\left(x,y\right)=\text{ln}\sqrt{{x}^{2}+{y}^{2}},P\left(3,4\right)$

$z={x}^{2}-2xy+{y}^{2},P\left(1,2,1\right)$

$-2\left(x-1\right)+2\left(y-2\right)-\left(z-1\right)=0$

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 $x-{x}_{0}=at,y-{y}_{0}=bt,z-{z}_{0}=ct.\right)$

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