# 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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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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biomolecules are e building blocks of every organics and inorganic materials.
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can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations? By JavaChamp Team By Melinda Salzer By Stephen Voron By Samuel Madden By Michael Nelson By Maureen Miller By Edgar Delgado By Briana Hamilton By Jessica Collett By Richley Crapo