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Chapter 4: Partial Differentiation
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Section 4.7: Approximations
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Essentials


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For a sufficiently well behaved function $f\left(x\,y\right)$, the change $\mathrm{\Δ}f\equiv f\left(x\+\mathrm{dx}\,y\+\mathrm{dy}\right)f\left(x\,y\right)$ is approximated by the differential $\mathrm{df}\equiv {f}_{x}\left(x\,y\right)\mathrm{dx}plus;{f}_{y}\left(xcomma;y\right)\mathrm{dy}$. Such a differential is often called either a total or an exact differential. The generalization to functions of more than two variables is obvious.

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Contrary to the logic in singlevariable calculus where the derivative is defined first, and the differential is defined in terms of the derivative as the derivative times an increment, for functions of several variables the differential must be defined first. If a function has a differential, then it is called differentiable!

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A sufficient condition for the existence of a differential, and hence for the function of several variables to be differentiable, is the continuity of the first partial derivatives.

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As for the function of one variable, a differentiable function is necessarily continuous.

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Functions of several variables can be approximated by Taylor polynomials. Table 4.7.1 lists Taylor polynomials of degrees one and two for the function $f\left(x\,y\right)$. The expansion point is taken as $\left(a\,b\right)$.

Degree

Taylor Polynomial

1

$f\left(a\+h\,b\+k\right)\=f\left(a\,b\right)\+{f}_{x}\left(a\,b\right)h\+{f}_{y}\left(a\,b\right)k$

2

$f\left(a\+h\,b\+k\right)\=f\left(a\,b\right)\+{f}_{x}\left(a\,b\right)h\+{f}_{y}\left(a\,b\right)kplus;\frac{1}{2}\left({f}_{\mathrm{xx}}\left(acomma;b\right){h}^{2}plus;2{f}_{\mathrm{xy}}\left(acomma;b\right)hkplus;{f}_{\mathrm{yy}}\left(acomma;b\right){k}^{2}\right)$

Table 4.7.1 Taylor polynomials of degrees one and two



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The equality of the mixed partial derivatives ${f}_{\mathrm{xy}}$ and ${f}_{\mathrm{yx}}$ is assumed in Table 4.7.1. Hence, instead of having two separate terms containing the product $hk$, both are combined into twice the one. It is sometimes useful to write the seconddegree terms as half the quadratic form

$\left[\begin{array}{cc}h& k\end{array}\right]\left[\begin{array}{cc}{f}_{\mathrm{xx}}& {f}_{\mathrm{xy}}\\ {f}_{\mathrm{xy}}& {f}_{\mathrm{yy}}\end{array}\right]\left[\begin{array}{c}h\\ k\end{array}\right]$
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where the row vector $\left[\begin{array}{cc}h& k\end{array}\right]$ is the transpose of the column vector $\left[\begin{array}{c}h\\ k\end{array}\right]$ and the intervening symmetric matrix of second partial derivatives is called a Hessian.
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A general representation of the $n$thdegree term is $\frac{1}{n\!}{\left[{\left(h\frac{\partial}{\partial x}plus;k\frac{\partial}{\partial y}\right)}^{n}f\right]}_{\left(acomma;b\right)}$, but unfortunately, no form of this expression is a valid Maple operator.

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The role existence and continuity of partial derivatives plays in differentiability of a function of several variables is examined at length in Section 4.11.



Examples


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Example 4.7.1

If $x$ changes from 3 to 3.04 and $y$ changes from $2$ to $2.2$, compare $\mathrm{\Δ}f$ and $\mathrm{df}$ for the function $f\left(x\,y\right)\=3{x}^{2}plus;4xy5{y}^{2}plus;7$.

Example 4.7.2

Approximate $\sqrt{4{\left(3.03\right)}^{3}\+5{\left(6.2\right)}^{2}}$ by using the total differential for some appropriate function $f\left(x\,y\right)$.

Example 4.7.3

Approximate $\frac{1}{2.03}\+\frac{1}{3.02}\+\frac{1}{4.97}$ by using the total differential for some appropriate function $f\left(x\,y\,z\right)$.

Example 4.7.4

Use the total differential to approximate the value of $f\left(x\,y\right)\={x}^{2}{e}^{2{\mathrm{xy}}^{2}}$ at $\left(x\,y\right)\=\left(1.03\,0.96\right)$.

Example 4.7.5

Use the total differential to approximate the value of $f\left(x\,y\,z\right)\=x\mathrm{ln}\left(\frac{yplus;z}{{x}^{2}plus;yz}\right)$ at $\left(x\,y\,z\right)\=\left(2.03\,3.12\,1.48\right)$.

Example 4.7.6

Use differentials to estimate the maximum error in determining the area of a rectangle measured (in cm) to be $4\times 5$, if each measurement is accurate to .1 cm.

Example 4.7.7

Use differentials to estimate the maximum error in determining the surface area of a closed rectangular box measured (in inches) to be $15\times 6\times 21$, if each measurement is accurate to .2 in.

Example 4.7.8

A closed box is constructed from lumber that is $3\/4$ in thick. The outside measurements are $9\times 7\times 5$. Use differentials to estimate the interior volume of the box.

Example 4.7.9

Show that the plane tangent to $f\left(x\,y\right)\=3{x}^{2}plus;5xy7{y}^{2}plus;4$ at $\left(x\,y\right)\=\left(2\,3\right)$ is the firstdegree Taylor polynomial constructed at the point of contact. Show further that approximating $\mathrm{\Δ}f$ by the total differential $\mathrm{df}$ amounts to a tangentplane approximation.

Example 4.7.10

At $\left(x\,y\right)\=\left(4\,1\right)$, construct the seconddegree Taylor polynomial for $f\left(x\,y\right)\=\sqrt{3{x}^{2}plus;{y}^{3}}$.




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