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180

Calculus Made Easy

The truck is a rectangular box open at the top. Let $x$ be the length and $y$ be the width; then the depth is ${\dfrac {V}{xy}}$ . The surface area is $S=xy+{\dfrac {2V}{x}}+{\dfrac {2V}{y}}$ .

$dS={\frac {\partial S}{\partial x}}\,dx+{\frac {\partial S}{\partial y}}\,dy=\left(y-{\frac {2V}{x^{2}}}\right)dx+\left(x-{\frac {2V}{y^{2}}}\right)dy$ .

For minimum (clearly it won't be a maximum here),

$y-{\frac {2V}{x^{2}}}=0,\quad x-{\frac {2V}{y^{2}}}=0$ .

Here also, an immediate solution is $x=y$ , so that $S=x^{2}+{\dfrac {4V}{x}}\quad$ , ${\dfrac {dS}{dx}}=2x-{\dfrac {4V}{x^{2}}}=0$ for minimum, and

$x={\sqrt[{3}]{2V}}$ .

Exercises XV. (See page 262 for Answers.)

(1) Differentiate the expression ${\dfrac {x^{3}}{3}}-2x^{3}y-2y^{2}x+{\dfrac {y}{3}}$ with respect to $x$ alone, and with respect to $y$ alone.

(2) Find the partial differential coefficients with respect to $x$ , $y$ and $z$ , of the expression

$x^{2}yz+xy^{2}z+xyz^{2}+x^{2}y^{2}z^{2}$ .

(3) Let $r^{2}=(x-a)^{2}+(y-b)^{2}+(z-c)^{2}$ .

Find the value of ${\dfrac {\partial r}{\partial x}}+{\dfrac {\partial r}{\partial y}}+{\dfrac {\partial r}{\partial z}}$ . Also find the value of ${\dfrac {\partial ^{2}r}{\partial x^{2}}}+{\dfrac {\partial ^{2}r}{\partial y^{2}}}+{\dfrac {\partial ^{2}r}{\partial z^{2}}}$ .

(4) Find the total differential of $y=u^{v}$ .

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