Page:Calculus Made Easy.pdf/200

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

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

 

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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 $y=u^{v}$.