Page:Calculus Made Easy.pdf/200

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

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

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

Exercises XV. (See page 262 for Answers.)

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

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

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

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

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