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 $$S=xy + \dfrac{2V}{x} + \dfrac{2V}{y}$$.

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

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

Exercises XV. (See page 262 for Answers.)

(1) Differentiate the expression $$\dfrac{x^3}{3} - 2x^3y - 2y^2x + \dfrac{y}{3}$$ 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

(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 $$\dfrac{\partial^2r}{\partial x^2} + \dfrac{\partial^2r}{\partial y^2} + \dfrac{\partial^2r}{\partial z^2}$$.

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