91Ó°ÊÓ

Determine whether \(z\) is a function of \(x\) and \(y\). $$ x^{2} z+y z-x y=10 $$

Find the total differential. $$ z=3 x^{2} y^{3} $$

The volume of an ellipsoid \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\) is \(4 \pi a b c / 3\). For a fixed sum \(a+b+c\), show that the ellipsoid of maximum volume is a sphere.

(a) evaluate \(f(1,2)\) and \(f(1.05,2.1)\) and calculate \(\Delta z\), and (b) use the total differential \(d z\) to approximate \(\Delta z\) $$ f(x, y)=9-x^{2}-y^{2} $$

Find both first partial derivatives. $$ z=x^{2} e^{2 y} $$

The parametric equations for the paths of two projectiles are given. At what rate is the distance between the two objects changing at the given value of \(t\) ? $$ \begin{aligned} &x_{1}=10 \cos 2 t, y_{1}=6 \sin 2 t \\ &x_{2}=7 \cos t, y_{2}=4 \sin t \\ &t=\pi / 2 \end{aligned} $$

Show that the rectangular box of maximum volume inscribed in a sphere of radius \(r\) is a cube.

Show that a rectangular box of given volume and minimum surface area is a cube.

(a) evaluate \(f(1,2)\) and \(f(1.05,2.1)\) and calculate \(\Delta z\), and (b) use the total differential \(d z\) to approximate \(\Delta z\) $$ f(x, y)=x e^{y} $$

A company manufactures two types of sneakers, running shoes and basketball shoes. The total revenue from \(x_{1}\) units of running shoes and \(x_{2}\) units of basketball shoes is \(R=-5 x_{1}^{2}-8 x_{2}^{2}-2 x_{1} x_{2}+42 x_{1}+102 x_{2}\), where \(x_{1}\) and \(x_{2}\) are in thousands of units. Find \(x_{1}\) and \(x_{2}\) so as to maximize the revenue.