91Ó°ÊÓ

Show that the mixed partial derivatives \(f_{x y y}\) \(f_{y x y},\) and \(f_{y y x}\) are equal. f(x, y, z)=x y z

If \(f(x, y)=0,\) give the rule for finding \(d y / d x\) implicitly. If \(f(x, y, z)=0,\) give the rule for finding \(\partial z / \partial x\) and \(\partial z / \partial y\) implicitly.

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

The temperature \(T\) (in degrees Celsius) at any point \((x, y)\) in a circular steel plate of radius 10 meters is $$T=600-0.75 x^{2}-0.75 y^{2}$$ where \(x\) and \(y\) are measured in meters. Sketch some of the isothermal curves.

Show that the mixed partial derivatives \(f_{x y y}\) \(f_{y x y},\) and \(f_{y y x}\) are equal. $$ f(x, y, z)=x^{2}-3 x y+4 y z+z^{3} $$

The electric potential \(V\) at any point \((x, y)\) is $$V(x, y)=\frac{5}{\sqrt{25+x^{2}+y^{2}}}$$ Sketch the equipotential curves for \(V=\frac{1}{2}, V=\frac{1}{3},\) and \(V=\frac{1}{4}\).

Prove that \(\lim _{(x, y) \rightarrow(a, b)}[f(x, y)+g(x, y)]=L_{1}+L_{2}\) where \(f(x, y)\) approaches \(L_{1}\) and \(g(x, y)\) approaches \(L_{2}\) as \((x, y) \rightarrow(a, b)\).

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

Show that the mixed partial derivatives \(f_{x y y}\) \(f_{y x y},\) and \(f_{y y x}\) are equal. $$ f(x, y, z)=e^{-x} \sin y z $$

Show that the mixed partial derivatives \(f_{x y y}\) \(f_{y x y},\) and \(f_{y y x}\) are equal. $$ f(x, y, z)=\frac{2 z}{x+y} $$