91Ó°ÊÓ

Show that the tangent plane to the surface of \(z=f(x, y)\) at a relative maximum must be parallel to the \(x y\)-plane.

Calculate \(d z\) for each of the following functions: (a) \(z=x^{2}+2 x y-y^{2}\). Ans. \(d z=2(x+y) d x+2(x-y) d y\). (b) \(V=\pi r^{2} h\). (c) \(z=\log \left(x^{2}+y^{2}\right)\). Ans. \(d z=\frac{2}{x^{2}+y^{2}}(x d x+y d y)\). (d) \(z=\sin x \sin y\) (e) \(z=e^{x y} \quad\) Ans. \(d z=e^{x y}(y d x+x d y)\)

Show that the values of \(\mathrm{z}\) in the function \(z=x^{2}-2 x y+y^{2}\) can never be negative.

Find the envelope of the following families of straight lines: (a) \(2 \alpha y=2 x+\alpha^{2}\). Ans. \(y^{2}=2 x\). (b) \(x \cos \alpha+y \sin \alpha=2\). (c) \(y=\frac{m}{a} x+a x\). Ans. \(y^{2}=4 m x\).

Find the envelope of the family of circles which have their centers on \(y=x^{2}\) and are tangent to the \(x\)-axis. Ans. \(y=0\) and \(x^{2}+y^{2}=y / 2\).

Find the envelope of the family of circles which pass through the origin and have their centers on the hyperbola \(x y=1\). Ans. The lemniscate \(\left(x^{2}+y^{2}\right)^{2}=16 x y\).

Show that the surface \(z=x y\) has neither a maximum nor a minimum point.

Find the direction at the point \((-1,1,7)\) for which the directional derivative of \(u=x y+y z+z x\) is a maximum and find the maximum value of the directional derivative.

Find the directional derivative of \(u=\log \sqrt{x^{2}+y^{2}+z^{2}}\) at the point \((3,4\), 8) in the direction of the point \((5,7,10)\). Ans. \(34 /(89 \sqrt{17})\).

Show that the ellipsoid \(3 x^{2}+4 y^{2}+8 z^{2}=24\) and the hyperboloid of two sheets \(4 x^{2}-4 y^{2}-z^{2}=4\) are orthogonal to each other at the common point \(\left(\frac{4 \sqrt{5}}{5}, \sqrt{2}, \frac{2 \sqrt{5}}{5}\right)\)