91Ó°ÊÓ

Show that any tangent plane to the cone \(z^{2}=a^{2} x^{2}+b^{2} y^{2}\) passes through the origin.

Heat-Seeking Path In Exercises 57 and \(58,\) find the path of a heat-seeking particle placed at point \(P\) on a metal plate with a temperature field \(T(x, y)\). $$ T(x, y)=400-2 x^{2}-y^{2}, \quad P(10,10) $$

Use a graphing utility to graph six level curves of the function. $$ f(x, y)=|x y| $$

True or False? In Exercises 61-64, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous for all nonzero \(x\) and \(y\), and \(f(0,0)=0\), then \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)=0\)

The function \(f\) is homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) Determine the degree of the homogeneous function, and show that \(x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)\) \(f(x, y)=x^{3}-3 x y^{2}+y^{3}\)

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

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 \(V=\frac{1}{4}\).

Volume \(\quad\) The radius \(r\) and height \(h\) of a right circular cylinder are measured with possible errors of \(4 \%\) and \(2 \%,\) respectively. Approximate the maximum possible percent error in measuring the volume.

Area \(\quad\) A triangle is measured and two adjacent sides are found to be 3 inches and 4 inches long, with an included angle of \(\pi / 4\) The possible errors in measurement are \(\frac{1}{16}\) inch for the sides and 0.02 radian for the angle. Approximate the maximum possible error in the computation of the area.

A propane tank is constructed by welding hemispheres to the ends of a right circular cylinder. Write the volume \(V\) of the tank as a function of \(r\) and \(l,\) where \(r\) is the radius of the cylinder and hemispheres, and \(l\) is the length of the cylinder.