91Ó°ÊÓ

Given that \(r=\sqrt{x^{2}+y^{2}},\) show that \(\frac{\partial r}{\partial x}=\frac{x}{\sqrt{x^{2}+y^{2}}}=\frac{x}{r}=\cos \theta .\) Starting from \(r=\frac{x}{\cos \theta}\) does it follow that \(\frac{\partial r}{\partial x}=\frac{1}{\cos \theta} ?\) Explain why it's not possible for both calculations to be correct. Find all mistakes.

Determine whether or not \(f(x, y)=x y^{2}\) is differentiable.

Show that the indicated limit exists. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y-x^{2}-y^{2}}{x^{2}+y^{2}}$$

Find the directions of maximum and minimum change of \(f\) at the given point, and the values of the maximum and minimum rates of change. $$f(x, y)=x^{2}-y^{3},(-1,-2)$$

Use a graphing utility to sketch graphs of \(z=f(x, y)\) from two different viewpoints, showing different features of the graphs. $$f(x, y)=y e^{x}$$

Find all points at which \(\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0\) and interpret the significance of the points graphically. $$f(x, y)=x^{2}+y^{2}$$

The heat equation for the temperature \(u(x, t)\) of a thin rod of length \(L\) is \(\alpha^{2} u_{x x}=u_{t}, 0

Use a graphing utility to sketch graphs of \(z=f(x, y)\) from two different viewpoints, showing different features of the graphs. $$f(x, y)=\ln \left(x^{2}+y^{2}-1\right)$$

Minimize \(2 x+2 y\) subject to the constraint \(x y=c\) for some constant \(c>0\) and conclude that for a given area, the rectangle with smallest perimeter is the square.

Show that the indicated limit exists. $$\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{3 x^{3}}{x^{2}+y^{2}+z^{2}}$$