91Ó°ÊÓ

Sketch the level curve of \(f(x, y)=y / x^{2}\) that goes through \(\mathbf{p}=(1,2) .\) Calculate the gradient vector \(\nabla f(\mathbf{p})\) and draw this vector, placing its initial point at \(\mathbf{p}\). What should be true about \(\nabla f(\mathbf{p})\) ?

Show that the surfaces \(x^{2}+4 y+z^{2}=0 \quad\) and \(x^{2}+y^{2}+z^{2}-6 z+7=0\) are tangent to each other at \((0,-1,2) ;\) that is, show that they have the same tangent plane at \((0,-1,2)\)

If \(w=u^{2}-u \tan v, u=x\), and \(v=\pi x\), find $$ \left.\frac{d w}{d x}\right|_{x=1 / 4} $$

Sketch the graph of \(\bar{f}\). $$ f(x, y)=e^{-\left(x^{2}+y^{2}\right)} $$

Express a positive number \(N\) as a sum of three positive numbers such that the product of these three numbers is a maximum.

Use the methods of this section to find the shortest distance from the origin to the plane \(x+2 y+3 z=12\).

If \(w=x^{2} y+z^{2}, x=\rho \cos \theta \sin \phi, y=\rho \sin \theta \sin \phi\), and \(z=\rho \cos \phi\), find $$ \left.\frac{\partial w}{\partial \theta}\right|_{\rho=2, \theta=\pi, \phi=\pi / 2} $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y^{2}}{x^{2}+y^{4}}\)

Find all first partial derivatives of each function. \(f(r, \theta)=3 r^{3} \cos 2 \theta\)

Show that the surfaces \(z=x^{2} y\) and \(y=\frac{1}{4} x^{2}+\frac{3}{4}\) intersect at \((1,1,1)\) and have perpendicular tangent planes there.