91Ó°ÊÓ

Consider the functions \(f(x, y)=6-x^{2}-y^{2} / 4\) and \(g(x, y)=2 x+y\) (a) Find a set of parametric equations of the tangent line to the curve of intersection of the surfaces at the point \((1,2,4)\), and find the angle between the gradient vectors. (b) Use a computer algebra system to graph the surfaces. Graph the tangent line found in part (a).

Describe the level curves of the function. Sketch the level curves for the given \(c\) -values. $$ z=x+y, \quad c=-1,0,2,4 $$

Find the angle of inclination \(\theta\) of the tangent plane to the surface at the given point.\(x^{2}-y^{2}+z=0, \quad(1,2,3)\)

Find the first partial derivatives with respect to \(x, y\), and \(z\). $$ w=\frac{3 x z}{x+y} $$

An annular cylinder has an inside radius of \(r_{1}\) and an outside radius of \(r_{2}\) (see figure). Its moment of inertia is \(I=\frac{1}{2} m\left(r_{1}^{2}+r_{2}^{2}\right)\) where \(m\) is the mass. The two radii are increasing at a rate of 2 centimeters per second. Find the rate at which \(I\) is changing at the instant the radii are 6 centimeters and 8 centimeters.

Discuss the continuity of the composite function \(f \circ g\). $$ \begin{aligned} &f(t)=\frac{1}{t} \\ &g(x, y)=3 x-2 y \end{aligned} $$

Find the absolute extrema of the function over the region \(R .\) (In each case, \(R\) contains the boundaries.) Use a computer algebra system to confirm your results. \(f(x, y)=x^{2}+2 x y+y^{2}, \quad R=\left\\{(x, y): x^{2}+y^{2} \leq 8\right\\}\)

The surface of a mountain is modeled by the equation \(h(x, y)=5000-0.001 x^{2}-0.004 y^{2} .\) A mountain climber is at the point \((500,300,4390)\). In what direction should the climber move in order to ascend at the greatest rate?

Find the path of a heat-seeking particle placed at point \(P\) on a metal plate with a temperature field \(T(x, y)\). $$ \begin{array}{ll} \underline{\text { Temperature Field }} & \underline{\text {Point}} \\ T(x, y)=400-2 x^{2}-y^{2}& \quad P(10,10) \end{array} $$

Use spherical coordinates to find the limit. [Hint: Let \(x=\rho \sin \phi \cos \theta, \quad y=\rho \sin \phi \sin \theta\), and \(z=\rho \cos \phi\), and note that \((x, y, z) \rightarrow(0,0,0)\) implies \(\left.\rho \rightarrow 0^{+} .\right]\) $$ \lim _{(x, y, z) \rightarrow(0,0,0)} \tan ^{-1}\left[\frac{1}{x^{2}+y^{2}+z^{2}}\right] $$