91Ó°ÊÓ

In Exercises \(1-8,\) find equations for the (a) tangent plane and (b) normal line at the point \(P_{0}\) on the given surface. $$ x^{2}+y^{2}-z^{2}=18, \quad P_{0}(3,5,-4) $$

In Exercises 1–3, begin by drawing a diagram that shows the relations among the variables. $$ \begin{array}{ll}{\text { If } w=x^{2}+y-z+\sin t \text { and } x+y=t, \text { find }} \\ {\text { a. }\left(\frac{\partial w}{\partial y}\right)_{x, z}} & {\text { b. }\left(\frac{\partial w}{\partial y}\right)_{z, t} \quad \text { c. }\left(\frac{\partial w}{\partial z}\right)_{x, y}} \\ {\text { d. }\left(\frac{\partial w}{\partial z}\right)_{y, t}} & {\text { e. }\left(\frac{\partial w}{\partial t}\right)_{x, z}} & {\text { f. }\left(\frac{\partial w}{\partial t}\right)_{y, z}}\end{array} $$

In Exercises \(1-12,\) (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded. $$ f(x, y)=\sqrt{y-x} $$

In Exercises \(1-10\) , use Taylor's formula for \(f(x, y)\) at the origin to find quadratic and cubic approximations of \(f\) near the origin. $$ f(x, y)=e^{x} \cos y $$

Find all the local maxima, local minima, and saddle points of the functions in Exercises \(1-30\) . $$ f(x, y)=x^{2}+3 x y+3 y^{2}-6 x+3 y-6 $$

Find \(\partial f / \partial x\) and \(\partial f / \partial y\). \(f(x, y)=x^{2}-x y+y^{2}\)

In Exercises 1–4, find the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point. $$ f(x, y)=\ln \left(x^{2}+y^{2}\right), \quad(1,1) $$

Find the limits in Exercises \(1-12\). $$ \lim _{(x, y) \rightarrow(0,4)} \frac{x}{\sqrt{y}} $$

Extrema on a circle Find the extreme values of \(f(x, y)=x y\) subject to the constraint \(g(x, y)=x^{2}+y^{2}-10=0\)

In Exercises \(1-10\) , use Taylor's formula for \(f(x, y)\) at the origin to find quadratic and cubic approximations of \(f\) near the origin. $$ f(x, y)=y \sin x $$