91Ó°ÊÓ

Find the directions in which the functions increase and decrease most rapidly at \(P_{0} .\) Then find the derivatives of the functions in these directions. \begin{equation}g(x, y, z)=x e^{y}+z^{2}, \quad P_{0}(1, \ln 2,1 / 2)\end{equation}

In Exercises \(1-22,\) find \(\partial f / \partial x\) and \(\partial f / \partial y\) $$f(x, y)=\sum_{n=0}^{\infty}(x y)^{n} \quad(|x y|<1)$$

Find the limits in Exercises \(13-24\) by rewriting the fractions first. $$\lim _{(x, y) \rightarrow(0,0)} \frac{1-\cos (x y)}{x y}$$

Minimum distance to the origin Find the point(s) on the surface \(x y z=1\) closest to the origin.

By about how much will $$ h(x, y, z)=\cos (\pi x y)+x z^{2} $$ change if the point \(P(x, y, z)\) moves from \(P_{0}(-1,-1,-1)\) a distance of \(d s=0.1\) unit toward the origin?

Draw a branch diagram and write a Chain Rule formula for each derivative. \(\begin{array}{l}{\frac{\partial w}{\partial p} \text { for } w=f(x, y, z, v), \quad x=g(p, q), \quad y=h(p, q)} \\ {z=j(p, q), \quad v=k(p, q)}\end{array}\)

In Exercises \(23-34,\) find \(f_{x}, f_{y},\) and \(f_{z}\) $$f(x, y, z)=1+x y^{2}-2 z^{2}$$

Find all the local maxima, local minima, and saddle points of the functions. $$ f(x, y)=y \sin x $$

In Exercises \(17-30,\) (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. $$ f(x, y)=\frac{1}{\sqrt{16-x^{2}-y^{2}}} $$

Find the directions in which the functions increase and decrease most rapidly at \(P_{0} .\) Then find the derivatives of the functions in these directions. \begin{equation}f(x, y, z)=\ln x y+\ln y z+\ln x z, \quad P_{0}(1,1,1)\end{equation}