91Ó°ÊÓ

Describe the change in accuracy of \(d z\) as an approximation of \(\Delta z\) as \(\Delta x\) and \(\Delta y\) increase.

Describe the domain and range of the function. $$ f(x, y)=\ln (x y-6) $$

Find an equation of the tangent plane to the surface at the given point.\(z=x^{2}-2 x y+y^{2}, \quad(1,2,1)\)

Find the gradient of the function at the given point. $$ g(x, y)=2 x e^{y / x}, \quad(2,0) $$

Find \(\partial w / \partial r\) and \(\partial w / \partial \theta\) (a) using the appropriate Chain Rule and (b) by converting \(w\) to a function of \(r\) and \(\boldsymbol{\theta}\) before differentiating. $$ w=\frac{y z}{x}, \quad x=\theta^{2}, \quad y=r+\theta, \quad z=r-\theta $$

Find the gradient of the function at the given point. $$ z=\cos \left(x^{2}+y^{2}\right), \quad(3,-4) $$

Describe the domain and range of the function. $$ z=\frac{x+y}{x y} $$

Examine the function for relative extrema and saddle points. \(h(x, y)=x^{2}-3 x y-y^{2}\)

Find the limit (if it exists). If the limit does not exist, explain why. $$ \lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x y+y z+x z}{x^{2}+y^{2}+z^{2}} $$

Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated point. [Hint: In Exercise 23, minimize \(f(x, y)=x^{2}+y^{2}\) subject to the constraint \(2 x+3 y=-1 .]\) $$ \begin{array}{ll} \underline{\text { Curve }} & \underline{\text {Point}} \\ \text { Line: } 2 x+3 y=-1 \quad (0,0) \end{array} $$