91Ó°ÊÓ

(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. $$f(x, y)=y-x$$

Find the derivative of the function at \(P_{0}\) in the direction of \(\mathbf{u}.\) $$g(x, y, z)=3 e^{x} \cos y z, \quad P_{0}(0,0,0), \quad \mathbf{u}=2 \mathbf{i}+\mathbf{j}-2 \mathbf{k}.$$

Find the limits by rewriting the fractions first. $$\lim _{(x, y) \rightarrow(2,2)\atop x +y \neq 4} \frac{x+y-4}{\sqrt{x+y}-2}$$

Find parametric equations for the line tangent to the curve of intersection of the surfaces at the given point. $$\begin{aligned}&\text { Surfaces: } x^{2}+y^{2}=4, \quad x^{2}+y^{2}-z=0\\\&\text { Point: }\quad(\sqrt{2}, \sqrt{2}, 4)\end{aligned}$$

(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. $$f(x, y)=\sqrt{y-x}$$

Find all the local maxima, local minima, and saddle points of the functions. $$f(x, y)=2 x^{3}+2 y^{3}-9 x^{2}+3 y^{2}-12 y$$

Draw a branch diagram and write a Chain Rule formula for each derivative. $$\frac{\partial w}{\partial x} \text { and } \frac{\partial w}{\partial y} \text { for } w=g(u, v), \quad u=h(x, y), \quad v=k(x, y)$$

Find the point on the sphere \(x^{2}+y^{2}+z^{2}=4\) farthest from the point \((1,-1,1).\)

Find \(\partial f / \partial x\) and \(\partial f / \partial y\). \(f(x, y)=\cos ^{2}\left(3 x-y^{2}\right)\)

Find the derivative of the function at \(P_{0}\) in the direction of \(\mathbf{u}.\) $$\begin{array}{l}h(x, y, z)=\cos x y+e^{y z}+\ln z x, \quad P_{0}(1,0,1 / 2) \\\ \mathbf{u}=\mathbf{i}+2 \mathbf{j}+2 \mathbf{k}\end{array}.$$