91Ó°ÊÓ

Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y, z)=x+y+z \text { subject to } x^{2}+y^{2}+z^{2}-2 x-2 y=1$$

Find the first partial derivatives of the following functions. $$f(w, z)=\frac{w}{w^{2}+z^{2}}$$

Find the following derivatives. \(z_{s}\) and \(z_{r},\) where \(z=x y-x^{2} y, x=s+t,\) and \(y=s-t\)

Find the points at which the following planes intersect the coordinate axes and find equations of the lines where the planes intersect the coordinate planes. Sketch a graph of the plane. $$3 x-2 y+z=6$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(3,1)} \frac{x^{2}-7 x y+12 y^{2}}{x-3 y}$$

Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y, z)=2 x+z^{2} \text { subject to } x^{2}+y^{2}+2 z^{2}=25$$

Tangent planes for \(z=f(x, y)\) Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). $$z=x^{2} e^{x-y} ;(2,2,4) \text { and }(-1,-1,1)$$

Compute the directional derivative of the following functions at the given point \(P\) in the direction of the given vector. Be sure to use a unit vector for the direction vector. $$f(x, y)=\sqrt{4-x^{2}-2 y} ; P(2,-2) ;\left\langle\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}}\right\rangle$$

Use what you learned about surfaces in Section 12.1 to sketch a graph of the following functions. In each case identify the surface and state the domain and range of the function. $$f(x, y)=3 x-6 y+18$$

Find the first partial derivatives of the following functions. $$g(x, z)=x \ln \left(z^{2}+x^{2}\right)$$