91Ó°ÊÓ

Find \(f_{x}(x, y)\) and \(f_{y}(x, y)\) $$ f(x, y)=\cosh (\sqrt{x}) \sinh ^{2}\left(x y^{2}\right) $$

Find the directional derivative of the function $$ f(x, y, z)=x^{3} y^{2} z^{5}-2 x z+y z+3 x $$ at \(P(-1,-2,1)\) in the direction of the negative \(z\) -axis.

Use appropriate forms of the chain rule to find the derivatives. $$ \begin{array}{l}{\text { Let } z=\ln \left(x^{2}+1\right), \text { where } x=r \cos \theta . \text { Find } \partial z / \partial r \text { and }} \\\ {\partial z / \partial \theta .}\end{array} $$

Find parametric equations for the tangent line to the curve of intersection of the cone \(z=\sqrt{x^{2}+y^{2}}\) and the plane \(x+2 y+2 z=20\) at the point \((4,3,5)\)

(a) Show that the second partials test provides no information about the critical points of the function \(f(x, y)=x^{4}-y^{4}\) (b) Classify all critical points of \(f\) as relative maxima, relative minima, or saddle points.

True-False Determine whether the statement is true or false. Explain your answer. Every level surface of \(f(x, y, z)=x+2 y+3 z\) is a plane.

True-False Determine whether the statement is true or false. Explain your answer. If \(D\) is an open set in 2 -space or in 3 -space, then every point in \(D\) is an interior point of \(D .\)

Evaluate the indicated partial derivatives. $$ f(x, y)=9-x^{2}-7 y^{3} ; f_{x}(3,1), f_{y}(3,1) $$

Find parametric equations for the tangent line to the curve of intersection of the cylinders \(x^{2}+z^{2}=25\) and \(y^{2}+z^{2}=25\) at the point \((3,-3,4) .\)

Sketch the graph of \(f\) $$ f(x, y)=3 $$