91Ó°ÊÓ

Find all critical points of the following functions. $$f(x, y)=x^{4}+y^{4}-16 x y$$

Find an equation of the plane tangent to the following surfaces at the given points. $$x y \sin z=1 ;(1,2, \pi / 6) \text { and }(-2,-1,5 \pi / 6)$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(-3,3)}\left(4 x^{2}-y^{2}\right)$$

Find the first partial derivatives of the following functions. $$g(x, y)=\cos 2 x y$$

Find the domain of the following functions. $$f(x, y)=\sqrt{25-x^{2}-y^{2}}.$$

Lagrange multipliers in two variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y)=y^{2}-4 x^{2} \text { subject to } x^{2}+2 y^{2}=4$$

Find an equation of the plane that passes through the point \(P_{0}\) with a normal vector \(\mathbf{n}\). $$P_{0}(2,3,0) ; \mathbf{n}=\langle-1,2,-3\rangle$$

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$f(x, y)=x e^{2 x y} ; P(1,0)$$

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$\begin{aligned} &d Q / d t, \text { where } Q=\sqrt{x^{2}+y^{2}+z^{2}}, x=\sin t, y=\cos t, \text { and } &z=\cos t \end{aligned}$$

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$f(x, y)=\sin (3 x+2 y) ; P(\pi, 3 \pi / 2)$$