91Ó°ÊÓ

Find the first partial derivatives of the following functions. $$f(x, y)=3 x^{2} y+2$$

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

Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$d w / d t, \text { where } w=x y \sin z, x=t^{2}, y=4 t^{3}, \text { and } z=t+1$$

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

Use Theorem 12.7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$\begin{aligned} &d Q / d t, \text { where } Q=\sqrt{x^{2}+y^{2}+z^{2}}, x=\sin t, y=\cos t, \text { and }\\\ &z=\cos t \end{aligned}$$

Find all critical points of the following functions. \(f(x, y)=x^{3} / 3-y^{3} / 3+3 x y\)

Find an equation of the plane that passes through the point \(P_{0}\) with a normal vector \(\mathbf{n}\). $$P_{0}(1,2,-3) ; \mathbf{n}=\langle-1,4,-3\rangle$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(2,-1)}\left(x y^{8}-3 x^{2} y^{3}\right)$$

\(\boldsymbol{F}(\boldsymbol{x}, \boldsymbol{y}, z)=\boldsymbol{0}\) Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). $$y z e^{x z}-8=0 ;(0,2,4) \text { and }(0,-8,-1)$$

Find the domain of the following functions. $$f(x, y)=\frac{1}{\sqrt{x^{2}+y^{2}-25}}$$