91Ó°ÊÓ

\(\boldsymbol{F}(\boldsymbol{x}, \boldsymbol{y}, z)=\boldsymbol{0}\) 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 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. $$\lim _{(x, y) \rightarrow(2,-1)}\left(x y^{8}-3 x^{2} y^{3}\right)$$

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

Find the domain of the following functions. $$f(x, y)=\sin \frac{x}{y}$$

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

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

Find all critical points of the following functions. \(f(x, y)=x^{4}-2 x^{2}+y^{2}-4 y+5\)

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(0, \pi)} \frac{\cos x y+\sin x y}{2 y}$$

\(\boldsymbol{F}(\boldsymbol{x}, \boldsymbol{y}, z)=\boldsymbol{0}\) Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). $$z^{2}-x^{2} / 16-y^{2} / 9-1=0 ;(4,3,-\sqrt{3}) \text { and }(-8,9, \sqrt{14})$$