91Ó°ÊÓ

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

Use Theorem 7 to find the following derivatives. When feasible, express your answer in terms of the independent variable. $$\begin{aligned}&d U / d t, \text { where } U=\ln (x+y+z), x=t, y=t^{2}, \text { and }&z=t^{3}\end{aligned}$$

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

Lagrange multipliers in three variables Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y, z)=x+3 y-z \text { subject to } x^{2}+y^{2}+z^{2}=4$$

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

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

Find the equation of the plane that is parallel to the vectors \langle 1,-3,1\rangle and \(\langle 4,2,0\rangle,\) passing through the point (3,0,-2)

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$h(x, y)=\ln \left(1+x^{2}+2 y^{2}\right) ; P(2,-3)$$

Find an equation of the plane tangent to the following surfaces at the given points. $$2 x+y^{2}-z^{2}=0 ;(0,1,1) \text { and }(4,1,-3)$$

Find the domain of the following functions. $$f(x, y)=\frac{12}{y^{2}-x^{2}}.$$