91Ó°ÊÓ

Find the four second partial derivatives of the following functions. $$p(u, v)=\ln \left(u^{2}+v^{2}+4\right)$$

Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility. $$f(x, y)=x^{4}+4 x^{2}(y-2)+8(y-1)^{2}$$

Graph several level curves of the following functions using the given window. Label at least two level curves with their \(z\) -values. $$z=x-y^{2} ;[0,4] \times[-2,2].$$

Find an equation of the plane parallel to the plane \(Q\) passing through the point \(P_{0}\). $$Q:-x+2 y-4 z=1 ; P_{0}(1,0,4)$$

Use differentials to approximate the change in \(z\) for the given changes in the independent variables. \(z=2 x-3 y-2 x y\) when \((x, y)\) changes from (1,4) to (1.1,3.9)

Given the following equations, evaluate \(d y / d x .\) Assume that each equation implicitly defines \(y\) as a differentiable function of \(x\). $$x^{2}-2 y^{2}-1=0$$

Find an equation of the plane parallel to the plane \(Q\) passing through the point \(P_{0}\). $$Q: 2 x+y-z=1 ; P_{0}(0,2,-2)$$

Given the following equations, evaluate \(d y / d x .\) Assume that each equation implicitly defines \(y\) as a differentiable function of \(x\). $$x^{3}+3 x y^{2}-y^{5}=0$$

Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility. $$f(x, y)=x e^{-x-y} \sin y, \text { for }|x| \leq 2,0 \leq y \leq \pi$$

Graph several level curves of the following functions using the given window. Label at least two level curves with their \(z\) -values. $$z=2 x-y ;[-2,2] \times[-2,2].$$