91Ó°ÊÓ

Find the limits. $$\lim _{(x, y) \rightarrow(0,0)} \cos \frac{x^{2}+y^{3}}{x+y+1}$$

Find the gradient of the function at the given point.Then sketch the gradient together with the level curve that passes through the point. \begin{equation}f(x, y)=\tan ^{-1} \frac{\sqrt{x}}{y}, \quad(4,-2)\end{equation}

Find all the local maxima, local minima, and saddle points of the functions in Exercises \(1-30 .\) $$ f(x, y)=x^{2}-4 x y+y^{2}+6 y+2 $$

Use Taylor's formula for \(f(x, y)\) at the origin to find quadratic and cubic approximations of \(f\) near the origin. \begin{equation}f(x, y)=\ln (2 x+y+1)\end{equation}

Find equations for the (a) tangent plane and (b) normal line at the point \(P_{0}\) on the given surface. $$ x^{2}-x y-y^{2}-z=0, \quad P_{0}(1,1,-1) $$

In Exercises \(1-6,\) (a) express \(d w / d t\) as a function of \(t,\) both by using the Chain Rule and by expressing \(w\) in terms of \(t\) and differentiating directly with respect to \(t\) . Then (b) evaluate \(d w / d t\) at the given value of \(t .\) $$ w=z-\sin x y, \quad x=t, \quad y=\ln t, \quad z=e^{t-1} ; \quad t=1 $$

In Exercises \(5-12,\) find and sketch the domain for each function. $$f(x, y)=\ln \left(x^{2}+y^{2}-4\right)$$

In Exercises \(1-22,\) find \(\partial f / \partial x\) and \(\partial f / \partial y\) $$f(x, y)=(2 x-3 y)^{3}$$

Find \((\partial u / \partial y)_{x}\) at the point \((u, v)=(\sqrt{2}, 1)\) if \(x=u^{2}+v^{2}\) and \(y=u v .\)

Constrained minimum Find the points on the curve \(x^{2} y=2\) nearest the origin.