91Ó°ÊÓ

Find equations for the (a) tangent plane and (b) normal line at the point \(P_{0}\) on the given surface. $$x^{2}+y^{2}-2 x y-x+3 y-z=-4, \quad P_{0}(2,-3,18)$$

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

Use Taylor's formula for \(f(x, y)\) at the origin to find quadratic and cubic approximations of \(f\) near the origin. $$f(x, y)=\cos \left(x^{2}+y^{2}\right)$$

(a) express \(\partial z / \partial u\) and \(\partial z / \partial v\) as functions of \(u\) and \(v\) both by using the Chain Rule and by expressing \(z\) directly in terms of \(u\) and \(v\) before differentiating. Then (b) evaluate \(\partial z / \partial u\) and \(\partial z / \partial v\) at the given point \((u, v)\). $$\begin{array}{l} z=\tan ^{-1}(x / y), \quad x=u \cos v, \quad y=u \sin v ;\\\ (u, v)=(1.3, \pi / 6) \end{array}$$

Find and sketch the domain for each function. $$f(x, y)=\frac{\sin (x y)}{x^{2}+y^{2}-25}$$

Find the limits. $$\lim _{(x, y) \rightarrow(1,1)} \ln \left|1+x^{2} y^{2}\right|$$

Find \(\partial f / \partial x\) and \(\partial f / \partial y\). \(f(x, y)=\left(x^{3}+(y / 2)\right)^{2 / 3}\)

Find \(\nabla f\) at the given point. $$f(x, y, z)=2 z^{3}-3\left(x^{2}+y^{2}\right) z+\tan ^{-1} x z, \quad(1,1,1).$$

Find and sketch the domain for each function. $$f(x, y)=\cos ^{-1}\left(y-x^{2}\right)$$

Use Taylor's formula for \(f(x, y)\) at the origin to find quadratic and cubic approximations of \(f\) near the origin. $$f(x, y)=\frac{1}{1-x-y}$$