91Ó°ÊÓ

In Exercises \(1-4,\) find the specific function values. $$ f(x, y, z)=\frac{x-y}{v^{2}+z^{2}} $$ $$ \begin{array}{ll}{\text { a. } f(3,-1,2)} & {\text { b. } f\left(1, \frac{1}{2},-\frac{1}{4}\right)} \\ {\text { c. } f\left(0,-\frac{1}{3}, 0\right)} & {\text { d. } f(2,2,100)}\end{array} $$

Find the limits in Exercises \(1-12.\) $$\lim _{(x, y) \rightarrow(3,4)} \sqrt{x^{2}+y^{2}-1}$$

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

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

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)=y \sin x\end{equation}

Find the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point. \(\begin{array}{c}g(x, y)=x y^{2}, \quad(2,-1)\end{array}\)

(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 \(t\). \(\begin{array}{c}w=\frac{x}{z}+\frac{y}{z}, \quad x=\cos ^{2} t, \quad y=\sin ^{2} t, \quad z=1 / t ; \quad t=3\end{array}\)

Find all the local maxima, local minima, and saddle points of the functions. $$ f(x, y)=x^{2}+x y+3 x+2 y+5 $$

Maximum on a line Find the maximum value of \(f(x, y)=49-\) \(x^{2}-y^{2}\) on the line \(x+3 y=10\) .

In Exercises \(1-22,\) find \(\partial f / \partial x\) and \(\partial f / \partial y\) $$f(x, y)=5 x y-7 x^{2}-y^{2}+3 x-6 y+2$$