91Ó°ÊÓ

In Problems 7-16, sketch the graph of \(f\). $$ f(x, y)=\sqrt{16-x^{2}-y^{2}} $$

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=\tan ^{-1}(4 x-7 y)\)

Find the maximum volume of a closed rectangular box with faces parallel to the coordinate planes inscribed in the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$

Use the total differential dz to approximate the change in \(z\) as \((x, y)\) moves from \(P\) to \(Q\). Then use a calculator to find the corresponding exact change \(\Delta z\) (to the accuracy of your calculator). See Example \(3 .\) $$ z=\ln \left(x^{2} y\right) ; P(-2,4), Q(-1.98,3.96) $$

\(\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}\)

In Problems 7-12, find \(\partial w / \partial t\) by using the Chain Rule. Express your final answer in terms of \(s\) and \(t\). $$ w=e^{x y+z} ; x=s+t, y=s-t, z=t^{2} $$

In Problems 1-16, find all first partial derivatives of each function. \(F(w, z)=w \sin ^{-1}\left(\frac{w}{z}\right)\)

In Problems 9-12, find a unit vector in the direction in which \(f\) increases most rapidly at p. What is the rate of change in this direction? \(f(x, y, z)=x e^{y z} ; \mathbf{p}=(2,0,-4)\)

Use the total differential dz to approximate the change in \(z\) as \((x, y)\) moves from \(P\) to \(Q\). Then use a calculator to find the corresponding exact change \(\Delta z\) (to the accuracy of your calculator). See Example \(3 .\) $$ z=\tan ^{-1} x y ; P(-2,-0.5), Q(-2.03,-0.51) $$

In Problems 11-14, find the gradient vector of the given function at the given point \(\mathbf{p}\). Then find the equation of the tangent plane at \(\mathbf{p}\) (see Example 1). $$ f(x, y)=x^{3} y+3 x y^{2}, \mathbf{p}=(2,-2) $$