Chapter 12

In Problems 7-16, sketch the graph of \(f\). $$ f(x, y)=6 $$

$$ f(x, y)=x y+\frac{2}{x}+\frac{4}{y} $$

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

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=\sqrt{x^{2}-y^{2}}\)

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}} $$

What are the dimensions of the rectangular box, open at the top, that has maximum volume when the surface area is 48 ?

In Problems \(1-8\), find the equation of the tangent plane to the given surface at the indicated point. $$ z=x^{1 / 2}+y^{1 / 2} ;(1,4,3) $$

In Problems 1-8, find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y, z)=x^{2}+y^{2}+z^{2} ; \mathbf{p}=(1,-1,2)\);. \(\mathbf{a}=\sqrt{2} \mathbf{i}-\mathbf{j}-\mathbf{k}\)

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

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

In Problems 1-16, find all first partial derivatives of each function. \(f(u, v)=e^{u v}\)

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y, z)=x^{2} y+y^{2} z+z^{2} x $$

Find the minimum distance between the origin and the plane \(x+3 y-2 z=4\).

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 \(3 .\) $$ z=2 x^{2} y^{3} ; P(1,1), Q(0.99,1.02) $$

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? \(f(x, y)=x^{3}-y^{5} ; \mathbf{p}=(2,-1)\)

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

\(f(x, y)=\cos x+\cos y+\cos (x+y)\); $$ 0

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

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y, z)=x^{2} y e^{x-z} $$

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 \(3 .\) $$ z=x^{2}-5 x y+y ; P(2,3), Q(2.03,2.98) $$

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? \(f(x, y)=e^{y} \sin x ; \mathbf{p}=(5 \pi / 6,0)\)

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

\(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-y^{4}}{x^{2}+y^{2}}\)

In Problems 1-16, find all first partial derivatives of each function. \(f(s, t)=\ln \left(s^{2}-t^{2}\right)\)

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y, z)=x z \ln (x+y+z) $$

Find the minimum distance between the origin and the surface \(x^{2} y-z^{2}+9=0\).

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 \(3 .\) $$ z=\ln \left(x^{2} y\right) ; P(-2,4), Q(-1.98,3.96) $$

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? \(f(x, y, z)=x^{2} y z ; \mathbf{p}=(1,-1,2)\)

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

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

\(f(x, y)=3 x+4 y\), $$ S=\\{(x, y): 0 \leq x \leq 1,-1 \leq y \leq 1\\} $$

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

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

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^{2} y-x y^{2}, \mathbf{p}=(-2,3)

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 \(3 .\) $$ z=\tan ^{-1} x y ; P(-2,-0.5), Q(-2.03,-0.51) $$

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? \(f(x, y, z)=x e^{y z} ; \mathbf{p}=(2,0,-4)\)

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 7-16, sketch the graph of \(f\). $$ f(x, y)=\sqrt{16-4 x^{2}-y^{2}} $$

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

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 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). $$ f(x, y)=x^{3} y+3 x y^{2}, \mathbf{p}=(2,-2) $$

Find the maximum volume of the first-octant rectangular box with faces parallel to the coordinate planes, one vertex at \((0,0,0)\), and diagonally opposite vertex on the plane $$ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 $$

Find all points on the surface $$ z=x^{2}-2 x y-y^{2}-8 x+4 y $$ where the tangent plane is horizontal.

In what direction u does \(f(x, y)=1-x^{2}-y^{2}\) decrease most rapidly at \(\mathbf{p}=(-1,2)\) ?

If \(z=x^{2} y, x=2 t+s\), and \(y=1-s t^{2}\), find \(\left.\frac{\partial z}{\partial t}\right|_{s=1, t=-2}\)

In Problems 7-16, sketch the graph of \(f\). $$ f(x, y)=3-x^{2}-y^{2} $$

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=y \cos \left(x^{2}+y^{2}\right)\)

Find a point on the surface \(z=2 x^{2}+3 y^{2}\) where the tangent plane is parallel to the plane \(8 x-3 y-z=0\).

In what direction u does \(f(x, y)=\sin (3 x-y)\) decrease most rapidly at \(\mathbf{p}=(\pi / 6, \pi / 4)\) ?