Chapter 12

Find the global maximum value and global minimum value of \(f\) on \(S\) and indicate where each occurs. \(f(x, y)=x^{2}-6 x+y^{2}-8 y+7 ;\) \(S=\left\\{(x, y): x^{2}+y^{2} \leq 1\right\\}\)

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

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). $$ f(x, y)=\frac{x^{2}}{y}, \mathbf{p}=(2,-1) $$

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

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}}\)

Find all first partial derivatives of each function. \(f(s, t)=e^{t^{2}-s^{2}}\)

Sketch the graph of \(\bar{f}\). $$ f(x, y)=2-x-y^{2} $$

Sketch the level curve of \(f(x, y)=y / x^{2}\) that goes through \(\mathbf{p}=(1,2) .\) Calculate the gradient vector \(\nabla f(\mathbf{p})\) and draw this vector, placing its initial point at \(\mathbf{p}\). What should be true about \(\nabla f(\mathbf{p})\) ?

Express a positive number \(N\) as a sum of three positive numbers such that the product of these three numbers is a maximum.

If \(w=u^{2}-u \tan v, u=x\), and \(v=\pi x\), find $$ \left.\frac{d w}{d x}\right|_{x=1 / 4} $$

Find the equation \(w=T(x, y, z)\) of the tangent "hyperplane" at \(\mathbf{p}\). $$ f(x, y, z)=3 x^{2}-2 y^{2}+x z^{2}, \mathbf{p}=(1,2,-1) $$

Show that the surfaces \(x^{2}+4 y+z^{2}=0 \quad\) and \(x^{2}+y^{2}+z^{2}-6 z+7=0\) are tangent to each other at \((0,-1,2) ;\) that is, show that they have the same tangent plane at \((0,-1,2)\)

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y^{2}}{x^{2}+y^{4}}\)

Find all first partial derivatives of each function. \(F(x, y)=2 \sin x \cos y\)

Sketch the graph of \(\bar{f}\). $$ f(x, y)=e^{-\left(x^{2}+y^{2}\right)} $$

Use the methods of this section to find the shortest distance from the origin to the plane \(x+2 y+3 z=12\).

If \(w=x^{2} y+z^{2}, x=\rho \cos \theta \sin \phi, y=\rho \sin \theta \sin \phi\), and \(z=\rho \cos \phi\), find $$ \left.\frac{\partial w}{\partial \theta}\right|_{\rho=2, \theta=\pi, \phi=\pi / 2} $$

Show that the surfaces \(z=x^{2} y\) and \(y=\frac{1}{4} x^{2}+\frac{3}{4}\) intersect at \((1,1,1)\) and have perpendicular tangent planes there.

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y^{2}}{x^{2}+y^{4}}\)

Find all first partial derivatives of each function. \(f(r, \theta)=3 r^{3} \cos 2 \theta\)

Find the directional derivative of \(f(x, y, z)=x y+z^{2}\) at \((1,1,1)\) in the direction toward \((5,-3,3)\).

The part of a tree normally sawed into lumber is the trunk, a solid shaped approximately like a right circular cylinder. If the radius of the trunk of a certain tree is growing \(\frac{1}{2}\) inch per year and the height is increasing 8 inches per year, how fast is the volume increasing when the radius is 20 inches and the height is 400 inches? Express your answer in board feet per year ( 1 board foot \(=1\) inch by 12 inches by 12 inches).

Show that $$ \nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}} $$

Find a point on the surface \(x^{2}+2 y^{2}+3 z^{2}=12\) where the tangent plane is perpendicular to the line with parametric equations: \(x=1+2 t, y=3+8 t, z=2-6 t\)

Describe the largest set \(S\) on which it is correct to say that \(f\) is continuous. \(f(x, y)=\frac{x^{2}+x y-5}{x^{2}+y^{2}+1}\)

Verify that $$\frac{\partial^{2} f}{\partial y \partial x}=\frac{\partial^{2} f}{\partial x \partial y}$$ \(f(x, y)=2 x^{2} y^{3}-x^{3} y^{5}\)

Sketch the level curve \(z=k\) for the indicated values of \(k\). \(z=\frac{1}{2}\left(x^{2}+y^{2}\right), k=0,2,4,6,8\)

Find the directional derivative of \(f(x, y)=e^{-x} \cos y\) at \((0, \pi / 3)\) in the direction toward the origin.

The temperature of a metal plate at \((x, y)\) is \(e^{-x-3 y}\) degrees. A bug is walking northeast at a rate of \(\sqrt{8}\) feet per minute (i.e., \(d x / d t=d y / d t=2\) ). From the bug's point of view, how is the temperature changing with time as it crosses the origin?

Show that the equation of the tangent plane to the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$ at \(\left(x_{0}, y_{0}, z_{0}\right)\) can be written in the form $$ \frac{x_{0} x}{a^{2}}+\frac{y_{0} y}{b^{2}}+\frac{z_{0} z}{c^{2}}=1 $$

Describe the largest set \(S\) on which it is correct to say that \(f\) is continuous. \(f(x, y)=\ln \left(1+x^{2}+y^{2}\right)\)

Verify that $$\frac{\partial^{2} f}{\partial y \partial x}=\frac{\partial^{2} f}{\partial x \partial y}$$ \(f(x, y)=\left(x^{3}+y^{2}\right)^{5}\)

Sketch the level curve \(z=k\) for the indicated values of \(k\). \(z=\frac{x}{y}, k=-2,-1,0,1,2\)

The temperature at \((x, y, z)\) of a solid sphere centered at the origin is given by $$ T(x, y, z)=\frac{200}{5+x^{2}+y^{2}+z^{2}} $$ (a) By inspection, decide where the solid sphere is hottest. (b) Find a vector pointing in the direction of greatest increase of temperature at \((1,-1,1)\). (c) Does the vector of part (b) point toward the origin?

A rectangular metal tank with open top is to hold 256 cubic feet of liquid. What are the dimensions of the tank that require the least material to build?

A boy's toy boat slips from his grasp at the edge of a straight river. The stream carries it along at 5 feet per second. A crosswind blows it toward the opposite bank at 4 feet per second. If the boy runs along the shore at 3 feet per second following his boat, how fast is the boat moving away from him when \(t=3\) seconds?

Find all points \((x, y)\) at which the tangent plane to the graph of \(z=x^{2}-6 x+2 y^{2}-10 y+2 x y\) is horizontal.

Find the parametric equations of the line that is tangent to the curve of intersection of the surfaces $$ f(x, y, z)=9 x^{2}+4 y^{2}+4 z^{2}-41=0 $$ and $$ g(x, y, z)=2 x^{2}-y^{2}+3 z^{2}-10=0 $$ at the point \((1,2,2)\). Hint: This line is perpendicular to \(\nabla f(1,2,2)\) and \(\nabla g(1,2,2)\).

Describe the largest set \(S\) on which it is correct to say that \(f\) is continuous. \(f(x, y)=\ln \left(1-x^{2}-y^{2}\right)\)

Verify that $$\frac{\partial^{2} f}{\partial y \partial x}=\frac{\partial^{2} f}{\partial x \partial y}$$ \(f(x, y)=3 e^{2 x} \cos y\)

Sketch the level curve \(z=k\) for the indicated values of \(k\). \(z=\frac{x^{2}}{y}, k=-4,-1,0,1,4\)

The temperature at \((x, y, z)\) of a solid sphere centered at the origin is \(T(x, y, z)=100 e^{ \left(x^{2}+y^{2}+z^{2}\right)}\). Note that it is hottest at the origin. Show that the direction of greatest decrease in temperature is always a vector pointing away from the origin.

A rectangular box, whose edges are parallel to the coordinate axes, is inscribed in the ellipsoid \(96 x^{2}+4 y^{2}+4 z^{2}=36\). What is the greatest possible volume for such a box?

Sand is pouring onto a conical pile in such a way that at a certain instant the height is 100 inches and increasing at 3 inches per minute and the base radius is 40 inches and increasing at 2 inches per minute. How fast is the volume increasing at that instant?

Describe the largest set \(S\) on which it is correct to say that \(f\) is continuous. \(f(x, y)=\frac{1}{\sqrt{1+x+y}}\)

Verify that $$\frac{\partial^{2} f}{\partial y \partial x}=\frac{\partial^{2} f}{\partial x \partial y}$$ \(f(x, y)=\tan ^{-1} x y\)

Sketch the level curve \(z=k\) for the indicated values of \(k\). \(z=x^{2}+y, k=-4,-1,0,1,4\)

Find the gradient of \(f(x, y, z)=\sin \sqrt{x^{2}+y^{2}+z^{2}}\). Show that the gradient always points directly toward the origin or directly away from the origin.

Find the three-dimensional vector with length 9, the sum of whose components is a maximum.

Find parametric equations of the line tangent to the surface \(z=y^{2}+x^{3} y\) at the point \((2,1,9)\) whose projection on the \(x y\) -plane is (a) parallel to the \(x\) -axis; (b) parallel to the \(y\) -axis; (c) parallel to the line \(x=y\).