Chapter 12

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

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

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

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

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

Show that the surfaces \(x^{2}+4 y+z^{2}=0\) 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 \((0,-1,2)\).

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 \(\nabla f(\mathbf{p})\) ?

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

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

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

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

In Problems 1-16, find all first partial derivatives of each function. \(F(x, y)=2 \sin x \cos y\)

In Problems 15 and 16, 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 \(z=x^{2} y\) and \(y=\frac{1}{4} x^{2}+\frac{3}{4}\) intersect at \((1,1,1)\) and have perpendicular tangent planes there.

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 $$ \left.\frac{\partial w}{\partial \theta}\right|_{\rho=2, \theta=\pi, \phi=\pi / 2} $$

In Problems 7-16, sketch the graph of \(f\). $$ f(x, y)=x^{2} / y, y>0 $$

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

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

In Problems 1-16, find all first partial derivatives of each function. \(f(r, \theta)=3 r^{3} \cos 2 \theta\)

In Problems 15 and 16, find the equation \(w=T(x, y, z)\) of the tangent "hyperplane" at \(\mathbf{p}\). $$ f(x, y, z)=x y z+x^{2}, \mathbf{p}=(2,0,-3) $$

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: \(x=1+2 t, y=3+8 t, z=2-6 t\).

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).

In Problems 17-22, sketch the level curve \(z=k\) for the indicated values of \(k\). $$ z=\frac{1}{2}\left(x^{2}+y^{2}\right), k=0,2,4,6,8 $$

In Problems 17-26, describe the largest set \(S\) on which it is correct to say that \(f\) is continuous. 17\. \(f(x, y)=\frac{x^{2}+x y-5}{x^{2}+y^{2}+1}\)

In Problems 17-20, verify that $$ \frac{\partial^{2} f}{\partial y \partial x}=\frac{\partial^{2} f}{\partial x \partial y} $$ \(f(x, y)=2 x^{2} y^{3}-x^{3} y^{5}\)

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

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 $$ \frac{x_{0} x}{a^{2}}+\frac{y_{0} y}{b^{2}}+\frac{z_{0} z}{c^{2}}=1 $$

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?

In Problems 17-22, sketch the level curve \(z=k\) for the indicated values of \(k\). $$ z=\frac{x}{y}, k=-2,-1,0,1,2 $$

In Problems 17-20, verify that $$ \frac{\partial^{2} f}{\partial y \partial x}=\frac{\partial^{2} f}{\partial x \partial y} $$ \(f(x, y)=\left(x^{3}+y^{2}\right)^{5}\)

Show that $$ \nabla\left(f^{r}\right)=r f^{r-1} \nabla f $$

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 \(\nabla g(1,2,2)\).

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 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?

In Problems 17-22, sketch the level curve \(z=k\) for the indicated values of \(k\). $$ z=\frac{x^{2}}{y}, k=-4,-1,0,1,4 $$

In Problems 17-20, verify that $$ \frac{\partial^{2} f}{\partial y \partial x}=\frac{\partial^{2} f}{\partial x \partial y} $$ \(f(x, y)=3 e^{2 x} \cos y\)

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.

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.

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?

In Problems 17-22, sketch the level curve \(z=k\) for the indicated values of \(k\). $$ z=x^{2}+y, k=-4,-1,0,1,4 $$

. 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?

In Problems 17-20, verify that $$ \frac{\partial^{2} f}{\partial y \partial x}=\frac{\partial^{2} f}{\partial x \partial y} $$ \(f(x, y)=\tan ^{-1} x y\)

In determining the specific gravity of an object, its weight in air is found to be \(A=36\) pounds and its weight in water is \(W=20\) pounds, with a possible error in each measurement of \(0.02\) pound. Find, approximately, the maximum possible error in calculating its specific gravity \(S\), where \(S=A /(A-W)\).

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.

In Problems 17-22, sketch the level curve \(z=k\) for the indicated values of \(k\). $$ z=\frac{x^{2}+1}{x^{2}+y^{2}}, k=1,2,4 $$

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

\(f(x, y)=\frac{x^{2}+3 x y+y^{2}}{y-x^{2}}\)

If \(F(x, y)=\frac{2 x-y}{x y}\), find \(F_{x}(3,-2)\) and \(F_{y}(3,-2)\).