Chapter 12

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 \(S=A /(A-W)\).

Describe the largest set \(S\) on which it is correct to say that \(f\) is continuous. \(f(x, y)=\frac{x^{2}+3 x y+y^{2}}{y-x^{2}}\)

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

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

Suppose that the temperature \(T\) at the point \((x, y, z)\) depends only on the distance from the origin. Show that the direction of greatest increase in \(T\) is either directly toward the origin or directly away from the origin.

Find the point on the plane \(2 x+4 y+3 z=12\) that is closest to the origin. What is the minimum distance?

Find parametric equations of the line tangent to the surface \(z=x^{2} y^{3}\) at the point \((3,2,72)\) 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 \(x=-y\).

Use differentials to find the approximate amount of copper in the four sides and bottom of a rectangular copper tank that is 6 feet long, 4 feet wide, and 3 feet deep inside, if the sheet copper is \(\frac{1}{4}\) inch thick. Hint: Make a sketch.

\(f(x, y)=\left\\{\begin{array}{cl}\frac{\sin (x y)}{x y}, & \text { if } x y \neq 0 \\ 1, & \text { if } x y=0\end{array}\right.\)

If \(F(x, y)=\ln \left(x^{2}+x y+y^{2}\right)\), find \(F_{x}(-1,4)\) and \(F_{y}(-1,4)\)

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

The elevation of a mountain above sea level at the point \((x, y)\) is \(f(x, y) .\) A mountain climber at \(\mathbf{p}\) notes that the slope in the easterly direction is \(-\frac{1}{2}\) and the slope in the northerly direction is \(-\frac{1}{4}\). In what direction should he move for fastest descent?

Find the maximum and minimum of the function f over the closed and bounded set \(S .\) Use the methods of Section \(12.8\) to find the maximum and minimum on the the interior of \(S ;\) then use Lagrange multipliers to find the maximum and minimum over the boundary of \(S .\) \(f(x, y)=x^{2}+y^{2}+3 x-x y\) \(S=\left\\{(x, y): x^{2}+y^{2} \leq 9\right\\}\)

Find the point on the paraboloid \(z=x^{2}+y^{2}\) that is closest to \((1,2,0)\). What is the minimum distance?

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

If \(f(x, y)=\tan ^{-1}\left(y^{2} / x\right)\), find \(f_{x}(\sqrt{5},-2)\) and \(f_{y}(\sqrt{5},-2)\)

Let \(T(x, y)\) be the temperature at a point \((x, y)\) in the plane. Draw the isothermal curves corresponding to \(T=\frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 0\) if $$ T(x, y)=\frac{x^{2}}{x^{2}+y^{2}} $$

Given that \(f_{x}(2,4)=-3\) and \(f_{y}(2,4)=8\), find the directional derivative of \(f\) at \((2,4)\) in the direction toward \((5,0)\).

Find the minimum distance between the point \((1,2,0)\) and the quadric cone \(z^{2}=x^{2}+y^{2}\).

Mean Value Theorem for Several Variables If \(f\) is differentiable at each point of the line segment from a to \(\mathbf{b}\), then there exists on that line segment a point \(\mathbf{c}\) between \(\mathbf{a}\) and \(\mathbf{b}\) such that $$ f(\mathbf{b})-f(\mathbf{a})=\nabla f(\mathbf{c}) \cdot(\mathbf{b}-\mathbf{a}) $$ Assuming that this result is true, show that, if \(f\) is differentiable on a convex set \(S\) and if \(\nabla f(\mathbf{p})=\mathbf{0}\) on \(S\), then \(f\) is constant on \(S\). Note: A set \(S\) is convex if each pair of points in \(S\) can be connected by a line segment in \(S .\)

The period \(T\) of a pendulum of length \(L\) is given by \(T=2 \pi \sqrt{L / g}\), where \(g\) is the acceleration of gravity. Show that \(d T / T=\frac{1}{2}[d L / L-d g / g]\), and use this result to estimate the maximum percentage error in \(T\) due to an error of \(0.5 \%\) in measuring \(L\) and \(0.3 \%\) in measuring \(g\).

Describe the largest set \(S\) on which it is correct to say that \(f\) is continuous. \(f(x, y)=\left(4-x^{2}-y^{2}\right)^{-1 / 2}\)

If \(f(x, y)=e^{y} \cosh x\), find \(f_{x}(-1,1)\) and \(f_{y}(-1,1)\).

If \(V(x, y)\) is the voltage at a point \((x, y)\) in the plane, the level curves of \(V\) are called equipotential curves. Draw the equipotential curves corresponding to \(V=\frac{1}{2}, 1,2,4\) for $$ V(x, y)=\frac{4}{\sqrt{(x-2)^{2}+(y+3)^{2}}} $$

The elevation of a mountain above sea level at \((x, y)\) is \(3000 e^{-\left(x^{2}+2 y^{2}\right) / 100}\) meters. The positive \(x\) -axis points east and the positive \(y\) -axis points north. A climber is directly above \((10,10)\). If the climber moves northwest, will she ascend or descend and at what slope?

Find the maximum and minimum of the function f over the closed and bounded set \(S .\) Use the methods of Section \(12.8\) to find the maximum and minimum on the the interior of \(S ;\) then use Lagrange multipliers to find the maximum and minimum over the boundary of \(S .\) \(f(x, y)=(1+x+y)^{2} ; S=\left\\{(x, y): \frac{x^{2}}{4}+\frac{y^{2}}{16} \leq 1\right\\}\)

The formula \(1 / R=1 / R_{1}+1 / R_{2}\) determines the combined resistance \(R\) when resistors of resistance \(R_{1}\) and \(R_{2}\) are connected in parallel. Suppose that \(R_{1}\) and \(R_{2}\) were measured at 25 and 100 ohms, respectively, with possible errors in each measurement of \(0.5\) ohm. Calculate \(R\) and give an estimate for the maximum error in this value.

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

Find the slope of the tangent to the curve of intersection of the surface \(36 z=4 x^{2}+9 y^{2}\) and the plane \(x=3\) at the point \((3,2,2)\)

If the temperature of a plate at the point \((x, y)\) is \(T(x, y)=10+x^{2}-y^{2}\), find the path a heat-seeking particle (which always moves in the direction of greatest increase in temperature) would follow if it starts at \((-2,1) .\) Hint: The particle moves in the direction of the gradient $$ \nabla T=2 x \mathbf{i}-2 y \mathbf{j} $$ We may write the path in parametric form as $$ \mathbf{r}(t)=x(t) \mathbf{i}+y(t) \mathbf{j} $$ and we want \(x(0)=-2\) and \(y(0)=1 .\) To move in the required direction means that \(\mathbf{r}^{\prime}(t)\) should be parallel to \(\nabla T\). This will be satisfied if $$ \frac{x^{\prime}(t)}{2 x(t)}=-\frac{y^{\prime}(t)}{2 y(t)} $$ together with the conditions \(x(0)=-2\) and \(y(0)=1\). Now solve this differential equation and evaluate the arbitrary constant of integration.

Find the minimum distance between the lines having parametric equations \(x=t-1, y=2 t, z=t+3\) and \(x=3 s\), \(y=s+2, z=2 s-1\)

A bee sat at the point \((1,2,1)\) on the ellipsoid \(x^{2}+y^{2}+2 z^{2}=6\) (distances in feet). At \(t=0\), it took off along the normal line at a speed of 4 feet per second. Where and when did it hit the plane \(2 x+3 y+z=49 ?\)

Describe the largest set \(S\) on which it is correct to say that \(f\) is continuous. \(f(x, y, z)=\ln \left(4-x^{2}-y^{2}-z^{2}\right)\)

Find the slope of the tangent to the curve of intersection of the surface \(3 z=\sqrt{36-9 x^{2}-4 y^{2}}\) and the plane \(x=1\) at the point \((1,-2, \sqrt{11} / 3)\).

Consider the Cobb-Douglas production model for a manufacturing process depending on three inputs \(x, y\), and \(z\) with unit costs \(a, b\), and \(c\), respectively, given by $$ P=k x^{\alpha} y^{\beta} z^{\gamma}, \quad \alpha>0, \beta>0, \gamma>0, \alpha+\beta+\gamma=1 $$ subject to the cost constraint \(a x+b y+c z=d\). Determine \(x, y\), and \(z\) to maximize the production \(P\).

Convince yourself that the maximum and minimum values of a linear function \(f(x, y)=a x+b y+c\) over a closed polygonal set (i.e., a polygon and its interior) will always occur at a vertex of the polygon. Then use this fact to find each of the following: (a) maximum value of \(2 x+3 y+4\) on the closed polygon with vertices \((-1,2),(0,1),(1,0),(-3,0)\), and \((0,-4)\) (b) minimum value of \(-3 x+2 y+1\) on the closed polygon with vertices \((-3,0),(0,5),(2,3),(4,0)\), and \((1,-4)\)

If \(T=f(x, y, z, w)\) and \(x, y, z\), and \(w\) are each functions of \(s\) and \(t\), write a chain rule for \(\partial T / \partial s\).

Show that a plane tangent at any point of the surface \(x y z=k\) forms with the coordinate planes a tetrahedron of fixed volume and find this volume.

Sketch the indicated set. Describe the boundary of the set. Finally, state whether the set is open, closed, or neither. \(\\{(x, y): 2 \leq x \leq 4,1 \leq y \leq 5\\}\)

Find the slope of the tangent to the curve of intersection of the surface \(2 z=\sqrt{9 x^{2}+9 y^{2}-36}\) and the plane \(y=1\) at the point \(\left(2,1, \frac{3}{2}\right)\).

Describe geometrically the domain of each of the indicated functions of three variables. \(f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}-16}\)

Find the minimum distance from the origin to the line of intersection of the two planes $$ x+y+z=8 \text { and } 2 x-y+3 z=28 $$

Let \(z=f(x, y)\), where \(x=r \cos \theta\) and \(y=r \sin \theta .\) Show that $$ \left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{\partial z}{\partial y}\right)^{2}=\left(\frac{\partial z}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial z}{\partial \theta}\right)^{2} $$

Find the most general function \(f(\mathbf{p})\) satisfying \(\nabla f(\mathbf{p})=\mathbf{p}\).

Sketch the indicated set. Describe the boundary of the set. Finally, state whether the set is open, closed, or neither. \(\left\\{(x, y): x^{2}+y^{2}<4\right\\}\)

Find the slope of the tangent to the curve of intersection of the cylinder \(4 z=5 \sqrt{16-x^{2}}\) and the plane \(y=3\) at the point \((2,3,5 \sqrt{3} / 2)\)

Describe geometrically the domain of each of the indicated functions of three variables. \(f(x, y, z)=\sqrt{x^{2}+y^{2}-z^{2}-9}\)

The temperature \(T\) in degrees Celsius at \((x, y, z)\) is given by \(T=10 /\left(x^{2}+y^{2}+z^{2}\right)\), where distances are in meters. A bee is flying away from the hot spot at the origin on a spiral path so that its position vector at time \(t\) seconds is \(\mathbf{r}(t)=\) \(t \cos \pi t \mathbf{i}+t \sin \pi t \mathbf{j}+t \mathbf{k} .\) Determine the rate of change of \(T\) in each case. (a) With respect to distance traveled at \(t=1\). (b) With respect to time at \(t=1\). (Think of two ways to do this.)

The wave equation of physics is the partial differential equation $$ \frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}} $$ where \(c\) is a constant. Show that if \(f\) is any twice differentiable function then $$ y(x, t)=\frac{1}{2}[f(x-c t)+f(x+c t)] $$ satisfies this equation.

For the function \(f(x, y)=\sqrt{x^{2}+y^{2}}\), find the secondorder Taylor approximation based at \(\left(x_{0}, y_{0}\right)=(3,4)\). Then estimate \(f(3.1,3.9)\) using (a) the first-order approximation, (b) the second-order approximation, and (c) your calculator directly.