Chapter 12

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 \(x=y\).

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.

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.

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

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

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

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 \(x=-y\).

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?

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 $$ T(x, y)=\frac{x^{2}}{x^{2}+y^{2}} $$

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

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 \(g\).

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 \((5,0)\).

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 $$ V(x, y)=\frac{4}{\sqrt{(x-2)^{2}+(y+3)^{2}}} $$

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

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

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 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.

An open gutter with cross section in the form of a trapezoid with equal base angles is to be made by bending up equal strips along both sides of a long piece of metal 12 inches wide. Find the base angles and the width of the sides for maximum carrying capacity.

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 \((3,2,2)\).

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 \(2 x+3 y+z=49\) ?

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 \(x=3 s\), \(y=s+2, z=2 s-1\).

\(f(x, y, z)=\ln \left(4-x^{2}-y^{2}-z^{2}\right)\)

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 \((1,-2, \sqrt{11} / 3)\).

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.

In Problems 27-32, describe geometrically the domain of each of the indicated functions of three variables. $$ f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}-16} $$

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 \((1,-4)\)

In Problems 27-32, sketch the indicated set. Describe the boundary of the set. Finally, state whether the set is open, closed, or neither. 27\. \(\\{(x, y): 2 \leq x \leq 4,1 \leq y \leq 5\\}\)

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 \(\left(2,1, \frac{3}{2}\right)\).

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 \(P\).

Find and simplify the equation of the tangent plane at \(\left(x_{0}, y_{0}, z_{0}\right)\) to the surface \(\sqrt{x}+\sqrt{y}+\sqrt{z}=a\). Then show that the sum of the intercepts of this plane with the coordinate axes is \(a^{2}\).

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

In Problems 27-32, describe geometrically the domain of each of the indicated functions of three variables. $$ f(x, y, z)=\sqrt{x^{2}+y^{2}-z^{2}-9} $$

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 \((2,3,5 \sqrt{3} / 2)\).

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

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

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.

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.

In Problems 27-32, describe geometrically the domain of each of the indicated functions of three variables. $$ f(x, y, z)=\sqrt{144-16 x^{2}-9 y^{2}-144 z^{2}} $$

The volume \(V\) of a right circular cylinder is given by \(V=\pi r^{2} h\), where \(r\) is the radius and \(h\) is the height. If \(h\) is held fixed at \(h=10\) inches, find the rate of change of \(V\) with respect to \(r\) when \(r=6\) inches.

For the function \(f(x, y)=\tan \left(\left(x^{2}+y^{2}\right) / 64\right)\), find the second-order Taylor approximation based at \(\left(x_{0}, y_{0}\right)=(0,0)\). Then estimate \(f(0.2,-0.3)\) using (a) the first-order approximation, (b) the second-order approximation, and (c) your calculator directly.

Let \(\mathbf{u}=(3 \mathbf{i}-4 \mathbf{j}) / 5\) and \(\mathbf{v}=(4 \mathbf{i}+3 \mathbf{j}) / 5\) and suppose that at some point \(P, D_{u} f=-6\) and \(D_{v} f=17\). (a) Find \(\nabla f\) at \(P\). (b) Note that \(\|\nabla f\|^{2}=\left(D_{\mathrm{a}} f\right)^{2}+\left(D_{\mathrm{v}} f\right)^{2}\) in part (a). Show that this relation always holds if \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular.

Show that if \(w=f(r-s, s-t, t-r)\) then $$ \frac{\partial w}{\partial r}+\frac{\partial w}{\partial s}+\frac{\partial w}{\partial t}=0 $$

In Problems 27-32, describe geometrically the domain of each of the indicated functions of three variables. $$ f(x, y, z)=\frac{\left(144-16 x^{2}-16 y^{2}+9 z^{2}\right)^{3 / 2}}{x y z} $$

Least Squares Given \(n\) points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\), \(\left(x_{n}, y_{n}\right)\) in the \(x y\)-plane, we wish to find the line \(y=m x+b\) such that the sum of the squares of the vertical distances from the points to the line is a minimum; that is, we wish to minimize $$ f(m, b)=\sum_{i=1}^{n}\left(y_{i}-m x_{i}-b\right)^{2} $$

The temperature in degrees Celsius on a metal plate in the \(x y\)-plane is given by \(T(x, y)=4+2 x^{2}+y^{3}\). What is the rate of change of temperature with respect to distance (measured in feet) if we start moving from \((3,2)\) in the direction of the positive \(y\)-axis?

Let \(w=x_{1} x_{2} \cdots x_{n}\). (a) Maximize \(w\) subject to \(x_{1}+x_{2}+\cdots+x_{n}=1\) and all \(x_{i}>0 .\) (b) Use part (a) to deduce the famous Geometric MeanArithmetic Mean Inequality for positive numbers \(a_{1}, a_{2}, \ldots, a_{n} ;\) that is, $$ \sqrt[n]{a_{1} a_{2} \cdots a_{n}} \leq \frac{a_{1}+a_{2}+\cdots+a_{n}}{n} $$

Let \(F(t)=\int_{g(t)}^{h(t)} f(u) d u\), where \(f\) is continuous and \(g\) and \(h\) are differentiable. Show that $$ F^{\prime}(t)=f(h(t)) h^{\prime}(t)-f(g(t)) g^{\prime}(t) $$ and use this result to find \(F^{\prime}(\sqrt{2})\), where $$ F(t)=\int_{\sin \sqrt{2} \pi t}^{t^{2}} \sqrt{9+u^{4}} d u $$

In Problems 27-32, describe geometrically the domain of each of the indicated functions of three variables. $$ f(x, y, z)=\ln \left(x^{2}+y^{2}+z^{2}\right) $$