Chapter 12

Sketch the indicated set. Describe the boundary of the set. Finally, state whether the set is open, closed, or neither. \(\left\\{(x, y): 0

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.

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

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 \(P, D_{\mathbf{u}} f=-6\) and \(D_{\mathbf{v}} f=17\) (a) Find \(\nabla f\) at \(P\). (b) Note that \(\|\nabla f\|^{2}=\left(D_{\mathbf{u}} f\right)^{2}+\left(D_{\mathbf{v}} f\right)^{2}\) in part (a). Show that this relation always holds if \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular.

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 Mean\(\begin{array}{llll}\text { Arithmetic } & \text { Mean Inequality for positive numbers }\end{array}\) \(a_{1}, a_{2}, \ldots, a_{n} ;\) that is, $$ \sqrt[n]{a_{1} a_{2} \cdots a_{n}} \leq \frac{a_{1}+a_{2}+\cdots+a_{n}}{n} $$

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 $$ f(m, b)=\sum_{i=1}^{n}\left(y_{i}-m x_{i}-b\right)^{2} $$ (a) Find \(\partial f / \partial m\) and \(\partial f / \partial b\), and set these results equal to zero. Show that this leads to the system of equations $$ \begin{aligned} m \sum_{i=1}^{n} x_{i}^{2}+b \sum_{i=1}^{n} x_{i} &=\sum_{i=1}^{n} x_{i} y_{i} \\\ m \sum_{i=1}^{n} x_{i}+n b &=\sum_{i=1}^{n} y_{i} \end{aligned} $$ (b) Solve this system for \(m\) and \(b\). (c) Use the Second Partials Test (Theorem C) to show that \(f\) is minimized for this choice of \(m\) and \(b\).

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

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.

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

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?

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

Maximize \(w=a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\), all \(a_{i}>0\), subject to \(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1\).

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 $$ F(t)=\int_{\sin \sqrt{2} \pi t}^{t^{2}} \sqrt{9+u^{4}} d u $$

Sketch the indicated set. Describe the boundary of the set. Finally, state whether the set is open, closed, or neither. \(\\{(x, y): x>0, y<\sin (1 / x)\\}\)

According to the ideal gas law, the pressure, temperature, and volume of a gas are related by \(P V=k T\), where \(k\) is a constant. Find the rate of change of pressure (pounds per square inch) with respect to temperature when the temperature is \(300^{\circ} \mathrm{K}\) if the volume is kept fixed at 100 cubic inches.

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

Find the maximum and minimum values of \(z=2 x^{2}+y^{2}-4 x-2 y+5\) (Figure 3) on the set bounded by the closed triangle with vertices \((0,0),(4,0)\), and \((0,1)\).

Call a function \(f(x, y)\) homogeneous of degree 1 if \(f(t x, t y)=t f(x, y)\) for all \(t>0 .\) For example, \(f(x, y)=\) \(x+y e^{y / x}\) satisfies this criterion. Prove Euler's Theorem that such a function satisfies $$ f(x, y)=x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y} $$

Sketch the indicated set. Describe the boundary of the set. Finally, state whether the set is open, closed, or neither. \(\\{(x, y): x=0, y=1 / n, n\) a positive integer \(\\}\)

Leaving from the same point \(P\), airplane \(A\) flies due east while airplane B flies \(\mathrm{N} 50^{\circ} \mathrm{E}\). At a certain instant, \(\mathrm{A}\) is \(200 \mathrm{miles}\) from \(P\) flying at 450 miles per hour, and \(B\) is 150 miles from \(P\) flying at 400 miles per hour. How fast are they separating at that instant?

Let $$ f(x, y)=\left\\{\begin{array}{ll} \frac{x^{2}-4 y^{2}}{x-2 y}, & \text { if } x \neq 2 y \\ g(x), & \text { if } x=2 y \end{array}\right. $$ If \(f\) is continuous in the whole plane, find a formula for \(g(x)\).

Show that the functions defined are harmonic functions. \(f(x, y)=x^{3} y-x y^{3}\)

Describe geometrically the level surfaces for the functions. \(f(x, y, z)=x^{2}+y^{2}+z^{2} ; k>0\)

Recall Newton's Law of Gravitation, which asserts that the magnitude \(F\) of the force of attraction between objects of masses \(M\) and \(m\) is \(F=G M m / r^{2}\), where \(r\) is the distance between them and \(G\) is a universal constant. Let an object of mass \(M\) be located at the origin, and suppose that a second object of changing mass \(m\) (say from fuel consumption) is moving away from the origin so that its position vector is \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Obtain a formula for \(d F / d t\) in terms of the time derivatives of \(m\), \(x, y\), and \(z\)

Prove that\(\begin{aligned} \lim _{(x, y) \rightarrow(a, b)}[f(x, y)+g(x, y)&] \\\=& \lim _{(x, y) \rightarrow(a, b)} f(x, y)+\lim _{(x, y) \rightarrow(a, b)} g(x, y) \end{aligned}\) provided that the latter two limits exist.

Describe geometrically the level surfaces for the functions. \(f(x, y, z)=100 x^{2}+16 y^{2}+25 z^{2} ; k>0\)

For the monkey saddle $$ z=x^{3}-3 x y^{2} $$ on \(-5 \leq x \leq 5,-5 \leq y \leq 5\), estimate the \(x y\) -coordinates of the point where a raindrop landing above the point \((5,-0.2)\) will leave the surface.

Find the maximum and minimum values of \(f(x, y)=10+x+y\) on the disk \(x^{2}+y^{2} \leq 9 .\) Hint: Parametrize the boundary by \(x=3 \cos t, y=3 \sin t, 0 \leq t \leq 2 \pi\)

Show that $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x y}{x^{2}+y^{2}} $$ does not exist by considering one path to the origin along the \(x\) -axis and another path along the line \(y=x\).

If \(F(x, y)=3 x^{4} y^{5}-2 x^{2} y^{3}\), find \(\partial^{3} F(x, y) / \partial y^{3}\).

Find the maximum and minimum values of \(f(x, y)=x^{2}+y^{2}\) on the ellipse with interior \(x^{2} / a^{2}+y^{2} / b^{2} \leq 1\) where \(a>b .\) Hint: Parametrize the boundary by \(x=a \cos t\), \(y=b \sin t, 0 \leq t \leq 2 \pi\)

Show that $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x y+y^{3}}{x^{2}+y^{2}} $$ does not exist.

If \(f(x, y)=\cos \left(2 x^{2}-y^{2}\right)\), find \(\partial^{3} f(x, y) / \partial y \partial x^{2} .\)

Describe geometrically the level surfaces for the functions. \(f(x, y, z)=9 x^{2}-4 y^{2}-z^{2}\)

37\. A box is to be made where the material for the sides and the lid cost \(\$ 0.25\) per square foot and the cost for the bottom is \(\$ 0.40\) per square foot. Find the dimensions of a box with volume 2 cubic feet that has minimum cost.

Let \(f(x, y)=x^{2} y /\left(x^{4}+y^{2}\right)\). (a) Show that \(f(x, y) \rightarrow 0\) as \((x, y) \rightarrow(0,0)\) along any straight line \(y=m x\). (b) Show that \(f(x, y) \rightarrow \frac{1}{2}\) as \((x, y) \rightarrow(0,0)\) along the parabola \(y=x^{2}\) (c) What conclusion do you draw?

Express the following in \(\partial\) notation. (a) \(f_{y y y}\) (b) \(f_{x x y}\) (c) \(f_{x y y y}\)

Describe geometrically the level surfaces for the functions. \(f(x, y, z)=4 x^{2}-9 y^{2}\)

A steel box without a lid having volume 60 cubic feet is to be made from material that costs \(\$ 4\) per square foot for the bottom and \(\$ 1\) per square foot for the sides. Welding the sides to the bottom costs \(\$ 3\) per linear foot and welding the sides together costs \(\$ 1\) per linear foot. Find the dimensions of the box that has minimum cost and find the minimum cost. Hint: Use symmetry to obtain one equation in one unknown and use a CAS or Newton's Method to approximate the solution.

Express the following in subscript notation. (a) \(\frac{\partial^{3} f}{\partial x^{2} \partial y}\) (b) \(\frac{\partial^{4} f}{\partial x^{2} \partial y^{2}}\) (c) \(\frac{\partial^{5} f}{\partial x^{3} \partial y^{2}}\)

Suppose that the temperature \(T\) on the circular plate \(\left\\{(x, y): x^{2}+y^{2} \leq 1\right\\}\) is given by \(T=2 x^{2}+y^{2}-y .\) Find the hottest and coldest spots on the plate.

Find the domain of each function. (a) \(f(w, x, y, z)=\frac{1}{\sqrt{w^{2}+x^{2}+y^{2}+z^{2}}}\) (b) \(g\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\exp \left(-x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}\right)\) (c) \(h\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\sqrt{1-\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right)}\)

Let \(f\), a function of \(n\) variables, be continuous on an open set \(D\), and suppose that \(P_{0}\) is in \(D\) with \(f\left(P_{0}\right)>0 .\) Prove that there is a \(\delta>0\) such that \(f(P)>0\) in a neighborhood of \(P_{0}\) with radius \(\delta\).

If \(f(x, y, z)=\left(x^{3}+y^{2}+z\right)^{4}\), find each of the following: (a) \(f_{x}(x, y, z)\) (b) \(f_{y}(0,1,1)\) (c) \(f_{z z}(x, y, z)\)

Sketch (as best you can) the graph of the monkey saddle \(z=x\left(x^{2}-3 y^{2}\right) .\) Begin by noting where \(z=0\)

Find the shape of the triangle of largest area that can be inscribed in a circle of radius \(r\). Hint: Let \(\alpha, \beta\), and \(\gamma\) be the central angles that subtend the three sides of the triangle. Show that the area of the triangle is \(\frac{1}{2} r^{2}[\sin \alpha+\sin \beta-\sin (\alpha+\beta)]\). Maximize.

The French Railroad Suppose that Paris is located at the origin of the \(x y\) -plane. Rail lines emanate from Paris along all rays, and these are the only rail lines. Determine the set of discontinuities of the following functions. (a) \(f(x, y)\) is the distance from \((x, y)\) to \((1,0)\) on the French railroad. (b) \(g(u, v, x, y)\) is the distance from \((u, v)\) to \((x, y)\) on the French railroad.

If \(f(x, y, z)=e^{-x y z}-\ln \left(x y-z^{2}\right)\), find \(f_{x}(x, y, z)\)

Let \((a, b, c)\) be a fixed point in the first octant. Find the plane through this point that cuts off from the first octant the tetrahedron of minimum volume, and determine the resulting volume.

Let \(f(x, y)=x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}} \quad\) if \(\quad(x, y) \neq(0,0)\) and \(f(0,0)=0\) Show that \(f_{x y}(0,0) \neq f_{y x}(0,0)\) by completing the following steps: (a) Show that \(f_{x}(0, y)=\lim _{h \rightarrow 0} \frac{f(0+h, y)-f(0, y)}{h}=-y\) for all \(y\). (b) Similarly, show that \(f_{y}(x, 0)=x\) for all \(x\). (c) Show that \(f_{y x}(0,0)=\lim _{h \rightarrow 0} \frac{f_{y}(0+h, 0)-f_{y}(0,0)}{h}=1\). (d) Similarly, show that \(f_{x y}(0,0)=-1\).