Chapter 12

Find the least-squares line (Problem 30 ) for the data \((3,2),(4,3),(5,4),(6,4)\), and \((7,5)\).

\(\\{(x, y): x>0, y<\sin (1 / x)\\}\)

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.

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

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 $$ f(x, y)=x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y} $$ Note: Let \(f(x, y)\) denote the value of production from \(x\) units of capital and \(y\) units of labor. Then \(f\) is a homogeneous function (e.g., doubling capital and labor doubles production). Euler's Theorem then asserts an important law of economics that may be phrased as follows: The value of production \(f(x, y)\) equals the cost of capital plus the cost of labor provided that they are paid for at their respective marginal rates \(\partial f / \partial x\) and \(\partial f / \partial y\).

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, A is 200 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?

Describe geometrically the level surfaces for the functions defined in Problems 33-38. $$ f(x, y, z)=x^{2}+y^{2}+z^{2} ; k>0 $$

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

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 \(z\).

Describe geometrically the level surfaces for the functions defined in Problems 33-38. $$ f(x, y, z)=100 x^{2}+16 y^{2}+25 z^{2} ; k>0 $$

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 \(\partial^{3} F(x, y) / \partial y^{3}\).

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 defined in Problems 33-38. $$ f(x, y, z)=4 x^{2}-9 y^{2} $$

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) \(f_{x y y y}\)

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 \(T=2 x^{2}+y^{2}-y_{\text {. Find }}\) the hottest and coldest spots on the plate.

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

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

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) \(f_{z z}(x, y, z)\)

Find the shape of the triangle of largest area that can be inscribed in a circle of radius \(r\). Hinti 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 \(\left.\frac{1}{2} r^{2} \mid \sin \alpha+\sin \beta-\sin (\alpha+\beta)\right]\). Maximize.

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

Identify the graph of \(f(x, y)=x^{2}-x+3 y^{2}+\) \(12 y-13\), state where it attains its minimum value, and find this minimum value.

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 \(f(x, y)=x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}}\) if \((x, y) \neq(0,0) \quad\) 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 \(f_{x y}(0,0)=-1\).

\(f(x, y)=y /\left(1+x^{2}+y^{2}\right) ;-5 \leq x \leq 5,-5 \leq y \leq 5\); global maximum point and global minimum. Check using calculus.

Plot the graphs of each of the following functions on \(-2 \leq x \leq 2,-2 \leq y \leq 2\), and determine where on this set they are discontinuous. (a) \(f(x, y)=x^{2} /\left(x^{2}+y^{2}\right), f(0,0)=0\) (b) \(f(x, y)=\tan \left(x^{2}+y^{2}\right) /\left(x^{2}+y^{2}\right), f(0,0)=0\)

Let \(A(x, y)\) be the area of a nondegenerate rectangle of dimensions \(x\) and \(y\), the rectangle being inside a circle of radius 10. Determine the domain and range for this function.

\(f(x, y)=-1+\cos \left(y /\left(1+x^{2}+y^{2}\right)\right) ;-3.8 \leq x \leq 3.8\), \(-3.8 \leq y \leq 3.8\); global minimum.

For each of the functions in Problems 43-46, draw the graph and the corresponding contour plot. \(f(x, y)=(\sin x \sin y) /\left(1+x^{2}+y^{2}\right) ;-2 \leq x \leq 2\), \(-2 \leq y \leq 2\)

\(f(x, y)=\exp \left(-x^{2}-y^{2}+x y / 4\right) ;-2 \leq x \leq 2\), \(-2 \leq y \leq 2\); global maximum and global minimum. Check using calculus.

Give definitions of continuity at a point and continuity on a set for a function of three variables.

The wave equation \(c^{2} \partial^{2} u / \partial x^{2}=\partial^{2} u / \partial t^{2}\) and the heat equation \(c \partial^{2} u / \partial x^{2}=\partial u / \partial t\) are two of the most important equations in physics ( \(c\) is a constant). These are called partial differential equations. Show each of the following: (a) \(u=\cos x \cos c t\) and \(u=e^{x} \cosh c t\) satisfy the wave equation. (b) \(u=e^{-c t} \sin x\) and \(u=t^{-1 / 2} e^{-x^{2} /(4 c t)}\) satisfy the heat equation.

\(f(x, y)=\exp \left(-\left(x^{2}+y^{2}\right) / 4\right) \sin (x \sqrt{|y|}):-5 \leq x \leq 5\), \(-5 \leq y \leq 5 ;\) global maximum and global minimum.

Show that the function defined by $$ f(x, y, z)=\frac{x y z}{x^{3}+y^{3}+z^{3}} \quad \text { for }(x, y, z) \neq(0,0,0) $$ and \(f(0,0,0)=0\) is not continuous at \((0,0,0)\).

\(f(x, y)=-x /\left(x^{2}+y^{2}\right), f(0,0)=0 ;-1 \leq x \leq 1\), \(-1 \leq y \leq 1\); global maximum and global minimum. Be careful.

Show that the function defined by $$ f(x, y, z)=(y+1) \frac{x^{2}-z^{2}}{x^{2}+z^{2}} \quad \text { for }(x, y, z) \neq(0,0,0) $$ and \(f(0,0,0)=0\) is not continuous at \((0,0,0)\).

A CAS can be used to calculate and graph partial derivatives. Draw the graphs of each of the following: (a) \(\sin \left(x+y^{2}\right)\) (b) \(D_{x} \sin \left(x+y^{2}\right)\) (c) \(D_{y} \sin \left(x+y^{2}\right)\) (d) \(D_{x}\left(D_{y} \sin \left(x+y^{2}\right)\right)\)

\(f(x, y)=8 \cos (x y+2 x)+x^{2} y^{2},-3 \leq x \leq 3\), \(-3 \leq y \leq 3\); global maximum and global minimum.

Give definitions in terms of limits for the following partial derivatives: (a) \(f_{y}(x, y, z)\) (b) \(f_{z}(x, y, z)\) (c) \(G_{x}(w, x, y, z)\) (d) \(\frac{\partial}{\partial z} \lambda(x, y, z, t)\) (e) \(\frac{\partial}{\partial b_{2}} S\left(b_{0}, b_{1}, b_{2}, \ldots, b_{n}\right)\)

\(f(x, y)=(\sin x) /(6+x+|y|) ;-3 \leq x \leq 3\), \(-3 \leq y \leq 3\); global maximum and global minimum.

Find each partial derivative. (a) \(\frac{\partial}{\partial w}(\sin w \sin x \cos y \cos z)\) (b) \(\frac{\partial}{\partial x}[x \ln (w x y z)]\) (c) \(\lambda_{t}(x, y, z, t)\), where \(\lambda(x, y, z, t)=\frac{t \cos x}{1+x y z t}\)

\(f(x, y)=\cos \left(|x|+y^{2}\right)+10 x \exp \left(-x^{2}-y^{2}\right)\); \(-2 \leq x \leq 2,-2 \leq y \leq 2\); global maximum and global minimum.

\(f(x, y)=2 \sin x+\sin y-\sin (x+y)\); \(0 \leq x \leq 2 \pi, 0 \leq y \leq 2 \pi\); global maximum and global minimum.