Chapter 13

Let \((a, b)\) be a fixed point in \(\mathbb{R}^{2}\) and let \(d=f(x, y)\) be the distance between \((a, b)\) and an arbitrary point \((x, y)\) a. Show that the graph of \(f\) is a cone. b. Show that the gradient of \(f\) at any point other than \((a, b)\) is a unit vector. c. Interpret the direction and magnitude of \(\nabla f\)

The output \(Q\) of an economic system subject to two inputs, such as labor \(L\) and capital \(K,\) is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b} .\) Suppose \(a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=1\) a. Evaluate the partial derivatives \(Q_{L}\) and \(Q_{K}\) b. Suppose \(L=10\) is fixed and \(K\) increases from \(K=20\) to \(K=20.5 .\) Use linear approximation to estimate the change in \(Q\) c. Suppose \(K=20\) is fixed and \(L\) decreases from \(L=10\) to \(L=9.5 .\) Use linear approximation to estimate the change in \(Q\) d. Graph the level curves of the production function in the first quadrant of the \(L K\) -plane for \(Q=1,2,\) and 3 e. Use the graph of part (d). If you move along the vertical line \(L=2\) in the positive \(K\) -direction, how does \(Q\) change? Is this consistent with \(Q_{K}\) computed in part (a)? f. Use the graph of part (d). If you move along the horizontal line \(K=2\) in the positive \(L\) -direction, how does \(Q\) change? Is this consistent with \(Q_{L}\) computed in part (a)?

Limits at (0,0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as \(r \rightarrow 0\) along all paths to (0,0) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)^{2}}{\left(x^{2}+y^{2}\right)^{3 / 2}}$$

Determine whether the following statements are true and give an explanation or counterexample. a. The plane passing through the point (1,1,1) with a normal vector \(\mathbf{n}=\langle 1,2,-3\rangle\) is the same as the plane passing through the point (3,0,1) with a normal vector \(\mathbf{n}=\langle-2,-4,6\rangle\) b. The equations \(x+y-z=1\) and \(-x-y+z=1\) describe the same plane. c. Given a plane \(Q\), there is exactly one plane orthogonal to \(Q\). d. Given a line \(\ell\) and a point \(P_{0}\) not on \(\ell\), there is exactly one plane that contains \(\ell\) and passes through \(P_{0}\) e. Given a plane \(R\) and a point \(P_{0},\) there is exactly one plane that is orthogonal to \(R\) and passes through \(P_{0}\) f. Any two distinct lines in \(\mathbb{R}^{3}\) determine a unique plane. g. If plane \(Q\) is orthogonal to plane \(R\) and plane \(R\) is orthogonal to plane \(S\), then plane \(Q\) is orthogonal to plane \(S\).

Level curves of a savings plan Suppose you make monthly deposits of \(P\) dollars into an account that earns interest at a monthly rate of \(p \% .\) The balance in the account after \(t\) years is $$B(P, r, t)=P\left[\frac{(1+r)^{12 t}-1}{r}\right], \text { where } r=p / 100 \text { (for }$$ example, if the annual interest rate is \(9 \%,\) then \(p=\frac{9}{12}=0.75\) and \(r=0.0075\) ). Let the time of investment be fixed at \(t=20\) years. a. With a target balance of \(\$ 20,000,\) find the set of all points \((P, r)\) that satisfy \(B=20,000 .\) This curve gives all deposits \(P\) and monthly interest rates \(r\) that result in a balance of \(\$ 20,000\) after 20 years. b. Repeat part (a) with \(B=\$ 5000, \$ 10,000, \$ 15,000,\) and \(\$ 25,000,\) and draw the resulting level curves of the balance function.

Consider the following surfaces \(f(x, y, z)=0,\) which may be regarded as a level surface of the function \(w=f(x, y, z) . A\) point \(P(a, b, c)\) on the surface is also given. a. Find the (three-dimensional) gradient of \(f\) and evaluate it at \(P\). b. The heads of all vectors orthogonal to the gradient with their tails at \(P\) form a plane. Find an equation of that plane (soon to be called the tangent plane). $$f(x, y, z)=x^{2}+y^{2}+z^{2}-3=0 ; P(1,1,1)$$

Two resistors in an electrical circuit with resistance \(R_{1}\) and \(R_{2}\) wired in parallel with a constant voltage give an effective resistance of \(R,\) where \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}.\) a. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by solving for \(R\) and differentiating. b. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by differentiating implicitly. c. Describe how an increase in \(R_{1}\) with \(R_{2}\) constant affects \(R\) d. Describe how a decrease in \(R_{2}\) with \(R_{1}\) constant affects \(R\)

Use the Second Derivative Test to prove that if \((a, b)\) is a critical point of \(f\) at which \(f_{x}(a, b)=f_{y}(a, b)=0\) and \(f_{x x}(a, b)<0

Evaluate the following limits. $$a.\lim _{(x, y) \rightarrow(0,0)} \frac{\sin (x+y)}{x+y}$$ $$b.\lim _{(x, y) \rightarrow(0,0)} \frac{\sin x+\sin y}{x+y}$$

Find an equation of the plane that passes through the point \(P_{0}\) and contains the line \(\ell\) a. \(P_{0}(1,-2,3) ; \ell: \mathbf{r}=\langle t,-t, 2 t\rangle\) b. \(P_{0}(-4,1,2) ; \ell: \mathbf{r}=\langle 2 t,-2 t,-4 t\rangle\)

One measurement of the quality of a quarterback in the National Football League is known as the quarterback rating. The rating formula is $$R(c, t, i, y)=\frac{50+20 c+80 t-100 i+100 y}{24}.$$ percentage of passes completed, \(t\) is the percentage of passes thrown for touchdowns, \(i\) is the percentage of intercepted passes, and \(y\) is the yards gained per attempted pass.a. In his career, Hall of Fame quarterback Johnny Unitas completed \(54.57 \%\) of his passes, \(5.59 \%\) of his passes were thrown for touchdowns, \(4.88 \%\) of his passes were intercepted, and he gained an average of 7.76 yards per attempted pass. What was his quarterback rating? b. If \(c, t,\) and \(y\) remained fixed, what happens to the quarterback rating as \(i\) increases? Explain your answer with and without mathematics.

Consider the following surfaces \(f(x, y, z)=0,\) which may be regarded as a level surface of the function \(w=f(x, y, z) . A\) point \(P(a, b, c)\) on the surface is also given. a. Find the (three-dimensional) gradient of \(f\) and evaluate it at \(P\). b. The heads of all vectors orthogonal to the gradient with their tails at \(P\) form a plane. Find an equation of that plane (soon to be called the tangent plane). $$f(x, y, z)=8-x y z=0 ; P(2,2,2)$$

Among all triangles with a perimeter of 9 units, find the dimensions of the triangle with the maximum area. It may be easiest to use Heron's formula, which states that the area of a triangle with side length \(a, b,\) and \(c\) is \(A=\sqrt{s(s-a)(s-b)(s-c)},\) where \(2 s\) is the perimeter of the triangle.

Let $$f(x, y)=\left\\{\begin{array}{ll}\frac{\sin \left(x^{2}+y^{2}-1\right)}{x^{2}+y^{2}-1} & \text { if } x^{2}+y^{2} \neq 1 \\\b & \text { if } x^{2}+y^{2}=1\end{array}\right.$$ Find the value of \(b\) for which \(f\) is continuous at all points in \(\mathbb{R}^{2}\).

Ideal Gas Law Many gases can be modeled by the Ideal Gas Law, \(P V=n R T,\) which relates the temperature \((T,\) measured in Kelvin (K)), pressure ( \(P\), measured in Pascals (Pa)), and volume ( \(V\), measured in \(\mathrm{m}^{3}\) ) of a gas. Assume that the quantity of gas in question is \(n=1\) mole (mol). The gas constant has a value of \(R=8.3 \mathrm{m}^{3} \cdot \mathrm{Pa} / \mathrm{mol} \cdot \mathrm{K}.\) a. Consider \(T\) to be the dependent variable and plot several level curves (called isotherms) of the temperature surface in the region \(0 \leq P \leq 100,000\) and \(0 \leq V \leq 0.5.\) b. Consider \(P\) to be the dependent variable and plot several level curves (called isobars) of the pressure surface in the region \(0 \leq T \leq 900\) and \(0< V \leq 0.5.\) c. Consider \(V\) to be the dependent variable and plot several level curves of the volume surface in the region \(0 \leq T \leq 900\) and \(0 < P \leq 100,000.\)

Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension (for example, a wave on a string) wave motion is governed by the one-dimensional wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u(x, t)\) is the height or displacement of the wave surface at position \(x\) and time \(t,\) and \(c\) is the constant speed of the wave. Show that the following functions are solutions of the wave equation. $$u(x, t)=\cos (2(x+c t))$$

Consider the following surfaces \(f(x, y, z)=0,\) which may be regarded as a level surface of the function \(w=f(x, y, z) . A\) point \(P(a, b, c)\) on the surface is also given. a. Find the (three-dimensional) gradient of \(f\) and evaluate it at \(P\). b. The heads of all vectors orthogonal to the gradient with their tails at \(P\) form a plane. Find an equation of that plane (soon to be called the tangent plane). $$f(x, y, z)=e^{x+y-z}-1=0 ; P(1,1,2)$$

Let \(P\) be a plane tangent to the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1\) at a point in the first octant. Let \(T\) be the tetrahedron in the first octant bounded by \(P\) and the coordinate planes \(x=0, y=0\), and \(z=0 .\) Find the minimum volume of \(T .\) (The volume of a tetrahedron is one-third the area of the base times the height.)

Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. $$g(x, y, z)=\frac{10}{x^{2}-(y+z) x+y z}.$$

Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension (for example, a wave on a string) wave motion is governed by the one-dimensional wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u(x, t)\) is the height or displacement of the wave surface at position \(x\) and time \(t,\) and \(c\) is the constant speed of the wave. Show that the following functions are solutions of the wave equation. $$u(x, t)=5 \cos (2(x+c t))+3 \sin (x-c t)$$

Consider the following surfaces \(f(x, y, z)=0,\) which may be regarded as a level surface of the function \(w=f(x, y, z) . A\) point \(P(a, b, c)\) on the surface is also given. a. Find the (three-dimensional) gradient of \(f\) and evaluate it at \(P\). b. The heads of all vectors orthogonal to the gradient with their tails at \(P\) form a plane. Find an equation of that plane (soon to be called the tangent plane). $$f(x, y, z)=x y+x z-y z-1 ; P(1,1,1)$$

Given three distinct noncollinear points \(A, B,\) and \(C\) in the plane, find the point \(P\) in the plane such that the sum of the distances \(|A P|+|B P|+|C P|\) is a minimum. Here is how to proceed with three points, assuming that the triangle formed by the three points has no angle greater than \(2 \pi / 3\left(120^{\circ}\right)\). a. Assume the coordinates of the three given points are \(A\left(x_{1}, y_{1}\right)\) \(B\left(x_{2}, y_{2}\right),\) and \(C\left(x_{3}, y_{3}\right) .\) Let \(d_{1}(x, y)\) be the distance between \(A\left(x_{1}, y_{1}\right)\) and a variable point \(P(x, y) .\) Compute the gradient of \(d_{1}\) and show that it is a unit vector pointing along the line between the two points. b. Define \(d_{2}\) and \(d_{3}\) in a similar way and show that \(\nabla d_{2}\) and \(\nabla d_{3}\) are also unit vectors in the direction of the line between the two points. c. The goal is to minimize \(f(x, y)=d_{1}+d_{2}+d_{3}\) Show that the condition \(f_{x}=f_{y}=0\) implies that \(\nabla d_{1}+\nabla d_{2}+\nabla d_{3}=0\). d. Explain why part (c) implies that the optimal point \(P\) has the property that the three line segments \(A P, B P,\) and \(C P\) all intersect symmetrically in angles of \(2 \pi / 3\). e. What is the optimal solution if one of the angles in the triangle is greater than \(2 \pi / 3\) (just draw a picture)? f. Estimate the Steiner point for the three points (0,0),(0,1) and (2,0)

Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. $$f(x, y)=\sin ^{-1}(x-y)^{2}.$$

Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension (for example, a wave on a string) wave motion is governed by the one-dimensional wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u(x, t)\) is the height or displacement of the wave surface at position \(x\) and time \(t,\) and \(c\) is the constant speed of the wave. Show that the following functions are solutions of the wave equation. \(u(x, t)=A f(x+c t)+B g(x-c t),\) where \(A\) and \(B\) are constants and \(f\) and \(g\) are twice differentiable functions of one variable

A snapshot (frozen in time) of a water wave is described by the function \(z=1+\sin (x-y),\) where \(z\) gives the height of the wave relative to a reference point and \((x, y)\) are coordinates in a horizontal plane. a. Use a graphing utility to graph \(z=1+\sin (x-y)\) b. The crests and the troughs of the waves are aligned in the direction in which the height function has zero change. Find the direction in which the crests and troughs are aligned. c. If you were surfing on this wave and wanted the steepest descent from a crest to a trough, in which direction would you point your surfboard (given in terms of a unit vector in the \(x y\) -plane)? d. Check that your answers to parts (b) and (c) are consistent with the graph of part (a).

Find an equation of the plane passing through the point (3,2,1) that slices off the region in the first octant with the least volume.

Show that $$\lim _{(x, y) \rightarrow(0,0)} \frac{a x^{2(p-n)} y^{n}}{b x^{2 p}+c y^{p}} \text { does }$$ not exist when \(a, b,\) and \(c\) are nonzero real numbers and \(n\) and \(p\) are positive integers with \(p \geq n\)

Find an equation of the plane passing through (0,-2,4) that is orthogonal to the planes \(2 x+5 y-3 z=0\) and \(-x+5 y+2 z=8\)

A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, $$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=e^{-x} \sin y$$

Generalize Exercise 75 by considering a wave described by the function \(z=A+\sin (a x-b y)\) where \(a, b,\) and \(A\) are real numbers. a. Find the direction in which the crests and troughs of the wave are aligned. Express your answer as a unit vector in terms of \(a\) and \(b\) b. Find the surfer's direction - that is, the direction of steepest descent from a crest to a trough. Express your answer as a unit vector in terms of \(a\) and \(b\)

Show that the following two functions have two local maxima but no other extreme points (thus no saddle or basin between the mountains). a. \(f(x, y)=-\left(x^{2}-1\right)^{2}-\left(x^{2}-e^{y}\right)^{2}\) b. \(f(x, y)=4 x^{2} e^{y}-2 x^{4}-e^{4 y}\)

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(1,0)} \frac{\sin x y}{x y}$$

Describe the set of all points at which all three planes \(x+3 z=3, y+4 z=6,\) and \(x+y+6 z=9\) intersect.

A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, $$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=x\left(x^{2}-3 y^{2}\right)$$

Potential functions arise frequently in physics and engineering. A potential function has the property that \(a\) field of interest (for example, an electric field, a gravitational field, or a velocity field is the gradient of the potential (or sometimes the negative of the gradient of the potential). The electric field due to a point charge of strength \(Q\) at the origin has a potential function \(V=k Q / r,\) where \(r^{2}=x^{2}+y^{2}+z^{2}\) is the square of the distance between a variable point \(P(x, y, z)\) and the charge, and \(k>0\) is a physical constant. The electric field is given by \(\mathbf{E}=-\nabla V,\) where \(\nabla V\) is the gradient in three dimensions. a. Show that the three-dimensional electric field due to a point charge is given by $$ \mathbf{E}(x, y, z)=k Q\left\langle\frac{x}{r^{3}}, \frac{y}{r^{3}}, \frac{z}{r^{3}}\right\rangle $$ b. Show that the electric field at a point has a magnitude \(|\mathbf{E}|=k Q / r^{2} .\) Explain why this relationship is called an inverse square law.

A function of one variable has the property that a local maximum (or minimum) occurring at the only critical point is also the absolute maximum (or minimum) (for example, \(f(x)=x^{2}\) ). Does the same result hold for a function of two variables? Show that the following functions have the property that they have a single local maximum (or minimum), occurring at the only critical point, but that the local maximum (or minimum) is not an absolute maximum (or minimum) on \(\mathbb{R}^{2}\). a. \(f(x, y)=3 x e^{y}-x^{3}-e^{3 y}\) b. \(f(x, y)=\left(2 y^{2}-y^{4}\right)\left(e^{x}+\frac{1}{1+x^{2}}\right)-\frac{1}{1+x^{2}}\) This property has the following interpretation. Suppose that a surface has a single local minimum that is not the absolute minimum. Then water can be poured into the basin around the local minimum and the surface never overflows, even though there are points on the surface below the local minimum.

Describe the set of all points at which all three planes \(x+2 y+2 z=3, y+4 z=6,\) and \(x+2 y+8 z=9\) intersect.

The closed unit ball in \(\mathbb{R}^{3}\) centered at the origin is the set \(\left\\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 1\right\\} .\) Describe the following alternative unit balls. a. \(\\{(x, y, z):|x|+|y|+|z| \leq 1\\}.\) b. \(\\{(x, y, z): \max \\{|x|,|y|,|z|\\} \leq 1\\},\) where \(\max \\{a, b, c\\}\) is the maximum value of \(a, b,\) and \(c.\)

Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. $$h(x, y, z)=\sqrt[4]{z^{2}-x z+y z-x y}.$$

Potential functions arise frequently in physics and engineering. A potential function has the property that \(a\) field of interest (for example, an electric field, a gravitational field, or a velocity field is the gradient of the potential (or sometimes the negative of the gradient of the potential). The gravitational potential associated with two objects of mass \(M\) and \(m\) is \(V=-G M m / r,\) where \(G\) is the gravitational constant. If one of the objects is at the origin and the other object is at \(P(x, y, z),\) then \(r^{2}=x^{2}+y^{2}+z^{2}\) is the square of the distance between the objects. The gravitational field at \(P\) is given by \(\mathbf{F}=-\nabla V,\) where \(\nabla V\) is the gradient in three dimensions. Show that the force has a magnitude \(|\mathbf{F}|=G M m / r^{2} .\) Explain why this relationship is called an inverse square law.

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(0,2)}(2 x y)^{x y}$$

Match equations a-f with surfaces A-F. a. \(y-z^{2}=0\) b. \(2 x+3 y-z=5\) c. \(4 x^{2}+\frac{y^{2}}{9}+z^{2}=1\) d. \(x^{2}+\frac{y^{2}}{9}-z^{2}=1\) e. \(x^{2}+\frac{y^{2}}{9}=z^{2}\) f. \(y=|x|\)

Potential functions arise frequently in physics and engineering. A potential function has the property that \(a\) field of interest (for example, an electric field, a gravitational field, or a velocity field is the gradient of the potential (or sometimes the negative of the gradient of the potential). In two dimensions, the motion of an ideal fluid (an incompressible and irrotational fluid) is governed by a velocity potential \(\varphi .\) The velocity components of the fluid, \(u\) in the \(x\) -direction and \(v\) in the \(y\) -direction, are given by \(\langle u, v\rangle=\nabla \varphi\) Find the velocity components associated with the velocity potential \(\varphi(x, y)=\sin \pi x \sin 2 \pi y\)

A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, $$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=\tan ^{-1}\left(\frac{y}{x-1}\right)-\tan ^{-1}\left(\frac{y}{x+1}\right)$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(0, \pi / 2)} \frac{1-\cos x y}{4 x^{2} y^{3}}$$

Identify and briefly describe the surfaces defined by the following equations. $$z^{2}+4 y^{2}-x^{2}=1$$

Prove that for the plane described by \(f(x, y)=A x+B y,\) where \(A\) and \(B\) are nonzero constants, the gradient is constant (independent of \((x, y)\) ). Interpret this result.

The flow of heat along a thin conducting bar is governed by the one- dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) $$\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u\) is a measure of the temperature at a location \(x\) on the bar at time t and the positive constant \(k\) is related to the conductivity of the material. Show that the following functions satisfy the heat equation with \(k=1.\) $$u(x, t)=10 e^{-t} \sin x$$

The domain of $$f(x, y)=e^{-1 /\left(x^{2}+y^{2}\right)}$$ excludes \((0,0) .\) How should \(f\) be defined at (0,0) to make it continuous there?

Identify and briefly describe the surfaces defined by the following equations. $$y=4 z^{2}-x^{2}$$