Math 53: Multivariable Calculus Worksheets, Exercises of Calculus

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Math 53: Multivariable Calculus Worksheets

7 th^ Edition

Department of Mathematics, University of California at Berkeley

i Math 53 Worksheets, 7 th^ Edition

Preface

This booklet contains the worksheets for Math 53, U.C. Berkeley’s multivariable calculus course. The introduction of each worksheet very briefly summarizes the main ideas but is not intended as a substitute for the textbook or lectures. The questions emphasize qualitative issues and the problems are more computationally intensive. The additional problems are more challenging and sometimes deal with technical details or tangential concepts. Typically more problems were provided on each worksheet than can be completed during a discussion period. This was not a scheme to frustrate the student; rather, we aimed to provide a variety of problems that can reflect different topics which professors and GSIs may choose to emphasize. The first edition of this booklet was written by Greg Marks and used for the Spring 1997 semester of Math 53W. The second edition was prepared by Ben Davis and Tom Insel and used for the Fall 1997 semester, drawing on suggestions and experiences from the first semester. The authors of the second edition thank Concetta Gomez and Professors Ole Hald and Alan Weinstein for their many comments, criticisms, and suggestions. The third edition was prepared during the Fall of 1997 by Tom Insel and Zeph Grunschlag. We would like to thank Scott Annin, Don Barkauskas, and Arturo Magidin for their helpful suggestions. The Fall 2000 edition has been revised by Michael Wu. Tom Insel coordinated this edition in consultation with William Stein. Michael Hutchings made tiny changes in 2012 for the seventh edition. In 1997, the engineering applications were written by Reese Jones, Bob Pratt, and Pro- fessors George Johnson and Alan Weinstein, with input from Tom Insel and Dave Jones. In 1998, applications authors were Michael Au, Aaron Hershman, Tom Insel, George Johnson, Cathy Kessel, Jason Lee, William Stein, and Alan Weinstein.

iii Math 53 Worksheets, 7 th^ Edition

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1. Curves Defined by Parametric Equations

As we know, some curves in the plane are graphs of functions, but not all curves can be so expressed. Parametric equations allow us to describe a wider class of curves. A parametrized curve is given by two equations, x = f (t), y = g(t). The curve consists of all the points (x, y) that can be obtained by plugging values of t from a particular domain into both of the equations x = f (t), y = g(t). We may think of the parametric equations as describing the motion of a particle; f (t) and g(t) tell us the x- and y-coordinates of the particle at time t. We can also parametrize curves in R^3 with three parametric equations: x = f (t), y = g(t), and z = h(t). For example, the orbit of a planet around the sun could be given in this way.

Questions

1. (a) Check that the graph of the function y = x^2 is the same as the parametrized curve x = t, y = t^2. (b) Using (a) as a model, write parametric equations for the graph of y = f (x) where f (x) is any function.
2. Consider the circle C = {(x, y) ∈ R^2 | x^2 + y^2 = 1}.

(a) Is C the graph of some function? If so, which function? If not, why not? (b) Find a parametrization for C. (Hint: cos^2 θ + sin^2 θ = 1.)

3. Consider the parametric equations x = 3t, y = t, and x = 6t, y = 2t.

(a) What curves do the two sets of equations describe? (b) Compare and contrast the motions for the two sets of parametric equations by interpreting each set as describing the motion of a particle. (c) Suppose that a curve is parametrized by x = f (t), y = g(t). Explain why x = f (2t), y = g(2t) parametrize the same curve. (d) Show that there are an infinite number of different parametrizations for the same curve.

Problems

1. Consider the curve parametrized by x(θ) = a cos θ, y(θ) = b sin θ.

(a) Plot some points and sketch the curve when a = 1 and b = 1, when a = 2 and b = 1, and when a = 1 and b = 2. (b) Eliminate the parameter θ to obtain a single equation in x, y, and the constants a and b. What curve does this equation describe? (Hint: Eliminate θ using the identity cos^2 θ + sin^2 θ = 1.)

2. Tangents, Areas, Arc Lengths, and Surface Areas

As we saw in the previous section, we can use parametric equations to describe curves that aren’t graphs of the form y = f (x). In Math 1A we learned how to calculate the slope of a graph at a point and how to evaluate the area underneath a graph. In Math 1B, we encountered the problems of calculating the arc length of a graph and the area of a surface of revolution defined by a graph. Here, we revisit these problems in the more general framework of parametrized curves. Here are some of the new formulas:

• The slope of a parametrized curve (when dx dt 6 = 0):

dy dx

dy/dt dx/dt

• The arc length of a curve parametrized for α ≤ t ≤ β:

∫ (^) β

α

dx dt

dy dt

dt (2)

• The area obtained by rotating a curve parameterized for α ≤ t ≤ β around the x-axis. ∫ (^) β

α

2 πy

dx dt

dy dt

dt (3)

Questions

1. A Geometric Proof of Equation 1. Let C be the curve given by the parametric equations x = f (t), y = g(t) and let (f (t 0 ), g(t 0 )) be a point on the curve. Let m(t) be the slope of the secant line connecting (f (t 0 ), g(t 0 )) to (f (t), g(t)) as in Figure 1.

(f (t 0 ), g(t 0 ))

(f (t), g(t))

Figure 1: A secant line

(a) Write a formula for m(t) in terms of f (t) and g(t). (b) Use l’Hospital’s rule to evaluate limt→t 0 m(t) and explain how this limit gives the slope of the tangent line.

2. In Question 3(d) on Worksheet 1, you found an infinite number of parametrizations for a single curve. Verify that Formula 1 yields the same tangent slope to the curve at a point, no matter which of the parametrizations is used.
3. Show that Formula 2 recovers the usual formula

arc length =

∫ (^) b

a

1 + [f ′(x)]^2 dt

in the special case when the curve is the graph of a function y = f (x), a ≤ x ≤ b.

Problems

1. Let C be the curve x = 2 cos t, y = sin t.

(a) What kind of curve is this? (b) Find the slope of the tangent line to the curve when t = 0, t = π/4, and t = π/2. (c) Find the area of the region enclosed by C. (Hint: sin^2 t = (1 − cos 2t)/2.)

2. Compute the arc length of the curve parametrized by x = cos(et), y = sin(et), 0 ≤ t ≤ 1. (Hint: Reparametrize.)
3. Consider the circle parametrized by x = 2 + cos t, y = sin t, 0 ≤≤ 2 π.

(a) What are the center and radius of this circle. (b) Describe the surface obtained by rotating the circle about the x-axis? About the y-axis. (c) Calculate the area of the surface obtained by rotating the circle around the x-axis. Why should you restrict the parametrization to 0 ≤ t ≤ π when integrating? (d) Calculate the area of the surface obtained by rotating the circle about the y-axis.

Additional Problem

1. Consider the surface obtained by rotating the parametrized curve x = et^ + e−t, y = et^ − e−t, 0 ≤ t ≤ 1 about the x-axis.

(a) Find the area of this surface by plugging x and y into Formula 3 and integrating by substitution. (b) Find the area of this surface by reparametrizing the curve before you plug into Formula 3.

2. The polar coordinate map x = r cos θ, y = r sin θ takes a line in the rθ-plane to a curve in the xy-plane. For each line ℓ, draw the corresponding curve in the xy-plane.

r

θ ℓ

(a)

x = r cos θ y = r sin θ x

y

r

θ ℓ

(b)

x = r cos θ y = r sin θ x

y

r

θ

ℓ

(c)

x = r cos θ y = r sin θ x

y

Problems

1. Sketch the curve given by r = 2 sin θ and give its equation in Cartesian coordinates. What curve is it?
2. Write an equation in polar coordinates for the circle of radius

2 centered at (x, y) = (1, 1).

3. Consider the curve given by the polar equation r = 3 + cos 4θ.

(a) Sketch this curve. (b) Find the slope of this curve at θ = π/4. (c) At which points does dr dθ = 0? Remember that this is not the slope. What is the geometric meaning of dr dθ = 0?

4. (a) Does the spiral r = 1/θ, π/ 2 ≤ θ < ∞ have finite length?

(b) Does the spiral r = e−θ, 0 ≤ θ < ∞ have finite length?

5. Sketch the lemniscate r^2 = a^2 cos(2θ) where a is a positive constant and calculate the area it encloses.

4. (a) Suppose that one side of a triangle forms the diameter of a circle and the vertex opposite this side lies on a circle. Use the dot product to prove that this is a right triangle.

(b) Now, do the same in R^3. (Hint: Let the center of the circle be the origin.)

Problems

1. Here we find parametric equations for the line in R^3 passing through the points a = (1, 0 , 1) and b = (2, 1 , −1).

(a) Find a vector u pointing in the same direction as the line. (b) Let c be any point on the line. Explain why c + tu gives parametric equations for the line. Write down these equations (c) Can you get more than one parametrization of the line from these methods?

2. Consider the plane x + y − z = 4.

(a) Find any point in the plane and call it a. Let x = (x, y, z) and show that (x − a) · (1, 1 , −1) = 0 is the equation of the plane. (b) Explain why i + j − k is a normal vector to the plane. (c) Show that if ax + by + cz = d is the equation of a plane where a, b, c, and d are constants, then ai + bj + ck is a normal vector.

3. Here we find the equation of a plane containing the points a = (0, 0 , 1), b = (0, 1 , 2) and c = (1, 2 , 3).

(a) Let u and v be vectors connecting a to b and a to c. Compute u and v. (b) Find a vector perpendicular to the plane. (c) Use the normal vector to find the equation of the plane. (Hint: First write the equation in the form given in Problem 2(a).)

Additional Problems

1. Suppose that you are looking to the side as you walk on a windless, rainy day. Now you stop walking.^1

(a) How does the apparent direction of the falling rain change? (b) Explain this observation in terms of vectors. (c) Suppose you know your walking speed. How could you determine the speed at which the rain is falling?

(^1) From Basic Multivariable Calculus by Marsden, Tromba, and Weinstein.

5. Quadric Surfaces

The quadric surfaces in R^3 are analogous to the conic sections in R^2. Aside from cylinders, which are formed by “dragging” a conic section along a line in R^3 , there are only six quadric surfaces: the ellipsoid, the hyperboloid of one sheet, the hyperboloid of two sheets, the elliptic cone, the elliptic paraboloid, and the hyperbolic paraboloid. We study quadric surfaces now because they will provide a nice class of examples for calculus.

Questions

1. For each of the six types of quadric surfaces listed above, which of the following is true?

¬ Every surface of this type can be formed by rotating some curve about an axis. Some surfaces of this type can be so formed and some cannot. ® No surface of this type can be so formed.

2. Which type of quadric surface is given by the equation x^2 + y^2 = 1? (Hint: It is not a circle.)

Problems

1. Which type of quadric surface is given by each of these equations?

(a) x^2 + 2y^2 + 3z^2 + x + 2y + 3z = 0 (b) x^2 + 2y^2 − 3 z^2 + x + 2y + 3z = 0 (c) x^2 − 2 y^2 − 3 z^2 + x + 2y + 3z = 0

2. Which type of quadric surface is z = xy?
3. Show that the quadric surface xy + xz + yz = 0 is a cone. (Hint: Begin by showing that if (a, b, c) is a point on this surface, then (ka, kb, kc) is another point on the surface for every number k.)

3. Describe what it means for: (a) r(t) to be constant. (c) |r(t)| to be constant. (b) r′(t) to be constant. (d) |r′(t)| to be constant.

Problems

1. Let r(t) = (cos t)i + (sin t)j + 0k.

(a) Is r(t) perpendicular to r′(t) for every t? Is r′(t) perpendicular to r′′(t) for every t? (b) If r were another function, would the two answers to (a) remain the same? If true, show why. If false, give a counter example.

2. Suppose that a particle’s position vector is given by r(t) = ti + t^2 j + t^3 k. Find its position, velocity, speed, and acceleration when t = 10.
3. Find the tangent line to the curve r(t) = t^2 i + t^3 j + t^4 k at the point (1, 1 , 1).

Additional Problems

1. An object starting at the origin and moving under the influence of gravity, but no other forces, follows the trajectory x(t) = at, y(t) = bt, z(t) = ct − 12 gt^2 , where g ≈ 9 .8m/s^2 is the gravitational acceleration constant and a, b, and c are constants depending on the particular motion.

(a) Find the velocity vector as a function of time. (b) What is the meaning of the constants a, b, and c? (c) Find the acceleration vector as a function of time. (d) Which way is “up?” (e) Find the equation of the plane containing the trajectory. (f) If b = 0, show that the trajectory is a parabola in the xz-plane. (g) Describe the trajectory in the general case.

7. Cross Products and Torque for Cyclists

The effect of a bicyclist’s foot on the pedal is measured by the torque produced at the center of the gear which is attached to the pedal crank. The physical notion of torque is so closely tied to the mathematical notion of cross product that Feynman introduces them both in the same chapter of his famous lectures on physics (Lecture 20, Volume I in the reference below). In this worksheet, we will use the cross product to analyze pedaling techniques, and we will see how this physical example helps us to understand the mathematics of the cross product.

Three vectors are relevant to our problem:

• F: the force applied by the rider’s foot to the pedal
• r: the “radius” vector from the gear’s center G to the point P on the pedal where the force is applied
• a: a unit vector along the rotation axis of the gear.

The torque at G produced by the force F applied at P is the cross product

τ = r × F. (4)

Since the gear can revolve only around its fixed axis, the effective part of the torque is its component in the direction of a, i.e. the scalar product

τeff = τ · a. (5)

1. Find τeff (θ) if the force is always applied downward, with a magnitude F , and then compute the total work done in a half revolution of the pedal.
2. Suppose now that the magnitude of the force is still F , but that it is now applied in the most efficient direction for each value of θ. As in Additional Problem 1, compute τeff (θ) and the total work done in a half revolution of the pedal.
3. How much is gained by pushing on the pedal in the most effective direction? Can you actually do this on a bicycle?
4. When is it reasonable to compute the work only for a half- revolution of the pedal?
5. Make some other assumptions about the direction of the applied force and compute the work done.

Reference

Feynman, Richard P., Leighton, Robert B., and Sands, Matthew. The Feynman Lectures on Physics. Addison-Wesley. 1964.

8. Functions of Several Variables

One way to understand the behavior of functions is visually. Here we investigate picto- rial ways to represent functions of the form f : Rn^ → Rm. A familiar way to represent a function is by its graph. Formally, the graph of f : Rn^ → Rm^ is the set of points {(x 1 ,... , xn, y 1 ,... , ym) ∈ Rn+m^ | (y 1 ,... , ym) = f (x 1 ,... , xn)}. For most of us, the graph is impossible to visualize if n + m > 3. For functions of the form f : Rn^ → R, we have another visualization tool at our disposal. A level set of f is the set {(x 1 ,... , xn) ∈ Rn^ | f (x 1 ,... , xn) = C} for some constant C. For example, if f (x, y) = x^2 + y^2 , then the level set f (x, y) = 4 is a circle of radius 2 as seen in Figure 5(a). Notice that a level set can be more complicated than a graph. In particular, it does not have to satisfy the vertical line test. A level set of f is actually the “projection” of a certain cross-section of the graph of f to the xy-plane. We see an example of this in Figure 5(b). The graph of a function can be obtained as the level set of another function by the following trick. The graph of f (x, y) is the level set g(x, y, z) = 0 where g(x, y, z) = f (x, y) − z. To see that this works, simply note that g(x, y, z) = 0 implies that f (x, y) − z = 0, so z = f (x, y).

(a) The level set f (x, y) = 4 when f (x, y) = x^2 + y^2

(b) The graph of f (x, y) = x^2 + y^2 and the level set x^2 + y^2 = 4

Figure 5:

Questions

1. Given functions f (x, y) and g(x, y) and two different numbers c and d, interpret each of the following phrases.

(a) the intersection of the level sets f (x, y) = c and g(x, y) = d. (b) the intersection of the level sets f (x, y) = c and f (x, y) = d. (c) the intersection of graph z = f (x, y) and the graph z = g(x, y).

2. Level sets are used in topographical maps to indicate the contour of the land. For instance, a single mountain is seen topographically in Figure 6(b). Sketch a topo- graphical map of the “mountain range” shown in Figure 6(c). Be sure to draw a level curve that includes the indicated “pass.”