Homework Assignments
MAC 3474, Section 3129 — Honors Analytic Geometry and Calculus III
Spring 2012

Last update made by D. Groisser Sun May 6 23:22:11 EDT 2012

Homework problems and due dates (not the dates the problems are assigned) are listed below. This list, especially the due dates, will be updated frequently, usually in the late afternoon or evening the day of class or the next morning. Due dates, and assignments more than one lecture ahead, are estimates; in particular, due dates may be moved either forward or back, and problems not currently on the list from a given section may be added later (but prior to their due dates, of course). Note that on a given day there may be problems due from more than one section of the book.
Exam dates and some miscellaneous items may also appear below.
If one day's assignment seems lighter than average, it's a good idea to read ahead and start doing the next assignment, which may be longer than average.
Unless otherwise indicated, problems are from our textbook (Stewart, edition 6e). It is intentional that some of the problems assigned do not have answers in the back of the book or solutions in a manual. An important part of learning mathematics is learning how to figure out by yourself whether your answers are correct.
Read the corresponding section of the book before working the problems. The author's "To the Student" remarks on p. xxiii are right on the money:
Some students start by trying their homework problems and read the text only if they get stuck on an exercise. I suggest that a far better plan is to read and understand a section of the text before attempting the exercises.
From the same page, another important bit of advice is this:
Reading a calculus book is different from reading a newspaper or a novel, or even a physics book. You should have pencil and paper and calculator at hand to sketch a diagram or make a calculation.

Date due Section # / problem #s

T 1/10/12
Read the home page and syllabus web pages, and the web handout Taking notes in a college math class.
12.1/ 1,2,4. In the first line of #1, "origin" means "origin in R³", of course, but the author should have said so explicitly.
Read Section 12.1 and start working on the problems that are due Wednesday.

W 1/11/12
12.1/ 3, 5-8, 10-18, 22, 23-27, 30-32,33,35,38,39
Read Section 12.2 and start working on the problems that are due Friday.
[Typo alert: The extra-credit problem I originally wrote down had the wrong section-number. The correct section is 12.1, as is now written below.] (This one is not part of what's due on Wednesday Jan. 11; I'm listing it here just to give you extra time to work on it.) Extra Credit, due Tuesday 1/17: 12.1/ 40. No consulting with anyone (except possibly me, in person [not by email]), or trying to find a similar problem worked out somewhere. If you plan to hand in this problem, ask me first about my rules for neatness and clarity of handed-in work.

F 1/13/12
12.2/ 1-3, 4abd,5acd,6ace,7,8,11,12,13,14,16,24-27,28,30,31,37-39,40-42.
In #25, the use of the word "quadrant" tells you that the author meant you to be working in R², not R³ (although he really should have said this explicitly).
In #42, "describe the set of all points ..." means "State what type of geometrical figure this set of points is," not "Find an equation in terms of x and y." The equation given in this problem says something about distances between various points. Think back to the analytic geometry you learned (I hope!) before calculus.
Using the algebraic definitions of vector addition and of multiplication by scalars, verify properties 2,3, and 5-8 in the "Properties of Vectors" box on p. 774.

T 1/17/12
[Typo alert: The section-number I originally put down for the extra-credit problem due today was wrong. The extra-credit problem is 12.1/40.]
12.2/ 4c,5b,6bdf,17-20,21,22,35,36,43

W 1/18/12
12.3/ 1,2,3,5-10,13,15,17-19,21,22,57,58,59. For all exercises in this section that ask you for an approximate number of degrees, you may use your calculator in the final step.

F 1/20/12
Finish reading Section 12.3, and get a head-start on the problems due Monday.

M 1/23/12
12.3/ 23,24c,25,27,28,35-37,40,41,42,44,51-55

T 1/24/12
12.3/ 45,49
12.4/ 1-4,7,9-12,13-15,17, 18, 19, 23-26,27,28, 46,47. Do #17, by computing both cross-products; don't just compute one and use anticommutativity. Do #18 by computing all the cross-products, not by using the formula given in class for a × (b × c).
In #47, you may notice that there are three other expressions of the form v₁ × (v₂ × v₃) (where {v₁, v₂, v₃} is some permutation of {a,b,c }) that do not appear in the equation, namely a × (c × b), b × (a × c) , and c × (b × a) . What's special about the expressions in #47 is that they are all cyclic permutations of each other: the second expression can be obtained from the first by the replacement-process "a → b → c → a" (i.e. in the product a × (b × c), replace a with b, replace b with c, and replace c with a), and the third expression can be obtained from the second the same way, and the first can be obtained from the third the same way. Similarly, the three expressions not appearing in #47 are cyclic permutations of each other.

W 1/25/12
12.4/ 29-38,43 (in part (a), you must assume Q≠R); 44 (in part (a), you must assume that Q,R,S are not collinear),45,48,49. Suggestion for #48: Use Theorem 8 (p. 790), parts 5 and 6.

F 1/27/12
12.5/ 1abj, 2-4,6,8-11,13, 14,15,17,18,19-22,64,66,67, 68, 75. Some additional instructions and comments:
In #1, assume all objects are in R³.
For algebraic simplicity, we regard a line as being parallel to itself. So in #13, if your answer is "yes", determine whether the lines are distinct or identical, and in #66, "two lines are identical" is a special case of "two lines are parallel".
In #67-68, the formula referred to is equivalent to the one derived at the end of Wednesday's class.
In #75, some geometric insight will be needed. Don't feel bad if this one gives you trouble. But looking at a solutions manual will defeat the purpose of the exercise, unless you've spent at least two hours on it without figuring out a way to do it. (The same goes for most exercises. You will learn more by struggling for hours and failing to find an answer, than by giving up after a few minutes and reading somebody else's solution.)

M 1/30/12
Finish reading Section 12.5. I will still go over some of it in class on Monday, but in order to preserve some chance of still having the first exam on Friday, I don't want to hold off assigning the problems below. For many of these problems, you'll need some material from Section 12.5 that I didn't get to in Friday's class.
12.5/1 (all parts not previously assigned; again assume all objects are in R³), 5,12,16, 23-34,36,37,43-45,46, 57,58,61 (assume a,b,c are all nonzero),62,63,65,66.
Do as many problems as you can from Section 12.5 that are listed below as part of the assignment due Tuesday 1/31.

T 1/31/12
12.5/ 49-54, 55 but with equation – x – y – z = – 1 for the first plane, 59, 69-72,73
Read Section 12.6.

W 2/1/12
12.6/ 9,10,12-18,29-36,45 (just find the equation; you may wait till the next assignment to graph it), 47. In 29-36, for now just figure out which of the equations can be put into a "standard form" that involve three squares with nonzero coefficients, and graph these. In your next assignment I'll have you graph the remaining equations.
   For three-variable quadratic equations that have at least two squares appearing on the same side of the equation, with the same sign (such as the equations in 11-18, but not 19 and 20), I use the "distorted surfaces of revolution" approach to identify the type of shape equation's graph has. I then use certain traces to refine my picture.
On p. 808, above Table 1, the book uses the phrase, "the six basic types of quadric surfaces in standard form". This, in combination with the definition of "quadric surface" on p. 805, is a little misleading. It is true that, by rotation and translation of the coordinate axes, any quadratic equation in three variables can be put into one of the standard forms at the top of p. 806, with a nonzero coefficient in front of at least one of the squares. (One or all of the remaining coefficients can be zero.) However, not all graphs of quadratic equations in three variables, even in standard form, are surfaces, and not all of the types of graphs that are surfaces are given in the Table 1. Sketch the graphs of the equations below, viewed as equations in x, y, and z. (If the graph has no points, say so, since in that case there is no graph.)
x² + 2y² + 3z² = 0
x² + 2y² + 3z² +4 = 0
5x² + 6y² = 0
5x² + 6y² +7 = 0
z² + 4 = 0
z² = 0
z² – 4 = 0
9x² – y² = 0

F 2/3/12
12.6/ 1-8,11,21-28,29-36 (just the equations that you didn't graph in the previous assignment),41-44, 45 (graph the equation you found in the previous assignment). In 1c, the words "as a set in R³" should have been inserted after "represent". The fact that the letter z occurs in an equation or formula doesn't mean you're in R³, although it may mean that in this book. For similar reasons, in 3-8 I would have preferred the instructions, "Describe and sketch the graph of the equation, viewd as an equation in three variables x, y, and z." However, the book's instructions for exercises 3-8 are adequate, even if not great, because the graphs of the given equations are surfaces only if the equations are viewed as equations in three variables.

General information A handout is now available that students would be wise to read before Monday's exam: Capital Crimes in Calculus III.)

M 2/6/12 First midterm exam (assignment is to study for it)

T 2/7/12
Hit yourself in the head for all the things you realized how to do after the exam, but not during it. Use only the palm of your hand to do the hitting. The use of rocks or sharp objects is especially inadvisable for this exercise.

W 2/8/12
13.1/ 1-6, 15-18, 35,41,43,45. #45 is the statement I made in class about an equivalent definition of the limit of a vector-valued function; the ``if and only if'' condition given in the book is exactly ``lim_{t→ a} || r(t)-b|| =0''. In #43 and #45, "This is true because my professor said so" is not a valid method of proof. You need a logical argument that proceeds from definitions and hypotheses to the conclusion.

F 2/10/12
13.1/ 7,8,10-12,27, 29-32 if you have a calculator or computer that can handle these (yes, I'm actually letting you use your graphing calculator, for once!)

M 2/13/12
13.1/ 11,19-24,25,26,28, 36-38

General information The grade scale for the first midterm is now posted on your grade-scale page, with a link to the list of scores so that you may see the grade distribution. Exams will be returned Monday 2/13/12.

T 2/14/12
13.2/ 2,3-8,9-16,23-26, 27-29 if you have a calculator or computer that can handle these, 30-32,41-43,45-51
13.2/ Re-do #45 and #46 by first computing the product function (the scalar-valued function u(t)•v(t) in #45; the vector-valued function u(t)×v(t) in #46) and then taking the derivative. Which method did you find more efficient?
The formulas in parts 4 and 5 of Theorem 3 are most often used to prove general facts about the derivatives of various products of vector-valued functions, regardless of any specific formulas for these vvf's, provided the vvf's are related to each other in some way (such as in #47 and #48). Exercises such as #47 and #48 show that in certain cases these parts of Theorem 3 allow you to compute an answer more efficiently when you do have formulas for the vvf's. But in most cases, when the vector-valued functions involved are just chosen randomly and are not related to each other, you will probably find it more efficient to compute the product function first, and then take the derivative of the result, than to use parts 4 and 5 of Theorem 3.

General information Tuesday's class will be regular lecture. Wednesday will be a HW Q&A day.

W 2/15/12
Re-do all the exam problems on which you did not get a perfect score.
Finish reading Section 13.2, so that you can do the remaining homework from this section. I haven't talked about the unit tangent vector field T yet, but you should be able to do the relevant parts of the problems based on your reading.
13.2/ 1, 21,22,33-35, 39,40

F 2/17/12 Read Section 13.3, and start working on the problems due Monday.

M 2/20/12
13.3/ 1-5, 13-16. In #s 2-4, the coefficients are fine-tuned so that the quantity you need to take the square root of is a perfect square. Any algebra/differentiation mistakes you make are likely to render the integral undoable. This is almost always the case for arclength problems; it is rare that
     (a sum of squares of common functions) = (square of another common function).
These problems require you to walk a tight-rope. There are hungry crocodiles below. One misstep and you are doomed.

In #s 13,14, and 16, let r_new denote the reparametrization with respect to arclength; i.e. r_new (s)= r(t(s)). (So in Example 2 on p. 831, r_new(s) is the right-hand side of the last equation in the example.) In each case, compute ||r_new'(s)||. If you do this correctly, your answer should be remarkably simple, and should be the same in all three cases.

Fun with the cycloid. Several parts of this problem are worked out in Example 5 on on p. 635. Try to do the problem yourself without looking at Example 5, but if you get stuck, you'll find some answers there.
     Let r(t)= (t– sin t)i +(1– cost)j. You may recall from last semester that this vvf parametrizes a curve called a cycloid in the xy plane. It is clear that this parametrization is continuously differentiable.
Show that || r'(t) || = 2 | sin(t/2) |, and hence that the parametrization is non-stop on any open interval of the form (2πn, 2π(n+1)), where n is an integer.
Show that the unit tangent-vector function T determined by r on the interval (2πn, 2π(n+1)) is given by T(t)=(– 1)ⁿ[sin(t/2) i +cos(t/2) j].
Compute lim_{t→ 2π–}T(t) and lim_{t→ 2π+}T(t) (the two one-sided limits of T(t) at 2π).
Sketch the cycloid over the interval 0 ≤ t ≤ 4π, trying to make your sketch properly reflect what you found in the previous part of this problem.
The portion of the curve over any interval of the form [2πn, 2π(n+1)] is called one arch of the cycloid. It is easy to see from the formula for r(t) that this arch can be obtained by translating the n=0 arch 2πn units to the right, so all arches have the same shape. Show that the arclength of each arch is 8. (It suffices to do this for the "first" arch, i.e. the one corresponding to 0≤ t ≤ 2π.)
The previous part of this problem shows that the portion of the curve from t=0 to t=4π has arclength 16 (since two arches are traversed). What would you have found for this arclength if you had incorrectly written "|| r'(t) || = 2sin(t/2)", instead of the correct "|| r'(t) || = 2 | sin(t/2) |", and integrated from 0 to 4π?
Let s(t) be the arclength function measured from t=0 (in the direction of increasing t). Show that for 0 ≤ t ≤ 2π,
     s(t) = 4(1– cos(t/2)),
and hence that for 0 ≤ s ≤ 8,
     t(s)=2 cos^–
1(1 – s/4).
Reparametrize the first arch of the cycloid by arclength. Simplify your answer; your final formula should not contain any expression that's a trig function of an inverse trig function. (Do not expect your final formula to be pretty, however.)
Let r_new(s) denote the reparametrization, just as in exercises 13, 14, and 16. Compute ||r_new'(s)||. A miracle should occur and you should get the same answer as in the earlier exercises. If not, you made a mistake.

T 2/21/12
For the parametrized curve in Example 4 (pp. 833-834), compute the curvature κ(t) directly from equation (9). (For this curve, this is also known as "the hard way".) This will help you appreciate the method used in Example 4 ("the easy way" for this curve). It will also give you practice computing derivatives and simplifying, which everyone needs. If you don't get the same answer the hard way as the easy way (for which the answer is given in the book), redo your work until you get the correct answer the hard way.
13.3/ 17-20, 22-25,30-31,33a.

In #30 you computed a certain formula κ(x) for the curvature for the graph of y=ln(x). Explain why, if you replace x by ln(x) in this formula, the result you get should be the formula for the function κ(x) that you computed in #31 for the graph of y=e^x. ("Should" means "unless there is something wrong with the way we defined curvature".) Generalize: for what other pairs of graphs would you expect to see some analogous relation?

W 2/22/12
13.3/ 21,27-28, 41-42, 52,59
(a) Compute the curvature κ(t) for the cycloid in the homework assignment that was due 2/20/12, for t not a multiple of 2π. Note that for this curve, you have already computed a simple, easily differentiated formula, for T(t), and a formula for the speed || r'(t) ||, so there is no reason not to use equation (9) in the book to compute κ(t). It's not always easier to use the formula in Theorem 10 or the formula in Exercise 40 than it is to use equation (9).
    (b) Find the limits of κ(t) as t → 0⁺, as t → 2π^–, and as t → 2π⁺ for the cycloid.
13.4/ 3,5,6,10-13,19,22

F 2/24/12
13.3/ 45-46 (osculating plane only)
13.4/ 15,16,33-36
Read Section 14.1 and get started on the problems due Monday.

M 2/27/12
14.1/ 6-30, 31-34,35-38, 39-43,45,46,47,48, 55-60.
   In 21-29, all of these graphs can be figured out using the techniques we used in Section 12.6. For 30e, while you should be able to select the correct graph from the choices, the domain over which the function plotted is a poor choice for displaying the shape of the graph. (It appears that the same domain—a square in the xy plane centered at the origin—was used in all six graphs, which is not an unreasonable thing to have done. The author may have felt that this was a good way to give the student an apples-to-apples comparison of the graphs.) The graph in 30e is a rotated version of one of the graphs we did when we discussed Section 12.6, but that's hard to tell from the picture. See if you can figure out what it's a rotated version of.
   In 35-38, if you are able to get the correct mental picture of a graph, but have trouble drawing it, I sympathize.
   In 55-60, the graph labeled VI is a little deceptive; there is no repeating pattern as you move further away from the origin than is shown. FYI, I did not find these trivial; I definitely had to think.
   In 61-64, "describe" does not mean "sketch". It means "Describe in terms of surfaces we dealt with in Section 12.6," and how the size, shape, location, and nature of the level sets changes as the level k changes.

T 2/28/12
14.1/ 61-64,65,66
Read Section 14.2.
    Warning: almost every statement in Stewart involving the word "discontinuous" is wrong. Statements such as "The function f is discontinuous at (0,0) because it is not defined there" (the first line of the solution of Example 6) are wrong, even though they permeate many calculus textbooks written in the last 20-30 years. You may even have taken exams in which a wrong answer was counted as right, and a right answer was counted as wrong. The word discontinuous has a precise meaning that has been oversimplified so much by recent textbook-authors that even many teachers of calculus misunderstand what the word means. In older calculus books, you will rarely even find the word "discontinuous", because there is no need for it in Calculus 1, 2, or 3, and because giving a correct definition in Calculus 1 would distract the student from understanding the much more important concept of what continuous means. In Example 6, it is true that f is discontinuous at (0,0), but not because f is not defined at (0,0).

W 2/29/12
14.2/ 1,2,5-14,16,17, 23 if you have a calculator/computer with 3D graphing capability, 25,26,29-36,39-41,46

F 3/2/12
14.2/ 15,18,19,25,26,37,38,39-41
Read Section 14.3

M 3/12/12
14.3/ 15,17,20-22, 26-30,35,37,39-42,49,50. (If you don't get these done before Monday's class, that's okay; just add them to the HW due Tuesday.)

(due-date not determined yet)
Determine whether each of the following limits exists. If a limit exists, find its value. (I have not yet decided the extra-credit status of these problems.)

lim_{(x,y) → (0,0)} (x⁴ + y⁴)/(x³ + y³)

lim_{(x,y) → (0,0)} (x⁵ + y⁵)/(x³ + y³)

lim_{(x,y) → (0,0)} (x² + y⁴)/(x + y²)

lim_{(x,y) → (0,0)} (x² – y⁴)/(x + y²)

Extra Credit. Let f and g be continuous one-variable functions, let a,b∈R, and assume that f(a) = 0 = g(b). Show that lim_{(x,y) → (a,b)} g(y)/f(x) does not exist.

T 3/13/12
14.3/ 45,47,51-53,55,57-59,61-64,71,72abd,73-75,87,91,95.
    In #55, simplify your answers (which is something you should always do anyway). You should then find that the formulas for f_x and f_y are very simple, and look just like the derivative of a familiar one-variable function. The reason is a trig identity that you may never have learned, that I leave you to discover or remember.
    #59, if you do it the hard way and then realize there's an easier way, should help you appreciate the value of simplifying formulas before you start calculating with them. Ditto for #64. That's why I've assigned these two.
    In #95, also compute f_xy and f_yx for (x,y) ≠ (0,0).
W 3/14/12 Second midterm exam (assignment is to study for it)

F 3/16/12
Read Section 14.4 through Theorem 8 on p. 895. Definition 7 is correct, but it obscures the meaning of differentiability. There is a much better definition, logically equivalent to the one in the book but providing a much better understanding. I will soon be posting some notes I'm writing about this.

M 3/19/12
14.4/ 1-6, 17-19
If your computer or calculator has 3D graphing capability, do 14.4/ 7-8.
Read Section 1 of these notes on differentiability.
FYI: I don't like the order in which the book covers the material in sections 14.4-14.6. I'll be covering this material in an order I prefer. Unfortunately, that means that there will be days like today in which there are hardly any book exercises I can assign. But don't fret; I'll make up for this with extra-long assignments soon enough.

T 3/20/12
Let's try this again: read Section 1 of these notes on differentiability. I'm not sure why the file I posted over the weekend was unreadable—I'd checked from home before posting it and found it readable, but I must have run the mathematical word-processor one more time, "broken" the file, and forgot to do a last check. But when I checked from my office after class, all I got was a row of dots (just as some of you had gotten) and a bunch of error messages. I just re-processed the file a little while ago, doing nothing different from before, and now the file is readable again.
14.4/ 11-16,21,31-34,38,40. In 11-16, replace the instructions "Explain why" with "Show that" or with "Explain why you can be sure that". In all of these, the point is to use Theorem 8 (p. 895), which is most definitely not an explanation of why a function is differentiable; it just gives a very useful criterion that guarantees differentiability. (I haven't yet gotten to Theorem 8 in class, but you may use it to do these HW problems anyway.) The explanation of why this criterion guarantees differentiability is the proof of this important theorem, which the author does not give till Appendix F, and which we won't have time to cover in class.
14.6 (yes, 14.6)/ 4-6 (here, "in the direction indicated by the angle θ" means "in the direction (cos θ)i + (sin θ)j" ).

W 3/21/12
14.6/ 7-9,11-13,15,16,28,30,37, 62 (in part (a), after "all other directions" insert "except for –i and –j")
Read Sections 2 and 3 of these notes on differentiability.

General information Your grade-scale page has now been updated.

F 3/23/12
14.6/ 21-25,29 (insert the words "the direction of" before i + j, since i + j is not a unit vector), 31,33,34,36
14.5/ 1-10,13,21,22,39,41,49,55
M 3/26/12
14.4/ 42
14.5/ 27-34, 45
Read pp. 917-919 (starting with "Tangent planes to level surfaces")
14.6/ 39-44,47-50,52-54,58 (recall from Calc 2 that the volume of a pyramid is 1/3 the area of the base times the height)

T 3/27/12
14.6/ 62 (in part (a), after "all other directions" insert "except for –i and –j").
Read Section 14.7. I'm not assigning any exercises due Tuesday because not a single one can be done until almost the entire section is covered. However, the assignment due Wednesday will be long, so you may want to try to start it, based just on your reading.

W 3/28/12
14.7/ 1,3,4,5-15,19,20,37,38 (computer part is optional in 37 and 38). Most of these problems require you to use the Second Derivatives Test (also known by the name I used in class, "Discriminant Test"), which I got to only at the very end of class on Tuesday. Ordinarily I would not assign these problems until I've done more examples, but the more of these you're able to do by Wednesday, the fewer you'll have left to do for Friday, which is the real due-date.
    In critical-point problems, usually the trickiest part is finding all the critical points. This involves solving two simultaneous equations, often non-linear, in two variables. (That's for the case of functions of two variables. For n variables, you have to solve n simultaneous, often non-linear, equations in n unknowns, and the algebraic complexity increases substantially for n ≥ 3.) The logic involved is straightforward but is unfamiliar to most students in Calculus III. I will do several examples in class.
    As you read Section 14.7, particularly pp. 928-930, be aware that Stewart uses the letter D in this section for two different things (three, if you count D_u): for discriminant, as in Theorem 3 (p. 924), and for a closed, bounded region in R², as in Theorem 8 (p. 928). I'm using R in class for "region in R² " to have separate letters for "discriminant" and "region".

F 3/30/12
14.7/ 39,42-47,50,53,55. These are "absolute" max/min problems, but in each case the absolute maximum or minimum is achieved at a critical point of an appropriate two-variable function. In each problem, try to give an argument for why the relative extremum you find is actually an absolute extremum.

Suppose g is a twice continuously differentiable function of one variable, and h is a twice continuously differentiable function of two variables, with range(h) contained in domain(g). Let f(x,y) = g(h(x,y)); i.e. f = g o h. (Observe that the function f in 14.7/ 8 is of this form, with g(u) = e^u and h(x,y) = 4y – x ² – y². But do not assume any particular formula for g and h in the current problem.)

Show that if g has no critical points (i.e. if g' is nowhere zero), then:
The critical points of f are the same as the critical points of h.
A point (x₀, y₀) is a (non)degenerate critical point of f if and only if it is a (non)degenerate critical point of h.
The function f has a nondegenerate relative maximum, minimum, or saddle at (x₀, y₀) if and only if h has, respectively, a nondegenerate relative maximum, minimum, or saddle there.
Check that your answer to 14.7/ 8 is consistent with the facts above.
Show that if there is a number c for which g'(c) = 0, then every point of the level-set {(x,y) | h(x,y) = c} is a degenerate critical point of f.

M 4/2/12
14.7/ 29-36
Do exercises 14.8/ 3-5, 7-12,18,19 without using the methods in Section 14.8. (These will later be reassigned for you to do with the methods of Section 14.8.)

T 4/3/12
Read this handout on algebra in Lagrange-multiplier problems.
Redo exercises 14.8/ 3-5, 7-12, 18,19 using the methods in Section 14.8. (If you're not able to do the three-variable problems 7-12 yet, that's okay; I will do at least one three-variable example in class on Tuesday.)

W 4/4/12
14.8/ 27,30-35,38,39

F 4/6/12
15.1/ 11-14,17 (see p. 951 for notation for rectangles)
15.2/ 1,2,3-5 (see p. 960 for notation for iterated integrals), 7-10,12-14,15-18,20-22,23,24

M 4/9/12
15.2/ 25-31,38.

T 4/10/12 Third midterm exam (assignment is to study for it)

W 4/11/12 No new homework.

F 4/13/12
15.3/ 1-6, 7-10. (Previously assigned with due-date of Monday 4/9; I should have removed it from that assignment earlier.)
15.3/11-18

General information Your grade-scale page has now been updated.

M 4/16/12
15.3/19-21,22,25,28,33,34,39-44 (in this group, you are not being asked to evaluate the integral, just to re-express it as an iterated integral in the order opposite to the given one), 52,55,56
Read Sections 15.4 and 15.6. We're going to skip 15.5.

T 4/17/12
15.3/ 45-50,58,60,61

15.4/ 1-4,5,6,7-11,14,15,16,19,21,22, 25-27,28,29-32, 35. In #7, try to predict the answer before doing any computations, then do the computation to check whether your prediction was right. If you're not able to make a prediction, then after you do the computation try to figure how you could have predicted the answer.

W 4/18/12
15.6/ 9,10,12,14,15,17,19,21,22,27,28,29-32,33,34,51,52
15.7/ 1-12,15,16,17

F 4/20/12
15.8/ 5,7,8 (hint for 7-8: rewrite the equation in Cartesian coordinates), 9,10,11-14,15-18,19,20 (in 19-20 choose the coordinate system in which it is easiest to express the domain), 21-24,26-28. In #28, for average value of a function defined on a solid region, review the definition that's just before 15.6/ 51. Explain qualitatively why, even before computing the answer to #28, you should expect the answer to be greater than a/2.

M 4/23/12
16.1/ 1-7,11-14,15-18,21-24,25,26
16.2/ 1-4,9,10,12

16.5 (yes, 16.5)/ 1-5,12

T 4/24/12
16.2/ 5-8,14-16,17,19-21,39,42
Read the definition of conservative vector field on p. 1032. A more precise version is: a vector field F is conservative on a region D if there is some differentiable, real-valued function f on D for which F is the gradient of f. In this definition, it is crucial that f be defined and differentiable at every point of D.
Read Section 16.3 through Example 5. Reading the proof of Theorem 4 is optional.
16.3/ 1,4,5,8,9,11,12,13,16,17,19,20

W 4/25/12
16.3/ 27,28,23,33,34a. Also, read the first sentences of 34b and 34c, to see a couple of reasons that 34a is important.
16.5/ 13-18, 19, 20
Read Section 16.4

Do these before final exam, preferably by Fri. 4/27/12
16.4/ 1-14

F 5/4/12 FINAL EXAM begins at 12:30 p.m. in our usual classroom.

After the exam, please do not email me with questions about your grade for the class, your performance on the exam, etc. I will not email any information relating to the final exam or grades. Course grades should be available from ISIS shortly after I submit them, which will be a few days after your final. I will post some exam statistics, and some other statistics, on your grade scale page.

Date due	Section # / problem #s
T 1/10/12	Read the home page and syllabus web pages, and the web handout Taking notes in a college math class. 12.1/ 1,2,4. In the first line of #1, "origin" means "origin in R³", of course, but the author should have said so explicitly. Read Section 12.1 and start working on the problems that are due Wednesday.
W 1/11/12	12.1/ 3, 5-8, 10-18, 22, 23-27, 30-32,33,35,38,39 Read Section 12.2 and start working on the problems that are due Friday. [Typo alert: The extra-credit problem I originally wrote down had the wrong section-number. The correct section is 12.1, as is now written below.] (This one is not part of what's due on Wednesday Jan. 11; I'm listing it here just to give you extra time to work on it.) Extra Credit, due Tuesday 1/17: 12.1/ 40. No consulting with anyone (except possibly me, in person [not by email]), or trying to find a similar problem worked out somewhere. If you plan to hand in this problem, ask me first about my rules for neatness and clarity of handed-in work.
F 1/13/12	12.2/ 1-3, 4abd,5acd,6ace,7,8,11,12,13,14,16,24-27,28,30,31,37-39,40-42. In #25, the use of the word "quadrant" tells you that the author meant you to be working in R², not R³ (although he really should have said this explicitly). In #42, "describe the set of all points ..." means "State what type of geometrical figure this set of points is," not "Find an equation in terms of x and y." The equation given in this problem says something about distances between various points. Think back to the analytic geometry you learned (I hope!) before calculus. Using the algebraic definitions of vector addition and of multiplication by scalars, verify properties 2,3, and 5-8 in the "Properties of Vectors" box on p. 774.
T 1/17/12	[Typo alert: The section-number I originally put down for the extra-credit problem due today was wrong. The extra-credit problem is 12.1/40.] 12.2/ 4c,5b,6bdf,17-20,21,22,35,36,43
W 1/18/12	12.3/ 1,2,3,5-10,13,15,17-19,21,22,57,58,59. For all exercises in this section that ask you for an approximate number of degrees, you may use your calculator in the final step.
F 1/20/12	Finish reading Section 12.3, and get a head-start on the problems due Monday.
M 1/23/12	12.3/ 23,24c,25,27,28,35-37,40,41,42,44,51-55
T 1/24/12	12.3/ 45,49 12.4/ 1-4,7,9-12,13-15,17, 18, 19, 23-26,27,28, 46,47. Do #17, by computing both cross-products; don't just compute one and use anticommutativity. Do #18 by computing all the cross-products, not by using the formula given in class for a × (b × c). In #47, you may notice that there are three other expressions of the form v₁ × (v₂ × v₃) (where {v₁, v₂, v₃} is some permutation of {a,b,c }) that do not appear in the equation, namely a × (c × b), b × (a × c) , and c × (b × a) . What's special about the expressions in #47 is that they are all cyclic permutations of each other: the second expression can be obtained from the first by the replacement-process "a → b → c → a" (i.e. in the product a × (b × c), replace a with b, replace b with c, and replace c with a), and the third expression can be obtained from the second the same way, and the first can be obtained from the third the same way. Similarly, the three expressions not appearing in #47 are cyclic permutations of each other.
W 1/25/12	12.4/ 29-38,43 (in part (a), you must assume Q≠R); 44 (in part (a), you must assume that Q,R,S are not collinear),45,48,49. Suggestion for #48: Use Theorem 8 (p. 790), parts 5 and 6.
F 1/27/12	12.5/ 1abj, 2-4,6,8-11,13, 14,15,17,18,19-22,64,66,67, 68, 75. Some additional instructions and comments: In #1, assume all objects are in R³. For algebraic simplicity, we regard a line as being parallel to itself. So in #13, if your answer is "yes", determine whether the lines are distinct or identical, and in #66, "two lines are identical" is a special case of "two lines are parallel". In #67-68, the formula referred to is equivalent to the one derived at the end of Wednesday's class. In #75, some geometric insight will be needed. Don't feel bad if this one gives you trouble. But looking at a solutions manual will defeat the purpose of the exercise, unless you've spent at least two hours on it without figuring out a way to do it. (The same goes for most exercises. You will learn more by struggling for hours and failing to find an answer, than by giving up after a few minutes and reading somebody else's solution.)
M 1/30/12	Finish reading Section 12.5. I will still go over some of it in class on Monday, but in order to preserve some chance of still having the first exam on Friday, I don't want to hold off assigning the problems below. For many of these problems, you'll need some material from Section 12.5 that I didn't get to in Friday's class. 12.5/1 (all parts not previously assigned; again assume all objects are in R³), 5,12,16, 23-34,36,37,43-45,46, 57,58,61 (assume a,b,c are all nonzero),62,63,65,66. Do as many problems as you can from Section 12.5 that are listed below as part of the assignment due Tuesday 1/31.
T 1/31/12	12.5/ 49-54, 55 but with equation – x – y – z = – 1 for the first plane, 59, 69-72,73 Read Section 12.6.
W 2/1/12	12.6/ 9,10,12-18,29-36,45 (just find the equation; you may wait till the next assignment to graph it), 47. In 29-36, for now just figure out which of the equations can be put into a "standard form" that involve three squares with nonzero coefficients, and graph these. In your next assignment I'll have you graph the remaining equations. For three-variable quadratic equations that have at least two squares appearing on the same side of the equation, with the same sign (such as the equations in 11-18, but not 19 and 20), I use the "distorted surfaces of revolution" approach to identify the type of shape equation's graph has. I then use certain traces to refine my picture. On p. 808, above Table 1, the book uses the phrase, "the six basic types of quadric surfaces in standard form". This, in combination with the definition of "quadric surface" on p. 805, is a little misleading. It is true that, by rotation and translation of the coordinate axes, any quadratic equation in three variables can be put into one of the standard forms at the top of p. 806, with a nonzero coefficient in front of at least one of the squares. (One or all of the remaining coefficients can be zero.) However, not all graphs of quadratic equations in three variables, even in standard form, are surfaces, and not all of the types of graphs that are surfaces are given in the Table 1. Sketch the graphs of the equations below, viewed as equations in x, y, and z. (If the graph has no points, say so, since in that case there is no graph.) x² + 2y² + 3z² = 0 x² + 2y² + 3z² +4 = 0 5x² + 6y² = 0 5x² + 6y² +7 = 0 z² + 4 = 0 z² = 0 z² – 4 = 0 9x² – y² = 0
F 2/3/12	12.6/ 1-8,11,21-28,29-36 (just the equations that you didn't graph in the previous assignment),41-44, 45 (graph the equation you found in the previous assignment). In 1c, the words "as a set in R³" should have been inserted after "represent". The fact that the letter z occurs in an equation or formula doesn't mean you're in R³, although it may mean that in this book. For similar reasons, in 3-8 I would have preferred the instructions, "Describe and sketch the graph of the equation, viewd as an equation in three variables x, y, and z." However, the book's instructions for exercises 3-8 are adequate, even if not great, because the graphs of the given equations are surfaces only if the equations are viewed as equations in three variables.
General information	A handout is now available that students would be wise to read before Monday's exam: Capital Crimes in Calculus III.)
M 2/6/12	First midterm exam (assignment is to study for it)
T 2/7/12	Hit yourself in the head for all the things you realized how to do after the exam, but not during it. Use only the palm of your hand to do the hitting. The use of rocks or sharp objects is especially inadvisable for this exercise.
W 2/8/12	13.1/ 1-6, 15-18, 35,41,43,45. #45 is the statement I made in class about an equivalent definition of the limit of a vector-valued function; the ``if and only if'' condition given in the book is exactly ``lim_{t→ a} \|\| r(t)-b\|\| =0''. In #43 and #45, "This is true because my professor said so" is not a valid method of proof. You need a logical argument that proceeds from definitions and hypotheses to the conclusion.
F 2/10/12	13.1/ 7,8,10-12,27, 29-32 if you have a calculator or computer that can handle these (yes, I'm actually letting you use your graphing calculator, for once!)
M 2/13/12	13.1/ 11,19-24,25,26,28, 36-38
General information	The grade scale for the first midterm is now posted on your grade-scale page, with a link to the list of scores so that you may see the grade distribution. Exams will be returned Monday 2/13/12.
T 2/14/12	13.2/ 2,3-8,9-16,23-26, 27-29 if you have a calculator or computer that can handle these, 30-32,41-43,45-51 13.2/ Re-do #45 and #46 by first computing the product function (the scalar-valued function u(t)•v(t) in #45; the vector-valued function u(t)×v(t) in #46) and then taking the derivative. Which method did you find more efficient? The formulas in parts 4 and 5 of Theorem 3 are most often used to prove general facts about the derivatives of various products of vector-valued functions, regardless of any specific formulas for these vvf's, provided the vvf's are related to each other in some way (such as in #47 and #48). Exercises such as #47 and #48 show that in certain cases these parts of Theorem 3 allow you to compute an answer more efficiently when you do have formulas for the vvf's. But in most cases, when the vector-valued functions involved are just chosen randomly and are not related to each other, you will probably find it more efficient to compute the product function first, and then take the derivative of the result, than to use parts 4 and 5 of Theorem 3.
General information	Tuesday's class will be regular lecture. Wednesday will be a HW Q&A day.
W 2/15/12	Re-do all the exam problems on which you did not get a perfect score. Finish reading Section 13.2, so that you can do the remaining homework from this section. I haven't talked about the unit tangent vector field T yet, but you should be able to do the relevant parts of the problems based on your reading. 13.2/ 1, 21,22,33-35, 39,40
F 2/17/12	Read Section 13.3, and start working on the problems due Monday.
M 2/20/12	13.3/ 1-5, 13-16. In #s 2-4, the coefficients are fine-tuned so that the quantity you need to take the square root of is a perfect square. Any algebra/differentiation mistakes you make are likely to render the integral undoable. This is almost always the case for arclength problems; it is rare that (a sum of squares of common functions) = (square of another common function). These problems require you to walk a tight-rope. There are hungry crocodiles below. One misstep and you are doomed. In #s 13,14, and 16, let r_new denote the reparametrization with respect to arclength; i.e. r_new (s)= r(t(s)). (So in Example 2 on p. 831, r_new(s) is the right-hand side of the last equation in the example.) In each case, compute \|\|r_new'(s)\|\|. If you do this correctly, your answer should be remarkably simple, and should be the same in all three cases. Fun with the cycloid. Several parts of this problem are worked out in Example 5 on on p. 635. Try to do the problem yourself without looking at Example 5, but if you get stuck, you'll find some answers there. Let r(t)= (t– sin t)i +(1– cost)j. You may recall from last semester that this vvf parametrizes a curve called a cycloid in the xy plane. It is clear that this parametrization is continuously differentiable. Show that \|\| r'(t) \|\| = 2 \| sin(t/2) \|, and hence that the parametrization is non-stop on any open interval of the form (2πn, 2π(n+1)), where n is an integer. Show that the unit tangent-vector function T determined by r on the interval (2πn, 2π(n+1)) is given by T(t)=(– 1)ⁿ[sin(t/2) i +cos(t/2) j]. Compute lim_{t→ 2π–}T(t) and lim_{t→ 2π+}T(t) (the two one-sided limits of T(t) at 2π). Sketch the cycloid over the interval 0 ≤ t ≤ 4π, trying to make your sketch properly reflect what you found in the previous part of this problem. The portion of the curve over any interval of the form [2πn, 2π(n+1)] is called one arch of the cycloid. It is easy to see from the formula for r(t) that this arch can be obtained by translating the n=0 arch 2πn units to the right, so all arches have the same shape. Show that the arclength of each arch is 8. (It suffices to do this for the "first" arch, i.e. the one corresponding to 0≤ t ≤ 2π.) The previous part of this problem shows that the portion of the curve from t=0 to t=4π has arclength 16 (since two arches are traversed). What would you have found for this arclength if you had incorrectly written "\|\| r'(t) \|\| = 2sin(t/2)", instead of the correct "\|\| r'(t) \|\| = 2 \| sin(t/2) \|", and integrated from 0 to 4π? Let s(t) be the arclength function measured from t=0 (in the direction of increasing t). Show that for 0 ≤ t ≤ 2π, s(t) = 4(1– cos(t/2)), and hence that for 0 ≤ s ≤ 8, t(s)=2 cos^– 1(1 – s/4). Reparametrize the first arch of the cycloid by arclength. Simplify your answer; your final formula should not contain any expression that's a trig function of an inverse trig function. (Do not expect your final formula to be pretty, however.) Let r_new(s) denote the reparametrization, just as in exercises 13, 14, and 16. Compute \|\|r_new'(s)\|\|. A miracle should occur and you should get the same answer as in the earlier exercises. If not, you made a mistake.
T 2/21/12	For the parametrized curve in Example 4 (pp. 833-834), compute the curvature κ(t) directly from equation (9). (For this curve, this is also known as "the hard way".) This will help you appreciate the method used in Example 4 ("the easy way" for this curve). It will also give you practice computing derivatives and simplifying, which everyone needs. If you don't get the same answer the hard way as the easy way (for which the answer is given in the book), redo your work until you get the correct answer the hard way. 13.3/ 17-20, 22-25,30-31,33a. In #30 you computed a certain formula κ(x) for the curvature for the graph of y=ln(x). Explain why, if you replace x by ln(x) in this formula, the result you get should be the formula for the function κ(x) that you computed in #31 for the graph of y=e^x. ("Should" means "unless there is something wrong with the way we defined curvature".) Generalize: for what other pairs of graphs would you expect to see some analogous relation?
W 2/22/12	13.3/ 21,27-28, 41-42, 52,59 (a) Compute the curvature κ(t) for the cycloid in the homework assignment that was due 2/20/12, for t not a multiple of 2π. Note that for this curve, you have already computed a simple, easily differentiated formula, for T(t), and a formula for the speed \|\| r'(t) \|\|, so there is no reason not to use equation (9) in the book to compute κ(t). It's not always easier to use the formula in Theorem 10 or the formula in Exercise 40 than it is to use equation (9). (b) Find the limits of κ(t) as t → 0⁺, as t → 2π^–, and as t → 2π⁺ for the cycloid. 13.4/ 3,5,6,10-13,19,22
F 2/24/12	13.3/ 45-46 (osculating plane only) 13.4/ 15,16,33-36 Read Section 14.1 and get started on the problems due Monday.
M 2/27/12	14.1/ 6-30, 31-34,35-38, 39-43,45,46,47,48, 55-60. In 21-29, all of these graphs can be figured out using the techniques we used in Section 12.6. For 30e, while you should be able to select the correct graph from the choices, the domain over which the function plotted is a poor choice for displaying the shape of the graph. (It appears that the same domain—a square in the xy plane centered at the origin—was used in all six graphs, which is not an unreasonable thing to have done. The author may have felt that this was a good way to give the student an apples-to-apples comparison of the graphs.) The graph in 30e is a rotated version of one of the graphs we did when we discussed Section 12.6, but that's hard to tell from the picture. See if you can figure out what it's a rotated version of. In 35-38, if you are able to get the correct mental picture of a graph, but have trouble drawing it, I sympathize. In 55-60, the graph labeled VI is a little deceptive; there is no repeating pattern as you move further away from the origin than is shown. FYI, I did not find these trivial; I definitely had to think. In 61-64, "describe" does not mean "sketch". It means "Describe in terms of surfaces we dealt with in Section 12.6," and how the size, shape, location, and nature of the level sets changes as the level k changes.
T 2/28/12	14.1/ 61-64,65,66 Read Section 14.2. Warning: almost every statement in Stewart involving the word "discontinuous" is wrong. Statements such as "The function f is discontinuous at (0,0) because it is not defined there" (the first line of the solution of Example 6) are wrong, even though they permeate many calculus textbooks written in the last 20-30 years. You may even have taken exams in which a wrong answer was counted as right, and a right answer was counted as wrong. The word discontinuous has a precise meaning that has been oversimplified so much by recent textbook-authors that even many teachers of calculus misunderstand what the word means. In older calculus books, you will rarely even find the word "discontinuous", because there is no need for it in Calculus 1, 2, or 3, and because giving a correct definition in Calculus 1 would distract the student from understanding the much more important concept of what continuous means. In Example 6, it is true that f is discontinuous at (0,0), but not because f is not defined at (0,0).
W 2/29/12	14.2/ 1,2,5-14,16,17, 23 if you have a calculator/computer with 3D graphing capability, 25,26,29-36,39-41,46
F 3/2/12	14.2/ 15,18,19,25,26,37,38,39-41 Read Section 14.3
M 3/12/12	14.3/ 15,17,20-22, 26-30,35,37,39-42,49,50. (If you don't get these done before Monday's class, that's okay; just add them to the HW due Tuesday.)
(due-date not determined yet)	Determine whether each of the following limits exists. If a limit exists, find its value. (I have not yet decided the extra-credit status of these problems.) lim_{(x,y) → (0,0)} (x⁴ + y⁴)/(x³ + y³) lim_{(x,y) → (0,0)} (x⁵ + y⁵)/(x³ + y³) lim_{(x,y) → (0,0)} (x² + y⁴)/(x + y²) lim_{(x,y) → (0,0)} (x² – y⁴)/(x + y²) Extra Credit. Let f and g be continuous one-variable functions, let a,b∈R, and assume that f(a) = 0 = g(b). Show that lim_{(x,y) → (a,b)} g(y)/f(x) does not exist.
T 3/13/12	14.3/ 45,47,51-53,55,57-59,61-64,71,72abd,73-75,87,91,95. In #55, simplify your answers (which is something you should always do anyway). You should then find that the formulas for f_x and f_y are very simple, and look just like the derivative of a familiar one-variable function. The reason is a trig identity that you may never have learned, that I leave you to discover or remember. #59, if you do it the hard way and then realize there's an easier way, should help you appreciate the value of simplifying formulas before you start calculating with them. Ditto for #64. That's why I've assigned these two. In #95, also compute f_xy and f_yx for (x,y) ≠ (0,0).
W 3/14/12	Second midterm exam (assignment is to study for it)
F 3/16/12	Read Section 14.4 through Theorem 8 on p. 895. Definition 7 is correct, but it obscures the meaning of differentiability. There is a much better definition, logically equivalent to the one in the book but providing a much better understanding. I will soon be posting some notes I'm writing about this.
M 3/19/12	14.4/ 1-6, 17-19 If your computer or calculator has 3D graphing capability, do 14.4/ 7-8. Read Section 1 of these notes on differentiability. FYI: I don't like the order in which the book covers the material in sections 14.4-14.6. I'll be covering this material in an order I prefer. Unfortunately, that means that there will be days like today in which there are hardly any book exercises I can assign. But don't fret; I'll make up for this with extra-long assignments soon enough.
T 3/20/12	Let's try this again: read Section 1 of these notes on differentiability. I'm not sure why the file I posted over the weekend was unreadable—I'd checked from home before posting it and found it readable, but I must have run the mathematical word-processor one more time, "broken" the file, and forgot to do a last check. But when I checked from my office after class, all I got was a row of dots (just as some of you had gotten) and a bunch of error messages. I just re-processed the file a little while ago, doing nothing different from before, and now the file is readable again. 14.4/ 11-16,21,31-34,38,40. In 11-16, replace the instructions "Explain why" with "Show that" or with "Explain why you can be sure that". In all of these, the point is to use Theorem 8 (p. 895), which is most definitely not an explanation of why a function is differentiable; it just gives a very useful criterion that guarantees differentiability. (I haven't yet gotten to Theorem 8 in class, but you may use it to do these HW problems anyway.) The explanation of why this criterion guarantees differentiability is the proof of this important theorem, which the author does not give till Appendix F, and which we won't have time to cover in class. 14.6 (yes, 14.6)/ 4-6 (here, "in the direction indicated by the angle θ" means "in the direction (cos θ)i + (sin θ)j" ).
W 3/21/12	14.6/ 7-9,11-13,15,16,28,30,37, 62 (in part (a), after "all other directions" insert "except for –i and –j") Read Sections 2 and 3 of these notes on differentiability.
General information	Your grade-scale page has now been updated.
F 3/23/12	14.6/ 21-25,29 (insert the words "the direction of" before i + j, since i + j is not a unit vector), 31,33,34,36 14.5/ 1-10,13,21,22,39,41,49,55
M 3/26/12	14.4/ 42 14.5/ 27-34, 45 Read pp. 917-919 (starting with "Tangent planes to level surfaces") 14.6/ 39-44,47-50,52-54,58 (recall from Calc 2 that the volume of a pyramid is 1/3 the area of the base times the height)
T 3/27/12	14.6/ 62 (in part (a), after "all other directions" insert "except for –i and –j"). Read Section 14.7. I'm not assigning any exercises due Tuesday because not a single one can be done until almost the entire section is covered. However, the assignment due Wednesday will be long, so you may want to try to start it, based just on your reading.
W 3/28/12	14.7/ 1,3,4,5-15,19,20,37,38 (computer part is optional in 37 and 38). Most of these problems require you to use the Second Derivatives Test (also known by the name I used in class, "Discriminant Test"), which I got to only at the very end of class on Tuesday. Ordinarily I would not assign these problems until I've done more examples, but the more of these you're able to do by Wednesday, the fewer you'll have left to do for Friday, which is the real due-date. In critical-point problems, usually the trickiest part is finding all the critical points. This involves solving two simultaneous equations, often non-linear, in two variables. (That's for the case of functions of two variables. For n variables, you have to solve n simultaneous, often non-linear, equations in n unknowns, and the algebraic complexity increases substantially for n ≥ 3.) The logic involved is straightforward but is unfamiliar to most students in Calculus III. I will do several examples in class. As you read Section 14.7, particularly pp. 928-930, be aware that Stewart uses the letter D in this section for two different things (three, if you count D_u): for discriminant, as in Theorem 3 (p. 924), and for a closed, bounded region in R², as in Theorem 8 (p. 928). I'm using R in class for "region in R² " to have separate letters for "discriminant" and "region".
F 3/30/12	14.7/ 39,42-47,50,53,55. These are "absolute" max/min problems, but in each case the absolute maximum or minimum is achieved at a critical point of an appropriate two-variable function. In each problem, try to give an argument for why the relative extremum you find is actually an absolute extremum. Suppose g is a twice continuously differentiable function of one variable, and h is a twice continuously differentiable function of two variables, with range(h) contained in domain(g). Let f(x,y) = g(h(x,y)); i.e. f = g o h. (Observe that the function f in 14.7/ 8 is of this form, with g(u) = e^u and h(x,y) = 4y – x ² – y². But do not assume any particular formula for g and h in the current problem.) Show that if g has no critical points (i.e. if g' is nowhere zero), then: The critical points of f are the same as the critical points of h. A point (x₀, y₀) is a (non)degenerate critical point of f if and only if it is a (non)degenerate critical point of h. The function f has a nondegenerate relative maximum, minimum, or saddle at (x₀, y₀) if and only if h has, respectively, a nondegenerate relative maximum, minimum, or saddle there. Check that your answer to 14.7/ 8 is consistent with the facts above. Show that if there is a number c for which g'(c) = 0, then every point of the level-set {(x,y) \| h(x,y) = c} is a degenerate critical point of f.
M 4/2/12	14.7/ 29-36 Do exercises 14.8/ 3-5, 7-12,18,19 without using the methods in Section 14.8. (These will later be reassigned for you to do with the methods of Section 14.8.)
T 4/3/12	Read this handout on algebra in Lagrange-multiplier problems. Redo exercises 14.8/ 3-5, 7-12, 18,19 using the methods in Section 14.8. (If you're not able to do the three-variable problems 7-12 yet, that's okay; I will do at least one three-variable example in class on Tuesday.)
W 4/4/12	14.8/ 27,30-35,38,39
F 4/6/12	15.1/ 11-14,17 (see p. 951 for notation for rectangles) 15.2/ 1,2,3-5 (see p. 960 for notation for iterated integrals), 7-10,12-14,15-18,20-22,23,24
M 4/9/12	15.2/ 25-31,38.
T 4/10/12	Third midterm exam (assignment is to study for it)
W 4/11/12	No new homework.
F 4/13/12	15.3/ 1-6, 7-10. (Previously assigned with due-date of Monday 4/9; I should have removed it from that assignment earlier.) 15.3/11-18
General information	Your grade-scale page has now been updated.
M 4/16/12	15.3/19-21,22,25,28,33,34,39-44 (in this group, you are not being asked to evaluate the integral, just to re-express it as an iterated integral in the order opposite to the given one), 52,55,56 Read Sections 15.4 and 15.6. We're going to skip 15.5.
T 4/17/12	15.3/ 45-50,58,60,61 15.4/ 1-4,5,6,7-11,14,15,16,19,21,22, 25-27,28,29-32, 35. In #7, try to predict the answer before doing any computations, then do the computation to check whether your prediction was right. If you're not able to make a prediction, then after you do the computation try to figure how you could have predicted the answer.
W 4/18/12	15.6/ 9,10,12,14,15,17,19,21,22,27,28,29-32,33,34,51,52 15.7/ 1-12,15,16,17
F 4/20/12	15.8/ 5,7,8 (hint for 7-8: rewrite the equation in Cartesian coordinates), 9,10,11-14,15-18,19,20 (in 19-20 choose the coordinate system in which it is easiest to express the domain), 21-24,26-28. In #28, for average value of a function defined on a solid region, review the definition that's just before 15.6/ 51. Explain qualitatively why, even before computing the answer to #28, you should expect the answer to be greater than a/2.
M 4/23/12	16.1/ 1-7,11-14,15-18,21-24,25,26 16.2/ 1-4,9,10,12 16.5 (yes, 16.5)/ 1-5,12
T 4/24/12	16.2/ 5-8,14-16,17,19-21,39,42 Read the definition of conservative vector field on p. 1032. A more precise version is: a vector field F is conservative on a region D if there is some differentiable, real-valued function f on D for which F is the gradient of f. In this definition, it is crucial that f be defined and differentiable at every* point of D*. Read Section 16.3 through Example 5. Reading the proof of Theorem 4 is optional. 16.3/ 1,4,5,8,9,11,12,13,16,17,19,20
W 4/25/12	16.3/ 27,28,23,33,34a. Also, read the first sentences of 34b and 34c, to see a couple of reasons that 34a is important. 16.5/ 13-18, 19, 20 Read Section 16.4
Do these before final exam, preferably by Fri. 4/27/12	16.4/ 1-14
F 5/4/12	FINAL EXAM begins at 12:30 p.m. in our usual classroom.
	After the exam, please *do not* email me with questions about your grade for the class, your performance on the exam, etc. *I will not email any information relating to the final exam or grades.* Course grades should be available from ISIS shortly after I submit them, which will be a few days after your final. I will post some exam statistics, and some other statistics, on your grade scale page.

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Homework Assignments MAC 3474, Section 3129 — Honors Analytic Geometry and Calculus III Spring 2012

Homework Assignments
MAC 3474, Section 3129 — Honors Analytic Geometry and Calculus III
Spring 2012