Homework Assignments
MAP 3474, Section 3129 - Honors Analytic Geometry and Calculus III
Spring 2010

Last update made by D. Groisser Wed Apr 21 17:10:54 EDT 2010

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

W 1/6/10
Read the home page and syllabus web pages, and the web handout Taking notes in a college math class.
12.1/ 1-8, 10-18, 22, 23-27. In the first line of #1, "origin" means "origin in R³".

F 1/8/10
12.1/ 30-32,33,35,38
12.2/ 1-3,4abd,5ac,6acde,7,8,11,12,13,14,16,37,39

M 1/11/10
12.2/ 4c,5bd,6bf,17-20,21,22,24-27,28,30,31,38,40-43
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 (hopefully) learned before calculus.
Using the algebraic definitions of vector addition and of multiplication by scalars, verify properties 1-8 in the "Properties of Vectors" box on p. 774. For property 1, just do this for n=3 (we did n=2 in class). For the remaining seven properties, do the verification in the n=3 case for three of them (whichever three you want), and in the n=2 case for the other four.

General Information Something I forgot to mention the first day of class: roughly every fourth or fifth class will be a "homework Q&A" day, in which I work out whatever homework problems you ask me to. The first of these will be Tuesday Jan. 12.
Here is a tentative schedule of lectures, which I will revise occasionally.

T 1/12/10
12.2/ 35,36
12.3/ 1,2,3,5-10,13,15,17-19,21,22,51,52. For all exercises in this section that ask you for an approximate number of degrees, you may use your calculator in the final step.

W 1/13/10 12.3/ 53-55,57,58

F 1/15/10
Read Examples 6 and 7 in Section 12.2 (pp. 783-784). This is an application that I did not have time to get to in Wednesday's lecture, and we'll fall farther behind schedule if I spend time on it Friday (Wednesday, we fell behind by some fraction of a lecture).
12.3/ 23,24c,25,27,28,35-37,40,41,42,44,45,49,59,60

T 1/19/10 12.4/ 1-4,7,9-12,13-15,17 (compute both cross-products; don't just compute one and use anticommutativity), 18 (do this by computing all the cross-products, not by using the formula given in class for a × (b × c)), 19, 21, 23-26, 46,47

W 1/20/10 12.4/ 27,29,30,33,35,44 (in part (a), you must assume that Q,R,S are not collinear),45,49

F 1/21/10 12.4/ 28,31,32,34,36,37,38,43 (in part (a), you must assume Q≠R)

M 1/25/10 12.5/ 1abj,2-4,6,8-11,13 (if your answer is "yes", determine whether the lines are distinct or identical), 14,15,17,18,19-22,63,64,66,67,68

T 1/26/10
12.5/ 5,12,16, 23-34,36,37,57,58,62,66,67
On this day, remind me to discuss whether to keep the first midterm on its originally scheduled day (Mon. Feb. 1), or move it to a later day.

W 1/27/10 12.5/ 1 (all parts not previously assigned),43-45,46,49-54, 55 but with equation -x-y-z=-1 for the first plane, 59,61,63,65,69-72,73,75

F 1/29/10 12.6/ 11-14,16,17,42-45,47

M 2/1/10
Finish reading Section 12.6 (hopefully you have already started), so that you can do all the assigned exercises below. The way the exercises are grouped makes it difficult to assign many of them usefully until you've read the entire chapter and have seen all the graphs in Table 1 on p. 808. I will go over the hyperboloids and the (double) cone on Monday, but I still want you to take your first crack at the exercises in involving these before Monday.
12.6/ 1-8,9,10,15,21-28,29-36,41

T 2/2/10 In observance of Groundhog's Day, no new homework!

W 2/3/10 Start studying for Friday's, exam, which will cover Chapter 12 but nothing from Chapter 13. I encourage you to look at (among other things) the Review questions at the end of Chapter 12.
The assignment relating to Tuesday's lecture is due Monday 2/8 (posted below).
If you ever want to know how my past classes did on their exams, here is a general way to find out:
Go to my home page, http://www.math.ufl.edu/~groisser
Under ``Course materials'', click on the ``Past classes'' link.
On the Past Classes page, click on the link to whatever class you want to look at. This will take you to the home page for that class, which has links to grade-scale pages. Each grade-scale page has some statistics on each exam, and (in more recent years, but not until the final exam in 2002) a link to the list of scores.

F 2/5/10 First midterm exam (assignment is to study for it)

M 2/8/10 13.1/ 1-6,7,8, 29-32 if you have a calculator or computer that can handle these, 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''.

T 2/9/10 13.1/ 9-12,15-18,19-24,25

W 2/10/10
13.1/ 26-28,36-38
Sketch well the intersection of the sphere x² + y² + z² and the plane y=z (Example 10 from today's class, which I struggled mightily with as a result of not having practiced it before class). The drawings of this circle that I gave in class deserve a grade no better than C. I have now re-done the example in the privacy of my office, and have succeeded in producing a drawing that I would give a B to. You should strive to be less drawing-impaired than your professor, and produce grade-A drawings.

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 Friday 2/12/10.

F 2/12/10
13.2/ 2,3-8,9-16,23-35,41-43,45-48,51.
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.

M 2/15/10
Read all parts of Section 13.2 that you have not read already. I will cover the material on integrals on Monday, but it's straightforward enough that you should be able to do the integral-related exercises in the list below.
13.2/ 1,21,22,33-38,39,40,49
Re-do all the exam problems on which you did not get a perfect score.
The treasure-hunt in exam problem #5 can be generalized to any number of landmarks. If there are n landmarks A₁, A₂, ..., A_n, where n≥ 2, the instructions are:
Step 1. Start at A₁. Walk straight towards A₂, but stop when you are 1/2 of the way there. Call this point B₂. (Set B₁=A₁ if you want to have a point named "B₁".)
Step 2 (if n ≥ 3). Walk from B₂ straight towards A₃, but stop when you are 1/3 of the way there. Call this point B₃.
Step 3 (if n ≥ 4). Walk from B₃ straight towards A₄, but stop when you are 1/4 of the way there. Call this point B₄.
.
.
.
Step n-1. Walk from B_n-1 straight towards A_n, but stop when you are 1/n of the way there. Call this point B_n. This is where the treasure is buried.
Find the (x,y) coordinates of the treasure in terms of the coordinates of the landmarks A₁, ..., A_n. You may yet save yourself from a watery fate.

T 2/16/10 No new homework.
I'm now two weeks behind the lecture schedule I'd originally planned, so I may not be able to have any HW Q&A days for a while. Sorry! Please use my office hours for homework questions.

W 2/17/10
Read the second half of p. 831, through the end of Example 2, so that you know what it means to (re)parametrize a curve by arclength when this comes up in the exercises below.
13.2/ 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.

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 &le 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 &le 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.

F 2/19/10
Read the material on curvature in pp. 832-834. In class I got as far as formula (9) on p. 832. In the book, you'll see that there are other ways to compute curvature that are often more efficient than formula (9). I may or may not have time to go over these in class (or may go over them very briefly).
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. In part (a), do one of the following: (i) Ignore the instruction to find N(t), which we have not yet defined in class. I expect to cover this on Friday, and will assign this part of problems 17-20 once I've covered it. (ii) Read the material in the book on N(t) and do these problems in their entirety.
    If you choose option (i), save your work so that you don't have to redo everything once I assign the remaining part of these problems.
13.3/ 21-25,27-28,30-31,33a,40-42.

In #42 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 #41 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?
(a) Compute the curvature κ(t) for the cycloid in the previous homework assignment, 0 < t < 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⁺ and as t → 2π^– for the cycloid.

M 2/22/10
13.3/ 17-20 (just compute N(t)),45-46 (osculating plane only), 51,52,59 ("length" means "arclength")
13.4/ 3,5,6,10-13,19,22,33-36

T 2/23/10
13.4/ 15,16
14.1/ 6-30. 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.

W 2/24/10
Read the material on level curves, pp. 860-864. In class I neglected to talk about the spacing of level curves in a contour map (what I called in class a topographical map). Usually we plot the level curves for C's with equal increments between successive values (e.g. C = 0,10,20,30,40, ...). The spacing of the level curves then carries information about the steepness of the graph.

14.1/ 31-34,35-38, 39-43,45,46,47,48, 55-60, 61-64,65,66.
   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 C changes.

F 2/26/10 14.2/ 1,2,5-14,16,17,19,20,25,26

M 3/1/10 14.2/ 15,18,21,22,29-38,39-41,46

T 3/2/10
Read the subsections "Higher Derivatives" and "Partial Differential Equations" on pp. 884-886.
14.3/ 15-23,26-30,35,37,39-42,49,50,73-75

W 3/3/10
14.3/ 45,47,51-53,55,57-59,61-64,71,72abd,87,91,95.
    In #55, you should 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).
14.4/ 1-6

F 3/5/10 Second midterm exam (assignment is to study for it)

M 3/15/10 Enjoy your spring break! Think about math the whole time!
Oh, I guess that's redundant.

T 3/16/10
Read Section 14.4. The definition of "differentiable function of two variables" in the book is equivalent to the one I gave in class, but the equivalence is not completely obvious. The book's definition is handy for proving certain theorems, but does not directly tell you what differentiability really means, which I'll be expanding upon more in class Tuesday. The definition I gave in class has other advantages that you aren't yet in a position to appreciate (for example, it generalizes usefully to functions of infinitely many variables, whereas the book's definition does not).
Note on terminology. We saw in class that a function f differentiable at r₀ has exactly one good linear approximation near r₀. This good linear approximation is called the linearization of f at r₀; class ended before I had a chance to tell you that. The book's definition of the linearization of a differentiable function of two variables is equivalent to the one I just gave. However, I do not like putting the definition of "linearization at r₀" before the defintion of "differentiable at r₀"; there is no useful definition of linearization for non-differentiable functions. Also, for some reason the book defines "linearization" only for the two-variable case; the definition above applies to any number of variables.

If your computer or calculator has 3D graphing capability, do 14.4/ 7-8.
14.4/ 42
14.6 (yes, 14.6)/ 37
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.

W 3/17/10 14.4/ 11-16,17-19,21,31-34,38,40. 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.

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.

F 3/19/10 14.6/ 4-6 (here, "in the direction indicated by the angle θ" means "in the direction (cos θ)i + (sin θ)j" ), 7-9,11-13,15,16,21-25,28,29 (insert the words "the direction of" before i + j, since i + j is not a unit vector), 30,31,33,34,36,62 (in part (a), after "all other directions" insert "except for –i and –j").

M 3/22/10
14.5/ 1-6,13,39,41,49,55
(Inspired by a question asked in class.) Let r,θ denote the usual polar coordinates in the xy plane. Recall that (-r,θ+π) represents the same point as (r,θ). Let h be a function of the polar angle θ satisfying h(θ+π) = -h(θ).

Show that the formulas
     f(x,y)= r h(θ)    if (x,y) ≠ (0,0),
     f(0,0)=0,
determine a well-defined function f. (Well-defined means that whichever of the possible pairs of polar coordinates you use for the point (x,y), the formula for f(x,y) yields the same value.)
Show that all directional derivatives of f exist at the origin.
Show that f is differentiable at the origin if and only if h(θ)= a cos θ + b sin θ for some constants a,b (in which case f(x,y) is simply a x + b y).
More generally, show that if g is any function of the radial variable r satsifying g(-r)= -g(r),
the formulas
     F(x,y)= g(r) h(θ)    if (x,y) ≠ (0,0),
     F(0,0)=0,
determine a well-defined function F;
all directional derivatives of exist F at the origin, provided that g'(0) exists;
no directional derivatives of F exist at the origin if g'(0) does not exist; and
F is differentiable at the origin if and only if
g'(0)=0 and the function h is bounded, or
g'(0) exists and is ≠ 0, and h(θ)= a cos θ + b sin θ for some constants a,b.
Hint for the "g'(0)=0 and h is bounded" case: show directly that the constant function 0 is a good linear approximation to F near the origin.

Let α be a real number and define g(r) = r^α for r > 0; g(r) = – | r |^α for r < 0; and g(0)=0. This function satisfies g(-r)= -g(r). For which exponents α and functions h is the function F defined above differentiable at the origin?

T 3/23/10 14.5/ 7-10, 21,22,45,46,50

W 3/24/10 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)

F 3/26/10
Read the material on implicit differentiation in pp. 905-907
14.5/27-34
(Completing an example started in class.) Let F(x,y,z) = x³ + y³ + z³ – xyz and let S be the graph of the equation F(x,y,z) = 1 (a level set of F).
Show that S contains at least one point, so that we're talking about a set that's "really there", unlike (say) the graph of the equation x² + y² = –1.
In class we saw that the equation F(x,y,z) = 1 determines z implicitly as a function of (x,y) near any point (x₀, y₀, z₀) of S for which 3z₀² – xy ≠ 0, but we postponed figuring out whether there are any points of S for which 3z₀² – xy = 0. Show that indeed there are such points, by finding at least one.
    In fact, with a little work, you should be able to show that the set of such points is the union of the graph defined by
x³ + y³ – (2/3^3/2)(xy)^3/2 = 1, z = (xy/3)^1/2 ,
and the graph defined by
x³ + y³ + (2/3^3/2)(xy)^3/2 = 1, z = – (xy/3)^1/2 .
With a little more work you should be able to show that each of these graphs is a smooth curve connecting the point (1,0,0) to the point (0,1,0). (Here, by "a little more", I mean "considerably more".) With some nontrivial work you should be able to show that the union of these two graphs is a smooth, simple closed curve. (Here, by "nontrivial", I mean "I will be impressed if you can do this.")
Show that the only point of R³ at which ∇F = 0 is the origin, and that the origin does not lie on S. From the discussion at the end of Wednesday's class, this implies that S is a smooth surface.
Show that the points of S at which ∂F/∂z = 0 are exactly the points of S at which the tangent plane is vertical. To what extent, if any, does this depend on the particulars of the function F?

M 3/29/10 Read Section 14.7. I'm not assigning any exercises due Monday because not a single one can be done until almost the entire section is covered, which we haven't done yet. However, the assignment due Tuesday will be long, so you may want to try to start it, based just on your reading.

T 3/30/10
14.7/ 1,3,4,5-15,19,20,37,38,39,42-47,50,53,55

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 = 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.

W 3/31/10 Read pp. 928-930 and do exercises 14.7/ 29-36.

F 4/2/10 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.)

General information Day/date for third midterm: Friday, Apr. 9.

M 4/5/10
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.
14.8/ 27,30-35,38,39

T 4/6/10
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
Note: Sections 15.1 and 15.2 are fair game for Friday's exam. As you will see in the exercises above, there are iterated integrals that require substantially more work than the one I did in class Monday. Also, you may sometimes find that you see how to carry out the iterated integration in one order but not the other.

W 4/7/10
15.2/ 25-31,38. Erinn was clairvoyant!
15.3/ 1-6, 7-10.
It is forgivable if you prioritize studying for Friday's exam, which goes only through 15.2, ahead of doing problems from 15.3 tonight. However, I'll be assigning more from 15.3 due Monday, so if you postpone tonight's assignment from 15.3, you'll have a busier weekend. Or a mathier weekend, if you like made-up words.
Note: Wednesday's class will be Q&A, not a new lecture.

F 4/9/10 Third midterm exam (assignment is to study for it)

M 4/12/10
15.3/11-18,19-21,25,28,33,34,39-44 (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), 52,52,55,56
15.3/ 22,25,28. I'm grouping these separately because you will probably find 22 more challenging than 19-21, and 25 and 28 quite a bit more challenging.

T 4/13/10
15.3/ 45-50,58,60,61
Did you finish all the problems due Monday? Yeah, that's what I thought. Try again.

W 4/14/10 Read Sections 15.4 and 15.6. We're going to skip 15.5.

F 4/16/10 15.4/ 1 (Ben's turn to be clairvoyant!), 2-4,5,6,7-11,14,15,16,19,21,22, 25-27,28,29-32, 35, 36d, 37. 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.

M 4/19/10
15.6/ 9,10,12,14,15,17,19,21,22,27,28,29-32,33,34,51,52
Read section 15.7. The method by which we did the last triple integral in class, "integrate with respect to z, then do the remaining xy-integral in polar coordinates," is the same as integration in cylindrical coordinates (with the order of integration the one that's the most common for cylindrical coordinates)—we just didn't define "cylindrical coordinates" before using them. So, combining what we did in class with what you'll read, you should be able to do the exercises below without much difficulty.
15.7/ 1-12,15,16,17

General information The grade scale for the third 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 4/19/10.

T 4/20/10 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, the average value of a function f defined on a solid region D is the integral of f over D, divided by the volume of D. Explain qualitatively why, even before computing the answer to #28, you should expect the answer to be greater than a/2.

W 4/21/10
16.1/ 1-7,11-14,15-18,21-24,25,26
16.2/ 1-6,9,10,12

Do these before final exam, preferably by Fri. 4/23/10
16.2/ 5-8,14-16,17,19-21,39,42
16.3/ 1,27,28

Date due	Section # / problem #s
W 1/6/10	Read the home page and syllabus web pages, and the web handout Taking notes in a college math class. 12.1/ 1-8, 10-18, 22, 23-27. In the first line of #1, "origin" means "origin in R³".
F 1/8/10	12.1/ 30-32,33,35,38 12.2/ 1-3,4abd,5ac,6acde,7,8,11,12,13,14,16,37,39
M 1/11/10	12.2/ 4c,5bd,6bf,17-20,21,22,24-27,28,30,31,38,40-43 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 (hopefully) learned before calculus. Using the algebraic definitions of vector addition and of multiplication by scalars, verify properties 1-8 in the "Properties of Vectors" box on p. 774. For property 1, just do this for n=3 (we did n=2 in class). For the remaining seven properties, do the verification in the n=3 case for three of them (whichever three you want), and in the n=2 case for the other four.
General Information	Something I forgot to mention the first day of class: roughly every fourth or fifth class will be a "homework Q&A" day, in which I work out whatever homework problems you ask me to. The first of these will be Tuesday Jan. 12. Here is a tentative schedule of lectures, which I will revise occasionally.
T 1/12/10	12.2/ 35,36 12.3/ 1,2,3,5-10,13,15,17-19,21,22,51,52. For all exercises in this section that ask you for an approximate number of degrees, you may use your calculator in the final step.
W 1/13/10	12.3/ 53-55,57,58
F 1/15/10	Read Examples 6 and 7 in Section 12.2 (pp. 783-784). This is an application that I did not have time to get to in Wednesday's lecture, and we'll fall farther behind schedule if I spend time on it Friday (Wednesday, we fell behind by some fraction of a lecture). 12.3/ 23,24c,25,27,28,35-37,40,41,42,44,45,49,59,60
T 1/19/10	12.4/ 1-4,7,9-12,13-15,17 (compute both cross-products; don't just compute one and use anticommutativity), 18 (do this by computing all the cross-products, not by using the formula given in class for a × (b × c)), 19, 21, 23-26, 46,47
W 1/20/10	12.4/ 27,29,30,33,35,44 (in part (a), you must assume that Q,R,S are not collinear),45,49
F 1/21/10	12.4/ 28,31,32,34,36,37,38,43 (in part (a), you must assume Q≠R)
M 1/25/10	12.5/ 1abj,2-4,6,8-11,13 (if your answer is "yes", determine whether the lines are distinct or identical), 14,15,17,18,19-22,63,64,66,67,68
T 1/26/10	12.5/ 5,12,16, 23-34,36,37,57,58,62,66,67 On this day, remind me to discuss whether to keep the first midterm on its originally scheduled day (Mon. Feb. 1), or move it to a later day.
W 1/27/10	12.5/ 1 (all parts not previously assigned),43-45,46,49-54, 55 but with equation -x-y-z=-1 for the first plane, 59,61,63,65,69-72,73,75
F 1/29/10	12.6/ 11-14,16,17,42-45,47
M 2/1/10	Finish reading Section 12.6 (hopefully you have already started), so that you can do all the assigned exercises below. The way the exercises are grouped makes it difficult to assign many of them usefully until you've read the entire chapter and have seen all the graphs in Table 1 on p. 808. I will go over the hyperboloids and the (double) cone on Monday, but I still want you to take your first crack at the exercises in involving these before Monday. 12.6/ 1-8,9,10,15,21-28,29-36,41
T 2/2/10	In observance of Groundhog's Day, no new homework!
W 2/3/10	Start studying for Friday's, exam, which will cover Chapter 12 but nothing from Chapter 13. I encourage you to look at (among other things) the Review questions at the end of Chapter 12. The assignment relating to Tuesday's lecture is due Monday 2/8 (posted below). If you ever want to know how my past classes did on their exams, here is a general way to find out: Go to my home page, http://www.math.ufl.edu/~groisser Under ``Course materials'', click on the ``Past classes'' link. On the Past Classes page, click on the link to whatever class you want to look at. This will take you to the home page for that class, which has links to grade-scale pages. Each grade-scale page has some statistics on each exam, and (in more recent years, but not until the final exam in 2002) a link to the list of scores.
F 2/5/10	First midterm exam (assignment is to study for it)
M 2/8/10	13.1/ 1-6,7,8, 29-32 if you have a calculator or computer that can handle these, 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''.
T 2/9/10	13.1/ 9-12,15-18,19-24,25
W 2/10/10	13.1/ 26-28,36-38 Sketch well the intersection of the sphere x² + y² + z² and the plane y=z (Example 10 from today's class, which I struggled mightily with as a result of not having practiced it before class). The drawings of this circle that I gave in class deserve a grade no better than C. I have now re-done the example in the privacy of my office, and have succeeded in producing a drawing that I would give a B to. You should strive to be less drawing-impaired than your professor, and produce grade-A drawings.
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 Friday 2/12/10.
F 2/12/10	13.2/ 2,3-8,9-16,23-35,41-43,45-48,51. 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.
M 2/15/10	Read all parts of Section 13.2 that you have not read already. I will cover the material on integrals on Monday, but it's straightforward enough that you should be able to do the integral-related exercises in the list below. 13.2/ 1,21,22,33-38,39,40,49 Re-do all the exam problems on which you did not get a perfect score. The treasure-hunt in exam problem #5 can be generalized to any number of landmarks. If there are n landmarks A₁, A₂, ..., A_n, where n≥ 2, the instructions are: Step 1. Start at A₁. Walk straight towards A₂, but stop when you are 1/2 of the way there. Call this point B₂. (Set B₁=A₁ if you want to have a point named "B₁".) Step 2 (if n ≥ 3). Walk from B₂ straight towards A₃, but stop when you are 1/3 of the way there. Call this point B₃. Step 3 (if n ≥ 4). Walk from B₃ straight towards A₄, but stop when you are 1/4 of the way there. Call this point B₄. . . . Step n-1. Walk from B_n-1 straight towards A_n, but stop when you are 1/n of the way there. Call this point B_n. This is where the treasure is buried. Find the (x,y) coordinates of the treasure in terms of the coordinates of the landmarks A₁, ..., A_n. You may yet save yourself from a watery fate.
T 2/16/10	No new homework. I'm now two weeks behind the lecture schedule I'd originally planned, so I may not be able to have any HW Q&A days for a while. Sorry! Please use my office hours for homework questions.
W 2/17/10	Read the second half of p. 831, through the end of Example 2, so that you know what it means to (re)parametrize a curve by arclength when this comes up in the exercises below. 13.2/ 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. 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 &le 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 &le 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.
F 2/19/10	Read the material on curvature in pp. 832-834. In class I got as far as formula (9) on p. 832. In the book, you'll see that there are other ways to compute curvature that are often more efficient than formula (9). I may or may not have time to go over these in class (or may go over them very briefly). 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. In part (a), do one of the following: (i) Ignore the instruction to find N(t), which we have not yet defined in class. I expect to cover this on Friday, and will assign this part of problems 17-20 once I've covered it. (ii) Read the material in the book on N(t) and do these problems in their entirety. If you choose option (i), save your work so that you don't have to redo everything once I assign the remaining part of these problems. 13.3/ 21-25,27-28,30-31,33a,40-42. In #42 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 #41 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? (a) Compute the curvature κ(t) for the cycloid in the previous homework assignment, 0 < t < 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⁺ and as t → 2π^– for the cycloid.
M 2/22/10	13.3/ 17-20 (just compute N(t)),45-46 (osculating plane only), 51,52,59 ("length" means "arclength") 13.4/ 3,5,6,10-13,19,22,33-36
T 2/23/10	13.4/ 15,16 14.1/ 6-30. 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.
W 2/24/10	Read the material on level curves, pp. 860-864. In class I neglected to talk about the spacing of level curves in a contour map (what I called in class a topographical map). Usually we plot the level curves for C's with equal increments between successive values (e.g. C = 0,10,20,30,40, ...). The spacing of the level curves then carries information about the steepness of the graph. 14.1/ 31-34,35-38, 39-43,45,46,47,48, 55-60, 61-64,65,66. 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 C changes.
F 2/26/10	14.2/ 1,2,5-14,16,17,19,20,25,26
M 3/1/10	14.2/ 15,18,21,22,29-38,39-41,46
T 3/2/10	Read the subsections "Higher Derivatives" and "Partial Differential Equations" on pp. 884-886. 14.3/ 15-23,26-30,35,37,39-42,49,50,73-75
W 3/3/10	14.3/ 45,47,51-53,55,57-59,61-64,71,72abd,87,91,95. In #55, you should 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). 14.4/ 1-6
F 3/5/10	Second midterm exam (assignment is to study for it)
M 3/15/10	Enjoy your spring break! Think about math the whole time! Oh, I guess that's redundant.
T 3/16/10	Read Section 14.4. The definition of "differentiable function of two variables" in the book is equivalent to the one I gave in class, but the equivalence is not completely obvious. The book's definition is handy for proving certain theorems, but does not directly tell you what differentiability really means, which I'll be expanding upon more in class Tuesday. The definition I gave in class has other advantages that you aren't yet in a position to appreciate (for example, it generalizes usefully to functions of infinitely many variables, whereas the book's definition does not). Note on terminology. We saw in class that a function f differentiable at r₀ has exactly one good linear approximation near r₀. This good linear approximation is called the linearization of f at r₀; class ended before I had a chance to tell you that. The book's definition of the linearization of a differentiable function of two variables is equivalent to the one I just gave. However, I do not like putting the definition of "linearization at r₀" before the defintion of "differentiable at r₀"; there is no useful definition of linearization for non-differentiable functions. Also, for some reason the book defines "linearization" only for the two-variable case; the definition above applies to any number of variables. If your computer or calculator has 3D graphing capability, do 14.4/ 7-8. 14.4/ 42 14.6 (yes, 14.6)/ 37 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.
W 3/17/10	14.4/ 11-16,17-19,21,31-34,38,40. 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.
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.
F 3/19/10	14.6/ 4-6 (here, "in the direction indicated by the angle θ" means "in the direction (cos θ)i + (sin θ)j" ), 7-9,11-13,15,16,21-25,28,29 (insert the words "the direction of" before i + j, since i + j is not a unit vector), 30,31,33,34,36,62 (in part (a), after "all other directions" insert "except for –i and –j").
M 3/22/10	14.5/ 1-6,13,39,41,49,55 (Inspired by a question asked in class.) Let r,θ denote the usual polar coordinates in the xy plane. Recall that (-r,θ+π) represents the same point as (r,θ). Let h be a function of the polar angle θ satisfying h(θ+π) = -h(θ). Show that the formulas f(x,y)= r h(θ) if (x,y) ≠ (0,0), f(0,0)=0, determine a well-defined function f. (Well-defined means that whichever of the possible pairs of polar coordinates you use for the point (x,y), the formula for f(x,y) yields the same value.) Show that all directional derivatives of f exist at the origin. Show that f is differentiable at the origin if and only if h(θ)= a cos θ + b sin θ for some constants a,b (in which case f(x,y) is simply a x + b y). More generally, show that if g is any function of the radial variable r satsifying g(-r)= -g(r), the formulas F(x,y)= g(r) h(θ) if (x,y) ≠ (0,0), F(0,0)=0, determine a well-defined function F; all directional derivatives of exist F at the origin, provided that g'(0) exists; no directional derivatives of F exist at the origin if g'(0) does not exist; and F is differentiable at the origin if and only if g'(0)=0 and the function h is bounded, or g'(0) exists and is ≠ 0, and h(θ)= a cos θ + b sin θ for some constants a,b. Hint for the "g'(0)=0 and h is bounded" case: show directly that the constant function 0 is a good linear approximation to F near the origin. Let α be a real number and define g(r) = r^α for r > 0; g(r) = – \| r \|^α for r < 0; and g(0)=0. This function satisfies g(-r)= -g(r). For which exponents α and functions h is the function F defined above differentiable at the origin?
T 3/23/10	14.5/ 7-10, 21,22,45,46,50
W 3/24/10	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)
F 3/26/10	Read the material on implicit differentiation in pp. 905-907 14.5/27-34 (Completing an example started in class.) Let F(x,y,z) = x³ + y³ + z³ – xyz and let S be the graph of the equation F(x,y,z) = 1 (a level set of F). Show that S contains at least one point, so that we're talking about a set that's "really there", unlike (say) the graph of the equation x² + y² = –1. In class we saw that the equation F(x,y,z) = 1 determines z implicitly as a function of (x,y) near any point (x₀, y₀, z₀) of S for which 3z₀² – xy ≠ 0, but we postponed figuring out whether there are any points of S for which 3z₀² – xy = 0. Show that indeed there are such points, by finding at least one. In fact, with a little work, you should be able to show that the set of such points is the union of the graph defined by x³ + y³ – (2/3^3/2)(xy)^3/2 = 1, z = (xy/3)^1/2 , and the graph defined by x³ + y³ + (2/3^3/2)(xy)^3/2 = 1, z = – (xy/3)^1/2 . With a little more work you should be able to show that each of these graphs is a smooth curve connecting the point (1,0,0) to the point (0,1,0). (Here, by "a little more", I mean "considerably more".) With some nontrivial work you should be able to show that the union of these two graphs is a smooth, simple closed curve. (Here, by "nontrivial", I mean "I will be impressed if you can do this.") Show that the only point of R³ at which ∇F = 0 is the origin, and that the origin does not lie on S. From the discussion at the end of Wednesday's class, this implies that S is a smooth surface. Show that the points of S at which ∂F/∂z = 0 are exactly the points of S at which the tangent plane is vertical. To what extent, if any, does this depend on the particulars of the function F?
M 3/29/10	Read Section 14.7. I'm not assigning any exercises due Monday because not a single one can be done until almost the entire section is covered, which we haven't done yet. However, the assignment due Tuesday will be long, so you may want to try to start it, based just on your reading.
T 3/30/10	14.7/ 1,3,4,5-15,19,20,37,38,39,42-47,50,53,55 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 = 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.
W 3/31/10	Read pp. 928-930 and do exercises 14.7/ 29-36.
F 4/2/10	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.)
General information	Day/date for third midterm: Friday, Apr. 9.
M 4/5/10	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. 14.8/ 27,30-35,38,39
T 4/6/10	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 Note: Sections 15.1 and 15.2 are fair game for Friday's exam. As you will see in the exercises above, there are iterated integrals that require substantially more work than the one I did in class Monday. Also, you may sometimes find that you see how to carry out the iterated integration in one order but not the other.
W 4/7/10	15.2/ 25-31,38. Erinn was clairvoyant! 15.3/ 1-6, 7-10. It is forgivable if you prioritize studying for Friday's exam, which goes only through 15.2, ahead of doing problems from 15.3 tonight. However, I'll be assigning more from 15.3 due Monday, so if you postpone tonight's assignment from 15.3, you'll have a busier weekend. Or a mathier weekend, if you like made-up words. Note: Wednesday's class will be Q&A, not a new lecture.
F 4/9/10	Third midterm exam (assignment is to study for it)
M 4/12/10	15.3/11-18,19-21,25,28,33,34,39-44 (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), 52,52,55,56 15.3/ 22,25,28. I'm grouping these separately because you will probably find 22 more challenging than 19-21, and 25 and 28 quite a bit more challenging.
T 4/13/10	15.3/ 45-50,58,60,61 Did you finish all the problems due Monday? Yeah, that's what I thought. Try again.
W 4/14/10	Read Sections 15.4 and 15.6. We're going to skip 15.5.
F 4/16/10	15.4/ 1 (Ben's turn to be clairvoyant!), 2-4,5,6,7-11,14,15,16,19,21,22, 25-27,28,29-32, 35, 36d, 37. 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.
M 4/19/10	15.6/ 9,10,12,14,15,17,19,21,22,27,28,29-32,33,34,51,52 Read section 15.7. The method by which we did the last triple integral in class, "integrate with respect to z, then do the remaining xy-integral in polar coordinates," is the same as integration in cylindrical coordinates (with the order of integration the one that's the most common for cylindrical coordinates)—we just didn't define "cylindrical coordinates" before using them. So, combining what we did in class with what you'll read, you should be able to do the exercises below without much difficulty. 15.7/ 1-12,15,16,17
General information	The grade scale for the third 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 4/19/10.
T 4/20/10	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, the average value of a function f defined on a solid region D is the integral of f over D, divided by the volume of D. Explain qualitatively why, even before computing the answer to #28, you should expect the answer to be greater than a/2.
W 4/21/10	16.1/ 1-7,11-14,15-18,21-24,25,26 16.2/ 1-6,9,10,12
Do these before final exam, preferably by Fri. 4/23/10	16.2/ 5-8,14-16,17,19-21,39,42 16.3/ 1,27,28

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Homework Assignments MAP 3474, Section 3129 - Honors Analytic Geometry and Calculus III Spring 2010

Homework Assignments
MAP 3474, Section 3129 - Honors Analytic Geometry and Calculus III
Spring 2010