Homework Assignments
MAC 2313, Section 7564 — Analytic Geometry and Calculus III
Fall 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. You are expected to have completed the homework assignment before class on the due date. 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 (Shabanov, Concepts in Calculus III).

Read the corresponding section of the book before working the problems! Here is a very important piece of advice from James Stewart, author of many calculus textbooks (including the one used at UF prior to our current textbook):

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.

Another important bit of advice, in Stewart's words, 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
F 8/24/12	Read the home page and syllabus web pages, and the web handout Taking notes in a college math class. Section 71.8/ 1–4,10
M 8/27/12	Section 71.8/ 5,6(i),6(iii),7. In 6(iii), change "touch" to "just barely touch". (This change should also be made in 6(ii).) Section 72.5/ 1 (all parts except (iv)). The instructions at the top should read: "Find the components and norms of each of the following vectors". Then: Part (i) should read, "The vector corresponding to the oriented segment that has endpoints A=(1,2,3) and B = (–1,5,1), and is directed from A to B." Part (ii) should read, "The vector corresponding to the oriented segment that has the same endpoints as in (i), but is directed from B to A." Part (iii) should read, "The vector corresponding to the oriented segment whose initial point is A and whose final point is the midpoint C of the line segment AB, where the points A and B are the same as in parts (i) and (ii)." In part (v), no rewording is necessary, but there are two vectors with the given properties. Find both of them. Note: Many of your homework assignments will be a lot longer than the ones I've given so far. I don't want anyone to feel after Drop/Add that he/she wasn't warned. Often, most of the book problems in a section aren't doable until we've finished covering practically the entire section, at which time I may give you a large batch to do all at once. Heed the suggestion above the assignment-chart: "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."
T 8/28/12	Section 72.5/ 2-6, 11–14. Challenge problem (optional): #16. Several of these problems need to be reworded or clarified as below.      2. Replace "Let a and b be two vectors" with "Draw two vectors a and b".      3. First sentence should read: "Draw three vectors a, b, and c in a plane, with none of them parallel to either of the others."      4. Replace "via" with "in terms of". (This applies to most or all uses of the word "via" in this textbook.) An example of a vector expressed in terms of a and b is 3a+2b. Note: in general, the phrase "in terms of" has a more general meaning than in the context of this problem.      5. The first sentence on p. 24 should read, "Find b if a is based at the origin, has its terminal point in the first quadrant, makes an angle π/3 with the positive x-axis, and has length 2." There are two vectors b with these properties. Find both of them.      6. "Use vector algebra" here means "Do not resort to writing vectors in component-form; just use properties of vector addition, subtraction, and multiplication by scalars."      11. Insert the word "geometrically" after "Describe". The shape of the given set is one you should know from high school.      14. In the last sentence, change "the condition" to "the most general condition".      16. Reword the problem this way: "A point object travels in the xy plane, starting from an initial point P0. Its trajectory consists of straight line-segments, where the nth segment starts at point Pn-1 and ends at point Pn (for n ≥ 1). When the object reaches Pn, it makes a 90° counterclockwise turn to proceed towards Pn+1. (Thus each segment, other than the first, is perpendicular to the preceding segment.) The length of the first segment is a. For n ≥ 2, the nth segment is s times as long as the (n-1)st segment, where s is a fixed number between 0 and 1 (strictly). If the object keeps moving forever, (a) what is the farthest distance it ever gets from P0, and (b) what is the distance between P0 and the limiting position P∞ that the object approaches?" The hint does not need to be reworded.
W 8/29/12	Section 73.7/ 1–7, 11–13, 17. Below are some additional instructions and rewordings.      1(iii). Replace "the unit vector" with "a unit vector".      3. Express your answer in radians.      6. Change the second sentence to: "Find the values of s for which the dot product a•b is maximal, minimal, or zero, if such values exist." Delete the third sentence.      12. Hint: Letting c=||a|| b + ||b|| a, compare the angle between c and a to the angle between c and b. (non-book problem)   Let a be a nonzero vector. Show that 0 is the only vector that is both parallel and perpendicular to a.
F 8/31/12	Finish reading Section 73.
T 9/4/12	Section 73.7/ 9,10,17. Read Section 74.
W 9/5/12	Section 74.5/ 1–4,7,11,17. Below are some additional instructions, rewordings, and comments. Some of these problems require material that you're supposed to have read, but that I haven't gone over in class yet (as of Tuesday 9/4/12). Note: I will not hold you responsible for the material on torque (the subsection that goes from near the bottom of p. 44 to near the bottom of p. 46).      2. For this assignment, do not use the "bac - cab" rule or the Jacobi identity (11.10) (p. 41); I want you to get practice with the determinant formula for cross-product. Altogether, there are six cross-products you should compute in this problem. Repetition builds retention; don't just tell yourself "I know this" after one or two computations. Once you're done, (i) save your answers for comparison with the next assignment, and (ii) as a consistency check, see whether your answers satisfy the Jacobi identity. If they don't satisfy this identity, then you definitely made a mistake. If they do satisfy this identity, then your answers are probably right, but not definitely right (you could have made errors that happened to cancel each other).      17. In addition to the information given about a, b, and c, assume a≠0. Feel free to start on the problems from Section 74.5 due Friday, based just on your reading. In Wednesday's class, among the things I'll be going over are the norm formula on p. 42, the right-hand rule, area formulas related to cross-product, and the "bac - cab" rule.
F 9/7/12	Section 74.5/ 2 (again), 5,6,11. Below are some additional instructions, rewordings, and comments.      2. For this assignment, do use the "bac - cab" rule (three times). I'll say again: repetition builds retention. Once you're done, (i) compare your answers with the ones you got in the previous assignment. If the answers don't agree, you can again use the Jacobi identity as a consistency check to see which set of answers is more likely to be correct. (Of course, if neither set satisfies the Jacobi identity, then both sets of answers are wrong.)      5. (i) Change "parallel to ê3" to "a positive multiple of ê3". (ii) Add this after the last sentence of the problem: "What if b is a negative multiple of ê3?"      6. None of the octants other than the first octant—the one in which x, y, and z are all positive—have standard numbers or names (in contrast to the quadrants of the xy plane, where the meaning of "quadrant II", "quadrant III", and "quadrant IV" are standard). To specify one of the other seven octants, you have to say something like ''the octant in which x>0, y<0, and z>0.'' In an exercise like this one in which you're going to refer to several octants, you can, if you like say something like this to be more efficient: "We will refer to the octant in which x>0, y<0, and z>0, as the (+, –, +)-octant, etc. for the other octants." But understand that this is just your notation for a specific problem; don't expect anyone else (including your professor) to understand the notation outside this problem unless you define it whenever you use it.
M 9/10/12	Read this handout: Capital Crimes in Calculus III 75.5/ 1–3, 6–8, 10 Read as much of Section 76 as you can, but at least read pp. 65–67 and Study Problem 11.24. 76.5/ 2, 5, 7. In #5, assume a,b and c are all nonzero. Do the following additional problems: Find an equation of the plane that is perpendicular to the line through (1, –1,1) and (2,3,4), and passes through the midpoint of the line segment between these two points. Find an equation of the plane that contains the points (0, –1,2), ( –3, 5, –7), and (11, –13, 17). Consider a general plane with equation n1x + n2y + n3z = d, where (n1, n2, n3) ≠ 0. (a) What is special about this plane, with regard to the coordinate axes, if one of n1, n2, n3 is zero? (b) What is special about this plane, with regard to the coordinate planes, if two of n1, n2, n3 are zero?
T 9/11/12	Finish reading Section 76, if you haven't already. No new exercises. If you are behind at all in the homework, use this opportunity to catch up. You are behind if you have not yet done every single problem that was assigned (except for the one problem I labeled "optional"), and have not read everything in Sections 71–76 other than material that I explicitly said you would not be held responsible for.
W 9/12/12	76.5/ 3, 7, 13 Read Section 77. If you look ahead to the next assignment, you'll see that there are several more problems from sections 76 and 75. These involve lines and/or distances between various objects, which are topics I wanted to cover in class before assigning problems on them. But if you feel ready to start tackling these based on your reading, do so. Otherwise, assuming you're all caught up in this course, use the time to get ahead in your other courses so that you'll have enough time to work on the next few assignments in this class. I expect the assignments due Friday, Monday, and Tuesday to be much longer than the recent assignments, plus you'll want to start reviewing for next week's exam.
F 9/14/12	76.5/ 1, 9. In #9, change "the closest" to "closer". 76.5/ 6, 8. These problems involve lines, but you should be able to do them based on what you've read (which should include Section 77 by now). 75.5/ 11
M 9/17/12	77.6 / 1, 3, 5, 6, 8–10. Below are some additional instructions and rewordings.      3. Delete the perpendicularity question; just find whether the lines intersect, and, if they do, find the point of intersection.      8. Change "perpendicular to the line x=1+t, y= –1+t, z=2 – 2t" to "perpendicular to the vector ⟨1, 1, –2⟩".
T 9/18/12	77.6 / 2, 4 75.5/ 13–15 Non-book problems: Let L be the line described by the vector equation r = ⟨3,2,1⟩ + t ⟨4,5,6⟩ . Find an equation for the plane that contains the line L and the point (1,2,3). Let L1 be the line described by the symmetric equations 2x – 7 = –(y – 3) =(z – 2)/4 , and let L2 be the line described by the parametric equations x = 1+t, y = 2 – 2t, z = 3 + 8t. Describe the relative positions of these lines, and determine whether there is a plane containing both lines. If there is such a plane, find an equation for it. Let L1 be the line parallel to the vector ⟨1, – 1, 2⟩ and passing through the point (1,1,0). Let L2 be the line parallel to the vector ⟨–1, 1, 2⟩ and passing through the point (2,0,2). In problem 77.6/ 10 you showed that these lines intersect at one point. Find an equation of the plane that contains both lines.
W 9/19/12	First midterm exam (assignment is to study for it)
F 9/21/12	Read Section 78.6, and get started on the problems due Monday.
M 9/24/12	78.6/ 1(v),(viii),(x); 2(i); 3(i),(ii); 7, 10. Do as many of the problems due Tuesday as you can, based on your reading.
T 9/25/12	78.6/ 1–12, all problems and problem-parts that were not due Monday, and that you are now able to do.
W 9/26/12	78.6/ 1–12, all problems and problem-parts that were not due Monday or Tuesday.
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 Wednesday 9/26/12.
F 9/28/12	79.5/ 1,2,9,11
M 10/1/12	79.5/ 8,12,13. In #8, replace "at which" with "for which".
T 10/2/12	79.5/ 3,4,5 Read Section 80 through subsection 80.1 (pp. 111-112). 80.5/ 1, 6, 14–16. In #16, at the end of the second sentence, insert "at all t for which r(t) ≠ 0".
W 10/3/12	80.5/ 2,4,5, 7–11. In 5(ii): At the end of the given information, insert "where ω is a positive constant". Notation and pronunciation: ω is the lower-case Greek letter "omega". (The omega used in fraternity names, Ω, is an upper-case omega.) It is not the English/Roman letter "w", and should not be pronounced "double-you", "curly double-you", or "that weird math-letter that looks kinda like a w". Change the z-coordinate of P0 to √3/2 .
F 10/5/12	80.5/ 3,12,13. Make sure you read Example 12.7 before starting #3. In #13, insert the following as a second sentence: "Assume that, among the points on the curve, there is one that is closest to the plane." Read Section 81 through the first line on p. 121. 81.4/ 1, 10. For #10, change "maximal" to "minimal". Also, speed is the norm of the velocity vector. This problem does not involve integrals; it really belongs in Section 80.
M 10/8/12	81.4/ 2–5,7 82.3/ 1 (all parts except (v)), 2–6. In 1(vi), change the y-component of r(t) to –sin(t)+t cos(t). I suggest ignoring the hint; for the r(t) in this problem it does not save time.
T 10/9/12	82.3/ 8 (all parts except (iii)), 9. Below are some additional instructions, rewordings, and comments. 8(iv). Restrict the t-interval to [0, 2π], and assume z(t)=0. 9. Assume that the helix is right-handed. (Otherwise there are two answers). Note: the answer does not simplify to anything nice. Look back at your work for 82.3/ 1, and in each case identify the velocity and the speed. Also compute the unit tangent vector field T(t) and the acceleration. (All of these are functions of t. Make sure you are clear on which is/are vector-valued functions and which is/are scalar-valued functions. "T" should have a hat over it, but I haven't figured out how to get it there in HTML.) For the parametrized curves in 82.3/ 1(i),(vi), (iii), and (iv), re-express T(t) as a function of s, where s(t) is the arclength function based at t=0 (i.e. measured from r(0)) in the direction of increasing t. I have ordered these "(i), (vi), (iii), (iv)" because this is the order of increasing difficulty. In each case, part of the work involves computing the inverse function t(s). Hint for part (iv): if you have computed s(t) correctly, you should be able to re-write the equation "s = (your formula for s(t))" as a quadratic equation in the quantity et. Solve this equation for et in terms of s, then take natural log of both sides to get t(s). Additional note for part (iv): when you express ||r'(t)|| as a function of s, the expression will initially look very messy, but it simplifies. Cutoff on new material for Friday's exam. In Tuesday's lecture, I will start Section 83, and expect to get at least as far as Theorem 12.7. As of now, I am not planning to include the content of pp. 140–145 as possible material for Friday's exam. If I get through Theorem 12.7 and examples quickly enough, I will jump over the rest of Section 83 for now, and cover the first half-page of Section 84 (through Definition 12.17). Friday's exam will go no further than this. After Tuesday's lecture I will be able to tell you how much of Sections 83 and 84 will be fair game for this midterm. Note for anyone who was not present at the end of Monday's class: Tomorrow (Tues. Oct. 9) I won't be able to hold my morning office hour, but I will have an office hour after class instead. I will also have my usual office hour Wednesday afternoon, and an additional office hour Wednesday morning (time TBA).
W 10/10/12	83.3/ 1 parts (i), (vi), (ix), and (x). For now, delete the words "and the curvature radius at a specified point P". In each case, determine whether there are any points at which κ(t) is undefined (in addition to finding a formula for κ(t) wherever it is defined). For the parametrized curves in 82.3/ 1(i),(ii), compute the curvature function κ(t).
General information	No material from Section 84 will appear on Friday's exam. The only part of Section 83 you're responsible for (for this exam) is the material up to but not including Theorem 12.7.
F 10/12/12	Second midterm exam (assignment is to study for it)
M 10/15/12	Read Section 83 from Theorem 12.7 through at least the first paragraph on p. 141. Wherever you see the phrase "curvature radius", it should be "radius of curvature". 83.3/ 1 (all parts except (viii)), 2, 9. For problem 1, change the instructions to: "Find the curvature of each of the following parametrized curves as a function of the parameter t, and find the radius of curvature at the indicated point P." For part (x) of problem 1, after computing κ(t), compute ρ(t), the radius of curvature as a function of t, for every t except the one value t0 that corresponds to the indicated point P. Then determine whether limt → t0 ρ(t) exists, and if so, what the value of the limit is.
T 10/16/12	83.3/ 1 (all parts except (viii)). This includes re-doing the parts that you did for the previous assignment; do them using Theorem 12.7 and compare them to your previous answers. If you now get different answers, at least one of your two computations has a mistake, which you should try to find. Again, change the instructions to the ones indicated for the previous assignment. 84.5/1 (all parts except (v)), 2,3,6,8. In the instructions for #1, there should be a comma after the word "time", and "a specified point" should be "the specified point".
W 10/17/12	Read Corollaries 12.1 and 12.2 on p. 140. I don't want to spend more time on curvature in class, but you will need this material to do several homework problems. Read the Remark on p. 153. 83.3/ 3–8, 11, 12. In #4, set a = 1.
General information	We are skipping most of Section 84. This material is lovely and is worth learning, but we will not have time for it. I will not hold you responsible for: the material in subsection 84.1 beyond Definition 12.18; subsection 84.3; the Study Problems in subsection 84.4 beyond Problem 12.17; or the "find the torsion" part of Problem 12.17.
F 10/19/12	85.6/ 1, 3, 4(i)–(v)
M 10/22/12	85.6/ 2, 5 Re-read the handout Taking notes in a college math class. Re-read the handout Capital Crimes in Calculus III. Re-read the Homework section of the syllabus.
General information	The grade scale for the second midterm is now posted on your grade-scale page. Exams will be returned Monday 10/22/12.
T 10/23/12	Read Section 86. In Definition 13.8, delete the word "corresponding". This word suggests that there's only one δ that works for each ε, which is false. Absolutely critical typo: on p. 27, in the second line of the Remark, the part in italics should begin with, "it does NOT depend". 86.4/ 1–3, 4(i)–(v),(vii),(viii),5(i)–(ii),6. Below are some additional instructions, rewordings, and comments. 1. In the instructions, change the part in parentheses to "i.e., for each ε > 0, find a δ (in terms of ε) satisfying the condition required in Definition 13.8". There is always more than one possible δ for each ε. For a given ε, if one δ works, so does any smaller, positive δ. For a given ε, usually there is a largest possible δ that works (call it δmax(ε)), but to establish that an asserted limit-statement is true, it is never necessary to find δmax(ε), so it's usually a waste of time to bother trying. Bearing in mind that this means that there's more than one correct answer to each part of problem 1, here are some correct answers some parts of this problem. (i) δ = ε/10. (iii) δ = min(1, 3ε/10), where min(a,b) is whichever of a and b is smaller (or just a if a = b). (v) δ = min(1, ε/12). 2. Change the instructions to: "Use the Simplified Squeeze Principle to prove the following limit-statements." 4. Answers to some parts: (i) (0,0). (ii) No points of discontinuity. (v) No points of discontinuity. (vii) No points of discontinuity. (viii) (a,a) for every real a ≠ 0. 5. In the second sentence, change "at which" to "for which". For (ii), the hint at the end of problem 2 is useful.
W 10/24/12	Read Section 87, skipping the subsection 87.5 and the material on "Repeated Limits" on p. 186. Don't skip the Study Problems (subsection 87.6). If you are ready to start tackling some problems, try the first eight parts of 87.7/ 2.
F 10/26/12	Redo 86.4/ 1, using the squeeze principle instead of finding a δ in terms of ε. You should find this easier than the first way you did these problems.    Note: a student told me yesterday that when I did part (ii) of this problem in class, somehow the "3x2 + 4y2" got changed to "x2 + y2". This was an unintentional "blackboard typo". For this example, the δ I found was still valid, because 3x2 + 4y2 ≥ x2 + y2 (a larger denominator means a smaller fraction: if A,B,C are positive numbers, and B > C , then A/B < A/C). Figure out why this inequality ensures that the δ I found (in terms of ε) still "works", i.e. satisfies the criterion given in Definition 13.8 (p. 175). The analysis would also have been valid if I had used the fact that 3x2 + 4y2 ≥ 3x2 + 3y2 = 3(x2 + y2); this would have led to a different, but valid, δ in terms of ε. However, applying the inequality 3x2 + 4y2 ≤ 4x2 + 4y2 would have been useless. (Figure out why.) 87.7/ 2(i)–(viii), 7(i),(ii),(v). Here are some hints for #2 and answers for #7. You should not look at these until you think you have solved the problem, or have gone through each step Shabanov lays out in Section 87.7 and still have gotten nowhere. Do not just look at a problem for a few minutes, decide "I don't know how to do this," and look at the hint; you will learn nothing that way. You need to experiment by trial and error, getting nowhere with some approaches until you find one that works, in order to develop some intuition about which limits exist and which don't. For the limit in 87.7/ 7(i), figure out how the answer changes (if at all) if x2 + y2 + z2 is replaced by each of the following: (a) x3 + y3 + z3 ; (b) x4 + y4 + z4 ; (c) x5 + y5 + z5 ; (d) x2 + y3 + z4 ; (e) x3 + y4 + z5 ; (f) x4 + y5 + z6 ; (g) x5 + y6 + z7 . If you have trouble, first do a simpler two-variable version of this problem by just erasing all the z's, then put the z's back in.
M 10/29/12	Read Section 88. 88.4/ 1 (all parts except (iii)), 2(i)–(vi), (viii). Compute the partial derivatives of f(x,y) = √x2 + y2 and g(x,y,z) = √x2 + y2 + z2. What are the domains of these partial derivatives?
T 10/30/12	89.4/ 1,2, 3(i)–(ii), 10. Below are some additional instructions and rewordings.     1. Here, "verify Clairaut's Theorem" means "see from your formulas that f''yx = f''xy."     2. In the second line, "that" should be "why".     10. The end of the first sentence should be "and f(0,0) = 0."
W 10/31/12	Read Section 90, with emphasis on the parts that I didn't get to in class on Tuesday (or got to only briefly): the last two lines of p. 215 through the end of p. 218. 90.5/ 1–4, 5(i)–(iv), 6, 7(i)–(ii), 8(i). Below are some additional instructions, rewordings, and comments.     2. A function g is called bounded if there is some number M such that |g(r)| ≤ M for all r in the domain of g. More generally, if S is a subset of the domain of g, we say that g is bounded on S if here is some number M such that |g(r)| ≤ M for all r in S.        The second sentence of the instructions means: "Determine whether the partial derivatives are continuous at (x_0,y_0) for (a) (x_0,y_0) ≠ (0,0) and (b) (x_0,y_0) = (0,0)."     5. Change the instructions to: "Find the largest domain in which the functions are continuously differentiable." Definition: a function is continuously differentiable on an open set D if its partial derivatives exist at every point of D, and are continuous at every point of D. The terminology "continuously differentiable" is justified because of Theorem 13.12: if a function meets the criteria to be called "continuously differentiable on D", then the function is differentiable on D.
F 11/2/12	No new homework
M 11/5/12	No new homework
T 11/6/12	Read Section 91, and Section 92 through Example 13.33. I plan to skip the rest of section 92 (meaning that you will not be responsible for it). 91.4/ 1–5, 9(i)(ii)(vi), 18, 19. Below are some reworded instructions.     9. Change the instructions to: "Find the first and second partial derivatives of the function g in terms of the first and second partial derivatives of the function f."     18. Change the instructions to: "Assume in each case below that the given equation determines z implicitly as a function of x and y. Find the first partial derivatives of this function z(x,y)."     19. Change the first sentence of the instructions to: "Let z(x,y) be the continuous function of x and y determined implicitly by the equation z3 – xz + y = 0 such that z(3, –2) = 2." 92.6/ 1
W 11/7/12	Read Section 93. 93.4/ 2,6,15,16. Below are some reworded instructions.     6. Change instructions to: "If f is a differentiable real-valued function of one real variable, and u is a differentiable real-valued function of m real variables, show that ∇(f(u(r)) = f '(u(r)) ∇u(r), where r = ⟨x1,x2, ..., xm⟩."     16. In parts (i) and (ii), change "the z-axis" to "the vector ê3".
T 11/13/12	93.4/ 3,6,14,18–20,24. Below are some reworded instructions.     3. The last sentence of the instructions should say: "Find the directions in which the function does not change to first order (i.e. the directions in which the linearization of the function does not change)." Read Section 94.1 (pp. 257–260). Skip Section 94.2. In Section 94.3, read Corollary 13.4 and Examples 13.43 and 13.44. In Corollary 13.4, "the hypotheses of Theorem 13.19" that are relevant are: "Let r0 be a critical point of a function f. Suppose that the second-order partial derivatives of f are continuous in an open set containing r0." 94.5/ 1(i)–(vi). Change the instructions to: "For each of the following functions, find all critical points. Determine which critical points are degenerate and which are nondegenerate. For each nondegenerate critical point, determine whether that point is a saddle point, a point at which a relative maximum is achieved, or a point at which a relative minimum is achieved. For each degenerate critical point, try to determine whether that point is a saddle point, a point at which a relative maximum is achieved, or a point at which a relative minimum is achieved."     Definition: A critical point (x0, y0) of a function f(x,y) with continuous second partial derivatives is degenerate if D(x0, y0) = 0, and nondegenerate if D(x0, y0) ≠ 0, where D(x,y) = fxx(x,y) fyy(x,y) – fxy(x,y)2.     Thus, degenerate critical points are exactly the ones for which the second-derivative test in Corollary 13.4 is inconclusive.     Definition: A saddle point of a function f is a critical point at which f does not achieve a local maximum or minimum value.     Thus, case (3) of Corollary 13.4 could also be worded: "If D < 0, then r0 is a saddle point of f." Redo part (iv) of the problem above with the last term of f(x,y) (the "y" at the end of the formula) replaced by y√2.
W 11/14/12	94.5/ 1(ix)–(xiv),(xxi). Change the instructions as in the previous homework assignment. Same instructions as the problems above, but for f(x,y) = xy – 2x2 – xy3. For this function, you should find that there are six critical points: (0,0), (1/2, 0), (0,1), (0,–1), all four of which are saddle points; (1/5, 1/√5), at which f has a relative minimum; (1/5, –1/√5), at which f has a relative maximum.
General information	The material for Friday's exam runs from the middle of Section 83 (Theorem 12.7) through the end of Section 94, minus anything that we officially skipped (see the "General information" between the assignments with due-dates 10/17/12 and 10/19/12, and the assignments due 10/24/12, 11/6/12, and 11/13/12). Of course, you should always be prepared to combine older material with the newer material to do some problems.
F 11/16/12	Third midterm exam (assignment is to study for it)
M 11/19/12	Read subsections 95.2 and 95.3. 95.5/ 6,8,10,l1
General information	The grade scale for the third midterm is now posted on your grade-scale page. Exams will be returned Monday 11/19/12.
T 11/20/12	Read Section 97
M 11/26/12	No homework due, but you may want to get started on the homework due Tuesday; it will be a long assignment.
T 11/27/12	Read Section 99 97.4/ 5(i),(ii) 98.1/ 1(i),(ii),(iv); 7(i). In 1(ii) assume k ≥ 0 for the volume interpretation. 99.3/ 1 (all parts except (ix), (xv)) 100.6/ 1; 2(i)–(iii),(v)–(viii),(x); 3(i),(ii),(v); 4(i),(ii)
W 11/28/12	100.6/ 5(i)–(viii) Read Section 101, but see correction below. 101.2/ 1; 2(i)–(iii); 3,5,6; 7(i),(ii); 8(i); 9(i)–(iii) Correction to line 2 of Section 101. The definition of θ is technically incorrect. Although Shabanov explains what he means by "θ = tan–1(y/x)", this is a highly non-standard meaning of the notation used (although it is geometrically well-motivated in this situation). The notation "tan–1" (or "arctan") has a universally accepted meaning: it is the inverse of the function obtained by restricting the function "tan" to the interval (–π/2, π/2). This restricted tangent-function has domain (–π/2, π/2) and range (–∞, ∞); correspondingly, the function "tan–1" has domain (–∞, ∞) and range (–π/2, π/2). Since ∞ is not a number, it is not in the domain of tan–1, and the expression tan–1(∞) is not defined. It is true that limu→∞ tan–1u = π/2, and that limu→ –∞ tan–1u = –π/2 . However, it is completely incorrect to say that "y/x = ∞ on the positive y-axis" or that "y/x = –∞ on the negative y-axis". It is completely incorrect to write "1/0 = ∞" or "–1/0 = –∞". 1/0 is no more "equal" to ∞ than it is "equal" to –∞. If we take a sequence of points in the xy plane approaching (0,1), then depending on the sequence, y/x may approach ∞, –∞, or neither. This is one reason that the expression "1/0" is undefined; it is not ±∞. It is not equal to anything. Polar coordinates used for integration are a restricted version of what you learned in Calculus 2. For purposes of integration, we always require r ≥ 0, and we usually require that θ take its values in an interval of length 2π, the most commonly-used intervals being [0,2π), [–π, π), [0,2π], and [–π, π]. If we use the either of the first two intervals, then every point in the xy plane, other than the origin, has exactly one pair of polar coordinates (r,θ). If we use the interval [0,2π], then each point (x,0) on the positive x-axis has two pairs of polar coordinates: (r = x, θ = 0) and (r = x, θ = 2π); all other points besides the origin have exactly one pair of polar coordinates. If we use the interval [–π, π], then each point (x,0) on the negative x-axis has two pairs of polar coordinates: (r = |x|, θ = –π) and (r = |x|, θ = π); all other points besides the origin have exactly one pair of polar coordinates. No matter what interval we choose, the origin always has infinitely many pairs of polar coordinates: (r = 0, θ = any value in the chosen interval). Fortunately, none of these coordinate-duplications has any effect on integration, because the set of points with more than one pair of polar coordinates has zero area. Therefore there is never any harm in using the closed interval [0,2π] or [–π, π] for θ in integration, and these choices make formulas look less strange than if we used half-open intervals. However, if we wish to assign every point of the xy plane, other than the origin, a unique pair of polar coordinates, we have to use a half-open interval such as [0,2π), or [–π, π). When we are using [–π, π) as our θ-interval, the correct formula for θ in terms of x and y is: θ = tan–1(y/x) if x > 0 (i.e. in quadrants I and IV, with the y-axis excluded) θ = π/2 if x = 0 and y > 0 (i.e. on the positive y-axis) θ = –π/2 if x = 0 and y < 0 (i.e. on the negative y-axis) θ = tan–1(y/x) + π if x < 0 and y > 0 (i.e. in the interior of quadrant II) θ = tan–1(y/x) – π if x < 0 and y ≤ 0 (i.e. in quadrant III, with the negative y-axis excluded) When we are using [0, 2π) as our θ-interval, the correct formula for θ in terms of x and y is: θ = tan–1(y/x) if x > 0 and y ≥ 0 (i.e. in quadrant I, with the positive y-axis excluded) θ = π/2 if x = 0 and y > 0 (i.e. on the positive y-axis) θ = 3π/2 if x = 0 and y < 0 (i.e. on the negative y-axis) θ = tan–1(y/x) + π if x < 0 (i.e. in quadrants II and III, with the y-axis excluded) θ = tan–1(y/x) + 2π if x > 0 and y < 0 (i.e. in the interior of quadrant IV) Note that none of the cases above define θ at the origin. As mentioned above, at the origin, θ is allowed to have all values in the chosen interval.
F 11/30/12	103.6/ 1(i)–(iv),(viii),(xi); 2(i),(ii),(x); 4, 5(i). Below are some additional instructions, rewordings, and comments.     1(ii) and (iii). Usually Shabanov writes the inequalities describing simple regions in the order opposite to what I've been using in class; his inequalities, when read from left to right, are like mine, when read from bottom to top. This is the case in (iii). However, in (ii), the order is not consistent with either this rule or the rule I use. My ordering of these inequalities would be:        0 ≤ z ≤ 1        0 ≤ x ≤ z        0 ≤ y ≤ x + z     1(iii). The integral should have a "dV" at the end.     5(i). The instructions mean: express the given iterated integral as an iterated integral five other ways. Note: We will not be covering Section 102.
M 12/3/12	Read Section 104. 104.5/ 1, 2, 3 except part (vi), 4, 5, 6 except part (vi), 7–9, 11(i). Below are some corrections and rewordings.     1(i) and (iii). There is a "≤" missing after θ.     3(i). There are two regions bounded by the given cylinder and sphere: E1, the region that is inside both surfaces (bounded on the sides by the cylinder, and bounded on the top and bottom by the sphere), and E2, the region that is outside the cylinder but inside the sphere. Compute the indicated integral both for E=E1 and for E=E2.     11(i). Reword the instructions as: "Assume b > a > 0. Find the volume of the region that is inside the cone z = √ x2 + y2 and between the spheres x2 + y2 + z2 = a2 and x2 + y2 + z2 = b2."
T 12/4/12	104.5/ 3(vi), 6(vi)
W 12/5/12	Read Section 107. (We're skipping Sections 105 and 106.) This is the last section you'll be responsible for on the final exam.
As soon as you can get these done.	107.3/ 1(i) – (viii),(x),(xiv),(xvi),(xvii); 2,3. In 1(x), one parametrization of the astroid is x = acos3t, y = asin3t, 0 ≤ t ≤ 2π. While there is no due-date for these exercises, the material is fair game for the final exam. The earlier you get these exercises done (or attempted), the earlier you'll be able to ask me about any difficulties you may have with them.
End-of-semester office hours	After Dec. 5, I will not have office hours at times I had them earlier in the semester. I will have the following office hours: Fri. 12/7, approximately 4:00-5:00, in our usual classroom (following the review session) Mon. 12/10, 11:45 a.m. - 1:15 p.m. Tues. 12/11, 11:45 a.m. - 1:15 p.m. Wed. 12/12, 11:45 a.m. - 1:15 p.m. If something unexpected comes up, I may have to change some of these hours, so re-check this page before coming to see me.
F 12/7/12	Review (Q&A) at 3:00 in our usual classroom.
W 12/12/12 (no more dates like that till the year 2101!)	FINAL EXAM begins at 5:30 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. If you have a question about your grade or would like to look at your graded exam, please see me in my office after the start of Spring semester. I will post my office hours when I know them.