Homework Assignments
MAP 3473, Section 3205 - Honors Analytic Geometry and Calculus II
Fall 2011

Last update made by D. Groisser Mon Dec 19 08:28:07 EST 2011

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

Date due Section # / problem #s

T 8/23/11
Read the home page and syllabus web pages, and the web handout Taking notes in a college math class.
6.1/ 1,3-5,8,9,15,19-21,26,28,31,33,40,44. Read the book, especially p. 419, before starting these exercises. For some of these, the region is most easily expressed as lying between two graphs of the form x=f(y), x=g(y), rather than as lying between the graphs of functions of x.

W 8/24/11
6.1/ 45ab,49,53
6.2/ 2-4,6,7,9,11,13,19,20, 28,31-35. Some of these problems involve cross-sections that are "washers" (the region between two concentric circles) instead of disks; see pp. 426-428.
6.ProblemsPlus/ 7,9. ("Problems Plus" are at the end of each chapter.)

F 8/26/11
6.2/ 41-44,46, 50-52,54,56,65,68,71
6.3/ 1-3,7,10,12,16,21,37-39. Note: I did not give a good answer on Wednesday to the question, "When is it better to use the cylindrical-shell method, rather than the cross-section method (disks and `washers'), to compute volumes of solids of revolution, or vice-versa?" There definitely are examples in which it's clear from the start that one method would be much easier than the other. For the region in Exercise 1, revolved about the y-axis, it's easy to compute the volume using cylindrical shells, and tricky to do it using cross-sections ("washers" in this case). For essentially the same reason, if the same region is revolved around the x-axis, it's easy to compute the volume using cross-sections (disks in this case). and tricky to do it using cylindrical shells.
   Tricky does not mean impossible. However, the trick in this case is to use a clever substitution (along with integration-by-parts, which you're not expected to know yet; that's Section 7.1) that essentially turns the integral you'd use for one method into the integral you'd use for the other.

M 8/29/11
6.3/ 5,22,26,28,29-32,40-42,43,44,46
6.ProblemsPlus/ 10

T 8/30/11
6.4/ 1-3, 5, 8, 15, 16 (assume the size of the bucket is negligible), 21, 27, 29
6.5/ 1-7, 9, 17, 18b (assume 18a), 23

W 8/31/11
7.1/ 1-6,12,19,22,25,31,33,34
∫ x^-1ln(x) dx can be done easily by substitution (what substitution?). It can also be done via IBP, but something interesting happens. Try it and see. Once you've seen what happens, can you think of other integrals for which the same thing would happen?

F 9/2/11
7.1/ 9-11,13-15,24,36

T 9/6/11
7.1/ 44-52,54,68a-d
Since Tuesday will be a HW Q&A session, we won't start covering the Section 7.2 techniques in class until Wednesday (unless you run out of questions before Tuesday's class ends). But to keep us on pace, and to avoid a triple-length assignment due Friday, I'm assigning you to read Section 7.2 and to start working on exercises there:
7.2/ 1-6,15,17,18,55,56

W 9/7/11
7.2/ 7-14,19-27,33,35,37
Let a > 0, f(t)= cos²(at). Find the average value of f over a cycle. In this case a "cycle" means an interval of the form [t₀, t₀+π/a], since f is periodic with period π/a.
Repeat the previous exercise with f(t)= sin²(at).

F 9/9/11
7.2/ 43-45, 53 (just evaluate the integral; don't bother graphing),58,59,62,63, 67-69,70. In #59, the icon next to the problem number means that this problem involves using a graphing calculator, but this is not a problem anybody should need a calculator for. BEFORE you do anything with your calculator, graph the function without that crutch. After you've done that, you may use a calculator to see whether your graph was correct.
  FYI: the sum in #70 is a "sine-only" finite Fourier series. General Fourier series, both finite and infinite, involve both sines and cosines.

M 9/12/11
7.3/ 1-3,4-10,12,22,33,40

Challenge problem (optional; no due date): 7.ProblemsPlus #3. Major hint: In class, several lectures ago, I very briefly referred to something I called the "desperation substitution". Get desperate. After that, you will still need to be clever; the desperation substitution doesn't solve the problem all by itself, and I've given you the only hint I'm going to give.
Work with classmates on this problem if you like, but you're not allowed to ask anyone outside the class for help. If you solve the problem, see how much you can generalize it. (For what other integrals would the method you used give the answer, and what would that answer be?)

T 9/13/11
7.3/11,13-19,23,24,26,27,29,30,34,43 (assume R > r)

W 9/14/11 Study for Friday's exam. The HW from the portion of Section 7.4 that I'd intended to covered in Tuesday's class is posted below, with due date Monday 9/19.

F 9/16/11 First midterm exam

M 9/19/11 Read Section 7.4 through Example 3, p. 476, and do the following problems:
7.4/ 7,9,10,12-14,16-18,47,48-50,63

T 9/20/11 Read "Case II" on pp. 476-477, and do the following problems. Identify which of the problems below fall into the book's "Case I" or "Case II" (and which fall into neither). Do those that can be done by the methods for these two cases. Those that can't be done by these two cases will be part of the next HW assignment, so feel free to read the rest of the section and get started on these early.
7.4/ 1-4,6,8,15,23,25-28,62,64

W 9/21/11
Do all the problems listed in the previous assignment that did not fall into "Case I" or "Case II".
7.4/ 29-32,34-37,39,40,42,46,50,57,58

F 9/23/11
7.5/ 2,3,6,7,9,12,14,18,22-24,35 (with the right observation, this can be done in 10 seconds, which would not be true if x⁸ were replaced by x⁷), 41,42,44,47 (hint: partial fractions come to mind, but there is a more efficient way to do this problem)

M 9/26/11
Read Sections 7.6 and 7.7.

General information The grade-scale page for your class now exists.

T 9/27/11 Re-do every problem on the exam that you did not get completely correct, until you're able to do it completely correctly.

W 9/28/11
Derive explicit formulas for cosh^–1x and tanh^–1x.
Derive formulas for sinh(A+B) and cosh(A+B) that are analogous to the formulas for sin(A+B) and cos(A+B). ("Derive" does not mean "look up". Fiddle with the algebra till you get something that works.)
7.3/ 31,32
7.4/ 53 (assume m≠0)
Read Section 7.8. Don't worry, I will cover this in class.

F 9/30/11
7.8/ 1-3, 5-23, 42, 71

M 10/3/11
7.8/ 24-40,45,46,60,61 [already done in class in answer to a question, but do it again yourself],74,75,78

T 10/4/11
7.8/ 55,56,58,59,79
Determine for which values of p the integral ∫₁^∞ (ln x)/x^p dx converges, in each of the following ways:
by making the (inverse) substitution x=1/y and using your answer to exercise 59;
by making the (inverse) substitution x=e^y.

Redo #59 by using the (inverse) substitution x=e^-y and using the integral you came up with in part 2 of the problem above.
7.Review/ 49,50
Use the definitions of improper integrals to give careful derivations of the linearity and additivity properties I stated in class, for "Type I" and "Type II" integrals.
Compute the value of ∫₀¹ (1/√x(1-x) ) dx.

W 10/5/11
7.8/ 49-54,80
Determine whether ∫₁^∞ (sin x)/x^3/2 dx converges, or whether you haven't learned enough techniques to tell.
From problems 7.Review/ 1-48, select three that would scare you if you were pop-quizzed on them, and do them. (If nothing would scare you, pretend.)

F 10/7/11
Re-examine the integrals in 7.8/ 8,11,17,18,23,26-33. Just using the comparison tests we've talked about, your knowledge of "1/x^p" integrals, try to determine which integrals converge and which diverge. I can answer the convergence/divergence question for all of these without taking any antiderivatives, and I want you to be able to too. This time around, don't try to evaluate the integrals that converge.

M 10/10/11
Same instructions as in previous assignment, but now for the integrals in 7.8/ 13,20,22,39,40, with the the growth hierarchy of functions added to your arsenal. (The growth hierarchy is useful for only a handful of the book's integrals, because Stewart has omitted this topic, but it's very useful in "real life" improper integrals.)
Some facts about the "In a battle, the stronger function wins" principle:
It applies only when there is a battle (e.g. in a product in which one factor is approaching infinity while the other is approaching zero). For example, as x→∞, there is a battle between numerator and denominator in e^x/x (the numerator is trying to make the fraction approach infinity, while the denominator is trying to make the fraction approach zero). But as x↓ 0, there is no battle between numerator and denominator in e^-x/x (the denominator is trying to make the fraction go to infinity, and the numerator isn't trying to stop it).
In the context of improper integrals that are improper for only one reason (infinite interval of integration, or exactly one bad endpoint), "winning" means: the integral behaves (i.e. converges or diverges) the way the stronger function wants it to. We'll see why on Monday, but for the current HW assignment, just take this on faith.
In 7.8/ 40 there is no battle in the integrand (make sure you understand why), so at first glance it's not clear how to get any mileage out of the the growth hierarchy. Here's a hint: compare the integrand to x^–3/4. If you see how to do the problem with this hint, next figure out what numbers other than "3/4" would have worked in this hint.

Re-re-examine 7.8/ 11, 13, 22, 23, and also #s 8 and 26 but with the interval of integration changed to (-∞ , ∞). For each of these six integrals, the interval of integration is the "symmetric" interval (-∞ , ∞), and the integrand is either an even function or an odd function. (Recall that a function f is even if f(–x)=f(x), and odd if f(–x)=& –f(x). Most functions are neither even nor odd.) For each of these integrals, one of the following three statements applies:
The integral converges, and its value is zero.
The integral converges, and its value is not zero.
The integral diverges.
Without taking any antiderivatives, determine which statement applies to each integral.

In 7.8/ 71, using your new wisdom on the growth hierarchy of functions, figure out what the domain of F is if f is any polynomial.
7.8/ 72. For those of you who eventually take Differential Equations (MAP 2302), you will see the Laplace transform again. These last two exercises will, I hope, make the topic a little less mysterious the next time you see it, and give you a leg up on the competition.

T 10/11/11
8.1/ 1-6,7-12,14,17 (the integral in #17 is way, way harder than the others; you are not going crazy),19,33,42.
Follow-up to #17: figure out why the arclength of the curve in this problem is the same as the arclength of the graph of y=ln(x) over the interval 1≤ x ≤ e. You should be able to do this even without being able to do problem 17.

W 10/12/11
7.8/ 63
(Optional) If your calculator does definite integrals, enter the Arc Length Contest at the top of p. 532. No time-limit, other than that anything you hand in has to be handed in by the last day of class (Dec. 7). The prize will be bragging-rights, probably worth millions on Ebay.

F 10/14/11
8.2/ 1-4, 5-8, 15,16,25,26,29,35. If you have trouble sleeping at night after doing #25, don't worry; that's a normal reaction. It will subside after a couple of years (maybe sooner, if your professor remembers to discuss this paradox in class).
   Note (added several days after assignment): the book's answer to 29b is wrong—in fact it's impossible, since the setup of the problem said "a > b", so b²–a² <0).

M 10/17/11 No new homework. Start preparing for Wednesday's exam.

T 10/18/11 Your choice: do the Section 8.5 problems listed below with a due-date of Friday, and have no new HW that you have to do for Friday, or use the early part of the week to prepare for the exam, and do the Section 8.5 problems on Wednesday or Thursday (or early enough on Friday).
Hmm (me thinking out loud), I wonder which option most students will choose ...

W 10/19/11 Second midterm exam (assignment is to study for it)

F 10/21/11
Read Section 8.5
8.5/ 1-5

M 10/24/11 For those of you who left before I noticed I'd made a mistake (you didn't leave early; I corrected myself late), the correct definition of "standard deviation" for a general probability distribution is the formula in exercise 8.5/18. In its definition of both the mean (p. 557) and the standard deviation of a general probability distribution, the book's author forgets to say, "provided the integral exist" (i.e. converges, in the truly-improper-integral case). The mean of a pdf f exists only if the integral in the formula "μ = ..." on p. 557 exists. The standard deviation exists only if the mean exists and the integral in 8.5/18 exists.

8.5/ 6,7.
Show that, for the probability density function in 8.5/5, the mean does not exist.
Come up with a probability density function for which the mean exists, but the standard deviation does not.
Assume that f is a probability density function for which both the mean and standard deviation exist. Show that the integral in 8.5/18 (whose square root is taken to define the standard deviation) equals (∫_-∞ ^∞ x²f(x) dx) – μ². (That's an integral from –∞ to ∞; I don't know how to get the formatting to look right in HTML.) I.e., show that ∫_-∞ ^∞ (x – μ)²f(x) dx = (∫_-∞ ^∞ x²f(x) dx) – μ².
Let f be a probability density function whose graph is "triangle-shaped", like the one in 8.5/8, but generalized by replacing the numbers 0,6,10 on the x-axis with a,b,c respectively, and with "0.2" on the vertical axis replaced by 2/(c – a) (so that the area of the triangle is 1).
Show that the mean of f is (a+b+c)/3.
Using the previous problem (the one referring to the integral in 8.5/18) to simplify the computation, show that the square of the standard deviation is (1/18)(a² + b² + c² – ab – ac – bc).

T 10/25/11
11.1/ 1,2,5,6,9-14,15,16,19-30,32-36,37 (be clever!), 45, 54,55,57,58. (I know this is a lot of problems, but you should find most of them very short.)
Let r₁ = (1+√5)/2, r₂ = (1-√5)/2. Define a sequence {a_n} by
a_n = (1/√5) (r₁ⁿ - r₂ⁿ),     n≥ 1.
Calculate a_n for 1≤ n ≤ 5. If your calculation is correct, you will find that, magically, these all work out to be integers, and in fact are the first five terms of one of the sequences used as an example in Monday's class. Prove that the sequence above is identical to that sequence from class for all n≥ 1.

General information The grade-scale page for your class has been updated with Exam 2 information. The exam will be returned Tuesday at the end of class.

W 10/26/11
11.1/ 71,72,74-77. For 72b, set your calculator in "radians" mode, enter "1", and press the "cos" button repeatedly until the display stabilizes. This is the limit of the sequence {a_n}, to within the precision of your calculator's display. Write down or store your answer. Then enter your UFID number, and again press the "cos" button repeatedly until the display stabilizes. It will be the same limit that you found the first time. You don't yet have the tools (and won't, unless you take Advanced Calculus) to explain why these sequences converge, but you should be able to come up with an explanation of why, if they converge, they must converge to the same limit.
You can play the same game with the sine function, but the limit is boring.
Re-do every problem on the exam that you did not get completely correct, until you're able to do it completely correctly.

F 10/28/11 Read any part of Section 11.1 you haven't read yet. After that, if you feel ready, get a head-start on the problems due Monday.

M 10/31/11
11.1/ 59,60-66,67-69,81. Hint for finding the limits in 67-69 and 81: use 72a.
11.1/ 79. After you've done part (c): let AGM(a,b) denote the arithmetic-geometric mean of a and b. Show that for all c >0, AGM(ca,cb)=c AGM(a,b). Use this to show that
AGM(a,b)=b AGM(a/b,1).
Therefore all properties of the two-variable AGM function can be deduced from properties of the one-variable function f(x)=AGM(x,1).
If you have a programmable calculator (or access to a device that can run mathematical programs) and enjoy playing with it, write a program to compute AGM(a,b) by computing as many terms of the sequences as necessary to get a_n – b_n < 10^–10. (Even if you don't enjoy playing this way, if your major's requirements include a course in programming in a scientific language, like C, C++, or MATLAB, this will be good practice for you.) Include an instruction to display the a_n and b_n each time the next of these numbers is computed. When you run the program, just set b=1, since we saw above that we have the simple relation AGM(a,b)= b AGM(x,1), where x=a/b. Run your program for a few values of x.
The sequences {a_n}, {b_n} converge amazingly fast. (That's why I'm suggesting you use as tiny a tolerance as 10^–10, so you can see that you get super-fast convergence to a very, very precise number.)
Let g(x)=(π/2)x/ln(x). Compute AGM(x,1)/g(x) for some VERY large values of x; you might want to start at around x=10⁷⁰. You should see that the ratio of these two functions is close to 1. It is a true fact that lim_{x→ ∞} AGM(x,1)/g(x)=1, but this limit is approached v-e-r-y s-l-o-w-l-y (an interesting contrast to the rapid convergence of the sequences used to compute AGM(x,1) for each individual x). Thus, AGM(x,1) is asymptotic to (π/2)x/ln(x) as x→ ∞.
The function x/ln(x) comes up elsewhere in mathematics: the Prime Number Theorem asserts that if N(x) denotes the number of prime numbers less than x, then N(x) is asymptotic to x/ln(x) as as x→ ∞.

11.1/ 80. The conclusion of part (b) is a fun fact. However, I could not figure out exactly how the author wanted you to use part (a) to compute the limit in part (b). Maybe you will see a shortcut that I missed. But in case not, here is one way to do the problem:
First, show that if a_n < √2 then a_n+1 > √2, and that if a_n > √2 then a_n+1 < √2. (In other words, the sequence keeps hopping from one side of √2 to the other.) Use this to show that all of the odd-numbered a_n are less than all of the even-numbered a_n.
   (Thank you, Russell, for showing me how to complete the square-root symbol in HTML!)
Second, show that a_n+2=(4+3a_n)/(3+2a_n).
Third, show that the odd-numbered subsequence {a₁, a₃, a₅, ... a_2n+1, ...} is an increasing sequence, and that the even-numbered subsequence {a₂, a₄, a₆, ... a_2n, ...} is a decreasing sequence. Then use the first step to deduce that both subsequences are bounded. Since they are monotonic, they therefore converge.

Let L_odd, L_even be the limits of the odd- and even-numbered subsequences. Using the second step, show that each of these limits satisfies the equation x=(4+3x)/(3+2x). Conclude from this that both limits are √2.
Now use part (a).

BOO!!!

T 11/1/11
11.2/ 1,2,9,10,11,12,14-16,19,20,65,75

W 11/2/11
11.2/ 21,22,35,37,38,40,42,46,47-51,55,57,59,63,68-71

M 11/7/11
11.2/23-34
11.3/1-8. For #8, the Integral Test is applicable but silly to use, in view of the Test for Divergence (which, as I mentioned in class, is often called the "n^th-term test", and for which I prefer the name "n^th-term screen") on p. 692.

T 11/8/11
Read the note I added at the end of the previous assignment. I meant to include it when I posted the assignment.
It's okay if you spread the rest of the assignment over two days, but you should do at least a significant portion before Tuesday's class.

11.3/ 9-26, 27,29,30. Remember that to apply the Integral Test in the form that's on p. 699 (essentially the same as the form given in class), you have to know that f is monotonically non-increasing on [1,∞). Recall from Calculus 1 that this condition is met if f '(x) ≤ 0 for all x ≥ 1. To do the problems above correctly, using only the tools presented up through Section 11.3, you need to verify monotonicity, either by this first-derivative criterion or by some other method. For the problems in this part of the assignment, "some other method" is the easiest approach except perhaps in #s 16, 18-20, 24, and 26; for these others, you should use the first-derivative criterion above. (In #20 you can simplify this to: use first-derivative considerations to show that the denominator is monotonically increasing on [2,∞).) Make a mental note of what a royal pain this can be, so that you'll better appreciate the comparison tests we'll be covering in a few days (which will make monotonicity-testing unnecessary).

The hypotheses given in class under which the integral test is applicable were: Assume that f is a continuous, non-negative, monotonically non-increasing function on [1,∞). (These are also the hypotheses in the textbook, p. 699, except that Stewart says "decreasing" where I say "non-increasing". "Non-increasing" is more general, but in practice most of the series to which you apply the integral test will involve a strictly decreasing f.) Explain why the hypotheses can be weakened to, "Assume that f is continuous, non-negative, monotonically non-increasing function on [N,∞) for some N," without altering the conclusion. (In other words, verify what Stewart said in the Note after the box on p. 699.) Then:
Guess a likely reason that, in #24, Stewart started the sum at n=3.
Given that Stewart started the sum in #24 at n=3, figure out what he probably overlooked in #20.
Figure out why the weakened-hypotheses version of the Integral Test above is sufficient to handle #20, and would have been sufficient to handle the series in #24 even had it started at n=1.

11.3/ 32,34. In these, you may use a calculator to do arithmetic. In #34, replace the requirement "correct to three decimal places" to "correct to within 10^–3 ". There is often no way of guaranteeing that an estimate is literally correct to three decimal places without knowing additional information. For example, if the true value of some quantity were between 0.1239991 and 0.1239999, and you estimated the value to be 0.124, the error would be less than 10^–6 and yet the third decimal place of your estimate would still be wrong.

W 11/9/11 In honor of the Gators having finally won a game, there is no new homework.

M 11/14/11
11.4/1-21,24,26 (I know that's a lot, but your experience from improper integrals should make most of these pretty short),37,38,40,42.

T 11/15/11
11.4/31,39,45,46. Hint for #39 and #46: the theorem on p. 692 is relevant.
11.5/ 2-20,35. Some comments:
Even more strongly than usual, I recommend reading the corresponding section of the book before starting these problems, since we spent only fifteen minutes in class Tuesday on Section 11.5, and you could probably use the reinforcement.
For some of these (e.g. 8,10,11) the method used in the book's Example 3 is useful. For 8 and 10, the masochists among you should take this advice literally. Non-masochists may want to make use of these observations: (i) a positive function is increasing/decreasing if and only if its square is increasing/decreasing (relevant to #8), and (ii) for a function f defined on [0,∞), f(x) is increasing/decreasing if and only if f(x²) is increasing/decreasing. These are special cases of the following (figure out why):
If g and h are functions for which the range of h is contained in the domain of g (so that the composition gοh is defined), and one of g,h is an increasing function, then the composition gοh is increasing/decreasing if and only if the other of g,h is increasing/decreasing. (Here (gοh)(x)=g(h(x)). I can't find a plain small circle in HTML to represent the composition symbol.

The exercise-set 2-20 is not a very rich source of examples. (After you've done them, see if you can figure out why I say this.) If I were writing a set of exercises for this chapter section, I'd put down maybe the first 10 of these, then give exercise 35, then give an exercise in which the series to test has {a_n = 1/n for odd n, a_n = –2/n for even n}. Then I'd give a longer follow-up set that includes the remainder of the 2-20 group, interspersed with some alternating series that diverge despite passing the n^th-term screen.

W 11/16/11
11.5/ 1,23,24,32-34,36. In 36a, I think you will find it easier to establish the desired formula if you write it, equivalently, as s_2n + h_n = h_2n. (I found that that helped me, at least.)
Read the material on the Root Test in section 11.6. I won't be going over this in class. Examples in which the Root Test is more useful than the Ratio Test tend to be contrived, and rarely arise in practice.
11.6/ 1-17, 27,28,29,31

F 11/18/11
Read Section 11.7. I also won't be going over that section explicitly in class but it will come up (probably) in Q&A when you ask me how to do various problems. Your exam-questions won't tell you which convergence test to use, so you'll want to practice figuring this out; that's what the exercises in 11.7 are for.
11.7/ 1-30

M 11/21/11 Third midterm exam (assignment is to study for it)

T 11/22/11
11.8/ 3-7,12-17. For now, modify the instructions to, "Find the set on which the Ratio Test guarantees absolute convergence." I will soon will re-assign these (as part of a larger assignment) with the instruction, with the book's instructions, "find the interval of convergence", restored, so hold onto your work on these problems (which will end up being most of the work needed to do the problems as worded in the book). The set on which a power series converges absolutely is, of course, a subset of the domain on which it converges (period), i.e. the set I called the "domain of convergence" in Wednesday's class. (Sometimes this subset is the whole domain of convergence, sometimes not.) My "domain of convergence" is exactly the book's "interval of convergence"; the latter is the standard term for this set. After we cover the theorem on p. 725 in class (which will probably be Tuesday), I'll start using the "interval of convergence" terminology, but I have my reasons for not wanting to use that terminology prior to covering that theorem.

W 11/23/11 For purposes of this assignment, assume the theorem on p. 725, even though I didn't finish proving it today. (The rest of the proof will take only a few minutes on Wednesday.)

11.8/ 2,3-7,12-24,33,35a,36,37,40,42. In 3-7 and 12-17, in the previous assignment you effectively computed the interior of the interval of convergence. (The interior is the open interval (a – R, a + R) if R, the radius of convergence, is positive and finite; the interior is the whole real line (–∞,∞) if the radius of convergence is ∞.) All that remains in these problems is to determine, in the finite-R case whether the series converges at one endpoint, both endpoints, or neither endpoint. (For 18-24, you have to start from scratch, since these weren't part of the previous assignment.)

M 11/28/11
11.8/ 29, 30, 32, 41
11.9/ 1-12,13,14,15,17,21,32,33a,34,35,36,38. Notes:
In the exercises for Section 11.9, all power series are power series in x; i.e., centered at 0.
Before doing #13, read Example 5.
Hint for #35b: using what you found in part (a), compute the derivative of e^–xf(x). (If you covered Taylor series in high school, you may be tempted to say, ' "Isn't the formula for f(x) just the Taylor series for e^x?" to which the answer is "Yes, but that's not the point of this problem." Taylor series are introduced in the next section of the book, 11.10. The intent of this problem is to have you show a different way, without the not-yet-introduced Taylor series, that the given power series converges to e^x.) My hint is not the only no-Taylor-series way to do this problem; if you think of another, that's fine.
Note that the series in #36 is not a power series (it's an example of something called a Fourier series), so statements that we made in the context of power series (like "domain of convergence is an interval" and "term-by-term differentiation is valid on the interior of the interval/domain of convergence") need not apply. This example illustrates that power series have some special properties not shared by all series of functions.

General information The grade-scale page for your class has been updated with Exam 3 information. The exam will be returned Monday at the end of class. Five minutes before the end of class, when it becomes obvious that I've forgotten all about handing back the exam, you may remind me.

T 11/29/11
11.9/ 16,18,23-25,27. Notes:
In #24, the integrand has the formula written in the problem for t≠0, and is defined at t=0 by taking the limit of the given formula as t→0. A similar comment applies to #25.
In #27, replace "to six decimal places" by "to within 10^–6 ", for the reason mentioned a few homeworks ago.

11.10/ 1 (find formula in terms of derivatives of f), 2,3,4, 5-12, 13-17, 70.
Some guidance for #70:
First show by induction that for x ≠ 0, and all n ≥1, f⁽ⁿ⁾(x) is e^–1/x² times a polynomial in 1/x.
Next show that lim_{x→ 0} f⁽ⁿ⁾(x) =0 for all n≥ 0. Hint: let y=1/x. To use this substitution, you will need to address the limits as x→ 0 from the left and right separately, but it's the same work for both cases.
Next show that f⁽ⁿ⁾(0)=0 for all n≥ 0.
From the preceding, it follows that every coefficient in the Taylor series of f is zero, so the series converges to 0 for all x. But f(x)≠ 0 for x≠ 0. Thus, for all x≠ 0 this Taylor series converges to a value different from f(x).

Show that if a power series Σ_n≥0 a_n xⁿ converges to zero for all x in an open interval centered at 0, then all the a_n must be 0. (Here "Σ_n≥0" means the sum from n=0 to ∞; my attempt to write the usual notation in HTML ended up looking like this: Σ_n=0^∞.) Note: "But wouldn't all the a_n have to be zero for this to happen?" is not an answer; it's a restatement of the question. "I don't see how a power series could converge to zero throughout an entire interval without all the coefficients being 0," is also not an answer. There's no such thing as "proof by lack of imagination".
You have all the preparation needed to do the "non-book" part of the assignment due Wednesday; start on it if you have time. Despite the space it takes up on this webpage, it shouldn't take you long to do, but I wasn't sure if including it in the assignment due Tuesday would overload you.

W 11/30/11
11.10/ 29,31,48,63-68,69
Let I be an open interval (possibly the whole real line) centered at 0. Recall that a function g defined on I is called even if g(–x) = g(x), and odd if g(–x) = –g(x). (The terminology comes from the behavior of integer powers: xⁿ is an even function of x if n is even, and is odd if n is odd. But unlike integers, most functions are neither even nor odd.)
Below, "function" always means "function defined on I".

(a) Show that the only function that is both even and odd is the constant function 0.
(b) Show that the sum or difference of two even functions is even, and that the sum or difference of two odd functions is odd.
For any function f on I, define f_ev(x) = (f(x)+f(–x))/2 and f_odd(x) = (f(x)–f(–x))/2.

(c) Show that f_ev is even, that f_odd is odd, and that f = f_ev + f_odd.
(d) Show that if f = g+h, where g is an even function and h is an odd function, then g = f_ev and h = f_odd. (Hint: use parts (a) and (b).) Thus f can be expressed as the sum of an even and an odd function in only one way. For this reason we can unambiguously refer to f_ev and f_odd as the even and odd parts of f, respectively.
(e) Show that cosh and sinh are the even and odd parts of the exponential function.
(f) Suppose f(x) = Σ_n≥0 a_n xⁿ on I. Show that f_ev(x) = Σ_{n even} a_nxⁿ and f_odd(x) = Σ_{n odd} a_nxⁿ. (HTML notation-problem again. In the first series, the sum is over all even n≥ 0; in the second, the sum is over all odd n≥ 1.)
(g) Use parts (e) and (f) to write down the Taylor series, based at 0, of cosh and sinh. They should look very similar to a couple of other series you know, but the sign-differences you'll observe make a world of difference.
Note: the Taylor series of cosh and sinh can be derived very simply from the definition of "Taylor series", without going through all the steps above. Parts (e) and (g) are included only to be illustrative, giving you some concrete examples pertaining to evenness/oddness of functions and how these properties are reflected in Taylor series.

F 12/2/11
11.10/33-34,38, 55-57,58,59 (read Example 12, p. 745, first), 61.

M 12/5/11
11.10/25-28,35,45,53,71.
10.1/ 1,2,5-7,10,11-14,19,22,27,34. In 1, 2, and 27 your sketches don't have to be very accurate, but should at least start and end at the correct points, go through the correct quadrants in the correct order, and have the correct extreme values of x and y.

T 12/6/11
10.2/ 1-4,7,9,12,14,15,29 (note: "points" here mean an ordered pairs (x,y), not values of t), 30,31,33, 37 (but using the instructions for 41-44 instead of the instructions for 37-40), 41,45,57 (just set up the integral). For 12-15 and 31 on, you'll need to read parts of Section 10.2 that I haven't covered yet (the equation at the top of p. 631, and the material on areas, arclength, and surface area in pp. 632-635). I'll go over some of this briefly on Tuesday, but probably not all of it.

W 12/7/11
10.3/ 1,3,5,7-9,11-12,21-26,27-28,29-33, 57,58,61,62,63,64,77.
   To do 57-64, first read "Tangents to Polar Curves", pp. 644-645, but don't get hung up on Example 9, part of which is done poorly. The key idea is that we can express x and y parametrically in terms of θ (i.e., θ plays the role played by t in Sections 10.1-10.2). Do not waste your time trying to memorize formula 3 on p. 644. The only real use of this general formula is to obtain information about tangent lines at the origin (if the curve happens to pass through the origin), using the last equation on p. 644 and the reasoning in the first three lines of p. 645. Do understand the Note at the top of p. 646; this is the way you should proceed in general (except for the use of the double-angle formula, which is not essential and is very unhelpful for Example 9.) In other words, the best way to do Example 9 is:
Figure out that x(θ) = cos(θ) + sin(θ)cos(θ), y(θ) = sin(θ) + sin²(θ).
Compute dx/dθ and dy/dθ, and use dy/dx = (dy/dθ)/(dx/dθ). Don't bother with the double-angle formula; write dy/dx as the last fraction preceding part (a) on p. 645. Beyond that step, proceed the way the book does.
   One further problem with Example 9: Figure 15 is misleading. The "pointy part" at the origin should be infinitely sharp (the curve has a single, vertical tangent line there, not the two oblique tangent lines that the picture suggests you'd have).

before final

10.3/ 34,37,39,41,42,43. In #43, note that on any θ-interval on which sin(2θ) is negative, there are no values of r that satisfy the equation, and hence no points on the graph for that θ-interval.
10.4/ 45,46,48,55 ("Formula 10.2.7" means the formula on p. 635 labeled by a boxed, colored "7").

The following curves are mentioned in your textbook. Which has the best name?

lemniscate
limaçon
astroid
cardioid
conchoid
cycloid
trochoid
strophoid
cissoid of Dioclese
nephroid of Freeth
oval of Cassini
witch of Maria Agnesi
hippopede

Thurs. 12/15/11 FINAL EXAM begins at 5:30 p.m. in our usual classroom.

General information The grade-scale page for your class has been updated with final exam and course-grade information. If you have a question about your grade or would like to look at your graded exam, please DO NOT EMAIL ME THE QUESTION; please see me in my office after the start of Spring semester. Tentatively, my spring office hours will be the same as they were in the fall. I will not send or discuss grades by email.

Date due	Section # / problem #s
T 8/23/11	Read the home page and syllabus web pages, and the web handout Taking notes in a college math class. 6.1/ 1,3-5,8,9,15,19-21,26,28,31,33,40,44. Read the book, especially p. 419, before starting these exercises. For some of these, the region is most easily expressed as lying between two graphs of the form x=f(y), x=g(y), rather than as lying between the graphs of functions of x.
W 8/24/11	6.1/ 45ab,49,53 6.2/ 2-4,6,7,9,11,13,19,20, 28,31-35. Some of these problems involve cross-sections that are "washers" (the region between two concentric circles) instead of disks; see pp. 426-428. 6.ProblemsPlus/ 7,9. ("Problems Plus" are at the end of each chapter.)
F 8/26/11	6.2/ 41-44,46, 50-52,54,56,65,68,71 6.3/ 1-3,7,10,12,16,21,37-39. Note: I did not give a good answer on Wednesday to the question, "When is it better to use the cylindrical-shell method, rather than the cross-section method (disks and `washers'), to compute volumes of solids of revolution, or vice-versa?" There definitely are examples in which it's clear from the start that one method would be much easier than the other. For the region in Exercise 1, revolved about the y-axis, it's easy to compute the volume using cylindrical shells, and tricky to do it using cross-sections ("washers" in this case). For essentially the same reason, if the same region is revolved around the x-axis, it's easy to compute the volume using cross-sections (disks in this case). and tricky to do it using cylindrical shells. Tricky does not mean impossible. However, the trick in this case is to use a clever substitution (along with integration-by-parts, which you're not expected to know yet; that's Section 7.1) that essentially turns the integral you'd use for one method into the integral you'd use for the other.
M 8/29/11	6.3/ 5,22,26,28,29-32,40-42,43,44,46 6.ProblemsPlus/ 10
T 8/30/11	6.4/ 1-3, 5, 8, 15, 16 (assume the size of the bucket is negligible), 21, 27, 29 6.5/ 1-7, 9, 17, 18b (assume 18a), 23
W 8/31/11	7.1/ 1-6,12,19,22,25,31,33,34 ∫ x^-1ln(x) dx can be done easily by substitution (what substitution?). It can also be done via IBP, but something interesting happens. Try it and see. Once you've seen what happens, can you think of other integrals for which the same thing would happen?
F 9/2/11	7.1/ 9-11,13-15,24,36
T 9/6/11	7.1/ 44-52,54,68a-d Since Tuesday will be a HW Q&A session, we won't start covering the Section 7.2 techniques in class until Wednesday (unless you run out of questions before Tuesday's class ends). But to keep us on pace, and to avoid a triple-length assignment due Friday, I'm assigning you to read Section 7.2 and to start working on exercises there: 7.2/ 1-6,15,17,18,55,56
W 9/7/11	7.2/ 7-14,19-27,33,35,37 Let a > 0, f(t)= cos²(at). Find the average value of f over a cycle. In this case a "cycle" means an interval of the form [t₀, t₀+π/a], since f is periodic with period π/a. Repeat the previous exercise with f(t)= sin²(at).
F 9/9/11	7.2/ 43-45, 53 (just evaluate the integral; don't bother graphing),58,59,62,63, 67-69,70. In #59, the icon next to the problem number means that this problem involves using a graphing calculator, but this is not a problem anybody should need a calculator for. BEFORE you do anything with your calculator, graph the function without that crutch. After you've done that, you may use a calculator to see whether your graph was correct. FYI: the sum in #70 is a "sine-only" finite Fourier series. General Fourier series, both finite and infinite, involve both sines and cosines.
M 9/12/11	7.3/ 1-3,4-10,12,22,33,40 Challenge problem (optional; no due date): 7.ProblemsPlus #3. Major hint: In class, several lectures ago, I very briefly referred to something I called the "desperation substitution". Get desperate. After that, you will still need to be clever; the desperation substitution doesn't solve the problem all by itself, and I've given you the only hint I'm going to give. Work with classmates on this problem if you like, but you're not allowed to ask anyone outside the class for help. If you solve the problem, see how much you can generalize it. (For what other integrals would the method you used give the answer, and what would that answer be?)
T 9/13/11	7.3/11,13-19,23,24,26,27,29,30,34,43 (assume R > r)
W 9/14/11	Study for Friday's exam. The HW from the portion of Section 7.4 that I'd intended to covered in Tuesday's class is posted below, with due date Monday 9/19.
F 9/16/11	First midterm exam
M 9/19/11	Read Section 7.4 through Example 3, p. 476, and do the following problems: 7.4/ 7,9,10,12-14,16-18,47,48-50,63
T 9/20/11	Read "Case II" on pp. 476-477, and do the following problems. Identify which of the problems below fall into the book's "Case I" or "Case II" (and which fall into neither). Do those that can be done by the methods for these two cases. Those that can't be done by these two cases will be part of the next HW assignment, so feel free to read the rest of the section and get started on these early. 7.4/ 1-4,6,8,15,23,25-28,62,64
W 9/21/11	Do all the problems listed in the previous assignment that did not fall into "Case I" or "Case II". 7.4/ 29-32,34-37,39,40,42,46,50,57,58
F 9/23/11	7.5/ 2,3,6,7,9,12,14,18,22-24,35 (with the right observation, this can be done in 10 seconds, which would not be true if x⁸ were replaced by x⁷), 41,42,44,47 (hint: partial fractions come to mind, but there is a more efficient way to do this problem)
M 9/26/11	Read Sections 7.6 and 7.7.
General information	The grade-scale page for your class now exists.
T 9/27/11	Re-do every problem on the exam that you did not get completely correct, until you're able to do it completely correctly.
W 9/28/11	Derive explicit formulas for cosh^–1x and tanh^–1x. Derive formulas for sinh(A+B) and cosh(A+B) that are analogous to the formulas for sin(A+B) and cos(A+B). ("Derive" does not mean "look up". Fiddle with the algebra till you get something that works.) 7.3/ 31,32 7.4/ 53 (assume m≠0) Read Section 7.8. Don't worry, I will cover this in class.
F 9/30/11	7.8/ 1-3, 5-23, 42, 71
M 10/3/11	7.8/ 24-40,45,46,60,61 [already done in class in answer to a question, but do it again yourself],74,75,78
T 10/4/11	7.8/ 55,56,58,59,79 Determine for which values of p the integral ∫₁^∞ (ln x)/x^p dx converges, in each of the following ways: by making the (inverse) substitution x=1/y and using your answer to exercise 59; by making the (inverse) substitution x=e^y. Redo #59 by using the (inverse) substitution x=e^-y and using the integral you came up with in part 2 of the problem above. 7.Review/ 49,50 Use the definitions of improper integrals to give careful derivations of the linearity and additivity properties I stated in class, for "Type I" and "Type II" integrals. Compute the value of ∫₀¹ (1/√x(1-x) ) dx.
W 10/5/11	7.8/ 49-54,80 Determine whether ∫₁^∞ (sin x)/x^3/2 dx converges, or whether you haven't learned enough techniques to tell. From problems 7.Review/ 1-48, select three that would scare you if you were pop-quizzed on them, and do them. (If nothing would scare you, pretend.)
F 10/7/11	Re-examine the integrals in 7.8/ 8,11,17,18,23,26-33. Just using the comparison tests we've talked about, your knowledge of "1/x^p" integrals, try to determine which integrals converge and which diverge. I can answer the convergence/divergence question for all of these without taking any antiderivatives, and I want you to be able to too. This time around, don't try to evaluate the integrals that converge.
M 10/10/11	Same instructions as in previous assignment, but now for the integrals in 7.8/ 13,20,22,39,40, with the the growth hierarchy of functions added to your arsenal. (The growth hierarchy is useful for only a handful of the book's integrals, because Stewart has omitted this topic, but it's very useful in "real life" improper integrals.) Some facts about the "In a battle, the stronger function wins" principle: It applies only when there is a battle (e.g. in a product in which one factor is approaching infinity while the other is approaching zero). For example, as x→∞, there is a battle between numerator and denominator in e^x/x (the numerator is trying to make the fraction approach infinity, while the denominator is trying to make the fraction approach zero). But as x↓ 0, there is no battle between numerator and denominator in e^-x/x (the denominator is trying to make the fraction go to infinity, and the numerator isn't trying to stop it). In the context of improper integrals that are improper for only one reason (infinite interval of integration, or exactly one bad endpoint), "winning" means: the integral behaves (i.e. converges or diverges) the way the stronger function wants it to. We'll see why on Monday, but for the current HW assignment, just take this on faith. In 7.8/ 40 there is no battle in the integrand (make sure you understand why), so at first glance it's not clear how to get any mileage out of the the growth hierarchy. Here's a hint: compare the integrand to x^–3/4. If you see how to do the problem with this hint, next figure out what numbers other than "3/4" would have worked in this hint. Re-re-examine 7.8/ 11, 13, 22, 23, and also #s 8 and 26 but with the interval of integration changed to (-∞ , ∞). For each of these six integrals, the interval of integration is the "symmetric" interval (-∞ , ∞), and the integrand is either an even function or an odd function. (Recall that a function f is even if f(–x)=f(x), and odd if f(–x)=& –f(x). Most functions are neither even nor odd.) For each of these integrals, one of the following three statements applies: The integral converges, and its value is zero. The integral converges, and its value is not zero. The integral diverges. Without taking any antiderivatives, determine which statement applies to each integral. In 7.8/ 71, using your new wisdom on the growth hierarchy of functions, figure out what the domain of F is if f is any polynomial. 7.8/ 72. For those of you who eventually take Differential Equations (MAP 2302), you will see the Laplace transform again. These last two exercises will, I hope, make the topic a little less mysterious the next time you see it, and give you a leg up on the competition.
T 10/11/11	8.1/ 1-6,7-12,14,17 (the integral in #17 is way, way harder than the others; you are not going crazy),19,33,42. Follow-up to #17: figure out why the arclength of the curve in this problem is the same as the arclength of the graph of y=ln(x) over the interval 1≤ x ≤ e. You should be able to do this even without being able to do problem 17.
W 10/12/11	7.8/ 63 (Optional) If your calculator does definite integrals, enter the Arc Length Contest at the top of p. 532. No time-limit, other than that anything you hand in has to be handed in by the last day of class (Dec. 7). The prize will be bragging-rights, probably worth millions on Ebay.
F 10/14/11	8.2/ 1-4, 5-8, 15,16,25,26,29,35. If you have trouble sleeping at night after doing #25, don't worry; that's a normal reaction. It will subside after a couple of years (maybe sooner, if your professor remembers to discuss this paradox in class). Note (added several days after assignment): the book's answer to 29b is wrong—in fact it's impossible, since the setup of the problem said "a > b", so b²–a² <0).
M 10/17/11	No new homework. Start preparing for Wednesday's exam.
T 10/18/11	Your choice: do the Section 8.5 problems listed below with a due-date of Friday, and have no new HW that you have to do for Friday, or use the early part of the week to prepare for the exam, and do the Section 8.5 problems on Wednesday or Thursday (or early enough on Friday). Hmm (me thinking out loud), I wonder which option most students will choose ...
W 10/19/11	Second midterm exam (assignment is to study for it)
F 10/21/11	Read Section 8.5 8.5/ 1-5
M 10/24/11	For those of you who left before I noticed I'd made a mistake (you didn't leave early; I corrected myself late), the correct definition of "standard deviation" for a general probability distribution is the formula in exercise 8.5/18. In its definition of both the mean (p. 557) and the standard deviation of a general probability distribution, the book's author forgets to say, "provided the integral exist" (i.e. converges, in the truly-improper-integral case). The mean of a pdf f exists only if the integral in the formula "μ = ..." on p. 557 exists. The standard deviation exists only if the mean exists and the integral in 8.5/18 exists. 8.5/ 6,7. Show that, for the probability density function in 8.5/5, the mean does not exist. Come up with a probability density function for which the mean exists, but the standard deviation does not. Assume that f is a probability density function for which both the mean and standard deviation exist. Show that the integral in 8.5/18 (whose square root is taken to define the standard deviation) equals (∫_-∞ ^∞ x²f(x) dx) – μ². (That's an integral from –∞ to ∞; I don't know how to get the formatting to look right in HTML.) I.e., show that ∫_-∞ ^∞ (x – μ)²f(x) dx = (∫_-∞ ^∞ x²f(x) dx) – μ². Let f be a probability density function whose graph is "triangle-shaped", like the one in 8.5/8, but generalized by replacing the numbers 0,6,10 on the x-axis with a,b,c respectively, and with "0.2" on the vertical axis replaced by 2/(c – a) (so that the area of the triangle is 1). Show that the mean of f is (a+b+c)/3. Using the previous problem (the one referring to the integral in 8.5/18) to simplify the computation, show that the square of the standard deviation is (1/18)(a² + b² + c² – ab – ac – bc).
T 10/25/11	11.1/ 1,2,5,6,9-14,15,16,19-30,32-36,37 (be clever!), 45, 54,55,57,58. (I know this is a lot of problems, but you should find most of them very short.) Let r₁ = (1+√5)/2, r₂ = (1-√5)/2. Define a sequence {a_n} by a_n = (1/√5) (r₁ⁿ - r₂ⁿ), n≥ 1. Calculate a_n for 1≤ n ≤ 5. If your calculation is correct, you will find that, magically, these all work out to be integers, and in fact are the first five terms of one of the sequences used as an example in Monday's class. Prove that the sequence above is identical to that sequence from class for all n≥ 1.
General information	The grade-scale page for your class has been updated with Exam 2 information. The exam will be returned Tuesday at the end of class.
W 10/26/11	11.1/ 71,72,74-77. For 72b, set your calculator in "radians" mode, enter "1", and press the "cos" button repeatedly until the display stabilizes. This is the limit of the sequence {a_n}, to within the precision of your calculator's display. Write down or store your answer. Then enter your UFID number, and again press the "cos" button repeatedly until the display stabilizes. It will be the same limit that you found the first time. You don't yet have the tools (and won't, unless you take Advanced Calculus) to explain why these sequences converge, but you should be able to come up with an explanation of why, if they converge, they must converge to the same limit. You can play the same game with the sine function, but the limit is boring. Re-do every problem on the exam that you did not get completely correct, until you're able to do it completely correctly.
F 10/28/11	Read any part of Section 11.1 you haven't read yet. After that, if you feel ready, get a head-start on the problems due Monday.
M 10/31/11	11.1/ 59,60-66,67-69,81. Hint for finding the limits in 67-69 and 81: use 72a. 11.1/ 79. After you've done part (c): let AGM(a,b) denote the arithmetic-geometric mean of a and b. Show that for all c >0, AGM(ca,cb)=c AGM(a,b). Use this to show that AGM(a,b)=b AGM(a/b,1). Therefore all properties of the two-variable AGM function can be deduced from properties of the one-variable function f(x)=AGM(x,1). If you have a programmable calculator (or access to a device that can run mathematical programs) and enjoy playing with it, write a program to compute AGM(a,b) by computing as many terms of the sequences as necessary to get a_n – b_n < 10^–10. (Even if you don't enjoy playing this way, if your major's requirements include a course in programming in a scientific language, like C, C++, or MATLAB, this will be good practice for you.) Include an instruction to display the a_n and b_n each time the next of these numbers is computed. When you run the program, just set b=1, since we saw above that we have the simple relation AGM(a,b)= b AGM(x,1), where x=a/b. Run your program for a few values of x. The sequences {a_n}, {b_n} converge amazingly fast. (That's why I'm suggesting you use as tiny a tolerance as 10^–10, so you can see that you get super-fast convergence to a very, very precise number.) Let g(x)=(π/2)x/ln(x). Compute AGM(x,1)/g(x) for some VERY large values of x; you might want to start at around x=10⁷⁰. You should see that the ratio of these two functions is close to 1. It is a true fact that lim_{x→ ∞} AGM(x,1)/g(x)=1, but this limit is approached v-e-r-y s-l-o-w-l-y (an interesting contrast to the rapid convergence of the sequences used to compute AGM(x,1) for each individual x). Thus, AGM(x,1) is asymptotic to (π/2)x/ln(x) as x→ ∞. The function x/ln(x) comes up elsewhere in mathematics: the Prime Number Theorem asserts that if N(x) denotes the number of prime numbers less than x, then N(x) is asymptotic to x/ln(x) as as x→ ∞. 11.1/ 80. The conclusion of part (b) is a fun fact. However, I could not figure out exactly how the author wanted you to use part (a) to compute the limit in part (b). Maybe you will see a shortcut that I missed. But in case not, here is one way to do the problem: First, show that if a_n < √2 then a_n+1 > √2, and that if a_n > √2 then a_n+1 < √2. (In other words, the sequence keeps hopping from one side of √2 to the other.) Use this to show that all of the odd-numbered a_n are less than all of the even-numbered a_n. (Thank you, Russell, for showing me how to complete the square-root symbol in HTML!) Second, show that a_n+2=(4+3a_n)/(3+2a_n). Third, show that the odd-numbered subsequence {a₁, a₃, a₅, ... a_2n+1, ...} is an increasing sequence, and that the even-numbered subsequence {a₂, a₄, a₆, ... a_2n, ...} is a decreasing sequence. Then use the first step to deduce that both subsequences are bounded. Since they are monotonic, they therefore converge. Let L_odd, L_even be the limits of the odd- and even-numbered subsequences. Using the second step, show that each of these limits satisfies the equation x=(4+3x)/(3+2x). Conclude from this that both limits are √2. Now use part (a). BOO!!!
T 11/1/11	11.2/ 1,2,9,10,11,12,14-16,19,20,65,75
W 11/2/11	11.2/ 21,22,35,37,38,40,42,46,47-51,55,57,59,63,68-71
M 11/7/11	11.2/23-34 11.3/1-8. For #8, the Integral Test is applicable but silly to use, in view of the Test for Divergence (which, as I mentioned in class, is often called the "n^th-term test", and for which I prefer the name "n^th-term screen") on p. 692.
T 11/8/11	Read the note I added at the end of the previous assignment. I meant to include it when I posted the assignment. It's okay if you spread the rest of the assignment over two days, but you should do at least a significant portion before Tuesday's class. 11.3/ 9-26, 27,29,30. Remember that to apply the Integral Test in the form that's on p. 699 (essentially the same as the form given in class), you have to know that f is monotonically non-increasing on [1,∞). Recall from Calculus 1 that this condition is met if f '(x) ≤ 0 for all x ≥ 1. To do the problems above correctly, using only the tools presented up through Section 11.3, you need to verify monotonicity, either by this first-derivative criterion or by some other method. For the problems in this part of the assignment, "some other method" is the easiest approach except perhaps in #s 16, 18-20, 24, and 26; for these others, you should use the first-derivative criterion above. (In #20 you can simplify this to: use first-derivative considerations to show that the denominator is monotonically increasing on [2,∞).) Make a mental note of what a royal pain this can be, so that you'll better appreciate the comparison tests we'll be covering in a few days (which will make monotonicity-testing unnecessary). The hypotheses given in class under which the integral test is applicable were: Assume that f is a continuous, non-negative, monotonically non-increasing function on [1,∞). (These are also the hypotheses in the textbook, p. 699, except that Stewart says "decreasing" where I say "non-increasing". "Non-increasing" is more general, but in practice most of the series to which you apply the integral test will involve a strictly decreasing f.) Explain why the hypotheses can be weakened to, "Assume that f is continuous, non-negative, monotonically non-increasing function on [N,∞) for some N," without altering the conclusion. (In other words, verify what Stewart said in the Note after the box on p. 699.) Then: Guess a likely reason that, in #24, Stewart started the sum at n=3. Given that Stewart started the sum in #24 at n=3, figure out what he probably overlooked in #20. Figure out why the weakened-hypotheses version of the Integral Test above is sufficient to handle #20, and would have been sufficient to handle the series in #24 even had it started at n=1. 11.3/ 32,34. In these, you may use a calculator to do arithmetic. In #34, replace the requirement "correct to three decimal places" to "correct to within 10^–3 ". There is often no way of guaranteeing that an estimate is literally correct to three decimal places without knowing additional information. For example, if the true value of some quantity were between 0.1239991 and 0.1239999, and you estimated the value to be 0.124, the error would be less than 10^–6 and yet the third decimal place of your estimate would still be wrong.
W 11/9/11	In honor of the Gators having finally won a game, there is no new homework.
M 11/14/11	11.4/1-21,24,26 (I know that's a lot, but your experience from improper integrals should make most of these pretty short),37,38,40,42.
T 11/15/11	11.4/31,39,45,46. Hint for #39 and #46: the theorem on p. 692 is relevant. 11.5/ 2-20,35. Some comments: Even more strongly than usual, I recommend reading the corresponding section of the book before starting these problems, since we spent only fifteen minutes in class Tuesday on Section 11.5, and you could probably use the reinforcement. For some of these (e.g. 8,10,11) the method used in the book's Example 3 is useful. For 8 and 10, the masochists among you should take this advice literally. Non-masochists may want to make use of these observations: (i) a positive function is increasing/decreasing if and only if its square is increasing/decreasing (relevant to #8), and (ii) for a function f defined on [0,∞), f(x) is increasing/decreasing if and only if f(x²) is increasing/decreasing. These are special cases of the following (figure out why): If g and h are functions for which the range of h is contained in the domain of g (so that the composition gοh is defined), and one of g,h is an increasing function, then the composition gοh is increasing/decreasing if and only if the other of g,h is increasing/decreasing. (Here (gοh)(x)=g(h(x)). I can't find a plain small circle in HTML to represent the composition symbol. The exercise-set 2-20 is not a very rich source of examples. (After you've done them, see if you can figure out why I say this.) If I were writing a set of exercises for this chapter section, I'd put down maybe the first 10 of these, then give exercise 35, then give an exercise in which the series to test has {a_n = 1/n for odd n, a_n = –2/n for even n}. Then I'd give a longer follow-up set that includes the remainder of the 2-20 group, interspersed with some alternating series that diverge despite passing the n^th-term screen.
W 11/16/11	11.5/ 1,23,24,32-34,36. In 36a, I think you will find it easier to establish the desired formula if you write it, equivalently, as s_2n + h_n = h_2n. (I found that that helped me, at least.) Read the material on the Root Test in section 11.6. I won't be going over this in class. Examples in which the Root Test is more useful than the Ratio Test tend to be contrived, and rarely arise in practice. 11.6/ 1-17, 27,28,29,31
F 11/18/11	Read Section 11.7. I also won't be going over that section explicitly in class but it will come up (probably) in Q&A when you ask me how to do various problems. Your exam-questions won't tell you which convergence test to use, so you'll want to practice figuring this out; that's what the exercises in 11.7 are for. 11.7/ 1-30
M 11/21/11	Third midterm exam (assignment is to study for it)
T 11/22/11	11.8/ 3-7,12-17. For now, modify the instructions to, "Find the set on which the Ratio Test guarantees absolute convergence." I will soon will re-assign these (as part of a larger assignment) with the instruction, with the book's instructions, "find the interval of convergence", restored, so hold onto your work on these problems (which will end up being most of the work needed to do the problems as worded in the book). The set on which a power series converges absolutely is, of course, a subset of the domain on which it converges (period), i.e. the set I called the "domain of convergence" in Wednesday's class. (Sometimes this subset is the whole domain of convergence, sometimes not.) My "domain of convergence" is exactly the book's "interval of convergence"; the latter is the standard term for this set. After we cover the theorem on p. 725 in class (which will probably be Tuesday), I'll start using the "interval of convergence" terminology, but I have my reasons for not wanting to use that terminology prior to covering that theorem.
W 11/23/11	For purposes of this assignment, assume the theorem on p. 725, even though I didn't finish proving it today. (The rest of the proof will take only a few minutes on Wednesday.) 11.8/ 2,3-7,12-24,33,35a,36,37,40,42. In 3-7 and 12-17, in the previous assignment you effectively computed the interior of the interval of convergence. (The interior is the open interval (a – R, a + R) if R, the radius of convergence, is positive and finite; the interior is the whole real line (–∞,∞) if the radius of convergence is ∞.) All that remains in these problems is to determine, in the finite-R case whether the series converges at one endpoint, both endpoints, or neither endpoint. (For 18-24, you have to start from scratch, since these weren't part of the previous assignment.)
M 11/28/11	11.8/ 29, 30, 32, 41 11.9/ 1-12,13,14,15,17,21,32,33a,34,35,36,38. Notes: In the exercises for Section 11.9, all power series are power series in x; i.e., centered at 0. Before doing #13, read Example 5. Hint for #35b: using what you found in part (a), compute the derivative of e^–xf(x). (If you covered Taylor series in high school, you may be tempted to say, ' "Isn't the formula for f(x) just the Taylor series for e^x?" to which the answer is "Yes, but that's not the point of this problem." Taylor series are introduced in the next section of the book, 11.10. The intent of this problem is to have you show a different way, without the not-yet-introduced Taylor series, that the given power series converges to e^x.) My hint is not the only no-Taylor-series way to do this problem; if you think of another, that's fine. Note that the series in #36 is not a power series (it's an example of something called a Fourier series), so statements that we made in the context of power series (like "domain of convergence is an interval" and "term-by-term differentiation is valid on the interior of the interval/domain of convergence") need not apply. This example illustrates that power series have some special properties not shared by all series of functions.
General information	The grade-scale page for your class has been updated with Exam 3 information. The exam will be returned Monday at the end of class. Five minutes before the end of class, when it becomes obvious that I've forgotten all about handing back the exam, you may remind me.
T 11/29/11	11.9/ 16,18,23-25,27. Notes: In #24, the integrand has the formula written in the problem for t≠0, and is defined at t=0 by taking the limit of the given formula as t→0. A similar comment applies to #25. In #27, replace "to six decimal places" by "to within 10^–6 ", for the reason mentioned a few homeworks ago. 11.10/ 1 (find formula in terms of derivatives of f), 2,3,4, 5-12, 13-17, 70. Some guidance for #70: First show by induction that for x ≠ 0, and all n ≥1, f⁽ⁿ⁾(x) is e^–1/x² times a polynomial in 1/x. Next show that lim_{x→ 0} f⁽ⁿ⁾(x) =0 for all n≥ 0. Hint: let y=1/x. To use this substitution, you will need to address the limits as x→ 0 from the left and right separately, but it's the same work for both cases. Next show that f⁽ⁿ⁾(0)=0 for all n≥ 0. From the preceding, it follows that every coefficient in the Taylor series of f is zero, so the series converges to 0 for all x. But f(x)≠ 0 for x≠ 0. Thus, for all x≠ 0 this Taylor series converges to a value different from f(x). Show that if a power series Σ_n≥0 a_n xⁿ converges to zero for all x in an open interval centered at 0, then all the a_n must be 0. (Here "Σ_n≥0" means the sum from n=0 to ∞; my attempt to write the usual notation in HTML ended up looking like this: Σ_n=0^∞.) Note: "But wouldn't all the a_n have to be zero for this to happen?" is not an answer; it's a restatement of the question. "I don't see how a power series could converge to zero throughout an entire interval without all the coefficients being 0," is also not an answer. There's no such thing as "proof by lack of imagination". You have all the preparation needed to do the "non-book" part of the assignment due Wednesday; start on it if you have time. Despite the space it takes up on this webpage, it shouldn't take you long to do, but I wasn't sure if including it in the assignment due Tuesday would overload you.
W 11/30/11	11.10/ 29,31,48,63-68,69 Let I be an open interval (possibly the whole real line) centered at 0. Recall that a function g defined on I is called even if g(–x) = g(x), and odd if g(–x) = –g(x). (The terminology comes from the behavior of integer powers: xⁿ is an even function of x if n is even, and is odd if n is odd. But unlike integers, most functions are neither even nor odd.) Below, "function" always means "function defined on I". (a) Show that the only function that is both even and odd is the constant function 0. (b) Show that the sum or difference of two even functions is even, and that the sum or difference of two odd functions is odd. For any function f on I, define f_ev(x) = (f(x)+f(–x))/2 and f_odd(x) = (f(x)–f(–x))/2. (c) Show that f_ev is even, that f_odd is odd, and that f = f_ev + f_odd. (d) Show that if f = g+h, where g is an even function and h is an odd function, then g = f_ev and h = f_odd. (Hint: use parts (a) and (b).) Thus f can be expressed as the sum of an even and an odd function in only one way. For this reason we can unambiguously refer to f_ev and f_odd as the even and odd parts of f, respectively. (e) Show that cosh and sinh are the even and odd parts of the exponential function. (f) Suppose f(x) = Σ_n≥0 a_n xⁿ on I. Show that f_ev(x) = Σ_{n even} a_nxⁿ and f_odd(x) = Σ_{n odd} a_nxⁿ. (HTML notation-problem again. In the first series, the sum is over all even n≥ 0; in the second, the sum is over all odd n≥ 1.) (g) Use parts (e) and (f) to write down the Taylor series, based at 0, of cosh and sinh. They should look very similar to a couple of other series you know, but the sign-differences you'll observe make a world of difference. Note: the Taylor series of cosh and sinh can be derived very simply from the definition of "Taylor series", without going through all the steps above. Parts (e) and (g) are included only to be illustrative, giving you some concrete examples pertaining to evenness/oddness of functions and how these properties are reflected in Taylor series.
F 12/2/11	11.10/33-34,38, 55-57,58,59 (read Example 12, p. 745, first), 61.
M 12/5/11	11.10/25-28,35,45,53,71. 10.1/ 1,2,5-7,10,11-14,19,22,27,34. In 1, 2, and 27 your sketches don't have to be very accurate, but should at least start and end at the correct points, go through the correct quadrants in the correct order, and have the correct extreme values of x and y.
T 12/6/11	10.2/ 1-4,7,9,12,14,15,29 (note: "points" here mean an ordered pairs (x,y), not values of t), 30,31,33, 37 (but using the instructions for 41-44 instead of the instructions for 37-40), 41,45,57 (just set up the integral). For 12-15 and 31 on, you'll need to read parts of Section 10.2 that I haven't covered yet (the equation at the top of p. 631, and the material on areas, arclength, and surface area in pp. 632-635). I'll go over some of this briefly on Tuesday, but probably not all of it.
W 12/7/11	10.3/ 1,3,5,7-9,11-12,21-26,27-28,29-33, 57,58,61,62,63,64,77. To do 57-64, first read "Tangents to Polar Curves", pp. 644-645, but don't get hung up on Example 9, part of which is done poorly. The key idea is that we can express x and y parametrically in terms of θ (i.e., θ plays the role played by t in Sections 10.1-10.2). Do not waste your time trying to memorize formula 3 on p. 644. The only real use of this general formula is to obtain information about tangent lines at the origin (if the curve happens to pass through the origin), using the last equation on p. 644 and the reasoning in the first three lines of p. 645. Do understand the Note at the top of p. 646; this is the way you should proceed in general (except for the use of the double-angle formula, which is not essential and is very unhelpful for Example 9.) In other words, the best way to do Example 9 is: Figure out that x(θ) = cos(θ) + sin(θ)cos(θ), y(θ) = sin(θ) + sin²(θ). Compute dx/dθ and dy/dθ, and use dy/dx = (dy/dθ)/(dx/dθ). Don't bother with the double-angle formula; write dy/dx as the last fraction preceding part (a) on p. 645. Beyond that step, proceed the way the book does. One further problem with Example 9: Figure 15 is misleading. The "pointy part" at the origin should be infinitely sharp (the curve has a single, vertical tangent line there, not the two oblique tangent lines that the picture suggests you'd have).
before final	10.3/ 34,37,39,41,42,43. In #43, note that on any θ-interval on which sin(2θ) is negative, there are no values of r that satisfy the equation, and hence no points on the graph for that θ-interval. 10.4/ 45,46,48,55 ("Formula 10.2.7" means the formula on p. 635 labeled by a boxed, colored "7"). The following curves are mentioned in your textbook. Which has the best name? lemniscate limaçon astroid cardioid conchoid cycloid trochoid strophoid cissoid of Dioclese nephroid of Freeth oval of Cassini witch of Maria Agnesi hippopede
Thurs. 12/15/11	FINAL EXAM begins at 5:30 p.m. in our usual classroom.
General information	The grade-scale page for your class has been updated with final exam and course-grade information. If you have a question about your grade or would like to look at your graded exam, please DO NOT EMAIL ME THE QUESTION; please see me in my office after the start of Spring semester. Tentatively, my spring office hours will be the same as they were in the fall. I will not send or discuss grades by email.

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Homework Assignments MAP 3473, Section 3205 - Honors Analytic Geometry and Calculus II Fall 2011

BOO!!!

Homework Assignments
MAP 3473, Section 3205 - Honors Analytic Geometry and Calculus II
Fall 2011