MAP 3473, Section 3205 - Honors Analytic Geometry and Calculus II

Fall 2009

Homework problems anddue dates(notthe 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:From the same page, another important bit of advice is this: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.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 8/25/09 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 ofx.W 8/26/09 6.1/45,49,53 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/9. ("Problems Plus" are at the end of each chapter.) F 8/28/09 6.2/ 41-44,46, 50-52,54,56,65,68,71 6.ProblemsPlus/ 7 6.3/ 1-3,7,10,12,16,21,37-39 M 8/31/09 6.3/ 5,22,26,28,29-32,40-42,43,44,46 6.ProblemsPlus/ 10 Let a,m,nbe positive real numbers withm < n,and letf(x) =x(^{m}/x). Use the First Derivative Test to show that the graph of^{n}+ay=f(x), forx≥ 0,is qualitatively similar to the one I drew in class fory=x/(x1) (i.e. that^{2}+fstarts at 0, rises monotonically to some maximum value, then falls off monotonically and asymptotically to zero). Do this without using the Second Derivative Test or L'Hôpital's Rule.Show the same thing, the same way, for f(x) =xwhere^{b}e^{-cx}bandcare positive.T 9/1/09 6.4/ 1-3, 5, 8, 15, 16 (assume the size of the bucket is negligible), 19, 21, 26-29 W 9/2/09 6.5/ 1-7, 9, 17, 18b (assume 18a), 22,23 F 9/4/09 7.1/ 1-4,6,9,10-15,19,22,24,25,31,33,34,36.

Useful tip: When doing adefiniteintegral via successive integrations-by-parts (IBP), as in #24, it's wise to check mentally whether either of the functionsH(x),g(x) that you get after each step, is zero at one or both endpoints of the interval of integration. (HereH(x) andg(x) are as I used in Wednesday's class.) IfH(x)g(x) is zero at an endpoint, you can often save a lot of time by doing the endpoint-evaluation before going on to the next IBP step, rather than carrying all the functions ofxpicked up at each IBP step through to the end of the indefinite integration, and only then doing the endpoint evaluations.There is at least one more of the assigned problems besides #24 that this idea is useful for, but I leave it to you to figure out which.

∫ x1ln(^{-}x)dxcan be done easily by substition (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?T 9/8/09 7.1/ 44-52 7.2/ 1-6, 15,17-20,58,59 W 9/9/09 7.1/ 54,58,68a-d Let a> 0,f(t)= cos^{2}(at). Find the average value offover a cycle. In this case a "cycle" means an interval of the form [t_{0},t_{0}+π/a], sincefis periodic with periodπ/a.Repeat the previous exercise with f(t)= sin^{2}(at).Read what's in the box on p. 465, and Example 9 below the box. 7.2/ 43-45, 53 (just evaluate the integral; don't bother graphing), 67-69, 70.

FYI: the sum in #70 is a "sine-only" finite Fourier series. General Fourier series, both finite and infinite, involve both sines and cosines.F 9/11/09 7.2/ 7-14,21-27,33,35,37,62,63 M 9/14/09 7.3/ 1-3,4-12,22,29,30,33,40

Challenge problem (optional; no due date): 7.ProblemsPlus #3. I don't think you'll find anything in Chapter 7 (including the parts we haven't gotten to yet) that will be of help, but don't let that stop you from trying all the techniques you know. [Note added later: Actually, thereissomething we've already covered that can be helpful, in conjunction with the "desperation substitution", but the step in which it can be useful is avoidable if you are sufficiently clever.]Major hint: In class, several lectures ago, I very briefly referred to something I called the "desperation substitution", which I said hardly ever helps. It actually does help here, but it doesn't solve the problem all by itself; you still need to be clever, 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.T 9/15/09 7.3/ 13-19,23,24,26,27,34,43 (assume R>r)W 9/16/09 Study for Friday's exam. The HW from the portion of Section 7.4 covered in Tuesday's class is posted below, with due date Monday 9/21. F 9/18/09 First midterm examM 9/21/09 7.4/ 1a,2a,7,9,10,12-14,16-18,47,48-50,63 T 9/22/09 7.4/ 1b,2b,3,4,6,8,15,23,25-28,62,64 W 9/23/09 7.4/ 29-32,34-37,39,40,42,46,50,57,58 F 9/25/09 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 x^{8}were replaced byx^{7}), 41,42,44,47 (hint: partial fractions come to mind, but there is a more efficient way to do this problem)

Factor these polynomials into linear and irreducible quadratic factors:

x^{3}+x^{2}-x+2x^{4}-1x^{6}-1M 9/28/09 Re-do all the problems that you got wrong on the exam. Also do the extra-credit problem. Remind me to give you the answers on Monday. 7.8/ 1-3, 5-15, 17-19 General information The grade-scale page for your class now exists. T 9/29/09 7.8/ 20-26,60,61,74,75,78 W 9/30/09 7.8/ 27-40,42,45,46,71 F 10/2/09 Use the definitions of improper integrals to give careful derivations of the linearity and additivity properties I stated in class, for all the subcases of "Type I" and "Type II" integrals. M 10/5/09 7.8/ 49-56,58,59,80. Determine for which values of pthe integral ∫_{1}^{∞}(lnx)/xconverges, in each of the following ways:^{p}dx

- by making the (inverse) substitution
x=1/yand using your answer to exercise 59;- by making the (inverse) substitution
x=e.^{y}Redo #59 by using the (inverse) substitution x=eand using the integral you came up with in part 2 of the problem above.^{-y}T 10/6/09 7.8/ 79 Determine whether ∫ _{1}^{∞}(sinx)/xconverges.^{2}dx7.Review/ 49,50 From problems 7.Review/ 1-48, select three that scare you, and do them. (If nothing scares you, pretend.) W 10/7/09 Compute the value of ∫ _{0}^{1}(1/√x(1-x))dx. (That square-root sign should extend over the "x(1-x)", but I don't know how to get that effect in HTML.)F 10/9/09 Re-examine the integrals in 7.8/ 8,11,17,18,20,22,23,26-33,39-40. Just using the comparison tests we've talked about, your knowledge of "1/ x" integrals, and/or the growth hierarchy of functions (the latter is useful for only a few of the book's integrals, because Stewart has omitted this topic, but it's very useful in "real life" improper 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.^{p}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 fiseveniff(-x)=f(x), andoddiff(-x)=-f(x). Most functions are neither even nor odd.) For each of these integrals, one of the following three statements applies:Without taking any antiderivatives, determine which statement applies to each integral.

- The integral converges, and its value is zero.
- The integral converges, and its value is not zero.
- The integral diverges.

In 7.8/71, using your new wisdom on the growth hierarchy of functions, figure out what the domain of Fis iffis 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. On Friday, we need to discuss the date of the next exam. Please remind me.

M 10/12/09 8.1/ 1-6,7-12,17 (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 ofy=ln(x) over the interval 1≤x≤e. You should be able to do this even without being able to do problem 17.T 10/13/09 8.1/ 14,38 Many integrals that can be done with trig substitutions can also be done with hyperbolic trig substitutions. Compute ∫ (1+ x^2)^{-1/2 }dxtwo ways: (1) using the substitutionx=tan(ϑ), and (2) using the substitutionx=sinh(y). For the second, at the end you will need to use the formula for sinh^{-1}(x) that we derived in class.Look here and here. W 10/14/09 7.8/ 63 (Optional) If your calculator does definite integrals, enter the Arc Length Contest at the top of p. 532. No time-limit or due-date. The prize will be bragging-rights, probably worth millions on Ebay. M 10/19/09 8.2/ 1-4, 5-8, 15,16,25,26,29,35 T 10/20/09 8.5/ 1-5 (the book's " X" is what I called "h" in examples today)W 10/21/09 8.5/ 6,7,12. In #12, just express your answers as integrals, unless your calculator allows you to compute numerical values for these integrals. The definitions of "mean" and "standard deviation" given in class were incomplete; both definitions should have ended with "provided the integral exists". The book also omits this fine point. (The book defines the standard deviation of a general probability-density function in 8.5/18.)

- 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 fis a probability density function for which both the mean and standard deviation exist. Show that the integral whose square root is taken in 8.5/18 (to define the standard deviation) equals∫ _{-∞}^{∞}x^{2}f(x) dx- μ^{2}.Let fbe 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 thex-axis witha,b,crespectively, 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
fis (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)( a).^{2}+ b^{2}+ c^{2}-ab-ac-bcF 10/23/09 Second midterm exam(assignment is to study for it)M 10/26/09 Start looking at Chapter 11; that's what we'll be covering next. Solve a major world problem. These are not easy, so you may work in pairs. T 10/27/09 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 them very short.) Let r_{1}=(1+√5)/2,r_{2}=(1-√5)/2. (Here "√5" is the square root of 5; I still don't know how to get the top of the square-root symbol in HTML.) Define a sequence {a} by_{n}

a=(1/√5)[_{n}r_{1}^{n}-r_{2}^{n}],n≥ 1.Calculate

afor 1≤_{n}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 today in class. Prove that the sequence above is identical to that sequence from class for alln≥ 1.W 10/28/09 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}, 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,_{n}ifthey converge, they must converge to the same limit.You can play the same game with the sine function, but the limit is boring.

General information Your exams will be returned in class Wednesday. The grade-scale page has been updated and includes a link to the score-distribution on the second midterm. F 10/30/09 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 ofaandb. Show that for allc>0, AGM(ca,cb)=cAGM(a,b). Use this to show that AGM(a,b)=bAGM(a/b,1). Therefore all properties of the two-variable AGM function can be deduced from properties of the one-variable functionf(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 geta< 10_{n}-b_{n}^{-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 theaand_{n}beach time the next of these numbers is computed. When you run the program, just set_{n}b=1, since we saw above that we have the simple relation AGM(a,b)=bAGM(x,1), wherex=a/b. Run your program for a few values ofx.

- The sequences {
a}, {_{n}b} converge_{n}amazinglyfast. (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 ofx; you might want to start at aroundx=10^{70}. 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) foreach individual x). Thus, AGM(x,1) is asymptotic to (π/2)x/ln(x) asx→ ∞.The function

x/ln(x) comes up elsewhere in mathematics: the Prime Number Theorem asserts that ifN(x)denotes the number of prime numbers less thanx, thenN(x)is asymptotic tox/ln(x) asx/ln(x) asx→ ∞.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< √2 then_{n}a+1 > √2, and that if_{n}a> √2 then_{n}a+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_{n}aare less than all of the even-numbered_{n}a._{n}- Second, show that
a+2=(4+3_{n}a)/(3+2_{n}a)._{n}- Third, show that the odd-numbered subsequence {
a_{1},a_{3},a_{5}, ...a+1, ...} is an increasing sequence, and that the even-numbered subsequence {_{2n}a_{2},a_{4},a_{6}, ...a, ...} is a decreasing sequence. Then use the first step to deduce that both subsequences are bounded. Since since they are monotonic, they therefore converge._{2n}

- 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 equationx=(4+3x)/(3+2x). Conclude from this that both limits are √2.- Now use part (a).
S 10/31/09 Wear garlic today. M 11/2/09 11.2/ 1,2, 10,65,75. In #10, among the possible answers to these questions are, "There is no difference" and "One of these makes sense; the other doesn't." T 11/3/09 Flannery O'Connor wrote a book entitled Everything That Rises Must Converge. (I have never read it, and know nothing about it other than the title and author.)Assuming this is a book about sequences of real numbers, what additional restriction should be put on " Everything That Rises" to make the title a correct mathematical statement?Read the book and tell me what the title actually refers to. W 11/4/09 11.2/ 11,12,15,16,19,20,35,37,38,40,42,46,55,57,59,63, 68-71 F 11/6/09 Read pp. 692-694. I meant to cover this before I started on the Integral Test. I'll go over it on Friday, but you should be able to do the problems below from 11.2 just based on your reading. 11.2/21-34, 47-51 11.3/1-8. For #8, the Integral Test is applicable but silly to use, in view of the Test for Divergence (sometimes called the " n^{th}-term test"; a better name might be "n^{th}-term screen") on p. 692.11.4/1,2,37,39 (hint: the theorem on p. 692 is relevant). M 11/9/09 11.4/ 31,40,42 T 11/10/09 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 isliterallycorrect 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.11.4/ 3-20 (I know that's a lot, but your experience from exercises with improper integrals should make most of these pretty short), 38,42,45,46 F 11/13/09 11.4/ 21,24,26 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
fdefined on [0,∞),f(x)is increasing/decreasing if and only iff(xis increasing/decreasing. These are special cases of the following (figure out why):^{2})If

gandhare functions for which the range ofhis contained in the domain ofg(so that the compositiongοhis defined), and one ofg,his an increasing function, then the compositiongοhis increasing/decreasing if and only if the other ofg,his 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 {
afor odd_{n}=1/nn,afor even_{n}=-2/nn}. 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 then^{th}-term screen.M 11/16/09 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. (I found that that helped me, at least.)_{2n}+ h_{n}= h_{2n}Read p. 719. 11.6/ 39,40. Extend the result of #40 by showing that the same statement is true with rreplaced by ∞ or-∞.T 11/17/09 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. Read Section 11.7. I also won't be going over that section explicitly in class, but it will come up (probably) in Wednesday's 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. 11.6/ 1-17, 27,28,29,31 11.7/1, 3-16 W 11/18/09 11.7/ 17-30 Read Theorem 3 and the subsequent paragraph on p. 725. Although you are notresponsible for knowing this theorem for Friday's midterm, you may find that it gives you a useful consistency check for examples in which you are asked to find the domain of convergence of a power series. You arenotresponsible, on the midterm, for knowing the terminology "radius of convergence" or "interval of convergence". (But FYI, we'll see Monday that the domain of convergence is an interval--possibly consisting of just one point, as in the last example we did on Tuesday--and from then on we'll use the terminology "interval of convergence" for the domain of convergence.)11.8/ 3-7,12-17. Modify the instructions for these to "Find the domain of convergence." Next week I will re-assign these (as part of a larger assignment) with the instruction, "Find the radius of convergence" (for which you are notresponsible for Friday's exam), so hold onto your work on these problems.F 11/20/09 Third midterm exam(assignment is to study for it)M 11/23/09 No new homework (but it appears that not everyone has done all the old homework ...) T 11/24/09 11.8/ 2,3-7,12-24,33,35a,36,37,40,42. In 3-7 and 12-17, you already found the interval of convergence in the last HW due before the exam, so just look back at your answers and state the radius of convergence. W 11/25/09 11.8/ 29, 30, 32, 41 11.9/ 1-12,13,32,33a,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.^{-x}f(x)M 11/30/09 11.9/ 14, 15-18,21,23-25,27. Notes:

- In #24, the integrand has the formula written in the problem for
t≠0, and is defined att=0by taking the limit of the given formula ast→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.Extra Credit:You may (re)do problems 7, 8, and 9 from the last exam, and get back up to half the points you missed on these problems. Rules:

- This work must be handed in on Monday, Nov. 30.
- If you received any credit for problems 8 and 9 on the exam, and you're handing in solutions to these problems, also hand your exam back in. (I have a record of how many points each person got on each problem, but not on each part of each problem.) If you answered a problem-part correctly on the exam, you're not eligible for any more points on that problem-part (but you may still have to hand that part in again, redone, in order to get credit for other parts; see below).
- What you hand in must be
neat, or I won't grade it. Work everything out on scrap paper first, rewrite the solutions carefully on clean sheets of 8.5" x 11" paper, leaving enough space for me to write comments, andstaplethe sheets together if there is more than one sheet. I won't grade anything that is messy, that looks like it's been erased and written over (written-over erasures are a struggle to read), that has shreds of paper dangling from it (for example, from being ripped out of a spiral-bound notebook), or that comes apart when I turn pages (which is what will happen if you use paper clips instead of staples, or if you attach your sheets of paper simply by folding and/or tearing the corners).- You may consult your textbook, notes, comments I wrote on your exam, etc., but you may not consult another person.
- For each problem you hand in, you must write a
solution, even if you received partial credit for that problem on the exam. For problems 8 and 9, this means you have to hand in complete solutions or answers to all three parts (and the solution/answer to each part must be complete), to get any new points for those problems.complete- There will be
nopartial credit within problem-parts. To get any credit for problem 7, your answer/solution must beperfect. To get any credit for part (a), (b), or (c) of problem 8 or 9, your answer/solution to that part must be perfect, AND your answer(s)/solution(s) to the earlier part(s) must be perfect. (So, for example, in order to get any credit for 8b, your solutions toboth8a and 8b must be perfect.)- Point-values for parts of problems 8 and 9 on the original exam were: 8 for 8a; 10 for 8b; 5 for 8c; 2 for 9a; 2 for 9b; 8 for 9c.
Some examples of how I will assign new points:

- You received 4 points (out of 8) on problem 7 on the original exam, and on Monday you hand in a complete, perfect solution to problem. You will receive 2 points (half of the remaining 4) added to your score on the exam.
- You did not attempt problem 8 on the original exam. On Monday you hand in solutions to 8a,8b, and 8c. Your solution to 8a is perfect, but there is a mistake or omission in your solution to 8b. You will receive 4 new points (half of the 8 that part 8a was worth), 0 for 8b, and 0 for 8c.
- You received the full 2 points for 9a on the original exam, but nothing for 9b or 9c. On Monday you hand in perfect answers/solutions to 9a, 9b, and 9c. You will receive 5 new points (half of the remaining 10 points for problem 9).
T 12/1/09 11.10/ 1 (find formula in terms of derivatives of f), 3,4, 5-12, 13-17 , 70.Some guidance for #70:

- First show by induction that for
x ≠ 0, and alln ≥1,fis^{(n)}(x)e1/^{-}x^2times a polynomial in 1/x.- Next show that lim
_{x→ 0}f=0 for all^{(n)}(x)n≥ 0. Hint: lety=1/x. To use this subsitution, you will need to address the limits asx→ 0 from the left and right separately, but it's the same work for both cases.- Next show that
f(0)=0 for all^{(n)}n≥ 0.- From the preceding, it follows that every coefficient in the Taylor series of
fis zero, sothe series converges to 0 for all x. Butf(x)≠ 0forx≠ 0. Thus, for allx≠ 0this Taylor series converges to a value different fromf(x).W 12/2/09 Note: Because we are supposed to cover essentially all of Chapter 10 between now and the end of the semester, I will probably not set aside any of our lectures for HW Q&A until the last day of class (Wed. Dec. 9). If enough students want, we can try to schedule one or more out-of-class review sessions.

11.10/ 2, 29,31,33,34,38, 48, 63-68,69 Show that if a power series Σ _{n≥0}aconverges to zero for all_{n}x^{n}xin an open interval centered at 0, then all theamust be 0. (Here "Σ_{n}_{n≥0}" means the sum fromn=0 to ∞; my attempt to write the usual notation in HTML ended up looking this way: Σ_{n=0}^{∞}.) Note: "But wouldn't all theahave 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"._{n}Let Ibe an open interval (possibly the whole real line) centered at 0. Recall that a functiongdefined onIis calledevenifg(-x) = g(x), andoddifg(-x) = -g(x). (The terminology comes from the behavior of integer powers:xis an even function of^{n}xifnis even, and is odd ifnis odd. But unlike integers, most functions are neither even nor odd.)Below, "function" always means "function defined on

I".For any function

- (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.
fonI, definef_{ev}(x) = (f(x)+f(-x))/2 andf_{odd}(x) = (f(x)-f(-x))/2.

- (c) Show that
f_{ev}is even, thatf_{odd}is odd, and thatf=f_{ev}+f_{odd}.- (d) Show that if
f=g+h, wheregis an even function andhis an odd function, theng=f_{ev}andh=f_{odd}. (Hint: use parts (a) and (b).) Thusfcan be expressed as the sum of an even and an odd function in only one way. For this reason we can unambiguously refer tof_{ev}andf_{odd}astheeven and odd parts off, respectively.- (e) Show that cosh and sinh are the even and odd parts of the exponential function.
- (f) Suppose
f(x)= Σ_{n≥0}aon_{n}x^{n}I. Show thatf_{ev}(x) = Σ_{n even}aand_{n}x^{n}f_{odd}(x) = Σ_{n odd}a. (HTML notation problem again. In the first series, the sum is over all even_{n}x^{n}n≥0; in the second, the sum is over all oddn≥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 12/4/09 11.10/ 25-28, 29,31,35, 45, 53, 55-57,58,59,61,71. Finish reading Section 11.10 before you start these problems. M 12/7/09 Read all of section 10.1 through the end of the material on the cycloid. I probably will not have time to cover any more of it in class than what I went over on Friday, but I'll be holding you responsible for it. The remainder of Section 10.1 (Example 8) is worth reading, but I won't hold you responsible for that example. Read all of Section 10.2. In order to have enough time to cover Sections 10.3-10.4, and hopefully 10.5-10.6, I probably will not go over Section 10.2 in class, but I'll still hold you responsible for it. (Eventually you will get to ask questions about it.) 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 xandy.10.2/ 1-4,7,9,12,14,15,29 (note: "points" here mean an ordered pairs ( x,y), not values oft), 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).Optional: 10.1/ 35,36. These look like they should be fun if (unlike me) you're handy with your graphing calculator. T 12/8/09 10.3/ 1,3,5,7-9,11-12,15-20,21-26,27-28,29-33 W 12/9/09 10.3/ 34,37,39,41,42,43,57,58,61,62,63,64,77. In #43, note that on any θ-interval on which sin(2θ) is negative, there are novalues ofrthat satisfy the equation, and hencenopoints on the graph for that θ-interval.

optional: 10.3/ 48. I'd never seen this one before. It's pretty interesting, and has a cool name; see #71 (#72 has an even cooler name). In order to see why the name in #71 applies to #48, figure out how the two graphs are related to each other. 10.4/ 2,3,5,6,8,9-13 (for #11, see note to 10.3/43 above), 17 but with the curve from #13 (so that most of your work is already done), 21,35,45,46,48,55 ("Formula 10.2.7" means the formula on p. 635 labeled by a boxed, colored "7"). Find a Cartesian equation for the lemniscate in 10.4/ 55b. Use this to solve for y^{2}in terms ofx, and then find the volume enclosed by the surface generated in 10.4/ 55b. (You should find thisvery messy, andlong, but doable. You need to combine a lot of things that you know, or once knew. There are a lot of steps, but every step is one that you should know how to do.)

Choose the answer:best

- lemniscate
- limaçon
- astroid
- cardioid
- conchoid
- cycloid
- trochoid
- strophoid
- cissoid of Dioclese
- nephroid of Freeth
- oval of Cassini
- witch of Maria Agnesi
- hippopede
Before the final exam No new homework! For the final exam, I consider us to have covered the following sections of the book:

6.1 - 6.5

7.1 - 7.5, 7.8

8.1 - 8.2, 8.5

10.1 - 10.4

11.1 - 11.10

T 12/15/09 FINAL EXAMbegins at 10:00 a.m. in our usual classroom.After the exam, please email me with questions about your grade for the class, your performance on the exam, etc.do notCourse grades should be available on ISIS shortly after I submit them, which will be a few days after your final. I will post some exam statistics, and perhaps some other statistics, on your grade scale page.I will not email any information relating to the final exam or grades.

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