# Math 222, Sections 365 and 367

## News

• 26 Dec: I forgot to copy down the final grades for transport to Ohio, so I can't send you your grades until I return to Madison around January 2. By that time, they may already be available on EASI. Sorry to make you wait!
• 21 Dec: I have computed the final grades. E-mail me if you want to know your grade. Also, as it stands now I will be teaching Math 234 for Marshall Slemrod next semester. My sections will be 301, 302, 311, and 312. Those are at 7:45 (boo!) and 1:20.
• 13 Dec: I have listed all of the review sessions being given by the various TAs below. You are welcome to attend any of them.
• 11 Dec: I have added an entire section containing information about our final exam below.
• 5 Dec: Information about the final exam has been posted below.
• 1 Dec: I have written a solution to Problem 25 in Section 10.9 below.
• 27 Nov: I have written a solution to Problem 33 in Section 10.8 below. It is based on the solution given in the solution manual, but my explanation is much more detailed than the manual's. In other news, note that our quiz this week will be on sections 10.8, 10.9, and 10.10.
• 11 Nov: I have posted the curve for the second midterm exam below.
• 2 Nov: I have added information regarding the second midterm exam below.
• 3 Oct: I have added a section for exam and quiz results below, so you can see how you're doing relative to your classmates.
• 22 Sep: I will be hosting a review session on Tuesday 26 September from 6 PM to 8 PM in B321 Van Vleck. Note that this is not our usual room. Bring questions to ask, or just come and work on your own or with other students.
• 20 Sep: There will be no quiz next week because of the exam. Furthermore, many details regarding the exam have been worked out. See below.
• 14 Sep: Two different solutions for problem 7.3 #35 (the lune problem) are available below. The solution I presented in my 11:00 section uses some ugly calculus. The other solution relies only on high school geometry and trigonometry.
• 12 Sep: The first quiz, on this Thursday, will cover only sections 7.1 and 7.2. You will not be responsible for 7.3. On future Tuesdays I'll try to give you more precise information.
• 8 Sep: Quizzes will now be given at the end of discussion, rather than at the beginning. Also, no graphing calculators are allowed on the exams and quizzes. See below.
• 5 Sep: My office hours have been determined. Every student should be able to attend at least one. See below.

## Introduction

Mathematics 222 is second-semester calculus. Our textbook is Calculus: Early Transcendentals, 3rd Edition, by James Stewart. The main topics to be covered are:
• techniques of integration
• basic differential equations
• arc length, parametrized curves, and polar coordinates
• sequences and series
• basic arithmetic and geometry of vectors
Professor Oh will lecture three days a week on new material, and I will lead discussion on the other two days. In discussion sections, students will be expected to ask questions on homework and on concepts covered in lecture. Often the answers to these questions will come from other students. My job is to encourage participation and keep us on track. In short, "discussion" will be a discussion.

## Course Requirements

• 20%: weekly quizzes, given at the end of most Thursday classes
• 20%: first midterm exam, given Wednesday 27 September 2000, 5:30-7:00 PM, 1310 Ingraham Hall
• 20%: second midterm exam, given Monday 6 November 2000, 5:30-7:00 PM, 165 Bascom Hall
• 40%: final exam, given Wednesday 20 December 2000, 5:05 PM
Attendance will not be taken in lecture or discussion, but all students will have to attend Thursday discussions in order to take the quizzes. Also, your homework will not be graded (unless you ask me to look at your work specifically). The weekly quizzes are intended to ensure that you are doing your work. They will require you to do problems similar to those on the homework, but under time pressure. So it is recommended that you practice by doing all of the homework problems (and even some extra). The exam questions will also be similar to those in the book, except that there might be one or two more challenging problems on each exam.

On the exams and quizzes, students are allowed to use any non-graphing calculator; on the other hand, use of a graphing calculator during an exam or quiz will be considered cheating, and will be dealt with severely. Students should familiarize themselves with the university's academic misconduct policies.

The first midterm exam will be given on 27 September 2000, 5:30-7:00 PM, in 1310 Ingraham Hall (next to Van Vleck). Students who do not have any conflict with this time must take this exam. Students who do have a conflict must arrange with me to attend the make-up exam, which is on 27 September 2000, from 12:00-1:00, in B130 Van Vleck. (This is our normal lecture room, but note the times carefully!) The exam will cover sections 7.1-7.4 and 7.8-7.9. No graphing calculators or "cheat sheets" are allowed. There will be no quiz on the next day, 28 September.

The second midterm exam will be given on 6 November 2000, 5:30-7:00 PM, in 165 Bascom Hall (next to Van Vleck). The exam will cover all course material since our previous exam, up to and including section 10.3. No graphing calculators or "cheat sheets" are allowed. I will be hosting a review session on 5 November, 4:00-5:30 in B113 Van Vleck.

For information on the final exam, see the next section.

## Final Exam

The final exam will be given on 20 December 2000, 5:05-7:05 PM, in room 125 of the "old" Biochemistry Building (near where Campus Dr. splits off University Ave.). The exam will be cumulative, but weighted heavily toward the material covered after the second midterm exam. No graphing calculators are permitted. Each student is allowed to bring in a "cheat sheet" with formulas, examples, etc. It should be a single, 8.5 x 11 (or smaller) piece of paper; you are allowed to use both sides. That's a lot of room!

I will be holding two review sessions for the final exam. The first will be Saturday 16 December, 2:45-4:45 PM, B113 Van Vleck. In it I hope to wrap up our discussion of the course material. That is, I envision it as an extra discussion section, in which students will ask questions from the recent homework, lectures, and reading. The second review session will be Monday 18 December, 7:25-9:25 PM, B113 Van Vleck. In it I will be happy to answer questions from Prof. Oh's review sheet, as well as any other questions you have encountered while studying. After this review session, I will take a walk down to the biochemistry buildings, to make sure I know where the final is. Students are welcome to accompany me on this remarkable journey of discovery.

Here is a schedule of review sessions being held by the various teaching assistants. You are welcome to attend any of them that you like. All times are PM.

DateTimePlaceTANotes
Wed 13 Dec5:00-8:003331 SterlingJoaquin
Sat 16 Dec2:45-4:45B 113 VVJosh
Sat 16 Dec2:45-4:45B 321 VVRob
Sun 17 Dec2:45-4:45B 219 VVGeircovering series only
Mon 18 Dec12:25-2:25B 135 VVBret
Mon 18 Dec7:25-9:25B 113 VVJoshwalk to 125 Biochemistry afterward
Tue 19 Dec7:25-9:25B 219 VVGeir
Tue 19 Dec12:25-2:25B 135 VVBret
Tue 19 Dec5:05-7:05B 321 VVRob

Here are my recommendations for studying for the final:

• The first third of the semester dealt with techniques of integration. Section 7.6 is just a summary of these techniques and how to use them. Read this summary, and think about why certain techniques are applicable to certain problems, and remind yourself of how each technique works. Work out an example or two for each technique.
• In our brief study of techniques for solving differential equations, we learned about separable, homogeneous, and first-order linear equations. For each type, there is a standard method that always works. Memorize and practice these standard methods a little.
• The middle of the semester was a mixture of various geometric topics, such as arc-length, parametrized curves, polar coordinates, and conic sections. Go back and read these sections, and memorize the formulas. Upon your second or third reading, you may find them to make much more sense than they did before. Try to understand what the formulas mean in English (or your native language) and how they are derived. For example, you should be able to look at the equation for an ellipse, immediately see the x- and y-intercepts, and sketch the ellipse.
• Toward the end of the semester we learned about sequences, series, and vectors. The exam will draw much (most?) of its material from these sections, so you want to focus your studying here!
• The first half of Chapter 10 deals mainly with series of numbers. Section 10.7 summarizes the various tests and when to use them. Just as when reviewing the techniques of integration, read this section carefully. Understand why certain tests are applicable to certain series. Make sure you can use each test by working out some examples. Focus on the concepts we used most in class (alternating series, the ratio test, etc.). If you aren't comfortable with computing limits, then practice or come talk to me.
• After learning about series of numbers, we learned about power series, which contain powers of a variable x. The key point about power series is that, if they converge, they describe a function, since we can plug in various numbers for x. But typically power series only converge on a specific interval, such as (-1, 1) or [-10, -2]. You should understand why the ratio test gives us the radius of convergence. Practice many examples. Practice integrating and differentiating power series. Remember that, if a function has a power series representation on a given interval, then that power series must equal the Taylor series of the function. Practice writing down the Taylor series for various functions centered at various points. (If you have enough time, then practice showing that the function equals its Taylor series by showing that the remainder goes to zero. But I think that this is less important for you to master right now than the other stuff.)
• Lastly we learned about vectors. This is the most important topic of the course, and it is easy to test, so I expect that the exam will have a lot of vector questions. Be familiar with the basic concept of vectors. Practice visualizing them: picture them adding, scaling, and projecting in your head. Master the computation of dot and cross products. Understand why a plane is determined by a point and a normal vector, and how they show up in the formula for the plane.
• Do all of the problems on Prof. Oh's final review sheet. Then invent your own problems along the same lines by changing the numbers and functions involved, and solve those problems. Look through the book for similar problems, and master those. There are only so many problems we can invent for you, and if you happen to do them ahead of time, you'll obviously do better on the exam.
• Do not stay up late studying the night before! In my opinion, it is far more important to be well-rested and clear-thinking on an exam, so you can use your knowledge quickly and effectively, than to memorize a couple more formulas that probably won't even appear. Furthermore, recent research suggests that the human brain organizes and remembers knowledge during sleep. So if you want to learn the material, sleep well. Sleep and eat regularly in the days leading up to the exam. On exam day, eat enough so that you won't be hungry, but not so much that you're tired.

## Contact Information

Here's how to get in touch with the professor:
Yong-Geun Oh (pronounced "yong gun oh")
Office: Van Vleck Hall, Room 715
Office Phone: (608) 262-2883
Office Hours: Tuesday and Thursday, 2:30-3:30 PM, or by appointment
E-Mail: oh@math.wisc.edu
Here's how to get ahold of me:
Joshua Davis (pronounced "josh")
Office: Van Vleck Hall, Room 518
Office Phone: (608) 262-3860
Office Hours: Monday 9:55-10:45, Thursday 1:20-2:10, Friday 2:25-3:15, or by appointment
E-Mail: jdavis@math.wisc.edu

If you can't attend my office hours, then consult my weekly schedule and make an appointment with me. If you have a quick question, then try catching me after discussion or lecture; I usually sit in the back of the room at lecture.

## Syllabus

WeekDateTextSummaryHomeworkNotes
15 Sep7.1-7.2integration by parts,
trigonometric functions
7.1: 1, 3, 4, 7, 13, 20, 29, 31, 38, 39, 62;
7.2: 2, 8, 17, 18, 29, 59-61
211 Sep7.3-7.4trigonometric substitutions,
partial fractions
7.3: 1, 4, 8, 10, 15, 21, 27, 35;
7.4: 2, 5, 9, 10, 17, 23, 26, 31, 52, 62
318 Sep7.8-7.9approximate integrals,
improper integrals
7.8: 1-4, 7, 15, 21, 29, 32;
7.9: 3, 11, 12, 17, 18, 23, 24, 41-43, 60, 66
425 Sep8.1-8.2review,
arc length
8.2: 4, 5, 9, 12, 27, 32, 36Midterm I
52 Oct15.1-15.2basics,
first order linear equations
15.1: 1, 3, 4, 5, 7, 11-14, 17-20, 33;
15.2: 1, 3, 7, 13, 17, 19, 23, 25, 33
69 Oct9.1, 9.3parametrized curves,
arc length
9.1: 1, 3, 4, 8, 11, 14, 26, 28;
9.3: 3, 5, 6, 13-16
716 Oct9.4-9.6polar coordinates,
conic sections
9.4: 13-17, 20, 24, 27, 30, 31, 34, 37, 43;
9.5: 1, 2, 5, 7, 10, 15, 19, 21, 23;
9.6: 1, 3, 6, 9, 13, 17, 20, 21, 25, 28, 31, 35
823 Oct10.1-10.2sequences,
series
10.1: 2, 4-6, 10, 15, 18;
10.2: 7, 8, 11, 14, 21, 30, 52, 54, 55, 58
930 Oct10.3-10.4integral test,
comparison test,
review
10.3: 1, 4, 7, 9, 14, 15, 21, 31, 32;
10.4: 1, 3, 6, 11, 14, 21, 27, 30, 37, 42
106 Nov10.6absolute convergence,
root test,
ratio test
10.6: 1, 3-5, 7, 8, 13, 17, 29 (just determine whether the series are absolutely convergent or not) Midterm II
1113 Nov10.8-10.10power series10.8: 3, 4, 7, 9, 12, 17, 22, 29, 33;
10.9: 1, 2, 5, 6, 11, 12, 21, 25, 27, 36
1220 Nov10.10-10.12Taylor series10.10: 2, 5, 7, 11, 17, 19, 33, 38, 53;
10.12: 9, 10, 13, 15, 17
Thanksgiving
1327 Nov11.1-11.3vectors,
dot products
11.1: 5, 7, 9, 10, 15, 17, 18, 22, 27;
11.2: 1, 3, 7, 9, 13, 14, 19, 24, 25, 29, 35;
11.3: 1-8, 12, 13, 19, 21, 25, 28, 53, 61, 63
144 Dec11.4-11.5cross products,
equations of planes
11.4: 1-7, 10, 12, 19, 21, 23, 27, 35;
11.5: 1, 3, 7, 11, 14, 15, 18, 20, 23, 27, 32, 35, 39
1511 Dec11.7motions in space,
review
11.7: 1-7, 10, 13, 23, 25, 28

## Exam and Quiz Results

Grades are not assigned according to a strict 90-80-70-60 percentage scheme. In order to estimate how well you are doing in comparison to your classmates, you should inspect the following data. Here are the approximate mean (average) scores for our in-class quizzes so far:
QuizDateSectionsMean
114 Sep7.1, 7.28.4/10
221 Sep7.3, 7.48.5/10
35 Oct7.9, 8.26.5/10
412 Oct15.1, 15.27.7/10
519 Oct9.1, 9.36.8/10
626 Oct9.4, 9.57.1/10
72 Nov10.1, 10.28.0/10
89 Nov10.3, 10.45.6/10
916 Nov10.69.2/10
1030 Nov10.8 - 10.109.8/14
117 Dec10.12, 11.18.9/10
1214 Dec11.2 - 11.48.8/10

Here are the rough letter grade equivalents for scores (out of 100) on the two midterm exams:
Letter GradeScore Range, Exam 1Score Range, Exam 2
A90-10075-91
AB85-8972-74
B75-8465-71
BC69-7462-64
C50-6850-61
D40-4939-49
F0-390-38

You have been given freedom over attendance and homework because it is recognized that you sometimes have other things to do - big papers, stressful days, dental appointments, etc. - and because you know your learning style and needs better than the teachers do. You will have to be self-motivated. Sometimes this is difficult for students in their first semester of college, who are accustomed to more structure. It can also be dangerous for students who have seen some of the material before, because the class suddenly becomes much harder when it starts covering new material.

Be careful not to fall behind. Attend class and do the homework. Have questions about homework problems and lecture material ready for discussion. Review each lecture before the next lecture arrives. Study with your fellow students. Visit me and the professor during our office hours. If you feel that you need additional help, you have a variety of options:

• At the beginning of each semester, the Mathematics Tutorial Program hosts a series of workshops to help you review material from earlier courses. Look for the yellow signs in the stairwells in Van Vleck.
• The Department of Mathematics runs a Math Lab in B227 Van Vleck, Monday through Thursday, 3:30-5:10 and 6:30-8:10. Math 222 students can drop in for help from miscellaneous math teaching assistants. (Note that Math Lab does not begin until a couple of weeks into the semester.)
• The Greater University Tutoring Service (GUTS) provides volunteer tutors for a variety of classes, and can also help with study skills and English speaking skills.
• The Department of Mathematics also maintains a list of tutors that you can hire for one-on-one tutoring sessions. See the department receptionist on the second floor of Van Vleck.
• The Mathematics Tutorial Program can provide more formal help from trained professional staff. Students who are having trouble in their math classes can be referred to this program by their teaching assistants.
Most importantly, if you're not understanding the material, then get help from me, the professor, or your fellow students as soon as possible. In math courses each concept builds on earlier ones, so it's important to understand the concepts as we go along.

## Extra Stuff

### Section 7.3, Problem 35

I now see two ways to do this pesky lune problem. The first expresses the area as a difference of two integrals, and then computes these using trig substitutions. This is undoubtedly how the author intended us to do the problem. I went through this solution laboriously in the 11:00 discussion today. The second method of solution requires no calculus and is much faster and easier. (In the following explanation, "*" denotes multiplication and "^" denotes exponentiation. For example, 3*3 = 3^2 = 9. Also, "sqrt" denotes square root.)

Here is a sketch of the first method. Place the center of the circle of radius r at the origin in the x-y plane. Then the function y = sqrt(r^2 - x^2) describes the top edge of the lune. Now the center of the circle of radius R lies at (0, -sqrt(R^2 - r^2)). This means that the function y = sqrt(R^2 - x^2) - sqrt(R^2 - r^2) describes the bottom of the lune. To get the lune's area, just compute the integral, from -r to r, of sqrt(r^2 - x^2) - (sqrt(R^2 - x^2) - sqrt(R^2 - r^2)) dx. This requires substitutions like x = r*sin(theta) and x = R*sin(theta).

Here is the second method. Let P and Q be the two points where the two circles intersect. Let C be the center of the circle of radius R. Draw the line segments PQ, PC, and QC. You end up with an "ice cream cone" consisting of the upper half of the radius r circle and the isosceles triangle PQC. Since you know how to compute the area of semicircles and triangles, you can easily verify that the area of this ice cream cone is pi*(r^2)/2 + r*sqrt(R^2 - r^2).

To get the area of the lune, we just need to subtract out the region of the ice cream cone that lies inside the big circle of radius R. Call this region A. Note that if theta is the measure (in radians) of the angle PCQ (at the bottom of the cone), then the area of A is theta/2pi * pi(R^2). To complete the computation, we just need to find theta. Splitting the triangle into two congruent right triangles, we see that sin(theta/2) = r/R, so theta = 2arcsin(r/R). Thus the final answer is p*(r^2)/2 + r*sqrt(R^2 - r^2) - (R^2)*arcsin(r/R).

### Section 10.8, Problem 33

We are asked to find the interval of convergence and an explicit formula for the series f(x) = 1 + 2x + x2 + 2x3 + x4 + 2x5 + .... To be more precise, the nth term in the series is cnxn, where cn is 1 for even n and 2 for odd n. It is possible to write down a formula for cn; in fact,

cn = (3 - (-1)n)/2.

But this formula does us little good in answering the problem. (Try it yourself; the ratio test fails.) The fact that the coefficients cn alternate between 1 and 2 makes this series difficult to analyze using our standard techniques. So we'll use an unusual approach. We will begin by analyzing the odd-numbered partial sums

s1, s3, s5, ..., s2n-1, ....

We'll show that this sequence of odd-numbered partial sums converges. Then we will analyze the even-numbered partial sums

s0, s2, s4, ..., s2n, ...,

and show that they converge to the same thing. Then we can conclude that the entire sequence of partial sums converges to this one thing. (You should stop to convince yourself of this fact. It requires you to understand well what it means for a sequence to converge.) Thus the series will converge. In the process of figuring this out, we'll also compute the interval of convergence.

So let's get started. First, we write down an odd-numbered partial sum s2n-1 and factor it:

s2n-1 = 1 + 2x + x2 + 2x3 + ... + x2n-2 + 2x2n-1

= (1 + 2x)(1 + x2 + x4 + x6 + ... + x2n-2).

Now (1 + 2x) is simple enough, but we need to understand the second factor, (1 + x2 + ... + x2n-2). We're going to need a bit more algebra. You've probably seen the following fact before; if not, then multiply it out yourself:

(1 - x)(1 + x + x2 + x3 + ... + xn-1) = 1 - xn.

In this equation we replace x with x2 to obtain the following identity:

(1 - x2)(1 + x2 + x4 + x6 + ... + x2n-2) = 1 - x2n.

Dividing both sides by (1 - x2), we obtain

1 + x2 + x4 + x6 + ... + x2n-2 = (1 - x2n)/(1 - x2).

The left-hand side of this equation is exactly what we were looking for. Combining this equation with our first equation, we get

s2n-1 = (1 + 2x)(1 - x2n)/(1 - x2).

So we have a nice formula for the odd-numbered partial sum s2n-1. Now the series converges if and only if the sequence of partial sums converges. For the sequence of partial sums to converge, we need at least the sequence of odd-numbered partial sums to converge. So when does it converge?

In our formula for s2n-1, n occurs only in the term x2n. So for our sequence of odd-numbered partial sums to converge as n goes to infinity, we just need x2n to converge as n goes to infinity. Thus we need |x| < 1. In fact, when |x| < 1, x2n goes to 0 as n goes to infinity. Thus, assuming that -1 < x < 1, we have that s2n-1 goes to (1 + 2x)/(1 - x2) as n goes to infinity.

Now let's analyze the even-numbered partial sums. We know that

s2n = s2n-1 + a2n = s2n-1 + x2n.

As before, if -1 < x < 1, then x2n goes to 0 as n goes to infinity. Therefore, as n goes to infinity, s2n goes to s2n-1. This means that the sequence of even-numbered partial sums converges to the same thing as the sequence of odd-numbered partial sums, which was (1 + 2x)/(1 - x2). Therefore the series converges to (1 + 2x)/(1 - x2) when -1 < x < 1.

A careful student will remember to check the endpoints -1 and 1. It is easy to check that the series diverges at both of these endpoints. (It diverges to negative infinity at -1, and to positive infinity at 1.) Our final answer is that the series converges to (1 + 2x)/(1 - x2) on the interval (-1, 1).

### Section 10.9, Problem 25

We want to compute the integral, from 0 to 1/5, of 1/(1 + x4), with respect to x, using a power series. First, from Problem 21 we already know that the antiderivative of 1/(1 + x4) for -1 < x < 1 is

C + sum(0 to infinity) (-1)nx4n+1/(4n + 1).

The definite integral we desire equals the antiderivative evaluated at 1/5, minus the antiderivative evaluated at 0. Fortunately, evaluation at 0 kills all terms (except C) in the power series, so the definite integral equals

sum(0 to infinity) (-1)n1/54n+1/(4n + 1).

This series converges to the exact value of the definite integral. We have been asked to approximate the value to six decimal places. I take this to mean "approximate the value, so that the first six decimal places in your answer are definitely correct". Regardless of what it means, I'm just going to approximate it to within 10-7 and consider that a good answer. So, anyway, how many terms of the series do we need to add up, and how do we know that this many makes the error small enough?

The key thing to notice now is that the series in question is alternating. This suggests that we should use the Alternating Series Remainder Theorem from Section 10.5. This theorem says that if you have an alternating series whose terms decrease in magnitude to zero, and you truncate the series somewhere (say, after the nth term, bn), then your error will be smaller than the first term you ignored (bn+1).

Our series satisfies the hypotheses of the theorem. Thus, if we just find a term bn+1 in the series that is smaller than 10-7, then we can cut off the series after bn, and our error in doing so is guaranteed to be smaller than bn+1, which is smaller than 10-7. Applying this idea to the series in question, we start computing the first few terms, which go (ignoring minus signs)

(1/5)1/1, (1/5)5/5, (1/5)9/9, ...

The third term is already less than 10-7, so we can take as our approximation the sum of the first two terms, which is 1/5 + 1/56.