Last modified 20 December 2001 by jdavis@math.wisc.edu
Math 221, Sections 390 and 391
 News
 Introduction
 Course Requirements and Grading
 Contact Information
 Getting Extra Help
 Exam and Quiz Results
 TakeHome Quizzes
 Writing Suggestions
 Study Guide
 Prof. Angenent's 221 Page with Homework Problems
 20 Dec: The final exam is tomorrow! It's from 5:05 to 7:05 PM, in room 270 of the Soils building (on Observatory Dr.).
 13 Dec: Next week I will have office hours Sun 1:003:00, Mon 12:003:00, Tue 12:003:00, Wed 12:003:00, and Thu 12:002:45. On Tuesday we will have our final "semester review session", on integrals, in Sterling 3331, 7:258:25, immediately followed by an extra discussion meeting, 8:259:25, to answer any questions people have about Chapter 6. The review session for the final exam will be Thursday, 2:454:45, in B231.
 12 Dec: Tomorrow we will discuss 55 and all of Chapter 6! (Really we'll finish Chapter 6 in a review session.)
 10 Dec: Tonight we have a review session, 89 PM, in B115, on applications of derivatives and on integrals. Tomorrow we will discuss the quizzes, 54, and 55. The takehome quiz for this week has been posted below.
 5 Dec: Tomorrow we will discuss 53 and 54, and then take two 10minute quizzes, as described two days ago:
 3 Dec: Tomorrow we will finish 52 and start 53 and 54. On Wednesday we have another review session, 89 PM, in B139, covering deltaepsilon limit proofs, derivatives, related rates, etc. This Thursday's quiz covers 52 and 53; immediately beforehand, we will take 10 minutes to "retake Midterm 2 Problem 6". That is, you will be given a similar problem, except that it will also involve the fundamental theorem. (I did an example like this last Thursday.) This retaken problem will count for four extra credit points on the midterm.
 28 Nov: Tomorrow we will finish Chapter 4 and move on to 52 and perhaps 53.
 26 Nov: Tomorrow we will discuss the exam and finish Chapter 4. This week's takehome quiz has been posted below. Preliminary results of the second midterm exam have been posted below.
 24 Nov: This Tuesday I will be hosting an extra review session covering limits. It will take place in B115, 89 PM. I will make a sheet with important limit facts available to everyone.
 19 Nov: We will indeed have discussion tomorrow. If the exams have been graded, then we will discuss them; even if not, we will forge ahead with 47 and 48.
 14 Nov: Tomorrow we will discuss the sigma notation of 45, and 47 and 48. We will take a quiz on 43 and 44. The review session for the second midterm will be held on Sunday 18 Nov, 12:002:00, in room B115.
 13 Nov: This Thursday's quiz will cover 43 and 44. The problems should resemble the homework problems from these sections. I have also added a small study quide below; currently it just talks about max/min problems. My extended office hours for this week include Wed 12:0512:55, 3:304:20, Thu 12:202:10, Fri 12:0512:55, Sat 3:004:00, Sun 2:003:00 (after the review session). I have decided to have our first "semester review session" two weeks from today, on limits.
 12 Nov: Tomorrow we will begin work on 45 and 46, which concern definite integrals. Our second midterm exam is Monday 19 November, 2:253:30 in lecture. I will be hosting a review session on Sunday from noon to 2 PM.
 7 Nov: Tomorrow we will answer any final questions from 42 and discuss 43 and 44. Also, we will vote on the time slots for our extra review sessions.
 5 Nov: Tomorrow we will do 39 #18, finish 42, and begin 43. Also, the takehome quiz due this Thursday has been posted below. Note that this time I am offering to proofread your composition before you hand in your paper.
Mathematics 221 is firstsemester calculus. Our textbook is Calculus and Analytic Geometry by Thomas and Finney, 5th Edition. In this course we will explore the fundamental concept of limit and then use it to define derivative and integral, two of the most prevalent ideas in calculus. We will also learn a variety of techniques and applications.
Professor Angenent will lecture three days a week on new material, and I (your teaching assistant) 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.
This web page will be updated frequently with class news, exam results, and other vaguely useful information. You will often find the answers to your classrelated questions here. You will find additional, lecturewide information at Prof. Angenent's 221 page.
Your course grade will be computed from the following components:
 20% first midterm examination
 20% second midterm examination
 40% final examination
 20% discussion activities
In my class sections, the discussion component of the grade will be determined by short weekly quizzes. Some of these quizzes will take place in class, while others will be given as homework. The takehome quiz policy is explained in detail below.
If you miss a quiz, you will probably receive a grade of zero, unless there are extenuating circumstances. I am reluctant to give makeup quizzes. For this reason, you should come to discussion on quiz days. At the end of the semester, I will drop each student's lowest quiz score, so you can miss or fail one quiz without penalizing your grade.
Homework and attendance do not count toward your grade, but I hope you will show up to discussion ready to talk about homework problems anyway. You are encouraged to work on homework problems (but not quiz problems) with your classmates.
Here's how to get in touch with the lecturer:
Prof. Sigurd Angenent
Van Vleck Hall, Room 617
(608) 2622883
Office hours Monday and Wednesday, 10:0011:00, and by appointment
angenent@math.wisc.edu
http://www.math.wisc.edu/~angenent/
Here's how to get ahold of me:
Joshua Davis (call me Josh)
Van Vleck Hall, Room 518
(608) 2623860
Office hours Monday 12:0512:55, Wednesday 3:304:20, Thursday 1:202:10, and by appointment
jdavis@math.wisc.edu
http://www.math.wisc.edu/~jdavis/
If you can't attend my office hours, then consult my weekly schedule, pick an open slot, 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.
I find it very helpful to talk to other people about mathematics. Get to know your fellow students; discuss concepts and practice problems with them. Also, visit the professor and me during office hours.
Even if you work diligently, it is easy to get lost in math courses, as I can testify. This is especially true in early calculus courses, where students with varying backgrounds and expectations are thrown together. If you do find yourself falling a little behind, consider these options for extra help:
 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, 3:305:10 and 6:308:10. Math 221 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 maintains a list of tutors that you can hire for oneonone 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 try to catch up 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.
You might find the following data useful in estimating how well you are doing in comparison to your classmates. If you are worried about your progress, come talk to me as soon as possible.
  Number of Students Receiving Each Score  Total  Mean


  20  19  18  17  16  15  14  13  12  11  10  9  8  7  6  5  4


Exam 1  My 2 Sections  3  4  2  8  5  6  7  6  0  2  2  1  1  1  0  1  0  49  14.8


6 of 8 Sections  8  11  11  15  18  19  14  15  6  7  7  6  3  2  0  1  1  144  ?


Approx. Grade  A  B  C  D  F  


Exam 2  My 2 Sections  0  0  1  0  2  4  3  5  8  9  4  2  0  4  2  0  0  44  12.5


All Sections  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?  ?


Quiz  Date  Topics  Mean


1  13 Sep  17, 18  8.2

2  20 Sep  110  6.9

3  27 Sep  22, 23  6.0

4  4 Oct  28, 25  7.6

5  18 Oct  29, 210, 32  5.6

6  25 Oct  36  8.2

7  1 Nov  38, 39  8.7

8  8 Nov  42  9.4

9  15 Nov  43, 44  7.2

10  29 Nov  46, 47  7.7

11  6 Dec  52, 53  6.4

12  13 Dec  54, 55  ?

You will be taking several takehome quizzes this semester. I like these because some students only get a chance to show what they know when they are given enough time to relax. Takehome quizzes count equally with quizzes taken in class, but typically they will be longer and more difficult.
There are several rules that you must understand and obey. You may take unlimited time to complete your quiz, but it must be turned in at the start of class on the day due. You may not discuss the quiz in any way with anyone else until it is handed in. You may consult your class notes and our class textbook freely, but you may not consult any other books or papers or electronic resources. You may use any calculator that is allowed on the exams. (Calculators with computer algebra systems are forbidden.) Your answers should be written clearly and concisely in complete, meaningful English sentences, as described below.
I am trusting you to obey these rules. Cheating will not be tolerated. If you have any questions regarding the rules, or any questions regarding the statement of the quiz problems, email me as soon as possible.
Quiz 2 (Due 20 Sep)
Let f(x) = x^{2} + x. It turns out that the limit, as x goes to 2, of f(x) is 6. I am asking you to prove this limit in the following two ways (neither of which is "plugging in"). (A) Prove this limit using an epsilondelta argument. You might find Example 2 in 110 helpful. (B) Prove this limit in detail using Theorem 1 in 110. Every step should use one of the parts of the theorem; show every step, and explain which part of the theorem applies.
Quiz 4 (Due 4 Oct)
(A) Do Problem 29 in 25. (B) Let f(x) = x^{12}, g(x) = x + 3, h(x) = 1/x, i(x) = 4x, and j(x) = sqrt(x) (that's the square root of x). First, write down the formula for the function f(g(h(i(j(x))))); then use the chain rule to find its derivative with respect to x. Remember to write up your work cleanly, in wellconstructed English sentences.
Quiz 6 (Due 25 Oct)
There is only one problem: 36, #10. This problem is worth 10 points. Do not forget to check every possibility necessary to show that your answer is correct. (For example, do not just check where the derivative is zero but also where it's undefined, and also find and check the boundary points, and make sure your answer is indeed a local maximum using the second derivative or some other argument.) You can check your answer in the back of the book; this means that I am more interested in your detailed, thorough work than in the answer itself.
Quiz 8 (Due 8 Nov)
Do Problems 4 and 25 in 42. In Problem 25, note that the top of the fraction looks like the derivative of the the expression under the cuberoot; this should lead you to the correct substitution. As always, you are required to write your answers in complete sentences, but now I am offering to proofread your papers (for English, not math) before you hand them in. Just visit me in my office hours or after the Wednesday lecture.
Quiz 10 (Due 29 Nov)
As before, I am offering to proofread your papers before you hand them in. (A) Do 46 #2. (B) Let f(x) = x^{3} + x^{2}. First, compute the integral, from 2 to 1, of f(x) dx. Second, compute the area of the region enclosed by the graph of y = f(x), the xaxis, and the lines x = 2 and x = 1.
Quiz 12 (Due 13 Dec)
As before, I am offering to proofread your papers before you hand them in. Remember that you are not allowed to discuss the quiz with anyone else! Your two problems are 54 #8 and 55 #5.
Since you have a lot of time to prepare your takehome quizzes, I expect a thorough, wellorganized exposition of your answers. Each problem should be worked out in detail, showing all of the steps necessary to derive the answer. That is, your solution should be a sequence of English sentences, in which each sentence follows logically from the sentences before it. Consider the following excerpt:
This is an English sentence; it reads, "The limit of x^{2}, as x goes to 3, is 9." Note that it contains a verb ("=", pronounced "equals" or "is") and uses punctuation. Before submitting your paper, read it aloud like this, and make sure that it consists of a sequence of grammatically correct sentences with proper punctuation.
Even if your sentences are grammatical, they can be hard to read if, like the one above, they are written entirely in mathematical notation. To help your reader, begin your sentences with additional words that convey how the sentence connects to the one before it, such as "therefore", "thus", "so", "on the other hand", "however", etc.
You do not need to write whole paragraphs of text explaining your strategy, unless what you are doing is highly unusual and would be hard to understand without explanation. On the other hand, you should indeed write in paragraphs, as you do when writing an essay for an English class. Do not use bullets or arrows to organize your sentences. For example, let's say you were doing a max/min problem (from 36). We typically solve these by a sequence of steps:
 Model the problem as a function to be maximized or minimized on an interval.
 Check where the derivative is zero.
 Check where the derivative is undefined.
 Check the boundary of the interval.
 Compare your answers from steps 2, 3, and 4 to arrive at your final answer.
Thus it would be natural for you to organize your exposition into five paragraphs, one for each step, with the first paragraph introducing the problem and the fifth paragraph concluding it. (But please do not number your paragraphs!)
Finally, you should be sure to submit a clean copy of your solutions, without significant erasures, crossingsout, or scratch work. You will probably want to figure out the problem in your own style and then recopy your work onto a new sheet of paper, organizing your thoughts as you go.
To get a better idea of the kind of clarity and organization that I'm hoping for, read your calculus textbook! Notice how the authors write in complete English sentences, even though mathematical notation is present.
Check out my onepage summary of limits and continuity here (PDF, 46 KB).
You can also view my onepage summary of derivatives here (PDF, 50 KB).
Maximization and Minimization
Max/min problems are often presented as story problems, and many students have difficulty in translating the story problem into mathematical terms. To practice this skill, I recommend that you go through all of the story problems in 36. Don't solve them all; just interpret them to the point where you know the function you want to maximize (or minimize) and the interval.
Once the max/min problem is clarified into the form ``We want to maximize (or minimize) f(x) on the interval [a, b],'' the solution becomes rather mechanical. You have to analyze all values of x where the derivative f'(x) is zero or undefined, and you have to check x = a and x = b. If you want to practice this skill, then you can again do book problems, but I have written a few extra ones where slightly unusual things happen:
 Maximize and minimize y = 4x^{2/3}  x^{8/3} on [1, 1], [2, 2], and [3, 3].
 Maximize and minimize y = tan x on [0, pi/4] and [0, pi/2]. (Notice that tan x is undefined at pi/2.)
 Maximize and minimize y = x^{2}/2  x on [1, 1] and [2, 2]. (To differentiate x, you have to break it up into the x > 0 case and the x < 0 case.)