Math 222, Spring 2008

8:50 MWF, B102 Van Vleck

Lecture 1, Wilson

Welcome to Math 222! This is the second semester in the 3-semester sequence of calculus courses for those who may use calculus in the physical and social sciences, those going on in mathematics or computer science or statistics, and those who enjoy careful thinking and mathematics!

We will cover quite a few topics, sometimes with not a lot of obvious connection from one to another, this semester:

• In your previous calculus course you were introduced to the (definite and indefinite) integral, and had some tools like u-substitutions, for making an integral resemble one of the anti-derivative formulas you knew: We will develop additional tools for evaluating integrals. In years past evaluating integrals was one of the major tasks someone in industry who was called an applied mathematician might be asked to do: Today, many of those integrals can be evaluated by software such as Maple, Matlab, Mathematica, etc., although there are still infinitely many integrals the software cannot do. So why do we still make you learn strategies that are built into software? There are many reasons: For one thing, note that no table of integrals or set of techniques can be sufficient to cover all the cases, so you might someday encounter an integral your software did not know how to do. But from my perspective the most important reason is this: When humans and computers interact, there should be no question who is the boss. You must know enough about what the computer is doing to retain control, even if you let it carry out the details. When I was a manager in industry, if I delegated a task to one of the people working for me and he/she screwed up, I was still responsible to my bosses. In the same way, saying “the computer screwed up” should never be an acceptable excuse, if you are the one giving it orders. (That does not make me popular with store clerks…)
• Do you think 1 is the same thing as 0.9999999…. or not? If you have an answer, either yes or no, you must have some idea what 0.9999999…. means! But what is it? You have used those patterns, e.g. saying 1/3 is the same as 0.3333333…., for years, but how would they have any precise meaning if the sequence of digits never stops? But it turns out that the very important topic of infinite series will give those a completely precise meaning! For a more practical application, suppose you are designing something (e.g. a new model microwave, or a cell ‘phone, or the next generation of your fancy calculator) that needs to be able to evaluate trig functions. You certainly can’t make its microprocessor or programmable controller “know” all the infinitely many values. We will get from our study of series ways to approximate something like sin(x) to any desired degree of accuracy, using just arithmetic operations that the processor does have.
• Remember the official name of Math 221 and 222, Calculus and Analytic Geometry? What is analytic geometry? You’ve done some of it for a long time, using numbers and algebra to do geometry when you plot the graph of some polynomial. But we will go both further and also a long way backwards, to some geometry that dates back more than 2400 years. It is still useful: For example, NASA could not plan the trajectory for a space mission without it.
• Perhaps the most successfully applied mathematics for the last three centuries has been, in one form or another, the solution of differential equations. In 221 you got a taste of that, but we will go on to be able to solve a lot more equations and initial value problems.
• To this point, virtually all of your calculus has had to do with a function like f(x) that has one input (x) and produces one output. So a graph is flat, inputs usually plotted horizontally and outputs vertically. But very little of real life is that simple. If I talk about the temperature in the room, there is only one output (the temperature at a point) but the inputs, the points in the room, require three coordinates to describe them. Or if I want to work with the flow of fluid in a stream, there are three coordinates needed to pick a point and then at least three to describe what is going on there, to tell what direction and how fast the water is moving at that point. Most of the calculus for space will be in Math 234, but we will build the tools for describing it: Geometry with coordinates like you are used to but with more than 2 dimensions.

Here some facts about how the course will run. Check the course web site, http://www.math.wisc.edu/~wilson/Courses/Math222/222spr08/index.htm, for the latest information.

• Grading: There will be two midterm exams and a final exam. Your grade will depend upon those exams as well as the grade from your TA-led discussion section.
• Exams: Midterm exams will take place in the evenings, so you can have more time and more room than in the regular class: 5:30-7:00 PM on Thursdays, February 28 and April 24. Since you know the exam schedule now you should clear these times well in advance rather than waiting until the last minute! (I will not be giving midterm exams on other days. I may be able to work around a few conflicts for time-of-day only.) Rooms for those exams (and for the final) have not yet been assigned. The final exam time, 7:25-9:25 PM on Wednesday, May 14, is the time given in the UW timetable: There should be no conflicts with that time. Requests for alternate final exam times will not be normally granted and, in keeping with UW policy, will never be granted for times earlier than the official time.
• On a separate page (see the class website) you can find the course schedule. The reading assignments should be read before we reach that point in class. There are suggested homework problems: Your TA may assign others, and may have you turn in some as part of the discussion grade As the saying goes, “Mathematics is not a spectator sport.” You only learn it by doing it. I can also refer you to additional problem sets if you need more practice.
• If a day does not have an assignment shown that does not mean we have no class! Mathematics 222 is a very crowded course, with a great deal of material we need to get through if you are to be prepared for subsequent courses in this and other departments. So we need to keep moving. You will need to keep up. Note that I have not scheduled review time for the midterm exams in the lecture! But I have in some cases allowed more than one day for a section, or included catch-up time in case we fall behind.
• You are not required to have a calculator for this course. A scientific calculator might be helpful, if you are used to using it : Some of the assigned problems require you to do a significant amount of arithmetic. On quizzes and exams you will be allowed to use calculators, but the questions will be designed to be fair to those not using calculators.
  Similarly, you are not required to use any mathematical computer software. But if you are going into a mathematics-using field, as is likely for a student in 222, you will want to learn to use such systems since they have become very much the standard way to carry out mathematical calculations. I will be using the Maple system in class fairly frequently. I am registering this course with the Maple Adoption Program which will entitle you to get a student version of Maple at a reduced price and to get online instruction in its use. There are also student versions of other packages such as Mathematica and Matlab available, and all three of these are installed on some of the computer systems around campus in DoIT infolabs. I would be glad to talk with you about use of Maple and similar systems both in connection with this class and in your other courses.
• This course will include both theory and practice. You will learn to solve problems, but you also need to understand how and why the solution methods work so as to use them correctly.
• The text for this course is Thomas' Calculus, 11th edition, special Wisconsin version with 2nd order differential equations. This is the same text that we have been using for 221 for over a year, so if you are continuing from that you do not need to get a new book. Be sure you have the special version, that includes chapter 17 in the back and has a note on the front cover saying it includes the extra material. We will be using that material this semester!
• Some people get copies of the solutions book showing worked out answers to odd-numbered problems: This can be helpful if you consult it after you have worked a problem, to check both your answer and the way you got it, but can be very harmful if look there before working out your own solution. Don’t buy the solution book unless you think you can control your use of it!
• I will post general announcements, sample problems, etc. at the course web site, which is http://www.math.wisc.edu/~wilson/Courses/Math222/222spr08/index.htm. I will send fairly frequent email messages to the class: The university maintains the email list based on your email address on record with the registrar. So I cannot add or change email addresses, you must do that yourself through MyUW.
• My schedule, including office hours, is posted at my web site, http://www.math.wisc.edu/~wilson. My office is 411 Van Vleck Hall, where the telephone is (608) 263-5944. My email address is wilson@math.wisc.edu.