Calculus I, II, and III: A Problem-Based Approach with Early Transcendentals:

Preface To the Student

How this Class Works

This class will be taught in a way that is likely to be quite different from mathematics classes you have encountered in the past. Much of the class will be devoted to students working problems at the board and much of your grade will be determined by the amount of mathematics that you produce in this class. I use the word produce because it is my belief that the best way to learn mathematics is by doing mathematics.

Therefore, just as I learned to ride a bike by getting on and falling off, I expect that you will learn mathematics by attempting it and occasionally falling off! You will have a set of notes, provided by me, that you will turn into a book by working through the problems. If you are interested in watching someone else put mathematics at the board, working ten problems like it for homework, and then regurgitating this material on tests, then you are not in the correct class. Still, I urge you to seriously consider the value of becoming an independent thinker who tackles doing mathematics, and everything else in life, on your own rather than waiting for someone else to show you how to do things.

A Common Pitfall

There are two ways in which students often approach my classes. The first is to say, “I'll wait and see how this works and then see if I like it and put some problems up later in the semester after I catch on.” Think of the course as a forty-yard dash. Do you really want to wait and see how fast the other runners are? If you try every night to do the problems then either you will get a problem (Yay!) and be able to put it on the board with pride and satisfaction or you will struggle with the problem, learn a lot in your struggle, and then watch someone else put it on the board. When this person puts it up you will be able to ask questions and help yourself and others understand it, as you say to yourself, “Ahhhh, now I see where I went wrong and now I can do this one and a few more for next class.” If you do not try problems each night, then you will watch the student put the problem on the board, but perhaps will not quite catch all the details and then when you study for the tests or try the next problems you will have only a loose idea of how to tackle such problems. Basically, you have seen it only once in this case. The first student saw it once when s/he tackled it on her/his own, again when either s/he put it on the board or another student presented it, and then a third time when s/he studies for the next test or quiz. Hence the difference between these two approaches is the difference between participating and watching a movie. Movies are filled with successfully married couples and yet something like 60% of marriages end in divorce. Watching successful marriages doesn't teach one to be successfully married any more than watching mathematics go by will teach you to do mathematics. I hope that each of you will tackle this course with an attitude that you will learn this material and thus will both enjoy and benefit from the class.

Board Work

Because the board work constitutes a reasonable amount of your grade, let's put your mind at ease regarding this part of the class. First, by coming to class everyday you have a 60% on board work. Every problem you present pushes that grade a little higher. You may come see me any time for an indication of what I think your current level of participation will earn you by the end of the semester for this portion of the grade. Here are some rules and guidelines associated with the board work. I will call for volunteers every day and will pick the person with the least presentations to present a given problem. You may inform me that you have a problem in advance (which I appreciate), but the problem still goes to the person with the least presentations on the day I call for a solution. Ties are broken randomly at the beginning of the course. Once the first test has been returned, ties are broken by giving precedence to the student with the lower test average. A student who has not gone to the board on a given day will be given precedence over a student who has gone to the board that day. To “present” a problem at the board means to have written the problem statement up, to have written a correct solution using complete mathematical sentences, and to have answered all students' questions regarding the problem. Since you will be communicating with other students on a regular basis, here are several guidelines that will help you.

Most importantly, remember that the whole class is on your side and wants to see you succeed, so questions are intended to help everyone, not to criticize you.

When you speak, don't use the words “obvious,” “stupid,” or “trivial.” Don't attack anyone personally or try to intimidate anyone. Don't get mad or upset at anyone. If you do, try to get over it quickly. Don't be upset when you make a mistake — brush it off and learn from it. Don't let anything go on the board that you don't fully understand. Don't say to yourself, “I'll figure this out at home.” Don't use concepts we have not defined. Don't use or get examples or solutions from other books. Don't work together without acknowledging it at the board. Don't try to put up a problem you have not written up.

Do prepare arguments in advance. Do be polite and respectful. Do learn from your mistakes. Do ask questions such as, “Can you tell me how you got the third line?” Do let people answer when they are asked a question. Do refer to earlier results and definitions by number when possible.

How to Study each Day

1. Read over your notes from class that day.

2. Make a list of questions to ask me at the beginning of the next class. (I love these!)

3. Review the recent problems.

4. Work on several new problems.

5. Write up as many solutions as you can so that you can copy these onto the board the next day.

What is Calculus?

The first semester of Calculus consists of four main concepts: limits, continuity, differentiation and integration. Limits are required for defining each of continuity, differentiation, and integration. All four concepts are central to an understanding of applications in fields including biology, business, chemistry, economics, engineering, finance, and physics. The second semester extends your study of integration techniques and adds sequences and series. The third semester of calculus is a repeat of these concepts in higher dimensions with a few other topics tossed in. All of these topics depend on a deep understanding of what might be called limiting processes, which is why limits are one of the topics that we will spend a lot of time with in the beginning and will recur throughout the entire course.

In addition to mastering these concepts, I hope to impart in you the essence of the way a mathematician thinks of the world, an axiomatic way of viewing the world. And I hope to help you master the important skill of solving some difficult problems on your own, communicating these solutions to your peers, posing questions respectfully, and responding to questions your peers have. These skills transcend calculus and will help you in all aspects of your life.