College of the Holy Cross Mathematics and Computer Science

Table of Contents

1. Is This The Right Course For You?
2. Course Description
3. Advice On How To Succeed In This Class
4. Notes on Calculators
5. Textbook
6. What Will Class Meetings Be Like?
7. Grading Policy
8. Course Schedule
9. Final Examination

Is This The Right Course For You?

This course is an intensive section of MATH 131 (Calculus for the Physical and Life Sciences 1). We will use the same textbook and cover the same material (roughly the first six chapters of the text) as the five sections of MATH 131. Both MATH 133 and MATH 131 are first calculus courses - neither assumes that you have any calculus background. Both are intended for students planning to major in mathematics or the sciences, or planning to enter the premedical program. (The department also offers another introductory calculus course, MATH 125, intended for students majoring in the social sciences.) The most obvious difference between MATH 131 and MATH 133 is that Intensive Calculus meets 5 days a week rather than 4. The extra hour allows us to spend a little more time on difficult topics, to review precalculus topics as necessary, to spend more time discussing homework problems, and to work on problems in groups. It is designed for those who feel that they could benefit from the extra class time.

Course Description

Calculus is the mathematics of change. First developed in the 17th century, it has been at the center of mathematics and science ever since. Calculus is important because it is the basis for a major portion of the science and technology that shape the contemporary world. Many of the techniques used to study motion of objects in physics, kinetics of chemical reactions, growth or decline of populations of organisms in biology, growth of national economies, and many other phenomena in the real world involve calculus. Although it sounds like an exaggeration to say it, calculus is also one of the crowning achievements of the human intellect. You are in for an exciting journey of exploration as you learn it!

Two men, Isaac Newton and Gottfried Leibniz, are given most of the credit for developing the calculus. Their contribution was primarily explaining the relation between finding the rate of change of a function (the derivative) and computing the "total accumulation" of a function over an interval (the definite integral). We will conclude this semester by studying their big result -- the Fundamental Theorem of Calculus.

The topics to be covered this semester are:

• Unit I: The "library of functions" -- the basic polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions, their properties and applications (about 3 weeks)
• Unit II: The conceptual meaning of the derivative of a function: rates of change, tangents, and applications (about 2 weeks)
• Unit III: "Shortcuts" for computing derivatives (about 3 weeks)
• Unit IV: Applications of derivatives to the study of functions, optimization problems, and so forth (about 3 weeks)
• Unit V: The conceptual meaning of integration of functions and the Fundamental Theorem of Calculus (about 3 weeks)

See the course schedule below for a more detailed week-by-week breakdown of the semester. This course continues to MATH 134 in the spring semester.

Advice On How To Succeed In This Class

A good "work ethic" is key. You do not need to be a "math genius" to master this material and do well. But you will need to put in a consistent effort and keep up with the course.

Come to class. Unless you are deathly ill, have a genuine family emergency, etc. plan on showing up here at 10:00 am every day this semester. If attending class wasn't important, all college courses would be by correspondence, and your tuition would be much lower!

Read the textbook. Don't just use it to look for worked problems similar to ones on the problem sets. You will find alternate explanations of concepts that may help you past a "block" in your understanding. Reading a math book is not like reading a novel, though. You will need to read very carefully, with pencil and paper in hand, working through examples in detail and taking notes. Make a list of questions to ask in office hours or at the next class. One thing to bear in mind while reading your text is that the result of an example is often secondary to the process used in obtaining the result.

Take notes and use them. This may seem obvious, but in my experience too many students diligently copy down everything on the board, and then never look at their notes again. Used intelligently, your notes can be a valuable resource as you work on problem sets and prepare for exams.

Set up a regular study schedule and work at a steady pace. It's not easy to play catch-up in a mathematics course, since every day builds on the previous one. Even though MATH 133 meets 5 hours/week, you should still expect to budget at least 6 hours in a typical week for work outside of class. The best way to use your time is to do a few problems, a little reading from the book, and reviewing of class notes every day.

Most importantly, if you are having difficulty learning something, get help as soon as possible. You can do this by asking questions during class (any time something isn't clear, but especially in the Thursday problem sessions), seeing me during office hours, or attending the Calculus Workshop in the evenings Sunday-Thursday.

Jodi Marani, class of '01 will be serving as Teaching Assistant for this course. She will be attending class, helping to answer questions on some group discussion days, and grading the weekly problem sets. If there is interest, we may be able to set up some review sessions with Jodi outside of class time as well.

Notes on Calculators

It is certainly true that a decent graphing calculator is a good investment for this and your other science courses. However a calculator will not be required and you will not be allowed to use one on several quizzes and exams when the goal is to make sure you know how to do certain things "by hand". The TI-82 and similar models are sufficient. The TI-86 contains some calculus features, but we will be using a program called Maple in the PC lab for many of these things, so it is definitely not necessary to have such a powerful calculator. The TI-89, TI-92 and other calculators with symbolic computation features are definitely not necessary. Because over-dependence on calculators can hinder your learning of mathematics, these will never be allowed on quizzes and exams.

Textbook

The text book for the course is Calculus, 2nd edition by Deborah Hughes-Hallett, Andrew Gleason, et al. (available in the H.C. bookstore). I think you will find reading and studying this book to be challenging, but ultimately rewarding. It is definitely not a standard calculus book --

• It consistently stresses qualitative understanding of the meaning and purpose of the mathematics, rather than mere proficiency at calculations.
• Many of the problems are non-routine; they require thought, not just calculation. Many are also open-ended -- there can be several different ``right answers''
• It is designed for use with mathematical computer software (or graphing calculators) that can perform routine tasks, freeing your time and effort for more interesting things!

What Will Class Meetings Be Like?

Most weeks, we will be following a schedule something like this:

• Monday, Tuesday and Wednesday will be devoted to lectures and group work on new topics. Some of these meetings will take place in the Haberlin 408 PC lab.
• Thursday will be devoted to questions on problems from the week's problem set, review for upcoming exams, or practice on material from the start of the week.
• The three in-class exams will be given on Fridays -- see the course schedule below for the dates. On most other Fridays, there will be 20 minute quiz. After that we might either go over the quiz questions or briefly introduce the topics for the following week.

So with these points in mind, regularly throughout the semester the class will break down into groups of 3 or 4 students for one or more days, and each group will work individually for most of those class periods on a group discussion exercise. I will be responsible for designing and preparing these exercises, and I will be available for questions and other help during these periods. At the conclusion of each discussion, the class as a whole may reconvene to talk about what has been done, to sum up the results, to hear short oral reports from each group, etc. Each group will keep a written record of their observations, results, questions, etc. which will be handed in. I will make copies of these, and return them with comments, for all members of the group.

Roughly once a week during the semester the class will meet in the Haberlin 408 PC laboratory for ``math lab'' classes. Some of these sessions will lead to a lab writeup assignment. (The due date will be announced when the assignment is given out.) We will be using an excellent, extremely powerful, general-purpose mathematical software system called Maple (the developers of the program come from the University of Waterloo in Canada, which explains the name). We will use Maple to produce graphs of functions, calculate numerical approximations to integrals, and compute symbolic derivatives and integrals of functions. But in fact Maple is so powerful that we will be using only a small fraction of what it can do. If you take additional mathematics courses at Holy Cross, you will probably use many other features of this same program. And programs like this are now a standard tool in many areas of science, engineering, and finance. Being able to use them effectively is also a valuable skill to have!

Grading Policy

Grading for the course will be based on

1. Three in-class tests, each worth 10% of the course grade,
2. A two-hour final exam, worth 20% of the course grade,
3. Weekly quizzes, worth 20% of the course grade. (There will be eight of these quizzes in all but I will use only the 5 highest scores.)
4. Written reports from small group discussions and computer labs -- one report from each group. Information regarding the expected format will be given out with the first assignment of this kind. Together, worth 15% of the course grade.
5. Weekly problem sets, worth 15% of your course grade. Most weeks, a problem set will be given out on Friday, covering the material in the next week's Monday, Tuesday, and Wednesday meetings. You should work on the problems through the early part of the week and bring questions to the Thursday problem meeting. The problem set will be due the next day (Friday) in class. No credit will be given for late homework, except in the case of an excused absence, or with my permission.

For all exams and quizzes, the topics to be covered and a list of practice problems will be given out well in advance of the date. The Thursday meetings of "exam weeks" will be used for review for the exam, since no problem sets will be due those weeks.

If you ever have a question about the grading policy or your standing in the course, don't hesitate to ask me.

Course Schedule

The following is an approximate schedule. Some rearrangement, expansion, or contraction of topics may become necessary. I will announce any changes in class and here.

Final Examination

The final exam for this course will be given Friday, December 8 at 8:30 a.m. in the regular classroom.