Math 131: Calculus 2

Offered:     Winter 2000
Instructor:  Kevin J. Mitchell
Office: Lansing 305 
Phone:  (315) 781-3619
Fax:    (315) 781-3860
E-mail: mitchell@hws.edu

Office Hrs:  M & W 1:40 to 3:00, Tues 10:30 to 11:45. Often available at other times by appointment.

Classroom:   M-W-F 8:00 to 9:10 in Napier 101; Lab: Thurs 9:30 to 12:00 in Eaton 111                    
Text:        Calculus: A New Horizon, Vol 2 by Howard Anton           
Calculator:  TI-82 or equivalent

About Math 131

First-year calculus is usually divided into two parts (terms): differential calculus and integral calculus. Briefly, differential calculus is concerned with rates of change ("the slope problem") and integral calculus is concerned with area, in particular the area enclosed by arbitrary curves. The Fundamental Theorem of Calculus shows how these different concerns are essentially the two faces of a single coin. Having completed a "tour" of differential calculus, you are now prepared to focus on integral calculus.

To prove the Fundamental Theorem of Calculus requires us to connect the process of antidifferentiation to the notion of area under a curve. You know how to find the areas of some regions: squares, rectangles, triangles (using triangles you can figure out the area of any polygon), and circles. While you might be able to justify the area formula for a rectangle, it is unlikely that you could give a satisfactory proof for the area formula of a circle. In fact, there is a more fundamental problem here: What is area?

In differential calculus, the motivating problem was the paradoxical notion of an "instantaneous" rate of change. What do we mean by speed "at an instant?" Average rates or speeds are familiar ideas: divide distance by time. But in an instant, no time passes and no distance is travelled. So there is no (rate of) change! We resolved the paradox by using average rates of change over smaller and smaller time intervals. In the end, the instantaneous rate of change (or derivative) was defined to be the limit of average rates of change. But note, the concept of an "instantaneous" rate had to be defined through a fairly long process and was not "obvious."

In the same way, area, though familiar, is not an obvious notion. Some area formulas are familiar, but what is area? What are its defining characteristics? We will start with this problem and see that its solution has a wide variety of applications. In learning how to find area, we will also learn how to find or define the length of a curve, the volume of a solid, the work done by a force applied over a distance, and so on.

As with derivatives, limits will be crucial to the solution of the "integral" or area problem. This time the "paradox" will be that we add up lots of (i.e., an infinite number in the limit) of small (0-sized in the limit) areas to obtain the area of a figure. There are lots of questions to resolve: How do you add an infinite number of things? How do you divide a region up into smaller pieces whose area you know? Answers to such questions are what will motivate our "definition" of area.

Text and Calculator

We will be using the sixth edition of Howard Anton's text, Calculus: A New Horizon (Vol 2), starting in Chapter 7. We will focus on the concepts of calculus, not just the algebraic and formulaic manipulations. Though there are not many proofs, there are several "plausibility arguments " which are important for you to understand. It's a very "readable" text, so please spend lots of time with it! Long after most of you have forgotten the "rules of integration," I hope you still remember what the process of integration is and how it is used.

Technology can take some of the drudgery out of complex computations, but you still need to understand the theory of calculus to get the most out of your calculator. A TI-82 graphing calculator will be extremely useful in the course and will help us explore various aspects of functions. By the way, the guidebook for the TI-82 calculator is available as a free download at http://www.ti.com/calc/docs/82guide.htm

Labs

Every Thursday our section will meet for a problem solving laboratory. Attendance is required. These labs are an excellent time for you to ask questions about the course material. Bring your text and notes. The labs will make use of the TI-82 calculator. At the end of each lab, there will be a brief open-book quiz.

Assessment

Homework, reading, and practice exercises will be assigned at the beginning of each class. I encourage working in small groups on practice problems. This can be very helpful in understanding the material. I have put a copy of the Student Resource Manual on Resrve in the Library. It gives a few hints for the odd numbered problems.

Once a week, there will be an assignment consisting of selected problems to hand in for grading. Unless otherwise stated graded assignments are to be your own work without collaboration. Your work will be due at the beginning of the next class. No late assignments, please.

In addition to the weekly lab quizzes, there will be three hour tests and a final exam. The dates are listed in the outline below. Tests will be cumulative but will concentrate on more recent material. It is impossible to construct fair makeup exams in mathematics. For your own protection, my policy is that there are no makeup examinations. If for some extraordinary reason you find you are unable to take an exam, let me know as soon as possible, certainly well before the exam is administered.

Your course grade will be calculated as follows. First I will make a list of your grades: Homework and Quiz Scores Combined, Test 1, Test 2, Test 3, Final Exam, Final Exam. Note: The final exam is listed twice. Next, I will toss out the lowest score. (If the final is your lowest grade, it is removed just once.) Then I will average the remaining five grades. I also reserve the right to take class participation and attendance into account in determining final grades.

Because of the nature of this course, its assignments, and its assessment, your attendance and participation is crucial. Mathematics is learned by regular, sustained, attentive effort over an extended period. Only when such effort has been invested will the concentrated study for an exam have any benefit. Therefore attendance at class is required. Unexcused absences may adversely affect your grade; certainly more than three absences will lower your grade. If you must miss a class for some reason beyond your control, talk to me about it in advance.

Finally, common courtesy demands that you be on time for class and that you do not leave the room during class (unless you are ill). This will help you, your classmates, and me to give our full attention to the course.

Caution

Calculus 1 is a prerequisite for this course. If you did not pass Math 130 or its equivalent with a grade of C- or better, this course is inappropriate for you and you should see me immediately. Thus, I assume that you understand how to use the notions of limit and derivative and are familiar with their applications and interpretations. To be successful in this course, you must be able to build on a solid understanding of differential calculus. Again, if you are concerned about your background, see me immediately!

Math Intern

Remember that Math Intern, Dana Olanoff, is available in Lansing 309 for extra help during the day and the evenings. I will pass along her office hours as soon as I have them. Please utilize this extra resource.

Tips for Success

Here are a few simple things that you can do to be more successful in the course

Outline of Weekly Readings

This assumes a fairly rapid pace through the listed materials. We may need to adjust this schedule based on our actual work in class.