Fall 2003

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# Mathematics 1a

Office: Science Center 430
Email: gottlieb@math.harvard.edu

Calculus is the mathematics of change and motion; it enables us to apply our knowledge of static situations to dynamic situations. We know how to find the slope of a line; calculus enables us to find the slope of a curve. We know how to find the area of a rectangle; calculus enables us to find the area between curves. We know how to find average rates of change; calculus enables us to calculate instantaneous rates of change. Derivatives, which measure instantaneous rate of change or the slope of a curve, and integrals, which can be used to compute areas between curves or net accumulation, together form the basis of our study of one-variable calculus this semester. Differentiation and integration are intimately related to one-another. This relationship is the focus of the Fundamental Theorem of Calculus. In Mathematics 1a we will explore the ideas of calclus and apply them to problems of optimization, graphing and problems from many disciplines.

The approach taken throughout the development of calculus is to get successively better and better approximations of a quantity we wish to compute. These successive approximations, when followed by a limiting process, allow us to get a handle on problems that originally may seem intractable. So in essense it is limits that enable us to apply our knowledge of static situations to dynamic ones.

The roots of calculus go back to the ancient Greek mathematicians Eudoxus and Archimedes, but it wasn't until the mid to late 1600s that Newton began to talk about limits, albeit informally, and that Leibniz introduced much of the notation in use today. Later mathematicians put Newton's notions on firmer ground.

Today the applicability of calculus is so broad that fluency in calculus is essential for not only mathematicians, but also for biologists, environmental scientists, physicists, economists, engineers, and many others. When you leave Math 1a, you will carry with you concepts and ideas from calculus that you can apply in your further studies, both in mathematics and in other fields.

The fundamental topics in this course, the derivative, the integral, and the limit, are presented and analyzed from several vantage points: symbolically, graphically, numerically, and as relationships between quantities. All of these perspectives are important to gaining a deep and thorough understanding of calculus. Your job, with the help of your instructor, is to learn calculus by reading about, thinking about, and experimenting with its ideas and practicing what you learn.

If this is a typical Math 1a group, many of you have had exposure to calculus before. However, those of you who haven't need not be alarmed: in the past students without a calculus background have done as well as those with it. Doing well in Math 1a requires a solid background in precalculus, as demonstrated by a score of 18 or more on part 1 of the Harvard Math Placement Test.

Single Variable Calculus: Concepts and Contexts by James Stewart. Second edition, Brooks/Cole 2001. This text is available at the Harvard Coop. There will be supplementary material available as well.

Math 1a is taught in sections that meet three hours per week. Each section of Math 1a has a Course Assistant who will be in class, collect and correct homework assignments, and hold weekly problem sessions. You are strongly encouraged to attend these problem sessions as they are an integral part of the course and will be generally be devoted to working problems and amplifying the lecture material. The pace of the course is rather fast, so these sessions should be particularly valuable to you in learning the material. A schedule of all problem sessions will be posted on the course web site; feel free to go to any Math 1a Course Assistant's Problem Session.

Problems are an integral part of the course; it is unlikely that you will learn the material and do well in the course without working through the homework problems in a thoughtful manner. Don't just crank through computations and write down answers; {\it think} about the problems posed, the strategy you employ, the meaning of the computations you perform, and the answers you get. It is often in this reflection that the greatest learning takes place.

An assignment will be given at each class meeting. Unless otherwise specified, the assignment is due at the following class meeting and will be returned, graded, at the subsequent class. If you miss a class, then you are responsible for obtaining the assignment and handing it in on time. Solutions put together by the course assistants will be available on the course website. When your homework assignments are returned to you, you can consult the solutions for help with any mistakes you might have made. Problem sets must be turned in on time. When computing your final homework grade, your lowest two homework scores will be dropped if you are in a TTh section and your lowest three homework scores will be dropped if you are in a MWF section.

Note that homework problems will sometimes look a bit different from problems specifically explicitly discussed in class. To do mathematics you need to think about the material, not simply follow recipes. (Following preset recipes is something computers are great at. We want you to be able to do more than this.) Giving you problems different from those done in class is consistent with our goals of teaching you the art of problem solving and applying ideas of calculus to different contexts. Feel free to use a calculator or computer to check or investigate problems for homework. However, an answer with the explanation  because my calculator says so" will not receive credit. Use the calculator as a learning tool, not as a crutch.

You are welcome to collaborate with other students on solving homework problems; in fact, you are encouraged to do so. However, write-ups you hand in must be your own work, you must be comfortable explaining what you have written, and there should be a written acknowledgement of collaboration with the names of you coworkers.

Odd-numbered problems are solved in the Student Solution Manual. After working on the problems on your own, you are free to consult this manual provided you acknowledge the use of this manual in your submitted work.

The following sources of math help are available without any appointment:

• weekly problem sessions lead by course assistants: go to as many as you like

• the Math Question Center: in Loker Common 8-10 pm every night except Friday and Saturday.

Exams are common and given in the evenings. Please keep these exam dates free from conflicts:

 Exam 1 Tuesday, October 21 7:00-9:00 pm SC C and SC D Exam 2 Monday, Nov. 24 6:00-8:00 pm Boylston 110 [Fong] and 2 Divinity Ave 18 [Yenching] Final Exam Tuesday, January 27, 2004 TBA TBA

There will also be a short quiz before the drop/add date; the quiz score will count towards your homework grade.
Calculators will not be allowed on examinations, due in part to equity issues. We will make sure that problems on the exams require minimal calculation to allow you to spend your time demonstrating your mathematical knowledge as opposed to your calculating ability. We expect you to express your ideas, line of reasoning, and answers clearly. If, for some reason you have an unmovable conflict with an exam time you are expected to e-mail the course head at least one full week in advance so we can make an attempt at accommodation. Please do not schedule Thanksgiving departures for before the date of the exam.

Your course grade will be determined as follows: Take the higher of

• 45% Final Exam + 20% Exam 1 + 20% Exam 2 + 15% Homework

• 35% Final Exam + 25% Exam 1 + 25% Exam 2 + 15% Homework

• 30% Final Exam + 25% Exam 1 + 25% Exam 2 + 20% Homework

You must pass the final to earn more than a C in the course.

Robin Gottlieb
Office: Science Center 430
Phone: (617) 495-7882
Email: gottlieb@math.harvard.edu