Course Head: Robert Winters, Sci. Ctr. 432, 495-8796, rwinters@math.harvard.edu.

Section Leaders: Bing Cheng, Peter Kronheimer, Greg Warrington, Sam Williams, and Robert Winters (2)

Goals: Math 1a is a first semester calculus course, covering differentiation, an introduction to integration, and applications. It emphasizes understanding as much as computation. When you leave Math 1a we want you to carry out with you the ideas that will enable you to use the concepts of the calculus later, both in mathematics and in other fields.

Prerequisites: Some of you will have had calculus before, some of you will have not. However, those of you who have not had calculus before need not be alarmed - in the past students without a calculus background have done as well as or better than those with this background. Doing well in Math 1a requires a solid background in precalculus, as demonstrated by an 18 on part 1 of the Harvard Math Placement Test.

Classes and Problem Sessions: Math 1a is taught in sections. Your section meets three hours per week. You will also be assigned to a problem session which meets once a week for 1 hour and is led by a Course Assistant (CA). Course Assistants grade homework assignments, attend classes, and hold weekly problem sessions. The problem sessions are in integral part of the course and will be devoted primarily to working problems and amplifying the material. Some course material not explicitly discussed in class may be taken up in the problem sessions. The pace of the course is quite fast, so these sessions should be particularly valuable to you in learning the material. You are strongly urged to attend. A schedule of all problem sessions will be posted on the math bulletin board on the first floor of the Science Center.

Homework: Homework exercises are an integral part of the course. It is unlikely that you will understand the material and do well on the exams without working through the homework problems in a thoughtful manner. Don't just crank through computations and write down answers - think about the problems posed, your strategy, the meaning your computations, and the answers you get.

We encourage you to form study groups with other students in the class so that you can discuss the work with each other. If you would like, your Section Leader or Course Assistant will distribute a list of names, phone numbers, and e-mail addresses of everyone in your section in order to help facilitate discussion. Although we encourage you to talk with your classmates and to work together, all work submitted must be written up individually.

Problems will generally be assigned in each class and are due at the next class. Assignments will be graded by your Course Assistant and will usually be returned at the following meeting.

Many of the problems for homework will look different from problems discussed in class and in the text. This is not an accident. We want you to think about the material and learn to apply it in unfamiliar settings and interpret it in different ways. Only if you understand the material (as opposed to merely knowing it) will you be able to go beyond the information you are given. In other words, the refrain "We didn't do a problem like this in class" will not wash in either the homework or the exams.

Your Course Assistant will put homework solutions on reserve in Cabot Library a few days after homework is due. You are encouraged to check the solutions. To allow extra practice (particularly before exams), we will be happy to provide the solutions to additional exercises in the textbook, not just those which were assigned.

Except in cases when an assignment is completely superseded by what was done in class or in the problem session, you may be allowed to turn in your homework late, but a penalty will be assessed. The decision whether to accept late homework and the penalties for lateness rest with your CA. Under no circumstances will any late homework be accepted beyond the last day of reading period.

Text: Calculus, Single Variable, 2^nd Edition, by Hughes-Hallett, Gleason, et al. Available at the COOP.

Exams: There will be two midterms and a final exam. The dates and times of the midterm exams are as follows:

Grading: The weights of the various parts of the course are as follows (subject to minor modification):

Each midterm: 20% Homework: 20% Final Exam: 40%

(A grade of C- or better on the final exam will guarantee a minimum passing grade for the course.)

Calculators: We encourage you to not rely too heavily on a graphing calculator as you work through your homework problems. Use the calculator to check your graphs if you must. That said, the use of a quality calculator can prove very helpful in understanding a good number of topics in the course from limits and successive approximation to graphing. While we recommend that you make use of these tools, we do not require them. Calculators will not be allowed during exams unless we explicitly state otherwise.

Syllabus: (in approximate order)

Basics - functions, domain, range, graphs, lines, slope, linear functions, etc.

Introduction to rates of change via examples with power functions and exponential functions

Velocity as a rate of change

Definition of the derivative - calculating numerically and algebraically

Tangent lines and linear approximation (part one)

Second derivatives

A few words from our sponsor - limits and successive approximation

Continuity and differentiability

Techniques of differentiation

Derivatives of the trigonometric functions

Derivatives of exponential functions

Rates of change and rectilinear motion

Composite functions and the Chain Rule

Implicit differentiation

Inverse functions and their derivatives

Related rates

Differentials and linear approximation (part two)

Newton's Method

Extreme Values of a Continuous Function

The Mean Value Theorem

First Derivative Test

Concavity and the Second Derivative Test

Infinite Limits and Asymptotes

Optimization

L'Hôpital's Rule

Antiderivatives

Area as the Limit of a Sum

Riemann Sums and the Definite Integral

The Fundamental Theorem of Calculus

The Mean Value Theorem for Integrals, Average Value

Numerical Integration: Trapezoid Rule, Midpoint Rule, and Simpson's Rule

Exponential and Logarithmic Functions

The Inverse Trigonometric Functions

A parade of applications