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Math 1a - Introduction to Calculus - Spring 2000
Information and Syllabus

Course Head: Robert Winters, Sci Ctr 435, 495-4744, rwinters@math.harvard.edu

Course Website: www.courses.harvard.edu/~math1a

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 which meet three hours per week. You will also be attending a problem session once per week led by a Course Assistant (CA). Course Assistants grade homework, 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 problem sessions will be posted on the math bulletin board on the first floor of the Science Center and on the course website.

Question Center: In addition to class, problem sessions, and office hours, the Mathematics Department operates a Question Center in Loker on Sunday, Monday, Tuesday, Wednesday, and Thursday evenings from 8pm to 10pm. The Question Center will be staffed by Course Assistants from Math 1a, 1b, 21a, and 21b and by graduate students and others. You are encouraged to use this resource as you do your homework and when questions arise. It is intended to supplement the office hours held by your Section Leader.

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 strategies, 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. Your Section Leader or Course Assistant can provide names and contact information for everyone in your section in order to help facilitate discussion. Although we encourage you to work together with your classmates, all work submitted must be written up individually.

Problems will generally be assigned in each class and due at the next class. Assignments will be graded by your Course Assistant and will usually be returned at the following class 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 the homework or on the exams.

Homework solutions will be posted at the Math 1a web site or put on reserve in Cabot Library a few days after homework is due. You are encouraged to check the solutions. Late homework will not, in general, be permitted. The decision to accept late homework in some circumstances and the penalties for lateness rests with your CA.

Text: Calculus, A New Horizon - Brief Edition by Anton. 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:

Exam 1: Monday, Mar 6, 4-5:30pm in Science Center Hall E

Exam 2: Tuesday, Apr 11, 4-5:30pm in Science Center Hall E

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

Exam #1: 20%
Exam #2: 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. Calculators will not be allowed during exams unless we explicitly state otherwise.

Syllabus (in approximate order of occurrence and subject to minor modification):

Basics Notions - functions, domain, range, graphs, lines, slope, linear functions, etc.
Limits, neighborhoods, approximation, continuity
Rates of change, slope of a graph at a point
Velocity – average and instantaneous
Definition of the derivative, differentiability.
Calculation of derivatives graphically, numerically and algebraically
Techniques of differentiation
Higher order derivatives
Derivatives of the trigonometric functions
Composite functions and the Chain Rule
Linear approximation, differentials
Inverse functions and their derivatives
Logarithmic and exponential functions
Implicit differentiation
The Inverse Trigonometric Functions
Related rates
Indeterminate forms and L’H˘pital’s Rule
Analysis of graphs: extreme values of a continuous function, derivative tests, inflection points,
         concavity, asymptotic behavior, relative and absolute extrema, symmetry, etc.
Optimization
Rectilinear motion
Newton’s Method
The Mean Value Theorem
Antiderivatives
Area as the Limit of a Riemann Sum, 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
Improper integrals (if there's time)