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Math 1b - Spring 2001

Calculus, Series and Differential Equations

Course Head: John Mackey, Science Center 331, 496-5211, jfm@math.harvard.edu

Course Outline: In Math 1b we will motivate ideas and techniques from three topics in calculus:

You should already be familiar with the definite integral, its interpretation as the limit of Riemann sums, and its calculation via antiderivatives and u-substitution. We will begin the course by using the notion of Riemann sums to calculate areas and volumes of objects by slicing them into smaller and simpler pieces. We will then prepare to solve problems concerning arc length and surface area by learning more sophisticated integration techniques.

We will then turn our attention to equations which relate a function to its rate of change and possibly higher order derivatives. Such differential equations are common in physics, chemistry and the life sciences. For example, under ideal conditions it is reasonable to assume that the rate of change of a population with respect to time is proportional to the size of the population itself. We will ponder some similar models and develop techniques to recover the underlying function from the differential equation. We will also develop techniques to analyze the behavior of the underlying function without actually finding a formula for it.

Lastly, we will show how one can replace complicated functions such as exponentials or sines with large polynomials. While the resulting expression will not be exact, it can frequently be made to be as exact as necessary by choosing a large enough polynomial. For example, the sine of x is very close to x whenever x is small. Even better approximations are x - x3/6 and x - x3/6 + x5/125. We will see how to continue getting better and better approximations. Finally, we will use these large polynomials to solve differential equations for which we previously had no technique.

Classes and Problem Sessions: Math 1b is taught in sections which meet three hours per week. Your Course Assistant (CA) will also hold a weekly problem session. I strongly encourage you to attend these problem sessions as they are an integral part of the course and will be devoted primarily to working problems and amplifying the material. A schedule of problem sessions will be posted 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 difficult to understand the material and do well on the exams without working through the homework problems in a thoughtful manner. Please think about the problems posed, your strategies, the meaning of your computations, and the answers you get.

Homework is due at the beginning of the class period following the one in which it has been assigned. Although discussion of the homework with your peers is encouraged, copying any part of another person's homework is not permitted. As a courtesy to the CAs late homework will generally not be accepted. If extreme circumstances cause an assignment to be late, the CA can determine whether to accept the homework. Homework solutions will be posted on the website each Friday afternoon.

Text: Calculus, Early Transcendentals by James Stewart. Available at the COOP.

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

Exam 1: Tuesday, March 6, 7-9 p.m. in Science Center Hall C

Exam 2: Tuesday, April 17, 7-9 p.m. in Science Center Hall C

Grading: Your course grade will be determined as follows:

Exam 1: 25%
Exam 2: 25%
Homework: 20%
Final Exam: 30%

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.

Week by week schedule (tentative):

Wednesday, January 31: Course Orientation, 8 a.m., Sc C

Week 1 (Feb 5-9):

Week 2 (Feb 12-16):

Week 3 (Feb 20-23):

Week 4 (Feb 26-Mar 2):

First Mid-Term March 6

Week 5 (Mar 5-9):

Week 6 (Mar 12-16):

Week 7 (Mar 19-23):

Week 8 (April 2-6):

Week 9 (April 9-13):

Second Mid-Term April 17

Week 10 (April 16-20):

Week 11 (April 23-27):

Week 12 (April 30 - May 4):

Reading Period (May 5 - 16)

Final Examination, Scheduled by the Registrar

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