Fall 2000	Mathematics 1b - Fall 2000 Calculus, Series and Differential Equations	Course Head:  John Boller Office:  SciCtr 320 Email: boller@math.harvard.edu

Course Head: John D. Boller, SC 320, 495-5377, boller@math.harvard.edu

Sections: The course will be divided into approximately ten sections, each of which meets three hours per week and is taught by a teaching fellows (TF). Each section has a course assistant (CA) who leads an additional weekly problem session.

Topics: Three main topics will be covered in this course: techniques of integration, sequences and series, and differential equations. It is expected that you are already familiar with differential Calculus, principles of integration, and the Fundamental Theorem of Calculus.

Differential equations are used in virtually all of the natural and social sciences to model the world around us, and their solutions range from the relatively simple to the exceedingly difficult. Indeed, the determination of what differential equations even have solutions may be a complex problem. The simplest differential equation is one of the form dy/dx = f(x), where fis some nicely behaved function of the independent variable x. And yet, the solution to this differential equation is equivalent to finding an anti-derivative for f. This is not always a straightforward problem, and so we develop many techniques of integration to help us solve it. Some differential equations have solutions that we may only be able to approximate, and for these, representing a function as an infinite series (for instance as a power series, though in later courses, you may see other types of infinite series such as Fourier series) is invaluable.

While differential equations will serve as a unifying theme throughout, they are not the sole motivation for studying the more purely Calculus ideas of techniques of integration and infinite sequences and series, and substantial portions of the course will be devoted to these topics.

Grades: Your course grade will be determined as follows:

• homework and quizzes 30%,

• two midterms 20% each,

• final exam 30%.

Midterms:
First midterm Tuesday, October 24, 7-9 P.M., Science Center C and E.
Second midterm Tuesday, December 5, 7-9 P.M., Science Center C and E.
No calculators will be allowed in the exams.
No make-up exams will be scheduled barring extraordinary circumstances.

Text: For most of the course we will closely follow the textbook ``Calculus Early Transcendentals'' by James Stewart, published by Brooks-Cole and available at the COOP. In addition there will be handouts for some of the material on differential equations.

There are three main topics in the course: differential equations, techniques of integration, and sequences and series, and we will spend approximately one-third of the course on each. These correspond roughly to chapters 9 (and 17), 7, and 11 in the text, respectively.

Reading the text is an integral part of the course. It is preferable to read through each section before it is covered in class and then again after it has been covered. On the tests you will be responsible for all the material discussed in the text and the handouts, unless specifically excluded.

Math Question Center: Another resource available for your benefit is the Math Question Center (MQC), which runs every Sunday through Thursday from 8-10 P.M. in Loker Commons. The MQC is staffed by CAs from all the Calculus courses, some upper-level CAs, graduate students, and others, and its main purpose is to help Calculus students with their Calculus courses. (In a happy coincidence for many of you, the Freshman Snack is at 10 P.M. each evening in Loker.)

Homework: Homework will be assigned daily and will be due at the start of the next class. Your CA will return your corrected homework to you the following class. He/she will also put homework solutions on reserve at Cabot Library, and we expect to have solutions available on the website. Once solutions are made public, late homework will not be accepted for credit, though of course, you are still encouraged to do those problems. In calculating the total grade for your homework your three (two for the TuTh sections) lowest individual homework grades will be dropped. You are strongly encouraged to keep up to date with the homework, otherwise you will find that you do not get all you should out of the classes. It will also become increasingly difficult to catch up again.

You are encouraged to discuss the course with other students, your CA and/or your TF. It is much easier to learn mathematics if you have other people who will help you test your understanding and surmount problems. You are encouraged to discuss homework problems with other students, but you should always write your homework solutions out yourself in your own words.

Homework problems are an integral part of this course. It is impossible to 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 of the computations you perform, and the answers you get. Nothing prevents you from trying a few more problems in a given section if you feel it may do you some good. More generally the only way to learn any topic in mathematics is to work out examples for yourself.

Many homework problems 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 to 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.

Some students seem to subscribe to the ``Ten Minute Rule'': If you can not solve it in ten minutes, you cannot solve it at all. Nothing could be further from the truth: you will learn most from those problems which keep you busy more than ten minutes, whether you ultimately solve them or not.

Technology: In some of the homework problems you will be asked not to use any technology (calculators or software packages). If no restriction is made, you may use the form of technology of your choice (for example, the TI-85 calculator or one of the packages MATLAB or Maple). Make sure to have access to some form of technology.

No calculators will be allowed in the exams.

Day-by-Day Syllabus:

Date	Section	Assignment
Sept. 25	9.1	Modelling
Sept. 27	9.2	Direction Fields and Euler's Method
Sept. 29	9.3	Separable Equations
Oct. 2	9.4	Exponential Growth and Decay
Oct. 4	7.1	Integration by Parts
Oct. 6	7.2	Trigonometric Integrals
Oct. 9		COLUMBUS DAY
Oct. 11	7.3	Trigonometric Substitution
Oct. 13	7.4	Rational Functions
Oct. 16	7.5	Strategy
Oct. 18	7.7	Approximation
Oct. 20	7.8	Improper Integrals
Oct. 23	REVIEW
Oct. 24	MIDTERM
Oct. 25	9.5	The Logistic Equation
Oct. 27	9.6	Linear Equations
Oct. 30	9.7	Predator-Prey
Nov. 1	11.1	Sequences
Nov. 3	11.2	Series
Nov. 6	11.3	Integral Test
Nov. 8	11.4	Comparison Test
Nov. 10		VETERANS' DAY (OBSERVED)
Nov. 13	11.5	Alternating Series
Nov. 15	11.6	Absolute Convergence; the Ratio and Root Tests
Nov. 17	11.7	Strategy
Nov. 20	11.8	Power Series
Nov. 22	11.9	Representations of Functions
Nov. 24		THANKSGIVING
Nov. 27	11.10	Taylor and Maclaurin Series
Nov. 29	11.11	Binomial Series
Dec. 1	11.12	Applications of Taylor Polynomials
Dec. 4	REVIEW
Dec. 5	MIDTERM
Dec. 6	17.1	Second-Order Linear Differential Equations
Dec. 8	17.2	Nonhomogeneous Linear Differential Equations
Dec. 11	17.3	Applications
Dec. 13	17.4	Series Solutions
Dec. 15	Ch. 17
Dec. 18	REVIEW