Harvard University,FAS
Fall 2001

Mathematics 1b
Fall 2001

Calculus, Series
and differential equations

Course Head: Robin Gottlieb

Office: SciCtr 429
Email: gottlieb@math

Head CA: Jonathan Yang
Email: jyang@fas
Math1b applied: a Biological system
Course Announcements

01/21/2002 - Review Material
These are the abbreviated solutions for the Fall 1999 Math1b Final. We hope it is of use to your studying. Note: these are the solutions only; the exam questions are here.
01/06/2002 - Review Information
Here is a list of the class reviews, coursewide reviews, and office hours that will take place over the next few weeks in preparation for the final on January 23.
12/28/2001 - Final Exam Practice
Here are some final exam practice problems with solutions (except for the Jan 2000 exam):
01/02/2002 - Second Exam Solutions
Here is the full solution set for the Second Midterm Exam.
12/07/01 - Exam 2 Solutions:
Here are some of the solutions for the second midterm. The remaining solutions will be posted shortly.
12/05/01 - Additional Problem:
Please note that there has been an addition to Wednesday's (or Thursday's) assignment, due this Friday or Tuesday of the coming week. It can be found on Handout H.
12/01/01 - Second Midterm Review Questions:
Here are two sets of review problems. Please note that the questions in the second set are more difficult than the questions you will likely see on Monday: Set 1 Set 2. Here are solutions for the first set of questions (except for the last problem on Probability). Solutions to the second set and the probability question will not be posted.
12/01/01 - Second Midterm Locations:
For the midterm this Monday, please take the exam in the following lecture halls depending on what class you are in:
If you are in Huan Yang's or Dmitry Tamarkin's class, please go to Science Center A to take the exam. If you are in Pete Clark's, Alina Marian's, or a MWF class, please go to Science Center C to take the exam.
11/21/01 - Second Midterm Review Material:
Here are some past exams for you to look at as practice for the upcoming second miterm. Because of the changes in course structure this year, some of the past exam questions may not be pertinent for our Second Midterm. Relevant questions are noted following each exam link (Note: The relevent questions are now accurate for each exam): Fall 1990 Exam 3 (Relevant questions: Part I #1-4 and Part II #1); Fall 1998 Exam 2 (Relevant questions: 1b, 2, 5-7); Spring 1999 Exam 2 (Relevant questions: 1-3, 4ab, & 7); Non-relevant problems are included so that you can use them to review for the final exam in January. Also, here are the solutions to the above exams: Fall 1990; Fall 1998; Spring 1999
11/16/01 - Technique Test Retest
There have been a handful of additions and corrections made to the integral practice problems and solutions. Please use these materials for the retest, which is on this coming Tuesday.
11/15/01 - Course Materials
Future homework assignments will come out of the handout received in class and a supplementary packet, in addition to the textbook. The supplement can be purchased at the Gnomen Copy on Mass Ave. across from Widener Library for about $3.00. This charge is simply to cover the cost of reproducing each book.
11/06/01 - Technique Test Prep IV
Here are some corrections on the solutions. Good Luck on the Test!
11/03/01 - Technique Test Prep III
The integral problems handout has been updated with a few additional problems and corrections to problems 22 and 32 - just go to the link listed in the 10/31 announcement to take a look. Here are the solutions; there may be a few errors remaining, but they should be accurate for the most part. If anyone comes across what they feel like is an error, please email me.
10/31/01 - Technique Test Preparation
Here is a list of integral problems from which at least 70% of the technique test questions will be chosen. Please check back on Friday in case there are additional problems posted or other changes. Solutions will be posted by Friday as well.
10/27/01 - Simpson's Rule as a Weighted Average
Please take a look at this handout that deals with Simpson's Rule.
9/26/01 - Taylor Series:
If you are interested in visualizing how Taylor series mimic complex functions, please visit one or both of the following websites and use their interfaces:
Previous Announcements
Announcements Regarding the First Midterm
Homework:      MWF       TTh

Course Information

Course Content and Goals:
About four hundred years ago, Galileo wrote
"The book of the universe is written in the language of mathematics."

Although the language of mathematics has evolved over time, the statement has as much validity today as it did when it was written. In Mathematics 1b you will become more well-versed in the language of modern mathematics and learn about its applications to other disciplines. Math 1b is a second semester calculus course for students who have previously been introduced to the basic ideas of differential and integral calculus. Over the semester we will study three (related) topics, topics that form a central part of the language of modern science:

  • infinite series and the representation of functions by infinite polynomials known as power series,
  • applications and techniques of integration,
  • differential equations.
The material we take up in this course has applications in physics, chemistry, biology, enviromental science, astronomy, economics, and statistics. We want you to leave the course not only with computational ability, but with the ability to use these notions in their natural scientific contexts, and with an appreciation of their mathematical beauty and power.

We will start the semester by studying infinite sums. You already are aware that a rational number such as $\frac{1}{3}$ can be represented by an infinite sum, ( $\frac{3}{10} + \frac{3}{100} + \frac{3}{1000}+ \frac{3}{10000}+ \cdots $, for the case at hand). Actually, irrational numbers such as e and $\pi $ have representations as infinite sums as well. In fact, we will find that many functions, such as $f(x) = e^x$ and $f(x) = \sin x$ can be represented by infinite polynomials known as power series. Polynomial approximations based on these power series representations are widely used by engineers, physicists, and many other scientists.

In your previous math courses you may have seen functions represented by integrals. For example, $\ln x$ can be represented by $\int_1^x \frac{1}{t} \, dt$. Integrals can be used in many contexts. The definite integral enables us to tackle many problems, including determining the net change in amount given a varying density. In the second unit of the course we will revisit integration. First we'll study the integration analogues of both the Product Rule and Chain Rule for differentiation and briefly touch on some alternative transformations of integrals that enable us to tackle them more efficiently. The goal is not to transform you into an integration automaton (we live in the computer age), but to have you acquire familiarity with the techniques and the ability to apply them to some standard situations. More important is the ability to apply the integration as appropriate in problem solving; we will devote time to developing your skill in doing this.

We will end with differential equations, equations modeling rates of change. Differential equations permeate quantitative analysis throughout the sciences (in physics, chemistry, biology, enviromental science, astronomy) and social sciences. In a beautiful and succinct way they provide a wealth of information. By the end of the course you will appreciate the power and usefulness differential equations and you will see how the work we have done with both series and integration comes into play in analyzing their solutions.

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.

Problem Sessions:
Each section of Math 1b has a Course Assistant who will be in class, collect and correct homework assignments, and hold weekly problem sessions. These problem sessions are 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 outside the Calculus Office (SC 308) and posted on the course web site; feel free to go to any Math 1b Course Assistant's Problem Session. Periodically there may will be group exercises scheduled during problem sessions - `homework' exercises meant to be worked on as a group and facilitated by a Course Assistant. You will be notified by e-mail when problem sessions will be utilized in this way.

Problems are an integral part of the course; it is virtually impossible to learn the material and to 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; 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 goal of teaching you the art of applying ideas of integration and differentiation 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. Calculators will not be allowed on examinations due in part to equity issues.

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

Odd-numbered problems are solved in the Student Solution Manual; some coies will be put on reserve in the Cabot Science Library. 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. (This is a standard rule of ethics.)

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

Exam 1 October 16 (Tuesday) 7:00 - 9:00 pm SC C and Jeff. 250
Technique Test November 7 (Wednesday) 6:15 - 7:15 pm. SC B
Exam 2 December 3 ( Monday) 7:00 - 9:00 pm SC C and SC A
Final Exam January 23 (Wedneday) TBA TBA

There will be an optional Technique re-Test available on Tues. Nov. 20th: 7:30 - 8:30 in SC C. The higher of your two scores counts in the computation of your course grade. The first test is not optional.

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 demostrating your mathematical knowledge as opposed to your calculating ability. We expect you to express your ideas, line of reasoning, and answers clearly.

Grading Policy:

Your course grade will be determined as follows:

exam score: Take the higher of
  • 25% first exam + 25% second exam + 10% technique test + 40% Final Exam
  • 15% first exam + 20% second exam + 5% technique test + 60% Final Exam
course score: Take the higher of
  • 85% exam score + 15% homework
  • 80% exam score + 20% homework

Sources of Academic Support:
In addition to your section leader's office hours (hours that you are free to come talk with him or her without appointment) and your Course Assistant's problem session, there is a Math Question Center in Loker Common. The Math Question Center is open from 8:00 to 10:00 PM every evening except for Fridays and Saturdays. The Math Question Center is staffed by both section leaders and course assistants. You can go there for help or simply to find other students with whom to discuss your work.

A schedule of all Math 1b problem sessions will be posted on the course website. You are welcome to go to any and as many problem sessions as you like.

Course Head
Robin Gottlieb     Science Center 429, (617) 495-7882, gottlieb@math.harvard.edu.

Tentative week-by-week syllabus

  • Week of Sept. 17 - 21 Geometric Sums and Geometric Series. Introduce the general idea of convergence of an infinite series.

    Infinite series in general. N'th term test for convergence. Harmonic series.

    Determining convergence by comparison to another series or to an improper integral.

  • Week of Sept. 24 - 28

    Alternating Series Test and accompanying error estimate. Absolute convergence. The Ratio Test.

    Power Series. Getting new power series from old ones by substitution, differentiation and integration.

    Representations of Functions as Power Series.

  • Week of Oct. 1 - 5

    Taylor polynomials and approximating functions by polynomials.

    The Taylor remainder and Taylor's Inequality.

    Taylor series and MacLaurin Series.

  • Week of Oct. 9-12

    Applications of Taylor Polynomials.

    Series Review and Recap

  • Tuesday, October 16th: Exam 1 in SC C and Jefferson 250 at 7:00 pm

  • Week of Oct. 15 - 19

    Functions as integrals. The Fundamental Theorem of Calculus.

    Integration by substitution.

    Integration by Parts.

  • Week of Oct. 22- 26

    Partial fractions and additional techniques of integration involving more sophisticated substitutions.

    Using series to integrate.

    Approximating definite integrals.

  • Week of Oct. 29 - Nov. 2

    Improper integrals.

    Applications of integration. Slicing problems: Total mass from density, total population from

    population density, etc.

    Areas and Volumes

  • Technique Test: 5:30 - 7:30 SC B on Wed. Nov. 7th

  • Week of Nov. 5- Nov. 9

    Applications of integration:

    volumes, arc length, average value, work, hydrostatic pressure and fluid force.

  • Week of Nov. 13- 16


    Modeling with differential equations.

  • Technique Test: Optional Take Two: 7:30 - 9:30 SC C on Tues. Nov. 20th

  • Week of Nov. 19 - 21

    What does it mean to solve a differential equation?

    Getting information without solving

    Directions fields and Euler's method.

  • Week of Nov. 26 - 30

    Autonomous first order differential equations: Qualitative analysis of solutions.

    Solving separable differential equations.

    Exponential growth versus logistic growth.

  • Monday, December 3: Exam 2 in SC C and SC A at 7:00 pm

  • Week of Dec. 3 - 7

    Using series to solve differential equations.

    Systems of differential equations: for example, predator-prey systems.

  • Week of Dec. 10 - 14

    Vibrating springs: second order linear homogeneous differential equations


Last update, 12/05/2001, math1b@fas.harvard.edu