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:
The material we take up in this course has applications in
physics, chemistry, biology, environmental 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.
- applications and methods of integration,
- infinite series and the representation of functions by
infinite polynomials known as power series,
- differential equations.
We will start the semester with integration. You should
already be familiar with the definite integral, its
definition as the limit of Riemann sums, and its calculation
using antiderivatives and u-substitution. The definite
integral enables us to tackle a multitude of problems in a
wide array of fields; we will begin the course by using the
notion of Riemann sums, (slicing, approximating, and
summing) to apply integration in various contexts. More
important than any one particular application is the ability
to apply the integration as appropriate in problem solving;
we will devote
time to developing your skill in doing this. We'll spend a
short time looking at methods of integration including the
integration analogues of both the Chain Rule and Product Rule
for differentiation. We'll look at some alternative
transformations of integrals that enable us to tackle them
In the second unit of the course we will approximate
familiar functions like exp(x), sin(x), exp(-x2), cos(x2)
by polynomials. The functions listed are challenging to evaluate
and some are challenging to integrate. By using polynomials of
increasingly large degree we can get increasingly good
approximations to the functions. In fact, we will find that
each of these functions has a representation as an infinite
polynomial. study infinite sums.
(You already are aware that a rational number such as 1/3
can be represented by an infinite sum,
for the case at hand). Actually, irrational numbers such as
have representations as infinite sums as well.)
We will end with differential equations, equations modeling
rates of change. Differential equations permeate
quantitative analysis throughout the sciences (in physics,
chemistry, biology, environmental 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
Text: 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.
Class and Problem Sessions: Math 1b is
taught in sections that meet three hours per week. Each
section of Math 1b has a Course Assistant who will be in
class, collect and correct homework assignments, and hold
weekly problem sessions. You are strongly encouraged to
attend these problem sessions as they are an integral 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 on the
course web site; feel free to go to any Math 1b Course
Assistant's Problem Session.
Homework: 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
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
You are welcome to collaborate with other students on solving
homework problems; in fact, you are encouraged to do so.
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. 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.
Math Help: The following sources of math help are
available without any appointment:
- weekly problem sessions lead by course assistants: go to as
many as you like
- office hours held by your section leader
- the Math Question Center: in Loker Common 8-10 pm every night
except Friday and Saturday.
Exams: Exams are common and given in the
evenings. Please keep these exam dates free from conflicts:
||Thurs. October 17
||Jefferson 250 plus a room tba
||7-9 pm (Jeff), 8-10 pm (SC)
||SC A and Jeff. 250
||Mon. November 25
||7-9 pm (Lib) 8-10 pm
||SC A and E, Yenching Library
||Mon. January 13
There will be an optional Technique re-Test
available on Mon. Nov. 5th: 6-7 in SC A and E. 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 demonstrating your mathematical knowledge as
opposed to your calculating ability. We expect you to
express your ideas, line of reasoning, and answers clearly.
Your course grade will be determined as follows:
- midterm score: 40% First Exam + 45% Second Exam +
15% Technique Test
- Course score: Take the higher of
- 45% Final Exam + 35% midterm score + 20%
- 30% Final Exam + 50% midterm score + 20% homework
Coursehead: Robin Gottlieb Science
Center 429, (617) 495-7882, email@example.com
TENTATIVE WEEK-BY-WEEK SYLLABUS:
- Sept. 23 - 27
- Definite integral as limit of Riemann Sums.
Applications: area under curve; area under rate curve
- Density and slicing. Total mass from density; total
population from population density.
- Slicing continued. Volumes by slicing.
- Sept. 30 - Oct. 4
- Volumes of revolution
- Average value; arc length
- Work: pulling, pushing, and pumping.
- Oct. 7 - 11
- Integration by substitution: the Chain Rule in Reverse
- Integration by Parts: the counterpart of the Product
- Partial Fractions
- Columbus Day: no class on Monday Oct. 14
- Oct. 15 - 18
- Recap plus with trig substitution - lightly.
- Improper integrals.
- Technique Test: Thursday, October 17
- Oct. 21 - 25
- Motivation for studying series. Geometric sums and
- Sequences and series. Infinite series: nth term test
for divergence and harmonic series.
- Exam 1: Thursday, October 24
- Oct. 28 - Nov. 1
- Introduction to comparison analysis. Direct comparison
(to geometric series).
- Comparison analysis continued. Integral test.
series. Comparison and limit comparison tests.
- Taylor polynomials: approximating a function by a
- Nov. 4 - 8
- Taylor series: representing a function by a power
- Absolute and conditional convergence. Alternating
Series Test and accompanying error estimate.
- Ratio test.
- Technique Retest: Nov. 5
- Veterans' Day: no class on Monday Nov. 11
- Nov. 12 - 15
- Power Series. Getting new power series from old ones by
substitution, differentiation and integration.
- Taylor polynomials and the Taylor remainder: Taylor's
Inequality. Taylor series and MacLaurin Series.
- Nov. 18 - 22
- More Taylor Series, including the Binomial Series.
Applications of Taylor Polynomials.
- Series Review and Recap
- Modeling with differential equations. Information
available from the differential equation itself.
- Nov. 25 - 27
- Exam 2: Monday, November 25
- Slope fields: dy/dt = 1, dy/dt = t, ty/dt = y, dy/dt = -t/y. Guess and check solutions.
- What does it mean to solve? Solving
dy/dt = ky and then dy/dt = ky + b.
What if you can't solve? Directions fields and Euler's
- Thanksgiving Vacation
- Dec. 2 - 6
- Separation of variables.
- Autonomous first order differential equations:
qualitative analysis of solutions.
- Solving first order linear differential equations.
- Dec. 9-13
- Systems of differential equations.
- Continue systems of differential equations.
- Series solutions to differential equations.
- Dec. 16 - 18
- Vibrating springs and second order linear homogeneous
differential equations with constant coefficients.
- Finish second order linear homogeneous differential
equations with constant coefficients.