Math
Xb
Introduction to Functions and Calculus
II
Spring 2003
Course Orientation Information
Introduction
Math X aims to provide you with a deep understanding of
topics from precalculus and calculus as
well as a strong sense of what mathematics is useful for
and how you can apply mathematics in
your work and studies. We want you to leave the course
with a solid set of mathematical skills and
a high degree of mathematical confidence so that you are
well-equipped for future studies, whether
in mathematics, biology, chemistry, economics, or other
disciplines. In order to do this we will use
multiple approaches to problem solving and we will stress
understanding the ideas behind the math
formulas and techniques that we learn.
The sequence Math Xa/Xb covers all of the material in
Mathematics 1a. In terms of prerequisites
for other courses at Harvard, Math Xa/Xb is considered
equivalent to Math 1a.
This semester, in Math Xb, will also cover some of the material from
Math 1b as well: geometric series (of particular interest to those
pursuing economics and finance) infinite series, integration and
differential equations (with
biological, medical and economic applications). The specific topics
that you will learn about are shown below. A day-by-day summary of
the course can be obtained by clicking
here.
- Quantities defined by rate of change and differential
equations
- Euler's Method
- Slope fields and equilibrium solutions
- Symbolic solutions of differential equations
- Finding symbolic solutions of differential equations
using the technique of separation of variables
- Series
- Geometric series
- Applications of geometric series - Mortgages and IRAs
- Infinite geometric series
- Infinite series in general
- Convergence tests for infinite series
- Area under a curve and the change in a quantity
- Euler's method re-visited
- Series, limits and the area under a curve
- Integrals and methods of integration
- Area under a curve and the definite integral
- Antiderivatives
- The Fundamental Theorem of Calculus
- Finding antiderivatives of functions
- The u-substitution method for finding
antiderivatives
- Applications of integrals - volumes of revolution
- Applications of integrals - slicing problems
- Functions defined by integrals
- Trigonometry
- Modeling periodic phenomena
- Circle and triangle definitions of trigonometric
functions
- Inverse trigonometric functions
- Derivatives of trigonometric functions
- Antiderivatives of trigonometric functions
- Derivatives and antiderivatives involving inverse
trigonometric functions
Remember to try to enjoy yourself this semester - often students
take math because they feel that they
have to take it. In the same spirit as last semester,
we will make a real effort to use examples, homework problems
and laboratory investigations to deal with phenomena and issues that
people actually care about. For example, the image shown below was
made by a Department of Defense Satellite over Siberia in October,
1978. What does this image show? In Math Xb you will learn
trigonometry by analyzing images such as this to try to determine
just exactly what was photographed by the military satellite in 1978.
Many of the examples, homework problems and laboratory projects
will be drawn from the biological sciences and economics,
so you will have a chance to examine
some of the other subjects that you are interested in from
the point of view of mathematics and
modeling.
Format of the Course
Math X is taught in small classes in order to provide an
environment where students are active
participation and dialogue is promoted. Small class sizes
allow us to tailor the classes to your needs
and to offer you more individual attention. Normally, the
size of each section is limited to 15
students.
There will be twice-weekly math lab sessions. These labs
are designed to focus both on problem
solving and on conceptual understanding. If you think of
mathematics as a science, you can think
of the labs as science labs where you work on problem solving.
The labs that you will be working on this semester are:
- Modeling the spread of a non-lethal disease.
- Can male performance be naturally enhanced?
- The global impact of the 1986 Chernobyl disaster.
- The lifting power of a lighter-than-air aircraft.
- Evaluating evidence for the existence of extra-terrestrial
life.
- The optimal firing angle for a cannon.
Course head
| Name |
Dale Winter |
| Office |
Room 430, Science Center |
| Phone |
(617) 495-5735 |
| e-Mail |
amanita@math.harvard.edu |
I am here to help ensure that the class runs smoothly for you.
My main responsibility is to
coordinate all of the sections of the class, so that they all
run uniformly. To this end, you should
feel free to contact me at any point during this semester if
any issues arise, such as a family
emergency, which might cause you difficulty in keeping up
with the class. In general, you should
contact your section leader first, to let them know what is going on.
This semester my office hours are:
- Monday: 2:30-3:30pm, 430 Science Center.
- Tuesday: 8:00-10:00pm, Loker Commons.
- Wednesday: 2:30-3:30pm, 430 Science Center.
- Thursday: 8:00-10:00pm, Loker Commons.
Any student (from any section) in Math Xb is welcome at
any of those times.
Teaching Fellows
Course Text
As was the case in Math Xa, there is no required text for
Math Xb. If you are interested in reading more about the
topics that we will look at during the semester, any book on
"Single Variable Calculus" or just "Calculus" that is at
least one inch thick is guaranteed to contain a lot of
relevant material.
Calculators
Any graphing calculator will be an tremendous
asset in this course. I strongly recommend that you
use a calculator and bring your calculator to class
and lab each day. You will be allowed to use
your calculators on all tests and examinations except
for the gateway tests.
Four notes about calculators:
- At a minimum, you should use a calculator with
the equivalent capabilities of a Texas Instruments
TI-83. As long as your calculator can do everything
that a TI-83 calculator can do, then you will not be at
any disadvantage in Math Xb.
- In Math Xb you will have to use your calculator to
evaluate very long sums and complicated known as
series. The commands that some calculators use
(especially the TI-85 and some of the non-TI models) can be
quite cryptic and very difficult for people who are not
experienced with graphing calculators to learn. The course
personnel in Math Xb are most experienced with the commands
used by the TI-83 and TI-83 Plus calculators and can give
you a lot of help with these calculators, and not very
experienced with any other kinds of calculators. If you
plan to use any other make or model of calculator, please
make sure that you have access to the instruction manual so
that you can figure out how to operate your calculator when
we start to use its more complicated capabilities.
- There are now some very powerful calculators (especially
the TI-89 and TI-92 models) that can perform many of the
symbolic manipulations found in courses like Math Xa and Math
Xb. For example, both the TI-89 and TI-92 can find formulas
for derivatives and many antiderivatives. You are very
welcome to use any make and model of calculator that you
would like. (As you probably noticed from Math Xa, a very
sophisitcated calculator does not necessarily give you any
advantage on exams - and can actually be a disadvantage if
you don't know how to use it very well).
- As was the case with Math Xa, we will write the tests so that
no one particular brand or model of calculator will give you an
advantage. So long as your calculator can do everything that a
TI-83 can do, you will be fine.
Math Question Center
During the semester, the Math Department operates and staffs a
drop-in center where you can go for help with your math courses.
This Math Question Center is staffed by the course assistants from
courses Xa through 21b, as well as some of the graduate students and
faculty from the Math department.
The Math Question Center is normally open between 8pm and 10pm, Sunday
through Thursday. (It is closed on Friday and Saturday night.) At
present the Math Question Center is located in Loker Commons. It will
open in the next few weeks. For the latest news on the Math Question
Center,click
here.
Grading, Homework, Tests and Exams
Grade Breakdown
Your semester grade is based on a weighted average of all of the
scores that you accumulate throughout the semester. The table given
below shows the weight that will be given to each of the parts of the
course.
| Component |
Percentage |
| Labs |
15% |
| Homework and Gateway Tests |
20% |
| Mid-terms1 |
35% |
| Final Exam |
30% |
| Total |
100% |
1 Your higher-scoring mid-term will be worth 20%, and the
other mid-term will be worth 15%.
If your grade on the final exam is higher than the grade from your
composite score, then your final grade for the course will be the same
as your grade on the final exam.
The Curve
As was the case with Math Xa, we will have a simple was to convert
numerical scores
into letter grades. This method is:
| Range of numerical values |
Corresponding Letter |
| 90-100 |
A |
| 80-89 |
B |
| 65-79 |
C |
| 50-64 |
D |
| 0-49 |
E |
When the course head calculates your final grade at the end of the
course, he will calculate a score on a 0-100 point scale using the
scores that you have obtained during the course, and using the grade
breakdown given above. Your course grade will then be obtained using
this table. In the event of a fractional score, the course head will
always round up to the nearest integer. The course head may modify
these letter grades with a "+" or a "-" if both he and your section
leader believe that your performance in the course warrants this.
There is only one set of circumstances under which the course head
will deviate from the policy outlined above. This will be to ensure
that at least 20% of the people in the class get grades of "A" or "A-"
and at least 30% of people in the class get grades of "B+," "B" or
"B-."
Homework
Each week you will be assigned three homework assignments.
Generally speaking, homework is due at the next
section meeting. For example, the homework assigned on
Monday is due
at the beginning of class on Wednesday, etc.
You can find the assignments and solutions by clicking here.
Solutions to the homework assignments will be posted on the course web
site very soon after the homework is collected. For this reason,
no late homework can be accepted except in the case of
unavoidable personal emergencies (such as hospitalization) or an
absence from class that is officially sanctioned by Harvard
University.
In much the same way as in Math Xa, at the end of the semester, your
three lowest homework assignments will be dropped.
Just as in Math Xa, each homework assignment will consist of five
questions. Over the
course of the semester, the composition of these questions will
average out to be something like:
- 65%: Mathematical operations, solving equations, amking
calculations. Fairly straight-forward uses of course content and
concepts that do not involve complicated applications or
modeling.
- 25%: Mathematical modeling of phenomena, starting with a
fairly precise and explicit description of the phenomena that can be
readily translated into mathematical symbols, graphs, etc. (Problems
like this will typically also involve calculations, solving equations,
etc.)
- 10%: Investigations into more complicated phenomena.
(This will likely also involve both modeling and mathematical
operations.)
Exams
The course-wide exams will be given on:
- Monday, March 3. 7:00pm-9:00pm. Science Center C.
- Monday, April 7. 7:00pm-9:00pm. Science Center
C.
- The final exam will be held somewhere between May 15
and May 23,2003. (We will update this when we know
the day for sure.)
If you find that you have an unavoidable conflict with either
of the first two exams times, please contact
the course head at the first opportunity. If you have a truly
unavoidable conflict with the final exam time,
then you must petition the registrar. Neither the course head
nor any of the teaching fellows can
change the time or day of the final exam, and are expressly
forbidden from making special
arrangements for individual students to take the final exam on
alternate days or at alternate times.
Gateway Tests
Just as in Math Xa, there will be gateway tests in Math Xb.
The idea
of the gateway tests is to provide you with a powerful incentive
to learn how to do the fundamental calculations and operations
in a particular subject area.
In Math Xb there will be three gateway tests during the
semester. As in Math Xa, these will be given during lab times and you
will have a large number of opportunities to make-up gateway tests
with no penalty whatsoever.
As in Math Xa, these tests will be straight-forward tests of
fundamental skills. You may take the gateway tests as many times as
you need to. Before each test, you will be provided with an extensive
collection of practice problems (complete with answers) that will
seem eerily similar to the problems that appear on the actual gateway
tests. In our experience, students who make an earnest effort to work
out all of the practice problems usually have little trouble passing
the gateway tests.
To take a look at a sample gateway test, click here.
(This is an example of a "Calculating Antiderivatives" gateway test.)
The subjects of the gateway tests and the times that they will be
given during class are:
- Gateway 1 (Concepts of series): Tuesday February
25.
- Gateway 2 (Calculating antiderivatives): Tuesday March
18.
- Gateway 3 (Trigonometry): Thursday April 24.
You can re-take the gateway tests during any Section Leader's office
hour or during any of the optional lab sessions. These optional labs
will be held on:
- Thursday March 6.
- Thursday March 20.
- Tuesday April 10.
- Tuesday April 29.
- Thursday May 1.
When you pass a gateway, it counts the same as one perfect score for a
homework assignment.
For every gateway test that you have not passed by 5pm on
Friday May 2, your final grade for the whole course will
be
reduced by one letter. For example, if you don't pass any of the
gateways by 5pm on Friday May 2, then your "A" would
turn
into a "D."
Day-by-Day Guide to Xb Spring 2003
To view this, click here.
Return to
MathXb Course Page