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\author{Marcus S. Cohen \and Edward D. Gaughan \and R. Arthur Knoebel \and Douglas
S. Kurtz \and David J. Pengelley\thanks{%
New Mexico State University}}
\title{Student Research Projects in the Calculus Curriculum }
\maketitle

\section{Student Research Projects in Calculus}

\section{New Mexico State University}

\medskip

\subsection{Contact:}

\textsc{David J. Pengelley}, Department of Mathematical Sciences, Las
Cruces, NM 88003-0001 \textsc{Phone:} 505-646-3901 \textsc{E-Mail:}
calculus@ nmsu.edu

\subsection{Institutional Data:}

New Mexico State University is a land grant university with an enrollment of
about 14,000. It has a graduate program offering doctoral degrees in
Agriculture, Education, Engineering, and the Sciences. Most undergraduates
are full time students and 28.4\% are minority students. Students in the
mainstream calculus courses are mainly engineering majors, of which 33.5\%
are minority students.

The Department of Mathematical Sciences has 36 full time faculty and 24
graduate assistants. There are 54 undergraduate majors, including students
from the College of Education with mathematics as a major teaching field.
The graduating class of mathematics majors this spring consists of 5 females
and 12 males. The department offers the B.S., M.S., and Ph.D. degrees.

\subsection{Project Data:}

The program at New Mexico State University began in the Fall of 1987 and has
expanded to include all courses in the calculus sequence. Six faculty and
five teaching assistants are teaching half of all our calculus classes with
our new materials, known as student research projects. All calculus classes,
both project and nonproject, have enrollments of 40-45 students. The text is 
\textit{Calculus and Analytic Geometry}, Stein, 4th edition. Funding has
been received from the National Science Foundation for the period June 1988
to December 1991.

\subsection{Project Description:}

Our program of instruction involves the use of ``student research projects''
in our mainstream calculus courses. A project is a meaningful and
substantial problem, harder and more involved than the usual homework
exercise found in the textbook. Students are usually given two weeks to work
on the problem. They must analyze the problem, decide on a strategy for the
solution, and bring appropriate mathematical ideas to bear on the problem.

Students in the calculus come from mathematics, engineering and the
sciences, and those in project classes are self selected in the sense that
they may choose between project and nonproject classes. In a typical project
class, there will be two two-week projects, quizzes and/or examinations, and
a comprehensive final examination. Class size and day-to-day organization of
the courses are similar for all classes. During the period when students are
working on a project, a ``project lab'' staffed by teaching assistants and
undergraduates is open 10-12 hours a week to offer assistance.

Students submit a written report giving their solution to the projects. The
project grade is based on mathematical content and written presentation. For
some projects, students work in groups of three or less.

Our department has been quite supportive and we have attracted other faculty
to our program. In addition, faculty at nine other schools have become
involved.

Assessment of the program is underway. Preliminary data suggests that in the
second and third semesters of calculus, pass rates are consistently higher
in the project sections than in the nonproject sections. Final examination
scores do not appear to be significantly different between project and
nonproject sections. In contrast, pass rates in the project sections of the
introductory calculus course have shown an inconsistent pattern. Some
30-45\% of students in project sections have indicated that projects were
the most valuable part of the course. Approximately 50\% of project section
students evinced a desire to take another projects course. These students
frequently cited reasons such as ``projects teach how to apply principles,''
``projects make you think,'' and preference for projects over exams. Another
student quote is ``I learned more from doing the project than I did from
cramming for an examination.''

We will be completing and publishing a book of projects this summer with
guidance for instructors on their use.

\bigskip

\section{Project Report}

\chapter{\medskip}

\subsection{History}

The calculus program at New Mexico State University began in the Fall of
l987. Two faculty members, Marcus Cohen and David Pengelley, decided to try
something different in their third semester calculus courses. Their aim was
to have students discover the excitement, intrigue, and beauty of calculus.
Consequently, during that semester, students were assigned three two-week
projects in place of the usual hour examinations. Professors Cohen and
Pengelley hoped that this experience would fundamentally alter their
students' view of what mathematics is all about, and simultaneously build
their self-confidence in what they could achieve through imaginative,
theoretical thinking. Students had to use the ideas of calculus, not just
cookbook techniques, to solve these problems. To succeed, students needed to
discover the calculus as the proper (and usually the only) tool that would
help them solve the problem. In a sense, the projects were theoretical---one
could not do them without an appreciation of the ideas behind the method.

The results were encouraging. Students were able to experiment with ideas,
ask relevant questions, and identify and read sections of the book that were
not covered in class. They were able to synthesize and unify ideas by
working on a complex problem over a long period of time. As one would
expect, good students found projects exciting and viewed them as an
individual challenge, while some of the failing students found it necessary
to seek intensive help from their professors. In many cases, the professors
were able to identify and correct plaguing misconceptions. As a result,
these students were able to get back on their feet in calculus.

In the spring of 1988, a faculty member, Douglas Kurtz, and a teaching
assistant, David Ruch, also started using student research projects in their
second semester calculus courses and the excitement started to spread. That
spring, Edward Gaughan and Arthur Knoebel joined the team and a proposal was
written to the National Science Foundation. In July 1988, the group was
awarded one of the first multiyear Calculus Curriculum Development Grants.
Since its modest beginnings, the program has expanded to include all the
courses in the calculus sequence, and next fall eight faculty and seven
teaching assistants will be using projects in half of the calculus classes
in the department. In addition, faculty at a number of other colleges and
universities have been using some of our projects in their calculus classes.

As a part of our grant funded by the National Science Foundation, four
faculty at neighboring institutions have worked closely with us in
developing our program. These faculty and their institutions are: David
Arterburn, New Mexico Institute of Mining and Technology; Adrienne Dare,
Western New Mexico University; Carl Hall, the University of Texas at El
Paso; and Richard Metzler, University of New Mexico. Their advice and
enthusiasm over the past two years has been very helpful.

\subsection{The Program}

Our program of instruction centers on the use of ``student research
projects'' in our calculus courses. A ``project'' is a meaningful and
substantial problem, harder, more open-ended and more involved than the
usual homework exercises found in textbooks. Students are given an extended
amount of time to work on the problem, usually two weeks. They must analyze
the problem, decide on a strategy for the solution, and bring appropriate
mathematical ideas to bear on it. The students need help in the beginning
but they are usually surprised at how quickly their problem-solving skills
develop. A sample project follows and more are included at the end of this
article.

\subsubsection{A Greenhouse Extension}

\begin{quote}
Your parents are going to knock out the bottom of the entire length of the
south wall of their house and turn it into a greenhouse by replacing some
bottom portion of the wall by a huge sloped piece of glass (which is
expensive). They have already decided they are going to spend a certain
fixed amount. The triangular ends of the greenhouse will be made of various
materials they already have lying around.

The floor space in the greenhouse is only considered usable if they can both
stand up in it, so part of it will be unusable, but they don't know how
much. Of, course this depends on how they configure the greenhouse. They
want to choose the dimensions of the greenhouse to get the most usable floor
space in it, but they are at a real loss to know what the dimensions should
be and how much usable space they will get. Fortunately they know you are
taking calculus. Amaze them.
\end{quote}

This project is a good example of one of our ``minimalist'' statements: only
the data essential to setting up a well-posed problem are given, and those
are given in words, with no clues as to how to turn them into mathematical
expressions. Instructors may expect to have to field many questions from
students as they first get started with this type of project. On the other
hand, some of our project statements tend more toward the ``roadmap''
approach, in which a student's progress through the project is guided in a
step-by-step process. Some such examples are included.

Projects may be used to enhance learning of material that students typically
find difficult. One such topic is vector algebra because of its placement in
our calculus sequence. Poor student performance on this material prompted us
to write projects related to vectors, vector algebra and the relations
between vectors and calculus. One such project involved the Starship
Enterprise being held captive in an elliptical orbit by the evil Klingons.
The students enjoyed this project and many entered into the spirit of it
with their own story line. Some students even took the time to learn about
the equations of ellipses in polar coordinates (which had not been covered
in class) and used polar coordinates to solve the problem.

Projects may also be used to cover important material that is not included
in the course. There have been proposals in recent years to eliminate
certain techniques of integration from the calculus course. Such topics can
be made a part of a project. Two of our projects have involved the use of
hyperbolic functions as a technique of integration. Students were asked to
read about these functions and then discover and prove the identities
necessary to make hyperbolic substitutions useful.

A few of our projects required computers and quite a number required a
calculator. However, our goal was to develop a program that could be used to
enhance the understanding of calculus without requiring sophisticated
equipment and software.

Writing is also a very important part of the process of completing the
projects. A written report must be presented to the instructor explaining
the solution, using correct English and correct mathematical language and
notation. Since the students have two weeks to work on the project, you can
expect to see their best work. A student's project grade will be determined
by the mathematical content and the written presentation.

The final task of a project is a discretionary ten to fifteen minute private
conference between student and instructor. One may choose to interview very
good students to encourage them to think about a career in mathematics. A
less successful student could be counselled on mistakes and what is needed
to improve on subsequent projects. These interviews play the additional role
of assuring us that the project solution was the student's own work.

We believe that students who are using projects are learning both critical
thinking and scientific writing, using calculus as their laboratory. We are
attempting to teach them the tools of mathematical thinking and application.

Mathematical thinking is, after all, at the foundation of science and
engineering, and the calculus sequence is the best place to learn it.

\subsection{The Setting}

New Mexico State University is the land grant university for the state of
New Mexico and has an enrollment of approximately 14,000. Most of our
calculus students come from the College of Engineering and the sciences. The
University has a large graduate program offering doctoral degrees in various
fields in Agriculture, Education, and Engineering, as well as Biology,
Chemistry, Computer Science, Mathematics, Physics and Psychology. Most
undergraduates are full-time students and university records show that
28.4\% of the undergraduates are minority students.

The Department of Mathematical Sciences has 36 full-time tenure-track
faculty and 24 graduate assistants. The department offers the B.S., M.S.,
and Ph.D. degrees in Mathematics.

Calculus classes at New Mexico State University enroll about 40 students
each, and the courses share a common syllabus. Each of the first two
semesters of calculus has a common final examination. A MWF class typically
meets 45 days a semester for 50 minutes each period. Because both project
and nonproject classes are offered to the student, we decided not to attempt
any changes in the course content of the three calculus courses. Students
may register for a project or for a nonproject class. Typically about half
of our calculus sections have been project-based.

\subsection{How It Works}

Individual project classes differ considerably from instructor to
instructor. For example, some instructors assign only two projects, while
others have assigned up to five projects in a semester. From our experience
we know that the project \underline{must} be a significant part of the
student's grade and that the length of time that students have to work on
the project must be specified and limited. These conditions are crucial for
creating the student commitment required to successfully complete a
challenging project.

What follows is a description of a typical class that assigns projects. In
this hypothetical course, there will be two in-class hour examinations, each
counting 15\% of the course grade; two two-week projects, each counting 15\%
of the course grade; graded homework worth 10\% of the course grade; and a
two hour comprehensive final examination worth 30\% of the course grade. The
first examination will be given sometime during the $4^{\text{th}}$ week and
the first project given out immediately after the first examination. The
timing of the project and the examination is adjustable based on what has
been covered in class, but we found it was not wise to have students working
on a project and preparing for an examination at the same time. The second
examination will be given during the $9^{\text{th}}$ week and the second
project given out shortly after the examination.

Before the project can be assigned, it must be developed and carefully
written. Writing a project, in contrast to writing an examination, may
extend over a period of several weeks. A typical project will begin with an
idea and both faculty and teaching assistants will have an opportunity to be
involved in the process of creation of the project. A large part of our
efforts over the past two years has been to create projects. We will publish
a collection of projects that includes some guidance for instructors on
designing and using them. The final writing and editing should be done in
the summer of 1990.

When our program began, we would distribute among the students three to six
different projects for each assignment. The idea was to discourage students
from copying each other's work. One of the main problems was the need to
assign projects of equal difficulty. Since we discovered that cheating was a
negligible problem, now we usually assign the same project to the entire
class. When the project is handed out, a cover sheet is attached that gives
advice and rules of the game. A copy of ``How to Work on Your Project: What
is Expected'' is included in the supplementary material. It is expected that
students will need assistance with the work and, in fact, they are
encouraged to seek counsel from their instructors, even if they think all is
well. As you might imagine, this can consume many hours of faculty time.

To ease the demand on faculty time, we have designed a laboratory where
students go and seek assistance on the projects. Initially, this lab was
staffed by teaching assistants who were also teaching a section of the same
course using projects. Now we have found that we can staff the lab with
senior undergraduates. This will make our program more practical at those
schools where teaching assistants are not available. Each of the three
calculus courses has a lab, open about twelve hours a week when students are
working on projects. We try to be open weekend and evening hours to
accommodate as many students as possible. In addition, we have some of the
lab hours set up for appointment only. This allows students with very tight
schedules to avoid long waits in the lab. The lab is closed the two days
prior to the day the projects are due. During these two days, students must
see their instructor, who has extended office hours, for help. The purpose
of this is to avoid unpleasant situations in the lab. Frantic students, who
waited until the last minute to begin, must face their own instructor.

The use of graduate teaching assistants in the projects lab and in teaching
their own calculus classes with projects has an added benefit: the teaching
assistants are trained in the method and contribute their own ideas and
energy to the program. One of our graduate students' innovative contribution
to the program proved valuable in his obtaining a university position. An
unexpected result of our program is the degree of camaraderie that has
developed within the group of faculty and teaching assistants involved with
projects. We meet regularly as well as talk informally among ourselves about
teaching and students' work on the projects. Teaching calculus has become a
common topic of coffee room conversation again.

We require students to write a report explaining their solution. Most
students are not accustomed to writing mathematics and our experience is
that their first efforts may be rather poor. It is a good idea to have
students show their instructor a preliminary version of the paper several
days before it is due.

For the second project of the semester students are encouraged to work in
groups of three or fewer. We allow the students to choose their own groups.
The only problem seems to be that some groups find their schedules don't
allow them to get together very often. We try to warn them about this
problem early enough to give them time to make up their groups. If the group
needs assistance, we allow a single representative to go to either the lab
or the instructor. In this way, that person must report back to the group,
encouraging mathematical dialogue among the students. Each group turns in a
single report and each member gives a confidential evaluation of the
relative contributions of all members of the group. All members of a group
will not necessarily get the same grade on a project.

Grading a project takes about the same amount of time as grading an
examination. It is less stressful since you are grading your students' best
work. They have thought long and hard about their problem, done background
reading, and discarded false starts. They have also eliminated algebraic
errors so the problem of determining partial credit is not a big issue.
Project grades are letter grades, and mostly A's and B's. If numbers of
students and other time constraints do not allow all students to be
interviewed, we interview selected students, usually those who did poorly
and need counselling or whose grades are borderline.

\subsection{Institutional Support}

Our department has been quite supportive. Since the beginning of the NSF
sponsored part of the program, the number of faculty teaching project
classes has grown from five to nine. Our department head has been very
encouraging and, with the approval of the college dean, she has given us
some released time to develop the program. The College of Engineering, our
primary source of calculus students, believes that we are teaching their
students valuable problem-solving skills and is supportive of what we are
doing. A few critics claim that students are working on projects at the
expense of other calculus skills. Results on common examinations seem to
refute that claim.

In the Fall of 1990, we will begin to integrate our program into the
department. The faculty of our department approved our proposal to give one
additional credit hour to students enrolled in calculus classes that assign
projects. This is partially in response to students' comments about the
amount of time they invest in a project. This will not involve any change in
the way project classes are handled.

\subsection{Is It Working?}

Assessment of the program is underway. Preliminary data suggests that pass
rates are consistently higher in the project classes than in the nonproject
classes, in the second and third semesters of calculus. Scores on final
examinations do not appear to be significantly different between project and
nonproject sections. In contrast, pass rates in the project classes of the
introductory calculus courses are inconclusive.

About 40\% of the students from sections which used projects have indicated
that projects were the most valuable part of the course. Approximately 50\%
of project section students evinced a desire to take another projects
course. These students frequently cited reasons such as ``projects teach you
how to apply principles,'' ``projects make you think,'' and ``I learned more
from doing the project than I did from cramming all night for an
examination.''

On the one hand, using projects in a calculus course is more work than
teaching in the traditional way. One has to find or design the projects,
someone has to arrange and schedule the lab, one spends more time talking to
students, and one may be involved in training other faculty to use projects.
But the extra time spent is high quality time. Students speak to you about
very specific questions that cut to the heart of what they are trying to do.
This is in contrast to the typical ``I don't understand anything'' type of
office visit. Good projects, often those with a story line, will excite the
students and their papers are a pleasure to read. One learns a lot more
about students using projects than using only examinations and we found that
our students are capable of much more demanding work than we ever imagined.
It is a unique and satisfying experience to compare notes on our students'
creative approaches to problems as opposed to bemoaning dismal performances
on examinations. Calculus is more fun to teach and, once hooked on projects,
it's hard to go back to the old ways.

\subsection{The Future}

As mentioned above, projects will become integrated into the departmental
program beginning in the Fall of 1990. Faculty participation in the program
is voluntary. So the long range future depends on us, the faculty, to make
the program attractive to both students and other faculty. The responses of
other faculty so far make us believe this can be done. We hope to export our
program to other colleges and universities by giving talks, workshops and
minicourses at meetings across the country. We will publish a book
containing about one hundred projects including guidance on constructing and
using projects.

Presently, we are experimenting with projects in differential equations, and
plans are in the works to include projects in the vector calculus course.
Projects have been tried in a college trigonometry course and in a high
school algebra course with encouraging results.

Beginning in the fall of 1990, we plan to work with several mathematics
teachers in the local high schools to incorporate projects into their
courses. This program will train teachers to create and use projects. We
hope to discover the manner in which projects may best be incorporated into
the high school curriculum.

\bigskip

\section{Sample Materials}

\medskip

\subsection{Instructions to TAs}

The purpose of this document is to describe our philosophy about your role
and obligations regarding ``calculus projects''.

You have a dual role in this department. You are both a student and a
teacher. It is important for all of us to not lose sight of this duality and
to keep the two roles in perspective. It is also important for you to know
that we as faculty recognize the importance of both roles. We do not want
your teaching to demand so much of your time that your graduate studies
suffer. On the other hand, you must be willing to devote the time necessary
to meet your responsibilities to your students.

As a teaching assistant involved in using projects, you have duties that
differ somewhat from those of other teaching assistants. Some of those are
detailed below.

\begin{enumerate}
\item  You must meet with your course coordinator before the semester
begins. Details, such as the number of projects, dates for projects, number
of examinations, homework, etc., must be discussed prior to the first class
meeting. Your syllabus should be designed in consultation with the course
coordinator and should include pertinent information relative to projects
and examinations. You should know enough about the use of projects to field
related questions on the first day of classes.

\item  At least a week before projects are assigned, you will receive copies
of the projects for you to solve. In order for you to be able to help
students with the project, you must understand it yourself, and the only way
to know the project is for you to work out a complete solution. If there are
several versions, it is your responsibility to work through each version.
You should discuss your solution with your course coordinator before the
projects are handed out.

\item  During the period that the projects are out, classes should continue
in the normal manner. Homework, quizzes, etc., occur as usual. There may be
occasions where it is appropriate to spend part of a lecture on a topic
directly related to the project.

\item  You will be assigned times to work in the lab for project students.
This is an important task and you should plan to be prompt and prepared for
the lab. You will be expected to keep a head count at the lab. Your course
coordinator will have discussed with you the types of hints that you can
give, since you will not actually solve the project for the students. An
important part of your job in the lab will be to help students understand
what is expected and keep them from heading down blind alleys. Do not agree
to proofread a project for someone else's student. This can cause problems.

\item  There will be a few days between the last day that the lab will be
open and the day the project will be due. During those days, you are
expected to have extended office hours for the purpose of helping your
students. Be careful that students don't camp in your office and work on
their projects. Perhaps you might want to post a sheet and have students
sign up for a time (perhaps at 10 minute intervals) to avoid a line in the
hall.

\item  After the projects are turned in, you should plan to read them and
return them promptly. The next class period is ideal; don't let more than
one class period pass before you return the papers.

\item  Read the project carefully, mark errors, note bad grammar and
incorrect use of language and notation. If the errors in language and
notation are excessive, make a note on the paper to that effect and don't
try to mark all the errors. You may want to correct some of the notation and
language errors, but don't correct the mathematics.

\item  Discuss with your course coordinator which students you should
interview. Return those papers without grades and ask the students to
schedule interviews. All other projects should be returned with grades
indicated. Grades for students to be interviewed will be assigned after the
interview and graded papers should be returned to the student at the first
class meeting after the interview if at all possible. Be prepared to discuss
the grade with the student if requested. It is not uncommon for students to
expect good grades on mediocre papers.

\item  After each project cycle is completed, be prepared to give your
course coordinator your impressions and comments.
\end{enumerate}

\subsection{Instructions to Students}

\subsubsection{Working on your Project: What is Expected}

Here are some words of advice and encouragement along with the rules of the
game.

First of all, this is a major, lengthy assignment. To do well you should
start immediately, and work on it every day. You will probably need all the
days you have been given in order to complete your project by the day it is
due.

\begin{enumerate}
\item  Start today. Let your subconscious work for you. It can do amazing
things. If you immerse yourself in the project, solutions will come to you
at the strangest times.

\item  Read the entire project to see what it's all about. Don't worry too
much about details the first time through. Do this today.

\item  Next, read the project very carefully and make a list of any
unfamiliar words or concepts you encounter. If concepts occur that you're
not sure about, you must understand those ideas before you can do the
project. Even if you understand all the words and terms, don't assume that
the project is easy. If you wait until the last few days or so to start, it
is doubtful you will be able to finish on time.

\item  You may need to do some outside reading. There are lots of books in
the library that contain information that might be helpful to you.

\item  After you have worked a bit every day on the project, you will find
certain parts easy and you will have completed those parts. You will have
identified the hard parts and have begun to zero in on the obstacles. You
are beginning to become the master of your project. You are so familiar with
it that it is easy to sit and work on it if you have a spare minute or two.
I recommend you work on the project some every day and keep a journal to
record your progress.

\item  While I expect you to work independently, I do not expect that you
can work through the project without some assistance. I encourage you to
come and talk to me about your project, and to get help in the special
Projects Lab for your course. Even if you think things are going along
smoothly, you should let me see what you are doing. This way I can head you
off if you are going in a wrong direction.

\item  There will be times when you need help and I'm not available. At
those times, you should go to the Projects Lab for assistance. It is open
several hours daily for most of the days you will be working on your
project. You will receive a separate schedule showing the Lab hours.

\item  Don't go to the Projects Lab expecting someone to work your project
for you. The purpose of the lab is to give you guidance and let you know if
you are on the right track. Come with specific questions, and be prepared to
show clear written work that you have prepared in advance.

\item  When you have done the work necessary to complete the project, you
need to prepare it in written form. The paper you turn in should have a mix
of equations, formulas and prose to support your conclusions. Use complete
sentences, good grammar and correct punctuation. Spelling is also important.
The prose should be written in order to convey to the reader an explanation
of what you have done. It should be written in such a way that it can be
read and understood by anyone who knows the material in this course. You
will be graded on your written presentation as well as the mathematical
content.
\end{enumerate}

\bigskip

\section{Sample Projects}

\medskip

\textit{In the following sections we provide a selection of sample projects
assigned at various times during the term.}

\subsection{Composing Functions}

Consider the two functions: 
\[
f(x) = 1 - x\text{ and }\ \ g(x) = {\frac{1}{x}}\text{ .} 
\]
We can compose them in two ways: 
\[
f(g(x))\text{ and }\ \ g(f(x))\text{ .} 
\]
We can go further and compose these two new functions with themselves, and
also with the old ones, in a number of ways. Keep composing these functions
with new ones as they are generated and figure out simplified formulae for
them in terms of the variable $x$ . (Don't forget to compose functions with
themselves, like $f(f(x))$.) You might think that more and more new
functions will be generated. Surprisingly only a finite number of new ones
are generated by composition, even though there may be many different ways
of composing $f$ and $g$ to get the same function. Remember that two very
different looking formulae may represent the same function.

\begin{itemize}
\item[a)]  How many distinct functions are there, including $f$ and $g$
themselves?

\item[b)]  List them.

\item[c)]  How is each one composed from $f$ and $g$?

\item[d)]  How do you know that these are all there are?

\item[e)]  For what real numbers are all these functions simultaneously
defined?
\end{itemize}

\subsection{A Continuous Additive Function}

In this project you will learn about additive functions and discover that
continuous additive functions have a very special form.

\textbf{Definition:} Let $f$ be a function whose domain is the set of all
real numbers. Then $f$ is \textit{additive} if $f(x + y) = f(x) + f(y)$ for
all real numbers $x$ and $y$ .

a) Give an example of an additive function and show that it is additive.
Give an example of a function that is not additive and show that it is not
additive.

b) Suppose that $f$ is an additive function and $m$ is any real number.
Define a new function $g$ by the formula $g(x) = f(x) - mx$. Show that $g$
is an additive function.

c) Let $m = f(1)$ and show that the function $g$ defined in part~(b) above
has the property that $g(x + 1) = g(x)$ for all $x$ .

\textbf{Definition:} A function $f$ is \textit{bounded} on a closed interval 
$[a,b] $ if there is a real number $B$ such that $|f(x)| \leq B$ for all $x$
in $[a,b]$. A function $f$ is \textit{bounded} if there is a real number $B$
such that $|f(x)| \leq B$ for all $x$.

d) Suppose that $f$ is an additive function that is bounded on [0,1] and
that $g$ is defined by $g(x) = f(x)-mx$ where $m = f(1)$. Show that $g$ is
bounded.

e) Let $g$ be a bounded additive function; that is, there is a real number $%
B $ such that $|g(x)| \leq B$ for all $x$. If there is a real number $a$
such that $g(a)$ is not zero, show that there is a real number $t$ such that 
$|g(t)| > B$, contradiction. What can you conclude about $g$? Explain. Hint:
what is $g(2a)$, what is $g(3a)$?

At the beginning of the project, you were told that you would discover that
continuous additive functions have a very special form. As you might
suspect, the preceding parts of this project are designed to help you in
that discovery. You will need to use some facts about continuous functions.

f) Show that if $f$ is additive and continuous, there is a real number $m$
such that $f(x) = mx$ for all $x$.

\subsection{Preparing for the 1990 Census}

The government census bureau is preparing to take the census of the United
States in 1990 and you have been hired to help them plan for the counting of
the American population. You job is to make a projection of the population
so that the bureau will know how many census takers to hire.

Your director, Dr. John Knowitall, wants you to proceed as follows. He wants
you to use the actual population data from two consecutive polls, along with
the actual rates of population growth at the times of the two polls. He
wants you to find a quadratic function which models the population as a
function of time and agrees with the actual populations and the rates of
growth of the two polls.

a) Your first task in this project is to show Dr. Knowitall that in general
it is impossible to find such a function. To do this, let $t_0$ be the year
of the first poll, and let $t_1$ be the year of the second poll. Also, let $%
p_0$ and $p_1$ denote the population values from the polls and let $r_0$ and 
$r_1$ denote the rates of growth from the two polls. To simplify your
computations, write your polynomial in the variable $t - t_0$. (If you do
not think this really does simplify the arithmetic, repeat part~(a) using a
polynomial in the variable $t$ and see what kind of computations are needed.)

Being the resourceful person that you are, you realize that the difficulty
was caused by the fact that the quadratic function was a polynomial of too
low a degree to allow you to include all the information that Dr. Knowitall
wants.

b) Show that by increasing the degree of the polynomial to three, it is
always possible to fit the data to a new polynomial (that is, to find a
polynomial which agrees with the populations and the rates of growth of any
two polls).

c) After completing parts (a) and~(b), use the actual census data below
along with your results from part~(b) to find population functions for each
pair of consecutive poll data. Use this to project the population of the
United States for 1990, as follows. First use your polynomial modeling
function for each two consecutive polls to project the population of the
succeeding census and compare the result to the actual census taken. Use all
this information to make your projection for 1990. How confident are you of
your projection? Discuss why.

\begin{center}
\begin{tabular}{|ccc|}
\hline
&  & Rate of growth \\ 
Year & Population & (people per year) \\ \hline
1950 & 151,325,798 & 2,357,930 \\ 
1960 & 179,323,175 & 2,598,812 \\ 
1970 & 203,302,031 & 2,360,967 \\ 
1980 & 226,542,518 & 2,324,049 \\ \hline
\end{tabular}
\end{center}

\subsection{Invariant Areas}

Consider the hyperbola $xy = 1$, with $x > 0$ and $y > 0$. We are going to
show that the lines tangent to this curve have a very special property.

\textbf{Definition:} The line tangent to the curve with equation $y = f(x)$
at a point $(a,f(a))$ on the curve is the line through $(a,f(a))$ with slope 
$f^{\prime}(a)$. We call this line a \textit{tangent line}.

The tangent line at any point on the hyperbola forms a right triangle with
the coordinate axes. Your first problem is to show that the area of this
triangle does not depend on the point chosen. To prove this, you may want to
proceed as follows. Expressing $y$ as a function of $x$, you are led to
consider the function $H(x) = {1/x}$. What is the equation of the line
tangent to the graph of $H$ at $(a,H(a))$? Draw a graph of this function, a
typical tangent line, and the triangle in question. What is the area of the
triangle? If the area depends on the point $(a,H(a))$ you have made a
mistake.

Now comes the interesting part. You are going to find all functions which
have this property. Let $f$ be any function such that $f(x) > 0$ whenever $x
> 0$, the derivative $f^\prime(x)$ is always negative, and the second
derivative $f^{\prime\prime}(x)$ is never zero. Consider a triangle formed
by the coordinate axes and the line tangent to the graph of $f$ at a point $%
(a,f(a))$. Assume that it has area which is independent of the point $%
(a,f(a))$.

As in the first part, find an explicit formula for the line tangent to the
curve $y = f(x)$ at the point $(a,f(a))$. Then, find the area of the
triangle formed by the tangent line and the coordinate axes.

Assume that the area is constant, which gives you an equation in the
variable $a$. Differentiate this equation in $a$, simplify, factor, and use
the conditions on $f$ to arrive at a simple differential equation. Solve
this differential equation to find all the solutions to the problem.

\subsection{Upper and Lower Riemann Sums}

Let $f(x)$ be a positive, continuous, increasing function on $[a,b]$. Your
goal is to develop a technique to estimate the integral $\int ^b_a f(x)\,dx$
within a given margin of error without evaluating the integral.

a) Your first step is to understand two special kinds of Riemann sums.
Partition $[a,b]$ into $n$ equal parts. What is the length of each
subinterval? For the first kind of Riemann sum, choose the height of each
rectangle to be the maximum value of $f$ over the subinterval. How do you
know the maximum exists? For what value of $x$ do you get the maximum value?
The Riemann sum defined by taking the maximum value of $f$ over each
subinterval is called an \textit{upper sum}. Explain why $\int ^b_a f(x)\,dx$
is less than or equal to any upper sum. Write explicitly the upper sum when $%
n = 4$.

b) For the second kind of Riemann sum, repeat part~(a), but replace the
maximum value of $f$ over each subinterval by the minimum value. This
Riemann sum is called a \textit{lower sum}. Explain why $\int ^b_a f(x)\,dx$
is greater than or equal to any lower sum. Write explicitly the lower sum
when $n = 4$.

c) Let $U_4$ be the upper sum and $L_4$ the lower sum for $n = 4$ found
above. The difference $U_4 - L_4$ is non-negative. Why? Simplify $U_4 - L_4$
as much as possible. Next, find a very simple expression for $U_n - L_n$.
Explain why your expression is correct.

d) Part~c) gives you a way to make the upper and lower sums as close to each
other as you wish. Suppose $\epsilon $ is a given positive real number. How
can you use the result of part~(c) to estimate $\int ^b_a f(x)\,dx$ with a
margin of error at most $\epsilon$? Explain.

e) Estimate $\int ^2_{0.5} (1 + x^2)^{1/3}\,dx$ within $10^{-1}$ using the
ideas from part~(d).

\textit{Extra credit:\/} Can you think of other ways (than the maximum or
minimum) to choose the value of $f$ on each subinterval that would give
better approximations to the integral than $U_n$ or $L_n$? Discuss your
ideas.

\subsection{Houdini's Escape}

Harry Houdini was a famous escape artist. In this project we relive a trick
of his that challenged his mathematical prowess, as well as his skill and
bravery. It will challenge these qualities in you as well.

Houdini had his feet shackled to the top of a concrete block which was
placed on the bottom of a giant laboratory flask. The cross-sectional radius
of the flask, measured in feet, was given as a function of height $z$ from
the ground by the formula 
\[
r(z) = {\frac{10}{\sqrt{z}}}\text{ ,} 
\]
with the bottom of the flask at $z = 1$ foot. The flask was then filled with
water at a steady rate of $22\pi$ cubic feet per minute. Houdini's job was
to escape the shackles before he was drowned by the rising water in the
flask.

Now Houdini knew it would take him exactly ten minutes to escape the
shackles. For dramatic impact, he wanted to time his escape so it was
completed precisely at the moment the water level reached the top of his
head. Houdini was exactly six feet tall. In the design of the apparatus, he
was allowed to specify only one thing: the height of the concrete block he
stood on.

a) Your first task is to find out how high this block should be. Express the
volume of water in the flask as a function of the height of the liquid above
ground level. What is the volume when the water level reaches the top of
Houdini's head? (Neglect Houdini's volume and the volume of the block.) What
is the height of the block?

b) Let $h(t)$ be the height of the water above ground level at time $t$. In
order to check the progress of his escape moment by moment, Houdini derives
the equation for the rate of change $dh/dt$ as a function of $h(t)$ itself.
Derive this equation. How fast is the water level changing when the flask
first starts to fill? How fast is it changing when the water just reaches
the top of his head? Express $h(t)$ as a function of time.

c) Houdini would like to be able to perform this trick with any flask. Help
him plan his next trick by generalizing the derivation of part~(b). Consider
a flask with cross sectional radius $r(z)$ (an arbitrary function of $z$)
and a constant inflow rate $dV(t)/dt = A$. Find $dh/dt$ as a function of $%
h(t)$.

\textit{Extra credit:\/} How would you modify your calculations to take into
account Houdini's volume, given Houdini's cross-sectional area as a function
of height?

\subsection{Undetermined Coefficients}

There are many integration problems which are straightforward to solve but
computationally difficult. Often, however, with a little thought we can
guess the form of the answer and then use differentiation to find the
antiderivative. In this problem, you are going to learn the technique itself
in one special case and then generalize it to other situations.

a) Let $A$ and $B$ be real numbers and consider the integral $\int \cos
(Ax)e^{Bx}\,dx$. From the integrals of $\cos (x)$ and $e^x$, we might guess
the antiderivative is a multiple of $\sin (Ax)e^{Bx} + C$. Show this is not
correct. By considering the derivative of $\cos (Ax)e^{Bx}$, we might guess
that an antiderivative involves a linear combination of trigonometric
functions times $e^{Bx}$. Use this idea to find the antiderivative of $\cos
(Ax)e^{Bx}$ and use this to evaluate 
\[
\int \cos (3x)e^{4x}dx\text{ .} 
\]

b) Using the technique of part~(a), derive and explain a technique for
evaluating the integral $\int \ \cos(Ax)\sin(Bx)\,dx$, where $A^2 \neq B^2$,
without performing any integration.

c) Many books have the following formula: 
\[
\int \cos (Ax)\sin (Bx)\,dx = {\frac{-\cos (B - A)x}{2(B - A)}} + {\frac{%
-\cos (B + A)x}{2(B + A)}} + C\text{ .} 
\]

Show that this is equal to your answer to part~(b).

d) Let $n$ be a positive integer and $P(x)$ a polynomial of degree $n$. Use
the technique above to derive a technique for evaluating $\dint
P(x)e^{Ax}\,dx$. Explain your technique in detail. Evaluate the following
integral: 
\[
\int (4x^6 - 2x^5 + 2x^3 - 10x^2 + 5)e^{4x}\,dx\text{ .} 
\]

\textit{Extra credit:\/} Can you think of other products of classes of
functions where we could use these ideas? Discuss the technique in these
cases.

\subsection{Life on the Hyperbolic Curve}

We have studied the use of trigonometric substitution in finding the
antiderivatives of certain special types of functions. The success of these
substitutions depends on certain trigonometric identities and the integrals
of special combinations of trigonometric functions. In this project, you
will learn about hyperbolic functions and how such functions may be used in
place of the trigonometric functions in the substitution method for finding
antiderivatives.

First, read about the hyperbolic functions. Any good calculus book can serve
as a reference. Then, you will be ready to find the antiderivatives of the
functions given below. You are to use substitutions with hyperbolic
functions to find the antiderivatives and you must justify any identities or
integration formulas that you use. Perhaps you might put all the
justifications in an appendix and give references to the results from the
appendix as you use them. Since you are using the hyperbolic functions only
as a tool to find the antiderivatives, your answers should involve neither
hyperbolic functions nor their inverses.

Find the indefinite integral of each of the functions given below.

a) $(9x^2 + 4)^{-3/2}$

b) $\sqrt{x^2-4}$

c) $\dfrac{1}{x^2 \sqrt{16-x^2}}$

d) $\dfrac{1}{\sqrt{1-x^2}}$

In part~(d), you probably expected to get arcsin$(x) + C$ . How do reconcile
this with your answer?

\subsection{Cubics and Coordinate Systems}

In this project you will discover some surprising facts about the graphs of
cubic polynomials.

a) Carefully graph the polynomial function $f(x) = x^3 - 3x^2$. Be sure to
find and identify critical points and inflection points.

b) By now you have discovered that $f$ has a relative maximum at $x = a$ and
a relative minimum at $x = b$. Let $P = (a,f(a))$ and $Q = (b,f(b))$. Draw
the line through $P$ and $Q$ and let $y = L(x)$ be the equation of that
line. Evaluate: 
\[
\int^b_a [f(x) - L(x)]\,dx\text{ .} 
\]

c) Repeat parts (a) and~(b) above for other cubic polynomials of your
choosing until you see a pattern. Make a conjecture based on that pattern.

d) Return now to the function $f$ given in part~(a). The chord PQ and the
graph of $f$ intersect in three points, $P$, $Q$ and a third point $R$. Two
regions are thus formed. Find the area of each. What do you notice? Now
explain the results observed in parts (b) and~(c) in terms of what you have
discovered.

e) Read about translations of axes and symmetry. Begin with the graph of $y
= x^3 - 3x^2$ and translate the axes in such a way that the new origin is at
the point $R$ you found in part~(d). What is the equation of the curve in
the new coordinate system? Use this equation and symmetry to further explain
your discoveries.

f) Make a conjecture concerning the symmetry of the graph of a general cubic
polynomial function $f(x) = ax^3 + bx^2 + cx + d$. Now you are to prove that
your conjecture is true. You will probably want to use translation of axes
as you did in part~(e).

\subsection{Sticking the Tank}

You have just been hired by the Environmental Protection Agency (EPA) under
the Superfund program to measure the level of toxic wastes in buried tanks
across New Mexico. Most of these tanks are cylinders, with their axes
horizontal. You are to ``stick the tank'' by inserting a stick through a
hole in the center of the top until it touches the bottom, then pulling it
out and reading off the liquid level showing on the stick. They have hired
you because you know calculus; they have faith that you can convert the
``height on the stick'' reading to ``filled volume in the tank.''

Assuming that the cross-sectional radius of the tank is $R$ and its length
is $L$, calibrate the stick for them. That is, convert height showing on the
stick to volume of liquid. Check your results by doing the calculation in
two separate ways:

a) Evaluate a definite integral that gives the filled volume in terms of the
height $h$ on the stick. Do not use tables! (Suggestion: place the origin of
your coordinate system at the center of the circular cross-section. Make a
sketch!)

b) Use elementary geometry and trigonometry (no calculus) to obtain the
volume.

c) Show that your results for a) and~b) are equal.

Since you have been so successful in such endeavors, the EPA sends you out
to stick a tank which has the shape of an elliptical cylinder, i.e.~whose
cross section is an ellipse instead of a circle. The major axis of the
ellipse is horizontal. Calibrate the stick for them, using calculus.

\textit{Extra credit:\/} Calibrate the stick for tilted tanks of circular
and elliptical cross sections.

\subsection{Space-Capsule Design}

You are part of a team of engineers designing the Apollo space-capsule. The
capsule is composed of two parts:

1) A cone with a height of 4 meters and a base of radius 3 meters;

2) A re-entry shield in the shape of a parabola revolved about the axis of
the cone, which is attached to the cone along the edge of the base of the
cone. Its vertex is a distance $D$ below the base of the cone.

Assume the capsule has uniform density $\rho $. Your project director has
specified that the center of mass of the capsule should be below the center
of mass of the displaced water because he believes this will give the
capsule better stability in heavy seas. He has given your team the task of
finding values of the design parameters $D$ and $\rho$ so that the capsule
will float with the vertex of the cone pointing up and with the water-line 2
meters below the top of the cone, in order to keep the exit port 1/3 meter
above water.

a) Show your project director that this task is impossible; i.e.~there are
no values of $D$ and $\rho$ that satisfy the design specifications.

b) Prove that you can solve this dilemma by incorporating a flotation collar
in the shape of a torus (doughnut). The collar will be made by taking hollow
plastic tubing with a circular cross section of radius 1 meter and wrapping
it in a circular ring about the capsule, so that it fits snugly. The collar
is designed to float just submerged with its top tangent to the surface of
the water. Show that this flotation collar makes the capsule plus collar
assembly satisfy the design specifications. Find the density $\rho$ needed
to make the capsule float at the 2 meter mark. Assume the weight of the
tubing is negligible compared to the weight of the capsule, that the design
parameter $D$ is equal to 1 meter, and the density of water is 1.

Your investigations will be guided by a physical principle (I), a formula,
which you will derive (II), and a theorem which you will state and prove
(III).

I) \textit{Archimedes' Principle:\/} A body floats in a fluid at the level
at which the weight of the displaced fluid equals the weight of the body.

II) Derive a formula for the center of mass of the solid of revolution
obtained by revolving the function $x = f(y)$ from $y = c$ to $y = d$ about
the $y$-axis. Explain why the center of mass is on the $y$-axis so you just
have to find $\bar y$.

III) Suppose a body is made up of two pieces, which are solids of revolution
about the $y$-axis. Conjecture and prove a result relating the center of
mass of the composite body to the centers of mass of the pieces.

\textsc{Hint}: Suppose one piece has center of mass at $\bar y_1$ and mass $%
m_1$ and the other has center of mass at $\bar y_2$ and mass $m_2$.
Calculate the center of mass of the composite object by assuming $m_1$ is
concentrated at $\bar y_1$ and all the mass $m_2$ is concentrated at $\bar
y_2$. The formula you obtain for $\bar y$ should be your conjecture. Proceed
with the proof using the definitions of $\bar y_1$, $\bar y_2$, and $\bar y$
in terms of integrals.

\subsection{Escape from the Klingons}

The starship Enterprise has been captured by the evil Klingons and is being
held in orbit by a Klingon tractor beam. The orbit is elliptical with the
planet Klingon at one focus of the ellipse. Repeated efforts to escape have
been futile and have almost exhausted the fuel supplies. Morale is low and
food reserves are dwindling.

In searching the ship's log, Mr. Spock discovers that the Enterprise had
been captured long ago by a Klingon tractor beam and had escaped. The key to
that escape was to fire the ship's motors at exactly the right position in
the orbit. Captain Kirk gives the command to feed the required information
into the computer to find that position. But, alas, a Klingon virus has
rendered the computer all but useless for this task. Everyone turns to you
and asks for your help in solving the problem.

Here is what Mr. Spock has discovered. If $F$ represents the focus of the
ellipse and $P$ is the position of the ship on the ellipse, then the vector $%
\overrightarrow {FP}$ can be written as a sum ${\overrightarrow{T}} + {\ 
\overrightarrow{N}}$ where $\overrightarrow{T}$ is tangent to the ellipse
and $\overrightarrow{N}$ is normal (perpendicular) to the ellipse. The
motors must be fired when the ratio $|\overrightarrow{T}|/|\overrightarrow{N}%
|$ is equal to the eccentricity of the ellipse. Your mission is to save the
starship from the evil Klingons.

\subsection{$\pi$ in the Sky}

You are the ``super-programmer'' for the world's largest supercomputer. Your
boss has taken you aside to give you your next assignment. He explains,
confidentially, that for several days now, Project SETI---which is concerned
with the search for extraterrestrial intelligent life---has been receiving a
string of digits from a powerful point source near Tau Ceti. He suspects
that these are the digits in the decimal representation of $\pi$. However,
he doesn't know when they started transmitting, so they may have gotten
quite far out, perhaps to the millionth decimal place or beyond. Your
assignment is to calculate $\pi$ to more decimal places than ever before, in
order to compare these computed digits with those from the extragalactic
transmission. What you need is an algorithm; that is what you will develop
here.

a) Express $\tan ^{-1} z$ as the definite integral of some function $f(t)$,
with $z$ as the upper limit. Show how, if you could obtain a numerical value
for this integral when $z = 1$, then you could get an exact numerical value
for $\pi$.

b) Now write the integrand $f(t)$ as a geometric series. Use the identity
for the sum of the first $n$ terms of a geometric series to express the
integrand as the sum of the first $n$ terms plus an ``error term.''

c) Integrate this sum term by term to get an expression for $\tan ^{-1} z$
consisting of a polynomial plus an error term. Show that the error term has
the form of an integral.

d) Give an upper bound on the error incurred when using only the polynomial
of part~(c) as your approximation to $\tan ^{-1} z$. How many terms do you
need to give a value for $\pi$ accurate to a hundred decimal places? To a
million? If the computation of each term and its addition to the previously
computed terms takes $1\mu$s on the supercomputer, then how many years will
it take to compute $\pi$ to a hundred decimal places? To a million? To
understand why this is a poor series to compute $\pi$, evaluate the first
six polynomials at $z = 1$.

e) You mention your problem over lunch to Sylvia, the mathematician down the
hall. She jots a couple of formulas down for you:

\begin{equation}  \label{eq1}
{\frac{\pi }{4}} = \tan ^{-1} {\frac{1}{2}} + \tan ^{-1} {\frac{1 }{3}},
\end{equation}

\begin{equation}  \label{eq2}
{\frac{\pi }{4}} = 4\ \tan ^{-1} {\frac{1}{5}} - \tan ^{-1} {\ \frac{1}{239}}%
.
\end{equation}

Verify the correctness of these identities. (It is not sufficient to punch
the \textit{arctan} button on your calculator!)

f) How would you convert Sylvia's formulas into algorithms to approximate $%
\pi$? Do her formulas yield better approximations (in terms of time and
money) than your original method in parts~(a)-(d)? Evaluate the first six
polynomials for both parts of (\ref{eq1}) and~(\ref{eq2}) and put these
beside the six term approximation in part~(d). Compare the number of terms
of your original series and the series generated by formulae (\ref{eq1}) and
(\ref{eq2}) needed to approximate $\pi$ to a million decimal places. How
long will these formulae take on the supercomputer to yield approximations
to $\pi$ correct to a million decimal places?

\subsection{Tribble Trouble}

You are the captain of the Starship Enterprise and tribbles have invaded the
hold of your ship. You are concerned enough to ask Spock, the Vulcan, for an
analysis. As usual, Spock takes a superior attitude. Vulcans are experts
with infinite series, although they never developed calculus. Hoping to
teach you something, he gives his report in a series of riddles, to which
you must supply the answers. Supply answers to questions (a) through (d)
below, and then go Spock one better by showing him how Earthlings would use
calculus to solve the problem in parts (e) through~(h).

\subsubsection{Spock Speaks}

Let $y_0$ be the population of tribbles at some initial time $t = 0$,
Captain. Then let $y_n$ be the population after $n$ intervals of length $%
\Delta t$ have elapsed, i.e.~at $t = n \Delta t$. If you allow one more
interval to elapse, the population will have increased during the next
interval by 
\[
\Delta y_n = y_{n+1} - y_n\text{ .} 
\]
Suppose $\Delta y_n$ is proportional to both $y_n$ and $\Delta t$: 
\[
\Delta y_n = k\ \Delta t\ y_n\text{ ,} 
\]
where $k$ is the growth rate. (You may neglect the death rate in your
calculations.)

a) Give an expression for $y_1$, the total number of tribbles after one
interval $\Delta t$, in terms of $y_0$, the initial population at $t = 0$.

b) Give expressions for $y_2$, $y_3$, and $y_n$, the total number after 2,
3, or $n$ intervals respectively, in terms of $y_0$. Make your expressions
as compact as possible and factor out $y_0$. Rewrite your expression for $%
y_n $ in terms of the total elapsed time $t$ by setting $\Delta t = {t/n}$.

c) Now expand the expression for $y_n$ you found in part~(b) using the
binomial theorem. In order to get better resolution and accuracy, let the
time scale of your analysis become finer and finer; that is, let $n
\rightarrow \infty$ as $\Delta t \rightarrow 0$. Assume that you can take
this limit term by term; your result is an infinite series. Please write
this series.

This series is your answer for the population at time $t$, Captain. You may
calculate the answer to any desired degree of accuracy by including enough
terms from the series.

\subsubsection{Captain Kirk's Comeback}

d) How do you know you can calculate the answer to arbitrary accuracy? For
example, how far off would I be if I just added up the first hundred terms?
The first $N$ terms? Give me an upper bound for the error after 100 terms,
and the error after $N$ terms, Spock, and then prove that the error goes to
0 as $N$ goes to $\infty $.

Now wipe that superior smile off Spock's face by showing him what can be
done with Earthling calculus.

e) Start with the same expression for the increment, $\Delta y = (k\Delta
t)y $, as Spock used in part~(a), and convert it into a differential
equation by taking the limit as $\Delta t \rightarrow 0$ and using the
definition of the derivative. Solve this equation by separation of variables
to find the function $y(t)$ which gives the population at time $t$ in terms
of the initial population $y_0$. It is your turn to smile. (Spock had to run
to the ship's library to look up separation of variables.)

\subsubsection{Spock's Comeback}

Captain, I claim that your solution is no different from mine. You have
simply renamed my infinite series as your function because Earthlings cannot
handle the concept of infinity. Allow me to demonstrate, Captain. In a book
in the ship's library, I came across something you Earthlings call the
`derivative.'

f) Differentiate my series term by term with respect to time and see if the
answer isn't $k$ times my series. Therefore, my series does satisfy your
differential equation, since its derivative is $k$ times itself. Now compare
your function from part~(e) with your answer in part~(c). Your function and
my series must be one and the same!

\subsubsection{Kirk's Second Comeback}

Not necessarily, Spock. The same differential equation may have many
different solutions. Allow me to demonstrate $\ldots$

At about this time, McCoy and Scotty burst into your cabin. Scotty says,
``You gentlemen can debate theory 'til you're blue in the face, but that
doesn't help us with our problem. \textit{I'm} prepared to use radiation
from the ship's engine just one time to exterminate 99\% of the little
beggars right now. Just give me the word, Captain!''

McCoy replies, ``I cannot condone such a waste of life, Captain. I've got a
drug that will cut their growth rate $k$ in half. I think you'll find this a
more effective and less brutal method than the one suggested by our
engineer.''

g) Decide between Scotty's and McCoy's strategies. Which is more effective
in the short run? In the long run? Convince them of your answers.

h) In the future, will these two treatments ever result in the same number
of tribbles again? When?

\subsection{Rearranging an Infinite Series}

The alternating harmonic series $\dsum_{n=1}^{\infty} {\frac{(-1)^{n+1}}{n}}$
converges to $\ln 2$. Now suppose $s$ is any given number. You are going to
prove the amazing result that the alternating harmonic series can be
rearranged (that is, its terms can be written in a different order) so that
the resulting series converges and has $s$ as its sum.

You will need to understand the precise definitions of the limit of a
sequence and the sum of a series.

Hint: Start by trying to reorder the terms to alternately overshoot and
undershoot $s$ with various new partial sums.

\subsection{Tetrahedra}

Start with the hyperbola $xy = k$, where $k$ is any arbitrary positive
constant. Draw several tangent lines to it at various points in the first
quadrant. Calculate the area of each triangle formed by the tangent line and
the coordinate axes. What do you notice? Make a conjecture about the areas
of these triangles and prove your conjecture.

Now generalize to three dimensions. Let $k$ be a positive constant and
consider the tangent plane to the surface $xyz = k$ at a point in the first
octant. Consider the volume of the tetrahedron formed by this plane and the
coordinate planes. Based on your investigation of hyperbolae, make a
conjecture about how this volume changes as the point of tangency varies?
Now prove your conjecture.

\subsection{An Ant in the Sugar Bowl}

An ant at the bottom of an almost empty sugar bowl eats the last few
remaining grains. It is now too bloated to climb at a vertical angle as ants
usually can; the steepest it can climb is at an angle to the horizontal with
a tangent equal to 1. The sugar bowl is shaped like a paraboloid, 
\[
z = x^2 + y^2 \quad\quad (0 \leq z \leq 4), 
\]
where the coordinates are in centimeters.

a) Find the path the ant takes to get to the top of the sugar bowl, assuming
it climbs as steeply as possible. Use polar coordinates ($r$, $\theta$) in
the xy-plane and think of the ant's path as parameterized by $r$; then find
a relation between the differentials $d\theta $ and $dr$, and integrate this
relation to get $\theta (r)$.

b) What is the length of the ant's path from the bottom to the rim? To
answer this, first discover a formula for arc length involving $dz$, $dr$,
and $d\theta $ in three dimensions.

c) Draw a graph of the sugar bowl and the path the ants takes to get out.
Hint: you may want to start with the projection $\theta (r)$ of the path in
the $r\theta $-plane.

\subsection{A Leaking Can}

A cylindrical can with constant density is filled with your favorite drink
and is sitting upright on the xy-plane. Unfortunately, it has a slow leak in
the bottom.

a) Where is the center of mass of the can plus drink before the drink starts
to trickle out? What do you think will happen to the center of mass as the
drink flows out? Will the center of mass continue to sink as the drink
empties? Where is the center of mass when the drink is all gone? Use
intuitive arguments to support your answers to these questions.

Now we will use calculus to answer these questions.

b) Suppose $R$ is an object in space divided into two pieces, $R_1$ and $R_2$%
. $R$ does not necessarily have uniform density. Let $M_1$ and $M_2$ denote
the masses of $R_1$ and $R_2$ respectively. Suppose $(\bar x_1$, \ $\bar y_1$%
,\ $\bar z_1)$ is the center of mass of $R_1$ and $(\bar x_2$, \ $\bar y_2$%
,\ $\bar z_2)$ is the center of mass of $R_2$. Find formulae for the
coordinates $(\bar x$,\ $\bar y$,\ $\bar z)$ of the center of mass of $R$ in
terms of $\bar x_1$, $\bar x_2$, $\bar y_1$, $\bar y_2$, $\bar z_1$, $\bar
z_2$, $M_1$, and $M_2$. Show that \text{$(\bar x$,\ $\bar y$,\ $\bar z)$}
lies on the line between the other two centers of mass.

c) Use the result of part~(b) to give a formula for the height of the center
of mass of the can plus drink when your remaining drink is at height $z$.
Find out how high the drink is when the center of mass of the can plus
remaining drink is lowest. Show that at that moment, the center of mass is
at the top of the drink. Explain what is going on.

\textit{Extra credit\/}: Can you develop an argument not using calculus
which shows that the center of mass must reach a minimum, and that the
minimum coincides with the top of the drink?

\end{document}
