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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\section*{What is calculus?}

As has probably been pointed out to you before, calculus is \emph{different},
different from any math that you would have seen before calculus. Why? There
are two fundamental things that make it different, and those are the things
that make it important for you to learn about. The first is that the whole point
of calculus is \emph{measurement} of things, things like slope, area, speed,
which we deal with all the time. Until calculus there hadn't been a way to describe
accurately, mathematically, those measurements except in very special (simple)
cases. Making the measurement of such quantities precise, mathematically, is
the fundamental idea of calculus. The other is the idea of modeling, of turning
a real situation into a mathematical structure. The mathematical approach to
understanding any situation is to strip away all the special circumstances of
the situation, so that it looks much like a number of other situations. Then,
solving that one general problem solves all the similar situations. We are going
to be worried about a lot of modeling situations that could not be dealt with
without calculus --- that is, without the measurements that calculus makes precise.

Calculus is also a watershed course in your education. It marks the end of courses
where finding an ``answer'' is the whole point of a problem. Often in this
course, what is important is showing how or why a certain fact is true. It is
often frustrating that the fact you have to prove is self-evident. It is also
the end of courses where the point is merely manipulation of expressions (that
is high-school algebra), or rote memorization. It is the first course in applied
mathematics, and the first course in much of pure mathematics. In later mathematics
courses the objects studied are often abstract, defined in that Alice-in-Wonderland
simplicity that permeates mathematics: a word means exactly what we say it means,
no more and no less. But in calculus the point is to measure concepts that we
already have definitions for (length of a curve, area, slope). In abstract mathematics
courses, the whole universe we consider is defined from the ground up, and we
can ascribe to it whatever rules and laws we think are interesting. Not so with
calculus. In calculus, we have to deal with the rules as they are.

Here are some more details about the course structure, the differences between
a high-school course (even a high-school calculus course) and a college course,
how this course will be run, and other administrative details. Course Particulars.

The topics listed here provide various insights into what calculus is, and how
we will be dealing with it in this course.


\subsection{The area problem}

This section explains how the idea of integration, one of the two major tools
of calculus, grew out of Greek mathematics.


\subsection{Tangent and Velocity Problems }

This section introduces the idea of the derivative of a function, in terms of
slopes of curves and changes of position with respect to time. 


\subsection{Achilles and the Tortoise}

Another Greek insight into mathematics, this time, infinite sequences and series.


\subsection{Sequences and series}

A bit more introduction to sequences and series. This idea is mentioned here
to give you an insight into the process of limits. You will study it seriously
in second-semester calculus.


\subsection{Functions}

A brief review of the main subject of study, functions. Everything in calculus
is expressed in terms of functions. While we presume that you have seen functions
before, this will refresh your memory a bit and introduce some of our notation.
These are called elementary functions, not because they are so simple, but because
they are the basic building-blocks for all functions.


\subsection{Modeling}

Modeling is the practical ``point'' of calculus. You express a given physical
situation abstractly, and the features of it which are just particulars are
removed, leaving only the relationships between the components. Those relationships
are the same for many different situations, and if we understand how one works,
we understand them all. Modeling is the process of moving from a particular
situation to its general relationships, which we then will solve mathematically. 

\end{document}
