
\section*{Introduction}

This module introduces the notion of a {\bf function} as well as
many of the most important kinds of functions.  We will meet linear,
polynomial, exponential, logarithmic, and trigonometric functions,
as well as many families of functions related to these.  We will use
these functions to model a large class of data as well as remind
ourselves of their properties.

The primary goal of this module is to take a series of measurements
from the real world--data--and translate them into a mathematical
object--a model.  Without being able to accomplish this, mathematics
cannot be applied to anything in the real world.  However, in the
mathematics you have studied in the past, it is unlikely that much
emphasis--if any--was placed on this process.  Hence, this kind of
mathematics may seem unfamiliar to you.  By its very nature, Applied
Mathematics cannot be learned by doing drill exercises from a
textbook, or by using pencil and paper to solve problems; applied
mathematicians are interested in being able to understand and
perform direct computations that arise in applications, not canned
problems used to illustrate ideas. For this reason, we will rely
heavily on Mathematica.  Mathematica is not to be considered a tool
to learn the review material; rather, it is the way we will be doing
mathematics.  You are strongly encouraged to seriously approach
learning Mathematica as a major goal for this course.

The data we will meet in this module will be ideal in the sense that
it will exactly fit the mathematical models.  Although this is not
frequently the case in actual applications, it can be (for instance,
in the case of interest, where the bank is using a function to
compute the interest rate).  Additionally, this will allow us to
study the functions themselves without worrying about the degree to
which they fit our data.  In later modules, you will use these kinds
of functions to model data that need not perfectly fit the model.

Many of the lectures in this module will be review of fundamental
prerequisite material from the text (Review Lectures).  It is
assumed that all of the students have met this material before, so
it will not be covered in depth.  Other lectures (the Data Lectures)
will begin with a specific set or sets of data from an application
of the mathematics to be discussed. We will look at plots of the
data and try to understand its behavior, and then see how to find
the appropriate function to model the data. Using the model, we will
see how to {\bf extrapolate} and {\bf interpolate} in order to
predict the behavior of the function for a larger domain than that
given.  In particular, we will be able to study the {\bf average
rate of change} of each function over ranges of the independent
variable and estimate the {\bf instantaneous rate of change} of the
function at given points.

In addition to the Data Lectures, there are two Supplemental Data
Lectures that you should read.  We will quickly see that the method
we use to fit data to a model does not depend much on the kind of
function we use.  However, these lectures offer additional examples
to supplement those we will cover in class.

For your final project, each group will be given a set of data and
will determine a function to match the data.  The group will then
apply the techniques we have learned in order to study the behavior
of the function.

The final week of the module (Monday, 10 September through Friday, 14 September) will be spent covering some additional prerequisite material for
this class.  There will be ample time for questions on the final
projects.
