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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*{Contents}

The ``chapters'', or ``units'' involved in this course correspond roughly
to the chapters in your text, but we won't be following the text exactly. You
really should work through the units in the order they are presented, though
in each one there is a lot of room to explore in the order that makes the most
sense for you.


\section{Introduction}

This chapter introduces the major ideas of calculus: slope of curves and velocity,
limits, and finding areas by successive approximation, so you can get an idea
of what is to come.


\section{Limits and continuity}

The notion of a limit is the single idea, more than any other, that separates
calculus from all other mathematics. It is a difficult idea to master completely,
and we will be coming back to this notion in various guises throughout the course.


\section{The derivative}

The first real use of this notion of a limit is to make sense of the ``slope
of a curve'', the derivative of a function. Although the idea of the slope
of a curve is not difficult, and simple tools like a speedometer can find the
derivative of the position of a car very easily, there are many serious applications
of this notion that help understand the behavior of functions.


\section{The Zoo}

Calculus is of course the study of functions. There is a collection of standard
functions that are the standard fare for calculus, which we call elementary
functions. These include all of the functions used to model most physical and
man-made phenomena. This chapter introduces, with reviews of the basics when
you need them, all the functions we will be encountering.


\section{Theory and Practice}

This chapter discusses the primary theoretical tool of differential calculus,
the Mean Value Theorem, and the many applications that actually owe their existence
(or at least their theoretical justification) to that tool, such as optimization
problems and qualitative descriptions of functions (curve-sketching). 


\section{The integral}

The theory of integration, the area under a curve, is as basic a notion in calculus
as the slope of a curve. It's introduced late in the course since the main method
of calculation of integrals, using the Fundamental Theorem of Calculus, is another
application of the Mean Value Theorem. This chapter also includes the ``standard''
techniques for computing integrals, which is a topic that will be covered in
greater depth in the second semester.


\section{Applications of Integration.}

Any computation that can be broken down into simplified sub-computations, and
then added up to solve the more complicated original question can be analyzed
in terms of integration. A simple example is the area of a plane region, which
can be sliced up into little strips, whose area is just the width times the
height. The sum of the areas of the strips gives the area of the original region.
We will go over several such applications of integration to various measurements.
More importantly, we will discuss how to go about turning a complicated computation
into an application of integration.


\section*{Review Material}


\subsection*{Algebra}

Factoring, quadratic formula, dealing with terms and factors in equations. Inequalities.


\subsection*{Functions}

What is an \( x \)? Domain and Range, graphing, composition and inverse.


\subsection*{Trigonometry}


\subsection*{Exponential and Logarithm functions}


\subsection*{Hyperbolic functions and hyperbolic trigonometry}


\subsection*{Curve sketching}


\section*{Explorations}


\subsection*{More on Differentiability and continuity}

Differentiability and continuity, integrability and continuity, uniform continuity, 


\subsection*{Sophisticated models}

Logistic growth, differential equations.


\subsection*{More integration theory}

Integrable functions. Step functions and impulses


\subsection*{Real-world applications}

Simple electric circuits, basic economic models, examples from physics


\subsection*{History}

Newton versus Leibniz, l'Hospital and Bernoulli. Cauchy and the advent of rigor.
What did the Greeks know? How close were they to developing calculus? What about
the Arabs, Phoenicians, or Meso-American mathematicians? 


\subsection*{Modern research}

Limits and function spaces, optimization and geometry, differential equations.

\end{document}
