University of Connecticut
College Enrichment Program
Calculus:
The Untold Story
Marc Corluy
Department of Mathematics
University of Connecticut
196 Auditorium Road
CT 06269, Storrs
[email protected]
Lecture Notes Accompanying the C.E.P. Course on Calculus  Summer Intersession 2003
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Preface
The first thought that quite understandably comes to mind when seeing these lecture notes is “Why?”.
Indeed, there is a multitude of calculus books on the market. You have the good, the bad and the
ugly.
They come in hardcover or paperback and range from affordable over pricey to downright
budgetsinking. And that is just for the English speaking market. Do I honestly believe that none of
these books is any good? Of course not. The main motivation for me to write these notes is to give a
very concise overview. Every self respecting publisher these days will only consider letting a calculus
book roll off the presses if it has at least four hundred pages. Every edition, a few dozen pages is added
and before anyone (including the author, I fear) realizes it, the book measures a grand total of 1300
pages and costs over a $100. Furthermore, the baffled student (and more than just one instructor) is
intimidated by the sheer volume and the fact that the memorization of about 300 formulas seems to
be encouraged.
These notes are not even seventy pages, and yet give a relatively complete overview of single variable
calculus. For every new concept, a few examples are discussed to give some intuitive understanding
of this new concept. In most cases, if a theorem is stated, it comes with a proof, yet the proof itself
is not a necessity to comprehend what follows after “Qed”. So proofs could be skipped during a first
reading. “First reading” of course implies that there has to be a second, and a third and so on.
The slenderness in volume, of course, comes with a price. Some of your favorite subjects (and mine)
have been skipped: there is nothing at all in here on implicit differentiation, the inverse function theo
rem is not even mentioned, and the myriad of existing integration techniques are only briefly discussed
on about two pages. And that is only the tip of a gargantuan iceberg. This is unfortunate but not
tragic; further exploration is needed and nobody ever mastered mathematics by reading mathematics.
Mathematics is mastered by doing mathematics.
Yet these few pages are a good starting point to
familiarize yourself with some of the key ideas of calculus. All the rest is “just a bit of fiddling around
with expressions”.
This text has been typeset using the L
A
T
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Xsystem (with several packages:
amssymb, latexsym,
amsmath, epsfig, wasysym
and
amsthm
).
The function plots have been generated by Maple VIII
and then edited using a handy graphical program called “The Gimp”.
All software used ran on a
Linux system with Gnome and KDE facilities.
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 Spring '11
 Prof.
 Derivative, Limit, Limit of a function, Inverse function, lim eln

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