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Calculus for the Millions lecturenotes

Calculus for the Millions lecturenotes - University of...

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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 budget-sinking. 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 E 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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