Calculus: A Complete Course, Eighth Edition + Solution
Book Preface
The word “tears” is found in mathematical titles surprisingly often. One reads of “mathematics without tears,” “geometry without tears ,” “to pology without tears,” “statistics without tears,” and, of course, “calculus without tears,” among others. Compare these juxtapositions of tears and mathematics with what the late Fields Medalist and member of Bourbaki, Laurent Schwartz, once famously wrote,”11 n’y a pas de mathematiques sans larmes … ” [There is no mathematics without tears …. ] It seems that there has been much weeping over mathematics, as well as disagreement over whether the weeping is necessary or not. Perhaps people have carried the tear metaphor too far, but the underlying sentiment behind tearlessness has also been long expressed in other ways too. Over the twentieth century we have had all manner of mathematics titles using words like , “o utline ,” “nutshell,” “simple,” or “dummies .”
For example, S. P. Thompson published Calculus Made Easy in 1910. It retains enough of a following today, more than a century later, that you can still buy fresh printings of the 1914 second edition on Amazon. Not even a more recent edition rewritten by Martin Gardener , no less, has been able to push the second edition into oblivion. Mathematics texts are like that. At the beginning of the twentieth century, some students were still taught from Euclid’s Elements, in the original ancient Greek. In mathematics the basics last. This makes the modern mathematics instructor a bit nihilistic about choosing which textbook to use in teaching calculus. “Does it really matter which text I use?” they ask. At some level it indeed does not, so they just employ whatever text was used last year. “More of the same” becomes the standard operating procedure. But this cannot always hold true or we might be using Thompson today instead of any number of modern texts.
One reason more-of-the-same doesn’t always hold is that the audience changes even when the basics don’t. We explained in the seventh edition that modern students, with alifelong exposure to modern graphical computer interfaces, cannot help but look at mathematics differently than previous generations. It is all too easy for them to have the impression that mathematics is an application of computer science, existing as a mere icon on desktop s. Why learn mathematics, they reason, when it seems we can have it all at a “cl ick?” Learning that computer science, not mathematics, is actually the application is a good first step. Learning that computers are not something to be believed , except in a conditional manner, is the definitive lesson for the computer age . That alone is reason enough to learn mathematic s. The seventh edition dealt with this through a new thematic topic called “N umerical Monsters” (marked by A. in this new edition). These represent a form of anti-numerical analysis-“anti” because the topic aims to make computer errors as large as possible instead of minimizing them. They provide natural, self-contained mathematical applications that play off the finite representation of number s within all computers. What is learned is fundamental, and qualitatively independent of code or platform. All of this was inconceivable to Euclid , Newton , or S. P. Thompson.
There is another reason why more-of-the-same is problematic . Much of the basic mathematics used in mathematically based fields was set about a century and half ago. At the time of Thompson , one hundred years ago, there was a gap between the application fields and the calculus exposition of that time . More-of-the-same has preserved that gap over all the subsequent years. When students move from calculus to the mathematic s of one of these fields, they enter a strange world with mathematical customs that may even contradict what they have been taught in their calculus courses. For example, what are physics , chemistry, or engineering students supposed to make of the famous equation, d E = JW + JQ? It depicts a differential equalling the sum of two things that are not actually differentials . Puzzling to say the least. But what are these thing s on the right side? Students are told not to be alarmed in their field-specific texts because E is not actually a function of W or Q. Well, since JW and bQ are not actually differentials anyway, maybe that is okay then , or is it? Sometimes the J’s are replaced by d’s with little bars through them to emphasize that some kind of uniqueto- thermodynamics “mat hematics” is in play. This representation is an anachronism dating from the nineteenth century, recalling dubious attempts to depict everything, in addition to functions, in terms of differentials alone . Calc ulus texts have simply not ventured to show how to proceed without the nineteenth century awkwardness still in play today. This is not the only example of this phenomenon by any means . Such things have generated more confusion and “tears” than any mathematics course ever has. A good introductory applied calculus textbook ought to lead the reader training. This not only helps to stave off unnecessary “tears” but ultimately can lead to a more lucid standard of exposition for the fields in question. We described these connections with the thematic title “Gateway Applications” in the seventh edition . We have marked them by the symbol O in the eighth edition. They should not be confused with “applications” that appear as tamed examples and staged problems typical in alJ textbooks. Instead, they take the reader from a calculus topic at hand directly to a mathematical tool often overlooked in calculus texts but crucial to an actual field, or they take the reader to an insight on how calculus sets the structure of an entire field, without actually pursuing the field. Now that’s application!
In the seventh edition we introduced a number of these gateway applications, from Liapunov functions to thermodynamics and Legendre transformations . We also sketched out why these things are important and how they are actually used. In the eighth edition we have added a calculus-based explanation of entropy as a gateway application, showing how it naturally arises from simple calculus properties and how it fits both into statistical mechanics in physics and information theory. Gateway applications are not meant to replace those traditional “applications.” They are meant to enhance the possibilities within a calculus course, either as source material for independent projects, or as enrichment for a course that an instructor may choose to explore to make a point to a class. Moreover , they are value-added when viewing the book as a future reference work. When students encounter the gaps between their calculus training and their subsequent courses , chances are they will find answers on how to bridge the gaps not available any place else. The eighth edition is no crib sheet to be discarded after the course is done.
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