Calculus Unlimited

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I'm always glad to see a calculus text that departs from the dull sameness of the standard treatments. It's also cool that the book is available online for free, although it's a shame that the authors have attached a noncommercial license to it, which means it's a dead end in terms of the free information movement.

I have to confess that I don't see the point of this book's approach. Part of the problem is that the authors seem vague on what they intend to fall under the rubric of "limit." In their preface, they ask, "Why, then, did Fermat, Newton, and Leibniz change the emphasis from the method of exhaustion to the method of limits?" Since the Cauchy-Weierstrass definition of the limit postdates the invention of the calculus by a century and a half, this seems like asking, "Why, then, did Leibniz take up a career as an airline pilot?" Leibniz used infinitesimals, not limits. I would have to infer from this that the authors intend some kind of extremely broad, all-encompassing definition of "limit." And yet in ch. 5, they define continuity using a definition that seems to me exactly equivalent to the Weierstrass epsilon-delta definition, and yet they claim that this is not a definition in terms of limits, because limits are being deferred until ch. 13. Huh? Now it seems as though they want the definition of "limit" to be so narrow that it applies to anything that isn't expressed using the Greek letters epsilon and delta.

They state that the purpose of the book is that it is for "instructors and students interested in trying an alternative to limits," but they never explain what the advantages of the alternative might be. The only hint of a motivation seems to be when they say that their approach allows the derivative to be introduced at the very first lecture. However, the problem is that the subsequent development of elementary techniques of differentiation becomes incredibly arduous. Page 32 (the second page of ch. 3) gives a proof of a lemma that is needed before they can show the linearity of the derivative and introduce the product rule, and the proof of the lemma is at least as complicated as any direct application of the Weierstrass definition of the limit. It seems as though the student is being asked to reinvent the wheel over and over again, rather than simply learning one all-encompassing definition of the limit and then putting it to work.

Information wants to be free, so make some free information.

The above review is copyrighted by its author, and is copylefted under the following license:
Open Publication License 1.0 without options A or B

The contents of this web page, except the parts contributed by members of The Assayer, are copyright (c) 2000 by Benjamin Crowell, and are copyleft licensed under the Open Publication License 1.0, without options A or B.