A Hitchhiker's Guide to the Calculus

Michael Spivak's Hitchhiker's Guide to the Calculus offers an interesting contrast with the book we discussed last month. Berlinski's Tour of the Calculus was almost entirely concerned with the intellectual structure behind the Calculus. In fact, Berlinski's outline is closer to the typical course on introductory real analysis than to the typical Calculus course. What Spivak gives us here is quite the opposite: the focus is on intuition and not on proof, and most ideas are introduced graphically before they are discussed more formally. Except for the fact that there are no exercises and no calculations, this is much like your "average" Calculus course.

So what's the point of the book? The author points out that since "the Calculus is a method of calculation (...) in your Calculus course you are going to be doing calculations, reams of calculations, oodles of calculations, a seeming endless number of calculations! (...) The trouble is, with all the time spent on these calculations, it becomes easy to lose sight of the basic ideas behind them. (...) So the aim of this guide is to provide a carefree jaunt that explores the new ideas of the Calculus." In other words, the book proposes to be an antidote to the focus on calculation that is oh-so-typical of our Calculus courses.

In this it succeeds relatively well. The basic facts about derivatives and integrals are discussed in a friendly and clear fashion, beginning from intuitive and graphical ideas and going on to some discussion of the technical issues. For example, the derivative is introduced by using the metaphor of a magnifying lens to get across the idea that if we look sufficiently closely every (differentiable) function looks like a straight line. There follow numerical calculations, symbolic calculations, and finally a formal definition of the derivative. Only afterwards does the author relate the derivative to speed and other physical rates of change.

It is quite intriguing to see that the basic topics of a first-semester calculus course all seem to fit into the 121 pages of this book. Add exercise sets, a few tables of derivatives and integrals, a detail here or there, and you'd be very close to something that could be used as a calculus text!

As it stands, of course, the book isn't meant as a text, but as a supplement to a textbook, and it is somewhat successful as such. At times one thinks that it is a bit too much like a textbook to be really successful as supplementary reading. On the other hand, many Calculus textbooks offer students an enormous quantity of material undifferentiated as to importance, and many students find it very hard to learn from such books. Spivak's book might serve to point out which ideas are really important.

There are a few problems with the book. On a minor note, there are a few typos and infelicitous expressions, as when Calculus is described as a "perhaps forbidding sounding new branch or mathematics." More serious is the fact that the exposition seems, at times, to go a little too fast than would be desirable given the target audience. For one thing, the book at times assumes a facility with algebra that most first-year Calculus students don't have. At other times, it is just very compressed. For example, discussing the parabola, the author says "This curve is symmetric about the y-axis, since (-x)^2=x^2." This is certainly correct, but how many students are able to understand that "since" without further explanation? To profit from this book, students will have to know already that reading mathematics involves actively "unpacking" such compressed arguments.

Our final comment is perhaps a matter of personal taste. This book is entirely devoid of history. It gives no hint about when and why the Calculus was developed, nor about the personalities (and the effort) involved. History often can act as motivation, and can also shed light on ideas and definitions. As it stands, the book assumes a reader that is already highly motivated to study and think about the Calculus and its meaning.

Overall, this book is a useful supplement to many Calculus texts, and many students may find it helpful. In fact, it is tempting to consider using it as the main textbook. But its rather compressed style and lack of motivation may make it harder for some students to profit from it.

Fernando Q. Gouvêa is the editor of MAA Reviews. This is the second review he wrote for the Read This! column, way back in the Dark Ages.