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What Is Calculus About? (New Mathematical Library)

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Whenever I teach calculus, my emphasis is always on the fact that the basic ideas of calculus can be understood by almost anyone. While I am not always successful in proving this to the students, Sawyer certainly would be. His explanations of the basics of a derivative are the clearest, most understandable that I have ever seen. There are many diagrams, and each one has a specific purpose and they are well integrated into the textual explanations.
This is not a book that you could use to teach a college calculus course. The mandatory epsilons and deltas that form the backbone of basic calculus are mentioned only as an incidental. Sawyer sets out to explain the foundation ideas of calculus in terms of everyday occurrences and for that reason it is better suited to someone who is curious about calculus. However, it could be used as a supplemental text in the foundations of science, as calculus is used in all areas of change, which describes almost all of nature.
While the notation of mathematics is concise, abstract and often appears esoteric, many of the ideas expressed in that notation are quite easy to follow. In this book, Sawyer explains what calculus is all about in terms that anyone who understands motion can follow. There needs to be more people like him writing books like this.

Sawyer's text introduces the concept of the derivative to students who are preparing to take a calculus course. Since Sawyer assumes only a solid foundation in algebra and some experience with graphing functions, the text is accessible to the lay reader. He takes an inutitive approach; formal definitions and proofs are not provided.

Sawyer begins his study of the derivative by looking at velocity. He describes the difference between average and instanteous velocity. He shows that instanteous velocity, the derivative of the position function, can be obtained as the limit of the average velocities as the time interval approaches zero. Sawyer then defines the derivative as the slope of the tangent line to a curve at a point, provided it exists. He shows that this definition is equivalent to the first. To keep things simple, he confines his attention to polynomial functions and reciprocal powers of the variable.

Sawyer uses his results to show how the derivative can be used to describe acceleration, curvature, and the behavior of graphs. The final chapters describe, without going into detail, how (integral) calculus can be used to compute areas and volumes and how calculus can lead to nonintuitve results.

Sawyer employs many diagrams and well-chosen examples to show why the formulas that he derives are reasonable. However, I think that formal definitions and proofs would have made his arguments far more convincing. Sawyer stresses conceptual understanding of the derivative, but the exercises, for which answers are provided in the back of the text, consist mostly of repetitive applications of the formulas he derives in his exposition.

The material in this text is covered in good calculus texts, making this text superfluous for students taking calculus. It is best suited to students who want to understand the basic concepts of calculus without having to deal with the formal machinery of calculus.

It all began one winter's night.

Forty years ago and one month on the February winter night of the lunar eclipse in North America in 1968 was the day I began to read W.W. Sawyer's "What is Calculus About?" And my life was forever changed.

I had never read a style of writing up to that point on any topic that exuded as much confidence, as much philosophy of learning and conveyed as much mystery, the mystery of how to handle infinity as that deceptively small book.

I had known about the concept of infinity but up to that point did not know that you could calculate with it. That is take limits. Take derivatives. Then work backwards. But when Sawyer explained it it was like watching a professional magician. There was it could be said a "fresh rebelliousness" in his attitude toward Mathematics understanding and its teaching. It was as if reading his words got you directly in contact with the minds of Newton and Leibnitz and other mathematicians of that "happy-go-lucky" era that Sawyer talks about in his book.

Everyone had warned me at the time that I wouldn't be able to learn Calculus on my own. That it was this mathematics that was so special that it was for many impossible to learn especially if you studied it by yourself. That is was somehow an unattainable goal. That I would somehow "ruin" my learning. It was amazing the degree of prejudice that at that time prevailed.

But they were all wrong. I ended up mastering it. And then the "fur really began to fly". I wasn't supposed to have understood it. I was as one science ( later to be my physics teacher) teacher said: "in way over my head...and if I didn't watch it I'd wind up ruining my career...life..you name it." It was the sixties, the 1960's and my parents generation were total reactionaries. My mother got all hysterical and dragged some of my mathematics books ( unbeknownst to me ) down to the school to the principal and was jumping up and down insisting that I might need psychiatric help. The mathematics teacher of the high school went absolutely berserk with much the same tirade as the physics teacher had. It was to say the least embarassing. All over something they reaqlly didn't understand.

And that is exactly what humanity at it's very best is like. If they don't understand it, "watch out". Because it's like Einstein said:" always the brilliant minds are persecuted by those minds who are of a simple, pedestrian nature." ( that's not a perfect quote but it will do)

But through it all there was W.W. Sawyer. You might say he helped me "get through" the sixties. ( I had other heroes too ) In a strange way you could compare him to Dylan at that time ( my other big hero).

(And it could be rightfully argued that Sawyer was in fact part of that 1960's "beat generation" of scholars who were trying to tell society and the Education Establishment that maybe as far as learning they the accepted teaching establishment had got it wrong.)

Then there was his bibliography and references at the back. That had me spell-bound as well. His fearlessness about mentioning taboo items like Tensor Analysis and General Relativity had me spellbound. ( again the so-called teachers around me assured me that only 12 people in the whole world understood relativity....what kind of scholarly statement does that sound like?). You see if you tried to understand it, Relativity, in their logic that is referring to the teachers and professors in my backwater little community, it meant you were crazy.

And then a few years later I would stop doing mathematics and physics for almost forty years. Only to pick it up recently once again.

The memories of that time are truly haunting.
Truly one of the greatest Mathematics teachers ever.

Simply mystic stuff. No less.

With Best Regards

Southern Jameson West

p.s. they hated me for liking Dylan as well.
p.p.s. one of the first to learn General Relativity and master it almost as soon as it was published was also a teenager. Wolfgang Pauli.

He lived in the 1800's and quickly became a legendary teacher with the ability to break down complex math concepts into information that made sense. W.W. Sawyer wrote several math books and this is one of his popular ones still sought after and very valuable today.

A friend of mine who recommended this book to me and has retires from career teaching math in public schools and was the head of the math department said, "I had taken calculus in college but this little book finally made it all make sense for me!"

When I was reading this book all along, I felt comfortable. I heard it as if my grandfather told an interesting story. When the story was over, I became a real university student. I had learned a vacant caculus, but I have known that the calculus 'is' our life.

The book was required for a master's in middle grade mathematics. It is a good introductory book into calculus.