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The Heart of Calculus: Explorations and Applications

Philip M. Anselone and John W. Lee
Mathematical Association of America
Publication Date: 
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Classroom Resource Materials
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

Like most large universities with a student population in the several tens of thousands, Iowa State offers a lot of calculus courses every semester. In addition to the specialized ones that are intended to service other departments (economics, biology, etc.) there are a great many sections designed for prospective majors in mathematics, physics, engineering, and other hard sciences. A number of these are denominated “honors” sections, open primarily to students in our honors program. This of course leads to the question: what should one do in these honors sections that isn’t done in the others? This book offers an answer to that question.

The book contains 16 chapters covering optional topics appropriate for inclusion in the usual three-semester calculus sequence (or possibly in a course on differential equations, the course that typically follows this sequence). The chapters are basically independent of one another (except for the last three, which seem to me to form an integrated unit), and are divided equally between applications that are too elaborate for a typical course (these are covered in the last eight chapters) and more theoretical discussions (in the first eight).

Many, but not all, of the topics covered in the theoretical chapters are ones that might be encountered in an introductory analysis course. In particular, these chapters cover: the intermediate value theorem for derivatives and applications to curve sketching; inverse functions (including a proof that one-to-one continuous functions defined on an interval are monotone); a derivation of the formula for the derivative of a general exponential function \(a^x\) (for \(a>1\)); Newton’s method and the fact that it converges quadratically; approximation by Taylor polynomials; extreme values for functions of two variables, with connections to Lagrange multipliers; angular velocity and curvature; and proofs of the irrationality of \(\pi\) and \(e\).

I must confess to a certain initial skepticism when I first glanced at these eight chapters. I was concerned that this part of the book would merely repeat an introductory analysis text, and, as we know, there are already plenty of those around. But as I read the chapters, my skepticism disappeared. The authors did a really good job of presenting the theory; the presentation was occasionally novel, and always done in a way that really could be presented to a calculus class populated by honors students.

The choice of topics was also impressive: the authors have made a careful selection of topics that, although they would not be out of place in an analysis course, would also not be out of place in an honors calculus course. The authors assume some knowledge of basic calculus, and proceed from there. The topics chosen are those that really emphasize the fact that analysis is rooted in calculus. More esoteric topics, like the development of the real numbers from the standpoint of Dedekind cuts, don’t appear here. The upshot of all this is that a student reader will not only get a deeper appreciation of basic calculus, but will be better prepared for a subsequent real analysis course.

My only criticism of this first half of the book is a fairly minor one: in some cases I thought the examples given by the authors were a bit too standard for a book of this nature. A worked-out example in the first chapter, for example, asks the reader to graph \(x^4-8x^3-18x^2-16x+21\); surely, however, this does not qualify as any kind of enrichment exercise, but is instead the kind of problem that appears in just about any calculus text. But this sort of thing happened only rarely.

The second half of the text looks at enrichment material of a more applied nature, discussing applications of calculus that are too involved to use in standard introductory courses, but nonetheless require nothing more than standard calculus and differential equations to understand. One thing that I found particularly interesting about these chapters is that some of the material in them is often discussed in books on other, seemingly unrelated, mathematical disciplines, like geometry and probability.

First, there is a chapter on hanging cables, a nontrivial but quite accessible introduction to mathematical modeling that uses calculus, some basic physics and differential equations. Next, intersections of calculus with probability are explored in a nice discussion of the Buffon Needle problem.

The next chapter looks at another optimization problem, this one geometry-based: given a triangle ABC, how do we find a point P such that the sum of the distances from P to the three vertices of the triangle is minimized? This is the so-called Fermat problem; for a discussion along different lines, see Isaacs’ Geometry for College Students. Isaacs’ discussion is purely geometric; the one in this text uses some calculus and successive approximation in addition to geometry.

The text returns to physics in the remaining chapters. The next two chapters, 12 and 13, discuss energy and springs and pendulums, respectively. The chapter on energy starts with a discussion of kinetic and potential energy and proceeds, via the principle of conservation of energy, to a derivation of the formula for the escape velocity of a particle launched vertically from the surface of the earth. Chapter 13 covers various kinds of oscillatory motion and exposes the reader, at its conclusion, to elliptic integrals.

The final three chapters of the text, as I indicated earlier, are related. The first of these chapters discusses Kepler’s laws of planetary motion, which are then used in the next chapter to derive Newton’s law of gravitation. This, in turn, is then used in the final chapter of the text to derive Newton’s generalization of Kepler’s laws, generalizing them from bodies orbiting the sun to two bodies orbiting each other.

All 16 chapters end with two sections, the first titled Problems and Remarks and the second titled Further Reading and Projects. The former provides a few remarks, (some of a historical nature, some designed to extend the material of the chapter) and a selection of homework exercises. (There is, to my knowledge, no solutions manual, either for students or faculty.) The latter section lists some more elaborate do-it-yourself problems, generally broken up into bite-sized chunks, and a bibliography, ranging from 2 to 13 entries depending on the chapter. Minor quibble: because of these references, there is no general end-of-text bibliography, which seems a pity; a bibliography, particularly an annotated one, would seem to go nicely with the general tenor of the book.

Although this review has so far emphasized the value of this book for people taking or teaching calculus or a calculus-related honors seminar, I should note explicitly that the book would seem to be useful for faculty members who aren’t teaching calculus at the moment. The chapter on the irrationality of \(\pi\) and \(e\), for example, is something that a person teaching analysis, number theory or some other course might want to look at; the material is available elsewhere, of course, but it’s always nice to have a source where a relatively elementary proof is made quickly accessible. Likewise, I thought, for the chapter on Buffon Needle Problem, which would make a great lecture in a probability course, or, for that matter, a talk to a math club. And finally, people (like me) who are more ignorant of physics than they should be might find it pleasant to just pick the book up and read one of the last eight chapters for fun. All told, this is a pleasant and useful book; I’m glad it’s now on my shelf.

Mark Hunacek ( teaches mathematics at Iowa State University. 

The Foundation on Which Calculus Stands
1. Critical Points and Graphing
2. Inverse Functions
3. Exponential and Logarithmic Functions
4. Linear Approximation and Newton’s Method
5. Taylor Polynomial Approximation
6. Global Extreme Values
7. Angular Velocity and Curvature
8. π and e are Irrational
9. Hanging Cables
10. The Buffon Needle Problem
11. Optimal Location
12. Energy
13. Springs and Pendulums
14. Kepler’s Laws of Planetary Motion
15. Newton’s Law of Universal Gravitation
16. From Newton to Kepler and Beyond