Preface

This book covers calculus of a single variable. It is suitable for a year-long (or two-semester) course, normally known as Calculus I and II in the United States. The prerequisites are high school or college algebra, geometry and trigonometry. The book is designed for students in engineering, physics, mathematics, chemistry and other sciences.

One reason for writing this text was because I had already written its sequel, Vector Calculus. A more important reason was my dissatisfaction with the current crop of calculus textbooks; they are far too bloated and filled with fluff, and become more so every year. Enormous textbooks send a bad signal—if the subject looks like a giant pain to slog through, then that’s exactly how students will treat it. Making matters worse, the trend that seems to have begun in the early 1960s to move calculus further away from its roots in physics appears to be accelerating.¹ In addition, many of the intuitive approaches and techniques from the early days of calculus—which I think often yield more insights for students—seem to have been lost. These are some of what I consider the fatal flaws in almost all current calculus texts, whether they take the “traditional”, “reform”, or so-called “rigorous” approach.²

I agree with the views of the late Russian mathematician V.I. Arnold on teaching mathematics, in particular the idea that “Mathematics is the part of physics where experiments are cheap.”³ The ties to physics are especially important in calculus, so this book tries to introduce new concepts with physical motivations (what other motivations can there be?). The book contains exercises and examples that I hope will adequately prepare students who continue on in physics and engineering.⁴

Perhaps controversially, the book uses infinitesimals, making it a bit of a “throwback” or “retro” calculus text. My justification for this heretical act was purely pedagogical: infinitesimals make learning calculus easier, and their use aligns more with the way students will see calculus in their physics, chemistry and other science classes and textbooks (where infinitesimals are employed liberally). This might ruffle some feathers among mathematical “purists,” but they are not the main audience for this book. That said, the book is still compatible with the usual limit-based approach, so an instructor could simply ignore the parts involving infinitesimals and teach the material as he or she normally would. I did not want to be dogmatic, so I used infinitesimals where I thought it made sense, and used limits where appropriate (e.g. in discussing continuity, series). Again, pedagogy was my priority.

The exercises at the end of each section are divided into three categories: A, B and C. The A exercises are mostly of a routine computational nature, the B exercises are slightly more involved, and the C exercises usually require some effort or insight to solve. A crude way of describing A, B and C would be “Easy”, “Moderate” and “Challenging”, respectively. However, many of the B exercises are easy and not all the C exercises are difficult. Appendix A provides answers and hints to many of the odd-numbered and some of the even-numbered exercises.

A few exercises require the student to write a computer program to solve numerical approximation problems (e.g. numerical methods for approximating definite integrals). Algorithms are presented in pseudocode, with code implementations in various languages (primarily Java, but also C, Python, Octave, Sage). I hope the code comments will help the reader figure out what is being done, regardless of familiarity with those languages. Students are free to implement solutions using the language of their choice. There are no dedicated “calculator exercises,” as those have been rendered pointless by modern computing (with which students need to become acquainted).

Stylistically I made a conscious effort to break from an unfortunate but all too common mode of writing in mathematics texts, lamented in the preface of a physics book: “Nothing is more repellent to normal human beings than the clinical succession of definitions, axioms, and theorems generated by the labours of pure mathematicians.”⁵ I have been guilty of that sin myself, but I have changed my ways and banished all traces of that sort of thing from this book. So you won’t find Definition 1.2, Theorem 3.3, Corollary 4.6, Lemma 5.7, Axiom 1B, etc. Instead, I tried to borrow the best of the styles from the physics and foreign languages textbooks I enjoyed so much in college. I also deliberately avoided what the author Gore Vidal called the “we-ness” that prevails in academic writing. There is no good reason for the “royal we” in a textbook, and it comes off as a bit pompous, so we won’t use it.

This book is released under the GNU Free Documentation License (GFDL), which allows others to not only copy and distribute the book but also to modify it. For more details, see the included copy of the GFDL. So that there is no ambiguity on this matter, anyone can make as many copies of this book as desired and distribute it as desired, without needing my permission. The PDF version will always be freely available to the public at no cost (go to http://www.mecmath.net/calculus). Feel free to contact me at mcorral@schoolcraft.edu for any questions on this or any other matter regarding the book. I welcome your feedback.

Schoolcraft CollegeMichael Corral
December 2020