This book under review here is the first volume of a two volume set, and is meant to be used for a standard single variable calculus course. I received this volume after having already read and reviewed the second (multivariable) volume . My overall impression of this first volume is as favorable as was for the second; the authors wrote a very stimulating text. For those instructors who wish to try something different, (and even for those who were not really aware of this desire), this text can provide a refreshing new start on the basic calculus course.
In this volume, too, Blank and Krantz clearly demonstrate their personal opinion about how calculus should be taught, and many times they come right out and spell it out for the reader. For instance on page 73, the student who bothers to read the supplementary but highly relevant Genesis and Development section following Chapter 1 will be told why a calculus instructor insists on proving assertions in class. The authors also use little blue boxes for something other than just to highlight formulas that the students should memorize. Most of these boxes are titled InSIGHT and speak to the student reader quite frankly, telling them what a typical instructor would like to say during lecture, but generally more stuffy calculus texts will not bother with. On page 113, for example, we find them emphasizing that the symbol for infinity is not a number, and should only be viewed as "a convenient piece of notation." This comes up so often in the classroom that I found it relieving to see it typed up explicitly in a nice blue box and certain to catch the students' eyes.
There are more diverse types of applications in this volume than in the second one; it is much easier to find problems here that may relate to students whose interest lies in a place other than that of the world of physics, (e.g., economics, baseball, chemistry, biology, pharmacology, genetics, etc). However there is also ample material for the student who is interested in mathematics proper; in fact this may finally be the calculus text which will still appeal to the applications-oriented student while enticing the more abstract-minded student into taking more and more math. The authors also manage to weave such material into the text in a way that flows very well. For instance, they explain the completeness of the real number system in detail, using three different formulations at various parts of the text. The reader may follow the thread or not, and either way the text will not lose its fluency.
I found the organization of the material quite innovative in its details and after the fact, quite natural. Most of the table of contents is parallel to any other text written for the same audience of first year college students taking a year long single variable calculus course. However, the authors make minor adjustments which will have interesting consequences. For instance, they introduce parametric curves in the first introductory chapter, and limits of sequences in the second chapter on limits. The parametric curves and the other material in the first chapter make it a chapter that should be covered, not skipped or left to the students to read on their own. This may help start the whole class at the same level, and thus set the tone of the course. The introduction of sequences and their limits in the chapter on limits is also a good idea, because then the unity of the whole concept of limits is preserved and the students have been exposed to the notion of sequences early on in the course. Then toward the end of the text, we find chapters on infinite series and Taylor series, and now this material, most likely corresponding to the second course in the sequence, follows more seamlessly from a study of integration.
Throughout the book are exercises that are to be solved by hand, some more computational and straight forward, others more theoretical and abstract. Each section also has quite a few problems that need a calculator or a computer. However, the authors make sure to note often that the ownership and use of such tools cannot make up for a lack of understanding the relevant concepts. Many times in the text, we see them refer to errors of approximation, and how one can lose information due to approximations made by calculators and computer algebra systems. This is a welcome feature in a modern text; clearly this book is not allergic to technology but also is ready with warnings about its possible abuse and consequences of relying completely on it.
In short Blank and Krantz have written a quite impressive textbook for single variable calculus which prepares the student for their quite impressive textbook in multivariable calculus. Instructors who consider this book can also keep in mind that their students will appreciate the choice of a relatively inexpensive text.
Gizem Karaali is assistant professor of mathematics at Pomona College.