Three-semester Calculus textbooks are well-known to be very similar to each other, sometimes to the point that section 6.3 starts on the same page in two different books. The material is the same, the teaching methods are the same, and the exercises are very similar. So the reviewer’s task is to find something that distinguishes the book at hand from the rest.
The title of this book, University Calculus, refers to the goal of the authors. As they state in the preface, many of the freshmen taking calculus today have already seen some of the subject in high school. While these students are likely to be able to solve medium-level problems, it is much less likely that they have a deep understanding of the concepts that lead to those solutions. Hence they need to be taught the same material again, this time with the focus being the concepts. One author is from MIT and another one from UC Davis, so the authors seem to be in a good position to know the specific difficulties that students of this kind have.
The text lives up to this promise, but the exercises at the end of sections not so much; they are no different from exercises in competing textbooks. However, at the end of each chapter we find a list of additional and more advanced exercises, which are in the spirit of the concept-oriented approach of the book.
The topics covered, not surprisingly, are very similar to those in other Calculus textbooks. A welcome addition is Chapter 7 (Integrals and Transcendental Functions), which has, among others, a section on the logarithm defined as an integral, which this reviewer has not seen elsewhere. There is also a section on the integrals of hyperbolic functions and their inverses, which most similar textbooks simply list. In addition to the 15 chapters that are included in the book, there are two chapters on differential equations provided as an online appendix. That is a good choice of topic for that purpose, since many schools have a separate course on differential equations, and so do not want that material to be included in Calculus.
To summarize, if you think that your students are clearly above the national average, (say, you teach at a research university, or a highly selective college with students who are seriously interested in science), then this book could be a good choice for you.
Miklós Bóna is Professor of Mathematics at the University of Florida.