Calculus with Early Transcendentals

In Calculus with Early Transcendentals Paul Sisson and Tibor Szarvas have written a decent text for a traditional Calculus I–III sequence. The book is due to come out in 2016 and offers a traditional exploration of Calculus seen in most texts. There are, however, different viewpoints presented and the placement of some topics are quite different than seen in other books. Both authors have gone into great depth to make sure all readers have a complete understanding of Calculus with historical references and varying levels of difficulty in problems.

I enjoyed reading the text and I have gained further ideas on how to make my own lectures better. This is always a great and positive feeling for an instructor reader to have. The exercises are some of the best I have seen: there are many problems to assign for homework, group work, and even exams. In addition, I liked that the problems varied from easy, medium, to difficult. The progression of challenging the student at every step of the learning process is wonderful. As Sisson and Szarvas point out: “This book arises from our deeply held belief that teaching and learning calculus should be a fascinating and rewarding experience for student and professor alike, playing a major role in the student’s overall academic growth.”

Indeed, while reading the text, I was reminded how much fun learning Calculus was and should be. I can see how much fun it would be for a professor to use this book. Sisson and Szarvas have gone back to how to learn mathematics properly, which is through practice. It is through this drill and practice approach that Sisson and Szarvas have written a book that takes the reader from problem solving to the rigor of concepts, definitions, and mathematical proofs in a well thought out way.

There are many applications, ranging from the sciences, business, and the social sciences, which adds a lot of value to the text. Many of these applications include real-world data to pull in the reader even further. At the end of each chapter are projects that go beyond what is taught in the classroom and make the students bring together concepts and apply them to more complex situations. In addition, as with other Calculus texts, there are chapter review exercises and many technology software packages are weaved throughout the text. The overall appearance of the text is very modern and eye appealing, with the sculpture Arabesque XXIX by the American artist Robert Longhurst on the cover. Inside, the book is very clean in the presentation of examples, technology, and pictures.

With any new text, there will always been ideas on how to further improve the text and it is always possible to question why the authors chose to do certain things. There are some aspects of the text I didn’t quite like. For example, I thought Chapter 1: A Function Primer was very wordy. I understand functions are the backbone of Calculus, but this is a Calculus book. At this point, students should have the knowledge of PreCalculus and the goal of chapter 1 should be a brief overview of functions and not be close to 100 pages in length.

In addition to Chapter 1, there are other places where the text could be shortened, such as the discussion of \( g(x)=(1-\cos x)/x\) on pages 129–130. This is also evident on page 152 for example 5, where Sisson and Szarvas are explaining the cancelation of the factors \(x + 3\). There are also many parts of the book where the authors’ tone seem wrong, which I believe takes away from the text. For example, on pages 152–153, examples 4a and 5, the phrase “first of all” sounds supercilious.

There are many parts of the book where there is an overuse of quotes. One of the prominent places is on page 161 in example 2. Words as “missing,” “punctured,” “wild,” and “squeezed” can be simply explained instead of the use of scare quotes. The definition of the Derivative, I believe, is very out of place in section 2.6 and should be moved to section 3.1. The text offers a very interesting perspective on differentiability through the discussion on one-sided derivatives and Darboux’s Theorem. This is rarely seen in other texts.

Section 3.6, on derivatives of inverse functions, starts out with the traditional derivation of \((f^{-1})’(x)\) but also presents logarithmic differentiation and derivatives of trigonometric functions. These concepts all work to make this section stand out, since most texts fall short on these important concepts. Section 3.7, “Rates of Change in Use,” is a good preview of Related Rates but it may be hard to get to with time constraints in a normal semester.

I am confused about how the authors decided to split up the topic of u-substitution for integration. I believe examples 1 and 2 in section 5.4 should be placed in section 5.3 and there should be examples of \(e^x\) substitutions for 5.4. I’m also not sure why the authors decided to talk about the area between two curves in 5.5, which still talks about u-substitution. Area between two curves should be a separate section in chapter 6: Applications of the Definite Integral. Then, the classical examples of producer and consumer surpluses can be presented.

Section 6.6 should include an example of finding tangent lines and limits for hyperbolic functions. Polar Coordinates are very well done; I especially like how the authors use the polar grid to accurately graph points and equations. Examples 8–9 are done via tables to graph \(r = 3 \cos (4\theta)\) and \(r = 1 + 2 \cos \theta\). These are a great way to show the graphs with the aid of a Texas Instruments graphing calculator and Mathematica.

Chapter 12 on Vector Functions is laid out in the traditional manner for a Multivariable Calculus III class with a nice addition of torsion \(\tau\). This is a refreshing addition to a very fundamental chapter. In addition, motivations on span sets are introduced through the osculating plane, normal plane, and rectifying planes, which makes a great preview for students who are Linear Algebra bound.

Section 14.5 is on Triple Integrals in Cylindrical and Spherical Coordinates. I was pleased to see Cylindrical and Spherical coordinate systems introduced within the section as some books introduce these systems early and students can forget the concepts when it comes to evaluating triple integrals. In addition, another very positive aspect of the text is Chapter 15: Vector Calculus. The initial section on Vector Fields is short and to the point. Topics such as conservative vector fields and finding potential functions are weaved into various parts of the chapter. This will make the chapter go by faster as it normally is the last chapter in a Calculus III class.

In conclusion, the text is well written and brings to the forefront the beauty of Calculus. The quantity of exercises is very impressive and one can tell the authors spent a great deal of time creating them along with the examples with solutions. Despite the parts of the book that are very lengthy and verbose, the book should be considered if looking for a solid rigorous Calculus text or if one’s mathematics department needs a textbook change.

Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College, an editor for Larson Texts and is the 362nd Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He has written several book reviews for the MAA and his research fields are in mathematics education, Cayley color graphs, Markov chains, and mathematical textbooks along with various software for online homework. He can be reached at pto2@psu.edu. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.