College Calculus: A One-Term Course for Students with Previous Calculus Experience

In College Calculus: A One-Term Course for Students with Previous Calculus Experience, Michael E. Boardman and Roger B. Nelsen have written a solid text for the serious Calculus student. It is intended for the students who had a successful experience with an introduction to calculus in high school or have scored a 4 or 5 on the AP Exam. The text begins with a review of some of the topics found in an AP Calculus course and provides readers with a preview of what to expect in their future mathematics courses. Chapter 0 is a preparation for college calculus, but is very brief, with a review of limits, continuity, derivatives, and integration. Chapters 1–11 contain material normally found in Calculus II, with topics from volumes of surfaces of revolution to infinite series.

Throughout the text, Boardman and Nelsen make references to readers who have had previous calculus experience (PCE) to complete the problems in the book, so that they are ready for Calculus II. At every stage of the book, from the derivations, examples, charts, graphs, and exercises, one can tell that Boardman and Nelsen have taken a great deal of time, thought, and consideration of various learning styles in the writing of the book. Everything is justified. The problems include some classics found in other texts and others to help students with more sophisticated critical thinking problems. For example, on page 34, Exercise 1 is to evaluate \[\int e^{-x}\left(x^4+2x^3+3x^2+4x+5\right)\, dx,\] but Boardman and Nelsen point out Theorem 1.6: If k is a nonzero constant and P is a polynomial, then \[\int e^{kx} P(x)\,dx = \frac{e^{kx}}{k}\left[ P(x) – \frac{P'(x)}{k} + \frac{P''(x)}{k^2}-\frac{P'''(x)}{k^3}+ \dots\right] + C.\] This is a very helpful note for students who wish to learn new integration techniques.

There are some topics presented in the text that make a nice connection to elementary topics in Calculus II but take one step further. For example, on pages 299–300, Boardman and Nelsen talk about the dilogarithm, which is a member of a class of functions known as the polylogarithms, all of which are defined in terms of integrals of a natural logarithm. The dilogarithm is defined as \(Li_2(x)=\int_0^x \frac{-\ln(1-t)}{t}\,dt\) for \(x \in (-\infty,1)\), which provides a nice connection to both series representations and intervals of convergence, since \[ \frac{-\ln(1-x)}{x} = 1+\frac{x}{2} + \frac{x^2}{3} + \frac{x^3}{4} + \dots = \sum_{n=0}^\infty \frac{x^n}{n+1}\] for \(-1\leq x < 1\).

There are many problems in the text based on applications obtained from various sources. Through these applications, Boardman and Nelsen want to expose students not only to the pure mathematical concepts but also to how they are used in real life examples. For example, chapter 3 on differential equations contains the Snowplow Problem in which the students are asked to find what time it started snowing given a snowplow started plowing at noon going 2 miles the first hour and 1 mile the second hour. Here, students must set up and solve the differential equation to arrive at 11:23 am, the time of which snow started falling. It should be mentioned that here the snowplow is a snow blower, which is pictured in the text.

The text is very direct and to the point. The examples clearly show each step of the solutions and are good practice for the exercises. There are some interesting topics not normally covered in any Calculus I text I have ever read but are interesting to explore. For example, on pages 145–146 in Chapter 6 (The Hyperbolic Functions), Boardman and Nelson talk about the angle \(\varphi\), which is called the Gudermannian of \(u\) (after German mathematician Christoph Gudermann.

Boardman and Nelsen have done a great job putting this text together, especially in the chapters on infinite series. They have split this large topic into two chapters, which is much better for the student to absorb. Most Calculus texts will have all the sequences and series topics along with all the tests for convergence and divergence along with Taylor and Maclaurin series in one big chapter. It has been my experience, by the time I have finished teaching sequences and series, students are burned out. With this split up, students will be able to study these concepts in smaller chunks. As with the other chapter exercises, the problems progress from easy to difficult. Chapters 10 and 11 have this same set-up with questions that can be used as project or group research projects. Among my favorites is Gabriel’s wedding cake, found on page 270. Here students must show Gabriel’s wedding cake is a cake that one can eat but cannot frost. In other words, like Gabriel’s horn, it has a finite volume but infinite surface area.

At the end of the book, there are five appendices, the first of which provides the instructor with a description of the AP Calculus AB Course. This serves as an information page so that instructors have a better understanding of the type of students they will have in their classroom. My only suggestion would be to add color to the pictures, charts, and graphs.

Simply put, this is a well-written text. Both Boardman and Nelsen have shown their knowledge and years of experience. I see this text being used in many different ways. Instructors can assign this text as summer reading as they progress to Multivariable Calculus, Linear Algebra, or Ordinary and Partial Differential Equations. It can also be used as a resource for tutors while tutoring or as supplementary or primary reading for Calculus II. It can also be used for instructor to use for in-class examples or test questions. No matter how used, the student is surely to gain the knowledge of Calculus II and have a leg up over other students. I highly recommend this book.

Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362^nd Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He can be reached at pto2@psu.edu.