[August 2019 update: I re-named the text “Active Prelude to Calculus” as part of recent summer 2019 editing. I won’t change the title again :-). All else below still applies, with updated links.]
Back in May, I wrote of my pending fall 2018 sabbatical and plans to write a textbook for the calculus-prep course that we now offer at GVSU. I’m pleased to share that a complete draft of my latest free, open-source text, Active Preparation for Calculus, is now publicly available in HTML, PDF, or its original PreTeXt source. I also have an activities workbook in PDF available upon request.
The text is written in the spirit and style of Active Calculus. Each section begins with a short introduction followed by a preview activity designed for students to complete prior to class, and proceeds with a mix of text and 3-4 activities that are designed for students to complete cooperatively during class. The number of worked examples is small, as students are expected to actively engage with the material in order to develop conceptual understanding.
I’ve written in a backward-looking fashion: from the perspective of someone who regularly teaches calculus, what are the most important ideas that I want my calculus students to know? What are the prerequisite concepts they often struggle with? Throughout, I kept in mind the fact that the audience for this text is students who will have seen related ideas in previous courses, so I endeavored to introduce new perspectives and approaches that will challenge students to think differently and develop new understanding. For example, linear functions are formally defined as functions whose average rate of change is constant.
The result is not a traditional “precalculus” book. When I read precalculus or college algebra books, I often find a considerable portion of the content devoted to topics that aren’t used much in calculus, as well as limited emphasis on central calculus-related ideas. In Active Preparation for Calculus, I chose to focus on helping students understand functions as processes (shaped considerably by the work of Marilyn Carlson et al — hat tip: Dave Kung), gain insight into a library of the most important basic functions (including how these lead to families of functions that depend on parameters), use average rate of change to interpret trends in function behavior, see how familiar functions model important phenomena in the world around us, and begin to comprehend the use of limits to describe key aspects of function behavior. Throughout, we work with functions from numerical, graphical, and algebraic perspectives, with an emphasis on the prominent role of inverse functions. The text includes a modest amount of trigonometry, with the primary focus being on the sine and cosine as circular functions, plus some key right triangle trigonometry. Further, through such problems as investigating water entering or leaving a tank with a certain shape or how constrained surface area of select containers enables us to write their volume as a function of a single variable, students encounter fundamental ideas they’ll see again in calculus in the settings of related rates and optimization problems, and themselves develop the functions that represent the quantities of interest. We also consistently make the distinction between exact and approximate values. I describe my overall goals and approach in more detail in the preface.
I hope that the text will not only serve as the basis for other calculus-prep courses, but also as a useful review resource for students currently in calculus who need to refresh select fundamental concepts and ideas.
Like the original version of Active Calculus: Single Variable, this first draft has 3-4 challenging exercises per section. For more routine exercises, instructors will need to supplement with WeBWorK or some other source of free, open problems. You might find Edfinity an option as well. I will be adding anonymous WeBWorK exercises to the text in the near future, and expect to have these (like the ones in Active Calculus) in place by August 2019. For now, I have some WeBWorK .def files that correspond loosely to the text that I’d be glad to share upon request.
I’m grateful to Grand Valley State University for the time provided by a sabbatical leave; to Rob Beezer for developing PreTeXt, the publishing language that allows the beautiful HTML output; to the American Institute of Mathematics for their support of free and open texts; to Mitch Keller for feedback, suggestions, technical support, and his usual production genius in creating the PDF; to David Austin for his help with graphics generally and fantastic interactives like Figure 1.8.10 specifically; and to each of you who wrote me back in May and June with ideas and requests.
As ever, I welcome hearing from you with your comments on errors, better approaches, and suggestions for additional topics and exercises.