Active Calculus

(4 reviews)


Matt Boelkins, Grand Valley State University
David Austin, Grand Valley State University
Steve Schlicker, Grand Valley State University

Pub Date: 2015

ISBN 13: 978-0-9898975-3-2

Publisher: Grand Valley State University

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Reviewed by Milos Savic, Assistant Professor, University of Oklahoma, on 1/13/2015.

I thought that the book was thorough in the subjects that were listed, including limits, derivatives, integrals, differential equations, and … read more



Reviewed by Carrie Kyser, Master Instructor, Clackamas Community College, on 1/8/2016.

This book is thorough and up-to-date in all areas of a single-variable differential and integral calculus course. I have been using it in my courses … read more



Reviewed by M. Paul Latiolais, Professor, Portland State University, on 1/8/2016.

PLEASE BEGIN BY READING THE "OTHER COMMENTS" SECTION AT THE BOTTOM FIRST. It seems to cover all of what we need for the first two quarters of … read more



Reviewed by Bethany Downs, Mathematics Instructor, Portland Community College, on 6/21/2017.

The book covers all major topics of differential and integral calculus. However, the emphasis is on "big-picture" understanding of the topics and … read more


Table of Contents

Chapter 1: Understanding the Derivative

Chapter 2: Computing Derivatives

Chapter 3: Using Derivatives

Chapter 4: The Definite Integral

Chapter 5: Finding Antiderivatives and Evaluating Integrals

Chapter 6: Using Definite Integrals

Chapter 7: Differential Equations

Chapter 8: Sequences and Series

About the Book

In Active Calculus, we endeavor to actively engage students in learning the subject through an activity-driven approach in which the vast majority of the examples are completed by students. Where many texts present a general theory of calculus followed by substantial collections of worked examples, we instead pose problems or situations, consider possibilities, and then ask students to investigate and explore. Following key activities or examples, the presentation normally includes some overall perspective and a brief synopsis of general trends or properties, followed by formal statements of rules or theorems. While we often offer a plausibility argument for such results, rarely do we include formal proofs. It is not the intent of this text for the instructor or author to demonstrate to students that the ideas of calculus are coherent and true, but rather for students to encounter these ideas in a supportive, leading manner that enables them to begin to understand for themselves why calculus is both coherent and true.

This approach is consistent with the following goals:

  • To have students engage in an active, inquiry-driven approach, where learners strive to con- struct solutions and approaches to ideas on their own, with appropriate support through questions posed, hints, and guidance from the instructor and text.
  • To build in students intuition for why the main ideas in calculus are natural and true. Often, we do this through consideration of the instantaneous position and velocity of a moving object, a scenario that is common and familiar.
  • To challenge students to acquire deep, personal understanding of calculus through reading the text and completing preview activities on their own, through working on activities in small groups in class, and through doing substantial exercises outside of class time.
  • To strengthen students’ written and oral communicating skills by having them write about and explain aloud the key ideas of calculus.

Features of the Text

Instructors and students alike will find several consistent features in the presentation, including:

  • Motivating Questions. At the start of each section, we list 2-3 motivating questions that provide motivation for why the following material is of interest to us. One goal of each section is to answer each of the motivating questions.
  • Preview Activities. Each section of the text begins with a short introduction, followed by a preview activity. This brief reading and the preview activity are designed to foreshadow the upcoming ideas in the remainder of the section; both the reading and preview activity are intended to be accessible to students in advance of class, and indeed to be completed by students before a day on which a particular section is to be considered. • Activities. A typical section in the text has three activities. These are designed to engage students in an inquiry-based style that encourages them to construct solutions to key examples on their own, working either individually or in small groups.
  • Exercises. There are dozens of calculus texts with (collectively) tens of thousands of exercises. Rather than repeat standard and routine exercises in this text, we recommend the use of WeBWorK with its access to the National Problem Library and around 20,000 calculus problems. In this text, there are approximately four challenging exercises per section. Almost every such exercise has multiple parts, requires the student to connect several key ideas, and expects that the student will do at least a modest amount of writing to answer the questions and explain their findings. For instructors interested in a more conventional source of exercises, consider the freely available text by Gilbert Strang of MIT, available in .pdf format from the MIT open courseware site via
  • Graphics. As much as possible, we strive to demonstrate key fundamental ideas visually, and to encourage students to do the same. Throughout the text, we use full-color graphics to exemplify and magnify key ideas, and to use this graphical perspective alongside both numerical and algebraic representations of calculus.
  • Links to Java Applets. Many of the ideas of calculus are best understood dynamically; java applets offer an often ideal format for investigations and demonstrations. Relying primarily on the work of David Austin of Grand Valley State University and Marc Renault of Ship- pensburg University, each of whom has developed a large library of applets for calculus, we frequently point the reader (through active links in the .pdf version of the text) to applets that are relevant for key ideas under consideration. ??
  • Summary of Key Ideas. Each section concludes with a summary of the key ideas encoun- tered in the preceding section; this summary normally reflects responses to the motivating questions that began the section.

About the Contributors


Matt Boelkins, Professor, Department of Mathematics, Grand Valley State University. PhD in College Teaching of Mathematics, Syracuse University. 

David Austin, Professor, Department of Mathematics, Grand Valley State University.

Steve Schlicker, Professor, Department of Mathematics, Grand Valley State University. PhD, Northwestern University, specializing in Algebraic K-Theory and the Cohomology of Groups.