# Active Calculus 2.0

Matt Boelkins, Grand Valley State University

David Austin, Grand Valley State University

Steve Schlicker, Grand Valley State University

Copyright Year: 2017

ISBN 13: 9781974206841

Publisher: Grand Valley State University

Language: English

## Formats Available

## Conditions of Use

Attribution-NonCommercial-ShareAlike

CC BY-NC-SA

## Reviews

Most of the main topics of calculus are present, but there are some noticeable exceptions. Some missing topics would be hyperbolic functions, Mean Value Theorem, initial value problems. Some ideas are explored in great detail (like... read more

Most of the main topics of calculus are present, but there are some noticeable exceptions. Some missing topics would be hyperbolic functions, Mean Value Theorem, initial value problems. Some ideas are explored in great detail (like velocity/position) and other areas are only covered superficially (like finding antiderivatives with a stated function).

The mathematical ideas are sound.

The textbook is presented in a way to promote Active Learning and to foster student activities for building mathematical ideas. The calculus ideas are sound, but pedagogical swings may contribute to another style of teaching.

The book is concise which may leave readers wanted to see more worked examples.

The book is highly consistent with its framework. All sections start with an activity to complete before class and an additional 3-4 activities for in-class discussions. There are a small amount of homework questions available at the end of the section.

Each section is broken up into smaller sub-sections. It would be nice to have them clickable in the provided online table of content to reduce the amount of scrolling necessary.

The order is fairly traditional. My one suggestion is to cover the Related Rates section currently at the end of Chapter 3 to the end of Chapter 2 closer to Implicit Differentiation.

The online interface of the this textbook is top notch. It is written in PreText which is ADA complaint and much easier to navigate compared to the PDF versions out there.

There are very few minor typos.

The textbook doesn't seem to have any mention of any race or ethnicities in their examples.

This book is a good source for activities for active learning in your classroom. It covers the big ideas in calculus, but doesn't necessarily dive into some of its complexities (more involved applications, hyperbolic functions, algebraic manipulations, more complex functions). There is no review of prerequisite material for a reference for students. However, it does have some short supplemental videos available to students for each section with the big ideas presented. There are few worked examples in the book. It is a good base for information/activities, but will need some supplemental work to form a complete course.

The text has the topics I expected to find. Perhaps your favorite application of integration, or your favorite test for the convergence of an infinite series etc is absent, but this is a small price to pay for a concise text. It should be noted,... read more

The text has the topics I expected to find. Perhaps your favorite application of integration, or your favorite test for the convergence of an infinite series etc is absent, but this is a small price to pay for a concise text. It should be noted, however, that this text does not contain the multitude of exercises at the end of each section that most traditional books do.

I didn't note any errors.

The content is standard, and the examples neutral.

Each section begins with motivating questions in easy to read language. The graphics are clear and illustrative.

The terminology is consistent, as is the style and organization of the sections.

I found the book easy to navigate, especially the online format.

This text is meant for an active learning approach to Calculus. The sections begin with motivating questions, followed immediately by an introductory activity for students to engage with before ideas are formally presented. There are other activities to engage with throughout the text as well. This is pedagogically sound, and a welcome addition to the literature.

I found the text easy to navigate, with graphics of good quality. Some of the problems in the sections use Webwork; while that interface didn't work perfectly for me, it did work well enough.

Typos are few, and the text is easy to read.

The author seems to have tried to bypass mention of race and gender as much as possible.

I am pleased to have the pre-activity and activity problems; this suits my teaching style, although it will not suit everyone. Likewise, the inlcusion of some Webwork problems right in the text is another facet that I particularly like seeing.

The text appropriately covers all areas and ideas of the Calculus 1 and Calculus 2 course topics we currently have at Drake University. Has effective and interactive index for the online version, A Short Table of Integrals, Answers to Activities,... read more

The text appropriately covers all areas and ideas of the Calculus 1 and Calculus 2 course topics we currently have at Drake University. Has effective and interactive index for the online version, A Short Table of Integrals, Answers to Activities, and Answers to Selected Exercises.

Content is accurate but seems to be geared toward “pure” math and physics students as there are very few examples from other disciplines.

It would be hard to make calculus content not up-to-date, and I am sure it will be relevant for years to come. However, I would certainly prefer to have more applied-oriented examples rather than pure math “analyze this random function” type of exercises. Most of the students who take our calculus 1 or even Calculus 2 are not actually math majors. For them, we would need to supplement with various “real-life” examples and projects to make sure their interest and motivation are sustainable.

It is an active calculus textbook, so I am sure some students will find it very unclear and “strange” format. The text will need great support from the instructor and a lot of explanation and motivation presented to students to defend the untraditional choice of presenting the material. It might be a very effective way to learn the concepts, and retention of the material might be much better this way, but it certainly will meet some resistance, especially from students that are not initially fond of mathematics but are required to take a Calculus class to fulfill other majors’ requirements. Each section starts with the motivating questions, then has a preview activity, after that, another activity or exercise that is usually explained (but not always). Ends with a summary and exercises.

The text is consistent.

The text is already divided into appropriate sections, and sections can be re-arranged if needed.

The topics in the text are presented in a logical, standard to calculus sequence order. I would prefer to have a little review of pre-calc material at the beginning of the text, but it can be easily supplemented from other sources.

The web version of the text is full of in-context links to previous activities and exercises whenever they are mentioned. Some of the exercises have buttons “Preview my answer”, “Check answer”, and finally “Show correct answers”. Examples have “hidden” solutions that you can unfold once ready. The interactivity of the interface is very impressive. Links to various visual applets seem to be very helpful for various topics.

I did not find any grammatical errors.

I didn’t find any examples that include a variety of races, ethnicities, and backgrounds. It is unfortunately common to calculus text, in general, not to have examples/discussions that contain any of these.

It might be an excellent text for inquiry-based learning classes or as supplemental material to lead classroom activities in more traditionally thought courses. The instructor would need to provide and supplement more motivational “real-world” examples for most topics. It can be very difficult to follow for the first-year students who are non-math or other STEAM majors, but for those interested in mathematics, it is a great resource.

For this review I am focusing on Calculus I versus the entire sequence. The issues that I see in the first course in the sequence, I have no doubt will persist through the remainder of the sequence. While the topic list addresses the majority of... read more

For this review I am focusing on Calculus I versus the entire sequence. The issues that I see in the first course in the sequence, I have no doubt will persist through the remainder of the sequence. While the topic list addresses the majority of the major concepts, there is not the same level of depth and exploration that I have seen in the Stewart and Thomas texts. Some of the learning outcomes that are required for MAT 201 based on GT Pathways are not included and would have to be supplemented.

There is also only a short table of integrals vs the standard 120 integrals that are include with the text. While only the first 20 integrals are used in MAT 201, it is useful have students get comfortable seeing the table and understand that they will eventually explore all 120 integrals through the sequence. Also missing are some Algebra and Trig quick reference pages. If students did not build course portfolios, these are quick reference guides for students who need to look something up vs search the web.

I am very confident in the book's accuracy as it has been in use for some time and it is the second edition. The problems that I worked through are level appropriate but there are not additional supplemental exercises if students need more practice. The problems are written from a mathematics lens are do have contextualized settings. It provides a very generic and unbiased approach to the content.

Mathematics texts in general are written in such a way that they can be easily updated. The content does not change only the context of some of the application problems. This text is the same. Calculus has been around since the 17th century. There is an order of topics that allows concepts to be connected and scaffolded. This does not change by author. Limits must be introduced first as derivatives are based on understanding rate of change and instantaneous rate of change. Derivative explore the slope of the tangent line for a family of curves. Once students can visualize the curves and connect how the curve changes based on first, second, and third derivatives, area under the curve can be explored. This leads us into integrals. I still have concerns about the depth, or lack there of, for many of the concepts. Limits at infinity and their connection to continuity is missing. Also missing is the precise definition of a limit and relationships within limit laws. Derivatives of inverse trig functions and logs are not introduced early and relate to limits at infinity. It also appears that hyperbolic functions are not explored at all in Calculus I.

The text is written so it does provide adequate context but is written in a way that is I feel is dry. There is not real introduction to each section of material. A real-world context is not shared to set the stage for the concept being explore. The lack of visual aids and applets do not make the concept come to life in a way other texts do. I like all of the applications and explorations within the material but feel there is more work to be done to make this rise to the same level of other publisher materials that are available. Active links to terminology would be helpful as would using color to highlight some of these terms.

Once students begin the course, they will see the framework is consistent in terms of both terminology and framework. The structure for each Unit, section, and preview is the same. The language of the course is used appropriately and consistently throughout the text but more could be done to enhance the introduction of each unit, creating relevant context within fields of study of the students enrolled in the course, and exploration across and within concepts. This is essential in Calculus I as we are trying to help students create a foundation for making more of these connections on their own as we move through the Calculus sequence. While we focus on it in class and within problem solving activities, having a text that supports the efforts is helpful as well.

There are large blocks of text with some included examples. The format of the modules could be structured better so that Preview item are highlighted differently to capture students' attention. Key concepts are in boxes with a different color background but could be set differently on the page to indicate significance. It would also be nice if the sub modules were easy to access within the menu of topics. There are section headings, but no sub headings under those to easily find key concepts within the section.

As mentioned previously, Calculus has been around since the 17th century. There is an order of topics that allows concepts to be connected and scaffolded. This does not change by author. Limits must be introduced first as derivatives are based on understanding rate of change and instantaneous rate of change. Derivative explore the slope of the tangent line for a family of curves. Once students can visualize the curves and connect how the curve changes based on first, second, and third derivatives, area under the curve can be explored. This leads us into integrals. Depth of some topics is my biggest concern.

The text does not contain enough images to add clarity to the mathematical concepts. The Rule of 4 in learning mathematics is fundamental. Table form, equation form, graph form, and written form all play a role in conceptual development and making connections across and within concepts. If students see more visual aids early on in the course, they learn to use them as part of their problem-solving approach. This text has minimal use of graphs, applets, and other engaging tech tools. It also is primarily a two-color text. To emphasize mathematical definitions, theorems, axioms, etc., it would be better if four colors were used to guide students through the text and to enhance the images that are included. The text feels monotonous to me.

The text is well written but a bit dry. I would say it is above the reading leveling I would normally select for a course as many of our students English is not their first language. Learning the vocabulary of the course is critical to success in the course. If the reading level is challenging along with learning the vocabulary of the course, students may have challenges understand key concepts, making deep conceptual connections, and completing assignments when they are unable to attend class.

I would say this section of the review does not apply well to this text. There are no examples written in a way that mentions races, ethnicities, or backgrounds. That may make text culturally insensitive overall as the only culture within the text is the culture of advanced mathematics.

I would use the text as a supplemental resource but not a primary text for the course. While the majority of the learning outcomes are included in the text, there are some that would have to be supplemented. In addition, there are no included homework exercises, extension questions, or technology explorations that support deep connection across concepts. These may be addressed in the activities workbook. I requested a copy of the workbook but it was never emailed to me.

The text covers all of the core ideas one would expect in a standard 2 semester calculus course. The authors clearly prioritized readability over exhaustiveness, and may be seen as skipping/missing material depending on your priorities. For... read more

The text covers all of the core ideas one would expect in a standard 2 semester calculus course. The authors clearly prioritized readability over exhaustiveness, and may be seen as skipping/missing material depending on your priorities. For example there are no sections on the Mean value Theorem or parametrization/polar coordinates.

The material is accurate.

The calculus concepts are timeless of course. I didn't notice any examples that will seem quickly dated. My only concern would be that the electronic text could end up with broken links and or links to broken or outdated programs. As long as it is supported there should be no problems. The active student centered approach is more aligned with modern pedagogy and so might keep Active Calculus relevant (and supported) for a long time.

I think this is where the text truly shines. The writing is clear and compelling. Each section has a clear narrative thread and all of the example are included to further the narrative. I think the author's big picture/idea driven approach more accurately reflects the nature of calculus than a traditional exhaustive reference like textbook.

The style and exposition are consistent from section to section. However, I would add that the activities vary greatly in difficulty can derail a class if you are not careful.

The sections cover an appropriate amount of material and are as independent as one could expect in a calculus course.

Another strong point for this book.

No issues.

Fewer typos/errors than the last two books our dept. required.

The text is not culturally insensitive or offensive in any way that I can see.

This book will work best if used in a student centered active classroom. Each section includes activities embedded in the exposition designed to get the students to engage the material and make conjectures. The activities are not great as lecture examples and so lecturing through them will fall flat. If you take the time to allow students to struggle through them, you will be rewarded.

Reviewer’s note: Please read my “Other Comments” in section #11 first, where I’ve written most of my review, and then return here to read my condensed, individual section comments. Comprehensiveness: The text covers most areas and ideas that I... read more

Reviewer’s note: Please read my “Other Comments” in section #11 first, where I’ve written most of my review, and then return here to read my condensed, individual section comments.

Comprehensiveness: The text covers most areas and ideas that I would expect, although there are some “missing” sections and additional exercises that I would like to see included, as described in the “Comments” section.

The text is very well-written and includes appropriate diagrams.

The subject matter is relatively timeless, but I would like to see more contemporary examples using recent real-world data.

The exposition is straightforward and comprehensible, at a level appropriate to my students.

The exposition and notation are internally consistent.

The modularity is generally good, except for the presentation of Antiderivatives being sprinkled throughout Chapter 4 rather than presented as a separate section.

The organization is generally good, except for the presentation of Antiderivatives being sprinkled throughout Chapter 4 rather than presented as a separate section.

The interface is generally good. I do wish that each “activity” could be linked as pop-up pdf page for easy printing when desired.

I found only a very few typographical errors when using the text for a full semester.

The text is not offensive or insensitive in any way, but includes very little mention of the history of the subject or biographical information. I understand the desire to “streamline,” but I would prefer greater inclusion of this context.

As implied by the textbook title, the design of Active Calculus 2.0 is based on the view that an active, inquiry-based approach is the best way to help students learn Calculus. The experience of the instructor and the students using this textbook will depend largely upon (1) the extent to which they agree with this inquiry approach, and (2) their interpretation of how “active learning” is most effectively implemented.

I used this textbook in a first semester Calculus course in the Spring 2018 semester. Students were asked to read the assigned section before class and attempt to complete the Preview Activities for each section via WeBWork exercises. In class, students worked in pairs through two to four “activities” from the text. (I required that students purchase a printed $10 “course pack” of the activity worksheets from our campus bookstore, available as a pdf file from the authors upon request, so that they could easily turn in their work at the end of each class period.) I circulated throughout the room checking students’ progress and answering questions, and then typically spent the last 10 to 15 minutes of each class summarizing possible approaches and results (with as much input as possible from the students), then collected their worksheets and gave credit for effort and participation (about 10% of their final grade). As the semester progressed, I adjusted this approach by starting most classes with a brief (10 minute) introduction to the material or sometimes completing the preview activity in lecture format, rather than simply starting the class with them working on worksheets. I also learned to break the class into smaller “chunks” of activity, so that by the end of the semester we were generally alternating between 5-10 minutes mini-lectures or recaps directed by me, interspersed with having students work on activities for only 10-15 minutes at a time rather than longer stretches.

The reason I go into such detail describing my pedagogical approach in this textbook review is to clarify that this textbook’s effectiveness and suitability will be highly dependent upon the instructor’s preferred pedagogical approach. If your own approach to teaching matches the active-learning approach that I have described, then you will find this text to be a valuable resource, especially compared to OER alternatives for Calculus. If you and your students prefer the lecture-discussion model, this textbook is not a good choice, at least not as your primary text.

Overall, using this textbook and this approach was a very positive experience for me and my students. I used a variety of resources associated with this text, generously provided by the primary author Matt Boelkins in response to my email requests. As previously mentioned, he sent me the pdf file of the activity worksheets only. (I would recommend using this separate pdf file, as I could not figure out a way to easily print the activities only from either the pdf or live versions of the online text.) Matt also shared with me sets of WeBWork exercises that align well with the Active Calculus sections. Instructors at Carroll College (MT) have developed a free “Chapter 0-Preliminaries” of Precalculus review as a supplement to this text. Instructors at Grand Valley State University (MI) have posted a series of instructional “screencasts” (videos) on YouTube that are matched to the sections of this text. The authors also suggest using the online Calculus applets developed by Marc Renault at Shippensburg College (PA). My students found these all these additional resources quite useful.

My comments so far have focused on the suitability of this text with an active-learning approach. I would next like to shift now to my second initially-posed question: How is “active learning” most effectively implemented?

My own experience with active learning has led me to believe that active learning using worksheets in groups is most effective when the following ingredients are present: engaging questions, student-led discovery, and supportive scaffolding.

Active Calculus succeeds for the most part in providing all three of these ingredients, to different degrees in different sections and different activities. If I use this text again, I would select more carefully which activities I reuse unaltered, which activities I use with some edits, and which activities I skip entirely or replace with my own personal worksheets.

Despite the active learning spirit embodied throughout the text, its contents are remarkably similar to a traditional text. I wish that the text included more real-world data and examples, in the spirit of a text such as the Hughes-Hallett Calculus. My students find such problems and examples more engaging and compelling than such questions as “How do we find the slope of the tangent line?” and “How do compute the area beneath a curve?”

I concur with the review comments of Professor M. Paul Latiolais (Portland State University) that some of the “motivating questions” at the beginning of sections could be reworded to better facilitate student discovery of the relevant concepts as opposed to just asking students “How do we define [fill in the standard Calculus term]?” and then proceeding to tell them in the following exposition. I also think that some of the activities could be improved by providing more scaffolding. Most of my students had difficulty correctly completing the Preview Activities (inWeBWork) before each class, to the point that I decided that scoring 50% on a Preview Activity would count as 100%, and any score above 50% would count as extra credit.

For example, Preview Activity 3.1.1 on critical numbers and extreme values was an especially effective activity because it breaks the concepts up into small enough questions and concepts that allow the students to proceed step by step to feel like they are discovering the notions of critical values and Fermat’s Theorem for Extrema on their own. Preview Activity 3.5.1 on related rates is also effective because it poses an interesting question and lays out a series of steps for the student to develop understanding of a procedure to answer the question.

On the other hand, some of the Preview Activities (1.8.1, 2.4.1, 2.7.1, to name a few) felt simply like homework problems and did not generate much student interest or feelings of discovery.

Several other features of the text could be improved, in my opinion. The text is missing several sections that I would expect to see in any Calculus text, even if included only as an optional section. The definition of “limit” is stated correctly but somewhat informally, in the sense that the epsilon-delta clarification of “sufficiently close” is never introduced. I agree that a strong case can be made for delaying and underemphasizing the formal definition and notation until students are better able to appreciate and understand it, but I would prefer to see the formal definition presented and explored intuitively. More surprisingly, the Mean Value Theorem is not included. Again, I am sympathetic to the view that most first semester Calculus students are not ready to appreciate rigorous proof or lengthy formal derivations, but I choose to introduce these notions to my students in intuitive visual contexts, so that they can gradually absorb their meaning and significance over several semesters. Of course, these “missing” sections are easily filled in using personal notes and/or other, more traditional texts (such as the OpenStax Calculus).

Also, I would prefer that the text contained both more examples in the exposition, and more exercises at the end of each section. Active Calculus does contain most of the standard examples found in any Calculus text, and any instructor using WeBWork has access to as many additional exercises as they could ever care to assign. I believe that the authors were intentionally trying to keep the text streamlined to force students to discover

This text contains all of the core ideas that I would include in Calculus I & II. It is not trying to be a comprehensive tome, which is for the best, especially because it allows for a text that is readable for learners of Calculus, which is... read more

This text contains all of the core ideas that I would include in Calculus I & II. It is not trying to be a comprehensive tome, which is for the best, especially because it allows for a text that is readable for learners of Calculus, which is the stated purpose.

Most of Calculus is in the canon at this point, so there is little controversy about the concepts. However, many of the root concepts, such as limits, are extremely subtle and are not usually discussed accurately until Real Analysis courses. I still like to talk with students about these subtleties, but I think there is almost no chance that they would be able to do that work by reading a written text (and I like to have them reinvent it anyway), so it's appropriate for limits and other similar-level concepts to be defined more heuristically, as they are in this text.

Like most mathematics texts, there is a Platonic bias

As above, the Calculus canon seems quite stable at this time, so I expect that this text will have great longevity. This text has been revised successfully, demonstrating that such work is sufficiently straightforward when desired.

Movements like integration-first, series-first, project-based, integrated STE(A)M, or community- and culturally-relevant approaches to Calculus would require large changes for this text, but that is true of almost every Calculus textbook. Moreover, I think it will be a very long time before there is not a sector of Calculus courses that would still want a text like this. Conversely, this text is much better aligned with active learning pedagogies than texts like Stewart's, so calls to incorporate active, constructivist teaching at all levels will make those other texts obsolete soon while making this text even more appealing to a wide audience.

The stated purpose of this text is as a readable resource for Calculus students, and it succeeds in being accessible while also maintaining concision and appropriate depth. Readers are asked to connect work across the text, which is hard for many Calculus students, but this work is not possible without it. An instructor will certainly have to teach students to read effectively (both in general and for mathematics specifically), but this is true of all university-level course work.

This text uses physical motion as a context regularly, though there are other contexts present. This text would be a better fit for a STEM population than a Business Calc population.

This text is extremely consistent. The concepts build on each other and are connected, making use of the consistency. The guiding goals and values of the text are also consistently employed; for example, the motivating questions and preview activities are used throughout and are central to the structure of the text.

This text uses subsections well; they are appropriate sizes and well titled. It would be easy to add a section for a special topic of interest to a course; most of the sections are important to later work, but it seems that the few topics that could be skipped are stand-alone sections, making that skip straightforward. It would also be easy to use a single section or some sections from this text as a module to replace other course materials in a course if, for example, an instructor preferred this development of a particular topic.

The flow of the concepts is logical and clear. Each section is consistently and appropriately structured with motivating questions, preview activities, development, examples, activities, summaries, and exercises.

The online interface for this book is REALLY strong. The table of contents is useful and fluid. The expandable content works well, especially the WeBWorK exercises, which are user-friendly and focused. The use of subsections keeps individual pages manageable. It might be nice to have PDF or printed versions for reading, in-class usage, and note-taking, but this text lives online.

This text is well edited. In some ways, Calculus is about learning to be precise with vague and intuitively defined terms such as approaching and smooth, and this text handles language professionally without a hint of pedantry.

This text is abstract and symbolic in a way that is normative and expected; many students and instructors will find it neutral and hence appropriate. In a larger context, the assumption that mathematics and its artifacts are culture-free is a deeply problematic issue with which we are not engaging. I am not aware of better models for university-level topics. This text is at least as relevant as texts for comparable courses.

What is the purpose of a Calculus textbook? In the authors' words: "It is our opinion that in the 21st century—an age where the internet permits seamless and immediate transmission of information—no one should be required to purchase a calculus text to read, to use for a class, or to find a coherent collection of problems to solve." This text is clearly a success for these goals.

I think, for many people, a textbook is a resource like an encyclopedia in which the concepts of a discipline are collected. Viewing a textbook in this way makes us focus on the ways that the textbook transmits information to a reader or structures their thinking. This text meets those needs and goes beyond by offerings easy on-ramps for educators to help students engage the concepts of Calculus actively.

However, I structure my courses using inquiry-based learning, and the presence of a pre-set authoritative summary of the concepts can be at odds with my goals. How does this text fit in this context? Well, the sections start with questions, which I think it EXCELLENT. The very first is "How do we measure velocity?", which is great. This question is a section title; however it is the only section title that is a question - the rest are topics. In the second section, the text asks "What is the mathematical notion of limit and what role do limits play in the study of functions?". In contrast to the first question, this question makes a lot of sense from the perspective of the expert and even the student who has learned a little Calculus. In my assessment, most of the motivating questions in this text are of this type; they frame Calculus as an existing body of knowledge that is to be learned rather than something that the reader is participating in building. This is not a critique of this text, since this isn't what it's trying to accomplish, but it does mean that I don't see how to use this text as the core element of an inquiry-based course.

If you are looking for a text you can ask students to read before class, such as a flipped/blended or project-based context, this text is a strong choice. If you are looking for a text students can read after class discussions and when reviewing for high-stakes assessments, this is an excellent option because it does a great job of making the core questions and goals that structure Calculus visible to the reader with enough experience to engage those questions from within the topic. In particular, if I had to have a text for a course but didn't intend to ask students to read it regularly (such as in my inquiry-based courses), this would be a top option because it focuses on conceptual questions, asks the reader to read actively, and is free! And if you are looking for a text from which you can pull active preview, exploration, and extension tasks to make your course active, this is a great choice.

The text was fairly comprehensive. The first portion of the book, which is dedicated to differential calculus, was very thorough. However, the sections on integral calculus was lacking in some of the integration techniques and methods commonly... read more

The text was fairly comprehensive. The first portion of the book, which is dedicated to differential calculus, was very thorough. However, the sections on integral calculus was lacking in some of the integration techniques and methods commonly taught in a Calc II course (such as trig sub, higher order partial fractions, etc.) and the Differential equations and Series sections were nice introductions.

Overall, the accuracy was decent in the book. I found some of the notations to be confusing. In the sequences and series sections, s_n was used to represent terms of the sequence, while later a_n was used for the terms of a series, which are just a sum of the terms of a sequence. Then the partial sum was S_n, which could easily be confused with s_n. It's also not consistent with other math texts I've seen that discuss this topic, which could also be confusing.

There is also a slight question mark over the topic of continuity as far as what constitutes being continuous over it's domain.

The text could be easily rearranged or updated if necessary. The examples were relevant to the topics at hand and there weren't any examples that were outdated.

The text was quite clear and should be accessible to most students. The only trouble I had was with some of the notation, as mentioned above.

The book was very internally consistent with it's terminology.

The text could be rearranged without much disruption to the reader, however can only be rearranged to a certain extent. (As with most math texts, certain topics have to go together.)

I really enjoyed the structure of this text. I liked that each section started with some motivational questions to start conversations, then progressed to a discussion of the topic and hitting the high points I appreciated that there were places for students to help build up the material themselves alone with the instructor.

I also enjoyed that there was a summary at the end of each section.

I had the e-book version of the text, and the only issue I had was that the table of contents page wouldn't take me to an individual section, it would only take me back to the cover page.

There were only a handful of errors that I saw in the wording of some of the questions. For example, in chapter one, there is a graph about calories burned and the question asks if the number of calories increased or decreased, instead of the number of calories used/ burned increased or decreased. Still, over 500 pages and a handful of grammatical errors is impressive!

There were no culturally insensitive examples that I found. These topics don't generally involve examples that could potentially be racially or culturally offensive, but those that were included were appropriate.

On the whole, I felt this was a great Calculus I (differential calculus) and introductory Calculus II (integral calculus) / Differential Equations text. I feel it would be a great addition to a Calculus I course, but would caution against using it in an upper level course or in a more in depth program (such a math or engineering majors.)

This textbook is intended for a two semester Single Variable Calculus sequence. I was mostly pleased with the textbook, although it lacks sections on the Mean value Theorem and parametrization/polar coordinates. However, the book is presented in... read more

This textbook is intended for a two semester Single Variable Calculus sequence. I was mostly pleased with the textbook, although it lacks sections on the Mean value Theorem and parametrization/polar coordinates. However, the book is presented in both online and pdf formats. The online version is somewhat interactive, so one simply has to click on the appropriate section and the web browser is directed to the respective section of the text. On the other hand, the book includes a section on the Qualitative Behavior of Solutions of Differential Equations, as well as a section on Euler's Method.

Overall, the book could be used for a standard 2-semester Calculus sequence, however, the instructor will have to introduce additional material and proofs.

The concepts are presented clearly and accurately. I did not find any errors on concepts, notation, or grammar.

As a Calculus textbook, this book will remain relevant for generations to come. It mostly follows in the typical layout of a traditional Calculus textbook. It's online format, though, has opened interesting venues since the book contains links to Java applets using GeoGebra in the author's website. This ability makes it outperform paper or PDF texts. Additionally, the book can be downloaded and customized (if the instructor is experienced in web publishing).

The material is presented clearly. Each sections begins with "Motivating Questions" as a means to introduce the material. In general, I found this text to be less "wordy" than a traditional Calculus textbook. However, I found the lack of proofs to be a considerable weakness in the presentation. Calculus without proofs transforms from an exercise in logic to a belief system.

The structure throughout the textbook is very consistent. All sections are organized in the same format and the notation is very clear and consistent. The exercises are appropriate, although the problems at the end of the text are few.

The essential ideas of Calculus cannot be segmented, and this book succeeds in presenting the material very cohesively. However, the material is grouped in a traditional format, so the sections in any one chapter are relevant to each other.

Every section in the book has the same structure and organization. The sections begin with a "Motivating Question" followed by conceptual explanations. In may of sections there are "Preview Activities" and WeBWorK exercises at the end of the section. Each section is properly referenced to relevant chapters, sections, and graphs throughout the text, so a simple click on a link will open a window with the information pertinent to the material being presented. I found this to be very valuable.

The interface works well and the full online version is available on github. The only problem I found was my computer not being able to open an Java Applet in the author's website in one instance. On a different day, I was able to open the same applet without a problem.

I did not find a single grammatical error. While I may have missed a run-on sentence, nothing distracted my attention to recognize as an obvious grammatical flaw.

Calculus has been relevant for 400 years and will continue to be relevant for generations to come. No issues here.

I liked the presentation in general. This book has a lot of potential. Using WeBWork exercises is great because the students can practice concepts on their own textbook with immediate feedback. As mentioned earlier, lack of mathematical proofs is a serious omission for a well-rounded Calculus textbook, which this one can be. The hyper links and applets make this book an interactive tool that can be continuously improved to become a fully interactive Calculus website.

The book covers all major topics of differential and integral calculus. However, the emphasis is on "big-picture" understanding of the topics and has relatively few (in comparison to other texts) formally stated theorems and even fewer proofs. ... read more

The book covers all major topics of differential and integral calculus. However, the emphasis is on "big-picture" understanding of the topics and has relatively few (in comparison to other texts) formally stated theorems and even fewer proofs. This could be seen as an asset, if the goal is introducing students to the overall ideas of calculus.

The text is accurate.

This is as up-to-date of a calculus textbook as I've seen. The examples and activities are interesting and draw from a number of disciplines (physical and social sciences, etc.) which gives the reader a sense that calculus has many applications. While specific, the examples will not feel dated in a decade or two and could be replaced with new examples or new data as the need arises.

This is a really well written book. The style is clear and easy to understand. The tone is conversational and uses appropriate vocabulary and description of the mathematics - it doesn't "dumb down" the topics, but also doesn't rely too heavily on "mathy" words when an explanation using regular language would suffice.

However, there is a lot of text between and explaining the very few examples and it is hard to say, having never taught with this text, whether or not students would enjoy reading it as much as I did.

The authors choose not to state many formal theorems or show detailed proofs. The emphasis is on overall understanding of the topics, not the underlying theory that makes the mathematics possible. This approach is successful for this textbook. If, however, you prefer a formal step-by-step build-up of theory and theorems, this is not the textbook for you. Also, the flow of this book and paucity of detailed examples and homework problems would make it difficult to use as a reference book.

Very consistent in layout and use or terms/vocabulary.

The chapters and sections of the book are of the appropriate length for breaking over class periods throughout the term.

The organization of each section is nicely done: "Motivating Questions", text with examples, activities for students, and the a "Summary" at the end followed by a homework set.

The interface is fine.

No errors found.

I have yet to see a culturally relevant calculus textbook! This particular text is in no way insensitive or offensive.

To take full advantage of the unique approach of this textbook would need a lot of instructor planning/preparation before class. The text has very few examples and the instructor would need to provide many of his or her own examples during lecture. This is especially true for the more algebra-intensive topics.

In class, students would need a lot of involvement from the instructor. It would be very important for the instructor to make sure that the students were completing and drawing the correct conclusions from the Activities within the text before moving on. With the right preparation and implementation by the instructor, this "lab"-style approach could be very successful and enjoyable for the students.

There are only a few homework problems listed for each section. Additional problems from elsewhere (either another textbook or something like WebWork) would be helpful.

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 for over a year now, and I haven't found it to be lacking any topic, theorem, or technique. It... read more

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 for over a year now, and I haven't found it to be lacking any topic, theorem, or technique.

It is current in its reduced emphasis on algebraic technique and greater attention to the underlying concepts and engineering-based applications. For example, integration techniques have been reduced in coverage and in emphasis in most calculus textbooks and this book is no exception. Substitution and Integration by Parts are featured, Partial Fractions gets a nod, and then students are introduced to the idea of a CAS (Computer Algebra System). This is in keeping with the reduced treatment of "by hand" integration techniques in most modern calculus textbooks.

There are no issues with the book's mathematical accuracy.

Another kind of accuracy, though, is how well an individual activity "hits its mark" in taking the student through an illuminating example of a topic. Generally, I think the text succeeds here, but there are some edits I might suggest.

For example, in Activity 1.15, after working through this activity in class with students, I altered the graph a bit to create more variation so the resulting discussion about displacement, velocity, and acceleration would be a bit more fleshed out. Teaching with this new version of this activity has had better results in terms of student understanding.

The content is not only up-to-date, but I think very forward-looking in its approach to the subject matter. The book has an almost conversational tone that I find very appealing.

However, to remain relevant going forward, I would like to see the "book" revised to take advantage of its medium. It's presented and used (especially by students, who are perhaps more open to using electronic resources than their older, more traditional instructors) as an *online* resource. To remain relevant, I hope that future editions will take advantage of the power of the computing devices on which this book is often read, and feature more video, applets, maybe some Desmos-type graphs with movable parts. When students want to learn how to do something, they are searching YouTube, not looking for a page of text that describes how they might do a thing. They want to try it on, see it in action, engage with it. We should encourage and provide more opportunities for students to do that.

My students did not much care for the text. I am teaching this course using a flipped model, so there is reading and also instructional videos that students are asked to do outside of class. Not surprisingly, most students prefer the videos. Some student comments (from an anonymous end-of-course survey) about the assigned reading in the text:

--I didn't find the textbook explanations very user-friendly, as they were much more difficult to comprehend than the videos. I don't know if there are textbooks with clearer explanations? About mid-way through the course, I also didn't find the reading to be necessary for most modules, as the videos and class explanations were clearer teachings of the same book concepts.

--The book tends to be confusing, as student that learns from examples, I find this book to be hard to understand.

--I liked that the textbook wasn't expensive, but I don't think the examples given were very helpful. I think they were a bit distracting from the point of the section at times.

--more videos less reading

I am not surprised that students prefer videos, but I don't think this is the fault of the text, but rather that they would prefer video explanations over ANY text. Nor am I surprised that they wanted more "example problems" from the text. Students have been taught that math is mostly manipulating expressions and equations. This book takes a very different approach. One student expressed his discomfort:

--...also there aren't very good examples and explanations in the text. For instance; A section has about a paragraph and then the preview activity.....there's no explanation or good examples of problems. We are kind of just thrown into a pit of fire.

...which is exactly the point of the text, that you learn this content by interacting with it.

The text is interspersed with Activities (as the book's title implies). I used most of the activities (either as-is or modified a bit) as group work in my Calc I and II classes. Students resoundingly preferred this "active" approach to learning calculus to the traditional lecture-based approach, and I think the quality of these activities was a big factor in students' satisfaction.

The book is consistent in terminology and framework, absolutely.

There is a bit of variation in the consistency of the relative difficulty of the activities, however.

For example, in the section on Implicit Differentiation--a topic that students often find challenging--Activity 2.20 features an expression that is algebraically quite complicated for students. I used this once in class. Students just laughed out loud, most refused to try it! I removed it from the set of activities I use.

At other times, there are questions that seem to confuse students because they are "too easy", like (d) in Preview Activity 1.3:

"Write a meaningful sentence that explains how the average rate of change of the function on a given interval and the slope of a related line are connected."

Students ask "Do they just want me to say that they are the same? Is that all?"

The text is easy to pull apart and put back together. It is suitably modular.

The text gets full points for organization/structure/flow.

I would like to suggest perhaps an alternate version of the text where it is organized more like a workbook, with more room left between the questions/problems where students might write their responses. I don't like asking students to copy down the text of a problem when they are working; their resistance to doing so is firm and vocal! But a bunch of answers on a piece of paper with no context is not good work product, nor very helpful as a study device.

A version of this text that invited that kind of "active" participation from the reader would be a marked improvement, I think.

The interface is fine; I've encountered no issues.

I think the book is not only grammatically correct, but very well-written. Not always the case with math textbooks!

I have encountered nothing even remotely insensitive or offensive in this text.

This book helped me to understand how I might teach calculus in a more learner-centered way, and for that I sing its praises! I recognize, though, that the "active" approach is a bit different from what most students are used to/expect, and they will need instructor support to make the most of this book and what it has to offer.

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 calculus except surface integrals, which we could add or move to the third term. read more

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 calculus except surface integrals, which we could add or move to the third term.

i found no errors

It is the most up-to-date book on Introductory Calculus that I have seen so far.

This is a book designed to teach. As such, it will not be a good resource for student who have already studied calculus. That would be a very different book.

consistent

not appropriate question for this subject.

Excellenet. See "other comments" for more details.

Very clear

I found no grammatical errors.

While that would be a great idea, no one has yet attempted to write a calculus textbook which was "inclusive". The closest thing was an environmental calculus book, but that included only covered the applied calculus material.

It is hard to get a good sense of how well a book will work before one has taught a class using it. Nonetheless, the approach articulated in the preface follows the the best of what is known about student learning as it relates to calculus. The approach would be challenging for graduate teaching assistants to accomplish, but possible with sufficient support and worth the effort toward the improvement of student learning.

I would do a "Dan Meyer" ( https://www.ted.com/talks/dan_meyer_math_curriculum_makeover?language=en ) on the activities and the initial questions. However, the formatting of questions and then activities seems a sound one. For example, I would not foreshadow the answer to the questions by using terminology too soon.

For example, I would change the question "How does the notion of limit allow us to move from average velocity to instantaneous velocity?" to "How do we manipulate average velocity to compute instantaneous velocity?"

Example 2: Instead of "What is sigma notation and how does this enable us to write Riemann sums in an

abbreviated form?", say "How can we write Riemann sums in an abbreviated form?"

I should have more examples, after testing this book in a class.

Our challenge is that this book would cover only 2 quarters, not the 3 quarters that we teach. We would be required to use a more traditional book (presumably open source) for the third term. Likely do-able, but challenging.

I thought that the book was thorough in the subjects that were listed, including limits, derivatives, integrals, differential equations, and sequences and series. I would have liked a few chapters on multi-variable calculus, but that wish should... read more

I thought that the book was thorough in the subjects that were listed, including limits, derivatives, integrals, differential equations, and sequences and series. I would have liked a few chapters on multi-variable calculus, but that wish should not degrade the comprehensiveness of the book. The book is hyperlinked throughout, so if on the PDF you look up a terming the index, clicking on the link will bring you right to the page that the term is introduced.

The book builds upon 400 years of calculus understanding, so most of the book is accurate and unbiased in terms of the content.

The text will not be obsolete for a long period of time. The topics covered, and the problems presented are relevant. I conjecture that if an application problem is ever out-of-date, it could be easily replaced.

This book is written contrary to many mathematics textbooks in a fresh, active, and accessible manner. The layout of each section of the text has a summary of what will be discussed, preview activities to get the reader situated, activities throughout the prose, and a summary of what was discussed prior to exercises. It seemed as though the activities and the mathematics had purpose and understanding built in, which I cannot say the same for some other textbooks. I was excited to move to the next section when reading.

The consistency of the textbook is fine. Every section has the same layout, and problems at the end of the section are probing no matter which section is discussed.

I think that it is slightly difficult to be modular with a mathematics textbook. With that being said, I thought that the authors had a different approach than other textbooks in terms of what they wanted to introduce first. For example, I have always learned to prove the fundamental theorem, I would need the interplay between derivatives and integrals. The authors prefer to conjecture the fundamental theorem from observations of velocity and position, and in the next chapter approach the proof.

I have already commented on the flow in the modularity section. I think that many parts flow in this textbook, but there were some parts that I had trouble with initially.

I was so pleased with the interface of this textbook. There are links to javascript modules where students can interact with the exact topics they are reading about. If there is any textbook that shows us the slight capabilities of the 21st century, it is this textbook.

I found little to no grammatical errors in this textbook.

I did not see any portion of this text that referred to any ethnicity or race, so technically it is inclusive of all races and ethnicities.

If you adopt this textbook in your classrooms, please adhere to the active learning modules in the text. They are written in a way that tries to empower the student mathematically. I really enjoyed the previews, recaps, and activities throughout each section. I enjoyed the references back to certain activities. Most of all, I enjoyed the javascript applets that accompanied this text, thus making it a textbook that takes advantage of the 21st century. I would highly recommend this textbook to any educator that wants their students to thoroughly understand the calculus material.

## Table of Contents

- 1 Understanding the Derivative
- 2 Computing Derivatives
- 3 Using Derivatives
- 4 The Definite Integral
- 5 Finding Antiderivatives and Evaluating Integrals
- 6 Using Definite Integrals
- 7 Differential Equations
- 8 Sequences and Series

## Ancillary Material

Submit ancillary resource## About the Book

*Active Calculus* is different from most existing calculus texts in at least the following ways: the text is freely readable online in HTML format and is also available for in PDF; in the electronic format, graphics are in full color and there are live links to java applets; version 2.0 now contains WeBWorK exercises in each chapter, which are fully interactive in the HTML format and included in print in the PDF; the text is open source, and interested users can gain access to the original source files on GitHub; the style of the text requires students to be active learners — there are very few worked examples in the text, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problems, and developing understanding of key calculus concepts; each section begins with motivating questions, a brief introduction, and a preview activity, all of which are designed to be read and completed prior to class; following the WeBWorK exercises in each section, there are several challenging problems that require students to connect key ideas and write to communicate their understanding.

## About the Contributors

### Authors

**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.