# Informal Calculus with Applications to Biological and Environmental Sciences

Tyler Seacrest, University of Montana Western

Debbie Seacrest, University of Montana Western

Rebekah Levine, University of Montana Western

Michele Anderson, University of Montana Western

Copyright Year: 2021

Publisher: TRAILS

Language: English

## Formats Available

## Conditions of Use

Attribution

CC BY

## Reviews

The book covers the basics well. It doesn't cover multi-variable functions, which is something we cover in our Brief Calculus at my institution, and this is the reason I rated it at a 4 instead of a 5. It has a good table of contents, but no index... read more

The book covers the basics well. It doesn't cover multi-variable functions, which is something we cover in our Brief Calculus at my institution, and this is the reason I rated it at a 4 instead of a 5. It has a good table of contents, but no index or glossary. I like the just-in-time algebra review. There is always a goodly number of students who need algebra review, and the topics are focused on what is needed for the next section of Calculus material. I also really like the projects connecting Calculus to present-day problems.

Everything seems to be covered accurately.

The basics of Calculus don't change, so the relevancy of that part of the book is not at issue. The projects are key, in my view, to how up-to-date the book is. The projects, such as tracking killdeer migration and modeling climate change, use current information and are relevant to the audience, which generally includes natural resources and life sciences students. There is also a project about Quake Lake, which was formed by an earthquake in 1959. On the time-scales involved in the problems, these are current. It would be relatively easy to update the projects as more current data becomes available or to add new projects of interest, since they are included in a modular way in the book. Each is a separate chapter.

The chapters are concise and give examples that are similar to the homework problems. There are a lot of short chapters which help make the material more manageable and customizable. You cover a focused topic and then complete exercises on that topic.

Notation and language that I would consider common are used and usage is consistent.

I like how each topic is presented in small chunks, effectively one objective at a time, with exercises for the students to practice on that topic. Each exercise set is organized as a separate chapter, too, so it will be relatively easy to adjust the sets as needed without having to modify the associated explanations and examples.

The story line for Calculus doesn't change much from book to book. This book has a typical outline, and it is a good one.

I looked at the pdf and the eText. The pdf doesn't seem to have active links, but that could be a matter of the computer and operating system I am using. The eText has a nice navigation tool in the form of the Table of Contents. I like how the answers for the homework problems expand.

I skimmed the whole book. I noticed very few.

Math has an advantage here. It tends to be objective. I'm not sure what the examples would be, but it would be nice to include some examples that specifically mention locations or phenomena outside North America. The projects are good general projects that any student of science can use.

I really like the book. It isn't quite a fit for the syllabus I use, but elements of it could be incorporated.

The textbook offers a review of algebra and covers all important topics of limit, derivative, and integration. The applications and modeling examples in this book, unlike the mainstream calculus textbooks, are specially designed and particularly... read more

The textbook offers a review of algebra and covers all important topics of limit, derivative, and integration. The applications and modeling examples in this book, unlike the mainstream calculus textbooks, are specially designed and particularly suitable for students in life sciences and environmental sciences. The author also provided six specially designed projects, as well as answers to all homework problems.

There is no index or formula table/sheet at the end of the book. It would be helpful if the author also included a preface, which provides some information such as the expectations of the course structure. For example, this book is suitable for a 3*50 mins/week course in a semester system, or a 4-unit course in a quarter system.

As the title, this is an informal discussion of calculus, and it is targeting non-math/physics/engineering students. Hence, it makes sense some topics, such as the discussion of inverse function and derivative of hyperbolic trig functions, are not covered in this book. However, the reviewer personally may add some content if to adopt this book. For example, the reviewer usually spends more time discussing continuity and discontinuity in calculus for life sciences. The reviewer would also cover limits at infinity, which is important for long-term behavior discussions, as well as both parts (instead of only one part) of the functional theorem of calculus and the definition of concavity. While the topics like "sum rule", “ constant coefficient rule", the "4-step methods", and the "continuity checklist" are all covered, the reviewer personally likes to introduce those names of students, which helps to build the system into students mind.

The authors are knowledgeable and experienced, and the topics discussed in the book are unbiased and free of errors.

The content is up-to-date. All projects included in this textbook are appropriate for students to major in life sciences and environmental sciences. Kudos to the authors for devoting several chapters to modeling and differential equations. Such a design makes the content highly relevant.

The context is clear. Important formulations are boxed. Newly defined terminologies are italic. The authors used different ways to demonstrate ideas, hence, the content is accessible. For example, this book uses graphical proof for the product rule. It uses color-coded proof for the chain rule. The authors also provide links for supplemental videos and online simulation tools. This book also includes a table to content and a summary of projects, so it is easy to "jump" to a particular chapter and a particular project.

The reviewer personally wishes there were more examples in some sections to demonstrate some ideas better. For example, chapter 31 (multirole derivatives) covers the most challenging topic in derivative calculations, but it only offers two examples. The other minor concern is that the long mathematical equations, especially those with denominators, are sometimes hard to read from time to time (in the pdf version). The authors may consider using software that is more friendly to mathematical formulations.

The text is consistent in terms of terminology and frameworks of a calculus I course for life science and environmental science students.

This book has a nice layout. The materials are all internally connected and properly ordered. Authors are considerate and provided algebraic reviews when needed. Any instructor using this book can easily tailor the sections to fix their needs.

The topics in the text are presented logically and clearly.

That being said, this text has seven parts and 69 chapters, and close to half of those chapters are homework sections. The reviewer personally wishes to combine the homework with the corresponding topic into one chapter, which will reduce the number of chapters to be about 30. The students may be less stressed if there were fewer topics to learn at a glance.

I was not able to use the XML version after downloading it to my computer (mac). The epub version is easy to navigate. I mostly used the pdf version. It would be easier to navigate if the bookmarks (in the pdf version) are numbered or has a hierarchy like in the epub version.

The book is well-written. I did not come across any grammatical or punctuation errors.

The examples, projects, and homework problems used in the book are related to life sciences and environmental sciences. I feel this book is culturally inclusive.

This book has done a great job introducing calculus to students in life sciences and environmental sciences in an informal but practical way. The authors spend significant efforts on graphics, providing supplemental material through hyperlinks, and designing the projects. Hope more universities consider adopting this book.

## Table of Contents

- Algebra Tips and Tricks: Part I
- Algebra Tips and Tricks Part I (Combining Terms, Distributing, Functions, Graphing)
- Homework for Algebra Tips and Tricks: Part I

- Derivative Introduction
- Ball Toss Project
- Position to Velocity
- Homework: Position to Velocity
- Algebra Tips and Tricks Part II (Piecewise Defined Functions)
- Limits
- Homework: Limits
- Algebra Tips and Tricks Part III (Factoring)
- Algebraic Limits
- Homework: Algebraic Limits
- Instantaneous Velocity
- Homework: Instantaneous Velocity
- Algebra Tips and Tricks IV (Tips for dealing with fractions)
- Definition of Derivative Examples
- Project: Hard Definition of Derivative Problems
- Homework: Examples of the Definition of the Derivative
- Project: Killdeer Migration Speed

- Rules for Derivatives
- Algebra Tips and Tricks Part V (Exponents)
- Power Rule
- Homework: Power Rule
- Algebra Tips and Tricks Part VI (Logarithms)
- Exponentials, Logarithms, and Trig Functions
- Homework: Exponents, Logs, Trig Functions
- Product Rule
- Homework: Product Rule
- Quotient Rule
- Homework: Quotient Rule
- Chain Rule
- Homework: Chain Rule
- Multirule Derivatives
- Homework: Multirule Derivatives
- Anti-derivatives
- Homework: Anti-derivatives

- More Derivative Intuition
- Derivatives and Graphs
- Homework: Derivative Graphs
- Second Derivatives and Interpreting the Derivative
- Homework: Second Derivatives and Interpreting the Derivative
- Optimization
- Homework: Optimization
- Derivatives in Space
- Homework: Derivatives in Space

- Differential Equations
- Recurrance Relations
- Homework: Recurrence Relations
- Introduction to Differential Equations
- Homework: Introduction to Differential Equations
- Understanding Differential Equations
- Homework: Understanding Differential Equations
- Initial Value Problems
- Homework: Initial Value Problems
- Growth and Decay
- Homework: Growth and Decay
- Exploring Graphs of Differential Equations
- Project: Modelling with Differential Equations

- Intuition for Integration
- Introduction to Integrals
- Numeric Integration Techniques
- Homework: Numeric Integration Techniques
- Fundamental Theorem of Calculus
- Homework: The Fundamental Theorem of Calculus
- Project: Measuring Streamflow
- Project: Quake Lake

- Rules for Integration
- Power, exponential, trig, and logarithm rules
- Homework: Power, exponential, trig, and logarithmic rules
- u-substitution
- Homework: u-substitution
- Integral Applications
- Homework: Integral Applications
- Integration by Parts
- Homework: Integration by Parts

- Acknowledgements

## Ancillary Material

## About the Book

This book is an approachable introduction to calculus with applications to biology and environmental science. For example, one application in the book is determining the volume of earth moved in the 1959 earthquake that created Quake Lake. Another application uses differential equations to model various biological examples, including moose and wolf populations at Isle Royale National Park, ranavirus in amphibians, and competing species of protozoa. The text focuses on intuitive understanding of concepts, but still covers most of the algebra and calculations common in a survey of calculus course.

## About the Contributors

### Author

**Tyler Seacrest** earned his B.S. in Mathematics from Harvey Mudd College, and his M.S. and Ph.D. in Mathematics from the University of Nebraska-Lincoln. Tyler Seacrest has taught at the University of Montana Western since 2011, and enjoys working with students at all levels. For freshman and sophomore courses, he integrates hands-on activities to make mathematics more concrete, and he is interested in making mathematics research accessible to junior and senior level students. His research area is discrete mathematics, especially graph theory, a subject with a strong history of undergraduate research.

### Editors

**Debbie Seacrest**,
University of Montana Western

**Rebekah Levine**, University of Montana Western

**Michele Anderson**, University of Montana Western