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APEX Calculus: for University of Lethbridge

Preface Preface

A Note on Using this Text.

Thank you for reading this short preface. Allow us to share a few key points about the text so that you may better understand what you will find beyond this page.
This text comprises a three—volume series on Calculus. The first part covers material taught in many “Calc 1” courses: limits, derivatives, and the basics of integration, found in Chapters 1 through 6.1. The second text covers material often taught in “Calc 2:” integration and its applications, including an introduction to differential equations, along with an introduction to sequences, series and Taylor Polynomials, found in Chapters 5 through 8. The third text covers topics common in “Calc 3” or “multivariable calc:” parametric equations, polar coordinates, vector-valued functions, and functions of more than one variable, found in Chapters 10 through 15. All three are available separately for free at
, and HTML versions of the book can be found at
These three texts are intended to work together and make one cohesive text, APEX Calculus, which can also be downloaded from the website.
Printing the entire text as one volume makes for a large, heavy, cumbersome book. One can certainly only print the pages they currently need, but some prefer to have a nice, bound copy of the text. Therefore this text has been split into these three manageable parts, each of which can be purchased for about $15 at

For Students: How to Read this Text.

Mathematics textbooks have a reputation for being hard to read. High—level mathematical writing often seeks to say much with few words, and this style often seeps into texts of lower—level topics. This book was written with the goal of being easier to read than many other calculus textbooks, without becoming too verbose.
Each chapter and section starts with an introduction of the coming material, hopefully setting the stage for “why you should care,” and ends with a look ahead to see how the just—learned material helps address future problems.
  • Please read the text.
    It is written to explain the concepts of Calculus. There are numerous examples to demonstrate the meaning of definitions, the truth of theorems, and the application of mathematical techniques. When you encounter a sentence you don’t understand, read it again. If it still doesn’t make sense, read on anyway, as sometimes confusing sentences are explained by later sentences.
  • You don’t have to read every equation.
    The examples generally show “all” the steps needed to solve a problem. Sometimes reading through each step is helpful; sometimes it is confusing. When the steps are illustrating a new technique, one probably should follow each step closely to learn the new technique. When the steps are showing the mathematics needed to find a number to be used later, one can usually skip ahead and see how that number is being used, instead of getting bogged down in reading how the number was found.
  • Most proofs have been omitted.
    In mathematics, proving something is always true is extremely important, and entails much more than testing to see if it works twice. However, students often are confused by the details of a proof, or become concerned that they should have been able to construct this proof on their own. To alleviate this potential problem, we do not include the proofs to most theorems in the text. The interested reader is highly encouraged to find proofs online or from their instructor. In most cases, one is very capable of understanding what a theorem means and how to apply it without knowing fully why it is true.

Interactive, 3D Graphics.

Versions 3.0 and 4.0 of the textbook include interactive, 3D graphics in the PDF version. Nearly all graphs of objects in space can be rotated, shifted, and zoomed in/out so the reader can better understand the object illustrated. However, the only pdf viewers that support these 3D graphics are Adobe Reader Acrobat (and only the versions for PC/Mac/Unix/Linux computers, not tablets or smartphones).
The latest version of the book, which is authored in PreTeXt, is available in HTML. In HTML, the 3D graphics are rendered using WebGL, and should work in any modern web browser.
Interactive graphics are no longer supported within the PDF, but clicking on any 3D graphic within the PDF will take you directly to the interactive version on the web.

APEX – Affordable Print and Electronic teXts.

APEX is a consortium of authors who collaborate to produce high quality, low cost textbooks. The current textbook—writing paradigm is facing a potential revolution as desktop publishing and electronic formats increase in popularity. However, writing a good textbook is no easy task, as the time requirements alone are substantial. It takes countless hours of work to produce text, write examples and exercises, edit and publish. Through collaboration, however, the cost to any individual can be lessened, allowing us to create texts that we freely distribute electronically and sell in printed form for an incredibly low cost. Having said that, nothing is entirely free; someone always bears some cost. This text “cost” the authors of this book their time, and that was not enough. APEX Calculus would not exist had not the Virginia Military Institute, through a generous Jackson—Hope grant, given the lead author significant time away from teaching so he could focus on this text.
Each text is available as a free .pdf, protected by a Creative Commons Attribution - Noncommercial 4.0 copyright. That means you can give the .pdf to anyone you like, print it in any form you like, and even edit the original content and redistribute it. If you do the latter, you must clearly reference this work and you cannot sell your edited work for money.
We encourage others to adapt this work to fit their own needs. One might add sections that are “missing” or remove sections that your students won’t need. The source files can be found at
You can learn more at

First PreTeXt Edition (Version 5.0).

Key changes from Version 4.0 to 5.0:
  • The underlying source code has been completely rewritten, to use the PreTeXt
    language, instead of the original \(\LaTeX\text{.}\)
  • Using PreTeXt allows us to produce the books in multiple formats, including HTML, which is both more accessible and more interactive than the original PDF. HTML versions of the book can be found at
  • The appendix on differential equations from the “Calculus for Quarters” version of the book has been included as Chapter 8, just after applications of integration. Chapters 8 — 14 are now numbered 9 — 15 as a result.
  • In the HTML version of the book, many of the exercises are now interactive, and powered by WeBWorK.
Key changes from Version 3.0 to 4.0:
  • Numerous typographical and “small” mathematical corrections (again, thanks to all my close readers!).
  • “Large” mathematical corrections and adjustments. There were a number of places in Version 3.0 where a definition/theorem was not correct as stated. See
    for more information.
  • More useful numbering of Examples, Theorems, etc. . “Definition 11.4.2” refers to the second definition of Chapter 11, Section 4.
  • The addition of Section 13.7: Triple Integration with Cylindrical and Spherical Coordinates
  • The addition of Chapter 14: Vector Analysis.