Preface About This Text
Our Goals.
Several fundamental ideas in calculus are more than 2000 years old. As a formal subdiscipline of mathematics, calculus was first introduced and developed in the late 1600s, with key independent contributions from Sir Isaac Newton and Gottfried Wilhelm Leibniz. Mathematicians agree that the subject has been understood rigorously since the work of Augustin Louis Cauchy and Karl Weierstrass in the mid 1800s when the field of modern analysis was developed, in part to make sense of the infinitely small quantities on which calculus rests. As a body of knowledge, calculus has been completely understood for at least 150 years. The discipline is one of our great human intellectual achievements: among many spectacular ideas, calculus models how objects fall under the forces of gravity and wind resistance, explains how to compute areas and volumes of interesting shapes, enables us to work rigorously with infinitely small and infinitely large quantities, and connects the varying rates at which quantities change to the total change in the quantities themselves.
While each author of a calculus textbook certainly offers their own creative perspective on the subject, it is hardly the case that many of the ideas they present are new. Indeed, the mathematics community broadly agrees on what the main ideas of calculus are, as well as their justification and their importance; the core parts of nearly all calculus textbooks are very similar. As such, it is our opinion that in the 21st century and the age of the internet, 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. Calculus belongs to humankind, not any individual author or publishing company. Thus, a primary purpose of this work is to present a calculus text that is free. In addition, instructors who are looking for a calculus text should have the opportunity to download the source files and make modifications that they see fit; thus this text is open-source.
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.
- 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-colorgraphics to exemplify and magnify key ideas, and to use this graphical perspective alongside both numerical and algebraic representations of calculus.
5
To keep cost low, the graphics in the print-on-demand version are in black and white. When the text itself refers to color in images, one needs to view the .html or .pdf electronically. - Summary of Key Ideas
- Each section concludes with a summary of the key ideas encountered in the preceding section; this summary normally reflects responses to the motivating questions that began the section.
How to Use this Text.
Because the text is free, any professor or student may use the electronic version of the text for no charge.
This text may be used as a stand-alone textbook for a standard first year college calculus sequence. Chapters 1–4 address the standard topics for differential calculus, while Chapters 5–8 correspond to integral calculus, including chapters on differential equations (Chapter 7) and infinite series (Chapter 8).
