Informal Calculus

Welcome to Informal Calculus! This book came from teaching Survey of Calculus at the University of Montana Western. You’ll find that this book reflects the needs of that course, which are:

• Intuition behind the main concepts of calculus,
• Short, to-the-point explanations about how to do calculations,
• Build-in review for algebra along the way, with trigonometry not being a pre-requisite
• Applications to biology and environmental science, the two main science majors at my university.

A huge thanks to my collaborators for this project: Michelle Anderson who helped with biology ideas, problems, and projects; Rebekah Levine, who assisted with environmental science ideas, problems, and projects; and Debbie Seacrest, who advised on mathematical content and edited the entire book. This work was partially supported by a grant from TRAILS Montana. Thanks to Christina Trunnell for helping with many aspects of this book directly, as well as supporting OER projects all over Montana.

Click on “contents” to start exploring the book.

FOR TEACHERS

Feel free to take, change, modify any materials from this book for your own class or other use. If your particularly interested in project ideas, I’ve collected links to the projects in this book below:

• Ball toss
• Hard Derivatives
• Killdeer Migration
• Modelling with Differential Equations
• Measuring Streamflow
• Quake Lake

Here are a few algebra tips and tricks to get you started. In later chapters, we will have some “just-in-time” algebra review, so you’ll review an algebra concept just before you need it.

COMBINING LIKE TERMS

A term is one or more things multiplied together: for example, is a term since it is times times , is a term, since it is times and is a term. If there is also a number multiplied in front of a term, that is called the coefficient (if no coefficient is present, the coefficient is ). Two terms are like terms if they have the same variables multiplied (but may have different coefficients). If two like terms are added together, they can be combined into one term by adding the coefficients.

When multiplying two sums, every term of the first must be multiplied by every term of the second. Thus, if there are two terms in the first sum and two in the second, there are four total terms in the product: the (f)irst two terms, the (o)utside terms, the (i)nside terms, and the (l)ast two terms. We can use the acronym “foil”: