Let’s be honest. Most people would like to learn calculus as much as they’d like to be kicked in the face by a mule. Usually, they have to take the course because it’s required or they walked too close to the mule, in that order. Calculus is dull, calculus is boring, and calculus didn’t even get you anything for your birthday.
It’s not like you didn’t try to understand calculus. You even got this bright idea to try and read your calculus textbook. What a joke that was. You’re more likely to receive the Nobel Prize for chemistry than to understand a single word of it. Maybe you even asked a friend of yours to help you, and talking to her was like trying to communicate with an Australian aborigine. You guys just didn’t speak the same language.
You wish someone would explain things to you in a language that you understand, but in the back of your mind, you know that the math lingo is going to come back to haunt you. You’re going to have to understand it in order to pass this course, and you don’t think you’ve got it in you. Guess what? You do!
Here’s the thing about calculus: things are never as bad as they seem. The mule didn’t mean it, and I know this great plastic surgeon. I also know how terrifying calculus is. The only thing scarier than learning it is teaching it to 35 high school students in a hot, crowded room right before lunch. I’ve fought in the trenches at the front line and survived to tell the tale. I can even tell it in a way that may intrigue, entertain, and teach you something along the way.
We’re going to journey together for a while. Allow me to be your guide in the wilderness that is calculus. I’ve been here before and I know the way around. My goal is to teach you all you’ll need to know to survive out here on your own. I’ll explain everything in plain and understandable English. Whenever I work out a problem, I’ll show you every step (even the simple ones) and I’ll tell you exactly what I’m doing and why. Then you’ll get a chance to practice the skill on your own without my guidance. Never fear, though—I answer the question for you fully and completely in the back of the book.
I’m not going to lie to you. You’re not going to find every single problem easy, but you will eventually do every one. All you need is a little push in the right direction, and someone who knows how you feel. With all these things in place, you’ll have no trouble hoofing it out. Oh, sorry, that’s a bad choice of words.
How This Book Is Organized
This book is presented in five parts.
In Part 1, The Roots of Calculus, you’ll learn why calculus is useful and what sorts of skills it adds to your mathematical repertoire. You’ll also get a taste of its history, which is marred by quite a bit of controversy. Being a math person, and by no means a history buff, I’ll get into the math without much delay. However, before we can actually start discussing calculus concepts, we’ll spend some quality time reviewing some prerequisite algebra and trigonometry skills.
In Part 2, Laying the Foundation for Calculus, it’s time to get down and dirty. This is the moment you’ve been waiting for. Or is it? Most people consider calculus the study of derivatives and integrals, and we don’t really talk too much about those two guys until Part 3. Am I just a royal tease? Nah. First, we have to talk about limits and continuity. These foundational concepts constitute the backbone for the rest of calculus, and without them, derivatives and integrals couldn’t exist.
Finally, we meet one of the major players in Part 3, The Derivative. The name says it all. All of your major questions will be answered, including what a derivative is, how to find one, and what to do if you run into one in a dark alley late at night. (Run!) You’ll also learn a whole slew of major derivative-based skills: drawing graphs of functions you’ve never seen, calculating how quickly variables change in given functions, and finding limits that once were next to impossible to calculate. But wait, there’s more! How could something called a “wiggle graph” be anything but a barrel of giggles?
In Part 4, The Integral, you meet the other big boy of calculus. Integration is almost the same as differentiation, except that you do it backward. Intrigued? You’ll learn how the area underneath a function is related to this backward derivative, called an “antiderivative.” It’s also time to introduce the Fundamental Theorem of Calculus, which (once and for all) describes how all this crazy stuff is related. You’ll find out that integrals are a little more disagreeable than derivatives were; they require you to learn more techniques, some of which are extremely interesting and (is it possible?) even a little fun!
Now that you’ve met the leading actor and actress in this mathematical drama, what could possibly be left? The love story, of course! In Part 5, Differential Equations, I weave a beautiful narrative detailing the intricate relationship between derivatives and integrals sharing their lives together in a small, rural suburban neighborhood. Well, that’s not entirely true, but you do get to play with fun things called slope fields and you end this part by taking an exam on all the content in the book and get even more practice! What could be better than that?
At the back of the book, I’ve included the solutions to all the practice problems as well as a glossary of helpful terms.
As a teacher, I constantly found myself going off on tangents—everything I mentioned reminded me of something else. These peripheral snippets are captured in this book as well. Here’s a guide to the different sidebars you’ll see peppering the pages that follow.
Critical Point
These notes, tips, and thoughts will assist, teach, and entertain. They add a little something to the topic at hand, whether it be sound advice, a bit of wisdom, or just something to lighten the mood a bit.
Definition
Calculus is chock-full of crazy- and nerdy-sounding words and phrases. In order to become King or Queen Math Nerd, you’ll have to know what they mean!
Kelley’s Cautions
Although I will warn you about common pitfalls and dangers throughout the book, the dangers in these boxes deserve special attention. Think of these as skulls and crossbones painted on little signs that stand along your path. Heeding these cautions can sometimes save you hours of frustration.
You’ve got problems
Math is not a spectator sport! After we discuss a topic, I’ll explain how to work out a certain type of problem, and then you have to try it on your own. These problems will be very similar to those that I walk you through in the chapters, but now it’s your turn to shine. Even though all the answers appear in Appendix A, you should only look there to check your work.
Dedication
This book is dedicated to Lisa, who is no Linda Ronstadt. Despite not knowing much, I know I love you. I also know that one day our hit single about ham that we dropped on the floor will be a worldwide phenomenon.
To my kids Nick, Erin, and Sara. I know that I will miss the noise (God in heaven, the noise) of you playing while I am working. One day soon you will be old enough to have loud children of your own. I love you very much. Now please shush—Dad is trying to write.
Finally, to Joe. It was a home run.
Special Thanks to the Technical Reviewers
Idiot’s Guides: Calculus I was reviewed by Robert Halstead, an expert who double-checked the accuracy of what you’ll learn here. He’s also the kind of super nice guy who helps you move furniture even when his shoulder is hanging out of its socket. The publisher would like to extend our thanks to Rob for helping us ensure that this book gets all its facts straight. We also thank Sue Strickland, who reviewed the previous editions and is still the best mathematics instructor that ever was.
Rob is a mathematics teacher at Northern High School in Calvert County, Maryland, with 22 years of teaching experience. He spent the last 15 of those years teaching Advanced Placement Calculus. He has served as the Mathematics Core Lead and department chair at his school, and he was chosen as the school’s Teacher of the Year in 2012.
Susan received a BS in Mathematics from St. Mary’s College of Maryland in 1979, an MS in Mathematics from Lehigh University in Bethlehem, Pennsylvania, in 1982, and took graduate courses in Mathematics Education at The American University in Washington, D.C., from 1989 through 1991. She was an assistant professor of mathematics and supervised student mathematics teachers at St. Mary’s College of Maryland from 1983 through 2001. In the summer of 2001, she accepted the position as a professor of mathematics at the College of Southern Maryland, where she expects to be until she retires! Her interests include teaching mathematics to the “math phobics,” training new math teachers, and solving math games and puzzles.