Calculus Simplified

Book Preface

What You Can Expect to Learn from This Book
This subheading’s title might seem silly, given that this is a book about calculus. But research showsthatstudentslearn best when they know beforehand what they are abouttolearnandwhattheyshouldbeabletodowiththatknowledgeattheendof the lesson. Rather than insert these learning goals and objectives at the start of each lesson, I have provided them in a supplemental document titled Calculus Simplified: Learning Goals and Objectives, available on the book’s website. (That document also maps this book’s learning goals and objectives to the curriculum framework used in AP Calculus, in case you’re enrolled in such a course.) I highly recommend you keep that document handy as you read through each section of the book.

How to Read a Math Book (Including This One)
Though I have done my best to infuse my writing with the elements of a novel— characters, a plot, etc.—this book is not a novel. One thing this implies is that you need to read this book differently than you would a novel. For example, simply reading this book won’t help you understand calculus. Rather, I recommend you work through this book—work out the examples, work out the solutions to the exercises, work through the supplemental content. By doing mathematics you will be helping yourself learn mathematics. Moreover, jot down questions and comments as you read and work through this book. This will ensure you are learning actively rather than passively.
Lastly, let me mention the special role that theorems play in mathematics, and how to ensure you’re getting the most out of them. Loosely speaking, a theorem is a statement that has been proven true. A typical theorem has the following structure: preamble, hypotheses, conclusion. Example:
Theorem (Pythagorean Theorem): Consider a right triangle in the plane. Let c denote the length of the hypotenuse of the triangle and a and b the lengths of the other two sides. Then c2 = a2 + b2.

To the Student

In this theorem the first sentence is the preamble; its role is to provide context for what the theorem says. The second sentence in the theorem contains some assumptions (as happened here, sometimes the preamble also contains assumptions). The last sentence contains the conclusion.
Echoing my earlier advice to work through this book, do the same with theorems—whenever you come across a theorem take a moment to understand what it is saying. Try to draw pictures, explain the theorem in words, and imagine removing some of the hypotheses to see how the conclusion might be affected. Doing all this will help you appreciate what the real use of the theorem is, help you remember it, and help you learn when it can (and cannot) be applied.

How to Become a Better Student
Ihaveonelastrecommendationforyou: employ the latest research from the science of learning while you study. Study strategies like retrieval practice and interleaving— both backed by cognitive science research—can supercharge your studying. You can read more about these and other research-backed study strategies in the supplemental document titled Evidence-Based Study Strategies,available on the book’s website.
Alright, that’s all I have for you at the moment. Let’s get started with your calculus adventure!