Calculus, Single Variable, Preliminary Edition

Book Preface

You learned at an early age that planets orbit the sun, and our sun orbits the center of the Milky Way because of a force called gravity. Many people today nd that fact unremarkable, but for most of human history celestial motion was something of a mystery. It wasn’t until the early 1600s that our modern understanding of the universe really began to take shape. That’s when the telescope was invented, Galileo Galilei saw the moons of Jupiter, and Johannes Kepler published his nding that planets move around the sun in elliptical orbits. After that, people knew what happened, and turned their attention to why. Men like Robert Hooke, Edmund Halley, and Christopher Wren struggled with the problem until 1684, when Halley paid a visit to Isaac Newton in Cambridge.

Halley asked if a force that obeyed an inverse square law would result in the It’s okay if the term inverse square law is unfamiliar to you. We’ll talk about it in more deorbits that Kepler had described. Newton’s ready response was, yes. In fact, he’d tail in a moment. already made the calculations . . . though he had misplaced them (see [29]). Newton promised to nd or redevelop the calculations, and three years later, he published his famous Philosophiae Naturalis Principia Mathematica in which he introduced the calculus. Since that time, it has become not only a tool for making calculations, but a way of understanding and talking about the world we live in.

This chapter is intended as a review of the basic functions that you will see in calculus|tools of the trade, as it were. The point of such a review is not to rehash the mechanics of how these functions work, but rather to help you begin thinking about mathematics in a new way.

– Whereas algebra and precalculus traditionally focus on the value of a function,
calculus is fundamentally about how functions change, so our discussion will
often come back to the topic of change.
– We’ll emphasize equations as a communication of ideas rather than as a collection
of symbols to be manipulated. Part of this emphasis will involve a
technique called dimensional analysis, which provides deeper understanding
of functions and change based on the units involved in a problem.
– While there are a lot of mathematical ideas that are intriguing, or beautiful,
or just plain fun, humans have been driven by the need to describe and understand
the world we live in. So our discussion will often include situations
in which particular types of functions or important ideas arise.

As you read, you’ll see that examples in the presentation are often followed by try it yourself” examples, to which an answer is provided here but not the full solution. Our hope is that you’ll try these examples, and learn by doing as you go. If you nd that you need help, you can nd a full solution on-line.