Copyright © 2004, by Bluejay Lispware, all rights reserved.
This material may be reproduced for personal use,
but may only be distributed within an institution if the
institution holds a current Mathwright Library Institutional License

The following 500 page book, which consists of 49 lectures and 43 Interactive Explorations, is presented in 10 Mathwright Microworlds. The Text version (see below) is free for all to read.

The Explorations are designed to encourage readers to pursue their own ideas by asking "What if?" questions. They also play a heuristic role to help the reader visualize new constructions, techniques and ideas, but for the reader for whom this book is written, that role is ancillary. This book is written for active (and aggressive) learners, who by now should have little trouble visualizing the constructions at the base of the Calculus. For those readers, the examples will be the important thing, and we bring the computer environment into a new relationship to the mathematics for them. In fact, this will likely strike Students as a new kind of mathematics book, unlike one they have ever seen, and certainly not like any Calculus Textbook, either in content or in form. Teachers: Be prepared for a few surprises! Library members are invited to correspond with the author on any questions about the material or the interactive explorations.

This book is designed to illuminate, and to give readers the chance to explore in some depth, the basic ideas of Calculus within the context of its first triumphal scientific success: Isaac Newton's deduction of Kepler's three laws of planetary motion from his single hypothesis of Universal Gravitation. This book differs from a Calculus textbook in several ways. While it assumes a basic understanding of Geometry and Algebra, it is designed around the theme of Gravitation, rather than any particular syllabus. It is an Interactive Story that invites you to explore a selected range of ideas from the Calculus that were inspired by this theme.

Unlike a text, the book does not attempt to give an encyclopedic account of all the standard techniques of calculation and problem-solving that readers might someday be called upon to know. We develop in some detail a great variety of techniques, but only as we require them in the telling of the story. To that end, each Microworld Section of each Chapter of the book first discusses a problem that we need to solve to deepen our understanding of the gravitation theme, and then recruits and explains the techniques that Calculus can supply to help us solve it. The problems are not easy ones, but we attempt in the lectures and interactive explorations to bring them to life, so that readers can experiment, and become familiar with them.

The book is written for readers who enjoy mathematics, and have the curiosity and the desire to see the small part of it that we develop here, as a whole: roots, branches, and leaves. As the title implies, this book is not meant to be read like a text, but is designed for you to learn by acting on the various stages of the story. If you would like to know what Calculus is really about, and how it came to be, this book may be for you.

But we begin with a word of caution. This book is a gradual ascent to a high place. We have tried to write it so that the early part will be immediately meaningful to the thoughtful and curious high-school student. The Pre-Calculus Introduction is fairly non-technical, dealing mainly with the circle of ideas that led to the invention of Calculus in the 17th century. Those ideas are what really matter, and will form the basis for all that follows in later chapters. And as we gather the analytic tools and techniques that Isaac Newton invented to solve Kepler's problem of planetary motion, we will adopt an increasingly "rigorous" tone. The level of discourse may then not be accessible to beginners, no matter how dedicated they are to the task. It is an unfortunate myth that a beginning student can master the calculus after only one or two years of study. When you have followed the story that this book tells, you will be well on your way to that mastery!

So we counsel patience. This book is a Story, not an encyclopedia. It is not a textbook. Take your time, and enjoy it. You will not find the answers neatly laid out in its pages, until you yourself ask the questions. And that requires both time and reflection. There is no "royal road" to the Calculus, and you may find that you return to this book many times in order to see a point clearly. But we believe that the book will offer you a new opportunity to formulate and to ask your questions. Each part of each section of each chapter invites you to experiment, and to bring the ideas to life in a way that is meaningful to you.

We introduce the key ideas of the Calculus only as we need them to solve the problem at hand, whether it is to calculate the trajectory of a baseball, to determine the escape velocity, or to place a satellite into geosynchronous orbit around the Earth. And like the Calculus itself, our problems all grow out of the questions: "How does an object fall ? How do the Moon and Planets move ?" That is the theme of the story, our main question, for which the Calculus is the language that provides our best answer.

In the hands of its creator, Isaac Newton, the Calculus was a musical instrument. He made it sing the song that told of Kepler's "harmony of the spheres," and he gave us an instrument of thought that he felt would reveal the deepest secrets of Nature. In this series of lectures, you will follow young Isaac, and will see for yourself what music Calculus can make in your own hands.

The microworlds that constitute the Chapter Sections will also invite you to roll up your sleeves and to use the Calculus to solve problems. Scroll down to the end of each lecture to enter its microworld. In fact, all of these lectures will be found in the microworlds themselves. Each page of each microworld, including the Calculator page has the story for that page under the   icon.  Just click on this icon to read the story for the page. You may (and probably should) print each lecture page so that you can read it separately. The best way to read this book is to download the Microworlds to your own computer and read them as you like offline in Mathwright32 Reader or Mathwright32 Net Reader. They will be peppier, and will not require online access to read them. They will be yours!

The author gratefully acknowledges the encouragement, advice, and sometimes the code given by several colleagues, Mathwright Authors all. Their contributions were both to the spirit and to the form of this book. In alphabetical order, these are: Margie Hale, Dan Kalman, Ravinder Kumar, Samad Mortabit, and Mike Pepe. Thank you all.

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