Introduction
Calculus ranks as one of the greatest accomplishments of the human mind. Virtually all of our modern technology, as well as modern scientific theories of the physical universe are based on a solid foundation of calculus. Calculus is more than just a collection of useful computational tools; it provides a conceptual framework within which one may perceive aspects of reality otherwise inconceivable. A good knowledge of calculus will open many doors. Millions of people from around the world have mastered this subject, and you can too.
A mastery of calculus entails knowledge of certain rules (theorems) and how they can be applied to solve various problems. In particular, you’ll need to know how to solve the problems you encounter on calculus tests. These lecture notes have as their primary focus, enabling you ace all the tests!
Calculus is often still taught much as it was in the 1800’s. If you would like to see a typical calculus text published in 1849, visit this old scanned calculus text from the Cornell Digital Mathbook Collection. My hope is that this course will end up looking more like a 22nd century course than it does a 19th century calculus course. Although a good understanding of the basic theorems and computations of calculus are as important as ever, students today need to also become competent in the use of powerful computer algebra systems [CAS] software that is now available. These software tools greatly extend our ability to visualize mathematical structures and to perform complex calculations (as oft arise in real-world applications). CAS are incredibly powerful and surprisingly easy to use, and perhaps (depending upon your criteria) the best of these is Maple. This class will make extensive use of Maple, which you should obtain for this class, in lieu of a text. By the time you complete this class, you’ll be competent in doing calculus with and without the assistance of Maple. Maple will prove very useful to you in future classes, future careers, and in your everyday life!
In addition to my ever-expanding lecture notes, other useful calculus resources can be found at Karl’s Calculus Tutor and other sources you'll find under the Links tab at the top of each page. And of course, you are welcome to email me your questions. Much of the growth of my lecture notes is motivated by students' questions.
So what is calculus, anyway? If you add but one new concept to precalculus mathematics, and stir vigorously, you get calculus! That new concept is limits. An intuitive notion of limits will suffice for this course, as it historically did for generations of mathematicians and scientists. Only in the 19th century was the notion of limit put on a solid foundation, and the formal definition of limits that finally crystallized is a bit difficult for students to grasp at first (don’t worry, you won’t need to fully master the formal definition for this course – this will be covered in a Real Analysis course, if you continue with your math studies beyond 2 years of calculus). The concept of limits allows us to compute the areas of strangely shaped regions, to compute instantaneous rates of change, and to add up infinite sums. The language of calculus allows us thereby to express how the value and rates of change of some quantities depend on the value and rates of change of other quantities. Physicists have found calculus to be the natural language for describing much of the physical universe…
If you’re in a philosophical mood, read some of these interesting musings concerning the surprisingly slippery question, “What is math?” (my comments are the last listed).
I look forward to helping you master this fascinating subject. It is my hope that this class will prove to be the most interesting, useful, and rewarding class you've ever taken. But I need your help to make this so! If you have any questions, comments, or suggestions, please don't hesitate to email me.
-Rafael Espericueta
|
©
1998-2003 by |
