This is a work in progress. Feel free to improve this lecture by rewording, clarifying statements, adding examples, and adding material relevant to the topic.
Lecture 0: An Introduction to Calculus
So How does this Course Work, Anyway?
Learning calculus takes determination that goes far beyond simply reading text. Make sure to always have paper and pencils ready to draw diagrams or make quick calculations when solving a problem; using a calculator can only get you so far in Calculus.
Some students will start learning calculus by trying the problems and will only resort to reading the text if they get stuck on a problem or do not know how to proceed. A better plan is to make sure that you know and fully understand the material before beginning work on problems. A link to problems relevant to the material in each lecture is located at the bottom of each page. The problems are separated into three levels. When first going through this course, it is recommended that you be able to complete all of the Level 1 problems and most of Level 2 before continuing to the next lecture. The Level 3 problems are for those who feel they have mastered this calculus course and still want a challenge. Separate links are included that show the answers to each problem and the worked out solutions to a few problems from each level.
A huge factor that will influence your understanding of calculus is your ability to think logically. These lectures will emphasize the importance of writing solutions to problems as a set of ordered steps rather than a hodgepodge of random formulas. You will find that many calculus problems become easier when the steps are written down on paper instead of constrained in your mind. And if you understand the material but still get stuck on a problem: start over. Throw away your work and get a clean sheet of paper with no writing on it. Ofttimes when starting a problem anew, an entirely different solution will come to light that you did not see before. If all else fails: come back later. That's right, you heard me: go outside, talk with a friend, watch television, get some rest, have fun. The God of Calculus will not smite you for leaving problems undone.
The authors of these lectures hope that you will not only be opened to the infinite applications that Calculus has to the modern world, but will find an intrinsic beauty that goes beyond what we as mere mortals can comprehend.
What is Calculus?
To define what calculus is, we should start with the origin of the name. The word calculus in Latin means "pebble". This is quite fitting because calculus deals with the very small. So inconceivably small, in fact, that it is ofttimes hard to differentiate the numbers we deal with from zero. In all, Calculus is the study of change. This is especially true with regards to the four major topics in calculus, which we will cover in this order:
Before we begin we should review the topics necessary for advancing to calculus.