Calculus for the Beginner— Building Calculus, Piece by Piece

Understanding the beauty of calculus

Having an understanding of calculus is integral to achieving a more clear and appreciative view not only of mathematics, but of the world. By adopting an intuition and comfort in working with infinitely large and small quantities, humanity can explore the world we inhabit with a greater accuracy and clarity than without. Unfortunately, because it may seem daunting, many are isolated from its beauty. In this article, with examples, diagrams, and explanations, we’ll begin with what we know and slowly build up calculus, piece-by-piece.

All images and diagrams created by author.

Calculus begins with limits. A good example to illustrate limits in action is to consider the question of why anything (with the exception of 0, which is a bit more complicated) divided by zero is undefined. In order to find the answer to this, one cannot simply directly “divide by zero” to find the solution. Instead, we can be more subtle and set up a test function of 1 / x, plotting out the function’s value as x nears zero and seeing the corresponding value of y at that point. In notation, we would write this as: