Calculus

Calculus is the branch of mathematics dealing with the study of continuous change.

Introduction

In calculus, everything is a function. We don't just define an equation, we turn it into one or more functions to work with. For instance, instead of saying that \(y = 3x\), we would say that \(y\) is a function of \(x\); we would then write it as \(y=f(x)\) where \(f(x)=3x\). At first, this may seem a bit daunting, but after a while, it becomes easy. The purpose for refering to functions instead of equations becomes clear once you delve a bit deeper into the calculus. The idea of a function replacing part of an equation is analagous to algebra substituting letters for numbers. And just like algebra, these substitutions make the math much easier in the long run.

I will say that reading any text on calculus can be a bit dry, and if you're just starting out, can be a bit confusing. I found that the key to learning is to simply read through confusing bits as if reading a novel. Don't stop to question it, and don't focus on the confusion. Then, use worked examples along with the text to figure it out. I personally tried to learn calculus for years, without success, until one day I found a small book that put it all in simpler terms using worked examples. I then asked myself, "why didn't they just say that in the first place?"

The sections below on differential and integral calculus both start with very technical descriptions of calculus in terms of limits. However, thanks to more modern approaches (mainly the fundamental theorem of calculus), it is possible to learn the basics of calculus without having to remember (or even ever touch) these sections.

Limits

Especially when working with the long-hand forms of calculus, we tend to work with a lot of limits. This is because calculus deals mostly with infinitessimally small slices, and limits are the best, if not the only way to do this.