Calculus the Basics

Calculus: A Brief Introduction

Calculus is a branch of mathmatics that was jointly created by Newton and Leibnitz and is based on the behavior of infintesimal changes. It has extremely far reaching capabilities and it used extensively in fields ranging from particle physics to economics. Calculus is divided into two basic subdivisions, the so-called differential and integral calculus.

Before discussing derivatives and integrals, a little background on functions will be helpful. A function (more specifically single variable function) is basically a “machine” that takes some arbitrary input value and gives a unique output. For example, in the function y=f(x), for every value of x that you feed the machine, you will get a ‘unique’ value for y. Lets look at an example of a simple function. Lets say that for every step you take you travel 2 feet. Then the distance that you travel for every amount of steps you take is given by the function f(x)=2x. It’s as simple as that, just plug in the amount of steps you take and out pops the distance you traveled. Now that we’ve got a basic example of functions under our belt, we can dig into the nitty gritty of elementary calculus.

Differential Calculus:
Differential calculus provides a way of determining the behavior of the ‘output’ of a function as you vary the ‘input’ by an extremely small amount. What you get when you do this is the rate of change of that function. That is the basic idea of differential calculus and its operator, the derivative. Therefore, when you take a function, f(x), and take its derivative, f'(x), you end up with the rate at which that function changes with respect to x. Lets go back to our previous example of a function. If you take the derivative of our walking function, you end up with a value for the speed at which you are walking. Think about this for second, the function tells you how far you’ve moved, the rate at which you move is simply your speed!

Integral Calculus:
Integrals in a very general nutshell are the opposite of derivatives. If you take the derivative of a function and then take the integral of the result, you end up with your original function. Say that you were given a function that describes how fast you are walking. If you were to integrate that function, you would then have a function that describes how much ground you are convering per step. The story runs a bit deeper than that however, but the derivative-independent definition of the integral is probably beyond the scope of this short introduction.

I hopes this helps and takes away a little bit of the stigma that people associate with the higher maths. It should also be noted, that before beginning any course in calculus, the reader will do well to have a solid background in geometry, algebra, and trigonometry.