Derivatives and Integrals

Consider a function 𝑓(𝑥) that is continuous and differentiable everywhere (that is, a curve that does not break in between and has a slope everywhere). Consider the transformation 𝑥+ℎ such that 𝑓(𝑥)−>𝑓(𝑥+ℎ). The derivative is defined as:

In essence, that a function which is continuous and differentiable everywhere has a slope f’(x) everywhere. This models a rate of change, for a small amount of f(x) over a small amount of x (hence, df(x), and dx, respectively). The reason why we introduce the derivative is to describe infinitesimal changes in practical cases.

Now, consider a function 𝑓(𝑥) bounded in between two points, 𝑎 and 𝑏, respectively. Suppose that the question posed to us is: what is the area underneath this curve between two points 𝑎 and 𝑏? Well first, consider a method in which we could first approximate the area, which is defining rectangles with a width Δ𝑥 and a height given by 𝑓(𝑥𝑛). Then it would be clear that the total approximate area under the curve is modeled by

Where Δ𝑥=𝑏−𝑎𝑛. Based on this definition, it becomes apparent that more rectangles mean a more precise approximation of the area. Eventually, when we take an infinite number of infinitesimally small rectangles with height 𝑓(𝑥𝑛), we will obtain the precise value for the area underneath a curve, bounded by two points 𝑎 and 𝑏. In other words: 

And we obtain the answer to our question. This process of defining an infinite number of infinitesimally small rectangles with width 𝑑𝑥 (which is our notation for really small width) is something called the integral.