Maths Times Infinity > Topics > Calculus

Calculus

Calculus is a really cool thing in maths. VERY useful. For scientists, engineers. It was a huge breakthrough at the time. It let Newton basically prove that gravity works. That it's the same everywhere and everything adds up correctly.

Okay, so the main idea of calculus is to calculate the gradient of curves. So, hopefully you know what the gradient of a straight line is. Well, for a curve, it's always going in a certain direction, but it keeps changing. So we need an actual formula for it.

What you can do is estimate it by taking two points close together and drawing a straight line through them. Then we find the gradient of that line, and it should be roughly the same as the gradient of the curve.

For example, let's look at #y=x^2#. Just about the easiest example of a curved function. What gradient does it have when #x=1#? Well, let's take the points #x=1# and #x=1.01#. Nice and close together. Well #1^2 = 1#. And #1.01^2 = 1.0201#. So the two points are #(1, 1)# and #(1.01, 1.0201)#. What's the gradient of the straight line between these two points? Let's use rise/run. As in #\frac{y_2-y_1}{x_2-x_1}#. The difference in #y# values divided by the difference in #x# values.

This gives #\frac{1.0201-1}{1.01-1} = \frac{0.0201}{0.01} = 2.01#. So it's about #2#. Cool. Turns out the limit as we approach #x=1# is *exactly* 2. So, if we pick two points even closer together and near #x=1#, we'll keep getting closer to #2#. Let's try to prove it!

Instead of #x=1#, let's do it for a general value of #x#. And instead of a point #0.01# to the right of #x#, let's make the distance #h#, where #h# is any number we pick, as small as we like.

The points are then #(x, f(x))# and #(x+h, f(x+h))#. Obviously the difference in #x# values is just #x+h-x = h#. So we want to calculate:

\begin{equation} f'(x) = \lim_{h->0}\frac{f(x+h)-f(x)}{h} \end{equation}