Calculus is one of the things that’s considered terrifying by most people. In fact, I’m sure a lot of people will consider me insane for trying to write a “basics” post about something like calculus. But I’m not going to try to teach you calculus – I’m just going to try to explain very roughly what it means and what it’s for.
There are actually two different things that we call calculus – but most people are only aware of one of them. There’s the standard pairing of differential and integral calculus; and then there’s what we computer science geeks call a calculus. In this post, I’m only going to talk about the standard one; the computer science kind of calculus I’ll write about some other time.
The first one – the more common one – is a branch of mathematics that uses limits
and/or infinitessimal values to analyze curves. Limits can be used in one way (the
differential calculus) to look at incredibly small sections of a curve to figure out
how it’s changing – and in particular, to find patterns in how it changes. Limits can be used in another way (the integral calculus) to compute the area under
a curve by adding up an infinitely large number of infinitely small values.
In differential calculus, what you’re usually doing is taking a curve described by an equation, and figuring out a new equation (the derivative of the curve) that describes how the first one changes. For example – look at a curve like y=x2. At any point in time, the curve has a slope – but it’s constantly changing. But what we can do is look at that curve, and say that at any point x, the slope of the curve will be 2x.
What does that mean? There are a lot of things that can be understood in
terms of rates of change. Suppose that you measured the position of a moving object,
and worked out an equation that described where it was along a line at any point in
time. Let’s say that that equation for the total distance moved in t seconds was f(t)=3t3+5t+11 meters. So:
Now, suppose I want to know how fast it was moving after 3 seconds, that is, at time t=3. How could I figure that out? It’s not moving at a constant speed. At any two moments, the speed is different. How can we know how fast it’s moving at a particular point in time?
The velocity that something is moving at some point in time is how much it changes its position divided by the length of the period of time; if position=p, then the
velocity v=Δp/Δt. Since the velocity is constantly changing, though, that equation isn’t too much good for us. We can’t say how fast it’s moving at t=3.
We can use it to start homing in. Between t=2 and t=3, it moved from (24+10+11)=45 to (81+15+11)=107 – so it moved 107-45=62 meters – so its average speed between t=2 and t=3 was 62 meters/1 second = 62 meters/second. Between t=2.5 and t=3, it moved from 70.375 to 107 – so its average speed for that half second was 36.625 meters/0.5 seconds = 73 1/4 meters per second. From t=2.9 to t=3, its average speed was 8.33meters/0.1 seconds=83.3 meters/second. From time t=2.9999 to time 3, it’s average speed was approximately 85.99 meters/second. To know exactly what speed it was moving at t=3, I need to know its velocity at precisely t=3 – an interval of length 0 at exactly t=3. The way that we can do that is to pull out a limit: the speed at time t=3 is limδ→0(f(3)-f(3-δ))/δ=86 meters per second.
We can do that symbolically on the original equation (I’m not going to go through the whole process), and end up with the velocity at time t=9t2+5. This second equation is called the derivative of the original equation.
In integral calculus, what you’re usually doing is taking a curve described by an
equation, and figuring out a new equation that tells you the area under the curve. (the
integral of the curve) So, again, taking the curve y=x2, we can ask
what’s the area under the curve between x=0 and x=6? The area under the curve
y=x2 is x3/3; so the area between 0 and 6 is 72. It can be used for the opposite of what we just did with the derivative – if we have an equation showing its velocity at different instants, we can figure out an equation for its position.
Differential calculus and integral calculus started out as two different (if conceptually related) fields – but they were tied together by something called the fundamental theorem of calculus. Stated very roughly and informally, what the fundamental theorem basically says is: if I start with some curve, and I take its derivative, and then I take the integral of the derivative, I’ll get back the same equation that I started with.
The history of calculus is really interesting – but to get into detail would be a whole post of its own. Basically, what we call calculus was invented roughly simultaneously by Isaac Newton and Gottfried Leibniz. Basically, Newton probably did work out the ideas of calculus first, but he didn’t publish it; Leibniz started later, but published first. The notations that we generally use for calculus are mostly those of Leibniz, as is the name calculus – Newton called it “the method of fluxions”. This conflict led to a huge feud between Leibniz and Newton, which expanded into a conflict between the mathematicians of England and the mathematicians of the European conflict.