Thursday, April 30, 2009

Calculus in a Nutshell

I'm here today to inform you of calculus.

If your gut reaction to that statement is panic and/or loathing, breathe easy. I'm not assigning homework or giving quizzes. Calculus is a useful tool, and it might help you some day.

Let's start with addition: in the equation, 5 + 6 = 11, the plus sign (+) is known as an "operator." It operates on the numbers 5 and 6. There are all kinds of operators, but they don't always operate just on numbers. Some operate on functions.

Now, let's talk about functions. Functions are just strings of numbers connected to other numbers. For example, on a long road trip from San Antonio to Provo, your distance traveled from San Antonio is a function of your time on the road. Let's say your average driving speed is 70 mph. You could represent your distance Y (in miles) as 70 times the time X (in hours).

Y = 70 * X

Y and X are variables, meaning they are numbers but their values can change. The amount that Y changes depends on how much X changes according to the relationship above. (Of course this relationship is overly-simplistic, but you get the idea).


Now calculus: Isaac Newton invented (discovered?) calculus operators that operate on whole functions instead of numbers. There are two main operators: the differential and the integrand.

The differential operator looks like this:

When you use it on a function it's called "taking the derivative", and you get another function out.

It allowed Isaac to look at the slope of a function at any point. In the distance vs. time function, the slope is easy, because it's constant (70 mph). But in a realistic scenario (see graph, below), you aren't driving constantly at 70 mph. The slope (which represents speed) changes as you go along. Your speedometer doesn't give you an average speed, it gives you your speed right now. Same idea with derivatives.


The integrand operator looks like a long, formidable S:

It's also the opposite of a differential operator; sort of the minus to the plus. So, if you were given the function of speed vs. time, you could extract the original distance vs. time function. Graphically, it let's you calculate the cumulative area beneath the speed curve, or in other words, the total distance traveled as a function of time.

It doesn't just give you the total distance between San Antonio and Provo (that's a number, not a function). It gives you the total distance traveled right now. At any point in your trip you could calculate the distance traveled up to that point.

That's it. That is calculus in a nutshell. Of course, there's the sticky matter of learning how to take derivatives and integrals, but I just wanted to relate the what. The rest is just details.

6 comments:

JethRobyn said...

you're such a nerd

JethRobyn said...

That was Jethro...but I think I agree :)

Jenette said...

Yay for Calculus! It almost seems like old times with the math lessons!

Jancisco said...

driving that distance is much less time consuming than trying to understand calculus.

I guess thats why I'm destined to be a trucker.

Beetle said...

I got bored/angry after the first 3 paragraphs. Sorry. I'm sure it was well-written.

Clinto Beans said...

Grr... I strongly dislike you for this. You left me with too many questions in my head. Knowledge needing to be found! I don't understand! Brain... shutting... down! *kaboom!* ...Calculus is next year for me, and this only established unsolved questions... I need the part two to this going in depth about those operators.
Yeah.