Calculus, the Way Forward

Hi, how are you doing? Just checking in? Or maybe you really want to read about this stuff. I'm kinda surprising myself here writing about calculus, that mathematical method of calculation of treating problems by a special system of algebraic notations. You know it, you may be an expert or else you may have come across it at school. Or maybe both. It consists of lots of squiggles and alien-like symbols. I'm not an expert, but calculus is cool, or at least I'm beginning to find out it's cool, I think.

You see, I'm more of a languages person. A sort of dichotomy developed in my brain, when I was about 15, between languages and maths-related subjects. It was all down to a Quebec youth leader at a French-speaking summer college. I'd gone there to learn French, obviously. Anyway, in one of the organized debates there to help us improve our French, she basically convinced me that Irish was not a dead language after all and that we should be speaking it in Ireland. "Look at the Aran islands", she said, "you can still find young kids there who speak Irish". So when I went back to school that year, I threw myself into learning Irish and rather neglected maths. I didn't complete honours maths as a subject for the Leaving Certificate. It was all too taxing for my brain. I kinda concentrated on the languages. In retrospect, it was probably a good thing that I dropped out of the honours maths class. I think teenage students are under too much pressure, as it is. Anyway, I find myself strangely drawn to maths now that I'm a little older (36) and want to learn about what I missed or what went over my head.

I've discovered that integration is the opposite of differentiation, I think. Don't ask me what integration is. I heard about differentiation at school, but it was never explained why we differentiate. It was just accepted as one of those things that you do. Our honours maths teacher, though a genius, used to call us all individually "scobey", a term I could never get to the bottom of. I think it was one more of endearment rather than of abuse. Anyway, the message about differentiation never really got through. Also, he didn't really differentiate between us, just called us each "scobey". Maybe in another time and place, things could have been better. Maybe, maybe not.

If only maths teachers would explain the 'why' of what you are learning, rather than just the 'what'. Something like the following interchange occurred one day.

- "Why are we learning this, sir? What use is it?"

- No satisfactory answer forthcoming, just an appeal to beauty.

Beauty? I didn't get it. I didn't see it. I didn't see the beauty of maths. Language, perhaps, there was beauty there alright, I suppose. Etymology, the history of words, interesting. But, maths? Beauty? No.

Anyway, differentiation is about finding something called the derivative. A derivative is a measure of how a function changes as its input changes. What's a function? Well, a function expresses the idea that one quantity (the input) completely determines another. So, for example, the speed (or velocity) of an object determines its position after a certain amount of time. The derivative of the position of a moving object with respect to time is the object's instantaneous velocity, (Wikipedia helped me out here - check out http://en.wikipedia.org/wiki/Derivative.) I hope I've got it right. Maths geniuses correct me!

Imagine an object moving at 1 metre per second (to keep it simple).

You can imagine that after 3 seconds say, it will be 3 metres along. After 4 seconds, it will be 4 metres along, etc.

So its position is equal to the amount of time which has elapsed multiplied by its velocity.

So (expressed rather unmathematically):

position = time elapsed x velocity

e.g. 3 m = 3 sec x 1 m per sec

and the derivative of this function is the velocity or, in the example given, 1 m per second.

Got that?

The position is dependent on the time elapsed multiplied by the velocity. It is a function.

If we call the position y , we can say that y is a function of x, where x is the time elapsed or (expressed mathematically):

y = f(x)

In our simple example, we can find the derivative by dividing the change in the position of the object by the change in the time (or the amount of time elapsed). It is written thus:

Δy / Δx

where / means "divided by".

The Δ is one of those alien symbols I was referring to earlier. It means "change in".

The derivative is equal to Δy / Δx.

So, our derivative is equal to the change in the position (3 metres) divided by the change in time (3 seconds).

3 divided by 3 is 1. So our answer is 1. It's 1 metre per second. (That's the speed the thing is travelling at.) Our answer is 1 metre per second. This is the derivative. There, we have differentiated! Good to know we can make a difference.

Of course, this is a very simple case. For more complication, check out Wikipedia.

And that's about as much calculus as I know or can muster at present.

Now that I think about it, there is beauty in maths, though. Think about a snowflake. It's quite mathematical when you break it down, possibly a fractal. Not sure if it's calculus.

I'm going to take calculus forward with me in my life, in a minor way, and learn a bit more about it. It's still mysterious to me, but I think I've broken the back of it through reading a little online for this hub.

I actually watched a good film about maths some time back on TV, "Proof" with Anthony Hopkins and Gwyneth Paltrow. It's very good. I didn't see it all, though, I think I came to it half way through or something. It got me thinking. I started seeing similarities between writing complex mathematical proofs and writing prose. (I'm trying to be a writer.) I realised that writing maths is just like writing English really, only using numbers and symbols instead of words. Interesting, yet slightly sad film. You should see it sometime.