Quite why calculus gets a definite article is a mystery to me, I mean I never studied "The Geometry" or "The Algebra" that I can remember.
Well the first thing that's presented is a demonstration of differentiation which, in essence, is finding out about the rate of how things change.
A nice way to think about this and one that helped me get my head around it is to think of a graph like this which shows distance travelled (on the y axis) versus time taken (on the x axis).
In this example the distance travelled in metres is the square of time taken in seconds so in maths-speak we say that distance (d) is a function of time (t) and that in this case the function is t squared; which you write like this
d = f(t) = t^2
OK so from this you can tell how far you have gone in how many seconds. But how fast are you going at any time? Well as speed is distance divided by time you can work out an average speed between two times, t and t + h by joining the points on the curve at t and t+h and dividing the distance covered by the time taken.
Now the time taken is (t+h) - t which is just h and the distance covered is the difference between the values of distance d at times t and t + h; but as we know that d = t^2 we can express these distances as (t+h)^2 and t^2.
So that gives us
(t+h)^2 - t^2
-------------
h
which simplifies to
t^2 + 2th + h^2
---------------
h
and more simply:
2th
---
h
Now what we do is start to reduce h and keep reducing it to a limit of 0
Ah, you can't divide by zero though, we went through that on the last post didn't we, you get a black hole if you do.
"Yes", says my OU tutor who has a brain so large there's nothing left for dress sense, but we don't go to zero, we TEND to the LIMIT of zero so think of h getting infinitessimally small, so small it doesn't count. So small you can just get rid of them leaving:
2t
And there you go, the first derivative of t^2. So after 5 seconds you've travelled 25 metres and you're going at 10m/s
Wel it does work but there's something inside of me that doesn't quite like it. I'm just a bit concerned that it's a bit of a fiddle throwing away these very tiny numbers. Apart from that at least I'm feeling pretty comfy with differentiation so far.
By the way if you type "Zero as a limit" into Google you get a load of song lyrics to a Human League tune from the 80's. Was the floppy haired one a mathematician I wonder?
