Integration.

MST121 has moved onto integration now. Integration conceptually seems pretty simple, it's like differentiation only seen from the back.

It's not too hard to do either: if you remember down there when I differentiated X^2 and ended up with 2x you can do a reverse operation and it turns out the formula, yielded by some relatively straightforward arseing about with algebra or geometry, to integrate a simple function of the form

      n
    ax

is

   a  1   n+1
     --- x
     n+1

which happily yields x^2

However the devil in the detail is that the result of integration needs to have an arbitrary constant added on as differentiating x^2 + a (where a is any constant) gives 2x so the reverse operation has to add this "constant of integration" back in. Apparently this makes these "Indefinite Integrals" but luckily for me you can get them to be a bit more definite when you subtract one from the other.

And subtracting the results of integrating a function for different values of the dependent variable is the one bit of calculus I vaguely remember being shown at school (I never did A-levels as I loathed the two maths teachers with a passion so I'm guessing this got touched on somewhere in AO level¹ pure maths) probably for finding areas under curves.

Oh and I got a book on Calculus as well (see - still a swotty dragon, reading around the course now!) called "Calculus made Simple (for scaly green firebreathing creatures like you)²" written by some guy in the early 1900's and apparently still the definitive text on the subject today. I will let you know how I get on




¹ This was an old UK exam that I think got dropped sometime in the 80's. I, being a swotty hatchling, was in an "advanced" stream at secondary school and we took our O levels (called GCSEs now) a year early at 15 and we also did a couple of these "midway between an O level and A level". I had a google and although I couldn't find a AO level paper I did come across a marking scheme for one here: www.edexcel-international.org/VirtualContent/49349/7362_PURE_MATHEMATICS_F.pdf and it does have a little bit of calculus in it.

² I think I might have imagined the subtitle.

Zero as a limit

Well MST121, which is the Open University maths course I'm doing, has now reached the dreaded "Chapter C" in which we get to study "The Calculus".

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?