Tuesday, December 25, 2007

Silly differentiation

Once upon a time last year I posted the most stupidest of things about logarithms. My brain had become saturated with mechanics was my excuse then; but it was indeed an embarrassing moment which made me ask the question about which degree I was studying for!

I am now giving you the wonderful opportunity for me to ask myself that question again (albeit reluctantly). I can't seem to differentiate. If anyone is reading this today then please do tell me what I have done wrong.

The following has to be differentiated and then evaluated at t=0:

M(t) = \displaystyle \frac{e^{tb}- e^{ta}}{t(b-a)}.

You can use the quotient rule or write the t at the bottom to a negative power and use the product rule. (I feel awfully dumb writing this in the sense that I am unsure of the "validity" of what I am saying.)

So we differentiate using the quotient rule on this instance to get:

\begin{align*} M'(t) &= \displaystyle \frac{t(b-a) \times (be^{tb}- ae^{ta}) - [(e^{tb}- e^{ta})(b-a)]}{t^2(b-a)^2}\\ & = \displaystyle \frac{t(be^{tb}- ae^{ta}) - (e^{tb}- e^{ta})}{t^2(b-a)}\\ \end{align*}

The million dollar question - what have I done wrong? The first line by the way can be read as follows: "Bottom times the derivative of the top minus the top times the derivative of the bottom. All over the bottom squared." (I remember that thanks to Prof D from last year!)

The second line is my attempt at simplifying things, but what is the dumbest thing that you have spotted in my attempted differentiation? Please there has to be something. My mind is honestly blank - I don't even know what day of the week it is! All day I have been thinking it is Saturday today (and I have only had three cups of tea today). Actually talking about tea - I am missing Thursday tea time it seems. (Well I had a dream and it was a Thursday. I had been drinking tea and discussing my worries about not caring anymore with the usual person!) Ah well, roll on next semester I hope. Thursdays were a good day.

Anyway why do I ask for someone to tell me the mistake? Well if we stick t=0 into M'(t) then the angry bees start buzzing "we can't divide by zero". I need that horrible looking derivative to simplify to (a+b)/2 when t=0. Can you do that for me? Can you give me a baseball bat for my head?

I don't really have a million dollars but if you tell me what's wrong, I can then tell you the mathematics behind the differentiation. (Which I am sure you are all desperate to hear!!) Pretty please...? (I have even broken a rule of grammar there (I think) -- just so that I can go to sleep with a clear and unburdened mind!)


steve said...

You have differentiated correctly and so M'(0) doesn't exist. BUT lim (t->0)M'(t)=(a+b)/2 which is probably what you are after. So somewhere in your notes must be a method for finding this limit - L'Hopital's rule perhaps?

Beans said...

Hi Steve,

You see this was a stats question, which is why I never thought to use L'Hopital's rule - though it is in my analysis notes. (I assumed that I was doing something incorrect and rightly so; for last year on three separate occasions I made the same mistake in a question involving the sieve of Eratosthenes!)

Thanks a lot though - I can get some sleep now! :) I am thinking that maybe I should apply the same procedure to the other similar problems I am encountering . (I feel quite stupid though and maybe should have realised this).

Beans said...

Before I forget - the mathematics behind this... M(t) is the moment generating function (mgf) of the Uniform distribution. One nice thing about mgf's is that the nth derivate of M(t), evaluated at t=0 is equal to E[X^n}. Hence the first derivate i.e. M'(t=0) = E[X} which is what I was after. There are other ways to work out the expectation but I was trying to use mgf's.

I said that maybe I should have seen what was required, because just before mgf's I had done probability generating functions. Here limits were taken (of something) to get E[X] (but not to 0 but 1)!

Anyway that is enough maths mumbo jumbo from me. I doubt anything made sense, but do feel free to ask!

I didn't finish 12 lectures today but only 2. That means I have 10 left for tomorrow! (If I am to stick to my rough timetable that is.)I know what the problem is though: I am expected to know my stuff from last year, which I successfully never learnt or understood. (How the heck am I supposed to remember the pdf of a normal distribution?)Stats drains one of energy.