Psychedelic Math Rock

I love learning about the sort of things most people don't want to know about, because their perception of, say, rocket science, is that it must be far too complicated for the average person to begin to comprehend. But I am the sort of person who needs to know how things work, which is probably what led to me buying a book on quantum physics a couple of years ago. However, even I have to admit that calculus takes some figuring out. Take this as an example;

Squeezing Theorem.

If f, g and h are functions such that

f(rhubarb) <= g(rhubarb) <= h(rhubarb)

for all values of rhubarb in some open interval containing custard and if lim_{rhubarb→custard} f(rhubarb) = lim_{rhubarb→custard} h(rhubarb) = L then

lim_{rhubarb→custard} g(rhubarb) = L

How to use the squeezing theorem?
Example 1: Find the limit
lim_rhubarb→0 rhubarb ² cos(1/rhubarb)
Solution to Example 1:

As rhubarb approaches 0, 1 / rhubarb becomes very large in absolute value and cos(1 / rhubarb) becomes highly oscillatory. However cos(1 / rhubarb) takes values within the interval [-1,1] which is the range of cos rhubarb, hence

-1 ≤ cos (1/rhubarb) ≤ 1

Multiply all terms of the above inequality by rhubarb ² (rhubarb not equal to 0)

- rhubarb ² ≤ rhubarb ² cos (1/rhubarb) ≤ rhubarb ²

The above inequality holds for any value of rhubarb except 0 where rhubarb ² cos (1/rhubarb) in undefined. As rhubarb approaches 0 both - rhubarb ² and rhubarb ² approach 0 and according to the squeezing theorem we obtain

lim_rhubarb→0 rhubarb ² cos(1/rhubarb) = 0
Example 2: Find the limit lim_rhubarb→0 sin rhubarb / rhubarb
Solution to Example 2:

Assume that 0 < rhubarb < Pi/2 and let us us consider the unit circle, shown below, and a sector ChickenFriedRice with central angle rhubarb where rhubarb is in standard position. Fried is a point on the unit circle and ChickenSoup is tangent to the circle at Chicken.

Point Rice is a point on the unit circle (radius = 1)and has coordinates (cos rhubarb, sin rhubarb). Let us find the areas of triangle ChickenFriedRice, sector ChickenFriedRice and triangle

area of triangle ChickenFriedRice = (1/2)*(base)*(height)

= (1/2)*(1)*(y coordinate of point C) = (1/2)(sin rhubarb)

Note: we have used base = radius = 1

area of sector ChickenFriedRice = (1/2)*(rhubarb)*(radius) ²

= (1/2) (1) ² rhubarb = (1/2) rhubarb

area of triangle ChickenFriedSoup = (1/2)*(base)*(height)

= (1/2)*(1)*(tan rhubarb) = (1/2) tan rhubarb Comparing the three areas, we can write the inequality

area of triangle ChickenFriedRice < area of sector ChickenFriedRice < area of triangle ChickenFriedSoup Substitute the areas in the above inequality by their expressions obtained above.

(1/2)(sin rhubarb) < (1/2) rhubarb < (1/2) tan rhubarb Multiply all terms by 2 / sin rhubarb gives

1 < rhubarb / sin rhubarb < 1 / cos rhubarb Take the reciprocal and reverse the two inequality symbols in the double inequality

1 > sin rhubarb / rhubarb > cos rhubarb Which the same as

cos rhubarb < sin rhubarb / rhubarb < 1 It can be shown that the above inequality hols for -Pi/ 2 < rhubarb < 0 so the the above inequality hold for all rhubarb except rhubarb = 0 where sin rhubarb / rhubarb is undefined. Since

lim_rhubarb→0 cos rhubarb = 1 and

lim_rhubarb→0 1 = 1 , we can apply the squeezing theorem to obtain

lim_rhubarb→0 sin rhubarb / rhubarb = 1 This result is very important and will be used to find other limits of trigonometric functions and derivatives

See what I mean? That makes no sense, whatsoever! ChickenFriedSoup? What the hell is that?

PCP for the soul

If you love what you do, you'll never work a day in your life. Or something like that. It's a nice thought; if you love your job, you probably won't actually dread going to work, won't be ever so slightly miserable while you're there, and won't spend the rest of your waking hours trying to cheer yourself up. Unfortunately, for most people, there is a certain element of 'aw, do I have to?' involved, but hey; it gives us something to bitch about, between weekends.

Of course, there are worse things than going to work on a Monday morning. Like watching The Jeremy Kyle Show on a Monday morning. Or any morning. Or afternoon. For those of you outside the UK, it's a bit like Springer, but with uglier guests and a self-righteous host. It mostly goes like this; an unmitigated waste of skin sits down and tells the host, Jeremy Kyle, a man with the cold, dead eyes of a killer and 12-gauge nostrils, what an unmitigated waste of skin they are, he shouts at them, the audience cheers and applauds like a bunch of trained seals, and I wonder what wrong turn I took in life to bring me to the point where I'm sitting in front of that TV in the first place. At some point in the show's one hour run time, there will inevitably be a DNA test to establish the parentage of some unfortunate child. There's also bound to be a lie detector in there, somewhere. As much as I hate the show itself, I must admit I like when the lie detector says someone is lying and they continue to protest their innocence, because then, Kyle points out that polygraphs are generally accepted to be about 96% accurate. As if this means the test can't possibly be wrong. If it's spot on 96% of the time, that means it has to be wrong in 4% of cases, and if that doesn't sound like much, bear in mind it means a 25 to 1 chance that the lie detector is indeed wrong. I don't know about you, but I've backed horses with worse odds than that.

While there was a time when I would endure this crap, probably because the batteries in the remote were dead, I now avoid it at all costs. To give you an idea, here are ten things I would rather do than watch The Jeremy Kyle Show;

1. Stare at a crack in the ceiling for an hour. Actually, two hours.
2. Run through the lion enclosure at the zoo, wearing a suit made of raw bacon and loudly suggesting that all male lions have homosexual tendencies.
3. Memorise pi to 100 decimal places.
4. Stand on top of the Eiffel tower during a thunderstorm, dressed head to toe in wet aluminium foil, using a powerful, and very well earthed, PA system to tell French jokes to the people below.
5. Walk up to the biggest, toughest guy in a really rough bar and say; “Damn right, I was looking at your woman. Turns out you really can put lipstick on a pig”.
6. Attempt to rollerblade down an oiled fire escape, while carrying two large glasses of Rioja.
7. Board a plane, then, as it takes off, sniff the air and ask; “Does anyone else smell burning?”
8. Sit nice and close to a camp fire, wearing a waistcoat made from firecrackers.
9. Go to the final of the next rugby world cup, and spend the entire match facing the wrong way.
10. Jump out of a plane at 5000 feet, strapped to an anvil, with a wet paper parachute, over a glass recycling plant, while singing auld lang syne backwards and trying to put the pin back in a grenade. While drunk. And wearing oven gloves.

I have a couple of ideas that I think would really improve the show, but they've rejected them all on health and safety grounds. I don't get it; surely if you were to get the guests to sign waivers, you could install all the hidden trapdoors you wanted. Cowards.