The Arithmetic vs Geometric Mean Trick

Posted by tpc at February 22nd, 2007

One classical trick is the following:
Since
(x -y)^2 \ge 0, we have
x^2 - 2 x y + y^2 \ge 0 \implies x^2 + 2 x y + y^2 \ge 4xy
Taking root we obtain (x+y)/2 \ge \sqrt{xy}

A variation of this trick can be used to show that
| a \sin x + b \cos x | \le \sqrt{a^2 +b^2}.
We start with (a\cos x - b \sin x)^2 \ge 0. Expanding the expression we get
a^2 - a^2 \sin^2 x - 2 a b \sin x \cos x + b^2 - b^2 \cos^2 x \ge 0.
A simple rearrangement completes the proof.

Posted in Calculus/Analysis| 1 Comment | 

Silly Riddle

Posted by tpc at February 22nd, 2007

What’s the difference between jumping down from the 2nd floor versus the 20th floor?

Ans:
Splat! Aaaaaaahhhhh! (2nd floor)
Aaaaaaaahhhhh! Splat! (20th floor)

What if you jump from the 10th floor? It all depends on how long you remain in the air. A little calculus (plus some physics) does the trick.

Let v be your velocity and u be the initial velocity which is assumed to be zero. That is you gently let yourself off the edge instead of making running jump. Of course a will be acceleration 10 m/s^2. Let’s not be fussy about precision.

Let t_s, i.e. time_splat, be the time you hit the ground and d the distance covered. We have
d = \int_0^{t_s} v dt = \int_0^{t_s} u +at \: dt = ut_s + \frac{1}{2} a t_s^2

Now let’s assume average height per floor is about 3 metres, so d = 30 gives
t_s = \sqrt{6} .
About 2.5 seconds, I guess enough for Aahh!Splat!
There you have it, who says calculus is useless?

Posted in Calculus/Analysis| No Comments |