Thursday, 27 January 2011

dA / dW

W = a day of the week: monday=1, tuesday=2, etc
Aw= the time of my first class on day W
Bw = the time I get to sleep on day W
Sw = the amount of sleep I get on day W
N = the amount of sleep I need nightly ?= 8 hours
Tw = the average amount of sleep I've gotten averaged over the week so far = sum from{i=1} to {w} of {Si / w}

Proof that my current class schedule SUCKS!:
We define suckiness (U) of my week as the total amount of deviation of Tw from N during the week. Namely:

U = sum from {i=1} to {7} of {N - Tw}

Lemma 1: Sw <= N
We know that I will stay up for long enough at night such that Sw cannot excede N

Lemma 2: Bw = B1 forall w We also know that when I go to sleep on the first night, sets when I go to sleep on subsequent nights.

And since: S = A - B (and B is constant from the above lemma 2) this implies that:

dS / dW = dA / dW
Sw = S1 + w * dA / dW

Tw = integral {Si / w} implies that Tw = (S1 * w + 1/2 * w^2 * dS/dW) / w, and pluggin in for the above, we get that:

Tw <= N + (1/2 * w^2 * dA/dW)

Breaking out definition of suckiness apart:

U = integral {N - Tw} = 5*N - integral{Tw}

Plugging in our inequality:

U = 5N - (5*N + 1/6 * 5^3 * dA/dW)

U = -37.5 * dA/dW

Thus we have shown that suckiness of a schedule is directly proportional to opposite of the rate of change of the time of your first class as the week progresses. In my case, I have class 1.5 hours earlier each day as the week progresses, so for me:

dA/dW = -1.5

Which implies that:

U = 56

Therefor: My schedule sucks by 56 (units?)

I did the calculus wrong here I think... please help me by letting me know how to fix it. I haven't slept enough in a while. . .

1 comment:

  1. Wow, the best (worst?) part about all this: You spelled therefore wrong.