But what about the major features of SMW already existing in the game? The only times powerups were used, their purpose was to get hit instantly and give Mario a few invincibility frames. Yoshi was also only used for a few seconds as an extra jumping platform. For me, it didn't really feel like a game to TAS. The levels were very linear, and a lot of them were on some kind of global timer (either an autoscroller, or waiting for a platform to reach you), making all optimizations useless in the end.
but I worry that GPW2 or whatever comes down the pipe in the near future will obsolete this extremely quickly.
Set u = df/dx. Apply the chain-rule and get f'' = du/dx = du/df * df/dx = u*du/df Now plug everything: uf = u du/df => f df = du => f² + c = 2u
So, the entire set of solutions should be: y = -2 * ln(cos(bx/2+k)) + a y = -2 * ln(cosh(bx/2+k)) + a y = -2*ln |x+k| + a y = ax + b
but for some reason you never make encodes anymore so I got to watch it through.
minus the lack of a hot spring scene.
All the games share the same money count, right? So surely it doesn't matter which game you play, the goal is just to hit the maximum amount of money, which is the end of content/best possible state to reach?
Suppose you have an amount of capital C you want to reach. Your plan is to do so by investing an amount A periodically. Suppose that, by the time you have to deposit this amount A, the capital you already had increased by x. That means, if you had M in your account, after a time unit passes, you'll have M*(1+x). Find the amount of deposits n of value A in order to reach the goal C. Prove that the formula tends to C/A when x->0
which simplifies to
(*)
Setting equal to C (the goal) and solving for n, we get:
(**)
For the example that you gave, plugging in C=1000000, A=2000, x=0.01 gives approximately 180.07 months, so it takes 181 months (~15.08 years) to reach the goal.
But that's not your actual question. The actual question is to prove that the formula tends to C/A when x->0. We could use L'Hopital's rule on (**).
Or instead we could just see that (*) approaches An as x->0. This is because ((1+x)^n - 1) / (1+x - 1) is precisely the value of the derivative of z^n at z=1, by definition. This gives An=C, so it follows that n=C/A in the case x->0.
This is just a F/A analysis right?
FractalFusion, just for you I made a 30 fps encode. That one should be roughly comparable in framerate with the slow encode of [2247] DOS Mega Man by DarkKobold in 02:23.55. Only the slowest encode has captions.
In one of his latest videos, blackpenredpen states that you can't use L'Hopital's rule in order to solve lim(x->0) sin(x)/x because "that would be circular reasoning". I didn't really understand why.