Don't Look Now! Advanced Math at the Gym

I have way too much time on my hands. Seriously, I head over to the local YMCA probably 4 or 5 times a week. I know, I know, there are other things that I could be doing with my time - like going to school, or taking some night classes at a college to improve some of my language or tech skills. I guess what I'm really focused on right now and until I can find another productive way of using my time, I am focused on just getting into the best shape of my life.

In any case, on one of those days I was at the YMCA, I was sitting on an exercise bike on an 'alpine hill' setting. I have been following the Tour De France lately and the cyclists really inspired me to push myself on the exercise bike. I know it's not the same but I couldn't help but feel like I was racing towards something on that bike as I tried to burn off calories. Going up and down mountains, it was just a normal stage on the Tour De France and it is probably what inspired me to try out the alpine hill setting on the bike.

This is what greeted me on the bike:

For the longest time, I always looked at that screen and figured that the bike would simulate the hills on the left of the bike screen.


But that's when I came to a pretty geeky discovery: they actually were first derivatives of the hill I was climbing.


Normally, when you're on an exercise bike, all you really should be thinking about is how to keep breathing and moving your legs (and possibly trying to keep pace on high difficulty settings). As I said, I figured that those hills were the ones that the bike was simulating, but as I got to the highest difficulty level (the middle peak), I realized that that was not what the bike was simulating at all. If you're already at the top of the hill, it would not be that difficult to pedal.

That's when I realized that the bike screen was actually showing a fairly rough graph of the first derivative of the height function of the hill. What I'm saying is that the graph on the bike screen is showing how difficult the climb is (ie. how steep the hill is). And I mean rough because I would assume that if the bike is simulating a hill, there would be an upwards ride and a downwards ride with flat rides as well (and if there were flat rides, then the first derivative would be 0; contrarily, the bike screen showing the 'rough' first derivative would never reach 0 unless it was set to 0 by the user).

Of course, I'm just assuming that there are flat rides, it is entirely possible that you are always climbing up the hill (in which case, the first derivative of the height function of the hill would then always be positive). This may be the more plausible explanation.



And of course, maybe I am the only one who really thinks about this kind of stuff at the GYM.


More geeky things that I could cover in this post but won't:

- Area underneath the first derivative = height function of the hill you are climbing
- Second derivatives and what it means in this context (acceleration - how fast you are ascending)
- Actual height function as depicted by the first derivative (most likely three 'hills' - think a three - step staircase and you have a rough idea of the hill function assuming you are always climbing the hill and never going down it. In the case that you are climbing down the hill at some point, the first derivative has a y-value of 0 at a certain level of the 'graph' )