Inspirado no Bolzano-Weierstrass Rap: https://www.youtube.com/watch?v=eM3S74kchoM _________________________________________________________________________________________ escrevi The Bisection Rap ----------------- Come on MS211 students if you wanna be free from stupid thoughts, then you listen to me Let f be continuous in [a, b] and sign(f(a)) be - sign(f(b)) Well, we know that this function has at least one root and we're thinking to ourselves "This is really good!" but how to find a root is what we ask so let's go ahead and deal with this task Well, are you down with that? Yo, are you down with that? Audience: Yes, we are down with that! (Refrain, can be used several times) To your right is b and to your left is a and in between a sequence tries to wind its way an infinity of x_k's this interval has in it still you don't know where to look to find the limit so you stand in the middle halfway in between let x_0 be (a+b)/2 if you know what I mean to your left a line segment half the big one's size to your right another one in the same way lies and at least one root lies in the other or the one but kid you'll never believe what this division has done you see, if a root lies in here then it is on the right or the left, is this clear? Well, are you down with that? Yo, are you down with that? Audience: Yes, we are down with that! (Refrain, can be used several times) So you slide to a side where f changes signs x_1 is the center of an interval of half the size this new interval [a_1, b_1] is half as long contains a zero if my logic ain't wrong now you do it again, divide the line in two and if you paid attention, you'll know just what to do you'll do it k times, things are getting really small (1/2)^k is your interval yet with this little space, with this little bound a root of f can still be found Well, are you down with that? Yo, are you down with that? Audience: Yes, we are down with that! You can do this forever until Tisha B'Av cause infinite recursion is what we love a chain of nested intervals, each inside the last like little Russian dolls, and they're getting smaller fast Well, are you down with that? Yo, are you down with that? Audience: Yes, we are down with that! You have to believe and then we're nearly done that soon b_k - a_k is smaller than epsilon in this case one often stops or if the modulus of the midpoint value below epsilon drops but if we generate intervals [a_k, b_k] on and on there's exactly one point that lies in every one see, all those left endpoints they have a supremum the same way that the right ones have an infimum well, this sup and this inf, they lie in each of these sets though the distance that's between them is as small as it gets They are both the same point, so I say "What the hell! I think it's a limit so let's call it L!" Well, are you down with that? Yo, are you down with that? Yes, we are down with that! Recall the kth interval and all the points that it spans well I only need one, 'cause that's how bad I am x_k's my name for that point, it lies in interval k which makes it quite close to L, see I planned it that way! Now you can pick epsilon as small as it wants to be 'cause I got nested intervals and they're working for me I will come back with an M so big it will do which is b - a divided by epsilon the log base 2 You see the thing with that M is that I picked it so good that after it the x_k's lie inside of L's hood L's hood is epsilon sized so all the intervals lie in it QED, you got a sequence that converges to that limit Well, are you down with that? Yo, are you down with that? Audience: Yes, we are down with that! ...............