calculus proof
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00:00Hello friends, welcome to Friday and I Love Funky, episode 203.4, yeah, okay, let's begin
00:16with two exercises.
00:30Let me get some water.
00:57Yeah, my toes are getting better, yeah, I can feel it.
01:25Yeah, it's slow recovery because it has to do with nerve, numb toe, but it's recovering.
01:32Okay, that's great.
01:34Alright, let's go.
01:37Massage and physical therapy for toes.
01:44I'm an Amazon Prime member, so it's like there's some annual fee, but that's fine.
01:57It has a lot of free movies in there, Amazon Prime.
02:04Free TV shows and I watched it yesterday while I was eating food.
02:10I watched Citadel.
02:12I watched it before, so I watched it again.
02:15Yeah, it's good.
02:16There's some parts I don't like.
02:18Excessible violence, I do not like that.
02:21But I like the martial arts.
02:24What I mean by violence is like a very good gun violence.
02:28Okay, I do not like that part.
02:30But martial arts is good.
02:32Because martial arts is not that dangerous.
02:35Gun is dangerous.
02:36Okay, so martial arts, yeah, it's not as dangerous as guns.
02:42So yeah, I like martial arts.
02:45Also great cinematography, great music, great casting like Priyanka Chopra.
02:52She's a former model, beauty pageant in India.
03:01Now great actress, martial artist, very charismatic.
03:08She carries this Indian charisma, you know, she's just fantastic.
03:13Yeah, great casting.
03:16Yeah, she was successful in Bollywood in India, but later on, Omega 2.
03:22Okay, nice.
03:31Okay.
03:38Okay, all right.
04:08Okay, healthy toes, healthy feet.
04:38Feet health is so important, toe health very important, yeah we will take care of our toes,
05:07we will take care of our toes, we will take care of our toes, we will take care of our toes,
05:36we will take care of our toes.
05:48yeah this fingernail massage is basically massage for the nerves
06:16nervous stimulation
06:23mm-hmm okay now two wave
06:38okay okay now let me wash my hands
07:01okay
07:13okay
07:27Yeah, I feel my toes are now strengthened, empowered.
07:35Okay, now physical therapy.
07:41Agh!
08:11Okay.
08:27Now stretching.
08:33Beautiful sunshine is someday.
08:46Agh!
09:03Okay?
09:20Okay.
12:00Now, let's do jump, knee kick.
12:14Yeah.
12:16Yeah.
12:18Yeah.
12:20Yeah.
12:22Yeah.
12:24Okay.
12:26Good.
12:28Five minutes break, please.
12:32Yeah, we get back to mathematics.
12:34I think I'm ready.
12:36Yeah.
12:38Yeah.
12:39It used to be like doing mathematics became boring because I did too much.
12:48Now, I've been not doing mathematics.
12:50Not doing mathematics became boring, okay?
12:52We get back to athletics.
12:54Okay.
12:58Yeah.
13:08Yeah.
13:10Yeah.
13:12Yeah.
13:14Yeah.
13:16Yeah.
13:18Yeah.
13:22Yep.
13:26Yeah.
18:04Yeah, we're not doing number 3 this day.
18:05We're doing, we'll get back to number 3 later, in the future, but we're doing calculus basically.
18:12So, why are we doing this? Basically, the hierarchy. Yeah, the order of infinity, like Hari said, we want to establish that order more analytically, okay? So, using this type loss, analytic geometry and stuff, and, yeah, cheers.
18:34Like, basically, 2 to the x is eventually bigger than x squared. That's the kind of thing we are going to do. So, we are doing basic proofs.
18:51Well, I mean, x squared, differentiated, that's 2x, right? 2 to the x, differentiated, nn2 times 2 to the x, okay?
19:08So, there comes a point where nn times 2 to the x is bigger than 2x, right? Yeah.
19:15So, those are the things we deal with, because we are saying, like, at some point, let's say x1, f1 is bigger than f2, also, for all x bigger than x1,
19:42f1 prime x is bigger than f2 prime. Then, prove that f1 x is bigger than f2 when x is bigger than x1, okay?
20:03It's, to me, it's not easy proof.
20:08So, there's one approach.
20:12We can do case by case.
20:17And,
20:18there is one more.
20:32That will be bigger than 0, okay?
20:34It's increasing function, OK?
20:42And we're going to grab a new whiteboard.
20:52And what I mean by case by case, like f1, 2 prime,
21:02f2, 2 prime, the second derivatives.
21:07Three cases, f1, 2 prime is bigger than 0,
21:12equal to 0, less than 0.
21:14f2, 2 prime, bigger than 0, equal to 0, and less than 0, OK?
21:18So 3 times 3, 9 cases.
21:21We can do that.
21:21But we'll deal with the easiest case out of nine cases,
21:35when f1, 2 prime is 0, and f2, 2 prime is 0, meaning they
21:44are both linear, straight line, upward, diagonal, OK?
21:50Let's create whiteboard, new whiteboard, and let's do that, OK?
21:59Yeah.
22:12And I discovered that whiteboard that I should not erase,
22:15because I think I forgot to write that part,
22:18but I put it as an appendix to the next paper, OK?
22:22Yeah.
22:23I wrote on that whiteboard, like, do not erase, and stop,
22:35so that they don't get erased.
22:40I wrote on that whiteboard, like, do not erase, and stop,
22:45so that they don't get erased.
22:49The next one is getting erased.
22:50Okay.
22:54I got erased.
23:28I like this one because this one got published in the appendix of the previous pen.
24:50I'm so hungry, but, well, I had a good sandwich yesterday.
24:54that we are watching that sitadoo show it's a TV show I say but it was nice
25:08yeah the spy political thriller action
25:15good cinematography rotation like upside down camera that's highly effective
25:24okay
25:42uh of course i'm not drinking now just the feeling
25:54you
25:55you
25:57you
25:58you
25:59you
26:00you
26:01you
26:03you
26:04you
26:05you
26:06you
26:07you
26:08you
26:09you
26:10you
26:11you
26:12you
26:13you
26:14yeah
26:15you
26:17complimented
26:18me
26:19they are honky you are a true artist
26:21oh thank you
26:22this beautiful compliment thank you
26:24yeah
26:25okay
26:27okay
26:41let's go
26:42case number one
26:45f one
26:46f one two prime
26:49f two two prime
26:52zero
26:53okay
26:54yeah
26:55yeah
26:56yeah
26:57yeah
26:58yeah
26:59yeah
27:00so we are dealing with situation where
27:03something called like dirt
27:08and dirt
27:10and dirt
27:11you
27:12you
27:14you
27:16you
27:18you
27:20you
27:22you
27:23you
27:24you
27:26you
27:28you
27:30You
27:55If one s one
28:00F1 X2. F2 X1. F2 X2.
28:16Yeah, so it's cramped, but this is the case, okay, so now F1 prime X is bigger than F2 prime X.
28:44Tipper slope and F1 X1 is larger than F2 X1.
29:05Now prove F1 X2.
29:17It's bigger than F2 X2 given if X1 is smaller than X2.
29:36Okay.
29:38Now if you want to prove it, go for it, okay?
29:43All right.
29:45I think I'm going to eat some peanuts, okay?
29:54I'm just hungry.
29:55Okay.
29:56Okay.
29:57Okay.
29:58Okay.
29:59Okay.
30:00Okay.
30:01Okay.
30:02Okay.
30:03Okay.
30:04Okay.
30:05Okay.
30:06Okay.
30:07Okay.
30:08Okay.
30:09Okay.
30:10Okay.
30:11Okay.
30:12Okay.
30:13Okay.
30:14Okay.
30:15Okay.
30:16Okay.
30:17Okay.
30:18Okay.
30:19Okay.
30:20Okay.
30:21Okay.
30:22Okay.
30:23Okay.
30:24Okay.
30:25Okay.
30:26Okay.
30:27Okay.
30:28Okay.
30:29Okay.
30:30Okay.
30:31Okay.
32:32And some peanuts and two pre-boiled, pre-peeled eggs with vitamin D in first milk.
32:42Okay.
32:44Good.
35:22There are a lot of movies about John Dillinger, he's the bank robber who escaped from prison
35:32so many times.
35:34The most reasonable is a 2009 one.
35:40Christian Bale as an FBI agent and John Dillinger as, I mean Johnny Depp as John Dillinger.
35:51Yes, watch the trailer.
35:53It looks really good.
35:55I think I was about to check out at some point.
35:57Sure.
35:59At some point.
36:01Yeah.
36:03Good casting and good acting.
36:05Good acting.
36:06Cinematography.
36:07Tommy Gunn and all that.
36:09Yeah.
36:11Okay.
36:13Okay.
36:14Yeah.
36:15Johnny Depp, when he was a teenager, I guess, he appeared in the movie The Nightmare on Elm Street.
36:23Yeah, I remember.
36:25But he became famous in the Pirates of the Caribbean, right?
36:33Yeah.
36:34Well, Chocolate Factory, I did watch that movie, Johnny Depp.
36:41Yeah.
36:42I did not like that movie at all.
36:43Okay.
36:44Tim Burton directed, but he didn't do a good job in that movie.
36:48Okay.
36:49Tim Burton movie, I liked the original Batman, 1980, something like that.
36:54It was in 1988 or something.
36:55Oh, so yeah, Tim Burton and Johnny Depp, so, seize your hands.
37:06Yeah, I did not like that movie at all.
37:09Okay.
37:10Okay.
37:11I'm checked.
37:12We should have left a moment?
37:16Okay.
37:17Time check.
37:23You should not have left a moment, guys.
37:37Okay.
37:39f1, we'll just call it y1, okay? y1 is equal to ax plus b, let's say 1x plus b1, y2,
38:00a2, maybe we have done this before, and a1 is bigger than a2, that will be bigger than 0.
38:30Oh, we'll be looking at x2, okay?
38:47F1, x2, f2, x2, okay?
39:05Now, f1, x1, a2, x1 plus b2,
39:32and we know that this is our assumption, okay?
39:47As you can see, it's not an easy problem, not that easy, not that difficult either,
39:53but we're dealing with the easiest case, but it's already a long story there, okay?
39:58Yeah.
40:08So, you subtract?
40:28And we know this is bigger than zero because of this, okay?
40:41And this is positive.
41:04Now, let's go here.
41:07Subtract.
41:11And our goal is to prove this is bigger than zero, okay?
41:14This is like an inequality operator variable.
41:18We'll call it, we'll use square instead because it looks like zero, okay?
41:21This is like an inequality operator variable.
41:25We'll call it, we'll use square instead because it looks like zero, okay?
41:31Okay?
41:32This is like an inequality operator variable.
41:34We'll call it, we'll use square instead because it looks like zero, okay?
41:38We'll call it, we'll use square instead because it looks like zero, okay?
41:45So that's la, okay?
41:48What's this?
41:49Yeah?
41:50Yeah.
42:11This feels like this gerekiyor.
42:12And x1, x2 is bigger than x1, okay.
42:28And we are also dealing with positive numbers, okay.
42:52Why?
42:53Because we are looking at the behaviorism when x goes to infinity, positive infinity, okay.
42:59So yeah, that will pose bigger than zero, okay.
43:20And then we have the proof.
43:24Your positive number, positive number, positive number, okay.
43:29And then b1 minus b2 is the same number, okay.
43:36And because x1, x2, both positive, okay.
43:43And we have same positive number as a1 minus a2, okay.
43:47And then we are multiplying a positive number that is bigger than x1, which is x2, okay.
43:54So yeah, so that's why this is like, okay, we have...
44:09Okay, now let's multiply positive number a1 minus a2 to all this.
44:30Okay, let's do this right.
44:51Okay.
44:53Okay.
45:03Now we have this.
45:33And for now, let's just add B1 minus B2, OK?
45:56In these two sides, OK?
46:11Yeah, just adding two numbers in this inequality, OK?
46:14Now, we know this is more than zero from here.
46:18That's an assumption, OK?
46:26Therefore, this is like associative law in inequality, OK?
46:54Which is f1, x2 minus f2, x2.
47:11Therefore, this is big therefore, OK?
47:16Put this guy to the outside inequality.
47:25f2, x2 is smaller than f1, x2, OK?
47:35Yeah, f2, x2 is smaller than f1, x2.
47:50There could be an easier proof than this.
47:54I do not know, but we proved it anyway.
47:59My guess is that there probably is easier way to prove this.
48:06Why is this obvious, isn't it?
48:13We have slow slope, steeper slope.
48:17At some point right here, we call it k, intersection.
48:25We call it x0, OK?
48:30So f1 is steeper, but it started slow, it's like rabbit, kind of lazy, kind of start running late, and slow slope, that's like slow, turtle, tortoise.
48:47Okay?
48:48And it only starts, but because rabbit is faster, although it started late, later, eventually catches up at point x0, time x0, OK?
49:02And they win the race.
49:09Aesop's f1, OK?
49:11Aesop's f1, that's like northern Africa, southern Europe, OK?
49:15Yeah.
49:16Aesop, that's like Greek name, OK?
49:19There's theory that Aesop was actually an African person.
49:23OK?
49:24Yeah.
49:25That's one theory that they have.
49:29They're not quite sure, surely know who Aesop is.
49:33Aesop's f1, OK?
49:36OK.
49:41So after rabbit catches up with turtle,
49:46the distance between turtle and rabbit gets bigger and bigger and bigger.
49:56And rabbit is always, always ahead of turtle, assuming that they maintain the same speed.
50:05Each of them, OK?
50:08Turtle?
50:09Land turtle, tortoise?
50:121 meter per second.
50:14Rabbit?
50:153 meter per second.
50:18OK?
50:19OK.
50:20Now, time check.
50:21It's been almost one hour.
50:22OK.
50:23Let's digitize this.
50:24And let's go to the timeline.
50:25OK?
50:26This is good.
50:27During the digitization process, I think of an easier way to prove this, OK?
50:28OK?
50:29OK.
50:30OK.
50:31OK.
50:32Yeah.
50:33Yeah.
50:34Yeah.
50:35See you soon.
50:37Nice.
50:38Good to be back to mathematics, huh?
50:39Thanks.
50:40Thanks.
50:41Sorry.
50:43Nice.
50:44Good to be back to mathematics, huh?
50:45Thanks.
50:46Okay.
50:48Thanks.
50:49Yeah.
50:50Good to be back to mathematics, huh?
50:51Thanks.
50:52Nice.