d the same thing back i bet the neighbors were celebrating because we'd never said a word to each other after that not even a whisper before i left what would you say if you had the chance i'm sorry my dad my mom to everyone and log addendum four conclusion interview 74905 date august 24th 2013 interviewer assistant researcher dr jessica miller interviewee scp 7490 begin log hi scp 7490 hey car michael's not with you he's in his office still trying to figure out your anomaly are you uh what is up with him you two are a lot of like how well for starters you both think you're responsible for other people's well-being so you both are trying to do what you think it's best for them think that's all we can do and you what brings you here just checking up on you well you checked no filter do you sorry apologize a lot too what's your point this anomaly is not kind that doesn't mean you're that too i'm serious thing is this whole time you've never been kind to yourself all the things people said and done to you you carry that alone maybe because i deserve it or do you want to feel like you do scp 7490 remain silent you've got horrible luck what happened to you when you were young wasn't your fault neither was gamma nine are these last two months silence this anomaly does not define you nothing does except you and i know damn well that you are a lot of things the question is do you know that's all i wanted to say hold on yes how are they jake and tommy the 2d class they're fine honest to god they're fine they are d66 9600 arm pain had finally gone away that's good that's really good come to think of it neither of them was that affected when they were with you you would think with the storm here normally would also work on them but it just locked the door maybe they got lucky i know you already said no more tests but i stand by that i understand that but i think there might be something about you and your anomaly that we missed maybe there's a way to control it honey's optimism is starting to rub off on you hear me out i had a theory going on in my head with the d13714 maybe it works similarly with these other two well what's the theory it's rough but i believe your anomaly is somehow tied to your state of mind if you think negatively bad things will happen but if you think positively then good things will happen yeah i don't know anything goes wrong i will take full responsibility but what if it doesn't work only one way to find out i don't know i can assure you everything will be fine trust me me i do i still think you shouldn't do it what if god forbid those sra shut down or a containment breach you're just gonna sit there and let it all happen i don't want any of that to happen it might sure we can stop it from getting worse but who's to say if anything we should be able to prevent breaches and if this test could serve that what's stopping you it's a theory an scp theory sorry i had to say that one that can probably save a lot of people if you want to you want to help people so badly this is how otherwise we're just gonna have to wait until that breach comes what time tomorrow at 11 30 a.m thank you 74 90 better be right and log subjects d all sevens and d 66 96 0 procedure scp 74 90 were to stand in between d 66 96 0 and d all sevens while two guards were to throw tennis balls at them results before continuing the procedure dr miller encouraged scp 74 90 that it will keep them safe after about 40 seconds of prep time the guards threw the tennis balls full force every ball the guards threw hit missed the three subjects until they ran out subject dr jessica miller procedure dr miller was to be shot in the head by one or two guards results once again dr miller had to encourage scp 74 90 that it will keep her safe scp 74 90 was hesitant but dr miller was able to persuade the subject it had another 40 seconds of prep time scp 74 90 was ready and both the guards aimed their handguns unintended fire however their weapons were jammed dr miller ordered the guards to continue while having the d class serve the tennis balls at her to which they all missed after 20 seconds dr miller called off the test from ad or taiga to dr car michael subject regarding the final test i saw the final test did miller honestly believe that scp 74 90 could have jam that guards weapon let alone two what she did could have got herself killed assistant director astin or taiga from dr car michael to or taiga regarding regarding the final test it was 50 50 to be frank she told the guards and me her plan before we got to testing which would let the weapons would have blank rounds in their magazines so in case the guards were successful pulling the triggers miller would have been fine either way regardless researcher miller's theory was correct dr tony call michael scp 74 90 finally accepted weekly sessions with dr connor quinn for psychological evaluation for a whole month since then the effects of scp 74 90 caused the following incidents incident 74 93 cathleen gray on a 13 director cathleen gray made a full recovery with no repercussions and has returned to her position within site 39 since her return site 39 has also returned to being systemic with next to no incidents regarding injury illness or death allison or taiga on august 30th 2013 assistant director allison or taiga was able to make an appointment with her doctors to remove her stomach cancer before it reached stage two after two weeks she made a full recovery and returned her position didn't even need chemo that's awesome luck sarah taylor on september 1st 2013 agent sarah taylor had fully recovered from her surgery as well as been promoted to senior security officer frankie castello on january 17 2014 junior researcher frankie castello won the power ball gaining $500,000 edward russo on june 5th 2014 dr edward russo had a call from his wife stating that his daughter had finally been accepted to the david eckles school of business in salt lake city utah they will be starting next year zavier mcqueen on 11 November 11 2014 after having a one-on-one interview with 05 3 as well as an extensive review of his record agent zavier mcqueen had been selected to be 05 3's personal bodyguard once he completes the six months of training he will have access to any secret detail of the foundation archives will be transferred to site one notice from director catlin gray of site 39 as you all know a lot of the favorable things a lot of favorable things have been having the last few weeks all due to scp 6490 however we must remember that it is only because of its state of mind and not because it will is it so because of that scp 7490 can either make site 39 fall or continue to keep it sanding for the sake of our stability we will continue with its containment along with these sessions please schedule a one-on-one meeting with me if anyone has any concerns or suggestions regarding its containment i've also heard rumors that scp 7490 will be used to develop site 39 prominence i'm here to disclose that we will do no such thing not only will we be painting a target on our backs but the notion of utilizing an scp that thinks and feels like us is unethical i will no longer hear such nonsense going forward other than that we will see what scp 7490 hasn't saw for us in the future in the meantime be cautious and treat scp 7490 with the utmost sensitivity scp 7490 was and still is one of us and so scp 7490 is our responsibility we will secure we will contain we will protect good luck to all and stay safe surface details that just annoyed me a little bit surface details that just annoyed me on this minor scale having tommy be the besty best friend who died in the car and then also be the name of one of the d-class uh jake and tommy are safe that weirded me out i got confused yeah i thought maybe there was some weird thing happening here some reality warping or something i don't know but yeah that threw me off too second the uh 7490 being based on in two different interviews this conversation with the d-class who goaded her with it's your fault you did it i feel that the d-class would not even know what the hell was going on like in most canons we don't really tell the d-class a whole lot of details yeah yeah very minimal for him to like goad her about the bad luck and it's all your fault right i agree with that yeah um a lot of there was some some tense issues i noticed too there was times when it would flip back and forth from present to past tense um like like weirdly too um and like you know there's misspellings and you know typical stuff it's it's not as long as i thought it was it's a six thousand word article um most of the bits dialogue that's why it ends up so long um like vis visibly um it's so wide too right that's not it's not like a complaint just there's notice it was wide in one of the uh yeah it's weirdly wide it's really hard to fit on this screen for the stream uh in one of these interview logs uh i know that dr miller gets changed to researcher miller for one interview and then it goes back in the next interview so that was just it's a it's a cameo it's a it's a it's a it's a hint it's an easter egg you know for vulgar yeah and like the i mean of course like the big thing for me is that this is basically just like a worst loser um in the sense that like the loser is yes there it is thank you yeah um it's not like you know it's not bad uh it's i just it's not bad but it it definitely highlights why the loser is like a contest winning level version of this yeah um yeah they what's missing here to me in the main so i went over my like little picky things and formatting errors but the big thrust of this to me is that it it hand waves at like i want to make sure this uh like ethics committee suicidality of humanoid anomalies also like there's a lot of like a lot of the articles you read today talked about suicide and like not a very like tasteful way and noticed like the last one and the black moon it's like i mean the black moon was a good article but it still was like kill yourself kill yourself you know like it's so it's a lot i hope the next one's a fun one i could use a fun one yeah the black moon i think i can get away with because it felt a little bit like the analog horror like yeah when you look at like even there's one of those that's like the moon is giving us messages or something and so it kind of like look like a link because i know that in one of the analog horror series i think it's like local 58 the something to do with uh things yeah and then in uh mandela catalogue the entities are blatantly trying to get people to end their own lives so like when i saw it in the black how black moon i was like oh it's telling them to kill themselves that's if they yeah they had a doppelganger's i would have definitely made that connection but then the next one was just like uh it was the one that was very unpleasant to read on purpose and it was just like a lot of on purpose and by accident i think a little bit of both yeah column a column b but uh yeah i just a lot of a lot of bummer article quite a streak we've been on in tonight's uh article list yeah whoo uh man yeah this one like the big structural issues for me are that the themes are kind of hand-waved at but not really dug into it feels two-dimensional yeah two-dimensional and the characters like there's so many interviews that feel like the same thing the plot points that we get in all of these interviews i don't feel like i get enough to humanize the characters or like get enough details to justify having this many logs yeah i also thought that was kind of weird that like i mean this one in the last one both had this thing where like you have a family member that you've had terrible experiences with but also you want to like forgive them or like apologize to them and like doesn't really sit well with me i don't know uh like this she had a very awful relationship with her dad and she's like i just want to apologize even though he told me that it should have been me that died instead of my mother i don't know i just it just rubs me the wrong way um and i don't know we didn't we didn't get to see her like talk about her service they set that whole thing up and then never they never paid off like i'm not gonna talk about that now you're right you're right they never did uh man oh and the last thing the last thing turning it around to being good luck with mood just like that at the end it came out like we had all of this going on and she suggests oh it's mood based and then it works and then the article's over i'm i'm cared now thank you i'll never be sad again yeah so yeah jar that's all i got jar yeah exactly uh yeah also the the choice to kill off an mtf uh group as a whole is very very bold that's usually looked down upon uh by critiquers because you know mtfs are the best of the best of the best in their chosen field if you kill off the mtfs you either gotta make the threat like commensurate for that or readers are going to think that the either the mtfs are incompetent or you wrote them to die and it feels more like the latter here uh but it did get away with it because it's a probabilistic anomaly that they weren't aware of you know it's fine just a it's a risky move risky move it's at what plus four so i don't know if it paid off but uh there's something here there's something here maybe it just needs a little rewrite yeah there's articles that i no vote because i'm neutral or downvote because i hate on x y and z this one i'm down voting just because i don't feel like it's finished and i feel like there's stuff there and i just want you to like flesh it out yeah yeah and get rid of some of the unnecessary stuff i think yeah let's let's take a look at the next one because i could use a palette cleanser let's see if scp 70 24 by jack kawoski is short enough to fit in tonight or if it's the start of next time all right 70 24 you said right correct we just did 70 42 say 70 24 lucky cola or the dirty secret of why it tastes so good that sounds fun i like that name um the article is short enough to do within our time limit so this will be our last one for the night excellent party on let me get that stream oriented and that is uh it's lucky cola by jack kawoski jack kawoski item number scp 70 24 object class safe special containment procedures when instances of scp 70 24 are found they should be confiscated and moved to the nearest storage depot for less hazardous materials for study and subsequent disposal depending on the storage capacity of each facility at least 20 of the instances should be stored no active efforts of memetic suppression will be made to the subjects affected by scp 70 24 publicly attributing the associated events to caustic or chance factors are causistic or chance factors for the cases in which these events have a high diffusion or notoriety disinformation campaigns will be carried out attributing them to unfounded rumors or hoaxes image caption home of redacted it's a long name scp 70 24 consumer and winner of the redacted international league of legends tournament hmm oh we do legalize this back again yeah the efforts of the foundation should be focused on discovery of the scp 70 24 manufacturing facilities and the capture of the workers for their subsequent interrogation and amnestization in cooperation with the local authorities these facilities will be shut down on charges of serious health problems distributors will be traced for the confiscation of scp 70 24 preventing its commercialization update one upon discovery of a new scp 70 24 production site a tactical team will be deployed to sabotage the machinery present and disable the site these events should be covered up with fires which will be reported as accidental see a dental three update two if viable instances of scp 70 24-1 are found their transport and delivery to the druidic secret society will be coordinated through the irish ministry of foreign affairs see a dental four interesting description scp 70 24 corresponds to a compound with ontokinetic properties of lower intensity commercially distributed under the name of lachicola scp 70 24 has in its composition ingredients usually found in energy drinks such as caffeine sucrose amino acids such as taurine and b vitamins together with a wormwood extract are artemisia absintheum other organic yeah it's got absintheum other organic substances of biological origin have also been found whose origin and composition have not yet been determined see that too biological origin have been found whose origin have not been determined okay scp 70 24 has been described as quote refreshing and energizing with a slightly bitter aftertaste similar to tonic water without observing adverse effects other than associated after consuming large quantities with the use of massive doses of caffeine and warmwood the anomalous effect associated with the use of scp 70 24 is related to the result of events with probabilistic outcomes in which the consumer participates especially in relation to games or other activities whose rewards are not tangible obtaining favorable results in a proportion much higher than what it could be expected according to chance or to the consumer's abilities or skills that was a long sentence additionally an increase in the frequency of events with negative or harmful outcomes has been observed among people in the family social or work environment of scp 70 24 consumers being the frequency and severity of these events directly proportional to the amounts consumed of scp 70 24 being originally many of these events associated to chance or to intentional actions on the part of scp 70 24 users another huge sentence the existence of a row or just one sentence each yeah the existence of the existence of scp 70 24 was discovered through the foundation's automatic monitoring system of social networks communications and news casandra upon which detecting a series of unusual domestic accidents established that all the homes were inhabited by a subject who had recently obtained a victory in a sports or recreational competition thank you for the rest seeking to rule out the presence of anomalous activities onto kinetic or the use of thaumatic rituals these subjects were interviewed establishing that the common factor was the use of scp 70 24 in a period close to the events described shh addendum one example of a subject affected by the use of scp 70 24 the following is a transcript of an interview with the with ann scp 70 24 user describing events that affected the subject and his family group interviewee tony fratti 55 years old an accountant from the city of redacted interviewer geronimo bozo police officer from redacted preamble mr fratti went to the redacted city police station on september 21st 2020 or september 20th 2021 to make a complaint against the casino redacted for possible criminal actions against his family the complaint was received by agent geronimo bozo of the state police redacted and the recording obtained was delivered to the foundation through a collaborating agent do you want to be agent bozo i would love to be agent bozo geronimo bozo is what you say when you kick someone out of a plane geronimo right yeah yes okay i'm back that name's been killing me i love it these names are like like comically italian like yeah good afternoon tell me how can i help you i didn't think they were capable but those damn bastards did it they just can't accept someone getting lucky and beating them fair and square no no sir because they're cheaters they think everyone else cheats excuse me the casino people they they clearly fleece whoever they can but if someone beats them they go ballistic i wonder uh is how they were able to get the get her to hurt her start something calm down please explain to me what happened excuse me i think i started in the middle what happened is that a while ago i won some money from the casino and apparently because they think i cheated and they couldn't prove it they retaliated against my wife i've always liked to gamble although i never do it in large amounts smiles the night of september 7th i went down to the casino to spend a few dollars and yes sir that was my night i started with the roulette and i turned 200 into 1000 then i went to the blackjack table in one hand after hand i folded when i hit 30k between you and me did you cheat of course not i'm a prestiburian cheating is a sin i don't drink alcohol either i only drink energy drinks that night i drank several cans of the only non-alcoholic thing they had a soda called lucky cola it's greener shit and kind of bitter but it's very refreshing i liked it so much that it took several cans at home with me that night i played it till about three or four a.m i was winning but i was afraid of losing everything so i went to change the chips when i got to the cashier and cashed out i was held and taken to a monitoring room the head of security told me that i was no longer welcome that my access to the casino was forbidden from that day on i i saw distrust and confusion in his eyes like he was trying to understand how i cheated they simply could not accept that i had been lucky i went back to my home in the next day i decided to spend some of that money that fell from the sky smiles to buy a new van from maria my wife her old truck was barely working and every day she has to commute an hour to work if you could have seen her now how happy she was when she went to pick it up it was brand new fresh from the dealer we went out to test it in the nearby road it was running perfectly i didn't make any strange noises just before reaching a junction the steering locked up we went through an intersection and ended up being hit by a car coming in the opposite direction the impact was on the driver's side i was on her but my wife got the worst part i think i had been in the hospital with her she gained regain consciousness a few days ago unfortunately her head is fortunately her head is fine but she suffered multiple fractures and the doctors don't know if she'll ever walk again it had given it a lot of thought but i can't help but think that the casino people did something to the van so that we would have had the accident but uh you know how did they know we would buy that vehicle from that place and how could they sabotage it either that or the money was cursed it's like my all good fortune has gone and one fell swoop end of record september okay it is okay 20 september 2021 at 1230 hours closing statement foundation technical personnel were able to access the wreckage of the wrecked vehicle finding no signs of malicious tampering on any of the van's components subsequent analysis determined that the accident was attributable to a very rare spontaneous failure of the hydraulic system not detected in the quality control tests addendum to access to a scp-7024 production plant though through interviews with distributors it was determined that the compound was produced by a u.s company called irish charm limited though it was possible uh though it was possible through a fund company to acquire a batch of 100 cans of 350 milliliter of scp-7024 the first attempts to find the production plant was unsuccessful not locating the indicated address or only finding disused industrial rooms the use of para technology was required so that an exploration team could access the plant and make a record of its production activities participating team members of mtf land the six snoopers participating personnel captain mario gutierrez land the one agent ali salamad lambda two an agent berry boil lambda three okay so there's one comically italian person in this group and one comic book superhero looks like one has the most uh lines i can do one you can do two three sounds good okay i'm not going to do the the accent anymore sorry i just it's tiring that's all right the three's getting an accent you can't be k called berry boil without having some funny voice that's fair guys this is the last building we need to expect today activate your delphos i don't want to encounter any unpleasant surprises okay captain although i doubt we can even enter the place it's about to collapse it looks like it's been abandoned since the prohibition era can't hear any noise from inside us see any steam coming out of the chimneys shall i go inspect if there are any service entrances no reconfigures his delfos equipment to a different wavelength how interesting we have the entrance right in front of us uh covered with camouflage glyphs do you think we have any arthomatoid just on site no i doubt it you know you can learn how to draw those things on the internet most likely they're just some junkies when to cook meth without anyone sticking their nose in way so far from the abnormal so far the only abnormal thing we found in the inspections are the abnormally large and hungry rats living in those abandoned factories let's check this dump fast and get back to base communicating by radio video transmission activated we're going in this place looks a lot bigger than i expected it also looks pretty clean like they've been using it until recently look what we have here points his camera at some stainless steel tanks either someone is reviving grandpa's illegal whiskey business or we finally found the factory i can report that there are sounds of machinery running and steam coming out at least one of the tanks but so far we don't see any of the operators i'll be controlled shall we send a support team no we have our weapons ready if hostiles are present also in the worst situation we will proceed to evacuate the compound copy that and the image caption is a section of scp 7024 production plant do you feel that smell it's like a decoction of medicinal herbs with a rat smell monkey smell shit smell i don't know it's like a smell i don't think it's relevant let's keep checking they find crates with sacks of sugar and dried grass plus other components they can't identify and they take samples for further analysis while checking another storage cellar allowed noises herd and steam and an amorphous mass of debris come out of one of the tanks being carried by a nearby conveyor belt to a trash container this is getting more and more interesting they continue with the search place accessing a warehouse with apparently several hundred units of scp 7024 this appears to be the finished product storage warehouse there's something that has me concerned we've been what 10 minutes checking the place so far and we haven't seen any workers it's impossible that the plant is fully automated hey i followed one of the scrap carts in an empty into the top of a warehouse the door is padlock it seemed to me that that's too much security for a single dump the three operatives approach the warehouse and lambda two manages using a lockpick to open the door they find several piles of boiled grass and peeking out of one of those piles they see a partially heat they see the partially heat destroyed remains of a small quadrupedal animal with a large skull shit and looks like they boiled a rat it's disgusting they continue exploring the place finding some offices with binders and accounting records more warehouses with raw materials and packaging materials they pass through the packaging section where the scp 7024 canning process occurs seemingly automatically hearing cries of distress coming from a warehouse at the far end of the plant they head over picking the lock to gain entry inside they find dozens of small sized cages several of which have small animals of an undetermined species inside which are in varying degrees of agitation oh shit what are those things they look like little monkeys i think this is where the stench of beasts we've been feeling all this time comes from since when are monkeys green also those things are whimpering i don't know if we should call in an animal recovery team from the foundation or wilson's guys i'm not a zoologist but i know those are not computians they are leprechauns listen good they're not whimpering it's like they're seeing a funeral dirge fuck that's really macabre lambda one approaches one of the cages crouching down what can we do for you little buddy cob hair cob hair cob hair help at that moment a hatch opens from the ceiling and a mechanical crane grabs the cage lifting up and carrying it quickly to the outside moments later the sound of steam rising and screams of pain can be heard no shit several subjects appear in the doorway wearing work clothes and carrying shovels and other heavy tools fuck me what the hell are they doing in here subjects rush towards the agents in lambda one proceeds to shoot them with his service weapon being hit in the head by one of the attackers lambda two and lambda three also open fire on the attackers with several of the cages being kicked and overturned during the struggle allowing several specimens of the unknown entity to flee the scene video contact is lost a few seconds later epilogue a tactical response team from the nearest site arrived 30 minutes later at the scene finding no trace of agents gutierrez salmott and boil or the attackers involved the incident three instances of the unknown biological entity were recovered in the vicinity along with the remains of at least a dozen cooked or partially dehydrated individuals the instances were located in bioterium of site redacted killing off the nts also i'm i'm getting kind of like i mean this is probably just me but i'm getting kind of metaphysically in vibes here except for the weird full well not the weird lucky stuff um i'm with sure i'm with the chat niki in there it's a slurm time it's all slurm every way down yeah i definitely i definitely get slurm vibes and i mentioned it yeah addendum three authorization level two or higher is required for access please enter password access authorized incident seven thousand seven thousand sash three dash 2022 on april third 2022 different format for the date now a consignment of several hundred cans of the new energy drink promoted as quote jinx cola that tastes the myth of misfortune a product of irish charm limited was detected on the market analysis of this formulation revealed in addition to caffeine sugar vitamins and plant extracts traces of the genetic material the foundation agents who disappeared in an incident uh of now the other way it's the hyphens too now uh january 15 2022 it proceeded to requisition all instances present on the market uh and update the level of response and discovery of sdb 74 production facilities addendum four authorization level two or higher required for access please enter password access authorized letter from the department of foreign affairs of ireland june 6 2022 through a high-level liaison of the foundation has come to our attention that they have under their control three instances of a species of crypto fauna belonging to the republic of ireland specimens which i must acknowledge with great regret we're along with many others taken by the citizens of our country who taken taken by citizens of our country who blinded by greed have been capable not only of jeopardizing the balance we have maintained the magical world for centuries but also the very well-being of the country itself those incident innocent creatures you that you called creatively i must admit uh as siebas viridus cantor we have known from at least the middle agents as leprechauns being protected for a very long time by a select group of men who knew agent secrets it was these same men who at the time preferred to form a black legend around the leprechauns so that they would be left alone the welfare of these creatures is literally our welfare and if they are happy luck will be with us and ireland will prosper on other occasions event with regret we have collaborated with you in the containment and even neutralization of other creatures of anomalous origin with which we had coexisted for centuries but which simply did not adapt to modern times getting out of control becoming noxious and unnecessarily attracting the attention of the world to our land on this occasion i ask you on behalf of the irish government just the opposite that you return these creatures to us so that they can return to live happily in their special corner of ireland where they belong a place unknown to most people because if they disappear so will we i'm hoping for your cooperation and that we do not have to resort to force measure measures to achieve our goal i take my best regards simon company minister of foreign affairs and defense of the republic of ireland hmm so once again there's a lot of spag errors and you know i kind of got like a vibe that that maybe this this person wasn't um like english is the first language but i'm not sure they do have uh they're like main first person that got crit from is an int person so that lends credence to that yeah um and like like the uh the like the feeling of smell like that kind of made so my first thought was the uh ghostbuster joke uh listen you smell that right but it didn't say smelly so if you feel that correct you feel the smell which you know could be a a translational issue which is fine okay here answer like the way that when you walk into a place that's rather like english is not is not is not jack kowalski's mother language he had to translate the article after writing it so that's probably what it is um so i will take i will judge this article specifically then on the content rather than the writing because you know it should be it should be cleaned up there should be someone in there to to uh to you know to improve that a little bit but i'm not going to hold that against the author um i don't know if i hmm most of the stuff is fine it is it is there it exists it's fine um i don't like the idea of killing off a off mtf um i don't like i don't know i don't like it so i don't like how it's a soda tasting like tonic water that just bothered me i'm just i wouldn't buy a soda that tasted like tonic water anyway um yeah i mean it's it kind of reminds me of the rainbow one yeah we have another like exploitation of the leprechauns yeah and then like leprechauns being kind of different than what you'd expect and then also like the fact that there's a lucky thing that when it wears off it makes you extra unlucky um because this one yeah it doesn't so it's lucky cola from steeping uh leprechauns in liquid like tea yeah and it grass also makes the people around you unlucky which doesn't make sense to me because if the liquid is giving you luck i don't feel like it saps luck from the area which there there could be an like they could have explained that because like it's similar thing happens in like the loser and stuff it's like you know if luck is a quantifiable thing then perhaps it pulls it from somewhere else and then there's a void that needs to be filled with bad luck like just just state that and i think that could be fine you know um also worst business decision ever is for a casino to have one non-alcoholic beverage and it's lucky cola yeah i don't know who uh who thought that was a good idea but you definitely want to pick any soda and you pick the one that was gonna lose you all the money like you got mad at this one guy i can't believe your casino is not closed yeah i was gonna say he cashed out and he was at 30k you got lucky you dodged a bullet there and what if he went into a heater and like like you know left with 10 mil and he'd be in trouble also i mean i guess because of like how often he was winning they suspected him but like 30 000 i don't know if that's enough to like attract the attention of casino security i mean that's also clef uh you know he'll he'll correct me if i'm wrong um but yeah we do have an expert so let us know what our feelings on casino beverage and winning street numbers are yeah at what point do uh the security be like okay this person is not just lucky you know i could have forgiven the uh mobile task force tpk if it like a little bit because it was going into the uh what's the principle for because they like made them into unlucky juice yeah at least like they like had a reason for it yeah desk vault of clef says there is a law that if you win or cash in 10k that the government gets involved so that's like taxes well yeah i mean like it's probably like you know i understand like they have to do an investigation are you saying and like is that what you're saying or like when you say government you just mean irs because like yeah irs is going to want to know but yeah um because i imagine that you go to vegas and you have that whole thing where like you start winning enough and they start comping your room so you stay longer and lose it back yeah yeah um and laundering investigation yeah okay so like the the fbi or something we get involved okay yeah i mean so i i could see that happening but it i don't know um maybe that would be a good way to get the foundation involved with this is like the guy won a bunch of money and then there's an investigation like you know ui you are you know juicy or not juicy but like you know like they get involved and then the foundation gets involved but um overall this is fine it's just like i said the run-on sentences were taxing the names sounded fake unfortunately um at least most of them was going down in history man i that i was very much i didn't want to acknowledge the actual pronunciation of that some like nope it's it's germino bozo because it's italian i was like it's italian you know yeah as for drinks they don't have a drink called lucky but the idea of non-alcohol and that more like you drink the more you spend yeah but you have to get too drunk then you can't spend so they get a balance it out that's what i get you know four locos or something no boy oh boy okay that was good that made me feel a little bit better yeah the vibe the vibes are definitely up ending on a high yeah i'm glad as a sneak preview that we're not reading it but we're getting ready for next time our next article will be scp 7123 by ross boss arts 7123 7123 we got dice on the opening uh double digitor um oh that font though i don't like the serifs on that and there's a lot there a little uh smudgy it looks like a lot of footnotes at least just in the beginning okay okay yeah okay there were seven footnotes but it's like all in the first paragraph okay uh i swear to christ if this if they say the house always wins i'm not gonna do anything i just well we got we got one of those last week right we got the house always wins in an article finally yeah we had them in at least two or three now um yeah oh man we should have had the luck related drinking game as we went through this all the luck cliches all the tropes in this very very narrow uh theme all right here's the dice here's the cards drink every time someone's cursed to be unlucky here's the luck deity here's the luck deity uh oh man did you see that there's a d-class with seven digits in its name in in 7123 oh crap i am serious d1971991 oh boy how many d-class does this foundation go through what population are they pulling from uh it's yeah it's the same timeline with d-2000 they just pull them out as scb 2000 whenever they want yo dog you are our one millionth d-class congratulations you get to do nothing different sorry this foundation is about to hit its two millionth d-class that's incredible for them i congratulations man that turnover rate we do we do daily terminations of the whole population we we got rid of that yearly or monthly a long time ago i mean the the black moon foundation was pretty brutal about that they were like i'm surprised they didn't they didn't they didn't put down that gold retriever in the beginning all right uh black moon five and a half billion people dead on day one go that was yeah shoo that's like um there's a couple of them that we've read where it's like it's killed seven billion people in the first three days like oh okay all right ah man good times good times indeed hey and we didn't crash a third time for our stream working yeah that's great let me pull up the final ad break we're closing out everybody and of course what do we want you to do we want you to support dr samarian so we can put those trees in the basement and do redacted and data expunge to them yes um some of you out there might be fans of maple syrup get it while you can that's all i gotta say oh no we're putting nanjima out of business all those metal trees everywhere we'll just have metal trees with maple syrup artificial maple syrup dispensers it's motor oil we couldn't really get close enough to yeah we tried on the texture a little bit yeah we tried also go contribute more to hymns for the road let's keep pushing them toward their stretch goals we got 35 hours to go on that yes and do that they worked very hard for that they deserve it and we'd love to see them succeed in the description you'll also find links for the scp redacted series with episodes one malo you'll find the steam link for scp info hazard don't play right now you'll have patreon and slow gramming and scp foundation files billus twitch we got everything all in the description lined up so you can support site 42 supply support our greater scp community and make cool art happen absolutely with that i'm gonna go stretch out grab some dinner and bill of the night we'll see you all in the next anomalous broadcast from site 42 thank you and good night anytime i need to see your face i'll just close my eyes and i am taking to a place for your crystal mind and magenta feelings take up shelter in the base of my spot sweet like a chicka cherry cola i don't even try to explain i just hold on time if it happens again i'm gonna move so slightly to the arms and the lips in the face of the human cannon ball that i need to or want to i really just want to get that chicka cherry coli line in that was the only one that was important we just did lucky soda


it's never the vegan diet the person is always at fault I have seen vegans who have done the vegan diet in every single way possible raw cooked eating all the foods they're supposed to be eating taking all the supplements they're supposed to be taking and they still have health problems but not once have I seen a vegan welcome someone else and say it's okay you can incorporate these foods if you're having health problems let's try to get you fixed no it's an immediate negative reaction oh my god you're not vegan anymore what's wrong with you you're doing the diet wrong what is you anyone can be healthy on a plant-based diet it's completely crazy and I have never seen anything as crazy as these vegans and their insistence in this diet which has proven time and time again to be something that is not feasible for human health unnatural vegan recently made a video titled my god this is getting old and it's about Khalil who is no longer going to be following a vegan diet Khalil is a very popular youtuber who has a bunch of health problems and got a lot of traction because she spoke about how she needed to stabilize her blood sugar and it's not possible on a high carb vegan diet now unnatural vegans argument to this was well you can follow a high fat vegan diet that's low in carbohydrates why not do that and then Khalil said oh she has anemia she has all these other health issues thyroid related issues and then natural vegan goes to say oh well to fix my anemia and a natural vegan admitted she's had anemia problems on and off for years little bit of a bad idea not a good look if you have health problems that other people have and you're giving advice on how to fix them don't admit that you can't really fix your own health problems but to fix her anemia she said I took an iron pill three times a day with a big glass of orange juice each time and you just said okay you can follow a high fat low carb vegan diet and then you said oh you need to drink three glasses of orange juice a day to fix your anemia what kind of hypocrisy is this and the whole basis of her video was you guys are all quacks these doctors are quacks these people saying plants have no bioavailability or quacks just take a multivitamin what is this bro science is everyone who doesn't agree with you a quack is that really what it is are you so intelligent and all knowing that you have all of these vegan answers with you and your god dr. Michael green bean the string bean dr. Michael Greger are you guys crazy I feel like I'm becoming crazy talking about this to be honest she brought up a couple points that I just wanted to pick apart she said conventionally raising animals and even farming animals on pasture isn't natural no it's not but the end product you're consuming was not made in a goddamn laboratory these animal products this meat this is food we've been eating for hundreds of thousands of years it's what fueled our evolution these nutrients that you claim are not important that you can get in a vegan diet apparently you have no understanding up you said something really silly like you said fat turns our blood into sludge if you actually understood the nutrients in animal foods you would understand that omega-3 fatty acids actually thin the blood and have a beneficial cardiovascular effect in regards to making platelets more flexible blood flow smoother a great example is the Eskimos and the Inuits and how their bleed time and how much lower their rates of cardiovascular events were one positive thing she did say was that we should be eating more oysters and there are many indigenous groups of people who subsisted all of their nutrition on seafood so even people not on a vegan diet even standard American dieters could benefit greatly from consuming more oysters great source of fat soluble vitamins great source of nutrition people don't really understand the importance of these fat soluble vitamins in general let alone specifically on a vegan diet to get a to convince a vegan that vitamin a retinoic acid is more important than carotenoids to convince them that vitamin K2 is way more important than vitamin K from plant foods to convince them that the bio availability of DHA EPA all of these things and animal foods is beneficial to you that's difficult to do even to a standard American dieter let alone a vegan and then she goes to say no major health authority she is a she contradicts herself she says that Khalil is appealing to authority talking about her doctor and this world-famous doctor and how this doctor is giving her advice and then she goes to call that doctor a quack but then she says no major health authority holds these views so she just criticized Khalil for appealing to authority and then she appealed to authority herself I watched this 37 minute video of unnatural vegan and I was like is this video a joke I'm serious I'm dead serious I was like did anyone with any semblance of intelligence watch this video and understand how much she contradicts herself to me it's completely crazy I was watching this thinking is this video real there are people actually watching this and listening to this woman what what is the level of intent is this what it takes to convince people is this why I don't have 10 million YouTube subscribers do I need to dump things down a bit like what do I have to do you guys want to understand more about the bioavailability of plant foods versus animal foods I will link that in the description or you could go to unnatural vegans channel and listen to her call every single person that doesn't agree with her a quack and to lift heavy and take a multi-bro man I swear to God the you guys are you really you guys really are like to me a bunch of comedians I could seriously listen to this vegan stuff all day and in you know when I was watching this for 37 minutes I was bored out of my mind but in hindsight I've been having fun all day laughing to myself about how out of their mind these people are but thank you guys for watching if any of you vegans would like some health advice from someone who knows more about plant foods than vegans do shoot me an email Frank a to Fano at gmail.com you guys can reach out to me through my website below if you guys want to check out my social media Twitter Instagram I made a vow to myself I'm gonna post something on Twitter and Instagram every day I don't know how long that's gonna last but um again thank you guys so much for watching let me know what videos you guys would like to see in the future this was at the request of a couple commenters and the drama is pretty big so let's see where it goes


Last time we have considered Romburg integration. Today we are going to show that first step in the Romburg integration is nothing but Simpson's rule. So, we assume that our function f is four times differentiable. We look at corrected composite trapezoidal rule. In the corrected composite trapezoidal rule when we remove or get rid of the term h square that is the first step of Romburg integration and that first step gives us Simpson's integration. So, our function f is four times continuously differentiable. We look at a uniform partition of interval a b. So, t 0, t 1, t n these are equidistant points h is length of subinterval which is going to be b minus a by n and in addition let us assume that n is even. Composite trapezoidal rule associated with this partition is given by at two end points the weight is h by 2 so h by 2 f a plus f b and at the interior partition points that means t 1, t 2, t n minus 1 the weight is going to be h. So, this is the composite trapezoidal rule. Now, integral a to b f x d x is t n plus term h square by 12 f dash a minus f dash b plus term of the order of h raise to 4. If you look at f dash a minus f dash b by 12 there is no h coming into picture. So, this term is independent of the partition. Now, you look at composite trapezoidal rule with partition of the length of the subinterval let it be 2 h and number of subinterval let them be n by 2. So, integral a to b f x d x is t n by 2 plus h will be replaced by 2 h square by 12 f dash a minus f dash b and term of the order of h by 2 raise to 4, but in the order the constant is there. So, we can one can say that it is of the order of h raise to 4. The first step of Rumberg integration is obtained by multiplying this equation by 4 subtracting this equation and dividing throughout by 3 and we have seen that integral a to b f x d x is equal to t n 1 plus term of the order of h raise to 4. So, now we are going to look at t n 1 and show that this t n 1 is nothing but Simpson's rule associated with the partition with n by 2 intervals and length to be equal to 2 h. So, it is a straight forward calculation, but what it shows is when you have f to be sufficiently differentiable, integral a to b f x d x minus t n, t n is the composite trapezoidal rule. We have got asymptotic series expansion with the terms of the type c 1 h square plus c 2 h raise to 4 plus c 4 h raise to 6 and so on. So, we have got only even degrees of h. Now, once we show that the first step of Rumberg integration is nothing but the Simpson integration that will mean that even for composite Simpson integration we have got this asymptotic series expansion. So, now let us look at t n 1. So, our t n is given by this formula in t n by 2. t n by 2 will consist of points t 0, t 2, t 4, t 6 and t n, only even order points. So, t n by 2 will be h times f a plus f b, the length of sub interval now it is going to be t 0 to t 2. So, that is going to be 2 h. So, it will be plus 2 h and now only even suffixes. So, it will be summation i goes from 2 to n minus 2 i even f t i. Our t n 1 is 4 times t n minus t n by 2 divided by 3. So, here you are multiplying by 4 and then subtracting. So, that will give you h by 3 f a plus f b plus odd terms they are going to come only from here. So, that is why you have 4 h by 3 summation i goes from 1 to n minus 1 i odd f t i plus there are even order terms here, even order terms here. You are multiplying this by 4 and then subtracting this and dividing by 3. So, that is why you get 2 h by 3 summation i goes from 2 to n minus 1 f t i i even. So, see our partition now is t 0, t 2, t 4, t 6, t n, t 1, t 3, t 5 these are going to be midpoints of our intervals and if you look at composite Simpson rule associated with partition t 0, t 2, t 4 it is nothing but this. So, thus we have shown that the first step of Humberg integration gives us composite Simpson rule. Now, we are going to start a new topic which is numerical differentiation. The idea is similar as in the case of numerical integration in order to integrate. We had we do not know how to integrate any continuous function. So, we look at interpolating polynomial, integrate it. So, there is some error involved. So, we obtain approximation. So, similar thing we are going to do here that if you have a polynomial then you can differentiate. So, approximate your function f by a polynomial and then hope that the derivative of polynomial at some point say a that will give you an approximation to f dash of a. Now, in numerical differentiation we will face some difficulties which we did not face in case of numerical integration. Numerical differentiation which I am going to describe now which allows us to find an approximation to the derivative f dash at a or the second derivative or higher order derivative. These formulae they are important in solution of differential equations. So, approximate solution of differential equations when one considers that is where we are going to use this formulae. As such as I said there is going to be some difficulty in the numerical differentiation rule. Suppose you want to find f dash of a then what should be done? So, instead of interpolating polynomial what one should do is look at some different polynomial approximation. Say look at least square approximation by polynomials. So, that part we have not considered so far, but we have considered cubic spline interpolation. So, instead of interpolating polynomial look at cubic spline interpolation of your function. So, in the cubic spline interpolation what we did was we looked at interval a b subdivided into equal sub intervals and then we tried to fit a piece wise cubic function. So, on each sub interval our function was a cubic polynomial or a polynomial of degree less than or equal to 3. And at the partition points because now we have got look at the two sub intervals. So, on one interval you have got a polynomial of degree less than or equal to 3 on another it is another polynomial of degree less than or equal to 3. So, we want that both of them they should join at in such a manner that overall we have got two times differentiable function. So, that gave rise to a tridiagonal system to solve and then we could obtain a cubic piece wise cubic polynomial which is overall two times differentiable which interpolates the given function at the partition points and in addition we had two end points. So, this cubic spline interpolation it gives acceptable approximation to the derivative of function. But today what we are going to do is we are going to look at our interpolating polynomial of appropriate order and get the derivatives and see what is the difficulty one faces in case of numerical differentiation. We are going to slightly make change notation the only change is our function earlier it was defined on interval a b. So, we will say that it is defined on interval c d and a will be interior point of our interval. So, our m is to find approximation to f dash of a. So, f is from c d to r real valued function a is interior point m is to approximate f dash of a. So, approximate f by an interpolating polynomial p n of degree less than or equal to n and f dash a is approximately equal to p n dash a. The simplest case is going to be take n is equal to 0 that means approximate your function by a constant polynomial. So, the derivative is going to be 0. So, then you get an approximation that is not really a very accurate approximation. So, instead of n is equal to 0 let us look at n is equal to 1. That means x 0 and x 1 we are going to choose two interpolation points fit the polynomial of degree less than or equal to 1 and then its derivative will give you approximation to f dash of a. And the starting point is going to be function f x is equal to interpolating polynomial plus an error take the derivative the derivative of the polynomial will give you approximation to f dash of a. And this approximation will depend which interpolation points you are going to choose. So, our interpolation points x 0 and x 1 they will give rise to two formally which we are going to consider. So, in one case we are trying to approximate f dash of a. So, x 0 and x 1 the points which you are taking as interpolation points it is logical that you should choose them in the vicinity of point a. So, suppose I choose x 0 is equal to a and x 1 to be equal to a plus h where h is going to be a small number. So, that will give us a formula which is known as forward difference formula. And another formula which we are going to consider there instead of choosing the points to be a and a plus h we will try to choose our points symmetrically. So, a is the point at which we want to find the derivative. So, we can choose point to be say a minus h and a plus h. And then we will see which one gives us a better error estimate x 0 and x 1 are distinct points in c d f x is equal to f x 0 plus divided difference based on x 0 x 1 into x minus x 0. And then this is the error term the divided difference based on x 0 x 1 x multiplied by w x w x is x minus x 0 x minus x 1 take derivative of both the sides. So, you will have f dash x is equal to p 1 dash x plus derivative of this error term p 1 dash x will be nothing but divided difference based on x 0 x 1 plus you apply product rule. So, you get d by d x is equal to f of x 0 x 1 x multiplied by w x plus the divided difference into w dash x the divided difference f x 0 x 1 is going to provide approximation to f dash of x and this is a error term. So, in the error term derivative of the divided difference x 0 x 1 x is appearing. We have already seen continuity of divided differences if your function f is once differentiable then the divided difference based on x 0 x 1 x this is a continuous function you are assuming x 0 and x 1 to be distinct point. So, this part we have already seen now we are going to show that if f is twice differentiable then the derivative of the divided difference based on x 0 x 1 x is going to be a divided difference based on points x 0 x 1 x x. So, this result now we will first prove so that we can estimate our error in numerical differentiation. So, first we look at divided difference based on x 0 and x x 0 is going to be a fixed point. So, this g I define g of x to be equal to divided difference based on x 0 and x which is going to be equal to f x minus f f x 0 divided by x minus x 0 if x not equal to x 0 and equal to f dash x 0 if x is equal to x 0. So, this is my function g and I want to find g dash x there will be two cases x not equal to x 0 and x is equal to x 0 when x is not equal to x 0 we have to we have a quotient of two functions. So, we will use standard formula for calculating the derivative when our point x is equal to x 0 then our function g is defined differently at x 0 it is f dash x 0 at x not equal to x 0 it is the quotient f x minus f x 0 divided by x minus x 0. So, when I want to calculate the derivative of g at x 0 I will have to go from the definition I will have to proceed from the definition that g dash x 0 will be limit as h tends to 0 of g of x 0 plus h minus g x 0 divided by h. So, we will consider these two special separate cases and show that the derivative of divided difference based on x 0 x is nothing but the divided difference based on x 0 x x. So, first the case x not equal to x 0 g dash x will be x minus x 0 square then denominator multiplied by derivative of the numerator x 0 is fixed. So, it is f dash x minus numerator f x 0 and then the derivative of the denominator that is going to be 1. Now, 1 x minus x 0 we cancel.  So, f x minus f x 0 divided by x minus x 0 that gives us divided difference of f based on x 0 x. Now, this is nothing but the divided difference based on x 0 x x that is the recurrence formula. This is nothing but divided difference of f based on x comma x. So, that is f dash x minus divided difference based on these two points and divided by x minus x 0. So, we have proved the claim for x not equal to x 0. Now, look at x is equal to x 0 and apply the definition. So, consider g of x 0 plus h minus g x 0 divided by h this is going to be equal to by our definition of g x it will be f of x 0 x 0 plus h minus g of x 0 is f dash x 0 divided by h this quotient is nothing but f of x 0 plus h minus f x 0 minus h times f dash x 0 and then divided by h square writing for f of x 0 x 0 plus h f of x 0 plus h minus f x 0 divided by h. We are assuming function to be two times differentiable. So, for the numerator we can apply extended mean value theorem. So, that extended mean value theorem gives us numerator to be equal to h square divided by 2 f double dash c and then this h square where t is going to lie between x 0 and x 0 plus h our h can be bigger than 0 or less than 0. So, our c will be in the interval x 0 to x 0 plus h if h is bigger than 0 or it will be in the interval x 0 plus h x 0 if h is less than 0. Now, we are going to let h tend to 0. So, when h tends to 0 c which lies between x 0 and x 0 plus h that will tend to x 0 and using that we get the f double dash c by 2 that will be tending to f double dash x 0 plus h by 2 which by definition of divided difference is f divided difference of f based on x 0 x 0 x 0 x 0 repeated thrice and thus g dash x 0 is f of x 0 x 0 x 0. We had already proved that g dash of x is a divided difference of f based on x 0 x x and now that formula is valid for x is equal to x 0. Now, what we wanted was the derivative of divided difference based on x 0 x 1 x because that is what comes into picture in the linear approximation. We have proved that the divided difference of f x 0 x its derivative is nothing but add 1 x x star. So, now, let us look at the divided difference based on x 0 x 1 x and try to find its derivative. So, x 0 and x 1 these are distinct points. You look at f of x 0 x 1 x our divided difference is symmetric. About its arguments that means the order of x 0 x 1 x will not matter. So, the divided difference based on x 0 x 1 x will be same as divided difference based on x 0 x x 1. So, now our g x is divided difference based on x 0 x x 1 by the current formula this is f of x x 1 minus f of x 0 x divided by x 1 minus x 0 and now we want to look at the derivative of this denominator is constant. Derivative of f x x 1 will be nothing but add 1 x x star. So, it is going to be f x 1 comma x comma x minus divided difference based on x 0 x 1 divided difference derivative of f of x 0 x that will be f x 0 x x and then divided by x 1 minus x 0. So, again by the recurrence formula we get g dash x to be the divided difference based on x 0 x 1 x x. So, that proves our claim and now we go back to our formula for finding approximate value of f dash of a. So, our function f is defined on interval c d takes real values a is an interior point x 0 and x 1 are two distinct points in the interval c d we are fitting a polynomial of degree 1 and then we have got error term you differentiate both the side. So, f x is equal to f x 0 plus f x 0 x 1 x minus x 0 and this is the error term. So, the error term has this divided difference and our function w x take the derivative of both the sides f x 0 is constant. So, f dash x will be equal to divided difference x 0 x 1 plus now the derivative of this we know that it is nothing but add 1 x star x. So, it will be f x 0 x 1 x x multiplied by w x we are applying product rule plus f of x 0 x 1 x into w dash x w dash x will be nothing but x minus x 0 plus x minus x 1. So, now we want to choose our x 0 and x 1. So, we are going to choose them in the vicinity of our point a. So, the first case is choose x 0 to be equal to a and x 1 is equal to a plus h. So, when you do that our w x is x minus x 0 x minus x 1 x is equal to x 0 is equal to a. So, w a is going to be equal to 0 and w dash a will be this will be 0 this will be a minus a minus h. So, it is minus h now f dash a is approximately equal to divided difference based on a and a plus h which is f of a plus h minus f a by h and the error is going to be equal to w at a is 0. So, there will be no contribution from this term that will be only from here. So, it will be f of a a plus h a into minus h now function f because it is 2 times differentiable it will be minus h f double dash c divided by 2. So, in fact this c it should be something say f double dash of psi this c and this c they are not the same it is minus h f double dash of psi divided by 2. So, this is discretization error and this is known as forward difference formula. So, now let us look at the case when you choose your 2 interpolation points symmetrically. So, our point will be a minus h and a plus h and we will see that in this case the discretization error is going to be of the order of h square so for forward difference we had only the error to be less than or equal to constant times h it will be now less than or equal to constant times h square. I recall that f dash x is the divided difference based on x 0 x 1 plus the term containing divided difference based on x 0 x 1 x x into w x plus divided difference based on x 0 x 1 x and w dash x w x is product of x minus x 0 x minus x 1 the derivative of w dash is x minus x 0 plus x minus x 1. If we choose x to be equal to a x 0 to be equal to a minus h x 1 it is to be equal to a plus h then w at a will not be 0 it will be equal to minus h square, but now w dash a is equal to 0 that gives us f dash a to be approximately equal to f of x 0 x 1 plus x 1 plus x 1 plus x 1 that is f of a plus h minus f of a minus h divided by 2 h and in the error w dash a is 0 so this term will not be there.  So, you have minus h square f 3 c third derivative again evaluated at some point divided by 6. Now, when one looks at these formulae which we obtained they are something expected like look at the forward difference formula.  Now, it is expected that from the definition limit of f of a plus h minus f a divided by h as h tends to 0 is equal to f dash a. So, may be I do not have to do all these interpolating polynomials and then obtain this formula. This formula is available from the definition. Similarly, the other formula the other formula was f of a plus h minus f of a minus h divided by 2 h.  So, this will tend to f dash a this will tend to f dash a and hence this will tend to f dash a because you are dividing by 2. So, both these formulae central difference formulae and forward difference formulae we could have written from the definition. The reason I went through this interpolating polynomial is that gives us a general method that now the same idea we are going to use for calculating the second derivative and we had idea about the discretization error like no doubt these quotients or these approximations they approximate to f dash a. So, for h small enough you are going to get a approximation, but we could tell that central difference formula in which case the interpolation points are symmetrically placed that formula is to be preferred because the discretization error in that case is h square as compared to the discretization error h in case of forward difference formulae. So, now let us look at the approximation and approximation of second derivative. Now, you are going to fit if you fit a polynomial of degree less than or equal to 1 and take it second derivative then the second derivative is going to be 0 and it is a crude approximation f double dash a. So, instead of a polynomial of degree less than or equal to 1 we should at least look at polynomial of degree less than or equal to 2. So, we will consider three distinct points x 0, x 1, x 2 fit a parabola take it second derivative and that will approximate f double dash of a. In this case again the points x 0, x 1, x 2 which we are going to choose we will choose them in the vicinity. So, one choice can be x 0 is equal to a x 1 is equal to a plus h x 2 is equal to a plus 2 a. So, that will give us forward difference formulae the other will be you place them symmetrically. Now, in this case you can choose a your points to be x 0 is a minus h x 1 is equal to a and x 2 is equal to a plus h. So, that will give you central difference formulae. Now, here when we try to look at the error is going to have a divided difference term which is based on x 0, x 1, x 2, x. Now, we are going to take two derivatives. So, the first derivative the proof is similar what we have proved is f of x 0, x the divided difference its derivative is nothing but a divided difference based on x 0, x x. If you consider the divided difference f of x 0, x 1, x its derivative is divided difference based on x 0, x 1, x x you have to just add 1 x. Similarly, divided difference of f based on x 0, x 1, x x 0, x 1, x 2, x 2, x 2 x, its derivative will be obtained by adding one more extra. Now, we are going to take the second derivative. So, we will need to take the derivative of divided difference based on x 0, x 1, x 2, x repeated twice. Its derivative will be the divided difference x 0, x 1, x 2, x repeated thrice. The proof is similar and what we are going to do is we will do it as a tutorial problem. So, at present assume that the derivative of f x 0, x 1, x 2, x it will be divided difference based on x 0, x 1, x 2, x x and it is the derivative of this divided difference based on x 0, x 1, x 2, x comma x that will be 2 times divided difference based on x 0, x 1, x 2, x repeated thrice. So, this derivative it is not only just add one x extra, but there is also term 2 coming into picture. Now, we are choosing three distinct points and the interval a, b should be interval c d, then this is the quadratic polynomial in the which is written using divided differences. This part is the error, take the first derivative. So, f dash x will be equal to p 2 dash x plus we need to take the derivative of this, use the product rule. So, first the derivative of this term. So, it is going to be x 0, x 1, x 2, x x, w x plus this term as it is multiplied by w dash x. So, that is our first derivative. The second derivative will be f double dash x is equal to p 2 double dash x plus derivative of this is going to be 2 times divided difference based on x 0, x 1, x 2, x repeated thrice multiplied by w x as it is. Then this multiplied by w dash x, derivative of this multiplied by w dash x. So, that together gives us 2 times divided difference based on x 0, x 1, x 2, x x into w dash x and lastly this divided difference multiplied by w double dash x. So, f double dash x is approximately equal to p 2 double dash x and this is going to be the error term. W dash x will be given by three terms x minus x 0, x minus x 1 plus x minus x 0, x minus x 2 plus x minus x 1, x minus x 2 and the second derivative is given by this formula. If I choose x 0 to be equal to a, then w at a is going to be 0. Now, if I choose x 1 is equal to a plus h, x 2 is equal to a plus 2 h, then w dash a will be minus 2 h square and w double dash a will be minus 6 h. If the x 1 and x 2 are chosen symmetrically, then w at a is already 0, w dash a will be equal to minus h square and w double dash a is going to be equal to 0. This is because w double dash x is given by this formula x minus x 0 is already 0, x minus x 1 will be minus h, x minus x 2 will be plus h. So, that is why w double dash is a is 0. This choice is going to give us forward difference formula and this is going to give us central difference formula. So, in the first case when we are choosing our points to be a, plus h, a plus 2 h, f double dash a is approximately equal to f a minus 2 times f of a plus h plus f of a plus 2 h divided by h square. The 2 gets cancelled with the 2 in the denominator for the divided difference and now for the error term, we have only w at a is equal to 0. So, we have this divided difference multiplied by 2 h square plus this divided difference multiplied by minus 6 h. So, you have f 4 c by 6 h square minus f triple dash d by into h and if you have chosen them to be symmetrically, then we had w at a to be 0, w double dash a to be 0. So, there is only one term and then you have got f 4 c divided by 12 into h square. So, when we look at the discretization error, if the points are a, a plus h, a plus 2 h, then the discretization error is less than or equal to constant times h because we have got 2 terms. One term contains h square, one term contains h, but then the term which contains h that is going to decide the rate. So, you have got this to be less than or equal to constant times h. If your points are symmetric, then the discretization error is less than or equal to constant times h square. So, we have got formally for the first derivative for the second derivative. This when we have fitted a polynomial of degree less than or equal to 2, that it can give us a formula for f dash a also, but what I wanted to do was illustrate that using interpolation points, we can obtain approximations to first derivative, second derivative and so on. And as I mentioned before, these approximations are going to be important in the numerical solution of differential equation. Now, look at our problem f dash a. I want to find f dash of a.  Both these quotients, they converge to f dash a as h tends to 0. So, what I have to do is choose my h to be small enough in order to have desired accuracy, because we have convergence. Both these divided differences, they tend to f dash a as h tends to 0. So, choose h small enough and then you will get the desired accuracy. Now, when you try to do it in practice, you face certain difficulty. Now, what are the difficulties?  You are going to make h small.  They are going to be near each other. So, you are going to subtract two numbers which are approximately equal. Now, as we have noticed that when you divide, when you subtract two numbers using computer, if the two numbers are approximately the same, then you will get a loss. Then there is loss of accuracy or there is loss of significant digits and then you are going to divide by h. So, you are dividing by a small number. So, this two facts they combine to make our divided difference approximation not as good as the average as we will like it to be. So, here for the divided difference approximation of f dash a, the two difficulties are you are dividing going to divide by a small number and you are going to subtract two numbers which are approximately the same. So, we have got f dash of a is equal to f of a plus h minus f of a minus h by 2 h and this was the discretization error.  So, instead of computing this quotient, what you will be computing will be f of a plus h plus e 1 minus f of a minus h plus e 2 divided by 2 h.    So, this term does not decrease, this is going into 0. So, this is the difficulty one faces. So, when you reduce h up to a certain stage, you will get better and better approximation, but after a certain stage, even if you reduce your h, instead of getting a better approximation, you will get worse approximation. So, in the next lecture, I will once again compare this phenomena with the phenomena in the composite numerical integration and then we will be considering solution of linear equations. So, thank you.