 Let's do the fundamental theorem of calculus on this X plus 2 let's write that down X plus 2 now fundamental theorem calculus says this right what you're doing is I'm going to give you a quick little intro on it We have a video on there if you do calculus into an intro fundamental theorem and whatnot It'll pop up because we did a live stream on this right so basically you're doing this if you have a function, right? What you want to do is you want to Let's say this is your function Let's say that's your function. You want to find the slope of a certain point, right? The rate of change at that point. So what you do is you Zoom in narrow down on that point So you first you find the slope between two two points and then you come closer closer closer closer you do this Right and the fundamental theorem of calculus says this if you have a function f of x and You want to take the limit? Let me find the proper terminology. I'm gonna write it down to see if I have any mistakes on it so you want to find the limit of f of x as X approaches a certain number h let's say I think it's h then what you do is you go f of x plus h oops h minus f of x over h right I think that's what it is Iceland snow storm here in southern LP of Minnesota But like yes Yes, sir. I came here on vacation with my wife and realized wow we have to live here I was originally born in the southern part of Italy. So I have the beach in my blood. Ah nice Went on vacation and decided to stay that's cool Or you must have come back and organized things and then came back again, right? Given the all liver break Yes, thanks. This is what I was referencing. Yeah, okay. Awesome. Marco. We do Think it should be limit as h approaches zero. Oh, that's h approaches zero. That's right. Thanks. Nice power My bad. So this is limit as h approaches zero and we're going to put down is equal to limit as h approaches zero And this is the derivative f of x limit. Do I have that right you put the limit on this side? Seriously We do this right f of x prime the derivative is limit as h approaches zero, right? Petra, how are you doing? Hello. Hello chicho. I think some of your neighbors might be vampires How come why? I love your type of stream. Awesome. Awesome. Petra. Love doing it, right? So take a look at this thing. What's the restriction we have in mathematics? What was the restriction we had in mathematics? The restriction we had in mathematics is we can't divide by zero So if h approaches zero if we set h is equal to zero We can't do that because then we're dividing by zero. Oh, how do we deal with this? Well We substitute in what we have here right Yes, yes, I did return. I was originally a police detective in italy So it took some time to get off the grid back on. Oh, man a great decision getting out of the Policing to go into jamaica live off the land It was so much better. I'm italian, but I want to hear you So take a look at this thing and welcome italy by the way salutations So let's do this when you're doing this Do this right and this basically means and by the way gang. Thank you for the follows. Thank you for the subs Appreciate them and the bits and the donations and the tokens and all that jazz, right? So f of x Plus h means just substitute x plus h For here, right? So what we're gonna do is hopefully I have enough room because this gets messy. Okay Whoa guy, how are you doing chicho? Hey Cholo, I didn't know you do math homies. Of course. What are they? Carnal On a graph if you are on the x-axis x is zero, right on the graph if you're on the x-axis No, y is zero on a graph if you're on the x-axis y is zero And if you're on y y is zero. No, no, no the other way around if you're on a On a graph x and f of x or if you want to call this y Right, if you're here This is y is equal to zero. That's y is one y is two y is negative one y is negative two. So y is zero and focus focus focus Right and the other way around Right. It's a common mistake. By the way, it's a common mistake It just Feels like you want to say x is zero. What's not it's y is zero, right? So substitute this in for x, right? So f of i'm just going to do it here And then we're going to erase it. So we're basically we're doing f of x plus h well f of x plus h just means substitute this in for x So that becomes one over x plus h plus two You're okay with that This is some breaking bad man Reminds me of eisenberg I don't make the blue stuff though Cheecho honestly, I highly Recommend this to anybody despite of their social status or age I have never been this happy in my life I've had so much more time to focus on my relationship with my partner and certain parts of me as an exam As an example my skin have become so clear and how oh brother. I doubt it not I seriously I believe you I believe you Everybody has to get out of the rat race and one of the ways you get out of a rat race is learn mathematics, right? Like the number line indeed Right, so let's plug that in. So that becomes f of x plus h is one over Let me write this down. Oh, we're gonna do the limit. I gotta write down the limit too, right? We can't forget this part limit as h approaches zero of one over x plus h plus two minus one over and that's just f of x x plus two all over H right What do we do now? We can't we still can't set h is equal to zero because we get a division by zero Well crunch crunch crunch add these two fractions. How do we add these two fractions? Well We've got to find a common denominator common denominator is x plus h plus two times x plus two, right? So what I'm going to do is I'm going to write this limit as h approaches zero I'm going to do a division over here divided by H right Because I'm going to do this part So common denominator For adding these two guys and it's just adding fractions, right? is x plus h plus two times x plus two Well, what did you multiply this by? To give you this when you multiply by x plus two So you multiply top by x plus two. So that just becomes one times x plus two, which is x plus two minus What did you multiply? x plus two by to give you this you multiply by x plus h plus two right So you go minus one times that x plus h plus two Right But yeah, it's nice. Nobody bothers me. My bills are dirt cheap. I have garden Everything is local. I buy my meat at a local farm and I can see the animals and food I am eating when they are alive. I've never used a barbecue so much nice nice Von dutchy been four years since I completed my math degree and I can maybe remember 30 percent of Brother Me too. I had to go back and relearn all this, right? This is the same type of stuff. We did a while back. Yeah So expand this well, this is just x plus two minus Minus in front of bracket means you subtract all these right all the sides change minus x minus h minus two So we just instead of rewriting it i'm doing this right now. We're going to simplify this Okay If we simplify this This becomes x minus x will that kill each other two minus two they kill each other so in top limit As h approaches zero At the top we got negative h over x plus h plus two times x plus two Divided by h means times one over h Right, that's what it means So chicho, what can you do with these math? I feel like this would be helpful for one in the for sure in business and engineering and philosophy and sociology and data analysis and Like everywhere really How I do I use calculus in my daily life? No, but I do look at The rate of change. I do look at graphs to see To get a visual of how function behaves right over time And then you can do the calculus i.e Doing a calculus means looking into the future or looking into the past Looking at the rate of change to see how a system behaves. So it's really analyzing systems, right? Just doing these sort of problems elevates your mind. It shows your ability to think Yeah, and it's problem solving it. It's critical thought is and you're you're able to If you know mathematics really here's one thing I tell everybody When you know math Like it can't do harm first of all if you know something you learn mathematics It's not going to do damage like it's one of the few things when you acquire knowledge over a certain thing It's not going to or tool it could only benefit you, right? But one thing mathematics does it allows you To see bs coming a mile away, right when people start throwing numbers at you and stuff like this You can you can do the calculus you can do the mathematics and go wait a second those numbers don't make sense, right? So it's a great bs detector Right, so it prevents you from falling for propaganda very important Cool. What is your current occupation my friend? You seem like an interest. I I do I create content online and I've been teaching mathematics for like 20 years Privately I can't function in an institution. I figured that one out a long time ago, right? I would get fired or I would I would not be happy. So I sort of do What you did I took myself off the grid, but I like I don't mind living I love living here actually top living in a rainforest for me is beautiful, right? I love the greenery Right All computers use a system of open close zero and one to make a graph computers Yeah, until we get to out the quantum computing, right big o notation Is the efficiency of the graph made by the data? So take a look at this thing. We're multiplying fractions Number one rule for multiplying fractions reduce before you multiply. So take a look at this thing I'm a little tangent, right? What if we had this 25 over 15 times 28 over 36 times 18 over 14 times I don't know 9 over 15, right? How do you simplify this? How do you multiply this? Well, if you're multiplying fractions, I've had people do this. I go, okay multiply these guys multiply multiply multiply What is this equal to? I've had students take a calculator when they first start working with and they start punching in numbers And I let them do this by the way. They punch in numbers this times this times this times this and they get one gigantic number Right, they got a gigantic number And then they go this times this times this times this and they get another gigantic number And then they go, okay, that's the answer. They tell me I go, well, you can't leave it like that You got to simplify reduce it and then they try to reduce This fraction to a smaller form, right? And then let them do this and they do this. I go, okay Uh, do you think there's an easier way to do this? They go, like, well, I just use a calculator. I just do this. I go, well, here's another easier way Simplify before you multiply My government failed calculus. I'm sure that'll help Simplify before you multiply anything from the top can kill anything from the bottom, right? As long as there's no plus and minus between them 5 goes into 25 5 times 5 goes into 15 3 times 3 goes into 9 3 times 3 goes into 15 5 times 5 kills 5 14 goes into 28 twice 6 goes into 36 6 times 6 goes into 18 3 times 2 goes into 6 3 times 3 kills 3. Oh the answer is 1 1 That's cool answers 1 Isn't that a lot easier than multiplying the numbers and then trying to figure out what they are afterwards You have to simplify the fraction first, right? So Keep that in mind All right. Well, we got negative h over this times 1 over h H kills h. So we got negative 1 up top We got rid of an h What can we do now? Well, what we can do is Multiply this out Right Yes, chicho, but now solve for the napkin ring paradox Now what you can do is FOIL this out or you got limit as h approaches 0 right of negative 1 over x plus h plus 2 times x plus 2 well Now it's just you can just sub in h is equal to zero Because you you're not going to get a division by zero So if you sub in h is equal to zero, this is what you get negative 1 over x plus 2 times x plus 2 Which is really equal to negative 1 over x squared plus 4x plus 4 so as for this function as f of f of x as h approach for the limit of h approaches zero of this thing, that's your function your equation of the tangent For this function, right? Does that make sense? So if you want to find out what The tangent line is for any point of x you just plug it in So here Let's find out what f of zero is so f of zero Is going to be 1 over 2 because you sub in x is equal to zero you got 1 over 2 What's the slope of the graph? at x is equal to zero Well, you just plug in because what we have is our function here now is f prime of x The derivative the derivative of this function is negative 1 over x squared plus 4x plus 4 Right So f prime of zero the slope Of this function at x is equal to zero you just plug in zero here. So it becomes One negative 1 over zero squared plus four times zero plus four is just negative 1 over 4 negative 1 over 4 That's the slope Of the function at x is equal to zero Right and see you'll be ready. Also make it ring o in big o notation This is that okay. I hope that answered your question Marco