Introduction to Calculus (2 of 2: First Principles)

by: Eddie Woo

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[0.06]
what's the gradient at that point because there are no two points to compare by definition so here's what they both do yes an Imagi so we can picture a tangent along here in fact that's where our language receives its first cultural now it's hard to do the maths with this thing because you just got that one point ah but if I can get towards this tangent and think of it in terms of something else that I can calculate something which does have two points then I can work with this for example if I put in a whole bunch of parallel lines here okay all of these guys these 1 2 3 4 that I've just drawn what they call they intersect the circle twice I think they're called cygnets these guys right these guys do have two points so I'm just going to think of the circle because that's where your secret in tangent language comes from and then we'll come over to a graph in a second can you see the length of the cord that's cut off by this secant right the length of this coin as I get closer and closer to becoming a tangent as I move this way right the lake of that cord is getting smaller your green right in fact eventually once you get to the tangent the lake of that cord the length of that cord is 0 is it not it's it's not actually a cord it's just a single point that's why it's actually Europe right so what Newton wanted to do was take this problem apart by thing here as one of these prongs he thought what if what if I could I can understand the behavior of this secant what if I could have imagined it as it approaches a chord of length 0 if it approaches a chord of length of 0 so both of these guys both of these guys said I need some language I need some language to talk about things that approach something which you know I can't actually get to but I can still do the calculations on it so they said let's talk about limits they introduced this notation something's getting closer and closer and closer until you actually there right you make conclusions also offer things that you know so you can draw something make that something you don't know okay so now here's where the rubber hits the road draw for me a new one you said axes let's just go first quadrant it's all you need drop me a curve drop me okay and let's call this some function of X some function of X now if what I'm after is the gradient of a tangent gradient of tangent okay what I'm going to start thinking about is the gradient of a secant and think about what happens to that as it gets closer and closer to becoming a tangent right so in order to get a second that's not hard you just pick any two points you like like here and here okay and can you sing like if the actual point I'm after is like somewhere in the middle here okay if you were to draw that tangent it's actually not that far off in terms of its gradient compared to this secret right like I'm in the right ballpark yeah so I've got these two points got these two points let's get some values on this alright if I imagine this is some x-coordinate who cares what it is I'll just call it X I want it to be anywhere that I like or they like input some value there okay and what I'll do is I'll think about going from that point and going a bit further right going like a distance of say they call it H okay you can think that H is kind of like for height but it's not that's not the best description because it's further anyway H is the convention there are other conventions as well sometimes you'll see it called Delta X we'll talk about why in a second now I hit that little distance there it's just a small distance is H then what's the coordinate of this point over here or I should say the exponent X percentage right it's just that Plus that that gets you over there okay so now if I want to work out the gradient of this secant here right the Green in the secret I've got x coordinates I'm going to need Y coordinates wanna come yeah and she okay now I don't know what this function is it could be anything it's f of X right so if this x coordinate is X just X right what am I going to be up here f of X good so for example if this were if this were x squared right y equals x squared then that would be X and this would be x squared right this is X plus h so my corresponding Y value over here will be f of X plus h whatever F happens to be okay so now what am I got here what have I got I have got what I working out is the gradient of the secret the greening of this guy here okay and it's literally just right everyone it's just lies over one okay so it's my rise okay it's the distance between these two points these two Y values is it not okay what is that distance I think that's good this take away this yeah that's all that's all that vertical distances okay so rise I've got that okay now run is the horizontal distance that I'm going there corresponds to that which of course I've defined as just H it's just H so if I have some value of x whatever you like okay and I have some function you define it to me whatever you like okay this will give me the gray or a secant that's anywhere you want okay but I don't want a secant really what I really want is that a single point where this distance in here I don't want it to be this big gaping gap in here okay I wanted to close in I wanted to come together just like these guys are coming together right so what I want is this guy here H I wanted to get very very very small does that make sense now I already have language for this right when it becomes zero I'm going to get not a secret not a secret but a tangent degree with that right that's what I want what I want is for H to be zero and H can't really be zero because look I have to divide by H I can't really be zero but I can think about what happens as I get closer to zero I just have to say tell me what the limit is right tell me what the limit is just like we looked at before if I gave you a trivial example


[415.69]
okay we looked at on we looked at evaluating limits like that okay now that function there that I've put over here it's got a hole at five doesn't it's like that you can't equal five because it's on the denominator so x equals 5 is not going to work okay but all I need to do is I just have to say well that top thing what can you do that topic I can factorize it it's the difference of squares so it's going to be X plus 5 X minus 5 and because of that a denominator goes no problem okay so at that point I could say oh that thing that thing that approaches 5 and I can actually put in 5 so the course is just going to be 10 right so even though this original thing I can't put five in I can see what happens as it approaches five it's going to go there okay from above and below you can go ahead and try it out so here I can't put in zero but I can think about what happens as I get ever closer to it okay so this thing here this guy is a big deal okay this was likenesses and Newton's big watershed moment they're like I can't I can't work out by just putting in 0 right at least into this which is what it is my definition rise over run but I can still do maths with it I can still do maths with it even if I can't actually put 0 in this is what we call the first principles of calculus the first principle what are we actually working out we're working out the gradient of the tangent okay but this started to get a bit awkward in terms of like talking about all this stuff so they introduced new language and they introduced a new notation okay so the new language they introduces they call this thing right like I could get it at a particular point but I get a particular point but I could get it at any point I like right and it's going to change all the way so run in this thing which has a gradient okay this thing over here is going to be a function changes right so it's actually called their gradient function because the gradient is going to take on different values depending on where you look depending on what value of x you look at okay now in the same way he said look this rise over run business rise over run it's not working because there's not really a run it's not really a roam okay so they said look this is just the change in Y divided by the change in X that's that's all it's really happy right so you guys know the symbol in science right for changes in Delta now use it and they say the change in Y that's rise right and you compare you gets ratio with its change in X that's the run except one is an inadequate work because there's not really any run happening here right so dy change in y over DX change in X it means rise over run but in that case where the rise of run are both tending towards 0 because looking at this infinitesimally small squat the rise is going to be zero the ones game



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