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Current time:0:00Total duration:9:07

AP.CALC:

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this is a picture of Isaac Newton super famous British mathematician and physicist this is a picture of Gottfried Leibniz super famous or maybe not as famous but maybe should be famous german philosopher and mathematician he was a contemporary of Isaac Newton these two gentlemen together were really the founding fathers of calculus and they did some of their most of their major work in the in the late 1600s and this right over here is a sane bolt and sprinter who's continuing to do some of his best work in 2012 and as of early 2012 he's the fastest human alive and he's probably the fastest human that has ever lived and you might not you might have not made the association with these three gentlemen you might not think that they have a lot in common but they were all obsessed with the same fundamental question and this is the same fundamental question that differential calculus addresses and the question is what is the instantaneous rate of change of something and in the case of Oh same old half-assed is he going right now not just what his average speed was well for the last second or his average speed over the next 10 seconds how fast is he going right now and so this is what differential calculus is all about instantaneous rates of change differential calculus calculus it's all canned i Newton's actual original term for differential calculus was the method of flux eons which actually sounds a little bit fancier but it's all about what's happening in this instant in this instant and to think about why that is not a super easy problem to address with traditional algebra let's draw a little graph here so on this axis on this axis I'll have distance so and I'll say Y is equal to distance I could have said D is equal to distance but we'll see especially in later on in calculus D is reserved for something else we'll say Y is equal to distance and in this axis we'll say time and I could say T is equal to time but I'll just say X is equal to time X is equal to time and so if we were to plot hussein bolts and distance as a function of time what times zero he hasn't gone anywhere he is right over there and we know that this gentleman is capable of traveling a hundred meters in 9.5 eight seconds so after nine point five eight seconds were assumed that this is in seconds right over here he's capable of going 100 meters 100 meters and so using this information we can actually figure out his average speed his average let me write it this way his average speed is just going to beat his change in distance his change in distance over his change in time over change in time and using the variables over here we're saying why is distance so this is the same thing as change in Y over change in X from this point to that point and this might look somewhat familiar to you from basic algebra this is the slope between these two points if I have a line that connects these two points if we if I have a line that connects these two points this is the slope of that line the change in distance is this right over here change in Y is equal to 100 meters and our change in time is this right over here so our change in time is equal to nine point five eight seconds we started zero we go to nine point five eight seconds another way to think about it the rise over the run you might have heard in your algebra class it's going to be a hundred meters over nine point five eight seconds so this is 100 meters 100 meters over nine point five eight seconds and the slope is essentially just the rate of change or you could view it as the average rate of change between these two points and you'll see if you even just follow the unit's it gives you units of speed here it would be velocity if we also specified the direction and we can figure out what that is let me get a calculator out let me so let me get T I'll get the calculator on the screen so we're going 100 meters in nine point five eight seconds nine point five eight seconds so it's ten point four I'll just write ten point four our round to ten point four so it's approximately ten point four and then the units are meters per second meters per second and that is his average speed and and we're gonna see the second is the average how a ver egde speed is different than instantaneous speed how it's different than what the speed he might be going at any given moment and just to have a concept of how fast this is let me get the calculator back this is in meters per second if you wanted to know how many meters he's going in an hour well there's 3,600 hours in a in a in a 3,600 seconds in an hour so he'll be able to go this many meters 3,600 times so that's how many meters he can if you were able to somehow keep up that speed in an hour this is how fast he's going meters per hour and then if you were to say how many miles per hour there's roughly 1600 and I don't know the exact number but roughly 1600 meters per mile so let's divided by 1600 and so you see that this is roughly a little over 23 about 23 and a half miles per hour so this is approximately I'll write it this way this is approximately this is approximately 23 point five miles miles let me scroll over so I don't have to bunch that miles per hour and relative to a car not so fast but relative to me extremely fast now to see how this is different than instantaneous velocity let's think about a potential plot of his distance relative to time he's not going to just go this speed immediately he's not just gonna go as soon as the gun fires he's not just gonna go 23 and a half miles per hour all the way he's going to have to accelerate so at first he's gonna start off going a little bit slower so his slope is going to be a little bit lower than the average slope he's gonna go a little bit slower then he's going to start accelerating and so his speed and he'll see the slope here is getting steeper and steeper and steeper and then maybe near the end may be near the end he starts tiring off a little bit and so his distance plotted against time might be a curve that looks something like this and what we calculated here is just the average slope across this this change in time but we could see it at any given moment the slope is actually different in the beginning he has a slower rate of change of distance then over here then he accelerates over here it seems like his rate of change of distance which would be roughly or you could view it as the slope of the tangent line at that point it looks higher than his average and then he starts to slow down when you all averaged out at Gustav twenty-three and a half miles per hour and I looked it up who say in volts instantaneous velocity his peak instantaneous velocity is actually closer to thirty miles per hour so the slope over here might be twenty-three whatever miles per hour but the instantaneous his fastest point in this nine point five eight seconds is closer to thirty miles per hour but you see it's not a trivial thing to do you could say okay let me try to approximate the slope right over here and you could do that by saying okay well what is the change in Y over the change of X right around this so you could say you could say well let me take some change of X let me take some change of X and figure out what the change of Y is around it or as we as we go past that so you get that but that would just be an approximation because you see that the slope of this curve is constantly changing so what you want to do is see what happens as your change of X gets smaller and smaller and smaller as your change of X gets smaller and smaller and smaller you're gonna get a better and better approximation your change of Y is going to get smaller and smaller and smaller so what you want to do and we're going to go into depth into all of this and study it more rigorously is you want to take the limit as Delta X approaches 0 as Delta X approaches 0 of your change in Y over over your change in X and when you do that when you do that you are going to approach that instantaneous rate of change you could view it as the instantaneous slope at that point in the curve or the slope of the tangent line at that point the curve or if we use calculus terminology we would view that as the derivative so the instantaneous slope is the derivative and the notation we use for the derivative is dy over DX and that's why I reserved the letter Y and this you say well how does this relate to the word differential well the word differential is relating it's this dy is a differential DX is a differential and one way to conceptualize it this is an infinitely small change in Y over an infinitely small change in X and by getting super super small changes in Y over change in X you're able to get your instantaneous slope or in the case of this example the instantaneous speed of a sane bolt right at that moment and notice you can't just put zero year if you just put change in X is zero you're gonna get something son defined you can't divide by zero so we take the limit as it approaches zero and we'll define that more rigorously in the next few videos

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