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  • This is a picture of Isaac Newton,

  • super famous British mathematician and physicist.

  • This is a picture of a Gottfried Leibnitz,

  • super famous, or maybe not as famous,

  • but maybe should be, famous German philosopher

  • and mathematician, and 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 late 1600s.

  • And this right over here is Usain Bolt, Jamaican sprinter,

  • whose 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 have not made the association with these three

  • gentleman.

  • 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 Usain Bolt, how fast is he going right now?

  • Not just what his average speed was 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.

  • Newton's actual original term for differential calculus

  • was the method of fluxions, which

  • actually sounds a little bit fancier.

  • But it's all about what's happening 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 I'll have distance.

  • I'll say y is equal to distance.

  • I could have said d is equal to distance,

  • but we'll see, especially 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.

  • And so if we were to plot Usain Bolt's distance

  • as a function of time, well at time zero

  • he hasn't gone anywhere.

  • He is right over there.

  • And we know that this gentleman is

  • capable of traveling 100 meters in 9.58 seconds.

  • So after 9.58 seconds, we'll assume

  • that this is in seconds right over here,

  • he's capable of going 100 meters.

  • And so using this information, we

  • can actually figure out his average speed.

  • Let me write it this way, his average speed

  • is just going to be his change in distance

  • over his change in time.

  • And using the variables that are over here,

  • we're saying y 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,

  • 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 9.58 seconds.

  • We started at 0, we go to 9.58 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 100 meters over 9.58 seconds.

  • So this is 100 meters over 9.58 seconds.

  • And the slope is essentially just 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 units,

  • 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 the calculator out.

  • So let me get the calculator on the screen.

  • So we're going 100 meters in the 9.58 seconds.

  • So it's 10.4, I'll just write 10.4, I'll round to 10.4.

  • So it's approximately 10.4, and then the units

  • are meters per second.

  • And that is his average speed.

  • And what we're going to see in a second

  • is how average 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 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 he 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 divide it by 1600.

  • And so you see that this is roughly a little over 23,

  • about 23 and 1/2 miles per hour.

  • So this is approximately, and I'll

  • write it this way-- this is approximately 23.5 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 going to go as soon as the gun fires,

  • he's not just going to go 23 and 1/2 miles per hour all the way.

  • He's going to accelerate.

  • So at first he's going to start off going a little bit slower.

  • So the slope is going to be a little bit lot lower

  • than the average slope.

  • He's going to go a little bit slower,

  • then he's going to start accelerating.

  • And so his speed, and you'll see the slope here

  • is getting steeper and steeper and steeper.

  • And then maybe 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 change in time.

  • What we could see 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 again.

  • When you average it out, it gets to 23 and 1/2 miles per hour.

  • And I looked it up, Usain Bolt's instantaneous velocity,

  • his peak instantaneous velocity, is actually

  • closer to 30 miles per hour.

  • So the slope over here might be 23 whatever miles per hour.

  • But the instantaneous, his fastest point

  • in this 9.58 seconds is closer to 30 miles per hour.

  • But you see it's not a trivial thing to do.

  • You could say, OK, let me try to approximate the slope right

  • over here.

  • And you could do that by saying, OK, well,

  • what is the change in y over the change of x right around this?

  • So you could say, well, let me take some change of x,

  • and figure out what the change of y is around it,

  • or 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 get smaller and smaller and smaller,

  • you're going to 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 of your change in y over your change in x.

  • And when you do that, you're 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 in 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 a dy over dx.

  • And that's why I reserved the letter y.

  • And then you say, well, how does this

  • relate to the word differential?

  • Well, the word differential is relating--

  • 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 Usain Bolt right at that moment.

  • And notice, you can't just put a 0 here.

  • If you just put change in x is zero,

  • you're going to get something that's undefined.

  • You can't divide by 0.

  • So we take the limit as it approaches 0.

  • And we'll define that more rigorously

  • in the next few videos.

This is a picture of Isaac Newton,

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