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This is a picture of Isaac Newton super famous British mathematician

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and physicist and this is the picture of Gottfried Leibniz super famous

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but maybe not as famous but maybe should be famous german

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philosopher and mathematician, he was a contemprary of Isaac Newton

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These two gentlemen together were really the founding

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fathers of the calculus and they did some of their, most of their

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major work in the late 1600's and

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this right over here is Usain Bolt

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Jamaican sprinter who's continuing to do some of his best work

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in 2012 and as early of 2012, he is the fastest human alive

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he would probably be the fastest human that has ever lived

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and you might not have made the association with these three

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gentlemen.Might not think they have lot in common but they

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were all obsessed with same fundamental question that differantial

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calculus addresses and the question is what is the instantaneous rate

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of change of something. In the case of Usain Bolt how fast is he going

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right now? Not just his average speed was for last second or his average

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speed in his next 10 seconds. How fast is he going right now?

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So this is what differential calculus is all about. Instantaneous rates of change

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differential calculus and Newton's term for differential calculus

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was the method of flexions which actually sound something fancier

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but its all about what's happening in this instant in this instant and to think about that

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why it not a super easy problem to address with the traditional

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algebra, lets draw a little graph here. So on this axis on this axis I have distance

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so now say y is equal to distance. I could have said d=distance we will see later

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in calculus d is reserved for something else.we will say y=distance

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and in this axis we will say time and i can say t=time but rather say x=time

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x=time, so if we had to plot Usain Bolt's distance as a function of time

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well in time 0 he hasn't gone anywhere he's right over there and we know at this

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gentleman is capable of travelling a 100 metres in 9.58 seconds

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so after 9.58 seconds we'll assume that this is in seconds over here, he is

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capable of going 100 metres. 100 metres.

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and so using this information we can actually figure out his average speed

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his average speed right this way his average speed is just going to be change in distance

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over his change in time

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and using the variable over here we are saying y is distance and this is change in y over

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change in x from this point to that point and this might look somewhat familiar to you

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from basic algebra.This is the slope between two points. If I have a line that connects these two points

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and if i have a line that connects two points this is the slope of that line.

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the change in distance is right over here.Change in y=100 m and our change in time is this right over

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here so our change in time is equal to 9.58 seconds we start with 0, we goto 9.58 seconds another way

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to think about it - the rise over the run

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you might have heard in your algebra class.

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It's going to be a 100 meters over 9.58 seconds.

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So this is 100 meters over 9.58 seconds.

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And the slope is essentially just the rate of change,

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or you can view it as the average rate of change,

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between these two points. And you'll see if you even follow

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the units it gives you units of speed here.

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It'd be velocity if we also specified the direction.

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And we can figure out what that is. Let me get a calculator out.

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Let me, so let me ... get the calculator on the screen.

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So we're going 100 meters in 9.58 seconds so it's around 10.4

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approximately 10.4 and then the units are meters per second.

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And that is his average speed. And what we're gonna see in a second is

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how average speed is different than instantaneous speed.

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How it's different that the speed that he might be going in any

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given moment. And just to have a concept of how fast this is

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let me get the calculator back. This is in meters per second.

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If you want to know how many meters he's going in an hour

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there's thirty-six hundred seconds in an hour.

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So he'll be able to go this many meters thirty-six hundred times.

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So that's how many meters he can, if he were able to somehow

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keep up that speed in an hour, this is how fast he is going

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in meters per hour. And then if you were to say how many miles

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per hour, there's roughly 16 hundred, and I don't know the exact number

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but roughly 16 hundred meters per mile so let's divide it

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by 1600. And so you see that this is roughly a little over 23

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about 23 and 1/2 miles per hour. This is approximately ...

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Let me write it this way, this is approximately 23.5 mph.

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Let me scroll over.

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Miles per hour.

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And relative to a car not so fast but

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relative to me extremely fast. Now, to see

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how this is different than instantaneous velocity,

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Let´s think about the potential plot of his distance relative to time

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He is not going to just go the speed immediately.

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He is not just gonna go as soon as the gun

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fires. He is not just gonna go 23 and 1/2 mph.

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All the way. He is going to have to accelerate.

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So at first, he starts off going a little bit slower.

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His slope is going to be a little bit lower.

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Than the average slope. He is going to be little bit slower.

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Then he is going to start accelerating. And so his

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speed and you see this slope here is getting steeper and steeper and steeper

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and then maybe near the end he starts tiring off a little bit.

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And so his distance part against time might be a curve

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that looks something like this.

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And what we calculated here is just the average slope across

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this change in time. We can see in any given moment

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the slope is actualy different.

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In the beginning he has a slower rate of change of

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distance. Then over here he accelerates,

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over here it seems like his rate of change of distance would be

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roughly -- or you could view it

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as a slope of the tangent line at that point --

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it looks higher than his average.

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And then he starts to slow down again.

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?? ... Average goes to 23.5 miles per hour.

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And I looked it up Usain Bolt´s instantaneous velocity

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his peak instantaneous velocity

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is actually close to 30 miles per hour.

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So the slope over here might be

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23... whatever miles per hour,

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but instantaneous, his fastest poin in this

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9.58 seconds is closer to 30 miles per hour.

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But you see it is not a trivial thing to do.

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You could say: OK, let me try to approximate

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the slope right over here and you

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could do that by saying : OK,

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well, what is the change in "y"

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over the change of "x" right around this.

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You could say: Let me take some change of x,

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and figure out what the change of y is around it.

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Or as we go past that. So we get that. But that would just be an approximation.

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Because you see that the slope of this curve is constantly changing.

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So what you want to do is see what happens as the change of x gets smaller and smaller.

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As the change of x gets smaller and smaller and smaller

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we are going to get better and better approximation.

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Your change in y is going to get smaller and smaller and smaller.

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So what you wanna do and we are going to get into depth of all of this

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and study it more rigorously,

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you want to take the limit as delta x aproaches zero.

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As delta x aproaches zero of

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change in y over your change in x

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and when you do that you are going to

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approach that instantaneous rate of change

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you could view it as instantaneous slope at that point of the curve.

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Or the slope of the tangent line at that point of the curve.

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Or if we use calculus terminology we would view it as the derivative.

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So the instantaneous slope is the derivative.

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And the notation we use for derivative is dy

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over dx. And that is why I reserved the letter y.

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How does this relate to word "differential"?

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This dy is differential.

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dx is a differential. And one way to conceptualize this. This is an

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infinitely small change in y, over an infinitely small change in x.

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And by getting super, super small changes in y or changes in x

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you are able to get this instantaneous slope

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or in the case of this example

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the instantaneous speed of Usain Bolt right at that moment.

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And notice, you can not just put zero hero.

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If we just put here change in x is 0 you are going to get something undefined.

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You can not divide by zeros. You take the limit.

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As it approaches zero and we will define that more rigorously in the next few videos.