Newton, Leibniz, and Usain Bolt Why we study differential calculus
Newton, Leibniz, and Usain Bolt
- This is a picture of Isaac Newton super famous British mathematician
- and physicist and this is the picture of Gottfried Leibniz super famous
- but maybe not as famous but maybe should be famous german
- philosopher and mathematician, he was a contemprary of Isaac Newton
- These two gentlemen together were really the founding
- fathers of the calculus and they did some of their, most of their
- major work in the late 1600's and
- this right over here is Usain Bolt
- Jamaican sprinter who's continuing to do some of his best work
- in 2012 and as early of 2012, he is the fastest human alive
- he would probably be the fastest human that has ever lived
- and you might not have made the association with these three
- gentlemen.Might not think they have lot in common but they
- were all obsessed with same fundamental question that differantial
- calculus addresses and the question is what is the instantaneous rate
- of change of something. In the case of Usain Bolt how fast is he going
- right now? Not just his average speed was for last second or his average
- speed in his next 10 seconds. How fast is he going right now?
- So this is what differential calculus is all about. Instantaneous rates of change
- differential calculus and Newton's term for differential calculus
- was the method of flexions which actually sound something fancier
- but its all about what's happening in this instant in this instant and to think about that
- why it not a super easy problem to address with the traditional
- algebra, lets draw a little graph here. So on this axis on this axis I have distance
- so now say y is equal to distance. I could have said d=distance we will see later
- in calculus d is reserved for something else.we will say y=distance
- and in this axis we will say time and i can say t=time but rather say x=time
- x=time, so if we had to plot Usain Bolt's distance as a function of time
- well in time 0 he hasn't gone anywhere he's right over there and we know at this
- gentleman is capable of travelling a 100 metres in 9.58 seconds
- so after 9.58 seconds we'll assume that this is in seconds over here, he is
- capable of going 100 metres. 100 metres.
- and so using this information we can actually figure out his average speed
- his average speed right this way his average speed is just going to be change in distance
- over his change in time
- and using the variable over here we are saying y is distance and this is 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 two points. If I have a line that connects these two points
- and if i have a line that connects two points this is the slope of that line.
- the change in distance is right over here.Change in y=100 m and our change in time is this right over
- here so our change in time is equal to 9.58 seconds we start with 0, we goto 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 a 100 meters over 9.58 seconds.
- So this is 100 meters over 9.58 seconds.
- And the slope is essentially just the rate of change,
- or you can view it as the average rate of change,
- between these two points. And you'll see if you even follow
- the units it gives you units of speed here.
- It'd 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 the calculator on the screen.
- So we're going 100 meters in 9.58 seconds so it's around 10.4
- approximately 10.4 and then the units are meters per second.
- And that is his average speed. And what we're gonna see in a second is
- how average speed is different than instantaneous speed.
- How it's different that the speed that he might be going in 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 want to know how many meters he's going in an hour
- there's thirty-six hundred seconds in an hour.
- So he'll be able to go this many meters thirty-six hundred 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 is going
- in meters per hour. And then if you were to say how many miles
- per hour, there's roughly 16 hundred, and I don't know the exact number
- but roughly 16 hundred 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. This is approximately ...
- Let me write it this way, this is approximately 23.5 mph.
- Let me scroll over.
- 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 the potential plot of his distance relative to time
- He is not going to just go the speed immediately.
- He is not just gonna go as soon as the gun
- fires. He is not just gonna go 23 and 1/2 mph.
- All the way. He is going to have to accelerate.
- So at first, he starts off going a little bit slower.
- His slope is going to be a little bit lower.
- Than the average slope. He is going to be little bit slower.
- Then he is going to start accelerating. And so his
- speed and you see this 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 part 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. We can see in any given moment
- the slope is actualy different.
- In the beginning he has a slower rate of change of
- distance. Then over here he accelerates,
- over here it seems like his rate of change of distance would be
- roughly -- or you could view it
- as a slope of the tangent line at that point --
- it looks higher than his average.
- And then he starts to slow down again.
- ?? ... Average goes to 23.5 miles per hour.
- And I looked it up Usain Bolt´s instantaneous velocity
- his peak instantaneous velocity
- is actually close to 30 miles per hour.
- So the slope over here might be
- 23... whatever miles per hour,
- but instantaneous, his fastest poin in this
- 9.58 seconds is closer to 30 miles per hour.
- But you see it is 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.
- You could say: 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 we 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 the change of x gets smaller and smaller.
- As the change of x gets smaller and smaller and smaller
- we are going to get better and better approximation.
- Your change in y is going to get smaller and smaller and smaller.
- So what you wanna do and we are going to get into depth of all of this
- and study it more rigorously,
- you want to take the limit as delta x aproaches zero.
- As delta x aproaches zero of
- change in y over your change in x
- and when you do that you are going to
- approach that instantaneous rate of change
- you could view it as instantaneous slope at that point of the curve.
- Or the slope of the tangent line at that point of the curve.
- Or if we use calculus terminology we would view it as the derivative.
- So the instantaneous slope is the derivative.
- And the notation we use for derivative is dy
- over dx. And that is why I reserved the letter y.
- How does this relate to word "differential"?
- This dy is differential.
- dx is a differential. And one way to conceptualize this. This is an
- infinitely small change in y, over an infinitely small change in x.
- And by getting super, super small changes in y or changes in x
- you are able to get this instantaneous slope
- or in the case of this example
- the instantaneous speed of Usain Bolt right at that moment.
- And notice, you can not just put zero hero.
- If we just put here change in x is 0 you are going to get something undefined.
- You can not divide by zeros. You take the limit.
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