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## Defining average and instantaneous rates of change at a point

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

# Newton, Leibniz, and Usain Bolt

AP Calc: CHA‑2 (EU), CHA‑2.A (LO), CHA‑2.A.1 (EK), CHA‑2.B (LO), CHA‑2.B.1 (EK)

## Video transcript

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.