>> [music] Let's fill up this bowl with
this green water.
We can graph the height of the water.
On the x-axis, I've plotted time in
seconds.
And on the y-axis, I've plotted the
water's height in pixels.
And here's the curve that I get.
Each of these little crosses is one of the
measurements that I made at some given
time, I measured the height of the water
and I get this curve.
What do you notice about this graph?
So, one thing I notice right off the bat,
is that the water level increases fairly
rapidly at first, but at the end of this
process, the water level is still
increasing but it's increasing more
slowly.
Of course, this makes sense since this
bowl is narrower at the bottom than it is
at the top, and the water is coming in at
a constant rate.
A bit more formally, alright, dv/dt, the
change in volume of water over time,
that's constant.
I'm pouring in water at a constant rate,
but dh/dt starts off very large, the
water's height is increasing very rapidly
at first.
And by the end of this process, dh/dt is
small.
So, I'd like a function that relates water
height to volume.
Well, if I take a look at the side of the
bowl, the bowl starts off fairly slanted,
and then sort of flattens out, and that's
what I'm graphing here.
I'm graphing the radius of the bowl.
On the x-axis, I'm plotting the height,
that's how far I am from the bottom of the
bowl.
And as I move up the bowl, the radius of
the bowl is increasing and that's what I'm
plotting on the y-axis.
And if you look at the bowl, right, the
bottom of the bowl starts off very slanted
and then it sort of flattens out.
And that's exactly what I'm representing
here with, with this broken line, right?
The bowl starts off fairly slanted and
then flattens out.
Once I know what the radius of the bowl is
at a particular height, I can then think
about the volume of water when then water
level is at some given height.
So, once I know the radius of the bowl at
a particular height, from the bottom of
the bowl, I can then make a graph that
shows the volume of water in the bowl,
given a particular height of water.
So, here on the x-axis, I'm plotting the
height of the water.
And on the y-axis, I'm plotting the volume
of water in the bowl.
And you can see that when the water is
deeper, there's more water in the bowl.
It's an increasing function.
Now, let's think a little bit about the
slope of the tangent line to this graph.
The tangent line initially has not very
large slope, compared to after a while,
the tangent line has a much higher slope.
And what that really means is that
initially, for a given change in volume,
there's quite a change in the height of
the water.
But after a while, the same change in
volume leads to a less dramatic change in
the height of the water.
That really makes sense from our original
experience, we poured all that water into
the bowl.
We think back to a graph of the water
level in the bowl.
Here, I've got time.
Here, I've got the height of the water.
The water is flowing in at a constant
changing volume per unit time.
And initially, aIright, the slope of this
tangent line is quite high, because the
water height is increasing very rapidly,
but after a while, because there's the
same change in volume during this entire
process, the change in the water height is
less dramatic, right?
It's the same amount of water coming in,
but the water's moving up the sides of the
bowl less quickly.