U2S2V05 Describing the problem.txt

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# Captions for MITx 18.01.1x module [IFeQuG1hxL4]
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So this is where we are.
We have our time variable t measured in seconds.
And then we were saying that f of t and g of t
were distances measured in meters, in which case
f prime and g prime, their derivatives,
are velocities which are measured in meters per second.
OK.
And then we had our h.
h was equal to f of t times g of t.
And so it's measured in meters squared, or it's an area.
We can visualize that with this rectangle here.
So f of t, we could say, is the width of this rectangle.
And g of t could be its height, in which case h is
the area of the rectangle.
Now, our goal in all of this is to figure out
what's going on with h prime.
Now, h's output was meter squared.
Its input is seconds.
And so we know that h prime has to be measured
in meters squared per second.
However, f prime times g prime is meters per second times
meters per second, which is meters
squared per second squared.
And we notice that these two things don't match.
And what that tells us is that h prime and f prime times g prime
can't be equal, because they measure different things.
So this is the fundamental reason
why the derivative of this product
should not be the product of the two derivatives.
But that leaves the question of, what
is then the derivative of h?
We know that h measures area.
So h prime should be measuring the rate of change of area.
Now, we can actually use our picture
to get a handle on what this rate of change of area
is going to be.
So let me clean up some of this stuff.
And let's put in some specific numbers
for some of our variables.
OK, so we've got maybe t is 3 seconds.
And then I'll erase this.
And we'll have f of 3 being 50 meters.
So f of 3, that's the width.
That's going to be 50 meters down here.
And then we have an f prime.
So let's say that f prime of 3 is 4 meters per second.
So what that means is that this width is increasing.
So maybe this right edge of this rectangle
is moving to the right at this rate of 4 meters per second.
OK?
Now, we also have g to deal with.
So let's say that g of 3 is 30 meters.
So that's the height of this rectangle at time t equals 3.
And then there's a g prime as well.
So maybe g prime of 3 is 2 meters per second.
So that's the rate of change of the height, which
we can think of as maybe this top edge of the rectangle
is moving up at this rate of 2 meters per second.
What we're trying to figure out is
h prime of 3, which is the rate of change of area.
So we can look at our picture of this rectangle
and kind of figure out how is the area changing,
or how fast is the area changing.
Well, we've got this region on the right.
We know this right edge is moving outwards.
And so we're gaining area over here.
Similarly, we're moving the top edge up at this 2 meters
per second.
And so we're going to be gaining some area
on the top as that edge moves.
And so we've got some questions for you
to try to quantify just how fast area
is being added to this picture.
So take a moment and think through those.
And then we'll come back and wrap everything up.