Welcome back.
In this presentation, I actually want to show you how
we can use the antiderivative to figure out the
area under a curve.
Actually I'm going to focus more a little bit more
on the intuition.
So let actually use an example from physics.
I'll use distance and velocity.
And actually this could be a good review for derivatives,
or actually an application of derivatives.
So let's say that I described the position
of something moving.
Let's say it's x.
Let's say that x is equal to, I don't know, 16t squared.
Right?
So s is distance.
Let me write this in the corner.
I don't know why the convention is to use s as
the variable for distance.
One would think, well actually, I know, why won't they use d?
Because d is the letter used for differential, I guess.
So s is equal to distance, and then t is equal to time.
So this is just a formula that tells us the position, kind of
how far has something gone, after x many, let's
say, seconds, right?
So after like, 4 seconds, we would have gone, let's say
the distance is in feet, this is in seconds.
After 4 seconds, we would have gone 256 feet.
That's all that says.
And let me graph that as well.
Graph it.
That's a horrible line.
Have to use the line tool, might have better luck.
It's slightly better.
Actually, let me undo that too, because I just want to do
it for positive t, right?
Because you can't really go back in time.
For the purposes of this lecture, you can't
go back in time.
So that'll have to do.
So this curve will essentially just be a parabola, right?
It'll look something like this.
So actually, if you look at it, I mean you
could just eyeball it.
The object, every second you go, it's going a little
bit further, right?
So it's actually accelerating.
And so what if we wanted to figure out what the velocity
of this object, right?
This is, let's see, this is d, this is t, right?
And this is, I don't know if it's clear, but this is
kind of 1/2 a parabola.
So this is the distance function.
What would the velocity be?
Well the velocity is just, what's velocity?
It's distance divided by time, right?
And since this velocity is always changing, we
want to figure out the instantaneous velocity.
And that's actually one of the initial uses of what made
derivatives so useful.
So we want to find the change, the instantaneous change
with respect to time of this formula, right?
Because this is the distance formula.
So if we know the instant rate of change of distance with
respect to time, we'll know the velocity, right?
So ds, dt, is equal to?
So what's the derivative here?
It's 32t, right?
And this is the velocity.
Maybe I should switch back to, let me write that,
v equals velocity.
I don't know why I switched colors, but I'll stick
with the yellow.
So let's graph this function.
This will actually be a fairly straightforward graph to draw.
It's pretty straight.
And then we draw the x-axis.
I'm doing pretty good.
OK.
So this, I'll draw it in red, this is this going
to be a line, right?
32t it's a line with slope 32.
So it's actually a pretty steep line.
I won't draw it that steep because I'm going to use
this for an illustration.
So this is the velocity.
This is velocity.
This is that graph, and this is distance, right?
So in case you hadn't learned already, and maybe I'll do a
whole presentation on kind of using calculus for physics, and
using derivatives for physics.
But if you have to distance formula, it's derivative
is just velocity.
And I guess if you view it the other way, if you
have the velocity, it's antiderivative is distance.
Although you won't know where, at what position,
the object started.
In this case, the object started at position of 0,
but it could be, you know, at any constant, right?
You could have started here and then curved up.
But anyway, let's just assume we started at 0.
So the derivative of distance is velocity, the antiderivative
of velocity is distance.
Keep that in mind.
Well let's look at this.
Let's assume that we were only given this graph.
And we said, you know, this is the graph of the
velocity of some object.
And we want to figure out what the distance is after, you
know, t seconds, right?
So this is the t-axis, this is the velocity axis, right?
So let's say we were only given this, and let's say we didn't
know that the antiderivative of the velocity function is
the distance function.
How would we figure out, how would we figure out what
the distance would be at any given time?
Well let's think about it.
If we have a constant, this red is kind of bloody.
Let me switch to something more pleasant.
If we have, over any small period of time, right, or if we
have a constant velocity, when you have a constant velocity,
distance is just velocity times time, right?
So let's say we had a very small time
fragment here, right?
I'll draw it big, but let's say this time fragment
it is really small.
And let's called this very small time fragment, let call
this delta t, or dt actually.
The way I've used dt is like, it's like a change in time
that's unbelievably small, right?
So it's like almost instantaneous, but not quite.
Or you can actually view it as instantaneous.
So this is how much time goes by.
You can kind of view this as a very small change in time.
So if we have a very small change of time, and over that
very small change in time, we have a roughly constant
velocity, let's say the roughly constant velocity is this.
Right, this is the velocity, so say we had over this very small
change in time, we have this roughly constant velocity
that's on this graph.
Actually, let me take do it here.
We have this roughly constant velocity.
So the distance that the object travels over the small time
would be the small time times the velocity, right?
It would be whatever the value of this red line is, times the
width of this distance, right?
So what's another way?
Visually I kind of did it ahead of time, but
what's happening here?
If I take this change in time, right, which is kind of the
base of this rectangle, and I multiply it times the velocity
which is really just the height of this rectangle, what
have I figured out?
Well I figured out the area of this rectangle, right?
Right, the velocity this moment, times the change in
time at this moment, is nothing but the area of
this very skinny rectangle.
Skinny and tall, right?
It's almost infinitely skinny, but it's, we're assuming for
these purposes it has some very notional amount of width.
So there we figured out the area of this column, right?
Well, if we wanted to figure out the distance that you
travel after, let's say, you know, I don't know, let's say
t, let's say t sub nought, right?
This is just a particular t.
After t sub nought seconds, right?
Well then, all we would have to do is, we would have to just
figure, we would just do a bunch of dt's, right?
You'd do another one here, you'd figure out the area of
this column, you'd figure out the area of this column, the
area of this column, right?
Because each of these areas of each of these columns
represents the distance that the object travels
over that dt, right?
So if you wanted to know how far you traveled after t sub
zero seconds, you'd essentially get, or an approximation would
be, the sum of all of these areas.
And as you got more and more, as you made the dt's smaller
and smaller, skinnier, skinnier, skinnier.
And you had more and more and more and more of these
rectangles, then your approximation will get pretty
close to, well, two things.
It'll get pretty close to, as you can imagine, the area
under this curve, or in this case a line.
But it would also get you pretty much the exact amount
of distance you've traveled after t sub nought seconds.
So I think I'm running into the ten minute wall, so I'm just
going to pause here, and I'm going to continue this in
the next presentation.