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