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So -- So, yesterday we learned
about the questions of planes

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and how to think of 3x3 linear
systems in terms of

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intersections of planes and how
to think about them

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00:00:38,000 --> 00:00:42,000
geometrically.
And, that in particular led us

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to see which cases actually we
don't have a unique solution to

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the system,
but maybe we have no solutions

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or infinitely many solutions
because maybe the line at

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intersection of two of the
planes happens to be parallel to

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the other plane.
So, today, we'll start by

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00:01:02,000 --> 00:01:08,000
looking at the equations of
lines.

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00:01:08,000 --> 00:01:18,000
And, so in a way it seems like
something which we've already

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00:01:18,000 --> 00:01:28,000
seen last time because we have
seen that we can think of a line

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as the intersection of two
planes.

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00:01:34,000 --> 00:01:37,000
And, we know what equations of
planes look like.

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00:01:37,000 --> 00:01:42,000
So, we could describe a line by
two equations telling us about

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00:01:42,000 --> 00:01:46,000
the two planes that intersect on
the line.

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00:01:46,000 --> 00:01:48,000
But that's not the most
convenient way to think about

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the line usually,
though, because when you have

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00:01:51,000 --> 00:01:53,000
these two questions,
have you solve them?

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00:01:53,000 --> 00:01:57,000
Well, OK, you can,
but it takes a bit of effort.

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00:01:57,000 --> 00:02:03,000
So, instead,
there is another representation

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00:02:03,000 --> 00:02:07,000
of a line.
So, if you have a line in

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00:02:07,000 --> 00:02:12,000
space, well, you can imagine may
be that you have a point on it.

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00:02:12,000 --> 00:02:14,000
And, that point is moving in
time.

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00:02:14,000 --> 00:02:19,000
And, the line is the trajectory
of a point as time varies.

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00:02:19,000 --> 00:02:33,000
So, think of a line as the
trajectory of a moving point.

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00:02:33,000 --> 00:02:44,000
And, so when we think of the
trajectory of the moving point,

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00:02:44,000 --> 00:02:51,000
that's called a parametric
equation.

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00:02:51,000 --> 00:03:01,000
OK, so we are going to learn
about parametric equations of

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00:03:01,000 --> 00:03:07,000
lines.
So, let's say for example that

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00:03:07,000 --> 00:03:13,000
we are looking at the line.
So, to specify a line in space,

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00:03:13,000 --> 00:03:18,000
I can do that by giving you two
points on the line or by giving

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00:03:18,000 --> 00:03:22,000
you a point and a vector
parallel to the line.

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00:03:22,000 --> 00:03:28,000
For example,
so let's say I give you two

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00:03:28,000 --> 00:03:35,000
points on the line:
(-1,2,2), and the other point

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00:03:35,000 --> 00:03:40,000
will be (1,3,-1).
So, OK, it's pretty good

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00:03:40,000 --> 00:03:43,000
because we have two points in
that line.

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00:03:43,000 --> 00:03:46,000
Now, how do we find all the
other points?

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00:03:46,000 --> 00:03:50,000
Well, the other points in
between these guys and also on

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00:03:50,000 --> 00:03:54,000
either side.
Let's imagine that we have a

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00:03:54,000 --> 00:04:00,000
point that's moving on the line,
and at time zero,

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it's here at Q0.
And, in a unit time,

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00:04:03,000 --> 00:04:05,000
I'm not telling you what the
unit is.

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00:04:05,000 --> 00:04:08,000
It could be a second.
It could be an hour.

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00:04:08,000 --> 00:04:12,000
It could be a year.
At t=1, it's going to be at Q1.

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00:04:12,000 --> 00:04:14,000
And, it moves at a constant
speed.

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00:04:14,000 --> 00:04:17,000
So, maybe at time one half,
it's going to be here.

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00:04:17,000 --> 00:04:19,000
Times two, it would be over
there.

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00:04:19,000 --> 00:04:21,000
And, in fact,
that point didn't start here.

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00:04:21,000 --> 00:04:24,000
Maybe it's always been moving
on that line.

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00:04:24,000 --> 00:04:29,000
At time minus two,
it was down there.

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00:04:29,000 --> 00:04:51,000
So, let's say Q(t) is a moving
point, and at t=0 it's at Q0.

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00:04:51,000 --> 00:04:55,000
And, let's say that it moves.
Well, we couldn't make it move

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00:04:55,000 --> 00:04:58,000
in any way we want.
But, probably the easiest to

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00:04:58,000 --> 00:05:02,000
find, so our role is going to
find formulas for a position of

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00:05:02,000 --> 00:05:06,000
this moving point in terms of t.
And, we'll use that to say,

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00:05:06,000 --> 00:05:08,000
well,
any point on the line is of

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00:05:08,000 --> 00:05:12,000
this form where you have to plug
in the current value of t

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00:05:12,000 --> 00:05:16,000
depending on when it's hit by
the moving point.

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00:05:16,000 --> 00:05:24,000
So, perhaps it's easiest to do
it if we make it move at a

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00:05:24,000 --> 00:05:31,000
constant speed on the line,
and that speed is chosen so

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00:05:31,000 --> 00:05:36,000
that at time one,
it's at Q1.

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00:05:45,000 --> 00:05:56,000
So, the question we want to
answer is, what is the position

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00:05:56,000 --> 00:06:03,000
at time t, so,
the point Q(t)?

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00:06:03,000 --> 00:06:08,000
Well, to answer that we have an
easy observation,

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00:06:08,000 --> 00:06:15,000
which is that the vector from
Q0 to Q of t is proportional to

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00:06:15,000 --> 00:06:23,000
the vector from Q0 to Q1.
And, what's the proportionality

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00:06:23,000 --> 00:06:27,000
factor here?
Yeah, it's exactly t.

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00:06:27,000 --> 00:06:34,000
At time one,
Q0 Q is exactly the same.

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00:06:34,000 --> 00:06:36,000
Maybe I should draw another
picture again.

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00:06:36,000 --> 00:06:43,000
I have Q0.
I have Q1, and after time t,

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00:06:43,000 --> 00:06:56,000
I'm here at Q of t where this
vector from Q0 Q(t) is actually

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00:06:56,000 --> 00:07:04,000
going to be t times the vector
Q0 Q1.

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00:07:04,000 --> 00:07:09,000
So, when t increases,
it gets longer and longer.

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00:07:09,000 --> 00:07:15,000
So, does everybody see this now?
Is that OK?

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00:07:15,000 --> 00:07:24,000
Any questions about that?
Yes?

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00:07:24,000 --> 00:07:26,000
OK, so I will try to avoid
using blue.

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00:07:26,000 --> 00:07:37,000
Thanks for, that's fine.
So, OK, I will not use blue

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00:07:37,000 --> 00:07:42,000
anymore.
OK, well, first let me just

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00:07:42,000 --> 00:07:46,000
make everything white just for
now.

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00:07:46,000 --> 00:07:49,000
This is the vector from Q0 to
Q(t).

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00:07:49,000 --> 00:07:56,000
This is the point Q(t).
OK, is it kind of visible now?

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00:07:56,000 --> 00:08:03,000
OK, thanks for pointing it out.
I will switch to brighter

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00:08:03,000 --> 00:08:09,000
colors.
So, OK, so apart from that,

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00:08:09,000 --> 00:08:13,000
I claim now we can find the
position of its moving point

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00:08:13,000 --> 00:08:15,000
because,
well, this vector,

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00:08:15,000 --> 00:08:19,000
Q0Q1 we can find from the
coordinates of Q0 and Q1.

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00:08:19,000 --> 00:08:26,000
So, we just subtract the
coordinates of Q0 from those of

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00:08:26,000 --> 00:08:30,000
Q1 will get that vector Q0 Q1 is

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00:08:32,000 --> 00:08:36,000
OK,
so, if I look at it,

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00:08:36,000 --> 00:08:44,000
well, so let's call x(t),
y(t), and z(t) the coordinates

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00:08:44,000 --> 00:08:50,000
of the point that's moving on
the line.

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00:08:50,000 --> 00:09:00,000
Then we get x of t minus,
well, actually plus one equals

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00:09:00,000 --> 00:09:07,000
t times two.
I'm writing the components of

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00:09:07,000 --> 00:09:13,000
Q0Q(t).
And here, I'm writing t times

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00:09:13,000 --> 00:09:19,000
Q0Q1.
y(t) minus two equals t,

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00:09:19,000 --> 00:09:28,000
and z(t) minus two equals -3t.
So, in other terms,

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00:09:28,000 --> 00:09:34,000
the more familiar way that we
used to write these equations,

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00:09:34,000 --> 00:09:42,000
let me do it that way instead,
minus one plus 2t,

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00:09:42,000 --> 00:09:53,000
y(t) = 2 t, z(t) = 2 - 3t.
And, if you prefer,

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00:09:53,000 --> 00:10:02,000
I can just say Q(t) is Q0 plus
t times vector Q0Q1.

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00:10:02,000 --> 00:10:07,000
OK, so that's our first
parametric equation of a line in

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00:10:07,000 --> 00:10:10,000
this class.
And, I hope you see it's not

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00:10:10,000 --> 00:10:13,000
extremely hard.
In fact, parametric equations

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00:10:13,000 --> 00:10:17,000
of lines always look like that.
x, y, and z are functions of t

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00:10:17,000 --> 00:10:22,000
but are of the form a constant
plus a constant times t.

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00:10:22,000 --> 00:10:26,000
The coefficients of t tell us
about a vector along the line.

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00:10:26,000 --> 00:10:33,000
Here, we have a vector,
Q0Q1, which is .

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00:10:33,000 --> 00:10:37,000
And, the constant terms tell us
about where we are at t=0.

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00:10:37,000 --> 00:10:41,000
If I plug t=0 these guys go
away, I get minus 1,2,

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2.
That's my starting point.

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00:10:46,000 --> 00:10:59,000
OK, so, any questions about
that?

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00:10:59,000 --> 00:11:05,000
No?
OK, so let's see,

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00:11:05,000 --> 00:11:12,000
now, what we can do with these
parametric equations.

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00:11:12,000 --> 00:11:26,000
So, one application is to think
about the relative position of a

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line and a plane with respect to
each other.

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00:11:36,000 --> 00:11:44,000
So, let's say that we take
still the same line up there,

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00:11:44,000 --> 00:11:53,000
and let's consider the plane
with the equation x 2y 4z = 7.

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00:11:53,000 --> 00:11:55,000
OK, so I'm giving you this
plane.

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00:11:55,000 --> 00:11:58,000
And, the questions that we are
going to ask ourselves are,

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00:11:58,000 --> 00:12:00,000
well, does the line intersect
the plane?

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00:12:00,000 --> 00:12:02,000
And, where does it intersect
the plane?

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00:12:22,000 --> 00:12:28,000
So, let's start with the first
primary question that maybe we

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00:12:28,000 --> 00:12:32,000
should try to understand.
We have these points.

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00:12:32,000 --> 00:12:35,000
We have this plane,
and we have these points,

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00:12:35,000 --> 00:12:38,000
Q0 and Q1.
I'm going to draw them in

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00:12:38,000 --> 00:12:42,000
completely random places.
Well, are Q0 and Q1 on the same

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00:12:42,000 --> 00:12:47,000
side of a plane or on different
sides, on opposite sides of the

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00:12:47,000 --> 00:12:50,000
planes?
Could it be that maybe one of

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00:12:50,000 --> 00:12:59,000
the points is in the plane?
So, I think I'm going to let

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00:12:59,000 --> 00:13:05,000
you vote on that.
So, is that readable?

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00:13:05,000 --> 00:13:08,000
Is it too small?
OK, so anyway,

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00:13:08,000 --> 00:13:12,000
the question says,
relative to the plane,

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00:13:12,000 --> 00:13:16,000
x 2y 4z = 7.
This point, Q0 and Q1,

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00:13:16,000 --> 00:13:22,000
are they on the same side,
on opposite sides,

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00:13:22,000 --> 00:13:29,000
is one of them on the plane,
or we can't decide?

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00:13:29,000 --> 00:13:41,000
OK, that should be better.
So, I see relatively few

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00:13:41,000 --> 00:13:46,000
answers.
OK, it looks like also a lot of

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00:13:46,000 --> 00:13:51,000
you have forgotten the cards
and, so I see people raising two

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00:13:51,000 --> 00:13:55,000
fingers, I see people raising
three fingers.

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00:13:55,000 --> 00:13:57,000
And, I see people raising four
fingers.

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00:13:57,000 --> 00:14:01,000
I don't see anyone answering
number one.

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00:14:01,000 --> 00:14:03,000
So, the general idea seems to
be that either they are on

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00:14:03,000 --> 00:14:07,000
opposite sides.
Maybe one of them is on the

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00:14:07,000 --> 00:14:10,000
plane.
Well, let's try to see.

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00:14:10,000 --> 00:14:14,000
Is one of them on the plane?
Well, let's check.

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00:14:14,000 --> 00:14:20,000
OK, so let's look at the point,
sorry.

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00:14:20,000 --> 00:14:25,000
I have one blackboard to use
here.

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00:14:25,000 --> 00:14:31,000
So, I take the point Q0,
which is at (-1,2,2).

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00:14:31,000 --> 00:14:37,000
Well, if I plug that into the
plane equation,

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00:14:37,000 --> 00:14:44,000
so, x 2y 4z will equal minus
one plus two times two plus four

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00:14:44,000 --> 00:14:48,000
times two.
That's, well,

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00:14:48,000 --> 00:14:52,000
four plus eight,
12 minus one,

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00:14:52,000 --> 00:14:54,000
11.
That, I think,

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00:14:54,000 --> 00:15:01,000
is bigger than seven.
OK, so Q0 is not in the plane.

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00:15:01,000 --> 00:15:07,000
Let's try again with Q1.
(1,3, - 1) well,

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00:15:07,000 --> 00:15:15,000
if we plug that into x 2y 4z,
we'll have one plus two times

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00:15:15,000 --> 00:15:20,000
three makes seven.
But, we add four times negative

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00:15:20,000 --> 00:15:23,000
one.
We add up with three less than

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00:15:23,000 --> 00:15:25,000
seven.
Well, that one is not in the

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00:15:25,000 --> 00:15:27,000
plane, either.
So, I don't think,

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00:15:27,000 --> 00:15:32,000
actually, that the answer
should be number three.

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00:15:32,000 --> 00:15:37,000
So, let's get rid of answer
number three.

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00:15:37,000 --> 00:15:42,000
OK, let's see,
in light of this,

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00:15:42,000 --> 00:15:50,000
are you willing to reconsider
your answer?

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00:15:50,000 --> 00:15:53,000
OK, so I think now everyone
seems to be interested in

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00:15:53,000 --> 00:15:57,000
answering number two.
And, I would agree with that

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00:15:57,000 --> 00:16:00,000
answer.
So, let's think about it.

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00:16:00,000 --> 00:16:02,000
These points are not in the
plane, but they are not in the

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00:16:02,000 --> 00:16:05,000
plane in different ways.
One of them somehow overshoots;

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00:16:05,000 --> 00:16:08,000
we get 11.
The other one we only get 3.

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00:16:08,000 --> 00:16:12,000
That's less than seven.
If you think about how a plan

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00:16:12,000 --> 00:16:15,000
splits space into two half
spaces on either side,

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00:16:15,000 --> 00:16:22,000
well, one of them is going to
be the point where x 2y 4z is

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00:16:22,000 --> 00:16:27,000
less than seven.
And, the other one will be,

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00:16:27,000 --> 00:16:32,000
so, that's somehow this side.
And, that's where Q1 is.

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00:16:32,000 --> 00:16:43,000
And, the other side is where x
2y 4z is actually bigger than

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00:16:43,000 --> 00:16:47,000
seven.
And, to go from one to the

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00:16:47,000 --> 00:16:53,000
other, well, x 2y 4z needs to go
through the value seven.

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00:16:53,000 --> 00:16:57,000
If you're moving along any path
from Q0 to Q1,

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00:16:57,000 --> 00:17:02,000
this thing will change
continuously from 11 to 3.

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00:17:02,000 --> 00:17:05,000
At some time,
it has to go through 7.

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00:17:05,000 --> 00:17:09,000
Does that make sense?
So, to go from Q0 to Q1 we need

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00:17:09,000 --> 00:17:12,000
to cross P at some place.
So, they're on opposite sides.

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00:17:31,000 --> 00:17:37,000
OK, now that doesn't quite
finish answering the question

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00:17:37,000 --> 00:17:43,000
that we had, which was,
where does the line intersect

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00:17:43,000 --> 00:17:46,000
the plane?
But, why can't we do the same

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00:17:46,000 --> 00:17:48,000
thing?
Now, we know not only the

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00:17:48,000 --> 00:17:51,000
points Q0 and Q1,
we know actually any point on

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00:17:51,000 --> 00:17:55,000
the line because we have a
parametric equation up there

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00:17:55,000 --> 00:17:57,000
telling us,
where is the point that's

198
00:17:57,000 --> 00:18:04,000
moving on the line at time t?
So, what about the moving

199
00:18:04,000 --> 00:18:08,000
point, Q(t)?
Well, let's plug its

200
00:18:08,000 --> 00:18:10,000
coordinates into the plane
equation.

201
00:18:10,000 --> 00:18:24,000
So, we'll take x(t) 2y(t) 4z(t).
OK, that's equal to,

202
00:18:24,000 --> 00:18:34,000
well, (-1 2t) 2( 2 t) 4( 2 -
3t).

203
00:18:34,000 --> 00:18:41,000
So, if you simplify this a bit,
you get 2t 2t -12t.

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00:18:41,000 --> 00:18:46,000
That should be -8t.
And, the constant term is minus

205
00:18:46,000 --> 00:18:54,000
one plus four plus eight is 11.
OK, and we have to compare that

206
00:18:54,000 --> 00:18:57,000
with seven.
OK, the question is,

207
00:18:57,000 --> 00:19:07,000
is this ever equal to seven?
Well, so, Q(t) is in the plane

208
00:19:07,000 --> 00:19:16,000
exactly when -8t 11 equals
seven.

209
00:19:16,000 --> 00:19:20,000
And, that' the same.
If you manipulate this,

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00:19:20,000 --> 00:19:27,000
you will get t equals one half.
In fact, that's not very

211
00:19:27,000 --> 00:19:30,000
surprising.
If you look at these values,

212
00:19:30,000 --> 00:19:32,000
11 and three,
you see that seven is actually

213
00:19:32,000 --> 00:19:35,000
right in between.
It's the average of these two

214
00:19:35,000 --> 00:19:39,000
numbers.
So, it would make sense that

215
00:19:39,000 --> 00:19:44,000
it's halfway in between Q0 and
Q1, but we will get seven.

216
00:19:44,000 --> 00:19:50,000
OK, and that at that time,
Q at time one half,

217
00:19:50,000 --> 00:19:59,000
well, let's plug the values.
So, minus one plus 2t will be

218
00:19:59,000 --> 00:20:04,000
zero.
Two plus t will be two and a

219
00:20:04,000 --> 00:20:11,000
half of five halves,
and two minus three halves will

220
00:20:11,000 --> 00:20:15,000
be one half, OK?
So, this is where the line

221
00:20:15,000 --> 00:20:16,000
intersects the plane.

222
00:20:43,000 --> 00:20:47,000
So, you see that's actually a
pretty easy way of finding where

223
00:20:47,000 --> 00:20:49,000
a line on the plane intersects
each other.

224
00:20:49,000 --> 00:20:52,000
If we can find a parametric
equation of a line and an

225
00:20:52,000 --> 00:20:55,000
equation of a plane,
but we basically just plug one

226
00:20:55,000 --> 00:20:59,000
into the other,
and see at what time the moving

227
00:20:59,000 --> 00:21:04,000
point hits the plane so that we
know where this.

228
00:21:04,000 --> 00:21:23,000
OK, other questions about this?
Yes?

229
00:21:23,000 --> 00:21:30,000
Sorry, can you say that?
Yes, so what if we don't get a

230
00:21:30,000 --> 00:21:32,000
solution?
What happens?

231
00:21:32,000 --> 00:21:36,000
So, indeed our line could have
been parallel to the plane or

232
00:21:36,000 --> 00:21:38,000
maybe even contained in the
plane.

233
00:21:38,000 --> 00:21:42,000
Well, if the line is parallel
to the plane then maybe what

234
00:21:42,000 --> 00:21:46,000
happens is that what we plug in
the positions of the moving

235
00:21:46,000 --> 00:21:48,000
point,
we actually get something that

236
00:21:48,000 --> 00:21:50,000
never equals seven because maybe
we get actually a constant.

237
00:21:50,000 --> 00:21:53,000
Say that we had gotten,
I don't know,

238
00:21:53,000 --> 00:21:56,000
13 all the time.
Well, when is 13 equal to seven?

239
00:21:56,000 --> 00:21:59,000
The answer is never.
OK, so that's what would tell

240
00:21:59,000 --> 00:22:02,000
you that the line is actually
parallel to the plane.

241
00:22:02,000 --> 00:22:06,000
You would not find a solution
to the equation that you get at

242
00:22:06,000 --> 00:22:13,000
the end.
Yes?

243
00:22:13,000 --> 00:22:16,000
So, if there's no solution at
all to the equation that you

244
00:22:16,000 --> 00:22:19,000
get, it means that at no time is
the traveling point going to be

245
00:22:19,000 --> 00:22:22,000
in the plane.
That means the line really does

246
00:22:22,000 --> 00:22:25,000
not have the plane ever.
So, it has to be parallel

247
00:22:25,000 --> 00:22:27,000
outside of it.
On the other hand,

248
00:22:27,000 --> 00:22:30,000
if a line is inside the plane,
then that means that no matter

249
00:22:30,000 --> 00:22:33,000
what time you choose,
you always get seven.

250
00:22:33,000 --> 00:22:37,000
OK, that's what would happen if
a line is in the plane.

251
00:22:37,000 --> 00:22:44,000
You always get seven.
So, maybe I should write this

252
00:22:44,000 --> 00:22:54,000
down.
So, if a line is in the plane

253
00:22:54,000 --> 00:23:10,000
then plugging x(t),
y(t), z(t) into the equation,

254
00:23:10,000 --> 00:23:18,000
we always get,
well, here in this case seven

255
00:23:18,000 --> 00:23:22,000
or whatever the value should be
for the plane,

256
00:23:22,000 --> 00:23:34,000
If the line is parallel to the
plane -- -- in fact,

257
00:23:34,000 --> 00:23:45,000
we, well, get,
let's see, another constant.

258
00:23:45,000 --> 00:23:49,000
So, in fact,
you know, when you plug in

259
00:23:49,000 --> 00:23:51,000
these things,
normally you get a quantity

260
00:23:51,000 --> 00:23:54,000
that's of a form,
something times t plus a

261
00:23:54,000 --> 00:23:57,000
constant because that's what you
plug into the equation of a

262
00:23:57,000 --> 00:23:59,000
plane.
And so, in general,

263
00:23:59,000 --> 00:24:01,000
you have an equation of the
form, something times t plus

264
00:24:01,000 --> 00:24:05,000
something equals something.
And, that usually has a single

265
00:24:05,000 --> 00:24:08,000
solution.
And, the special case is if

266
00:24:08,000 --> 00:24:11,000
this coefficient of t turns out
to be zero in the end,

267
00:24:11,000 --> 00:24:14,000
and that's actually going to
happen,

268
00:24:14,000 --> 00:24:20,000
exactly when the line is either
parallel or in the plane.

269
00:24:20,000 --> 00:24:24,000
In fact, if you think this
through carefully,

270
00:24:24,000 --> 00:24:26,000
the coefficient of t that you
get here,

271
00:24:26,000 --> 00:24:30,000
see, it's one times two plus
two times one plus four times

272
00:24:30,000 --> 00:24:33,000
minus three.
It's the dot product between

273
00:24:33,000 --> 00:24:37,000
the normal vector of a plane and
the vector along the line.

274
00:24:37,000 --> 00:24:41,000
So, see, this coefficient
becomes zero exactly when the

275
00:24:41,000 --> 00:24:44,000
line is perpendicular to the
normal vector.

276
00:24:44,000 --> 00:24:46,000
That means it's parallel to the
plane.

277
00:24:46,000 --> 00:24:51,000
So, everything makes sense.
OK, if you're confused about

278
00:24:51,000 --> 00:24:55,000
what I just said,
you can ignore it.

279
00:24:55,000 --> 00:25:03,000
OK, more questions? No?
OK, so if not,

280
00:25:03,000 --> 00:25:09,000
let's move on to linear
parametric equations.

281
00:25:09,000 --> 00:25:13,000
So, I hope you've seen here
that parametric equations are a

282
00:25:13,000 --> 00:25:18,000
great way to think about lines.
There are also a great way to

283
00:25:18,000 --> 00:25:22,000
think about actually any curve,
any trajectory that can be

284
00:25:22,000 --> 00:25:34,000
traced by a moving point.
So -- -- more generally,

285
00:25:34,000 --> 00:26:00,000
we can use parametric equations
-- -- for arbitrary motions --

286
00:26:00,000 --> 00:26:15,000
-- in the plane or in space.
So, let's look at an example.

287
00:26:15,000 --> 00:26:20,000
Let's take, so,
it's a famous curve called a

288
00:26:20,000 --> 00:26:23,000
cycloid.
A cycloid is something that you

289
00:26:23,000 --> 00:26:27,000
can actually see sometimes at
night when people are biking If

290
00:26:27,000 --> 00:26:31,000
you have something that reflects
light on the wheel.

291
00:26:31,000 --> 00:26:33,000
So, let me explain what's the
definition of a cycloid.

292
00:27:05,000 --> 00:27:07,000
So, I should say,
I've seen a lecture where,

293
00:27:07,000 --> 00:27:10,000
actually, the professor had a
volunteer on a unicycle to

294
00:27:10,000 --> 00:27:13,000
demonstrate how that works.
But, I didn't arrange for that,

295
00:27:13,000 --> 00:27:17,000
so instead I will explain it to
you using more conventional

296
00:27:17,000 --> 00:27:23,000
means.
So, let's say that we have a

297
00:27:23,000 --> 00:27:31,000
wheel that's rolling on a
horizontal ground.

298
00:27:31,000 --> 00:27:34,000
And, as it's rolling of course
it's going to turn.

299
00:27:34,000 --> 00:27:40,000
So, it's going to move forward
to a new position.

300
00:27:40,000 --> 00:27:45,000
And, now, let's mention that we
have a point that's been painted

301
00:27:45,000 --> 00:27:47,000
red on the circumference of the
wheel.

302
00:27:47,000 --> 00:27:51,000
And, initially,
that point is here.

303
00:27:51,000 --> 00:27:53,000
So, as the wheel stops
rotating, well,

304
00:27:53,000 --> 00:27:57,000
of course, it moves forward,
and so it turns on itself.

305
00:27:57,000 --> 00:28:02,000
So, that point starts falling
back behind the point of contact

306
00:28:02,000 --> 00:28:07,000
because the wheel is rotating at
the same time as it's moving

307
00:28:07,000 --> 00:28:12,000
forward.
And so, the cycloid is the

308
00:28:12,000 --> 00:28:21,000
trajectory of this moving point.
OK, so the cycloid is obtained

309
00:28:21,000 --> 00:28:27,000
by considering,
so we have a wheel,

310
00:28:27,000 --> 00:28:38,000
let's say, of radius a.
So, this height here is (a)

311
00:28:38,000 --> 00:28:47,000
rolling on the floor which is
the x axis.

312
00:28:47,000 --> 00:28:53,000
And, let's -- And,
we have a point,

313
00:28:53,000 --> 00:29:01,000
P, that's painted on the wheel.
Initially, it's at the origin.

314
00:29:01,000 --> 00:29:04,000
But, of course,
as time goes by,

315
00:29:04,000 --> 00:29:13,000
it moves on the wheel.
P is a point on the rim of the

316
00:29:13,000 --> 00:29:21,000
wheel, and it starts at the
origin.

317
00:29:21,000 --> 00:29:27,000
So, the question is,
what happens?

318
00:29:27,000 --> 00:29:32,000
In particular,
can we find the position of

319
00:29:32,000 --> 00:29:37,000
this point, x(t),
y(t), as a function of time?

320
00:29:37,000 --> 00:29:42,000
So, that's the reason why I
have this computer.

321
00:29:42,000 --> 00:29:48,000
So, I'm not sure it will be
very easy to visualize,

322
00:29:48,000 --> 00:29:54,000
but so we have a wheel,
well, I hope you can vaguely

323
00:29:54,000 --> 00:30:00,000
see that there's a circle that's
moving.

324
00:30:00,000 --> 00:30:05,000
The wheel is green here.
And, there's a radius that's

325
00:30:05,000 --> 00:30:09,000
been painted blue in it.
And, that radius rotates around

326
00:30:09,000 --> 00:30:12,000
the wheel as the wheel is moving
forward.

327
00:30:12,000 --> 00:30:23,000
So, now, let's try to paint,
actually, the trajectory of a

328
00:30:23,000 --> 00:30:26,000
point.
[LAUGHTER]

329
00:30:26,000 --> 00:30:30,000
OK, so that's what the cycloid
looks like.

330
00:30:30,000 --> 00:30:37,000
[APPLAUSE]
OK, so -- So the cycloid,

331
00:30:37,000 --> 00:30:47,000
well, I guess it doesn't quite
look like what I've drawn.

332
00:30:47,000 --> 00:30:52,000
It looks like it goes a bit
higher up, which will be the

333
00:30:52,000 --> 00:30:57,000
trajectory of this red point.
And, see, it hits the bottom

334
00:30:57,000 --> 00:31:01,000
once in a while.
It forms these arches because

335
00:31:01,000 --> 00:31:04,000
when the wheel has rotated by a
full turn,

336
00:31:04,000 --> 00:31:07,000
then you're basically back at
the same situation,

337
00:31:07,000 --> 00:31:09,000
except a bit further along the
route.

338
00:31:09,000 --> 00:31:13,000
So, if we do it once more,
you see the point now is at the

339
00:31:13,000 --> 00:31:18,000
top, and now it's at the bottom.
And then we start again.

340
00:31:18,000 --> 00:31:23,000
It's at the top,
and then again at the bottom.

341
00:31:23,000 --> 00:31:40,000
OK.
No.

342
00:31:40,000 --> 00:31:48,000
[LAUGHTER]
OK, so the question that we

343
00:31:48,000 --> 00:31:58,000
want to answer is what is the
position x(t),

344
00:31:58,000 --> 00:32:05,000
y(t), of the point P?
OK, so actually,

345
00:32:05,000 --> 00:32:07,000
I'm writing x(t),
y(t).

346
00:32:07,000 --> 00:32:10,000
That means that I have,
maybe I'm expressing the

347
00:32:10,000 --> 00:32:13,000
position in terms of time.
Let's see, is time going to be

348
00:32:13,000 --> 00:32:15,000
a good thing to do?
Well, suddenly,

349
00:32:15,000 --> 00:32:20,000
the position changes over time.
But doesn't actually matter how

350
00:32:20,000 --> 00:32:24,000
fast the wheel is rolling?
No, because I can just play the

351
00:32:24,000 --> 00:32:27,000
motion fast-forward.
The wheel will be going faster,

352
00:32:27,000 --> 00:32:29,000
but the trajectory is still the
same.

353
00:32:29,000 --> 00:32:32,000
So, in fact,
time is not the most relevant

354
00:32:32,000 --> 00:32:36,000
thing here.
What matters to us now is how

355
00:32:36,000 --> 00:32:39,000
far the wheel has gone.
So, we could use as a

356
00:32:39,000 --> 00:32:44,000
parameter, for example,
the distance by which the wheel

357
00:32:44,000 --> 00:32:46,000
has moved.
We can do even better because

358
00:32:46,000 --> 00:32:49,000
we see that, really,
the most complicated thing that

359
00:32:49,000 --> 00:32:50,000
happens here is really the
rotation.

360
00:32:50,000 --> 00:32:55,000
So, maybe we can actually use
the angle by which the wheel has

361
00:32:55,000 --> 00:32:57,000
turned to parameterize the
motion.

362
00:32:57,000 --> 00:33:02,000
So, there's various choices.
You can choose whichever one

363
00:33:02,000 --> 00:33:04,000
you prefer.
But, I think here,

364
00:33:04,000 --> 00:33:07,000
we will get the simplest answer
if we parameterize things by the

365
00:33:07,000 --> 00:33:10,000
angle.
So, in fact,

366
00:33:10,000 --> 00:33:23,000
instead of t I will be using
what's called theta as a

367
00:33:23,000 --> 00:33:36,000
function of the angle,
theta, by which the wheel has

368
00:33:36,000 --> 00:33:50,000
rotated.
So, how are we going to do that?

369
00:33:50,000 --> 00:33:57,000
Well, because we are going to
try to use our new knowledge,

370
00:33:57,000 --> 00:34:03,000
let's try to do it using
vectors in a smart way.

371
00:34:03,000 --> 00:34:07,000
So, let me draw a picture of
the wheel after things have

372
00:34:07,000 --> 00:34:12,000
rotated by a certain amount.
So, maybe my point,

373
00:34:12,000 --> 00:34:18,000
P, now, is here.
And, so the wheel has rotated

374
00:34:18,000 --> 00:34:21,000
by this angle here.
And, I want to find the

375
00:34:21,000 --> 00:34:23,000
position of my point,
P, OK?

376
00:34:23,000 --> 00:34:29,000
So, the position of this point,
P, is going to be the same as

377
00:34:29,000 --> 00:34:35,000
knowing the vector OP from the
origin to this moving point.

378
00:34:35,000 --> 00:34:39,000
So, I haven't really simplify
the problem yet because we don't

379
00:34:39,000 --> 00:34:43,000
really know about vector OP.
But, maybe we know about

380
00:34:43,000 --> 00:34:47,000
simpler vectors where some will
be OP.

381
00:34:47,000 --> 00:34:50,000
So, let's see,
let's give names to a few of

382
00:34:50,000 --> 00:34:52,000
our points.
For example,

383
00:34:52,000 --> 00:34:54,000
let's say that this will be
point A.

384
00:34:54,000 --> 00:34:58,000
A is the point where the wheel
is touching the road.

385
00:34:58,000 --> 00:35:02,000
And, B will be the center of
the wheel.

386
00:35:02,000 --> 00:35:07,000
Then, it looks like maybe I
have actually a chance of

387
00:35:07,000 --> 00:35:12,000
understanding vectors like maybe
OA doesn't look quite so scary,

388
00:35:12,000 --> 00:35:16,000
or AB doesn't look too bad.
BP doesn't look too bad.

389
00:35:16,000 --> 00:35:27,000
And, if I sum them together,
I will obtain OP.

390
00:35:27,000 --> 00:35:35,000
So, let's do that.
So, now we've greatly

391
00:35:35,000 --> 00:35:39,000
simplified the problem.
We had to find one vector that

392
00:35:39,000 --> 00:35:42,000
we didn't know.
Now we have to find three

393
00:35:42,000 --> 00:35:47,000
vectors which we don't know.
But, you will see each of them

394
00:35:47,000 --> 00:35:50,000
as fairly easy to think about.
So, let's see.

395
00:35:50,000 --> 00:35:56,000
Should we start with vector OA,
maybe?

396
00:35:56,000 --> 00:36:04,000
So, OA has two components.
One of them should be very easy.

397
00:36:04,000 --> 00:36:06,000
Well, the y component is just
going to be zero,

398
00:36:06,000 --> 00:36:10,000
OK?
It's directed along the x axis.

399
00:36:10,000 --> 00:36:15,000
What about the x component?
So, OA is the distance by which

400
00:36:15,000 --> 00:36:21,000
the wheel has traveled to get to
its current position.

401
00:36:21,000 --> 00:36:23,000
Yeah.
I hear a lot of people saying R

402
00:36:23,000 --> 00:36:25,000
theta.
Let me actually say a(theta)

403
00:36:25,000 --> 00:36:28,000
because I've called a the radius
of the wheel.

404
00:36:28,000 --> 00:36:33,000
So, this distance is a(theta).
Why is it a(theta)?

405
00:36:33,000 --> 00:36:36,000
Well, that's because the wheel,
well, there's an assumption

406
00:36:36,000 --> 00:36:38,000
which is that the wheel is
rolling on something normal like

407
00:36:38,000 --> 00:36:40,000
a road,
and not on, maybe,

408
00:36:40,000 --> 00:36:45,000
ice, or something like that.
S So, it's rolling without

409
00:36:45,000 --> 00:36:48,000
slipping.
So, that means that this

410
00:36:48,000 --> 00:36:53,000
distance on the road is actually
equal to the distance here on

411
00:36:53,000 --> 00:36:57,000
the circumference of the wheel.
This point, P,

412
00:36:57,000 --> 00:37:01,000
was there, and the amount by
which the things have moved can

413
00:37:01,000 --> 00:37:06,000
be measured either here or here.
These are the same distances.

414
00:37:06,000 --> 00:37:15,000
OK, so, that makes it a(theta),
and maybe I should justify by

415
00:37:15,000 --> 00:37:22,000
saying amount by which the wheel
has rolled,

416
00:37:22,000 --> 00:37:30,000
has moved, is equal to the,
so, the distance from O to A is

417
00:37:30,000 --> 00:37:37,000
equal to the arc length on the
circumference of the circle from

418
00:37:37,000 --> 00:37:40,000
A to P.
And, you know that if you have

419
00:37:40,000 --> 00:37:42,000
a sector corresponding to an
angle, theta,

420
00:37:42,000 --> 00:37:45,000
then its length is a times
theta, provided that,

421
00:37:45,000 --> 00:37:48,000
of course, you express the
angel in radians.

422
00:37:48,000 --> 00:37:58,000
That's the reason why we always
used radians in math.

423
00:37:58,000 --> 00:38:01,000
Now, let's think about vector
AB and vector BP.

424
00:38:30,000 --> 00:38:39,000
OK, so AB is pretty easy,
right, because it's pointing

425
00:38:39,000 --> 00:38:45,000
straight up, and its length is
a.

426
00:38:45,000 --> 00:38:55,000
So, it's just zero, a.
Now, the most serious one we've

427
00:38:55,000 --> 00:39:00,000
kept for the end.
What about vector BP?

428
00:39:00,000 --> 00:39:04,000
So, vector BP,
we know two things about it.

429
00:39:04,000 --> 00:39:17,000
We know actually its length,
so, the magnitude of BP -- --

430
00:39:17,000 --> 00:39:23,000
a.
And, we know it makes an angle,

431
00:39:23,000 --> 00:39:29,000
theta, with the vertical.
So, that should let us find its

432
00:39:29,000 --> 00:39:34,000
components.
Let's draw a closer picture.

433
00:39:34,000 --> 00:39:40,000
Now, in the picture I'm going
to center things at B.

434
00:39:40,000 --> 00:39:44,000
So, I have my point P.
Here I have theta.

435
00:39:44,000 --> 00:39:49,000
This length is A.
Well, what are the components

436
00:39:49,000 --> 00:39:57,000
of BP?
Well, the X component is going

437
00:39:57,000 --> 00:39:59,000
to be?
Almost.

438
00:39:59,000 --> 00:40:03,000
I hear people saying things
about a, but I agree with a.

439
00:40:03,000 --> 00:40:04,000
I hear some cosines.
I hear some sines.

440
00:40:04,000 --> 00:40:07,000
I think it's actually the sine.
Yes.

441
00:40:07,000 --> 00:40:10,000
It's a(sin(theta)),
except it's going to the left.

442
00:40:10,000 --> 00:40:18,000
So, actually it will have a
negative a(sin(theta)).

443
00:40:18,000 --> 00:40:23,000
And, the vertical component,
well, it will be a(cos(theta)),

444
00:40:23,000 --> 00:40:27,000
but also negative because we
are going downwards.

445
00:40:27,000 --> 00:40:46,000
So, it's negative a(cos(theta)).
So, now we can answer the

446
00:40:46,000 --> 00:40:52,000
initial question because vector
OP, well, we just add up OA,

447
00:40:52,000 --> 00:40:57,000
AB, and BP.
So, the X component will be

448
00:40:57,000 --> 00:41:09,000
a(theta) - a(sin(theta)).
And, a-a(cos(theta)).

449
00:41:09,000 --> 00:41:25,000
OK.
So, any questions about that?

450
00:41:25,000 --> 00:41:29,000
OK, so, what's the answer?
Because this thing here is the

451
00:41:29,000 --> 00:41:35,000
x coordinate as a function of
theta, and that one is the y

452
00:41:35,000 --> 00:41:39,000
coordinate as a function of
theta.

453
00:41:39,000 --> 00:41:44,000
So, now, just to show you that
we can do a lot of things when

454
00:41:44,000 --> 00:41:48,000
we have a parametric equation,
here is a small mystery.

455
00:41:48,000 --> 00:41:54,000
So, what happens exactly near
the bottom point?

456
00:41:54,000 --> 00:41:57,000
What does the curve look like?
The computer tells us,

457
00:41:57,000 --> 00:41:59,000
well, it looks like it has some
sort of pointy thing,

458
00:41:59,000 --> 00:42:02,000
but isn't that something of a
display?

459
00:42:02,000 --> 00:42:12,000
Is it actually what happens?
So, what do you think happens

460
00:42:12,000 --> 00:42:19,000
near the bottom point?
Remember, we had that picture.

461
00:42:19,000 --> 00:42:24,000
Let me show you once more,
where you have these

462
00:42:24,000 --> 00:42:28,000
corner-like things at the
bottom.

463
00:42:28,000 --> 00:42:31,000
Well, actually,
is it indeed a corner with some

464
00:42:31,000 --> 00:42:34,000
angle between the two
directions?

465
00:42:34,000 --> 00:42:38,000
Does it make an angle?
Or, is it actually a smooth

466
00:42:38,000 --> 00:42:42,000
curve without any corner,
but we don't see it because

467
00:42:42,000 --> 00:42:46,000
it's too small to be visible on
the computer screen?

468
00:42:46,000 --> 00:42:50,000
Does it actually make a loop?
Does it actually come down and

469
00:42:50,000 --> 00:42:55,000
then back up without going to
the left or to the right and

470
00:42:55,000 --> 00:43:01,000
without making an angle?
So, yeah, I see the majority

471
00:43:01,000 --> 00:43:05,000
votes for answers number two or
four.

472
00:43:05,000 --> 00:43:08,000
And, well, at this point,
we can't quite tell.

473
00:43:08,000 --> 00:43:10,000
So, let's try to figure it out
from these formulas.

474
00:43:10,000 --> 00:43:17,000
The way to answer that for sure
is to actually look at the

475
00:43:17,000 --> 00:43:23,000
formulas.
OK, so question that we are

476
00:43:23,000 --> 00:43:34,000
trying to answer now is what
happens near the bottom point?

477
00:43:52,000 --> 00:43:58,000
OK, so how do we answer that?
Well, we should probably try to

478
00:43:58,000 --> 00:44:03,000
find simpler formulas for these
things.

479
00:44:03,000 --> 00:44:06,000
Well, to simplify,
let's divide everything by a.

480
00:44:06,000 --> 00:44:08,000
Let's rescale everything by a.
If you want,

481
00:44:08,000 --> 00:44:12,000
let's say that we take the unit
of length to be the radius of

482
00:44:12,000 --> 00:44:15,000
our wheel.
So, instead of measuring things

483
00:44:15,000 --> 00:44:18,000
in feet or meters,
we'll just measure them in

484
00:44:18,000 --> 00:44:25,000
radius.
So, take the length unit to be

485
00:44:25,000 --> 00:44:32,000
equal to the radius.
So, that means we'll have a=1.

486
00:44:32,000 --> 00:44:35,000
Then, our formulas are slightly
simpler.

487
00:44:35,000 --> 00:44:45,000
We get x(theta) is theta -
sin(theta), and y equals 1 - cos

488
00:44:45,000 --> 00:44:49,000
(theta).
OK, so, if we want to

489
00:44:49,000 --> 00:44:52,000
understand what these things
look like, maybe we should try

490
00:44:52,000 --> 00:44:56,000
to take some approximation.
OK, so what about

491
00:44:56,000 --> 00:45:00,000
approximations?
Well, probably you know that if

492
00:45:00,000 --> 00:45:07,000
I take the sine of a very small
angle, it's close to the actual

493
00:45:07,000 --> 00:45:12,000
angle itself if theta is very
small.

494
00:45:12,000 --> 00:45:18,000
And, you know that the cosine
of an angle that's very small is

495
00:45:18,000 --> 00:45:21,000
close to one.
Well, that's pretty good.

496
00:45:21,000 --> 00:45:23,000
If we use that,
we will get theta minus theta,

497
00:45:23,000 --> 00:45:26,000
one minus one,
it looks like it's not precise

498
00:45:26,000 --> 00:45:29,000
enough.
We just get zero and zero.

499
00:45:29,000 --> 00:45:31,000
That's not telling us much
about what happens.

500
00:45:31,000 --> 00:45:39,000
OK, so we need actually better
approximations than that.

501
00:45:39,000 --> 00:45:50,000
So -- So, hopefully you have
seen in one variable calculus

502
00:45:50,000 --> 00:45:57,000
something called Taylor
expansion.

503
00:45:57,000 --> 00:46:14,000
That's [GROANS].
I see that -- OK,

504
00:46:14,000 --> 00:46:17,000
so if you have not seen Taylor
expansion,

505
00:46:17,000 --> 00:46:21,000
or somehow it was so traumatic
that you've blocked it out of

506
00:46:21,000 --> 00:46:24,000
your memory,
let me just remind you that

507
00:46:24,000 --> 00:46:27,000
Taylor expansion is a way to get
a better approximation than just

508
00:46:27,000 --> 00:46:32,000
looking at the function,
its derivative.

509
00:46:32,000 --> 00:46:42,000
So -- And, here's an example of
where it actually comes in handy

510
00:46:42,000 --> 00:46:52,000
in real life.
So, Taylor approximation says

511
00:46:52,000 --> 00:47:01,000
that if t is small,
then the value of the function,

512
00:47:01,000 --> 00:47:04,000
f(t), is approximately equal
to,

513
00:47:04,000 --> 00:47:07,000
well, our first guess,
of course, would be f(0).

514
00:47:07,000 --> 00:47:12,000
That's our first approximation.
If we want to be a bit more

515
00:47:12,000 --> 00:47:15,000
precise, we know that when we
change by t,

516
00:47:15,000 --> 00:47:17,000
well, t times the derivative
comes in,

517
00:47:17,000 --> 00:47:23,000
that's for linear approximation
to how the function changes.

518
00:47:23,000 --> 00:47:28,000
Now, if we want to be even more
precise, there's another term,

519
00:47:28,000 --> 00:47:32,000
which is t^2 over two times the
second derivative.

520
00:47:32,000 --> 00:47:37,000
And, if we want to be even more
precise, you will have t^3 over

521
00:47:37,000 --> 00:47:41,000
six times the third derivative
at zero.

522
00:47:41,000 --> 00:47:43,000
OK, and you can continue,
and so on.

523
00:47:43,000 --> 00:47:49,000
But, we won't need more.
So, if you use this here,

524
00:47:49,000 --> 00:47:53,000
it tells you that the sine of a
smaller angle,

525
00:47:53,000 --> 00:47:57,000
theta, well,
yeah, it looks like theta.

526
00:47:57,000 --> 00:48:01,000
But, if we want to be more
precise, then we should add

527
00:48:01,000 --> 00:48:06,000
minus theta cubed over six.
And, cosine of theta,

528
00:48:06,000 --> 00:48:12,000
well, it's not quite one.
It's close to one minus theta

529
00:48:12,000 --> 00:48:16,000
squared over two.
OK, so these are slightly

530
00:48:16,000 --> 00:48:21,000
better approximations of sine
and cosine for small angles.

531
00:48:21,000 --> 00:48:28,000
So, now, if we try to figure
out, again, what happens to our

532
00:48:28,000 --> 00:48:31,000
x of theta, well,
it would be,

533
00:48:31,000 --> 00:48:36,000
sorry, theta minus theta cubed
over six.

534
00:48:36,000 --> 00:48:44,000
That's theta cubed over six.
And y, on the other hand,

535
00:48:44,000 --> 00:48:53,000
is going to be one minus that.
That's about theta squared over

536
00:48:53,000 --> 00:48:57,000
two.
So, now, which one of them is

537
00:48:57,000 --> 00:49:01,000
bigger when theta is small?
Yeah, y is much larger.

538
00:49:01,000 --> 00:49:03,000
OK, if you take the cube of a
very small number,

539
00:49:03,000 --> 00:49:06,000
it becomes very,
very, very small.

540
00:49:06,000 --> 00:49:09,000
So, in fact,
we can look at that.

541
00:49:09,000 --> 00:49:15,000
So, x, an absolute value,
is much smaller than y.

542
00:49:15,000 --> 00:49:17,000
And, in fact,
what we can do is we can look

543
00:49:17,000 --> 00:49:21,000
at the ratio between y and x.
That tells us the slope with

544
00:49:21,000 --> 00:49:27,000
which we approach the origin.
So, y over x is,

545
00:49:27,000 --> 00:49:35,000
well, let's take the ratio of
this, too.

546
00:49:35,000 --> 00:49:38,000
That gives us three divided by
theta.

547
00:49:38,000 --> 00:49:45,000
That tends to infinity when
theta approaches zero.

548
00:49:45,000 --> 00:49:53,000
So, that means that the slope
of our curve,

549
00:49:53,000 --> 00:50:00,000
the origin is actually
infinite.

550
00:50:00,000 --> 00:50:05,000
And so, the curve picture is
really something like this.

551
00:50:05,000 --> 00:50:07,000
So, the instantaneous motion,
if you had to describe what

552
00:50:07,000 --> 00:50:09,000
happens very,
very close to the origin is

553
00:50:09,000 --> 00:50:12,000
that your point is actually not
moving to the left or to the

554
00:50:12,000 --> 00:50:17,000
right along with the wheel.
It's moving down and up.

555
00:50:17,000 --> 00:50:20,000
I mean, at the same time it is
actually moving a little bit

556
00:50:20,000 --> 00:50:24,000
forward at the same time.
But, the dominant motion,

557
00:50:24,000 --> 00:50:29,000
near the origin is really where
it goes down and back up,

558
00:50:29,000 --> 00:50:33,000
so answer number four,
you have vertical tangent.

559
00:50:33,000 --> 00:50:37,000
OK, I think I'm at the end of
time.

560
00:50:37,000 --> 00:50:44,000
So, have a nice weekend.
And, I'll see you on Tuesday.

561
00:50:44,000 --> 00:50:47,000
So, on Tuesday I will have
practice exams for next week's

562
00:50:47,000 --> 00:50:50,000
test.