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PROFESSOR: Hi.
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Our lecture for today probably
should be entitled it
00:00:39.810 --> 00:00:42.690
should've been functions, but
it's analytic geometry
00:00:42.690 --> 00:00:46.950
instead, or a picture is
worth a thousand words.
00:00:46.950 --> 00:00:50.930
What we hope to do today is
to establish the fact that
00:00:50.930 --> 00:00:54.410
whereas in the study of calculus
when we deal with
00:00:54.410 --> 00:00:57.340
rate of change we are interested
in analytical
00:00:57.340 --> 00:01:01.600
terms, that more often than not,
we prefer to visualize
00:01:01.600 --> 00:01:05.770
things more intuitively in
terms of a graph or other
00:01:05.770 --> 00:01:10.160
suitable visual aid, and that
actually, this is not quite as
00:01:10.160 --> 00:01:14.950
alien or as profound as it
may at first glance seem.
00:01:14.950 --> 00:01:17.270
Consider, for example,
the businessman who
00:01:17.270 --> 00:01:20.780
says profits rose.
00:01:20.780 --> 00:01:21.770
Profits rose.
00:01:21.770 --> 00:01:25.870
Now, you know, profits don't
rise unless the safe blows up
00:01:25.870 --> 00:01:26.940
or something like this.
00:01:26.940 --> 00:01:31.650
What profits do is they increase
or they decrease.
00:01:31.650 --> 00:01:35.940
The reason that we say profits
rise is that when the profits
00:01:35.940 --> 00:01:39.920
are increasing, if we are
plotting profit in terms of
00:01:39.920 --> 00:01:46.690
time, the resulting graph
shows a rising tendency.
00:01:46.690 --> 00:01:50.670
As the profit increases,
the curve rises.
00:01:50.670 --> 00:01:54.080
And in other words then, we
begin to establish the feeling
00:01:54.080 --> 00:01:58.510
that we can identify the
analytic term increasing with
00:01:58.510 --> 00:02:00.700
the geometric term rising.
00:02:00.700 --> 00:02:04.410
And this identification, whereby
difficult arithmetic
00:02:04.410 --> 00:02:08.100
concepts are visualized
pictorially, is something that
00:02:08.100 --> 00:02:12.270
begins not only very early in
the history of man, but very
00:02:12.270 --> 00:02:15.560
early in the development of the
mathematics curriculum.
00:02:15.560 --> 00:02:19.820
Oh, as a case in point, consider
the problem of 5
00:02:19.820 --> 00:02:24.210
divided by 3 versus
6 divided by 3.
00:02:24.210 --> 00:02:26.920
I remember when I was in grade
school that this particular
00:02:26.920 --> 00:02:30.290
problem always seemed more
appealing to me than this
00:02:30.290 --> 00:02:34.440
problem, that 6 divided by 3
seemed natural, but 5 divided
00:02:34.440 --> 00:02:35.920
by 3 didn't.
00:02:35.920 --> 00:02:38.810
And the reason was is that in
terms of visualizing tally
00:02:38.810 --> 00:02:44.370
marks, it was much easier to see
how you divide six tallies
00:02:44.370 --> 00:02:47.840
into three groups than five
tallies into three groups.
00:02:47.840 --> 00:02:52.240
However, as soon as we
pick a length as our
00:02:52.240 --> 00:02:54.640
model, the idea is this.
00:02:54.640 --> 00:02:59.510
Either one can divide a line
into three parts of equal
00:02:59.510 --> 00:03:02.210
length or one can't
divide the line.
00:03:02.210 --> 00:03:05.135
Now, if I can geometrically
divide this line into three
00:03:05.135 --> 00:03:09.010
equal parts, and in plain
geometry we learn to do this,
00:03:09.010 --> 00:03:12.940
then the fact is that I can
divide this line segment into
00:03:12.940 --> 00:03:15.570
three equal parts regardless
of how long this
00:03:15.570 --> 00:03:16.930
line happens to be.
00:03:16.930 --> 00:03:22.760
Oh, to be sure, if this line
happens to be 6 units long,
00:03:22.760 --> 00:03:25.310
this point is named 2.
00:03:25.310 --> 00:03:29.440
And if, on the other hand, the
line happens to be only 5
00:03:29.440 --> 00:03:37.130
inches long, the resulting
point is named 5/3.
00:03:37.130 --> 00:03:41.290
But notice that in either case,
I can in a very natural
00:03:41.290 --> 00:03:47.110
way define or identify either
ratio as a point on the line.
00:03:47.110 --> 00:03:52.240
And this idea of identifying
numerical concepts called
00:03:52.240 --> 00:03:56.170
numbers with geometric concepts
called points is a
00:03:56.170 --> 00:04:00.130
very old device and a device
that was used and still is
00:04:00.130 --> 00:04:03.210
used in the curriculum today
under the name of the number
00:04:03.210 --> 00:04:07.470
line, under the name of graphs,
and what we will use
00:04:07.470 --> 00:04:11.620
as a fundamental building block
as our course develops.
00:04:11.620 --> 00:04:14.940
Now, you know, in the same way
that we can think of a single
00:04:14.940 --> 00:04:19.640
number as being a point on the
line, we can think of a pair
00:04:19.640 --> 00:04:23.380
of numbers, an ordered pair
of numbers, as being a
00:04:23.380 --> 00:04:25.210
point in the plane.
00:04:25.210 --> 00:04:29.150
This is Descartes geometry,
which we can call coordinate
00:04:29.150 --> 00:04:33.200
geometry, the idea being that
in the same way as we can
00:04:33.200 --> 00:04:37.170
locate a number of along the
x-axis, shall we say, we could
00:04:37.170 --> 00:04:41.970
have located a numbered pair
as a point in the plane.
00:04:41.970 --> 00:04:45.850
Namely, 2 comma 3 would mean the
point whose x-coordinate
00:04:45.850 --> 00:04:49.230
was 2 and whose y-coordinate
was 3.
00:04:49.230 --> 00:04:53.610
By the way, the reason that we
say ordered pairs is, if you
00:04:53.610 --> 00:04:57.490
observe, 2 comma 3 and
3 comma 2 happen to
00:04:57.490 --> 00:04:58.950
be different pairs.
00:04:58.950 --> 00:05:02.580
And notice again how vividly the
geometric interpretation
00:05:02.580 --> 00:05:03.640
of this is.
00:05:03.640 --> 00:05:07.130
Namely, in terms of locating a
point in space, it's obvious
00:05:07.130 --> 00:05:10.870
that the point named 2 comma 3
is not the same as the point
00:05:10.870 --> 00:05:12.760
named 3 comma 2.
00:05:12.760 --> 00:05:14.790
Again, the important
thing is this.
00:05:14.790 --> 00:05:18.400
When I think of 5 divided by 3,
when I think of the ordered
00:05:18.400 --> 00:05:23.460
pair 2 comma 3, I do not have
to think of a picture.
00:05:23.460 --> 00:05:26.010
I can think of these things
analytically.
00:05:26.010 --> 00:05:29.590
But the picture gives me certain
insights that will
00:05:29.590 --> 00:05:33.000
help me with my intuition, an
aid that I don't want to
00:05:33.000 --> 00:05:34.120
relinquish.
00:05:34.120 --> 00:05:36.790
For example, going back
to the graph again.
00:05:36.790 --> 00:05:41.190
Thinking of the analytic term
greater than, notice how much
00:05:41.190 --> 00:05:49.470
easier it is to think of, for
example, higher than, see, one
00:05:49.470 --> 00:05:53.308
point being higher than another
or to the right of.
00:05:58.680 --> 00:06:02.650
You see, geometric concepts to
name analytic statements, or
00:06:02.650 --> 00:06:04.150
instead of increasing,
as we mentioned
00:06:04.150 --> 00:06:06.450
before, to say rising.
00:06:06.450 --> 00:06:10.080
And there will be many, many
more such identifications as
00:06:10.080 --> 00:06:12.260
we go along with our course.
00:06:12.260 --> 00:06:14.660
At any rate, let's continue
to see then what is the
00:06:14.660 --> 00:06:18.180
relationship then between
functions that we talked about
00:06:18.180 --> 00:06:19.720
and graphs?
00:06:19.720 --> 00:06:23.270
The idea is something
like this.
00:06:23.270 --> 00:06:27.060
Let's return to our friend of
the first lecture: s equals
00:06:27.060 --> 00:06:28.730
16t squared.
00:06:28.730 --> 00:06:32.720
We can think of a distance
machine being the function
00:06:32.720 --> 00:06:40.590
where the input will be time and
the output will be what?
00:06:40.590 --> 00:06:44.130
The square of the input
multiplied by 16.
00:06:44.130 --> 00:06:49.150
Observe that from this, I do not
have to have any picture
00:06:49.150 --> 00:06:51.590
to understand what's
happening here.
00:06:51.590 --> 00:06:55.820
Namely, I can measure an input,
measure an output, and
00:06:55.820 --> 00:06:58.420
observe analytically
what is happening.
00:06:58.420 --> 00:07:02.620
However, as we saw last time,
our graph sort of shows us at
00:07:02.620 --> 00:07:05.820
a glance what seems to be
happening, that we can
00:07:05.820 --> 00:07:10.380
identify rising and falling with
increasing and decreasing
00:07:10.380 --> 00:07:11.930
and things of this type.
00:07:11.930 --> 00:07:15.020
We will explore this, of course,
in much more detail as
00:07:15.020 --> 00:07:17.820
we continue in our course.
00:07:17.820 --> 00:07:21.730
By the way, there's no reason
why the input has to be a
00:07:21.730 --> 00:07:22.750
single number.
00:07:22.750 --> 00:07:25.450
For example, why couldn't
the input be an
00:07:25.450 --> 00:07:27.350
ordered pair of numbers?
00:07:27.350 --> 00:07:30.310
Among other things, let's take
a simple geometric example.
00:07:30.310 --> 00:07:35.380
Consider, for example, finding
the volume of a cylinder in
00:07:35.380 --> 00:07:38.720
terms of the radius of its
base and the height.
00:07:38.720 --> 00:07:41.790
We know from solid geometry
that the volume is
00:07:41.790 --> 00:07:44.220
pi r squared h.
00:07:44.220 --> 00:07:47.970
We could therefore think of a
volume machine where the input
00:07:47.970 --> 00:07:53.210
is the ordered pair r comma h,
and the output is the single
00:07:53.210 --> 00:07:56.310
number pi r squared h.
00:07:56.310 --> 00:07:59.650
By the way, notice here the
meaning of ordered pair.
00:07:59.650 --> 00:08:04.030
You see, if the pair 2 comma 3
goes into the machine, notice
00:08:04.030 --> 00:08:05.790
that the recipe here
says what?
00:08:05.790 --> 00:08:08.120
You square the first
member of the pair.
00:08:08.120 --> 00:08:11.420
In other words, if 2 comma 3
is the input, we square 2,
00:08:11.420 --> 00:08:15.790
which is 4, multiplied by 3,
which is 12, and 12 times pi,
00:08:15.790 --> 00:08:17.950
of course, is 12 pi.
00:08:17.950 --> 00:08:21.820
On the other hand, if the input
is 3 comma 2, the first
00:08:21.820 --> 00:08:22.810
number is 3.
00:08:22.810 --> 00:08:25.480
Our recipe squares
the first number.
00:08:25.480 --> 00:08:31.640
That would be 9, times 2 is
18, times pi is 18 pi.
00:08:31.640 --> 00:08:35.659
But again, observe that I at
no time needed a picture to
00:08:35.659 --> 00:08:37.750
visualize what was
happening here.
00:08:37.750 --> 00:08:40.789
Of course, if I wanted a
picture, I could try to plot
00:08:40.789 --> 00:08:44.940
this also, but notice now that
my graph would probably need
00:08:44.940 --> 00:08:46.710
three dimensions to draw.
00:08:46.710 --> 00:08:48.820
And why would it need
three dimensions?
00:08:48.820 --> 00:08:52.930
Well, notice that my input has
two independent measurements r
00:08:52.930 --> 00:08:57.250
and h, and therefore, I would
need two dimensions just to
00:08:57.250 --> 00:08:58.860
take care of r and h.
00:08:58.860 --> 00:09:02.380
Then I would need a third
dimension to plot v.
00:09:02.380 --> 00:09:04.750
And by the way, notice
the next stage.
00:09:04.750 --> 00:09:08.520
If I had an input that consisted
of three independent
00:09:08.520 --> 00:09:12.210
measurements, this would still
make sense, but now I would be
00:09:12.210 --> 00:09:14.870
at a loss for the picture.
00:09:14.870 --> 00:09:17.190
In other words, what I'm trying
to bring out next is
00:09:17.190 --> 00:09:21.010
the fact that whereas pictures
are a tremendous help, maybe a
00:09:21.010 --> 00:09:24.900
second subtitle to our lecture
should've been a picture is
00:09:24.900 --> 00:09:27.505
worth a thousand words
provided you
00:09:27.505 --> 00:09:28.970
can could draw it.
00:09:28.970 --> 00:09:32.610
Because, you see, if we needed
three independent dimensions
00:09:32.610 --> 00:09:36.570
to locate the input and then
a fourth one to locate the
00:09:36.570 --> 00:09:37.995
output, how would we
draw the picture?
00:09:37.995 --> 00:09:40.820
By the way, this happens in high
school algebra again, if
00:09:40.820 --> 00:09:42.390
you want to see the analogy.
00:09:42.390 --> 00:09:44.140
Look at, for example,
the algebraic
00:09:44.140 --> 00:09:46.020
equation a plus b squared.
00:09:46.020 --> 00:09:50.230
We learned in algebra that this
is a squared plus 2ab
00:09:50.230 --> 00:09:51.920
plus b squared.
00:09:51.920 --> 00:09:56.020
Observe that we do not need to
have a picture to understand
00:09:56.020 --> 00:09:57.460
how this works.
00:09:57.460 --> 00:10:03.360
Oh, to be sure, if we had a
picture, we get a tremendous
00:10:03.360 --> 00:10:05.450
amount of insight as to
what's happening here.
00:10:05.450 --> 00:10:10.060
Namely, let's visualize a square
whose side is a plus b.
00:10:10.060 --> 00:10:13.920
On the one hand, you see, the
area of the square would be a
00:10:13.920 --> 00:10:17.080
plus b squared, you see,
the side squared.
00:10:17.080 --> 00:10:20.020
On the other hand, if we now
subdivide this figure this
00:10:20.020 --> 00:10:23.910
way, we see that that same
square is made up of four
00:10:23.910 --> 00:10:29.020
pieces having one piece of area
a squared, two pieces of
00:10:29.020 --> 00:10:32.840
area ab, and one piece
of area b squared.
00:10:32.840 --> 00:10:35.270
And so we see, on the other
hand, that the area of the
00:10:35.270 --> 00:10:40.470
square is a squared plus
2ab plus b squared.
00:10:40.470 --> 00:10:43.990
And so again, observe that
whereas this stands on its own
00:10:43.990 --> 00:10:47.020
two legs, the picture helps
us quite a bit.
00:10:47.020 --> 00:10:51.430
By the way, we could continue
this with a cube over here.
00:10:51.430 --> 00:10:55.950
Namely, it turns out that a plus
b cubed is a cubed plus
00:10:55.950 --> 00:11:01.600
3a squared b plus 3ab squared
plus b cubed.
00:11:01.600 --> 00:11:04.450
And again, if we wanted to,
and we won't take the time
00:11:04.450 --> 00:11:08.970
here, but if we wanted to, we
could now draw a cube whose
00:11:08.970 --> 00:11:10.670
side is a plus b.
00:11:10.670 --> 00:11:14.020
Namely, we could take the same
diagram that we had before and
00:11:14.020 --> 00:11:15.680
now make a third dimension
to it.
00:11:15.680 --> 00:11:18.210
And if you did that,
you would see what?
00:11:18.210 --> 00:11:21.850
That the cube whose volume is a
plus b cubed is divided into
00:11:21.850 --> 00:11:27.080
eight pieces, one of size a by
a by a, three of size a by a
00:11:27.080 --> 00:11:31.740
by b, three of size a by
b by b, and one of
00:11:31.740 --> 00:11:34.660
size b by b by b.
00:11:34.660 --> 00:11:38.540
Again, the picture is a
tremendous visual aid.
00:11:38.540 --> 00:11:40.620
Now, the key step is this.
00:11:40.620 --> 00:11:44.410
If we were to now give in to
our geometric intuition and
00:11:44.410 --> 00:11:47.720
say, lookit, why worry about
algebra when it's so much
00:11:47.720 --> 00:11:51.110
easier to do this thing by
geometry, the counterexample
00:11:51.110 --> 00:11:54.410
is consider a plus b to
the fourth power.
00:11:54.410 --> 00:11:57.800
Now, by the binomial theorem,
and I might add, the same
00:11:57.800 --> 00:12:01.150
binomial theorem that allowed
us to get these results, we
00:12:01.150 --> 00:12:02.880
can also write that
this is what?
00:12:02.880 --> 00:12:08.840
It's a is the fourth plus 4a
cubed b plus 6a squared b
00:12:08.840 --> 00:12:14.760
squared plus 4ab cubed
plus b to the fourth.
00:12:14.760 --> 00:12:17.730
Now, it's not important how
we get this result.
00:12:17.730 --> 00:12:20.640
The important point is that
analytically, we can raise a
00:12:20.640 --> 00:12:23.770
number to the fourth power just
as easily as we can to
00:12:23.770 --> 00:12:26.160
the third power or
the second power.
00:12:26.160 --> 00:12:30.800
The only difference is that in
the case of the third power,
00:12:30.800 --> 00:12:32.910
we had a picture that
we could use.
00:12:32.910 --> 00:12:36.910
In the fourth power case, we
didn't have a picture.
00:12:36.910 --> 00:12:39.660
And what a tragedy it would
have been to say, hey, we
00:12:39.660 --> 00:12:41.740
can't solve this problem
because we
00:12:41.740 --> 00:12:43.240
can't draw the picture.
00:12:43.240 --> 00:12:46.350
And by the way, as a rather
interesting aside, notice the
00:12:46.350 --> 00:12:49.230
geometric influence on
how we read this.
00:12:49.230 --> 00:12:52.260
This is called a plus b
to the fourth power.
00:12:52.260 --> 00:12:55.490
But somehow or other, we don't
call this one a plus b to the
00:12:55.490 --> 00:12:56.310
third power.
00:12:56.310 --> 00:13:01.440
We call it a plus b cubed,
suggesting the geometric
00:13:01.440 --> 00:13:03.110
configuration of the cube.
00:13:03.110 --> 00:13:06.650
And here we don't usually say a
plus b to the second power.
00:13:06.650 --> 00:13:10.220
We say a plus b squared.
00:13:10.220 --> 00:13:14.240
You see, the idea is that when
the picture is available, it
00:13:14.240 --> 00:13:17.740
gives us a tremendous insight as
to what can be done, and it
00:13:17.740 --> 00:13:21.440
helps us learn to visualize
what's happening analytically.
00:13:21.440 --> 00:13:24.830
In fact, what usually happens
is we use the picture to
00:13:24.830 --> 00:13:27.730
justify what's happening
analytically when we can see
00:13:27.730 --> 00:13:30.490
the picture and then just
carry the analytic part
00:13:30.490 --> 00:13:33.690
through unimpeded in
the case where we
00:13:33.690 --> 00:13:35.140
can't draw the picture.
00:13:35.140 --> 00:13:37.050
Let me give you another
example of this.
00:13:37.050 --> 00:13:38.800
Let's look at the
following set.
00:13:38.800 --> 00:13:41.790
That also should review our
language of sets for us.
00:13:41.790 --> 00:13:45.710
Let S be the set of all ordered
pairs x comma y such
00:13:45.710 --> 00:13:49.150
that x squared plus y
squared equals 25.
00:13:49.150 --> 00:13:54.910
Question: Does the ordered pair
3 comma 4 belong to S?
00:13:54.910 --> 00:13:56.470
Answer: Yes.
00:13:56.470 --> 00:13:57.360
How do we know?
00:13:57.360 --> 00:13:59.460
Well, we have a test
for membership.
00:13:59.460 --> 00:14:00.940
We're supposed to do what?
00:14:00.940 --> 00:14:05.640
Square each of the entries, each
of the numbers, add them,
00:14:05.640 --> 00:14:08.880
and if the answer is 25, then
that ordered pair belongs to
00:14:08.880 --> 00:14:14.610
S. 3 squared plus 4 squared is
25, so this pair belongs to S.
00:14:14.610 --> 00:14:16.120
How about 1 comma 2?
00:14:16.120 --> 00:14:21.260
Well, 1 squared plus 2 squared
is 1 plus 4, which is 5.
00:14:21.260 --> 00:14:26.920
5 is not equal to 25, so 1 comma
2 does not belong to S.
00:14:26.920 --> 00:14:30.070
Well, did we need any geometry
to be able to
00:14:30.070 --> 00:14:31.860
visualize this result?
00:14:31.860 --> 00:14:35.730
Hopefully, one did not
need any geometry to
00:14:35.730 --> 00:14:37.380
visualize this result.
00:14:37.380 --> 00:14:40.150
On the other hand then, what
does it mean in analytic
00:14:40.150 --> 00:14:44.100
geometry when we say that x
squared plus y squared equals
00:14:44.100 --> 00:14:46.580
25 is a circle?
00:14:46.580 --> 00:14:51.040
As badly as I draw, x squared
plus y squared equals 25 looks
00:14:51.040 --> 00:14:54.590
less like a circle than the
circle I drew over here.
00:14:54.590 --> 00:14:57.030
You see, what we really
mean is this.
00:14:57.030 --> 00:15:01.670
Consider all the points in the
plane x comma y for which x
00:15:01.670 --> 00:15:04.390
squared plus y squared
equals 25.
00:15:04.390 --> 00:15:08.480
These are precisely the points
on this particular circle.
00:15:08.480 --> 00:15:11.200
And the easiest way to see that,
of course, is that since
00:15:11.200 --> 00:15:15.850
the radius of the circle is 5
and the point x comma y means
00:15:15.850 --> 00:15:18.980
that this length is x and this
length is y, notice that from
00:15:18.980 --> 00:15:22.800
the Pythagorean theorem, we see
it once, that x squared
00:15:22.800 --> 00:15:26.970
plus y squared equals 25.
00:15:26.970 --> 00:15:31.940
Now again, the solution set to
this equation is our set S
00:15:31.940 --> 00:15:34.510
whether we're thinking of this
thing algebraically or
00:15:34.510 --> 00:15:35.640
geometrically.
00:15:35.640 --> 00:15:39.400
On the other hand, watch what
our picture seems to give us
00:15:39.400 --> 00:15:40.890
that we didn't have before.
00:15:40.890 --> 00:15:44.380
Let's return to our point 1
comma 2, which we saw didn't
00:15:44.380 --> 00:15:47.010
belong to S. Well,
look at this.
00:15:47.010 --> 00:15:48.630
Where would 1 comma 2 be?
00:15:48.630 --> 00:15:52.560
1 comma 2 would be inside
the circle.
00:15:52.560 --> 00:15:53.890
Why is that?
00:15:53.890 --> 00:16:01.080
Because, you see, if we take
the point 1 comma 2, if we
00:16:01.080 --> 00:16:04.470
take, say, for example, the
point 1 comma 2, notice that
00:16:04.470 --> 00:16:07.930
the distance from the origin
to the point 1 comma 2
00:16:07.930 --> 00:16:09.080
is less than 5.
00:16:09.080 --> 00:16:12.460
In other words, the distance is
less than 5 so the square
00:16:12.460 --> 00:16:14.660
of the distance is
less than 25.
00:16:14.660 --> 00:16:18.550
In other words, not only can we
say that 1 comma 2 does not
00:16:18.550 --> 00:16:21.420
belong to S, which we could have
said without the picture,
00:16:21.420 --> 00:16:24.080
we can now say what?
00:16:24.080 --> 00:16:26.410
1 comma 2 is--
00:16:26.410 --> 00:16:28.880
and notice the geometric
language here--
00:16:28.880 --> 00:16:30.780
is inside the circle.
00:16:34.590 --> 00:16:36.310
In other words, the study of
00:16:36.310 --> 00:16:39.210
inequalities can now be reduced.
00:16:39.210 --> 00:16:42.325
Instead of talking about less
than and greater than, we can
00:16:42.325 --> 00:16:46.060
now talk about such things
as inside and outside.
00:16:46.060 --> 00:16:48.520
You see, inside the circle,
which is a simple geometric
00:16:48.520 --> 00:16:50.340
concept, just means what?
00:16:50.340 --> 00:16:53.530
A set of all points for which
x squared plus y squared is
00:16:53.530 --> 00:16:54.920
less than 25.
00:16:54.920 --> 00:16:58.250
Outside that circle, x squared
plus y squared is
00:16:58.250 --> 00:16:59.700
greater than 25.
00:16:59.700 --> 00:17:04.040
On the circle, x squared plus
y squared equals 25.
00:17:04.040 --> 00:17:09.490
Again, a nice identification
between numbers and pictures,
00:17:09.490 --> 00:17:11.890
analysis and geometry.
00:17:11.890 --> 00:17:16.510
Well, this then shows us why
we want to study pictures
00:17:16.510 --> 00:17:18.160
rather than functions.
00:17:18.160 --> 00:17:21.300
Now, if we look at any textbook
in which we deal with
00:17:21.300 --> 00:17:24.500
graphs, it always seems that
we start with graphs of
00:17:24.500 --> 00:17:25.319
straight lines.
00:17:25.319 --> 00:17:28.359
And the question is what is so
great about a straight line?
00:17:28.359 --> 00:17:31.360
After all, pictures in general
are going to be much more
00:17:31.360 --> 00:17:32.480
complicated than that.
00:17:32.480 --> 00:17:36.180
What advantage is there in
starting with straight lines?
00:17:36.180 --> 00:17:39.680
And again, we begin to realize
how straight lines are the
00:17:39.680 --> 00:17:43.540
backbone of all types of
analytical procedures and all
00:17:43.540 --> 00:17:44.840
types of curve plotting.
00:17:44.840 --> 00:17:47.910
For example, let's suppose we
were studying this particular
00:17:47.910 --> 00:17:50.860
curve, and we wanted to know
what was happened to that
00:17:50.860 --> 00:17:55.310
curve in the neighborhood
around the point p.
00:17:55.310 --> 00:17:59.320
Let's draw in the tangent line
to the curve at the point p.
00:17:59.320 --> 00:18:03.090
Notice that this line that
we've drawn serves as a
00:18:03.090 --> 00:18:07.740
wonderful approximation curve
itself if we stay close enough
00:18:07.740 --> 00:18:09.520
to the point of tangency.
00:18:09.520 --> 00:18:12.590
In other words, notice how much
we can deduce about this
00:18:12.590 --> 00:18:16.910
curve if we study only the
straight line segment at the
00:18:16.910 --> 00:18:18.000
point of tangency.
00:18:18.000 --> 00:18:20.660
Of course, the approximation
gets worse and worse as we
00:18:20.660 --> 00:18:22.180
move further and further out.
00:18:22.180 --> 00:18:25.000
But in the neighborhood of the
point of what's going on,
00:18:25.000 --> 00:18:27.660
notice again then that the
straight line is an important
00:18:27.660 --> 00:18:28.820
building block.
00:18:28.820 --> 00:18:32.170
By the way, again we use
straight lines in a rather
00:18:32.170 --> 00:18:35.520
subtle way in something
called interpolation.
00:18:35.520 --> 00:18:38.370
For example, let's suppose I go
to a log table and I look
00:18:38.370 --> 00:18:40.030
up the log of 2.
00:18:40.030 --> 00:18:44.270
I find that the log
of 2 is 0.301.
00:18:44.270 --> 00:18:45.770
I look up the log of 4.
00:18:45.770 --> 00:18:48.900
I find that's 0.602.
00:18:48.900 --> 00:18:52.080
Now I look for the log of 3,
and I see that it's been
00:18:52.080 --> 00:18:53.660
obliterated.
00:18:53.660 --> 00:18:54.710
I don't know what it is.
00:18:54.710 --> 00:18:57.670
So I say, well, let me guess.
00:18:57.670 --> 00:19:00.640
3 is halfway between 2 and 4.
00:19:00.640 --> 00:19:04.670
Therefore, I would suspect that
the log of 3 is halfway
00:19:04.670 --> 00:19:07.570
between the log of 2
and the log of 4.
00:19:07.570 --> 00:19:10.030
And so, halfway between here
would be about what?
00:19:10.030 --> 00:19:13.880
0.452, roughly speaking.
00:19:13.880 --> 00:19:16.160
All of a sudden, the
obliteration on
00:19:16.160 --> 00:19:18.080
my book clears up.
00:19:18.080 --> 00:19:21.760
And I look, and I don't
find 0.452.
00:19:21.760 --> 00:19:24.660
Instead I find--
00:19:24.660 --> 00:19:27.120
well, let's write
it over here.
00:19:27.120 --> 00:19:32.960
What I find is 0.477.
00:19:32.960 --> 00:19:36.030
Now, you know, this is a pretty
big error to attribute
00:19:36.030 --> 00:19:38.800
to slide rule inaccuracy
or trouble in
00:19:38.800 --> 00:19:40.030
rounding off the tables.
00:19:40.030 --> 00:19:42.010
What really went wrong
over here?
00:19:42.010 --> 00:19:44.770
And the answer comes up again
that unless otherwise
00:19:44.770 --> 00:19:49.260
specified, the process known
as interpolation hinges on
00:19:49.260 --> 00:19:53.320
replacing a curve by a straight
line approximation.
00:19:53.320 --> 00:19:56.690
In fact, you see, if we were
to draw the curve of the
00:19:56.690 --> 00:19:59.940
logarithm function, we would
find that the picture is
00:19:59.940 --> 00:20:01.570
something like this.
00:20:01.570 --> 00:20:05.910
And when we looked up the log
of 2, this height is what we
00:20:05.910 --> 00:20:07.000
found in the table.
00:20:07.000 --> 00:20:11.400
When we looked up the log of
4, this height is what we
00:20:11.400 --> 00:20:15.670
would have found in the table.
00:20:15.670 --> 00:20:18.800
If we had looked up in the table
the log of 3, this is
00:20:18.800 --> 00:20:20.240
the height that we
would have found.
00:20:20.240 --> 00:20:23.500
This is the height
that's 0.477.
00:20:23.500 --> 00:20:27.630
Notice that in general, if we go
halfway from here to here,
00:20:27.630 --> 00:20:29.900
we do not go halfway
from here to here.
00:20:29.900 --> 00:20:31.770
It depends on the shape
of the curve.
00:20:31.770 --> 00:20:34.750
The only time you can be sure
that you have proportional
00:20:34.750 --> 00:20:38.600
parts is if the curve that
joined these two points was a
00:20:38.600 --> 00:20:41.620
straight line.
00:20:41.620 --> 00:20:44.860
And notice, by the way, that
by the shape of this curve,
00:20:44.860 --> 00:20:49.490
the straight line falls below
the curve, and therefore, the
00:20:49.490 --> 00:20:51.550
height that we found
was to the straight
00:20:51.550 --> 00:20:53.080
line, not to the curve.
00:20:53.080 --> 00:20:55.810
That was the point 0.452.
00:20:55.810 --> 00:20:58.470
So, in other words, notice that
we got smaller than the
00:20:58.470 --> 00:21:01.760
right answer because we
approximated as if it was a
00:21:01.760 --> 00:21:03.930
straight line that was
joining the curve.
00:21:03.930 --> 00:21:06.990
You see, what interpolation
hinges on is that the size of
00:21:06.990 --> 00:21:10.920
the interval is very small and
that you can assume that for
00:21:10.920 --> 00:21:13.850
the accuracy that you're
interested in that the
00:21:13.850 --> 00:21:17.120
straight line approximation to
the curve is sufficiently
00:21:17.120 --> 00:21:20.710
accurate to represent
the curve itself.
00:21:20.710 --> 00:21:22.200
Well, enough about that.
00:21:22.200 --> 00:21:25.990
Once we've talked about why
straight lines are important,
00:21:25.990 --> 00:21:28.850
the next thing is how do we
measure straight lines?
00:21:28.850 --> 00:21:31.770
See, another interesting point
to something like this.
00:21:31.770 --> 00:21:35.280
Many times we know what
something means subjectively,
00:21:35.280 --> 00:21:37.900
but we don't know what
it means objectively.
00:21:37.900 --> 00:21:41.350
For example, one way of finding
a line is to know two
00:21:41.350 --> 00:21:42.730
points on the line.
00:21:42.730 --> 00:21:45.280
Another way is to know
one point and the
00:21:45.280 --> 00:21:46.480
slant of the line.
00:21:46.480 --> 00:21:48.650
The question that comes up
is how do you measure
00:21:48.650 --> 00:21:50.180
the slant of a line?
00:21:50.180 --> 00:21:53.350
In other words, shall you say
that the line is very slanty?
00:21:53.350 --> 00:21:55.820
And if the answer to that is
yes, how do you distinguish
00:21:55.820 --> 00:22:01.460
between slanty and very slanty,
steep and very steep,
00:22:01.460 --> 00:22:03.240
very steep and very,
very steep?
00:22:03.240 --> 00:22:05.670
We need something
more objective.
00:22:05.670 --> 00:22:08.450
And the way we get around
this is as follows.
00:22:08.450 --> 00:22:11.850
Given a line, we define
the slope as follows.
00:22:11.850 --> 00:22:14.960
We pick any two points
on the line.
00:22:14.960 --> 00:22:18.880
And from those two points, we
can measure what we call the
00:22:18.880 --> 00:22:22.490
run of the line, in other words,
how far you've gone
00:22:22.490 --> 00:22:26.680
this way, and the rise
of the line, how much
00:22:26.680 --> 00:22:28.020
it's risen this way.
00:22:28.020 --> 00:22:32.780
And what we do is we define
the slope to be the rise
00:22:32.780 --> 00:22:38.790
divided by the run, or without
the delta notation in here, y2
00:22:38.790 --> 00:22:42.680
minus y1 over x2 minus x1.
00:22:42.680 --> 00:22:45.050
By the way, there are little
problems that come up.
00:22:45.050 --> 00:22:48.020
After all, our answer should
not depend on the picture.
00:22:48.020 --> 00:22:50.520
It should be sort of
self-contained analytically.
00:22:50.520 --> 00:22:53.790
The question comes up is what
if I had labeled this point
00:22:53.790 --> 00:22:58.350
x2, y2 and this point x1, y1?
00:22:58.350 --> 00:22:59.620
What would have happened then?
00:22:59.620 --> 00:23:03.090
And observe that as long as we
keep the pairs straight, it
00:23:03.090 --> 00:23:06.570
makes no difference whether you
write this or whether you
00:23:06.570 --> 00:23:07.430
write this.
00:23:07.430 --> 00:23:09.600
Because, you see, in each
case, all you've done is
00:23:09.600 --> 00:23:13.540
change the sign, and negative
over negative is positive.
00:23:13.540 --> 00:23:17.060
So certainly our answer to what
a slope is is objective
00:23:17.060 --> 00:23:19.930
enough, so it does not depend on
how the points are labeled.
00:23:19.930 --> 00:23:22.240
A second objection that most
people have is they say
00:23:22.240 --> 00:23:25.140
something like, well, who are
you to say that we pick these
00:23:25.140 --> 00:23:25.770
two points?
00:23:25.770 --> 00:23:29.050
What if I came along and picked
these two points and I
00:23:29.050 --> 00:23:33.460
now computed the slope by taking
this as my delta y and
00:23:33.460 --> 00:23:35.450
this as my delta x?
00:23:35.450 --> 00:23:38.250
Obviously, it would be a tragedy
if the answer to the
00:23:38.250 --> 00:23:41.315
problem depended on which pair
of points you picked since a
00:23:41.315 --> 00:23:43.730
line should have
but one slope.
00:23:43.730 --> 00:23:46.750
Again, notice that our high
school training in geometry,
00:23:46.750 --> 00:23:50.540
similar triangles, motivates
why we pick ratios.
00:23:50.540 --> 00:23:54.460
Namely, while this delta y and
this delta y may be different
00:23:54.460 --> 00:23:57.510
and this delta x and this delta
x may be different, what
00:23:57.510 --> 00:24:02.160
is true is that the ratio of
this delta y to this delta x
00:24:02.160 --> 00:24:05.830
is the same as the ratio of this
delta y to this delta x.
00:24:05.830 --> 00:24:08.040
And that's why we
pick the ratio.
00:24:08.040 --> 00:24:11.110
By the way, another way of
talking about ratio is if you
00:24:11.110 --> 00:24:14.460
look at delta y divided by delta
x and you've had some
00:24:14.460 --> 00:24:17.330
trigonometry, it reminds
you of a trigonometric
00:24:17.330 --> 00:24:19.090
relationship.
00:24:19.090 --> 00:24:23.000
Namely, you look at delta y, you
look at delta x, and you
00:24:23.000 --> 00:24:25.510
say, my, isn't that just
the tangent of
00:24:25.510 --> 00:24:26.960
this particular angle?
00:24:26.960 --> 00:24:28.650
Couldn't I define the slope?
00:24:28.650 --> 00:24:31.480
And by the way, the general
symbol for slope, for better
00:24:31.480 --> 00:24:33.550
or for worse, just happens
to be letter m.
00:24:33.550 --> 00:24:39.280
Why couldn't I define m to be
the tangent of phi, where phi
00:24:39.280 --> 00:24:41.590
is the angle that the straight
line makes with
00:24:41.590 --> 00:24:43.050
the positive x-axis?
00:24:43.050 --> 00:24:45.180
And, of course, there is a
little subtlety here that we
00:24:45.180 --> 00:24:46.600
should pay attention to.
00:24:46.600 --> 00:24:49.970
This would be an ambiguous
definition if the scale on the
00:24:49.970 --> 00:24:52.380
x- and the y-axis were
not the same.
00:24:52.380 --> 00:24:55.360
In other words, notice that by
changing the scale here, I can
00:24:55.360 --> 00:24:58.330
distort the same analytic
information.
00:24:58.330 --> 00:25:02.180
So if I agree, however, that the
unit on the x-axis is the
00:25:02.180 --> 00:25:05.980
same as the unit on the y-axis,
then I can say, OK,
00:25:05.980 --> 00:25:09.300
the slope is also tangent
of the angle phi.
00:25:09.300 --> 00:25:12.670
I much prefer to say it's delta
y divided by delta x,
00:25:12.670 --> 00:25:14.400
because then if I forget
the scale,
00:25:14.400 --> 00:25:16.010
I'm still in no trouble.
00:25:16.010 --> 00:25:20.180
On the other hand, if we use the
tangent definition, we can
00:25:20.180 --> 00:25:23.670
utilize all we know about
trigonometry to get some other
00:25:23.670 --> 00:25:25.190
interesting results.
00:25:25.190 --> 00:25:27.800
Namely, the question that might
come up is can we study
00:25:27.800 --> 00:25:31.330
the slopes of two different
lines very conveniently in
00:25:31.330 --> 00:25:33.540
terms of our definition
of slope?
00:25:33.540 --> 00:25:35.330
And the answer is this.
00:25:35.330 --> 00:25:39.630
If we imagine now that our lines
are drawn to scale here,
00:25:39.630 --> 00:25:42.560
and here are two different
lines, which I'll call l1 and
00:25:42.560 --> 00:25:46.400
l2, and we'll call the angle
that l1 makes with the
00:25:46.400 --> 00:25:50.280
positive x-axis phi 1, the angle
that l2 makes with the
00:25:50.280 --> 00:25:52.760
positive x-axis phi 2.
00:25:52.760 --> 00:25:55.220
Therefore, what? m1
is tan phi 1.
00:25:55.220 --> 00:25:57.060
m2 is tan phi 2.
00:25:57.060 --> 00:26:01.100
Notice that our formula for the
tangent of the difference
00:26:01.100 --> 00:26:03.760
of two angles-- you see,
notice that this
00:26:03.760 --> 00:26:04.890
angle here is what?
00:26:04.890 --> 00:26:08.960
Since this angle is the sum of
these two, this angle here is
00:26:08.960 --> 00:26:13.580
phi 2 minus phi 1 or the
negative of phi 1 minus phi 2.
00:26:13.580 --> 00:26:16.220
I should have had this phi 2
minus phi 2, but since that
00:26:16.220 --> 00:26:19.020
just changes the sign, that will
not have any bearing on
00:26:19.020 --> 00:26:20.300
the point I want to make.
00:26:20.300 --> 00:26:22.220
Let's continue this way.
00:26:22.220 --> 00:26:27.230
Tangent of phi 1 minus phi 2 is
tan phi 1 minus tan phi 2
00:26:27.230 --> 00:26:30.570
over 1 plus tan phi
1 tan phi 2.
00:26:30.570 --> 00:26:33.500
On the other hand, by our
definitions of m1 and m2, this
00:26:33.500 --> 00:26:37.080
is m1 minus m2 over
1 plus m1 m2.
00:26:40.220 --> 00:26:44.360
Now, this tells me how to find
the angle between two lines
00:26:44.360 --> 00:26:46.730
just in terms of knowing
the slope.
00:26:46.730 --> 00:26:50.670
Two very special interesting
cases as extremes suggest
00:26:50.670 --> 00:26:52.110
themselves right away.
00:26:52.110 --> 00:26:55.760
One case is what happens if
the lines are parallel?
00:26:55.760 --> 00:27:00.020
If the lines are parallel, you
see, phi 1 equals phi 2, in
00:27:00.020 --> 00:27:04.350
which case phi 1 minus phi
2 is 0, in which case the
00:27:04.350 --> 00:27:08.110
tangent of phi 1 minus phi
2 had better be 0.
00:27:08.110 --> 00:27:12.060
But the only way a fraction can
be 0 is for the numerator
00:27:12.060 --> 00:27:16.120
to be 0, and that means
that m1 must equal m2.
00:27:16.120 --> 00:27:19.420
In other words, in terms of
slopes, we can study parallel
00:27:19.420 --> 00:27:22.310
lines just by equating
their slopes.
00:27:22.310 --> 00:27:25.590
A less obvious relationship
that's equally important is
00:27:25.590 --> 00:27:29.200
how do you measure whether two
lines are perpendicular?
00:27:29.200 --> 00:27:31.700
The answer is if they're
perpendicular, the angle
00:27:31.700 --> 00:27:33.870
between them is 90 degrees.
00:27:33.870 --> 00:27:37.330
The tangent of 90 degrees is
infinity, as we learned.
00:27:37.330 --> 00:27:39.430
That's equivalent
to saying what?
00:27:39.430 --> 00:27:42.270
That the denominator is 0.
00:27:42.270 --> 00:27:44.230
See, the only way a fraction
blows up is for the
00:27:44.230 --> 00:27:45.520
denominator to be 0.
00:27:45.520 --> 00:27:50.400
But the only way that 1 plus
m1 m2 can be 0 is for what?
00:27:50.400 --> 00:27:54.260
m1 m2 to be equal to minus 1.
00:27:54.260 --> 00:27:57.780
And this gives us the other very
well-known result that in
00:27:57.780 --> 00:28:00.250
terms of slopes, to study
whether two lines are
00:28:00.250 --> 00:28:03.510
perpendicular, all we need to
investigate is whether one
00:28:03.510 --> 00:28:07.570
slope is the negative reciprocal
of the other.
00:28:07.570 --> 00:28:10.070
Well, again, the textbook
will bring out
00:28:10.070 --> 00:28:11.630
slopes in more detail.
00:28:11.630 --> 00:28:14.220
The next question that we'd like
to bring up in terms of a
00:28:14.220 --> 00:28:19.000
picture is worth a thousand
words is how do you identify
00:28:19.000 --> 00:28:21.780
the straight line with an
algebraic equation?
00:28:21.780 --> 00:28:24.690
What do we mean by the equation
of a straight line?
00:28:24.690 --> 00:28:27.790
Well, again, there are
two possibilities.
00:28:27.790 --> 00:28:30.150
The first possibility
is that the line is
00:28:30.150 --> 00:28:33.020
parallel to the y-axis.
00:28:33.020 --> 00:28:36.580
If the line is parallel to the
y-axis, if the line goes
00:28:36.580 --> 00:28:40.710
through the point a comma 0,
notice that the only criteria
00:28:40.710 --> 00:28:43.940
for the point to be on
that line is that its
00:28:43.940 --> 00:28:46.530
x-coordinate equal a.
00:28:46.530 --> 00:28:49.940
By the way, this is often
abbreviated in the textbook as
00:28:49.940 --> 00:28:52.030
the line x equals a.
00:28:52.030 --> 00:28:54.270
Many a student says how do
you know this is a line?
00:28:54.270 --> 00:28:56.740
Why isn't this the
point x equals a?
00:28:56.740 --> 00:28:59.750
And here again is a good review
of why we stress the
00:28:59.750 --> 00:29:01.480
language of sets.
00:29:01.480 --> 00:29:05.030
Here again is a good reason why
we express the language of
00:29:05.030 --> 00:29:06.530
sets so strongly.
00:29:06.530 --> 00:29:09.060
Namely, go back to the universe
of discourse here.
00:29:09.060 --> 00:29:12.900
When you see the set of all
ordered pairs x comma y for
00:29:12.900 --> 00:29:15.730
which x equals a, this gives
you the hint that you're
00:29:15.730 --> 00:29:18.760
talking about pairs of points,
and that tells you that you
00:29:18.760 --> 00:29:22.710
have numbers in the plane, not
on the line, not on the
00:29:22.710 --> 00:29:26.650
x-axis, a two-dimensional
interpretation over here.
00:29:26.650 --> 00:29:30.400
You see, if this said the set of
all x such that x equals a,
00:29:30.400 --> 00:29:31.850
it would just be a point.
00:29:31.850 --> 00:29:33.900
But notice the hint over here.
00:29:33.900 --> 00:29:37.030
At any rate, this then becomes
the equation of a straight
00:29:37.030 --> 00:29:40.660
line if the straight line is
parallel to the y-axis.
00:29:40.660 --> 00:29:42.710
Of course, the other possibility
is what if the
00:29:42.710 --> 00:29:45.590
line isn't parallel
to the x-axis?
00:29:45.590 --> 00:29:48.870
And here, too, we say OK,
suppose we know a point on the
00:29:48.870 --> 00:29:51.500
line and suppose we know
the slope of the line.
00:29:51.500 --> 00:29:54.750
What we will do is pick any
other point on the plane,
00:29:54.750 --> 00:29:58.640
which we will label arbitrarily
x comma y, and see
00:29:58.640 --> 00:30:02.100
what the equation x comma
y has to satisfy.
00:30:02.100 --> 00:30:05.340
How do the coordinates have to
be related to be on this line?
00:30:05.340 --> 00:30:09.820
Well, we already know that slope
does not depend on which
00:30:09.820 --> 00:30:11.200
two points you pick.
00:30:11.200 --> 00:30:15.690
Consequently, since the slope
of this line is m, the slope
00:30:15.690 --> 00:30:17.020
must also be what?
00:30:17.020 --> 00:30:21.250
y minus y1 over x minus x1.
00:30:21.250 --> 00:30:24.790
And this becomes the fundamental
definition for the
00:30:24.790 --> 00:30:29.260
equation of a line which is not
parallel to the y-axis.
00:30:29.260 --> 00:30:34.280
And by the way, again, I think
m's and x1's and y1's tend to
00:30:34.280 --> 00:30:37.080
give you a bit of hardship
at first until
00:30:37.080 --> 00:30:38.250
you get used to them.
00:30:38.250 --> 00:30:42.300
Let's illustrate this thing
with a specific example.
00:30:42.300 --> 00:30:46.090
Suppose I say to you I am
thinking of the line whose
00:30:46.090 --> 00:30:50.600
slope is 3 and which passes
through the point 2 comma 5.
00:30:50.600 --> 00:30:52.280
And notice the language
of sets here.
00:30:52.280 --> 00:30:55.600
To say that 2 comma 5 is on the
line is the same as saying
00:30:55.600 --> 00:30:58.910
that 2 comma 5 belongs
to the set of points
00:30:58.910 --> 00:31:00.440
determined by the line.
00:31:00.440 --> 00:31:03.520
Drawing a rough sketch
over here--
00:31:03.520 --> 00:31:06.340
and by the way, notice something
very important here.
00:31:06.340 --> 00:31:09.680
I never have to draw to scale.
00:31:09.680 --> 00:31:12.370
Because, you see, all I'm going
to use is the analytic
00:31:12.370 --> 00:31:16.370
terms, and 2 comma 5 is still 2
and 5, no matter how I draw
00:31:16.370 --> 00:31:17.280
the picture.
00:31:17.280 --> 00:31:20.670
So, for example, if I say OK,
let's see what it means for
00:31:20.670 --> 00:31:24.020
the point x comma y to belong
here, I say, well,
00:31:24.020 --> 00:31:25.020
what does that mean?
00:31:25.020 --> 00:31:28.330
My slope is going to have
to be what? y minus 5.
00:31:28.330 --> 00:31:29.640
That's my rise.
00:31:29.640 --> 00:31:35.170
My run is x minus 2, and
that must equal 3.
00:31:35.170 --> 00:31:38.780
And if I clear this of
fractions, I get what? y is
00:31:38.780 --> 00:31:47.610
equal to 3x minus 1.
00:31:47.610 --> 00:31:48.490
By the way, does
this check out?
00:31:48.490 --> 00:31:52.680
If x is 2, 2 times 3
is 6, minus 1 is 5.
00:31:52.680 --> 00:31:54.960
2 comma 5 is on the line.
00:31:54.960 --> 00:31:56.120
You see, here's the thing.
00:31:56.120 --> 00:31:58.630
We talked about the line
geometrically.
00:31:58.630 --> 00:32:00.850
Now I have an algebraic
equation.
00:32:00.850 --> 00:32:02.950
I no longer have to refer
to the picture.
00:32:02.950 --> 00:32:04.730
I have something analytic now.
00:32:04.730 --> 00:32:07.450
For example, suppose a person
says to me I wonder if the
00:32:07.450 --> 00:32:10.810
point 8 comma 23 is
on this line?
00:32:10.810 --> 00:32:12.470
I don't have to draw
a picture to scale.
00:32:12.470 --> 00:32:14.040
I don't have to waste
any time.
00:32:14.040 --> 00:32:18.650
I know that the equation of my
line is y equals 3x minus 1.
00:32:18.650 --> 00:32:25.720
By the way, if y equals 3x minus
1, as soon as x is 8,
00:32:25.720 --> 00:32:27.550
what must y equal?
00:32:27.550 --> 00:32:31.310
y must equal what?
00:32:31.310 --> 00:32:34.260
23?
00:32:34.260 --> 00:32:35.540
Is that right?
00:32:35.540 --> 00:32:42.130
And so is the point 8 comma
23 on the line?
00:32:42.130 --> 00:32:42.900
Yes.
00:32:42.900 --> 00:32:46.350
How about 8 comma 12?
00:32:46.350 --> 00:32:51.560
8 comma 12 isn't on the
line because 3 times 8
00:32:51.560 --> 00:32:53.370
minus 1 is not 12.
00:32:53.370 --> 00:32:56.800
But notice that we can even see
algebraically that 8 comma
00:32:56.800 --> 00:33:00.170
12 must be below the line.
00:33:00.170 --> 00:33:04.390
In other words, our study of
equations allows us not only
00:33:04.390 --> 00:33:07.800
to visualize lines as equations,
but we can also
00:33:07.800 --> 00:33:11.230
visualize inequalities
as pictures.
00:33:11.230 --> 00:33:18.000
In other words, if we have the
equation of a line, if this is
00:33:18.000 --> 00:33:24.570
the line y equals, say, 3x plus
1 or something like this,
00:33:24.570 --> 00:33:26.770
then what is this region here?
00:33:26.770 --> 00:33:30.310
These are all those values which
lie-- whose heights lie
00:33:30.310 --> 00:33:32.710
below the height to
be on the curve.
00:33:32.710 --> 00:33:36.140
Again, not a very clear example
in the sense of
00:33:36.140 --> 00:33:39.090
drawing the picture neatly for
you, but our main aim is not
00:33:39.090 --> 00:33:40.580
to draw neat pictures here.
00:33:40.580 --> 00:33:43.950
Our main aim is to show how
analytical terms can be
00:33:43.950 --> 00:33:47.350
studied very conveniently
in terms of pictures.
00:33:47.350 --> 00:33:50.560
In fact, perhaps to conclude
today's lesson, what we should
00:33:50.560 --> 00:33:54.550
talk about is an old algebraic
concept called
00:33:54.550 --> 00:33:56.390
simultaneous equations.
00:33:56.390 --> 00:34:00.270
Suppose you're asked to solve
this pair of equations.
00:34:00.270 --> 00:34:03.490
You say, well, let's see, if y
equals 3x minus 1 and it's
00:34:03.490 --> 00:34:08.699
also equal to x plus 1, that
says that x plus 1
00:34:08.699 --> 00:34:11.159
equals 3x minus 1.
00:34:11.159 --> 00:34:13.510
I now solve this thing
algebraically.
00:34:13.510 --> 00:34:18.050
I get 2x equals 2,
so x equals 1.
00:34:18.050 --> 00:34:23.630
Knowing that x equals 1, I can
see that y equals 2, and I see
00:34:23.630 --> 00:34:28.030
that 1 comma 2 is my solution.
00:34:28.030 --> 00:34:30.620
In other words, if I wound up
with this thing algebraically,
00:34:30.620 --> 00:34:35.409
1 comma 2 is the only member
that belongs to both of these
00:34:35.409 --> 00:34:37.520
two solution sets.
00:34:37.520 --> 00:34:41.690
Now, again, notice how I can
solve this purely analytic
00:34:41.690 --> 00:34:45.530
problem without recourse
to a picture.
00:34:45.530 --> 00:34:48.510
On the other hand, if I want
to think of this thing
00:34:48.510 --> 00:34:52.989
pictorially, notice that y
equals 3x minus 1 is the
00:34:52.989 --> 00:34:58.040
equation of a particular
straight line, and y equals x
00:34:58.040 --> 00:35:02.450
plus 1 is also the equation
of a line.
00:35:02.450 --> 00:35:05.850
Notice that since these two
lines are not parallel, they
00:35:05.850 --> 00:35:08.420
intersect at one particular
point.
00:35:08.420 --> 00:35:12.530
And the geometric problem that I
solved on the previous board
00:35:12.530 --> 00:35:16.820
turns out to be that the point
1 comma 2 is the point that
00:35:16.820 --> 00:35:19.040
both of these lines
have in common.
00:35:19.040 --> 00:35:23.640
In fact, if we call this line
as before l1 and if we name
00:35:23.640 --> 00:35:30.270
this l2, in the language of
sets, 1 comma 2 is what?
00:35:30.270 --> 00:35:33.560
The point which is the
intersection of the two lines
00:35:33.560 --> 00:35:38.190
l1 and l2, again, a geometric
interpretation
00:35:38.190 --> 00:35:39.830
for an analytic problem.
00:35:39.830 --> 00:35:43.380
In fact, notice also how much
mileage I can get out of the
00:35:43.380 --> 00:35:44.910
geometric picture.
00:35:44.910 --> 00:35:49.170
For example, notice that this
region here has a very nice
00:35:49.170 --> 00:35:50.870
geometric interpretation.
00:35:50.870 --> 00:35:52.050
It's the set of what?
00:35:52.050 --> 00:35:57.620
All points which are below this
line and above this line.
00:35:57.620 --> 00:36:00.010
In other words, what?
00:36:00.010 --> 00:36:04.970
To be below this line, y must be
less than x plus 1, and to
00:36:04.970 --> 00:36:09.880
be above this line, y must be
greater than 3x minus 1.
00:36:09.880 --> 00:36:13.430
Notice then that a pair of
simultaneous inequalities,
00:36:13.430 --> 00:36:16.510
which may not be that easy to
handle, are very easy to
00:36:16.510 --> 00:36:19.890
handle in terms of regions
in the plane.
00:36:19.890 --> 00:36:23.580
Notice also that since two lines
can either be parallel
00:36:23.580 --> 00:36:27.660
or not parallel, we also get a
nice geometric interpretation
00:36:27.660 --> 00:36:30.270
as to why simultaneous
equations
00:36:30.270 --> 00:36:32.080
may have one solution.
00:36:32.080 --> 00:36:34.240
Namely, the lines are
not parallel,
00:36:34.240 --> 00:36:35.540
and hence, they intersect.
00:36:35.540 --> 00:36:38.610
Or no solutions, the lines
could've been parallel without
00:36:38.610 --> 00:36:39.440
intersecting.
00:36:39.440 --> 00:36:43.500
Or infinitely many solutions,
the two lines could have been
00:36:43.500 --> 00:36:45.350
different equations.
00:36:45.350 --> 00:36:47.490
In effect, I should say what?
00:36:47.490 --> 00:36:50.130
The two equations could've been
different equations for
00:36:50.130 --> 00:36:51.290
the same line.
00:36:51.290 --> 00:36:55.300
Again, this may seem a little
bit sketchy and rapid, but all
00:36:55.300 --> 00:36:57.250
we want to do is give
the overview.
00:36:57.250 --> 00:37:00.460
The reading assignment in the
text goes into great detail on
00:37:00.460 --> 00:37:02.370
the points that we've
mentioned so far.
00:37:02.370 --> 00:37:05.760
But again, in summary, what our
lesson was supposed to be
00:37:05.760 --> 00:37:09.640
today was to indicate the
importance of being able to
00:37:09.640 --> 00:37:13.880
visualize and to identify
analytic results with
00:37:13.880 --> 00:37:15.830
geometric pictures.
00:37:15.830 --> 00:37:17.680
And so, until next
time, goodbye.
00:37:20.840 --> 00:37:24.040
Funding for the publication of
this video was provided by the
00:37:24.040 --> 00:37:28.090
Gabriella and Paul Rosenbaum
Foundation.
00:37:28.090 --> 00:37:32.270
Help OCW continue to provide
free and open access to MIT
00:37:32.270 --> 00:37:36.460
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