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PROFESSOR: Hi.
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Our lecture for today probably
should be entitled it
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00:00:39,810 --> 00:00:42,690
should've been functions, but
it's analytic geometry
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00:00:42,690 --> 00:00:46,950
instead, or a picture is
worth a thousand words.
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00:00:46,950 --> 00:00:50,930
What we hope to do today is
to establish the fact that
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whereas in the study of calculus
when we deal with
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00:00:54,410 --> 00:00:57,340
rate of change we are interested
in analytical
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00:00:57,340 --> 00:01:01,600
terms, that more often than not,
we prefer to visualize
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00:01:01,600 --> 00:01:05,770
things more intuitively in
terms of a graph or other
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00:01:05,770 --> 00:01:10,160
suitable visual aid, and that
actually, this is not quite as
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00:01:10,160 --> 00:01:14,950
alien or as profound as it
may at first glance seem.
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00:01:14,950 --> 00:01:17,270
Consider, for example,
the businessman who
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00:01:17,270 --> 00:01:20,780
says profits rose.
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00:01:20,780 --> 00:01:21,770
Profits rose.
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00:01:21,770 --> 00:01:25,870
Now, you know, profits don't
rise unless the safe blows up
25
00:01:25,870 --> 00:01:26,940
or something like this.
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00:01:26,940 --> 00:01:31,650
What profits do is they increase
or they decrease.
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00:01:31,650 --> 00:01:35,940
The reason that we say profits
rise is that when the profits
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00:01:35,940 --> 00:01:39,920
are increasing, if we are
plotting profit in terms of
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00:01:39,920 --> 00:01:46,690
time, the resulting graph
shows a rising tendency.
30
00:01:46,690 --> 00:01:50,670
As the profit increases,
the curve rises.
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00:01:50,670 --> 00:01:54,080
And in other words then, we
begin to establish the feeling
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00:01:54,080 --> 00:01:58,510
that we can identify the
analytic term increasing with
33
00:01:58,510 --> 00:02:00,700
the geometric term rising.
34
00:02:00,700 --> 00:02:04,410
And this identification, whereby
difficult arithmetic
35
00:02:04,410 --> 00:02:08,100
concepts are visualized
pictorially, is something that
36
00:02:08,100 --> 00:02:12,270
begins not only very early in
the history of man, but very
37
00:02:12,270 --> 00:02:15,560
early in the development of the
mathematics curriculum.
38
00:02:15,560 --> 00:02:19,820
Oh, as a case in point, consider
the problem of 5
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00:02:19,820 --> 00:02:24,210
divided by 3 versus
6 divided by 3.
40
00:02:24,210 --> 00:02:26,920
I remember when I was in grade
school that this particular
41
00:02:26,920 --> 00:02:30,290
problem always seemed more
appealing to me than this
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00:02:30,290 --> 00:02:34,440
problem, that 6 divided by 3
seemed natural, but 5 divided
43
00:02:34,440 --> 00:02:35,920
by 3 didn't.
44
00:02:35,920 --> 00:02:38,810
And the reason was is that in
terms of visualizing tally
45
00:02:38,810 --> 00:02:44,370
marks, it was much easier to see
how you divide six tallies
46
00:02:44,370 --> 00:02:47,840
into three groups than five
tallies into three groups.
47
00:02:47,840 --> 00:02:52,240
However, as soon as we
pick a length as our
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00:02:52,240 --> 00:02:54,640
model, the idea is this.
49
00:02:54,640 --> 00:02:59,510
Either one can divide a line
into three parts of equal
50
00:02:59,510 --> 00:03:02,210
length or one can't
divide the line.
51
00:03:02,210 --> 00:03:05,135
Now, if I can geometrically
divide this line into three
52
00:03:05,135 --> 00:03:09,010
equal parts, and in plain
geometry we learn to do this,
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00:03:09,010 --> 00:03:12,940
then the fact is that I can
divide this line segment into
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00:03:12,940 --> 00:03:15,570
three equal parts regardless
of how long this
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00:03:15,570 --> 00:03:16,930
line happens to be.
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00:03:16,930 --> 00:03:22,760
Oh, to be sure, if this line
happens to be 6 units long,
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00:03:22,760 --> 00:03:25,310
this point is named 2.
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00:03:25,310 --> 00:03:29,440
And if, on the other hand, the
line happens to be only 5
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00:03:29,440 --> 00:03:37,130
inches long, the resulting
point is named 5/3.
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00:03:37,130 --> 00:03:41,290
But notice that in either case,
I can in a very natural
61
00:03:41,290 --> 00:03:47,110
way define or identify either
ratio as a point on the line.
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00:03:47,110 --> 00:03:52,240
And this idea of identifying
numerical concepts called
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00:03:52,240 --> 00:03:56,170
numbers with geometric concepts
called points is a
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00:03:56,170 --> 00:04:00,130
very old device and a device
that was used and still is
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00:04:00,130 --> 00:04:03,210
used in the curriculum today
under the name of the number
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00:04:03,210 --> 00:04:07,470
line, under the name of graphs,
and what we will use
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00:04:07,470 --> 00:04:11,620
as a fundamental building block
as our course develops.
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00:04:11,620 --> 00:04:14,940
Now, you know, in the same way
that we can think of a single
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00:04:14,940 --> 00:04:19,640
number as being a point on the
line, we can think of a pair
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00:04:19,640 --> 00:04:23,380
of numbers, an ordered pair
of numbers, as being a
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00:04:23,380 --> 00:04:25,210
point in the plane.
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00:04:25,210 --> 00:04:29,150
This is Descartes geometry,
which we can call coordinate
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00:04:29,150 --> 00:04:33,200
geometry, the idea being that
in the same way as we can
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00:04:33,200 --> 00:04:37,170
locate a number of along the
x-axis, shall we say, we could
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00:04:37,170 --> 00:04:41,970
have located a numbered pair
as a point in the plane.
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00:04:41,970 --> 00:04:45,850
Namely, 2 comma 3 would mean the
point whose x-coordinate
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00:04:45,850 --> 00:04:49,230
was 2 and whose y-coordinate
was 3.
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00:04:49,230 --> 00:04:53,610
By the way, the reason that we
say ordered pairs is, if you
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00:04:53,610 --> 00:04:57,490
observe, 2 comma 3 and
3 comma 2 happen to
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00:04:57,490 --> 00:04:58,950
be different pairs.
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00:04:58,950 --> 00:05:02,580
And notice again how vividly the
geometric interpretation
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00:05:02,580 --> 00:05:03,640
of this is.
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00:05:03,640 --> 00:05:07,130
Namely, in terms of locating a
point in space, it's obvious
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00:05:07,130 --> 00:05:10,870
that the point named 2 comma 3
is not the same as the point
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00:05:10,870 --> 00:05:12,760
named 3 comma 2.
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00:05:12,760 --> 00:05:14,790
Again, the important
thing is this.
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00:05:14,790 --> 00:05:18,400
When I think of 5 divided by 3,
when I think of the ordered
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00:05:18,400 --> 00:05:23,460
pair 2 comma 3, I do not have
to think of a picture.
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00:05:23,460 --> 00:05:26,010
I can think of these things
analytically.
90
00:05:26,010 --> 00:05:29,590
But the picture gives me certain
insights that will
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00:05:29,590 --> 00:05:33,000
help me with my intuition, an
aid that I don't want to
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00:05:33,000 --> 00:05:34,120
relinquish.
93
00:05:34,120 --> 00:05:36,790
For example, going back
to the graph again.
94
00:05:36,790 --> 00:05:41,190
Thinking of the analytic term
greater than, notice how much
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00:05:41,190 --> 00:05:49,470
easier it is to think of, for
example, higher than, see, one
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00:05:49,470 --> 00:05:53,308
point being higher than another
or to the right of.
97
00:05:53,308 --> 00:05:58,680
98
00:05:58,680 --> 00:06:02,650
You see, geometric concepts to
name analytic statements, or
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00:06:02,650 --> 00:06:04,150
instead of increasing,
as we mentioned
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00:06:04,150 --> 00:06:06,450
before, to say rising.
101
00:06:06,450 --> 00:06:10,080
And there will be many, many
more such identifications as
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00:06:10,080 --> 00:06:12,260
we go along with our course.
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00:06:12,260 --> 00:06:14,660
At any rate, let's continue
to see then what is the
104
00:06:14,660 --> 00:06:18,180
relationship then between
functions that we talked about
105
00:06:18,180 --> 00:06:19,720
and graphs?
106
00:06:19,720 --> 00:06:23,270
The idea is something
like this.
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00:06:23,270 --> 00:06:27,060
Let's return to our friend of
the first lecture: s equals
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00:06:27,060 --> 00:06:28,730
16t squared.
109
00:06:28,730 --> 00:06:32,720
We can think of a distance
machine being the function
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00:06:32,720 --> 00:06:40,590
where the input will be time and
the output will be what?
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00:06:40,590 --> 00:06:44,130
The square of the input
multiplied by 16.
112
00:06:44,130 --> 00:06:49,150
Observe that from this, I do not
have to have any picture
113
00:06:49,150 --> 00:06:51,590
to understand what's
happening here.
114
00:06:51,590 --> 00:06:55,820
Namely, I can measure an input,
measure an output, and
115
00:06:55,820 --> 00:06:58,420
observe analytically
what is happening.
116
00:06:58,420 --> 00:07:02,620
However, as we saw last time,
our graph sort of shows us at
117
00:07:02,620 --> 00:07:05,820
a glance what seems to be
happening, that we can
118
00:07:05,820 --> 00:07:10,380
identify rising and falling with
increasing and decreasing
119
00:07:10,380 --> 00:07:11,930
and things of this type.
120
00:07:11,930 --> 00:07:15,020
We will explore this, of course,
in much more detail as
121
00:07:15,020 --> 00:07:17,820
we continue in our course.
122
00:07:17,820 --> 00:07:21,730
By the way, there's no reason
why the input has to be a
123
00:07:21,730 --> 00:07:22,750
single number.
124
00:07:22,750 --> 00:07:25,450
For example, why couldn't
the input be an
125
00:07:25,450 --> 00:07:27,350
ordered pair of numbers?
126
00:07:27,350 --> 00:07:30,310
Among other things, let's take
a simple geometric example.
127
00:07:30,310 --> 00:07:35,380
Consider, for example, finding
the volume of a cylinder in
128
00:07:35,380 --> 00:07:38,720
terms of the radius of its
base and the height.
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00:07:38,720 --> 00:07:41,790
We know from solid geometry
that the volume is
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00:07:41,790 --> 00:07:44,220
pi r squared h.
131
00:07:44,220 --> 00:07:47,970
We could therefore think of a
volume machine where the input
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00:07:47,970 --> 00:07:53,210
is the ordered pair r comma h,
and the output is the single
133
00:07:53,210 --> 00:07:56,310
number pi r squared h.
134
00:07:56,310 --> 00:07:59,650
By the way, notice here the
meaning of ordered pair.
135
00:07:59,650 --> 00:08:04,030
You see, if the pair 2 comma 3
goes into the machine, notice
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00:08:04,030 --> 00:08:05,790
that the recipe here
says what?
137
00:08:05,790 --> 00:08:08,120
You square the first
member of the pair.
138
00:08:08,120 --> 00:08:11,420
In other words, if 2 comma 3
is the input, we square 2,
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00:08:11,420 --> 00:08:15,790
which is 4, multiplied by 3,
which is 12, and 12 times pi,
140
00:08:15,790 --> 00:08:17,950
of course, is 12 pi.
141
00:08:17,950 --> 00:08:21,820
On the other hand, if the input
is 3 comma 2, the first
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00:08:21,820 --> 00:08:22,810
number is 3.
143
00:08:22,810 --> 00:08:25,480
Our recipe squares
the first number.
144
00:08:25,480 --> 00:08:31,640
That would be 9, times 2 is
18, times pi is 18 pi.
145
00:08:31,640 --> 00:08:35,659
But again, observe that I at
no time needed a picture to
146
00:08:35,659 --> 00:08:37,750
visualize what was
happening here.
147
00:08:37,750 --> 00:08:40,789
Of course, if I wanted a
picture, I could try to plot
148
00:08:40,789 --> 00:08:44,940
this also, but notice now that
my graph would probably need
149
00:08:44,940 --> 00:08:46,710
three dimensions to draw.
150
00:08:46,710 --> 00:08:48,820
And why would it need
three dimensions?
151
00:08:48,820 --> 00:08:52,930
Well, notice that my input has
two independent measurements r
152
00:08:52,930 --> 00:08:57,250
and h, and therefore, I would
need two dimensions just to
153
00:08:57,250 --> 00:08:58,860
take care of r and h.
154
00:08:58,860 --> 00:09:02,380
Then I would need a third
dimension to plot v.
155
00:09:02,380 --> 00:09:04,750
And by the way, notice
the next stage.
156
00:09:04,750 --> 00:09:08,520
If I had an input that consisted
of three independent
157
00:09:08,520 --> 00:09:12,210
measurements, this would still
make sense, but now I would be
158
00:09:12,210 --> 00:09:14,870
at a loss for the picture.
159
00:09:14,870 --> 00:09:17,190
In other words, what I'm trying
to bring out next is
160
00:09:17,190 --> 00:09:21,010
the fact that whereas pictures
are a tremendous help, maybe a
161
00:09:21,010 --> 00:09:24,900
second subtitle to our lecture
should've been a picture is
162
00:09:24,900 --> 00:09:27,505
worth a thousand words
provided you
163
00:09:27,505 --> 00:09:28,970
can could draw it.
164
00:09:28,970 --> 00:09:32,610
Because, you see, if we needed
three independent dimensions
165
00:09:32,610 --> 00:09:36,570
to locate the input and then
a fourth one to locate the
166
00:09:36,570 --> 00:09:37,995
output, how would we
draw the picture?
167
00:09:37,995 --> 00:09:40,820
By the way, this happens in high
school algebra again, if
168
00:09:40,820 --> 00:09:42,390
you want to see the analogy.
169
00:09:42,390 --> 00:09:44,140
Look at, for example,
the algebraic
170
00:09:44,140 --> 00:09:46,020
equation a plus b squared.
171
00:09:46,020 --> 00:09:50,230
We learned in algebra that this
is a squared plus 2ab
172
00:09:50,230 --> 00:09:51,920
plus b squared.
173
00:09:51,920 --> 00:09:56,020
Observe that we do not need to
have a picture to understand
174
00:09:56,020 --> 00:09:57,460
how this works.
175
00:09:57,460 --> 00:10:03,360
Oh, to be sure, if we had a
picture, we get a tremendous
176
00:10:03,360 --> 00:10:05,450
amount of insight as to
what's happening here.
177
00:10:05,450 --> 00:10:10,060
Namely, let's visualize a square
whose side is a plus b.
178
00:10:10,060 --> 00:10:13,920
On the one hand, you see, the
area of the square would be a
179
00:10:13,920 --> 00:10:17,080
plus b squared, you see,
the side squared.
180
00:10:17,080 --> 00:10:20,020
On the other hand, if we now
subdivide this figure this
181
00:10:20,020 --> 00:10:23,910
way, we see that that same
square is made up of four
182
00:10:23,910 --> 00:10:29,020
pieces having one piece of area
a squared, two pieces of
183
00:10:29,020 --> 00:10:32,840
area ab, and one piece
of area b squared.
184
00:10:32,840 --> 00:10:35,270
And so we see, on the other
hand, that the area of the
185
00:10:35,270 --> 00:10:40,470
square is a squared plus
2ab plus b squared.
186
00:10:40,470 --> 00:10:43,990
And so again, observe that
whereas this stands on its own
187
00:10:43,990 --> 00:10:47,020
two legs, the picture helps
us quite a bit.
188
00:10:47,020 --> 00:10:51,430
By the way, we could continue
this with a cube over here.
189
00:10:51,430 --> 00:10:55,950
Namely, it turns out that a plus
b cubed is a cubed plus
190
00:10:55,950 --> 00:11:01,600
3a squared b plus 3ab squared
plus b cubed.
191
00:11:01,600 --> 00:11:04,450
And again, if we wanted to,
and we won't take the time
192
00:11:04,450 --> 00:11:08,970
here, but if we wanted to, we
could now draw a cube whose
193
00:11:08,970 --> 00:11:10,670
side is a plus b.
194
00:11:10,670 --> 00:11:14,020
Namely, we could take the same
diagram that we had before and
195
00:11:14,020 --> 00:11:15,680
now make a third dimension
to it.
196
00:11:15,680 --> 00:11:18,210
And if you did that,
you would see what?
197
00:11:18,210 --> 00:11:21,850
That the cube whose volume is a
plus b cubed is divided into
198
00:11:21,850 --> 00:11:27,080
eight pieces, one of size a by
a by a, three of size a by a
199
00:11:27,080 --> 00:11:31,740
by b, three of size a by
b by b, and one of
200
00:11:31,740 --> 00:11:34,660
size b by b by b.
201
00:11:34,660 --> 00:11:38,540
Again, the picture is a
tremendous visual aid.
202
00:11:38,540 --> 00:11:40,620
Now, the key step is this.
203
00:11:40,620 --> 00:11:44,410
If we were to now give in to
our geometric intuition and
204
00:11:44,410 --> 00:11:47,720
say, lookit, why worry about
algebra when it's so much
205
00:11:47,720 --> 00:11:51,110
easier to do this thing by
geometry, the counterexample
206
00:11:51,110 --> 00:11:54,410
is consider a plus b to
the fourth power.
207
00:11:54,410 --> 00:11:57,800
Now, by the binomial theorem,
and I might add, the same
208
00:11:57,800 --> 00:12:01,150
binomial theorem that allowed
us to get these results, we
209
00:12:01,150 --> 00:12:02,880
can also write that
this is what?
210
00:12:02,880 --> 00:12:08,840
It's a is the fourth plus 4a
cubed b plus 6a squared b
211
00:12:08,840 --> 00:12:14,760
squared plus 4ab cubed
plus b to the fourth.
212
00:12:14,760 --> 00:12:17,730
Now, it's not important how
we get this result.
213
00:12:17,730 --> 00:12:20,640
The important point is that
analytically, we can raise a
214
00:12:20,640 --> 00:12:23,770
number to the fourth power just
as easily as we can to
215
00:12:23,770 --> 00:12:26,160
the third power or
the second power.
216
00:12:26,160 --> 00:12:30,800
The only difference is that in
the case of the third power,
217
00:12:30,800 --> 00:12:32,910
we had a picture that
we could use.
218
00:12:32,910 --> 00:12:36,910
In the fourth power case, we
didn't have a picture.
219
00:12:36,910 --> 00:12:39,660
And what a tragedy it would
have been to say, hey, we
220
00:12:39,660 --> 00:12:41,740
can't solve this problem
because we
221
00:12:41,740 --> 00:12:43,240
can't draw the picture.
222
00:12:43,240 --> 00:12:46,350
And by the way, as a rather
interesting aside, notice the
223
00:12:46,350 --> 00:12:49,230
geometric influence on
how we read this.
224
00:12:49,230 --> 00:12:52,260
This is called a plus b
to the fourth power.
225
00:12:52,260 --> 00:12:55,490
But somehow or other, we don't
call this one a plus b to the
226
00:12:55,490 --> 00:12:56,310
third power.
227
00:12:56,310 --> 00:13:01,440
We call it a plus b cubed,
suggesting the geometric
228
00:13:01,440 --> 00:13:03,110
configuration of the cube.
229
00:13:03,110 --> 00:13:06,650
And here we don't usually say a
plus b to the second power.
230
00:13:06,650 --> 00:13:10,220
We say a plus b squared.
231
00:13:10,220 --> 00:13:14,240
You see, the idea is that when
the picture is available, it
232
00:13:14,240 --> 00:13:17,740
gives us a tremendous insight as
to what can be done, and it
233
00:13:17,740 --> 00:13:21,440
helps us learn to visualize
what's happening analytically.
234
00:13:21,440 --> 00:13:24,830
In fact, what usually happens
is we use the picture to
235
00:13:24,830 --> 00:13:27,730
justify what's happening
analytically when we can see
236
00:13:27,730 --> 00:13:30,490
the picture and then just
carry the analytic part
237
00:13:30,490 --> 00:13:33,690
through unimpeded in
the case where we
238
00:13:33,690 --> 00:13:35,140
can't draw the picture.
239
00:13:35,140 --> 00:13:37,050
Let me give you another
example of this.
240
00:13:37,050 --> 00:13:38,800
Let's look at the
following set.
241
00:13:38,800 --> 00:13:41,790
That also should review our
language of sets for us.
242
00:13:41,790 --> 00:13:45,710
Let S be the set of all ordered
pairs x comma y such
243
00:13:45,710 --> 00:13:49,150
that x squared plus y
squared equals 25.
244
00:13:49,150 --> 00:13:54,910
Question: Does the ordered pair
3 comma 4 belong to S?
245
00:13:54,910 --> 00:13:56,470
Answer: Yes.
246
00:13:56,470 --> 00:13:57,360
How do we know?
247
00:13:57,360 --> 00:13:59,460
Well, we have a test
for membership.
248
00:13:59,460 --> 00:14:00,940
We're supposed to do what?
249
00:14:00,940 --> 00:14:05,640
Square each of the entries, each
of the numbers, add them,
250
00:14:05,640 --> 00:14:08,880
and if the answer is 25, then
that ordered pair belongs to
251
00:14:08,880 --> 00:14:14,610
S. 3 squared plus 4 squared is
25, so this pair belongs to S.
252
00:14:14,610 --> 00:14:16,120
How about 1 comma 2?
253
00:14:16,120 --> 00:14:21,260
Well, 1 squared plus 2 squared
is 1 plus 4, which is 5.
254
00:14:21,260 --> 00:14:26,920
5 is not equal to 25, so 1 comma
2 does not belong to S.
255
00:14:26,920 --> 00:14:30,070
Well, did we need any geometry
to be able to
256
00:14:30,070 --> 00:14:31,860
visualize this result?
257
00:14:31,860 --> 00:14:35,730
Hopefully, one did not
need any geometry to
258
00:14:35,730 --> 00:14:37,380
visualize this result.
259
00:14:37,380 --> 00:14:40,150
On the other hand then, what
does it mean in analytic
260
00:14:40,150 --> 00:14:44,100
geometry when we say that x
squared plus y squared equals
261
00:14:44,100 --> 00:14:46,580
25 is a circle?
262
00:14:46,580 --> 00:14:51,040
As badly as I draw, x squared
plus y squared equals 25 looks
263
00:14:51,040 --> 00:14:54,590
less like a circle than the
circle I drew over here.
264
00:14:54,590 --> 00:14:57,030
You see, what we really
mean is this.
265
00:14:57,030 --> 00:15:01,670
Consider all the points in the
plane x comma y for which x
266
00:15:01,670 --> 00:15:04,390
squared plus y squared
equals 25.
267
00:15:04,390 --> 00:15:08,480
These are precisely the points
on this particular circle.
268
00:15:08,480 --> 00:15:11,200
And the easiest way to see that,
of course, is that since
269
00:15:11,200 --> 00:15:15,850
the radius of the circle is 5
and the point x comma y means
270
00:15:15,850 --> 00:15:18,980
that this length is x and this
length is y, notice that from
271
00:15:18,980 --> 00:15:22,800
the Pythagorean theorem, we see
it once, that x squared
272
00:15:22,800 --> 00:15:26,970
plus y squared equals 25.
273
00:15:26,970 --> 00:15:31,940
Now again, the solution set to
this equation is our set S
274
00:15:31,940 --> 00:15:34,510
whether we're thinking of this
thing algebraically or
275
00:15:34,510 --> 00:15:35,640
geometrically.
276
00:15:35,640 --> 00:15:39,400
On the other hand, watch what
our picture seems to give us
277
00:15:39,400 --> 00:15:40,890
that we didn't have before.
278
00:15:40,890 --> 00:15:44,380
Let's return to our point 1
comma 2, which we saw didn't
279
00:15:44,380 --> 00:15:47,010
belong to S. Well,
look at this.
280
00:15:47,010 --> 00:15:48,630
Where would 1 comma 2 be?
281
00:15:48,630 --> 00:15:52,560
1 comma 2 would be inside
the circle.
282
00:15:52,560 --> 00:15:53,890
Why is that?
283
00:15:53,890 --> 00:16:01,080
Because, you see, if we take
the point 1 comma 2, if we
284
00:16:01,080 --> 00:16:04,470
take, say, for example, the
point 1 comma 2, notice that
285
00:16:04,470 --> 00:16:07,930
the distance from the origin
to the point 1 comma 2
286
00:16:07,930 --> 00:16:09,080
is less than 5.
287
00:16:09,080 --> 00:16:12,460
In other words, the distance is
less than 5 so the square
288
00:16:12,460 --> 00:16:14,660
of the distance is
less than 25.
289
00:16:14,660 --> 00:16:18,550
In other words, not only can we
say that 1 comma 2 does not
290
00:16:18,550 --> 00:16:21,420
belong to S, which we could have
said without the picture,
291
00:16:21,420 --> 00:16:24,080
we can now say what?
292
00:16:24,080 --> 00:16:26,410
1 comma 2 is--
293
00:16:26,410 --> 00:16:28,880
and notice the geometric
language here--
294
00:16:28,880 --> 00:16:30,780
is inside the circle.
295
00:16:30,780 --> 00:16:34,590
296
00:16:34,590 --> 00:16:36,310
In other words, the study of
297
00:16:36,310 --> 00:16:39,210
inequalities can now be reduced.
298
00:16:39,210 --> 00:16:42,325
Instead of talking about less
than and greater than, we can
299
00:16:42,325 --> 00:16:46,060
now talk about such things
as inside and outside.
300
00:16:46,060 --> 00:16:48,520
You see, inside the circle,
which is a simple geometric
301
00:16:48,520 --> 00:16:50,340
concept, just means what?
302
00:16:50,340 --> 00:16:53,530
A set of all points for which
x squared plus y squared is
303
00:16:53,530 --> 00:16:54,920
less than 25.
304
00:16:54,920 --> 00:16:58,250
Outside that circle, x squared
plus y squared is
305
00:16:58,250 --> 00:16:59,700
greater than 25.
306
00:16:59,700 --> 00:17:04,040
On the circle, x squared plus
y squared equals 25.
307
00:17:04,040 --> 00:17:09,490
Again, a nice identification
between numbers and pictures,
308
00:17:09,490 --> 00:17:11,890
analysis and geometry.
309
00:17:11,890 --> 00:17:16,510
Well, this then shows us why
we want to study pictures
310
00:17:16,510 --> 00:17:18,160
rather than functions.
311
00:17:18,160 --> 00:17:21,300
Now, if we look at any textbook
in which we deal with
312
00:17:21,300 --> 00:17:24,500
graphs, it always seems that
we start with graphs of
313
00:17:24,500 --> 00:17:25,319
straight lines.
314
00:17:25,319 --> 00:17:28,359
And the question is what is so
great about a straight line?
315
00:17:28,359 --> 00:17:31,360
After all, pictures in general
are going to be much more
316
00:17:31,360 --> 00:17:32,480
complicated than that.
317
00:17:32,480 --> 00:17:36,180
What advantage is there in
starting with straight lines?
318
00:17:36,180 --> 00:17:39,680
And again, we begin to realize
how straight lines are the
319
00:17:39,680 --> 00:17:43,540
backbone of all types of
analytical procedures and all
320
00:17:43,540 --> 00:17:44,840
types of curve plotting.
321
00:17:44,840 --> 00:17:47,910
For example, let's suppose we
were studying this particular
322
00:17:47,910 --> 00:17:50,860
curve, and we wanted to know
what was happened to that
323
00:17:50,860 --> 00:17:55,310
curve in the neighborhood
around the point p.
324
00:17:55,310 --> 00:17:59,320
Let's draw in the tangent line
to the curve at the point p.
325
00:17:59,320 --> 00:18:03,090
Notice that this line that
we've drawn serves as a
326
00:18:03,090 --> 00:18:07,740
wonderful approximation curve
itself if we stay close enough
327
00:18:07,740 --> 00:18:09,520
to the point of tangency.
328
00:18:09,520 --> 00:18:12,590
In other words, notice how much
we can deduce about this
329
00:18:12,590 --> 00:18:16,910
curve if we study only the
straight line segment at the
330
00:18:16,910 --> 00:18:18,000
point of tangency.
331
00:18:18,000 --> 00:18:20,660
Of course, the approximation
gets worse and worse as we
332
00:18:20,660 --> 00:18:22,180
move further and further out.
333
00:18:22,180 --> 00:18:25,000
But in the neighborhood of the
point of what's going on,
334
00:18:25,000 --> 00:18:27,660
notice again then that the
straight line is an important
335
00:18:27,660 --> 00:18:28,820
building block.
336
00:18:28,820 --> 00:18:32,170
By the way, again we use
straight lines in a rather
337
00:18:32,170 --> 00:18:35,520
subtle way in something
called interpolation.
338
00:18:35,520 --> 00:18:38,370
For example, let's suppose I go
to a log table and I look
339
00:18:38,370 --> 00:18:40,030
up the log of 2.
340
00:18:40,030 --> 00:18:44,270
I find that the log
of 2 is 0.301.
341
00:18:44,270 --> 00:18:45,770
I look up the log of 4.
342
00:18:45,770 --> 00:18:48,900
I find that's 0.602.
343
00:18:48,900 --> 00:18:52,080
Now I look for the log of 3,
and I see that it's been
344
00:18:52,080 --> 00:18:53,660
obliterated.
345
00:18:53,660 --> 00:18:54,710
I don't know what it is.
346
00:18:54,710 --> 00:18:57,670
So I say, well, let me guess.
347
00:18:57,670 --> 00:19:00,640
3 is halfway between 2 and 4.
348
00:19:00,640 --> 00:19:04,670
Therefore, I would suspect that
the log of 3 is halfway
349
00:19:04,670 --> 00:19:07,570
between the log of 2
and the log of 4.
350
00:19:07,570 --> 00:19:10,030
And so, halfway between here
would be about what?
351
00:19:10,030 --> 00:19:13,880
0.452, roughly speaking.
352
00:19:13,880 --> 00:19:16,160
All of a sudden, the
obliteration on
353
00:19:16,160 --> 00:19:18,080
my book clears up.
354
00:19:18,080 --> 00:19:21,760
And I look, and I don't
find 0.452.
355
00:19:21,760 --> 00:19:24,660
Instead I find--
356
00:19:24,660 --> 00:19:27,120
well, let's write
it over here.
357
00:19:27,120 --> 00:19:32,960
What I find is 0.477.
358
00:19:32,960 --> 00:19:36,030
Now, you know, this is a pretty
big error to attribute
359
00:19:36,030 --> 00:19:38,800
to slide rule inaccuracy
or trouble in
360
00:19:38,800 --> 00:19:40,030
rounding off the tables.
361
00:19:40,030 --> 00:19:42,010
What really went wrong
over here?
362
00:19:42,010 --> 00:19:44,770
And the answer comes up again
that unless otherwise
363
00:19:44,770 --> 00:19:49,260
specified, the process known
as interpolation hinges on
364
00:19:49,260 --> 00:19:53,320
replacing a curve by a straight
line approximation.
365
00:19:53,320 --> 00:19:56,690
In fact, you see, if we were
to draw the curve of the
366
00:19:56,690 --> 00:19:59,940
logarithm function, we would
find that the picture is
367
00:19:59,940 --> 00:20:01,570
something like this.
368
00:20:01,570 --> 00:20:05,910
And when we looked up the log
of 2, this height is what we
369
00:20:05,910 --> 00:20:07,000
found in the table.
370
00:20:07,000 --> 00:20:11,400
When we looked up the log of
4, this height is what we
371
00:20:11,400 --> 00:20:15,670
would have found in the table.
372
00:20:15,670 --> 00:20:18,800
If we had looked up in the table
the log of 3, this is
373
00:20:18,800 --> 00:20:20,240
the height that we
would have found.
374
00:20:20,240 --> 00:20:23,500
This is the height
that's 0.477.
375
00:20:23,500 --> 00:20:27,630
Notice that in general, if we go
halfway from here to here,
376
00:20:27,630 --> 00:20:29,900
we do not go halfway
from here to here.
377
00:20:29,900 --> 00:20:31,770
It depends on the shape
of the curve.
378
00:20:31,770 --> 00:20:34,750
The only time you can be sure
that you have proportional
379
00:20:34,750 --> 00:20:38,600
parts is if the curve that
joined these two points was a
380
00:20:38,600 --> 00:20:41,620
straight line.
381
00:20:41,620 --> 00:20:44,860
And notice, by the way, that
by the shape of this curve,
382
00:20:44,860 --> 00:20:49,490
the straight line falls below
the curve, and therefore, the
383
00:20:49,490 --> 00:20:51,550
height that we found
was to the straight
384
00:20:51,550 --> 00:20:53,080
line, not to the curve.
385
00:20:53,080 --> 00:20:55,810
That was the point 0.452.
386
00:20:55,810 --> 00:20:58,470
So, in other words, notice that
we got smaller than the
387
00:20:58,470 --> 00:21:01,760
right answer because we
approximated as if it was a
388
00:21:01,760 --> 00:21:03,930
straight line that was
joining the curve.
389
00:21:03,930 --> 00:21:06,990
You see, what interpolation
hinges on is that the size of
390
00:21:06,990 --> 00:21:10,920
the interval is very small and
that you can assume that for
391
00:21:10,920 --> 00:21:13,850
the accuracy that you're
interested in that the
392
00:21:13,850 --> 00:21:17,120
straight line approximation to
the curve is sufficiently
393
00:21:17,120 --> 00:21:20,710
accurate to represent
the curve itself.
394
00:21:20,710 --> 00:21:22,200
Well, enough about that.
395
00:21:22,200 --> 00:21:25,990
Once we've talked about why
straight lines are important,
396
00:21:25,990 --> 00:21:28,850
the next thing is how do we
measure straight lines?
397
00:21:28,850 --> 00:21:31,770
See, another interesting point
to something like this.
398
00:21:31,770 --> 00:21:35,280
Many times we know what
something means subjectively,
399
00:21:35,280 --> 00:21:37,900
but we don't know what
it means objectively.
400
00:21:37,900 --> 00:21:41,350
For example, one way of finding
a line is to know two
401
00:21:41,350 --> 00:21:42,730
points on the line.
402
00:21:42,730 --> 00:21:45,280
Another way is to know
one point and the
403
00:21:45,280 --> 00:21:46,480
slant of the line.
404
00:21:46,480 --> 00:21:48,650
The question that comes up
is how do you measure
405
00:21:48,650 --> 00:21:50,180
the slant of a line?
406
00:21:50,180 --> 00:21:53,350
In other words, shall you say
that the line is very slanty?
407
00:21:53,350 --> 00:21:55,820
And if the answer to that is
yes, how do you distinguish
408
00:21:55,820 --> 00:22:01,460
between slanty and very slanty,
steep and very steep,
409
00:22:01,460 --> 00:22:03,240
very steep and very,
very steep?
410
00:22:03,240 --> 00:22:05,670
We need something
more objective.
411
00:22:05,670 --> 00:22:08,450
And the way we get around
this is as follows.
412
00:22:08,450 --> 00:22:11,850
Given a line, we define
the slope as follows.
413
00:22:11,850 --> 00:22:14,960
We pick any two points
on the line.
414
00:22:14,960 --> 00:22:18,880
And from those two points, we
can measure what we call the
415
00:22:18,880 --> 00:22:22,490
run of the line, in other words,
how far you've gone
416
00:22:22,490 --> 00:22:26,680
this way, and the rise
of the line, how much
417
00:22:26,680 --> 00:22:28,020
it's risen this way.
418
00:22:28,020 --> 00:22:32,780
And what we do is we define
the slope to be the rise
419
00:22:32,780 --> 00:22:38,790
divided by the run, or without
the delta notation in here, y2
420
00:22:38,790 --> 00:22:42,680
minus y1 over x2 minus x1.
421
00:22:42,680 --> 00:22:45,050
By the way, there are little
problems that come up.
422
00:22:45,050 --> 00:22:48,020
After all, our answer should
not depend on the picture.
423
00:22:48,020 --> 00:22:50,520
It should be sort of
self-contained analytically.
424
00:22:50,520 --> 00:22:53,790
The question comes up is what
if I had labeled this point
425
00:22:53,790 --> 00:22:58,350
x2, y2 and this point x1, y1?
426
00:22:58,350 --> 00:22:59,620
What would have happened then?
427
00:22:59,620 --> 00:23:03,090
And observe that as long as we
keep the pairs straight, it
428
00:23:03,090 --> 00:23:06,570
makes no difference whether you
write this or whether you
429
00:23:06,570 --> 00:23:07,430
write this.
430
00:23:07,430 --> 00:23:09,600
Because, you see, in each
case, all you've done is
431
00:23:09,600 --> 00:23:13,540
change the sign, and negative
over negative is positive.
432
00:23:13,540 --> 00:23:17,060
So certainly our answer to what
a slope is is objective
433
00:23:17,060 --> 00:23:19,930
enough, so it does not depend on
how the points are labeled.
434
00:23:19,930 --> 00:23:22,240
A second objection that most
people have is they say
435
00:23:22,240 --> 00:23:25,140
something like, well, who are
you to say that we pick these
436
00:23:25,140 --> 00:23:25,770
two points?
437
00:23:25,770 --> 00:23:29,050
What if I came along and picked
these two points and I
438
00:23:29,050 --> 00:23:33,460
now computed the slope by taking
this as my delta y and
439
00:23:33,460 --> 00:23:35,450
this as my delta x?
440
00:23:35,450 --> 00:23:38,250
Obviously, it would be a tragedy
if the answer to the
441
00:23:38,250 --> 00:23:41,315
problem depended on which pair
of points you picked since a
442
00:23:41,315 --> 00:23:43,730
line should have
but one slope.
443
00:23:43,730 --> 00:23:46,750
Again, notice that our high
school training in geometry,
444
00:23:46,750 --> 00:23:50,540
similar triangles, motivates
why we pick ratios.
445
00:23:50,540 --> 00:23:54,460
Namely, while this delta y and
this delta y may be different
446
00:23:54,460 --> 00:23:57,510
and this delta x and this delta
x may be different, what
447
00:23:57,510 --> 00:24:02,160
is true is that the ratio of
this delta y to this delta x
448
00:24:02,160 --> 00:24:05,830
is the same as the ratio of this
delta y to this delta x.
449
00:24:05,830 --> 00:24:08,040
And that's why we
pick the ratio.
450
00:24:08,040 --> 00:24:11,110
By the way, another way of
talking about ratio is if you
451
00:24:11,110 --> 00:24:14,460
look at delta y divided by delta
x and you've had some
452
00:24:14,460 --> 00:24:17,330
trigonometry, it reminds
you of a trigonometric
453
00:24:17,330 --> 00:24:19,090
relationship.
454
00:24:19,090 --> 00:24:23,000
Namely, you look at delta y, you
look at delta x, and you
455
00:24:23,000 --> 00:24:25,510
say, my, isn't that just
the tangent of
456
00:24:25,510 --> 00:24:26,960
this particular angle?
457
00:24:26,960 --> 00:24:28,650
Couldn't I define the slope?
458
00:24:28,650 --> 00:24:31,480
And by the way, the general
symbol for slope, for better
459
00:24:31,480 --> 00:24:33,550
or for worse, just happens
to be letter m.
460
00:24:33,550 --> 00:24:39,280
Why couldn't I define m to be
the tangent of phi, where phi
461
00:24:39,280 --> 00:24:41,590
is the angle that the straight
line makes with
462
00:24:41,590 --> 00:24:43,050
the positive x-axis?
463
00:24:43,050 --> 00:24:45,180
And, of course, there is a
little subtlety here that we
464
00:24:45,180 --> 00:24:46,600
should pay attention to.
465
00:24:46,600 --> 00:24:49,970
This would be an ambiguous
definition if the scale on the
466
00:24:49,970 --> 00:24:52,380
x- and the y-axis were
not the same.
467
00:24:52,380 --> 00:24:55,360
In other words, notice that by
changing the scale here, I can
468
00:24:55,360 --> 00:24:58,330
distort the same analytic
information.
469
00:24:58,330 --> 00:25:02,180
So if I agree, however, that the
unit on the x-axis is the
470
00:25:02,180 --> 00:25:05,980
same as the unit on the y-axis,
then I can say, OK,
471
00:25:05,980 --> 00:25:09,300
the slope is also tangent
of the angle phi.
472
00:25:09,300 --> 00:25:12,670
I much prefer to say it's delta
y divided by delta x,
473
00:25:12,670 --> 00:25:14,400
because then if I forget
the scale,
474
00:25:14,400 --> 00:25:16,010
I'm still in no trouble.
475
00:25:16,010 --> 00:25:20,180
On the other hand, if we use the
tangent definition, we can
476
00:25:20,180 --> 00:25:23,670
utilize all we know about
trigonometry to get some other
477
00:25:23,670 --> 00:25:25,190
interesting results.
478
00:25:25,190 --> 00:25:27,800
Namely, the question that might
come up is can we study
479
00:25:27,800 --> 00:25:31,330
the slopes of two different
lines very conveniently in
480
00:25:31,330 --> 00:25:33,540
terms of our definition
of slope?
481
00:25:33,540 --> 00:25:35,330
And the answer is this.
482
00:25:35,330 --> 00:25:39,630
If we imagine now that our lines
are drawn to scale here,
483
00:25:39,630 --> 00:25:42,560
and here are two different
lines, which I'll call l1 and
484
00:25:42,560 --> 00:25:46,400
l2, and we'll call the angle
that l1 makes with the
485
00:25:46,400 --> 00:25:50,280
positive x-axis phi 1, the angle
that l2 makes with the
486
00:25:50,280 --> 00:25:52,760
positive x-axis phi 2.
487
00:25:52,760 --> 00:25:55,220
Therefore, what? m1
is tan phi 1.
488
00:25:55,220 --> 00:25:57,060
m2 is tan phi 2.
489
00:25:57,060 --> 00:26:01,100
Notice that our formula for the
tangent of the difference
490
00:26:01,100 --> 00:26:03,760
of two angles-- you see,
notice that this
491
00:26:03,760 --> 00:26:04,890
angle here is what?
492
00:26:04,890 --> 00:26:08,960
Since this angle is the sum of
these two, this angle here is
493
00:26:08,960 --> 00:26:13,580
phi 2 minus phi 1 or the
negative of phi 1 minus phi 2.
494
00:26:13,580 --> 00:26:16,220
I should have had this phi 2
minus phi 2, but since that
495
00:26:16,220 --> 00:26:19,020
just changes the sign, that will
not have any bearing on
496
00:26:19,020 --> 00:26:20,300
the point I want to make.
497
00:26:20,300 --> 00:26:22,220
Let's continue this way.
498
00:26:22,220 --> 00:26:27,230
Tangent of phi 1 minus phi 2 is
tan phi 1 minus tan phi 2
499
00:26:27,230 --> 00:26:30,570
over 1 plus tan phi
1 tan phi 2.
500
00:26:30,570 --> 00:26:33,500
On the other hand, by our
definitions of m1 and m2, this
501
00:26:33,500 --> 00:26:37,080
is m1 minus m2 over
1 plus m1 m2.
502
00:26:37,080 --> 00:26:40,220
503
00:26:40,220 --> 00:26:44,360
Now, this tells me how to find
the angle between two lines
504
00:26:44,360 --> 00:26:46,730
just in terms of knowing
the slope.
505
00:26:46,730 --> 00:26:50,670
Two very special interesting
cases as extremes suggest
506
00:26:50,670 --> 00:26:52,110
themselves right away.
507
00:26:52,110 --> 00:26:55,760
One case is what happens if
the lines are parallel?
508
00:26:55,760 --> 00:27:00,020
If the lines are parallel, you
see, phi 1 equals phi 2, in
509
00:27:00,020 --> 00:27:04,350
which case phi 1 minus phi
2 is 0, in which case the
510
00:27:04,350 --> 00:27:08,110
tangent of phi 1 minus phi
2 had better be 0.
511
00:27:08,110 --> 00:27:12,060
But the only way a fraction can
be 0 is for the numerator
512
00:27:12,060 --> 00:27:16,120
to be 0, and that means
that m1 must equal m2.
513
00:27:16,120 --> 00:27:19,420
In other words, in terms of
slopes, we can study parallel
514
00:27:19,420 --> 00:27:22,310
lines just by equating
their slopes.
515
00:27:22,310 --> 00:27:25,590
A less obvious relationship
that's equally important is
516
00:27:25,590 --> 00:27:29,200
how do you measure whether two
lines are perpendicular?
517
00:27:29,200 --> 00:27:31,700
The answer is if they're
perpendicular, the angle
518
00:27:31,700 --> 00:27:33,870
between them is 90 degrees.
519
00:27:33,870 --> 00:27:37,330
The tangent of 90 degrees is
infinity, as we learned.
520
00:27:37,330 --> 00:27:39,430
That's equivalent
to saying what?
521
00:27:39,430 --> 00:27:42,270
That the denominator is 0.
522
00:27:42,270 --> 00:27:44,230
See, the only way a fraction
blows up is for the
523
00:27:44,230 --> 00:27:45,520
denominator to be 0.
524
00:27:45,520 --> 00:27:50,400
But the only way that 1 plus
m1 m2 can be 0 is for what?
525
00:27:50,400 --> 00:27:54,260
m1 m2 to be equal to minus 1.
526
00:27:54,260 --> 00:27:57,780
And this gives us the other very
well-known result that in
527
00:27:57,780 --> 00:28:00,250
terms of slopes, to study
whether two lines are
528
00:28:00,250 --> 00:28:03,510
perpendicular, all we need to
investigate is whether one
529
00:28:03,510 --> 00:28:07,570
slope is the negative reciprocal
of the other.
530
00:28:07,570 --> 00:28:10,070
Well, again, the textbook
will bring out
531
00:28:10,070 --> 00:28:11,630
slopes in more detail.
532
00:28:11,630 --> 00:28:14,220
The next question that we'd like
to bring up in terms of a
533
00:28:14,220 --> 00:28:19,000
picture is worth a thousand
words is how do you identify
534
00:28:19,000 --> 00:28:21,780
the straight line with an
algebraic equation?
535
00:28:21,780 --> 00:28:24,690
What do we mean by the equation
of a straight line?
536
00:28:24,690 --> 00:28:27,790
Well, again, there are
two possibilities.
537
00:28:27,790 --> 00:28:30,150
The first possibility
is that the line is
538
00:28:30,150 --> 00:28:33,020
parallel to the y-axis.
539
00:28:33,020 --> 00:28:36,580
If the line is parallel to the
y-axis, if the line goes
540
00:28:36,580 --> 00:28:40,710
through the point a comma 0,
notice that the only criteria
541
00:28:40,710 --> 00:28:43,940
for the point to be on
that line is that its
542
00:28:43,940 --> 00:28:46,530
x-coordinate equal a.
543
00:28:46,530 --> 00:28:49,940
By the way, this is often
abbreviated in the textbook as
544
00:28:49,940 --> 00:28:52,030
the line x equals a.
545
00:28:52,030 --> 00:28:54,270
Many a student says how do
you know this is a line?
546
00:28:54,270 --> 00:28:56,740
Why isn't this the
point x equals a?
547
00:28:56,740 --> 00:28:59,750
And here again is a good review
of why we stress the
548
00:28:59,750 --> 00:29:01,480
language of sets.
549
00:29:01,480 --> 00:29:05,030
Here again is a good reason why
we express the language of
550
00:29:05,030 --> 00:29:06,530
sets so strongly.
551
00:29:06,530 --> 00:29:09,060
Namely, go back to the universe
of discourse here.
552
00:29:09,060 --> 00:29:12,900
When you see the set of all
ordered pairs x comma y for
553
00:29:12,900 --> 00:29:15,730
which x equals a, this gives
you the hint that you're
554
00:29:15,730 --> 00:29:18,760
talking about pairs of points,
and that tells you that you
555
00:29:18,760 --> 00:29:22,710
have numbers in the plane, not
on the line, not on the
556
00:29:22,710 --> 00:29:26,650
x-axis, a two-dimensional
interpretation over here.
557
00:29:26,650 --> 00:29:30,400
You see, if this said the set of
all x such that x equals a,
558
00:29:30,400 --> 00:29:31,850
it would just be a point.
559
00:29:31,850 --> 00:29:33,900
But notice the hint over here.
560
00:29:33,900 --> 00:29:37,030
At any rate, this then becomes
the equation of a straight
561
00:29:37,030 --> 00:29:40,660
line if the straight line is
parallel to the y-axis.
562
00:29:40,660 --> 00:29:42,710
Of course, the other possibility
is what if the
563
00:29:42,710 --> 00:29:45,590
line isn't parallel
to the x-axis?
564
00:29:45,590 --> 00:29:48,870
And here, too, we say OK,
suppose we know a point on the
565
00:29:48,870 --> 00:29:51,500
line and suppose we know
the slope of the line.
566
00:29:51,500 --> 00:29:54,750
What we will do is pick any
other point on the plane,
567
00:29:54,750 --> 00:29:58,640
which we will label arbitrarily
x comma y, and see
568
00:29:58,640 --> 00:30:02,100
what the equation x comma
y has to satisfy.
569
00:30:02,100 --> 00:30:05,340
How do the coordinates have to
be related to be on this line?
570
00:30:05,340 --> 00:30:09,820
Well, we already know that slope
does not depend on which
571
00:30:09,820 --> 00:30:11,200
two points you pick.
572
00:30:11,200 --> 00:30:15,690
Consequently, since the slope
of this line is m, the slope
573
00:30:15,690 --> 00:30:17,020
must also be what?
574
00:30:17,020 --> 00:30:21,250
y minus y1 over x minus x1.
575
00:30:21,250 --> 00:30:24,790
And this becomes the fundamental
definition for the
576
00:30:24,790 --> 00:30:29,260
equation of a line which is not
parallel to the y-axis.
577
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
578
00:30:34,280 --> 00:30:37,080
give you a bit of hardship
at first until
579
00:30:37,080 --> 00:30:38,250
you get used to them.
580
00:30:38,250 --> 00:30:42,300
Let's illustrate this thing
with a specific example.
581
00:30:42,300 --> 00:30:46,090
Suppose I say to you I am
thinking of the line whose
582
00:30:46,090 --> 00:30:50,600
slope is 3 and which passes
through the point 2 comma 5.
583
00:30:50,600 --> 00:30:52,280
And notice the language
of sets here.
584
00:30:52,280 --> 00:30:55,600
To say that 2 comma 5 is on the
line is the same as saying
585
00:30:55,600 --> 00:30:58,910
that 2 comma 5 belongs
to the set of points
586
00:30:58,910 --> 00:31:00,440
determined by the line.
587
00:31:00,440 --> 00:31:03,520
Drawing a rough sketch
over here--
588
00:31:03,520 --> 00:31:06,340
and by the way, notice something
very important here.
589
00:31:06,340 --> 00:31:09,680
I never have to draw to scale.
590
00:31:09,680 --> 00:31:12,370
Because, you see, all I'm going
to use is the analytic
591
00:31:12,370 --> 00:31:16,370
terms, and 2 comma 5 is still 2
and 5, no matter how I draw
592
00:31:16,370 --> 00:31:17,280
the picture.
593
00:31:17,280 --> 00:31:20,670
So, for example, if I say OK,
let's see what it means for
594
00:31:20,670 --> 00:31:24,020
the point x comma y to belong
here, I say, well,
595
00:31:24,020 --> 00:31:25,020
what does that mean?
596
00:31:25,020 --> 00:31:28,330
My slope is going to have
to be what? y minus 5.
597
00:31:28,330 --> 00:31:29,640
That's my rise.
598
00:31:29,640 --> 00:31:35,170
My run is x minus 2, and
that must equal 3.
599
00:31:35,170 --> 00:31:38,780
And if I clear this of
fractions, I get what? y is
600
00:31:38,780 --> 00:31:47,610
equal to 3x minus 1.
601
00:31:47,610 --> 00:31:48,490
By the way, does
this check out?
602
00:31:48,490 --> 00:31:52,680
If x is 2, 2 times 3
is 6, minus 1 is 5.
603
00:31:52,680 --> 00:31:54,960
2 comma 5 is on the line.
604
00:31:54,960 --> 00:31:56,120
You see, here's the thing.
605
00:31:56,120 --> 00:31:58,630
We talked about the line
geometrically.
606
00:31:58,630 --> 00:32:00,850
Now I have an algebraic
equation.
607
00:32:00,850 --> 00:32:02,950
I no longer have to refer
to the picture.
608
00:32:02,950 --> 00:32:04,730
I have something analytic now.
609
00:32:04,730 --> 00:32:07,450
For example, suppose a person
says to me I wonder if the
610
00:32:07,450 --> 00:32:10,810
point 8 comma 23 is
on this line?
611
00:32:10,810 --> 00:32:12,470
I don't have to draw
a picture to scale.
612
00:32:12,470 --> 00:32:14,040
I don't have to waste
any time.
613
00:32:14,040 --> 00:32:18,650
I know that the equation of my
line is y equals 3x minus 1.
614
00:32:18,650 --> 00:32:25,720
By the way, if y equals 3x minus
1, as soon as x is 8,
615
00:32:25,720 --> 00:32:27,550
what must y equal?
616
00:32:27,550 --> 00:32:31,310
y must equal what?
617
00:32:31,310 --> 00:32:34,260
23?
618
00:32:34,260 --> 00:32:35,540
Is that right?
619
00:32:35,540 --> 00:32:42,130
And so is the point 8 comma
23 on the line?
620
00:32:42,130 --> 00:32:42,900
Yes.
621
00:32:42,900 --> 00:32:46,350
How about 8 comma 12?
622
00:32:46,350 --> 00:32:51,560
8 comma 12 isn't on the
line because 3 times 8
623
00:32:51,560 --> 00:32:53,370
minus 1 is not 12.
624
00:32:53,370 --> 00:32:56,800
But notice that we can even see
algebraically that 8 comma
625
00:32:56,800 --> 00:33:00,170
12 must be below the line.
626
00:33:00,170 --> 00:33:04,390
In other words, our study of
equations allows us not only
627
00:33:04,390 --> 00:33:07,800
to visualize lines as equations,
but we can also
628
00:33:07,800 --> 00:33:11,230
visualize inequalities
as pictures.
629
00:33:11,230 --> 00:33:18,000
In other words, if we have the
equation of a line, if this is
630
00:33:18,000 --> 00:33:24,570
the line y equals, say, 3x plus
1 or something like this,
631
00:33:24,570 --> 00:33:26,770
then what is this region here?
632
00:33:26,770 --> 00:33:30,310
These are all those values which
lie-- whose heights lie
633
00:33:30,310 --> 00:33:32,710
below the height to
be on the curve.
634
00:33:32,710 --> 00:33:36,140
Again, not a very clear example
in the sense of
635
00:33:36,140 --> 00:33:39,090
drawing the picture neatly for
you, but our main aim is not
636
00:33:39,090 --> 00:33:40,580
to draw neat pictures here.
637
00:33:40,580 --> 00:33:43,950
Our main aim is to show how
analytical terms can be
638
00:33:43,950 --> 00:33:47,350
studied very conveniently
in terms of pictures.
639
00:33:47,350 --> 00:33:50,560
In fact, perhaps to conclude
today's lesson, what we should
640
00:33:50,560 --> 00:33:54,550
talk about is an old algebraic
concept called
641
00:33:54,550 --> 00:33:56,390
simultaneous equations.
642
00:33:56,390 --> 00:34:00,270
Suppose you're asked to solve
this pair of equations.
643
00:34:00,270 --> 00:34:03,490
You say, well, let's see, if y
equals 3x minus 1 and it's
644
00:34:03,490 --> 00:34:08,699
also equal to x plus 1, that
says that x plus 1
645
00:34:08,699 --> 00:34:11,159
equals 3x minus 1.
646
00:34:11,159 --> 00:34:13,510
I now solve this thing
algebraically.
647
00:34:13,510 --> 00:34:18,050
I get 2x equals 2,
so x equals 1.
648
00:34:18,050 --> 00:34:23,630
Knowing that x equals 1, I can
see that y equals 2, and I see
649
00:34:23,630 --> 00:34:28,030
that 1 comma 2 is my solution.
650
00:34:28,030 --> 00:34:30,620
In other words, if I wound up
with this thing algebraically,
651
00:34:30,620 --> 00:34:35,409
1 comma 2 is the only member
that belongs to both of these
652
00:34:35,409 --> 00:34:37,520
two solution sets.
653
00:34:37,520 --> 00:34:41,690
Now, again, notice how I can
solve this purely analytic
654
00:34:41,690 --> 00:34:45,530
problem without recourse
to a picture.
655
00:34:45,530 --> 00:34:48,510
On the other hand, if I want
to think of this thing
656
00:34:48,510 --> 00:34:52,989
pictorially, notice that y
equals 3x minus 1 is the
657
00:34:52,989 --> 00:34:58,040
equation of a particular
straight line, and y equals x
658
00:34:58,040 --> 00:35:02,450
plus 1 is also the equation
of a line.
659
00:35:02,450 --> 00:35:05,850
Notice that since these two
lines are not parallel, they
660
00:35:05,850 --> 00:35:08,420
intersect at one particular
point.
661
00:35:08,420 --> 00:35:12,530
And the geometric problem that I
solved on the previous board
662
00:35:12,530 --> 00:35:16,820
turns out to be that the point
1 comma 2 is the point that
663
00:35:16,820 --> 00:35:19,040
both of these lines
have in common.
664
00:35:19,040 --> 00:35:23,640
In fact, if we call this line
as before l1 and if we name
665
00:35:23,640 --> 00:35:30,270
this l2, in the language of
sets, 1 comma 2 is what?
666
00:35:30,270 --> 00:35:33,560
The point which is the
intersection of the two lines
667
00:35:33,560 --> 00:35:38,190
l1 and l2, again, a geometric
interpretation
668
00:35:38,190 --> 00:35:39,830
for an analytic problem.
669
00:35:39,830 --> 00:35:43,380
In fact, notice also how much
mileage I can get out of the
670
00:35:43,380 --> 00:35:44,910
geometric picture.
671
00:35:44,910 --> 00:35:49,170
For example, notice that this
region here has a very nice
672
00:35:49,170 --> 00:35:50,870
geometric interpretation.
673
00:35:50,870 --> 00:35:52,050
It's the set of what?
674
00:35:52,050 --> 00:35:57,620
All points which are below this
line and above this line.
675
00:35:57,620 --> 00:36:00,010
In other words, what?
676
00:36:00,010 --> 00:36:04,970
To be below this line, y must be
less than x plus 1, and to
677
00:36:04,970 --> 00:36:09,880
be above this line, y must be
greater than 3x minus 1.
678
00:36:09,880 --> 00:36:13,430
Notice then that a pair of
simultaneous inequalities,
679
00:36:13,430 --> 00:36:16,510
which may not be that easy to
handle, are very easy to
680
00:36:16,510 --> 00:36:19,890
handle in terms of regions
in the plane.
681
00:36:19,890 --> 00:36:23,580
Notice also that since two lines
can either be parallel
682
00:36:23,580 --> 00:36:27,660
or not parallel, we also get a
nice geometric interpretation
683
00:36:27,660 --> 00:36:30,270
as to why simultaneous
equations
684
00:36:30,270 --> 00:36:32,080
may have one solution.
685
00:36:32,080 --> 00:36:34,240
Namely, the lines are
not parallel,
686
00:36:34,240 --> 00:36:35,540
and hence, they intersect.
687
00:36:35,540 --> 00:36:38,610
Or no solutions, the lines
could've been parallel without
688
00:36:38,610 --> 00:36:39,440
intersecting.
689
00:36:39,440 --> 00:36:43,500
Or infinitely many solutions,
the two lines could have been
690
00:36:43,500 --> 00:36:45,350
different equations.
691
00:36:45,350 --> 00:36:47,490
In effect, I should say what?
692
00:36:47,490 --> 00:36:50,130
The two equations could've been
different equations for
693
00:36:50,130 --> 00:36:51,290
the same line.
694
00:36:51,290 --> 00:36:55,300
Again, this may seem a little
bit sketchy and rapid, but all
695
00:36:55,300 --> 00:36:57,250
we want to do is give
the overview.
696
00:36:57,250 --> 00:37:00,460
The reading assignment in the
text goes into great detail on
697
00:37:00,460 --> 00:37:02,370
the points that we've
mentioned so far.
698
00:37:02,370 --> 00:37:05,760
But again, in summary, what our
lesson was supposed to be
699
00:37:05,760 --> 00:37:09,640
today was to indicate the
importance of being able to
700
00:37:09,640 --> 00:37:13,880
visualize and to identify
analytic results with
701
00:37:13,880 --> 00:37:15,830
geometric pictures.
702
00:37:15,830 --> 00:37:17,680
And so, until next
time, goodbye.
703
00:37:17,680 --> 00:37:20,840
704
00:37:20,840 --> 00:37:24,040
Funding for the publication of
this video was provided by the
705
00:37:24,040 --> 00:37:28,090
Gabriella and Paul Rosenbaum
Foundation.
706
00:37:28,090 --> 00:37:32,270
Help OCW continue to provide
free and open access to MIT
707
00:37:32,270 --> 00:37:36,460
courses by making a donation
at ocw.mit.edu/donate.
708
00:37:36,460 --> 00:37:41,204