1
00:00:00,000 --> 00:00:09,510
DAVID JORDAN: Hello, and welcome
back to recitation.

2
00:00:09,510 --> 00:00:12,500
Today what we want to work on
is drawing level curves.

3
00:00:12,500 --> 00:00:15,520
This is for all the artists
out there in the audience.

4
00:00:15,520 --> 00:00:21,180
We have three functions here: z
is 2x plus y, z is x squared

5
00:00:21,180 --> 00:00:24,680
plus y squared, and z is x
squared minus y squared, and

6
00:00:24,680 --> 00:00:26,990
we want to get some practice
drawing their level curves.

7
00:00:26,990 --> 00:00:30,000
Now, just to remind you, the
level curves are not drawn in

8
00:00:30,000 --> 00:00:30,730
three dimensions.

9
00:00:30,730 --> 00:00:35,220
They're drawn in the xy-plane
and they're constructed by

10
00:00:35,220 --> 00:00:38,130
setting z to be a constant and
then graphing the curve that

11
00:00:38,130 --> 00:00:41,190
we get, so we can think about a
relief map that we might use

12
00:00:41,190 --> 00:00:42,440
if we were hiking.

13
00:00:42,440 --> 00:00:45,030
So why don't you get
started on that.

14
00:00:45,030 --> 00:00:48,110
Pause the video, and we'll check
back, and I'll show you

15
00:00:48,110 --> 00:00:49,360
how I solve this.

16
00:00:49,360 --> 00:00:57,330

17
00:00:57,330 --> 00:00:58,440
Welcome back.

18
00:00:58,440 --> 00:01:02,370
So over here, we've got
the equation for part

19
00:01:02,370 --> 00:01:04,330
a already set up.

20
00:01:04,330 --> 00:01:08,070
So z is 2x plus y.

21
00:01:08,070 --> 00:01:12,300
So now, what we need to
do to get started is

22
00:01:12,300 --> 00:01:13,550
just draw the xy-axis.

23
00:01:13,550 --> 00:01:20,610

24
00:01:20,610 --> 00:01:24,480
And, you know, there's really
not a precise science for

25
00:01:24,480 --> 00:01:25,780
drawing these level
curves out.

26
00:01:25,780 --> 00:01:28,050
We just need to choose some
values of z that we feel are

27
00:01:28,050 --> 00:01:30,830
representative and then
just draw them in.

28
00:01:30,830 --> 00:01:35,640
So one thing we notice about
this is that if we choose z to

29
00:01:35,640 --> 00:01:37,930
be a constant, then the
equations that we're going to

30
00:01:37,930 --> 00:01:42,150
get is 2x plus y equals
some constant, right?

31
00:01:42,150 --> 00:01:47,110
So, you know, these are just
going to be lines.

32
00:01:47,110 --> 00:01:49,600
The level curves in this case
are just going to be lines.

33
00:01:49,600 --> 00:01:54,360
So, for instance, if we take the
level curve at z equals 0,

34
00:01:54,360 --> 00:01:59,730
then we have just the equation
2x plus y equals 0.

35
00:01:59,730 --> 00:02:02,230
And so that has intercept--

36
00:02:02,230 --> 00:02:05,660
so we're looking at--

37
00:02:05,660 --> 00:02:15,090
so 0 equals 2x plus y, so that's
just y equals minus 2x.

38
00:02:15,090 --> 00:02:19,150
So that's this level curve.

39
00:02:19,150 --> 00:02:21,980
That's the level curve
at z equals 0.

40
00:02:21,980 --> 00:02:24,080
Now, if you think about it, all
the other level curves,

41
00:02:24,080 --> 00:02:26,500
we're just going to be varying
the constant here, and so

42
00:02:26,500 --> 00:02:28,170
we're just going to be
shifting this line.

43
00:02:28,170 --> 00:02:36,080
So all of our level curves
in this case are

44
00:02:36,080 --> 00:02:37,520
just straight lines.

45
00:02:37,520 --> 00:02:41,290
So let's see if we can make
some sense out of that by

46
00:02:41,290 --> 00:02:42,510
thinking about the
graph in three

47
00:02:42,510 --> 00:02:45,170
dimensions of this function.

48
00:02:45,170 --> 00:02:46,815
So over here, I'm
going to draw.

49
00:02:46,815 --> 00:02:55,690

50
00:02:55,690 --> 00:03:00,500
So this function z equals 2x
plus y, if we draw its graph,

51
00:03:00,500 --> 00:03:02,270
it's just a plane, right?

52
00:03:02,270 --> 00:03:05,560
So it's just a plane, which
I'll just kind of draw in

53
00:03:05,560 --> 00:03:09,870
cartoon form, something
like that.

54
00:03:09,870 --> 00:03:12,210
And now when we do level curves,
what we're doing is

55
00:03:12,210 --> 00:03:15,920
we're slicing this plane with
another plane, which is the

56
00:03:15,920 --> 00:03:18,480
horizontal values where
z is a constant.

57
00:03:18,480 --> 00:03:22,590
And so, for instance, if we
take the level curve here,

58
00:03:22,590 --> 00:03:28,390
then we're just intersecting
these two planes, and their

59
00:03:28,390 --> 00:03:31,290
intersection is just a line, and
that's exactly the lines

60
00:03:31,290 --> 00:03:32,540
that we're drawing here.

61
00:03:32,540 --> 00:03:36,820

62
00:03:36,820 --> 00:03:39,090
So it's not surprising that
we were graphing a linear

63
00:03:39,090 --> 00:03:42,520
function and that our contour
lines, our level curves, were

64
00:03:42,520 --> 00:03:43,790
just straight lines.

65
00:03:43,790 --> 00:03:46,500
So let's go on to a slightly
more interesting example,

66
00:03:46,500 --> 00:03:51,210
which is part b, which I
have written up here.

67
00:03:51,210 --> 00:03:54,630
So this is the function
z equals x

68
00:03:54,630 --> 00:03:56,030
squared plus y squared.

69
00:03:56,030 --> 00:03:58,660
Actually, this is even easier
to get started drawing the

70
00:03:58,660 --> 00:03:59,910
level curves for.

71
00:03:59,910 --> 00:04:07,800

72
00:04:07,800 --> 00:04:11,500
Well, if you think about it, if
I fix the value of z, then

73
00:04:11,500 --> 00:04:15,440
this is exactly the equation
for the circle with radius

74
00:04:15,440 --> 00:04:16,880
square root of z.

75
00:04:16,880 --> 00:04:29,510
So level curves, level curves
for the function z equals x

76
00:04:29,510 --> 00:04:33,340
squared plus y squared,
these are just

77
00:04:33,340 --> 00:04:35,890
circles in the xy-plane.

78
00:04:35,890 --> 00:04:40,130
And if we're being careful and
if we take the convention that

79
00:04:40,130 --> 00:04:44,190
our level curves are evenly
spaced in the z-plane, then

80
00:04:44,190 --> 00:04:48,940
these are going to get closer
and closer together, and we'll

81
00:04:48,940 --> 00:04:51,830
see in a minute where
that's coming from.

82
00:04:51,830 --> 00:04:55,885
So let's draw what's going
on in three dimensions.

83
00:04:55,885 --> 00:05:02,570

84
00:05:02,570 --> 00:05:06,040
So if we graph z equals x
squared plus y squared in

85
00:05:06,040 --> 00:05:13,420
three dimensions, this is just
a paraboloid opening up.

86
00:05:13,420 --> 00:05:17,740
And now what you can see is that
if we slice this through

87
00:05:17,740 --> 00:05:22,730
the constant, through z equals
constant planes, then we're

88
00:05:22,730 --> 00:05:28,180
just getting these circles, and
those are precisely the

89
00:05:28,180 --> 00:05:30,520
circles that we're drawing
on the level curve.

90
00:05:30,520 --> 00:05:35,270
And because the parabola gets
steeper and steeper, that's

91
00:05:35,270 --> 00:05:42,220
telling us that these circles,
if we keep incrementing z in a

92
00:05:42,220 --> 00:05:45,760
constant way, that's telling us
that the circles, which are

93
00:05:45,760 --> 00:05:49,220
the shadows below here, are
going to get closer and closer

94
00:05:49,220 --> 00:05:50,580
and closer.

95
00:05:50,580 --> 00:05:52,720
This reflects the fact
that this is

96
00:05:52,720 --> 00:05:54,050
getting steeper and steeper.

97
00:05:54,050 --> 00:05:55,640
In fact, this is
generally true.

98
00:05:55,640 --> 00:06:00,310
If you're looking at a contour
plot where the intervals

99
00:06:00,310 --> 00:06:05,650
between level curves are at
regular distances, then very

100
00:06:05,650 --> 00:06:09,060
close contour lines means
that the function

101
00:06:09,060 --> 00:06:10,850
is very steep there.

102
00:06:10,850 --> 00:06:13,290
So that's something
to keep in mind.

103
00:06:13,290 --> 00:06:16,480
Let's look at one
more example.

104
00:06:16,480 --> 00:06:21,210
This is z equals x squared
minus y squared.

105
00:06:21,210 --> 00:06:32,120
So to get started with this,
well, again, if we start

106
00:06:32,120 --> 00:06:35,590
choosing constant values of
z, this is just giving us

107
00:06:35,590 --> 00:06:38,340
hyperbola, hyperbolas
of two sheets.

108
00:06:38,340 --> 00:06:42,570
So, for instance, if we take--

109
00:06:42,570 --> 00:06:45,230
so let's see what happens
if we take z equals 0.

110
00:06:45,230 --> 00:06:47,290
So if we take z equals
0, then something a

111
00:06:47,290 --> 00:06:49,040
little special happens.

112
00:06:49,040 --> 00:06:57,210
This becomes x plus y times
x minus y equals 0.

113
00:06:57,210 --> 00:07:00,620
We can factorize x squared minus
y squared as x plus y

114
00:07:00,620 --> 00:07:04,240
times x minus y, and if this is
0, then that means either

115
00:07:04,240 --> 00:07:07,190
plus y is 0 or x minus y is 0.

116
00:07:07,190 --> 00:07:13,650
So that tells is that the zero
level curves for this graph

117
00:07:13,650 --> 00:07:18,565
are the lines y equals minus
x and the lines y equals x.

118
00:07:18,565 --> 00:07:21,070
OK.

119
00:07:21,070 --> 00:07:26,820
And now, if we move z away from
that, then what we're

120
00:07:26,820 --> 00:07:45,170
getting are hyperbolas, and
these hyperbolas will approach

121
00:07:45,170 --> 00:07:50,120
this asymptotic line y equals
minus x or this line--

122
00:07:50,120 --> 00:07:52,560
sorry, this line y equals
x or this line equals y

123
00:07:52,560 --> 00:07:53,690
equals minus x.

124
00:07:53,690 --> 00:07:55,650
They'll approach this as they
go down, but they'll never

125
00:07:55,650 --> 00:07:57,190
quite reach it.

126
00:07:57,190 --> 00:07:59,600
So the level curves here
are just hyperbolas.

127
00:07:59,600 --> 00:08:04,000
So now let's see, what is this
telling us about the

128
00:08:04,000 --> 00:08:05,255
three-dimensional graph
of this function?

129
00:08:05,255 --> 00:08:15,850

130
00:08:15,850 --> 00:08:19,500
OK, so, first of all, we have
these level curves when y

131
00:08:19,500 --> 00:08:22,970
equals x and when y equals minus
x, and so those level

132
00:08:22,970 --> 00:08:24,220
curves we can kind of draw.

133
00:08:24,220 --> 00:08:30,130

134
00:08:30,130 --> 00:08:31,040
OK, so I want you to think that

135
00:08:31,040 --> 00:08:32,420
that sits in the xy-plane.

136
00:08:32,420 --> 00:08:35,810
It's kind of hard to draw
in three dimensions.

137
00:08:35,810 --> 00:08:39,610
And so this is where our
function is going to be zero.

138
00:08:39,610 --> 00:08:43,663
Now, if we take x
to be positive--

139
00:08:43,663 --> 00:08:48,970

140
00:08:48,970 --> 00:08:49,470
sorry.

141
00:08:49,470 --> 00:08:58,770
If we take x to be greater than
y and both positive, then

142
00:08:58,770 --> 00:09:01,410
this is a positive number and
this is a positive number.

143
00:09:01,410 --> 00:09:04,690
So if we look in the region
where x and y are both

144
00:09:04,690 --> 00:09:08,140
positive, that's in here, and
where x is greater than y,

145
00:09:08,140 --> 00:09:09,790
then our function comes up.

146
00:09:09,790 --> 00:09:15,490
So it looks like this, and then
it dips down and goes

147
00:09:15,490 --> 00:09:23,220
down, comes back up,
and goes back down.

148
00:09:23,220 --> 00:09:26,980
And now at the middle here, it
has to dip down to zero, so we

149
00:09:26,980 --> 00:09:29,710
have something like this.

150
00:09:29,710 --> 00:09:35,040
So what we end up getting in the
end, this is a saddle, so

151
00:09:35,040 --> 00:09:36,160
it's a bit hard to draw.

152
00:09:36,160 --> 00:09:39,000
It's a bit hard to see on this
so let me draw a sketch of it

153
00:09:39,000 --> 00:09:40,600
off of the axes for you.

154
00:09:40,600 --> 00:09:48,010
So we have a rise, and then a
drop, and then a rise in the

155
00:09:48,010 --> 00:09:52,770
back, and then a drop, and then
down the middle it dips

156
00:09:52,770 --> 00:09:58,030
down in this direction and it
rises up in this direction, so

157
00:09:58,030 --> 00:10:00,380
it's a saddle like you
could put this on a

158
00:10:00,380 --> 00:10:01,630
horse and ride it.

159
00:10:01,630 --> 00:10:05,750

160
00:10:05,750 --> 00:10:09,700
And so we can see that the
three-dimensional contours of

161
00:10:09,700 --> 00:10:14,550
the saddle when we look at their
projection down onto the

162
00:10:14,550 --> 00:10:16,340
contour plot become
these hyperbolas.

163
00:10:16,340 --> 00:10:20,480
So a saddle is sort of like a
hyperbola stretched out into

164
00:10:20,480 --> 00:10:22,790
three dimensions.

165
00:10:22,790 --> 00:10:25,020
And I think I'll leave
it at that.

166
00:10:25,020 --> 00:10:25,425