1
00:00:12;02 --> 00:00:13;02
Change of plans

2
00:00:13;03 --> 00:00:16;07
OK Carlos

3
00:00:19;06 --> 00:00:21;02
[What did you put
for the first one?]

4
00:00:21;12 --> 00:00:22;05
[For what?]

5
00:00:22;10 --> 00:00:25;09
[For this one. That one.]

6
00:00:57;00 --> 00:00:58;13
[Where's the next
point after this?]

7
00:01:01;11 --> 00:01:03;13
[You're looking for
the next point?]

8
00:01:03;14 --> 00:01:04;00
[Yeah]

9
00:01:15;03 --> 00:01:18;10
Let's chit-chat just
for a second.

10
00:01:18;11 --> 00:01:22;03
In fact, I'm going to...

11
00:01:32;03 --> 00:01:34;14
At negative 4, at the
X value of negative 4,

12
00:01:34;15 --> 00:01:36;01
What's my rate
of change again?

13
00:01:37;05 --> 00:01:37;11
[Negative 7]

14
00:01:37;12 --> 00:01:39;06
Negative 7. Where is that?

15
00:01:41;03 --> 00:01:42;14
[I so can't see my numbers]

16
00:01:44;04 --> 00:01:50;04
2, 4, 6, 7, so rate
of change is right there.

17
00:01:50;05 --> 00:01:52;14
At negative 3, what
was my rate of change?

18
00:01:52;12 --> 00:01:54;02
[Negative 4]

19
00:01:54;10 --> 00:01:55;06
Negative...

20
00:01:55;07 --> 00:01:56;00
[5]

21
00:01:56;01 --> 00:02:09;11
5. Yeah? No. 2, 4...2, 4, 5.
Found it. Yeah.

22
00:02:12;14 --> 00:02:14;02
[The one before]

23
00:02:14;03 --> 00:02:15;07
The one before is off?

24
00:02:20;00 --> 00:02:22;12
[The first point that
you did was wrong]

25
00:02:22;14 --> 00:02:24;00
[Ryan, stop
giving directions]

26
00:02:24;01 --> 00:02:24;07
Is that better?

27
00:02:24;08 --> 00:02:26;07
[Yeah, that's good]

28
00:02:26;15 --> 00:02:29;13
No, I want it to look correct.

29
00:02:29;14 --> 00:02:31;12
At negative 2,
the rate of change was?

30
00:02:31;13 --> 00:02:32;12
[Negative 3]

31
00:02:32;14 --> 00:02:39;05
So 2, 3. At negative 1?

32
00:02:39;06 --> 00:02:40;04
[Negative 1]

33
00:02:40;05 --> 00:02:43;13
Negative 1. At zero?

34
00:02:45;05 --> 00:02:44;12
[1]

35
00:02:45;12 --> 00:02:52;05
It's 1. So zero comma
1 is right there. At 1?

36
00:02:53;12 --> 00:02:54;13
[1]

37
00:02:54;15 --> 00:02:57;05
At 1 it's 1?

38
00:02:57;06 --> 00:02:58;01
[Oh, whoops.]

39
00:02:58;02 --> 00:02:59;03
No? What is that one?

40
00:02:59;04 --> 00:02:59;12
[3]

41
00:02:59;13 --> 00:03:06;12
3? K, at 2?

42
00:03:06;13 --> 00:03:08;00
[5]

43
00:03:12;11 --> 00:03:13;09
At 3?

44
00:03:13;10 --> 00:03:14;08
[7]

45
00:03:15;09 --> 00:03:20;10
7. And then, we actually
don't have a 4, do we?

46
00:03:22;12 --> 00:03:26;10
OK, you all get dots that
resemble at least? Mike?

47
00:03:28;09 --> 00:03:32;03
The question is what
about in between the dots?

48
00:03:32;05 --> 00:03:33;11
[Draw a line.]

49
00:03:34;14 --> 00:03:37;01
So just, whoosh,
draw a line?

50
00:03:37;02 --> 00:03:39;12
[No, like, a horizontal line]

51
00:03:39;13 --> 00:03:47;08
A horizontal line. Like
stairs? This rate of change,

52
00:03:47;09 --> 00:03:50;13
how long was this particular
rate of change valued for?

53
00:03:50;15 --> 00:03:52;05
[1]

54
00:03:52;06 --> 00:03:56;10
1 whole unit. That
rate of change,

55
00:03:56;11 --> 00:03:58;04
How long was it valued for?

56
00:03:58;05 --> 00:03:59;00
[1]

57
00:03:59;01 --> 00:04:03;05
1 whole unit. Then
I can continue that?

58
00:04:03;06 --> 00:04:04;00
[Yeah]

59
00:04:10;14 --> 00:04:14;00
So, it's your steps. Carlos,

60
00:04:14;01 --> 00:04:15;09
What did you say I could
do with your little thingy?

61
00:04:17;07 --> 00:04:21;06
Where did it go? OK,
what could I have it do?

62
00:04:24;01 --> 00:04:28;03
[N to 2 should be
their function]

63
00:04:30;01 --> 00:04:31;05
And then, step size 1?

64
00:04:31;06 --> 00:04:31;12
[Yeah]

65
00:04:44;14 --> 00:04:45;13
I'll hide these

66
00:04:52;13 --> 00:04:55;04
Let's see, you guys saw more
X values than this

67
00:05:01;08 --> 00:05:05;02
Nonetheless, what
we drew looks like that?

68
00:05:05;03 --> 00:05:05;10
[Yeah]

69
00:05:05;11 --> 00:05:11;08
alright good stuff. Is the
most accurate rate of

70
00:05:11;09 --> 00:05:12;29
change graph I could
ever draw my whole life?

71
00:05:13;00 --> 00:05:13;04
[No]

72
00:05:13;05 --> 00:05:14;06
No, why not?

73
00:05:14;10 --> 00:05:18;02
[You can't draw the
stack of the curve]

74
00:05:21;01 --> 00:05:24;04
Why isn't this the
most accurate one

75
00:05:24;05 --> 00:05:25;04
In the whole wide world?

76
00:05:25;05 --> 00:05:26;10
[Step size 1]

77
00:05:26;11 --> 00:05:27;14
Yeah, good Sabrina,

78
00:05:27;15 --> 00:05:31;04
Because our intervals
were step size 1.

79
00:05:31;05 --> 00:05:33;07
If I wanted a more accurate version of

80
00:05:33;08 --> 00:05:36;00
this rate of change graph,
what could I do?

81
00:05:36;01 --> 00:05:37;02
[Smaller step size]

82
00:05:37;03 --> 00:05:40;01
Step size .1. What if I wanted
more accurate?

83
00:05:40;11 --> 00:05:41;08
[.01]

84
00:05:41;09 --> 00:05:43;07
.01. I want more accurate.

85
00:05:43;08 --> 00:05:44;10
[.001]

86
00:05:44;11 --> 00:05:46;05
.001. OK, we can play
this game forever, right?

87
00:05:46;06 --> 00:05:51;07
In terms of the rate of
change for that curve that

88
00:05:51;08 --> 00:05:55;03
you guys were given, if I did
take step sizes smaller, and

89
00:05:55;04 --> 00:05:58;08
smaller, and smaller, what
would happen to my step

90
00:05:58;09 --> 00:06:04;06
function? What would it
start to do? Go ahead.

91
00:06:04;05 --> 00:06:05;04
[Make them linear]

92
00:06:05;05 --> 00:06:07;11
Approach in a linear
fashion. Exactly.

93
00:06:07;12 --> 00:06:10;03
It's that whole idea we just
went through with that

94
00:06:10;04 --> 00:06:11;05
scenario of approaching
this smoother and smoother

95
00:06:11;06 --> 00:06:21;02
and smoother. Why? So,
good, really good. The

96
00:06:21;03 --> 00:06:27;07
moral of the story is
that when you have...

97
00:06:27;06 --> 00:06:29;12
Gonna go back up,
ahh too far.

98
00:06:34;11 --> 00:06:37;00
Just by looking at that
curve, ladies and gents,

99
00:06:37;01 --> 00:06:39;02
does this have a constant rate of change?

100
00:06:39;15 --> 00:06:43;00
Does this have a
constant rate of change?

101
00:06:43;01 --> 00:06:43;08
[No]

102
00:06:43;09 --> 00:06:44;07
Why not?

103
00:06:44;09 --> 00:06:47;02
[Because it's a parabola]

103
00:06:47;14 --> 00:06:50;00
Linear functions have
constant rates of change.

104
00:06:50;01 --> 00:06:51;12
Good. This is not linear.

105
00:06:51;13 --> 00:06:56;08
But, if I took step
size infinitely small,

106
00:06:56;09 --> 00:07:00;02
what kind of rate
of change does it have?

107
00:07:00;03 --> 00:07:02;13
How could you describe it
to me? It's not constant.

108
00:07:02;14 --> 00:07:06;03
But in a perfect world of
step size infinitely small,

109
00;07:06;04 --> 00:07:07;09
How could you describe to

110
00:07:07;10 --> 00:07:09;07
somebody the rate of
change for this function?

111
00:07:14;06 --> 00:07:15;03
[It's positive]

112
00:07:15;04 --> 00:07:21;11
This one is increasing. Yes.
Increasing what though?

113
00:07:21;12 --> 00:07:23;07
Remember, you can look
at yours if you want to,

114
00:07:23;08 --> 00:07:24;13
but that's step size 1.

115
00:07:24;14 --> 00:07:27;12
Imagine infinitely small step
size in a perfect world.

116
00:07:27;13 --> 00:07:31;02
How would you describe
it to me in a word?

117
00:07:33;08 --> 00:07:34;07
[A line]

118
00:07:34;08 --> 00:07:38;05
Linear. A line, exactly.
Again, you guys don't have

119
00:07:38;06 --> 00:07:40;04
a line because
we took step size 1.

120
00:07:40;05 --> 00:07:42;00
We've got
segmented pieces,

121
00:07:42;01 --> 00:07:44;14
but if Carlo's
program showed us

122
00:07:44;15 --> 00:07:48;07
If you took smaller, and smaller,
and smaller steps

123
00:07:48;08 --> 00:07:49;14
in a perfect world, to
make them infinitely small,

124
00:07:49;15 --> 00:07:54;14
The rate of change for this
bad boy is- what is it again?

125
00:07:54;15 --> 00:07:55;11
[Linear]

126
00:07:55;12 --> 00:08:03;02
Linear. It's a line. And
it is increasing, good.