1
00:00:04;02 --> 00:00:14;07
I want to talk about what this function is going to do.
I want to start looking at around X equal to .9.

2
00:00:14;08 --> 00:00:19;02
We're not going to go crazy on numbers, I just want to talk about a couple things,

3
00;00:19;03 --> 00:00:22;05
And if we really need exact, we've got
computers or calculators to help us do that.

4
00:00:22;06 --> 00:00:30;12
If I pick approximately .9, will this first factor
be positive or negative if I plug in .9?

5
00:00:33;00 --> 00;00:33;11
[Negative]

6
00:00;34;04 --> 00:00;37;00
.9 minus 1, is that a positive or negative?

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

8
00;00:37;14 --> 00:00:40;11
It will be negative. What about .9 minus 2?

9
00:00:40;12 --> 00:00:44;06
[Positive. Positive.]

10
00:00:44;07 --> 00:00:45;06
.9, take away 2?

11
00:00:45;07 --> 00:00:46;03
[Positive]

12
00:00;48;12 --> 00:00:49;07
[Negative]

13
00:00:49;08 --> 00:00:52;00
It will be negative. I don't know what it will- it's negative.

14
00:00:52;01 --> 00:00:53;05
[Because it's .9...]

15
00:00:53;06 --> 00:00:55;12
Will that be positive or negative if I plug in .9?

16
00:00:55;13 --> 00:00:56;10
[Negative]

17
00:00:56;11 --> 00:00:57;09
They're all going to be negative

18
00:00:57;10 --> 00:01:05;08
And what about this one? Negative. Things are getting multiplied, so do you multiply two negatives?

19
00:01:05;09 --> 00:01:07;02
[Positive]

20
00:01:09;04 --> 00:01:10;12
Here's two more negatives

21
00:01:10;13 --> 00:01:11;12
[Positive]

22
00:01:12;08 --> 00:01:14;11
So I have a positive times a positive

23
00:01:14;12 --> 00:01:16;02
Positive

24
00:01:16;03 --> 00:01:23;06
So, we're expecting a positive number. If I plug in .9,
I'll get a positive. A big positive, a small positive

25
00:01:28;07 --> 00:01:30;09
[Are going to be doing the thing that Carlos showed us?]

26
00:01:31;07 --> 00:01:40;09
No. Here I plugged in .9 and it's positive that. It does it for me because it's cool like that.

27
00:01;40;10 --> 00:01:45;02
We already knew it was positive just
by thinking about signs in general.

28
00:01:45;03 --> 00:01:49;06
We knew it would be a positive number,
turns out it's a positive kind of small number.

29
00:01:49;07 --> 00:01:56;09
At about .9, I'm about .7, so I'm going to go, let's see, .8, .9,

30
00:01:57;06 --> 00:02:02;02
I'm about right there. I'm a small, positive
number at X equals approximately .9.

31
00:02:02;03 --> 00:02:13;10
What about X equals .99? Will it be
a positive or negative number, you think?

32
00:02:14;11 --> 00:02:15;07
[Positive]

33
00:02;15;08 --> 00:02:19;02
Why positive again? What's this sign going to be if I plug in .99?

34
00:02:19;13 --> 00:02:20;06
[Negative]

35
00:02:20;07 --> 00:02:21;08
Still negative. This one?

36
00:02:21;09 --> 00:02:22;03
[Negative]

37
00:02:22;04 --> 00:02:23;02
Still negative. That one?

38
00:02:23;03 --> 00:02:23;10
[Negative]

39
00:02:23;11 --> 00:02:24;12
Still negative.

40
00:02:24;13 --> 00:02:25;05
[Negative]

41
00:02:25;06 --> 00:02:30;06
Still negative. We can think about signs the same
way as the one above, so we are expecting a positive.

42
00:02:30;07 --> 00:02:34;11
The question is are we expecting a
bigger or smaller positive than before?

43
00:02:34;12 --> 00:02:35;04
[Bigger.]

44
00:02:35;15 --> 00:02:36;09
[Smaller.]

45
00:02:37;05 --> 00:02:43;14
I heard bigger and smaller...why do you think smaller?
Katie said smaller kind of quick, why smaller?

46
00:02:43;15 --> 00:02:47;03
[.99 is less than .9]

47
00:02:48;05 --> 00:02:51;12
She said because .99 is less than .9.

48
00:02:51;13 --> 00:02:52;14
[Yeah]

49
00:02:55;05 --> 00:02:56;05
[The same exact number]

50
00:02:56;06 --> 00:03:00;13
They're the same exact number? 99
cents is the same number as 90 cents?

51
00:03:00;14 --> 00:03:03;14
[No, because it's closer]

52
00:03:06;08 --> 00:03:13;13
This is actually a bigger number than
that one, but it's getting closer to zero.

53
00:03:13;14 --> 00:03:21;06
Er, no, it's getting closer to 1, so what
does that mean for this factor?

54
00:03:22;00 --> 00:03:23;02
[It's getting smaller]

55
00:03:23;03 --> 00:03:26;13
This is what's getting smaller, yeah. Let's just see.

56
00:03:26;14 --> 00:03:40;09
You can wrestle with numbers in your head, I do
think it will be smaller. So .06, is it smaller?

57
00:03:40;10 --> 00:03:41;04
[Yeah]

58
00:03:41;05 --> 00:03:49;08
Yes. Smaller. I could keep doing this,
but I don't want to plug it in every time.

59
00:03:49;09 --> 00:03:53;12
I'm going to change all these values that I want to plug in to just A's,

60
00:03:53;13 --> 00:04:00;01
That way I can tell it what I want A to be.
If I want A to be .9, I'll just say A equals .9.

61
00:04:00;02 --> 00:04:05;09
If I want A to be .99, I'll just say A equals
.99, so this will save me time.

62
00;04:05;10 --> 00:04:11;12
We just looked at .99 and that was the
number we got. OK, what about .99?

63
00:04:11;13 --> 00:04:15;10
I don't want to hit it yet, because it's going to calculate
it right away, so what do you think about .99?

64
00:04:15;11 --> 00:04:23;12
999, sorry, I'm adding another one. There we go.
Will it still be an overall positive value?

65
00:04:23;13 --> 00:04:24;10
[Yeah]

66
00:04:24;11 --> 00:04:26;10
But smaller or bigger than the one over here?

67
00:04:26;11 --> 00:04:27;08
[Smaller]

68
00:04:27;09 --> 00:04:28;12
Smaller, why?

69
00:04:28;13 --> 00:04:31;00
[It's getting closer to 1.]

70
00:04:31;01 --> 00:04:36;02
Because it's getting closer to 1, which
means this factor is getting closer to zero.

71
00:04:36;03 --> 00:04:39;02
So, let's find out. Did it do what you thought it would?

72
00:04:39;03 --> 00:04:39;12
[Yes]

73
00:04:39;13 --> 00:04:42;02
Yeah, it took it down by a decimal point, didn't it?

74
00:04:42;03 --> 00:04:49;06
Because this is a change from tenths to
hundredths, and hundredths to thousandths.

75
00:04:49;07 --> 00:04:56;13
It's counteracting it by a factor of 10. As this
number approaches 1, we're seeing it get smaller.

76
00:04:56;14 --> 00:05:03;14
What about just 1 exactly? What about when it's 1 exactly?

77
00:05:03;15 --> 00:05:09;09
I can put N equals 1, but I don't want to yet.
What about when it's 1 exactly? Samantha?

78
00:05:09;10 --> 00:05:10;12
[It would be zero]

79
00:05:10;13 --> 00:05:12;10
Why will it be zero?

80
00:05:12;11 --> 00:05:19;07
[Because when you plug it into the first one, it will be 1
minus 1, which is zero, then times the rest, it will be zero.]

81
00:05:20;02 --> 00:05:24;14
Let's just make sure everyone's on the same page with Samantha. If you plug in a 1, what's 1 minus 1?

82
00:05:24;15 --> 00:05:26;01
[Zero]

83
00:05:26;02 --> 00:05:28;14
Zero. So, will it be zero here?

84
00:05:28;15 --> 00:05:29;13
[No]

85
00:05:30;13 --> 00:05:38;00
Here? No. Here? No. So you've got a number.
This will still give you a number if you plug in 1.

86
00:05:38;01 --> 00:05:44;01
But this will give you a zero, so whatever this is
if you times it by zero,  what are you going to get?

87
00:05:44;02 --> 00:05:44;11
[Zero]

88
00:05:44;12 --> 00:05:56;06
Zero. So, at 1, it is exactly zero. At 1 it is exactly zero. What about just past the 1?

89
00:05:56;07 --> 00:06:06;12
We'll look at about 1.1. Just past 1. What
will this factor be- positive or negative?

90
00:06:06;13 --> 00:06:07;11
[Positive]

91
00:06:07;12 --> 00:06:10;11
1.1 minus 1 will give me?

92
00:06:10;12 --> 00:06:11;06
[.1]

93
00:06:11;07 --> 00:06:14;14
A positive, yeah. What about on that factor?

94
00:06:14;15 --> 00:06:15;14
[Negative]

95
00:06:15;15 --> 00:06:17;04
A negative still?

96
00:06:17;05 --> 00:06:18;00
[Yeah]

97
00:06:18;02 --> 00:06:19;11
That factor?

86
00:06:19;12 --> 00:06:20;07
Negative

87
00:06:20;08 --> 00:06:22;00
That factor?

98
00:06:22;01 --> 00:06:22;08
[Negative]

99
00:06:22;12 --> 00:06:27;03
Help me group my things. Here's two negatives, so

100
00:06:27;04 --> 00:06:28;14
[Positive]

101
00:06:28;15 --> 00:06:37;01
A positive. This positive and this positive can make
a positive. What's this last negative going to do?

102
00:06:37;02 --> 00:06:38;02
[Negative]

103
00:06:38;03 --> 00:06:41;06
Overall, it's negative