1
00:00:00,000 --> 00:00:00,000

2
00:00:00,000 --> 00:00:07,420
PROFESSOR: Hi.

3
00:00:07,420 --> 00:00:08,970
Welcome back to recitation.

4
00:00:08,970 --> 00:00:12,680
I have here a little bit of
a strange problem for you.

5
00:00:12,680 --> 00:00:16,040
So let me just tell it to you,
and then I'll give you some

6
00:00:16,040 --> 00:00:16,640
time to work on it.

7
00:00:16,640 --> 00:00:19,540
So I want to define a function,
g of x, and I want

8
00:00:19,540 --> 00:00:21,390
to define it piecewise.

9
00:00:21,390 --> 00:00:26,020
So when x is positive, I just
want g of x to be 1 over x.

10
00:00:26,020 --> 00:00:31,320
But when x is negative, I want
g of x to be 1 of x plus 2.

11
00:00:31,320 --> 00:00:33,510
So I've got a little graph
here of the function.

12
00:00:33,510 --> 00:00:36,120
So you've got, you know, when x
is positive, it's just your

13
00:00:36,120 --> 00:00:37,800
usual, y equals 1 over x.

14
00:00:37,800 --> 00:00:40,230
But when x is negative,
I've taken, I've

15
00:00:40,230 --> 00:00:41,170
shifted it up by 2.

16
00:00:41,170 --> 00:00:42,510
So this is a perfectly
good function.

17
00:00:42,510 --> 00:00:43,627
It's not defined as 0.

18
00:00:43,627 --> 00:00:45,130
OK?

19
00:00:45,130 --> 00:00:48,090
So what I would like you to do
is to compute the derivative

20
00:00:48,090 --> 00:00:50,550
of this function, wherever
it's defined.

21
00:00:50,550 --> 00:00:54,331
And you'll notice when you get
there that you'll have some,

22
00:00:54,331 --> 00:00:58,440
you'll get some answer, and
maybe you'll notice something

23
00:00:58,440 --> 00:01:00,730
a little weird about
that answer.

24
00:01:00,730 --> 00:01:03,130
So if you notice something weird
about it, what I want

25
00:01:03,130 --> 00:01:05,710
you to do is try and explain
why this is true.

26
00:01:05,710 --> 00:01:07,340
And if you don't notice
something weird, then, you

27
00:01:07,340 --> 00:01:11,000
know, come back and we'll
talk about it together.

28
00:01:11,000 --> 00:01:14,350
So why don't you pause the
video, go do that computation,

29
00:01:14,350 --> 00:01:16,950
and think about what, if there's
something strange

30
00:01:16,950 --> 00:01:18,370
going on here.

31
00:01:18,370 --> 00:01:20,453
And then come back and we can
talk about it together.

32
00:01:20,453 --> 00:01:28,510

33
00:01:28,510 --> 00:01:29,180
Welcome back.

34
00:01:29,180 --> 00:01:31,460
Hopefully you had some fun
working on this problem and

35
00:01:31,460 --> 00:01:32,200
thinking about it.

36
00:01:32,200 --> 00:01:34,480
So let's do the first part,
which is just the

37
00:01:34,480 --> 00:01:36,020
computational part.

38
00:01:36,020 --> 00:01:39,120
Let's have a go at it.

39
00:01:39,120 --> 00:01:41,170
So, because this function is
defined piecewise, when we

40
00:01:41,170 --> 00:01:43,950
compute a derivative, we can
just compute the derivative on

41
00:01:43,950 --> 00:01:44,630
the different pieces.

42
00:01:44,630 --> 00:01:47,740
So the function isn't defined at
0, so of course, it doesn't

43
00:01:47,740 --> 00:01:48,628
have a derivative at 0.

44
00:01:48,628 --> 00:01:51,600
And so, but then we can compute
a derivative when x is

45
00:01:51,600 --> 00:01:54,580
positive, and we can compute a
derivative when x is negative.

46
00:01:54,580 --> 00:01:57,320

47
00:01:57,320 --> 00:02:05,630
So when x is bigger than 0, g
prime of x, well, that's just

48
00:02:05,630 --> 00:02:10,020
d over dx of 1 over x.

49
00:02:10,020 --> 00:02:11,730
So that's something we're
familiar with.

50
00:02:11,730 --> 00:02:14,355
Its minus 1 over x squared.

51
00:02:14,355 --> 00:02:15,875
So that's for x positive.

52
00:02:15,875 --> 00:02:19,870

53
00:02:19,870 --> 00:02:29,140
When x is less than 0, g prime
of x is d over dx of 1 over x

54
00:02:29,140 --> 00:02:32,760
plus 2, because that's
what g of x is.

55
00:02:32,760 --> 00:02:35,690
And, OK, and so this is, well,
the plus 2 gets killed, and so

56
00:02:35,690 --> 00:02:37,910
then we have the derivative
of 1 over x.

57
00:02:37,910 --> 00:02:41,350
That's minus 1 over x squared.

58
00:02:41,350 --> 00:02:44,100
So one thing you've noticed is
that this is minus 1 over x

59
00:02:44,100 --> 00:02:46,120
squared here, and it's minus
1 over x squared here.

60
00:02:46,120 --> 00:02:48,960
So although we defined this
piecewise, we could, we can

61
00:02:48,960 --> 00:02:55,950
summarize this by saying so the
derivative is minus 1 over

62
00:02:55,950 --> 00:03:02,850
x squared always, so for all
x0 equals-- you know, it

63
00:03:02,850 --> 00:03:04,430
doesn't have a derivative
at x equals 0.

64
00:03:04,430 --> 00:03:07,360
It's not defined at 0, it can't
have a derivative there.

65
00:03:07,360 --> 00:03:11,720
So, but we don't need the
piecewise definition, anymore.

66
00:03:11,720 --> 00:03:14,780
So that was kind of interesting
that we can

67
00:03:14,780 --> 00:03:18,320
summarize the derivative of this
piecewise function in a

68
00:03:18,320 --> 00:03:19,685
non-piecewise way.

69
00:03:19,685 --> 00:03:24,610

70
00:03:24,610 --> 00:03:27,980
Now, the thing is, we've
learned what the

71
00:03:27,980 --> 00:03:30,240
anti-derivative of
this function is.

72
00:03:30,240 --> 00:03:40,820
So we know that the
anti-derivative of minus 1

73
00:03:40,820 --> 00:03:48,290
over x squared dx is 1 over
x plus a constant.

74
00:03:48,290 --> 00:03:51,710
So we know that the functions
whose derivative is minus 1

75
00:03:51,710 --> 00:03:53,670
over x squared are
of the form, 1

76
00:03:53,670 --> 00:03:55,740
over x plus a constant.

77
00:03:55,740 --> 00:03:59,480
The thing is, this function g
that we just talked about,

78
00:03:59,480 --> 00:04:01,125
this function g isn't
of that form.

79
00:04:01,125 --> 00:04:02,020
Right?

80
00:04:02,020 --> 00:04:06,080
You don't get this function by
taking the function 1 over x

81
00:04:06,080 --> 00:04:07,880
and just shifting
it up or down.

82
00:04:07,880 --> 00:04:09,825
You, something weird happens.

83
00:04:09,825 --> 00:04:12,110
You've shifted it up on
one piece and not

84
00:04:12,110 --> 00:04:12,970
on the other piece.

85
00:04:12,970 --> 00:04:17,220
And yet, it's still true that
the derivative of g is equal

86
00:04:17,220 --> 00:04:19,170
to minus 1 over x squared.

87
00:04:19,170 --> 00:04:21,749
So this is a little bit
of a head scratcher.

88
00:04:21,749 --> 00:04:25,450
And I wanted to talk about
why this happens.

89
00:04:25,450 --> 00:04:27,680
And the thing is that there's
a sort of theoretical reason

90
00:04:27,680 --> 00:04:30,390
for this, which is that you
remember that the reason that

91
00:04:30,390 --> 00:04:34,750
we know that anti-derivatives
have this form, a function

92
00:04:34,750 --> 00:04:38,500
plus a constant, is because we
know that constants are the

93
00:04:38,500 --> 00:04:40,640
functions with derivative 0.

94
00:04:40,640 --> 00:04:44,900
And so we were able to apply
the mean value theorem in

95
00:04:44,900 --> 00:04:48,690
order to show that if two
functions have the same

96
00:04:48,690 --> 00:04:52,030
derivative, then they differ
by each other, differ from

97
00:04:52,030 --> 00:04:53,500
each other by a constant.

98
00:04:53,500 --> 00:04:55,560
If two functions have
the same derivative,

99
00:04:55,560 --> 00:04:57,870
they differ by a constant.

100
00:04:57,870 --> 00:05:01,870
And we used, as a really crucial
step in that proof,

101
00:05:01,870 --> 00:05:03,830
the mean value theorem.

102
00:05:03,830 --> 00:05:06,170
Now the thing is, the main value
theorem has, as one of

103
00:05:06,170 --> 00:05:09,880
its assumptions, as one of its
hypotheses, that the functions

104
00:05:09,880 --> 00:05:12,840
that you're working with are
continuous and differentiable

105
00:05:12,840 --> 00:05:13,377
in some interval.

106
00:05:13,377 --> 00:05:15,620
OK?

107
00:05:15,620 --> 00:05:18,830
So what's happened here is that
the functions that we're

108
00:05:18,830 --> 00:05:23,090
talking about, the function 1
over x and the function minus

109
00:05:23,090 --> 00:05:26,250
1 over x squared, those
functions are continuous and

110
00:05:26,250 --> 00:05:28,160
differentiable on certain
intervals.

111
00:05:28,160 --> 00:05:31,280
So if we look, if we go back
to this picture here we see

112
00:05:31,280 --> 00:05:35,620
that this function g of x, just
like the function 1 over

113
00:05:35,620 --> 00:05:39,570
x, it's continuous and
differentiable for positive x,

114
00:05:39,570 --> 00:05:42,040
it's continuous and
differentiable for negative x,

115
00:05:42,040 --> 00:05:44,960
but at 0, there's
a discontinuity.

116
00:05:44,960 --> 00:05:48,360
So there's no interval that
crosses 0 on which this

117
00:05:48,360 --> 00:05:51,260
function is continuous
or differentiable.

118
00:05:51,260 --> 00:05:55,070
As a result, the mean value
theorem can't tell us anything

119
00:05:55,070 --> 00:05:58,080
about intervals that cross 0.

120
00:05:58,080 --> 00:06:00,850
So if the mean value theorem
doesn't tell us anything in

121
00:06:00,850 --> 00:06:03,620
that case, it means the
conclusion isn't true and we

122
00:06:03,620 --> 00:06:05,200
get a situation--

123
00:06:05,200 --> 00:06:05,710
sorry.

124
00:06:05,710 --> 00:06:07,070
I should rephrase that.

125
00:06:07,070 --> 00:06:10,120
It means the conclusion doesn't
have to be true.

126
00:06:10,120 --> 00:06:13,210
Our proof doesn't work in
a case where we have a

127
00:06:13,210 --> 00:06:15,150
discontinuity.

128
00:06:15,150 --> 00:06:18,360
And what happens, in fact, is
right what we have here, which

129
00:06:18,360 --> 00:06:21,210
is that when you have a
function that has a

130
00:06:21,210 --> 00:06:25,090
discontinuity and you look at
its anti-derivatives, what you

131
00:06:25,090 --> 00:06:28,410
can do is that, in addition to
shifting the whole thing up

132
00:06:28,410 --> 00:06:31,870
and down, you can shift the
pieces on either side of the

133
00:06:31,870 --> 00:06:33,660
discontinuity separately.

134
00:06:33,660 --> 00:06:37,460
Just like in this case we can
shift the piece to the left of

135
00:06:37,460 --> 00:06:39,960
0 separately from the piece to
the right of 0 and get a

136
00:06:39,960 --> 00:06:42,690
function whose derivative is
still what we started with.

137
00:06:42,690 --> 00:06:47,130
So this function, g of x we, get
by shifting part of 1 over

138
00:06:47,130 --> 00:06:49,730
x up, and it gives us a function
whose derivative is

139
00:06:49,730 --> 00:06:53,020
still minus 1 over x squared.

140
00:06:53,020 --> 00:06:55,650
So this is true anytime you
have a function with a

141
00:06:55,650 --> 00:06:55,735
discontinuity.

142
00:06:55,735 --> 00:06:58,590
So one consequence of this--

143
00:06:58,590 --> 00:07:00,910
I'm going to go back over here
and just write down one

144
00:07:00,910 --> 00:07:02,790
special case of this--

145
00:07:02,790 --> 00:07:11,540
is that to say, we say that the
anti-derivative of 1 over

146
00:07:11,540 --> 00:07:19,010
x dx is equal to ln of the
absolute value of x plus c.

147
00:07:19,010 --> 00:07:24,840
What this really means is that
when x is positive, we have a

148
00:07:24,840 --> 00:07:27,700
single kind of anti-derivative,
and they're

149
00:07:27,700 --> 00:07:30,470
of the form, ln x
plus a constant.

150
00:07:30,470 --> 00:07:33,115
And when x is negative, we have
a single anti-derivative,

151
00:07:33,115 --> 00:07:37,370
that's, or single family of
anti-derivatives, of the form

152
00:07:37,370 --> 00:07:40,630
ln of minus x-- remember,
absolutely value of x is minus

153
00:07:40,630 --> 00:07:41,770
x when x is negative--

154
00:07:41,770 --> 00:07:42,740
plus c.

155
00:07:42,740 --> 00:07:45,040
But if we consider x to be
positive and negative at the

156
00:07:45,040 --> 00:07:47,330
same time, the two
constants don't

157
00:07:47,330 --> 00:07:48,670
necessarily have to agree.

158
00:07:48,670 --> 00:07:50,770
You can have the same situation
that you had before

159
00:07:50,770 --> 00:07:53,960
where one side can shift up and
down independently of the

160
00:07:53,960 --> 00:07:57,490
other because there's that
discontinuity at 0 there.

161
00:07:57,490 --> 00:07:59,710
So this is just something
to keep in mind.

162
00:07:59,710 --> 00:08:01,980
It also means you have to
be careful with certain

163
00:08:01,980 --> 00:08:02,820
substitutions.

164
00:08:02,820 --> 00:08:04,970
You don't want to do
substitutions that have

165
00:08:04,970 --> 00:08:06,290
discontinuities.

166
00:08:06,290 --> 00:08:08,880
If you do substitutions that
have discontinuities, you

167
00:08:08,880 --> 00:08:13,030
might accidentally introduce a
discontinuity and bad things

168
00:08:13,030 --> 00:08:16,020
can happen that I won't
go into now.

169
00:08:16,020 --> 00:08:19,100
You can make, end up with
statements that don't make any

170
00:08:19,100 --> 00:08:23,020
sense by making a substitution
where the function that you're

171
00:08:23,020 --> 00:08:25,920
substituting has a discontinuity
in it.

172
00:08:25,920 --> 00:08:27,720
So you, or another way of
saying it is you have to

173
00:08:27,720 --> 00:08:33,630
restrict to some interval on
which it really is continuous.

174
00:08:33,630 --> 00:08:36,370
And then on each of those
intervals it makes sense, but

175
00:08:36,370 --> 00:08:38,170
bad things could happen
when you cross those

176
00:08:38,170 --> 00:08:39,850
discontinuities.

177
00:08:39,850 --> 00:08:42,130
So this is a little bit
theoretical, but I think it's

178
00:08:42,130 --> 00:08:45,370
a nice thing to be aware of, a
nice thing to keep in mind

179
00:08:45,370 --> 00:08:49,460
when you're working with some
of these expressions.

180
00:08:49,460 --> 00:08:51,270
So I'll end there.

181
00:08:51,270 --> 00:08:51,464