1
00:00:06,867 --> 00:00:07,450
PROFESSOR: Hi.

2
00:00:07,450 --> 00:00:09,050
Welcome back to recitation.

3
00:00:09,050 --> 00:00:11,710
We've been talking in lecture
about antiderivatives.

4
00:00:11,710 --> 00:00:13,850
So I have here a
problem for you.

5
00:00:13,850 --> 00:00:16,310
Just an exercise about
computing an antiderivative.

6
00:00:16,310 --> 00:00:18,710
So the question is to
compute an antiderivative

7
00:00:18,710 --> 00:00:20,510
of this big fraction.

8
00:00:20,510 --> 00:00:24,180
So on top it's got x to
the eighth plus 2x cubed

9
00:00:24,180 --> 00:00:30,330
minus x to the 2/3 minus 3,
that whole thing over x squared.

10
00:00:30,330 --> 00:00:33,760
So just a quick
linguistic note about why

11
00:00:33,760 --> 00:00:37,430
I said an antiderivative
instead of the antiderivative,

12
00:00:37,430 --> 00:00:39,370
and then I'll let you
work on it a little.

13
00:00:39,370 --> 00:00:40,660
So an antiderivative.

14
00:00:40,660 --> 00:00:43,772
There are many functions whose
derivative is this function.

15
00:00:43,772 --> 00:00:44,630
Right?

16
00:00:44,630 --> 00:00:49,450
So they all differ from
each other by constants.

17
00:00:49,450 --> 00:00:53,230
So I would be happy with any one
as an answer to this question.

18
00:00:53,230 --> 00:00:57,350
That's why I chose the
word an antiderivative.

19
00:00:57,350 --> 00:00:59,810
So I'm looking for a
function whose derivative

20
00:00:59,810 --> 00:01:01,600
is equal to this function.

21
00:01:01,600 --> 00:01:04,620
So why don't you take a
couple minutes, work this out,

22
00:01:04,620 --> 00:01:07,350
come back and you can check
your answer against my work.

23
00:01:16,880 --> 00:01:17,680
OK, welcome back.

24
00:01:17,680 --> 00:01:21,290
So we were just talking about
this antiderivative here.

25
00:01:21,290 --> 00:01:24,024
So one thing you'll
notice about this function

26
00:01:24,024 --> 00:01:26,190
is that I've written this
in a sort of a silly form.

27
00:01:26,190 --> 00:01:29,120
And it's probably
a lot easier to get

28
00:01:29,120 --> 00:01:31,060
a feel for what this
function is if you

29
00:01:31,060 --> 00:01:33,790
break this fraction apart
into its several pieces.

30
00:01:33,790 --> 00:01:37,060
So for example, x to the
eighth over x squared

31
00:01:37,060 --> 00:01:38,960
is just x-- So,
well OK, so let me,

32
00:01:38,960 --> 00:01:41,130
this antiderivative
that I'm interested in,

33
00:01:41,130 --> 00:01:46,340
antiderivative of x to the
eighth plus 2x cubed minus x

34
00:01:46,340 --> 00:01:52,620
to the 2/3 minus 3,
over x squared dx.

35
00:01:52,620 --> 00:01:54,440
So I've written
this in a silly form

36
00:01:54,440 --> 00:01:56,210
and you can get
it in a nicer form

37
00:01:56,210 --> 00:01:58,170
if you just realize
that, you know,

38
00:01:58,170 --> 00:02:00,760
this is just a
sum of powers of x

39
00:02:00,760 --> 00:02:04,740
that I've put over this
silly common denominator.

40
00:02:04,740 --> 00:02:07,560
So our life will be a little
simpler if we write this out

41
00:02:07,560 --> 00:02:11,440
by splitting it up into
the separate fractions.

42
00:02:11,440 --> 00:02:14,460
So if I do that, this is just
equal to the antiderivative

43
00:02:14,460 --> 00:02:18,000
of well, x to the
eighth over x squared.

44
00:02:18,000 --> 00:02:19,830
That's x to the sixth.

45
00:02:19,830 --> 00:02:23,320
And 2 x cubed over
x squared is just x.

46
00:02:23,320 --> 00:02:29,160
So I have x-- sorry,
it's 2x-- plus 2x.

47
00:02:29,160 --> 00:02:34,170
Now, OK so x to the
2/3 over x squared.

48
00:02:34,170 --> 00:02:37,610
So that's x to the 2/3 minus 2.

49
00:02:37,610 --> 00:02:44,150
Which is x to the minus 4/3.

50
00:02:44,150 --> 00:02:47,860
And minus 3 over
x squared, so OK,

51
00:02:47,860 --> 00:02:51,270
so we could write that as minus
3 over x squared, or maybe it's

52
00:02:51,270 --> 00:02:55,210
a little more convenient
to write it as minus 3

53
00:02:55,210 --> 00:02:59,250
x to the minus 2, dx.

54
00:02:59,250 --> 00:03:01,580
So far I haven't really
done anything, you know.

55
00:03:01,580 --> 00:03:03,930
A little bit of algebra here.

56
00:03:03,930 --> 00:03:08,180
OK, but now we know
that we've seen

57
00:03:08,180 --> 00:03:10,860
a formula for
antidifferentiating

58
00:03:10,860 --> 00:03:11,980
a single power of x.

59
00:03:11,980 --> 00:03:13,950
I mean we know how
to differentiate

60
00:03:13,950 --> 00:03:16,640
a single power of x, and
so to do an antiderivative

61
00:03:16,640 --> 00:03:19,780
is just the inverse process.

62
00:03:19,780 --> 00:03:24,730
And we also know that when you
have the derivative of a sum,

63
00:03:24,730 --> 00:03:26,110
it's the sum of derivatives.

64
00:03:26,110 --> 00:03:30,420
And so consequently, if you have
the antiderivative of a sum,

65
00:03:30,420 --> 00:03:32,420
it's just the sum of
the antiderivatives.

66
00:03:32,420 --> 00:03:36,510
So we can write this out
into its constituent parts.

67
00:03:36,510 --> 00:03:41,110
So it's the antiderivative
of x to the sixth dx

68
00:03:41,110 --> 00:03:44,890
plus-- now of course you
don't have to do this.

69
00:03:44,890 --> 00:03:48,030
You could probably proceed
just from this step onwards,

70
00:03:48,030 --> 00:03:53,390
or, but I don't see any harm
in actually splitting it

71
00:03:53,390 --> 00:03:54,120
up myself.

72
00:03:54,120 --> 00:03:59,340
So antiderivative
of 2x*dx minus, OK,

73
00:03:59,340 --> 00:04:10,470
x to the minus 4/3 dx plus
minus 3 x to the minus 2 dx.

74
00:04:10,470 --> 00:04:12,470
So I've just split it up
into a bunch of pieces.

75
00:04:12,470 --> 00:04:15,034
I guess this one I sort of
pulled the minus sign out

76
00:04:15,034 --> 00:04:15,950
and this one I didn't.

77
00:04:15,950 --> 00:04:18,090
But you know, whatever.

78
00:04:18,090 --> 00:04:20,020
Either way.

79
00:04:20,020 --> 00:04:21,980
OK so now we just
need to remember

80
00:04:21,980 --> 00:04:26,740
our formulas for taking the
antiderivative of a power of x.

81
00:04:26,740 --> 00:04:31,140
So in order to that, so
when you take a derivative,

82
00:04:31,140 --> 00:04:34,510
the power goes down by one.

83
00:04:34,510 --> 00:04:36,760
So if you take an
antiderivative to the power

84
00:04:36,760 --> 00:04:38,660
will always go up by one.

85
00:04:38,660 --> 00:04:42,290
So in this case
you get, so you're

86
00:04:42,290 --> 00:04:45,880
going to get x to the seventh.

87
00:04:45,880 --> 00:04:49,410
And now when you differentiate
x to the seventh,

88
00:04:49,410 --> 00:04:51,170
a 7 comes down in front, right?

89
00:04:51,170 --> 00:04:52,940
You get 7 x to the sixth.

90
00:04:52,940 --> 00:04:55,360
So in order to get
just x to the sixth,

91
00:04:55,360 --> 00:04:58,320
we have to also divide
by that 7 there.

92
00:04:58,320 --> 00:05:00,250
So x to the sixth,
the antiderivative

93
00:05:00,250 --> 00:05:02,720
is x to the seventh over 7.

94
00:05:02,720 --> 00:05:09,560
2x, so that's going to give
us plus 2 x squared over 2.

95
00:05:09,560 --> 00:05:12,500
Or if you like, you could just
recognize right away that 2x

96
00:05:12,500 --> 00:05:16,100
is the derivative of x squared.

97
00:05:16,100 --> 00:05:18,870
Minus-- OK now we've
got minus powers.

98
00:05:18,870 --> 00:05:22,600
Rather, negative powers, so
that always is a little trickier

99
00:05:22,600 --> 00:05:24,094
to keep track of.

100
00:05:24,094 --> 00:05:25,760
So again, the same
thing is true though.

101
00:05:25,760 --> 00:05:28,390
You have to, you add
one to the exponent.

102
00:05:28,390 --> 00:05:30,890
The exponent goes up by one
when you take an antiderivative.

103
00:05:30,890 --> 00:05:33,440
It goes down by one when
you take a derivative.

104
00:05:33,440 --> 00:05:38,270
So when you add 1 to minus
4/3 you get minus 1/3.

105
00:05:38,270 --> 00:05:43,240
So we have x to the minus 1/3.

106
00:05:43,240 --> 00:05:48,100
And now I have to
divide by minus 1/3.

107
00:05:48,100 --> 00:05:52,140
When I take a derivative here,
we get-- of x to the minus 1/3,

108
00:05:52,140 --> 00:05:56,990
I get minus 1/3 x
to the minus 4/3.

109
00:05:56,990 --> 00:05:58,580
So I need to divide
by that minus 1/3.

110
00:05:58,580 --> 00:06:00,040
OK.

111
00:06:00,040 --> 00:06:04,620
And finally here so
minus 3 x to the minus 2.

112
00:06:04,620 --> 00:06:06,100
So OK, so just like
this first one,

113
00:06:06,100 --> 00:06:08,250
you might recognize
that right off

114
00:06:08,250 --> 00:06:09,962
as the derivative
of x to the minus 3.

115
00:06:09,962 --> 00:06:14,760
So this is plus-- Oh!

116
00:06:14,760 --> 00:06:15,990
Ha ha!

117
00:06:15,990 --> 00:06:20,180
You could do that if you were
completely confused like me.

118
00:06:20,180 --> 00:06:25,470
So right, so x to the minus
2, it increased by one.

119
00:06:25,470 --> 00:06:26,570
Increases by 1.

120
00:06:26,570 --> 00:06:28,876
So when it increases by 1,
you get minus 1 not minus 3.

121
00:06:28,876 --> 00:06:29,376
Oh!

122
00:06:29,376 --> 00:06:30,810
OK, good.

123
00:06:30,810 --> 00:06:39,730
So this is minus 3 times x
to the minus 1 over minus 1.

124
00:06:39,730 --> 00:06:40,480
OK.

125
00:06:40,480 --> 00:06:41,271
That's much better.

126
00:06:41,271 --> 00:06:43,260
And if you like,
right, so, OK, so

127
00:06:43,260 --> 00:06:45,380
we could-- any constant
we add to this,

128
00:06:45,380 --> 00:06:47,400
it'll still be an
antiderivative.

129
00:06:47,400 --> 00:06:49,320
And now we can do a
little bit of arithmetic

130
00:06:49,320 --> 00:06:52,380
to arrange this into
nicer forms if you wanted.

131
00:06:52,380 --> 00:06:55,560
So you could
rewrite this as say,

132
00:06:55,560 --> 00:07:02,770
x to the seventh
over 7 plus x squared

133
00:07:02,770 --> 00:07:14,540
plus 3 x to the minus 1/3
plus 3 x to the minus 1

134
00:07:14,540 --> 00:07:15,370
plus a constant.

135
00:07:15,370 --> 00:07:18,350
Now, suppose you
got here and suppose

136
00:07:18,350 --> 00:07:21,850
that you did the same
mistake that I just made.

137
00:07:21,850 --> 00:07:24,084
And you had accidentally
thought that this

138
00:07:24,084 --> 00:07:26,250
was going to be a minus 1/3
power instead of a minus

139
00:07:26,250 --> 00:07:27,280
first power.

140
00:07:27,280 --> 00:07:30,120
So how would you,
is there any way

141
00:07:30,120 --> 00:07:32,430
that you can prevent
yourself making that mistake?

142
00:07:32,430 --> 00:07:33,730
Well there actually is.

143
00:07:33,730 --> 00:07:36,037
So one nice thing
about antiderivatives

144
00:07:36,037 --> 00:07:37,870
is that it's really
easy to check your work.

145
00:07:37,870 --> 00:07:40,530
After you've computed an
antiderivative, or something

146
00:07:40,530 --> 00:07:42,072
that you think is
antiderivative,

147
00:07:42,072 --> 00:07:44,530
you can always go back and take
the derivative of the thing

148
00:07:44,530 --> 00:07:46,200
that you've computed
and check that it's

149
00:07:46,200 --> 00:07:47,630
equal to what you started with.

150
00:07:47,630 --> 00:07:51,240
So if you, if
you're ever worried

151
00:07:51,240 --> 00:07:53,830
that you made a mistake
computing an antiderivative,

152
00:07:53,830 --> 00:07:56,050
one thing you can always
do is take a derivative

153
00:07:56,050 --> 00:07:57,300
of what you've got at the end.

154
00:07:57,300 --> 00:07:58,716
So if we take a
derivative here we

155
00:07:58,716 --> 00:08:05,800
get x to the sixth plus 2x minus
x to the minus 4/3 minus 3 x

156
00:08:05,800 --> 00:08:06,575
to the minus 2.

157
00:08:06,575 --> 00:08:08,240
OK?

158
00:08:08,240 --> 00:08:11,855
So that was just using our
rule for powers one by one.

159
00:08:11,855 --> 00:08:14,960
And OK, so you say that
out loud or write it down

160
00:08:14,960 --> 00:08:16,060
and then you just check.

161
00:08:16,060 --> 00:08:16,560
Right?

162
00:08:16,560 --> 00:08:19,526
So I said that, so that's
exactly the same thing

163
00:08:19,526 --> 00:08:20,400
we've got right here.

164
00:08:20,400 --> 00:08:21,310
Yeah?

165
00:08:21,310 --> 00:08:25,340
So x to the sixth plus 2x
minus x to the minus 4/3

166
00:08:25,340 --> 00:08:27,490
minus 3x to the minus 2.

167
00:08:27,490 --> 00:08:31,620
So one of the nicest things
about antiderivatives,

168
00:08:31,620 --> 00:08:34,330
they can be difficult to
figure out in the first place,

169
00:08:34,330 --> 00:08:36,220
but after you've got
something that you think

170
00:08:36,220 --> 00:08:39,580
is antiderivative it's very
easy to go back and check

171
00:08:39,580 --> 00:08:42,300
whether you did it correctly
by taking the derivative

172
00:08:42,300 --> 00:08:46,250
and making sure that it
matches the thing that you

173
00:08:46,250 --> 00:08:48,750
were trying to antidifferentiate
at the beginning.

174
00:08:48,750 --> 00:08:50,097
So that's that.