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

2
00:00:00,000 --> 00:00:05,820
PROFESSOR: Hi.

3
00:00:05,820 --> 00:00:07,400
Welcome back to recitation.

4
00:00:07,400 --> 00:00:09,560
In lecture you've been learning
about applications of

5
00:00:09,560 --> 00:00:10,170
integration.

6
00:00:10,170 --> 00:00:11,860
And one of them is
how to compute

7
00:00:11,860 --> 00:00:13,420
average values of a function.

8
00:00:13,420 --> 00:00:15,800
So I have a nice average value
problem for you here.

9
00:00:15,800 --> 00:00:20,410
So Christine was out jogging
one day and she saw a bear,

10
00:00:20,410 --> 00:00:22,360
and so she broke
into a sprint.

11
00:00:22,360 --> 00:00:25,260
She starts speeding up, and her
velocity t seconds after

12
00:00:25,260 --> 00:00:30,600
she started sprinting is given
by v of t equals 1500 divided

13
00:00:30,600 --> 00:00:33,790
by the quantity 100 plus t minus
5 squared, the whole

14
00:00:33,790 --> 00:00:34,960
thing minus 7.

15
00:00:34,960 --> 00:00:38,280
And it's in meters per second.

16
00:00:38,280 --> 00:00:42,150
So OK, so she starts at t equals
0, she's going 1500

17
00:00:42,150 --> 00:00:45,740
divided by 125, so that's
12 minus 7, so

18
00:00:45,740 --> 00:00:47,250
5 meters per second.

19
00:00:47,250 --> 00:00:51,280
And she starts to speed up and
after 5 seconds she's going

20
00:00:51,280 --> 00:00:52,490
1500 divided--

21
00:00:52,490 --> 00:00:54,160
well this is 0--

22
00:00:54,160 --> 00:00:57,460
so she's going 8 meters per
second at 5 seconds.

23
00:00:57,460 --> 00:00:59,460
And then she looks over her
shoulder and she sees, oh no,

24
00:00:59,460 --> 00:01:01,810
it wasn't up a bear, it was just
an unkempt math graduate

25
00:01:01,810 --> 00:01:04,590
student, so then she starts
to slow down again.

26
00:01:04,590 --> 00:01:08,280
She slows back down and 10
seconds later she's back to

27
00:01:08,280 --> 00:01:10,140
her original velocity.

28
00:01:10,140 --> 00:01:13,630
So the question is, over the 10
seconds of this sprint, she

29
00:01:13,630 --> 00:01:17,040
sped up, she'd slowed down, from
5 meters per second to 8

30
00:01:17,040 --> 00:01:18,680
meters per second and
then back to 5.

31
00:01:18,680 --> 00:01:21,380
What was her average velocity
over this time?

32
00:01:21,380 --> 00:01:23,880
So why don't you pause the
video, take some time to work

33
00:01:23,880 --> 00:01:26,100
that out, come back and we
can work on it together.

34
00:01:26,100 --> 00:01:34,090

35
00:01:34,090 --> 00:01:36,460
So hopefully you had some luck
working on this problem.

36
00:01:36,460 --> 00:01:40,400
I have over on my left here a
little picture of the graph of

37
00:01:40,400 --> 00:01:42,000
this function v of t.

38
00:01:42,000 --> 00:01:45,270
I described it to you earlier,
but just you see, OK, so it

39
00:01:45,270 --> 00:01:49,330
starts at 5, after 5 seconds
it's at 8, and then it goes

40
00:01:49,330 --> 00:01:52,390
back down to 5 at 10 seconds.

41
00:01:52,390 --> 00:01:55,560
So this is roughly the picture,
so we have a feel for

42
00:01:55,560 --> 00:01:56,570
what we're working with.

43
00:01:56,570 --> 00:01:59,210
Now what we want to compute
is the average

44
00:01:59,210 --> 00:02:00,890
velocity over this time.

45
00:02:00,890 --> 00:02:03,240
So anytime you have to compute
an average value of any

46
00:02:03,240 --> 00:02:06,400
function what you want to do,
you always do the same thing.

47
00:02:06,400 --> 00:02:09,990
So you want to compute the total
contribution of that

48
00:02:09,990 --> 00:02:11,670
function over the interval
in question.

49
00:02:11,670 --> 00:02:13,900
So you want to add it up, you
want to integrate it over the

50
00:02:13,900 --> 00:02:14,930
interval in question.

51
00:02:14,930 --> 00:02:16,940
But then you want to divide by
the length of the integral.

52
00:02:16,940 --> 00:02:20,160
So in particular, when you have
a velocity function what

53
00:02:20,160 --> 00:02:21,950
that means is, you integrate
the velocity.

54
00:02:21,950 --> 00:02:24,400
So that gives you the total
distance traveled, and then

55
00:02:24,400 --> 00:02:26,350
you divide by the
time interval.

56
00:02:26,350 --> 00:02:28,200
S you're just taking
total distance

57
00:02:28,200 --> 00:02:29,950
divided by total time.

58
00:02:29,950 --> 00:02:33,130
So in this case we can write
down this average value.

59
00:02:33,130 --> 00:02:35,380
So I'm going to use just
a made up notation.

60
00:02:35,380 --> 00:02:40,810
I'm going to write avg
of v, so what is it?

61
00:02:40,810 --> 00:02:44,590
Well the first thing I have to
do is I want to not forget to

62
00:02:44,590 --> 00:02:47,420
divide by the length of
the interval here.

63
00:02:47,420 --> 00:02:50,870
So in this case, the interval
is from 0 to 10, so it has

64
00:02:50,870 --> 00:02:53,510
length 10 minus 0,
which is 10.

65
00:02:53,510 --> 00:02:57,810
So I want 1/10 in front and what
I want to multiply this

66
00:02:57,810 --> 00:03:01,650
by, is I want to multiply by the
integral over this entire

67
00:03:01,650 --> 00:03:03,610
interval of my function v.

68
00:03:03,610 --> 00:03:05,980
Sp I just take the formula that
I had over there for v

69
00:03:05,980 --> 00:03:07,950
and I plop it down
into my integral.

70
00:03:07,950 --> 00:03:18,490
So it's the integral from 0 to
10 of 1500 divided by 100 plus

71
00:03:18,490 --> 00:03:25,210
t minus 5 quantity squared
minus 7 dt.

72
00:03:25,210 --> 00:03:27,200
So this integral is the average
value that we're

73
00:03:27,200 --> 00:03:29,050
looking for.

74
00:03:29,050 --> 00:03:29,130
OK.

75
00:03:29,130 --> 00:03:31,030
So now we have to go about
evaluating it.

76
00:03:31,030 --> 00:03:33,080
Now, this looks kind of
ugly, but actually

77
00:03:33,080 --> 00:03:34,210
it's not that bad.

78
00:03:34,210 --> 00:03:36,880
I mean, one thing you can
notice, so the minus 7 part,

79
00:03:36,880 --> 00:03:38,600
that's going to be easy
to take care of.

80
00:03:38,600 --> 00:03:41,310
And then in this more
complicated part, well there's

81
00:03:41,310 --> 00:03:43,100
a kind of obvious substitution
here.

82
00:03:43,100 --> 00:03:46,920
You can set u equal
t minus 5, that'll

83
00:03:46,920 --> 00:03:48,450
simplify it a little bit.

84
00:03:48,450 --> 00:03:51,100
and then what you'll see is that
you have an integrand of

85
00:03:51,100 --> 00:03:56,160
the form something divided
by 100 plus u squared.

86
00:03:56,160 --> 00:03:59,160
And so that's 10 squared
plus u squared.

87
00:03:59,160 --> 00:04:01,450
So that should keep-- and the
something as a constant.

88
00:04:01,450 --> 00:04:04,380
So that should put you in the
mind of a tangent substitute,

89
00:04:04,380 --> 00:04:06,790
or just even not by
substitution, you can just

90
00:04:06,790 --> 00:04:09,780
remember that that's an arc
tangent type of thing that

91
00:04:09,780 --> 00:04:10,730
you're going to get out.

92
00:04:10,730 --> 00:04:14,450
so I'm not going to go through
all the steps of doing that,

93
00:04:14,450 --> 00:04:19,060
but we should get at the end--
well OK, so I've got this 1/10

94
00:04:19,060 --> 00:04:21,780
out in front, so I have to
not forget about that.

95
00:04:21,780 --> 00:04:26,060
I get 1/10 times, I've gone
ahead and I've computed the

96
00:04:26,060 --> 00:04:34,270
anti-derivative and it's 150
times arctan of the quantity t

97
00:04:34,270 --> 00:04:37,880
minus 5 over 10.

98
00:04:37,880 --> 00:04:42,160
So here's that t minus 5, that's
the same t minus 5 over

99
00:04:42,160 --> 00:04:44,010
there that's coming out.

100
00:04:44,010 --> 00:04:46,670
Minus 7t.

101
00:04:46,670 --> 00:04:51,210
And you want to take that
between t equals 0

102
00:04:51,210 --> 00:04:54,440
and t equals 10.

103
00:04:54,440 --> 00:04:55,710
So this is, I've just, I've gone
ahead, I've computed the

104
00:04:55,710 --> 00:04:57,780
anti-derivative for us.

105
00:04:57,780 --> 00:05:00,770
There's my 1/10 because I'm
doing an average value and

106
00:05:00,770 --> 00:05:03,650
then I'm, it's a definite
integral and so I'm using

107
00:05:03,650 --> 00:05:06,210
fundamental theorem of calculus
there on my balance.

108
00:05:06,210 --> 00:05:07,535
OK, so now we just have to
plug in and evaluate.

109
00:05:07,535 --> 00:05:07,575
So this is equal to,
so let's see.

110
00:05:07,575 --> 00:05:14,650
So we've got when we put in t
equals 10, well the 1/10 and

111
00:05:14,650 --> 00:05:25,120
the 150 gives me a 15 arctan of
1/2 minus, well it's 1/10

112
00:05:25,120 --> 00:05:30,140
of 7 times 10, it's just 7.

113
00:05:30,140 --> 00:05:31,120
That's from the first one.

114
00:05:31,120 --> 00:05:33,380
And from the second one
I guess minus--

115
00:05:33,380 --> 00:05:41,180
so it's going to be 15
arctan of minus 1/2.

116
00:05:41,180 --> 00:05:43,940
When I put in t equals 0, and
when I put in t equals 0, the

117
00:05:43,940 --> 00:05:45,380
7t is just 0.

118
00:05:45,380 --> 00:05:46,240
OK.

119
00:05:46,240 --> 00:05:47,110
This is a little messy.

120
00:05:47,110 --> 00:05:49,760
We can simplify it a little
bit, because, remember

121
00:05:49,760 --> 00:05:51,940
arctangent is an odd function.

122
00:05:51,940 --> 00:05:56,230
So arctan of minus x is
minus arctangent of x.

123
00:05:56,230 --> 00:06:00,050
So minus 15 arctan minus
1/2 is the same as

124
00:06:00,050 --> 00:06:01,940
just 15 arctan 1/2.

125
00:06:01,940 --> 00:06:03,610
And so we can combine those.

126
00:06:03,610 --> 00:06:04,430
So we can rewrite that.

127
00:06:04,430 --> 00:06:05,860
Let's go all the
way over here.

128
00:06:05,860 --> 00:06:14,760
We can rewrite this as 30
arctan 1/2 minus 7.

129
00:06:14,760 --> 00:06:16,830
So that's about it as
nice a form as you

130
00:06:16,830 --> 00:06:17,740
can make this take.

131
00:06:17,740 --> 00:06:20,380
If you wanted to plug it into
a calculator, I think you'd

132
00:06:20,380 --> 00:06:25,480
see that this is approximately
equal to 6.9 and I guess the

133
00:06:25,480 --> 00:06:28,720
units there are going to
be meters per second.

134
00:06:28,720 --> 00:06:32,340
So her average velocity over
this time is about 6.9 meters

135
00:06:32,340 --> 00:06:33,310
per second.

136
00:06:33,310 --> 00:06:36,360
Let's just quickly recap
what we did.

137
00:06:36,360 --> 00:06:39,240
So we started all the way back
here with this velocity

138
00:06:39,240 --> 00:06:42,720
function v, and we wanted to
compute it's average value

139
00:06:42,720 --> 00:06:45,266
over the intervals 0 less than
or equal to t, less than or

140
00:06:45,266 --> 00:06:46,890
equal to 10.

141
00:06:46,890 --> 00:06:49,840
So we did what we always do
in problems of that sort.

142
00:06:49,840 --> 00:06:52,385
So we came over here, and
whenever you want to compute

143
00:06:52,385 --> 00:06:56,080
an average value, you take 1
divided by the length of the

144
00:06:56,080 --> 00:06:58,060
interval in question, times--

145
00:06:58,060 --> 00:07:00,460
then just you integrate the
function who's average value

146
00:07:00,460 --> 00:07:03,000
you want over the interval.

147
00:07:03,000 --> 00:07:03,750
OK?

148
00:07:03,750 --> 00:07:07,740
And so then we computed, well
we didn't show any of the

149
00:07:07,740 --> 00:07:11,290
steps, but we computed the
definite integral by taking

150
00:07:11,290 --> 00:07:13,910
the anti-derivative and plugging
in and it happened

151
00:07:13,910 --> 00:07:16,900
that the answer worked
out to be about 6.9.

152
00:07:16,900 --> 00:07:17,600
So there you go.

153
00:07:17,600 --> 00:07:19,610
I'll stop there.

154
00:07:19,610 --> 00:07:19,639