1
00:00:01,133 --> 00:00:02,670
Now, let's do something pretty interesting, this

2
00:00:02,670 --> 00:00:04,933
in some degree will be one of the easiest functions to find

3
00:00:04,933 --> 00:00:07,407
the maclaurin series representation of

4
00:00:07,407 --> 00:00:11,192
let's try to approximate e^x

5
00:00:11,192 --> 00:00:13,200
f(x) is equal to e^x

6
00:00:13,200 --> 00:00:14,861
and what makes this really simple

7
00:00:14,861 --> 00:00:16,000
is when you take the derivative

8
00:00:16,000 --> 00:00:17,333
and frankly this is one of the amazing things

9
00:00:17,333 --> 00:00:18,867
about the number e

10
00:00:18,867 --> 00:00:20,067
is that when you take the derivative

11
00:00:20,067 --> 00:00:23,568
of e^x, you get e^x

12
00:00:23,568 --> 00:00:25,867
so this is equal to f ' (x)

13
00:00:25,867 --> 00:00:29,200
this is equal to f 2nd derivative of x

14
00:00:29,200 --> 00:00:31,933
this is equal to f 3rd derivative of x

15
00:00:31,933 --> 00:00:34,102
this is equal to the nth derivative of x

16
00:00:34,102 --> 00:00:36,270
it's always equal to e^x

17
00:00:36,270 --> 00:00:40,867
that's the first mind-blowing thing about e

18
00:00:40,867 --> 00:00:42,867
you can just keep taking it's derivative

19
00:00:42,867 --> 00:00:44,443
the slope at any point on that curve

20
00:00:45,021 --> 00:00:48,762
is the same as the value of the point on that curve

21
00:00:48,762 --> 00:00:50,533
that's kind of crazy

22
00:00:50,533 --> 00:00:51,432
with that said

23
00:00:51,432 --> 00:00:52,933
let's take its maclaurin representation

24
00:00:52,933 --> 00:00:54,067
so we have to find

25
00:00:54,067 --> 00:00:56,610
f(0), f ' (0), the 2nd derivative of 0

26
00:00:56,610 --> 00:00:59,188
and when we take e^0

27
00:00:59,188 --> 00:01:01,733
e^0 is just equal to one

28
00:01:01,733 --> 00:01:04,267
and so this is going to be equal to f(0)

29
00:01:04,267 --> 00:01:07,595
this is going to be equal to f ' (0) it's going to be

30
00:01:07,595 --> 00:01:10,194
equal to any of the derivatives evaluated

31
00:01:14,086 --> 00:01:17,200
the nth derivative is going to be valued at zero

32
00:01:17,200 --> 00:01:20,467
and that's what makes the Maclaurin series

33
00:01:20,467 --> 00:01:23,000
fairly straightforward

34
00:01:23,000 --> 00:01:24,200
if I want to approximate

35
00:01:24,200 --> 00:01:27,669
e^x using a maclaurin series

36
00:01:27,669 --> 00:01:28,722
so e^x

37
00:01:28,722 --> 00:01:30,467
and I'll put a little approximately over here

38
00:01:30,467 --> 00:01:31,672
and we'll get closer and closer

39
00:01:31,672 --> 00:01:34,691
to the real e^x and as we keep adding more and more terms

40
00:01:34,691 --> 00:01:37,231
especially if we add an infinite number of terms

41
00:01:37,231 --> 00:01:38,590
it would look like this:

42
00:01:38,590 --> 00:01:41,680
f(0) and we do it in

43
00:01:41,680 --> 00:01:44,467
what colors did I use for cosine and sine?

44
00:01:44,467 --> 00:01:46,754
pink and green

45
00:01:47,754 --> 00:01:49,954
I'll use the yellow here

46
00:01:51,000 --> 00:01:53,600
f(0) is 1

47
00:01:53,600 --> 00:01:55,467
plus f ' (0) times x

48
00:01:55,467 --> 00:01:57,333
f ' (0) is also one

49
00:01:57,333 --> 00:01:58,508
plus x

50
00:01:59,400 --> 00:02:01,067
plus this is also one

51
00:02:01,067 --> 00:02:03,518
so x^2 / 2!

52
00:02:06,318 --> 00:02:08,476
all of these things are going to be one

53
00:02:08,476 --> 00:02:09,533
this is one

54
00:02:09,533 --> 00:02:10,733
this is one

55
00:02:10,733 --> 00:02:11,933
when we're taking about e^x

56
00:02:11,933 --> 00:02:13,267
so you go to the third term

57
00:02:13,267 --> 00:02:14,667
this is one

58
00:02:14,667 --> 00:02:17,067
you just have x^3 / 3!

59
00:02:17,067 --> 00:02:19,467
plus x^3 / 3!

60
00:02:19,467 --> 00:02:20,800
and I think you see the pattern here

61
00:02:20,800 --> 00:02:22,200
we just keep adding terms

62
00:02:22,200 --> 00:02:22,942
x^4 / 4!

63
00:02:22,942 --> 00:02:31,667
plus x^5 / 5! plus x^6 over 6!

64
00:02:31,667 --> 00:02:34,200
and something pretty neat is starting

65
00:02:34,200 --> 00:02:35,800
to emerge

66
00:02:35,800 --> 00:02:38,000
this is just really cool

67
00:02:38,000 --> 00:02:40,357
that e^x can be approximated by

68
00:02:40,357 --> 00:02:44,026
1 plus x plus x^2 / 2!

69
00:02:44,026 --> 00:02:45,933
plus x^3 / 3!

70
00:02:45,933 --> 00:02:48,133
once again, e^x is starting to look like

71
00:02:48,133 --> 00:02:49,467
a pretty cool thing

72
00:02:49,467 --> 00:02:51,758
this also leads to other interesting results

73
00:02:51,758 --> 00:02:54,684
like if you want to approximate e

74
00:02:54,684 --> 00:02:58,600
you just evaluate this as x is equal to one

75
00:02:58,600 --> 00:03:06,533
so this is, well, so you just say e is e ^ 1

76
00:03:06,533 --> 00:03:08,000
and this is going to be approximately equal

77
00:03:08,000 --> 00:03:10,682
to this polynomial evaluated at 1

78
00:03:10,682 --> 00:03:13,067
if x is 1 here, then we make x equal to one

79
00:03:13,067 --> 00:03:13,667
over here

80
00:03:13,667 --> 00:03:18,241
so it will be, one plus one

81
00:03:18,256 --> 00:03:21,590
plus one over 2!

82
00:03:21,590 --> 00:03:24,933
plus one over 3!

83
00:03:24,933 --> 00:03:27,800
plus one over 4!

84
00:03:27,800 --> 00:03:30,744
and so on and so forth all the way into infinity

85
00:03:30,744 --> 00:03:33,467
and you could also do this as one

86
00:03:33,467 --> 00:03:35,933
over 1 ! as well

87
00:03:35,933 --> 00:03:39,103
but what's really cool is that

88
00:03:39,103 --> 00:03:41,267
this is another way to represent e

89
00:03:41,267 --> 00:03:42,467
it shows that e

90
00:03:42,467 --> 00:03:43,608
once again shows up as

91
00:03:43,608 --> 00:03:49,133
kind of 2 + 1/2 + 1/6

92
00:03:49,133 --> 00:03:51,271
if you keep doing this, you get close

93
00:03:51,271 --> 00:03:53,121
to the number e

94
00:03:54,429 --> 00:03:56,193
but that by itself isn't the only

95
00:03:56,193 --> 00:03:58,353
fascinating thing

96
00:03:58,353 --> 00:04:00,200
if we look back at the Maclaurin Representation

97
00:04:00,200 --> 00:04:02,400
of these other functions.

98
00:04:02,400 --> 00:04:03,533
cosine (x)

99
00:04:03,533 --> 00:04:06,600
let me copy and paste cosine(x)

100
00:04:06,600 --> 00:04:09,867
so, cosine (x) right up here

101
00:04:09,867 --> 00:04:14,026
let me do my best to copy and paste the whole thing

102
00:04:18,272 --> 00:04:24,600
so that is cosine(x) and let's do the same thing

103
00:04:24,600 --> 00:04:27,773
for sine(x) that we did last video

104
00:04:27,773 --> 00:04:30,887
so the sine of x

105
00:04:31,533 --> 00:04:37,825
let me copy and paste that

106
00:04:40,533 --> 00:04:44,770
so do we see any relationship between

107
00:04:44,770 --> 00:04:46,348
these approximations?

108
00:04:46,348 --> 00:04:48,067
so before, you probably would have

109
00:04:48,067 --> 00:04:49,181
guessed there's some relationship

110
00:04:49,181 --> 00:04:50,342
between cosine and sine

111
00:04:50,342 --> 00:04:52,733
but what about e^x?

112
00:04:52,733 --> 00:04:54,429
so what you see here

113
00:04:54,429 --> 00:04:58,144
is that cosine(x) looks a lot like this term

114
00:04:58,144 --> 00:05:00,800
plus this term, although we probably want

115
00:05:00,800 --> 00:05:02,347
to put a negative out front here

116
00:05:02,347 --> 00:05:04,400
so we have a negative version of this term

117
00:05:04,400 --> 00:05:04,900
right here

118
00:05:04,900 --> 00:05:06,200
plus this term right here

119
00:05:06,200 --> 00:05:09,200
plus a negative version of this term

120
00:05:09,200 --> 00:05:11,067
right over here

121
00:05:11,067 --> 00:05:14,667
and sine (x) looks just like

122
00:05:14,667 --> 00:05:19,019
this term plus a negative version of

123
00:05:19,019 --> 00:05:21,527
this term

124
00:05:21,527 --> 00:05:23,533
plus this term

125
00:05:23,533 --> 00:05:24,600
plus a negative version of the next term

126
00:05:25,172 --> 00:05:27,200
so if we could somehow reconcile the

127
00:05:27,200 --> 00:05:30,067
negatives in some interesting way

128
00:05:30,067 --> 00:05:31,800
it looks like e^x

129
00:05:31,800 --> 00:05:34,867
at least the polynomial representation

130
00:05:34,867 --> 00:05:36,271
of e^x

131
00:05:36,271 --> 00:05:39,336
is somehow related to a combination

132
00:05:39,336 --> 00:05:41,533
of the polynomial representations of

133
00:05:41,533 --> 00:05:43,423
cosine of x and sine of x

134
00:05:43,423 --> 00:05:44,467
so this is starting to get

135
00:05:44,467 --> 00:05:46,200
really really really cool

136
00:05:46,200 --> 00:05:47,400
we're starting to see a connection

137
00:05:47,400 --> 00:05:48,764
between something related to

138
00:05:48,764 --> 00:05:50,180
compound interests

139
00:05:50,180 --> 00:05:52,400
or a function whose derivative

140
00:05:52,400 --> 00:05:53,802
is always equal to that function

141
00:05:53,802 --> 00:05:55,544
and these things that come out of the unit circle

142
00:05:55,544 --> 00:05:57,680
and oscillatory motion and all kinds of things

143
00:05:57,680 --> 00:06:00,267
there starts to seem a kind of pure

144
00:06:00,267 --> 00:06:01,867
connectiveness there. but i'll leave you

145
00:06:01,867 --> 00:06:03,276
there on that video and in the next video

146
00:06:03,276 --> 00:06:07,333
I'll show you how to reconcile these three

147
00:06:07,333 --> 99:59:59,999
fascinating functions