1
00:00:00,000 --> 00:00:00,700


2
00:00:00,700 --> 00:00:02,589
I had dinner between the last video and this one.

3
00:00:02,589 --> 00:00:05,299
So I might have forgotten what I just did.

4
00:00:05,299 --> 00:00:08,740
I think I was about to-- if what I see on my board makes

5
00:00:08,740 --> 00:00:13,009
sense-- I was about to use the Taylor series, or in this

6
00:00:13,009 --> 00:00:15,990
specific example the Maclaurin series approximation, to figure

7
00:00:15,990 --> 00:00:21,410
out a polynomial version, a sum of polynomial terms to

8
00:00:21,410 --> 00:00:23,480
approximate e to the x.

9
00:00:23,480 --> 00:00:27,199
And remember let me write here what the definition of

10
00:00:27,199 --> 00:00:31,379
the Maclaurin series was.

11
00:00:31,379 --> 00:00:42,560
That we said that f of x is equal to the sum from n equals

12
00:00:42,560 --> 00:00:49,890
0 to infinite of the nth derivative of f evaluated at

13
00:00:49,890 --> 00:00:52,460
0-- I don't know if I remembered to put it evaluated

14
00:00:52,460 --> 00:00:55,840
at 0 last time I wrote this down-- times x to the

15
00:00:55,840 --> 00:00:57,990
n over n factorial.

16
00:00:57,990 --> 00:00:59,429
And hopefully that makes sense to you.

17
00:00:59,429 --> 00:01:01,920
This might seem really confusing and strange.

18
00:01:01,920 --> 00:01:04,930
But now that we're going to apply it to e to the x, this

19
00:01:04,930 --> 00:01:06,780
should maybe be a little bit more concrete.

20
00:01:06,780 --> 00:01:09,650
And I think at the end of the last video I said that if f of

21
00:01:09,650 --> 00:01:15,140
x is e to the x, f of 0 is e to the 0, which is 1.

22
00:01:15,140 --> 00:01:21,180
And f prime of x, any derivative of e to the x

23
00:01:21,180 --> 00:01:23,810
is equal to e to the x.

24
00:01:23,810 --> 00:01:30,090
So you take any derivative of at 0, and it equals 1 for e to

25
00:01:30,090 --> 00:01:32,490
the x for this particular case of f of x.

26
00:01:32,489 --> 00:01:33,199
And that's really neat.

27
00:01:33,200 --> 00:01:37,770
That means that the rate of change of y with respect to x

28
00:01:37,769 --> 00:01:43,299
is for every 1 you move in x, you move 1 in y at e to the 0.

29
00:01:43,299 --> 00:01:46,609
But that also means that the rate of change of the

30
00:01:46,609 --> 00:01:47,709
rate of change is also 1.

31
00:01:47,709 --> 00:01:49,059
At the rate of change at the rate of change at the

32
00:01:49,060 --> 00:01:50,030
rate of change is also 1.

33
00:01:50,030 --> 00:01:57,469
So at either the 0 or x equals 0 of e to the x, the slope of

34
00:01:57,469 --> 00:01:59,469
the slope of the slope of the slope of the slope of the slope

35
00:01:59,469 --> 00:02:00,709
of the slope, they're all 1.

36
00:02:00,709 --> 00:02:05,419
Which to me tells me something mysterious is happening.

37
00:02:05,420 --> 00:02:08,039
It's another reason why you should just sit and ponder e.

38
00:02:08,039 --> 00:02:10,239
But anyway, back to what we were trying to do.

39
00:02:10,240 --> 00:02:11,370
So how would we do this?

40
00:02:11,370 --> 00:02:12,805
How would be write the approximation?

41
00:02:12,805 --> 00:02:16,659


42
00:02:16,659 --> 00:02:17,680
Let me write the approximate.

43
00:02:17,680 --> 00:02:18,800
I'll call that p of x.

44
00:02:18,800 --> 00:02:19,620
Because it's going to be a polynomial.

45
00:02:19,620 --> 00:02:22,740


46
00:02:22,740 --> 00:02:30,040
Well in this particular case, what's the derivative of any

47
00:02:30,039 --> 00:02:33,590
derivative evaluated at f of 0?

48
00:02:33,590 --> 00:02:34,490
Well that term is 1.

49
00:02:34,490 --> 00:02:37,290
We wrote that down right here.

50
00:02:37,289 --> 00:02:39,889
f of 0 is 1.

51
00:02:39,889 --> 00:02:41,489
The first derivative at 0 is 1.

52
00:02:41,490 --> 00:02:42,830
The second derivative at 0 is 1.

53
00:02:42,830 --> 00:02:43,010
Right?

54
00:02:43,009 --> 00:02:44,409
That's what's special about e to the x.

55
00:02:44,409 --> 00:02:48,680
So all of these terms are going to equal 1.

56
00:02:48,680 --> 00:02:56,740
So this polynomial simplifies to the sum from n equals 0 to

57
00:02:56,740 --> 00:03:02,469
infinite of x to the n over n factorial.

58
00:03:02,469 --> 00:03:03,919
That to me is very neat.

59
00:03:03,919 --> 00:03:06,009
Remember these are all 1 in every term.

60
00:03:06,009 --> 00:03:07,959
So that's why I took it out.

61
00:03:07,960 --> 00:03:08,810
So what does that mean?

62
00:03:08,810 --> 00:03:11,960
Well that tells us that e to the x can be approximated.

63
00:03:11,960 --> 00:03:13,629
And actually, I don't prove it here.

64
00:03:13,629 --> 00:03:16,659
But it actually turns out that we take the infinite sum that

65
00:03:16,659 --> 00:03:19,030
the Maclaurin series not only approximates e to the

66
00:03:19,030 --> 00:03:21,245
x at x equals 0.

67
00:03:21,245 --> 00:03:23,620
When you take the infinite series, it actually

68
00:03:23,620 --> 00:03:25,460
equals e to the x.

69
00:03:25,460 --> 00:03:30,490
So when you take a Maclaurin series at 0, and the resulting

70
00:03:30,490 --> 00:03:33,670
function, the resulting polynomial actually converges--

71
00:03:33,669 --> 00:03:35,559
and that's something we'll learn a little bit more

72
00:03:35,560 --> 00:03:43,340
rigorously hopefully later when we start doing analysis-- but

73
00:03:43,340 --> 00:03:46,090
it can actually converge to the function at all points.

74
00:03:46,090 --> 00:03:48,490
And it actually is the case with e to the x.

75
00:03:48,490 --> 00:03:51,240
So we can actually say that e to the x is equal.

76
00:03:51,240 --> 00:03:52,050
I didn't prove this.

77
00:03:52,050 --> 00:03:54,330
But you can take my word for it.

78
00:03:54,330 --> 00:03:56,100
And you can even test it out with some numbers.

79
00:03:56,099 --> 00:03:57,340
It equals this sum.

80
00:03:57,340 --> 00:03:59,060
Well what is this sum?

81
00:03:59,060 --> 00:04:08,020
It's x to the 0 over 0 factorial plus x to the 1 over

82
00:04:08,020 --> 00:04:12,659
1 factorial plus x-squared over 2 factorial.

83
00:04:12,659 --> 00:04:13,780
And you keep going.

84
00:04:13,780 --> 00:04:14,780
Of course that's equal.

85
00:04:14,780 --> 00:04:17,949
So e to x is equal to-- x to the 0 is 1.

86
00:04:17,949 --> 00:04:19,870
0 factorial, I said in the last video is 1.

87
00:04:19,870 --> 00:04:26,740
So it's 1 plus this is just x, plus x-squared over 2 factorial

88
00:04:26,740 --> 00:04:32,610
plus x to the third over 3 factorial plus x to the

89
00:04:32,610 --> 00:04:34,810
fourth over 4 factorial.

90
00:04:34,810 --> 00:04:36,790
And you just keep going on forever.

91
00:04:36,790 --> 00:04:38,480
And that's e to the x.

92
00:04:38,480 --> 00:04:40,490
And to me that is amazing.

93
00:04:40,490 --> 00:04:44,810
Because this strange number e, this 2.7 whatever, whatever,

94
00:04:44,810 --> 00:04:48,129
that we got from compound interest, it can be written as

95
00:04:48,129 --> 00:04:51,600
an infinite polynomial, this polynomial series or this

96
00:04:51,600 --> 00:04:54,730
Maclaurin series, that actually has a certain beauty to it.

97
00:04:54,730 --> 00:04:56,870
This number is kind of ugly when you write it out,

98
00:04:56,870 --> 00:04:58,379
2.7 whatever, whatever.

99
00:04:58,379 --> 00:05:01,560
But when you write it to an exponent power as an infinite

100
00:05:01,560 --> 00:05:04,959
sum, it kind of has a nice rhythm to it.

101
00:05:04,959 --> 00:05:06,680
It's very patterned.

102
00:05:06,680 --> 00:05:08,540
Who would have guessed that you could have written it

103
00:05:08,540 --> 00:05:10,390
in such a simple form.

104
00:05:10,389 --> 00:05:15,459
And even more, what happens when x is equal to 1.

105
00:05:15,459 --> 00:05:15,769
Right?

106
00:05:15,769 --> 00:05:16,769
So what's e to the 1?

107
00:05:16,769 --> 00:05:19,125
Well then we set x equal to 1 on both sides.

108
00:05:19,125 --> 00:05:25,819
And I think I have space to do it here, e to the 1,

109
00:05:25,819 --> 00:05:27,349
which is equal to e.

110
00:05:27,350 --> 00:05:28,810
We just set all the x's to 1.

111
00:05:28,810 --> 00:05:35,569
So we get that's equal to 1 plus 1 plus 1 over 2 factorial

112
00:05:35,569 --> 00:05:40,339
plus 1 over 3 factorial plus 1 over 4 factorial plus

113
00:05:40,339 --> 00:05:42,069
1 over 5 factorial.

114
00:05:42,069 --> 00:05:44,949
That to me, once again, is amazing.

115
00:05:44,949 --> 00:05:48,170
That the number e, and we've just stumbled on another

116
00:05:48,170 --> 00:05:56,400
definition of e, e is equal to the sum from n equals 0 to

117
00:05:56,399 --> 00:06:03,489
infinite of 1 over n factorial.

118
00:06:03,490 --> 00:06:05,269
That is amazing.

119
00:06:05,269 --> 00:06:06,789
So now we have two definitions for e.

120
00:06:06,790 --> 00:06:08,520
We have this one that we stumbled on, and of course we

121
00:06:08,519 --> 00:06:10,319
had the ones from compound interest that I will

122
00:06:10,319 --> 00:06:11,750
do in magenta.

123
00:06:11,750 --> 00:06:17,149
The limit as n approaches infinite, 1 plus 1

124
00:06:17,149 --> 00:06:19,689
over n to the n.

125
00:06:19,689 --> 00:06:22,509
That also is equal to e.

126
00:06:22,509 --> 00:06:25,279
This is starting to give me chills.

127
00:06:25,279 --> 00:06:29,659
Because this very strange, bizarre number is popping out.

128
00:06:29,660 --> 00:06:31,210
This might not seem so natural to you.

129
00:06:31,209 --> 00:06:32,339
But it's neat.

130
00:06:32,339 --> 00:06:34,519
And it comes out in compound interest, and continuously

131
00:06:34,519 --> 00:06:35,919
compounding interest.

132
00:06:35,920 --> 00:06:38,170
But this is even simpler.

133
00:06:38,170 --> 00:06:40,550
I just keep picking one over the factorial of numbers and

134
00:06:40,550 --> 00:06:41,420
I add them all together.

135
00:06:41,420 --> 00:06:45,009
And if I take every number really in existence, and I

136
00:06:45,009 --> 00:06:48,319
sum them all up, I get e.

137
00:06:48,319 --> 00:06:50,170
That to me is amazing.

138
00:06:50,170 --> 00:06:52,890
1 over n factorial of essentially every integer

139
00:06:52,889 --> 00:06:54,199
from 0 to infinite.

140
00:06:54,199 --> 00:06:57,860
If I sum them up, I get the number e.

141
00:06:57,860 --> 00:07:01,740
You hopefully are getting chills right now.

142
00:07:01,740 --> 00:07:05,170
Well anyway, let's do the Maclaurin series for a

143
00:07:05,170 --> 00:07:05,990
couple more functions.

144
00:07:05,990 --> 00:07:10,090
And then we'll get to something that is even more mind-blowing.

145
00:07:10,089 --> 00:07:11,139
I'll see you in the next video.

146
00:07:11,139 --> 00:07:12,899