1
00:00:00,122 --> 00:00:04,184
In the last video we took the Maclaurin Series of Cosine of x

2
00:00:04,184 --> 00:00:06,738
we approximated it using this polynomial

3
00:00:06,738 --> 00:00:08,334
and we saw this pretty interesting pattern.

4
00:00:08,334 --> 00:00:10,782
Let's see if we can find a similar pattern if we try

5
00:00:10,782 --> 00:00:14,384
to approximate sine of x using a Maclaurin series.

6
00:00:14,384 --> 00:00:17,293
And once again a Maclaurin series is really the same thing

7
00:00:17,293 --> 00:00:21,595
as Taylor Series where we are centering our approximation

8
00:00:21,595 --> 00:00:27,108
around x is equal to 0 .so this is a special case of a Taylor Series

9
00:00:27,108 --> 00:00:32,762
so let's take f of x in this sitiuation to be equal to sine of x

10
00:00:32,808 --> 00:00:37,877
f of x is now equal to sine of x and lets's do the same thing

11
00:00:37,877 --> 00:00:40,256
that we did with cosine of x . Let's just take the different

12
00:00:40,256 --> 00:00:44,754
derivatives of sine of x really fast. so if you have the first

13
00:00:44,754 --> 00:00:50,348
derivative of sine of x is just cosine of x . The second derivative

14
00:00:50,348 --> 00:00:56,441
of sine of x is derivative of cos x which is negative sine of x

15
00:00:56,441 --> 00:00:59,302
the third derivative is going to be the derivative of this

16
00:00:59,302 --> 00:01:01,098
so I will just write 3 in parentheses there in stead of doing

17
00:01:01,098 --> 00:01:04,231
all the prime prime prime. so the third derivative is

18
00:01:04,231 --> 00:01:09,021
derivative of this which is negative cosine of x . the fourth

19
00:01:09,021 --> 00:01:12,975
derivative the fourth derivative is the derivative of this

20
00:01:12,975 --> 00:01:17,467
which is positive sine of x again. so you see just like cosine of x

21
00:01:17,467 --> 00:01:20,436
it kind of cycles after you take the derivative enough time.

22
00:01:20,436 --> 00:01:23,252
and we care in order to the Maclaurin series we care

23
00:01:23,252 --> 00:01:29,169
about evaluating the function and each of these derivatives at x is equal to 0 .

24
00:01:29,169 --> 00:01:32,215
so let's do that. so for this let me do this in a different

25
00:01:32,215 --> 00:01:37,800
color not that same blue. so i'll do it in this purple color.

26
00:01:37,800 --> 00:01:40,329
so f that's hard to see I think. so let's do this in the

27
00:01:40,329 --> 00:01:47,733
other blue color. so f of 0 in this situation is 0 and f the

28
00:01:47,733 --> 00:01:51,938
first derivative evaluated at 0 is 1. cosine of 0 is 1

29
00:01:52,846 --> 00:01:59,267
negative sine of 0 is going to be 0. so f prime prime

30
00:01:59,267 --> 00:02:01,652
the second derivative evaluated at 0 is 0.

31
00:02:01,652 --> 00:02:06,944
the third derivative evaluated at 0 is negative 1.

32
00:02:06,944 --> 00:02:10,800
cosine of 0 is 1 you have a negative out there it is

33
00:02:10,800 --> 00:02:15,421
negative 1 and the fourth derivative evaluated at 0 is

34
00:02:15,421 --> 00:02:19,733
going to be 0 again. we could keep going once again seems like

35
00:02:19,733 --> 00:02:22,169
there is a pattern 0 1 -1 0 then you are going to go back

36
00:02:22,169 --> 00:02:27,000
to positive 1 so on and so forth . so let's find it's

37
00:02:27,000 --> 00:02:30,001
polynomial representation using Maclaurin Series.

38
00:02:30,001 --> 00:02:33,995
just a reminder this one up here this was approximately

39
00:02:33,995 --> 00:02:36,148
cosine of x and you will get closer and closer

40
00:02:36,148 --> 00:02:38,615
to cosine of x I am not rigorously showing you how close

41
00:02:38,615 --> 00:02:41,843
and that is definitely the exactly the same thing as cosine of x

42
00:02:41,843 --> 00:02:43,441
but you get closer and closer and closer to cosine of x

43
00:02:43,441 --> 00:02:46,046
as you keep adding terms here and if you to infinity

44
00:02:46,046 --> 00:02:49,179
you are going to be pretty much at cosine of x

45
00:02:49,179 --> 00:02:52,435
now let's do the same thing for sine of x . so i'll pick

46
00:02:52,435 --> 00:02:56,518
a new color. this green should be nice. so this is

47
00:02:56,518 --> 00:02:58,827
our new p of x. so this is approximately going to be

48
00:02:58,827 --> 00:03:02,067
sine of x as we add more and more terms

49
00:03:02,067 --> 00:03:07,133
and so the first term here f of 0 that's just going to be 0

50
00:03:07,133 --> 00:03:10,467
so we are not even going to need to include that. the next term

51
00:03:10,467 --> 00:03:15,333
is going to f prime 0 which is 1 times x . so it's going to be x

52
00:03:15,841 --> 00:03:19,901
then the next term is f prime the second derivative at 0

53
00:03:19,901 --> 00:03:23,436
which we see here is 0. let me scroll down a little bit

54
00:03:23,436 --> 00:03:27,133
it is 0 so we won't have the second term

55
00:03:27,133 --> 00:03:30,862
this third term right here the third derivative of sine of x

56
00:03:30,862 --> 00:03:34,831
evaluated at 0 is negative 1 so we are now going to have

57
00:03:34,831 --> 00:03:40,333
a negative 1 . let me scroll down so you can see this

58
00:03:40,333 --> 00:03:44,877
negative 1 this is negative 1 in this case times x the third

59
00:03:44,877 --> 00:03:50,836
over 3 factorial. so negative x the third over 3 factorial

60
00:03:50,836 --> 00:03:54,446
and then the next term is going to be 0 because that's

61
00:03:54,446 --> 00:03:57,748
the fourth derivative . that's the fourth derivative evaluated

62
00:03:57,748 --> 00:04:01,892
at 0 is the next coefficient. we see that that's going to be 0

63
00:04:01,892 --> 00:04:04,712
so it's going to drop off and what you are going to see here

64
00:04:04,712 --> 00:04:06,823
and actually maybe I haven't done enough pep terms

65
00:04:06,823 --> 00:04:08,917
for you. for you to feel good about this let me do

66
00:04:08,963 --> 00:04:13,379
one more term right over here just so it becomes clear

67
00:04:13,379 --> 00:04:17,067
f of fifth derivative of x is going to be cosine of x

68
00:04:17,067 --> 00:04:20,249
again. the fifth derivative let me do that in the same color

69
00:04:20,249 --> 00:04:23,200
just so that it's consistent. the fifth derivative

70
00:04:23,492 --> 00:04:28,608
the fifth derivative evaluated at 0 is going to be 1

71
00:04:29,685 --> 00:04:34,148
so the fourth derivatives evaluated at 0 is 0. then you

72
00:04:34,148 --> 00:04:38,133
go to the fifth derivative evaluated at 0 is going to be

73
00:04:38,133 --> 00:04:41,923
positive 1 and if I kept doing this it would be positive 1 times

74
00:04:41,923 --> 00:04:47,184
I would have to write 1 as a coefficient times x to the fifth

75
00:04:47,184 --> 00:04:49,867
over 5 factorial so there is something interesting going

76
00:04:49,867 --> 00:04:54,475
on here and for cosine of x I had 1 essentially 1 times

77
00:04:54,475 --> 00:04:58,890
x to the zero then I don't have x to the first power

78
00:04:58,890 --> 00:05:01,349
I don't have x to the odd powers actually then I just

79
00:05:01,349 --> 00:05:04,089
have x to all the even powers and whatever power it is

80
00:05:04,089 --> 00:05:07,339
I am dividing it by that factorial and then the sign

81
00:05:07,339 --> 00:05:10,590
keeps switching and this is ,I shouldn't say this is an

82
00:05:10,590 --> 00:05:13,562
even power because 0 really isn't , well I guess you

83
00:05:13,562 --> 00:05:16,333
can view it as an even number cuz it.. I won't go into

84
00:05:16,333 --> 00:05:21,733
all of that but it's essentially 0 2 4 6 so on and so forth

85
00:05:21,733 --> 00:05:25,451
so this is interesting specially when you compare to

86
00:05:25,451 --> 00:05:28,619
this . this is all of the odd powers this is x to the first

87
00:05:28,619 --> 00:05:31,387
over 1 factorial I didn't write it here there's x to

88
00:05:31,387 --> 00:05:34,385
the third over 3 factorial plus x to the fifth over

89
00:05:34,385 --> 00:05:36,831
5 factorial. ya zero would be an even number . anyway

90
00:05:36,831 --> 00:05:40,033
I don't.almost. my brain is in a different place right now

91
00:05:40,033 --> 00:05:42,866
and you could keep going if we kept this process up

92
00:05:42,866 --> 00:05:45,672
you would then keep switching signs x to the seventh

93
00:05:45,672 --> 00:05:49,460
over 7 factorial plus x to the ninth over 9 factorial

94
00:05:49,460 --> 00:05:51,067
so there is something interesting here you once

95
00:05:51,067 --> 00:05:56,400
again see this kind of complimentary nature between

96
00:05:56,400 --> 00:05:59,467
sine and cosine here. you see almost this..they kind of

97
00:05:59,467 --> 00:06:01,395
they are filling each other's gaps over here cosine of x

98
00:06:01,395 --> 00:06:05,714
is all of the even powers of x divided by that power's factorial

99
00:06:05,714 --> 00:06:09,133
sine of x when you take it's polynomial representation

100
00:06:09,133 --> 00:06:12,867
is all of the odd powers of x divided by it's factorial

101
00:06:12,867 --> 00:06:17,195
and you switch signs. In the next video I'll do e to the x

102
00:06:17,195 --> 00:06:20,477
and what's really fascinating is that e to the x starts

103
00:06:20,477 --> 00:06:24,430
to look like a little bit of a combination here. but not quite

104
00:06:24,430 --> 00:06:27,462
and you really do get the combination when you involve

105
00:06:27,462 --> 00:06:30,333
imaginary numbers and that's when it starts to get

106
00:06:30,333 --> 99:59:59,999
really really mind-blowing.