00:01
Hello, numerate.
00:02
We're back.
00:03
Okay, so now we're on question of 16, where f of x is sine x.
00:13
They want you to expand this around a equals 5 over 6 up to the fourth degree polynomial.
00:19
And to an error, a taylor series error term, so we need basically five derivatives.
00:28
So let's see.
00:33
Can i do it this way? f prime like this.
00:36
We know what it is.
00:38
It's the, i'm going to write out diff of f of x with respect to x.
00:47
Okay.
00:49
What did i do? diff of f of x.
00:56
There you go.
00:57
Let's show this so far.
00:59
I usually do fp, fpp, fpp, p, p, p, p, p.
01:01
That's too many p.
01:02
So i'm do f1, f2, f, f, f, f, f, f, f, f, f, f, f, f, f, f, f, triple prime.
01:06
How about that? so show.
01:09
There's so many derivatives here, and we're evaluating i think this at pi over six, so many radicals.
01:13
I'm just going to do this straight on here.
01:16
So show, koko comma, f of x, is f of x.
01:23
And let's do the same thing over here.
01:29
And we'll generate the fourth degree taylor polynomial near pi over six.
01:33
Yes.
01:34
Okay, so this is f1.
01:36
There's f1 to the derivative of the sign.
01:37
Of course, it's a cosine.
01:38
Let's just make sure this is all working out.
01:41
Okay.
01:42
So let's do f double prime, based on f a single prime and display f double prime now of course you know that the derivative the sign is the kiosin derivative of kittalitkine is negative the sign so there should be negative the sign this is not the hard part just writing this whole thing out little radicals and stuff okay this is working out nicely all right so i need up to f5 right okay so here we go third derivative is the derivative of the second derivative of the second derivative x and display the third let's see here.
02:30
So the derivative sign is cosine, derivative the cosine is negative the sine, derivative the negative cosine, perfect, this is nice.
02:35
All right, two more.
02:37
Those are you used the fifth derivative for the error term.
02:53
Oh, what was that? so the derivative of negative cosine is positive sign, perfect.
03:07
And the derivative sign should be cosine from the last one.
03:10
We'll evaluate this at pi over six.
03:12
So the first one is one half.
03:14
The next one is right through with two and so forth, right? 5 of 6 radiance 30 degrees.
03:21
All right, you know your trig values, do you? unit circle and all that.
03:35
Yep, there you go.
03:37
All right, now, we want to evaluate this at pi over 6.
03:41
So let's do that.
03:47
All right, pi is defined.
04:04
And let's evaluate it.
04:05
Those were labels.
04:06
This is the evaluation over here.
04:07
What's not in quotes? is evaluated? what's in quotes is just a label gets printed out verbatim.
04:14
It's a string for computer science people.
04:18
So let's see.
04:19
I have one half, brad 3 over 2, negative 1 1 .5, negative right 3 over 2, 1 half again.
04:26
We really don't need this value.
04:27
We want to maximize this value.
04:29
Now think about it.
04:30
We're doing this.
04:31
We're evaluating this from 0 to pi over 3, isn't it? 0 to 5 with 3, i think.
04:46
So isn't the maximum value of cosine? so this one just going to leave f of x it was cosine was it max value of cosine on the domain zero to pi over three is at zero cosine of zero is one right so the max of it's cosine from right because it's zero one right so then that's the max okay all right so now let's write out the p4 piece of four the fourth the fourth degree tail polynomial well all right you know what wait i'm going to start with p zero they usually do it's going to be um just this while evaluating times x minus a which is power with six to the zero over zero factor over just one okay so now what we're going to find is that p1 is going to be p0 well zero no no plus all right the first degree turn oh this is no this is wait this is just this is just f this is just this is just f okay but then so you can think of it as if zero no derivatives right all right so now i'm going to take this and i'm going to modify it for the first degree term which is the first derivative and to the one over one factorial okay then okay so then i'm going to copy this i'm going to say all right i want to get to p4 right okay so p2 is going to be p1 which is over the first two terms the 0th term the 1th term the first degree term right plus f prime at 2 if sorry f double prime and to the second over 2 there you go that's how i'm going to do it and then p3 is going to be everything up to p2 plus the third derivative at pi over 6 times x minus a to the third over 3 factorial which is 6 and that nice and then p 4 is going to be everything up to p 3 all those terms plus the 4th derivative at 5 6 this is a default rating mode by the way the system here to the 4th over 4 factorial which is 24 and we really don't need to show all of these, but what the heck? let's see.
07:56
What if i do this? p0.
08:16
1, 2, 3, 3.
08:29
Let's see what this looks like.
08:38
Whoops, i meant 1, 2, 3, and 4.
08:43
There you go.
08:45
That's where we go.
08:47
So, p4 is one big, huge, complicated mess.
08:52
Oh, wow...