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00:00 Yeah, mm hmm. Now we go so and I start to talk

00:08 uh huh. Well, normal is particular data serious was mentioned last time

00:15 a way of um figuring autopsy manipulate in order not to lose significance in

00:22 case stable, serious, what's And as I mentioned, they were

00:28 . It's kind of used as a to understand approximation accuracy, pretty much

00:35 the chapters of the book. So some of you are very familiar with

00:44 chinese cities. Let's see if we get this to work so already and

00:58 time. So move on to actually about the specifics obtained a serious.

01:06 this is kind of formal way in the correction ah look something on how

01:13 can also construct it. So it's a series of terms of higher order

01:24 terms of the degree, but the . So it starts with the constant

01:31 then so the first order and their and 3rd order were power of um

01:40 the variable extract as they are Trying to understand how to approximate the

01:46 at some point X. Resuming that is kind of a known entity and

01:53 the terms involves the derivatives are higher that corresponds start of the expression while

02:03 far away from tests effectively want to how far away from C. You

02:10 to evaluate the function. So this this is a deviation from from

02:16 And do you have this expression? , so it brings the ball,

02:23 the taylor series converges, then either is quality design or it's so

02:31 It's also approximation of the function and the better the more terms you

02:37 the better approximation. Forget. So not nothing diplomatic think about that.

02:46 um so we'll play it on with too. There was all the chapters

02:49 the book when we want to figure the accuracy of approximations. And so

02:58 particular thing in order to construct okay, expression or the series than

03:06 need the value is called the relatives successively higher order depending on the terms

03:12 have and that they were serious and special case when you're trying to expand

03:21 , We'll look at some something around equals zero. They've got a different

03:25 . McLaurin Services. So It's quite spectacle. Say this year's special

03:36 So the next line, I have few examples here. Good. Not

03:42 difficult. So again, the expression the table series up there Now and

03:48 examples of c equals zero. Not worry. So hopefully you remember that

03:57 the derivative E to the X. the function itself, no matter how

04:03 times you take the derivative into the part of the plan. And if

04:08 don't want to look at the function as well as it's derivative two C

04:16 zero. So each of the X X0 is evolving. So that means

04:23 the derivatives that have crime that goes front of the powers Of X&C equals

04:30 . It's always the one that's supposed see and then you have the denominator

04:36 is the correspondent back to work. it's a fairly straightforward and simple,

04:43 . Yeah. Remember thanks to the questions on that. As long as

04:55 remember that the derivative of the function , it's just probably X equals zero

05:03 get one more time. Don't want do everything into the water. That's

05:10 it. Okay, so another comment there's a serious expansion for again,

05:25 biggest force equals zero. So remember you take the relative of science they

05:36 co sign and effect the river that cosign, they get signed. Whether

05:42 organized within pluses minuses. Remember. but then if you apply in for

05:53 they uh the derivative for the first of zero, that's that's the first

06:03 . Therefore C. zero. So sine of zero is 0. So

06:07 is no constant in this particular case the constant safe. And then the

06:12 one is tech derivative of sine. can get cosine X and co sign

06:19 equals zero is the one functions. then we just get them. And

06:26 so you get every other term disappears that's symbols scientists. So you take

06:35 derivative of course and you can sign concerned again still zero deficits for

06:40 So they've got this um serious suggests there were power X. And the

06:47 alternating paul Simon is ones so it's stiff, correctly constructed. And I'll

07:00 back to this particular form film is there's some nice properties what else did

07:08 have sign but those signs we do us actually for zero. But then

07:19 you get all the even terms even of X instead of all the and

07:27 else are a little bit trickier today of is the one by the sex

07:33 it same idea to take the derivative this part. Yeah a negative sign

07:41 when you take the derivative of minus in other words it's kind of a

07:46 the world whatever it takes denominated one -X to the father of -1

07:55 them for a -2. That's the stuff. Okay. And I think

08:02 and another one tip. So these very commonly used in particular the church

08:08 and its financials and we're trying to When the particular one X. is

08:16 important. And you look at these so much different approximation. Well that's

08:27 one in this case. So and questions on that let's start we're also

08:40 um just an example we'll see in um just the convergence can be

08:49 Great if that's a small number faced a number of zeros to the vine

09:02 decimal points. Keep sort of doubling her powers. So they get pushed

09:07 very quickly as um the terms increase question and fun again this is another

09:20 where the sign of it in terms alternate and that in fact makes the

09:32 generally quicker than of the science are the same or Alex fall damage

09:47 So you can drive or e to X. Mm hmm. So as

09:58 can see in this case there is alternating science and that also means that

10:04 convergence is not very quick. So this case intriguing to the 8th,

10:11 do you mean? six stars? quite far off only after what so

10:24 sort of various colors as we exponential, what kind of work?

10:32 it's a small change. It's magnified to increase the argument for X in

10:41 place. So it takes many terms try to follow the exponential through to

10:49 foot approximation that precinct. So so is kind of like bottom line and

10:58 all measures from the straits expansions that depending upon again what the series

11:08 The approximation accuracy is quite sensitive to wire away from. That's that's not

11:15 point of expansion of the policy. are you doing with this? So

11:24 And here is kind of one of cases like the sine function that is

11:29 soft, wow sign terrorists. Kind dashed line and then the first term

11:36 just X itself and that's yeah, to seriously good but you don't have

11:41 go very far before it gets bad but when you add terms like this

11:47 term for instance, you get the numbers match up pretty well and then

11:51 get the for the part if you remember sign was X. And the

11:58 term is executed and the next terms fine. So you get more of

12:03 curvature of the photo this time they out. So that was that.

12:14 now it's a little bit of a here too. And I have to

12:22 , we'll figure that out too. the taylor series function in this case

12:27 polynomial simple expression anywhere that sometimes it be useful too. And then do

12:36 famous years expansion. If you're interested the approximation of the function posted armament

12:44 in this case. So we needed derivatives and the first function by itself

12:50 then the various derivatives. So in case is pretty simple detective derivative.

12:55 the function of the polynomial exists. try to be straightforward and takes the

13:02 selves affected for the first time And gets to the five. That means

13:07 five points down and you get four five students 50 and so on for

13:13 doing this for the eastern the term then you can just follow through and

13:20 a little bit for this. But higher order derivatives of the that was

13:28 polynomial in his face and they are to first evaluate For actually. So

13:38 seeing equals two. So the system plug into and I get that and

13:42 the derivative at x equals two. you get all these numbers for the

13:49 of higher order. They're a protection you have that. You can write

13:55 now the polynomial expression practice. So nothing magic. AH 207 was the

14:06 evaluation. Right? The function around too. The first estimate it was

14:13 and that should be delighted by one , which is so just in 96

14:19 next one Um 2ndly, did find divided by You two will just have

14:25 try to find So the district forward construct. And the one thing that

14:32 notice that again, if you wanted evaluate things goes to except for stew

14:43 all these expressions, they are fairly and they powers of some very small

14:48 gets exceedingly small, atlantic critically. that means if you're very close to

14:54 , you don't need to many of terms in the taylor series expansion before

14:59 have numbers are so small. You need to worry. So that's that's

15:05 to trying to do it directly from pulling over expression that have Now you

15:11 something that exposes and also managed That's questions on that. What just

15:28 let's remind this impetus of the famous I had before. Again, depending

15:35 how the argument value that. Is the racist? It's on power things

15:42 practically. Yeah. All right. the next thing was uh please uh

15:55 construct the coefficients in the Davis here approximation itself again, starting in this

16:06 again the polynomial is a function. the last lecture we talked about This

16:13 scheme the DNA was this is called factor in this case. They want

16:22 fact there are so ex miners are far as the point. Oh expansion

16:31 you want to look at the function have close to Mexico so far they

16:37 to pull out. Exponents are part this uh functional polynomial and this particular

16:45 . So you want to invest rewrite before the normal on this form in

16:50 to get powers, distance and or you try to evaluate Yeah. So

17:02 time I remember when they have this process using the I think it's the

17:12 pulling off one of them or why pulling over the Q. Is one

17:17 lower than the enormity because next time power this is this is just one

17:23 episode that makes issues one lower order for the normal ones. And that

17:30 or is not the root of the of But it also means that then

17:39 you have done this the transformation on polynomial to reduce its degree that they

17:46 want to for the normal view. you are look at the coefficients and

17:55 expression So if your problem exit polls everything will disappear. Success C

18:02 And if you look at this factor that means the remainder in the

18:07 Ations The Coefficients C0. So by this division then you have the way

18:16 figuring out uh the coefficients C And this form of writing the

18:25 Mm hmm. So and then that's you can repeat this process and then

18:34 get the successive professions. So the one to get is c.

18:41 And if you look at that there's rewrite this equation and what they have

18:47 for the remainder to the left hand and factory does. So then this

18:51 a few. So now they have that looks work they started with except

18:57 is now one border lower. So next time when you do the same

19:02 on huge Trying to get C. that is the next position in let's

19:08 this year. So you can just this deflation process and we have two

19:15 profession's and get services. Um mr you have to do is follow what

19:30 last time and then I think I a little concrete example wherever it has

19:36 people in all member 4th degree. this corner steamer will start with the

19:42 total term or the coefficient from the school the term that is one and

19:46 -4 successive provisions in the polynomial And they wanted Vibrates, if there are

19:58 Then they take RS three. And Schema was at 1st to go column

20:04 column from left to right. And you move from one corner to the

20:10 to modify what to their previous column The value that you want,

20:16 So 23 and then yeah they are way. When you move to the

20:21 13 times -1 is 43 and keep and these are now the coefficients.

20:29 you for calling all. And then have the reminder and are ready for

20:37 which is not the first coefficient. a serious form of the polynomial.

20:44 so they just stopped I said And you can continue to get C1.

20:51 now we still want to have four three. So yeah. Now operating

20:57 two. So The first thing you it down 3.13 acceptance of the

21:03 So now we got the remainder a in terms he said that what I

21:09 call a normal stuff. It gets less term every time to do this

21:14 and now we have C- 136. I want some then repeat this a

21:22 of times until I don't want to valentines. Ah Yeah and with space

21:32 that's all the all the coefficients. right now this polynomial in a fatal

21:40 form. So it's in a way the same polynomial in the sense that

21:50 to give me an X. You get the same rather if you plug

21:54 in and specifically for your but it's written in the way, if you

21:59 to evaluate something for exposed to then this is succeeding with this

22:06 So it only needs sort of a of the charges, but it's not

22:12 , whatever. So this is kind a procedure to construct the politicians For

22:23 years expansion. So give me some big thing about just everything's coming.

22:35 any questions on that? Thanks. . Yes. What are the what

22:47 the next after the So that's the -1. Foreign service. This

22:55 I'm sorry, the 3630. Ah so going next, the next step

23:08 happened. So now investments are here the coefficients of Q. And this

23:15 right? So the next time I want to start to Q.

23:21 then when there is like you they're out expanses are, so now we

23:26 this is the next lower or the . So that's good. So that's

23:35 so here everybody que fortifications. So they kind of more or less forget

23:43 what's on here. So now we come to work with you. So

23:48 going to try to generate coefficients for . So that means now here is

23:55 coefficient with you. I still want do things for our April three.

24:01 now we do the same thing as this part. But they kind of

24:05 forget it. So the expense this the operation to get things for

24:10 So here's the first coefficient Q. look here we just sort of

24:17 Next one is to generate the next . Yes. Now take this For

24:24 2nd proficient for Q. Three times is 3 and then add them up

24:28 get to. And so now we developed the politicians can be best

24:38 That is one lower that one worked the world. So he has And

24:48 K five proficiency constant and and coefficients increasing powers of X. To uh

25:01 with the best of these are now wow um powers or no disrespect to

25:10 power for the ex uh the highest of you. That should be seven

25:20 one constant. And then this is the bar. So that's why we

25:33 it's one less proficient as you move towards the floor people normally. So

25:42 the end I think that's mhm. expression for the highest order.

25:49 The next location. The 8th. their nuclear successive before what is successively

26:00 order for converse with you start with the first as an anger. Second

26:08 third remained Probably the 1-1 systematic Mind Yes, one wonder wonder

26:24 But it's the same schema. These the profession's that was actually like all

26:34 , quantum expansion. And my work to regenerate that we want from left

26:39 right now. So this is kind it. Mm hmm. What

27:03 Famous years expansions um for when countries useful and that is that once.

27:17 hmm. A certain number of terms thanks for the sign. And then

27:24 is some error in the approximation of exact representation of the function. So

27:30 is from here. So if I something about the function, yeah,

27:36 possible to get an estimate of what do ever. Sorry. Yes,

27:43 that's about it too. And that's you were doing some of it.

27:48 something that you want to give and to figure out how many terms do

27:51 need to get together? So that's important aspect and practice. And what

28:01 else has to do with this? in the term store needs to get

28:05 decent approximation. Um for that one I want to use is if it

28:15 or the function has the derivative of least one higher order than the number

28:21 terms you include your expectations. Then can get an estimate for oh,

28:29 they haven't correct and so on. the one of these daniel is nice

28:36 this suppression so soon. I But they expounded okay, it's not

28:43 of growing. That's nuts. Then factorial grows pretty quickly. So that

28:50 denominator grows very quickly. And also the distance from sea is small then

28:58 the this symptom C c is raised the barrel and puts one in this

29:04 it also gets diminished. So first this function has derivatives are thousands

29:14 badly. You and you should get know still number of terms of low

29:22 and just about for bigger. And I'll give the exception. So

29:31 is kind of the basic, this the form just written down generally was

29:36 derivatives and the factorial on the corresponding of the derivative and the accommodated child

29:42 power. Um The difference from seed we moved on number of cards they

29:56 . So so yeah, this kind 71. So we talked together and

30:09 about the tennis years example. So now so the X is kind

30:17 nice. His example of course the is just a function himself all the

30:23 . So we know what is that A return. So to remember

30:29 this was the form of mhm David expansion. There was one plus six

30:34 six squared over two. You it's stewed over, uh huh three

30:40 etc. So This is the general and 90 there's the value of the

30:47 of the function B to the X zero and X in this case.

30:56 it's in the interval between zero and where they wanted. So they know

31:04 see And somewhere in the interval between and X. Um forget this

31:12 But depending on so it should put bounds. This impression. You

31:19 that's the one maximize the get some air. So now the question is

31:33 of the role that this in fact that it actually does converged so so

31:42 can look at you know what how this expression of the error term um

31:48 has increased the number of terms and for the given argument. So they

31:59 about this. Mm hmm three value , value for which this exact is

32:08 in the interval between there are actually necessarily with this addiction but we know

32:14 no longer. So this is now he tries to put an upper down

32:22 this is a question and we certainly that into the explodes the largest day

32:33 , E or the argument, the is similar ex you know was expect

32:39 than the power also grows so we that and after bound for this expression

32:48 fully support if you know the maximum interested is So but this S is

32:57 than the interval of experts are interested . So then you know that this

33:05 definitely is smaller than this because this has to be less than S.

33:11 this is the largest and mr and X is a lot smaller than

33:17 So this is a term. So like this one. So you said

33:24 was much larger than X. So not dealing So regardless of what 10

33:35 for any given success or X. This is true. The value of

33:43 expression depends on a large access because football has to be imagined. That

33:49 big enough. Absolutely. So it's as a function of X. So

34:03 were given an X. And then try to find an approximation of each

34:08 the excusing favorite series XS and No . But and the job is to

34:14 out how many turns going the number terms with your pants that will depend

34:25 that because uh this is the value this expression. They're all sweet.

34:36 , that's that's a positive number to the absolute value of X. So

34:42 is possible numbers in order the access larger this expression is that means the

34:48 and ending for this impression to this . That's our show. So the

34:56 grows faster than the power. So why eventually it will be fun.

35:12 , so this is one way I catch from the shows up. The

35:19 eventually killed. That's fine. So one thing is, what's the

35:25 test You take one drum here when take the next turn. So it's

35:33 This is the same. So we need to worry about this one.

35:36 we need to worry about these So for ah got your and say

35:44 K. What at this part of game? And there's Evan whatever X

35:49 the K infrastructure. Okay, Astoria one minute and the next term is

35:57 Incriminating Things. Part one. So you look at two successive terms,

36:04 turns out, but the ratio between two or the next term And you

36:09 from Kate, Kate Plus one and plus one then eventually, regardless of

36:17 . As long as it's a fixed , this thing is supposed to

36:23 So eventually in terms of their better . So, so this is

36:42 Um just one definitely represents the access but it depends on the, on

36:53 attraction. So we had it before to the east of the eighth

37:03 Okay. Uh on the side I it's five terms and it was a

37:10 bad as possible from to the X X is a very small number.

37:17 to get away from zero. I know. Yes. It's basically like

37:24 proving that he was actually right. . If you just take enough terms

37:33 you will get think about somebody for but the number of terms indeed.

37:44 eventually it will be a good Yes. And So Mark three on

37:59 next one. So another function that's focuses on it evaluating a lot of

38:09 . Natural law functions. So instead your one and then you touch

38:14 take the derivatives and then for they just depression and plugging in the

38:21 um, That for X0, that The value of one square and from

38:31 what X? That's zero. So may be confusing but the arguments of

38:36 function one and then some and the expressions that we have right, excellent

38:51 or That's part of myself. So that fixed again, That's one

38:59 by 1, 1 divided by but I'm going to sign on

39:05 Yeah um this success and the terms standard procedures before so yeah, that's

39:13 alternative. Thank serious for this part then we can estimate the Eritrea the

39:25 of the derivative. So this is of discretion. So in this case

39:31 sectorial follow something in this in our . Yeah, so please in this

39:50 um that's kind of the way So in this case can depending on

40:02 X is that is sufficiently calls for then there is uh the convergence on

40:14 other hand, if it is greater one then things do not converge.

40:25 so you can do the same kind asian test example here which are done

40:29 the next trying to prove that this the case that unfortunately, so as

40:34 as um excess ah Between plus and , so it's small sensing cells converge

40:44 , but that's not so Between 1001 , no, that was, I

41:00 this is a couple of examples of of suspensions and have some more examples

41:11 can still figure out figure out the how many times we have such sights

41:19 up. So one Bernie um important guess is what's known as the and

41:34 is the way to get somebody there derivatives and sometimes it's useful and the

41:45 itself. Um doctors and for functions . My personal sense is that if

41:54 have a continuous function Between two points Somewhere between the two points A and

42:07 . The derivative assumes the values that equal to the snow Of a straight

42:14 between the two. A function value A and B. And function

42:20 Okay somewhere everything these two points, there is a continuous function from A

42:26 B, the function has a derivative to the, that's the way of

42:34 some idea of you put in your On the day that they have about

42:41 . So there's just a picture of I've been talking about. So there

42:45 a play and functionally and the functionality yes that's so online within a week

42:54 then in this case it's a simple . So this case went up and

42:59 it comes down to come up with . So somewhere The derivative of the

43:04 of the function between the two it is the same with attempt.

43:09 is dependence that is parallel to the . That is an estimate of and

43:24 course it can be more than one it just tells you that at least

43:30 one ah savannah that it doesn't say necessarily it's an actual derivative. Mhm

43:45 , I tried derivative in between depending how the function but sometimes it's a

43:54 yeah, expression about. Mhm. , alright, so this is just

44:12 different form of writing expression that this probably what they use most of the

44:20 in the following. And basically focused are close to appointed fashion. We

44:31 do uh the approximation. So if simply replacing X minus C with the

44:42 . So this is, it's just deviation from ah I see. So

44:56 if it's only that by way of the very seriously function this is X

45:05 C which is H. And then have both expansion itself in terms of

45:11 deviation from C. As well as air return expressed in terms of that

45:20 very convenient for local that expansion from point playing around but so far

45:29 No go fancy. Mm hmm. . Mm hmm. Yeah. So

45:40 this so that's why I would have it explaining it. I want my

45:46 about the order of approximation weather the power of age. That is

45:54 and the error to so morbid, complete examples. So this is notation

46:06 you said. So it tells you the innocent optically quickly there are the

46:15 that's a function from the airport and one critical started proportional to the power

46:26 the distance to the first part. , so this is not just but

46:37 this got the biggest one in place is no dependence on a job embedded

46:44 this shopping rotation. So it's a , the cost of not being very

46:50 small which is nice but it could be very large. No the magnitude

46:57 this expression depends entirely on this concept upon how the functions yes behaves was

47:09 the previous slide. So so this basically what's hidden in see the higher

47:18 derivatives. It's very hard see you taking very well sure. So what

47:27 see is entirely dependent on but this does not depend on how far away

47:34 you directly interrupted mind because the value city in this case is related to

47:45 because she is that some in the between president exports but there is no

47:54 dependence in terms of. Okay, . So that's a big so that's

48:13 . Oh if you just had the story term and the proportion to it's

48:19 the concept and the sports derivative then this case an error for approximation is

48:27 border age. That is the first term that is not included in the

48:35 . So it's just a constant plus truck and then here is there is

48:42 squared because that's the the expected see between X and expose it. So

48:52 is a second order expression the third etcetera and those are the oceans will

48:58 used throughout the course and approximations polynomial derivatives or integral. It's all about

49:11 . Look the air return and the on the air term or the power

49:18 change the distance from the court for values all that stuff questions on this

49:32 . Okay. Mm hmm. So . But now I have some concrete

49:45 things on this fence effect is quite good function of X. So let's

49:51 the derivative in person, 2nd and derivative squirrel to Becks. And then

50:02 notion Uganda instead of this. And actually wanted to be in this case

50:10 the value of one. So their , um, neighborhood, so to

50:15 from one in which they want to at this point. This is a

50:20 expression. So we have the derivatives then they're talking if I do you

50:28 see on this one and this So we have for X. About

50:36 ? So this is basically this spiral one of us were also once it's

50:41 half the next term Right now this the power of one soul.

50:51 The So we have a second term this 1 -1 quarter and then two

50:58 . So that's why we get eight then the next star. Um,

51:06 , you're right To the 3rd power that helps. That's what That wants

51:19 have the first guy and this and the next one. Yes.

51:24 So we have three and the So that goes away. So it's

51:29 third factorial. so that that's been touch too, that's in the

51:34 the Johnsons are nominated. So and if it again the value of age

51:44 mine is fine. Alright. This using the depression then we have let's

51:56 there 1/2 times 10 to the Ah and then 1 8 times the

52:05 attempted to stand. So thanks for workers please. Okay. Um mm

52:18 . Yeah, I can do And the other direction to if it's

52:22 instead of positive, that's minus that's another one. Yes.

52:30 So any questions on that, it's senator. Do sponges derivatives and write

52:38 take a seat depression. Anything by by the correspondence victoria the power or

52:46 this case the deviation to the point the evaluation. That's what's wrong.

52:54 . No. We talked about the functions, the sine and cosine and

53:04 famous functions that were conducted. Um those serious expansions at the form that

53:14 um signed between successful term was alternating and those are known as alternating serious

53:25 the size of them. Because the often it in that case it's a

53:33 central form of estimating the error in to because the signs are alternating that

53:44 is subject for guaranteed the some of rest of terror, his old,

53:52 father and they absolutely nothing on the turn it's positive. The next part

54:01 that, what is the negative. reduce some of the error of the

54:07 . And even if the other ones builds up but there since morning it's

54:14 the case that when you're alternating seriously converge then you can estimate the

54:23 Oh the currency do not include. the first term. Absolutely. For

54:29 first time you don't have to go look at the testimony of the derivative

54:34 be between. Yeah so that's the of alternating serious. So it's something

54:46 this that's the error and and turns the true value and authenticity discussion in

55:00 the absolute areas although smaller than the the next term. So coming back

55:14 the sine function and in this case believing this is not the proof that

55:22 claim that the area is always less the first time to be ignored.

55:30 that is true. It is true if you stop that. Including in

55:38 of the next during that you do think the best player as the

55:44 One over the last two and last . Yeah as you can see that

55:49 are the ones all the time. the denominator all of these forms.

55:57 again it's an alternating series. So only need to make sure that the

56:01 time we did nothing to is less the area we will um or can

56:10 . So in this case if they the error to be for the results

56:13 the corrective safe six decimal digits and need to ensure that's the first term

56:21 is not included is less than the . So now you have an expression

56:29 you can figure out what should MB order to meet this condition. So

56:36 gives you again a way to figure how many firms in the state of

56:42 you need to get the desired So in this case so it

56:48 So that's as well. Many fire be stuff questions like that. Mm

57:08 . So let's see what else we . We also have this Welcome one

57:13 Sex. That is also often mating and that is a similar case and

57:22 out what it is and now it's the factorial. So it was obviously

57:27 it ends up forgiven error. You more terms because they don't decrease as

57:34 huh? But the sign the Well it's a pictorial expression so that

57:42 very quickly and then your term does go as quickly. So that means

57:47 need a lot more charge. For this type of paranormal take a

57:54 expansion approximation. Okay one. So some other kind of bad example but

58:09 got up and he was nice. yes thanks for face value that this

58:19 one of the way of approximating By power of four divided by 90.

58:28 in this case if you are to things. Use the same idea.

58:35 was a log of 1.6 um since just the power and then nominate your

58:43 terms. Like terms. Look at form. So we just assumed that

58:50 that the last sermon this case should less than they wanted. Um you

58:54 particular about 37 but it turns out this is not a good approximation does

59:02 need again, this is not an series in this case about the same

59:09 . So it doesn't have the benefit alternating series that they I am a

59:14 expression and for the all the terms positive so things kind of accumulate,

59:24 not that the next term healthy So in this case things are not

59:31 . So there's ways to kind of at this and try to figure out

59:36 you know, I wanted to do approximation. Then the remainder of the

59:44 series is text of his form. was basically run over the Value of

59:52 to the power of four. So need not to basically put the balance

59:58 the some of these variables and some be banded by the first term anymore

60:03 they're all having success the work I'm opening it up for this approximated by

60:12 an integral and stuff. But some these terms what the actual song is

60:22 , there's some of them wow defending boxes for Saint Brother is following the

60:29 . So the area of the curve mr Jordan and some of the

60:35 So in that sense, this sum bounded by the integral on the same

60:42 function. Now this is an easy to evaluate in this particular case.

60:47 in this case, you know that error, that's the way less than

60:52 of this form, that's the Why did you stop? That's a

60:57 serious. So in this case, , there's that expression That tells you

61:03 you actually into terms and not the sermon that since you just gambled and

61:10 it on the last term for the time of the day mark, the

61:16 says that it's not an alternating You cannot depend on the first time

61:22 the group, which was the best errors accumulate as opposed to trying to

61:30 each other out. And then, know, in this case was particularly

61:35 simple way of trying to find something is larger than the sum and that

61:42 the easiest with value. So this fun where you have to be creative

61:47 and figuring out what uh yeah, approximation of rebound On the part of

61:54 stadium series and one would be to at the derivative of the function and

61:59 mhm And explicit functions that was just serious human to us. So

62:13 so that was pretty much, but will take this area somewhere, start

62:19 about it. And there was a of most of them, I remember

62:24 everybody. So again it's general form the VCS expansion and the convenient form

62:33 is often used this to look at taylor series and functions relative to some

62:39 to the realization X. Moving around little bit from X. Um,

62:44 it is a portable X. That are. So basically the same thing

62:48 with stand up to have them the of estimating the area by trying to

62:59 , you can put the C. in the interval which can see and

63:02 going to see. So it's within range of pitch and this impression and

63:09 upon the function the natives. Ah . It is possible that they're putting

63:18 down. It's a highly function I to really make sense to protect should

63:25 very high. And that's definitely but aren't there nothing need a lot more

63:34 and expressions. Mm hmm. That's . But then you go back to

63:46 last ride. Next one. Yeah . Yeah. What does you know

63:59 ? So this is the gun. is using this big O notation.

64:06 , so this is identical to this . Just replace expectancy with H.

64:10 . Tells you how far away from . U. R. So I'm

64:16 uh basically just a notation difference and thinking of it as being approximating function

64:25 to see and using age as a of. And then there's the big

64:32 notation. Uh, that we will about talk about and use a

64:38 It's best to sorrow kind of ignorant what this is kind of sweeping it

64:45 the rug. But it and it shows that's how seductively how things change

64:52 age because of street and transfers power . And so that means for a

65:01 interval around see, you know, mind is within the dad the more

65:12 cool generally decreases possess. So I some of the best doesn't. So

65:22 particular finding exactly what this is but knows that as and increases this is

65:31 thing that I still take so quickly Arabs increases. It has been

65:38 This is the specific forms. If want to really captain America you have

65:44 put the balance and that was the example that wants to make things

65:52 The next term That puts 1st Otherwise you have to look at ah

66:01 continuing the example of the T. . You know that regardless of what

66:06 derivative is, they know that it's the XO wants to know x.

66:12 can put the bounds on what is the interval. I will get that

66:27 . So a lot of the I want to talk about differentiation and

66:33 later wrong is have expressions on his to understand how quickly who heritage pieces

66:43 a function of obviously dodging for approximate . So I want to use the

66:49 series expansion and try to figure out . Um he evaluated the great americans

66:57 instance, size. This is kind maybe DiMaggio Aaron the second pick two

67:06 function A you know A and And the corresponding function value. And

67:10 all starts with me. Well that's of one of approximating and if you

67:18 something of the conference that's that's the between main beats bad news. Ah

67:27 that sort of this is not perfectly because error is proportional to the distance

67:33 and otherwise you're doing the derivatives as well very much more quickly and similar

67:46 will happen in terms of very what we talk about integration methods? Well

67:59 the function happens to be kind of normal construction degree no, you then

68:09 there will be no error. So can use this notion of the

68:17 What functions Yeah, exactly. Strong . So this notion of order is

68:25 important. That one especially wants to with in terms of and that talk

68:33 is distance between integration points. That's happened this particular slide. Years of

68:43 . H when it comes to integration is maybe the width on these

68:48 So in that case you can see smaller the rectangles are the better approximation

68:59 taxes, you'll come back and show that there's a lot of places trying

69:03 understand but then it's just kind of kids with me. Okay,

69:15 This was about the alternating serious part I said that. Okay, so

69:23 was now I want to switch to elimination in the next probably repetition.

69:29 for all of it. Oh, any more questions or serious expansion.

69:36 it's important, you know, lead is in the book or other material

69:41 define, I'm not sure that the here is coaching is something that mhm

69:54 Safe from future, assume that that's . All right, can I say

70:06 based on the slide? There's something up but so um just a couple

70:14 motivational examples, they've never won. of the most of you are done

70:24 . Um first and slow and trying figure out carbon city network. Both

70:30 went away. Yeah, a very time to go on tour. Uh

70:37 in that case, trying to go and write down, it's quite changeable

70:41 the door, you know, the across resistance if that this simple network

70:49 the resistance from the garden goes through and it will follow the rope around

70:53 come back to the same point. the rest of getting the question that

70:57 up in that place and also a , you cannot a bunch of

71:05 they should try to figure out what currents are. Um and they kept

71:10 to the current from this resistor is this current and this, but you

71:18 figure out the current in the you know, the resistance in No

71:23 wants about this from the battery. make sure that things would settle in

71:28 the currents will do what they are the system of equations with politics on

71:35 battery and the resistance in the You'll find that the current so about

71:42 , writes the best. But here's you're like when you're on it.

71:45 bestest so expanded would not be the current in that particular look and it

71:53 the current that flows piece wires. these other resisters depends on the other

71:59 and the other circles about the systems . So only for um this nuclear

72:12 that can identify the terms, but this case. Okay, so 300

72:16 batteries. All the specifics. Thanks doing the next one though. The

72:22 ones have no batteries. So they basically the seer of all things.

72:27 you go around, there is no force. So let's see if we

72:34 figure it out that we have somewhat . Um, ah Because it's a

72:45 up there and says 15 here, shouldn't be. But I have gone

72:48 this. Your sister would do with resistor with sex and things like these

72:52 terms. So you can test So therefore kind of loops that you

72:59 in this particular circuit. So that's planned for each one of the loops

73:04 terms of the car. So now have a set of equations and let's

73:11 . So these things that we can out what the currents are. So

73:15 is just an example, what's informally you go, son. Just some

73:22 equations to figure out what they want part. Um so yes, Now

73:36 thinking, okay, so 15 comes the fact that X one runs along

73:41 seven plus two is nine per six this year's assistance for X one.

73:46 and then we have the other So we have six is For X

73:50 , which is this time going through minus peoples and force in the opposite

73:55 . And then we have to root the extremely interesting. So I was

74:00 following the loop and adding up the for them current domestic service and then

74:07 get this question. So besides that , so I suppose not about circuits

74:13 . Um so here's another one where kind of more hello serious things.

74:22 once upon a time this was actually was involved in. But firstly to

74:30 up also solving linear systems of My my look at Select a minute

74:36 for instance, radar profile of aircraft can get huge systems of equations but

74:45 need to solve. And so so this is just, yeah,

74:57 a bunch of examples. Okay, that's something. So a lot of

75:05 . So my best general reform is like to see it's a linear system

75:11 . Again, the politicians from the example is what a contains an action

75:18 case with the current So I'm working that system of equations. Former former

75:27 equals three. And of course uh you can just say okay, just

75:33 by the inverse of it on left right side and then texas solution but

75:40 you don't want to form the interest that's not operations. Um, so

75:48 very most basic method is called elimination now just to do that. Um

75:57 not necessarily the most commonly used patterns various least and it's not that but

76:05 in many applications okay is not for corporate Dennis magic supplements uh many

76:14 Most of the coefficients And the natives are known to be zero and then

76:21 main using the social emulation may not computational position but anyway, those elimination

76:30 for example all the questions that the of direct methods um and the attention

76:37 the property that they need to go the whole procedure for you know anything

76:41 the whereas you just think that this the property that against normally successive approximations

76:50 the solution and not the model chronically improving but the principal interview installations and

77:00 have a way of that or I also found the error. That's

77:04 function of the number. So we'll about that. We need her.

77:10 the one following the book of in later chapter. So we just talked

77:15 ourselves in relation as an example of separate for the moment. So and

77:25 as you may remember what viciously happened make up for flights next. There's

77:35 implicit of work too. And we're is while in fact construct two other

77:43 and then use such that their product equal to the matrix A where the

77:47 is what's known as lower triangular and . No, that's up for

77:53 What they don't talk about anything through direct methods, but just a householder

78:00 , but it's also very commonly And another one is not, let's

78:04 expectations. And those to Memphis has set of properties. The elimination and

78:17 properties in the sense that um, are still preserving, so to

78:27 So things doesn't blow up, which can do is one used cars in

78:33 nation and I'll give examples of that let me see if I can do

78:41 more before they need to quit because getting there. So anyway, suggestion

78:49 when it does work without having to too much. And when systems are

78:54 known as selective positive definite, it that the errors are actually London and

79:01 it means is stagnant or dominant and definite is the predominant means that the

79:10 value of what you have on the or not. And it's larger than

79:14 me throw our columns. Some the of the elements in rows of

79:21 Well, let me just remind you before we end this today. So

79:26 was the guards in the nation. right, well, let's figure it

79:31 there is that can manipulate this um a way that preserves the solution.

79:39 manipulation is in this case produced the job so called favorite role. So

79:44 think some multiple of the first goal subjected multiple of the first two from

79:50 the other roads And they choose to on the first goal is to multiply

79:54 by two And subjected from the 2nd . Then the entry in this role

80:00 here in this column Become zero and you have the updates for the

80:05 Um what it was, it was minus two times of efficient Tuesday,

80:10 minus 42 times two is four, . And then they do a different

80:16 . So I looked at the third here, I want to buy the

80:20 over the half and then three et cetera. So after you've done

80:26 , you have eliminated the countries in first column below the first job and

80:33 your best to have a smaller struggling is. Now this little problem

80:39 So the rest of the stuff you of forget about this first growing and

80:42 you want to work on this and you repeat the same procedure, Take

80:45 multiple of this role and at least interests and then you get the smaller

80:50 then you work on the smaller and so you get down to having something

80:57 looks like this. So when you done this procedure. So now this

81:03 sort of factory ization and formal elimination it's known. So now you have

81:09 equation here with one unknown. And you can solve for X four and

81:14 you know X four you can stick into this. Are there any

81:17 And if you do then in this you will have a new unknown and

81:22 has no. So you can solve extreme this and they use that also

81:26 these two etcetera. So that's the substitution. So that's the way they

81:30 some elimination works. And this is example there plugging in numbers. This

81:40 just a simple code. I got so I will continue to fight for

81:46 life. All right. The pitfalls other things. They've got some mhm

82:07 of 10%. Yes. Great. a good question so far here I

82:47 . But very

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