| 00:00 | Yeah, mm hmm. Now we go so and I start to talk |
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| 00:08 | uh huh. Well, normal is particular data serious was mentioned last time |
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| 00:15 | a way of um figuring autopsy manipulate in order not to lose significance in |
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| 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 |
|
