Welcome to class on nonlinear control. So, let us go back to where we were. We had started talking about feedback linearization alright and we were actually doing a couple of of course I mean we introduced some notation yeah which probably was a little bit complicated, but then we also did an example. So, hopefully you know there is a little bit of clarity and you also hope that some of you did go back and actually try to look through this material. I posted it already on Moodle yeah. So, the lectures are already on there. So, this is what I have been using and I plan to use subsequently also alright. So, the only two real notations and then of course extended versions of this there is the Li derivative which is basically the derivative the directional derivative of a scalar valued function along a vector field and then there was the Li bracket which basically takes two vector fields and gives you a sort of a solve a skew symmetric operation yeah it is like a dgf minus dfg and this kind of an operation. And then there was of course the add notation which is basically just a different way of writing iterated Li brackets right. We had used this to of course look at systems of this form with output which looks like this and we want to sort of come up with this diffeomorphism this transformation. So, if you remember the feedback linearization is two pieces there is a state linearization or a state transformation and there is a feedback which uses which is then finally used to linearize the system or make the system look linear yeah I would not say linearize yeah. So, the idea is you just take successive derivatives of this output yeah and you try to see in which derivative the input shows up and that is called the relative degree yeah and that is basically qualified with this sort of expression yeah because as you take derivatives of y the time derivatives of y you will start seeing the dynamics again and again and again right and that is what happens you have y is h and then you have the first derivative is Lfh if you assume that the you know that that Lgh is 0 right similarly the second derivative is becomes Lf squared h if you assume that Lglfh is 0 and so on and so forth yeah and this does happen because you assumed relative degree R. So, this happens until k equal to R minus 2 ok after that you get something nonzero all right. So, notice that we are still working with a scalar valued control ok this is a single input system yeah the idea is again work for the multi input case but this things is become a little bit more complicated yeah. So, easier to work here with the single input case ok alright. So, so that is the idea we essentially try to compute the relative degree by seeing how many derivatives of the output you need to get to the input right and these then say sort of say that this is how this is the part of the dynamics that you can actually linearize by a feedback ok. So, this R the dynamics of size R is what you can linearize. So, it is like a partial feedback in addition and the rest of it is some additional dynamics ok. So, in order to get to this sort of nice linear sort of structure we need a few inequalities that is in order to verify that these things this y equal to h its derivative with second derivative they do form new coordinates we need some equalities. The first one basically said that if you have these qualities to be 0 that is Lglh Lglf Lglf square all the way to Lglf k to be 0 then it is equivalent to saying that Lgl add fg L add kg all the way is 0 ok. These two are equivalent ok. So, and the key the very, very key identity that we used to prove all of this is basically this guy ok this is what you need alright this is what you need. So, I can even highlight this yeah. So, this is really what you will require in order to complete all of this proof we actually looked at the proof ok. So, once you have that this sort of an iterated relationship yeah there is an equality that is between Lglf k to L add f kg yeah there is a similarity here there is an equivalence here you can actually start talking about independence of these vectors ok. What are these vectors this is just the Jacobian corresponding to this new coordinates that is the new coordinates are h Lfh all the way to this Lfr minus 1h ok. And so, if you take the d which is just a partial you get a Jacobian structure alright. So, because dh is a row vector d Lfh is a row vector. So, you essentially get you know a Jacobian and we used we want to prove that we want to prove that this is linearly independent ok these are row vectors are in fact linearly independent. And how did we do that? We did that by looking at the multiplication of this guy with some other vector ok which was very smartly chosen ok. Why was it smartly chosen? Because once I do the multiplication here we actually did it very carefully right once I do the multiplication here you can see that every row basically just contains you know things like this this guy right and then this guy and so on and so forth. So, basically you start seeing a lower triangular structure alright why this is because of the first equality that we proved ok this is just coming from the first equality right. In this case you have this last term to be non-zero in the second row you have last two terms to be non-zero and so on and so forth. So, you have a lower triangular form alright and it is well known that the lower triangular form whatever is the number of non-zero rows is the rank ok. So, the product of this matrix these two matrices is an r by r matrix and we have proven that r by r matrix has rank r ok which means individually each of these guys also have to have rank r right and therefore these are linearly independent and these are also linearly independent ok. So, you simultaneously proved that all of these are in fact linearly independent ok alright. So, this is where we were yeah of course with the linear independence you once you have this linear independence you can construct this change of coordinates right and since this gives me only r number of coordinates I actually add a few more what is how many more n minus r more because I started with an n dimensional you know state space. So, I have to go to another n dimensional state space. So, I have this r and then I have this n minus r where these phi are simply chosen to make sure that this whole map phi capital phi is a diffeomorphism ok which essentially means that the Jacobian is invertible ok has to be an has to have an invertible Jacobian right. So, that is essentially what it this says this guy has full rank ok yeah full rank means invertible in this case yeah because you can see that phi is its n dimensional x is also n dimensional. So, when I take the Jacobian it is an n by n matrix right. So, full rank means it is an invertible matrix your del phi del x or d phi however you want to write it is supposed to be an invertible matrix. Now, in order to help with that we already know that this the Jacobian of this is invertible or full rank sorry not invertible in this case maximal rank right why because of the lemma for 0.2 right lemma 0.2 basically just said that that this is maximal rank or this is rank R if these rows are rank R then obviously the Jacobian of this is exactly this guy right and that is also rank R ok. So, this helps us obviously. So, lemma 0.2 is what helps us in claiming that these are in fact independent coordinates this is what it means for the coordinates to be independent I mean we are not used to I mean we are not used to this because we just say that x 1 x 2 x 3 are our coordinates and we think they are independent yes because they are in orthogonal frames right. So, we never think about it, but now if I transform this x 1 x 1 x 2 x 3 non-linear to some other function other 3 functions ok.  Now, what is the guarantee that these new coordinates are in fact coordinates that is the linearly independent ok the only way to claim that is by checking the Jacobian ok you have to take the d phi ok and then you have to see that it is full rank right and if it is you are good and that is really what we are trying to do. So, because of these are independent coordinates because it gives me maximal rank not full rank in this case, but maximal rank. So, all I have to do is to make sure that the d phi of these that is partials of these are also linearly independent ok that is it ok. And we actually looked at an example right this DC motor example right what did we do we we actually verified both the lemmas right. First is this equality lemma which is basically saying that you know this L G L F and L G L F all the way to L G L F R minus 2 is 0 which means that you want to prove that L add F R minus 1 is 0 ok sorry L add F R minus 2 is 0 ok. And that is what we did basically it is easy to see that L add F K is 0 in this case because what does R turn out to be what was the relative degree of the system the relative degree was just 2 it is very simple we just we did all the computations what did we do we computed L G L F first we computed L G H it was 0. Then we computed L F H which was theta x 1 x 2 then we computed L G L F H which was theta x 2. So, this itself came out to be non-zero right. So, this is just the first power. So, so basically you have R equal to 2 right because this 1 is equal to R minus 1. So, R is basically equal to 2 ok. So, we got relative degree 2 right. And so all we had to prove was that L add F 0 G H is equal to 0 and that is what we proved it is very easy already done yeah. So, lemma 0.1 was easy to prove then we wanted to look at the rank of you know the lemma 0.2 which is basically saying that the H and L F H which are our new coordinates. So, in this case H is basically your x 3 and L F H is theta x 1 x 2 these are the 2 new coordinates right. And we wanted to see if their Jacobian has maximal rank. So, we computed that right we actually computed the Jacobian it is d H is 0 0 1 because H is x 3 right and then L F H is theta x 1 x 2. So, d L F H is basically this guy and for this to be maximal rank we just need for x 1 x 2 to be non-zero ok. Then the way we chose our phi that is the third coordinate was just to make sure that I get a full rank Jacobian ok. And what did I recommend I well I thought I I just thought I will make the third row orthogonal to the second row because this first and the second row already orthogonal notice because the dot product of first and second is 0 already right. So, the first and second are already orthogonal. So, I just made sure that the second and third are orthogonal. So, to do that I chose this sort of a y 3 right. So, everything looks non-linear and of course now I know that if x 1 x 2 is not equal to 0 0 then this is a full rank matrix right. So, I am good to go this is a valid new set of coordinates ok. The other choice which I had sort of used earlier was taking the third coordinates as x x 2 minus b right. In that case I would have got 0 1 0 let actually I have written it here yeah corresponding to this I would get the third row as 0 1 0 yeah this is fine to the only problem is in this case we saw that you need x 2 to be non-zero for this to be full rank yeah. If x 2 is 0 then you see that the second and third rows are exactly the same alright and that is a problem ok. So, if x 2 is non-zero all of this works no problem this new coordinate also works there is value in looking at this we will look at we will see this immediately subsequently ok alright. So, now great we now are in a position to sort of talk about the transform system what does it look like. So, if you look at this this sort of new representation or new variables in which we are writing the dynamics ok and you start computing the derivatives right you start computing these derivatives this is basically well I mean this is written in terms of this guy right that is z is basically looked upon as all these coordinates these new coordinates are just written as z I used y that is not a big deal ok alright. So, this is just writing in terms of the capital phi this is not very useful to us just look at this part ok z 1 is basically the actual output of the system. So, z 1 dot is actually Lfh has to be because Lgh is 0 that is how we have been doing right and that is equal to z 2 itself because z 2 is Lfh. Now, if you take the derivative again of z 2 it is d dt Lfh, but that is going to be Lf squared h because again Lglfh is 0 right this is again how we have got the relative degree right this is by the relative degree assumption right. So, therefore, you have z 2 dot is Lf square h which is z 3 ok and you can keep on going like this you essentially form an integrator a chain of integrators ok z 1 dot is z 2 z 2 dot is z 3 and so on and so forth until you get to z r dot right which is now the last coordinate this guy. Now, when I take the derivative of this I will get what Lfrh plus Lglfr minus 1h right just by taking the standard derivative right and plugging in the dynamics ok. Now, you know that this is not 0 because of a again relative degree r assumption right. So, I am going to write this as some Bz plus Azu where Az is not 0 ok. So, that is what the linear part looks like right it is just a bunch of integrators and then the final state has some nonlinearity in its derivatives ok. So, notice I started with a nonlinear system which was nonlinear everywhere right in all the states probably yeah just like in the DC motor case, but now I have because of my state transformation I have reached a stage where I actually have linear integrators everywhere and only in the last state there is a nonlinearity but I will just define this as my new input V right and that is possible because U is actually V minus V divided by A and A is nonzero right. Therefore, I can do this assignment from V I can compute U no problem there is no singularity if you are anything because A is nonzero by my relative degree assumption ok. Therefore, I just had a chain of integrators here. So, this is the partial linearization that we have achieved ok you cannot do anything more whatever is the relative degree of your output that is the best you can do. If you can find an output for which your relative degree is n then this entire thing will look like a bunch of integrators ok. I hope that is clear that if relative degree is equal to n then this is just all going to be looking like a bunch of integrators ok. So, you effectively completely linearize the system ok, all right great. Now, for the rest of the dynamics if we assume that the phi is ok until now that the way we were choosing phi was just to ensure that you got a diffeomorphism that is you got Jacobian full rank which is what we did here right. We just gave a row which sort of made sure that you know this is a full rank Jacobian and from that I went back and constructed the additional phi states right. But here we are saying suppose I have another assumption on phi that in the phi dynamics the control does not show up ok. So, that is the assumption. So, that is what is the normal form. If control does not appear in phi i dot ok, if the control does not appear in phi i dot equations then that is how you choose the phi which means what does it mean? It means that l g phi i x is equal to 0, is that is the way control will not appear right because this is the control vector field right.  So, if l g phi i is 0 then no control appears in the phi equations right and with such choice of phi if you wrote this dynamics you have what is called the normal form ok. And the d dt phi i is l f phi i and we just call it q i z because q i is just a new notation because now we are writing in terms of the new variables z that is all alright. So, what do I have then in the normal form? In the normal form I again have this bunch of integrators that does not change here control does not appear anyone ok. Control does not appear here because I choose the phi's in a smart way right. So, that phi dot does not have the control alright does that make sense ok. Now, if you go back now to our example ok you look at this choice of y 1, y 2 and then y 1, y 2 is what they are we cannot really play with them yeah we chose this guy what is y 3 dot this guy yeah but control appears in that equation right. So, this choice of transformations does not give me the normal form ok does not give me the normal form it just gives me a linearized form, but it is not the normal form usually in feedback integration control theories prefer to work with the normal form why because it is just nice right you have a non-linear system right whatever non-linearities there are, but there is no control in that, but then you have a linear system sort of driving this non-linear system alright and this driving system is linear ok. So, you can do a lot of things with the driving system right remember again with the cas go back to the cascade idea right there was a driving system or a driven system. So, here this will become the driving system and the output of this will go to this right because we will look at it how that happens yeah, but the point is there is non-linearities here with no control and then there is this linearity here where you can control these states very well basically you can make sure these days do whatever you want yeah because it is linear and has a very nice structure ok. So, this is the normal form what I am trying to say is what we chose as y 3 does not get us to the normal form this on the other hand gives us the normal form why because if you look at z 3 dot ok what is z 3 dot z 3 dot is just x 2 dot right what was x 2 dot no control no control in x 2 dot ok. So, but remember this was a restrictive choice right because this was a valid choice only when x 2 is non-zero ok this is a valid transformation only if x 2 is non-zero ok. So, it is a slightly more restrictive choice here we had the freedom of having either x 1 or x 2 non-zero anything being non-zero was good enough for here, but here no we definitely need x 2 to be non-zero, but although this is a restrictive choice it still gives us the normal form ok which is why this is also an important sort of transformation I think that is what I wrote here yeah this this transformation for the DC motor system is what gives you the normal form ok it gives you z 1 dot is z 2 z 2 dot is this guy and z 3 dot is this no control here notice the control does not appear here ok alright. So, that is how you do it you get to the normal form in addition to trying to get to a diffeomorphism you make sure that the in the derivative of the extra states or the phi states the control does not appear alright. So, that is how you get to the normal form alright. So, now you have essentially what is called a partially linearized system yeah because like I said with this choice of v you have the z 1 to z r states being a linear system right just bunch of integrators actually not any linear system, but a very specific linear system and then you have a bunch of non-linear systems alright we will see what can be done in these cases yeah how to control in these cases, but before we do that we want to define the notation of a 0 dynamics alright. So, we are denoting by psi these partially linearized states which is z 1 to z r and by eta the rest of the states ok just for notation sake. So, the psi dot system is again in this form integrator form right. So, in this integrator form yeah. So, and the eta system is some non-linear form right we do not know what it is, but it is some non-linearity the only thing that we know is a because of it is a normal form the control does not appear right. So, it is some function of psi and eta. So, what we have done is we have split the z states into psi and eta states ok. So, I have split the z states into the psi states and the eta states that is all yeah the psi states correspond to the linear part and the eta correspond to the non-linear part yeah. Now, suppose that the output is identically 0 then all its derivatives are also identically 0 right which means psi is identically 0 ok. What this does to the psi dynamics or the whatever I mean the way. So, if you look at the z r dynamics. So, z r dot is also 0 ok and that essentially is equal to this guy I mean it was just plugging in psi equal to 0 in this right hand side yeah no because this is this can this is depending on all the states see if you go back here let us go back here all this entire state right entire z I cannot control that that is coming from all this L F H and L G H that you cannot control it is not just depending on these states yeah it can depend on both. So, all I am doing is I am splitting the z into these two pieces alright right. So, anyway this is anyway too much detail to get to the basic point. The basic point is that what you call the 0 dynamics is when psi is 0 here ok. This is remember this is the one on linear part and if you put psi equal to 0 here what you get is the 0 dynamics ok. Why this is of interest why this is of value is because this psi system is the linear system right the assumption is that I can do anything with it. So, I can even drive it to 0 as fast as I want. So, if you remember even in the cascade case right what did we say that we had a stable system which is being which is an additive term which is the which is basically coming out of a passive system which is the output of a passive system. So, we were putting a non-linear stable system in cascade with a right you had this guy. So, here what did you have you had a stable system here right and you had in addition to it this basically this wise guy that was coming out of a passive system right. It is very similar to that here if I make this 0 right that was the whole idea right. If y is 0 right then this system is just z dependent and this is a stable system right. But if but we also know that I can do nice things with y because of passivity in this case in this case it was passivity and that is what is driving this system. So, similarly you have this idea that this size states not passive in this case but they are basically a linear system coming from a linear system. So, the assumption is I can do whatever I want with it therefore it is important to actually study the 0 dynamics that is what happens or how does this system the non-linear system behave when the linear part goes away or decays or dies down to 0 what happens ok and that is important alright ok.


Senator Bernie Sanders and Representative Alexandria Ocasio-Cortez unveiled sweeping new legislation earlier this month that would impose a federal cap of 15% on credit card interest rates. The bill would also allow state governments to set interest rate ceilings even lower than the federal mandate. Naturally, Sanders and Ocasio-Cortez are framing the bill as something designed to help ordinary people, but in reality the legislation will only act as to reduce access to credit for low-income and other high-risk borrowers. Hello, this is Ryan McMacon and you're listening to Radio Rothbauer. Credit card companies don't attach high interest rates to credit cards because they are mean and cruel. Credit cards with especially high interest are that way because the borrowers have been determined to be in especially high credit risk. Credit card companies want people to borrow money from them so if they can make loans at lower rates they will in order to undercut the competition. But these companies also must make sure they're likely to cover their costs. Thus the high interest rate exists to ensure the lender can make consumer loans while still accounting for the high risk of default by borrowers based on a risk profile. Given that interest rates are similar to a price of money, if Sanders and Ocasio-Cortez manage to slap a new limit on credit card interest rates they will be essentially imposing a price ceiling on credit cards and price ceilings are sure to lead to shortages. That is, they'll lead to shortages in consumer credit for high-risk borrowers, many of whom will be low-income borrowers also. If lenders cannot price their product in a way that allows them to recover costs they'll simply stop providing that service. Rather than face lower interest rates on credit cards as Sanders and Ocasio-Cortez imagine will happen, high-risk borrowers are more likely to not be able to borrow using credit cards at all. Given that default rates are generally higher for low-income borrowers, the cost of collecting payments is higher. Lending to high-risk groups then is only possible if the price of those loans is higher. Without the higher price, the service will go away. On the other hand, maybe this is exactly what Sanders and Ocasio-Cortez want. One way to claim to have done something about high levels of debt is to simply cut off potential borrowers from credit. After all, there is an implicit paternalism in efforts to place roadblocks between low-income high-risk consumers and the products these consumers may wish to purchase. In the minds of a government planner, the solution to the problem of people borrowing too much money is to pass a law preventing them from doing so. This, of course, is inherently unfair to those people who are, for now, in the high-risk category but who do pay their bills most of the time. They might simply be in their category because they are young and have never established much of a credit history. Moreover, many people who missed payment in the past may now be much more reliable and less prone to default. As people who fit a certain high-risk profile at first, they're likely to face high rates. One of the best ways these people can build good credit, though, is to first gain access to credit at high interest rates. Over time, they will increasingly gain more access to credit on better terms. Should these people then be punished and cut off from credit because they can't qualify for more moderate interest rates right away? The effect of the Sanders and Ocasio-Cortez legislation would be to do exactly that. Meanwhile, lenders who offer loans to high-risk groups are themselves being blamed for the proliferation of credit card debt among American consumers of all types. In his essay on payday lending, an issue very similar to that of high-interest credit cards, Tom Lieman analyzes the accuracy of these sorts of claims. Quote, The allegation that payday lending causes chronic or habitual borrowing may ignore the old adage that, correlation does not equal causation. As indicated above, it is a well-known fact that payday loans appeal to a clientele that face numerous financial difficulties, many of them self-induced, quite independent of the payday lending industry itself. Most of these households have failed to establish good credit, have poor credit histories, are not known for their timely bill-paying habits, frequently bounce checks, frequently change jobs, and may relocate often. In short, they are the type of people who are going to be frequently short of cash and who borrow chronically when given the opportunity. Because payday lending institutions provide them with this opportunity to borrow, when other institutions will not, does not mean that payday lenders cause their behavior, they simply provide an opportunity for this behavior to be exhibited more often than otherwise. Unquote. As a so typical of politicians, the answer offered by this new legislation, though, is to limit the options available to the most at-risk populations. A better approach is to allow freedom for both borrowers and lenders, to treat borrowers like adults and to not assume they are incapable of managing their own money. Thank you for listening to Radio Rothbard. Have a wonderful day.


So now we can talk about the main idea for lecture three. So what does it mean for two triangles to be the same? Or what does it mean for two triangles to be similar? So we say that two triangles ABC and DEF are congruent if all of the corresponding parts are congruent. That is the triangles ABC is congruent to DEF. If angle A is congruent to angle D, angle B is congruent to angle C, angle E, excuse me. Angle C is congruent to angle F. The side AB is congruent to DE. The side AC is congruent to DF. And the side BC is congruent to EF. And so if we think about the possible drawings we could get from such a thing, we say that two triangles are congruent, maybe something like this. If when labeled, so we have like labels, A, B, C, D, E, F. So we require that angle A be congruent to angle D. We have that angle B is congruent to angle E. We have angle C is congruent to angle F. So they have the exact same angles. And also in terms of side lengths, we have AB is congruent to DE. We have that CB is congruent to FE. And we have that side AC is congruent to DF. That's what it means for two triangles to be congruent to each other. All of the corresponding parts are congruent as angles and line segments. And so note that congruent triangles are essentially just the same triangle but possibly in different locations in the plane. So like the difference between ABC and DEF is just translation, I moved triangle ABC. You could also throw reflections and rotations in the mix there, but the two triangles are essentially the same although they might be oriented in different places in the plane, okay? A weaker notion than congruence is the idea of similarity. We say that triangles ABC and DEF are similar if just the angles are congruent to each other. So if angle A is congruent to angle D, angle B is congruent to angle E, and angle C is congruent to angle F, we say the triangles are congruent to each other. Excuse me, they're similar to each other. And we don't use the usual congruent symbol, we just draw a little squiggle to say that two triangles are similar to each other. Well, if two triangles are congruent, they definitely are similar, but it's not necessarily the case that two triangles are similar that they have to be congruent. For example, we could draw a picture like the following. So drawing back our labels here, so we have ABC and we have DEF. And I apologize that this picture is not perfectly drawn to scale, but you'll notice that angle A is still congruent to angle B, angle E is still congruent to B, and angle C is still congruent to F. So these triangles are similar to each other, but they're not congruent. You'll notice that the side lengths of the two triangles are different. The DEF triangle is much bigger than ABC. But even though the side lengths are bigger, they have still the same basic shape because the shape of a triangle is determined by its angles, not by the side lengths. And so when it comes to similar triangles, similar triangles have corresponding parts that are proportional. So AC and DF are proportional to each other. That is the factor which DE is larger to AB is the same as the factor as AC as well. And so what we see here is if we look at the proportions here, if you take AC, the length there and divide it by DF, this is equal to the same thing as AB divided by DE, which is the same thing as BC divided by EF. And these are all equal to some constant. So like maybe the triangle's twice as big or three times as big or one and a half times as big. This is what we get by proportionality. These ratios are always the same. And so because of this, similar triangles using this notion of proportionality is a very, very useful tool in trigonometry and calculus and beyond. And so I wanna demonstrate some examples of how you can set up proportions for similar triangles. So consider the two triangles you see on the screen, ABC and DEF. These triangles are similar to each other because their angles are congruent. A and D are congruent angles. B and E are congruent angles and F and C are congruent angles. So these triangles are similar. We know the three side lengths of triangle ABC. The side AC is length 16, side AB is 24 and CB is 32. But we only know the proportion for FD. Well, because they're proportional, if I take the segment AC over DF, this is gonna equal the same fraction for the other corresponding parts. So we get that AB divided by DE would equal that. And we also get that the last one AC over what's the corresponding side there, FD, these are all gonna equal each other. Oh, we already did AC. Starting again in three, two, one. And BC over FE, these are all gonna equal each other, some constant. And so let's fill in the information we know. So AC is 16, DF is eight. Which if you simplify just that fraction right there, you get a two. And so that's that constant K. This triangle ABC is twice as big as DEF. And so all the other sides are gonna satisfy that same proportion. So you get that if we take, for example, AB. AB was equal to 24. And if we take DE, which we don't know what that is, we get that DE like so. Two is equal to 24 over DE. That's to say that two times DE is equal to 24 or DE equals 12. So the missing side here was 12. The other one as well. If we just cut this in half, that's gonna give us the side length for FE as well. So that means the remaining side must be 16 because there's a factor of two proportionality between these triangles. Let me give you some examples that I use in my actual calculus class. You can actually see the links to these videos on the page right now. So these are genuine calculus problems. And I just wanna demonstrate not the calculus part but why similar triangles become essential to some of these calculus problems. So imagine a spotlight is on the ground and it shines on a wall that's 12 meters away. So here's our spotlight, here's the wall and the distance between them is gonna be 12 meters. So we know about that. So there's a man who's between the wall and the spotlight and he himself is two meters tall, which I remember from Jurassic Park, that's how tall a velociraptor is as well. So he's a pretty tall dude, not like unhumanly tall but he's on the taller side of things. So if he's two meters tall and he's standing between the wall and the spotlight, there's gonna be a shadow of the man on the wall. So like as you see in this diagram here, the light shines but as it hits the man it obstructs the light and so there's a shadow on the wall here. Let's call Y the height of the shadow. If the man is exactly four meters from the building, how tall is the shadow? How tall is the shadow here? And so if he's four meters from the building, that means since the spotlight and the wall are eight meters apart, he'd be eight meters away and so you can take 12 minus four, that's where this eight came from, just so you're aware. So what we wanna consider here is consider this triangle right here and I'm gonna use a different color to emphasize it. We're gonna draw this one in red. Let's look at the triangle that's formed between the spotlight and the man and we're just gonna extrapolate it for a second over here. If we just look at that triangle, then notice that this side right here is eight, this side right here is two and let me mention that this is a right angle. This is a right triangle because the man is standing upright. Now let's consider the triangle that's formed between the spotlight and the wall. So this green triangle over here, let's bring it over to the side. So if we draw that again, I'm just gonna draw it much bigger. The distance between the spotlight and the wall is gonna be 12 and then the height over here is y, we don't know what it is. But what we do know is that this is also a right triangle but it's not just that. We know that this angle formed from the spotlight is the angle in both of these triangles. So we have a right triangle which have two angles that are congruent. Then because the angle sum of a triangle always equals 180 degrees, if you have two triangles where two of the angles are the same, then the third angle must be the same as well. In a right triangle, the two non-right angles are necessarily complementary to each other. So because of that, the angle or the triangles have to be similar. If a right, if two right triangles share a common angle, then they have all angles the same, they're similar. So we can set up a proportionality argument here. So notice that y over two corresponds to 12 over eight. So y and two go together and 12 and eight go together. So y over two corresponds to 12 over eight. So if you times both sides by two to clear the denominators, well, at least to clear the denominator on the right-hand side, you're gonna, cause we're trying to solve for y the height of the shadow, you get y equals 24 over eight which eight goes into 24 three times. And we see that the height of the shadow would be three meters. Let me give you another example coming directly from my own calculus course. We have a tank of water that itself is a inverted circular cone right here. So it's kind of like an ice cream cone. It points downward like this. Although this diagram right here, we're looking straight onto it and we just see a triangle as this cross-section. The height of this cone is gonna be 12 meters. The radius of the cone is gonna be four meters as labeled right here. This water, there's water that's filled into the tank. And so the height of the water is gonna be eight meters. So the tank is not completely filled. The height is gonna be eight meters. And so particularly there's a gap of two meters at the top of this that we could add two more meters of water into this tank right here. So let's figure out what the radius of the water is at this specific height. So this unknown value is r. What would be the radius? Cause when you fill the tank with water, the tank, the water as a fluid will take the shape of its tank. So what's the radius of the water that has this conical shape right here? Well, again, we could make a similar triangle argument. So the first one I want you to think about is that we look at the whole tank. We could take this whole triangle right here but to make life a little bit easier, I'm actually gonna take half of the triangle. This is an example of an Asosceles triangle and we're gonna cut the Asosceles triangle in half which always gives you a right triangle. So let's look at this triangle right here, pull it off over here. So the triangle in yellow, this side length is gonna be the height of that triangle which is 10 meters. This side length is gonna be the radius which is four meters. So we have that. It's a right triangle and notice that this angle at the bottom right here is just this angle right here. Next, let's look at just the triangle involving the water and again, we're just gonna take half of it. Pull this over here to the side. We get a triangle that looks like this. The height is gonna be eight that's given to us. The radius we do not know. And again, this is the same angle in play right here. So these angles are congruent. Right angles are congruent to each other. So the other two angles have to be congruent to each other. These triangles are similar to each other so we can set up proportions. So this tells us that R over four corresponds to eight over 10 for which we can times both sides by four. This would give us that R equals eight times four over 10 which is gonna give you 32 over 10 which gives you 3.2 meters as the radius of the water at this point in the tank. And we could do this for greater generality, right? We don't just have to stop right here. What if the height is unknown? What if the height is say X meters at any moment in time? Well, we can set up these exact same proportions like we did a moment ago because what's so special about the number, what's so special about the number eight right here? Nothing really much. If we just said like, hey, what if the height is X? Well, then all of the subsequent eights have to get taken away. So you end up with a four X right here in which case then the general solution, let's start again in three, two, one. Suppose we wanna next find the radius R when the height of the water is itself a variable, okay? What if we allow the gap between the top of the tank and the water level, what if we call that X? So in the example we did first, X, if the height was eight, then X was equal to two, okay? So what if we allow that variable to come into play right here? Then the height of the water, the height of the water is just gonna be 10 minus X, whatever the gap turns out to be. And so that then leads to the proportion that R over four corresponds to the height which is 10 minus X all over 10. In which case then we can solve for R easy enough. We get R is equal to four times 10 minus X all over 10, four and 10 both have a common factor of two which you can cancel out. And so we see that the radius is equal to two fifths times 10 minus X, which we can then check our answer. If we take X to be two like we saw before that would look like two fifths take 10 minus two which is gonna be eight, two over five times eight, two times two, excuse me, two times eight is 16 over five which 16 over five is the same thing as 32 over 10 which is the same thing as 3.2. This gives us the answer we did before. And so this is the idea, why similar triangles can be very important in the calculus setting. Calculus cares about when we take these quantity and treat it like a variable. How does the system change? Well, no matter how you change the system it's always a similar triangle. So we can find a formula for the radius dependent on the height of the water or in this case the gap of the water. And it's always, this formula is always set up by a similar triangle argument. So for calculus students being able to set up and use a similar triangle argument is very, very important which is why we wanna talk about it in this trigonometry class.


And it's time for entertainment news. Actress and humanitarian Angelina Jolie spoke in support of the Senate's renewal of the Violence Against Women Act, her voice breaking as she acknowledged the women and children for whom this legislation comes too late. Almost a year ago, the House of Representatives approved the Violence Against Women Act, a law that strengthens domestic violence protections. It was originally sponsored in 1994 by then-Senator now-President Joe Biden, but expired in 2019, and Joe Biden campaigned on renewing it. She said she wants a solution that won't just be a political talking point for one side or the other, but a bill that can gain bipartisan support past the Senate and truly deliver for her fellow survivors of the life-altering abuses. I can think only of everyone who's been made to feel powerless by their abusers, by a system that failed to protect them, parents whose children have been murdered by an abusive partner, women who suffered domestic violence, yet are not believed, children who've suffered life-altering trauma and post-traumatic stress at the hands of people closest to them. Anyone who's been in those situations will tell you just how far they feel from the power concentrated here in this building, the power to pass laws that might have prevented their pain in the first place. So, folks, I'm a survivor. I know firsthand what happens when someone you trust abuses you. I want to come to a solution that won't just be a political talking point for one side or the other, but a bill that can gain bipartisan support needed to pass the Senate and truly deliver for my fellow survivors of these life-altering abuses.


In this presentation, we will take a look at budgetary controls and reporting. In a prior course, in prior presentations, we took a look at the creation of the master budget, a static budget, and the components of the master budget. First a word from our sponsor. Yeah, actually we're sponsoring ourselves on this one because apparently the merchandisers they don't want to be seen with us. But that's okay whatever.  Like our Accounting Rocks product line. If you're not crunching cords using Excel, you're doing it wrong. A must-have product because the fact as everyone knows of accounting being one of the highest forms of artistic expression means accountants have a requirement, the obligation, a duty to share the tools necessary to properly channel the creative muse. And the muse, she rarely speaks more clearly than through the beautiful symmetry of spreadsheets. So get the shirt because the creative muse, she could use a new pair of shoes. If you would like a commercial free experience, consider subscribing to our website at accountinginstruction.com or accountinginstruction.thinkific.com. Here we're going to be focusing more on the flexible budget, but we want to review some of the components of the budgeting process and then go into those components more in depth concentrating on the flexible budget. So the budgeting process will start off with the creation of the budget. We're going to have to compare the actual numbers to the budgeted numbers. So we're going to create the budget of course, then we'll have the actual period. We'll run by the time period, the month, the year, then we'll do the comparison. What did we expect to happen in the budget? What actually happened? What are the actual numbers? We'll do that comparison. Then we'll analyze the differences and then we'll strategize. We'll look at the differences, of course, the variances, and then we'll strategize based on those make changes based on those. And then of course, implement another budget. We're going to revise the objectives for the new budget. And then we're going to create, of course, the new budget again. This is going to be the standard cycle that we will be going through within the budgeting process. Of course, we plan, we budget, we project out with that budget. Then the time period passes. We compare what happened to what our projection was. We look at the differences and then we make changes based on those differences. Budget report timing, typically as we saw in prior course in prior presentations when just looking at the static master budget, we're considering a budgeting for a year's time period. And then we could break that down into quarters and then often into months. That would be the standard types of budgeting process that we would have. We'd say, hey, I'm going to budget this out into a year. We want to break that out into quarters so that we can then review each quarter and possibly break that out into months so that we can have a review each month comparing budgeted numbers to actual numbers on that basis. The budget promotes future focus, analysis, thought and direction. So remember that when we're considering the budget, we're focusing out into the future and we're getting that kind of vision statement out. Well, the vision is going to be the long term focus. The budget really takes that long term vision and puts it into digestible chunks, actionable chunks, yearly chunks, quarterly chunks, monthly chunks. And it allows everybody in the organization then to be kind of focused towards that immediate goal. So our operations from a day to day standpoint, then as we make day to day decisions, we can align to the immediate goal and have all of our conversations basically within the decision making process on a day to day basis in alignment with those future objectives. Communicate the plans to the organization. Once we have the budget put together, remember that that's going to be our communication tool. That's going to be our roadmap type tool that we're going to use to communicate to the organization. And that's the better the budget is put together, the more clear that roadmap should be. And again, it should be those actionable steps that we can kind of go to and help us to get to the long term vision. We'll have the long term vision and then be able to communicate what our action plan is in the short term in the next year, the next month in order to achieve the objectives that go towards that long term vision. Makes long term strategy plans into short term financial plans. So remember that we do have again, that long term type of vision. What we're trying to do is take that long term vision, which is something that's going to be seem like it's not obtainable really in the immediate presence, of course, because it's a long term vision. We need to break that down into immediate strategies, financial strategies because financial strategies are those that we can measure easily. And so we want to break down those long term vision into the strategies that people can then focus in on work towards within a specific time period within the framework of the long term vision provides a benchmark for evaluation. So of course, we're going to take the budgeted numbers and then we're going to run the process and look at the actual numbers and we'll have the comparison of the budgeted numbers to the actual numbers. We need to have that benchmark because if we do not have that benchmark, then people don't have anything really to shoot for. There's no way to evaluate how well people did. As we put that benchmark in place, of course, we need to be careful that we don't make the benchmark too low, too easy for people to clear, not too high so that it kind of discourages people. We want to basically set the appropriate type of benchmark that's enough that it's challenging, but hopefully achievable and therefore that will inspire people. And of course, then once the actual numbers happen, once the time period has passed, we'll do that comparison. We'll do that evaluation. Motivates employees through participation. Remember that the budgeting process should be a bottom-up type process, especially more and more for larger types of companies given the fact that we're going to have to get the numbers, the data, from the departments. So the departments are going to have to provide that information logistically. And it's also going to be a great opportunity for us to communicate with other people in the organization if we have more communication in the process of the creation of the budget, focusing in on the process when we're aligning and leading the budgeting process, meaning the focus isn't so much from the leaders on making the actual decision themselves, but in making sure that the process is in alignment so that the decision can be made well so that everybody communicates within that decision. Once the budget has then been created then, that process of the creation and the decision-making is basically done at that point and now the budget is over. It's a roadmap now. Now the communication is going to be there to communicate what the goals and objectives that have been decided on are through the budget. So remember that the budget-wide budget, the process is going to be promote future focus analysis, thought and direction for the organization, communicates the plan to the organization, makes long-term strategic plans into short-term financial plans, provides benchmark for evaluation and motivates employees through participation.


Oh, tonight we're making salmon patties, cream potatoes, and pinto beans. Get ready! We're making salmon patties, cream potatoes, and I'm heating up some pentos I made the other day. I've already got the on for my potatoes, because Amy is almost here, and she's drove for six hours, and I don't even have supper ready. Because I've been in there watching the news. We're making some double-queue salmon, pink salmon. I don't have any green onion like I typically put in mine, so I'm just slicing up a small portion of an onion, and I like to do it thin, because I don't like real big pieces in them. Okay, so what I'm gonna do is I'm gonna put just a little bit of liquid from the salmon in here. I'm gonna drain this over here, and I am going to leave the skin in this bowl. Now a lot of people eat it all. They eat the skin, they eat the bones, and yes, they say it's healthy and it's good for you, and it probably is, but I still take them out. I don't get every single one, but the majority of them I take off. And y'all can do what you like. Of course you can. That's the great thing about cooking and eating at home. You get to do it your way. I try to tell people there's no right way and no wrong way when it comes to what you like. Everybody's tastes are different. Some people like to have 15 spices in every dish, and then some people just like salt and pepper.  Delish, delish. All right, for a whole can. I've got a recipe in my first cookbook, and so we put in one egg, some salt and pepper, some self-rising flour and cornmeal, and then cayenne pepper, a teaspoon of salt, about a quarter teaspoon of pepper, a teaspoon of flour, cornmeal, that makes just put in a little cornmeal, an egg, and the onion. That's it, simple and easy, great way to cook. And now our salmon is around $4 to $5 a can. There are some that are, what's available in the grocery stores. So break this in here, stir it up. The egg is the only thing wet in here, besides the little bit of salmon juice that I left in there as well. Now, the flour makes some more like cakes. Texture's different. They're not just real fishy-tasting. They're really good. So let's put in our little bit of cayenne pepper, and I'm just putting in a little bit. Now put in what you want. You can put in that, you can put in some creole seasoning. You can put in what you want to, especially if you're eating these more often now that there's not a lot of meat at the grocery store. But it makes a lot. This is one can, and you can see that it makes quite a bit of salmon patty. I guess we could call it. Yum-yum. Let's get out here. Let's get a jingle, jingle jingle, because I have started my cream potatoes. These are some pentose that we're gonna warm up on the back. And this is already on the low temperature. I'm gonna add the oil to it. This is some oil that we use the other day to fry fish in, and I just strained it and kept it because I knew I would be making some salmon patties. So you're gonna let your heat put a little bit of batter in it, and when it starts to sizzle really good, then you know it's hot enough. And typically, with this electric stove, we've noticed that things get pretty and golden brown around number six. So I'll set it on six, and we'll wait till this starts to sizzle. Our potatoes are boiling. Our pentose are warming. So it's all looking good. All right, it's starting to get pretty warm.  You don't wanna make them real big because you want them to get done on the inside. And then I've had other people say, if you're gonna make a salmon patty, make it round. And let me just say, mama always just dipped it out with a spoon and stuck it in the skillet. So we didn't like, make sure they were perfectly round or anything. They all taste the same, right? Just how mama did it. And that's how I do it because that's how mama did it. Good supper. A nice fishy supper. So we're gonna let these get nice and brown before we flip them. It's gonna take a good, probably three minutes at least, okay? Them look good in there fried, don't they? They're nice and golden brown, I can tell. So we're gonna get up under them with this scraper. Flip them over. Well, they're not really. Sometimes you need to flip them with something smaller. They're gonna get a good handle on them. I'm gonna put some of the towels on it so that I wouldn't burn my sink with my paw. I do drain them. Potatoes in there, so I'm putting in three pats of butter. Tablespins, really. We're gonna put in some sour cream, do it while I'm mixing. And if per person, and if you do a good job peeling them and don't waste a lot of it, you really do have a lot of good potatoes. I'll put another pat of butter on there just to make it pretty by the time everybody gets to the table. So we're gonna pour up our pentose at the first of the week. And then I saved them for tonight for the salmon patties. Look how pretty my broth looks in those pentose. And y'all know my secret, a lot of y'all don't like my secret, but I think it's wonderful. Makes great beans. It's pentose beans, and I need some cornbread, but I didn't make any.  So this is a good supper, easy. Only takes a minute to make. And yes, this is how I eat it. I take me a bite of everything. Delicious supper. Homemade goodness. Good old Southern cooking. I hope you've enjoyed watching Card Valley Cooks, where we cook like mama did. Bye y'all, love ya.


Tyreece man Tyreece is uh he flipping out real good right now you know um Hey man, he's having issues up there with that what he was held in contempt in court Check it man. Check it. Hey man. Listen man. So a couple of things that's been brewing uh right now um The most important thing to do is like and subscribe to our channel That's more important than anything else man If you want to see more news you can use man, you gotta like subscribe share our content Make sure everybody know what's going on man Uh boss talk 101. We hear uh, it's a real thing man. Appreciate everybody that's uh subscribed Make sure you share our content make sure you go tap in with mr. Mako money modes is follow their uh platforms as well But let's get down to it man.  It's it's not for us It's not fooboo. It's not for us by us. That's for sure.   Tyrese you my boy. I rock with you Uh, I really love everything you do.  You my guy, but hey, listen, man They're not playing man.    This guy's been locked up for a minute now Tory Lane's has been locked up he inside that Jail sale now Do some push-ups Try to figure it out You're not going to be able to you can't get out right now.  It's not the time for that right now So Megan the stallion, you know, she's pretty much it's a campaign that says that now you look in real Like like you've done something even if you didn't or even if you couldn't beat the court system But just out speaking talking about the all alter to dna They they took the dna and these are some very very slippery slopes that you're trying to climb. So I've been caught eyes. I'm saying hey, man Come on man.   You don't want to mess that up.  Um, you definitely dope, man When it come down to your music, but that was a clown move You had two beautiful women or it was women around you everywhere money everything you had at all And I always tell y'all and I've been speaking this for a long time You have to make sure that you don't Basically turn around and and and mess up the millions of dollars that you're being blessed with Now you done messed that up. You had the women. You had the money. You had the power. You had the fame Now you got to sit down for a little bit and they ain't talking no little time So Tory, I'm telling you right now, man quick calling Talk to your your lawyer the appeal write your appeal court Throw yourself up on the mercy of the court, but make sure you don't talk Another again another brother trying to say too much let it go man, um I want to talk, you know one more thing I want to talk about man And this is something that I'm very very serious about and this is something that it messes with me But it's a celebration I seen ad I seen ace boy pun. I seen them pull out, you know, they they they they got over 100,000 subscribers I just interviewed ad last week He got a hundred thousand subscribers He was proud of that And what he done was he did his unboxing so he takes and get this right here He grabs his plaque see it right there He grabs his plaque and he said hey man. Thank you guys man. Thank my subscribers I appreciate y'all y'all doing a great job man. Thank y'all for you know helping me to get there and all that and he told the story Um, I'm proud of my people that helped us to get here too And so when I see that it's motivation To now we're trying to head for a million We got to get to a million our guys But the thing I want to point out is If you got the plaque Then it's official But when you you don't have a plaque and you've never showed it It's something to that because Like ad and like uh Ace boy pun did they're going to show it. They're proud of it And they're going to say hey man, I made it and that's what it's all about so that we can motivate others That they don't want to do the same thing But they deserve everything they get because they work it. I think uh t-rail They're part of they're at 90 something thousand subscribers. They'll be getting their plaque soon within probably this month They'll be applying for it. But this is what it's all about guys. No, listen man women lie Me and lie numbers don't and youtube is not going to send that plaque out to you again I'm going to send it out to you unless and the numbers is correct And a lot of people that got a hundred thousand subscribers do not show this plaque Because they know they didn't legitimately get it They they they basically fudge the numbers And so they can't show it to you because they don't have it because youtube has a real reflection of the numbers that are being Calculated but congratulations to ad Congratulations to them community them guys Dope like I said I interviewed him last week. It was just funny that he put his plaque out and started showing it this week Proud of you guys man. I hope we can do a lot more interviews together When i'm in la when you in texas when you guys are In houston or dallas or colleen or san antonio and shout out to the people that contacted me yesterday from uh cali man Love cali. Love louisiana. Shout out to gd again. He hit me up this morning. Love louisiana Just love all of the people that support boss talk 101 Hey, man, I can say subscribers. Let's go to this million. Let's go get to go We got the silver black now. Let's go get the gold one. Let's go get the million subscribers, man We can do it man together man right now hit that like button hit that comment button Hit that subscribe button. Let's go to the top man. Thank you so much, man This is the news you can use man. Appreciate the love. Appreciate it. Man. Hold up