So there is another type of problem and I'll introduce that by an example. Splits the briquettes problem with variable endpoints. Okay? So what it means is the following. Again, I'm going to draw the y axis. You fix one end here, and you pick another point, and then you have a parabola. It said draw a parabola. Let's actually use a different color if I can. Okay, this is, remember Y axis pointing down through this as y equals x minus three squared. Okay? So the problem becomes the following. Let's use a green color. Ok? So you're going to draw curves. One end is here and the other end is on this. What is at the origin. And the other end is on the parabola. So you're allowed to choose this. And you're allowed to choose. This could be some, any curve you like. Find the curve from a to be parabola. So that the bead slides from a to the parabola in police type. So I'm giving you a little more freedom actually, last time, things had fixed endpoint. Now you're allowed to move your end point b on the parabola. What do you want to do? What's the shape of the curve? And where should the point b? What should I choose the point B2B. Okay, so there are two things here and what's the shape? And where should I go to athletes going backwards, actually, which point B should I choose? And then what curve should I, should it be? Is the question clear right? Now? You can answer part of this question based on what we've already done. Can anyone tell me the shape of the curve? Which family of curves it'll be. Okay cycler, why will it be the sigmoid? You can give a very good reason for it. Why should the shape AB will be some psych loaded with some constants? We can figure out the constants, we can figure out the location of B. But why will this Kirby a sitewide? You're clearly making a connection to the fixed endpoint breakfast or corn problem, correct? What's the connection? Can you convince me that? Because I know the answer to that fixed endpoint breakfast a cone problem. I know that this will be a psych Lloyd and me at the end of the day, aren't they kind of the same problems as once you fix your you're just solving that same thing anyway. Exactly, that's the point. So once you've figured out your b, then amongst all the curves joining a to B, you are picking the curve. It takes a list of time, so it's a fixed endpoint brushstroke Ron problem. Then, once you figured out where your b is, right, once you figured out where your b is, it's fixed. Then amongst all the curves, the shortest one is the brac has grown from a to b. And that we've already solve this cycline. So with the variable end point problem, the OD is going to be the same as the fixed end point problem. Only thing new will be. Where do you go on this variable curve? That is a new thing I need to figure out. Correct? So that is what I'm going to tell you. So I'm going to do a general problem. So let me write this down actually. So instead of actually writing this down problem, I'm just going to just do it. So here's a general problem. Okay? So I have some point X1, Y1. Some point x2. Now, okay? So I'm going to vary both left hand endpoint. Okay. So let me redo this and stick to the picture I have. Yeah. So I was using red for the two curves. So what it is is suppose I have two curves. Ok? This curve is given by the equation g of x, y. Let's drop the access. I don't need the access. Some curve g of x, y. Okay? And there is another curve. I call it h of x equal to 0. And what you have is, you're allowed to look at curves. One endpoint is here, and the other endpoint is there. Ok? And the left-hand point has coordinates x l, and then you don't know what the why is. And the right endpoint critical x r and y x bar. Okay? Until the problem becomes maximize and minimize some integral. Now only thing is I don't know the left endpoint and the right endpoint. I don't know that. Okay. Subject to okay. Yx is on left-hand point and right endpoint. Unknown. Only thing is that this left-hand point is on this curve, g, x, y, which means the left-hand point is 0. And the right endpoint is on this curve, H, x, y. Sorry, I can write this properly. So the right endpoint is undamped. Okay? So these are the new things before the endpoints will fix, but now these are flexible. Okay. So as I think it's Andrew who said the OD will be exactly the same because the moment you have located the points, these two points, then amongst all the curves, you will choose the curve which optimizes that's the same OD. Only thing is I need help finding the XL and the Tsar. Okay? So theorem, optimum curve is solution of same, same thing except exactly this Him. Along with this condition, these are the new conditions. F minus y prime, F y prime t sub y minus F y prime g sub x is 0 at the left endpoint. And the right-hand point, the function h plays a role. And of course, be there. The points lie on the endpoints with g of x. Ok. So it's the, which allow you to determine next Helen. Okay. There's the same OD. Okay. As I said, I'm not showing you any derivations, I'm just showing you what the result is. Alright? So this is the new piece of information here. This stuff. Let's just backtrack for a second. So same problem, same type of integral. Only thing is the left and the right endpoints are unknown. Okay? All I know of that is that the left and the right point points are on. This curve and this curve. That's all I know. This equation. These three things help us. These two things allow us to determine the endpoints. Okay, I have five minutes. I'll start the problem, but I will continue to read the notes and I will finish the problem there. So let's do this for the record. I was going to do a specific problem. Yeah. This one. Find the ROC is to Chrome who with left and point at 0 and right end point on this parabola. So I'm going to actually write it another way. I'm going to write it in the form h x y equal to 0. Okay? And the curve that I'm looking for is because Excel is fixed. This is x and y are, I don't know what that is. Okay. So for this problem, the tie was. Okay, it was minimize the function TY, the time taken, which was from 0 to x, right? And the time taken was square root one plus y prime square. And this is Y minus Y1. Y1 was 0. Ok, that's what it was. This was the time taken. 10 heroes. Hero. Her right endpoint. Right endpoint on the CRH. So your fix y prime. Is this. Okay, independent of x case. We've already done this. So can you get, you've got this, if you recall, is a times phi plus P. Y is a times one minus cosine phi. And phi is between left and right. Okay? And then now, still take one minute. So XML is of course 0. Okay? And to determine r, we use this condition here. This one. Actually I'll stop here. This will take time so you can read the notes. It's all written there. I'm going to actually rewrite that a little bit, but it's there in the notes. Okay? So you can start working on these problems even with the variable n points, right? Okay, let me stop sharing and let's do a quick well, if you don't mind. Okay? Alright, good. Please. As I said, you know, do the homework problems and really do contact me, alright, if you have any issues. Okay guys, I should stop recording also.
Variable end points - general theory
From Rakesh Rakesh October 22, 2020
8 plays
8
0 comments
0
You unliked the media.