 Hi there folks! Sorry about that. My internet has decided to die exactly at 8 a.m., which is super fun. Okay, so I'm doing this on a data plan. Hopefully this won't cut out at any point. Yeah, just maybe let me know if there are any thankful issues during the interview. Just let me know in chat or just feel free to shout out. Alright, so let's jump right into things. I will share my screen. Cool, so just some quick announcements right to the pre-classes sort of going on as usual. Just laid out an order pretty much for the rest of the semester so you can just work through those. They're all sort of available on Gradescope right now. So if you want to try to work ahead a little bit on any of them, if you have extra time, then you can do that. Just kind of clear as many out as you're able to. The rest of the semester is really just everything we're doing now, all of this trigonometry. I think the last section is 5.2 or 5.1 or 5.2. I think a lot of that is simplifying trig functions. So really, I think you're probably equipped at this point to do all of these kinds of problems. It's just a matter of practicing some specifics. Sorry, I just realized my volume is super low. Alright, so pre-classes, yep, just have a number of them left. Let's see, in terms of the project, I think I kind of sent out sort of a status summarizing where you should be. I think what we want to do is go ahead and have some kind of rough draft due, I think I said at the end of this week, maybe Friday. Let me just pull up the announcements so I can double check the exact dates. But just as a rough idea of where you should be at this point, hopefully you've met with your partner and at least kind of discussed your understanding of the project and sorted out if there are any parts that you don't understand. This would be the week to get any questions that you have answered. So I'll be in office hours later today and I'll try to hold some extra office hours throughout the week. And this would be the best time to just clarify things. Hopefully you've read the project handout. Just see if there's anything outstanding that you don't understand about the project at this point. Sort of what's required and what needs to go into it. So we'll talk about it today more in more detail in Thursday. So hopefully you've already met. If you haven't met, then at this point you probably are a little bit behind. So please do try to meet with your partner ASAP just to avoid missing easy points on the project. If you end up behind on these kind of things, then it's hard to absorb the finer points we get into in these later. So get closer to the things that we discussed. They're meant to be more fine points on things you've already looked at. Hopefully I want to discuss the structure with your partner a little bit. Just going back to the announcement on ELC here. Just how many overall sections do you think you want? You definitely have to have an intro section, some kind of middle piece and some kind of conclusion. So there should be some kind of like macro structure to your document that you've sorted out so far. But you might want to also decide like in that introduction, do you want to combine this like explanation of the physical situation and just make it one sort of introduction that covers everything. You want to break it down into two pieces. Some people have done this for previous projects where one of them is an introduction and discussing kind of what the goals are and what the mathematics you'll be doing will be. And then there's a separate section for like the actual physical situation. You want to discuss a little bit how you want to break up this middle analysis. So we're discussing some stuff in class. So there's some stuff from last class, some stuff from today and some stuff from Thursday. These will definitely need to be in the project in some form or another. And just a quick disclaimer, if you're doing the derivations for stuff that you have to include in the project, you really do have to walk through these steps yourself. So that's really not super great to, you know, if we just like go through something in class, we just get to the final answer. Like in your project, you don't want to just like start with the initial thing and then say, okay, and here's the answer. Like last time we did some like derivation of solving for M and this kind of stuff. And I think a lot of people for Project 2 just said, okay, here are the initial equations. Here's what happens when you solve for M. But this being a math class, that's kind of the part that you really need to show or like that's, you know, that's one of the more important parts. The analysis is important, but being able to walk through these derivations yourself is also really key. But yeah, so discuss with your partner how, you know, how much of that analysis you guys understand and make sure to try and get to office hours. If you don't discuss how you want to break up this intermediate analysis section. So in our case, you know, we at least have two things to look at. We have like one situation is where this leaf is stationary in one situation where the leaf is rotating throughout the day. So maybe you want to do one now one analysis of both together, maybe you want to break this into two subsections. Maybe you want to do one and your partner do that's the other that kind of thing. Definitely have have these discussions. So ideally that's where you'd want to be at by today, having met with your partner, having discussed the structure and having discussed your understanding of this analysis in the middle just based on what you've read in the handout which hopefully you've read at this point. By Thursday, I would aim to have some kind of rough draft completed based on just what we discussed today will go through a lot of the analysis you need for the stationary leaf case. But there are some decisions you as a group have to make. We talked a little bit about this there's going to be coming up with some kind of angle function, some sort of periodic trigonometric function that models this like energy density throughout the day. I'll say a little bit more about that when we're talking about these details. But there's some choice here that you have to make. It's kind of a difficult choice, I think. Like you have to now turn all over the math that we've been doing on its head and you have to like yourself come up with a function to model this thing. And I think that's that's a little bit tough. Yeah, by Thursday, you want to have some of this analysis for the stationary leaf at least copied down written out. And make sure you understand it and then make sure you get to office hours feed help with the derivation. And then I would also aim to like have, you know, just things like the abstract introduction. I usually recommend writing these last, because you kind of don't know how much you need to introduce or what the summary is until you've already done all the work. I think in this case, it's maybe easier to have an abstract or an intro or both to have a rough draft for them. I think they're pretty easy to write in this case, just describing the situation overall, there's not really too much to say about those. It's complicated as project to a project one, in terms of the physical situation. But maybe just have a rough draft of those and then after you're done with the project come back and do a final final draft of the intro abstract sections. And then by Friday, go ahead and turn in something on great scope should just be some version of your project which hopefully includes abstract introduction. And this analysis of the stationary leaf. And some indication that you've started looking at the rotating leaf, which is what we'll be talking about on Thursday. And I think that's pretty much it and then we'll try to go ahead and have the final draft you next week. How are you going to have an actual sort of fall break of some sorts anyways. Alright, so that being said, I think that's pretty much it for announcements. I'm going to say stuff about office hours a little bit later too. Okay, so. Professor Garza. I saw in great scope that there was like a pre class and like work sheet do like Thursday of next week and I was wondering if that's still, if we still have those assignments to you. Right because we have holiday break starts Wednesday, I think. Yeah, that's why. Yeah, I'll probably go ahead and move those. Okay, thank you. You know which, which assignment it was off end. No, but I'll look it up right now. Okay. Yeah, if you want to just drop it in chat or something that's, that's really fine. And so whatever the pre class and whatever the worksheet was for that I'll probably just push it back to the next week. It was the 4.5 b worksheet. Yeah, it was like the pre class and I think the worksheet too. Yeah, so I'll try to just push everything back. Make that do the following Tuesday. Yes. Okay. And so we see what we are looking at today. So I think today is probably one of the more, more fun days we just get to look at a lot of a lot of demonstrations which is, which is nice. I just want to remind, set out some reminders of things that are kind of important at this point. So maybe want to review. So things that are coming up on pre classes are these special angles. And I just remember that there were a few, maybe like a handful of these that we want to know. Maybe it's actually better to say special like reference angles. It's kind of the best way to think about them. So these were all of these angles in the first quadrant. I won't list them out. I'll just say in words you can kind of go back and look at the notes from a previous lecture, if you want to see a discussion of them. And the important point is that there is huge complicated diagrams on Google where they list out like every single special angle on the circle. And there's something like a 69 plus four 13 of them. And the important point there is do not memorize that circle. I think that's something that's pushed in a lot of trigonometry or other pre calculus classes. The idea here was that we had this sort of flipping process where you kind of pick any angle you want, you find this reference angle. And this reference angle is going to tell you what triangle to look at in the first quadrant to sort of determine sine and cosine and all of that. And then you can just kind of flip things around change the x and y coordinates between plus and minus to get the right thing. And I think to memorize, I think it was six angles, you can go back and look at the table but it's essentially in the first quadrant, you know, something that's like zero radians. And then you have by six by fourths by thirds. And so I guess that's only five to zero by six by fourths by thirds, and I have. I have to emphasize that we really do want to memorize what the values are for these or have a way of like easily remembering how to come up with them. And just remember we had this table trick where you can kind of list out sine and cosine and kind of do like square root of one over four square root of two over four, kind of work your way up for the the sine values and work your way down for the cosine values. How do you remember which one is up or down we have this. And this demonic that sine values should kind of look like why coordinates cosine values should look like x coordinates. It's not exactly true unless you're on the unit circle it's not that cosine of theta sine of theta is the x y coordinate. It's taken to account the radius for this like polar transformation. But loosely this is kind of how you want to think about these things. And then so when you're trying to remember these angles you can just remember that if you're at, you know, zero degrees or something, the sine is starting out at zero. It's the y coordinate. And as you're, you're kind of going through this quarter rotation up to I have radians, the y coordinate is increasing. So the sine is the one you want to increase when you're doing this table process. And the cosine is the x coordinate so the cosine is the one that's decreasing. Okay. So yeah, just remembering the special angles and remembering this demonic about signs and cosines corresponding to y and x coordinates. Maybe I'll just say remembering polar coordinates. This is one of those things where coming to class is kind of useful, because this is not necessarily the way Torian approaches it in her videos and it's not really the way it's approached on the worksheets. But I think that this is worth doing a little bit of extra effort up front to understand these like this this back and forth process between x y coordinates and polar coordinates are theta. I guess somehow you can just do all of this trigonometry. Most of the stuff in in chapter four with this like one single idea. So it's kind of a hard idea to think about internalize just how do we do these polar coordinates. But I think it's definitely worth the effort up front because it makes the last five to six units. It's much easier. Just as a quick reminder of how this goes. We kind of choose our little reference frames we've sat down. We've decided that someplace in the real world is the origins we've marked that we've picked two orthogonal directions y hat x hat. And let's say we pick a pair are in theta. The first thing I do is I measure theta. So maybe it gives me something like this. That's the angle theta. Remember that data determines the rate. This is an important idea. But if we choose an R, then we choose a point on that rate. So now we get a line segment of radius R. See if I can switch to a different research. Okay, cool. Once we've done that. It's a line segment or another way to think about this is as a maybe I'll make this polar coordinates slash slash vectors. Sorry, I just saw that something was in chat. So let me just double check 4.5 p. Oh, from the pre class. Okay, so we are starting to switch the switch the eraser back here. Okay. So we are thinking about our theta determining a vector in the plane. So it's an arrow with a direction in the magnitude, the direction of the state of the magnitude is R. And we get some coordinate out of it x, y. And the formula for that coordinate is given by following x coordinate is our cosine of theta. And I guess I should mention here there is a way to kind of go back from this. So if you this is kind of like going from this is like going from our theta to x, y values. And there's a way to go backwards, which we didn't talk about. I'll just tell you what the formula is. If you're given x and y, and there's a formula for our name, the R is just square root of x squared plus y squared. And hopefully I'll go back up to the triangle in a second. I'll just tell you what the theta is. And the theta is a little complicated. It's the best way to put it. Instead, when I can say. So everyone has an easy description is telling you what the reference angle theta is. And this is the arc tangent or inverse tangent of y over x. Okay, so hopefully we try to point out why it's hopefully believable that the R thing would work. We try to point out why it's hopefully believable that the R thing would work. So right if we had this coordinate x, y, we could drop a perpendicular. Down to the axis. That would be a right angle. The length of this segment would be x. The length of this segment would be y. So we would just be doing the Pythagorean theorem to get R. This is just R squared equals x squared plus y squared. So no real funny business there. Except that we've kind of gone through this process of breaking a vector into components kind of without explicitly saying so. We've broken it into a horizontal component and a vertical component and the length of the horizontal component was x and the length of the vertical component was y. And so, okay, we do Pythagorean theorem to get R. No big deal to get theta. Well, we can just note that if we pick this beta here, we have an adjacent opposite and I bought news in this triangle that we found. And tangent of theta. It would be equal to the opposite over the adjacent side. And for us that's y over x. These are just lengths. And so if I apply, so maybe apply arc tan to both sides. And the key thing to remember here is that just like the exponential in the log and arc trig function in the original one so arc sine and sine or cosine and cosine and so on. These are a functional, a pair of functional inverses. And so we're not really like we're not multiplying them we're actually taking tangent of theta and plugging it into arc tan as a function. The other thing about functional inverses is that they, when you compose them together like this it's like you did nothing at all. So it's the identity function it just gives you the argument back. So the argument, full argument there is just theta. And so this would tell you that theta is arc tangent of y over x. And the side here, kind of motivating this stuff. But well, the problem this is that this only really works in the first quadrant, just because of the way we've drawn our picture. So here's kind of a warning here is that up here is to always draw a picture. This goes for virtually everything in this unit trying to find some way to draw a picture. This is a sort of a sanity check. The algebra we're doing at least makes sense, saying like the first quadrant or something. The problem is is that we can't actually use this picture for proof. It's just one of the tools in our tool bag. And what goes wrong is that if I'm in quadrant two or three, I think our problem areas is our tangent is going to come back with an incorrect sign. So if you just plug this into your calculator, you're not going to quite get the right thing. So we aren't going to have to worry about this too much because we're never going to do this direction of it. We're not going to start with an XY and go to an R theta. They'll usually be the other way around. The kind of the thing to keep in mind here is that if you ever do need this comes up in engineering and physics. Then you just have to be a little bit careful because our tangent is just giving you the reference angle. And so you need to actually play this flipping game to figure out. You have to sit down and draw a picture to figure out what quadrant to put it in and maybe you have to. What happens here is that theta is equal to a ref reference angle. But you might have to adjust it by pi. So what'll happen is that whatever you get out of arc tan will either be pointing in the correct direction, in which case you're fine. Or it'll be pointing in exactly the opposite direction of what you want. So it determines the same sort of line segment, the same way it's just pointing in the wrong direction. I shouldn't say so it's not the same way, but it's determining the same line. So you just need to flip the X and Y coordinates a little bit. I guess this is useful like if you ever end up doing like computer science or something to you have to do this all the time because your computer monitor is a two dimensional plane. And, you know, sometimes you have to do this kind of transformation to like draw something on a monitor. Okay, so there's a little bit of how to go back and forth between those two representations. Just keep in mind, we won't need this one much. Another important concept was vectors and breaking into components. Just because we're pressed on time we have to say a lot about the project. I won't say much more about this, but we want to look through a little bit the notes from last time where we had one vector and we broke it into an X component and the Y component. And you'll need that for the project so try to make sure you understand that or if not, bring it to the office hours. Oh, and I see somebody's asking a chat if this was the method we were supposed to use for question 12 or sorry quiz 12 on the first question. And the answer there is not quite. Sort of, I mean you can use this process to do it. What you kind of wanted to do and in that question was to essentially if you just draw a picture, like what's happening here, then you can just reason about these sign links. And if you think what happens there is if you plug in these side links, you get something that's like a standard angle. So maybe you do the inverse tangent of one or something like that. And then you know that just from like this geometric ratio of business and that's like a sign over a cosine. So you need some angle where the sign is equal to the cosine. If the tangent is going to be equal to one. So maybe just like what I'm saying here is like an easier sort of general way to go about this but you can always kind of go to this kind of picture and draw the triangle and reason about the actual ratios. So, you have this business about breaking a vector into components. You know just do a quick sketch of what this is supposed to look like something like this and like this. So the vector. Very important we've chosen a coordinate system so we should tell a reader. The details of the system. The idea here was that we could break this into horizontal piece by kind of projecting it down. And we could break it into a vertical piece by projecting it onto y axis. We can kind of move vectors around freely as long as we keep your links. And so if we had some, let's just say this is a vector, be with a little arrow is just telling you that it's a vector. Then there are the red one is a V sub X. And the blue one was a B sub Y. And there were kind of formulas to determine what the like how do you actually express these vectors. Like if you knew that this vector was given by a point X, Y, then you could find coordinates for these and it essentially boils down to using polar coordinates to get the VX and the VY. Okay, so that being said, we're hopefully ready to talk about something slightly new. Let me see if I can pull this up. Okay, it works. Okay, so hopefully this animation is visible to everyone. Just one sec here. So this is kind of a new situation we want to think about. Not really so new. But it'll be something new that comes out of it. So we want to think about sort of what's happening here is, let me see if I can simplify this a little bit. So this is a new situation to start out with. And it's just like a little particle traveling around in a circle. Over time right now it's just on the unit circle. And so what I'm doing here is these are literally just, I'm plotting cosine of theta and sine of theta as data ranges between zero and two pi. This is giving you a little point on the circle as data increases from zero to two pi. This is kind of traveling in a counterclockwise fashion around the circle. And this is kind of the basic model of periodic behavior. And so we want to use this to model other things that are periodic in the world. So there's sort of two things that'll be important here. One of them is that you will measure if I can here. This one. And that one. Okay. So there are two things that we could measure on this particle. One of them is that we've chosen our origin. And we can measure the height of this thing. So this is just the Y coordinate as we go around. And you can kind of see that if you're just thinking about the height, it's starting out at zero. It's increasing to one and then it's decreasing back to zero. And then it kind of continues decreasing once you're all the way over on the left-hand side. Sorry, let me do it like this. So we start here at zero as we travel from, for just increasing theta from zero to pi halves. The Y coordinate is just getting bigger and bigger until we max out at one. And at which point starts decreasing again. And so this is going from pi halves to pi. At some point we hit zero again. After we increase past pi, then now we have a Y coordinate's decreasing. And eventually it minns out. And then it increases again. So that's one thing that we could measure. It's just the height of this particle over time. And then another thing we could measure here. I turn these two off. These two on. It's not the greatest way to see it. But we could measure the, so hopefully you can see the kind of blue thing swinging back and forth from left to right. We could measure the X coordinate as we go around. And we remember that the X coordinate literally was cosine of theta. So what we're trying to do is understand cosine is like a function of time instead of a function of theta. So change the domain a little bit. And so you can just see that as the X coordinate, or sorry, as we go around the circle, the X coordinate is just changing in sort of a similar way. It is starting out at one. And as you go over time, just following the blue dot on the projection down to the X axis, it is decreasing. It eventually hits zero, continues decreasing and hits negative one. And at some point it starts increasing again and goes back to one at the end of the day. And then starts over. So these are kind of two phenomena we'd like to understand. Just as we vary this coordinate around the circle, we want to measure the X coordinate and the Y coordinate separately. Okay, so let's start off by looking at what the sine function looks like when you do this. And so hopefully this is, hopefully everybody can see the animation here. What we're doing again, this is just the sine function on the unit circle. And now we're thinking of it as a function of time instead of theta. So if you think about the Y coordinate as a function of time, all I'm really doing is just measuring the height as I left this particle sort of run around the circle over time. And I'm just measuring the height at every, every point in time. And what I'm getting out is some kind of graph. And the XY plane that looks like this sort of thing where you can tell there's there's some kind of periodicity to it, right? Because as soon as I've done two pi revolutions, right, I'm just now back to square zero, right? I can just start. You know, I've kind of measured all the possible ways the height could change in that full revolution. This is the kind of animation to keep in mind. We're just measuring the Y coordinate. Now let's think about what happens when you measure the X coordinate. This one's a little bit trickier to keep track of. So let me maybe walk through this one a little bit slower here. So if we're starting off here at zero, at time zero, we're just measuring a height of one as we increase from zero to pi halves in terms of theta. And just at each time we're just recording the height of the Y coordinate. You can see the graph here is just measuring that height and it's maxing out. The exact same behavior. As we continue past pi halves to pi, we're getting a decreasing of the Y coordinate, decreasing of the height until it zeroes out, continues. It mins out at some point. At two pi comes back to zero. And then we're just playing the same game again. It's periodic. So it's periodic with zero to two pi in this case. And we just measure the heights again. So we're doing the same thing now with the X coordinates instead of the Y coordinates, which you end up with is something, right? So if we're starting at time zero, you just maybe let this run a few times first. Theta is increasing. And then over time we're just asking you each point in time. So let me just run this forward. At each point in time, what is the X coordinate of that particle on the unit circle? Well, the X coordinate literally is just cosine of theta. How is that varying over time? Starting at one, we are here. So it's a cosine starting at one there. As we go up to pi halves, we're decreasing to zero. And because you're just pointing straight up, there's no X coordinate next to zero. As you go, continue on to pi. The X coordinate goes all the way out to negative one on the unit circle. So we're minning out in terms of this cosine graph. Again, the cosine graph is really just measuring the X coordinate. And then as you go past pi radians into the third quadrant, the X coordinate starts decreasing again down to zero. So eventually it's increasing again. And then in two pi, we're periodic. On the project, are we only looking at zero to pi or something because of like the sun? Like it rises. Yep. Yep. It sets or something like that. Yeah, it's a little bit complicated because. Yeah, you'll be looking at angles between zero. And pi. And what you'll want to do is come up with what it'll be is the angle theta as a function of T. So it's going to be a little bit weird. So you won't necessarily get a cosine graph like this out. Yeah, maybe it's easier just to say what that'll actually be. Yeah, so essentially what'll happen is that when you do this kind of thing, you'll just be starting at zero. You'll start at pi in terms of angles. But what'll happen is that for the this like function you're coming up with the angles will be the range of your function. And instead what you'll want to do is have a T going from zero to 24. And it'll go and your thing will go from like zero pi. And then it'll stay zeroed out for a while for the rest of the day. And then back at the start of the next day, you'll start back at angle zero. So yeah, it's it's essentially the right idea except for the the angles are the the outputs instead of the inputs and what you're doing in the project. Okay, so hopefully that takes things and microphone went out for a sec. Okay, so these are the sort of things to keep in mind for these trick functions that you really want to think of sort of a particle moving around the circle. Cosine is measuring the x coordinate over time. And sine is measuring the y coordinate over time. And that's kind of where these these usual these other graphs you've probably seen elsewhere have come up. Yes, I'll say a quick word about sort of the the general form for these things. The general form. So what I'll just say is of a wave. So there's an identity, like somehow I think it just suffices to think about cosine. It's kind of the easiest way to approach it is that every wave is just a cosine and a sign is just like a shifted cosine. If you think about that. Using the one of that your identity or something so if you understand cosine really well, and it's not too difficult to transfer all of your knowledge to a sine function. What works is that if you have so a sign. So start with the picture. So it's a little bit tricky because we're graphing things now where the dependent variable isn't really an x, but it's a like a time or something as we're thinking about this particle moving around as a function of time. This will be kind of you can call them really anything you'll just call them w's or something. And if we're thinking about like some kind of wave, the general form will be f of t was a cosine. I guess W wasn't great here. Let's call this L or something. The general form is a cosine omega t minus five. And what does this correspond to essentially extend this down a little bit so we have some room. There's a plus a minus a. So if you have a cosine I guess this is starting at one in general. So you get one for like the parent function cosine here. This would be starting in a where you've shifted over by some phase shift five. And I'll just roughly sketch what this does. It's something like this where the there's a period here of since that being two pi over omega things called the period. And that was the phase shift. So we're just looking at one period but this is a periodic function which just means that if we just know this one little region we kind of know it everywhere because it just repeats itself. After that continues in essentially the same way. And I think that's pretty much it for this. So there's just three important parameters one of them is to say amplitude. It's very important to recognize that your amplitude is not just the height of the graph or anything like that. It's, you have to take the midpoint of the heights. So it's like the average height or something like that. So, yeah, so it should be going plus a units above the the sort of you just draw this line in here. And one one analogy here is that electricity for example is modeled by waves and the red line would be like a like a ground current or something like that. But the ground current doesn't necessarily have to be zero. It could be like, you know, you'd have like something shifted up by 12 or something, just some baseline. And then you oscillate around the baseline and the a is just measuring how far are you away from the baseline and even time. So we don't have to worry about this for the moment. This like amplitude business but you'll see this on later free classes. So some sort of common mistakes there. Okay, and you may need this to as you're doing the modeling of your function for the project, something like this may be useful, but you may want something like a sign instead, depending on how you set it up. But the sign really just works the same way, except for you might have to change this would not start at, you know, one there and said it would start to help. Okay, so let's say some stuff about the project. So number one has to see. Here's an uploaded video from Cal who designs the project. You know, he sort of does this for all of the pre calculus classes. So he sees put up a video where he kind of walks through the example calculation in the project handout. And I think that's that's worth going through. See a lot of the same stuff I'm saying today showing up there. I need a reminder of kind of what's happening. It's probably one of the best resources. Let's just kind of remember what was going on. Let me grab a screenshot here to save us some time writing. Okay, that's visible. Remember, we had this kind of situation. It is we want to sort of the goal for this is to we have we have some kind of leaf sitting on the earth, absorbing some sunlight over the day. So how much sunlight could theoretically possibly absorb throughout the entire day. And so eventually this will, you know, we were going for like the theoretical maximum is kind of a phenomenon here where only these light rays that are hitting the leaf in a perpendicular manner contribute to the energy that it absorbs. So we have to do some business about fiddling with how do we break up this light ray into components and find the component that's hitting this in a perpendicular way. So what we'll do first is analyze this in a stationary way so the leaf is in the same place. The sun is moving over it. And so that'll give us some amount of energy that we can expect. But then you can also imagine, you know, if leafs were if leaves were trying to, you know, or trees were trying to optimize the amount of energy they absorb in a given day, you know, they might like shift their leaves a little bit so that they're always perpendicular to the sun to maximize that energy. So that's kind of the behavior we want to understand for this is like, if tree was doing everything in its power to keep itself flying with the light rays at all time. What is the total possible energy it could absorb over a 24 hour period. Okay. And yeah, maybe this will help answer that question from earlier if I just drop this picture in something like this. This is kind of our simplified model. What's happening? She didn't preserve the ratio, unfortunately. Okay. So the model was that this yellow line was like an incoming light vector. And what'll happen is that this light vector will also is like an energy density or something coming coming from the sun. There are two phenomena to keep in mind here. One of them is that this yellow vector depends on time. So time zero, it's, you know, sitting out horizontally. As time increases, it's, you know, kind of sweeping out the horizon and ranging from zero to pi. And then, you know, once the day starts again, it kind of resets and goes goes back to zero. One of the angles to think about here, one of them was this angle that the light ray was making with this plane of the leaf. And that's the, that's something that's changing over time. And the other one, the one that's important for us is one angle is this light ray making with the, we've kind of made this green kind of vector and pointing out of the leaf sort of sort of indicating what the orthogonal direction is. The component of the yellow vector that's lining up with the green vector, once we do this decomposition will be the component that's being absorbed. So you can see that that's, you know, here in this first situation, it's not maximized because some component is kind of lost in the horizontal direction. This yellow breaks up into something that goes that way a little bit, something that goes that way. So if it's a fixed amount of energy, some of it's getting lost because those rays are more parallel in the situation in the middle. All like all the right light rays. It's exactly lining up to the sort of the agreement of these angles is maximizing the amount of the energy density there. And then we kind of go back to them in again. Okay, and what's happening here is that we want to think of as a function of time L of t. So it's a little bit weird to think about but this is something. If you give me a time bill value I tell you what the vector is. So I'm telling you what the full yellow arrow is. That's, that's a little bit hard to think about. So just simplify things. Maybe I'll just write Theta sub L as a function of time. Theta sub LN as a function of time. I haven't labeled these super well but let's call this angle. Theta sub LN, so just the angle between whatever orthogonal you have on the leaf and the light ray coming in. I'm just remembering that this changes over time because the sun is moving. And then Theta L, which is this angle here. So you get two angles that depend on time. And maybe it's actually, maybe it's helpful to kind of think about what these look like in terms of graphs. So you're going to have to be a little bit careful with domains and co-domains. What we're sending into these functions will be T values and what we're getting out will be Theta values. So what happens here is that there's going to be some, let's say we're doing Theta L of T. And the first thing I would say to do is like let's determine the domain of Theta L of T. And this is just asking like what are the T values we should be expecting. And these will be from zero to maybe 24. You just include one of the endpoints to make sure you get the full day. What is the range of Theta sub L of T? And this is, if we go back up to this picture, this is exactly the, so all of the possible values this angle could take. And so we'll kind of artificially restrict this to say zero to pi. And I'll just mention there is a way to make this work where you don't have to make this restriction to where like it would more naturally model the fact that you know that at some point the sun is kind of, or kind of the earth is turned away from the sun or something like that. So this angle, if you think about this light angle coming in. Okay, let's go to the last picture here. At some point the angle is pi. And if you think about just what would happen if you continued rotating, while at some point this would be coming in like that. And you would get some kind of like negative angle or some obtuse angle between the light vector and the perpendicular vector. And you could kind of make that work with equations, but for us, since we have a choice over the function, it's maybe easier just to say it's zero. Or to say it stops at pi or something like that. So you could have this be zero to pi, it would just be more complicated to work out. And so what this graph will look like is if we just go back up to this picture, we know it's something that should start at zero. And you'll mark off where 12 noon is and then I'll mark off where 24 hours will be. So what I want is that as in this picture I want when the, let's see, so we're looking at theta sub LFT. So this should be something that increases from zero to pi will be the max. And so we know we have two points on this graph, something at zero zero and something here at 24 hours and pi. And okay, I'll just draw one possible function that might make sense here, which would just be a linear function. So this is an angle that increases over time and it just increases at a constant rate. This is just a linear kind of growth. And let me just indicate really quickly, we've only specified one period but we want this to exist for all time. So what you can do is just say, okay, this function is going to be periodic. So it's just going to repeat itself there. And it's going to repeat itself there. And you can kind of work with it this way. But you can see this is a little bit of a problem, right? Because we can't have, for it to be a function, we can have two outputs for one input and sometimes zero is going to be a little bit of an issue, time 24, time 48 and so on. So you kind of need to delete a point somewhere. So maybe you delete this point. So you're saying that theta of 24 is equal to zero. It's also equal to theta. Sorry, this is theta L times zero. You just have to be a little bit careful. Maybe delete some points to make sure you get a function. And you just make it periodic everywhere. But actually, you get to choose what this function should be. So maybe you might think that there's, you know, is it, is it true that the sun just sort of gradually increases angle at a constant rate? Maybe that's true. You might think that, you know, maybe it's something more like this where sun increases. I'll just draw another function in here randomly. You might think it's more gradual like this green thing. And you could have something kind of more like that. It increases really quickly in the early part of the day and then kind of levels off. The angle doesn't change as quickly later in the day. So you have a lot of choice over what function to use for this. But in any case, you'll have at least two points to model your function on. And so you can, you know, go into Desmos and do a fit if you want, kind of mess around, see what kind of functions you might like there. So you can, you know, do things like they'd have t equals a not t squared, where a not is like a parameter plus a one t plus. And maybe make these match up a to t plus a one to a to t squared plus a one t plus a zero. You might fit it to like a, some kind of parabola or something in this, this range or maybe you shift it around a little bit. You might do data. These are all theta L's of t. Maybe you want to do a one of t plus a not, which would be this like linear function situation. Maybe you think it's an exponential change. You have to pick some kind of function and then in your, in which you're writing up you should sort of justify your choice with some like what physical intuition are you using from the situation to decide on what function to use. Okay. So this is, we've looked at theta L of t. Let's look at theta L of t. Just remembering what these are in the physical situation theta L is just the angle of the sun where we haven't even talked about the leaf yet. L is the component. So the angle between the light ray and this normal vector for the leaf. So we'll need to model this as a function of time to. Again, we have to be a little bit careful with domains and ranges. The domains are again time values. The outputs are angles theta. And now we have to kind of fit this to our intuition from the picture or at least start there to figure out what kind of function to use. So we're examining the this kind of angle here between the green and the yellow vector. And if we like plot this on Desmos and just sort of thought of this like particle running around the circle sort of thing. See that the angle starts off being big. So it starts off being a right angle as L sort of, you know, traces out the sky. This angle closes up to zero. And as L continues past like 12 noon. The angle is opening back up to something like pi halves. So maybe you get something like, there's pi halves. You're at time zero. Think about what happens at 12. What happens at 24. So it seems like we're at zero. Or sorry, we're at pi halves at time zero. We're at pi halves at time 24. Let me just double check something here. Yeah, should be fine. So the angle between these two start something like this. Maybe you think it's just kind of linearly increases if you chose the linear function for this one. You might think it's something more like this if it sort of changes the rate of change changes over time. Maybe it's something more like a parabola or something. You might think it kind of changes more gradually to something like that. So all of this is you have to choose sort of what function to use here. And okay, once you have all of this done, you are essentially ready to go for the last sort of the last part of the analysis for the stationary relief. So what will happen here is that you've come up with these functions theta ln of t, theta l of t. And these could be related in some way like you can maybe just do theta l of t and then do something to it to get theta ln of t. If you use some geometric considerations from this picture here, you might expect that these angles are related somehow. So maybe their functions over time are related to. But whatever you end up doing, what will happen is that you found out and see if I could find what we concluded from last time. Okay, so I can paste this in. Okay, so this is, I'm sorry, it's super blurry, but it's a little bit about what we talked about last time we were talking about. So f and g you're talking about here, these are corresponding to our theta, theta l of t, theta ln of t we're talking about today. The point is they can just be any functions. We concluded somehow that there was this energy density that depended on the vector, this light ray vector l. We found that it was proportional to the sign of theta one. We kind of did some derivation to get there. So you'll want to try to reproduce this sort of derivation when you're writing up your analysis. So you'll need some way of drawing geometric pictures and kind of arguing about, you know, why is this, why is this sign showing up? That should really be something you explain in the project. So once you have this, I guess I should say something a little bit more about what happened here. We found that E of l was proportional to sign of theta. And then we came up with this function that just gave us the vertical component. So I'm saying call that function g of t. And we made an assumption. This was assuming at some point we assumed that the vector had radius one. And this kind of made the analysis a lot simpler for us. The reason we did that is because we can actually go through. And at the end of the day, scale up that vector by whatever we want and we can scale it up by a function. So this function E of t is going to be something else you figure out from. You'll find energy density function. Sorry, realize we just have one minute left here. So I would look at the video for some more details on how this part will go and we'll talk about it more a little bit later. But the idea here is that the energy density hitting it. So this is energy density, say E on this axis as a function of time, it will max out like 136 or something. And you want some kind of function that does this kind of thing, probably more symmetric. And this will depend on time and should just, yeah, so you actually get to choose this too. So you have to kind of argue about. So the energy density kind of depends on this angle. So you're whatever sort of functions you chose up here for your angles, your angle functions. So theta ln was measuring the angle between the light vector and the plant. So this should be somehow related to the energy density. Right, because if you're off at an angle, you get much less energy coming in than if you're at a, you know, an angle that's lined up. So these two analyses should be tied pretty closely together. And what'll happen at the end is that total energy, let's say in a time period, delta T will be equal to essentially E of T times G of T times the area of belief times a little delta T. This is getting into a kind of a more complicated part of the analysis. So I'll say more about that. But kind of what you want to have to, you know, started on is definitely finding these angle functions and then starting to choose a synergy density function. Okay, so that video is up in. You'll see right now on your projects, definitely try to look that over and then I'll be in office hours, department office hours of four to five. Yep, that'll let you go. I'll post these notes up online. So to find like the light intensity, you do just like, could you use your time on your, as you're like kind of your X and your angle, because the angles change constantly right on the first part. And that could you like find the like vector or the, I don't know, it'd be like the hypotenuse, I guess. Would that be your light intensity? Essentially, when you go back to this picture here, the yellow vector coming in. In the first part we kind of assumed it was length one so we could find out what the component was. And then this EFT function you're doing is making this vector longer or shorter based on time. Okay, it's a little bit complicated. I'll try to have a demo for next time and hopefully make it a little bit more clear. Also my partner, like for the, we met on zoom and he found the leaf and stuff, and he was going to use, like the project says to you divide it into rectangles or something and then, but he was going to just do it like some sort of other way, I don't know, like some sort of other way to find the area that's more accurate. Like, and I was going to like ask, because I was like maybe we should ask if that's like, I don't know if maybe like they want us to use the rectangles, I don't know. Probably. Yeah, I think that's that's the intended way for this project. If you look at the video part of this. Part of what Kelly has lined up for people is to put this info in a spreadsheet. Yeah, so what this is is this is doing an approximation of something from calculus where we're measuring like little rectangles instead of doing something called the integral which does that precisely. Yeah, I remember like the Ramon sums or something like that from when I took calculus in high school, but I don't remember. Yeah, like I don't remember much of it, but I remember the Ramon sums. Yeah, exactly that same thing. And then at the end of the day what you're going to do is chunk up the day into like little two hour blocks, and then just evaluate this function at every two hours. And then we'll approximate it and if you imagine if you chunked it up into one hour blocks or like 30 minute blocks or 15 minutes, the approximation would get better and better and better. But we're just doing kind of a very rough approximation. Okay. Thank you. Thank you.