 before I jump into that a few quick announcements about the quiz results that those are graded. So just some notes on the quiz. So one of them is just that we should be thinking of processes versus solutions. So this has been one of the big themes in the class. We've talked about this at least a couple of times. Somehow in these kinds of questions I'm not really so concerned with people getting like the exact right answer for everything. Although that's of course important in mathematics. One of the things I stress for this class is that we want to essentially come out of this class with a good sense of what it means to communicate mathematics to somebody else and to like write it down in a way that somebody else can read and comprehend it. Which seems minor but I think it's sort of one of the better skills to pull away from this class because I know you'll run into it sort of whatever major you're doing. If you use mathematics in any way you may or may not use this exact mathematics that we're studying right now. And so you know maybe at some point you'll have to get help from somebody or you'll have to look something up. And I think in modern times like you know that the places you get help might be like the internet. There are places like math stack exchange and math overflow which we saw in the previous class. Or you know you might be asking if you're doing research or something you might ask your principal investigator to help you with a mathematical derivation or something. And so what you'll actually be tasked with doing is sort of communicating what you've already done sort of what you've tried and what process you've gone through. And then you know that puts you in the best position to get help because then you're kind of getting the most efficient help possible. You know people can kind of laser it on and focus on exactly which part might need to be changed or improved or whatever. So it's great for this class of course because it makes it easier to catch exactly where you know it makes it easier for me to catch exactly where mistakes are. But I think it's also like the important life skill if you're just communicating in any mathematics there's a nice way to do it clearly. And then of course there's like the method too like you want to be able to explain the method to other people not necessarily just write down the right solution. I think there are a few people so like again most of this class I think is review for a lot of people. If you've taken some kind of I mean definitely if you've taken like a calculus class in high school or if you've taken a pre-calculus but at least in this unit a lot of this should have should have come from a trigonometry class. If you haven't seen that that's that's totally fine but that's kind of how a lot of this is pitched is that you know maybe you've seen these trig functions or something before in a previous class and we're kind of you know reviewing it because you need a lot of these things for calculus. I'll show one example but also kind of looking at a nicer way to sort of remember it. Learning a few new things hopefully you know these things like polar coordinates which give you a different way to look at things and again would make calculus a lot easier when you go on to take that. So processes over solutions. I want to say something just about using exact ratios whenever possible. What I mean by this is a lot of people on the quizzes so that entire quiz you could have done without a calculator and actually like 90% of this class you can do without one. It's actually sort of preferable to actually just keep the like if you're doing a problem where you have you're finding sine of theta or something and you have the actual side lengths and they're given to you as like integers so maybe it's you know something like this kind of triangle here and you're trying to find sine of theta and you're given and then maybe that's six maybe that's seven has to be longer then I would just go ahead and make that six over seven and leave that as it as is without evaluating that or anything. I think it's much nicer to have the actual exact value because then somebody can plug it into you know if this was somehow like you were doing trigonometry for NASA or something and you were sending a space shuttle off and you told them this was equal to let's see what actually what is that six sevens we approximate that and this floating point number on our computer it's something that's like 0.85 so if I just told somebody that sine of theta was equal to 0.86 this could be a problem right it sort of depends on the accuracy needed you know if you're on a space shuttle and you're trying to like fix your trajectory or whatever this could be the difference between I mean I don't know hitting two destinations kind of it depends on how far you go so you would want to either communicate that this is an approximation or just leave it in the exact form and then you know whenever you turn your solution over to NASA for their space shuttle flight path then they can plug it into their super computer and you know approximate it exactly as well as they'd like but so that's that's kind of why that's sort of a preferable thing to do for trig functions quick note on polar coordinate stuff just so remembering that you have this kind of situation in a theta this is your y hat this was your x hat this was an r when you go to look at this this point you get some like x and y value out of it it's really important to remember that this is actually r cosine of theta and r sorry r sine of theta and it's only actually equal to so like for example right so what is this quantity this is x this is y and let's say you're given that some people put you know like tangent of theta or what are they let's see all right so some people were like going to look at this x coordinate and they said all right x is equal to cosine of theta or something like that and this only is true if um if r is equal to 1 right so that's only if you're on the unit circle does cosine of theta exactly correspond to the x coordinate and does sine theta exactly correspond to the y coordinate otherwise you have to take into account this radius that's scaling you off the unit circle and I wanted to point out I guess this might be coming up on a pre-class but why why do any of this so the answer of course we're in a pre-calculus class and you know a lot of this is geared towards things you may need for calculus and so what happens there is you're trying to solve something called an integral which you may or may not see in a first calculus class you probably do it the the class of UGA for example um and what happens is that some of these integrals like going back to these are just some kind of mathematical equations they're really difficult to solve in general um if you just write down a random equation like this you will probably not be able to solve it so there's there's this big bag of techniques for solving these these things um and one of them is doing a trigonometric substitution which is essentially letting one of your variables where it's like an x or something imagining that x is the cosine of an angle or the sine of an angle this lets you bring in all of the tools we're looking at now with trigonometry to solve this this kind of problem and at the end of the day so if you're solving some integral I'll just leave integral in quotes because it's just something we don't know about yet but maybe by the end of today's class we'll say something about it since it's sort of related to what we're doing in the project you'll do some stuff and you'll end up at a a solution that where you need to know like the cosine so this is just an example uh maybe the cosine of the sine of uh theta let's say here I guess I should do it like this should be the cosine of the arc sine 37 fifths is equal to something and that something will be you know your sort of solution from the problem and the question is uh what is this thing you're evaluating one trig function at the functional inverse of another trig function and I mean I guess maybe you could plug this into your calculator um which might be useful all right so this is this is kind of why we're talking about you'll see in three classes and in the videos that we're talking about domains and ranges of science cosines are sines are cosines things like that because um if we this is a periodic function it's just over all of the real numbers it doesn't actually have an inverse if you kind of allow all numbers in the domain so we have to do something kind of familiar from maybe the first month of class or something where we restrict the domain um and only allow angles within a certain range certain uh interval um as angles in our domain and that'll give us a function that's injective provided we're always restricted to that domain um so that's at least why we're covering some of that like arc sine doesn't make sense everywhere we have to restrict the domain down and then why we're doing the kind of geometry part is that um what you usually want in most problems like say if you're doing physics or econ or bio or chemistry and you have an integral that you want to solve you actually don't want the numerical approximation most of the time if you're trying to like establish the theory like somehow whenever this thing evaluates to as a number doesn't really tell you much about the underlying situation um you can't like analyze the function and do any like prediction so remember the function is kind of like your model and you kind of want to know what the actual model is in terms of an equation and so you want um something that doesn't involve plugging into a calculator for this and so the way that you might do a problem of this form you would have some triangle you have some angle theta that you're thinking of and theta is really this right so arc sine's any arc function takes in a number and gives you back a theta so it's it's asking for what theta is the sine 37 fifths so you would just go in and label a triangle um you would see that c is of sine is 37 fifths so there's kind of two ways to do this i'll do them in different colors one of them is um so let's see if theta equals arc sine of 37 fifths then i can apply sine to both sides sine of theta equals sine of arc sine 37 fifths this will imply sine of theta well it equals something but the moral of the story here is that trig functions are complicated like they aren't as nice to work with as some of the other functions we've looked at like exponential and log where we're a nice inverse um but so i mean what i'm saying here is that like i wish we could just cancel these two right and just say that this is equal to 37 fifths and that would be great because then we could continue with the problem but this is um i guess on a restricted domain so the problem is is there are infinitely many angles that satisfy this that sine of theta is 37 fifths i mean if nothing else if i find one angle i could just take that angle and then push it by two pi and get a different number for the angle it's it's bigger i just added two pi to it um but now this is a problem because it's not a function right i have two numbers mapping to one or one number mapping to two numbers so you're going to either have a problem with injectivity or you're going to have a problem with just being a function so that's kind of why we're looking at this restricted domain stuff too is because you'd like you'd like to make a step like this where you sort of just cancel sine and arc sine but provided you can do that um then you can write you know that this is like opposite over hypotenuse or whatever so you just do i guess this isn't realistic in 537th it's the better one so the one i had was kind of it was something outside of the domain of arc sine so you have to do some trickiness to figure out what it is but just for simplicity just flip those um flip it here too this was 5 over 37 so it's a small number that's just saying that that's 5 that's 37 would have been a problem otherwise because the hypotenuse has to be longer than either of the legs um in which case if you had this situation you could find out what this is and i guess in this case it's 5 squared plus a squared equals 37 squared so this thing is square root of whatever it is 37 squared minus 5 squared you know great now what were we trying to do we had this thing this is saying cosine of theta is equal to question mark okay but now we we don't have to evaluate anything we just have cosine of theta so there's adjacent in the opposite opposite sorry the adjacent in the hypotenuse and whatever this is this is root 37 squared minus 5 squared all over 37 so we get just a nice exact answer out of that um and things are good so saying there's another way to approach this one thing you can do is scale this all down to the unit circle so you could just say that the opposite side was 5 37s here in blue you could say that this hypotenuse was just like one because this is still the same ratio right if i just take the opposite over the hypotenuse i just get 5 over 37 and this becomes something different uh with the relation uh you get at the end of the day ends up being this thing okay so that's all i wanted to say about that is that there is some reason for doing this that comes up in a certain type of calculus problem you do um and it's good to learn sort of both methods you want to have some like you need this like geometric way of doing it when you run into the calculus problem um and having this this polar thing in your back pocket makes some of these problems easier too okay and then just the last note just on project stuff is uh there's a video under the uh ELC content projects and everybody should definitely go and watch that and i'm going to somehow incorporate this into the rough draft grade just because this is like a full explanation of most of what you have to do for this project that the person who wrote the project actually made so this is like your number one best source for anything to do with the project definitely make sure you've watched the video before you turn in any kind of draft or write anything up um and yeah i i mean i don't know if this is something that like students are super aware of but like any lc it tells you as the instructor like if people have like gone and looked at a resource that you've put up which is uh for the most part kind of creepy i don't actually use it for much um but in this case i'll probably just go in and check that one thing just to make sure that people are sort of right if you're turning in a rough draft and haven't watched the video that tells you what you have to do for the project that kind of puts you on a on a problematic route for the the project so just want to make sure we catch that early okay so let's talk about what's actually going on in the project some project details and sorry i actually just heard my coffee finished brewing so let's take a quick like one minute breather feel free to like stretch your legs or something be right back okay hopefully everybody uh it's back now um okay let's see so uh just quick note one thing is maybe i'll just put this as a big warning sign uh refer to class notes explanation video and the handout refer to those closely as you're writing i think this honestly isn't a problem for most most projects but um if you're losing points on the project these are the easiest points to lose just if you're missing something that um you know we covered in a class lecture or something that's like explicitly listed out in explicitly listed out in the rubric that you're attaching to your project um or something that's explicitly listed out in the handout that has to be included in the project um just make sure you're consistently referring back to that and just double check everything before you submit anything makes my life easier because i could just go through and check all the boxes yes this is there yes this yes that is there um yeah so so definitely refer to all of the resources you have as you're writing things and just make sure um you have as much as is required and then also just be referring to that consistently so that if you have questions you can bring them to office hours i'll maybe say if i don't say um so i'll just put a reminder here so i'll have some office hours today if i don't say what they are at the end of class please do remind me um so right we had this this kind of model we were talking about hopefully everybody's familiar with the situation now of you know this leaf sitting and having sunlight hitting it at some angle and this was a complicated three-dimensional situation that we turned into a two-dimensional one and what i've done here is just pick a spot where the leaf will be there was this normal vector emanating from the leaf which is telling you the orientation of the leaf kind of what what plane it sits in and we had this uh light vector maybe i'll do it in orange coming in at some angle and right remember we did this funny thing where i just changed the direction of it so that way they're kind of two vectors sitting at the origin this isn't too big of a deal because we're just thinking of a line segment really or some like direction with some magnitude um so this vector only differs from the light vector the light is coming into the leaf it differs from it by a negative sign or something like that um but we don't have to worry about it for anything we're doing because we're going to decide what the what scaling we're using anyways so we'll force it to be in the right direction so what happens here is that you have a theta sub l and a theta sub l n sorry i forgot to say that this is a vector we're calling l l for light and for normal and okay so hopefully the situation is familiar to everyone there are sort of two angles to worry about here and this l vector is traveling i guess we should say that l depends on time so it is a vector that depends on time so if i plug in one time maybe time zero i get a horizontal vector if i plug in time 12 noon i should probably get like a vertical vector and somewhere in between midnight and noon i should be getting this orange vector like the way it's drawn here okay and here are some things that we want sort of need to be saying like the rough draft okay so the first one is going to be a graph of i guess theta l of t is the easiest one to start with it's right these angles are changing over time so we should be able to come up with some kind of function that models that changing behavior labeling the axes is going to be extremely important in this project so if you if you aren't like really clear and precise about your graph labeling you probably lose easy points the reason it's so important here is because we have so many different functions with completely different domains and ranges that we really have to be precise about exactly what we're plotting and where so there's zero sorry this is a time so there's 12 there's 24 and what we have is that we know that at time zero it should be an angle of zero so maybe we know that there's one point here and also at time let's see how does this angle work so go back up to this model it's theta sub l so this is one that's traveling from zero to pi and after it gets to pi I guess it just stops or resets to zero so we want to although I guess there will be two important points one of them is pi halves right just a vector pointing straight up and one of them is pi and I imagine that at pi halves we want that to happen at noon so maybe just you know pointing directly up halfway through the day and then by the time we reach 24 hours we want this to have reached an angle of pi and this is just the angle that the sun is making with the the plane of the ground okay and then you just need to find some kind of function that hits these lines so you could do some kind of linear function and maybe what they do in the example maybe you do something maybe you think it's like it doesn't change for a while very much but then all of a sudden it kind of changes a lot and then kind of slopes out to something like this so this is a situation where it's kind of like it's spending a lot of time near zero so maybe for the first six hours of the day the angle isn't changing that much or something but then between six and nine a.m. or something all of a sudden it's sweeping out from like zero to that so it's kind of making some more quick movement and it's like really quickly going here and then it kind of once you get to sunset or something it like slowly descends to that this is like at what speed is this vector moving and is what this data L of t is um yeah you might think it's something totally different you might think it's like it really quickly ramps up and then maybe it flattens out here or something like that during the day it doesn't change much and then really quickly sets sort of a lot of options for functions here um and you have to pick okay and then similarly and I will post these notes up so don't worry too much if you aren't able to write it down the other thing we'll need is data ln of t so just going back up to the the model this is now the angle between n and l and so I would just like you guys to come up with some function that models this behavior this may not be explicitly mentioned in the project handout but this is something I'd like to see in the analysis just to make sure you have a good handle on what the problem um is and sort of this whole vector situation that we've been talking about sorry I should have labeled this coordinate here this last coordinate was 24 pi and this middle coordinate was 12 pi halves in this situation data ln we know is going to be actually it's going to be so it's going to start at yes this is going to be a little bit yes let's do something like this where it's between zero and pi halves for example so zero would mean these two vectors are lining up directly pi halves would mean they're exactly orthogonal and I guess what's starting at time zero is they're orthogonal so there's a n zero pi halves point we know that at time 12 they are completely lined up so there will be point here at 12 zero and we know there will be time 24 this will be 24 in the t coordinate and then pi halves again um just coming from again this this is all just coming from this model that we've set up what's happening at time 24 is that the vector is rotated all the way to the left so the green vector is pointing straight up the orange vector is pointing straight to the left the angle between them should be um you know like a 90 degree angle so they're orthogonal so the angle is pi halves and you need to come up with some kind of function that um interpolates these data points somehow so some function that just hits these particular points you might say okay maybe it's um maybe it's linear but it's kind of what I've drawn here like this piecewise function or piecewise linear function um so you might want to try that you might say okay maybe it's somehow related to theta um l this here is like a theta l n axis up here was like a theta l axis and this function that we're drawing here was theta l of t and this function we're drawing here is theta l n of t and you might expect that these functions are related somehow from the geometry in which case you can do that maybe it's just this is some modification of the previous function that would be totally fine um or okay so you might you might expect it's something totally different too like you might expect it's um maybe you think it's more like a parabola or something or part of a sine wave um I guess there are a lot of things you could do here maybe you think it's something like this where it's like mostly constant for most of the day and then it kind of does something like that and it kind of comes back up or something sort of a lot of options that I could happen here and these are up to you um if you want some ideas for how to for like things that you could use here we just pull up a quick um I can find it okay so here's here's a um this is something you actually do like in real life all of the time or at least in my real life so I've talked a bit about how I used to do like computer science stuff uh an industry before I started graduate school um so one thing you do is the CSS is this language that you use for making websites and you can do all these kinds of like animations in them um so this is like you know if you're doing some work for a client they want like a logo that's animated you know this is kind of the easiest way to do it and so you have to use this kind of function you know this modeling function stuff all the time to determine um so usually you want animations like a periodic or sinusoidal behavior um and these are kind of like the basic building blocks of them so let me let me see if there's one that really illustrates kind of what we're doing maybe something like this I don't know if you can see kind of how this like height is changing over time in this particular function so the little thing on the right hand side is supposed to indicate um you know like maybe how an animation is coming into the screen over time you can see that it comes in like if we have this kind of sloping end business here on the this like first little leg of the function if it's kind of flat at the beginning it's saying you know move really slowly at first and then if it's like a really steep um incline this is saying you know speed up so you can see it kind of like pauses at the bottom and then it moves really quickly to the top and then you know it kind of bounces back with this other function here this is like a piecewise sort of function um so you can just kind of use this something like this these are called easings so e i e a s i n g s you can just search these on google this is easings dot net um you can see that there are sort of various um just to get an idea of like how these these functions behave here's something that's more cubic in nature you can see it starts off really slow which is kind of like the rate of change is slow near the beginning so the speed is slow because it's sort of flat in this little area down here and then as you get towards the middle the slope kind of gets you know the slope increases and so it sort of speeds up once you're about halfway through um so yeah i don't know it's worth let me see maybe one more here where it's kind of like really steep at the beginning to compare that something really steep is going to like come in really fast and then once it's kind of like leveling out here this is saying that once you're close to the destination kind of slow the speed down a little bit and you can see it kind of goes up and down so you don't have to use these in any way shape or form but just to help give you like an intuition of like how the how the like curvature of the graph so the curvature is like somehow measuring a rate of change um like how it affects the speed of this data angle changing you can get some ideas from that okay um let's talk about just some things that sort of need to be oops so need to uh slip to some things that definitely need to to make it into the project hopefully in the first first draft um we're definitely in the final one so for the above actually sorry let me do one more function and then we'll i'll say um what you need to do for all of them so let's think of this as like a part one is determining these data data functions part two is modeling this like energy over time and actually i wonder if i have well let me just draw the graph and then hopefully it'll make sense so what's happening is that you have some energy kind of hitting leaf over time and the time zero at zero you know it kind of depends on the time of day you know that there's sort of the most energy that could ever happen and it happens when the vectors are lined up completely and this is like i think it's like 168 milliwatts or something if we get the exact number this is time 12 this was time 24 so we know a few points on this we know zero zero you know that it maxes out at 12 it's a 12 168 and we know that by the end of the day it's back down to 24 zero and so this one you want to um this one's not linear so you don't want to do the same thing we did before it was kind of like this triangle um hat shape thing um instead what we want now is the energy density should be like more smoothly varying over time so we don't want any of these like sharp changes in the graph and if you've seen like calculus before what's happening like what's happening in a graph like this is that you have a tangent line at this point or you want to have a tangent line because the tangent line is the slope of it is measuring a rate of change and you want that rate of change to be uh continuous so you don't want it to like jump or anything you know all the you know you're going two miles an hour and then all of a sudden you're going 200 miles an hour it's not like physically reasonable for that to happen in like an instant um so you want to like have a well-defined tangent line there and the problem with this kinks um with these kind of uh cusp sort of points is that we don't know what the tangent line is if you're coming in from this way it kind of looks like the tangent line shaped like that if you're coming in from the left but then if you're coming in from the right it kind of looks like the tangent line is shaped like that so there's no well-defined tangent line there and usually this is telling you that like it's somehow corresponding like it's not really an accurate model model of physical situations because in the real world nothing kind of changes instantaneously like that it like gradually changes even if it's quickly um so which one here is that maybe a function that looks more like this so this one you'll have to sort of uh sorry I haven't drawn that very symmetrically it's still not super symmetric but okay um just some kind of like function smoothly not having um so no no points like uh this one so no spots where the function abruptly and suddenly and instantaneously just changes direction in the graph um so there's a lot of ways you could do this this one I won't say as much about because this is like coming up with this is kind of uh the one of the key pieces of the project I'll say that maybe you could use a sine wave here like you could take a sine function or a cosine function and kind of shift it up um and maybe you'll end up with something that looks ask what would that look like you get something like this like you get one half period of the sine wave happening here and of course this function exists everywhere else too so you might have stuff outside of that in which case you might need to make a piecewise definition to say maybe it's well I don't know yeah you want to define it at least on 0 to 24 so you might get something like this from like a sine or a cosine um you might decide that's something like a parabolas works here you might find some totally different function but this is up to you to come up with and then a question here is is this a function of let's say theta sub l or theta sub l m what I mean here is that so this is some graph of a function e of t and maybe it's the case that e of t is just equal to f of theta ln of t so maybe I just obtain this graph by coming up with some function f and then plugging theta l in into it maybe so f would be a function of theta instead so you can kind of analyze it that way maybe you think that the energy depends on the theta in which case you could come up with some energy as a function of instead of t down here you would have a theta and that's totally fine and then the theta you could plug in would be well theta ln of t you can plug t's into the theta function so then you could get a graph like this where it's e versus t but this is a this is sort of an arbitrary choice you're going to make I'll say a little bit about what has to be included for these so I need to include the following so you will definitely need to let me do it this this way for theta l of t theta ln of t and e of t formulas for all of these things graphs just remember that the labels are important maybe say some words about what the domains and ranges are explain the periodicity and so what I mean here is that maybe if your domain is say zero to 24 right so these are the times you could plug into a theta for example um and your your range is maybe if you're in that first case of theta sub l your range is like zero to pi um you should explain so you've defined a function on one domain zero to 24 um but now think about how we would apply it to the real world we would want this this would just be repeating every day right so after 24 it's kind of like the function resets but you're at like hour 25 and it's the same thing as being an hour one so just one hour after the starting point in both cases so you want to explain how do you you've defined this function on a restricted domain and range you want to now define how would you extend that out periodically to make it work for all time so how to extend domain to all real numbers whereas here you might have like the domain of theta sub l of t was maybe this interval zero to 24 so you've given me a function that I can only evaluate I can only plug in numbers between zero and 24 in this step if I try to plug in 25 it's like the computer throws an error the function doesn't know how to handle it um but how can you how can you take this function you've defined on one little chunk of the real line and make it work everywhere so I can plug in 25 okay so I think that's mostly it for this these definitely need to be included for these uh for these three functions you come up with so this was sort of a let's see what was this part two maybe a part three is driving the vector components so just a recap of the situation here it's something like this and this n vector perpendicular and so what's happening here that's going to be new is that we don't assume that the magnitude of this l vector is equal to one this is something that we did in the I think maybe the first pass at looking at how to pull out these components um so what we want now is a vector yeah but instead let's uh I guess that we should think of this as a cell of t instead let's let the norm vary over time since the vector l depends on t and the way it varies is just going to be that e of t that we already came up with so we'll want to model two things kind of separately is one of them is what is the angle of this l vector over time and then we can just think about you know one fixed vector of a fixed length kind of moving through that angle over time and now e of t is just going to tell us how do we scale that vector up at individual times so maybe a time near time zero you'll have like a little this is like maybe l of three or something but then by the time you get to l of six maybe it's kind of a bigger vector maybe if you're up here in like l of nine it's an even bigger one and hopefully I could just draw it this way this works yeah so you think of like a small vector I do if I can pull all of this out yeah so it's kind of an animation to think about here so orange is like the the l vector and you can imagine that at lower times it's like a small vector maybe it's even zero as you kind of rotate it through it grows maybe it's just it gets to its like longest point here and then maybe it kind of shrinks back down as you go back through the rest of the angle eventually by the time you get down here it's back to zero then it kind of resets the next day and just does the same thing something it kind of sweeps out something like that that makes sense so just the point is that the vector length depends on time or really depends on the angle and the way it'll depend is that the length will just be your e of t okay and so we can choose so we'll choose a coordinate system let's say a long in this should be familiar from previous class so we'll want to do something like this you've chosen an origin and we've chosen it so that kind of lines up where they start so there's our n vector and there's our y hat direction and we kind of want these to line up so we don't have to worry about the n hat and this is the next hat direction I'm sorry they just see we don't have to worry about the n direction at least for this and we're thinking about this vector LFT but now it's not radius one so here is LFT we have a radius that now depends on time and the important thing from before was that we know that some some components of this vector just won't contribute to the amount of energy that's absorbed so this red thing is not going to be contributing anything because the light rays are sort of parallel to the leaf and this green one we'll be doing all of the contribution this green one is maybe L sub y I should be labeling these with vectors everywhere this green one is L sub y now depending on t and this red one is L sub x now depending on t and as a as a vector we have LFT is equal to L of y of t remember plus is this kind of funny thing here it's not like an addition of numbers it's an addition of vectors where we do this tail to tip kind of addition on them um and it's just equal to the sum of these two component vectors so what'll happen here is that while the radius as it depends on t what we've been calling the radius kind of all along and all of these geometric things we've been doing the trig functions this is literally the same thing as the norm of the vector just now we we've chosen a coordinate system so the radius actually makes sense because we can measure you know a distance from zero because we've chosen a zero um and we've set that so this this is just kind of a fact that the radius in this picture will be the length of that vector and we're setting that equal to E of t and the question is what is the norm of L y of t so we have some vector L sub y this y component is changing in time this one you kind of have to imagine but maybe you can draw something that makes it more clear um yeah so you can imagine this vector being swept out over time and if this vector is like down here at a really small time then this green component is really short and the right component is maybe long maybe if it's sweeping out some kind of circle maybe it's growing changing in size over time and you get up here then now the green portion is very long and the red portion is very short and if you're you know directly up the green portion is everything and there's no red portion kind of the same story over here in quadrant two so the lengths of the red and the green kind of change over time and we want to know what the length of the green one is and here we'll just use the standard kind of trig stuff we've been using before so here's this thing we said was theta sub l we can label an adjacent opposite in hypotenuse we know the sign of theta sub l it all depends on t now so we have to kind of keep track of that everywhere we go well I just know from the geometry of the situation if I just fix my time t it's just the opposite over the hypotenuse and what is the length of so I guess I should say the opposite is the length of l of y hypotenuse is the length of l adjacent is the length of lx and I just mean here the vector lx isn't the same thing as like the length of the vector or that the number for that line segment like a vector has a magnitude and a direction and number is just a number so like adjacent opposite opposite hypotenuse these are all just numbers but they're kind of associated vectors that have those likes so if I have this geometric relationship well I know that the opposite is given by the length of l y t which is the thing I'm looking for and the hypotenuse is given by well as the radius as a function of time but then we saw that this was just we're going to make this e of t and so one thing you can conclude from this by kind of moving the e of t to the other side set the length of l y of t is equal to e of t sign of I guess in this case it was theta sub l of t this is an important thing to work out in the project again you can't really just put in this this final answer you need to like somehow go through some of this derivation of like having the components of this vector identifying them in a graph saying what all like saying what the geometric picture is and then doing a little bit of this this kind of derivation of how do you find the the contribution I guess of the the y component the component that lies along the um the like in the vector in direction okay with just a few minutes left so maybe I can say finally what what else has to go into this so this is I guess part three or four this we now want to compute total energy density and this will be some function gamma of t and what it'll be I'll just say here is it's going to be the component or sorry the the length of the component that's lined up with the leaf um in this picture above so that incorporated like the the energy of t incorporated the kind of angle and the idea is we just multiply this by an area and what this ends up being say for what we've just done above is e of t times sign of theta l of t times the area okay so you'll get some kind of function that depends on t and it should somehow look like the function you've chosen for e of t but also you see there's this kind of sinusoidal component coming into it too in the area it's just a constant it's just the the surface area of the leaf um so you'll get a new graph you want to graph this thing so this is going to be gamma as a function of t and I can't really say anything about what this graph will look like you have to it'll it'll depend heavily upon what you choose um for your e of t but I don't know it maybe looks something like this probably not exactly like that it may not increase and decrease in this way but I think it should be zero at time zero and it should be zero at time 24 and so you want to graph this on this interval I'm sorry I just saw somebody mention something in chat you care about the angle the leaf hangs from the tree do we assume it's parallel to the ground um I have to think I think for this first part yeah so I think in the first part you're going to assume that it's parallel to the ground then there's this second part about yeah maybe the leaf is at an angle or maybe it's rotating throughout the day so I think where that would come into play is yeah sort of when you set up this picture is that your this picture might be skewed a little bit um and like at some angle in which case yeah so if you imagine that this whole picture was at this angle instead which I think is probably what's what's in the handouts um there's just some third angle down here uh call it psi and then the only difference would be that this new thing would be theta sub L plus some constant angle so essentially all of this work the same except for maybe you have to add some constant angle psi which is measuring the yeah maybe it's the angle of the leaf from the ground and what I think what that'll do is it'll probably just take these graphs here and it'll shift them up vertically a little bit but also I don't mind if in this first part if you just make the assumption that things are just level with the ground for the first part that's totally fine okay so what we want to do here um so what'll happen is that this is like some kind of energy density and so if we want to compute like total energy this will be equal to like a density times uh like a time interval okay and this is like essentially area in this graph okay so what happens here is that you'll you'll start at some like whole time t0 you go up to t1 and you will measure whatever the uh let me do it in a slightly different place let's do it over here t0 t1 so this is going to be the total energy used in this time period is something like this so you just think about this region following the graph there okay so that's the actual energy being used in that time is just all of everything that's in here so what we're going to do is approximate that by saying okay well we know the function value at this point so we're just going to make kind of a rectangle here and we're going to measure in green what is the area of this rectangle that's going to be our approximation and there's going to be some error associated to it this stuff in red we're not counting but okay so the way this will go is that this uh interval will be with delta t equals t1 minus t0 and so this thing our approximation will essentially be gamma at t0 that's the height of this this rectangle just evaluating it here t0 gamma of t0 and then times delta t and then to get the total energy what you want to do let's say so this is total energy in one like a interval of the form t0 to t1 so you want to break up 0 24 sorry I realized we're at time now so I'll wrap this up really quick let's say what what needs to go here um what you're going to do is break this up into say 0 to t1 and then some other union t1 to t2 so on and so forth all the way up to tn to 24 so I forget exactly what the project hands out assumes here maybe it's like these are 15 minute intervals or maybe they're one hour or two hour intervals in the example and then the formula will be essentially delta 0 times delta t well sorry delta t0 delta t plus delta t1 delta t plus so and so forth so again you just evaluate the function at the first coordinate that's like the height of the rectangle you multiply by your delta t you add all of these things up and then you get an approximation okay so I think that's probably all I have for today I will try to jump in office hours let me so let me pull up the time because so between 11 and 12 I will aim for and then also three to four so office hours 11 to 12 three to four today and I will try to post them up for tomorrow as well although I don't know the exact times yet but I'll try to post those on ELC and then just email me if you I'll do some buy appointment outside of that if you need it so all right thank you guys very much