For those of you that are here and like if your partner isn't here, you will probably want to make careful notes and relay any project stuff that we're talking about today related to your partner. I think just because the last project, people missed out on just a ton of easy points on just things we mentioned in class that had to be in the project. So definitely don't don't get caught in that. Anything I'll try to mention today sort of how I'm going to break up the actual submission so like for the last one, you know, in the grade scope submission there were, I think three or four sort of sections I put for you to submit your work under. And so I'll try to at least outline what that'll look like today so you can kind of hear, you know, start thinking about what your what pages you're going to put under which which part of the submission. Right so just seeing people kind of an indication of where you should be so just remember that the the due date on the project handout I want to say it's, let me check here. Yeah, they have it due today. Okay, so I've decided to give us a little bit more leeway on it. So I think we'll probably have it due next week. But you really should have met with your partner at this point you should have started working on the first section. I'm just like looking at these two functions that are showing up and maybe trying to plot them or solve for m or solve for tea or something. I'm just going to go way more in detail what has to go into that section today we'll move on from. We looked at the example on Tuesday the example calculation and today since we're getting closer to when it's actually do look at the actual equation and I can help walk you through how the derivation for those goes. But that's kind of where you should be like you should have definitely started looking at this at this point. So you may want to get up pretty lost. So you may want to spend the next couple of days trying to take it back up to speed, you know plan a meeting with your partner. You know start working through the first part of the handout, and probably start, you know, making graphs and typing things up at this point. I wanted to make a quick note let me share my screen. Okay. And just because I've gotten a lot of people emailing me since we're six weeks from the end of the semester, maybe five something like that. People have been emailing me about grades and how to revise assignments and all this kind of stuff. And so I have to make a blanket statement that grades for this class are extremely difficult to compute in the middle of the semester. So today we have a couple of projects that are worth, you know, there's there's still at least one more and then a final. Also, there's at least a final at least a final potentially one more project in addition to that. And these are worth a huge chunk of your grade I think they add up to like 35% or 40% or something. So it's kind of impossible to say you know what anyone's great is it, you know this point in the semester. But if you want some indication of sort of where you are the best way to see that is to just go into grade scope and see what you've submitted and kind of what your point totals are. If you're missing any points at all on free classes that's maybe grounds to be a little bit worried the free classes are supposed to be kind of on the easy side, and really just line up with the videos. So if people, like if you're missing points on that, I would guess it's just because, like maybe you spent through the video or you skipped a video or something in the middle, and maybe just miss some key concept that toy and talks about one of those, or potentially there's like a mathematical writing sort of issue that you're running into that I've probably commented on in previous assignments and just hasn't been fixed yet. You know, they will also be taking off like 2% or 3% or something for the early assignments, and now since we're three quarters of the way into the semester I'm taking off more like 10% or 15%. Just to indicate that, you know, these are things that you have to look at and work on for math writing style, in order to be, you know, to be considered a successful student on the, when you're coming out of this class. And so some people have also asked me like what constitutes good math writing style and I just wanted to point out like I have all these solutions posted on my web page. You can just go to the class page, go down, look through three classes. And so for example I have one up here from 1.7. So the main thing you want to keep in mind is that in your solutions you should not be, you know, writing down a solution, like it's not really, it's not a race to collect up points for getting the right answers, this is. We talked about this a couple of times in class where like somehow the solution is the least important part of the problem. You know, the method, the process, the algorithm, that's the valuable part because that's the part another human being can use to solve a problem they care about instead of this one specific problem we're looking at in the assignment. So you should gear your solutions not toward writing the answer, although of course you do have to have the answer somewhere, but you should try to explain to your reader, imagine that your reader is just some pre-calculus student who doesn't know this yet or imagine you're finding it to yourself six weeks ago or something when you didn't know about it. And sort of, you know, just indicate what you're doing, what the method is, what things you're using from class. So you can see in my homeworks I've done this in my write-ups for the solutions I've done this to the extreme where I've almost, you know, got the entire concept in the homework. So here this is homework about even and odd functions. And I've said something about, you know, here's what it means to be an even function, that's an important thing to include. Later on I've said here's what it means to be an odd function. And then I've told the reader, you know what I'm doing before I start any calculation I've said here we're checking what is f of a negative x. And you should be able to read it out loud as a sentence well f of negative x is equal to four times the quantity negative x cubed minus negative x, which is just equal to something else just equal to something, so on and so forth. And then I've indicated, you know, in words and in like written English words, you know, when I'm computing next to what is it negative f of x, and I've computed it. I'm like logically reading, leading the reader to my conclusion I'm saying look I computed f of negative x I computed negative f of x, and I found that they were equal, so I can conclude f is odd, because I, you know, I told you the reader just a second ago what it means for a function to be odd. So, you don't have to, you know, go out with fancy graphs and explaining things in that way. That's definitely like, it's probably a little bit too much time investment for the homeworks but you know do again say say what you're using from class say that the general form of an equation if you're using that or say that the definition. If you're talking about something like function being even in a rod. You know, what you're computing and words like a lot of a lot of students will just kind of write out a calculation. And this is a confusing jarring thing for the reader because the reader doesn't know what you're calculating at that point they just see, you know, a bunch of math symbols on the page. So try to like augment what you're writing with a natural writing with, you know, words. And most importantly, well, not most importantly but like, you should indicate what the actual solution to the problem is. So here I've like boxed it I've highlighted it I put a period after it, because it you know at some point the, you have to connect it back to the original question homework is asking a question, and you want to indicate what is what is the answer to that question. So remind yourself at the end of the problem what question was asked, am I am I answering that question if so box highlight, put a period, squiggle, put a cloud around it and like anything you want to do but like somehow highlight or indicate that that's like the main point or the punchline or the answer. And you can see, so this is this is from an older assignment this is like 1.7. This is from a newer one where it's more of just a calculation. So I've used far fewer words to describe what's going on because the problem is really just solve this equation. Right. But I've tried to like set everything out in a series of logical steps to kind of lead the reader through the method that I'm using. So like in this first line here we're computing some we're doing some exponential formula and I've used this arrow to say okay look I'm taking the log of both sides now. And here you know maybe now I'm bringing down a power, I don't necessarily have to explain every single algebraic step in words. Usually it's pretty clear if you're just looking at the equation if you've just taken small steps between each line, it should be clear kind of which step you've taken. And so, here I've just said, I'm taking the log of both sides, here I've kind of like moved the power down from the log. And here you know I've just isolated X. And here maybe doing some simplification. And then at the end of the day I have a solution to the problem which I've highlighted I've boxed I put a period on it, just to indicate that you know this is, this is the destination we were hoping to end up at. So those are kind of two two different, two different ways to approach writing up solutions. You could also off to the side say what you're doing. Here, off to the right hand side of this first line I could write taking the logs of both sides in words. But I see a lot of students will like modify an equation in place like maybe they'll just take this equation here and start writing logs on it and start, you know, cancelling things out and stuff like that. So try to try to avoid that because it's it's confusing for the reader and it's not good for bookkeeping for your own purposes. If you're very clear about doing one step for each, each line. It's much easier to catch, you know where you might be making a mistake. And, you know, in which case I can help you much much more with, you know, with the actual mathematics if I can find, like some specific mistake where you've gone from one line to another but that's, you know, there's something and so you can get better feedback out of it that way too. Okay, so that's all I'll say about homework just read the announcements for set some stuff about revising if you want to get points back and then just make sure you're you're writing things up nicely. Okay. Speaking of which a lot of this writing math business will be important for the project. So, one thing I've linked to, if you go to content project to you may want to go ahead and pull up the handout now because we're going to start going through this in a second. But there's some other links in there. One of them is this writing mathematics well sort of handout. Not for me but from someone else at Harvey mud. And so what I'd recommend is like if you want to get a super good grade on this project I would a read through all of these extra resources they're extremely valuable. Some of them are kind of a long read. But if you really try to like sit down and take maybe an hour or two. And sort of see what points they're discussing in these articles and make sure that your project here so to those suggestions then it'll improve your grade quite a lot so it's worth, it's really worth the investment here. There's, you know, there's a lot of hints in here for writing mathematics well and it sort of talks about the difference between, you know, sort of not so great or poorly written solution and like a nice well written solution. And you can use these these ideas in your project as well. So there's this is the writing math well hand handout. There's this. This is some example of mathematical writing just for this is mainly intended like if you haven't written many like scientific reports that include mathematics in them. This is maybe a good way to get an idea of kind of just what what that means what that includes. You know, writing your your project sort of like a research article kind of like this. And I don't mean that in the sense that you have to format it this exact way, or you have to include every kind of section there including. I'm just pointing out that this is, you know, just some kind of guide for, you know, what you what you might include or how you might structure it. And that sort of, you know, be clear on what's important here. You know that the art the thing has a, this article has, you know, a title down here we'll talk about this in a second and it has sort of an abstract in the section here. I'll kind of say why it's an important thing to have. There's this introduction section where this is familiar from project one where they're kind of laying out the physical background describing things. You know, in a standalone format which you definitely don't have to use but you're totally free to. And you know, after that they have this other section on materials and methods and it's like it's clearly delineated from the first section you know there's a little bit of space between this and the previous paragraph to indicate that. You know if somebody is like scheming through the article and just trying to find, you know what were the methods for this experiment or something they would know where to find it. So if you include a graphs things like that, you know they've labeled them with, you know, saying what they are figure one some description of it. If you refer to it in the text you might refer to it as you know, see figure one or see figure two. And so if you include a graph, you know, for example in your project, then this is something you might want to do is label it with a figure number and refer to it by by its number. So these, these aren't so important. You can see here some mathematics is starting to be included. And what they've done is for inquiry equations that are important or somehow like if it's a, if it's the punch line of part of your analysis or it's some fundamental piece of it. You should try to highlight that or indicate that in some way here they've chosen to, you know, center the equation put it on its own line. You know, they've given the equation a number labels so they can refer to it in the text as like, you know, an equation one, this thing is equal to that thing. An equation to this to know it's that. So you can do stuff like that. And importantly what they've done in the text is they've, they've mentioned what the variables are what they mean. You know, especially in this project we have a lot of unknown variables floating around. So the reader really needs to know what these things represent and these you can find these all in the project handout like what the interpretation is. And what else. I think that's that's mostly it just kind of glance through the rest of this to see how they're how they're approaching it. But notice that there's not really a huge amount of the calculations that they're doing necessarily. For us, you know you'll want to include some of the calculations, you might want to like have a small CC here they've like denoted a subsection here in the bottom right.  You might want to have a subsection about the derivation or the calculation that you're using. But usually you can you can kind of minimize that because somebody reading the article may not be as interested in repeating every single step of the calculation you did, but they'd want to have enough information that they could reproduce it because you know you imagine somebody is in a lab on the other side of the world you want to give them enough information to reproduce the experiment. While also not just like drowning in in mathematical notation. So yeah there's there's a little difficult balance on that and since this is a math paper you can air on the side of including more calculations. But there was, there were at least a few papers, not just in this class and in a lot of classes where you know people would just have one page and it's just like all calculations just line by line by line with no words or context or anything breaking it up. So you definitely want to avoid that if you're doing any long calculations, you should try to like stop in the middle and explain to the reader what you're doing. So this is less about like recording what calculation you did and more about like telling a story to your reader and if the calculation is an important part of that story then include it. Okay. So that's that and then there's this, just some more general writing tips here in the scientific reports link, which just gives you a lot of like, sort of in depth details about like what each section could include a lot of this applies to things more like your physics, but you can take take away a lot of this and use it for mathematics reports to, especially since this one is like science based in some way we have this like zooplankton thing going on. It's really important what they you'll see that they mentioned drafting in this the same this handout. So definitely go and read the whole thing. I think one thing that's important for this project is to separate kind of the same way in the homeworks that you have like your scratch work, and you have the solution you present. And we should really be thinking about these two different things like the scratch work is what you do to come up with the solution. What you present is what you would like how you would tell another human about that process. And so with the report, you know, maybe you have a draft where you're like sort of going through the analysis and you're kind of discovering what the story is. But at the end, you might want to repackage it in a different way to tell a story more clearly or like rearrange the analysis or like omit parts or include part extra parts, things like that. So you really have like a sort of a first draft where you're just kind of like working through and figuring things out. And then there should be a second draft where you can kind of try to rewrite it and restructure it to more effectively communicate. Yeah, just definitely go through and read this I think maybe for all three of these it would take maybe an hour or two to read through all of these just make, you know, make a couple of quick notes on things that stand out as important try to incorporate incorporate them in your project if you do that. You'll probably be fine. In terms of like points for writing style, things like that. I'm sorry and there's one last thing I've linked in here. If anybody does want to use this. Sorry, you can type up your things in Microsoft Word. But so like in mathematics we have sort of a funnels like a programming language we do for this like if we're writing papers or submitting them. And you might run into this if you're in like chemistry or physics, maybe, maybe in biology kind of depends you gone. It's probably about it maybe computer science as well. So it's it's a programming language called Lee Tech, and I've included a report for this is the IEEE's view what it actually stands for it's like one of the big engineering organizations, and they have, you know, they publish journals and things like that and here's like an actual template they use for their journal papers, sort of laid out.  So maybe a little bit too much definitely don't worry about this if you're strapped for time at this point. But you can go in there and if you want to like try to modify the actual text. So to see what happens. Let's see. Yeah so here, there's this thing that says section introduction, and all of this text here it's just corresponding to what's written in this introduction section. So you can literally just go in there and start typing other things out, you know whatever it is, and recompile, and now the introduction says something different. Okay, and you can look up, you know it's if you want to learn how to use this you can just sort of Google around for things that you need as you need them. But for example if you want to do mathematics, you can do a dollar sign in text. So, you can do like f of x equals mx plus b. Okay, recompile that. I don't quite see it there but that's included some like really nicely formatted mathematics in in line. If you want to put something on its own line you just use two dollar signs, sort of like this, you do g of x equals x squared plus c, recompile that. You can use this equation on its own line. And you can figure out if you Google how to number equations or sort of whatever you need you can sort of sort that out if you'd like. This is just available if you have a little bit of extra time and want to like format it nicely and get used to, you know like writing up super super professional reports like this is actually what you'd use if you were submitting a paper to the IEEE. There you go. But don't worry about it if you're if you're just using word or something that's totally fine. Just make sure you're figuring out how to format the mathematics correctly. Okay. Let's talk about project two. Before I start going into so we have some kind of gnarly computation to do today. Does anybody have any specific questions they want to try to look at sometime in the next 45 minutes. So I'm not hearing anything. If anybody has said something and like I'm not hearing it then just mention it in chat. I'll try to keep an eye on it, or just if any questions come up. So we talked a little bit last time about what the physical setup for the project is so I won't skip. I want to mention too much of that right now. And I do want to say a bit about is how we'll chunk up the project and kind of what this computation for the first part will look like. Okay, so essentially is going to be three, three sections that you'll be working with on the grade scope submission. So it's going to be the abstract. These are the main sections abstract and introduction, and you can just sort of look through the project's handouts sort of see what needs to be included in the introduction I'll say something about the abstract in the second. Number two is when I'm calling some kind of parallel analysis, which we talked a little bit about last time there are two equations and we want to use some kind of structured analysis of both of them. I say parallel because we want to structure them in such a way that we're kind of doing the same analysis in the same order on both. And number three, which we'll talk about more today will be this intersection analysis. And that's pretty much it's really just these three sections. Okay, so I should say what needs to go into the abstract. Right so in terms of looking at part zero. So you'll want to have some kind of like title page and title.  If you want to have some kind of abstract. Okay, so see the example writing that report we looked at a second ago, it's linked on ELC under project to to get an idea of the difference between an abstract and an introduction. But these should be short, something like maybe one half of a page at most. And the whole point of this is just to, to just include enough wording to state your main conclusion. So you don't really need to set up as much of the physical background in the abstract you just want to condense as much as much as much of it as you can. Exactly the minimal background necessary to state like what your results are, what they mean how you interpret them. Like what's what's the punchline of your whole paper what's the story you're trying to tell that should be encapsulated in the abstract, you'll have some intro. And I think everybody pretty much knows how this goes from the last one, I don't think anybody had too many issues with that but just to say in words, kind of what you need. So if you're setting up the physical situation. You can also do a separate section if you want to set up the physical situation so you can have an abstract intro and a physical situation section, where you can just do it all in the intro if you'd like. So, set up the physical situation. And I just want to mention that you probably should write this last. Maybe it seems more logical to write the intro first and then do everything else.  But the intro really should be setting up. You know it's like laying the groundwork for your story. It's like putting a setting up. And so you really won't even know what needs to be introduced until after you've done the analysis. So I would just I would write this last and then at the very end of the day go back to the intro and think okay I've done all this work. And we write something here to keep in mind is what is the major question answer. I would like to start this three times because this is sort of the point of the introduction is to sort of the abstract was a very condensed version of like, what's your question and answer and here's where you want to like dive into it a little bit. Maybe explain the question in a more nuanced way and explain the conclusion you've come to. So this is why I say right this last because you might not know what the question is or what the answer is until after you've done the analysis. I just want to write here and just to remember the target audience. So it's very important here to remember that the person you're writing for might have like a background in pre calculus, you can sort of assume that from the reader so they have the mathematical sophistication to understand the equations you're presenting to them. But you have to imagine like they've never seen this project before they know nothing about the methods they know nothing about this physical situation. You're kind of building it all from the ground up. So if you introduce anything you really do have to explain to the reader what it is what it means. This is especially true like mathematical equations. You don't want to just like lay out an equation and be like here's our analysis that we're doing on it. Let's explain a little bit what the equation means if there are variables, tell the reader what the variables, you know mean maybe G for generation time or something. And try to like motivate to the reader where these are coming from. Okay, so this is all section zero. Let's get into part one. Okay, so this was like the calmness of parallel analysis. Okay. Yeah, maybe I'll give people a quick break since we're about to do a fun calculations you might want to like brace yourself for it. I'm just going to grab some coffee. Okay, so I would at least try to follow along with some of these calculations keep in mind that like if you're taking notes or something. You don't necessarily have to write everything down I'm going to post these notes that I'm writing after class, but it's really really important to understand the derivation and how you're solving and why you're solving. So the first equation we have is something that looks like this, the log of G equal to our log of M minus AT plus C. Okay, and I'm going to underline here in green. So there are a lot of letters floating around. So let me actually, let me do this first. So this thing is equal to that. Then let me define a function let's call it F one. Let's see what'll it depend on it'll depend on G, R, M, A, T, and C is a six variable function. So let me select everything up. Let me move this down. So if I have this equation here. Sort of the immediate thing I can do is, let's move this to the other side, the negative sign set this as zero equal to that. So I've just collected everything up on one side. And I call everything on that right hand side, a function F depending on all of these, these variables. And so we're interested in this F one being equal to zero. Okay, so hopefully this isn't too intimidating intimidating at this point we've seen some crazy functions in the previous quizzes. We're just thinking of all of these as unknown variables is kind of what we're giving. So this is some crazy six dimensional surface. It's not something we can easily, you know, visualize a reason about. So what we're going to do is we're going to do this dimension reduction technique. And the way we're going to do that is we're going to. So we talked a little bit about this in the last class, we're going to fix a bunch of variables, and we're just going to let one vary at a time or something like that. So what we're showing up here is that we're going to have numerical estimates for our AC and sort of G, G we at least know is, we'll have some restriction on the range, or that the domain for that. And these will have numerical estimates. And so the things that are remaining will be independent variables. And this is something we can actually work with there are only two variables floating around. So this is something we think we can like graph on a plane. So we've kind of just gone out into the literature and we found enough estimates to reduce the six dimensional thing down to a two dimensional thing. And we'll analyze that instead. So what you will want to do is take this equation, and that equals zero and you want to solve for M or T. So just one of the red variables from above. And this is again just the motivation for this is that we want to put this on a graph. And so we need one of them to be a function of the other one, if we want to graph it. So let us do, let's solve for M in this equation. Okay, so if I have this whole thing is equal to zero. Let's move everything that doesn't involve an M to the other side. So I guess I move me out a log of G to the other side. It becomes positive. If I move all of this to the other side, this becomes negative. Okay. And let me move this R so I'm going to do the sort of reverse exponent rule and move this up like that. I'm going to do the log of M to the R instead. Okay, so I've done a lot of operations, all at once here if you're writing out your derivation you would really want to do these and like separate lines. But this is just for the sake of understanding and running through the argument right now this is not how you would want to write it up. So if that is true. And so the first line logically implies this that line logically implies this. So I want to isolate M right so I can exponentiate both sides, and see what will happen here. Let's see. So we're taking E to the log of something, and that just gives us the something back and something here is this. So we kind of excise that out. And we're just left with that. So here on the right hand side I want to use the sort of addition rule and exponents and I want to move this off into another exponent. We all do that is having any down here and then these will be multiplied. Okay, and I will surgically extract that G because we're taking E to the log of something it just gives you the something back. So this is just G, and maybe I will distribute this negative sign into this goes to a plus this goes to a minus, I can erase that. And this is what we're left with. Okay, and now I would just take the earth root of each side. It's all I'm doing is just taking another version of this equation. Both sides to the one over our power. And I sort of cooked it up so that on the left hand side is just keep canceling out that are. And they have an M left. And so I've extracted the M from this equation using exponents and logs and I will want to save that box it highlighted, put a period on it because this will be an important quantity for us a little bit later. Okay, just before I do anything does anybody have questions about that derivation or any confusion about any steps, like, we can revisit it. Okay, so like if you're including this in your report. You would not necessarily want to go from the first equation all the way to this directly. You would want to include this this is maybe part of a derivation you might want to include just walking walking through some of these steps. And if you don't walk through the steps, you know you should at least, you should explain to the reader kind of how you got from the first line to the last line. So you know explain in words that you know by collecting things to one side and then exponentiating and then taking roots, we are able to isolate M, so on and so forth. Okay, so we just zoom out. Let's see what happened here started at that thing equals zero and we solve for M. You could also run through solving for T if you want, in fact that might be easier for this step but I think it's harder for the later step. So I would go with M personally here. And what you're going to do now is now that you have this. So let me think of this now as it's really like M is a function. So now we're going back to thinking of it as a function instead of a number kind of depends on G. It depends on a T, C, and R. So we're down to something five dimensional. And we're going to have an estimate for the G for the A the C in the R, and then read will be the only independent variable. So we can really think of this as like M of T, or maybe even just call it like F one of T. So we're just, we're just giving a name to the function that has this formula in the middle. But now if we have some function of T, we can do all sorts of things. You want to graph it on a plane like this. This should be M for the dependent variable T for the independent variable. And let me actually call this F zero of T, if not of T, because this is like we've, we've not done any modifications to it. So let's make one way to remember it. The shape of this function is going to be a positive exponential, it looks like, except maybe I've lost the sign somewhere. Let me double check. Okay, so I'll have to double check this really quick. So I want to make sure we got the right, right sign before we jump into other stuff. L and G equals R, L and M. Let's see. Whoops. Yeah, I think something, something went slightly wrong in the first step. Let me just double check here. This should be fine. So what'll happen here is you get some positive exponential like this as your F zero of T. And what'll happen here is now you'll do some analysis of, let's see, go back to the green variables, it'll be like change G by a plus or minus 1%. So you have this F zero of T is like when you've said G equal to some specific number. And then maybe F1 of T is where you modify G by plus or minus 1%. And maybe F2 of T will be where you modify the second variable A by plus or minus 1%. I'll essentially repeat this for all of the other parameters that we've estimated because sort of the point of this part is that we want to see which parameter affects the outputs the most. So we're going to wiggle each parameter individually while holding all the rest constant and see what happens.  Try drawing them in a different way. So they might not intersect, you know, like right along the axis or something. And so these are supposed to represent the F1 of T corresponding to like plus 1%. This is the minus 1%. This is like F1 of T for the plus 1% or something. And so you definitely want to have some kind of graph like this. You have to worry a little bit about it getting too cluttered, but like sort of like one graph for each variable. And right if you if you go in, you might do a plus plus 10% and then you get something that's more like that. And that this would be like an F and call whatever you call it F10 if you want F10 of T, which would be like a minus 10%. F10 of T for a plus 10%. But somehow just try to indicate to your reader how much does this graph change as you change that one parameter G. And then you just, okay, that's for just doing this, this one parameter. And you just do the same thing for the second one. So this is why it's kind of nice to solve this in the abstract form with all of these parameters. Just kind of floating around in the expression because now you can easily put it into Desmos or something, make these sliders, set some of them to specific numbers and then just like slide one at a time, generate a bunch of graphs, slide a different one at a time, generate a bunch of graphs, so on and so forth. Okay, so now we want to, that's really all you have to do for that function, but now you want to repeat for the second function. Okay, let me just write down what that actually was. It was something like just going back to the project handout I'm just writing down. This is an equation two. So this is given to us as I equals I not m to the d e to the negative e over k t and d is equal to three quarters in a box and highlight this because we'll need to remember it for later. So this is something I would recommend doing in these kind of situations if you ever run into some huge equation and like in this case it's really like m to the three quarters. We're already dealing with four letters, it's probably not that much more difficult to deal with five. But it's kind of easier to track where this one letter goes than to try to worry about where this fraction three quarters goes because that's kind of two things to write at each step. So we'll need to remember this. So we'll want to back substitute at the end. Okay, and I can just play the same game where take this whole thing. I just want to collect everything on one side. So I will subtract I from both sides and that's just zero on the left hand side. I will just call this whole thing, maybe F2 of, okay, what does it depend on I zero m d e k t and I this now depends on seven variables looks like. And what will happen is that will essentially have an estimate for we have an estimate for I not over I or I over I not. So we'll have some way to get a number out of that. We'll have a numerical estimate for D, E, and K. These are again the miracle estimates, which brings us down to two variables, M and T that we can work with. Okay, so now we want to take this sort of just like the last one solve for M. I'm just going to move that back over and I back over to the other side because it's going to be easier for this part. Okay, so if we have that equation logically implies that I can have this equation. So what will I do on the first thing I'll do is maybe divide through by I not I want to isolate the M. So maybe I will multiply through by each of the positive this thing. The effect that that has is moving this to the other side with a positive sign now. And now I want to take the roots. And now move this over here. Again, this is not the way you would want to write it up this is just kind of one way to communicate as we're speaking face to face and you can sort of see things happening in real time kind of what operations are going on. But if you were actually writing up the derivation you maybe you might want to do more steps in between. Here I can raise this to the one df power, raise this to the one df power. And okay at the right hand side, that exactly cancels that out. And I have isolated him. Yeah, it does seem like I lost a sign on the first one. So I'll revisit that in a second. So what you'll do here is you can think of this is now, let's call this thing. We're thinking of it now as like in as a function of, let's do it this way. Take this whole thing. Let's say I over I not let's call it I prime, save ourselves some writing where I prime equal to I over I not that will be important to remember. And we'll have a numerical estimate for that so we're fine. And so what we can think of this whole thing as M as a function of I prime. E, K, T and D. So we're down from six variables to five. We'll have numerical estimates for I prime, E, K, D. So we can really think of this as like M as a function of just T, once we've plugged in a bunch of numbers. So maybe we'll just call this something like g zero of T. This is some other thing you can plot. Like this kind of the whole reason we solve for M in terms of T is because now we can compare these two functions on the same coordinate axis. So we can have an M here. T here. And sort of the same situation will play out. G not of T will be this. And now you vary. I prime by plus or minus 1%, and maybe call this thing like a G one of T. And what is this going to do, you're just going to go in, you're going to get a slightly different graph, same sort of business. And if you vary it by 10%, you'll get more of a change. So you do all of that with just I prime and then ditto but now you do it with E. And it's 10%. Maybe you call this thing. G two of T or you'll have to come up with some kind of labeling to indicate which one you're looking at any given moment. And so on and so forth, one for each variable. So essentially one, I guess this amounts in four plots. Looks like one for each variable up here in green. So each one should have some number of different graphs showing how you're varying these at different percentages. And you might also want to, you know, you might want to like fix, fix an M here, you can do this in the first one to fix an M here and compute the modified values of M. This is like a T not M not. And maybe if you vary that I prime parameter by 1% it moves that M on the first line up to this M on the green line, maybe M, M one. And so you can compute delta I prime. So this is, yeah. Yeah, delta I prime so how much you changed it just remembering this would be like 1% or something or 10% over this delta and M's, which is just the difference between this M one and M zero. This is just giving you some, some relative rate of change. So you change I by 1%. How much did M change as a response for just some fixed M, and you can compare these in a table if you'd like. So I need to look back at the first part because we lost a negative sign somewhere. Just need to see where that happened. So let's see if log G equals our log M minus a T plus C, that looks good. And so just double check one sec. Okay, so I have to double check this later. And if you're doing the assignment you should go ahead and try to redraw this for yourself. So just think what happens here is that you're supposed to get a negative exponent. So I'd have to run through the calculations again just to make sure I didn't lose it somewhere. Let me say something about the how the calculation goes for part three assuming that part. This one's a little bit more difficult. So I had something like this for first one. And I had something like this for the second equation. And so first thing you want to do is plot these together on one plot. And roughly speaking you should get something that looks like this is going to be M T. This will be the graph of one. This will be the graph of two. I'm sorry I just now saw that there's there's stuff in chat. What numbers have numerical estimates for F2. Yeah, so I will try to upload this recording later today. If you want to solve for T in the first part you totally can. So the only effect that would have is you would get different equations. This year, you would be T equals something and T equals something else above. And your grass might look slightly different, but you would go through the exact same process so you can pick one or the other to solve for him or to solve for T. Because the analysis would be this wiggling the parameters would be the same in either case. What numbers have estimates for F2. Let's see. Yeah, that is right. Sorry, is this one. Yeah, this is one. So for F2 the ones you'll have estimates for will be sorry I'm just scrolling all over the place here. Here we go. So it's I zero. I, D, E, K and T, or sorry, not T really really in both cases maybe the easier way to think about it is that you'll have numerical estimates for everything that isn't M and T. M and T are going to be the only unknowns throughout the whole problem of the whole project. We're just set to be a little bit careful on to because it's not like we don't really have I and I not we have the ratio I over I not, which I'm calling I prime, but that ends up being okay. Okay, so let's take a look at this last computation. What you're wanting to do here is set these two things equal to each other and solve for what T is setting these equal to each other is exactly the same as finding this intersection point. This intersection point is going to be the thing that we're calling the optimal like their optimal mass and temperature. So we need to solve for that intersection point, which means solving this equals this and solve. We just double check to make sure this is kind of a tricky calculations me to be a little bit careful to achieve the negative 18 minus C whatever our looks good. Whatever D that's good. The only thing I did differently was I called this I prime, and that's to the, it should be one over D on the right hand side. Okay, seven minutes. Can we do it? Let's let's see. Okay, so math goggles. What I want to see is that T isolated by itself I want to solve for T equals something. So there are really a lot of ways you could solve this. I'm going to do what I think is probably the easiest one. The first thing I will do is take this whole expression. I will raise both sides to the D of power. Okay, so if I raise this whole thing to the D on the left hand side that will be equal to this whole thing on the right to the D power. I can simplify a little bit on the left hand side by just bringing that D to the numerator of the exponent. And I kind of cooked it up so on the right hand side this stuff is all canceling. Okay, not too bad. I still want to isolate T's, I see T's and exponents so I know that I log is going to be what I want to do. So I can take this whole thing. And so if that thing was true, this thing is true, then this thing is true. Here I will take the log, I have to be a little bit careful. I have to take the log of the entire left hand side. It has to include that power of D over R. Okay, and I have to take the log of the entire right hand side. Okay, and how does this help us? Let's see. So on the left hand side, the first thing we're going to do is use this power rule to bring that thing out. So we get something that looks like D over R, the log of all of that stuff. And on the right hand side, I now have a, this is a log of a product of numbers. And I know I can write that out as a sum of logs. So in other words, I can take this and I see multiplication inside so I know I can kind of split that out. Be a little bit careful. On the right hand side I have, on the left hand side I have D over R. I have this log here. I'm just kind of systematically working through it just applying all the log rules from class. I see another multiplication so I'm going to split it out. This thing is equal to, let's keep the log of I prime. And now I'm going to use the exponent rule here to pull that E over KT out. And then I'm going to do this with a log of E on the inside. But okay, that's just one so I can get rid of that. Okay, I think I'm almost home free. I just want to get that T out of the log. So I can take this entire thing. Let's multiply the right hand side by D over R. Let's multiply both sides by R over D, rather. That and that. I'm going to split this guy while we are at it. Let's go ahead and subtract off that log G. Just remember you would want to spell out some more of these steps if you were writing it up yourself. Okay, now we have some room on the left hand side. I see something in the exponent for the entire argument of the log so I know I can move that out. And log of E is just one so I know that can go. What do I need to do now? I need to collect everything up. I think what I want to do here is move everything to the right hand side.  And I'm just left with zero on the left hand side. Okay, so it's a lot of steps all in one, but if you just kind of break it out into a bunch of little pieces and ready sign separately. It ends up just mostly being bookkeeping and then just making sure you're very careful with how you apply the exponent and log rules. Okay, so now I'm going to, I want to solve for T. Remember that was the goal. So I see a T in the denominator. I'm going to multiply through by T. So on the right on the left hand side still just a zero. Let me do this. Actually R log I prime over D plus RE over KT D. I'm distributing that R over DN first and then minus log G plus AT minus C. Okay, now I'll multiply through by T. So I get a T there. I cancel out this T in the denominator. I get a, let's see if we need to spread some things out. Easier just to rewrite. Okay, so minus T, T log G. Plus A and then it got an extra T squared. And there's a minus C with a T on it. Okay. Sorry, I know we're just out of time, but there's just one last step here. I now want to collect up all the things I see with T squared. So it's zero equals T squared. Okay, so I see an A and that's it. I collect up everything with a T. So I see this whole thing here. We've got that. I see a log G there coming in with a negative sign from that. I see a minus T or minus C there coming from that. And I guess plus a T to the zero term. It's just a constant. It's just this thing that's left. So as you now, let's take this and think of it as zero equals a T squared plus BT plus C and do the quadratic formula.   And that will give you the solution for T equals something. So that's like the main part of the derivation for that. I'll let you guys go. I'll try to announce office hours a little bit later and I'll post these up and the video.