 I want to talk a little bit about, like, what's the set up or motivation for this project? So the goal is that we are given a model. Well, let's maybe frame this as the setup. Okay, I'm going to write some things down. It's just a little bit of notation and I'll tell you what it means. We're going to have some function which takes n real numbers as inputs, as independent variables. And it's some measurement of those numbers. So it outputs one real number. And we'll denote this as something like f of x1, x2, dot, dot, dot, dot xn equals maybe just y. And all I mean by this is that there are these in input variables into the function. What does that mean? You know, if we have our function black box, like this, f. Okay, well, it still just has one output. These y values. But now it has n many, it's n is in undetermined numbers. We don't know how many there are. But it has n many inputs, maybe drawing a copy of the real line here. But it kind of goes into it in times zero and then y is also something that lives as a real number. So what this means is like, the way I like to think about this is f is some measurement based on some number of explanatory variables. So like in, in my own, you know, if I did sort of data science work for a fair, fair while. And the problem you would see with this kind of situation is like maybe you have a location somewhere in the United States. And, and then maybe you're some big retailer, some large coffee company or something, and you want to, let's say you want to put down a new location in some new spot. Okay, and so you might have like a number of variables you could measure at sort of all the spots in the US, like it might be, I mean, two obvious ones might be like, what's the position in the US? What's the latitude and longitude? So that would be two of these input variables. But other things might be like, what's the population density at this exact location? Or, I don't know, what is, what is the the gender makeup? Or what is the, what are the different like education levels or, you know, just sort of like all the things that you might put on like a census, for example, these could all be, you know, millions and millions of input variables you can include in your model. And there's a separate, separate question of like, okay, so which ones do you choose? We kind of ran into this issue in the last project of like, when you have the option of two explanatory variables, but somehow one of them is already explaining the other one. So in this previous case, it was like temperature and altitude, or two of the potential explanatory variables, but we knew they were already kind of linked. So what's happening in this case is sort of similar, we have, you know, in many explanatory variables, we're taking some measurement of it. In the case that like the physical situation I was talking about, it might have been, you know, latitude, longitude, a bunch of demographic information, and then the output would have been like some estimate for say the revenue that you might get at that location you know, a year after building it or something. So that's kind of like a, I don't know, an economic application, but you could also imagine this in the sciences where like the output is some measurement like an introvert interval, but now it depends on maybe many more variables. And then the question we want to answer, or the goal question is, which X of I has the most impact? So in my situation that I worked in, this was like a very common question, you know, you would go do some mathematics, you would build this huge mathematical model, which takes in all of these input variables and predicts their revenue. And the immediate next question they would ask would be like, okay, well, which variable do I change to increase revenue the most, right? Because that's, you know, at the end of the day, it's, it's, you know, human or a business doing mathematics or something. But you might imagine this popping up in the sciences to, you know, if you're cared about the introvert interval, and it depends on hundreds of variables, you might want to know, well, which one should we be paying attention to before like trying to preserve an ecosystem or something like that. And which sort of which one has the most impact, you know, if I change this temperature by a little, does it change the interval by a lot? And if that's the case, then maybe we need to be careful with what temperatures we put them in. And okay, so how are we going to do this? The answer is going to be the goal is to analyze this question. And so I won't say that there's like an answer to it. That's not so clear that, you know, one is the most impactful. The idea is to somehow consider different combinations and variables, see how changing their inputs changes the outputs, and do some kind of systematic analysis of this. So I want to set up what the actual physical situation is. So this should be on the handout if you want to follow along. Starting in section four. And okay, so the setup is it's another sort of biological situation. And this this time we have, so I want to say this is pronounced as zooplankton, but I'm not entirely sure. Maybe somebody, somebody can correct me if they know how it's actually pronounced. But I guess the idea here is to go through some development sort of life cycle, which includes this embryonic and post embryonic stage. And we're trying to measure this thing we're calling the generation time, which is somehow the total time it takes for them to go through these these stages of their life cycle. We know that it depends on two things coming from some paper that did the analysis and found out that these were two explanatory variables, namely, how much mass does this little zooplankton have, you can just go measure it weight on the scale. And also, this generation time depends on the temperature. So you can imagine if you're incubating these things in a lab or something, then, you know, you might incubate more over a fixed period of time if the temperature is higher or lower than room temperature or something. Okay, so these are coming from paper. So it's linking. I don't know how easy to see if I, okay, I zoom in to this paper one in the reference. You can sort of take a look at that if you want. But it's not super important for the immediate setup it's following, but maybe important as you're writing up your project and want to, like, if you're doing your introduction and your conclusion, you want to explain the physical situation, you might read the abstract of that paper for some ideas. Okay. So we know that the generation time depends on the body mass and the temperature. And we have this other factor. So here's the problem is we want to predict this generation time but it depends on two input variables. Maybe, maybe it's, you know, okay, time to draw a picture of this. Well, okay, I'll come back to this in a second. But, right, so the problem with this as we saw in sort of the previous thing is that if we had one equation that depended on two input variables, this is some surface in space, it was kind of difficult to reason about. So we tried to come up with a constraint equation to like have this surface and like intersect it with a plane or something and get some smaller small dimensional surface that now we can do recalculus and algebra on. So it's quite the same idea here. We have this generation time, which is the thing we care about, but it depends on two variables, the body mass and the temperature. So we're going to go out and search now for some other equation because this equation depends on two variables. If we can find some other equation which depends on the same two variables, then we have a system of two equations and two unknowns and this is something we can solve. Or rather, it's like an objective and a constraint function. So we can bring it down to one variable. So the constraint we go out and find is this business about measuring the metabolic rate of these zooplankton. We find that it's also limited by body and temperature. That's kind of what we were looking for. And we have some equations that give us the generation time as a function of M and T and the metabolic rate as a function of M and T. And so at least that's the physical setup. I want to say a little bit about how we're going to analyze this question. So in our case, we have, let me show you the same notation from last time. Let's see what the actual, so I'll come back to the actual equation in a second. So I think I'm mostly going to be walking through the equations from the sample. It's because I don't want to give away too much of how to actually solve the equation for the first part, but I'll at least write it down for you guys so you can see how to start working with it. But okay, how are we going to analyze this question? We're going to consider the quantities. Okay, so I'm going to use some notation here and then I will tell you what it means. This is what they're actually called. They're called the partial derivatives. So we have this function f for member, which is depends on input variables, x1 up through xn. And we're going to measure the partial derivative of f with respect to one of the x sub i's. Some people might, if you've seen any differential calculus before, you might have seen this as like d, df dx or something. For our purposes, it'll suffice to consider these delta f's over delta x sub i's, which is maybe more familiar. This is more like a slope of a line or something. But all this notation means is it's just a rate of change, a rate of change. It's not a fraction. It's just this whole thing, I should put in quotes, just means something. This whole thing means something else. And what does it mean? Well, it means like take these small increments in f, take a small increment, so the f, take a small increment in the output variables, take a small increment in the input variables and just compare what these look like. And this is giving you some indication about how important that variable is. If I change every x variable by one, so this is the marginal rate of change, maybe mentioned a little earlier in the class. Then I can just vary each parameter by one, like plus or minus one, and I can see how the outputs change. And if I change the x parameter by one and the output changes by three, then this is just a number. It's three over one, three. If I change, you know, a different x parameter, and now it's the output changes by 10, and I change the x parameter by one, then this is just a number of 10. And this is somehow saying that that variable is more important because I changed it by the same amount in the inputs, just plus or minus one, but it disproportionately affected outputs. And so just to kind of give you an idea of what this is supposed to look like, if you want to geometrically interpret this. Again, part of the reason we're doing this kind of analysis is because these things are really hard to draw and think about in general. The idea is we're used to living in this flat xy plane in the real world. We live in a three dimensional sort of space. We have, we can go out anywhere, just forwards, backwards, left and right. We can also potentially like move up and down. So that's all this coordinate system is saying. And if we go out to some position here, something like this, a function f that depends on, so this is f of just two variables x1, x2. And the z coordinate is now what's measuring instead of the y's, the z coordinate is measuring the outputs. And I guess really I should call these instead of x1 and x2, here we're just calling them x and y. And so now y is one of the input variables. It's just a name. And if I go out and measure some function of my position, that just outputs real number, this is like walking out to this point here, I've drawn with the dotted lines and looking up and measuring some height of like a ceiling above you. So what you get is some kind of, I can actually draw this. Okay, that's not so bad. Okay, something like this. So not the most amazing drawing in the world, but I can imagine this is like a blanket or a sheet or something kind of suspended up in space. And you go out to this position, you go up, you find the point on the space that corresponds to that point downstairs. And so what we're doing here is, let's say we want to consider first f, sorry, let me actually change this labeling, labeling a little bit. This is equals f of x, y, this blanket thing in space is literally the graph of a function. So it's the exact same kind of graph we've been looking at in the rest of class, or you know, we graphed on the plane, this is just how grass generalizes to higher dimensions. What we're looking at here is let's look at df dx. And all this is saying is that if I move a little bit in the x direction, say one unit, here I've drawn to this to two grid points. Well, I go up and so I've just changed the x, I haven't actually changed the y coordinate. Remember, y is the one that's kind of pointing back into the page. So I go up and look on the surface, and I get some different point here. And I get some little difference in x values, right? So maybe I've changed this from x naught, x one. And here this is just going to be f of, sorry, maybe this is y naught over here. We'll consider a y one in a minute. So there's a df, it's going to be x naught and y naught, minus f vet x one y naught over x one minus x dot. So the thing to notice here is that I kept the y value of it constant, it's just the same y naught and I'm only changing the x direction. So I have this big complicated thing and I'm just going to analyze it one variable at a time by just changing one variable and leaving all of the rest of the variables fixed to some number. Okay, so this will be part of our analysis. And then let's now think about it's not going to be easy to draw because it's kind of back in the surface. But if we go enough, I can, you can really see that. Okay, so all I've done is I've kept now I'm staying at x naught, and I'm changing the y naught to y one. And maybe that moves me to this point up here on the surface. Maybe there's kind of a, maybe it slopes up a little bit to get there. Then this thing is like a, let me draw it down here, df dy or delta x delta y. So then here we kept the x variable constant, we've just wiggled the y parameter a little bit. And it's going to be f at x naught, y naught, minus f of now we leave the x alone, and we change the y a little bit to y one. And now we're measuring the change over this other variable y. And that's all we're doing, like the way the way to think about this. And so this is this is pretty abstract at the moment. Let me put a put a magnifying glass on this little region of the surface here. And it's kind of like zoom in on it. So what happens here is that, you know, if I'm an ant, you know, I imagine walking around on this, this surface, the surface is telling me something about something I care about. You know, it's a function measuring something about input parameters. By zooming on the surface, it starts here. And I'm just looking at, you know, maybe I can go in the y direction, maybe I can go in the x directions. This is like y direction. This is the x direction. And I know, like I'm an ant on the surface, so it's like it's really flat near where I am. But I know that if I go a little bit in the x direction, I just want to know, like, am I going downhill? Am I going uphill? And if so, like, what's what's the incline? And same thing with the y value, if I instead pivot and go in the y direction instead, am I going downhill? Or am I going uphill? And if so, what's the incline? And so you can imagine, like, if this was some, some cost function or something like that, that was measuring revenue or something, like, you would want to find all of the places that are, you would want to find all of the places that are like highest on it, like maybe something up there. But you might also be interested in, like, what are all of the lowest, just depending on what your function is measuring. So the way I think about this is that if you're an ant and you're looking for the highest or lowest place, okay, what are you going to do? You're going to, like, look in one direction, you're going to measure the slope in the incline, you're going to look in the other direction, measure the slope in the incline. And let's say you're going for the highest, you're going to compare those two measurements and you're going to go whichever way increases your height the fastest, right? If you go one way and it increases your height by one foot per minute, and you go the other way and increases your height by 100 feet every minute. If your aim is to get to the top of the mountain, you're going to go with the trail that's 100 feet per minute increasing altitude. Okay, so that's at least like some, some sketch of the sort of geometry that's going on in the background. You won't really have to worry about this for this assignment, but it's just to kind of contextualize it. I'm going to work through kind of what's happening in the example. In this case, we're given sort of the function we have interest, the function of interest for us is something like A. So they just write it as A, but I want to emphasize that it's a function of mass and temperature. If you want to follow along, I am down here at this 5.1 section, or maybe just slightly above it, just five. So this is given to us by some equation. In this case, it's like E for ME to be CT minus 270. And this is supposed to be some analog of this generation time for the zooplankton. It's somehow the thing that the study is most concerned with. And then we go out and we find some constraint. This is where I remember from we're just looking at this, this first equation, it depends on two input variables. We're not going to be able to analyze it so easily. So we're going to cut down the dimensions. We find this other equation to do that for us, where it depends on the same input variables, but it's a different formula. Something like that, 3M squared E to the negative B, T minus 270. And just a heads up, when you're doing your analysis for this, you'll want to say something about... So one of these functions is increasing and the other function is decreasing. A little bit like we saw on the quiz question. So you might want to start, at this point, graphing things and seeing what's the point of intersection, maybe try to figure out why is this function increasing versus decreasing, things like that. Okay, so step one is going to be to solve for M after setting all A will be some number. P will also be some. So we'll have this function A depending on M and T, but instead of regarding it as a function, we're going to regard it as some number A. It's going to be our first step. And you could either... I mean, you could plug in A at this point, but if you're thinking about doing this on Desmos or something and you want sliders, it's good to do the algebra completely abstractly at first and then plug in A at the very end. So you want to solve this for E or for M rather. So maybe I won't say too much about what goes into this, but you end up getting something like this. The process for this ends up being like multiply both sides by E to the negative C times T minus 270. That'll cancel out this second, the C in the first equation you're seeing and put a negative on the other side and then I just rearranged. Okay, and you'll do the same thing for P and this one is going to be more complicated. We're going to play the same game where first we thought of P as the function, checks inputs and outputs. Now we want to change our mindset and think of it as a number and we want to solve for M. And well, I'll have to leave it to you to look for the example to see how this is computed. It takes a couple of steps, but you end up getting something like so I wrote it slightly differently, but it should be equivalent. It is equivalent. So I think the way they've written it in the handout is something like P over three E to the bT minus 270 all over two or something. But all I've done is just pull out that factor of one half in the exponents, remembering that square root is the same thing as raising to the half power. And there's that little bit of a choice here like you had to choose a plus or a minus, but here we're using the physical consideration of the situation of thinking of this as representing mass to restrict ourselves to only positive masses. Okay, so you'll have these two equations. And maybe I want to think of these now. Now instead of thinking of it as a number, we can think of M as a function of A and T. Similarly, this one, let me call this, this is a different function, let's call it M1 of A and T. Let's call this function M2 of P and T. Okay, so that's like the next sort of perspective shift is that we've thought of things as numbers, we solve for things, one in terms of the other. Now we want to shift our perspective back and think about the masses as a function of A and T. Maybe what we'll do is restrict A to be an actual number. So that'll kind of bump this down from being a function of A and T to just being a function of say T, we'll do the same thing with M2 instead of thinking of it as a function of P and T, we'll kind of fix the number P. And we're just going to consider this to be a function of T. And now we have two things that we can compare because this is, both of these are outputting the same sort of value that we're interested in, in this case it's past. And both of them depend on the same independent variable temperature. So you can do some kind of plot of B2. This will be the M axis. This will be the T axis. And let's see what happens. You get, let's do an orange, this one. So this is a decreasing exponential. So it's something like this. And the other one is an increasing exponential. So it's something like this. And you can sort of plot them together. For this part of the analysis though, what I think you'll want to do is now split it up into two sort of parallel analysis analyses. So two, so on M1 and two. And let me just say in words what you're going to do here. For M1, you have it as a function of A and a function. So there are two parameters kind of floating around. There's this A parameter and this C parameter, which will have been fixed to numbers at some point to make this just a function of a clear some estimates for these numbers coming from the research paper or something. But what you might want to do is see how sensitive it is to those inputs. So maybe you will change. Let me, let me write it down on the table first. So let's say M1 depends on a bunch of things. So it looks like C is A, C, and T. Is there something else? No, yes, that's it. So I think what you'll want to do here is essentially do some kind of plot of, how do I want to say this? So you want to do some table of, if you're changing, let's say A a little bit, and you get some change in M1. You can, you can think of different ways to organize this data. The idea here is you're just going to wiggle these parameters around and see what the outputs, how the outputs change. So maybe you start at something like plus or minus 1% of whatever your original value of A was. So if your original A was 100, you might change A to 101. You might change A to 99. You're going to get two different graphs out of that. So you might want to just put those two on the same graph, along with your original A. So let me draw this out. Let's see our original M1, what was this orange thing? It's like M1, first one. And so if you change A by 1%, maybe you'll get some slightly different. It's not so visible. Let me try this. So if you change A by 1%, you might get some slightly different graph like that. If you change A by minus 1%, maybe you'll get some graph like this. And you just want to do some analysis of like, what is the change of output when you fix? So one is just plotting these different graphs. If you change A by 1%, if you change A by 10%, try to sort of plot these in some way that makes sense and try to think about like, how are you telling a story to your reader about how these graphs change depending on the input parameter? And you might want to also like fix a t value, like saying, hello, here's one. Or sorry, fix that. Yeah, so I guess this is, should be clear. This is a t down here. This is an M down here. So fix some t value, say t0. And kind of look at what happens to it on these other graphs. And that'll be what your your delta M's are, or comparing the outputs, right? So here on this green line, you'll get some M. You're on this orange line, you already have some M. So you look at the difference of the two M's when you change that at A value by 1%. And then maybe it's not so and it's like, maybe it's not so meaningful to compare the actual numbers of M. Like maybe you don't want to say it went from 560 kilograms to or 560 grams to 575 grams. It's maybe more useful to do this in relative terms. So this should maybe be like plus or minus some question mark percentage might be an easier way to go about this or more clear. You might tell a more clear story if it's in relative terms. So you'll do this on M1. And then you'll basically play the same game. So now same thing with, so we changed the A parameter in M1. So now we can think about changing the C parameter. It's gonna do delta C and delta M1. Again, this is just notation. All this means is that you're changing the C parameters by a little bit. It's good to have these as a slider or something. And over here, you'll get some kind of plus or minus question mark percent. And in both of these cases, maybe you'll do this for some interesting number of percentages, maybe four or five. So maybe you do like one percent, two percent, three percent, five percent, 10 percent or something. And just do this measurement of how the output's changing. So this is all for like M1. Well, and then you just keep going. So maybe you also do this with maybe with T2. So this is more like the usual. If you do it with T, T is more like an average rate of change. So this is more like the usual thing you've seen. So you do all of this with M1 and then repeat with M2. Essentially all the same stuff. So you can organize this in a couple of ways, but I would try to keep it like have some method or do it in some systematic way. Either do all of the variables for M1 in some order and then repeat it in the same order for M2. Or like do M1 and M2 and just wiggle the A parameter on both and compare the two M1 and M2 and then go to the C parameter and then do M1 and M2. But in some how you want these to be like two sort of parallel analyses. So I'll leave it up to you to sort out somebody to organize that. Okay. So vary these things in some systematic way. You want to compare the graphs. Here is what I think is one of the maybe the most important piece of and any piece of mathematical or technical writing that you do anywhere in college is that there should be a story behind it somehow or maybe ask what's the story. So at the end of the day and I was just having a bunch of equations and graphs and calculations and stuff in front of your eyes. Even people that are professional mathematicians. You know it's not the most meaningful thing in the world to just be staring at a bunch of things like that. Just remember that it's like it's actual humans reading your mathematics and the audience that you would be pitching this to is someone who has never seen these calculations doesn't really know this analysis but they're kind of interested in you know what is the result of this analysis. What is you know some kind of story they're trying to extract from it. And to some lesser extent like what what are the methods you might do to reproduce that analysis for yourself. And so the story here is something like we have to go back to our original question of why why were we even looking at this in the first place. It's because we wanted to know like what impact these variables have by change or sort of which which input variable has the most impact. So as you're looking at these graphs you might want to start and telling the reader a story about like how like what kind of conjecture you're formulating here. Like are you seeing that if you vary A you're seeing a lot of change in the output in which case maybe you're thinking A is kind of the more important parameter in this setup. And again to kind of hark back to why you're looking at this is because you know it could be that these equations are given to you as a paper or something like that and you're on the theoretical side and maybe there's an experimental scientist going out and repeating repeatedly collecting this data or whatever. So the problem is that there's some error in the data you collect like it could just be you know some some instruments have built in sort of margin of error like a scale for example has a plus or minus you know whatever it is 1% or 0.05% or whatever margin of error. You can imagine anything that you're you're measuring could could it could introduce some error you know could be you know the weather changing on different days changes the electromagnetic frequencies in the air and it throws off your equipment just a tiny bit like who knows there's a zillion things that can happen. But so we want to know which which variables do we have to be careful about you know like if if a tiny change in A produces a huge change in the outputs well then we should be really really careful when we measure A to not introduce a bunch of error is even a slight amount of error is going to change our conclusion drastically. So how much you act on this right this is where you might like you know ask your department for better measurement devices or you might like if you have some grant funding or something you might say well here's the the most error prone or most impactful measurements we're going to spend a lot of money on some really expensive piece of equipment to measure that super super accurately because we know we need to for this experiment to be accurate at all so that's at least kind of where where this motivation is coming from um and you should sort of try to to weave that through your analysis a little bit um sort of hint at the reader what like what what is the result what is the conjecture what is the data actually telling you sorry I just realized there's something in chat you just double check you just randomly pick a t value to fix I think that's that should be fine here and going up up to this this graph here um yeah I think you might be a little bit restricted because maybe there's some there's you have to do some kind of like domain restriction based on the physical situation kind of like we did for project one so maybe pick a t that's actually meaningful in this problem like not uh negative 10 000 or something but yeah I think as long as you're picking something something that's yeah that makes sense then you can kind of pick it arbitrarily although I'll have to double check when I go back in there might it might be they might say in the handout somewhere like pick a specific value have to look through okay so this is kind of all of part one and this is the kind of stuff you should be looking at uh this week I just want to give you a hint of what happens in part two so it's kind of this part two uh let me see if I can find these equations again so we'll we'll have these two equations that we solve for m sorry kind of scrolling scrolling all over the place here okay so we have these two equations for m1 and m2 what we will want to do is take these and set m1 equal to m2 I really I guess we should be thinking of these as functions in one of t is equal to m2 of t sorry we remember that we have this this interpretation now kind of on the algebraic side setting functions equal um corresponds to on the geometric side if we go back up to the graphs uh let's see where are we yeah this one here setting these functions equal is exactly asking where this this point of intersection is for these and so we'll find that you know these are just two graphs and we set them equal there should just be one point of intersection or potentially zero points so we'll need to actually find this intersection point and this will go pretty much like the quiz question where we set them equal to each other and then solve one of these exponential equations using methods from this section so what you will get from this okay well in this case you'll be looking at something like this a e to the negative c t minus 270 equals one third p e to the b t minus 270 one half and what you will want to do here is solve for t and this this might be this might take a little while um right so we have two exponential equations set to each other I mean okay first thing you're going to do is like square both sides for sure um but then you essentially you can sort of pass that square into the exponent just using exponent rules and you have something if I put on math goggles what I see here is something that looks like e to the negative c t or something like e to the negative two c t equals roughly so I'll just put approximately e to the b t so I'm kind of ignoring all of all of the constants and parameters that are floating around and if I just kind of put on the the rose colored goggles here this is what I see just looks like two exponential equations being set equal to each other so we kind of know that we can sort of hit both sides with a log this is going to move the t's down out of the exponents and then we can just solve for it but of course there are more details that go into that and in that case you really do have to track the the exponents and the constants and everything but this is just some indication of like what technique would you apply to this problem where I see something that looks like an e to the t on the left hand side and I see something that looks like an e to the t on the right hand side so logs are probably the way to go okay so yeah I'll just I'll just say what you get from this just you can have some idea of what it looks like so it looks like you get just gonna write this in kind of a weird way but maybe I'll save some space 540 c plus 270 plus 2 natural log root of p thirds or maybe 3 over p and a outside of the root and there's going to be something in the denominator which I'm just going to write in this way the negative one to indicate that that's uh downstairs and I missed uh brand here we go so you get some some function of as I don't involve a p c and b this depends on a p c b so this is another another time we're just keeping the the variables set to these abstract letters it's kind of a nice nice thing to do because then you can kind of trace out where they go and the equation when you solve it for t and now when you want to go to analyze this function you can make this these four variables sliders in your function and you can just really easily generate a lot of graphs that way so a little bit of hard algebra work but it saves us a lot of replotting a bunch of functions with a bunch of different numerical values and what you'll be doing there is sort of similar analysis of now it'll be a delta t as you vary a delta t as you vary p and so on so it's going to be the same kind of sensitivity analysis where you're going to have and maybe start off with like a table you'll have the delta t's I'm sorry so t is the output now so it's the t-vented variable that you go on the right the a's of the inputs maybe you change this by plus or minus one percent and you get a plus or minus some percent change in the output and so remember now that this is this is where that high-dimensional surface picture comes in handy because we have a function that depends on four variables so I can't even draw it this time the first one only depended on two I was lucky because two plus one is three so I can I can visualize a function of two variables and its output a third in three-dimensional space I'm okay but this is four input variables and one output variable so this is like something living in five-dimensional space so we have no chance of like actually visualizing it in this level of generality so this is the kind of algebraic analysis we have to do to kind of get a handle on what that crazy high-dimensional surface actually looks like and we're really asking the same question in this crazy surface the outputs are now t values so these are temperatures and we're asking because there's kind of four different directions you can go there's an a direction a p direction a c direction and then some other v directions these are somehow intertwined and you're just asking if I move a little bit in the a direction how much does my t value change is it uphill or downhill and if so like what's the incline so same same sort of analysis and you're just going to do this with so same thing it's delta p delta t maybe plus some graphs if they help tell your story so again in this this project you have a little bit more leeway like it's not it's not as set in stone for this one like what you have to include in each each section or anything like that in this one I'd like you guys to focus more on you know once once you've done your analysis and you've reached your conclusion it should be so you should kind of do two drafts so I'm not really requiring a second or you know first draft draft or anything like that because I think it's maybe just too much extra work um you know doing this like submitting and grading and doing feedback and all of that but in terms of just writing anything in general I think it's very good to have a rough draft and to do the analysis as like one chunk of your project and then at the end you should do a second re rewrite of the entire thing so you're actually organizing all of your analysis in some way to like have some kind of narrative to it so I have a question is part two kind of just like a different way to do part one because um you said we could kind of do it one at a time like the rich change the variables one at a time instead of doing that big long equation with like all of them because I don't think I could really do that so could we also just do this the long way yeah so the first part like yeah you definitely have to include this kind of analysis for the first part right where you have the two different equations and you're just like frame one you're only changing A and C and separately for M2 you're changing P and B I'm trying to see if there's and I think I think somehow so this this section in the in the handout the 5.2 analytic solution I think it's really just like a separate analysis where you have to combine them all okay I think so do you think it's would the issue be doing this this um solving this equation here you think yeah I would probably need help for that but I can just worry about it when I get there that's no problem I think what I'll do then is I'll try to work through the derivation of this and then we can go through it in class on Thursday and yeah so that should that should make it a little bit easier to follow I think okay thank you we can go over that in class down here at the end so for this solving this we'll go ahead and cover this in class okay zoom out really quick just to make sure so just a quick quick recap on kind of what you should be looking at between now and Thursday so try to add to me with your your group partner or partners preferably between now and Thursday because you'll definitely want to start on this this first section here I'm just doing this analysis of writing these equations solving for n and plotting them what I would do at this stage is maybe start making a bullet list where you're kind of outlining your project and like what do you want the sections and subsections to be you don't have to start writing it but kind of do an outline draft um we'll cover a little bit of what goes into part two on Thursday after that you should probably like actually do this analysis you know record it down in like a first draft something like that and then at some point you should plan to meet up with your partner again to do a second draft where you maybe use all of the the tables and the graphs or the analysis you did from your rough draft but kind of organize it in a way that um you know frames frames what is your conclusion and how does sort of each section or subsection of your project like what is it telling the reader about your conclusion um so there might be like one one narrative you might go with here is that you know as you're looking at the graphs you might have some conjecture about which one is influencing the outcome more and so you might hint at that through your paper like oh here in this graph we're seeing that as we change A the outputs and in change quite a lot so maybe A is an important variable all right now that I think about it in yeah so part one is really an analysis of the changes in mass as a function of the input parameters and I think part two is actually now kind of flipping that and it's saying now let's look at the change in temperature as near the input parameters since we know mass and temperature are two things that go into dislike generation time that we're concerned with okay so I will go ahead and let you guys go please yeah just feel free to write down your questions or email me or bring them to class on thursday and good luck