 What's on the agenda today? So I guess we'll kind of look through a little bit. We'll talk about the last quiz. I think people mostly did pretty well on that. I want to point out just a couple of things to keep thinking about as we're going forward because we'll of course be revisiting these concepts as time goes on. I talked a little bit last time about this like this idea of space repetition. So, you know, the first time you saw these concepts was maybe in the pre-calculator, the pre-video, the pre-class videos. Maybe the second time you see them is either in like lecture in class or on one of these quizzes. And then maybe we'll do a third time you see them on some sort of like later review. So they might show up on a later take-home quiz. Right. So I wanted to say something really quick about, right, so this whole quiz is on domains and ranges. So I guess number one is, well, okay, number zero is if you haven't looked at the quiz grades yet, try to do that as soon as like today if you can, just while it's still fresh in your mind, as always be sure to like go through and try to find the feedback. If I left feedback on any specific part, I think that's probably one of the most valuable things I can give you as a part of this class. You know, there's always sort of mechanical things they have to do through Alex and through the worksheets and stuff, but kind of the human component of it is like doing work and like having somebody else reading your work and giving you feedback on it. So I think it's like one of the most important pieces of this class. So do read those. And if you go through and look at the points, one thing you'll notice, this is like maybe point number one is that there were a lot of points given for this like explanation aspect of it. So like on most of the questions, I put something that said, you know, write a sentence explaining or justifying your answer. And so I think there were some, so that it varied between questions like it wasn't just a set number of points for explaining sort of depends on how important the explanation was for the question, but in most cases the explanation was worth more than half of the points. So even if you somehow got the mechanical part of the answer, like if you got a domain or a range completely wrong, if you were kind of explaining your work and your thought process and how you got to where you did, then you could get almost half of the points for that. And so I'm doing this kind of for two reasons. One is that this semester is very different, right? You know, we don't really have exams per se. We don't, now there's not a lot of sort of group work in class. Normally these would be big components of the class. And so to sort of make up for that, I'm trying to sort of introduce for more writing into the class and sort of trying to like write down an explanation to it. So you'll see more of this as we go forward. And then secondly, this is like all, this is also a huge part of what I think mathematics actually is, you know, so sometimes, you know, media is like a working mathematician. There's sort of two aspects to the job. One of them is like sitting down and doing some calculation or something, you know, figuring out the domain or the range of a function. But then the other half of the job is like going out and giving a talk about it or explaining to your peers or, you know, teaching a lesson about some topic. Just maths is this huge, vast, there's so many things to learn, not everybody knows everything. So it's really important to be able to sort of like go and learn something complicated and then find some way to like synthesize what you've learned and teach other people about it or explain what you've learned. So that's kind of why the emphasis is there. And I think outside of maths, this is also something that's just really a useful skill to have and sort of anything if you're going into like any science discipline, of course, that's half of any science job is, you know, on one hand, you do the science, you work in the lab or whatever and find your results. And then you spend six months to a year, something, you know, writing up a paper about it to tell your peers. And so this is like peer reviewed research. It's the, you know, the center post of that job. And of course, like if you're thinking about like business or things like this too, that's an everyday occurrence, you know, analyzing some complicated data and finding some way to simplify it and tell other people who are non-experts about it. So if you lost a lot of points for explanations, don't worry about it too much. This is something, you know, we'll be working on as we go through the semester, just be sure to like read the feedback and try to like incorporate into your next assignments. Okay, I want to share my screen really quick. I'm sorry, I guess there's a obligatory cat break. Camille wants to say hi, I guess. Say hi, Camille. She likes to help with the lectures, so she like climbs up on the desk every five minutes here. Okay, so hopefully I'm sharing the right screen. Oh, that's a little bit weird. Yeah, so my pen kind of cuts out when I share the screen. It's a little weird. Sorry, one sec. Yeah, so I wanted to say something about just how to think about domains and ranges coming away from this quiz. So like, I think a lot of the explanations talked about. So there's a lot of, I guess, variation in the explanations for domains and ranges. I want to kind of say something about just how to think of them going forward and then also I'm mentioning this now because, so we'll talk about this a little bit later in the class, there's this first project coming up, right? Instead of having a first exam, we have this sort of writing project to do and domains and ranges will be an important piece of that. So I want to kind of set up a little bit there. So like the one thing we've been kind of looking at is you have some sort of function like this. Y, there's an X axis and this is some function given by f of X, maybe we had things where it was like deleted at points, that kind of thing. And then we talked about the domain and ranges and a lot of it was like, well, we can just kind of look at the graph and we can reason about this pictorial representation of our function and sort of figure out what the domain and range is from there. And this works really, really well for the kind of work that we're doing specifically in the worksheets or in Alex or something where we're given this function that takes in X values that are numbers and gives you maybe this is X equals three or something and maybe this is Y equals two and that point is like three, two. And this whole pictorial representation is just telling us that when I put in an X equals three I should be getting out of Y equals two. Same sort of deal over here. It's just telling you, if you put in these inputs X what do you get as the output? And we can sort of reason this way. But in a lot of situations we don't necessarily have a graph of the function. Maybe it's something more complicated where we're trying to, like we just don't know anything about this function and we're trying to ourselves analyze it and learn about it in which case we won't have this tool of like looking at the graph available to us. And so may not be available or sometimes this question is flipped on its head where there are many, many different choices of graph to use and we ourselves have to decide the domain and range. So it's a little bit tricky in terms of a concept but this is what's gonna happen on the first project where there's sort of a lot of data you could be working with but somehow you have to yourself restrict what data you wanna push into your function and what kind of things you want to see coming out of it. So you may have choice is what I'm trying to say here. So you may have choices. So like instead of some function having a domain and range you have multiple functions and you yourself get to decide what the domain and range is. So I want to kind of say, I think this viewpoint is somewhat nice the way we've been looking at things before is that we have this F and we have sort of a number line this universe of things we can send into the function. I guess maybe there's zero, negative infinity infinity and we're thinking of this function as a black box maybe we have a formula for it but maybe it's something out there in the world that we just know is some function of the input. So we know it has this like deterministic property that if I send in unique inputs and then I get unique outputs or rather yeah, for every input I send in I get a unique outputs one X value goes to one Y value. The situation we've been kind of working in thus far is this kind of thing. So we send in, you think of these as like X values and Y values but then we also had kind of a maybe we had functions that were sort of restricted and like maybe if this function was given by the formula F Fax equals square root of X for example then we had to take this universe on the left hand side and sort of cut it down. So I have to think about deleting all of this stuff that's less than zero and so the domain of this function is just let's see so we can include zero we can, well we can't include infinity but we can include everything sort of in between and this thing was the domain but it's kind of important to remember that it sits inside of this larger universe in this case this function like what are the things that could even what could you potentially send into this function what type of function is it? Well, it's something that takes in some real numbers and the domain is just saying okay now what real numbers does it make sense to put into this function? And in this case we saw so I guess what happens here is that the all like the universe of potential outputs for this function what is it? Well, it's all of the real numbers except that it doesn't actually hit all of the real numbers really we find that it only hits the positive ones. That's the range. I just mentioned this again to point out that the range too can sort of live in some bigger universe like what are all of the possible outputs this function could reduce if we didn't know anything about the function well we know without knowing anything about the function this particular function we know it gives you real numbers in some way but then we do some more analysis on the function and we find out oh well it's not all real numbers it's some very specific subset of the real numbers and we'll call that the range of the function and so this all matches up nicely with kind of this like graph picture where like the universe of inputs is kind of going on this axis it's like x-axis right? And this universe of outputs is going on the vertical axis. So we can kind of recover the previous picture of like sort of reasoning about graphs you know maybe we somehow knew that this was what the function looks like we could reason about the domain and range but if we didn't know anything about this function then we would kind of have to go back to the drawing board on this and sort of start asking ourselves what does it make sense to put into the function where is it undefined and what are all of the possible y values it could actually take on? And so the point is here is to definitely think about this in terms of like inputs and outputs and one reason that this is kind of nice to mention here is that we saw that sometimes we just have a relation so a relation is really set up in the same way as this function business where we again have like inputs and outputs and I think most people got this right on the quiz this relation can fail to be a function if you send in one input and now you have kind of two choices of output to do and so this is really the same data as the this like vertical line test so maybe there's a vertical line on this function and we're kind of good in this situation because this vertical line only intersects the graph once but like what is this telling us in words this is telling us I've like picked some x value maybe x not and the line is asking what are all of the y values that could be associated to that x not then intersecting it with the graph is just saying here's some output of the function that takes on that for that x value takes on a y value so this is where thinking through a little bit why the vertical line test and this business about like sending in an x and getting a unique y out are really just two ways of saying the same thing and in fact something else to think about too is that so this is the relation of the form y equals square root of x so this is the function of the form like this we can also think of the relation of the form it's a good way to notate this maybe y squared minus x equals zero or y squared equals x so this is worth thinking about a little bit you've usually seen this in the other sort of order that y equals x squared and there's kind of no problem with that y equals x squared thing is a parabola like that this one's y equals x squared and this has kind of no problems with the vertical line tests everywhere you go you're just finding one point on the graph like that the so if we flip this relation to y squared equals x instead what we get is a parabola so it traces out the same thing as y equals square root of x when the y's are positive but it also has this other branch down here and this relation fails the vertical line tests because essentially what's happening here is maybe you're fixing maybe this is x not equals two or something and okay well the y's are the well let's do this sorry just to make our lives a little bit easier let's make this something like y equals four or x equals four and then let's do let's see so y equals square root of x is kind of the plus two part of that when we're thinking about this y equals square root of x but if we're thinking about the orange relation where these you square the y value and you get the x value back then there's this other point down here y equals negative two and so in this relation y squared equals x we can think about a lot of y values that's square to the same x value in this case two and negative two both square to four and so that's kind of a problem if we want a function of like y is a function of x because it sort of fails this vertical line test there are multiple y values that square to an x value and so in some sense what we actually do is we restrict the domain or the range of this relation by sort of deleting this entire branch to sort of force it to be a function to force it to pass the vertical line test and that's what y equals square root of x is so like we have this relation y squared equals x it's a problem because it's not really a function it fails the vertical line test and so we actually throw out part of the domain of this relation namely all of these like negative y values to come up with something that's actually a function and that's just to emphasize that like I know maths isn't really like given to us like somehow you have to kind of like make decisions as you go to get things to be functions I'll just say really quick just things to remember of course these two functions will be showing up all the time these are functions where so the square root of x has domain issues right because we can't send in we don't want to send in negative values right so let's just saying that you know there's none of this none of these negative x values can be sent into the square root function the relation kind of doesn't live in those quadrants definitely want to remember that this has sort of issues when x is strictly less than zero and so any negative number is going to cause an issue at the square root but x equals zero is totally fine and this is something to kind of remember when you're doing these checking domain range problems is like kind of like check the end points somehow like if we know that negative infinity maybe up until zero could cause problems then it's worth checking an endpoint like that like x equals zero and you can just sort of do that manually by just like plugging this into the function and then of course the one over x business is well it's the same deal but just when x equals zero and what you want to keep in mind here is that we can kind of now compose these in arbitrary ways so I can put sort of a lot more stuff under the square root and it can put a lot more stuff in the denominator and what happens here is I just have to check if that whatever that more stuff was that I put in it's really the same business I just have to check if that is less than zero I have to check if that is equal to zero and so for these two steps you might have to actually like go and solve some equation to figure out what for what x values is that true? Okay, so I wanna say a quick word about math modeling and then I'll say something about the project because the project is going to be really closely tied to modeling and so hopefully you remember this from our distant past of last week before the long weekend that we were doing these like linear models and I like to think of modeling as some kind of like math wizard here some somehow like the closest we get to doing something relatively magical in mathematics and what I mean by that is like a lot of times in real world situations we'll have things that are functions of time or something like that and modeling lets us kind of observe the data we've looked at in the past and project out into the future and somehow try to like predict what will happen at some point and not the best artist but try to draw this is supposed to be like a very happy wizard very excited to be doing mathematics today at 8 a.m. So let me start with like a situation. So let's say I've just like moved into a house and maybe I'm planning to be there for 20 or 30 years or something so I'm going to plant a garden or something like that and so I'm going to plant a bed of petunias or something okay so that's the situation but there's kind of a problem if say the average yearly temperature is maybe I'll call this like T sub average and what do I mean by this? I mean that somehow I go to theweatherchannel.org and I pull down data and I look at 2019 or something and I look at all 365 days I look at all the temperatures I add them all up and I divide by 365 and I get some average temperature through the year and the problem is that if the average temperature is say above 85 degrees Fahrenheit then the petunias will, so not a great situation. So this is something I'm worried about maybe this year the average temperature was only 82 degrees so I'm good but I need to know if like is this a good thing to go and do? So what I do is I go to weatherchannel.com or something, I think about this graph where it's going to be some function of time and now I'll call this like T average as a function of time. This is a little bit weird, right? Like now it's not an X and it's not an F of X or a Y or something like that. It's a T which is just some variable, some input we can send into some function in this T average which for that year calculates the average temperature of that year. So it's a little bit more abstract than the situations we've been in before but we still know that these are functions that operate on numbers. I can send numbers in, I get numbers out so we have at least the graph to work with. And let's say before I do this, let's say maybe I measure it here for, let's do this is something like 1980, it's there. And maybe this is 2000, maybe it didn't change too much between 2000 and 2020 like this, maybe there's 2040, but okay, it's only 2020. So this is, I only have data up until now. We're kind of only looking at the past and I don't know, maybe this is, see what numbers I actually used here. Let's do, this is like, it was 70 degrees. That was the average temperature in 1980. It was 79, maybe this was 82, something like that. So I just go out, I get all this data, I just plot it. I don't necessarily have a function yet to think about but I want to sort of come up with a function. And so this is, we've seen at least one way to do this is like to fit a line to this guy. And so we find that, see hopefully I can line this up, okay, something like that. I guess this actually demonstrates something you'll run into with the project. Is that sometimes you'll fit a line to something and it may not fall exactly on the data points. This is kind of something to be expected because your data might not actually all lie on one line. So you can ask for the line that's kind of like closest to it. Okay, so what happens here? It's like, let's say I go out and I somehow fit this line to these three points and I get something that looks like, sorry, it shouldn't be Y, it should be T average equals M, now times T plus B. So it's kind of the same form but now the outputs are these T averages, the inputs are the T's. And I fit some line to it and this is where like domain and range issues start becoming something to think about. Like, does it make sense to plug in, I don't know, a time like this? Well, in this case, maybe. So this is corresponding, I guess, to like zero temperature or something like that. And maybe this happened back in, well, according to the scale of our graph, you know, maybe this was like 1940 or something, who knows. So this model may not like actually be accurate outside of the range of our data. This is kind of a, I guess you call this like extrapolation issues. So now you might just like restrict your domain to say, well, maybe I don't believe this function makes sense outside of say 1960 to 2060 or something. So you have this function which kind of lives everywhere and the domain and range is all of ours, just a line. But now we're kind of making this choice that we want to restrict the domain of our function, this function that we've come up with. We said, okay, well, it just doesn't make sense to send in points between our sort of before 1960 because we don't believe that this is a good model for that time period. And we just make sense to plug in points after 2060 because we don't believe it's going to be a good model after that. So this is something that can happen. And right, the range is a little bit restricted too, does it make sense for the average yearly temperature to be 8,000 degrees? Maybe not on earth. It doesn't make sense for it to be negative 3,000 degrees. Well, it's like not physically possible. So now we need to like artificially think about like how do we restrict the range of this too? Okay, so why do I call this wizardry? Because now I sort of know something. Let's say here's my 85 degree mark. Remember this was the temperature I was worried about. If as soon as the average temperature goes that high, then right, so all of the flowers will not survive the year. And so I can kind of make a prediction about when this will happen. Sort of based off of this model, I can line it up. So I've just looked at the Y value, this like 85 degrees. And I've asked myself, according to my model, when will that actually happen? I'm observing a pattern in the data that the temperature is going up year to year. And so now my question is like, how long do I have before I should worry about this? And according to this model, it's apparently not too long. This is like 20, 25 or something. So I'm not being like super accurate with the scaling here, but just to communicate the idea that you have some model and you can sort of extrapolate a little bit from it and sort of predict some future behavior based off of this model. And so, okay, that's when I want to start worrying is in five years. So maybe I can now make some decision based on this model. Don't play in petunias, okay. So they won't last five years. Maybe not a sound investment or something. I'll leave that up for a second. So really quick, I'll pause here to start. Are there any questions on just anything about the modeling aspect of this so far? So hopefully this looks somewhat familiar based on some of the linear modeling we were doing last week. In that case, we had something that was like two points and we knew we could use things like point slope formulas and well, we had a slope intercept formula. But the important thing to remember here was that two points determined a unique line and we had some formulas to work with that. And kind of the situation we'll see going forward is people have some sort of huge collection of points and the chances of all of these points lying on like one single line are pretty slim. But we can sort of still make this simple line model work we can ask for like a best fit line. All right, I should say one more thing here about this and it's just the meaning of the slope. So we have this like, right? So this is something of the form, the output variable is like some M times some slope times the input variable plus some Y intercept. In this case, it's a T average intercept or something. So the intercept still sort of means the same thing the V is just telling you, well, this graph is a little bit misleading because this should maybe be like T equals zero should be what I'm thinking of here. And then this T should be like years after 1960. It's another kind of weird situation where we have ourselves a choice of how to plot this data. We can just call 1960 like the zero year but we can also call like zero AD the zero year we call negative 2000 BC the zero year. So it's like, you have a lot of choices here but the intercept is still telling you the same thing whatever you call zero, it's just telling you what is the output value at that time? So there's some intercept to it. We have this like rise over one business this is rise over runs, this is delta T average. This is a delta T. And so what is this actually telling you? This is telling you that for every like T increments of time, so maybe in this case, well, okay, so there's kind of two ways to think about it. One is that if you had say, M, right, and again, this is like delta T average over delta T. And let's say this was something like kind of two sevenths or something. And so this is telling you that for every, so every delta T, delta is like, it says the Greek letter, this is using mathematics to denote a small change. So like a delta and T is like a difference in T values like a T two minus the T one kind of a thing. And so this is telling you that for every change of seven years in this case, and this is telling you how much the Y values change in that span of time. You get a change of two degrees plus two degrees in temperature. So this is one way to interpret the slope, right? It's rise over a run or a delta in the outputs over a delta in the inputs, how much do your outputs change with respect to your inputs? You can sort of play a silly game here where you move everything upstairs in the fraction and just put a one downstairs. And this is telling you now something about, so if you ever take like an economics class or something they'll call this like a marginal change, marginal here just means that if you change your inputs by one unit, what is the change in the output? And so this is telling you something like every one year, the temperature changes by, well, whatever the numerator is, and it's two sevenths I guess of a degree, Fahrenheit. So really it's just the same information, just a little bit of algebra moving things around. But interpreting the slope is a very useful piece of information here, is telling you something about the rate of change of the problem you're thinking about. And just to say a quick word about something that's, I think I'll say, yeah, I won't say too much about the situation, but at least point it out, at least point it out because this is something you'll run into on the project a little bit. So we've been thinking about these sort of functions where, and so maybe previously you had a function that was like this kind of business where you sent in kind of one input X and you got one output Y. And so something we'll sort of be thinking about now is maybe I'll think of this as like an F of an X. So we can expand this sort of definition a little bit by now asking about things where, okay, let's say it still takes some kind of F or so it's still some kind of F, it can still take X inputs, but maybe now it takes some other input T. So now it's a function of two real numbers. This is a little bit tricky to think about, but somehow the black box takes in an X value, takes in a T value and these are both just real numbers and it gives you, it does some, mixes up X and T in some way that we don't know and just outputs some measurement about those inputs. So some output Y value. And so maybe a situation where this could happen is, all right, let's say we're trying to predict, maybe I am a salesman and I want to predict how many pools I can sell maybe this year or something like that. Like I want to go install pools in people's backyards and I want to know like is this a good prospect this year and maybe I expect that that's something like the revenue or something from this depends on two things. Maybe like before I expected, it depends on like the average yearly temperature. I don't know if I can move this down. Yes, okay, cool, okay, so sorry. It depends now one on the average temperature and maybe two, it depends on some like demographic information of some kind, maybe it's population density in this case. And maybe I expect that, so maybe I'll call this revenue R will depend on X and T and maybe my T will be this input and my X will be that input. Okay, so I expect now that somehow the output of this and again, the output of this function is just some number. Maybe it's just some expected revenue from a year of sales or something. But now I expect it depends on multiple inputs. And so there's sort of a lot of ways I could do this. I could try to like come up with one function. So I could have maybe like an F1 that just depends on T and okay, this is getting some revenue that just depends on the temperature. I could give some F2, maybe that just depends on X and this is some revenue function that only depends on this population density. But somehow like these aren't enough to capture the information. Somehow I expect that the T and the X kind of mix up in some complicated way and somehow not captured by just looking at these one variable functions. So we'll think of this like revenue now instead of X and T. And choices for I guess maybe just like types of functions to consider. And so we'll use this new sort of concept of having a function of two variables because it lets us model sort of more complicated situations or somehow take into account the two variables at once. And so maybe if we were doing this kind of like modeling with F1 of T, we would have some situation like this where we could fit some linear model to it. We could fit some line if we had maybe some data points like this and these are T's down here. And this is like revenue. So something like that. And what is the picture for this new situation? Well, this is where I'm sort of claiming we should probably move away from the sort of like reasoning based off of graphs because now we have something that's like higher dimensional. And what I mean is like this whole thing lives in a two space. It lives in the plane. And that's just because it's like taking one input and giving you one output. So you have one plus one, okay, two sort of dimensions to think about the input dimension and the output dimension. But for this RX of T business, we now have two inputs and one output. And if you add these all up you have something that's like three dimensional. So now it's just something that's a lot harder to draw and reason about. But sort of all of our algebraic techniques will still go through just fine. Maybe I'll just point out kind of what this looks like in case this is not something people have seen before. And I'm not the greatest artist. So hopefully I can try to draw something here. What this ends up looking like is, let's see if I can, this is supposed to represent like a three dimensional space or something. And so now instead of a line, what we have here is a plane going through space that represents our function. It's kind of supposed to look like sort of a flat sheet of some sorts. So maybe something like that. So this is just like, something the shape of a piece of paper or something like that. And it just somehow sits in three space. And what happens? How do we sort of use this to reason about things? Is that maybe this is the X coordinate. This is the Y coordinate. And we have this sort of other coordinate Z up here. And then this new, so this is still, I claim it's really the same thing we've been looking at. This is again, just the graph of a function. But now we have sort of more variables floating around. This is R of X and T. Instead of being a line, it's this plane how you use it while you go out and you measure some X value. And then you go out on the, sorry, I guess we're using T's instead of Y's here. Sorry about that. So you go out and you measure some T value. And let's see, maybe I want to consider this X value and this T value. And what happens is kind of walk along that value. You find the corresponding T value. So now you're somewhere down here, like on the floor of this in the XT plane. And you just kind of go up to the hyperplane that's sitting above it. Sorry, the plane. So you walk out in some X and T direction. You look above you, you see some kind of like plane sitting up there and the graph or like what is the output of this function while it's the Z value corresponding to the part of the plane that's like right above your head if you look directly up. And it's, so as you can see, this is like kind of more difficult to reason about. It's not easy to ask like, what is the domain and range of this kind of thing? Sort of the previous techniques of like, just look at the graph, end up being more difficult. So we kind of do this thing where we can kind of like break it up by like bringing the dimension down and start plotting things of like two variables at once and sort of see how things go that way. And I want to, oh yeah, I actually have a better image of this that I can probably throw in. Let me see. Okay, yeah. Yeah, so this is kind of, I don't know if people can see this. It's kind of the situation that's happening here. This is free space, the blue and the red are like our X and T coordinates and the screen is like the Z coordinate. And this purple plane is somehow like measuring the graph of this function R of X and T. And so you can kind of see that if you, see if I can angle it just right. Like if you look at it from some directions, you end up getting something that looks like a line on sort of one of the axes. So here like, this is a line on the red and the green axis. And if I kind of spin it around, do something like this, I guess this is now a line on the green and blue axes. So somehow this is combining like two linear models into one and you get this sort of plane kind of thing as the output. I'm sorry, somebody mentioned something in chat and they can see it. Draw or interpret a plane for the project. No, no, no. Yeah, you won't have to worry about that. For the project, essentially, you're just gonna be working with things algebraically. And we're actually gonna do this, this sort of dimension reduction thing where instead of considering a plane, so we'll have a bunch of variables to work with, but we'll just sort of consider them in pairs instead of having this like function of two variables. We'll consider a lot of different functions of one variable. So that way we can use the techniques that we have from this class. And just pointing this out for like, I don't know, mathematical edification. It's kind of a fun thing to see that, you know, these things are, they exist. Okay, yeah, so let's talk a little bit about the project here. Actually, let me do a quick break here. I'll stop. Does anybody have any questions about any of the modeling stuff just before we talk about what the project is going to entail? All right, so yeah, let's talk about the project a bit. So this is going to be due in, I guess, 20 days. You have just, let's see, I guess just under three weeks to do it. Hopefully it shouldn't be too bad. There's always sort of one new thing you'll really have to learn to do this. I think most of it is just going to be in sort of understanding this problem, playing around with it a little bit and sort of writing something up. All right, so what's the situation for the project? The situation here is that we have some sort of scientific data that we're given and I'll scroll down to it later. We're given this data in the form of a table and this data is relating, I actually have some notes that actually say what's specific. Okay, here we go. Okay, so this is a case study, I guess, from biology and I guess what we're looking at is this interbirth interval. A certain species of baboons, it's poppyo baboon species. Okay, and so the interbirth interval, I guess, is some measure of like how many, like how much time it takes between births of like new baboons and say like one, if you're following one group of these baboons or something, you might see that like a new one is born every two years or something like that or maybe it's one year. And so this sort of affects all kinds of like population dynamics and sort of a very complicated ecosystem we're trying to just pick out one aspect and study it and that is what factors influence this interbirth interval. So what factors make it so that there's a longer interval between births and which factors make it so there's a smaller interval. And in this case, we're gonna see that there are, so we've picked out two variables that we think are sort of explanatory in describing this interval. So this interval, again, it's just some number. Maybe it's two years, maybe it's 1.73 years, just some real number of time. And we are expecting that it depends on two inputs. And the one input will be the temperature of say where the group or the colony is living. And the other input will be the altitude at which they're living. Actually, let me just wanna write a few things down. Let's see if we can move this. Oh wait, sorry, juggling windows here, sorry about that. Okay, so yeah, we'll have this interbirth interval and maybe we'll call it this like capital T and we're expecting this T will be some function of T and actually, sorry, let me use different variables here. Let's call it I, we're expecting that it'll be some function of T and A. So this is the interbirth interval. This T is a temperature and this A is an altitude. So this is kind of what we expect. And we want to have some idea of like what drives this process. And so what we maybe observe, sorry, let me switch sides here. So what we maybe observe from experimental data is that if I plot the interbirth interval versus the temperature, then we don't know what sort of function it is, but we expect that it increases as the temperature goes up. And so what's the meaning here is that this interval between births is getting longer and higher temperatures. And we expect actually that because we expect this because as the temperature goes down, the wait, so hold on, let's see, right, right. So yeah, as the temperature goes down, we expect that there's sort of less stress on the colony overall. Maybe there's less competition for resources. Okay, so that's the interval as a function of temperature. And we can sort of separately consider now the interval as a function of altitude. And we kind of expect the same behavior whereas the altitude, as you go to higher and higher altitudes, the interval goes up maybe because there's sort of less food available or maybe it's a more competitive environment or it's harder to survive because of lower oxygen levels or something. So we expect some relationship like this, but then also we have this other kind of relationship between the two variables themselves, the two inputs. Temperature as a function of altitude. And so I actually expect that as I increase my altitude, kind of don't know what the function is, but we expect that the temperature goes down. Hopefully that makes sense, right? So you go to higher areas, you get lower temperatures. And so we essentially, what this project will be doing is fitting models to sort of these, all of these combinations of variables and sort of trying to understand the relationships between them. Let's see. So I've uploaded this PDF doc to ELC. So you'll definitely wanna go through and like read the fine prints on everything to see sort of what all the details are here. But essentially it's going to be, you go to this middle section here. So starting here at section four, it sort of describes the situation I've just mentioned. And this 4.1 is kind of like, what are the actual details of the project? What needs to be in what you submit? And the idea is you're going to write up some kind of paper that includes some analysis of these like different ways to model these relationships. And it's giving you like a couple of very specific relationships to investigate. So the first one is to describe a model between the temperature and the altitude. So that's this third one here. And if you can see here, we have this table down in the PDF. And so you're giving a bunch of introvert intervals here in the table. And you're giving some altitudes and some temperatures. And so for the first one, you're just considering like, don't worry about the introvert interval at all, kind of block that out. And we just look at the temperature and the altitude and we try to figure out how these are related. And I'll show you in a second how we can model these. You can use this desmos.com tool to do fitting of these models. So I'll show you how to do that in a second. But this is going to be the first thing to model is, you'll plot all this data. And then you'll ask yourself, okay, of all of the functions that are in my toolbox that we know of from PreCalculus, so it's not a huge amount, but we just have like things like, we have lines, we have things like parabolas. I guess from the last somewhere, if we have cubics, things that look like X cubed, maybe things like square root of X, things like that. And so we're just going to try to see, see experiment with fitting the data we see out in the world to some of the functions that we know of and sort of see which function best models that behavior. And so, okay, so you'll be putting these table entries into a tool on Desmos and sort of fitting that, coming up with some kind of model. So I guess here's what the plot of temperature and altitude looks like. So you can see it's not really quite a line, but maybe a line is a good model for this kind of thing because it sort of looks almost linear. Like it looks like it could lie on a line, maybe up to like some experimental error in the measurement of the temperature or the measurement of the altitude. Maybe this altitude was measured to be slightly higher because, I mean, you know, it was maybe the equipment was slightly off that day. Maybe this, sorry, I don't know if you can see the mouse, this dot that kind of lies off of the line here. So this could just be like an outlier for the situation. And so the data may well just be a line, but you can sort of fit it and then you'll get some equation of a line. And then the idea is you'll be discussing kind of what does this equation mean? Like what conclusions can you make from it? You know, it'll be something like y equals mx plus b if it's a line and you'll have some value for m. Maybe it's, I guess maybe it's like negative 1.731 or something. And so the question is, what does that number mean, right? In the context of this actual problem, what is that slope telling you? And what is the intercept actually telling you? Okay, so you'll have the altitude, this relation between the altitude and the temperature. And maybe you'll plot it as like the altitude is a function of the temperature or the temperature is a function of the altitudes. You have some choice in which one's the x and which one's the y. So you maybe do both and explain which one more closely captures this physical situation. Right, so you wanna like take some graphs of whatever you're modeling here and include these. Again, you're just gonna sort of write up a paper and sort of explaining your analysis of each of these different parts. And it might include like the equation of the model you come up with and a graph of what that equation looks like. Some explanation of what the parameters mean. Again, if it's like a slope, like say what does that slope mean in the context of this real problem? And some other sort of things like, okay, where is the function increasing? Where is it decreasing? Just some like analysis of this function you've come up with. We haven't talked so much about this yet. This part is determining an objective function that should be minimized or maximized. And so what objective function means here is that we have something we want to measure. In this case, this is this introvert interval and we expect it depends on a bunch of other variables. And this is a very common situation where we'll ask which combination of input variables maximizes or minimizes the output of the function that we care about. And maybe this tells us some interesting information about the situation. So here the objective function will be this introvert interval. And it'll be some function here again like of temperature and altitude, but it's kind of up to you to come up with some combination, some formula of T's and A's that somehow best measures the situation. So this is kind of the tricky bit. And so there's a lot of choices in what this function could be. One thing could be maybe as an example, it could be three and you cube the T and then maybe you do plus A squared minus square root of A plus T at the end or something like that. So this is probably not a good model. Probably does not capture the data. So I just sort of picture it randomly. But it's some formula that involves the input variables that gives you some output. In this case, the introvert interval. And then you'd wanna start plugging in the actual values from this table to see if I plug in 23.4 for the temperature and I plug in 1127 for the altitude here, does this formula spit out something that's close to 24, like the actual observed data? Okay, so that's the second piece of it. It'll be exploring these relationships, graphing the various pairs of variables like that, coming up with this objective function and then doing some kind of fitting to it. And at the end, it'll also be discussing a little bit, like what's the domain and range of this function that you've come up with? Like, what does it make sense to put into this function you've made up and sort of what are all the possible outputs it could take on? And same sort of business, where is it increasing? Where is it decreasing? Maybe some kind of graph of, right? And again, this is a little bit difficult because this is some kind of very complicated thing sitting in three space, but maybe you can think about what does it look like if you set T equals to zero, now it's just some function of A. So you can try plotting that and then if you set A equals to zero, now it's just some function of T, you can try plotting that separately. It's somehow just like finding some way to get a handle on this complicated function. Yeah, so let me, yeah, there's a little bit more, like this is, this section here is just some analysis of this I, this I function you come up with, this function of T and A. So do some, you know, graphs, explaining kind of the behavior of the graph, finding out where this like optimum altitude and temperature is that either maximizes or minimizes this introvert interval. Yeah, so describing a trend for how things change. So if I increase T, what happens if I decrease T, what happens? So it'd be some like exploration of how this, how this function works. And then this last thing here is a little bit tricky, determine how sensitive your approximation is by making a small change to the parameters. And so what this means is like if I, so I have this, this equation here, what happens if I replace this by three point zero one T cube plus A minus square root of A plus T. So I've changed one of the, the input parameters just a tiny bit. And now I want to like send in all of my input values in, in again and ask, well, is it still giving me like approximately the right outputs for the introvert intervals? And if I go back to this table, if I plug in say 23.4 and 1127 from this first row into this new equation with 3.01, do I still get something that's close to this 24 output from the first row? Okay, so that's, that's the crux of the project. I want to say a little bit about how to actually do this, this modeling part of it. And again, I would, I would really recommend, so I posted this up on ELCs to definitely go and like read through, read through all of the different sections, try to make sure you have like a good understanding of what the project is asking, bring any questions you have about requirements or anything for the project to the next class. I definitely tried to like read through it in detail. We'll do some sort of, we'll make it sort of a group project. I think what I'll do is pair people up through ELC. The idea is you'll be working in groups of two, you can sort of meet outside of classes to sort of work on the project, collaborate on it and two groups of up to two or three and you'll be able to submit it as a group. Okay, so I'm going to say something about how we actually do this sort of fitting business. So this is, if you guys haven't been here before, this is Desmos.com and this is a good place to just sort of explore different functions, plug them in and play with parameters. And so if you just go to Desmos.com and I don't even think you have to sign in, but the idea here is this thing on the left is like a function, you can sort of input some specification of a function. So there's f of X equals X squared. We have X squared plus two, shift it up. Minus two, you can shift it down. So you can kind of play with this sort of stuff. You can let things be variables, which is really nice. So if I just do, let's do plus C for example and I'll add a slider for C. And so this lets you sort of change the value of C. So here it's equals one and there's C equals negative one or something like that. So you can see how your function changes as you swing these parameters around. So this is really good too as if you're trying to understand these like shifts. So maybe I'll do X plus C squared. Now I can see how these shifts go. I can add in a second parameter D and I can change that one too. So C parameter is the horizontal thing. The D parameter now is doing the vertical shifts. And so this is a good way just to like plug in, put in some functions and then just see what they look like and how they behave as you change these parameters. You need this one. If you want to plot data, there's this table thing. You go to the plus, you bring up a table, you can put in X values and Y values and it will plot them on the graph. Here's two and four and one, one, two, four, three, nine, something like that. So you can plot a bunch of values and then you can also plot a function and maybe this is X squared. Oh, well geez, funny how that worked out, right? So this function exactly matches my data. Okay, what if there's some experimental error? It's T four point time, 0.5, 11.3. Okay, so now my things don't exactly fall on that parabola, but maybe I can sort of play with a function to get one that closely matches in. It's okay, maybe something like that seems like a good fit, maybe not. There's one that sort of goes through exactly one point. Maybe I won't want to go through the bottom point instead. So you can kind of mess around with this in an actual function. One thing you can do, and this will be probably the most important piece of it, is you can do something called regression on your function. And so what this does is you specify some general form of a function. So some way the Ys are expressed in terms of Xs and it will find the best function of that form that fits your data. So let me explain what I mean here. The way you do it is you do Y one. And so that's referencing the Y values in this column here. Do this little like till the twiddle thing. And let's say we expected it landed on a line. Let's do M as a parameter. And the input variables are the X ones. So these are the things in the column on the left. And so we're doing like Y equals MX plus B. And so what this is doing is it's finding the best fit line to your data. So we have three data points and it's finding the line that somehow is the best line that's close to all three points simultaneously. And so it'll be up to you to determine what does this R squared thing mean? Here it's telling you the parameters are 5.4. What does the slope of 5.4 mean in this context? What is this intercept of negative 5.233 actually mean? But I'll also say that you don't have to use just lines. You can do something like Y equals X one squared plus M times X. Does that actually work? Okay, so you can do something like this. So here right now, instead of doing a line, I'm hoping that it's maybe something of this form. Y equals MX squared plus B. All right, so instead of just a line and some parabola or something. And we see that we actually get a function that seems to be a pretty good fit. Like it almost passes through all three data points. But now it's a question of like, what does this M mean now? Like it's definitely not a slope because we don't have a line. What does this B mean now? It's not so clear. So that's how you do the modeling. The idea is right there's a stable in the actual project. So you wanna put that data from the table in here and then experiment with like different types of functions to put on the right-hand side here to see what kind of gives you just visually looking at it. What is a good fit? You can also look at this R squared statistic if you wanna read up on it a little bit. It's also some measure of how good the fit is. Okay, so I'll leave it at that. Again, be sure maybe before Thursday to go ahead and look through just the fold, all of the sections of the project, read through all of the fine print there. And then on Thursday, just bring any questions you have about that. And I will probably assign the groups here within a, maybe within a day or two. So you should be getting some notification of who you're paired with pretty soon. All right, thank you all. Bye, thank you. Yep, see ya.