Good afternoon, folks. This is Steve Rhodes coming to you live from the shores of sunny Delray Beach, Florida with your 2 p.m. update. We've got a slightly mixed bag out here. The Dow's off 40 points, about a little over one tenth of a percent. The S&P is up 15, four tenths. The NASDAQ is up nine tenths or 109 points. Russell's off 10 points. That's a half a percent. Semi's are up six tenths of 18 points. So it is a truly mixed bag. It's not mixed in the Goldilocks area of 12 bucks. Shaking off that move lower earlier this morning. Trade out at 1712. Silver is flat at 1867. Lights recruit. It's off $3.00. Trade at 9680.  A pump of three pennies. And the 30th Treasury up one point and 2130 seconds. She's printed out at 140. 15. Let's take a look at some of our indices out here. See what they're communicating to us. It's the Dow that is struggling. The Dow is struggling up at a prior level of resistance. That is at the 31, 885 level. The price can close above that. Then we've got an A to B equal CDT upside with the 32, 375 being its target. The A to B on the S&P 500 cash is 4030. That's the one to one level. The NDX100 is already above the one to one area. Its next destination would be the 1.272 expansion range. And that's at the 12771 area. We see the ESL Russell 2000. This made the one to one. You're seeing a pullback today. What's that mean, Stevie? Well, what you have is an inside bar. And as we end with an inside bar today, an inside bar's meaning to us is that the trend that's in place will continue. That suggests that the Russell 2000 would go target the 1872 level. That's its 1.272 expansion. The semis, they're already up at the 1 to 1.618 level. So you certainly want to watch for any kind of reversal candle there. Typically, if we get a reversal candle in the semiconductor, that should pull the markets lower. The transport is also struggling today. We mentioned that, or I should have mentioned that. We took a look at the charts, but it too has an inside day. Now we don't have an A to B equals CD pattern or anything along those lines here. But prices likely go target. It's descending trend line. And that's in the 14-115 area. We'll switch over. Take a look at the white background version set of these charts out there. There may be a few others to take a look at. But just see if there's anything else that sticks out at us. And the answer is not really. So you can see the transports. They also are in bar number six. They're the most advanced with regard to their TD9 count patterns. But that's not going to mean a thing until sometime next week. It may not mean a thing at all because the TD9 count pattern may not unfold at all. Folks, stay tuned for your favorite polar bear. David Whitey is up next. Tom O'Brien will take us on home. I'll see you back here tomorrow at 1 o'clock sharp. Have a terrific Thursday, folks.


Right now, I'm with HP. I have the front-end team here, so it's a group of designer development guys, people, so they have the skillsets of a designer as well as development, front-end development. So, this is something new we are trying here, you know, putting engineers into the design group and actually getting them to write front-end code. Usually, the standard process is to get engineers within engineering to write, right? So, the priorities of an engineer obviously is different. It's scalability, architecture, optimization and all that, but when you put them under the context of design, the priorities change. It's more how pixel-perfect can you make the design to be and all that. So, it's an experiment. We have tried it. I've tried this in past in Yahoo. It fairly worked out. If you look at the latest Yahoo mail-in calendar, that's where I learned this model of working and I have had the opportunity to work with some of the best guys in the industry, you know, guys like Jonathan Snooke who is a known guru of CSS and prototyping and all that. So, I learned some of the, you know, tricks of trade from them and now I've tried to improvise on top of it and try it here. That's great. That's exactly what we're trying to do at the conference because we deliberately didn't try to sort of polarize in one direction but we intended to understand how you build things and not just how you imagine and sort of put it on a paper or on a storyboard. But tell us what you're going to speak about. So, you know, in the context that I'm talking about right now, so it's, so when you're getting front-end engineers to think about design, think about pixel-perfect designs and all that, there's a lot of effort and there's a lot of change in your shift in thinking pattern that's happening, right? Now, traditionally there's a way user experience designers engage with engineering. The way is they design something and then they throw it over the wall to engineers and the engineers try to make some sense out of it and then they do something and then end result is a faculty monster, right?  So, for that we use a bunch of new technologies and frameworks to get there. It's not just, you know, human beings, right? We need the infrastructure to support us get there. So, I think it's this bunch of frameworks and technologies. So, a lot of people have heard about it on the internet. I'm pretty sure a lot of people would have read about it on smashing magazine and all that but I've rarely seen people actually using it. So, for example, when I was trying to hire here, I spoke to like 200, 300 people in Bangalore and none of them had used any of these technologies. So, I thought, you know, this is one of the good forums to actually, you know, familiarize the community towards such frameworks and technologies and how to use them and get off effectively transition the whole prototyping mystery out there into a proper scientific front-end development process. Right. And will this be a lecture, a demo, a workshop? What format you have? I was thinking of a 80% lecture, 20% code kind of thing because I have always felt that, you know, if you do slides, people get it then, but then by the time they are out of the room, it's washed. Right. So, I often get mails back from other conferences where people go back and ask the same stuff that I have, you know, told, covered in the slides. So, I think a little bit of code would help. So, that's my plan.  So, that's what I'm hoping to do. I've still not figured out which is the right way to do it, but that's, that's kind of what I'm doing. No, sure. It sounds good. I guess the question I have, I'm curious about is how you, when you hire people for your team, how do you kind of bring them up to speed with where the team is and how they do business as in how you manage what your process, the design process as well as engineering process for the team is, how do you bring them up? Right. So, one of the first things I have to fight out is here kind of, so I get two kinds of people, right? When I try to hire, since I'm part of this user experience studio, it attracts a lot of designers who are from either school of arts in ID, ID, IDZ from ID Kanpur, Mumbai. And then there's other bunch who, who are primarily engineers in some by education and they have spent the last three, four years doing front end work. And there is this group who are like completely back end guys. I mean, they don't really care about the presentation layer. They think about the controller, they think about the database and that's where they're comfortable. So these are three kinds of people I usually who apply to this job. And so the first step is to actually convince them, you know, that they are nice niche area. Because most guys who are in front end, I've seen that the HTML series of us, they're trying to transition to the front end. And if you talk to a bunch of designers, you'll find them trying to transition to front end work. So it's a very niche area, right? So you're ideally positioned between, you know, technology and creativity. I mean, this is the best place in the whole software industry you can be in, right? So it gives you the power of actually building something new, you can influence the product, and you can make it work. So you won't get that you're working very closely with product management. And you can actually shape the product. Back in engineering guys rarely get that chance to do. They can improvise, they can make it scalable, they can, you know, make make it work faster. But they can't make the user go off.  No matter what's happening in the background, and that's your engineering genius, it could be, but the user doesn't see and doesn't care. So do you kind of feel the totally totally that that's absolutely to see feature is a given, right? Without a feature, it's not a product like let's take the key that you have, for example, if it won't unlock a car, the key is useless. It's not a key, right? The feature is a given. But the question is, you have these kind of keys, and then you have the micro kind of keys where you just have to put the key in your pocket and press the button. So these are this is user experience, right? The feature unlocking the car has to happen. Without that, there's no product. But once you have the product, how do you differentiate between product X and product Y? If you take the example of MP3 players, this is what I, you know, obviously, you know, MP3 players have been there since the 90s, 94, 95, not of MP3 players have been there. But in the late 90s or early 2000s, Apple came with an iPod. Now, what was an iPod? It was just an MP3 player, but it redefined MP3 players, right? So that that that user experience what makes a product. So I totally agree with the idea. This has been great. Thank you so much.


Hello everybody. Hope you're all doing well. Thank you for joining me today to talk about why product managers shouldn't be striving for perfection in the workplace. Let's get started. My name is Samantha Nara Harisetti. I'm a product manager at House. For those of you that are unfamiliar with House, we are a home design inspiration platform. We also have a e-commerce marketplace at House that sells home goods and furniture, and this is the team that I'm part of. Here at our marketplace, I specifically focus on repeat purchase behavior and the overall customer lifetime value. If you'd like to connect with me later or ask any questions, you can find me on LinkedIn. Just look for me by my full name. A little bit of background about me. I started my career as an iOS developer for a banking client with TCS. Here I was working on building iPhone and iPad applications for this banking client. I left that role to pursue my MBA. I went to business school at the University of Arizona with the hopes of actually moving into tech consulting. While I was in school and throughout my internship, I was introduced to the field of product management and decided to pursue that path instead. Out of school, I landed a product manager position at Overshark.com on their mobile apps team, thanks to my iOS Dev background. At Overshark, I was focused on customer acquisition and conversion on the app, so essentially getting people to download the app and make a purchase on the app. After Overshark, I moved on to Carfax to launch a new product for them from scratch. This is a new web-based product. It's a web-based marketplace, and my focus there was organic traffic growth and user engagement on the platform. Here's the agenda for today. We'll start off by defining who a perfectionist is and how this might translate to your everyday life and work. We'll also talk about how a product manager's role is far from that of a perfectionist, and why you should and need to embrace perfection as a product manager. So, are you a perfectionist? By definition, a perfectionist is someone who has a personality that strives for lawlessness. This often occurs by fixating on imperfections, trying to control situations, working really hard, and or being critical of themselves or others around them. Do you relate to any of these, or at least in some degree? I know these seem a bit extreme, but these behaviors can actually translate to your everyday work and life in a few ways. So, let's ask ourselves a few questions. Do you put off doing something until the time is right or you have it all figured out and you can start doing it and getting it done perfectly? Or, when you think of an idea or a task, you either do it all or you don't do it at all. There's just no in between, like there's no progressive steps. Or, do you get upset when you don't get the results that you wanted, even though you've got some of the results, but not all? And lastly, do you expect yourself to give your best in every situation and nothing short of perfect is acceptable? If you answer yes to any or most of these, then you are a perfectionist to some degree. Not that being a perfectionist is a bad thing, but as a product manager, it is a required skill to be able to work in chaos, in uncertainty and to react and pivot really quickly on the job. Perfection, for us at least, has been glamorized since our childhood and as you progress in your journey as a product manager, you need to learn to embrace and practice in perfection. So let's look at how a perfectionist and a product manager skills are completely opposite. There you go. So as you can see, just as you go down the line, some of the characteristics of a perfectionist are pretty much different from that of a product manager and what they're expected to do on their job every day. And today we'll just kind of go down the list, just one by one and talk a little bit in detail about how you can tackle each of these perfectionist traits. As a product manager, just some big little tips that you can do at work that could help you just kind of combat some of these traits. So the first one we'll talk about is overcoming procrastination. This honestly needs a little bit of rewiring of the brain. So for me, I am someone that I do procrastinate at times and usually when I don't know the right way to do something or I don't have all the details. So one way this translates to my job is in the form of writing product requirement documents or PRDs, I will wait till I knew all the details to get started and not shared with the team until it was all flushed out. So this is not ideal because it might take a while before you know every single thing about that project. So one of the best advices to tackle this and I read this online is to simply get started. So start to document, fill out the headers and all the info that you would need as part of the document what questions you have, what other information you would need, things like that.  This way you're tracking the document in by size pieces versus taking the whole chunk as one. So just starting, just getting started helps in a couple of ways. One, you understand the gaps in your knowledge of the initiative itself. And two, it helps estimate the amount of work that is needed by you to fill out the doc and you will accordingly plan your next steps. Three, it really reduces the anxiety within you to start and finish the document because you're working on it, you're making progress. And four, a lot of the features that you build as a product manager, they require like cross-functional work with other teams. So they can look at this document as you're working on it and start adding their inputs, their questions. And essentially the work can get started a lot quicker because work is continuously happening on the document as you're working on it. Now, another example that I gave you was very specific. It's around a product requirements document, but this could be essentially any task you have. All you have to do is just get started and tackle in by size pieces. The next one is overcoming a all or nothing mindset. As a product manager, you're responsible for launching new and exciting features. Sometimes these products have multiple elements and each of these elements could be adding to the overall effort and the time needed to launch this feature. Now, the problem is as a product manager, these new features that you're looking to build and launch, they're often based on an understanding of your customer, of your product, of the data behind your product. So essentially you're hypothesizing that launching this feature is going to be impactful and beneficial to both your product and your customer. You don't know that for sure most of the time. So what we should be doing is testing our hypothesis in small increments, taking an iterative approach, meaning we should only be creating a MVP version of these products. So if you're not familiar, an MVP or a minimal viable product is basically the most minimum version of your feature that you can build and launch to test your hypothesis. So once your hypothesis proves right, you can go on adding other bells and whistles to your feature and keep optimizing it to where you want it to the ideal state. This is also the most cost efficient approach to building a product because imagine the opposite. Imagine you went all in on this new feature that took several weeks or months to build and it didn't create the impact that you thought it would or it is actually dragging down the metrics in a negative way, we don't want that. So what you wanna do is take these small steps while building your product. So again, taking an iterative approach. The third one is overcoming being heavily disoriented. Yes, of course, as a PM, you show impact by through the results you're gaining, but like we talked about in the previous slide, our ideation of what we're going to build next or the next feature is typically based on a hypothesis. Again, around our understanding of our customer, of our product and how our product performs. Seldom, we would know for sure that this product or feature is going to launch and drive up the metrics. So with the information that we have, we launch a product, sometimes it might not do well and that's okay because being a product manager is all about experimentation. If you don't get the results that you wanted, it is our responsibility as a product manager to pause, go back and understand why we didn't get the results that we wanted, what could have gone wrong, is the feature not working as expected, is some things that are friction in the flow or in the user experience. So learn from that and pivot your strategy based on what you just learned rather than following the plan that you already had before. So make a plan for the follow-up builds and test again, again, being a product manager is all about experimentation. So most of the time we will not be going from zero to hero, it's not a zero to one approach, it's always about making progress as we build the feature out. The next one, let's talk about learning to not be self-critical and appreciate yourself. Like the example in the last slide, when a feature or its results don't go as planned, remember, just be kind to yourself. Even if the feature doesn't win, take it as a learning opportunity and celebrate it. You should focus on the process from that launch, learn from it and move forward. Remember that the destination is always not easy to reach, especially for a product manager, but when you enjoy the journey, the destination just becomes that much sweeter. And don't forget to tell yourself that you did a good job and just focus on learning forward. And I want to emphasize on this because as a product manager, this will happen to you so many times in your career, possibly potentially on a daily basis, given just you're working on multiple features at a time, or things can go wrong, due to factors that are external to you and your company, so you just have to take it with us, right? The fifth one is overcoming imposter syndrome. Just have you ever felt like you didn't belong in a job or you didn't deserve a job? It just felt really guilty about it. So this is called imposter syndrome and it is loosely defined as doubting your abilities to do a certain job and basically feeling like a fraud. As a product manager, it is very common to feel this way, mostly because our job is very chaotic, it's very vague, and it's hardly defined with the structure. You have ownership over a lot of different aspects of your product and it changes day to day, minute to minute, feature to feature. So the lack of that structure sometimes makes you feel like you don't know what you're doing. One day you know everything, another day you know nothing about your job. And you need to remember that you're not alone, this happens to all product managers, this happens to everybody, no matter who they are, what job they're in, everybody's learning on the job. And secondly, recognize when you feel guilt and manage it. Tell yourself again that you're deserving of this job because which is why I had that in the first place. So just be kind to yourself. And the next one is how to overcome overcorrection. So everybody, no matter who they are, they receive feedback in their job or as part of their performance review, from time to time they do receive feedback. But for a product manager, it's a little bit different because we receive feedback almost every day, whether it is from our customers not making our feature or another team not being able to prioritize our info, seeming brighter than the features or not having all the info for our stakeholders not feeling aligned with their roadmap. Feedbacks comes to us in all different shapes and forms. It can get overwhelming but it's important again to remember just not to react immediately, not react emotionally or over correct immediately after getting the feedback. Understand what the feedback is and where it's coming from. And just take the feedback constructively, listen to what is being said and apply it in a way that makes sense for what you're doing for your product. Don't just blindly apply the feedback for you at work. You need to take a step back, reflect on the feedback and then react accordingly. And the last one to overcome is expecting perfection from others. If anything, this is the most important trait to have as a product manager is not expecting perfection from everybody else because 100% of your tasks are cross functional. So you work with many, many different teams, many different people. You will rarely work on something by yourself, just by yourself. So when we work with so many different people and different teams, each team and each person, they have their own processes, they have their own working styles. So some ways to manage expectations for yourself and for everybody else is when working with different teams or people, be clear in what you're asking for. So what do you need from them? Why do you need it? When do you need it by? Is there a specific format or template to watch you're asking? Et cetera, things like that. Two, if it is a long task that is going to take several days or weeks, have intermittent touch points with the folks on the team to see the progress of the task. So in case it's not going the right direction, you can always stop and pivot the team in the right direction. This is better than waiting till the end and realizing that it hasn't gone the way that you wanted it. Three, and you need to manage your own expectations. Again, like I said, not everybody works like you or in the same pace as you, some people are pastors, some people slower than the way they work. So accepted things will be a bit different and you need to communicate with your team to work in harmony with them. And lastly, be patient while you're going to work with your team because as you're learning, they're also learning to work with you. So that is the end. Thank you all for your time today. I'd encourage each and every one of you to think about what perfectionist traits you have even if it's a fraction of it and think about how that affects your everyday life and work. Should we have a message on LinkedIn if you have any questions that you'd like to chat? Thank you, everybody.


Welcome back, Math 241 Calculus II at NC State University. We are on day 20, lecture 20 for this course. We are embarking in chapter 7 and we started it with the idea that we're going to look at some things called differential equations. Equations that have possibly some unknown original function and some derivatives. First derivative, second derivative, third derivatives. The order of the differential equation is based on the nature of the derivative that is in that equation. So today we're going to look at kind of the visual image of what a differential equation might look like if we graphed it in the x, y plane. So we're going to be given something about a differential equation. And we're in the second section of chapter 7 now. They call these things direction fields or slope fields. So a differential equation, let's say it has a first derivative in it. And let's say it's just something really, really simple. Let's say the derivative of y with respect to x is the x value. You can't get a whole lot simpler than that. We'll look at the whole slope field for this in a moment. But at any point in the plane, if we want to know what the slope of the tangent line is, if the curve happens to contain that point, all we have to do is look at the x value. And this tells us that the slope at any point is the same as the x value. We're not going to generate lots of slope fields ourselves, but we do know when we see a slope field, we do need to know how to handle that. So let's say we're at the point 2, 1. If we're at the point 2, 1. And if our curve, ultimately our curve goes through that, remember we had a family of curves yesterday, so we want to pick the one that goes through this point. The slope at this point should be the x value. Therefore the slope, if it goes through this point, would be what? It would be 2. So we would go up 2 and over 1.  Have you dealt with slope fields before? Okay. So we're not going to generate lots of these pictures, but that's how each individual slope is generated. We know the derivative. It's described in terms of x or x and y or just y alone. We end up with this mass of little tiny line segments in the plane. All they are is these slope fields are kind of sign posts or guide posts that if the curve goes through there, then the curve does something like this. It kind of lays up against there in that fashion, or maybe the curve is down here and it lays up against that little segment that constitutes the slope of the tangent line at that point. It has to be that. That's the way the differential equation has described it. So we would expect then if we come out here to 3, it doesn't really matter what the y value is. It matters what the x value is. That's the slope. At 3, 1 it would be a little bit steeper. At 4, 1 it would be even steeper than that. And we generate this thing which has a bunch of these line segments on it. And let's take a look at if I can find it quickly what this would look like. And we'll come back to this. If we were to put all these together, this would be the slope field for dy over dx equals x. So we might as well talk about this. This was generated in Maple. This little package, I think you've used repeatedly the with student package, right? In Maple. So this is another package with DE tools. So that activates a lot of the stuff that you would do with differential equations. So here is the differential equation itself. DEQ, that's the name of it. It is diff y of x with respect to x. So that's basically dy over dx equals x. So there's the statement of the differential equation. And then this is a DE plot that what we named this up here. We're going to plot that from x equals negative 4 to 4 and y equals negative 4 to 4. So we end up with this slope field. So here's the kind of the Maple ease version of this. Its name is DEQ. What is it? It's derivative of y with respect to x equals x. So then it generates this. We were at the point 2, 1, and we knew that the slope was 2. Well, look at every place where the x value is 2. What's the slope at every one of these points where the x value is 2? It's 2. Jump over here to x equals 3. The slope at every one of these points where x is 3 is 3 and so on. So you can see the commonalities of the graph based on the fact that it's a pretty simple differential equation. The slope is always the x value. Where is it flat? Where is this particular family of curves? If we were to put some in here, where would they be flat? On the y axis, because on the y axis the x value is 0, therefore the slope is 0. So at all these points in here, the slope is actually 0. Can you visualize the curves that would fit in this particular family of curves? So as long as we stipulate, we want the one that goes through the point 2, 1. Well, 2, 1 is right here. So what would that look like? That might look something like that, right? And wouldn't it have a mirror image over here? The same thing. Is that what you would expect for something that has x for its derivative? Well, let's anti-differentiate and see what we would expect. If y prime is x, what is it that has x for its derivative? One-half x squared. One-half x squared, right? So y would be one-half x squared plus c. Is that what you visualize in here?  So the one-half, does that make it a little fatter than our old friend y equals x squared or a little skinnier? It makes it a little fatter, right? Which these ought to look a little fatter. So if we want the one that goes through this point, maybe it looks like this. So all these little slope fields, these little line segments kind of tell us where the curve should be going if the curve, in fact, is in that vicinity. If we've got one that's down here, then we've got these, again, these signposts that tell us where the curve is if it's down here. So it can be that simple. D y over dx equals the x value, and we know what these look like. They're falling into this format. If we force it to go through a certain point, then that allows us to focus in on a specific value for c. Other than that, it is a family of curves. All right, this will change things drastically. I think this is from our book somewhere. Direction field for this differential equation. So we've looked at a real simple one. D y over dx equals x. The slope is the x value at every point. That's significantly different, but that is a description of what the slope is at any specific point. Can you visualize some different curves in this particular slope field? If it comes up through this point, it looks like it dips down, right, and comes up here. Maybe it dips down not quite so far. I don't know. Let's graph these. So there's a whole bunch of solutions. Let's suppose that the y of 0 is 1. That means when x is 0, y is 1. So there's 0, 1. What's that look like? Right there, it's flat, right? Is that true? By the way, at the point 0, 1, 0 for x, it doesn't really matter what y is, right? If x is 0, the slope is 0. So if you'll notice that x equals 0, the slope is always 0. Shouldn't it, what, head upward and then flatten out something like that? If that's the slope field that represents this set of points. Looks like we've got a symmetric image over here. That's one of the members, kind of rough, of this family of curves. How about this one? Y of 0 equals 2. So it goes through the point 0, 2. Is that a different looking curve? It's slightly different looking. Looks like it comes up here and then is asymptotic there again. But it doesn't dip down quite so far. Different looking, but kind of similarly shaped. The next one's very different looking. Y of 0 equals pi. What's that look like? It's linear, isn't it? And that's an equilibrium solution. Why would that be true? Because don't we have the sine of y, right? And the sine of pi is 0. And all of these isn't the y value going to be pi, right? So we're always going to get 0. Well, if the rate of change is 0, it's not changing at all. So it's flat all the way across. So this is an equilibrium solution. How about y of 0 equals 4? Now we're up here. Again, a very different looking curve from the first two because these are asymptotic but coming in from the other side. And y of 0 equals 5. We end up with a curve that looks like that. So you've got a whole bunch of possible curves, including an equilibrium solution. We found one of them. Is there another one? Zero. At zero, right? So down here it's flat all the way across. There's an equilibrium solution. Why is that? Because the x value is 0 all the time. It doesn't really matter what the y value is because 0 times the sine of whatever y is is going to be 0. Are there other equilibrium solutions that aren't pictured here? 2 pi, same thing going the other direction, every pi units, right? So we're going to have equilibrium solutions. First one is at zero and what? Every pi radians from there? How would we write that? n times pi where n is what? Not all real numbers. We want it to be an integer, right? Some integer multiple of pi. Integer normally is a capital Z. That'd be a good little research project. Why it's called capital Z for integers. I is irrational. But that'll take care of all of them, right? Does that take care of the ones down here also? Because that'd be zero plus negative one times pi. So we're going to capture all the equilibrium solutions in that statement. Questions about that before it moves? So you're not going to have to generate these by hand, but they're very easy to do in Maple with that d e plots command. I think this is in your book. Here's a slope field. We have no idea what the equation is, but can you visualize a lot of different curves in this particular slope field? This is on page 506. I think it's the same thing. This is actually from the previous edition, but I think it's the same equation. Looks like if they're coming up here, they're right, and then go back up. So here's a variety of solutions from this particular slope field. So if we're forcing it to capture this point right here, how would it do that? Follows the slope field. A little signpost, that's all they are that tell the graph where to go if it goes through that point. Looks like these are almost simultaneous here, but this one does something radically different from the one that's slightly above it, simply because it captured this point.  One that goes through this point. So we have different looking curves. They have some similarity, but they all have the same differential equation. That differential equation, let's see if I can find it. I don't see it. It's not all that relevant. We're looking at a slope field. These are some different members of that family of curves. All right, here is a maple version differential or the diff of y of x with respect to x equals y. There's the command. So here's our equation. Derivative of y with respect to x is y. Here's what they look like. It goes through this point. It comes over here. It comes over here and goes through this point. Looks like they're all asymptotic to the x-axis. They might be down here. A lot of different members of this family of curves. I bet you know what the solution is to this. What is a y thing that we can take the derivative of y and it's the same thing as what we started with? e to the x would be a solution, right? Can you see e to the x in here? Something that looks like this, right? And other variations of that? Well, how could we get other variations of this? We could have some constants involved, right? And the derivatives of the constants wouldn't appear. How else could we have a coefficient out in front that would change it? Could we have a negative 1 out in front? Could we have an e to the negative x, possibly? Would that work? Is the derivative of e to the negative x also e to the negative x? Not quite, right? It's the negative of itself, so that's slightly different from this. But we could have other coefficients which will take this image here and reflect it down, right? So we could have some coefficients out in front. So we've got a whole bunch of different shapes, but all of them above the axis are similarly shaped to this and all of them below the axis are similarly shaped to that. We'll have a technique to solve. We won't have to make a guess at this. Our first technique of solving differential equations will be those that are called separable differential equations. We'll isolate or separate the x's and the dx's from the y's and the dy's, and we'll be able to solve for y. Okay, another one. Here's the differential equation. It looks like dy over dx is the x value minus the y value. Now, that looks like a pretty simple differential equation. That actually is a pretty difficult problem to do in terms of a separable differential equation. Things that are added and subtracted are going to be difficult to deal with. Things that are multiplied and or divided will be relatively easy to deal with because we can move them around where we want to. To isolate the x's and dx's from the y's and the dy's. So if we take the x value, subtract away the y value, we get the slope at any point. So where do you see the flat places here? Where would the slope of the tangent lines be zero? When x and y are the same, right? If x and y are the same and you subtract one from the other, you get zero. So look at a point where the x and the y are the same. And here's the point two, two, right? Slope of the tangent is zero. So we've got a little flat line segment right here. We could come to the point three, three, if I can find it, right here. The slope of the tangent is zero. So you notice some similarities as you work your way up. If the x and the y value are the same and we subtract one from the other, we should expect to get zero. So there's the flat places on this graph. So zero, zero, it's flat. One, one, it's flat. So you see all these flat places all the way up the diagonal. What could the curve look like? It could look something like this, right? And we've got it asymptotic in that fashion to the line y equals x and a whole bunch of different curves that could be illustrated down here as well. So we do have a family of curves for the differential equation. All right, one more of these, and we'll talk about how we can use Euler's method to approximate solutions. So here's one where the derivative of y with respect to x is the x value divided by the y value. Let's just pick another number other than zero. Where would the slope be? Negative one, okay? One, negative one, negative one, one. So in general, those are some specifics. In general, where would the slope be negative one? When x and y are equal. Right, so they're equal in magnitude. One that's positive and one is negative, right? The slope would be negative one. So if you check out some points like that, we had some examples. You'll see that the slope is negative one. Where is the slope positive one? Well, the x and the y value would have to be the same, right? So at the point two, two, we should expect the slope to be what? One, up one and over one. At the point negative two, two, we should expect the slope to be negative one. Where would the slope be zero? For a fraction to be zero, what has to be zero? The top. So as long as x is zero, the slope is zero, right? Now what about zero, zero? What about the solution that actually contains the origin? It's got an undefined slope, right? If the y value is zero, in fact, probably worse than that, isn't zero over zero worse than just good old-fashioned undefined? What is zero over zero? Eleven? Sure. Is that a word? Fourteen? Negative seven? It's whatever you want it to be, which is kind of worse than undefined. It's awfully defined. It's too defined. We've got way too many descriptions or definitions for zero over zero. What do you think? Do you think we're going to have a solution that goes through the origin? I don't know. What are you seeing when you look at this slope field? Everything's going away from it. Everything's kind of going away from it. So if we had a solution, let's say, that's up here, is that what you're seeing? And we maybe have its image down here. What is that? Hyperbola, right? So maybe we've got a family of hyperbolas. Do you think that's the only position that we could have in the plane? I guess we could have some over here, right? Maybe a hyperbola over here or a branch of it over here and its symmetric image over here. So this could very well be a family of hyperbolas where the slope of the tangent is the x value divided by the y value. Obviously we're not going to have both of these simultaneously because then we would be talking about something that's not a function. So if we had the green set, we would have its image but not these two. So we've got either a hyperbola with branches opening up and down or scrap that another hyperbola or a set of hyperbolas that open left and right. All right, we're going to have an approximating technique called Euler's Method to come up with a sketch of the real curve based on the differential equation. Euler's Method uses linear approximation. We're going to approximate what is going on along the curve by what's happening along a line that's tangent to the curve. Why would we want to use tangent lines having started with the differential equation? That's what the differential equation is, right? It's a description of the slope of the tangent. So we're not going to have the luxury all the time of having the exact curve, but let's say that the exact curve looks something like this. We know for a fact that the curve goes through this point. We're going to use the differential equation which is a description of the slope of the tangent to say I want to try to keep track of that curve, but I can only do so incrementally by going along the tangent line to the curve. Not quite increments like this big, so we're going to use the slope of the tangent and say I know I don't know where the curve itself is, but I know where the tangent to the curve is and if I use my increments small enough if I choose a point on the tangent line it's going to be pretty darn close to the actual curve itself. So we really want to choose way closer than what I'm going to do here, but we're going to come out here some delta x and we're going to say that this is our approximation. For where the curve actually is. Of course we have the luxury here of having the curve and we know that we're not on the curve we're on the tangent line to the curve which begins to separate from the curve. Then with Euler's method we reload we determine the slope of the tangent again so maybe the slope of the tangent now comes up here and instead of choosing a point on the curve because we don't know where the curve is we choose a point that is over delta x units along the tangent to the curve and you can see how they begin to separate because we're using an approximation to do a further approximation so we get further and further away from the actual curve but eventually we end up with something that has a similar shape to what the real curve looks like in absence of some exact method of solution this is not that. If you keep delta x small I'll put that in quotes because what is small for one problem may not be small enough for another problem. If the curve goes up very steeply this kind of changes what small is. We would need our increments to be very, very small because the curve changes so rapidly. If the curve is fairly flat we don't have to be quite so small with the delta x values. Here's some pictures of what it is we're going to do again we're not going to have the real curve the luxury of having the real curve in most cases. Here's the real solution curve we're going to approximate along a line that's tangent to the curve and we'll do that iteratively so we'll do that here reload, get a new tangent line go along the new tangent line reload, get a new tangent line again to look something like this. So what we have is a bunch of points that are on line segments but you could connect them and make a curve out of them. It ought to approximate the solution curve itself and we can compare and contrast once we get some methods to find the actual solution curve. In some cases we already know what it is. So how will we do that? How does Euler's method actually work? We start with some initial x value and some initial y value a starting point just like this point right here we know it's on the curve because we're forcing the curve to contain that point what we don't know is any other points that are necessarily on the solution curve. So we get our x0 and we get our y0 that's our starting point that's handed to us for the most part we start with that then we want to find our next x value so we're going to use the tangent line eventually to get to our next point well to get to our next x value we just simply add this change in x to it. That's easy. So each time you want to crawl along in terms of x add delta x to your previous x value and you're there. Here's what we really want where's the new y value that's the one we have to work at and that's the one we have to use the differential equation for. So we'll go back to our old y value and here's kind of Euler's method using the differential equation we're going to take what we know dy over dx to be that's going to be given to us the differential equation it'll be x over y or the x value x minus y we're going to use the derivative and we're going to multiply that by delta x so the new x is the old x plus the change in x the new y is the old y plus well it'd be nice if we knew the exact change in y, we don't so we have to approximate I don't know if this helps it helps me, I don't know if it helps you I'm going to think of that dx and that delta x kind of knocking each other out and then we're left with dy it kind of sounds like a change in y doesn't it dy is actually the change in y along the tangent to the curve not along the curve itself but this is our differential equation it gets plugged into that spot that's our slope, that's our dy over dx and we continue that process to get x2, we'll take our previous x value those are always easy just keep adding delta x to the previous x value to get y2, we'll take our previous y so this iterative process is called Euler's method it's flawed, it has its flaws we've seen it in the picture why is it flawed, because first of all we're using a tangent line to the curve and not the curve itself and we're using an approximated value to start to approximate further so we actually end up getting further and further away from the curve even though it probably has a somewhat similar shape so how far do we go with this it kind of depends on what we're directed to do or maybe we know the y of 0 is a certain value and they tell us to approximate until we get 1 or the y of 2 and how many increments we have how many steps we have along the way alright, first example pick one that we've already looked at let's start with this one I'm going to take this diagram away so we kind of have an idea what the finished picture is going to look like or might look like let's take the one that goes through the point I don't know let's pick one so we're going to pick this curve that's going to come through here so our differential equation pretty simple, it's just the y value now it's not the new y value it's the previous y value we're always using the previous data to generate new data and we want x sub 0 to be 0 and y sub 0 to be 1 and let's just pick the delta x in this problem to be 0.25 and let's just keep going until we get we want to see what the y value is when x is 1 so we want to go through 4 steps with this crawling along 0.25 each time we go in terms of x so we have our initial point what's our x1 quarter to it, that's easy old x is 0 plus 0.25 so the x value of our new point we hope is pretty close to the solution curve is 0.25 so our new y, y1 is the old y, y0 plus the derivative well what is the derivative on this problem it's the y value okay so the old y is 1 the derivative is the y value so we're pulling that directly from the differential equation that's given to us so the old y value is what we use all this data to generate x1, y1 so we're using x0, y0 to generate x1, y1 so what's the old y 1 is that right so there's the old y value there's y0 times delta x so we get 1.25 does that work so there's our new point because we're using we can't use the new y yet because we're trying to generate the new y so we have to use the previous data so we're going along the tangent to the curve to find a new point we don't know where that point is yet so we're using the old point as our starting point so we have no idea what y1 is so the only y value that we know anything about is this one the only x value we know anything about is this one so we use the old data to generate the new data all the way through the process mainly because we don't know the new y yet we're trying to find the new y so from our picture we've looked at the slope field we started out with the 0.01 now we have a new point that we think is pretty darn close to the solution curve it's over a quarter and up one and a quarter does that work so let's generate a new point so we want x2 x's are always easy previous x 0.25 we're going to add the change in x 0.5 y2 is y1 I'm going to go ahead and write the first derivative what is dy over dx it's the y value what y value the previous y value times this delta x value so really this becomes y1 plus the previous y value that's the slope of the tangent times delta x y1 is 1.25 plus this previous y value 1.25 times the change in x so this is 5 fourths times 1 fourth is 5 sixteenths added to one and a quarter that's it anybody got a calculator out and working so there's our x2 y2 no mystery about the x we started at 0, we went a quarter and then we went over another quarter so we're at 0.5 the y value so back to our picture we're now over here at a half and we're up 1.5625 is that kind of what we expected it's probably underneath where we should be but it's relatively close you don't have to write this out each time I'm just writing it out because it's our first example you can just add the increment each time to x what's the slope the slope is the y value times delta x so what is y2 0.5625 plus that same value since it is also the slope times the change in x Katie we're trusting your calculator here 0.9531 so there's our new point 0.75 1.9531 so we go back to our picture 0.75 1.95 so you can see that we're ending up with what we thought we would get not exactly now how do you know when to stop well we're going to be given something like this we want the y of 1 what is y when x is 1 which tells us to go one more step we want x4 in this case so we are at x equals 1 that tells us we're at the place we want to stop in terms of x we want y4 which is y3 which is the slope the slope is the y value so what was that number 1.9531 plus that same number which is the slope times delta x so that should be our approximation for the y value when x is 1 what's that is that alright with the process looks like euler's method he'd be offended I think if we called him euler let's look at one other alright little bit more to it we'll do a couple iterations through the slope this time is x minus y so it's not just the y value we've got to take the previous x minus the previous y that generates the slope we multiply that by delta x let's see what we want this to look like let's make it go through the point so let's make that now that's actually on the final solution curve when we get to choose the point that's the one we want the curve to go through now from this point though we're going to approximate what we think the curve does based on this initial point let's just get a couple of iterations it's probably all we have time for today so delta x I don't know the smaller we make delta x the better we're going to get I don't know if I showed this earlier have I showed that yet? these are approximations to a curve you can see the solution curve which is the top one up here and as you choose different values for delta x you do a better job of approximating so if we let delta x be smaller you can see the difference in the proximity to the actual solution curve if delta x is chosen fairly wisely or sometimes we don't have the choice it's just given to us so let's say that just to make it different let's say it's 0.2 what's x1? 1.2? so the beginning x, x0 is 1 we're going to add delta x to that so it's 1.2 in fact we're just going to add 0.2 all the way through to the x so the new y is the old y what's our slope this time? isn't the slope x minus y? new differential equation we're going to take that times delta x what are the numbers for that? y0 is 2 so we're going to use the previous point to generate this new point so the previous x minus corresponding y value so that's what? 1 minus 2 there's the slope times delta x which is 0.2 so what do we get? this one's not increasing it's decreasing isn't it? 1.8 so x1 y1 we went over 0.2 in terms of x but the y value fell from 2 to 1.8 let's do one more x2 previous x plus delta x that's 1.2 plus 0.2 y2 is y1 plus slope slope here is x minus y y1 1.8 aren't we using x1 y1 to generate y2 we're using the previous values to generate our new value so x 1.2 minus y 1.8 times delta x we're using x1 y1 to generate x2 y2 so let's see what we have here there's a negative 0.6 times 0.2 which is negative 0.12 is that right? 1.68 so our new value which will be our last value today now would we expect these values to be falling let's see let's see if we're right here so we started out at the 0.2 no at the 0.12 which is somewhere in here shouldn't we expect according to what the slope field says shouldn't we expect our y values to fall for a while and then for them to pick back up right? so it seems to be matching what the slope field actually looks like alright we'll continue with this section some circuit problems tomorrow and I will see you then


Dear colleague, in my presentation I'm going to show a social role of prestige copper objects in the Middle Corporate Society's north of Carpathians, an example of the cemetery in Książnice in the southeast of Poland. Around 4500 BC, the process of formation of the social stratification in the early corporate cultures in Balkans, Peninsula and Carpathian basing began. One of the visible aspects of this process was the custom of setting up of the cemeteries outside of settlements. So, sorry, could you speak a little bit louder? Okay, the second was the practice of burial differentiation depending on sex of the dead and last was the appearance of the prestige copper and gold artifacts in funerary practice, for example in Varna and Hamadze culture and Tisa-Polgar culture. Between 4300 to 4200 BC, under Transcarpathian influence of late Langell and early Bodrok Kereslu culture, the process of social stratification and formation of the local elites in southwest of Poland in Jordan of culture and in the middle of Poland in Dreszko-Jewski Grove began. Circa about 4100 to 4000 BC, under influences of late Bodrok Kereslu and Hunyadi-Halon cultures from the Tisa river valley, the same process began in southeast of Poland in environment of the late Danubian units called the Lubin-Volinyan culture and Wichong-Zwutniki Grove. The Lubin-Volinyan culture developed mainly in the less regions of southeastern Poland and western Ukraine in the period since circa about 4100 until 3600 BC. The Wichong-Zwutniki Grove developed in a small region in the upper part of Vistula river in both sides of Vistula river in the same period. This figure shows the location of cemeteries and single graves of the both cultural unit under discussion, with special regard to the cemeteries where prestige copper objects were found, red and yellow points. We can clearly observe two zones where the burials with copper objects occurred. The first on the left bank of the Vistula river and the second one covering the northern part of the Lubin-Apland and the western-Volinyan-Apland. In the light of the radiocarbon dating obtained in the recent years, the rich in copper cemeteries in the western zone, for example the Zvota and Książnice, seem to be a slightly older than those in the eastern zone. And now I show you archeological data from one of the cemeteries from Książnice site 2. Książnice, site 2 in Książnice lies on the summit of Les covered hill, excavation carried out by me in 2001 to 2017, have uncovered the biggest cemetery of Lubin-Volinyan culture. On the cemetery, 22 copper artifacts were discovered in 7 burials, which constitute the greatest number of copper prestige artifacts in the southeast of Poland. And now I will present the men and female burials containing prestige copper artifacts. Grave number 3 is a symbolic partial burial with shut-hole copper acts of shiria type and with relics of human hand. We found in this grave only a few fingers. Grave number 4 belonged to the 50 years old male, skeleton which was lying in a flexed position on the right side with the skull directed toward the soul, discovered in a rectangular burial pit. Copper chisel was found slightly above the skull. In this burial we found also 3 vessels and numerous set of chocolate flint, microlights which were interpreted as arrowheads. Grave number 5 belonged to the 55, 60 years old male, skeleton which was lying in a flexed position on the right side with the skull directed toward the soul, discovered also in a rectangular burial pit. The grave goods comprised 3 otter vessels, copper hatchet of the Sakalhat type and 16 flint objects including reduced flint blade dagger which was lying in the position of the chest between left arm and the ribs. And this is position of copper hatchet. In grave number 17, poorly preserved skeleton in a flexed position on the right side belonged to the specimen in the age of adultus, most probably a male was discovered. The burial goods consisted of 3 clay vessels, 5 flint artifacts including a massive chocolate flint reduced blade also dagger, as well as an earring made of copper wire discovered at chest height at this point. Grave number 2 belonged to the 9 to 11 years old child, probably female because we have DNA data from this grave. Grave goods consist of 3 clay vessels, 4 flint artifacts and 3 copper objects. On the base of the skull a massive wire necklace was found, under the skull there lay a massive wire earring. A unique bracelet decorated with dimples and made of copper plate was found on the right walnut. A very rich collection of gifts consists of 4 clay vessels and 5 copper objects were excavated also in grave number 7, belonged to the 9 to 10 years old child, female, we also have a DNA data for this burial. Under the skull lay the massive wire earring, additionally at the level of thorax a cylindrical bead from a rolled sheet of copper was found. Another such bead was found in a nearby water pipe excavation which partly destroyed the grave, while the wire bracelet and the small ring were discovered approximately 30 centimeters to east of grave. Grave number 8 is probably one of the richest female burial in all younger Danubian cultures in the territory of Poland, a skeleton of a 35 years old female lying in a flexible position on the left side with the skull directed towards the soul. It was discovered in a partly destroyed rectangular burial pit. The unusually rich grave goods consisted of 10 ornaments made of copper wire, 2 necklaces with spectacle shaped pendants, 2 bracelets, 2 earrings and 2 rings, a chocolate flint retouched blade and a blade as well as fragments of 2 clay vessels. This is a close up of a situation of necklaces. And copper jewelry from this burial after conservation process. Now have a quick look at the chronology of graves containing copper artifacts. The Bayesian analysis of 4 radiocarbon dates shows the period of using of the prestige copper artifacts in Książnice necropolis most probably between 4030 to 3820 BC by medians. Now I am going to present the problem of the reconstruction, how people of the copper age wore their prestige objects. This is very difficult because we don't know much more about dress of these people and second because we have only data from a special kind of context from burials. And this is an attempt of reconstruction of the way of wearing. The prestige copper on the base of the richest male burial of the 60 years of old men. In the burial we found flint retouched blade, dagger in the position of the chest and we reconstruct it as a flint blade mounted in the organic handle put into organic sheet attached to the belt which was worn across the chest. We found a copper hatchet which was located next to the skull. We suppose that this hatchet was mounted in the wooden handle and this man held the hatchet in his right hand. There were also a numerous set of flint tools which were lying next to the right pelvic. They think that there were in a bag made of organic material. And the way of dressing in this picture is based on the Otzi reconstruction of dress force. And this is an attempt of reconstruction of the way of wearing the copper jewelry on the base of the richest female burial of 35 years old women. In this burial we found 10 artifacts made of copper wire. Two necklaces with a spectacle shaped pendants. First was located under the skull, second on the lever of the belly. Two earrings bigger and smaller, both of them laid under the skull. Two copper rings which were found on the lower part of the chest and two bracelets on the wrists, on the left and right hand. However we have a problem with interpretation of the way of wearing the earrings which we discovered under the partly destroyed skull and two rings which were found on the lever of the upper part of the chest. We suppose that the bigger of the earring was worn on a textile bandage or other head gear but smaller open earring was put on her ear. Similarly we have some doubts with the case of the two rings. If it's an element of the belt or application of the dress we don't know. And conclusion. And cemetery outside, too, in Chiang Nhi Tse is a perfect example of the changes taking place in the social structure of the younger Danugian cultures on the turn of 5th and 4th millennium BC. The necropolis consists of two separate burial fields situated several meters apart. The eastern field, a field intended for the local elite, consisted of two parallel rows of graves. The eastern row comprised an adult female burial and two burials of female children. The western row included two adult male burials, a putative male burial and a cenotaph. With this layout there was a partial burial accompanied by a shut-hole copper acts of Sharia night. The western field has more irregular layout and contains eight poorer burials. Grave goods consisted mainly of vessels and flint artifacts. Moreover, in this area we discovered a cremation grave and second cenotaph. On the base of the anthropological analysis and arrangement of skeleton right or left side and DNA data would give the grounds for us assuming that in the grave number 12, 15 and probably 14 men were buried, but in the graves number 10 and 11 on. In between of the both burial fields there was a male burial number 17, with copper earring and flint dagger, which are considered symbols of the high social status of the dead. One of the most interesting problems which we can observe in the Ekshonshnitsa is presence of rich female and female children burials accompanied by a prestige objects. It confirms that in the middle copper age in south of Poland certain women from that environment had attained special social status, presumably through an inheritance of other kinds of social relationships. Rich copper weapons and tools show close references to the late Bodrokera Stur culture, but copper jewelry is a specific mix of late Langell culture tradition, for example, spectacle-shaped pendants. And local tradition with copper wire necklaces with triple culture patterns, for example with bracelets made of plate of copper. Another kind of archaeological sources from this cemetery, a spotter and presence of cremation burial have a strong connection to the Hunyadi Halon-Naginani horizon. Thank you.