Good morning, everyone. Thank you, Andrea. And many thanks to the International Institute of Communications and to Dr Hesse and ICT Qatar for the kind invitation to give the regulators perspective on digital communications literacy at today's forum. I will start. I will start with why the Australian Communications and Media Authority thinks that digital communications literacy is important and tell you about our approach in the area. Firstly, a little background. The ACMA is a converged regulator with responsibilities for broadcasting, the internet, telecommunications and radio communications, including a range of consumer protection measures. Convergence poses many challenges for the current media and communications legislative framework. In response, the ACMA, while concentrating on its core regulatory tasks, is also focused on future trends such as the development of next generation networks and issues relating to regulating new services and cross-platform content in an increasingly digital world. The ACMA's research arm plays a vital role in that focus to identify and understand changing consumer, citizen and business uses of communications technologies and inform the ACMA with evidence that contributes to its understanding of regulatory issues in this dynamic environment. Beginning in 2007, the ACMA committed to a digital media literacy research program and I lead the researchers responsible for that work. What is the ACMA's view of the meaning of digital communications literacy or is it sometimes called digital media literacy, digital literacy or literacies for the 21st century? Terms or expressions which are often used interchangeably these days. They are the subset of a broader umbrella concept of media literacy which is understood as the ability to use, understand and create media and communications in a variety of contexts. People who have digital communications literacy, our explicit interest today are able to use, understand and create digital media and communications. This term overlaps nicely with digital media literacy which is the expression that the ACMA uses as it reflects our converged regulatory responsibilities for both internet, broadcasting and telecommunications. So forgive me if I sometimes fall back on this expression. The Australian government sees digital communications as enabling technologies and promotes their wider use for participation in the digital economy and inclusion in society generally. To quote from our portfolio department's website. Thank you. The internet is an essential tool for Australians including children. The ability to use online tools effectively provides both a skill for life and the means to acquire new skills. The ACMA's framing of digital communications literacy as set out on this slide aligns with our government's key strategic priorities and also recognises our responsibilities to help Australians deal with security and safety risks on the internet and mobile phones. It also helps to shape the ACMA's thinking on new regulatory challenges in an IP based communications environment. Areas such as public interest protections and social concerns about going online including protection from cyber stalking and cyber bullying and safeguarding financial and personal data. When the ACMA refers to digital media literacy we generally mean the skills and capabilities needed for effective participation in the digital economy and to encourage social inclusion in a networked society. These skills and capabilities include use including the access to infrastructure and devices and the information skills to find content and services. Understanding and interpretation. The ability to understand, evaluate and control aspects of media content to judge the quality and authority of sources and trust in various forms of media found online. Creation and participation for example the ability to participate in social media and contribute to user generated content and we've heard how important those developments are. Consumer protection and security is also an essential aspect of digital communications literacy. Understanding online security risks and how to protect oneself and those for whom one is responsible. Why is digital communications literacy important? In July 2009 the government published Australia's digital economy future directions report providing the vision for taking strategic and enabling action to ensure all parts of Australia benefit fully from the digital economy. This includes investment in a national broadband network to deliver super fast broadband to Australian homes and workplaces. The report specifically identified the need for a digitally empowered, confident and literate community to maximise the social, cultural and economic value of this infrastructure. The social value to individuals of digital communications is fairly clear. Internet users report increasingly positive impacts in many areas including pursuing hobbies and interests, convenient and cheaper shopping, increased work readiness, employment and further education opportunities, easier access to healthcare and financial information and enhanced communications and interactions with friends and family and community. Less immediately obvious is that digital communications literacy is becoming fundamental to being a competent participant in today's society. Digital media and communications technologies provide new means for creative expression, innovation, collaboration, emerging Web 2.0 platforms, tools such as blogs, wikis, social networking provide additional ways for community engagement across many cultural spheres. Digital technologies have a role in preserving and supporting cultural heritage. For example, museums and libraries across Australia are implementing digital preservation policies and taking the collections online. Important for Australia and Australian heritage is the digital recording of traditional stories and cultural practices of indigenous Australians. As well as traditional owner group languages and their knowledge of natural resource management. There are great economic advantages deriving from online participation across the broad community. Employment opportunities such as employment opportunities. An IBM study on the economic potential of high speed broadband in Australia estimates that investment in smart technology supported by broadband in electricity, irrigation, health and transport could add more than 70,000 jobs in Australia in 2014 alone. The growing value of e-commerce is all around us. A simple statistic. A quarter of Australian businesses in 2007 and 2008 were generating income from selling goods and services online and cost savings of course. The Australian government has invested heavily in e-government initiatives benefiting through administrative cost savings in areas such as online tax returns, Medicare benefits and other benefit claims. And the ability for government to better target its services and more easily communicate with citizens. Governments around the globe increasingly recognise the importance of their citizens being able to effectively use new media and communications technologies. As they seek to ensure that countries reap maximum benefit from their developments. They're looking to their own agencies as well as industry and the community sector to develop strategies and programs to help them achieve their goals. The Australian government looks to the Australian communications and media authority and its research to provide guidance and advice in the area of digital communications literacy. Unlike the UK regulator Ofcom, the ACMA has no specific mandate to promote media literacy. However, our legislation does provide us with responsibilities in key areas of research, public education and facilitating cooperation between stakeholders. In Australia, 83% of households have the internet. Mobile phone penetrations even higher at almost 90%. And nearly 60% of Australians own 3G capable phones according to research commissioned by the ACMA in late 2009. While the great majority of Australians are online, this leaves almost 20% of Australian adults having never used the internet. Much of the quantitative research around adoption and use of digital media and communications points to a generational divide. The older people are the less likely they are to be using digital media and communications currently. This often overlooks the fact that many people in younger or middle age groups are not online and also many people are very marginal or limited users of the internet and mobiles. For example, maybe just using mobiles for voice calls or for SMS. These are individuals for whom using digital media might have significant benefits. They might wish to undertake further education, re-enter the workforce and more easily access government services or generally be able to keep up with peers in communicating and accessing information and entertainment. We also recognise that effective and confident use of digital communication involves a process of adapting learning strategies as new devices and applications become available and also continuous learning as people move through different life stages with needs for different skills and competencies. So in 2009, when the ACMA commissioned qualitative research to better understand the barriers to digital communications literacy we made sure we included limited users of the internet and mobile phones as well as those who were not using. Our research identified two key factors that were clearly affecting people's attitudes and behaviour. Their existing competencies and their level of motivation to learn more about digital communications and to use more. Many of our non and limited users had not been required to use technology on a day-to-day basis in their job or education or hadn't been able to explore and experiment with computers and the internet. They did not understand the underlying assumptions of how digital media works or the associated commonplace language that's used by those who are comfortable with digital media. Most importantly, they had not developed the transferable skills to allow them to become more competent. Instead, they might memorise, by rote, some steps to get them to their favourite website.


When chronic obstructive pulmonary disease, we always have to distinguish between two different pathophysiological mechanisms that are going on here. One of them is emphysema and the other one, chronic bronchitis. So let's look at emphysema first. So emphysema is caused by in-health pollutants and most commonly that is smoke or any kind of other environmental pollutants such as smoke, living close to factories, a lot of agriculture, those kind of things. And these in-health pollutants over time destroyed the elastin that the alveoli are made of. Therefore, when the elastin is destroyed, the alveoli lose their recoil ability after exhalation. So basically what that means, I've drawn it out here. So you have the airway and on the alveoli at the end and typically the air comes in, the O2 comes in, gas exchange happens at the alveoli and then CO2 is exchanged. Well when the alveoli here lose their elasticity, that elastic recoil, they're kind of a little bit more distended to begin with and because they can't really kind of contract as they're trying to get rid of the CO2, they're not able to exhale or eliminate the CO2. And that means that the CO2 gets trapped here in the alveoli. And that's right here, the small airways then collapse and leading to alveolar distension and that air trapping that I just explained over here. Which means that the CO2 can't exit and the O2 can't enter and therefore the patient will become hypoxemic, so less oxygen concentration in the blood. In comparison chronic bronchitis is also a chronic obstructive pulmonary disorder, but it works a little bit differently. The causative organism or mechanism again is chronic exposure to smoke with the environment. But in chronic bronchitis, the bronchi and bronchioles get inflamed. Now this inflammation as we know typically anywhere, especially in the lungs causes increase in mucus production and therefore the airway walls thicken causing an airway obstruction. So now we're having the problem where the air also can't escape because of the mucus and the thickened airway walls. And a chronic bronchitis diagnosis is usually made if there's a daily productive cough for three months for two consecutive years. So that's a long time exposure to any of these pollutants or smoke that cause all these things going on over time. And keep in mind both of these are chronic diseases. So nothing happens really quickly here. This is a chronic exposure and these changes happen very slowly over time.


Okay, we're going to get started today by just recalibrating the geometry and the best way to start this out is to show you how to access the internal service menu for This and any other in series Sony PVM monitor that you might be working on so the first thing I'm going to do is I'm having it on a Different line One that's got nothing on it no input into it right now just so you can See on screen better The menu options I'll go through that a little bit and then we'll pull up a test suite and use that test suite to calibrate So this is your general normal menu menu one the normal menu when you press menu. That's what you're going to see it's just this short menu with a few controls and That's it and you can change your aspect ratio your RGB sink to sync on green if you need to and That's Pretty much all there is to it for that So there is however a way to get into the service menu, which is a sub menu inside of this That lets you make a lot more adjustments. So let's start the way to access that is To have nothing on screen simultaneously press menu plus enter and Then that will come up and then you press enter once Enter twice and it gets you to this the first thing you'll see is 001 destination and then some more numbers down here Now this is the sub menu or the service menu whatever you want to call it. This is this is where we are right now So to change this destination or this is this is 001 setting I need to hold down the menu button and go either up or down on The arrows so if I go up it's going to bring it to the second setting And if I go down it'll bring it all the way down to the very last setting, which is a hundred and three So there's a hundred and three different settings in there And then when you get to a setting that you want to make a change to Press up or down to change the value on it So once you change and enter your new value if you want to write your data that you've changed You need to press menu and enter together and it'll say save and Then you press it again while that save is listed and you'll see if you saw it there But if you press menu and save and then press it again, you'll see two musical notes and that means That your save has been written to the memory And then you don't have to go and make an adjustment again or it will save the adjustments So now I'm going to change it over to RGB and we're going to use the 240p test suite to calibrate again for geometry Okay, so I've got my brightness turned down I've got my test suite pulled up and I'm going to pull up some test patterns and one of the first test patterns I like to start with is the 256 by 224 grid pattern Now to start my calibration, I want to make sure that I see every edge line in my picture of my grid here, so I can there are some Spots on the corners where it is pushed into the actual bezel blind area, which is fine But for the most part I want to see these lines Before I finished with the geometry now this one is in really good condition So we're not going to have to really go in and make a lot of adjustments really at all to the geometry settings for like the pin cushion settings and other Linearity settings. I will show you those settings, but this one is just pretty well adjusted already But this is what I'm shooting for to begin with is this Putting this grid pattern in the very center of my screen so I can see every edge and most of every corner and Then I can make some adjustments from there So if we go ahead now and we press our menu and enter together and get into our service menu So there are some adjustments now that we're running into that are our first actual Settings that we could change for geometry for example horizontal corner pin cushion settings horizontal pin amp settings pin phase size horizontal Vertical linearity we've got a few settings for that size correction 16 by 9 size our regular size our vertical center our angle our bow and Then we've got some other random Settings thrown in here that we will not be worried about but there is one more lower down here So the horizontal center-ness you have to adjust the horizontal center-ness on this monitor for the specific input that you're using as far as 60 Hertz or PAL 50 Hertz So there's those two and then the RGB is what we'll be using And we'll actually be using 60 Hertz. So that's would be our example for us And then again if we wanted to make a change would go up or down on the arrows and Then we'd write it by pressing menu and enter together twice So if we start here on our horizontal center and we have it on 60 Hertz RGB If I press up or down, it's on 23 right now, but if I press it, you know this way it'll go that way But that right there will adjust your horizontal center-ness and then if we move up in our menu items Let me show you what the bow looks like If I were to increase that There's it's it's only goes up to the number 15 and then it resets back to zero so It's not a wide range of ability to change that but there is at least some ability to change that We've got our angle setting here Which is the next one a vertical center Okay You get the idea here Vertical size and then 16 by 9 and then there's size correction side correction and linearity Which will need different test patterns to go through the linearity settings. So Horizontal size is the same as vertical size, you know, you just increase it and it increases the size of your screen Pin phase is something that we can show here So that's the actual angle of the screen and Our pin amp, which is going to be our corner pin cushion settings And this is our entire screen So this actually has a larger amount of adjustment you can make to that and then we'll move to corner pin amp Which is just tighter controls on your corners Not so much the whole entire screen just concentrating on the corners for the most part You can go in here and look at some of these other settings. These are going to be like color controls Most of these is color drives Sub-brightness you can also change sub-brightness in here in the middle of it and then sub-contrast Sub-chroma. So there's a lot of adjustments inside this menu Now what I'm going to do now is show you my preferred setting for this is to take this screen and make sure It's perfectly centered. Once I have it centered. I'm going to slowly Expand the screen out to the point where I am half way between the red lines Which is the edge line and then the red dots And that's going to be the same both vertically and horizontally so that way the red line and About you know, just a couple centimeters over that is going to actually be In our over scan area and it's not going to show up on our picture You know when we're running a program through either a video game or anything else that we're running on screen through RGB So that would be about my desired Horizontal size and then if I go to vertical Size I'm going to be slow to expand this one But I've got you know the line disappears pretty much the same time in the top and the bottom And everything else is looking very good on here. So To again save we're going to press menu and enter twice And then we're going to get out of our service menu and check our results So to get out of the service menu While I have the service menu up I'm going to press enter and it's going to go this raster center adjustment mode and then press enter Again and then when nothing's on the screen We're going to press menu and enter one more time and it'll show us the RGB and our input And that just means we're back to our normal menu and we're out of service mode That's how we operate through that service menu. Let's just take a quick moment here Well, we've got the screens these screens pulled up to show off some of the settings This is our linearity and if we wanted to go through and make some linearity adjustments We could use those linearity settings in that service menu to make sure these are all rounded as possible and Positioned properly one of the other things you'll always want to check is just to make sure that you don't have any color Purity by checking each color make sure every color is evenly Showing can always do this check for the color bars just to make sure everything looks fine there and The last thing I like to do is set it on the scroll test Just to see that you're not having any problems with linearity or Warping or screen tearing or really anything just that's a really good backdrop to show what is going on throughout the entire screen and You can also see how we've expanded it to the point where a lot of the edge Is in the overscan area, which is the way it's intended to be that way We're not going to see any edges of imperfection on either the program end or on the edges That might be imperfect on our calibration Because again a lot of this is just due to magnetism and we do have a lot of you know Ability to calibrate it to a certain point But then there's also going to be a certain point of error that's automatically built into every CRT Simply because it's an analog video system with a lot of Magnetism that can be impacted by your environment. That's really the entire service menu run-through Hey, thanks again for watching today. I just want everybody know what I was using to calibrate first I was using a super nintendo with a rom cart and then I was using our team EO or being a software Which is the 240p test suite and this is the best? Software you can use to calibrate any old CRT or really video display monitor Anything that you need to make a adjustment to for either like color or even calibration for geometry This is really the best to use for like 240p and 480i and older analog video signals So I just want to let everybody know that that's what it is in case you don't know and you can just Google 240p Test suite and you will find Junker HQ, which is our team EO's website big shout out to our team EO He's a great guy and this software is incredible and it is open source and free So go make sure you download it and use it on whichever system you have capable to run the program on Thanks again for watching and I'll see you guys next time with some more retro content


The Federal High Court, sitting in Port Harcourt River State, has struck out the suit challenging the competence of the Governor-May Malabuni-led Keataka Extraordinary Convention Planning Committee of the All Progressive Congress, the APC. Delivering judgment, the presiding judge, Justice Fibi Musung Ayua, struck out the name of National Chairman of the All Progressive Congress, the National Working Committee of the APC, as well as the National Secretary, Keataka Extraordinary Convention Planning Committee, sued as second, third and fourth defendants for being non-juristic persons. The court held that the action is non-justiceable and incurably incompetent on the ground that the substratum of the matter borders on the internal affairs of the All Progressive Congress. Recall that on the 15th day of October 2021, Delta State High Court of Justice struck out an action filed by Elvis Ayamano and others challenging the APC ward, LGA and state congresses. The Federal High Court ruling has put to rest issues surrounding the legality of the Malabuni-led Keataka Extraordinary Convention Planning Committee.


Hey everyone, this is Rob Greenfield at the TD Greenie and we did it! We turned my one house into 10 tiny houses and this is Beth Lindley. She's the one who bought it and won it in the auction and I also want to give a big thank you to everyone who made donations as well as especially Sheena Singh who donated $3,000. So with Beth and with Sheena and everyone who donated, we raised $10,000 which means building 10 tiny houses. So you can follow the adventure of the tiny house community that's being built. That is through the nonprofit down here. Their website is homeless2housed.org. That's homeless, T-O, housed, H-O-U-S-E-D, and then you can follow the tiny house community. I am heading down to Central America with my girlfriend Cheryl who's filming right now and this is the last you'll see of me at the tiny house. This is the last video here. Beth is taking it up to Northern California. So my plan now, Cheryl and I are going to travel indefinitely first in Central America for about three months then probably come back to the USA and I'm giving up almost every possession that I have and the plan is to live simply and live in the service of others. So keep your watch out. I'll be possibly in your country, possibly in your city.  And this is goodbye to the tiny house but it is by no means goodbye to me. So stay tuned and I love you all.


Alright, good evening everyone. I'm going to be speaking about drip irrigation tonight, primarily for home gardening and be applied to other larger agriculture as well. So what I'm going to be talking about here, just from an agenda point of view, I'm going to do a quick introduction, kind of explain why I was interested in drip irrigation, why I've kind of pursued this, why you might also want to pursue it. Then I'm going to go into what are the basic components of drip irrigation, what does it actually take to install this, and then finally we're going to talk a little bit about what it takes to design a system for your own home. So who am I? To start with, I am definitely not an expert on this subject at all. I am an IT consultant that just happened to want to install a garden in my backyard. I live in a very suburban environment and I work a lot of hours. I travel a lot, I do a lot of these type of things, and I found in my first year of gardening, I probably spent more time watering my plants than I did anything else in my garden. More time watering than I did weeding, more time watering than I certainly did harvesting, and I wasn't very pleased with the harvest that I got at the end of the year as well. So the season wasn't that good. So I prefixed this by saying I'm not an irrigation expert by any means. I am speaking about this just from my own personal experience and what I've been able to research on it. But I'm sure that there's others out there that can give you a lot more detailed information about how this works. As I had mentioned, why I'm interested in this is that I was looking to start a garden. Personally, it was more about getting closer to my food. I wanted to be more familiar with where my food is coming from, the quality of my food, and so forth. And fitting in with the Oak Keeper side of this, it was about prepping as well in the sense of being prepared to supply my family with food beyond what I can just get in a grocery store. As I already kind of explained, I am in a very suburban environment. So I have a small backyard. I have neighbors on all sides of me. I also live in an area of St. Louis County that is plagued with a lot of pests, deer, rabbits, squirrels and so forth. So for me, when I was designing my garden, I was looking for something that I could do that was a little more resilient to those type of things. So I ended up going with a container garden.   Eventually, I kind of graduated to a larger garden and I now have a more raised bed style garden. And I'm using some other interesting techniques that keep the pests away. But having a container garden brings with it a number of, you know, some challenges in terms of watering. If you're using a standard taricata type of a pot or you're using a plastic type of pot, it's very, very easy to overwater your plants and soak them to the point that their roots rot and the plants die. That was definitely a concern of mine. So I researched some ways of being able to handle this. And one of the ways that we went with is decided to use essentially a landscaping fabric-based pot instead of a taricata pot. That essentially makes your pot more like a raised bed than an actual container type garden. And so the water in it wicks into the ground below it. And again, it becomes more like an extension of the ground than just a container. But all of this was kind of leading me to I needed a way of being able to manage my water beyond just going out there with a hose and watering it every night during the hot summer. So that is really what I was, you know, what kind of led me to this decision to investigate irrigation. So what is drip irrigation? Really it's just a simple tubing system. It really is just that.  I actually have one here to show. So this is a drip emitter. There's several different types. This is the type that goes on a little post so it's slightly raised above the ground. And it has a very tube that connects to it and it has a very predictable drip rate. So this particular one right here will drip a half gallon for every hour that is running. And that drip rate is really important when we talk about designing our system, which we'll get into here in a minute, because we have to be able to make sure we're not over-watering our plants and that we're not under-watering them. And so the fact that drip irrigation allows us to have a very predictable rate of flow is what makes it very advantageous. The other side of it though is that it's a pretty cheap system. Most of the plastic parts that are part of this are the cost range of cents, not dollars. That little emitter was about 10 cents. Other emitters maybe around 30 cents. The tubing is usually a few cents a foot, so you can buy 100 feet of tubing for just a handful of dollars. It's not an expensive irrigation type of system. It's also very easy to do. And that kind of takes us into what are some of those benefits. The main one that gets pushed tends to be around water efficiency, however. When you have a sprinkler or you're watering with like a watering can or a hose, a lot of that water just sits on the surface and evaporates or runs away from the root system of your plants. And so you're not getting the water right where it needs to be. This little emitter, however, will just slowly drip and it keeps the ground around it moist. So there really isn't a lot of water waste. Water's not running away, water's not really evaporating. Just a few drips every few seconds as it kind of dribbles out of the end of it. So water efficiency is a big part of this. And water efficiency goes into savings. If you live in a metropolitan area, you're paying for your water. And so if you're having to water your plants, you add up pretty quickly to cost as well. So again, not just from a component perspective, you know that this is a cheap part. It's also cheap from an ongoing maintenance point of view of actually running this system. In general, from my personal experience of doing this, I didn't notice any difference in my water bill by having this on the entire summer than I did when I didn't have it on. It was very marginal in terms of a cost increase. But the other advantage is I'd say it's also very versatile. So when I started, I explained I had a container-based garden that was literally on my deck. So I was able to run the tubing to each one of my potted plants, have an emitter in each one of the potted plants, and keep it moist. Now that I have changed, though, to more of a bed-based garden, I can still use the same techniques that I use to do the individual pots for that larger bed. Actually, doing it in the bed is easier, but the purpose of it is exactly the same. And so it's very versatile. It's very easy to install. So here's an example of some of the tubing. So this is a half-inch. This is a half-inch nylon tubing. This is what your standard carrier tube would be. And we'll talk a little bit more about this when we get into the various components. But this is a very thin walled tube. So in terms of being able to install this, all you really need is a scissors. You don't need any expensive special tools. The various drip irrigation companies, though, have plenty of specialized tools that they would like to sell you, but you really don't need it. A scissor is more than good enough to be able to cut this. And the only tool to be able to punch a hole in this early, you could use just a nail. It's very, very simple. So getting into some of the basic components. So when we're thinking about designing one of these irrigation systems, we need to understand our water source. So for me, my initial water source was a hose bib, just a normal hose outlet on the outside of my home. So I needed to be able to understand where that is, how far away from it is from my plants. And we'll talk about that in a lot of detail when we get into the designing. But there's other ways of hooking this up.  I tap directly into that now for my water. So it's always on. There's a timer associated with it, though, to handle my cycles of the water, but I'm no longer connected to a hose bib. The other main component is a head assembly.  And it's really not that complicated. We'll go through each of these parts here, but this is a very typical head assembly. So I'll break this down and talk about the individual components of it. Then you have the tubing. The tubing comes in a range of sizes. And we'll talk about how you decide what size is important. And then finally you have a number of fittings to connect the tubes, to do T-intersections, to get it to all of the places that you need to. And then, of course, you have the drip emitters and some other miscellaneous. As I already mentioned, though, your typical water source is different. And so with your hose bib, most of the reactors are already designed to fit a standard hose that you just literally screw it on. Unfortunately, though, your tubing is not a standard hose size. It isn't a standard hose at all. You have other connectors that help you basically bridge between this size hosing and your normal garden hose. But again, there's a number of ways of supplying water to it. Most of them are far outside the scope of what I'm going to talk to you tonight. I can tell you that for me personally, what I am switching to is a rain catchment system for feeding my garden. So it's completely off of the MSD type water system or whatever. I don't want the public water source for it. So I'm switching over to rain water. But that brings some challenges as well because it's not pressurized. And now you have to figure out how you're going to supply enough pressure to push the water through your hose. So breaking down the head assembly, the first thing you would typically see is your timer. And that's this top piece right here. There's a lot of different types of timers. This is a simple battery operated timer. It allows you to program a schedule into it for how frequently it will water on and how many times a day. This particular one can turn the water on and off four times in a 24 hour period. And you can also program different schedules for weekdays to the weekend and so forth. So it gives you a lot of flexibility to do a little bit. There's also an override switch on this particular one where I can hook this up to a rain detection system where if it recently rained it will override the timer and it won't turn on. But all of that is optional. You do not need this. In a very basic system, you can just hook it up and just turn it on periodically as you need water. I would suggest however you do have some sort of timer, it makes it much, much easier. Pretty slowly, that kind of half gallon an hour, maybe a gallon. One of these type of systems is turning on and off several times in a day and running for fairly long periods of time. The next piece in the head assembly, and this is not an optional component, this is your back flow preventer. So it prevents any inside of your irrigation system from back flowing into your home so you don't get any contamination into your water supply. Very simple little valve that basically just stops the water from flowing. The next piece that you would typically see is a filter. There's a lot of different types of filters. But why a filter is important is because the outlets on your emitters are very small and it's very, very simple to get them jammed and plugged with a little grain of sand or any sort of debris. So you really do require some sort of filter to be able to keep the water clean. Again, this is just a very, very simple filter. You do have to clean this periodically, but this particular one is designed to be able to take the filter out while the whole system is installed. A lot of the cheaper ones are in line and you literally have to disconnect everything to clean up the filter. And so as a kind of a tip, I would definitely go with something you can clean without having to disconnect your system. The next major piece that you will see is a pressure regulator. This is just another valve that has a spring inside of it. This will wear out over time. Just as a note, but it's also another fairly cheap piece. I think this is like $1.25 for one of these. Your irrigation system does require a pretty normalized pressure. Most drip irrigation wants between 15 pounds per square inch to 25 pounds per square inch. This particular one is a 25 PSI pressure regulator. Again, it just has a spring inside of it. Usually your home pressure is way higher than that. So we're actually lowering the pressure down, but we want to try to regulate it to keep it average. But this is exactly also why a typical rain catchment system that isn't pressurized would have problems being able to push it because it can't get the 25 pounds of pressure unless you had like a little makeshift water tower or something where you were raising it up in there to provide the pressure. And the last typical piece that you're going to see is the part that takes it from the hose connection and brings it down into your actual tubing for your system. And that's what this is, is it just converts it. There's one more piece that's optional that you will see as part of many head assemblies, and that's a fertilizer injector. And I do have a fertilizer injector, but I would warn that they're quite pricey. And when I say quite pricey, I think they range about $2 to $400 for a irrigation fertilizer injector. It also does require specialized fertilizer. It has to be a liquid fertilizer, again, because you can't use a water-soluble standard fertilizer with most of these screens. It'll clog it up. So that's just a quick note that if you're really, really lazy like I am, you would probably want to do some sort of fertilization injection as well as part of your system. So your plants are not just getting watered, but every single watering is giving them a minute amount of fertilizer at the same time. And again, from a liquid fertilizer point of view, it is available in organic forms, inorganic, and all of that sort of thing. So if you are interested in organic gardening, you can still use liquid fertilizer for it. I would also warn, though, that there's some very cheap fertilizer injectors on the market, and I have tried them, and they are very much not reliable in terms of how much fertilizer they inject into your system. And that can be very troublesome if you inject too much, and now you burn your plants and they're killing them. I just warn against them they're not worth it. You're better off having no fertilizer injector than growing another cheap one. So as mentioned, your next major component is the tubing. It's available in a poly or a vinyl. There really isn't any difference between the two in terms of performance. The main difference is the thickness of the wall. The vinyl is a little bit thicker than the poly. The poly tends to be thinner. The poly tends to be a little stiffer as well. So it's a little harder to maneuver around your garden than the vinyl is. The thing that's common to all of these irrigation systems is you have some sort of main line to deliver your water. And when we get into the designing part, we'll talk about why that is, but essentially you need a larger tube that goes into a bunch of smaller tubes. That's really what it means, and your main line tube is either going to be a half inch at minimum or up to a full inch if you have a very large supply of water that you're trying to do. The other side of it is the microtubing. That's typically what will go to the individual plants. If you're doing a bed, however, usually you would still use a main line tubing, just do it in a very long line and have emitters all the way along. In this picture up here, this is an example of the microtubing being stuck into the main line. So it's just a branch off of it essentially. And how you install this is there's a little barb fitting. You literally just pop a hole in it, push the barb into the tube, and it provides enough of a seal when it's pressurized that it won't leak. Which brings us to the fittings themselves. There is a lot of different fittings. It's Y fittings, T fittings, L fittings, every kind of fitting you can think of. The next thing I would mention here, there was the types. This is what's called a perma-lock fitting that I have right here. So basically what you do with this type of fitting is you would push your tubing onto the end of it. Just like that. And then you screw this down until it's tight. And that's what provides your seal. And so this is now sealed. It won't come off as long as this is tight. And it's reusable. So the other type is a compression fitting. Compression fittings are a little bit cheaper than the perma-locks. But the compression fitting is a once use. And like any plastic, even though this has UV protectant in it, it will eventually wear out in the sunlight. And you will need to replace this at some point. And so to me, the difference in cost for a compression fitting is not worth it. Because you end up having to replace all of your fittings as well as the tubing. So just going to keep that in mind. And then the last one is the barbed fitting. The barbed fitting that you really only see in the microtubing. And this is just an example of a barbed fitting right there.  And it pushes in, expands the tubing and essentially locks it. Very simple. So the watering devices, these are your emitters essentially. But there's a few different types of emitters. All of the emitters I have in my system are simple drip emitters like this. But there are micro sprinklers that literally will have a little pop-up and they sprinkle the water all over. There are various sprayers as well. There's also misters. And so it depends on the type of, you know, plants that you're growing. Some flowers prefer to have the water fall down on top of them. Others don't like to be touched by water. As an example, lettuce is okay with being wet. Tomatoes are not. So in general, though, this goes directly to the roots where the sprayers and the sprinklers end up in the air and there is more waste. They also require a lot more water and a lot more pressure to run as well. I guess the last one I would say is the drip tape. It's not really an emitter per se. It's really just a lighter version of this with holes already pre-insolved in it. The problem I have with most drip tape is it can really only be laying in a line. I cannot move it around the plants. It has to go in a straight line. It's also quite a bit thinner wall than this is. And so if you have pressure beyond about 15 pounds per square inch, you can actually burst your line. So it's not very durable either, but it's extremely cheap. It's pennies of foot and so you can get hundreds of feet of your thing for just a few dollars. So it is definitely an option. I don't personally think it's a very good option. So a few more things on the pressure, or on the dripper emitters themselves is they do come in a couple different types. This is what's called a pressure compensating dripper. The idea behind the pressure compensating is that your pressure in your tubing will vary as it goes up and down over your terrain.  And so a drip emitter that's like this is able to consistently drip at the same rate regardless of the pressure as long as it's within a range. So I mentioned earlier that it's about 15 pounds or 25 pounds is kind of an ideal range for these. So as long as it's within that range it will always drip consistently. The non compensating one will not end up with some very strange results of when you're using them they are cheaper than these but when I say cheaper instead of 10 cents it's like 8 cents. So it's really a big difference in cost. So the other miscellaneous components you would see are various stakes, various risers to hold the tubing. This tubing is fairly stiff it does not like to hold still it doesn't like to stay where you put it. So I definitely would suggest that you get some sort of stakes to keep it in place because as the sun hits this it will expand and it will move and it will do all sorts of things that you really don't want it to do so stake it down. So those are really the only other components though that are part of your system. As I did mention earlier though in terms of tools there are a number of specialized tools. If you're doing a lot of this it probably is worth buying one of those tools. The holes in here to put the micro tubing they also have specialized cutters and so forth but in general you literally can get away with a little nail and a scissors. So into designing your system I guess the first thing I'd warn intimidated by this all of this really truly is easy but it does require some math it does require some planning we need to work out how much water we are going to need for our plants and because of that we have to do some calculations but it is all very basic stuff so I would suggest if you are interested in sticking with it not getting too intimidated. So what do we actually need to know? The main thing this will turn is we need to know where our water source is all of this tubing has a certain capacity we will talk about it in the next slide but it is very important to know where we are going from and where we are going to so we really have to identify where our water source is and at distance to our garden we then need to know how much water our plants actually need this is where it gets a little complicated because we have to deal with things like well what if it is raining what if the temperature difference in things like water our plants require I do have some charts in here that we can go through to help with that also need to understand the number of plants we are actually going to use so each one of the emitters you have is taking a little bit of water out of the tube so if I have a small microtubing I can add max supply 30 gallons of water down that tube so if I have one gallon an hour emitters 30 of them before I am out of capacity and then I have no more water so it is just part of your planning you need to understand how many plants I am going to run to and then you also do need to understand a little bit about your soil type so in terms of the water source considerations the main thing that you want to be aware of is if you are using a hose bib and I am installing this head assembly to it I don't really have enough room to install my hose assembly to here when I first did my planning and I purchased all my little parts and was very excited to go to install it just to find out that I was about six inches short before I hit the ground so I ended up having to make some modifications to make all of this work but that is the main thing I would say that you have to be considerate of this because you actually have enough space to have all these little pieces and all the parts hanging underneath your hose bib I would say that actually most standard hose bibs do not have enough room to install this and so what you do in that case is well get creative I essentially created a little piece that came down with PVC piping that was connected and made it go right back up so it is now taller on the other side though that is our garden hose spot too so again consider using some sort of Y adapter or something like that just to keep that in mind so how much water do you need this is literally in my opinion the most difficult part to figure out if you have a standard garden it is not so bad because we can do it just based on kind of the square footage of the garden but in my case where I had all of these pots that I was trying to supply water to each one of them has a different plant and it requires a different amount of water and so there was a lot of calculation that went into it to figure out how long can I actually run this with a half gallon emitter and this one five half gallon emitters and this other one so it would balance out and they would all get an equal amount of water it really was kind of a pain to figure all of that out but again it is fairly simple to understand basically your vegetables need one to two inches a week that does vary by plant type some plants are more water hungry than others some are also less tolerant of over watering than others definitely suggest you spend a little time researching your plants for the most part they're all in that range temperature though does make a difference and we'll talk about that in a chart here in a minute so in order to figure it out though we have to understand what is our actual flow rate in one of these tubes how much water can we get through and we'll talk about that when we get to the tubing but we also then need to understand when you're talking about inches every one of these components is measured in gallons per hour so how do I go from gallons per hour to inches of rainfall and I guess the important part here is to understand that when they're talking about inches of rainfall you're essentially talking about a cubic inch and so then when we're talking about a cubic inch we're talking about how that will spread out over a piece of ground and so if we can calculate the volume of soil we can figure out how many cubic inches do we need to be able to cover that with water and again we'll go in and do some math here in a second but essentially one of water one cubic inch cubic inches so let's do some math so if we take some basic let's say we have a one cubic foot root ball which is rather small by the way but we'll just use this to keep the math simple and we require two inches of water a week and we have a half gallon emitter just like this one how much water do we need to do really what we're trying to figure out is what is the duration of the day in order to get the right volume of water so one cubic foot is 144 cubic inches two cubic inches of water is 288 cubic inches so we just divide that and now we know we need 1.25 gallons that's how much water we need every week in order to supply this plant with that desired two inches of water so we take that we divide it out by our emitter five hours or 150 minutes a week that this needs to be running we can divide that by our number of days in the week and so forth what you can find out here though is that in many cases when you have a plant having a single emitter is just not enough you might need two or three of these emitters to be able to get it down to a time period that is reasonable so for my garden for instance I'm running my timer four times a day for about two and a half minutes for each run so it's a very short period of time just comes on a few times a day and runs at I don't have any periods of time for 21 minutes half hour or anything like that it's usually short durations that does though depend a little bit on your soil type which we'll talk about so the things to keep in mind if we are talking about that average or even a raised bed we don't need to have exact per plant calculations we can average it out for the entire bed we're just going to figure out what the volume of soil is in my plant bed, my garden bed or figure out what is the volume of soil in a typical row of crops in my garden as I mentioned though if you're using a container garden like I was you have a lot more math to do you also need more than one emitter and also make sure to factor in your average rainfall so plants need two inches of rain a week and I just received an inch well I need to factor that into my schedule because if I go ahead and add another two inches to that in that week I am now over watering and again depending on the plants they might not be tolerant to that some plants might be completely okay and the plants I have are alright with that so I usually ignore the rainfall because I am within my range that's appropriate but that isn't always the case amounts does vary this is a pretty standard gardening chart that you will find you just search online for watering needs by temperature when they typically say that vegetable gardens need an inch that is an average temperature of 60 degrees in the week so when we say average temperature here it's literally just taking the high plus the low divided by two so our average temperature in the summer is not 60 degrees here in Missouri, here in St. Louis so we need quite a bit more water than that so as you can see as our temperature goes up our water needs also go up so you do need to keep that in mind with this system so in the springtime you may be perfectly okay with running an inch a week in the middle of summer and again basics what that actually equals in terms of gallons for a school soil does matter though for me all of my soil is completely tailored what I had in my pots and now what I have in my raised bed so I have an ideal soil situation so I don't have to worry about soil type at all I don't have to average it out but the things to keep in mind if you have very sandy soil it rains very very and so you're going to want your drippers closer together because the water won't get a chance to spread out it basically goes straight down in your soil and if you have clay in dry soil it's going to do the exact opposite it's going to spread out and take quite a bit of time to actually sink in I think I have the next chart that tells us a little bit about that but how much in order to get down 6 inches of depth inside of a clay base soil you need between 8 and 10 inches of rainfall to actually seep that far down in dry soil so you need quite a bit more water than you do for the depth and you can see in the sand we don't even need 6 inches of water to be able to sink down that far because it just starts running straight down so again these are some of the things that you need to understand when you're designing this is what is my soil like for a typical garden we're trying to get this kind of loamy type of soil where we're somewhere in the middle where we have good drainage but we also have good water retention and that's what's ideal is another thing to understand though is how deep is the root systems for our plants and this is just a chart that shows some common vegetables so if you look at something like a tomato plant a mature tomato plant can literally have its feeder roots down up to 4 feet into the soil they dig very far in search of water and so that's again keep in mind when you're watering this is make sure that you are getting the depth for your volume when you're calculating volume not just there you can't just expect that you need to have it moist down a foot you might need it moist down several feet the next thing that you're going to want to consider is the timing of when are you going to run your water how frequently you're going to run your water and so on so I mentioned that I personally run mine 4 times a day that is more just to kind of balance it out I do not run it in the dead of the day though I don't try to go at the hottest point of the day because I do want to try to avoid evaporation so I'm usually doing it in the morning early evening and I do it very late evening and also once in the middle of the night is how my personal schedule is going to set up but typically though morning is the best time to water so tube capacity so I already alluded to this a little bit but these tubes do have a fixed volume they can only have so much water flowing through them and so I mentioned with the micro tubing you do have a limitation of 30 gallons per hour that also from a pressure point of view you can have a maximum run of 30 feet before it loses its pressure that is actually much easier to break than you would think most of my beds are only about 10 feet long but after I have all the little side runs to each individual plant I easily am past the 30 foot mark and so for my beds I'm not even able to use the micro tubing I have to use more of a tube like this to carry my water but this is also where I said it's important to understand where your water source is it's a few hundred feet away from your house while we're going to make sure we need to have a higher capacity main line to carry the water out there so you may very well need to go with a 3 quarter inch tubing instead as the tubing gets bigger it does get more expensive and it's not a wildly expensive thing no matter what you're doing but it does get up there in price as you start getting into bigger tubes usually when you're talking about the one inch tubes that is typically what you would see in commercial irrigation most of the consumer irrigation sources don't even sell the one inch tubing but it is available if you have a need so this is kind of reiterating what I've already said but really the maximum run length why that's important is because we want to keep a consistent pressure all the way across in our tubing and once we breach that maximum length our pressure starts to drop pretty rapidly and then obviously the maximum gallons per hour we just need to make sure that we have enough water to feed every one of the emitters we have nothing too magical about that so how do you get started well I think the easiest way to get started is to grab a piece of paper and literally draw out your plan draw where your house is draw where your garden is note down how many feet it is note how many bends you have so you can understand how many little connectors you're going to need to have to make this work again this stuff is not all that flexible if I bend it it kinks so if I want to go around a corner I need a connector to go around that corner so you need to understand where all of your corners are where every place you're going to have to tee off of this any of those type of things and in my opinion the easiest way of doing that is just to draw it out make it very simple and then you can turn that drawing essentially into your parts inventory so you know what to buy and price it out very easily as well so it looks basically something like this you have all of your beds laid out and this is a case this case is a raised bed type garden so literally we have a main line that goes to each one of the beds that main line has a single line that goes on the side and then we have our drip emitters going down these feeder lines a very simple system something like that shouldn't take more than even a few minutes to lay out design again if you're doing more of a container based it is definitely more complicated but for gardens for raised beds this is extremely simple so some tips and tricks one this will come coiled up and as I mentioned it's stiff if you want to work with it let it sit out in the sun a little bit it becomes soft and pliable you can now straighten it out and get it mostly straight but as I also mentioned you're going to want to clip it down and hold it down because as the sun hits it it's going to want to bend back and do things like that at the same time though it is much much easier to punch this when it's cold so put your holes in it before you let it heat up in the sun or do it early in the morning or something like that when you're actually going to put the punched holes in this warm trying to punch holes in it is not a fun task it just will bend yep well I'd actually probably just use some ice water as opposed to an ice cube but yeah you can definitely cool it off and that's actually one of the tips here for tubing microtubing can be a little bit hard to get on those barbed ends but if you just take a cup of warm water or hot water dip it in there for 10 seconds it's now pliable it will go right on there's no reason to try to force it on these things and give yourself a headache it's very easy just put a little bit of warm water and another I think really important thing is do not over tighten these connectors they're all plastic I have yet to see anyone that really makes these in any sort of metal components or anything so if you over tighten them it's very easy to wreck it so do not over tighten you seal quite well you just have to have it finger tight and that's enough to keep it from dripping so there's no need to tighten it down to the point where it wrecks the thing the other thing I would definitely highly suggest is when we go if I go back here actually and we look at this bed you potentially have multiple beds that are feeding off of this and multiple lines if I was designing this system I would definitely install some valves in here so I can do some maintenance every now and then so I can turn off the water here and not have to worry about it still feeding this garden so that's another thing just to keep in mind from a tip is consider installing valves wherever you think that it would fit so you can turn off the water source very easily so what resources do you need to get started there's a number of companies that sell this stuff this is one that I've used it's drip works I think I actually probably purchased most of the stuff from drip depot these companies that specialize in it definitely have the best prices they do tend to sell some of these things in bulk though you usually can't buy one of these 10 cents emitters one at a time you usually have to buy like a pack of 10 or a pack of 20 of them but if you're doing a garden chances are you're buying 20 of them anyways that said though Home Depot and Lowe's carries all of these components they have them usually in their plumbing section I don't know why it's there and not in their gardening section but that's where they are is in the plumbing area I would say though my experience with what you can get at Lowe's and Home Depot is it tends to be not nearly as good quality as the stuff you're buying from these specialized dealers most of the stuff that you're buying from these dealers is also made in the USA it's not Chinese type parts I don't know that that really makes a big difference on these because they're all pretty cheap anyhow but they are mostly made in America where a lot of the Home Depot and Lowe's stuff is cheap Chinese plastic the last thing I suggest is when you're looking for more plant information University of Missouri extension they have a lot of information on their website probably one of the most atrocious websites I've had to try to navigate and find things but it does have a lot of really really good information that's very specific to our area in terms of watering schedules in terms of what the plants actually need for care here in St. Louis area so it's definitely worth looking there and also if you're really really interested they have a master gardener program so you can learn to be a master gardener well thank you absolutely


Welcome to this quick recap of section 8.5 on Taylor polynomials and Taylor series. This is a high point of chapter 8, and so to make sure that it's as clear as possible, we're going to start with an extended example of what Taylor polynomials and series are, how you might use them, and why we care. To begin, let's look at the function e to the x, a very common function from algebra and calculus. We've plotted the one and only point on this function that we know how to evaluate by hand. So how can we evaluate other values on this function? Well, in calc 1 we learned one way to do that by using a tangent line plotted here. Using the first derivative, we were able to calculate the slope of the function at x equals 0 and come up with what we call the tangent line or local linear approximation for e to the x at near x equals 0. The tangent line is a close approximation for e to the x as long as we stay near x equals 0, although as you can see it gets farther and farther away from the curve as we get farther away from x equals 0. We can use this to approximate the value of e. If we substitute x equals 1 into both sides of this approximate equality, we get e on the left and an estimate for it, 1 plus 1 on the right. We've plotted these two values on the curve. The black dot is the location of the exact value of e. It's y value is e because it's on the function e to the x. The red dot is the approximation that is given to us by this tangent line. You can see that they're not particularly close together because we're not close to the point of tangency. That's the local part of a local linear approximation. So although we can get an estimate for e, it's not particularly good. But we don't need to stop here. We can find not only a tangent line but even a tangent quadratic. Here this quadratic shares the same point, 0, 1, the same slope as e to the x, but it also has the same concavity as e to the x at x equals 0. That gives it a much better match for the shape of the curve e to the x. By substituting x equals 1 into this quadratic, we get a better estimate for the value of e, and you can see that the red and black dots have gotten closer together. We don't have to stop there either. We can find a tangent cubic with the same point, slope, second derivative, and third derivative at x equals 0, making it an even better match for the shape of the curve. And again, we get a better estimate for the value of e. If we keep repeating this over and over, we can get an infinitely long polynomial following this pattern. It has the same derivative as e to the x at x equals 0, as well as the same second, third, fourth, and so on derivatives, all at x equals 0. By substituting x equals 1 into this infinitely long polynomial, we get the exact value of e, written out as an infinite series. So why would we want to do this? The first thing to notice is that, while e to the x is on the left, written in the familiar way, this infinitely long polynomial is exactly the same thing as e to the x. It's just another way to write it. It's okay for functions to have more than one way to be written. The reason that we like this is that polynomials are much easier to work with than mysterious functions like e to the x. In particular, we know how to evaluate polynomials at any x value that we're interested in. In fact, by doing that, we can get the exact value of the number e. We could even evaluate e to the x using this infinite polynomial at any other x value that we want, and we can calculate e to any number at all as an infinite series. In fact, any calculator or computer that allows you to calculate e to any power uses either this method or one very similar to it to actually do the calculations. This infinitely long polynomial we found is actually called a Taylor series for e to the x centered at x equals 0, and we can use it to find any value that we're interested in. The polynomials that we found on the previous screens are called Taylor polynomials, and they are estimates for e to the x in very much the same way that a tangent line is an estimate or approximation. Now that we have that idea, let's take a look at the definition of Taylor polynomial. The nth order Taylor polynomial of a function f centered at x equals a is given by this formula, which can be simplified to the summation below, or rewritten in this way. Well, this looks intimidating. A couple of things you should notice are that the first part of it right here is exactly the same as the formula for a tangent line, and every term after that uses a higher derivative of the function f evaluated at this number a called the center. This is just an extension of the idea of a tangent line, making it so that we can calculate tangent polynomials of any degree. Some things to know about this is that these are tangent polynomials. They act just like tangent lines, but they share more than just the same slope. In fact, they share all of the same derivatives as the function that we are interested in at this point a. And so this polynomial piece of n, the nth order Taylor polynomial, approximates f of x near the center x equals a, just like a tangent line approximates f of x near the point of tangency. And the kth derivatives of p and f at this point x equals a all agree, which gives these polynomials the same shape, the same local shape, as the function that we're approximating, as long as you are close to x equals a. Polynomials are much easier to work with than other functions, but Taylor polynomials can be used to calculate values for functions that we don't otherwise know how to work with.  This formula is exactly the same as the one on the previous slide, but it goes on forever. These are infinitely long Taylor polynomials, and as such, they can be perfect matches for the function, not merely approximations. If a equals zero, we have a special name for these. If the center is zero, we call them Maclaurin series, after the person who first worked with them. Here's another visual example to show what we mean by all of these. Here's a Taylor polynomial of order one for cosine of x centered at pi over two. Notice that when we say the center is pi over two, that turns out to be the point where the tangent line is tangent to it. A first order Taylor polynomial is a tangent line. Here's the third order Taylor polynomial, or a tangent cubic. Again centered at pi over two, it's a better match for the function. The fifth order Taylor polynomial is an even better match, as is the seventh and the ninth. If we let the order go off to infinity, we get an infinitely long Taylor polynomial called the Taylor series for cosine of x, and you can see here that it is a perfect match for cosine of x. For each x value, a Taylor series becomes a normal numerical series that converges or diverges. We saw this with the example of e to the x before. When we substituted x equals one, we went from a polynomial to a normal numerical series. All of the x values together where the series converges is called an interval of convergence. This interval turns out to always be centered at x equals a, called the center of the series. This interval goes from a minus some number r to a plus some number r, and this value r is called the radius of convergence. The endpoints of this interval might be open or closed. You'll need to use convergence tests from sections 8.3 and 8.4 to decide. And the interval might even be infinite. Two examples are shown below. Visually, you can see on the left how the Taylor series does not appear to match the function at all once we get outside of the interval from negative one to one, but it's a perfect match as long as we're inside that interval. Similarly, the series that we saw for cosine is shown on the right, and it's a perfect match for cosine of x at all x values, and so we say its radius of convergence is infinite. Finally, one last idea from this section is the Lagrange error bound. This tells us how good of an approximation a Taylor polynomial of order n is. So if we want to approximate f of c for some number c, remember f is the real function that we're interested in, and if we can find this number m, which is a positive number so that the n plus 1th derivative of our function f is always between m and negative m on the interval from a to c. That's the meaning of these absolute values. Then the difference between our polynomial at c and the real function's value at c is less than or equal to this formula given on the right over here. And although this looks complicated, this allows us to calculate an estimate for how close our Taylor polynomial really is to the function. It turns out that we can use this to show that many Taylor series actually converge perfectly to the function that the Taylor polynomials are approximating. Now that we've seen these, let's take a look at many examples of how to calculate Taylor polynomials and series and how to work with them.


Let's do a celebrity boxing match. Me and you. How much? We get the money. Oh, Ian, come on, let's get the sponsors. We get the money. Me and you, we'll get the debt together. We're getting the debt together. How much you want, Gary? How much? How much are we doing it for? I'm about peace. No, but I'm saying that. You said, you said, how much? But y'all said, how much? Nick, you talked about a window in my bedroom, nigga. We doing a celebrity boxing match, nigga.   Yo, how much you want, Joe? How much you want, Ian? We're gonna get what, I'ma call Al Heyman. Al Heyman, I'm calling you, man. Absolutely. A lot of people can't call Al. We getting this debate. You already agreed to it. I don't know if we're playing that. But this is big business. I know. This is big business.   We call it Al. Stream. We fighting right before Jake Paul and the white boys. Listen, I'ma let you know.  Anybody, listen. Why don't we go, why don't we got to take a bit? Open up, Jake Paul.  Oh, we don't all know, cuz. We got to own you. Can you talk to me?


Well, hi there, my name is Sandy Alnok. I'm an artist and paper crafter and I bring you a pull tabs slider card made for the Hero Arts blog hop celebrating their summer catalogue and they sent me a bunch of stamps and these two are the ones I'm going to use for these cards and they're generally I think of them as planner stamps because they'd be great to use with your planner when you're getting your summer vacation plans together, but I'm going to use them on a card and I've got this piece of watercolor paper that I drew two lines on This one is it one and three quarters. The other one is it one and a half and the one and three quarters one I'm going to turn into a slit and the slit has to be just kind of wide enough for a brad to fit through and I'm not going all the way down the edge of the card because I want the pull tab slider to stop I don't want it to pull all the way out and Be removed from the card. I'm gonna on a separate piece of paper I'm going to create a panel that's going to be popped on top and that line is one and a half inches So it matches the one and a half inches on the other piece the entire width of this or the entire outside piece by the way is Five and a quarter by four and a quarter because I want my tab to stick out a little bit on one side And you'll see how that plays out They use some just sticky notes to do some quick masking and then fill it in with a sharpie Which will work with my watercolors Because that won't run and fill in some of the areas that I didn't get stamped really well because I can't stamp where the darn if you Haven't heard that from me before Then you're learning that now. I just struggle with stamping But I'm going to add just a bunch of items on either side of this panel because I'm creating something To hide some things that are behind it So I'm just gonna stamp a bunch of things on the left and the right and we're gonna cut away the portion at the top to pop onto the card But I want to create a little section here with that's just gonna be the the signposts for all of my little signs that I'm gonna have and draw them in there with my Sharpie pen and do the same thing on the other side to add those those little lines in there straighten up anything that's a Little bit crooked that sort of thing and then I'm gonna add a line across all of it because that's gonna be my baseline Across the card. I'm gonna cut against that one but I'm also gonna have it continue on the left and right hand side and You could either make the line completely across some of the stamps that you use you could drop them down below Like I'm dropping the the gas tank down a little bit lower And then some of them will be sort of merging into this There's like a little place for some of them to merge into the bottom and others are just gonna hide behind that line Next I'm gonna do my fussy cutting and I'm gonna cut off that whole portion that has that X on it and I'm gonna whack off different portions of it and then go in with my detail scissors after I get this big chunk off Because it's just gonna make it easier for my scissors to get in there and do the fussy cutting So once I've got that big section out of there It's much easier to zoom in here with my little scissors and you can leave a little white outline around this or there's some areas Like you can take a pen and fill them in so in between where some of these signs are I'll add that and as well if I don't get my fussy cutting done all the way to the very very edge I can go in with a pen and repair some of that Once that panel is finished you can start to see how it's going to appear on the card that little track is above it and Then there's the one and a half inch line So I'm just going to mark for myself a few places that I want to make sure I get some background in there Because I'm going to stamp along this and I want these things to hide behind that panel that's in the front So I'm going to take the mountains from one of the sets and then I wanted to have some trees as well But before I put the trees in I thought wait a minute. I can put a speed limit sign in here So I stamped that and made a little mask for it You could also Just put a little sticky note over it But I wanted to do the trees a couple times just to make sure I have enough trees going behind everything And then even down at the bottom there So I could have some at the that bottom section You're not going to see a whole lot of it when the Vehicle is going past but now you can start to see how the image comes together and when the Little van is going to travel along there. It'll be really cute So I'm going to take my Daniel Smith watercolors and just do some quick painting The background I wanted that the mountains to look like they were kind of melting into trees as they went down So I did some blues at the top and some greens down at the bottom I'm using my Daniel Smith watercolors, but you could use any watercolors You could actually do this on regular card stock and color them with copics or with colored pencils Whatever kind of thing you'd like to do I would just recommend using some sturdy card stock because on any interactive card You just want to make sure that you have enough heft to the pieces of it that they don't fall apart Especially if you're going to send them to small children So on the front panel, I'm going to do some water coloring I put some green to match the trees In the the background portion So that it it looks like you're looking through the signage at the trees back there Then I'm just kind of throwing some extra colors on some of the other parts Mixing around deciding which colors I wanted to do And then I stamped on a piece of scratch watercolor paper the van So I decided that this little van was going to be the one that's going to pull along here There's I think four different vehicles And you'll see more of them at the end because I made some other cards with these two That I'll show you but there's a bunch of different vehicles So you can pick which one looks like your family vacation car And this is this would also be a fun card for somebody going on an exciting road trip Just to wish them well would be kind of fun or something to send To somebody as a souvenir of you having gone on a trip with them It'd be a fun memento And then there's also the option of doing something like this in your scrapbook page All right, let's look at the construction of the pull tab slider And this is my own invention I didn't look this up so I don't have anybody to blame for how hard this is So the panel that's going to go on the inside that's going to move That one is Three and a half by five because I wanted to give it as you saw enough room around the back side That I have room for adhesive and then I poked a hole in it and I Wanted to make this go through both The front piece all the way through that tabbed piece and have Two little brads sticking through it and I'm going to stick my car to it So you can see I've got that that little lip around the whole edge And that's where I'm going to put my a creative tape. This is I think Quarter inch tape, but whatever size you have left on your card Then use tape that will fit along there. You need really strong adhesive and this stuff once it grabs It's not going to let go And then I have another panel for the back side that is the same size as this And that panel is just going to stick to all of that around the edges So now I've got the base construction and you can see the slider does move And you don't necessarily have to have that other panel that I've created Except I wanted to hide some of the things and make it look like this is going through Through the scenery I'm using two layers of dimensional adhesive on that front panel because it needs to pop up in front of the van And I attached the van to the brads using some more of the be creative tape Didn't catch that on film. Sorry But now you can see when I pull that tab out When that has a little piece of twine on it the van moves along that little track Which is kind of cute and then I stamped one of the little cars On the tab Panel that pulls out Which I thought was just a really fun little crazy card Now here's another one that I made and this one involves some different stamps It has more of a southwest theme to it and it's got not only a van But it's pulling another little trailer along with it So I've got some Some suitcases packed stacked up on one side and a teepee on the other teepee tent So you can create all different kinds of scenes using this same kind of an idea And there are some card type sentiments in there now just for fun. I tried this on a vertical card With the Seattle set as well And around this right around where I'm pointing is where the be creative tape is And I have this little tiny panel that's holding onto The boat the little ferry boat. So there's a lot of different ways you can use a lot of their travel sets I'll have them all listed so you can see if your city is in their collection. I'll have them all listed on my blog So Here are some other videos if you're interested the left is more daniel smith watercolor The other two are interactive cards Be sure to go to the blog hop and check out what everybody's creating And you know hero always has some goodies to give away So you want to enter or to have a chance to win some fun things and I will see you guys next time. Take care. Bye