If we know f is differentiable over an interval, we know quite a bit more about its behavior over that interval. Suppose f is differentiable over some interval between a and b, then we know... well, let's see what we know. So for example, suppose f' of x is continuous over the interval between 0 and 5, must f of x be continuous at x equals 0. And so we might say something like, well, we know the differentiability requires continuity, so we must be continuous at the endpoints. But let's be a little bit more careful about this, and the thing to remember is that if you don't find the flaws in your reasoning, someone else will. Now, in order for f to be continuous at x equals 0, then we know the limit and the function value have to exist. So if f of 0 isn't defined, then f won't be continuous. Well, a simple function for which f of 0 is undefined is f of x equals 1 divided by x, and we see that the derivative is equal to... and this is defined and continuous over the interval between 0 and 5, as long as we exclude x equals 0. So it's possible for a function to be differentiable over an interval, but not continuous at the endpoints. So this leads to two useful results. Actually, one pre-result and one actually useful result. The pre-result is something called Rohl's theorem. This is named after a French mathematician by the name of Michel. I can't remember his last name. I'll think of it in a moment. Anyway, he discovered the following. Suppose our function f satisfies the following. f of a is equal to f of b. f is continuous over the closed interval between a and b, that includes both a and b, and f is differentiable over the open interval from a to b, which does not necessarily include x equals a or x equals b. Then there's some c in this interval and the derivative of c is equal to 0. And one important thing to note about Rohl's theorem is it's what mathematicians call an existence theorem. It's saying that something exists, but it doesn't tell you how to find it. And so we don't know the actual value of c, only that it must exist. And it's also worth reading the fine print, which in this case doesn't exist. In particular, while this says there's some c, it doesn't preclude the possibility that there may be more than 1. So there could be more than one place where our derivative is equal to 0. Let's build up a little bit of intuition for what Rohl's theorem tells us. Since f of x is continuous, and f of a equals f of b, the graph of y equals f of x starts and ends at the same level, that's this equality of the function values of our endpoints, with no breaks. That's our continuity requirement. And since f prime of x exists for all x in the interval, there are no cusps or points of non-differentiability. We always have a tangent line. And since Rohl's theorem claims that there is some place where f prime of c is equal to 0, this says that somewhere there's a place where the tangent line is horizontal. Now, Rohl's theorem by itself isn't all that useful because the conditions we have to have in order for it to work are a little bit restrictive. We have to start and end at the same function value. But it does give us a much more important result known as the mean value theorem. And we'll start off in the very similar way. Suppose f is continuous over a closed interval and differentiable over an open interval. Then there's some c in that interval where the derivative is equal to this expression. Now, that's a weird expression. I guess it uses the beginning and the ending points, but what does it actually mean? Well, let's see if we can make sense out of this. So we have a function that is continuous and differentiable. Let's take a look at the graph. So our interval runs from x equals a to x equals b. And so let's graph our function. We start somewhere, we end somewhere, and we're continuous between the two points. And remember, if it's not written down, it didn't happen. Let's go ahead and label these points and the graph. And if we sit and stare at this, we realize that the value of this expression is the slope of the straight line between the end points on the graph of y equals f of x. So the mean value theorem says that there's some point where the tangent line parallels the straight line. So for example, suppose my function is continuous over the closed interval between 5 and 10 and differentiable over the open interval between 5 and 10. If we know where we are at 5 and at 10, what do we know about both our function and our derivative in the interval? Well, since this is in the video about the mean value theorem, you could probably guess that the mean value theorem is applicable. But again, to get a feel for what this really means, let's go ahead and use this information to construct a graph. We know the function values at 5 and at 10. So those give us two points on the graph. And again, if it's not written down, it didn't happen. Label. f of x is continuous and differentiable, so there's no cusps or corners, and it's an unbroken graph. And so the mean value theorem guarantees there is some c for which the derivative at c is equal to the slope of the straight line connecting the endpoints. And that slope will be... But wait, there's more. We can find some more about this. We know that f of x is continuous over the interval, and so the intermediate value theorem tells us that for any value m between f of 5, 10 and f of 10, negative 3, there is some value d for which f of d is equal to m. And, well, this is positive, this is negative, so we know that f of x equals 0 has some solution x equals d, where d is between 5 and 10.


Hi, I'm Sivirin Givolf, and I would like to show Java and Red Hat Enterprise Linux 8. As a Red Hat Enterprise Linux 8 user, I would like to create and run simple Java applications. The way to do this is to install relevant packages. Yum install Java 1.8.0 OpenJDKDevelop for OpenJDK8 and Yum install Java 11 OpenJDKDevelop for OpenJDK11. The system I'm demoing this on is REL 8 Beta. When trying to run Java, I notice that I don't have it installed. It can be installed by Yum via the following command. Yum install Java 1.8.0 OpenJDKDevelop. Once JDK8 is installed, the Java version command works. I've prepared a simple Hello World Java application as displayed on the screen. It prints the string Hello World to standard output when run. I'm compiling the program with Java C. Once compiled, the program is ready to run. Java Hello World, Hello World. Next, I'm going to show you a small but cool new feature of JDK11. Launch single file source code programs. Again, I don't have JDK11 installed on my REL 8 system, so I installed the relevant package with Yum. Yum install Java 11.0 OpenJDKDevelop. Once installed, I see JDK11 is available. What I'm demonstrating here is that with JDK11, it's now possible to put Java code in a file with an arbitrary name. In this case, the file is called factorial. The first line of that file contains a shebang line, which tells the JDK to interpret the code in this file as a single file source code program. When looking at the entire file factorial, we see that it's just a regular Java class file with the implementation of factorial. Once the file is made executable and run, that tells the JDK to compile class factorial and run, and run it transparently. Factorial 6, 720. Nice. Thanks for watching.


Hello, Spotlight friends. Adam Navas here with wonderful Liz Wade, and we are doing another conversational podcast program. How are you, Liz? Ah, I'm good. Good. If you don't know Liz or myself, maybe this is your first time stumbling across this program. We hope you check out our Spotlight YouTube channel. You can go to our website, which is spotlightenglish.com, and know that this program, this conversation, kind of falls into a, it's not a pattern. It's a spectrum of programs. We have a classic program that's in slow English. We have an advanced version. We do some live events that are kind of like this, but responding to listener comments. And then we have this program where we're talking about one of our programs. It's kind of a extra behind the scenes, a little, you get to hear what two native English speakers think about the program. And this week's program is about tips for, I'm not gonna remember the title exactly, but it's tips for recovering from injury. It's part of a series of tips about different stages of life, right? Can I say it's tips for a healthy adult. How to recover from an injury. Perfect. Yeah, I have that on my screen. I don't know why I didn't just read it. I got it. You know why, Adam? Cause you're getting old. Because I'm getting old. I am not, I make, I'm just gonna have a moment and decide I'm getting old. I make a lot of lists. I don't remember things. I don't know if I don't remember them because I make the list, but I am a list maker. And I imagine as I get older, I will make more and more lists. Yeah, I make lists too, but I think that's just like a part of who I am. Yeah, that could be.  I've always- You know why you make lists, Adam? Cause I'm awesome. Because you are, because you listened to the program, what is it about? How to manage your time better. Yeah, I- That's why. And because that's one of the tips in that how to manage your time better. And to get more done. Now we're gonna have to link that in the show notes and people can go manage their time better. Just like Adam. Just like- It's not about getting old. Yeah. It's just- It's about managing your time well. Yeah, but there is something about getting old. So this program begins with like a funny scenario, right? Yes. Or have you ever heard somebody get into a chair like this? Oh! Yeah, the sounds we make. Adam, I'm telling you. When I sat down in this chair, I said- You made those sounds. Ugh! Yeah. I mean, we're getting- Not on purpose. Yeah, they just kind of escape you. Like, you start- Oh! Yeah. Well, when you get old, let's be honest, your body goes through certain changes. And we, for a lot of people, it becomes harder to move their bodies either through the same kind of range of motion or with the same kind of strength that they used to have. It's harder to- Yeah. I know you don't struggle with this, but it is harder for people like me to lose weight if you're trying to stay fit. Oh yeah, I struggle with that as well. With the exercise thing, though, that's one of the reasons that I do exercise a couple times a week, because the more that you move your body and the more you exercise your joints and make your body strong, the better it will be years down the line. So that is, I exercise for many reasons, my mental health and just being stronger. So you're saying you're looking 10, 20 years into the future and you're saying, how can I invest now that can make me, yeah. No, I think I do that a lot too to say, I don't wanna hit 60 or 70 years old and not be able to, if I ever have grandchildren, for example, to play with them on the floor, I'm sure my body will have its creaks and its aches, but yeah, to be able to do that and stay healthy, that's a big deal. That's a real big deal. It is, yes, that's one of the main reasons that I exercise. But let's talk about, there's this aspect of this particular program, it's not just avoiding injury, but it's saying what do you do if and when you get injured from exercise or anything or you have any kind of sickness. Because when you're young, you do recover from injury faster, your body heals, you sleep better, you do a lot of things better. So have you had, I know I have, I could tell the story, but have you had an athletic injury or an acute injury that you had to really try to heal? Well, I will say, so I have had, like in my way past, I know I've talked about it in the live shows before, but I was hit by a car when I was 15, and I healed pretty well. I did have to do some physical therapy and some exercises that helped rebuild those muscles after not using them for many months. But even so, it doesn't mean my leg is perfect. So sometimes, excuse me, sometimes when I walk, I have a limp or it's difficult for me to, I can't run. It just doesn't work with my leg. But it also affects other parts of your body and right now I have an injury that is in my neck and shoulder, it's just so tense and I've gone to physical therapy a little bit for it, but it's just, I don't know. Part of it is that I'm so set in my ways that I don't think about it. And part of it is it's a lot of work to do all of the things that you need to do to make yourself better. Well, you named the biggest, hardest thing to get to healing is admitting that you need to do those things. It's fine, it's normal, everybody has these things.  It's not okay that it's this way. And it's really hard mentally. There's a, we talk in the program about that mental aspect of having an injury is draining but also the process of healing takes a lot of effort and mental effort as well, right? And some changes that maybe even as an adult you don't want to make. Yes. You're set in your ways and it's a little bit, yeah, it's more difficult to change what you want to do. Yeah, well, my injury was from a few years ago. I was training for a very long trail race, a 50 mile trail race. And it took me about a year of training to build up to that distance. And I felt a little ache in my Achilles tendon in my ankle. And I knew that I was about six weeks or two months from this race. It didn't like hurt like I couldn't walk on it. But like as I ran, it wasn't healing and it just, it was always there in my head. I could always sense it. But I had another goal, I wanted to fit it. I had invested so much time. So I ran through it and then I didn't run for like a year after that because it just, it took that long to heal. And maybe it would have been better if I had just stopped and let it heal. Well, that's very difficult because you don't know, sometimes the pain is like, do you run through it and it will get better? Or do you stop and treat your injury right then? Do you, like what we would say, baby it? Do you baby it? Yeah, take it and give yourself a little self care as we say. Yeah, yeah. But then you might not hit your goal of running that race. Right, or you wanna keep exercising just because it's good for your mental health, but you need to take a few days off. But you don't like how you feel. You don't like, you know, you might not sleep as well at night or you know, exercise does a lot of things for people. That thing about sleep, that one's interesting to me because that is, that's one of the things that this program recommends, right? That you should get a lot of sleep. But that's one of the things that I struggle with a lot. When I get to the end of the day, I maybe want to not go to bed on time. I wanna do something fun. I've been working all day. Maybe we did something as a family. Maybe I did an activity. But then, when I get to the end of the day, I just wanna have some alone time where I read or watch TV or something like that. And then I think, oh, now I gotta go to bed? I don't wanna, you know, get all the way to bed. It's like one of your favorite parts of the day where it's just you, you can control. Oh yeah, that's tough. It's tough to say, I'm just gonna sleep. But you know how healthy sleep is for you. That's true. I mean, all in all, I think this program is an amazing program in that it just focuses on our bodies are amazing at healing themselves and so much of what you have to do, rest, sleep, eat good foods is just letting your body do what it already wants to do, right? There are some times you need medicine or surgery. But most of the time or a lot of the time, you just need to, and I don't like thinking about medical stuff, that's not my thing. But just that, even that our fingernails and our skin replace itself is such an amazing thing. Yeah. There was something in this program. Oh, there's one more thing I wanted to talk about. Oh, is it about the study? That part was really interesting. Yeah, go ahead. Okay, so there was, this program talks about a study where they studied like a group of adults, right? So they were ages 55 to 77 and they were all healthy. And so it's a little more complicated than this, but essentially what they did is they gave each adult in this study, well, first of all, they started a month before and they said, half of these people should exercise three times a week, right? The other people, again, were healthy, they just didn't exercise at all. So then a month after they started the study, then they gave each person a wound. I know, when they said a small but deep cut, I'm like. Yes, exactly, it was like, it was a real actual like harmful wound, right? It was not like a pinprick or whatever. I did, I read this and I was like, oh, it makes me a little bit. I would never volunteer for that. Anyway, so they took, then they checked in with these people. I don't remember if it was weekly or daily or whatever, but they took a picture of the wound so they could track the progress on everybody, okay? And they found that the people who exercised consistently, even just three days a week and not even crazy exercise, but their exercise routine was something like a five minute warmup and then 20 minutes of exercise and then a five minute cool down and stretching, okay? This was not strenuous, difficult exercise. It was just basic exercise. The people who exercised, I think they healed something like 11 days faster. That is, I mean, yeah, it's insane. Yeah, it is insane. Like 11 days fast, like on average, right? So some people healed faster and some people healed a little slower, but just the fact that they were exercising and keeping their bodies moving healed their bodies faster. Yeah, it's just a little crazy that a cut would heal faster, right? That your overall, and this is the thing that I think sometimes we forget that our bodies are all one piece. Like we have heart doctors, we have skin doctors, we have bone doctors, but all the pieces that even if you're like, oh, I have a cut on my arm, if you're eating well and drinking well and sleeping well, your body will be healthier and that will heal faster. Yep. Oh, well. But it's making all those little choices, right? Exactly. It's choosing to exercise, even if it's just a little bit, it's choosing to eat healthy, it's choosing to go to sleep a little earlier. Just drink enough water, yeah, all those things. Right, but the thing is too that, like especially this year when we've had the pandemic and I don't know if I'm speaking for everybody else, but it's been a lot more difficult to make those healthy choices, I think. Oh, yeah. Because you're making the choice to go grocery shopping or to go to a store and all of those choices, if you have a limited ability to make good choices. Well, I'm trying to remember if we've talked about this in a spotlight program or if maybe you and I have just talked about it, but we've talked about willpower and that you have a limited amount of choices, good choices in a day. And if early in the day, if someone comes into your office and brings like a donut or a cake or something and you say, no, no, no, I don't want that.  And you say, no, no, no. When it gets to nighttime, it's really you have less good choices to make. And it's really hard when you're saying, you know what, I'm gonna do these good choices, good choices. It's hard to do that all the time. Maybe if you're thinking, okay, so I have this wound and I know that exercise can help my whole body. Maybe that will help make that choice for you. Yeah, yeah. I guess, I mean, I will say one of the, so this is me growing old, Adam, is these glasses right here. I don't actually wear glasses, but I found since I work at the computer all day, that my eyes were becoming strained and it was too much work for my eyes to just look at the computer screen or TV or whatever my phone, phones are bad habits. But so these are not actually, they don't help me see well, but they block out blue light and I have found that that really helps and that was a good choice that I made. Well, good. Well, I would love for you guys in the comments to write two things. One, how did you know that you were growing old? And two, like Liz, what is the thing you did about it? Did you wear glasses that block out blue light? Did you try to get a little extra sleep? Did you try to, you know, what do you do to heal and make sure that you're living a great life? Or, Adam, we also have younger viewers. Maybe you think that age will never come for you. Oh, man. But we hope that you grow very old. What will you be like when you're older? Oh, man. Have you thought about it?   Just enjoy it while it's there. That's my first advice to my past self. And then just stay healthy, right? It's so much easier to stay healthy than to get healthy, right? Yeah, that's true. Well, you're making your life better. If you've made it this far into this rambling program, you're doing amazing because you're learning English, you're practicing English with us. And as we always say, if you're part of the spotlight community, you're the best. You guys are kind. That is true. And we hope that you join us, visit our website, follow us on Twitter and Facebook and all the places. We've got a lot of... Yeah, don't forget to subscribe. Subscribe, get the notification bell. Yeah, we've got lots of content. I know you'll get that subscription, but go back into some playlists. Check it out. We really do appreciate it. It helps when you tell people about spotlight. Get the word out there. And we really do appreciate that.  Yeah, sure. I always love the last word. I'm Liz Wade saying we'll see you later.