For the past two years the marketing department at TaylorMade has been in overdrive. Carbonwood was the future, Red was the color and stealth was its name. Not any longer, Carbonwood face remains but is now blue and the re-brand to QI-10 sees a new chapter in driver technology. And with one model the quest for inertia has hit new heights. We now have three models QI-10 LS. The max and standard model mean there is a driver for every golfer but the QI-10 max is the one that I feel could be the best ever produced by TaylorMade and soon you will see why. Right so three models available in the lineup but for me there is one standout product that is really really interesting and that is the QI-10 max. This is the one that's got the huge amount of suggested forgiveness and I say suggested because we're going to find out if that's true very very shortly. I'm here at the Berkshire we're going to hit a few balls to warm up then we're going out on the golf course but this could be a really interesting driver for 2024.  Wow interesting right away is the ball flight and I'm talking about launch doesn't seem to be affected in terms of spin and that ball is carrying a long way out of here. A few more on the driving range let's get on to the fairways. It's a high launching ball you know. Do you know that's the best example of why this driver is so good. I felt like I slipped a little bit slightly icy ground at the moment early on in the morning. It was a terrible swing and somehow the ball has gone straight and it's gone long at the same time. I'm amazed at what that's just done based on the swing I put on it.  They're literally a couple of yards apart. I felt that second ball was a lot lot firmer but I'm going to tell you in today's video just why as I said earlier this QI-10 max is something that is really really interesting. It is all about that 10k again in the forgiveness MOI stakes and I've got to say out on the golf course which is the all-important bit this has been doing some phenomenal things. The QI-10 max model is the highest combined inertia MOI in tailor-made history and that is achieved by strategic weighting and the far rear position carries an incredible 30 grams. So you've seen plenty of close-ups now no doubt what's plenty of videos about this QI-10 lineup. For me like I said I'm just pushing the other two for one side for the time being because it's that max product that's of real interest but I want to know what you think in terms of aesthetics. Big change. Red has gone to my disappointment and been replaced by blue that almost looks like very much an Everton blue but we'll forget about that for the time being. The profile on the max in particular has been it's big. It's a 460 cc head but they've achieved sort of a bigger overall profile visually from the top by removing quite a lot of the underneath. A lot of the sole has disappeared and it's a real clever piece of engineering. From the shelf appeal perspective me I think this is potentially a real winner and for what I've seen so far from the other strivers it's competing against this one for me is popping off the shelf. There's also another big difference that they've made in this QI-10 lineup. Today's video is sponsored by our friends at Hot Golf, the online golf retailer for all major brands and if you want new golf gear then please support us by supporting them. The crown of the dress especially in the sun is absolutely glistening. I much prefer a matte crown. I don't know why every driver isn't finished with a matte crown. Please tell me why it's not. However it's a little bit of a turn to the right there. We've just held on to the right side of the fairway but just another really good high-launching ball flight. Again what I wanted to reference I've got my Pro Force HL shaft which has been again one inch shorter all about control. This was a shaft that I was fitted for quite some time ago with TaylorMade so that's that nine degree head and that combination is working really really well. The other big difference I mentioned in the previous clip is the sound and feel has again sort of evolved from what we've heard from the previous iterations of that carbon face. It's very much like a normal driver, titanium face driver. It seems to be getting a little bit sharper and it just feels as though it's absolutely zipping out. Great feel, great sound and an address. Although that's for getting the glossy crown the actual physical shape of it I've got no problem with it. It's oversized but for some reason it's very unnoticeable let's say. So why is it that I think this driver is so special? Well apart from the fact that it's performed out on a fairway today has been very good. I also played it at the Glen a few weeks back. It's doing what a lot of drivers are doing in terms of distance and I've pretty much tested everything right now with my clubhead speed I'm generating probably around a 230-235 carry. Then you've got some numbers you're looking at in terms of optimal spin numbers and launch numbers and all the rest of it but essentially ball speeds off the center of the club face remain the same for every driver. We know that. But what's happened with this driver and maybe one or two others is a big concentration as we know on forgiveness. Now forgiveness for me all it means is that we're getting the same ball speeds out of all of the club face and don't forget you've watched reviews from a lot of other channels right now they're all professional golfers they'll generally hit the club face the center of the club face a lot more often than you or I. I am the best man to test this theory and believe me as an average golfer I hit all over that club face and I can assure you that this driver is performing incredibly well from all aspects of it heel toe center high and low and if you look at my dribble data for those who want it it's that 230 odd carry it's a spin number in around 2-2 launching between 14 and 15 they are real every parameter is hitting the button but what I like when you get on the fairways is you see in that consistency and the ball I just hit off this previous tee right back into the wind we're not getting a spin number that's hovering but we're still launching really high and we're still getting that carry distance this driver trust me is well worth giving a go but you know what I then will finish there even move that ball a little bit from right to left and that one's absolutely gone bounding down the fairways the big deal for any driver that's been released in the last few days has been the real focus on forgiveness and not necessarily about ball speed but what they've tried to achieve is no compromise in that ball speed and that spin parameter in launching the ball but still having forgiveness right across the club base it's one of those kind of almost a myth how do you believe and how do you prove whether or not this is forgiving across the club face well I think the only way to do it is by trying out for yourself and seeing what your thoughts are mine are on this particular driver really really impressed it's going a long way it's been fairly straight today and in the last couple of rounds I've played to be quite honest with you and I think tailor-made I've got something like I said pretty half decent on their hands but the only important thing is like I said go out get custom fit try it for yourself and if you're in the market for a new driver this year then the qi 10 max in my opinion it's got to be on the list right thanks for watching thanks to the Barksha Golf Course for having us and I'll see you all soon


Hello. Welcome to this lecture on Biomathematics. We have been discussing integration. We talked about a bit about definite, indefinite and definite integrals. So, today we will continue discussing that and we will discuss a few more things later to calculus. So, what we said previously. So, today's lecture is basically integration part two. We will discuss integration. Towards the end, we will bit, we talk a bit about something related to calculus, something related to differentiation and we will see when we reach there. So, what we basically discussed in the previous lecture is two things. We discussed indefinite integrals and definite integrals. So, we said in the class that if you have a function which has an integral, let us say integral g of x, the answer could be like, let us say this is g prime of x, g prime and the answer could be like g of x plus some constant c. So, what does this says is g of x g x, what does this say is that if this tells you to find a curve which has a slope g prime x. For example, if g prime of x is 2, integral 2 d x is basically 2 x plus c, which is basically a curved line with slope 2. So, this basically, this kind of integral basically telling you to find a curve that has a slope g prime of x. We also said that you can have, you can ask, you can calculate definite integrals. So, such integrals are called indefinite integrals. We discussed this. Definite integrals, we could have two limits like, let us say within this limits, let us say a to b, some function which is d f pi d x into d x. So, your function is f prime. So, on other words a to b, f prime of x d x. And the answer to this we said is that d f pi d x, f prime is d f pi d x. So, d f pi d x is integral f. So, the answer is f of b minus f of a. So, we said this and the answer this is nothing but a number. And this represents area under this particular curve. So, this is, this kind of definite integral basically tells you, if you have a curve given by this particular function. So, let us say, let us call this f prime of x as some, some, some curve. And then what will be the area under that curve between a and b, this is what this tells us, because you can see this is a number. So, this is what we saw for example, we said that 0 to 1 f of x d x can be written as, it can be divided into many paths like this, which has width d x 1, d x 2, d x 3, d x 4. And if you do this f of x i, d x i, sum over i, you will get the area under this, because area is nothing but this width in this height. So, the width is d x 1 here and the height is the function f of x here. So, f of x into d x. So, this is f of x into x, f of x 1 into d x 1 plus f of x 2 into d x 2 plus f of x 3 into d x 3 plus f of x 4 into d x 4 will give this area. And this is nothing but this integral. So, this integral is area under this curve. But then, this is, we understood this for a simple curve, which is like a constant value, because area here is length into breadth, which is area of a rectangle, which is easy. But you can think of more complex curve, which is, which has some curvature. At that point, this kind of thing will be little more tricky to do. So, let us think of some, some kind of a curve, some kind of a wiggly curve like this. So, let us think of this curve, some curve like this, some curve, which is like this. Now, we wanted to have some integral of this function. So, this function, let me call this function f of x. And we want to integrate this function between this point and this point. So, this, let us call this a and this, this call this b. So, what we want to calculate is integral f of x from a to b. We want to calculate this integral from a to b of this function f of x, yes. What is this? So, from what we learnt last time, it is nothing but the area under this curve between these two points. If we can calculate the area between these two points, we will get the answer. So, to, so the area of this state, so let me shade this region. So, area of this shaded region, essentially will give you this answer. So, if you can calculate this area of this region, that will give you this answer. Now, how do we calculate the area? We can divide this into many parts like we did. So, let us go ahead and draw this little more carefully. So, again if we divide this into many parts, so we had some function. This looked like this and we had a and b here, a and b, a and b. And now, we can divide this into many parts. Now, how many parts will we divide? That is a question. Can we divide just to two, three parts or four parts? Is that enough? So, let me divide this to like a few n parts, like I am not going to count this, but let us say there are n. I divide this to n parts. Let us assume that all this weight are equal. What I have drawn here is not clearly equal, but we could draw this equal. So, let us say you divide this into n equal parts. Then, this integral f of x d x. So, this is our f of x from a to b can be written as sum over i f of x i d x i. I is 1 to, let us say I divide into n parts. So, this is 1 to n. Now, what should be the value of n? Should I divide this into like 5 intervals, 10 intervals, 20 intervals? So, we will answer this question, but assume that you divide to n interval. This is the answer. Now, you will get the correct answer if the n is very large. So, in the limit n goes to infinity. That means, the larger the better. So, you will get more accurate answer when, when you let us say you do for a practical curve. Practically, you have a curve and you really want to divide this. The more you divide, you will get the more accurate answer. The real, the correct answer will be in the limit n going to infinity. Like, if you take a million intervals, if you divide this into a million intervals, you will get pretty accurate answer for such a curve. But, if it is much more complex curve with lot of wavy natures, you will have to divide this into more than a million maybe sometime. So, the correct statement to make is that, as the number n goes to infinity, the sum will be equal to this integral. So, integral and the sum. So, what we wrote? We wrote that integral a to b f of x dx will be equal to sum over n, sum over i is equal to 1 to n f of x i dx i. So, if you take all equal intervals, you will have again here n of such intervals. So, at anyway let me write this dx i. So, if you do this, in the limit, in the limit n going to infinity, this will be equal. So, this will be equal when n goes to infinity. When n is very large, this two will, this two, this sum will be equal to this integral. So, now, we will see some example.  So, this is equal in the limit dx tending to delta x tending to 0. So, dx tending to 0, when the dx is very small. So, when this quantity is very small, then this two will be equal. So, similarly here the limit n tends to infinity, this will be equal. So, the correct thing is to divide this into large number of this and then calculate the area. So, right now, we will do some example for this curve. So, let us think of a simple function that you know and see whether we can calculate, divide this into many parts and do the sum and when the n goes to very large, will we get the answer we know. So, let us think of this function f of x is equal to e power x. So, we learn that integral of e power x dx is e power x itself. So, now, let us divide, let us say this is, now we want to calculate integral of 0 to 2, let us say e power x dx. Now, what is this? So, this we said that this integral in the limits, this is e power x in the limit 0 to 2. So, this is the answer. Now, how do we apply the limit? This is, what does this mean? This means that this is equal to you substitute x equal to 2 first, that is e power 2 minus e power 0. So, you put substitute x equal to 2, that is the e power minus the evaluate the function at, first evaluate the function at the upper limit, that is x is equal to 2, then evaluate the function at x equal to 0 and this difference is the answer here. So, this is nothing but e 2 e power 2 minus e power 0 is 1. So, this is the answer. So, we know this answer, we know the final answer. Given that we know the final answer, let us see, we can think of this like as dividing this into many parts and can we do this? Can we divide, write it as a sum of many, can we divide this into many intervals and can we find the sum and get the same answer? That is what we want to check. So, we want to check, if we can divide this e power x into many parts and we find the sum of this function, will we get the same answer? So, let us draw e power x for a minute here. So, let us draw e power x. So, let us say we have e power x here. So, let us say e power x. So, the e power x at x equal to 0 will be 1. So, this will be 1 here and let us go like this. So, at 2 it will have some value here. So, this is 2 and this is 0. So, this is your x and this is your e power x. Now, basically what we want is the area between these two. So, we can divide this into many intervals. So, let us divide this into like n intervals equal. So, let us call this 1, 2, 3, 4, 5, 6, etcetera. So, what does this mean? What does this mean? What did we say? We, according to what we said, integral 0 to 2 e power x d x will be equal to. So, you divide into many parts. First, take at 0. So, this is e power 0 and let us divide this to equal intervals. So, we divide this to n intervals between 0 and 2. So, there are n intervals between 2 and 0. So, the each interval will be 2 by n because this will let me call h, which is the length of this interval. So, let me call this d. d is the length of the width of this interval. So, this d will be 2 by n because between 0 and 2, I divide into n intervals. So, each interval will have width. All of them are equal intervals. So, each interval will have a width 2 by n. So, e power 0 into 2 by n plus. Now, next part, this will be at 2 by n. So, this will be e power 2 by n into 2 by n plus. This will be 4 by n e power 4 by n into d x is 2 by n. The width is 2 by n plus dot dot. If you do this sum for large n n tends to infinity, you will get this. So, this is what we know. So, let us try doing this. So, we can take this 2 by 2 by n common outside. So, look at this carefully for a minute. So, I can take this 2 by n out in common. So, let me do this. Let me take this out here. So, this, if I do this here, I take 2 by n out in common and what I have is e power 0 is 1 and 1 plus e power 2 by n plus. This is e power 4 by n which is basically e power 2 by n whole square. So, let me write it as e power 2 by n whole square. The next term is going to be e power 6 by n. So, which is going to be e power 2 by n. Sorry, this is going to be e power 2 by n. So, let me write the term here e power 2 by n whole cube plus dot dot dot. This is going to be this term. So, think about it. So, you can, you try this yourself. This sum like this can be rewritten as in this particular way. So, if you look at this, look a minute. What do you have? 1 plus e power 2 by n plus e power 2 by n whole square plus e power 2 by n whole cube plus I can write e power 2 by n whole power 4 plus e power 2 by n whole power 5. So, this is something that you learned in school. This is something called geometric series. So, this is a series. This is like a geometric series.  So, this is a geometric series where R is nothing but e power. So, R here is e power 2 by n. So, this is a geometric series with R is equal to 2 power n. So, if you do this sum of this geometric series, we should get e power x. So, we know how to do sum of geometric series. We learned in school. So, sum of geometric series, the answer of sum of geometric series is like what we call s n up to n term is basically 1 minus R power n divided by 1 minus R. So, we have learned this kind of a formula. So, or I can I can rewrite this. So, what here in our case here R is e power 2 n. So, we have a 2 by n here. So, let us rewrite this 2 by n into 1 minus R power n 1 minus e power 2 by n whole power n divided by 1 minus e power 2 by n. So, this is the sum and this in the limit n going to infinity, we should get the integral. So, this is the theorem we said. This is what we said that the integral is nothing but the sum in the limit n going to infinity. So, we know this. So, you look at here e power 2 by n whole power n is nothing but e by 2. So, I can take a minus sign from both numerator and denominator outside and rewrite this as. So, let me rewrite this term slightly and you can see the answer emerging from this. So, let me rewrite this term a bit. So, let me take this minus sign out from here and minus sign out from here. So, I can write this e power 2 n by n whole power n minus 1. So, let me write this e power 2 by n whole power n minus 1 divided by e power 2 by n minus 1. So, I took a minus sign out from both and I have a 2 by n here. This is my S n, the sum of n terms. Now, what is this? e power 2 pi n whole power n. This is nothing but. So, this we have 2 by n into e square minus 1 divided by e power 2 by n minus 1. So, see like you know we know the answer is e square by n and this is emerging from this. And what we have to show now is that the rest of it goes to 1 when n goes to infinity. So, this is our S of n. Now, we have to see when n goes to infinity what is this. So, let us see this. So, what we have here is this. So, let me rewrite this in a slightly different way. So, let me rewrite the whole thing in a slightly different way. So, I can rewrite the whole thing as e square minus 1 divided by e power 2 by n minus 1 e power 2 by n minus 1 divided by 2 by n. So, I wrote 2 by n here is divided the whole thing. Now, this is my S of n and I want to do this limit n going to infinity. So, it turns out that when limit n going to infinity this term will go to 1. So, then you will have answer e square minus 1. So, we saw previously that integral 0 to 2 e power x dx is basic nothing but e power x 0 to 2 which is e square minus 1. So, we got the answer both the answers by doing a sum we got the answer e square minus 1 which is basically the area under this curve area under the curve that we saw like we drew this such a curve and this area under this we divided into many parts and the area we found as a sum and we found that the answer is e square minus 1. So, what did we find now we found that if you sum up to infinity you will get exactly the same answer. So, this is one way of proving that or saying you could do it yourself of some other function that you know and see that if you take large number of divided large number of intervals and calculate this the sum you will get this answer. So, we know that definite integral is nothing but area under this particular curve. So, we will do now one or two examples of let us say one simple example of some integral and we will see some example of. So, let us do a function which is so we did many example for. So, let us do one simple example. So, we already know by examples like x power n. So, we said that x power n dx is x power n plus 1 by n plus 1. Now, it can be like root x. So, the root can be written as x power half. So, then similarly you can use the you substitute n is equal to half here. So, this is you can substitute n equal to half here and you will get the answer here for this particular function. . So, you can think of some other example like let us say we will do more complex example as the coming lecture, but as a one example of area under the curve let us think of a circle something that we already know. So, we know this that this circle and we want we know the area under this. So, let us say you have a circle of radius a. So, let us say this circle has a radius a then we already know that the area of this circle is pi r square which is pi a square. So, we know that area of the circle is pi a square. Now, let us check that by doing integral do we get the same thing do we get the same thing or do we get something different.  So, what is the equation of a circle? The equation of a circle is x square plus y square is equal to the radius square is the equation of a circle. So, we know that equation of a circle is x square plus y square is equal to let us say you have a radius a which is a square. In other words I can write y square is a square minus x square. In other words y of x is root of a square minus x square. So, this is my y of x. So, now let us take a small section. Let us think of a circle we just we did it. So, let us think of the circle like I did not draw it very nicely, but now this as in. So, this is where x axis and this is a y axis. So, any point here will have a y value given by root of x square minus a root of a square minus x square. So, we just said that y is root of a square minus x square. So, depending on. So, this is at this value of x you will have this particular y value at some other value we will have some other value. Now, we want to calculate the area of this circle and show that this is nothing but the integral of this function. Now, how do you calculate area? So, we can divide let us say this into many small parts like this. So, let us first do area of this quadrant and we know that 4 times of this. So, if you do area of this part and same as area of this same as area of this. So, the area here is same here for a circle. So, assume that this is a circle like I have not drawn it very nicely actually it is a circle let us assume. So, area here will be equal to area here will be equal to area here and the say it will be the area here. So, if you calculate the area here and multiply by 4 we will get the total area of a circle. So, we know this y and if you can do this integral y dx from here to here. So, y can start from here and y will go along this point up to here. So, this is x equal to 0 and this is x equal to a. So, sorry this is x equal to 0 and this is x is equal to a. So, sorry this is y equal to 0 from y equal to a. So, let us integrate this from 0 to a. So, if you do this y going from height y equal to 0 to y is equal to a which is this. So, y here the y at this point y is nothing but the radius like a here the height is y is nothing but the height here. Here x is equal to a, but y is equal to 0. So, here x is equal to a sorry x is equal to a and y is equal to 0. Here y is equal to a here y x is equal to 0, but y is equal to a. So, these are this point is 0 a and this is a 0. So, integral 0 to a y dx if you can calculate this it has to be area according to what we said it has to be area under this and it has to be pi a square 1 by 4. This has to be pi a square 1 4th because 4 of this has 4 times this has to be pi a square. So, now let us think about what is integral y dx for a minute and let us do this. So, we said that our equation is x square plus y square is equal to a square and y is root of a square minus x square. So, integral y dx 0 to a will be integral 0 to a root of a square minus x square dx. So, this is what you have to calculate root of a square minus x square dx 0 to a. So, how do we calculate? We can do this many ways whatever way you do you will at the end of it you will find that the answer of this is nothing but pi by 4 a square. So, now let us do one way of doing it like. So, let us do this x square let us substitute. So, one simple way of doing it is substitute x is equal to let us say sin theta. So, let us substitute x is equal to sin theta. So, this is 1. So, let us take x is equal to. So, what we have to calculate? We want to calculate 0 to a root of a square minus x square. So, and let us substitute x is equal to sin theta. What does this mean? This means dx is derivative of sin theta is cos theta d theta because dx by d theta will be cos theta. So, dx is cos theta d theta. It is clear for you that if x is equal to sin theta dx by d theta has to be cos theta. The derivative of this with respect to theta has to be sin theta as a derivative cos theta. So, that means dx is equal to cos theta d theta. So, we can rewrite this integral. So, we can write this integral root of a square minus substitute x as. So, it is more sensible to substitute x is equal to a sin theta. So, this will have dx is a cos theta d theta. So, you will have. So, let us say here we have a square. So, x is equal to a sin theta. You have a square sin square theta and dx is a cos theta d theta. So, you will have a here and you will have cos theta d theta. Now, we have to think of the limit. So, when. So, let us think of the limit a minute. So, what we have is x is equal to sin theta. When x is equal to 0, this limit lower limit means x is equal to 0. So, when x is equal to 0, this means sin theta is equal to 0. That means theta is theta has to be theta has to be 0 because when x is equal to 0, theta 0 sin 0 is 0. So, this is value, this is correct. When x is equal to a, when x is equal to a sin theta has to be 1, then only like x. So, when x is equal to a means sin theta has to be 1. So, sin theta is 1, when theta is equal to pi by 2. Otherwise, also we know that we have to integrate like half of the circle, which is 0 to pi by 2. So, this limit has to be 0 to pi by 2 here. So, now think about this. What do we have here? You have an a square a square common. So, I can take out a square outside. So, root of a square is a. So, you will have. So, if I rewrite this, what do I get? If I rewrite. So, let us rewrite here. You have an a here. So, let us say 0 to pi by 2. I have this a, which I am taking out. I have a square here, a square here. I take it out common. So, a square and there is a root into 1 minus sin square theta d theta cos theta. So, there is cos theta. So, let me write it cos theta d theta. So, I have this here. So, root of a square I can take out is a and a times this a is a square. So, I have a square 0 to pi by 2. 1 minus sin square theta is cos theta cos square theta. Root of cos square theta is cos theta. So, this whole part here, this root part is cos theta. So, you have 1 cos theta here and you have another cos theta here. So, cos theta into cos theta is cos square theta. So, what you have is cos square theta d theta. So, 0 to pi by 2 cos square theta d theta. So, you can do this. So, if you think of, how do you plot cos square theta? So, again you can plot cos square theta. So, little plots will be like. So, 1, 0 to pi by 2. So, this is 0 to pi. 0 to pi by 2 only will be a part of this. So, at 0 cos theta is 1. So, 0 to pi by 2 will be one part of this. So, this will be like 1 quarter of a circle again. So, this is another way of thinking about it. But, you can do this. You will see again this answer is a square into pi by 4. So, we can do this in different ways. You will end up seeing that the answer is pi a square by 4. So, always you will see that the answer of 1 quarter of a circle will have a area pi a square by 4 and you multiply with 4, you will get the whole circle. So, you can make this in other words, 0 to 2 pi cos square theta and you will get the answer as pi and the full answer will be pi a square. So, what we saw right now is that area under this circle is pi a square or 1 quarter of a circle is one-fourth pi a square. So, this is same, this is consistent with what we have been saying that area under the curve integral is nothing but area under the curve that we have. So, by doing integration we found that area under a circle is pi a square. So, with this we will go to a different. So, we could do, we can now do simple integrals and what you have to do is to practice this by going to some books and looking at many examples and doing them as much as you can because the more you do the better you will be and you will have, you will know more things about integrals. One another way of doing integral is something very useful and important. So, that is called integration by parts. So, we will learn how to do integration by parts because this is going to be very useful. So, this is something that we are going to learn next is called integration by parts. So, let us do this. So, while to do this, so let us recall something we learned in during different, when we learn differential calculus.  So, it is good to recall this because we will use this idea now to think of integration by parts. So, we learn that if you have a product of two functions, this could be F and G if or u and v two functions, the product of this derivative of this product is first function times the derivative of the second function plus the second function times the derivative of the first function. So, from this we can get some interesting formula for integral of a, for doing some certain kind of functions like whenever in you have a product of two function, any functions in particular, we can, we can get a interesting formula. . So, let us have a look at this for a minute. We will make use of this formula that we learned that for the integration by parts. So, let us look at what we can extend this formula here. So, we had learned this formula earlier. So, now, let us see go ahead from here. Let us go ahead from here and write this as an integral. So, let us, let us write this as an integral. So, let us write integral d by d x of u v. So, I wrote an integral here integral of this which has to be integral of this, integral of u d v by d x.  So, what did I do? I just put integral here, here, here. So, I integrate on both sides of this equation. This equation is something that we had learned. Then what do we get? We get something like this. So, when I integrate I have to put a d x everywhere. So, I have a d x here and a d x here. So, what is d by d x u v d x? So, the integral, this has to be u v itself. So, this has to be u v itself. So, let us say if we are doing this between a and b, a and with two limits a and b. Then you will have to apply a limit a and b here in the limit a to b is equal to integral u d v d s o.  So, we have an interesting formula here. What do we have in formula? So, let me write this in a slightly different way. So, if I write in a slightly different way, what do I get? So, let me write this first.  So, this is an interesting formula which will be very useful and we will be using it later.  So, what did I do? I did not use much. I just used the fact that the way we learn derivative of a product of a function. So, here also you have two functions and let us say you want to find the integral of this. We can use this kind of a formula. So, let us do this in example. So, if you do not want to take the limit, you can also write this as integral u d v is u v minus integral v d u. If you, this is a definite integral, this is an indefinite integral. Now, so let us do an example to be clear here. .. So, let us do an example, integral x sin x. So, let us say in the limit a to b d x. So, this is what you want to do. So, what is sin x d x? So, this sin x d x you can think of as, so you can think of this as u and this as d v. So, this is, this is like integral u d v. So, if this is d v, the integral of sin x that is v is nothing but cos x. So, we have integral of this which is cos x. So, actually it is minus cos x. So, we will put a minus sin. So, we can use the formula now here. The formula that we learnt that integral u d v is integral.  So, here what we have now here is that u is x. So, this is our u and sin x d x is d v. Let us take it as d v. So, then what you have is integral u d v. So, you have to find out v. So, d v is sin x d x. So, v has to be minus cos x because what does this mean? This means that v, so this would imply that v is minus cos x. So, that d v can be sin x d x. If v is cos x minus the derivative of this is sin x d x.  So, let us do this. So, this is u into v. So, u is x and v is cos x with the minus sin in the limit a to b minus integral v. v here is minus cos x. So, minus minus will become plus cos x and the derivative d u. So, d u is you can take it as here derivative of x which is 1. So, this is nothing but integral of cos x which we already know the answer. So, this is a to b. So, this will be integral of cos x d x. So, this can be done and this answer we know. So, this integral can be easily done. So, this is the trick of doing this integral. This is the you can use this particular idea which we will use in the coming lectures. So, with this like we have some good look at the integrals. Now, we will quickly say something about like we said that integral can be written as sum of many, sum of many parts like sum of area under the curves can be written as sum of things with equal areas f of x i d x i sum over i. Now, if we go back to this derivative and think a bit about derivatives and let us say you are you are given a function. Instead of a function what you have given is a table let us say x and f of x.  So, this is a function and this x and f of x. So, how do we calculate the derivative of such data? So, for this if you could know the discrete form of. So, we know if you know the functional form if you know the f of x is a sin x we know how to calculate d f by d x. So, how do we calculate the derivative of such functions? So, what we are going to discuss quickly is that derivative given data. So, if you are given data like this how do we calculate the derivative? So, it turns out that this is easy we can extend the results that we already learnt.  So, the same thing can be used here. So, here d x.  So, what we have here this is x 1, x 2, x 3, x 4 and what we have is f of x 1, f of x 2, f of x 3 and f of x 4. So, if you have this what we can do is that to find the derivative between the slope between these two what I can do is that I can do f of x 2 minus f of x 1 divided by x 2 minus x 1. . Similarly, the slope between this two will be f of x 3 minus f of x 2 divided by x 3 minus x 2. So, what essentially what you have is basically this formula f of x i plus 1 minus f of x i divided by x i plus 1 minus x i. So, this can be used you can extend this to higher derivative. So, you can just say that higher derivative.  So, you can write this as m at x i plus 1 minus m of x i divided by x i plus 1 minus x i. So, where m is the slope. So, I can substitute for m of x i plus 1 and m of x i because this itself is derivative and you can go ahead and do this. So, similar things can be done for a given a data you can also integrate by the same rule that we said integration of a data can be written as like as we as we did i is equal to 1 to n f of x i d x i. So, in other words sum over i is equal to 1 to n f of x i into x i plus 1 minus x i. So, if we have data like given here x 1, x 2, x 3 we can use this formula to do the integral of this numerically also. So, this ideas can be extended to numerical part which is actually a different course in itself that numerical methods like how do you use numerical methods to do the integration there are like very very efficient. We can make this methods very efficient by doing various techniques which is a course in itself. So, I would not go into that. . So, what we learnt today is two three things like we talked about integration we discussed integration. We talked about area under the curve, area under the curve f of x for definite integrals. So, this will give you definite integrals and we saw couple of examples like we saw e power x and we saw the area under the circle and we also said that quickly said something about derivative and integral of data. So, if you are given some data you can use the formula to do the derivative and integral of this data. So, with this I will stop this integral and in the coming classes we will see something all differential equations and use this integrals. So, we will discuss the integrals in details there as an examples of solving differential equations. So, integrating is nothing but solving some kind of different equations. So, which is very useful in biology like which is you can see very often in biology. So, we will discuss in the coming classes the differential equations and how do we use integration to integrate the differential equation and get something that is very useful in biology. With this I will stop today's lecture. Bye.


Here are ten characteristics of what I call high-maintenance people that I've observed over the years. You can check to see then if you are one or are doing life with one and need to adjust that high-maintenance relationship. Number one, they always have urgent needs. Number two, they have a sense of entitlement. Number three, they cling to stories of wrongs done to them. Number four, they talk a lot about themselves. Number five, they crave a listening ear but don't listen to others. Number six, they live in a state of perpetual drama. Number seven, they're obsessed with details. Number eight, they're quick to criticize without offering any alternatives to what they're critical of. Number nine, they resent change. Number ten, they actually have no idea at all how high-maintenance they are. Zero self-awareness that makes it even more exhausting to try and say anything about it.  Love you guys, hope that helps. Cheers!


Today we're doing tracking. When puppy grows I like to play with the puppy by dropping some treats on the grass on my lawn to teach him sniff and that's a really fun exercise for him and I do it occasionally so naturally when they very little they just learn that it's fun game to do and then later on when the dog is ready I start teaching proper tracking. Tracking is great with doberman's they have great nose sense of smell they like sniffing and I think if you have a pet and you want to do any type of training for nose work this is really interesting and challenging at the same time thing to teach your dog because we humans we don't know we don't use nose for something but for dogs it's the major sense and so we humans have to figure out how to teach a dog something that we don't understand so for that it is very important in tracking to take your time do not rush first let the dog figure it out secondly never progress too fast repetition is the key you have to repeat and do it and over and over and over again and if you see that the dog struggles step back in your process and start repeating again so unlike with my obedience exercise when I see when the dog is ready I move to the next step with tracking I really like to practice and practice and practice repetition when we track it's better to go in the morning you want to find maybe a park or another place mornings are usually best because over the night nobody was running on it there were no footprints so it's fresh and it's easier for the dog to start figuring out what you want from him because if you go to the public park where there are a lot of smells and sense it's very distracting to the dog plus they won't really understand what what you're doing with him especially when the dog is so young so we start usually early in the mornings if for example depending on the season of the year if I start with grass which is what I'm doing with my dog currently I stay with grass for a while until the dog knows what we do in until he understands to follow my scent not the grass and that takes a while if you live in the area where you have the plowed dirt and you start in that just stick to that until the dog understands before start changing the terrains or changing the sanding area that they have to work with so first the treats they always have to be soft because you don't want your dog to choke when they are excitingly searching for it and and eating so I use hot dogs and it has to be a high reward for the dog something that they will eat for sure will love to eat also another reason why we do it in the morning because the dog is not fed yet and they they eager to have their food but instead of breakfast you kind of go and do the tracking first and then you go home and give the dog his food I I'm showing in the video how I step on the grass so the idea the very first exercise that we have to do is scent pads because that's the way how we transfer our knowledge to the dog that the scent that the sausage associated with the scent of the grass where you stepped on the kind of stepped on grass and so what you have to do is that you can see behind I have four flags I usually select a right kind of an area of six feet in radius about six feet in radius the Indian diameter sorry and then I try to cover this whole area with my steps so every grass has to be kind of stepped on because it changes how the grass smells and that's what we need to for the dog to learn eventually to identify the grass that is untouched versus the area that was suppressed by the step that's what they follow they are able to identify that and so at the beginning we obviously try to stimulate it was with the reward which is hot dog and then later on you fade it out and then the dog follows that son that they understand so I always do two sand pads every day or not every day every training about the same size I cover the whole area and then I drop the hot dogs and I step on hot dogs a little bit because I don't want them to be on the top for the dog to be visible because I want the dog to use nose not not his eyes and I always have a little beginning pad right there it's right by the flag it's the size of my two footprints and I put four or five pieces of hot dog there that's my starting point so the dog is always eager to start that teaches the dog it's fun here it is and then the dog starts then goes and search when the dog searching you will see I do not say anything I do not help the dog I don't control the dog I let him figure out let the dog do do the natural work and then when I see that he's nearly done with the hot dogs because I know how much are there or when I see that he's getting distracted I pull him off the Sun pad immediately because I want the dog to be only concentrated if the dog lost concentration I move away that's it and that's for that reason I always have two sandpads because puppies are small and they get distracted easily so if I see that the first one didn't go well I always have the second one and at the same time if the dog is happy and excited to do one I stop it I never wait until the dog is ran out of rewards and he's searching searching there is nothing you don't want that you want to pull the dog of the tracking pad before the here and out of rewards and then if the dog wants to continue then you use the second pad and my dog is very happy to do this so I always do too and so it's a double exercise for him every time we do this he loves it and then one other thing to keep in mind before you start send but make sure you take the dog to do his business to party because again you want the dog to have very good positive experience and if the dog was sitting in the car while you do in the sand pad and then you pull it out to do the nose work but the dog needs to pee he'll pee on the sand pad that's not a very good experience for the dog plus it's not what you want your dog to do so make sure his needs are taken care of if he needs water give him water and then let him concentrate on this work let's see how my dog at us will do today you can see that the dog is wearing flat color and I have him on line you can see he's eager to start I'll help him find find the first pad which is right here usually hold him to make sure he picks up everything and then I let him go you can see four flags that's the boundaries of my circle you'll see how the dog steps out of the boundaries but then he gets back he figures it out by himself I don't give any commands and we'll be doing this probably five six or maybe a dozen of times just the circles before we move on to the next step okay he's very off he figures out good boy I'm gonna pull him off because he's nearly out of treats good that's good that's good good boy you did well you did well good boy you did well you know there's a second one you do oh good boy good boy good boy what you want to start your second one yeah yeah and on him and come on just hey hey no pay attention pay attention pay attention here here sometimes I give him a break in between but I figure it all just film it together so you guys can see how he does and if you if during the training the dog got distracted by something like another dog running or a bird or something do take a break between starting the second circle because you want the dog to be focused and concentrated not distracted and he's got excited by something different than he needs time to come down he's on fire today likes it when this is done I always praise him okay that's not good good boy good boy he quit by himself that's not what I wanted but he's since he quit I'm not letting him finish it so that's it good boy good boy you're finished now you did well good boy good boy and I usually if the dog is very excited then he's already happy you don't need to do much you just walk him and let him sniff around or play maybe a little bit and they usually put him back in the car so he knows that this is the routine the exercise he keeps sniffing thinking there is another one and so for us in the morning that's the routine you do tracking and that's it so that's why the dog kind of by default he learns that if you're going somewhere in the morning that's gonna be tracking so he's already excited so hope you can teach your dogs to track training


Hi, I'm Zor. Welcome to Indizor Education. We continue talking about conditional probabilities and this lecture is about a very interesting theorem which was proved by very a long time ago living individual. I can't even tell that he is a mathematician, Thomas Bias and he was Englishman He was actually theologist, but he was really very good mathematician as well. I shouldn't really say he was not and so this very very simple theorem has his name It's very simple. I will prove it to you in just almost like one line or two lines but it has a very interesting practical applications I do recommend to listen to this lecture from Unizor.com because this is basically a part of the whole course of advanced mathematics for teenagers So you can read the notes to this lecture. It's basically written as a textbook and then listen to the lecture if there is anything which you might not really understand from the first reading So this is basically the suggested way of working with this website. Anyway, so let me go straight to this Bias theorem. Bias, that's the name of the guy Sometimes it's called formula. Bias formula or Bias theorem doesn't really matter So what I will do is I will actually present this theorem as a purely mathematical statement, which is very very easy and then we will talk about how to apply it in different cases So here is the theorem. Let's consider You have certain sample space Well, I can use a Greek letter omega, which sometimes is used for this Which contains certain elementary events I know E1, E2, etc. E n doesn't really matter And if you have two different Events, which is basically a combination of elementary events, right? Whatever the indices are doesn't really matter Then I have introduced the concept of conditional probability of event A under condition B And we have agreed that this is a good expression for the probability of this conditional probability Now, I really hate formulas which are not really supported by good understanding of what it is So let me just deviate from my main Theme of this particular lecture and again talk about this particular formula This is an extremely simple formula and all it does it basically states that if you have certain number of elementary events This is everything whatever is possible to occur There are elementary events here certain number Let's talk about finite case when you have a finite number of points inside of this area And then you have certain event which encompasses only part of these elementary events Then geometrically speaking the probability of this event is a ratio of number of points which are inside relative to the entire number of points in the area or If you're not talking about finite number of Elementary events and if you are not talking about Elementary events which which have equal chances. It's still exactly the same thing Whatever number of points here this area. Let's use the geometric term area divided by this area represents the probability of This particular event So it's nothing but really very very plain geometric interpretation of the probability Now if for whatever reason You know that this is Event a now if you know that this event b actually has happened which means All the probabilities are concentrating now from the entire space Only into this sub set Then obviously if you are looking for The probability of happening of the event a All which is theoretically possible to happen Are these elementary events which are inside the intersection, right? Because everything outside of b is not happening. We definitely know this is our knowledge Before we even maybe conduct an experiment or we conducted some other experiment that this event b is Happening always which means that the probability of all the points outside of this b is zero There is no measure allocated And all the measure is allocated to this particular Sub set b then obviously you have to divide only the intersection by The area of the entire b Intersection divided by the area of b to know what's the probability of occurrence of of event a Under condition that basically only events from the b Actually can happen, right? So That's why i'm saying that this is a formula which you should not really remember as a formula You should remember it as a picture that would be the great That would be really the great way to to understand it better It's always the ratio of the areas in this case It's area of intersection divided by the whole area because otherwise You cannot have any other event outside of b happening So it's only the intersection between a and b which actually Constitutes events a under condition that b is happening, right? So that's my deviation And that's how I if you wish derive the formula for conditional probability But now let's talk about this way if my conditional probability Of a under condition b equals to this The intersection of a and b the probability divided by probability of b then I can equally say That probability of b under condition a is a b and a divided by p of a Right, it's exactly the same formula just a and b changed places now obviously a Intersections b and b intersections a is exactly the same thing because the set theory operation of intersection is commutative, right? so a Intersect b is equal to b intersect a so these probabilities are the same Which means what I can do is I can resolve it for this And substitute it into this so the probability of a intersections b is probability of b times the conditional probability so I have So I have this formula Equals So instead of this I will use this expressed as p b times p a of b divided by p a So it's basically one liner Derivation right and this is the bias formula. It combines two different conditional probabilities b under condition a and a under condition b Why is it so important? Well to demonstrate how important it is I will just go into out a concrete problem and In that problem, I will try to demonstrate the importance of this Okay, so let me Go to concrete example Now for this concrete example I do need One very small Formula Which which is not really a theorem. It's just basically kind of an introduction into this problem Which I'm going to do. Let's say again geometrically. This is your Omega this is your Sample space so all the elementary events are here Let's call it e1 e2 em And let's consider that they are of the same chances to occur So every elementary probability is exactly the same and equals to 1 over n If it's not that It's exactly the same kind of logic. It's just simpler to assume it this way. Now. Let's consider that this entire Sample space is subdivided into two different events x and y So let's say x has elementary events Up to k and y Has elementary events from k plus 1 To n Doesn't matter The most important they are Not intersecting They do not have anything in common. They divide basically into some Unequal generally speaking parts, right? That divide the whole the whole set of elementary events into two into two parts And I also would like to say that their union Is equal to entire space Entire sample space Now under these conditions sometimes I will use Different notation instead of intersection I will use the multiplication sign and instead of union. I will use The plus sign now Why is it justifiable? well, um Basically, it's just easier to to to write the books in this particular case and the text But in general you will see that the probabilities are obviously Supposed to be added. You see the probability of x plus probability of y It's the area of this and the area of this it's supposed to be the probability of the entire space, right? Which is omega so the union x as a Edition Now the intersection is not really acting as as multiplications. So don't don't Um, don't assume that but the addition is probably the most important, right? All right, so let's assume that my Entire uh sample space is subdivided into these two mutually exclusive mutually exclusive events For instance Well, I flip the coin and there is either Cat cats or or tails, right? There are no others. So these are two Events or if you roll the dice it can be either Odd or even, right? These are mutually exclusive events or it can be less than two Or greater or equal than two on the dice. I mean, there are many different ways how any space Any sample space can be subdivided into two events. Now now let's consider you have some event a Which also encompasses certain elementary events some of them belong to x some of them belong to y, right? now obviously It's obvious from the set theory um I can always represent my a as a union of two different subset a Intersections with x. It's this one And a intersection with y. It's this one and obviously their union or I like to really put plus sign here because this is a union of non intersecting Non-intersecting parts. This is not intersecting with that Right. So similarly the probability of a Is equal to probability of a and x plus probability of a and y Right Now let's remember our conditional probability formula. Let's do it again That's the one which we just talked about right Okay So what does it mean in this particular case? Well, if b is actually x I can replace it with So this probability of their intersection is probability of b which is x in this case Times probability of A under condition Uh, b is x actually. So let's just use x Plus in this case instead of b. That's y. So that's probability of y Times probability of a under conditioned y And this is two p of a So you see if my Initial um sample space is subdivided into x and y I can always represent the probability of some event as a sum of probabilities Which are calculated using this thing. So Look at it this way. Well event a Can happen Under some circumstances, right? So sometimes it happens if x is happening, right? And that's these things And then obviously the probability of this piece which is intersection of a and x Is the probability of happening of the x. That's what That's what conditional probability is times The uh conditional probability of A under condition x same thing this this piece That's basically represents the probability of a under condition of y, right? So the area is equal to probability of y times Uh, probable conditional probability of a under condition of y Why is it important To to to do this type of manipulation? Because sometimes we don't know the probability of a but we do know conditional probability of a Under certain conditions and this is exactly the example which I would like to present to you Let's consider I am in a business of selling the tennis balls, okay um, I have two suppliers One supplier Oh, by the way, this is called a formula of total probability, right? So one supplier supplier x manufactures white and green tennis balls, right? So it manufactures White And green and let's say it's 50 50 That's how it's produced Now the supplier y Manufactures also two kinds of uh tennis balls two different colors yellow 90 of the production goes to yellow And green 10 Now I'm buying from both suppliers. I'm supply. I'm buying 40 of this guy And 60 of my entire stock from this guy now Here comes a customer And he wants a green ball Now what I do with all the balls which I am receiving from those guys. Well, I just put them in one very big box And there are whites yellow Green another white another green.  Well, I know that 40 of this box comes from supplier x And out of these 40 percent 50 percent are white and 50 percent are green Now 60 percent of the entire box belongs to the second supplier and out of these 60 percent I know that 90 percent is yellow and 10 percent green Now if I just randomly pick the bar the the ball From from this particular box. What's the probability of this ball to be? the green one This is a perfect example Of this formula Look at this formula. What do I know? I know this The probability of getting the ball Manufactured by supplier x is actually 40 0.4 I also know that If This is a tennis ball produced by the manufacturer x Then the probability of the event a that this ball is green is equal to 50 percent I also know the probability of getting the ball from um The second supplier That's 60 percent i'm getting from it, right? So this is 0.6 And finally, I do know That the probability of getting a green tennis ball Manufactured by the supplier y is equal to 10 percent 0.1 And that would give me the total probability of Getting the green ball, which is what 20 26 0.26 So If I just randomly pick up the ball the probability of this tennis ball to be green is 0.26 or 26 percent That's where this total probability actually is applied so notice that This is a pure conditional probability problem. I have not yet Used the bias theorem in the beginning of this lecture when i'm basically reversing the usage A under condition b And b under condition of a and they are expressed one over another For this particular problem. I didn't use it. But now let's consider that this problem is solved But what i'm interested in the real In the real case i'm interested in the following My customer Wants actually a a tennis ball, which is manufactured by well, let's say the first manufacturer Now if i'm picking up the white ball, then obviously this is manufactured by the first Supplier because white ball is only from this guy if I Pick the yellow It's obviously from the second supplier white, right? But what if I pick the green one? And this is my next problem and that's where i'm going to use the bias theorem so What is the probability that this tennis ball is manufactured by the first Supplier if I picked the green one Because again the probability of this if I picked white Is one right because there are no more suppliers which are manufacturing the white So if I pick the white ball, I know with the probability of 100 that that's manufactured by x if I pick the yellow The probability of this yellow to be manufactured by the first is equal to zero right because all 100 goes to What but if it's green it needs some calculations so What exactly are the calculations? So that's my next problem, okay Let's just reverse all these calculations. I know What basically I'm supposed to get is the probability of x under condition a right the probability that my Tennis ball is manufactured by x if it's green Okay That's what I need to know now Now let's recall the bias theorem Well, obviously I don't remember the formula, but I do remember that p of a over x is equal to p of a and and x over p of x right and p of x over a Is equal to p of again a and x divided by p of a so this one is equal to p a and x which is p x times p of a over x divided by p of a Okay So as you see, I just decided to derive the formula on the fly Rather than remember. Okay. I know that now Is that sufficient? Oh, well, let's think about it p of x. I do know that's 0.4 40 percent is manufactured by the first guy conditional probability of getting green from x is 50 percent right so it's 0.5 And p of a that's a total probability which includes both manufacturers this and this and I have just calculated it That's this formula and there is a result which is 0.26 So the whole thing is equal to what 0.20 divided by 0.26 And I think I calculated it somewhere Right at 0.77 So if I fooled the green tennis ball from the box Randomly and it appears to be green Then the probability of this particular tennis ball to be manufactured by my x supplier is 0.77 which is 77 percent And obviously the corresponding probability of this particular guy to be manufactured by the y supplier Should be the opposite, right? Let's do it the probability of y under condition of me picking the green ball is equal to p of y times p of a under condition y divided by p of a Which is equal to the probability of getting y it's 60 percent which is 0.6 conditional probability of the green tennis ball If it's manufactured by y is 10 percent 0.1 And this is still the same 0.26. This is the total probability of getting the green ball And that's equal to 0.06 divided by 0.26 Which is approximately 0.23 which is 23 percent. So under all these conditions all these numbers if I pick the green ball my probability of Having this tennis ball manufactured by the first guy is equal to 77 percent and by the second supplier 23 percent If I have picked any other tennis ball then everything is obviously If it's white then it's 100 percent to this guy and if it's yellow it's 0 percent for this guy So i'm talking about only the probability for the first one. All right That's it. I suggest you maybe to read again these All these calculations and formulas are in the notes for this lecture And again, it's always good to register on the website on unizor.com. It's completely free And that would enable you just just to go through a complete course With Enrollment and taking exams, etc. So that's what I definitely recommend you to do All right. Thank you very much and good luck


Guitar and Excel, open chords, C major scale, G major chord, intervals. Get ready and don't fret. Remember, the board's totally fretted already, so you need to be the calm one in the relationship. Here we are in Excel. If you don't have access to this workbook, that's okay because we basically built this from a blank worksheet, but we started in a prior presentation. So if you want to build this from a blank worksheet, you may want to begin back there. However, you don't necessarily need access to this workbook if looking at this from a music theory standpoint, because we will simply use it as a tool to map out the fretboard, give us the scale, and chords we're focused in on. If you do have access to this workbook, though, there's currently five tabs down below. We've got three example tabs, an OG tab and a blank G tab. The OG tab representing the original worksheet we put together in a prior section, you can see it now acting as the starting point going forward, mapping out the entire fretboard, giving us the entire musical alphabet and letters, numbers, and combining them for letters and numbers. Having an adjustable key where we can change the scale we're focused in on by changing the green cell, providing or creating the worksheets on the right hand side, giving us the scale that we're focused in on as well as the chords. We then wanted to focus on the key of C in this section and the chords that can be constructed from the notes in the key of C, starting of course with the C major chord, which is in the example C tab on the left hand side. We're focused on the open positions, which I'm defining as positions zero on the fretboard to three on the fretboard. We mapped out the one three five of the C major scale in that position and discussed it from multiple different angles. We then did a similar process for the F chord, jumping from the one chord to the four chord, instead of going from the one to the two to the three and then to the four. Why? Because the one four five are going to be the major chords constructed from a major scale, as can be seen from the lettering or the uppercase Greek numbers. And so we went to the F and now we're going to do a similar process going to the key or the chord of G. So we have that here. That's the one we're going to be working on the right hand side in this blank G cell. This is what we've constructed thus far, copying over from the OG tab and then hiding everything except the open positions, which I'm representing as the first three frets, remembering that the heavy string, the one closest to the ceiling when you're holding the guitar is going to be the top string. The way we're visualizing this, we then have our scale on the right hand side, which is still in the key of C. But now we're focusing in on the five chord that has been constructed, which is going to be a G major chord. So what we want to do this time is think about the interval. So in prior presentations, we mapped it out. We looked at the fingering of it. We've mapped it out on top of the pentatonic scale in the key of C, as well as the major scale in the key of C. Now we want to think about the intervals in more depth. Remembering this is often the area that confuses a lot of people because people get mixed up in what the terminology means when they're using different numbering systems. So I highly recommend doing this like in the morning before work possibly or something like that. When your mind is still working, spend 15 to half hour in the morning trying to differentiate and really understand what you're doing when you're holding down these chords. And then you'll plant the seeds so that when you're just strumming around, it'll start to solidify itself in the proper places, hopefully. First, a quick list of some of the numbering systems we need to keep straight and differentiated in our mind because when our mind mixes these things up, it leads to confusion. So first, we need some kind of system to name all the notes in the musical alphabet. We can use either letters or numbers to do that. We typically want a numbering system that's going to name the relative position within a scale of a particular note. And we could have a slightly different numbering system which still names the position of a note relative to a scale but gives us more information about the kind of chords that we can construct from that note. Will they be major, minor, diminished? We could have a numbering system helping us to name the notes in a chord, not in relation to the scale we're looking at, but rather in relation to the root note of the chord and its scale. And we could have a numbering system helping us to name the absolute intervals when looking at the notes in a particular chord in relation to the root of the chord. So let's give a little bit more detail on some of these items. Let's first go to the OG tab. If I go to the OG tab, we're working in the key of C, so I'm going to change this key over here to a 4 so we're looking at the key of C. This over here represents our musical alphabet. So we have our musical alphabet A and then A-sharp is being represented by a lowercase A and a B because we could name it either A-sharp or B-flat depending on which way we are going. We might get into that in more detail in the future, but just note that it's still, at this point, the same note tone-wise. So A-sharp, B-c-c-sharp, D-d-sharp, E, and then there's no sharp and then there's an F-sharp, G-sharp, and then it starts back over at A. Now remember, one of the benefits of this system is that, at least in the key of C, you can count the musical alphabet using our nice little song here at A-B-C-D and so on because there's no sharps or flats when we're looking at the key of C. But that gets more confusing when we're looking at other keys. It has its benefits because you can still construct any major or minor by using like every note in the alphabet, although there could be sharps and flats in it. So that's kind of neat, but it's still going to be a little bit difficult to try to count the musical alphabet forwards and backwards. If you try to sing the musical alphabet backwards from G, even if you didn't have sharps or flats and worrying about whether it should be a sharp or a flat, and go back to A, that's a lot more difficult than numbering where you can go forwards and backwards. So I think it's useful to memorize, since you're going to memorize a lot of stuff anyways, the absolute position of the names of the notes. So an A would be a 1, A-sharp would be 2, a B would be 3, C-4, C-sharp 5, D-6, D-sharp 7, E-8, F-9, F-sharp 10, G-11, and G-sharp 12, and then it starts back over as an A as a 1. By doing that, then you can easily count forwards and backwards, and you can start to count intervals using simple math. So I think that is really worthwhile to do. That's why over here, we have both the letter and the number so that we can see it both ways and get the benefits of using both of those kinds of systems to name the musical alphabet. So if I pull that, so notice when we construct our scale then, what we're doing is we're starting on a C, which I'm saying is an absolute position 4, and we're applying our formula, whole step, which is 2 notes, whole step, 2 notes, half step, 1 note, whole step, whole step, whole step, half step. So if I apply that out, we start at a 4, which is a C, and then 4, 5, 6 to a D, see how the intervals work, 6, 7, 8, 8 is an E, 8, 9, half step is an F, 9, 10, 11, 11 is a G, 11, 12 goes around the horn back to 1, which is an A, there's only 12 notes in the musical alphabet, 1 plus 2, whole step gets us to 3, which is a B, and then 1 gets us back to the 4. So there's our musical alphabet, so that means if I apply that formula, that is how we create our notes in our scale. So we took the 12 notes, now we dropped it down to 7 using that formula. I'm not going to get into why we use that formula, that's just the formula. So I'm just going to accept that a priori now, and that's the formula. So now we've constructed our major scale based on that formula with these notes, and that's going to be our baseline worksheet. So then if I go back to the blank G tab over here, now given that we're looking at this numbering system and we're going to say, okay, what note are we focused in on? We're focused on the 5 note. So the 5 note is our point of focus, how can we construct a chord from the 5 note? Well, if I go over here and look at my chords in a circle, which I think is easier to do sometimes, and I start at a G, we just do the same thing we do every time, we just take every other note. So we skip the A and we go to the B and then we skip the C and we get to the D. That means we've constructed something from the C major scale starting at the 5 note, which is a G, B, and a D. However, when I name these kind of intervals in here, we usually call them a 1, 3, 5 of a G major. Now I know this is going to be a G major, one because I can see it over here in my cheat sheet, right? Because I put it as a capital or uppercase Roman numeral. That means that it's going to construct a major chord from it. But we constructed it the same way that we would construct this minor chord up here. We just took every other note in the scale that we're in. So how do I know it's major or minor? Well, one way is that the interval, the absolute interval, will have a two-hole step or a four-note-away interval, which we'll talk about that. So the intervals are what's giving us the differentiation. Notice that when I name the notes in relation to the chord, however, I'm not going to call it a 5, 7, 2. You could, you can see it that way because we constructed it from the C major scale. But I'm going to name the intervals a 1, 3, 5 because I'm naming it in relation to its major chord. So if I say that's a major chord, I'm going to name the notes from the intervals related to its major chord. So if I go back to the OG tab, for example, and I go up top and I say this is an 11, which is a G. Now I have my worksheet for a G instead of a C. The G is now the 1, which is also a major chord. If I was working from the G scale instead of the C scale, I still get a GBD. But I can see now in relation to the scale, it's the 1, 3, 5. See, it's the 1, 3, 5 of its related scale. GBD is the same GBD we get here. So when I'm looking at these notes up top, the chord scale, you know, like intervals within its related scale, I'm not looking at the scale I'm currently built it from. I'm thinking of it in terms of its related scale. But remember that I don't really need to flip my mind and know all the notes in the G major scale. What I need to know is the intervals. I can also map this out in my mind in terms of the intervals that create a major scale and a minor scale. And that's what these are going to be up top. These are going to be the intervals up top. Okay, so given that then, let's, when we map this out on our fretboard, we have our fretboard in open position, which I'm naming as from 0 to 3. The G is a classic open position shape because it utilizes many of the open strings here to ring it out in that position. We talked last time that there's different ways that you can basically finger this position. What I would practice now then is thinking about the intervals as we finger the position first using the worksheet and then trying to not use the worksheet as a crutch. So you could just kind of finger the position and think about what you're playing. So if I use my normal fingering of a G, which is the most common fingering, which looks like that of the G, I can start to say, first, let me just think about these intervals, the 1, 3, 5 of the related chord that we're working, which is a G. So if this is the 1, that's going to be a G. So I'm going to say, okay, that's a G. And then down one back one, you'll start to see if you looked at these other ones, the C and the F, which were also major, that's going to be the major third that is going to be back there. That's that position down one and back one will start to look familiar. And then this one is an open. So that one's open right here. So I've got the open D. That's the fifth I can see right here. That's the fifth and let's actually do another one. Let's do this copy and let's paste and let's make this one. Red not the inside of it. I want to make the outside red. So this one. So that's going to be the fifth. And then if I look at this one, that's an open string on back to an open G. That's the one. And then if I look at this one, I have once again another. It's repeating itself. I don't have anything but these three notes in it. That's going to be another three. That's a third. And then I have this one, which is going to be another G or one. So I can start to map that out when I'm looking at it and say, OK, this is a one. This is a major third. This is going to be a five. This is going to be the one again. And you can start to see those relative positions. And that helps you when you move it up or think about basically different things that you're going to be playing in relation to are they going to be major or minor and what are the notes within it. So I'm going to move this one up here. So then I would map it out a little bit more in depth. Once I do that and start to say, OK, let me think about this in terms of the intervals. So this one again is the one. So I'll actually say out loud. If you have the ability to say out loud, this is the relative position one because I'm saying it's relative position relative to the chord, not relative to the scale in this case, right? It's relative position one to the chord that we are in, which is a 11 note number 11, which is a G, right? And so then I'm going to go, OK, that one's straightforward because it's the one note. And then I'll go to this one and say, OK, this one that I'm playing is going to be, I would call it then a four note away major third. Now, remember, I have to say distinguish somehow, and I want to distinguish both with the number of intervals, as well as with my words, that it's a major third as opposed to a minor third because this third is the differentiating factor between a major and a minor chord. So I'm going to distinguish that in my mind by looking at the notes away, that it is the interval, absolute interval, not just the scale interval as well as the position relative to the scale. So let's give a quick recap. What does that mean? It's the third. So you can see, well, it's not the third right here of the C, right? It's the third if I went to the two, it's related scale. If it was then one, the third would be the B. But I don't need to think about it like that. I can think about it as though it's two whole steps or four notes away. I can think about it in terms of intervals. So let's pull out the trusty calculator here. And this is why numbering the notes is useful. So I'm going to say it's a four note away major third of note 11, which is a G, right? So if I use my math here, I'm going to say, well, 11 is a G plus four notes. That would get us to 15. There's only 12 notes in the musical alphabet, so I can say minus 12. And that gets us to three. Three is a B. So notice how that little bit of math can be useful. I can do it this way too. I could say 11 plus four is 15. Usually in my mind, if it's between 13 and 19, I'm just going to drop the one and subtract two because that's easier to do in my mind. So I'm just going to say, okay, that's like five, which would be minus 10. And now I just have five minus two. Five minus two is three. And that's going to be a B. And then this one over here, if I look at this note, that's the open note, which is a D. So that's a D. I can say, okay, I can see it's the fifth. Now, what does it mean that it's the fifth? Well, if I looked at the G major scale, it would be the fifth, a D of the G major scale. But I want to think about it in terms of intervals over here, absolute intervals, not intervals related to the scale. It's seven notes away. So I'm going to say this is a seven note away fifth of note 11 G. Now, when I say seven notes away note, I don't have to say that it's major or minor because whether this G be major or minor, the fifth will be seven notes away. Now, again, you got to keep that kind of straight in your mind because you're like, well, it's a fifth. That means it's the five note. Why are you saying it's a seven note away fifth? Because the five is relative to its scale, which only has seven notes in it out of the 12. And when I say it's seven notes away, it's absolute, meaning I can use my math and say, well, if I was on 11 before, 11 plus seven brings me up to 18, drop the one or minus 10 gets you to eight minus two gets us to the six, which is a D, or you say 11 plus seven minus 12 gets you to six, right? So it's a seven. So that way you can kind of easily do the math in your head. It takes, I mean, if you know these notes, you could do the math. It's not easy at first, but you could do it, right? So you can also count up this way. You can say, okay, well, if that was, what does it mean to be seven notes away? Sometimes it's useful to count this up on one string. So for example, if this is my one position, that's my G, and I count up seven frets, which are seven notes. One, two, three, four, five, six, seven, it gets me to there. That would be then my D. That's useful to know. And I could still use that when I'm playing. I can be doing this and then go up there and play that note. But if I'm trying to make a chord in open positions from zero to three frets, it's going to be difficult to hold that D down. So we're looking for the relative positions in this case underneath because the one is the string that we're holding out up top. And by looking at those relative positions, you'll be able to recognize what those patterns are when you start moving up the fretboard, for example. And if we see this one, let's go back to this one real quick. This was the third, which was the B. That relative position is really useful to be able to recognize. If this is your one, then your third is down one and back one unless you're in between these two strings where they have that weird kind of relationship. If I was to count up four frets because this was four notes away, I could also do it this way. If this is my G, one, two, three, four would get me to there, there's my B. That's useful to kind of see. But obviously I want to play it in open position. You can also kind of see the relation between these two strings by saying, well, that's the same note as this. So if I count this up, that's a B, one, two, three, four, five strings up this way and one up that way gets you to another B. So you can kind of see the pattern of the fretboard when you start to look at these intervals and mapping out what you're actually just playing. Just take a little bit of time to map it out. If I go then back here, this is a G. So this is an open position G here on this string. So I would just say that that's going to be relative position one of note 11 G, which is a note 11 G. This is an open B right here. So there's an open B. And I'd say that would be relative position here. I would say four note away major third of note 11, which is a G. I would just kind of repeat that in my mind. And then here, once again, we're back to the one. So I'd say, okay, and then this note is relative position one of note 11 G, which is, of course, note 11, which is a G. And then you can kind of switch up your fingerings and say, okay, well, what if I didn't hold that one down and I held this one down instead? That means that I wouldn't be ringing this out and I'm not going to ring this bottom string out at all. And I'm just going to hold this one down. And that means that down here I have another seven note away, a seven note away fifth of note 11, which is a G. So let's just go through all of these in this position now. By the way, you could hold this one down and this one, right? I could put these two fingers like that. Let's go through these and just count them a little bit faster this time. So if I get to the point, you can do this without the cheat sheet, right? So I'd say, okay, but I'll use the cheat sheet right now. So I say this is going to be the one relative position one of note 11 G, which is note 11 G, right? And then I'm going to go to this one and say, okay, this finger down one and back one is relative position four note away major third. I have to say it's major because that's the differentiating factor between major and minor of note 11, which is a G. And then I can do my math and say, okay, that means that 11 plus four is 15 minus 12 or drop it down to five minus 10 minus two, five minus two is three, which is three, which is a B, right? So I'm going to do a little math in my head to get down to the number of three and then say, well, three is a B and then I hope me to memorize that a three is a B. And then I can say, okay, if I go to the next one, this one is the D. So I'm going to say we're here now. This is relative position seven note away fifth, which I don't have to say major or minor because it wouldn't matter if the one was major or minor. Seven note away fifth of note 11, which is a G. And then I could say, well, note 11, 11 plus seven, not 11 plus five, 11 plus seven is 18, 18 minus 12 or drop it down to eight, which is minus 10, eight minus two is going to be, wait a second, eight minus two is going to be six. And that's going to be my D, which is a six and a six is a D. Right? So because I'm going to start to memorize the absolute notes. All right. And then I'm going to say, well, okay, now here's another one. Obviously another one right here when I'm looking at this note relative position relative position one of note 11 G is note 11 G. And then this one is going to be another third. Now this one isn't the third because I'm looking over here now. So now I'm holding this one down and I'm going to say, okay, now that's another seven note away. This is another seven note away fifth of note 11 G. I'm going to do my math and say, okay, 11 plus seven 18 or eight minus two minus 10 gives me eight minus two is six and six is a D. That's a D. And then I could do my last one here and this is going to be another G. So I could say, okay, that's going to be relative position one of note 11 G, which is note 11 G. So so you can look at all these different positions that we looked at last time. You can just hold down these two strings and you can see you have this one, this one and this one. And you can start to see that that's you know what that that makes the the chord because you have the one for the the one three five in there. And you can play just parts of the chord that way. And that also shows you why you can play this this bottom one like this or like this or like this or if you want to finger it this way. And you can you could see that you're still just playing all the notes that are in the key of G. Now we also said that because you're in the key of G and we're thinking about it as related to the C major that you could lift up any of your fingers here. Right. So I could do this.  Well now I'm revealing an A. So I'm going to say I'm going to reveal an A and say, okay, what am I actually doing there. This is going a little bit beyond what we're what I don't want to get too much into this. But if I reveal an A, if I can if I look at this chord that's been constructed from the key of C and a if I just repeated my my process. I started on a G. I took every other note to a B and then a D and then I keep on doing that I would get to the F and then I would get to the A. So the A we're saying is the ninth. Now this is where the numbering gets a little bit confusing here because you might say, well, wait a sec. I thought the one three five were numbering in relation to the G's relative scale. If I go to my G, there's the one three five. There is no nine because they don't there's only seven notes in the chord. So so this so how you get a nine, right? So because that a like if I looked at that if I looked at an A on its relative chord, it would be a two, not a nine. But but we usually but the so what we usually do is we skip every other note. So we don't we don't want to call it a two. We want to continue with our pattern of saying that we're starting here and kind of taking every other note. And so the next note, even though you end up with what would be like the two. If I looked at its relative scale and we call it a nine, right? But in any case, you can start to map that out. You could say, well, this is going to be a nine. Now notice that these notes up top are relative to this first are mapping out in relation to the C. So these intervals are not always going to be correct for everything else, right? Because I only mapped it out. We only mapped it out up top for the first chord that's constructed. And then we'll look at all the other chords and see how they differ, how the intervals will differ and try to try to recognize that. Now, when you look at the first three notes, the one, three, five, then the only note that's going to differ is the three. That's the differentiating factor between the majors, the one, four, five and the minors, the two, three and six. When you start to get to the sevens through the 13s, then it gets a little bit more weird because because like if I construct the seven note, then you'll note that the one and the four will have an interval that's going to be different than the five. So we'll start to have different differentiating intervals other than the one you need to know for sure, which is the major and minor when we start to add sevens, which is the next most important thing. And then you'll have other weird stuff that happens when you go to a nine and 11 and a 13. But you could start to map those out, same kind of concept in your mind and start to think, well, why is this seven different when I play it on the one, the four and the five? It's because the intervals are different. And so now I can determine which interval is different on which relative chord that's being constructed so that I can start to fit these things into what I'm playing. And so it basically makes sense. Also, when you move this up now, you can see these patterns move up. So when you when you see this pattern, if I was to move that up the fretboard and play it like on an A here, I'd have to switch my fingering, right? But then you could see, so I have my knees in a weird position, but you could see you have that or you could play it, you know, this way. And you could see that pattern one, three, five, one, three, five. So you'll start to see that pattern. And when you see that kind of A position, if I play like an A, this is a C right here. But I have these three notes, which people often think of as an A kind of position, although it's kind of in between between A and a G because an A bar chord would look like this. And the G, you know, is in between here. But you could say that's going to be if if I was to play it down here, this would be the one which in this case would be a C and the five is above it. So it would be a five, one major third. And you'll start to recognize those positions as we as you kind of move up the fretboard when we talk about that stuff later. Now also just remember that when you when you think about the entire scale, when we looked at the scale down here, the major scale, for example, we constructed the major scale around this from the key of C. And remember that we constructed this chord from the key of C. So this fits in the entire shape of the C major scale. And when you pick up any of these notes, what you're really doing is picking up the remaining notes within the C major scale. So you can do that. And that's that's when you're thinking of it more as the fifth, or you can switch your mind and say, I'm now going to be moving when I play this chord entirely, not only to the three notes that fit in the C major and the G major scale, but I'll switch entirely in my mind to the G major, which means it would be the one note. And so you'll still have the same notes in the chord, but you'll have different notes around it, right? And that's why you end up to having different notes when you're thinking about the 7, 9, 11, and 13th if you're thinking about it as the fifth note of a major scale as opposed to the one note of its own major scale. And so we'll so right now we're thinking about everything in terms of constructing it from the C major scale. Later on, hopefully we'll get to other scales and we'll get to a G scale because it's a great position to play on the guitar, very user-friendly to play on, so maybe we'll look at that later. But for right now, you just want to make sure that you have in your mind that as we map this out, these three notes will be in the G major and in the C major scale, but we're constructing it as though it's the fifth right now and therefore everything around it we're still seeing as being in the C major scale.


Good Morning. You can't talk about food supply without talking about population. When many of you were born, global population stood at about 5 billion people. By the middle part of this century, population will rise to about 9 billion people. And that's an astonishing increase. And the increase in population will surely impact food supply. It's really hard standing here in the middle of a corn field in rural Pennsylvania to think that there are hungry people everywhere. And there are hungry people just a few miles away from where I'm standing. The main issue with food supply that we'll discuss in this module is how climate change will impact it into the middle part of this century. And there's nowhere more devastating than sub-Saharan Africa, where drought and increased temperature will impact food supply and the ability of countries to feed people. The main thing that I really worry about with food supply is the ethical issue. The countries that are emitting the most CO2 into the atmosphere, the United States in particular, is a country that can feed its people if it tries to. Whereas those countries that are not emitting as much CO2 into the atmosphere are the countries that are going to be impacted by food supply issues in the most devastating way into the middle part of this century.