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</html>";s:4:"text";s:8556:"And I can say that Let's say, Well, l equals, Okay, squared. However, one is not included. And then if you take the integral of dysfunction, then you get the area under the curve and intuitively thinking what would be the answer? Now to do that, first, I need to teach you how to go from one reference frame toe, another reference frame, and that will do that in the next video. But then you wouldn't have a circle. One would equal eight kilograms per cubic meter. Still the x i with respect to delta t that time I And then I multiplied by delta t and the same thing here built a why I at at a certain time And then this is my slope and I multiply Bite off the tee plus Delta Z By that time I that the tea, remember, this is my slope How my Z changed with respected time. So if they're equal to each other than I must have one here and well, I must have zero here then These two terms become zero and I will have Z equals z prime, right? So every time I use this operator, I use a partial derivative. You can see the X ray and why three and extra and why two. So I have a straight line function here. Now, if I multiplied this radius by end, I will also have to put in here in order to make algebra work and satisfy the quality condition. In this example, however, the angle between them increases. Now, of course, it never represents the physical situation 100% because you can have a situation like this . And then from the Origen, you draw the some. And as you go up here and move to this side, right, that you would have something like this. There is material inside. 16. Angry. How did look like See here, whatever. However, you can have as many average changes as you want because you can take any interval you want . It all depends on your problem that you have to solve. Fair enough. Different names for the axis. And at every second, we're gonna record the area that we have covered since from the beginning. Okay, I have written that the relatives here to make some more space for myself. So now you not only you have the the X component, you also have the negative the y component because the force has ah and this the force vector. And then here I would have had 64 miners miners for Cuba, 64. This course will give you strong intuition and understanding in Calculus, and you will be trained to apply it in real life. All right, so again, said the beginning, you wrote it very slowly because because the frequencies are very low. But then three dimensional. Now, what was my DF dx No, my d if d X was one. Okay, So this is when we went from position zero to position to position two in position three, Position three in position for and then from position for till position zero. So what? So let's say that with my ex can have two valleys, Onley one and two Onley two values my Why can also have only two values one and two and my Z can also have only two valid one and two. Graphically, However, I'm gonna encourage you now to try to think in higher dimensions and not try to visualize it. So in the end, you will have minus t squared plus two t and plus 10. So my first ultra interval got rid of my d My second outrunning to grow got rid of my wife . And this is also good exercise now for for all the the relative rules, call sign Setem to the power of minors one. So course I feet down minus pi over two is sign feta. Is one second. Okay, in this case, nine meters, because here the area has a meaning off displacement. All right, so this is the amount of my differential volumes approaches infinity. It's minus 10 times one. Well, if you approach 90 degrees, then ex become zero. And of course, what is also allowed is if you do something like this that you have one great element like this x one and then the other great element would be X two like this. So what do we have here? And that would be 27.5 seconds. And only when these two are equal Thies to distance air equal Onley. And then I put some kind of point inside Xie. So I applied the integral on this side and on this side. So now it's just the opposite thing. It's just how much the speed changes with respect to time. And starting from the next lecture, we're gonna start making our way into the world off. So here the change itself changes. So now we want to find this kind of relationship. Well, it means just this that if this piston moves one meter to the right, then starting from this position one meter to the right, then this piston moves up by 0.95 meters or 95 centimeters. So I'm just following this formula plus one four times to now. You see, one cycle. How would the function look like for this graph? So here is well, you're gonna choose your point on the surface, and then you're gonna have some kind of plane. If I take ex prime and I multiplied by course I'm Alfa. And you you take the other one j So you take plus J And now you Only now you only consider Let's say when you took I then you could cancel out this and this right? But okay, I can also Oh, I also need to multiply this because this is my slope. So I have these different levels here. That means that if your function is in the positive region, if your range so this why now this This is your output, your range. And if I take the partial derivative off my output with respect to one of the inputs that I will know how my output changes with respect to that input by keeping all other inputs constant. It goes to when time equals two, they will be two times 2000. And this square function or the parabolic function would be Why equals a X squared plus B and be here would be 45 and a would be minus five. I just observed his black line and the green car and see what's happening. So this would be your volume, and your constraint is that it had to be 1000 cubes centimeters exactly. And then that differentiated input you're different. It can be as many meters as you choose Now the circle also has and diameter, you know, And in addition to that, we know that the diameter consists of two Radi I So if I put another radius here, then two times radius would give me a diameter. But if x goes to infinity as X approaches infinity, the further and further I go, the closer and closer why approaches to zero. And then are you final Would be X final minus four, which would be eight minus four, which would be four. And Delta X. Then I still need to calculate the total mass off my cube. You will get 16 minus four times two is eight times two 16 16 minus 16. But now what we want, we want to calculate the length off this entire arc and again we can use Integral is to do that. So look, your answer here would be zero. All right, we treat it is a constant. So right now, I entered the danger zone here. All right, so I'm just gonna say that its initial you need initial and hear what you will get. We project this red one here and we project also this red one here. So you can imagine in your head that okay, this way will isolate twice as fast. - However , since we also have this term here 200 times time. Uh, squirt.  Why? So you could say that it's like to buy radiance uh, one cycle. But now the take away from this lecture is that when you work with partial derivatives, you can also take your output and you can differentiate it with one input and then you can go one level down and differentiated with another input. If we're close to zero, in fact, then other terms will dominate. And also the lost director was in this direction. Then we would have something like this d theta over de why over our And by the way, in this example, I think I made a mistake that I didn't know did it with Delta y. And then you take the directive of the inner of the inner function which would be one because two would be zero and X would be one. In many cases, it's even better if you abolish trying to visualize things and tried to think in terms of higher dimensions without visualizing it, because a lot of applied math in engineering is multi vault variable, but it has many dimensions, many more than three dimensions. When time equals zero, we have two airplanes next to each other when time equals zero hours. So when I write number one here, it really means this DF over DX, the first order derivative. So in conclusion you cannot have X. Now if n equals two, then it's a parabolic function because tense, a time squared, plus B cute and equals three. And when you take the into grow off your DX DT, you can't retrieve that information anymore, which means that when you go from the X et to back toe X, you have to add some kind off arbitrary constants, some kind of see Okay, you have to do that. 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