Friends, ever since humans have started observing this world, they have started observing this universe, some things have attracted them towards themselves, such as our Sun, Moon, Stars, one thing was common in all these things that they change their position with time. Now they used to see the birds flying in the sky, they also change their position and man always wanted to make such things which change their position, well today we have achieved this technology. We have already done this, we cover huge distances in a very short time by using airplanes, we send satellites into space, we are thinking of going to Mars, but why is it taking so much time for us to achieve this technology? The answer given is quite simple. We did not have the science to make these things. We did not have the mathematics to make them. Equations did not exist. If you look at this universe, you will see everything moving. You also read this in the smaller classes. It would be that in this world there is no object at perfect rest, all things are in motion, so to create such a technology that changes its position with time and moves, for that we will need a science that can give us a description of the change. We will need Mathematics of Change. We will need a tool that will give us information about change and you all know that mathematical tool is Calculus. Now this tool was first used in Physics because in Physics we mostly study changes. Let us study this Mathematics of Change today from a historical perspective and why humans needed it. So first of all let me take you to the time of Greek Civilization. They told us a lot about geometry. Told us about shapes but Aristotle believed that we cannot model our universe with these geometric shapes like circle, triangle, rectangle because if the universe was made of these geometric shapes then it would have been a quite ideal universe and an abstract universe in it. There are no irregular shapes, but you know this very well that we cannot describe the events taking place in the universe with the help of any geometric shape, which means at that time geometry was not able to describe the mathematics of change. You could not describe the events taking place in the universe, but as time passed, it became clear that you could model the universe with mathematical tools. Now let me take you to the 17th century. Here I will meet you in a I get it from a French philosopher, his name is René de Cards. Actually, Cards always thought about how we can locate the position of an object. He thought of an idea. He made two lines, a horizontal and a vertical line and the intersection of these two lines. He considered it as his reference frame, he considered it as the origin whose coordinates were 0 0, from this reference you can measure the position of any point, so suppose you have a point which is moving with time, then consider its initial position. Let's say 5 20, after some time its position becomes 10 20 then 15 20 then 20 20 meaning you have stored the position of that point in the form of The plane, which was a problem facing us for many 100 years, solved this problem between geometry and algebra. Geometry and algebra, which we used to study separately, the Cartesian plane brought them together. Algebra gives us equations and geometry gives us shapes. It gives shape, so the function of Cartesian plane was that it actually showed us the shapes of algebra. It showed us the shape of algebra like y = x is an algebraic equation but if you plot it in a Cartesian plane then actually it is a straight line. It means that now you have a tool through which you can plot any equation, whether it is quadratic, straight line equation or Paulino Miele equation, in the Cartesian plane and you can see it like now. A few days ago, in my Algebra video, I told you that earlier we used to write Algebra in the form of word problems like Euclid's. Look at this definition which describes the circle, it is in word form but when we wrote the circle. When we made the equation algebraic and plotted it on the Cartesian plane, we came to know that the shape of the circle is something like this, meaning from here you can see that mathematics got a lot of benefit from the introduction of Cartesian plane, otherwise till now we have not known mathematics. If you keep reading it only in the form of words, keep reading it in the form of text, then now let me take you to 1646 and you meet a German philosophy. Actually, the dream of Lebanon was that he could somehow measure the changes. He created a basic calculator. He also invented a computer which could do some basic operations like addition and subtraction but he wanted to create a tool by which he could know the future and past of anything through mathematics and in fact the tool he gave us was The name was integration. Well, I have already talked about integration in full detail in my video, the link of which you will find in the description box below. So in integration, basically you calculate the area under any curve. You calculate the area under the curve. Now because you do not have any mathematical formula to calculate the area of a curve, then what you do is you divide that curve into small rectangular strips and you know how to calculate the area of the rectangle and then You add the areas of all the rectangles together, meaning if the curve is changing then the area under the curve will also change, so from this we can calculate the change, measure the change, so like look at this image, first here. Large rectangles were used in the image but then these rectangles were cut into smaller pieces and finally we used very small rectangles, so now you can see here that the accuracy in the first image is less because this area is under The curve is being missed but when we use smaller rectangles, we get a smoother curve and the accuracy increases and the errors start decreasing. Look at this last image, you can get a very good approximation of the area under the curve. You can see and we use integral to measure the future and past, so if you do integral then you can predict that if this thing was like this in the past then how will this thing be in the future and see. If we go to physics, this is what we do, by looking at the past record of an object, we try to predict its future and whenever we do this, we definitely come across integration because integration is the only tool which helps in predicting the past and future of an object. If you measure the future, then in this way you can measure the change, but along with this, now let me introduce you to Isaac Newton. Actually, he also wanted to study the change and wanted to define the moving object. But he did not have any mathematics that could represent change. In fact, Newton used to look at moving objects like the trajectories of a comet or an apple falling from a tree. So he wanted to model these things in mathematics but he He found that such mathematics does not exist, so he thought of inventing a mathematical tool that would define the change at any point, like if an apple is falling from a tree, then at a given point of time its change would be Integration cannot tell us what the speed will be. Another tool can tell you this, which we call derivative. Actually, derivative tells the slope at any given point. Derivative gives us the slope at an instant new point by calculating it. dydekop.org Meaning, with these tools, we can model all the things that are changing and Newton has described this thing very well in his book, The Method of Flux, meaning Newton had invented Calculus. He also shared calculus with the Lebanese. He also wrote some letters to the Lebanese in which he described the calculus. Now here was another thing that the Lebanese were also working on the study of change. They were also working on integrals. And Newton had also published this differentially. He told that to measure the change of a moving object, we need the acceleration of that object which is nothing but the rate of change of velocity. But now there is a debate here. The thing was that Newton used to say that he had first discovered the derivative and created calculus and he had also published it before Leibniz, but Leibniz used to say that he had created the integral, that means he had also discovered calculus and that Both of them wanted that they should get the rights of Calculus. Now this case reached the Royal Society at that time and Bernoulli was sitting in the Royal Society. Bernoulli sent a problem to all the famous scientists which could be solved only by Calculus, so the question was this. That if you roll a ball from a high object and you have to take that ball from point A to point B, then which trajectory will it follow, which equation will it follow so that it reaches the bottom first? Will it be a straight line, will it be a curved path, and if there is a curved path, then what type of curved path will it be or will it be a high degree Paulino mean? Actually, you can tell this only from calculus and when Newton read that letter, Newton solved this problem and replied back to Bernoulli. Newton mathematically told that the ball will reach down first through cycloids. He also told about cycloids and when this thing was checked, it was found that Newton was right and then Newton Was given the title of Inventor of Calculus and today we know very well that Calculus is used in many fields,
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