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| 21 Sep 2017 07:56 AM |
You remember talking about xD in a precalculus course. It represents a distance along the x-axis; or, to put it another way, the difference between any two values of x. Well, dx means exactly the same thing, with one difference: it is a differential distance, which is a fancy way of saying very, very, very small. In technical terms, xD is what happens to Dx in the limit when Dx approaches zero.... Now, when you have a quantity whose value is virtually zero, there's not much you can do with it. 2+dx is pretty much, well, 2. Or to take another example, 2/dx blows up to infinity. Not much fun there, right?
But there are two circumstances under which terms involving dx can yield a finite number. One is when you divide two differentials; for instance, 2dx/dx=2, and dy/dx can be just about anything. Since the top and the bottom are both close to zero, the quotient can be some reasonable number. The other case is when you add up an almost infinite number of differentials: which is kind of like an almost infinite number of atoms, each of which has an almost zero size, adding up to a basketball. In both of these cases, differentials can wind up giving you a number greater than zero and less than infinity: an actually interesting number. As you may have guessed, those two cases describe the derivative and the integral, respectively. So let's talk a bit more about those, one at a time.
There
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| 21 Sep 2017 07:58 AM |
when i see ppls using "xd" word.
i cringe |
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hahh12
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| Joined: 02 Jun 2013 |
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mcque12
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| Joined: 18 Jan 2016 |
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| 21 Sep 2017 08:07 AM |
You remember talking about xD in a precalculus course. It represents a distance along the x-axis; or, to put it another way, the difference between any two values of x. Well, dx means exactly the same thing, with one difference: it is a differential distance, which is a fancy way of saying very, very, very small. In technical terms, xD is what happens to Dx in the limit when Dx approaches zero.... Now, when you have a quantity whose value is virtually zero, there's not much you can do with it. 2+dx is pretty much, well, 2. Or to take another example, 2/dx blows up to infinity. Not much fun there, right?
But there are two circumstances under which terms involving xD can yield a finite number. One is when you divide two differentials; for instance, 2dx/xd=2, and dy/dx can be just about anything. Since the top and the bottom are both close to zero, the quotient can be some reasonable number. The other case is when you add up an almost infinite number of differentials: which is kind of like an almost infinite number of atoms, each of which has an almost zero size, adding up to a basketball. In both of these cases, differentials can wind up giving you a number greater than zero and less than infinity: an actually interesting number. As you may have guessed, those two cases describe the derivative and the integral, respectively. So let's talk a bit more about those, one at a time.
To start off, remember how you define the slope of a line. You take any ### ###### on the line, and define the slope of the line as xD/Dx: the change in y divided by the change in x, or "the rise over the run." The slope physically represents how fast the graph is going up. The great thing about lines is, it doesn't matter where you pick your two poin####### slope will always be the same.
Now, when you want the slope of a curve, you might try to define it the same way. The problem is, the slope varies from point to point. In the curve below, I have labeled thre#####n####### you can see that if we calculated xD/Dx from A to B we would get a negative slope, from B to C would give us a positive slope, and from A to C might give us zero!
So, we're going to invoke a limit, to get "infinitely close" to A. We will talk about xD/Dx at points very close to A and see what happens to that ratio when Dx approaches 0. Dy also approaches 0, o##c##r####### the ratio of these two tiny numbers approaches the exact slope at that point.
If you followed that last example, you have gotten out of this paper exactly what I wanted you to get. A whole host of problems in math and physics follow that same approach: Divide the problem into differential amounts Solve the problem for each differential amount Integrate to sum up all the differential amounts, and get your answer Of course, there are a lot of things I haven't explained. The biggest one is why you sum up things by taking an antiderivative: maybe I'll write another paper on that some day. But once you do a few problems like this, you will find that a whole world of previously insoluble problems are now within your reach. |
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