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May 14th, 2016, 03:25 AM   #1
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What's the point of differentiation and antidifferentiation?

Are there any actual applications for shuffling numbers around like that? It's just subtracting/adding to exponents and the multiplying/dividing the whole thing by the same amount. What is the purpose?
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May 14th, 2016, 05:19 AM   #2
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Math Focus: tetration
It is fun!
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May 14th, 2016, 05:20 AM   #3
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It is fun!
Manus please...
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May 14th, 2016, 06:01 AM   #4
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So? Can't I have a little fun!
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May 14th, 2016, 06:20 AM   #5
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So? Can't I have a little fun!
It was quite funny yes...
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May 14th, 2016, 03:19 PM   #6
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Perhaps looking at the reason for the development of Calculus would help. A major problem at the time of Newton and Leibniz was the motion of the planets. Newton (as well as Leibniz) believe that some force, centered toward the sun, getting weaker as the distance from the sun increased, was the cause.

The problem was this- if that force was a function of the distance from the sun, then at any given instant, one could (theoretically) measure that distance and find the force at that instant. But "force= mass times acceleration". If we could find the force "at an instant" and the mass remains constant, then we could find the acceleration at that instant.

But what does that even mean? Acceleration is "change in speed over a given change in time". And even further, speed is "change in distance over a given change in time". But at an instant, there is NO "change in time"! In order to over come that Newton introduced the notion of "infinitesimal" changes in time so that the speed is the "infinitesimal change in distance over an infinitesimal change in time" and acceleration is the "infinitesimal change in speed over an infinitesmal change in time. Later, the idea of "infinitesimal change" was replaced by the more mathematically rigorous "limit" concept leading to the "derivative of today".

And, or course, any time you have a new "operation" you want to be able to reverse it, leading to the "anti-derivative" as the reverse of the derivative.

(The "integral" arose, originally and earlier, from a completely different problem, that of finding areas of unusually shaped regions, developed largely by Fermat, although it roots go back to Archimedes. One reason why Newton and Leibniz are recognized as the "founders of Calculus" is their recognition that the "integral" was the same as the anti-derivative.)
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May 14th, 2016, 06:24 PM   #7
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Perhaps looking at the reason for the development of Calculus would help. A major problem at the time of Newton and Leibniz was the motion of the planets. Newton (as well as Leibniz) believe that some force, centered toward the sun, getting weaker as the distance from the sun increased, was the cause.

The problem was this- if that force was a function of the distance from the sun, then at any given instant, one could (theoretically) measure that distance and find the force at that instant. But "force= mass times acceleration". If we could find the force "at an instant" and the mass remains constant, then we could find the acceleration at that instant.

But what does that even mean? Acceleration is "change in speed over a given change in time". And even further, speed is "change in distance over a given change in time". But at an instant, there is NO "change in time"! In order to over come that Newton introduced the notion of "infinitesimal" changes in time so that the speed is the "infinitesimal change in distance over an infinitesimal change in time" and acceleration is the "infinitesimal change in speed over an infinitesmal change in time. Later, the idea of "infinitesimal change" was replaced by the more mathematically rigorous "limit" concept leading to the "derivative of today".

And, or course, any time you have a new "operation" you want to be able to reverse it, leading to the "anti-derivative" as the reverse of the derivative.

(The "integral" arose, originally and earlier, from a completely different problem, that of finding areas of unusually shaped regions, developed largely by Fermat, although it roots go back to Archimedes. One reason why Newton and Leibniz are recognized as the "founders of Calculus" is their recognition that the "integral" was the same as the anti-derivative.)
Something I've never understood about theoretical math concepts is that they're unprovable. How can anyone determine that the formulas for concepts like speed and acceleration and infinitesimal limits actually reflect what happens in the real world?

For instance, a square root is just a symbol with a number on paper. It doesn't exist in real life. How can people be sure that the idea represented by a square root is valid? Someone can count 2 + 2 = 4 by counting and see that the addition operator is valid, but that can't be done with a square root. There are machines that calculate square roots, but people program those machines to say anything, so it doesn't count.
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May 14th, 2016, 06:51 PM   #8
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Originally Posted by Incomprehensible View Post
How can anyone determine that the formulas for concepts like speed and acceleration and infinitesimal limits actually reflect what happens in the real world?
This isn't the right answer, but just stop and think for a second about all modern science/mathematics and where it has gotten us physically. Seriously, just think about it. They are provable because they have been shown to work in reality numerous times.

Did you ever do the simple pendulum experiment in school where you need to measure the period experimentally, and then compare it theoretically? You'd get a string of length L (and of negligible mass), and attach a mass to the end, then you'd cause the pendulum to swing and record how long a period is. Then you'd compare it with the results from the following expression,

$\displaystyle T = 2\pi\sqrt{\frac{L}{g}}$

What do you find? You find that your experimental results are very comparable with the theoretical, is it by chance?
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May 14th, 2016, 07:41 PM   #9
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Quote:
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This isn't the right answer, but just stop and think for a second about all modern science/mathematics and where it has gotten us physically. Seriously, just think about it. They are provable because they have been shown to work in reality numerous times.

Did you ever do the simple pendulum experiment in school where you need to measure the period experimentally, and then compare it theoretically? You'd get a string of length L (and of negligible mass), and attach a mass to the end, then you'd cause the pendulum to swing and record how long a period is. Then you'd compare it with the results from the following expression,

$\displaystyle T = 2\pi\sqrt{\frac{L}{g}}$

What do you find? You find that your experimental results are very comparable with the theoretical, is it by chance?
That looks interesting. I might try that. What does g stand for?

Is this a high school experiment? I was barely able to do anything science or math related in school. My teachers almost always failed to accommodate for my hearing and visual disabilities, and their usual solution was to just exempt me from assignments involving group work or labs. That kind of experiment seems like something I would've been exempted from.
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May 14th, 2016, 08:02 PM   #10
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g stands for the acceleration due to gravity (about 9.81m/s^2). This value is determined through experimentation also (but is a little more complicated).

Sure, you could try the pendulum experiment at home, it's fairly simple. But remember the equation I provided in my previous post is only valid for small oscillations.
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