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- 09-21-2016, 01:01 PM #21
## Re: Functions

I am showing the ERROR in John's definition of the slope. The ERROR comes about because m 0.

To CORRECT the error, one sets m = 0. The point in ERROR moves to (x',f(x)) where it becomes (x,f(x)), and thus is on the line y=Ax + b which is the correct equation of the line. At that point, one replaces x' with x (at the left end of the green line).

The correct diagram for the equation of a line can be found in many sources on the internet. (Try "Math is Fun")

When m = 0, John's equation then becomes the standard prototype for the derivative (a line with slope = A):

Note that at this point, the function has not been specified; only that it runs through the two points (x,f(x)) and ((x+n),f(x+n))

Notice that the "run" (n) is parallel to the x axis, so can be represented as n = x for the LINE y = Ax + b

To allow for the slope of an arbitrary function f(x) (i.e., without specifying the function explicitly) one replaces n = x with the catchall term h, because now the slope rotates along the curve of the function (whatever it is) as the limit is reached. (There is a .gif of this in the "toast" link below below from "Math Is Fun")

See the first post in the thread "Derivatives" (and ignore the rest)

The equation then becomes the standard definition of the derivative:

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For physics, a potential energy is represented as a function V(x); if h is the quantum of action, the derivative represents the case where h -> 0 (at 0, A(x) = 0 if A(x) = V(x) = , t = 1 (one period of the DeBroglie wave function, and h is the action of a single photo-electron).

(When the quantum potential energy goes to 0 (the electron doesn't exist), classical physics (Newton's equations of force and , and Maxwell's equations apply (only masses and EM fields) - and time-space Relativity where x=ct is a "ruler creation process").

(This last result is a topic for another thread I'm working on. This is just to show its relevance to physics - Newton used his to calculate orbits.)- Share
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