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Flowreckoning is a kind of scorelore. Flowreckoning anent how things shift.

Unknownworthings and WorthlinksEdit

An unknownworthing is a worth that shifts through scores. For byspel, x is a widespread unknownworthing. This worth, x, can be any score. However, if we have another unknownworthing, such as y, we can link the unknownworthings with a worthlink. For byspel, this is a worthlink:

$ y = x $

This worthlink says that for each worth of y, x is the same worth. If we have a y worth of 2, the x worth is 2 also. Here is another worthlink:

$ y = 2x $

This worthlink says that for each worth of y, x is two times that worth. If we have a y worth of 2, the x worth is 4, since 2 * 2 = 4.

Worthdrawings and WorthspeedEdit

Once we have a worthlink, we can draw a seenly worthdrawing of that worthlink, handling each worth of one unknownworthing with the worth of the other unknownworthing.

On a worthdrawing, two threads at a right nook stand for the worths of x and y. As the unknownworthing y shifts from one score to the next, a nib is marked at the worth of x, getting these worths from the worthlink. For byspel, here is the worthdrawing for the worthlink $ y = x $:

Worthlink y=x

Worthlink $ y = x $

The worthdrawing makes a straight thread since each worth of the unknownworthing y is the same as the unknownworthing x. This means that the worthdrawing is shifting from one score to the next at an unshifting speed. The worthspeed is the speed at which one unknownworthing's worth is shifting with heed to the other. The worthspeed of the worthlink $ y = x $ is 1, since the unknownworthing x is shifting at the same speed as the unknownworthing y.

Straightforwardly, the worthspeed is the shift in the unknownworthing y over the shift in the unknownworthing x. We can then work with the following worthlink when finding worthspeed: Δy / Δx.

See the worthdrawing $ y = 1/2x $:

Worthlink y = 1over2x

Worthlink $ y = 1/2x $

The worthspeed of this worthdrawing is again Δy / Δx, so the worthspeed is $ 1/2 $.

Shifting WorthspeedsEdit

In the byspels above, the worthspeed has been unshifting. However, let us see the worthdrawing of a worthlink with a strength, $ y = x^2 $:

Worthlink y=x^2

Worthlink $ y = x^2 $

Here, the worthspeed is shifting, it is not like the above, where the worthspeed is always 1, or always 2. We can find the speedshift of this worthdrawing, or how fast its worthspeed is shifting.

The worthspeed at any nib on the worthdrawing is the worthspeed of a grazing thread, a thread that hits the worthdrawing at only one nib. To find the worthspeed of this grazing thread, we will again use our worthlink, Δy / Δx. However, this worthlink behoovely needs two nibs of x and y in order to be handy; this worthlink would give us the worthspeed of a crossing thread, a threat that hits the worthdrawing at two nibs. Thus, how could we find the worthspeed at just one nib?

As the Δx between two nibs becomes smaller and smaller, the worthspeed of the crossing thread becomes more and more aright to the worthspeed of a grazing thread. This means the Δx must be naught, 0, for the worthspeed of the grazing thread to be reckoned. We must then take the bound as Δx atsteps naught.

Worthlink y=x^2 grazing thread

Worthlink $ y = x^2 $; the dashed thread represents the grazing thread

In this bound, however, Δx will be h. We will be finding the bound as h atsteps naught of the crossing line, to find the worthspeed of the grazing thread. We will abide by the same worthlink to find worthspeed: Δy / h. If $ y = x^2 $ is our worthspeed, the shift in y would be $ (x+h)^2 - x^2 $. Putting this together, the bound is:

:$ \lim_{h \to 0}\frac{(x+h)^2 - x^2}{h} $

After working the bound, we would find that this bound actually evens to $ y = 2x $. This means that at any nib on the worthlink $ y = x^2 $, the worthspeed is $ y = 2x $. This second worthlink that tells us the worthspeed of any nib on a first worthlink is called the speedshift.

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