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Somehow this seems important
Jul. 10th, 2007 08:56 am(Sin)' = Cos
(Cos)' = -Sin
When Sin = 1, then (Sin)' = Cos = 0
When Sin = 0, then (Sin)' = Cos = 1 or -1
When Sin = -1, then (Sin)' = Cos = 0
What does this mean?
In a sine wave, the derivative (meaning "rate of change") of Sin is greatest when it is closest to equilibrium.
The closer to the middle ground that a wave is, the greatest its rate of change.
This occurs in many places in nature, but my favorite is the Sine wave that describes the length of the day and night over the course of the year. The length of the day and that of night changes every day. The days and nights are of equal length on the equinoxes and at their longest/shortest at the solstices.
However the change in the length of the day and night is greatest when they are closest to equal. It is at the Fall equinox that the days shorten at the greatest rate, and in the Spring that they grow at the greatest rate - the date before the Spring solistice, the day is about four minutes shorter than on the solstice, and again about four minutes longer on the day after.
Conversely, at the summer soltstice, when the day is longest, there is very little change in the length of the day - the date before the solstice, the sun is up perhaps thirty seconds less than on the solstice, and after the solstice, the sun is up about half a minute less.
I'm sure there's some lesson here.
In any case, Calculus is awesome.
(Cos)' = -Sin
When Sin = 1, then (Sin)' = Cos = 0
When Sin = 0, then (Sin)' = Cos = 1 or -1
When Sin = -1, then (Sin)' = Cos = 0
What does this mean?
In a sine wave, the derivative (meaning "rate of change") of Sin is greatest when it is closest to equilibrium.
The closer to the middle ground that a wave is, the greatest its rate of change.
This occurs in many places in nature, but my favorite is the Sine wave that describes the length of the day and night over the course of the year. The length of the day and that of night changes every day. The days and nights are of equal length on the equinoxes and at their longest/shortest at the solstices.
However the change in the length of the day and night is greatest when they are closest to equal. It is at the Fall equinox that the days shorten at the greatest rate, and in the Spring that they grow at the greatest rate - the date before the Spring solistice, the day is about four minutes shorter than on the solstice, and again about four minutes longer on the day after.
Conversely, at the summer soltstice, when the day is longest, there is very little change in the length of the day - the date before the solstice, the sun is up perhaps thirty seconds less than on the solstice, and after the solstice, the sun is up about half a minute less.
I'm sure there's some lesson here.
In any case, Calculus is awesome.
