Guided functionsBelow is a simplified, conceptual graph of a damped harmonic oscillator (e.g. attach a spring to a wall, attach a ball to the end of the spring, mark that position as 0, tug on the spring, and let it go, measuring the distance (y-axis) from the initial position over time (x-axis)). Maybe a better example would be a pendulum? It's ultimately the same math. In any case, notice that as time increases, the frequency of the oscillations stays the same, but the distance the thing oscillates continues to die down until eventually it is no longer moving. But it does not die down randomly! No, there are invisible guides that predictably decrease the height of the oscillations according to a simple exponential decay formula. Thus one might think of this graph as a guided sine wave: the exponential function guides it to 0, even though the exponential function is not as visible as the blue sinusoidal function. Understand?
Consider the adage: "the rich are getting richer and the poor are getting poorer." Many people believe this! Many people with refrigerators, television, a long life-expectancy where they can actually retire instead of working until death, educated children... in short, many luxuries their very recent impoverished ancestors lived most or all of their lives without! If the poor truly were getting poorer, we should expect to see a systematic decline from wherever one draws the line of "poor". I invite any reader to find such a decline.
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