Saturday, June 27, 2009

Exponent is a center in power functions

Anybody learning calculus know the power functions: xⁿ where positive, rational and negative n results in parabolic, square and hyperbolic functions. The function passes from square to parabola at n = 1 (this is a straight line y=x) and into hyperbole at n=0. There is one power function -- the exponent. One should obviously wonder how it is related with power functions especially studying
integration, it seems to break the rule: (xⁿ)'=n*(xⁿ-1). This rule is true for all n, but an exception exists for reverse when for n=0. That is, (1 = x°)' = 0*(x-1) = 0 but integration of x-1 is not equal to 1; rather, integral (x-1) = ln (x).

I have found that exponent is a solution of dy/dx = y, while parabolic growth is defined by dy/dx=y1/2 and dy/dx=y2 is a hyperbolic grows. So, exponent is in a midpoint spliting two power realms. Formally, dy/dx = yⁿ for different n has the following solution

n: -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5
y: (3x)1/3 (2.5x)1/2.5 (2x)1/2 (1.5x)1/1.5 x (x/2)2 (1/-0.5x)-2 1/-x (-1/1.5x)3/2

Notably, 0° is indefinite by this the general formula. The solution in range n=0..1 is a parabola which becomes more and more steep. Because of the factor, it starts along x axis then rapidly goes high overcoming slower parabolas. At n=1, it is absolutely vertical in the far right x=∞. It does not seem to converge to exponent. Yet, it is known as ex.

If bank pays a complex interest proportional to the deposit, the growth is exponential. If it pays less, say the square root of the deposit, the parabolic growth results. If more is payed, say square of deposit, hyperbolic growth results.

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