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Monday, October 15, 2018

Base of differential fixed points

Consider the definition of the difference quotient in terms of the quotient produced by an interval of length h. \frac{f(x+h)-f(x)}{h} We want to study the fixed points of this difference quotient, supposing that h is fixed. In order to do so we can equate h with f(x). To simplify this equation we can multiply both sides by h and then add f(x) to both sides. \frac{f(x+h)-f(x)}{h}=f(x) f(x+h)-f(x)=hf(x) f(x+h)=(h+1)f(x) At this point we can raise h+1 to the power of 1/h to get the base of the fixed point of difference quotient with fixed length h. b = (h+1)^{\frac{1}{h}} This formula works in general to compute the base of the fixed point of a difference quotient of any given length. We tend to want to reduce our consideration to the case of unit fractions 1/n in which case we get the following familiar formula for the base of the differential of the fixed point of a given unit fraction length: {(\frac{n+1}{n})}^n Actual values of this sequence of rational numbers are provided below. The first case 2 is the fixed point of the difference quotients of length 2 or the difference quotients of length 1, the next number 9/4 is the base of the difference quotients of length 1/2, and the next is the base of the difference quotients of length 1/3, and so on. 2, \frac{9}{4}, \frac{64}{27}, \frac{625}{256}, \frac{7776}{3125} ... As a result, in the case of finite differences it is possible to use 2^x as a fixed point. The consecutive differences of the 2^x sequence are the values of 2^x itself. But this 2^x value does not suffice when taking difference quotients of smaller lengths. 1,2,4,8,16,32,64,128,256,... In order to get the fixed point of the difference quotients of smaller lengths it is necessary to use the formula {(\frac{n+1}{n})}^n we derived earlier to get a different base. If we then continue this process to infinity we get the base of the fixed point of the derivative which is e which leads to the function e^x which is equal to itself under differentiation. This demonstrates that the limit {(\frac{n+1}{n})}^n is not simply a random means of generating e it is actually a formula for the different fixed points of the difference quotient. So actually the identity of e is as the fixed point of the derivative and the limit definition is just computing different fixed points of the difference quotient.

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