chemistry_geek
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So I was watching Steven Spielberg's "Taken" mini-series when one of the project leaders says the Aliens are using Fibonacci Numbers for something I don't remember now. Being the nerd that I am, I look up the Finonacci Sequence in my very own personal "The VNR Concise Encyclopedia of Mathematics, 2nd Ed." and found the sequence definition on page 381. The Fibonacci sequence is defined as a(1) = 0, a(2) = 1, and for n ≥ 3: a(n) = a(n-1) + a(n-2). I started M$ Excel and generated the Fibonacci sequence from index n=1 to n=1477. M$ Excel tanks out at 1.3069892237634E+308 for n = 1477. When looking at the first 30 numbers, the sequence kinda/sorta looks like e^x. The numerical derivative approximations (*) for the first and second derivatives of the data provided some interesting results. The first derivative approximation kinda/sorta resembles the sequence, however, the 2nd derivative approximation IS the sequence but 3 index numbers later. From freshman calculus, the derivative of e^x = e^x, i.e, d(e^x)/dx = e^x. Attached is the M$ Excell spreadsheet for your review. If I recall correctly, e^x is a transcendental function; I think it is fascinating that taking the numerical second derivative of a sequence of integers results in the exact same sequence of numbers but three points later in the index. Calculus was a long time ago. I wonder if this is a property of Fibonacci numbers.
If you're wondering how numerical derivatives work and are generated/derived, go to college, take EVERY Calculus course offered, then take Differential Equations, Linear Algebra, and a finally a full year course in Numerical Analysis.
* 5-point centered 1st-order 1st derivative taken from "Numerical Analysis" 6th Ed. By Richard L. Burden and J. Douglas Faires. Chapter 4.1 "Numerical Differentiation", page 173.
* 3-point centered 1st-order 2nd derivative taken from "An Introduction to Numerical Analysis" 2nd Ed. By Kendall E. Atkinson. Chapter 5.7 "Numerical Differentiation", pp. 315-320.
If you're wondering how numerical derivatives work and are generated/derived, go to college, take EVERY Calculus course offered, then take Differential Equations, Linear Algebra, and a finally a full year course in Numerical Analysis.
* 5-point centered 1st-order 1st derivative taken from "Numerical Analysis" 6th Ed. By Richard L. Burden and J. Douglas Faires. Chapter 4.1 "Numerical Differentiation", page 173.
* 3-point centered 1st-order 2nd derivative taken from "An Introduction to Numerical Analysis" 2nd Ed. By Kendall E. Atkinson. Chapter 5.7 "Numerical Differentiation", pp. 315-320.
Christmas break begins at the end of the day tomorrow, except that my mind has already shut down...oh well - math is cool.