/sci/ - Science & Math

Consider now a sequence defined not at the points [math] 1,2,3,..., [/math] but on a very fine grid of points spaced [math] \epsilon [/math] apart, so that the n-th point is at position [math] n\epsilon [/math] All the ideas of the previous section transfer to this situation, since all you have to do is rescale everything to make the unit of length [math] \epsilon [/math] and the points lie on top of the integers.
In this case, the sequence [math] A(n) [/math] turns into a function [math] A(x) [/math] defined on all x's on the lattice. The derived sequence is [eqn] \Delta_\epsilon A = A(x+\epsilon) - A(x) [/eqn] And you can see that as [math] \epsilon [/math] goes to zero, it goes to zero. For a typical function, like multiplication or raising 2 to a power, we can ask, how does it go to zero? [eqn] \Delta x^2 = (x+\epsilon)^2 - x^2 = 2x \epsilon + \epsilon^2 [/eqn] This is just the rescaled version of the first difference of [math] n^2 [/math] (you can work it out directly, and it is good if you do). The lesson is that the thing tends to vanish linearly, meaning that if [math] \epsilon [/math] is small, and it is made twice as small, your derived sequence is generally made about twice as small.
So you can take out this scaling and define the derivative of f [eqn] {df\over dx} = {\Delta f \over \epsilon} = {\Delta f\over \Delta x} [/eqn] Where the idea here is that you define the derivative for each [math] \epsilon [/math], and let [math] \epsilon [/math] become so miniscule that the derivative stops changing. This never happens for any finite nonzero positive value of [math] \epsilon [/math], so it is formally useful to introduce the concept of an infinitesimal [math] \epsilon [/math]

>>11447888
Seven hundred ten.
Seven hundred eleven.
Seven hundred twelve.