Quote:
Taylor Series expansion of y = sin(x + cos(x)) at x = Pi/2 (up to the x7 term, say)
While it's easy to get the result with a calculator that is able to perform symbolic differentiation I'm trying to show that this isn't needed. The arrays contain the first coefficients of the Taylor Series at x = Pi/2.
x = [Pi/2, 1, 0, 0, 0, 0, 0, 0]
cos x = [0, -1, 0, 1/6, 0, -1/120, 0, 1/5040]
Thus:
u = x + cos x = [Pi/2, 0, 0, 1/6, 0, -1/120, 0, 1/5040]
sin u = [1, 0, -1/2, 0, 1/24, 0, -1/720, 0]
v = u - Pi/2 = [0, 0, 0, 1/6, 0, -1/120, 0, 1/5040]
v2 = [0, 0, 0, 0, 0, 0, 1/36, 0, -1/360]
We only have to take the 2nd coefficient (-1/2) of sin u into consideration, since v4 is O(x12) already.
So we end up with:
sin(x + cos x) = 1 - v2/2 + ...
= 1 - (x - Pi/2)6/72 + (x - Pi/2)8/720 + ...
All we need are operations that interpret an array correctly as a Taylor Series. This shouldn't be too difficult to implement. Though I have no idea whether this would be possible within the WP-34s project, it's far from symbolic differentiation. Maybe as an idea for a follow-up project then?
Kind regards
Thomas
Edited: 21 Jan 2013, 2:36 a.m.