Taylor's Series is based on the fact that, if a function is continuous and differentiable, The value of that function a small distance, h, from point x will be equal to the value of the function at x, plus a "fudge factor," or really a series of fudge factors. This is stuff you should know, because it is used extensively in math, physics, and geophysics.
Taylor's series can be written in several forms. The first is:
The Incremental Form assumes one is evaluating the function f(h) at f(h+x).
Usually, in problems in applied physics/math, x is a small number, so only the first two terms are kept:
This amounts to saying that the value of the function at h+x, namely, f(h+x) will be equal to f(h), plus a term that represents the slope of the function at h, times the distance, x:
The MacLaurin series is a special case of the Taylor series where h, above, is zero:
In Homework Set 4, problem 5, we have:
Since f, the flattening is small (0.003) compared to 1, we essentially have the term:
Using Taylor's Series (note that x in the Taylor Series is our -x in the equation above), we get:
