1. When one is doing certain numerical computations, an approximate solution may be computed by any of several heuristic methods, then refined to a final value. By using the starting point of a first approximation of the answer, one can write an algorithm that converges more quickly to the correct result.
2. In jargon, a preface to any comment that indicates that the comment is only approximately true. The remark "To a first approximation, I feel good" might indicate that deeper questioning would reveal that not all is perfect (e.g. a nagging cough still remains after an illness).
[Jargon File]