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One should be careful though, because the multiplicative function will be defined for the whole interval. If only three data points are available, one has no knowledge about the rest of the [[Interval (mathematics)|interval]], which may be a large part of it. This means that ''y'' could have another component which equals 0 at the ends and in the middle of the interval. A number of functions having this property are known, for example ''y'' = sin π''x''. [[Taylor series]] is useful and helps predict an [[Closed-form expression|analytic solution]], but the approximation alone does not provide conclusive evidence.
===Second-order===
''Second-order approximation'' is the term scientists use for a decent-quality answer. Few simplifying assumptions are made, and when a number is needed, an answer with two or more significant figures ("the town has {{val|3.9|e=3}}, or ''thirty-nine hundred'', residents") is generally given. In [[mathematical finance]], second-order approximations are known as [[Convexity (finance)|convexity corrections]]. As in the examples above, the term "2nd order" refers to the number of exact numerals given for the imprecise quantity. In this case, "3" and "9" are given as the two successive levels of precision, instead of simply the "4" from the first order, or "a few" from the zeroth order found in the examples above.
A second-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a [[quadratic polynomial]], geometrically, a [[parabola]]: a polynomial of degree 2. For example:
: <math>x = [0.00, 1.00, 2.00],</math>
: <math>y = [3.00, 3.00, 5.00],</math>
: <math>y \sim f(x) = x^2 - x + 3</math>
is an approximate fit to the data. In this case, with only three data points, a parabola is an exact fit based on the data provided. However, the data points for most of the interval are not available, which advises caution (see "zeroth order").
===Higher-order===
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