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==Usage in science and engineering==
In formal expressions, the [[English_numerals#Ordinal_numbers|ordinal number]] used before the word [[Order_(mathematics)#Analysis|order]] refers to the highest [[Power function|power]] in the [[series expansion]] used in the [[Approximation#Etymology_and_usage|approximation]]. The expressions: a '''''zeroth-order''' approximation'', a '''''first-order''' approximation'', a '''''second-order''' approximation'', and so forth are used as [[fixed phrase]]s. The expression a ''zero -order approximation'' is also common. [[Cardinal numeral]]s are occasionally used in expressions like an ''order -zero approximation'', an ''order -one approximation'', etc.
The omission of the word ''order'' leads to [[Phrase|phrasesphrase]]s that have less formal meaning. Phrases like '''first approximation''' or '''to a first approximation''' may refer to ''a roughly approximate value of a quantity''.<ref> ''first approximation'' in Webster's Third New International Dictionary, Könemann, {{ISBN|3-8290-5292-8}} .</ref><ref>[http://www.webster-dictionary.org/definition/to%20a%20first%20approximation ''to a first approximation''] in Online Dictionary and Translations Webster-dictionary.org].</ref> The phrase '''to a zeroth approximation''' indicates ''a wild guess''.<ref name=":0">[http://www.webster-dictionary.org/definition/to%20a%20zeroth%20approximation ''to a zeroth approximation''] in Online Dictionary and Translations Webster-dictionary.org].</ref> The expression ''order of approximation'' is sometimes informally used to mean the number of [[significant figure]]s, in increasing order of accuracy, or to the [[order of magnitude]]. However, this may be confusing, as these formal expressions do not directly refer to the order of derivatives.
The choice of series expansion depends on the [[scientific method]] used to investigate a [[Phenomenon#Scientific|phenomenon]]. The expression '''order of approximation''' is expected to indicate progressively more refined approximations of a [[Function_(mathematics)|function]] in a specified [[Interval_(mathematics)|interval]]. The choice of order of approximation depends on the [[Research|research purpose]]. One may wish to simplify a known [[Closed-form_expression#Analytic_expression|analytic expression]] to devise a new application or, on the contrary, try to [[Curve_fitting|fit a curve to data points]]. Higher order of approximation is not always more useful than the lower one. For example, if a quantity is constant within the whole interval, approximating it with a second-order [[Taylor series]] will not increase the accuracy.
In the case of a [[smooth function]], the ''n''th-order approximation is a [[polynomial]] of [[degree of a polynomial|degree]] ''n'', which is obtained by truncating the Taylor series to this degree. The formal usage of ''order of approximation'' corresponds to the omission of some terms of the [[Series_(mathematics)|series]] used in the [[Series_expansion|expansion]] (usually the higher terms). This affects [[Accuracy_and_precision|accuracy]]. The error usually varies within the interval. Thus the numbers ''zeroth'', ''first'', ''second'' etc. used formally in the above meaning do not directly give information about [[percent error]] or [[significant figures]].
=== Zeroth-order ===
===First-order===
<ref name=":0" />''First-order approximation'' is the term scientists use for a slightly better answer.<ref name=":0" /> Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has {{val|4|e=3}}, or '''four thousand''', residents"). In the case of a first-order approximation, at least one number given is exact. In the zeroth -order example above, the quantity "a few" was given, but in the first -order example, the number "4" is given.
A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a linear approximation, straight line with a slope: a polynomial of degree 1. 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 + 2.67\,</math>
is an approximate fit to the data.
In this example there is a zeroth -order approximation that is the same as the first -order, but the method of getting there is different; i.e. a wild stab in the dark at a relationship happened to be as good as an '"educated guess'".
===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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