Editing Precalculus

{{Short description|Course designed to prepare students for calculus}}
{{more footnotes needed|date=March 2022}}
[[File:Diagram showing how to derive the power reducing formula for sine.svg|thumb|Diagram for the deriving the power-reducing formula for the sine function|260x260px]]
{{calculus|expanded=miscellaneous}}
In [[mathematics education]], '''precalculus'''  is a course, or a set of courses, that includes [[algebra]] and [[trigonometry]] at a level that is designed to prepare students for the study of [[calculus]], thus the name precalculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework.<ref>{{cite book |last=Cangelosi |first=J. S. |title=Teaching mathematics in secondary and middle school, an interactive approach |publisher=[[Prentice Hall]] |date=2012}}</ref>

==Concept==
For students to succeed at finding the [[derivative]]s and [[antiderivatives]] with [[calculus]], they will need facility with [[algebraic expression]]s, particularly in modification and transformation of such expressions. [[Leonhard Euler]] wrote the first precalculus book in 1748 called ''[[Introductio in analysin infinitorum]]'' ([[Latin]]: Introduction to the Analysis of the Infinite), which "was meant as a survey of concepts and methods in analysis and analytic geometry preliminary to the study of differential and integral calculus."<ref>{{cite book |last=Bos |first=H. J. M. |author-link=H. J. M. Bos |date=1980 |chapter=Chapter 2: Newton, Leibniz and the Leibnizian tradition chapter 2 |page=76 |title=From the Calculus to Set Theory, 1630 – 1910: An Introductory History |editor-first=Ivor |editor-last=Grattan-Guinness |editor-link=Ivor Grattan-Guinness |publisher=[[Duckworth Overlook]] |isbn=0-7156-1295-6}}</ref> He began with the fundamental concepts of [[variable (mathematics)|variable]]s and [[function (mathematics)|function]]s. His innovation is noted for its use of [[exponentiation]] to introduce the [[transcendental function]]s. The general logarithm, to an arbitrary positive base, Euler presents as the inverse of an [[exponential function]].

Then the [[natural logarithm]] is obtained by taking as base "the number for which the hyperbolic logarithm is one", sometimes called [[Euler's number]], and written <math>e</math>. This appropriation of the significant number from [[Grégoire de Saint-Vincent]]’s calculus suffices to establish the natural logarithm. This part of precalculus prepares the student for integration of the monomial <math>x^p</math> in the instance of <math>p = -1</math>.

Today's precalculus text computes <math>e</math> as the limit <math>e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^{n}</math>. An exposition on [[compound interest]] in financial mathematics may motivate this limit. Another difference in the modern text is avoidance of [[complex number]]s, except as they may arise as roots of a [[quadratic equation]] with a negative [[discriminant]], or in [[Euler's formula]] as application of [[trigonometry]]. Euler used not only complex numbers but also [[infinite series]] in his precalculus. Today's course may cover arithmetic and geometric sequences and series, but not the application by Saint-Vincent to gain his hyperbolic logarithm, which Euler used to finesse his precalculus.

==Variable content==
Precalculus prepares students for calculus somewhat differently from how [[pre-algebra]] prepares students for algebra. While pre-algebra often has extensive coverage of basic algebraic concepts, precalculus courses might see only small amounts of calculus concepts, if at all, and usually involve covering algebraic topics that might not have been given attention in earlier algebra courses. Some precalculus courses might differ from others in terms of content. For example, an honors-level course might spend more time on [[conic section]]s, [[Euclidean vector]]s, and other topics needed for calculus, used in fields such as medicine or engineering. A college preparatory/regular class might focus on topics used in business-related careers, such as [[Matrix (mathematics)|matrices]], or [[power function]]s.

A standard course considers [[function (mathematics)|function]]s, [[function composition]], and [[inverse function]]s, often in connection with [[set (mathematics)|set]]s and [[real number]]s. In particular, [[polynomial]]s and [[rational function]]s are developed. Algebraic skills are exercised with [[trigonometric functions]] and [[trigonometric identities]]. The [[binomial theorem]], [[polar coordinate]]s, [[parametric equation]]s, and the [[limit (mathematics)|limit]]s of [[sequence]]s and [[series (mathematics)|series]] are other common topics of precalculus. Sometimes the [[mathematical induction]] method of proof for propositions dependent upon a [[natural number]] may be demonstrated, but generally, coursework involves [[exercise (mathematics)|exercise]]s rather than theory.

==Sample texts==
* Roland E. Larson & Robert P. Hostetler (1989) ''Precalculus'', second edition, [[D.C. Heath and Company]] {{ISBN|0-669-16277-9}}
* Margaret L. Lial & Charles D. Miller (1988) ''Precalculus'', [[Scott Foresman]] {{ISBN|0-673-15872-1}}
* Jerome E. Kaufmann (1988) ''Precalculus'', PWS-Kent Publishing Company ([[Cengage Learning|Wadsworth]])
* Karl J. Smith (1990) ''Precalculus Mathematics: a functional approach'', fourth edition, [[Brooks/Cole]] {{ISBN|0-534-11922-0}}
* Michael Sullivan (1993) ''Precalculus'', third edition, Dellen imprint of [[Macmillan Publishers]] {{ISBN|0-02-418421-7}}

===Online access===
* Jay Abramson and others (2014) [https://openstax.org/details/precalculus Precalculus] from [[OpenStax]]
* David Lippman & Melonie Rasmussen (2017) [http://www.opentextbookstore.com/precalc/ Precalculus: an investigation of functions]
* Carl Stitz & Jeff Zeager (2013) [http://www.stitz-zeager.com/szprecalculus07042013.pdf Precalculus] (pdf)

==See also==
{{Portal|Education|Mathematics}}
* [[AP Precalculus]]
* [[AP Calculus]]
* [[AP Statistics]]
* [[Pre-algebra]]
* [[Mathematics education in the United States]]
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==References==
{{reflist}}

==External links==
{{Wiktionary}}

* [http://mathworld.wolfram.com/classroom/classes/Pre-Calculus.html Precalculus information at Mathworld]
{{Mathematics education}}
[[Category:Mathematics education]]