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Precalculus

In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level which 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.[1]

Diagram for the deriving the power-reducing formula for the sine function

Concept edit

For students to succeed at finding the derivatives and antiderivatives with calculus, they will need facility with algebraic expressions, 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."[2] He began with the fundamental concepts of variables and functions. His innovation is noted for its use of exponentiation to introduce the transcendental functions. 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  . This appropriation of the significant number from Gregoire de Saint-Vincent’s calculus suffices to establish the natural logarithm. This part of precalculus prepares the student for integration of the monomial   in the instance of  .

Today's precalculus text computes   as the limit  . An exposition on compound interest in financial mathematics may motivate this limit. Another difference in the modern text is avoidance of complex numbers, 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 edit

Precalculus prepares students for calculus somewhat differently from the way that 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 often involves covering algebraic topics that might not have been given attention in earlier algebra courses. Some precalculus courses might differ with others in terms of content. For example, an honors-level course might spend more time on conic sections, Euclidean vectors, 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 matrices, or power functions.

A standard course considers functions, function composition, and inverse functions, often in connection with sets and real numbers. In particular, polynomials and rational functions are developed. Algebraic skills are exercised with trigonometric functions and trigonometric identities. The binomial theorem, polar coordinates, parametric equations, and the limits of sequences and 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 exercises rather than theory.

Sample texts edit

  • 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 (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 edit

  • Jay Abramson and others (2014) Precalculus from OpenStax
  • David Lippman & Melonie Rasmussen (2017) Precalculus: an investigation of functions
  • Carl Stitz & Jeff Zeager (2013) Precalculus (pdf)

See also edit

References edit

  1. ^ Cangelosi, J. S. (2012). Teaching mathematics in secondary and middle school, an interactive approach. Prentice Hall.
  2. ^ Bos, H. J. M. (1980). "Chapter 2: Newton, Leibnitz and the Leibnizian tradition chapter 2". In Grattan-Guinness, Ivor (ed.). From the Calculus to Set Theory, 1630 – 1910: An Introductory History. Duckworth Overlook. p. 76. ISBN 0-7156-1295-6.

External links edit

  • Precalculus information at Mathworld

precalculus, this, article, includes, list, general, references, lacks, sufficient, corresponding, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, march, 2022, learn, when, remove, this, template, message, mathem. This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations March 2022 Learn how and when to remove this template message In mathematics education precalculus is a course or a set of courses that includes algebra and trigonometry at a level which 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 1 Diagram for the deriving the power reducing formula for the sine function Contents 1 Concept 2 Variable content 3 Sample texts 3 1 Online access 4 See also 5 References 6 External linksConcept editFor students to succeed at finding the derivatives and antiderivatives with calculus they will need facility with algebraic expressions 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 2 He began with the fundamental concepts of variables and functions His innovation is noted for its use of exponentiation to introduce the transcendental functions 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 e displaystyle e nbsp This appropriation of the significant number from Gregoire de Saint Vincent s calculus suffices to establish the natural logarithm This part of precalculus prepares the student for integration of the monomial xp displaystyle x p nbsp in the instance of p 1 displaystyle p 1 nbsp Today s precalculus text computes e displaystyle e nbsp as the limit e limn 1 1n n displaystyle e lim n rightarrow infty left 1 frac 1 n right n nbsp An exposition on compound interest in financial mathematics may motivate this limit Another difference in the modern text is avoidance of complex numbers 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 editPrecalculus prepares students for calculus somewhat differently from the way that 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 often involves covering algebraic topics that might not have been given attention in earlier algebra courses Some precalculus courses might differ with others in terms of content For example an honors level course might spend more time on conic sections Euclidean vectors 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 matrices or power functions A standard course considers functions function composition and inverse functions often in connection with sets and real numbers In particular polynomials and rational functions are developed Algebraic skills are exercised with trigonometric functions and trigonometric identities The binomial theorem polar coordinates parametric equations and the limits of sequences and 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 exercises rather than theory Sample texts editRoland E Larson amp Robert P Hostetler 1989 Precalculus second edition D C Heath and Company ISBN 0 669 16277 9 Margaret L Lial amp Charles D Miller 1988 Precalculus Scott Foresman ISBN 0 673 15872 1 Jerome E Kaufmann 1988 Precalculus PWS Kent Publishing Company 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 7Online access edit Jay Abramson and others 2014 Precalculus from OpenStax David Lippman amp Melonie Rasmussen 2017 Precalculus an investigation of functions Carl Stitz amp Jeff Zeager 2013 Precalculus pdf See also edit nbsp Education portal nbsp Mathematics portalAP Precalculus AP Calculus AP Statistics Pre algebra Mathematics education in the United StatesReferences edit Cangelosi J S 2012 Teaching mathematics in secondary and middle school an interactive approach Prentice Hall Bos H J M 1980 Chapter 2 Newton Leibnitz and the Leibnizian tradition chapter 2 In Grattan Guinness Ivor ed From the Calculus to Set Theory 1630 1910 An Introductory History Duckworth Overlook p 76 ISBN 0 7156 1295 6 External links edit nbsp Look up precalculus in Wiktionary the free dictionary Precalculus information at Mathworld Retrieved from https en wikipedia org w index php title Precalculus amp oldid 1174303521, wikipedia, wiki, book, books, library,

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