Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
In Which We Give the Rules of This Calculus Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

3. Use of the Differential Calculus for Finding the Greatest And the Least Ordinates, to Which Are Reduced Questions De Maximis & Minimis

verfasst von : Robert E. Bradley, Salvatore J. Petrilli, C. Edward Sandifer

Erschienen in: L’Hôpital's Analyse des infiniments petits

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In Chapter 3 of the Analyse, l’Hôpital turns his attention to problems of maxima and minima. In a manner analogous to the modern method of finding critical numbers of a function, l’Hôpital solves these problems by finding values of the abscissa x for which either dy = 0 or dy is infinite. Among the problems that l’Hôpital considers is finding the maximum ordinate on the closed loop of the Folium of Descartes, maximizing or minimizing surface area of solids, and the so-called Pulley Problem.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Use of the Differential Calculus for Finding the Tangents of All Kinds of Curved Lines
Nächstes Kapitel Use of the Differential Calculus for Finding Inflection Points and Cusps
Fußnoten
1
In L’Hôpital (1696) this italicized Latin expression meaning “of maximums and minimums” was used within the French text. See Problem XII on p. 213 for the treatment of these types of problems.
 
2
This is an implicit use of the Intermediate Value Theorem; the author clearly considers this to be self-evident.
 
3
The case of an extremum at a cusp was omitted from the Lectiones. Bernoulli alerted L’Hôpital to this case in Letter 22 (p. 259).
 
4
We note that the case where dy is infinite is not considered, presumably because this does not correspond to a greatest or least ordinate.
 
5
In other words, we wish that ANB: BF: : CB: CE.
 
6
This is the first of many places in L’Hôpital (1696) where the expression un plus grand (a greatest) is used, with the last two words italicized. We translate this as “a maximum,” preserving the emphasis. The words “maximum” or “minimum” are not used in this chapter after Definition II.
 
7
The is the first of many places in L’Hôpital (1696) where un moindre (a least) is used, with both words italicized. We translate this as “a minimum.”
 
8
I.e., surface area not including the base.
 
9
Compare to Problem XVI on p. 215.
 
10
In L’Hôpital (1696) the word campagnes was used, which could mean “countries,” or simply “fields.”
 
11
This is the notation used in L’Hôpital (1696) for \((a^{2} -b^{2})x^{4} +(-2a^{2}f +2b^{2}f)x^{3} +(a^{2}f^{+}a^{2}g^{2} -b^{2}f^{2} -b^{2}h^{2})x^{2} -2a^{2}\mathit{fg}^{2}x+a^{2}f^{2}g^{2} = 0\).
 
12
Two quantities are incommensurable if their ratio is irrational; “removing the incommensurables” means rationalizing the equation.
 
13
Compare to Problem XIX on p. 219. For a modern discussion of this problem, see Hahn (1998).
 
14
In L’Hôpital (1696), this literally says “the complements within two right angles.” We consistently translate this construction as supplementary.
 
15
Compare to Problem XX on p. 220.
 
16
I.e., the shortest twilight. Evening twilight is defined as the length of time from sunset until the time that the center of the sun is 18∘ below the horizon. Morning twilight is defined similarly for the period before sunrise.
 
17
This right parenthesis was missing in L’Hôpital (1696).
 
18
The term “total sine” is a synonym for the radius.
 
Literatur
Bernoulli, Johann, Opera Omnia, vol. 3, Bousquet, Lausanne, 1742.
Bernoulli, Johann, Der Briefwechsel von Johann I Bernoulli, vol. 1, ed. O. Spiess, Birkhäuser, Basel, 1955.
Bernoulli, Johann, Der Briefwechsel von Johann I Bernoulli, vol. 2, ed. P. Costabel, J. Peiffer, Birkhäuser, Basel, 1988.
Bernoulli, Johann, Der Briefwechsel von Johann I Bernoulli, vol. 3, ed. P. Costabel, J. Peiffer, Birkhäuser, Basel, 1992.
Bradley, Robert E., “The Curious Case of the Bird’s Beak,” International J. Math. Comp. Sci., 1 (2006), pp. 243–268.
Bossut, Charles, Histoire Générale des Mathématiques, vol. 2, 2nd ed., F. Louis, Paris, 1810.
Burton, David, The Hstory of Mathematics: An Introduction, 6th ed., Mc Graw Hill, Boston, 2007.
Cohen, I. Bernard, A Guide to Newton’s Principia, in The Principia, Newton, Isaac, University of California Press, Berkeley and Los Angeles, 1999.
Suzuki, Jeff “The Lost Calculus (1637–1670): Tangency and Optimization without Limits,” Mathematics Magazine, 78 (2005), pp. 339–353.
Descartes, René, trans. , Smith & Latham, The Geometry of René Descartes, Dover, New Yrok, 1954.
Eneström, Gustav, “Sur le part de Jean Bernoulli dans la publication de l’Analyse des infiniment petits” Bibliotecha Mathematica, 8 (1894), pp. 65–72.
Fontenelle, Bernard de, Histoire du renouvellement de l’Académie royale des sciences, Boudot, Paris, 1708.
Hahn, Alexander, “Two Historical Applications of Calculus,” The College Mathematics Journal, 29 (1998), pp. 99–103.
Hall, A. Rupert, Philosophers at War, Cambridge U. Press, Cambridge, 1980.
Huygens, Christiaan. Horologium Oscillatorium sive de motu pendulorum, F. Muguet, Paris, 1673. English translation by Richard J. Blackwell, The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks, Iowa State University Press, Ames, 1986. Page references are to the 1986 translation of Huygens’ Horologium Oscillatorium.
Katz, Victor, History of Mathematics: An Introduction, 3rd ed., Addison-Wesley, Boston, 2009.
Leibniz, Willhelm G. von, “Nova methodus pro maximis et minimis,” Acta eruditorum, 3 (1684), p. 467–473.
l’Hôpital, Guillaume F. A. de, “Méthode facile pour déterminer ler points des caustiques …,” Mémoires de mathématique et de physique, tires des registres de l’Académie Royale des Sciences, 1693, pp. 129–133.
Anonymous (Guillaume François Antoine, Marquis de l’Hôpital), Analyse des infiniment petits, Imprimerie Royale, Paris, 1696.
l’Hôpital, Guillaume F. A. de, Traité analytique des sections coniques, Boudot, Paris, 1707.
l’Hôpital, Guillaume F. A. de, Analyse des infiniment petits pour l’intellignece des lignes courbes, 2nd ed., Montalant, Paris, 1715.
l’Hôpital, Guillaume F. A. de, Analyse des infiniment petits pour l’intellignece des lignes courbes, 2nd ed. [sic], Montalant, Paris, 1716.
l’Hôpital, Guillaume F. A. de, Analyse des infiniment petits pour l’intellignece des lignes courbes, new ed. with a commentary by l’abbé Aimé-Henri Paulian, Didot le jeune, Paris, 1768.
l’Hôpital, Guillaume F. A. de, Analyse des infiniment petits pour l’intellignece des lignes courbes, new ed., revised and augmented by Arthur LeFevre, A. Jombert, Paris, 1768.
Lockwood, E. H. (1971). A Book of Curves. Cambridge, England: Cambridge University Press.
Montucla, Jean F., Histoire des Mathématiques, second ed., vol. 2, Agasse, Paris, 1799.
Schafheitlin, Paul, “Johannis (I) Bernoullii Lectiones de calculo differentialium,” Verhandlungen der Naturforschenden Gesellschaft in Basel, 34, pp. 1–32.
Stone, Edmund, An Analytick Treatise of Conick Sections, Senex et al, London, 1723.
Stone, Edmund, The Method of Fluxions, both Direct and Inverse, Innys, London, 1730.
Varignon, Pierre, Eclaircissemens sur l’analyse des infiniment petits, Rollin, Paris, 1725.
Metadaten
Titel
Use of the Differential Calculus for Finding the Greatest And the Least Ordinates, to Which Are Reduced Questions De Maximis & Minimis
verfasst von
Robert E. Bradley
Salvatore J. Petrilli
C. Edward Sandifer
Copyright-Jahr
2015
Buch
L’Hôpital's Analyse des infiniments petits

Print ISBN: 978-3-319-17114-2

Electronic ISBN: 978-3-319-17115-9

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-319-17115-9_3