Guillaume de l’Hôpital a historical perspective to mathematical analysis

Overview Personal life Socioeconomic context We will provide background from l’Hopital’s personal life, the socioeconomic context of 17th century France, and his contribution to mathematics relating to the topics covered in Advanced Calculus. Socioeconomic context Mathematical contributions

Personal life:

Military background high ranking military family captain in a cavalry regiment

Influence from Johann Bernoulli Congregation of the Oratory Private lessons with Johann Bernoulli “You have only to let me know your definite wishes, if I am to publish nothing more in my life, for I will follow them precisely and nothing more by me will be seen.”

First textbook on differential calculus “I must own myself very much obliged to the labours of Messieurs Bernoulli, but particularly to those of the present Professor at Groningen. I have made free use of their discoveries, as well as those of Mr Leibniz, so that I frankly return to them whatever they please to claim as their own. I must here in justice own (as Mr Leibniz himself has done in 'Journal des Sçavans' for August 1694) that the learned Sir Isaac Newton likewise discovered something like the Calculus Differentialis ... But the method of Mr Leibniz is much more easy and expeditious, on account of the notation he uses, not to mention the wonderful assistance it affords on may occasions.”

Académie des Sciences After the reorganization of the Academy of sciences by Louis XIV in 1699, l’Hôpital was given the status of honorary member.

Socioeconomic context:

The reign of Louis XIV and the Age of Reason 1643 1699 1715 1682 Beginning of the reign of Louis XIV Foundation of the Royal Academy of Sciences 1715 1682 Death of the Sun King Versailles became the permanent home for royalty and nobility

Political power Absolute monarchy Administrative and fiscal reforms French influence in the world through commerce and trade Old military aristocracy ceased to have a monopoly over senior military positions and rank Revocation of the Edict of Nantes

Sciences in Versailles Construction of The Versailles Castle Education of Princes Creation of Academies

Mathematical contribution: L’Hôpital’s Rule

Background Information Leibniz’s (as well as Newton’s but to a smaller extent) invention of infinitesimal calculus was needed before L’Hôpital could do the work that led to the theorem that is known as L’Hôpital’s Rule. L’Hôpital mentions preferring Leibniz’s calculus for its simple and consistent notation. The same notation that is most commonly used today. Leibniz Notation y’=dy/dx ∫ for integration

Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes Published in 1696 l’Hôpital’s Rule first appeared in Section IX of this book Largely debated whether or not the math depicted in this book is l’Hôpital’s or Bernoulli’s, including l’Hôpital’s Rule.

L’Hôpital’s Rule The rule states that for the differential functions f and g on an open interval. g’(x)≠0 for all x on the interval and the limit of f’(x)/g”’(x) exists.

Proof: Special Case Since many functions have continuous derivatives, this case is worth mentioning. Assume that f and g be continuously differentiable at the point c and f(c) = g(c) = 0. And g’(c) ≠ 0

Proof (cont.) The proof follows from the difference-quotient definition of the derivative and the continuity of the derivatives at c. Since g’(c) ≠ 0 the limit is not indeterminate.