Calculus A/B

The origins of L’Hospital’s rule:

This is a nice historical project on the mathematical origins of l’Hospital’s Rule. It would be good as a longterm
project for a liberal arts course, or as a shorter project in which just the historical aspects are emphasized.
Details about directions for investigation are given in the text (p. 310 ).
Plenty of information about the historical figures can be found on the World Wide Web, but urge the students
to use caution — information from the Web may not be meticulously researched, and may therefore be

Remember that only the first 4 responses from each period will receive credit

6 thoughts on “Calculus A/B

  1. In the 1980’s when the first AP Calculus exams were proctored, many students have since been relieved to find out rather think using the long method to find a limit, they could take (f’/g’) also known as L’Hospital’s Rule to find the limit. However, as most of us today in AP Calculus have reviewed proofs theorems and their relations to formulas we use,, it is rare that any of us have understood the backgrounds of the inquisitive mathematicians who actually discovered these rules. In an attempt to investigate one of the most useful tools in AP Calculus, this paper will examine the lives both Marquis de L’Hospital and Johan Bernoulli and demonstrate the dilemma in the publication of L’Hopsital’s rule.
    Born in 1661 Marquis de L’Hospital also known as Guillaume-François-Antoine Marquis de L’Hôpital, Marquis de Sainte-Mesme, Comte d’Entremont and Seigneur d’Ouques-la-Chaise, was born in France and grew up in both an upper-class and military oriented family. As expected, L’Hospital’s did enlist into the French army but was soon reunited with his passion after he resigned due to his near-sightedness. It followed that in the year 1691, L’Hospital met up with Johan Bernoulli in Paris, the man who later team up with L’Hospital to derive L’Hospital’s Rule. Born in Basel, Switzerland a few years after L’Hospital, Johan Bernoulli was likewise born into a wealthy family and chose to study mathematics over both business and medicine. After studying at Basel University and working with other famous mathematicians, which included L’Hospital.
    In the 17th century, Calculus started to spread worldwide and was soon supported by L’Hospital’s 200 pages work discussing new finding in Calculus. Because L’Hospital did publish his work after collaborating with Bernoulli, through a series of letters, Bernoulli argued that L’Hospital stole many of his new findings including those of integral calculus when working together, and did not give him permission to publish them. On the other side, L’Hospital did not explicitly state that he stole Bernoulli’s ideas, but did state in the preface of his work that “he expresses his indebtedness” to Bernoulli. Though seemingly enough, what angered Bernoulli the most was the fact that within L’Hospital’s text, he gives special acknowledgments to half a dozen of mathematicians, but not one to Bernoulli. After few mutual agreements on whose ideas were whose, Bernoulli suddenly refrained from aiding L’Hospital in his future works. Communications were halted between these two mathematicians until L’Hospital sent a letter to Bernoulli stating, “W will be happy to give you a retainer of 300 pound, beginning with the first of January this year… I promise shortly to increase this retainer, which I know is very modest, as soon as my affairs are somewhat straightened out… I am not so unreasonable as to demand in return all your time, but I will ask you to give me at intervals some hours of your time to work on what I request and also to communicate to me your discoveries, at the same time asking you to disclose any of them to others. I ask you even not send here to Mr. Varignon or to others any copies of the writing you have left with me; if they are published, I will not be at all pleased. Answer me regarding all this”. Unfortunately the response letter from Bernoulli was lost, but the second letter coming from L’Hospital allowed others to surmise that he had agreed to the deal.
    We accept today that the rule attributed to L’Hospital really belongs to Bernoulli, but
    L’Hospital wrote a text that kindled an enthusiasm and excitement for mathematics that
    lasted a long time and served as a model for future texts. The rule may indeed belong to
    Bernoulli, but L’Hospital gave it to mathematics. L’Hospital said it himself, “I am very sure that there is scarcely a geometer in the world who can be compared to you”.

  2. Johann Bernoulli was a Swiss mathematician and is know for his contributions to infinitesimal calculus. He was born in Basel and studied at Basel University. He worked with his older brother Jacob on infinitesimal calculus and applied caluculs to different problems. He was hired by Guillaume de L’Hospital as a math tutor and they both signed a contract which gave the latter the right to use his discoveries. Therefore, L’Hospital wrote the first textbook on infinitesimal calculus in 1696, which actually consisted mostly of Bernoulli’s work (L’Hospital’s rule was in this book). After the book was published, Bernoulli said he hadn’t received enough credit for his contributions even though L’Hospital recognized Bernoulli’s work in the preface of the book stating, “I recognize I owe much to Messrs. Bernoulli’s insights, above all to the young (John), currently a professor in Groningue. I did unceremoniously use their discoveries, as well as those of Mr. Leibniz. For this reason I consent that they claim as much credit as they please, and will content myself with what they will agree to leave me.”
    Guillaume was a French mathematician and he is famous for publishing L’Hospital’s rule for calculating limits with 0/0 and infinity/infinity (the indeterminate forms). His book was the first book for differential calculus. The riff between him and Bernoulli caused a lot of controversy. Here is the rule:

    as found in Struik’s textbook. The L’Hospital’s rule in our textbook states that if we have an indeterminate form of type 0/0 or infinity/infinity, then the limit as x approaches a of f(x)/g(x) equals the limit as x approaches a of f’(x)/g’(x). This is essentially the same rule as the one shown above in Struik’s textbook because in the first 2 parts of the equation, it is saying that the function will be indeterminate f(a)=0 and g(a)=0, therefore it equals the primes of each function divided by each other.

  3. L’Hopital was born into a military family, however, because of poor eyesight, he pursued a career in mathematics. He was elected to the French Academy of Sciences, where he was elected vice president twice. His name is firmly associated with l’Hôpital’s rule for calculating limits involving indeterminate forms 0/0 and ∞/∞.
    l’Hospital’s rule, also known as Bernoulli’s rule, uses derivatives to help evaluate limits involving indeterminate forms (an algebraic expression obtained in the context of limits). The rule is named after the 17th-century French mathematician Guillaume de l’Hôpital. An earlier letter by John Bernoulli gives both the rule and its proof, so it seems likely that Bernoulli discovered the rule first. Bernoulli was hired by Guillaume de L’Hôpital for tutoring in mathematics. Bernoulli and L’Hôpital signed a contract which gave l’Hôpital the right to use Bernoulli’s discoveries as he pleased. L’Hôpital authored the first textbook on infinitesimal calculus, Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes in 1696, which mainly consisted of the work of Bernoulli, including what is now known as L’Hôpital’s rule.
    The general form of l’Hôpital’s rule covers many cases. Let c and L be extended real numbers (i.e., real numbers, positive infinity, or negative infinity). The real valued functions f and g are assumed to be differentiable on an open interval with endpoint c, and additionally g'(x) does not equal 0 on the interval. This rule makes it easier to evaluate the limit by taking the derivative of the numerator and the denominator separately.

  4. L’ Hospital’s rule is named after a French nobleman named Marquis de l’ Hospital, but was discovered by Bernoulli. Marquis de l’ Hospital was born in 1661 and was foremost among the French nobility intrigued by the development of calculus. Johann Bernoulli was a Swiss mathematician and was known for his contributions to infinitesimal calculus. He earned a masters degree in philosophy, and later a medical license from the University of Basel. A little while after, Bernoulli went to Paris to teach the material to l’ Hospital, in a series of public lectures. L’ Hospital then hired Bernoulli as a private tutor. L’ Hospital wrote to Bernoulli in 1694 offering him a yearly income of 300 “livres” in exchange for help with math questions, and a promise that they would be published under his own name. Bernoulli sent l’ Hospital what we call today L’ Hospitals rule. In 1696, l’ Hospital published the very first book on calculus.
    L’ Hospital’s rule is a shortcut for doing some limit problems. It is a method of differentiation to solve indeterminate limits, which are limits of functions where both the function in the numerator, and the function in the denominator are approaching 0, or positive or negative infinity. This rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives, provided that the given conditions are satisfied. When using l’ Hospital’s rule, you differentiate the numerator and denominator separately. If you wanted to calculate the limit of f(x)/g(x), as x approaches some value of a, which could be infinity, That limit is the same as the limit of f'(x)/g'(x). In the textbook, when there is an indeterminate form of type 0/0 or infinity/infinity, the limit as x goes to a of f(x)/g(x) is equal to the limit as x goes to as of f'(x)/g'(x) if the limit on the right side exists, or is positive or negative infinity.

  5. L’Hopitals rule is a shortcut that can be used when evaluating limits. If the limit is in “indeterminate forms”, such as 0/0 or infinity/infinity, then we can apply L’Hopitals rule. As stated similar in in Struik’s Source Book. This rule involves taking the derivative of the numerator and the denominator, which converts it to determinate form. This rule is named after a French mathematician Guillaume de l’Hopital, but was discovered by a Swiss mathematician named Johann Bernoulli. L’Hopital actually purchased this formula from Bernoulli and published it in the first differential calculus textbook, “ L’Analyse des infiniment petits pour l’intelligence des lignes courbes.” They had a contract that gave l’Hopital the right to use any of Bernoulli’s discoveries. This contract included an income of 300 pounds for Bernoulli.
    L’Hopital was born into a military family in Paris, France, in which he was meant to continue on the legacy and take on a position in the military. Due to his poor eyesight, he began to pursue a field in mathematics. He eventually served as the vice president of the French Academy of Sciences twice. In 1691, he met Johann Bernoulli and hired him as his private tutor. Johann Bernoulli was born in Basel, Switzerland. Despite what his father wanted, Bernoulli turned away from studying medicine and began pursuing mathematics. Bernoulli discovered how to determine the lengths and areas of curves and proposed the “cycloid” to help determine the equation for the path that follows a particle.

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