Abstract
After having learned to form the derivatives and differentials of functions of a single variable, we now indicate how we can make use of them for the solution of various problems.
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- 1.
Lecture Six is the only lecture within Cauchy’s Calcul infinitésimal textbook that is devoted to exploring applications of the calculus.
- 2.
Problem I considers the classic application of the derivative in which the sign of the first derivative indicates whether the function is increasing or decreasing.
- 3.
Here, Cauchy is using the term “limits” to denote bounds.
- 4.
Cauchy is setting the stage for his next problem by arguing extrema cannot occur unless the derived function is zero or it does not exist.
- 5.
Problem II deals with using the derivative to locate extrema.
- 6.
Problem III addresses the use of the derivative to find the slope of a line tangent to a curve at a point. Cauchy has taken care to distance himself from anything geometric as well as possible thus far in his text. This is the first time we see him directly connecting the derivative to this geometric interpretation.
- 7.
This is the placement of the first of the two footnotes Cauchy makes of his own in the original text of 1823. It reads, “We indicate here the points with the help of their coordinates included between two parentheses, that we will always use hereafter. Often, we also indicate the curves or curved surfaces by their equations.”
- 8.
Problem IV is the first of several times Cauchy deals with the important indeterminate form \(\frac{0}{0}.\)
- 9.
This result is essentially the weaker version of what today is known as l’Hôpital’s Rule. The rule is named after Guillaume de l’Hôpital (1661–1704) ; however, the result is actually due to Johann Bernoulli (1667–1748) from around the year 1694 while he was developing the content for a new textbook. Interestingly, this textbook will eventually be published under l’Hôpital’s name due to a mutually agreed arrangement the two had made allowing l’Hôpital to be credited for Bernoulli’s work in exchange for payment. The text, Analyse des infiniment petits pour l’intelligence des lignes courbes (Analysis of the Infinitely Small for the Understanding of Curved Lines), was published in 1696 and is recognized as the first major textbook on differential calculus. Cauchy will extend the idea developed in this problem to a more general case at the very end of his Calcul infinitésimal text.
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Cates, D.M. (2019). USE OF DIFFERENTIALS AND DERIVED FUNCTIONS IN THE SOLUTION OF SEVERAL PROBLEMS. MAXIMA AND MINIMA OF FUNCTIONS OF A SINGLE VARIABLE. VALUES OF FRACTIONS WHICH ARE PRESENTED UNDER THE FORM \(\frac{0}{0}\).. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_6
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DOI: https://doi.org/10.1007/978-3-030-11036-9_6
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