Representing the curve (the witch) of Agnesi
Personatges:
Tema: Change. Graphic study of the growth and decrease of functions in everyday life contexts with the support of technological tools: absolute, relative, and average rates of variation.
Competències
Competència Matemàtica, en ciència, tecnologia i enginyeria
Competència personal, social i aprendre a aprendre
Competència en consciència i expressions culturals
Matèries i cursos per Sistema Educatiu
Espanya > Matemàtiques > 3r ESO > Sentit algebraic
Espanya > Matemàtiques > 3r ESO > Sentit socioafectiu
Espanya > Matemàtiques > 4t(A) ESO > Sentit de la mesura
Espanya > Matemàtiques > 4t(A) ESO > Sentit algebraic
Espanya > Matemàtiques > 4t(A) ESO > Sentit socioafectiu
Enunciat
Maria Gaetana Agnesi was a famous 18th century mathematician who published the first complete book on differential and integral calculus. The book featured a famous function that would then receive the name of "the witch of Agnesi".
We are going to try to represent the witch of Agnesi, which has the following equation.
$$f : y = \frac{125}{x^2 + 25}$$
a) To do so, we must complete the following data table first (with just two decimals):
$$\begin{array}{|c|c|} \hline x & \ \ \ y \ \ \ \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline 5 & \\ \hline 10 & \\ \hline 20 & \\ \hline 100 & \\ \hline -1 & \\ \hline -2 & \\ \hline -5 & \\ \hline -10 & \\ \hline -20 & \\ \hline -100 & \\ \hline \end{array}$$
b) Now, represent it.
c) Study the function in the graph. Is it symmetrical? For what value of x does it reach its maximum? And what about its minimum? Do you think that it will cross the OX axis or that it will take negative values? Why? What is the interval of growth? What is the interval of decrease?
Observacions i context
- Alternatively, we can tell students how this curve was mistranslated as "the witch of Agnesi" instead of "Agnesi’s curve" and explore what connotations or motivations there could be behind that mistranslation (by a man) of one of the best texts on mathematics at the time, written by a woman.
- Maria Gaetana Agnesi was a child prodigy that, from a very young age, was able to communicate in several languages and to have deep philosophical and scientific conversations. Her sister, Maria Teresa Agnesi, was a musician and composer.
- Some of Maria Gaetana Agnesi's forerunners in mathematics, philosophy and astronomy are Theano of Crotone (c. 546 BC - c. 450 BC), Hypatia of Alexandria (c. 370 – c. 416) and, in the early modern period, Sophia Brahe (1556-1643), Maria Cunitz (1610-1664), and Elena Lucrezia Cornaro Piscopia (1646 -1684).
- Some of her contemporaries were renowned scientists such as Margaretha Kirch (1703-1744), astronomer; Faustina Pignatelli Carafa (1705-1785), physicist and mathematician; Émilie du Châtelet (1706-1749), mathematician, physicist and philosopher; Laura Maria Catharina Bassi (1711-1778), scientist, poet, and philosopher; Angelique-Marguerite le Boursier du Coudray (1712-1794), midwife; Dorothea Christiane Leporin (1715-1762), doctor; Anna Morandi Manzolini (1716-1774), anatomist; Marie-Geneviève-Charlotte Thiroux d’Arconville (1720-1805), chemist, anatomist, and biologist; María Juana Rosa Andresa Casamayor de la Coma (1720-1780), mathematician; Nicole-Reine de la Brière Lepaute (1723-1788), astronomer; Marie Anne Victoire Pigeon (1724-1767), mathematician, or Maria Angela Ardinghelli (1728-1825), mathematician and physicist, among others.
- Other relevant scientists in the 18th century were Maria Christina Bruhn (1732-1808), chemist and inventor; Claudine Picardet (1735-1820), chemist, mineralogist and meteorologist; Jeanne Baret (1740-1807), botanist and explorer; Caroline Lucrecia Herschel (1750-1848), astronomer; Marie-Anne Pierrette Paulze (1758-1836), chemist; and Sophie Germain (1776-1831), mathematician and physicist.
Descripció
In this activity, students will be asked to represent the witch of Agnesi for a=5 with the help of the teacher; then, they will comment on its most relevant characteristics.