The curve (the witch) of Agnesi
Characters:
Theme: Elementary functions
Competencies
Mathematical competence in science, technology and engineering
Personal, social and learning to learn competence
Competence in cultural awareness and expressions
Subjects and year by Educational System
Spain > Mathematics > 4th(A) ESO > Algebraic sense
Spain > Mathematics > 4th(A) ESO > Socio-affective sense
Spain > Mathematics > 4th(B) ESO > Algebraic sense
Spain > Mathematics > 4th(B) ESO > Socio-affective sense
Enunciation
Maria Gaetana Agnesi, an 18th-century Italian mathematician, worked with "the witch of Agnesi". This curve was discussed by Fermat in 1703. It has been recently established that it is an approximation of the distribution of X-rays and optical rays energy spectrum, as well as dissipated power in high-frequency resonance circuits.
The equation for the curve is:
$$y = \frac{a^3}{x^2+a^2}$$
Draw the graph for the following rational functions:
$$\mathit{a)}\quad y = \frac{1}{x^2+1}\qquad \mathit{b)}\quad y = \frac{8}{x^2+4}\qquad \mathit{c)}\quad y = \frac{27}{x^2+9}$$
They are particular cases for the witch of Agnesi.
Note: Use the following data table for each function.
$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline x & -10 & -8 & -6 & -4 & -2 & \ 0\ & +2 & +4 & +6 & +8 & +10 \\ \hline y \\ \hline \end{array}$$
Observations and context
- 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.
Description
Graphical representation of rational functions that are particular cases of the witch of Agnesi.