Computer Experiments with Newton's Method

J. Orlando Freitas (1) , J. Sousa Ramos (2)

1 ESCOLA SECUNDáRIA DE FRANCISCO FRANCO
2 IST - Instituto Superior Técnico, Universidade Técnica de Lisboa

We study the chaotic behaviour of Newton's method with graphic calculators and computers. Through repeated experiments the students can explore the behaviour of the sequence Xn=Nf(Xn)=x-f(x)/f´(x) where f(x) is a quintic polynomial. If we expand our study of the dynamics of Newton's method to the complex plane, we find lots of interesting properties: fractals, chaos, attracting periodic cycles and Julia sets. Our study illustrates a symbiotic relationship between technology and mathematics. Technology is used to develop our intuition, and mathematics is used to prove our intuition is correct. Much of what is known about dynamical system was discovered using technology, and it is natural to use technology to study the Newton's method. Almost of our examples were done with the graphic calculators Voyage 200.

Keywords: Education -- Data processing; Information and communications technology<br />Educational technologies; Education<br />Computer-assisted education

Keywords: Education et informatique<br />Nouvelles technologies de l'information et de la communication

Domains: Humanities and Social Sciences; Education

J. Orlando Freitas, J. Sousa Ramos. Computer Experiments with Newton's Method. Jun 2003, Reims, France. ⟨edutice-00001330⟩