LANÇAMENTO OBLÍQUO: UMA ABORDAGEM MATEMÁTICA

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛

Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠

▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧

▲❛♥ç❛♠❡♥t♦ ❖❜❧íq✉♦✿ ❯♠❛ ❆❜♦r❞❛❣❡♠

▼❛t❡♠át✐❝❛

❋r❛♥❝✐s❝♦ ❋❛❜✐♦ ▼♦♥t❡✐r♦ ❞❡ ❆❧♠❡✐❞❛

TERMO DE CIÊNCIA E DE AUTORIZAÇÃO PARA DISPONIBILIZAR AS TESES E

DISSERTAÇÕES ELETRÔNICAS (TEDE) NA BIBLIOTECA DIGITAL DA UFG

Na qualidade de titular dos direitos de autor, autorizo a Universidade Federal de Goiás (UFG) a disponibilizar, gratuitamente, por meio da Biblioteca Digital de Teses e Dissertações (BDTD/UFG), sem ressarcimento dos direitos autorais, de acordo com a Lei nº 9610/98, o do-cumento conforme permissões assinaladas abaixo, para fins de leitura, impressão e/ou down-load, a título de divulgação da produção científica brasileira, a partir desta data.

1. Identificação do material bibliográfico: [ x ] Dissertação [ ] Tese

2. Identificação da Tese ou Dissertação

Autor (a): Francisco Fabio Monteiro de Almeida E-mail: ffmatematica@yahoo.com.br

Seu e-mail pode ser disponibilizado na página? [ x ]Sim [ ] Não Vínculo empregatício do autor Professor de Matemática

Agência de fomento: Secretaria de Estado de Educação do Distrito Federal

Sigla: SEDF

País: Brasil UF: DF CNPJ: 00.394.676/0001-07

Título: Lançamento Oblíquo: Uma Abordagem Matemática

Palavras-chave: Lançamento Oblíquo, Matemática, Função Quadrática, Movimento Uni-formemente Variado, Velocidade Média.

Título em outra língua: Oblique Launch: A Mathematical Approach

Palavras-chave em outra língua: Oblique launch, Math, Quadratic function, Uniformly Varied Movement, Average speed.

Área de concentração: Matemática do Ensino Básico Data defesa:(dd/mm/aaaa) 31/03/2016

Programa de Pós-Graduação: Mestrado em Matemática - PROFMAT Orientador (a): Dra. Lidiane dos Santos Monteiro Lima

E-mail: lidymaths@gmail.com

*Necessita do CPF quando não constar no SisPG

3. Informações de acesso ao documento:

Concorda com a liberação total do documento [ x ] SIM [ ] NÃO1

Havendo concordância com a disponibilização eletrônica, torna-se imprescindível o en-vio do(s) arquivo(s) em formato digital PDF ou DOC da tese ou dissertação.

O sistema da Biblioteca Digital de Teses e Dissertações garante aos autores, que os ar-quivos contendo eletronicamente as teses e ou dissertações, antes de sua disponibilização, receberão procedimentos de segurança, criptografia (para não permitir cópia e extração de conteúdo, permitindo apenas impressão fraca) usando o padrão do Acrobat.

________________________________________ Data: 27 / 04 / 2016. Assinatura do (a) autor (a)

1

❋r❛♥❝✐s❝♦ ❋❛❜✐♦ ▼♦♥t❡✐r♦ ❞❡ ❆❧♠❡✐❞❛

▲❛♥ç❛♠❡♥t♦ ❖❜❧íq✉♦✿ ❯♠❛ ❆❜♦r❞❛❣❡♠

▼❛t❡♠át✐❝❛

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦✳

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❛✳ ❉r❛✳ ▲✐❞✐❛♥❡ ❞♦s ❙❛♥t♦s ▼♦♥t❡✐r♦ ▲✐♠❛✳

Ficha catalográfica elaborada automaticamente

com os dados fornecidos pelo(a) autor(a), sob orientação do Sibi/UFG.

Almeida, Francisco Fabio Monteiro de

Lançamento Oblíquo: Uma Abordagem Matemática [manuscrito] / Francisco Fabio Monteiro de Almeida. - 2016.

lxv, 65 f.

Orientador: Profa. Dra. Lidiane dos Santos Monteiro Lima.

Dissertação (Mestrado) - Universidade Federal de Goiás, Instituto de Matemática e Estatística (IME) , Programa de Pós-Graduação em Matemática, Goiânia, 2016.

Bibliografia.

Inclui lista de figuras.

❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳ ➱ ♣r♦✐❜✐❞❛ ❛ r❡♣r♦❞✉çã♦ t♦t❛❧ ♦✉ ♣❛r❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ s❡♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❞♦ ❛✉t♦r ❡ ❞❛ ♦r✐❡♥t❛❞♦r❛✳

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s q✉❡ ❡stá s❡♠♣r❡ ❞♦ ♠❡✉ ❧❛❞♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥✲ t♦s✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ❉é❜♦r❛ ❡ ❛♦ ♠❡✉ ✜❧❤♦ ❉❛✈✐ ❋❡❧✐♣❡ ♣❡❧♦ ❛♣♦✐♦ ❡ ❝♦♥✜❛♥ç❛ ❞✉r❛♥t❡ ❡ss❡s ✷✭❞♦✐s✮ ❛♥♦s ❞❡ ❝✉rs♦ ❡ q✉❡ ♠❡ ♠♦t✐✈❛♠ s❡♠♣r❡ ❛ s❡r ✉♠ ❡s♣♦s♦ ❡ ♣❛✐ ♠❡❧❤♦r✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s ❏♦sé ■✈❛♥❞✐ ❡ ❚❡r❡s❛ ❞❡ ❋át✐♠❛ ❡ ❛♦s ♠❡✉s ✐r♠ã♦s ❋❛❜✐❛♥❛✱ ▼❛r✐❛ ❈❡❝í❧✐❛ ❡ ❈❧é❜❡r ♣♦r t✉❞♦ q✉❡ ♠❡ ♦❢❡r❡❝❡r❛♠ ❡ q✉❡ ✜③❡r❛♠ ❡✉ ♠❡ t♦r♥❛r ❛ ♣❡ss♦❛ q✉❡ s♦✉ ❤♦❥❡✳

❆❣r❛❞❡ç♦ ❛♦s ❞❡♠❛✐s ❢❛♠✐❧✐❛r❡s ✭s♦❣r❛✱ s♦❣r♦✱ ❝✉♥❤❛❞♦s✱ s♦❜r✐♥❤❛s✱ ❡t❝✮ q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯❋● q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❡ ❛♣r✐♠♦r❛r❛♠ ❛ ♠✐♥❤❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ❝♦♠♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ❉r❛✳ ▲✐❞✐❛♥❡ ❙❛♥t♦s q✉❡ s❡ ❞❡❞✐❝♦✉ ♥♦ ❛✉①í❧✐♦ à ❝♦♥❝r❡t✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ♠❡ ❞❛♥❞♦ ❝♦♥s❡❧❤♦s✱ ❝♦♥tr✐❜✉✐çõ❡s ❡ ♦r✐❡♥t❛çõ❡s✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ❝♦❧❡❣❛s ❞♦ P❘❖❋▼❆❚✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛♦s ❝♦❧❡❣❛s ❆❢♦♥s♦✱ ❱❛✲ ♥❡ss❛ ❡ s❡✉ ❡s♣♦s♦ ❊❞s♦♥ ✭♥♦ss♦ ♠♦t♦r✐st❛ ♣❛rt✐❝✉❧❛r✮✱ q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❝♦♠♣❛r✲ t✐❧❤❛r❛♠ ❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ q✉❡ ❡st❡ s♦♥❤♦ ♣✉❞❡ss❡ s❡r r❡❛❧✐③❛❞♦✳

❆❣r❛❞❡ç♦ ❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭❙❇▼✮ ♣❡❧❛ ✐♥✐❝✐❛t✐✈❛ ❡♠ ❝r✐❛r ♦ ♣r♦❣r❛♠❛ P❘❖❋▼❆❚ q✉❡ ♣r♦♣♦r❝✐♦♥❛ ❛ ✈ár✐♦s ♣r♦❢❡ss♦r❡s ❡ ♣r♦❢❡ss♦r❛s ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ s❡ q✉❛❧✐✜❝❛r❡♠✳

➚ ❈❆P❊❙✱ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳

❘❡s✉♠♦

❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡ss❡ tr❛❜❛❧❤♦ é ✈❡r✐✜❝❛r q✉❡ ♣♦❞❡♠♦s ❛tr✐❜✉✐r ❛♦ ❧❛♥ç❛♠❡♥t♦ ♦❜❧íq✉♦ ✉♠❛ ❛❜♦r❞❛❣❡♠ ♠❛t❡♠át✐❝❛✳ P❛r❛ ❝❤❡❣❛r♠♦s ❛ ❡st❛ ❝♦♥❝❧✉sã♦✱ ❛❜♦r❞❛♠♦s ❛ss✉♥t♦s ✐♠♣♦rt❛♥t❡s ❞❛ ▼❛t❡♠át✐❝❛ ❡ ❞❛ ❋ís✐❝❛✱ ❝♦♠♦ ❢✉♥çõ❡s ❛✜♠ ❡ q✉❛❞rát✐❝❛✱ ❡ ♠♦✈✐♠❡♥t♦s ✉♥✐❢♦r♠❡ ❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ✈❛r✐❛❞♦✳ ❚❛✐s ❝♦♥❝❡✐t♦s ❢♦r❛♠ ❛♣r❡s❡♥t❛❞♦s ♣❛r❛ ❢❛❝✐❧✐t❛r ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ✐♠♣❧❡♠❡♥t❛r ✉♠❛ ❡q✉❛çã♦ q✉❡ ♣❡r♠✐t❛ ❞❡t❡r♠✐♥❛r ♦ ✐♥st❛♥t❡ ❞❡ s✉❜✐❞❛✱ ❛ ❛❧t✉r❛ ♠á①✐♠❛ ❡ ♦ ❛❧❝❛♥❝❡ ❤♦r✐③♦♥t❛❧✱ ❝♦♥❤❡❝❡♥❞♦ ✉♠ â♥❣✉❧♦ ❞❡ t✐r♦✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧ ❡ ❛ ❛❝❡❧❡r❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡✳

P❛❧❛✈r❛s✲❝❤❛✈❡

▲❛♥ç❛♠❡♥t♦ ❖❜❧íq✉♦✱ ▼❛t❡♠át✐❝❛✱ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛✱ ▼♦✈✐♠❡♥t♦ ❯♥✐❢♦r♠❡♠❡♥t❡ ❱❛r✐❛❞♦✳

❆❜str❛❝t

❚❤❡ ♠❛✐♥ ❣♦❛❧ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ ✈❡r✐❢② t❤❛t ✇❡ ❝❛♥ ❛ss✐❣♥ t❤❡ ♦❜❧✐q✉❡ ❧❛✉♥❝❤ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ❛♣♣r♦❛❝❤✳ ❚♦ r❡❛❝❤ t❤✐s ❝♦♥❝❧✉s✐♦♥✱ ✇❡ ❛❞❞r❡ss ✐♠♣♦rt❛♥t ✐ss✉❡s ♦❢ ♠❛t❤❡♠❛t✐❝s ❛♥❞ ♣❤②s✐❝s✱ ❛s ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ❛♥❞ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s ✱ ❛♥❞ ✉♥✐❢♦r♠ ❛♥❞ ✉♥✐❢♦r♠❧② ✈❛r✐❡❞ ♠♦✈❡♠❡♥ts✳ ❚❤❡s❡ ❝♦♥❝❡♣ts ✇❡r❡ ♣r❡s❡♥t❡❞ t♦ ❢❛❝✐❧✐t❛t❡ ✉♥❞❡rst❛♥❞✐♥❣ ❛♥❞ ✐♠♣❧❡♠❡♥t ❛♥ ❡q✉❛t✐♦♥ t❤❛t ❛❧❧♦✇s t♦ ❞❡t❡r♠✐♥❡ t❤❡ ♠♦♠❡♥t ♦❢ r✐s❡✱ t❤❡ ♠❛①✐♠✉♠ ❤❡✐❣❤t ❛♥❞ ❤♦r✐③♦♥t❛❧ r❡❛❝❤✱ ❦♥♦✇✐♥❣ ❛♥ s❤♦t✬s ❛♥❣❧❡✱ t❤❡ ✐♥✐t✐❛❧ ✈❡❧♦❝✐t② ❛♥❞ ❣r❛✈✐t②✬s ❛❝❝❡❧❡r❛t✐♦♥✳

❑❡②✇♦r❞s

❖❜❧✐q✉❡ ❧❛✉♥❝❤✱ ▼❛t❤✱ ◗✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✱ ❯♥✐❢♦r♠❧② ❱❛r✐❡❞ ▼♦✈❡♠❡♥t✳

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❉❡✜♥✐çã♦ ❞❡ ❋✉♥çã♦ ♣♦r ❉✐❛❣r❛♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ❋✉♥çã♦ ♣♦r ▼❡✐♦ ❞❡ ❉✐❛❣r❛♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❉✐❛❣r❛♠❛s q✉❡ ◆ã♦ ❘❡♣r❡s❡♥t❛♠ ❋✉♥çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✹ ❈♦♥❥✉♥t♦s ❉♦♠í♥✐♦✱ ❈♦♥tr❛❞♦♠í♥✐♦ ❡ ■♠❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✺ ◆ã♦ ❘❡♣r❡s❡♥t❛ ✉♠❛ ❋✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✻ ●rá✜❝♦ ❞❛ ❋✉♥çã♦ ❆✜♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✼ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✽ ●rá✜❝♦s ❞❛ ❋✉♥çã♦ y=ax2 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺}

✷✳✶ ❘❡t❛ ❚❛♥❣❡♥t❡ ♥♦ P♦♥t♦ a ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ❊q✉❛çã♦ ❍♦rár✐❛ ❞♦ ❊s♣❛ç♦ ❡♠ ❋✉♥çã♦ ❞♦ ❚❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✸✳✶ ❚r❛❥❡tór✐❛ ❞❡ ✉♠ ▼ó✈❡❧ ❡♠ ▼♦✈✐♠❡♥t♦ ❯♥✐❢♦r♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ●rá✜❝♦ ❞♦ ❊s♣❛ç♦ ❡♠ ❋✉♥çã♦ ❞♦ ❚❡♠♣♦ ♥♦ ▼♦✈✐♠❡♥t♦ ❯♥✐❢♦r♠❡ ✳ ✳ ✸✶ ✸✳✸ ●rá✜❝♦s ❞❛ ❱❡❧♦❝✐❞❛❞❡ ❡♠ ❋✉♥çã♦ ❞♦ ❚❡♠♣♦ ♥♦ ▼♦✈✐♠❡♥t♦ ❯♥✐❢♦r♠❡ ✸✶ ✸✳✹ ➪r❡❛ ❂ ❱❡❧♦❝✐❞❛❞❡ ① ❚❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✺ ❱❛r✐❛çã♦ ❞❛ ❱❡❧♦❝✐❞❛❞❡ ❡♠ ❋✉♥çã♦ ❞♦ ❚❡♠♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✻ ●rá✜❝♦s ❞❛ ❱❡❧♦❝✐❞❛❞❡ ❡♠ ❋✉♥çã♦ ❞♦ ❚❡♠♣♦ ♥♦ ▼♦✈✐♠❡♥t♦ ❯♥✐❢♦r♠❡✲

♠❡♥t❡ ❱❛r✐❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✼ ●rá✜❝♦ q✉❡ ❘❡♣r❡s❡♥t❛ ❛ ❱❛r✐❛çã♦ ❞♦ ❊s♣❛ç♦ ∆s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✽ ●rá✜❝♦s ❞♦ ❊s♣❛ç♦ ❡♠ ❋✉♥çã♦ ❞♦ ❚❡♠♣♦ ♥♦ ▼♦✈✐♠❡♥t♦ ❯♥✐❢♦r♠❡♠❡♥t❡

❱❛r✐❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✾ ❘❡t❛ ❚❛♥❣❡♥t❡ à ❈✉r✈❛ ♥✉♠ ■♥st❛♥t❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✶✵ ●rá✜❝♦s ❞❛ ❆❝❡❧❡r❛çã♦ ❡♠ ❋✉♥çã♦ ❞♦ ❚❡♠♣♦ ♥♦ ▼♦✈✐♠❡♥t♦ ❯♥✐❢♦r♠❡✲

♠❡♥t❡ ❱❛r✐❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✹✳✶ ▲❛♥ç❛♠❡♥t♦ ❖❜❧íq✉♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✹✳✷ ❚r❛❥❡tór✐❛ ❞♦ P♦♥t♦P ❆♥❛❧✐s❛♥❞♦ ♦s P♦♥t♦s Px ❡ Py ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✸ ❱❡❧♦❝✐❞❛❞❡ ❡♠ ✉♠ P♦♥t♦ ◗✉❛❧q✉❡r ❞❛ ❚r❛❥❡tór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✹ ❚❡♠♣♦ ❞❡ ❙✉❜✐❞❛✱ ❆❧t✉r❛ ▼á①✐♠❛ ❡ ❆❧❝❛♥❝❡ ❍♦r✐③♦♥t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ ❖ ❊st✉❞♦ ❞❛ ❋✉♥çã♦ ❆✜♠ ❡ ❞❛ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ✸

✶✳✶ ❊st✉❞♦ ❞❛s ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❋✉♥çã♦ ❆✜♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷ ❱❡❧♦❝✐❞❛❞❡ ❡ ❆❝❡❧❡r❛çã♦ ♣♦r ▼❡✐♦ ❞❡ ❉❡r✐✈❛❞❛s ✶✽

✷✳✶ ❘❡t❛ ❚❛♥❣❡♥t❡ ❡ ❉❡❝❧✐✈❡ ❆♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ■♥t❡r♣r❡t❛çã♦ ●❡♦♠étr✐❝❛ ❡ ❈✐♥❡♠át✐❝❛ ❞❛ ❉❡r✐✈❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✸ ▼♦✈✐♠❡♥t♦ ❯♥✐❢♦r♠❡ ❡ ▼♦✈✐♠❡♥t♦ ❯♥✐❢♦r♠❡♠❡♥t❡ ❱❛r✐❛❞♦ ✷✽

✸✳✶ ▼♦✈✐♠❡♥t♦ ❯♥✐❢♦r♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✷ ▼♦✈✐♠❡♥t♦ ❯♥✐❢♦r♠❡♠❡♥t❡ ❱❛r✐❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✹ ▲❛♥ç❛♠❡♥t♦ ❖❜❧íq✉♦ ✹✸

✹✳✶ ❚❡♠♣♦ ❞❡ ❙✉❜✐❞❛✳ ❆❧t✉r❛ ▼á①✐♠❛✳ ❆❧❝❛♥❝❡ ❍♦r✐③♦♥t❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✷ ❊q✉❛çã♦ ❞❛ ❚r❛❥❡tór✐❛ ✲ ❯♠❛ ❆❜♦r❞❛❣❡♠ ▼❛t❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✺ Pr♦❜❧❡♠❛s ❙♦❜r❡ ▲❛♥ç❛♠❡♥t♦ ❖❜❧íq✉♦ ✺✵

✻ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✻✷

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✹

■♥tr♦❞✉çã♦

➱ ❝♦♠✉♠ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ s❡ ❞❡♣❛r❛r ❝♦♠ ❛ss✉♥t♦s ❞❡ ❞✐❢í❝✐❧ ❝♦♠♣r❡❡♥sã♦ ♣♦r ♣❛rt❡ ❞♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦r ♣❛rt❡ ❞❛q✉❡❧❡s q✉❡ ✐♥✐❝✐❛♠ s❡✉s ❡st✉✲ ❞♦s ♥❡st❛ ♥♦✈❛ ❡t❛♣❛✳ ❆ss✉♥t♦s ❝♦♠♦ ❢✉♥çõ❡s✱ ❡♠ ▼❛t❡♠át✐❝❛✱ ❡ ♠♦✈✐♠❡♥t♦ ✉♥✐❢♦r✲ ♠❡♠❡♥t❡ ✈❛r✐❛❞♦✱ ❡♠ ❋ís✐❝❛✱ ♣❛r❡❝❡♠ t✐r❛r ♦ s♦♥♦ ❞❛ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦s✳

❈♦♠♦ ♣r♦❢❡ss♦r❡s✱ ✜❝❛♠♦s ❢r✉str❛❞♦s ❡♠ ❞❡t❡r♠✐♥❛❞❛s s✐t✉❛çõ❡s ❡ ❝♦♠❡ç❛♠♦s ❛ ♣r♦❝✉r❛r ❝✉❧♣❛❞♦s ♣❛r❛ t❛♠❛♥❤❛ ❞✐✜❝✉❧❞❛❞❡ ❛♣r❡s❡♥t❛❞❛✳ P♦rt❛♥t♦✱ t❛❧ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♠❡t❛✱ ❛❧é♠ ❞♦s ♦❜❥❡t✐✈♦s ❛❜❛✐①♦ ❛♣r❡s❡♥t❛❞♦s✱ ❛✉①✐❧✐❛r ♦ ❛❧✉♥♦✱ ♥❛ s✉❛ ❝❛♣❛❝✐✲ ❞❛❞❡ ❝rít✐❝❛✱ q✉❡ é ♣♦ssí✈❡❧ ❞❡s❡♥✈♦❧✈❡r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝✐❡♥tí✜❝♦ ♣♦r ♠❡✐♦ ❞❡ ♦✉tr❛s ❡str❛té❣✐❛s✳

❙❡❣✉♥❞♦ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✭P❈◆s✮ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ✉♠❛ ❞❛s ❝♦♠♣❡tê♥❝✐❛s ❛ s❡r❡♠ ♣❡rs❡❣✉✐❞❛s ❞✉r❛♥t❡ ❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ❡ ❝♦♠♣❧❡♠❡♥t❛r ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ é ❛ ❝♦♥t❡①t✉❛❧✐③❛çã♦ ❞❛s ❝✐ê♥❝✐❛s ♥♦ â♠❜✐t♦ só❝✐♦ ❝✉❧t✉r❛❧✱ ♥❛ ❢♦r♠❛ ❞❡ ❛♥á❧✐s❡ ❝rít✐❝❛ ❞❛s ✐❞❡✐❛s ❡ ❞♦s r❡❝✉rs♦s ❞❛ ár❡❛ ❡ ❞❛s q✉❡stõ❡s ❞♦ ♠✉♥❞♦ q✉❡ ♣♦❞❡♠ s❡r r❡s♣♦♥❞✐❞❛s ♦✉ tr❛♥s❢♦r♠❛❞❛s ♣♦r ♠❡✐♦ ❞♦ ♣❡♥s❛r ❡ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝✐❡♥tí✜❝♦✳

P♦r ✐ss♦✱ ♦s ♣r✐♥❝✐♣❛✐s ♦❜❥❡t✐✈♦s ❞❡st❡ tr❛❜❛❧❤♦ sã♦✿

✶✳ ❉❡s♣❡rt❛r ♦ ✐♥t❡r❡ss❡ ❡ ❛ ❝✉r✐♦s✐❞❛❞❡ ❞♦ ❛❧✉♥♦ ♣❡❧❛ ▼❛t❡♠át✐❝❛❀

✷✳ ▼♦str❛r ❛ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r✐❡❞❛❞❡ q✉❡ ❛ ▼❛t❡♠át✐❝❛ t❡♠ ❝♦♠ ♦✉tr❛s ár❡❛s✱ ♥❡st❡ ❝❛s♦ ❛ ❋ís✐❝❛❀

✸✳ ❆♠♣❧✐❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❛❞q✉✐r✐❞♦ ❡♠ ❛♠❜❛s ❞✐s❝✐♣❧✐♥❛s✱ ♠♦str❛♥❞♦ q✉❡ ♣♦❞❡♠♦s r❡s♦❧✈❡r ✉♠ ❡①❡r❝í❝✐♦ ❞❡ ❋ís✐❝❛ ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❝♦♥❝❡✐t♦ ♠❛t❡♠át✐❝♦✳

❊st❡ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

◆♦ ❈❛♣ít✉❧♦ ✶ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ✐♥tr♦❞✉çã♦ ❛♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s✱ ❛❜♦r❞❛♥❞♦ ❛ s✉❛ ❞❡✜♥✐çã♦ ❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❣rá✜❝♦✱ ❞❛♥❞♦ ❧♦❣♦ ❛♣ós ✉♠❛ ♠❛✐♦r ê♥❢❛s❡ ♥♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s ❛✜♠ ❡ q✉❛❞rát✐❝❛✳

◆♦ ❈❛♣ít✉❧♦ ✷ s❡rá ❛❜♦r❞❛❞❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ❞❡r✐✈❛❞❛s t❛♥t♦ ♣❡❧❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦✲ ♠étr✐❝❛ q✉❛♥t♦ ♣❡❧❛ ✐♥t❡r♣r❡t❛çã♦ ❝✐♥❡♠át✐❝❛✱ ❛♣❡s❛r ❞❡ t❛❧ ❝♦♥t❡ú❞♦ ♥ã♦ s❡r ❛❜♦r❞❛❞♦ ♥♦ ❡♥s✐♥♦ ▼é❞✐♦✳ ❆❧é♠ ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❢ís✐❝♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳

◆♦ ❈❛♣ít✉❧♦ ✸ ❛❜♦r❞❛♠♦s ❝♦♥❝❡✐t♦s ❞❛ ❋ís✐❝❛✱ ❝♦♠♦ ♦s ♠♦✈✐♠❡♥t♦s ✉♥✐❢♦r♠❡ ❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ✈❛r✐❛❞♦✳

◆♦ ❈❛♣ít✉❧♦ ✹ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ▲❛♥ç❛♠❡♥t♦ ❖❜❧íq✉♦ ✲ ❛❜♦r❞❛♥❞♦ ♦s ❛ss✉♥t♦s tr❛t❛❞♦s ♥♦ ❈❛♣ít✉❧♦ ✸✱ ❛❧é♠ ❞❡ ❞❛r♠♦s ✉♠ ❡♥❢♦rq✉❡ ♠❛t❡♠át✐❝♦ ❛ ❡st❡ ❛ss✉♥t♦✳

❊✱ ♣♦r ✜♠✱ ♥♦ ❈❛♣ít✉❧♦ ✺✱ ♠♦str❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s s♦❜r❡ ❧❛♥ç❛♠❡♥t♦ ♦❜❧íq✉♦ ❡ s✉❛s r❡s♣❡❝t✐✈❛s r❡s♦❧✉çõ❡s ✉s❛♥❞♦ t❛♥t♦ ❛ ❛❜♦r❞❛❣❡♠ ❞❛ ❋ís✐❝❛ q✉❛♥t♦ ❛ ❛❜♦r❞❛❣❡♠ ❞❛ ▼❛t❡♠át✐❝❛✳

❱❛❧❡✱ t❛♠❜é♠ s❛❧✐❡♥t❛r✱ q✉❡ ♦ ❧✐✈r♦ ❬✶✵❪ t❡✈❡ ✉♠ ✐♠♣♦rt❛♥t❡ ♣❛♣❡❧ ♥♦ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✳

❊s♣❡r♦ q✉❡ t❛❧ tr❛❜❛❧❤♦ s✉♣❡r❡ ❛s ❡①♣❡❝t❛t✐✈❛s ❛❧❝❛♥ç❛♥❞♦✱ ❛ss✐♠✱ ♦s s❡✉s ♦❜❥❡t✐✈♦s✳

❈❛♣ít✉❧♦ ✶

❖ ❊st✉❞♦ ❞❛ ❋✉♥çã♦ ❆✜♠ ❡ ❞❛ ❋✉♥çã♦

◗✉❛❞rát✐❝❛

◆❡st❡ ❝❛♣ít✉❧♦✱ ✈❛♠♦s r❡✈❡r ♦s ♣r✐♥❝✐♣❛✐s tó♣✐❝♦s s♦❜r❡ ❢✉♥çõ❡s✱ ❝♦♠♦ ❞❡✜♥✐çã♦✱ ❞♦♠í♥✐♦✱ ✐♠❛❣❡♠ ❡ ❣rá✜❝♦✳ ❚❛♠❜é♠ ✈❛♠♦s r❡✈❡r ❛s ♣r♦♣r✐❡❞❛❞❡s r❡❢❡r❡♥t❡s às ❢✉♥çõ❡s ❛✜♠ ❡ q✉❛❞rát✐❝❛✳

✶✳✶ ❊st✉❞♦ ❞❛s ❋✉♥çõ❡s

◆❡st❛ s❡çã♦✱ ❞❡✜♥✐♠♦s ❢✉♥çã♦ ♣♦r ♠❡✐♦ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ ❞♦✐s ❝♦♥✲ ❥✉♥t♦s q✉❛✐sq✉❡r✱ ❛❧é♠ ❞❛ ♥♦çã♦ ❞❡ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦✳ ❆ ❞❡✜♥✐çã♦ ❞❡ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❡ ❞❡ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ t❛♠❜é♠ é ❡①♣♦st❛ ♥❡st❛ s❡çã♦✳

❈♦♥s✐❞❡r❡ ❞♦✐s ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s A ❡ B✳ P♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ❢✉♥çã♦ f ❞❡ A ❡♠ B✱ ❝♦♠♦✿

❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❬✸❪ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s ❆ ❡ ❇✱ ✉♠❛ r❡❧❛çã♦f ❞❡ ❆ ❡♠ ❇ é ✉♠❛ ❢✉♥çã♦ s❡ ♣❛r❛ t♦❞♦ x_∈A✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ y _∈B✳

❆❧❣✉♥s ♣♦♥t♦s ❞❡✈❡♠♦s ❧❡✈❛r ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❡♠ r❡❧❛çã♦ ❛ ❉❡✜♥✐çã♦ ✶✳✶✳✶✳ Pr✐✲ ♠❡✐r♦✱ q✉❡ ✉s❛♠♦s f : A _→ B ♣❛r❛ ❞❡♥♦t❛r q✉❡ f é ✉♠❛ ❢✉♥çã♦ ❞❡ ❆ ❡♠ ❇✳ ❊ ❛ ✈✐s✉❛❧✐③❛çã♦ ❞❡ ✉♠❛ ❢✉♥çã♦ f :A _→B ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❞✐❛❣r❛♠❛s ❡①♣r❡ss❛ ❜❡♠ ❛ ❉❡✜♥✐çã♦ ✶✳✶✳✶✱ ♣♦✐s ❝❛❞❛ s❡t❛ ✐♥❞✐❝❛ q✉❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ x _∈ A ❡stá ❛ss♦❝✐❛❞♦ ❛ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ y_∈B ✭❋✐❣✉r❛ ✶✳✶✮✳

❋✐❣✉r❛ ✶✳✶✿ ❉❡✜♥✐çã♦ ❞❡ ❋✉♥çã♦ ♣♦r ❉✐❛❣r❛♠❛s

❉❡♥♦t❛♠♦s✱ ❡♥tã♦ ❛ss✐♠✱ ♦ ❡❧❡♠❡♥t♦ y ❛ss♦❝✐❛❞♦ ❛ x ♣♦r y = f(x)✱ s❡♥❞♦ y ❞❡♥♦✲ ♠✐♥❛❞♦ ✐♠❛❣❡♠ ❞❡ x♣❡❧❛ ❢✉♥çã♦ f✳

❋✐❣✉r❛ ✶✳✷✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ❋✉♥çã♦ ♣♦r ▼❡✐♦ ❞❡ ❉✐❛❣r❛♠❛s

❖✉tr♦ ♣♦♥t♦ q✉❡ ❞❡✈❡ s❡r ♦❜s❡r✈❛❞♦ é q✉❡ ❛ ❞❡✜♥✐çã♦ é ❜❡♠ ❡①❛t❛ ♥♦ t❡r♠♦ ♣❛r❛

t♦❞♦✱ ♠♦str❛♥❞♦ ❛ss✐♠ q✉❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❞✐❛❣r❛♠❛s ❝♦♥❢♦r♠❡ ♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r ♣♦❞❡ ♦✉ ♥ã♦ s❡r ✉♠❛ ❢✉♥çã♦✳ P♦❞❡♠♦s t❡r ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ❆ ❛ ♣❛rt✐r ❞♦s q✉❛✐s ♥ã♦ ♣❛rt❡ ♥❡♥❤✉♠❛ s❡t❛ ✭❋✐❣✉r❛ ✶✳✸✭❛✮✮✱ ❝♦♠♦ ♣♦❞❡ ❤❛✈❡r ♠❛✐s ❞❡ ✉♠❛ s❡t❛ ♣❛rt✐♥❞♦ ❞❡ ✉♠ ♠❡s♠♦ ❡❧❡♠❡♥t♦ ❞❡ ❆ ✭❋✐❣✉r❛ ✶✳✸✭❜✮✮✳ ◆❡st❡s ❞♦✐s ❝❛s♦s✱ ♦ ❞✐❛❣r❛♠❛ ❝♦❧♦❝❛❞♦ ♥ã♦ r❡♣r❡s❡♥t❛ ✉♠❛ ❢✉♥çã♦✳

✭❛✮ ✭❜✮

❋✐❣✉r❛ ✶✳✸✿ ❉✐❛❣r❛♠❛s q✉❡ ◆ã♦ ❘❡♣r❡s❡♥t❛♠ ❋✉♥çõ❡s✳

❆♥t❡s ❞❡ ❞❡✜♥✐r♠♦s ❛ r❡❧❛çã♦ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s✱ ✈❛♠♦s ❞❡✜♥✐r ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s ❆ ❡ ❇✱ s❡✉ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦A_×B é ♦ ❝♦♥❥✉♥t♦ A_×B❂④(x, y)❀ x_∈A ❡ y_∈B⑥✳

❈♦♠ ❜❛s❡ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❞✐❛❣r❛♠❛s ❞❛ ❋✐❣✉r❛ ✶✳✶✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ✉♠❛ ❢✉♥çã♦ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❛ s❡❣✉✐r✿

❉❡✜♥✐çã♦ ✶✳✶✳✸✳ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s ❆ ❡ ❇✱ ✉♠❛ r❡❧❛çã♦ ❞❡ ❆ ❡♠ ❇ é ✉♠ s✉❜❝♦♥❥✉♥t♦ F ❞♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ A_×B✱ ♦✉ s❡❥❛✱ F é ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s (x, y)✱ ❝♦♠ x_∈A ❡ y_∈B✳

❖ ❡①❡♠♣❧♦ ❛❜❛✐①♦ ♥♦s ♠♦str❛ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❛ q✉❛❧ ♣♦❞❡♠♦s r❡❝♦rr❡r ♣❛r❛ ❝♦♥s✲ tr✉✐r♠♦s ✉♠❛ r❡❧❛çã♦F ❡♥tr❡ ❝♦♥❥✉♥t♦s ♥ã♦✲✈❛③✐♦s✳

❊①❡♠♣❧♦ ✶✳✶✳✹✳ ❈♦♥s✐❞❡r❡ ♦s ❝♦♥❥✉♥t♦s ❆❂④✵✱ ✶✱ ✷⑥✱ ❇❂④✶✱ ✷✱ ✸✱ ✹⑥ ❡ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ F =_{(x, y)_∈A_×B;y=x+ 1_} ❡ ✈❛♠♦s ❞❡t❡r♠✐♥❛r A_×B ❡ ♦ ❝♦♥❥✉♥t♦ F✳

❖ ❝♦♥❥✉♥t♦ A_×B ❂_{(0,1),(0,2),(0,3),(0,4),(1,1),(1,2),(1,3), (1,4),(2,1),(2,2),(2,3),(2,4)_}.

❖ ❝♦♥❥✉♥t♦F ❞❡✜♥✐❞♦ ❛❝✐♠❛ é ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s q✉❡ ♦❜❡❞❡❝❡♠ ❛ r❡❧❛çã♦ y=x+ 1✳ ▲♦❣♦✱ F❂_{(0,1),(1,2),(2,3)_}.

❖❜s❡r✈❡ q✉❡ F _⊂ A_×B✱ ❝♦♥❢♦r♠❡ ❡①♣♦st♦ ❛❝✐♠❛✳ ➱ ❢á❝✐❧ t❛♠❜é♠ ✈❡r✐✜❝❛r ♣❡❧❛ ❉❡✜♥✐çã♦ ✶✳✶✳✶ q✉❡ ♦ ❝♦♥❥✉♥t♦ F ❂ _{(0,1),(1,2),(2,3)_} r❡♣r❡s❡♥t❛ ✉♠❛ ❢✉♥çã♦

f :A_→B✱ ❥á q✉❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❆ ❡stá r❡❧❛❝✐♦♥❛❞♦ ❛ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❇✳

❆♦ tr❛❜❛❧❤❛r♠♦s ❝♦♠ ✉♠❛ ❢✉♥çã♦ f : A _→ B✱ ❞❡♥♦♠✐♥❛♠♦s ♦s ❝♦♥❥✉♥t♦s A ❡ B✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ ❞♦♠í♥✐♦ ❡ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦f✳ ▲♦❣♦✱ ♥♦ ❊①❡♠♣❧♦ ✶✳✶✳✹✱ ♦ ❝♦♥❥✉♥t♦A é ❞♦♠í♥✐♦ ❞❡ f ❡ ♦ ❝♦♥❥✉♥t♦ B é ❝♦♥tr❛❞♦♠í♥✐♦ ❞❡f✳

❉❛❞❛ ✉♠❛ ❢✉♥çã♦ f : A _→ B✱ ♦ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠✱ ♦✉ s♦♠❡♥t❡ ✐♠❛❣❡♠ ❞❡ f✱ ❞❡♥♦t❛❞♦ ❛q✉✐ ♣♦r Im(f)✱ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s y_∈B t❛❧ q✉❡ ❡①✐st❡ x_∈A ❝♦♠ y=f(x)✱ ♦✉ s❡❥❛✱

Im(f) =_{f(x)_∈B;x_∈A_}. ✭✶✳✶✮

◆♦t❡ q✉❡ Im(f)_⊂B✳

❋✐❣✉r❛ ✶✳✹✿ ❈♦♥❥✉♥t♦s ❉♦♠í♥✐♦✱ ❈♦♥tr❛❞♦♠í♥✐♦ ❡ ■♠❛❣❡♠

➱ ❝♦♠✉♠ tr❛❜❛❧❤❛r♠♦s ❝♦♠ ❢✉♥çõ❡s f : A _→ B t❛✐s q✉❡ A, B _⊂ R✳ ◆❡st❡s ❝❛s♦s✱ ✐♥❞✐❝❛♠♦s ♦ ❡❧❡♠❡♥t♦ f(x) _∈ B ❛ss♦❝✐❛❞♦ ❛ ✉♠ ❡❧❡♠❡♥t♦ ❣❡♥ér✐❝♦ x _∈ A✱ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❡①♣r❡ssã♦ ❡♠ x✱ t❛❧ q✉❛❧ r❡♣r❡s❡♥t❛ ✉♠❛ r❡❣r❛ q✉❡ ❛ ❢✉♥çã♦ ❞❡✈❡ s❛t✐s❢❛③❡r✳ ❖❜s❡r✈❡ ♦ ❡①❡♠♣❧♦ ❛❜❛✐①♦✳

❊①❡♠♣❧♦ ✶✳✶✳✺✳ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ f : R _→ R✱ ❞❛❞❛ ♣♦r _{f(x) = 2x}✳ ❱❛♠♦s ❞❡t❡r✲ ♠✐♥❛r f(0)✱ f(1) ❡ f(2)✳

❖❜s❡r✈❡ q✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ é ✈❡r✐✜❝❛❞❛✱ ♣♦✐s ♣❛r❛ ❝❛❞❛ x _∈ R✱ t❡♠♦s ❛s✲ s♦❝✐❛❞♦ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ r❡❛❧ y = f(x)✳ ❉❡ ❢❛t♦✱ f(0) = 2.0 = 0✱ f(1) = 2.1 = 2 ❡ f(2) = 2.2 = 4✳

❖ ❡①❡♠♣❧♦ ❝✐t❛❞♦ ❛❝✐♠❛✱ t❛♠❜é♠ ♣♦❞❡ s❡r ❞❡♥♦t❛❞♦ ♣❡❧❛ s❡❣✉✐♥t❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛

f :A_→B

x_7→2x.

➱ ❝♦♠✉♠ tr❛❜❛❧❤❛r♠♦s ❝♦♠ ❢✉♥çõ❡s f : A _→ R t❛✐s q✉❡ _{A ⊂} R✳ ◆❡st❡ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡ f é ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✳

P❛r❛ ✜♥❛❧✐③❛r♠♦s✱ ❞❛♠♦s ✉♠❛ ❞❡✜♥✐çã♦ s♦❜r❡ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✳

❉❡✜♥✐çã♦ ✶✳✶✳✻✳ ❬✸❪ ❉❛❞❛ ✉♠❛ ❢✉♥çã♦ f :A _→B✱ ♦ ❣rá✜❝♦ ❞❡ ❢ é ♦ s✉❜❝♦♥❥✉♥t♦ Gf ❞♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ A_×B✱ ❞❡✜♥✐❞♦ ♣♦r Gf =_{(x, y)_∈A_×B;y=f(x)_}✳

◗✉❛♥❞♦A❂B❂R✱ t❡♠♦s q✉❡_A❡_B r❡♣r❡s❡♥t❛♠ r❡t❛s ♥✉♠❡r❛❞❛s ❡_A×Br❡♣r❡s❡♥t❛ ✉♠ ♣❧❛♥♦✱ ♠✉♥✐❞♦ ❝♦♠ ✉♠ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ❚❛❧ s✐st❡♠❛ é ❞❡♥♦t❛❞♦ ♣♦rx0y✳

➱ ✐♠♣♦rt❛♥t❡ ❝♦♠♣r❡❡♥❞❡r q✉❡ ♥❡♠ t♦❞♦ s✉❜❝♦♥❥✉♥t♦ ❞♦ ♣❧❛♥♦ ✭s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ xOy✮ r❡♣r❡s❡♥t❛ ✉♠ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✳ ❙✉♣♦♥❤❛ ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧ f :A _⊂R_→R ❡ ❝♦♥s✐❞❡r❡ q✉❡_{(x0, y0)∈Gf}✳ ▲♦❣♦✱ ♣❡❧❛s ❉❡✜♥✐çõ❡s ✶✳✶✳✶ ❡ ✶✳✶✳✻ t❡♠♦s q✉❡ ♣❛r❛x0 ∈A ❡①✐st❡ ❛♣❡♥❛s ✉♠ y0 ∈B t❛❧ q✉❡ y0 =f(x0)✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡

❛ r❡t❛ ✈❡rt✐❝❛❧ x=x0 ✐♥t❡rs❡❝t❛ ♦ ❣rá✜❝♦ ❞❡ f ❛♣❡♥❛s ✉♠❛ ú♥✐❝❛ ✈❡③✳

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ G é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦ ♣❧❛♥♦ t❛❧ q✉❡ ❛ r❡t❛ ✈❡rt✐❝❛❧ x = x0

✐♥t❡rs❡❝t❛ G ❡♠ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ♣♦♥t♦s✱ ❡♥tã♦ G ♥ã♦ r❡♣r❡s❡♥t❛ ♦ ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ❛❧❣✉♠❛✳ P♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ t❡♠♦sy0 =f(x0) =y1 ❝♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✶✳✺ ❛ s❡❣✉✐r✳

❋✐❣✉r❛ ✶✳✺✿ ◆ã♦ ❘❡♣r❡s❡♥t❛ ✉♠❛ ❋✉♥çã♦

✶✳✷ ❋✉♥çã♦ ❆✜♠

◆❡st❛ s❡çã♦ ❢❛③❡♠♦s ✉♠ ❡st✉❞♦ ❞❛ ❢✉♥çã♦ ❛✜♠✱ ❛♥❛❧✐s❛♥❞♦ ❛ s✉❛ ❞❡✜♥✐çã♦ ❡ s❡✉ ❣rá✜❝♦✳ ❆ ❢✉♥çã♦ ❛✜♠ ❞❡s❡♠♣❡♥❤❛ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ❡♠ ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐✲ ♠❡♥t♦✱ ❝♦♠♦ ❛❞♠✐♥✐str❛çã♦✱ ❡❝♦♥♦♠✐❛✱ ❡♥tr❡ ♦✉tr❛s✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❬✶❪ ❯♠❛ ❢✉♥çã♦ ❢:R _7→R ❝❤❛♠❛✲s❡ ❛✜♠ q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦ _x∈ R ♦ ✈❛❧♦r ❢✭①✮ é ❞❛❞♦ ♣♦r ✉♠❛ ❡①♣r❡ssã♦ ❞♦ t✐♣♦ f(x) = ax +b✱ ♦♥❞❡ a ❡ b _∈ R sã♦ ❝♦♥st❛♥t❡s ❡ a₆= 0✳

❖ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝❧áss✐❝❛ ❞❛ ❢✉♥çã♦ ❛✜♠✳

❊①❡♠♣❧♦ ✶✳✷✳✷✳ ❯♠❛ ❝♦rr✐❞❛ ❞❡ tá①✐ ❝✉st❛ar❡❛✐s ♣♦r ❦♠ r♦❞❛❞♦ ♠❛✐s ✉♠❛ t❛①❛ ✜①❛ ❞❡ b r❡❛✐s✱ ❝❤❛♠❛❞❛ ✧❜❛♥❞❡✐r❛❞❛✧✳ ❊♥tã♦ ♦ ♣r❡ç♦ ❞❡ ✉♠❛ ❝♦rr✐❞❛ ❞❡x❦♠ é f(x) =ax+b r❡❛✐s✳

P❡❧❛ ❉❡✜♥✐çã♦ ✶✳✷✳✶ é ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ ♥✉♠❛ ❢✉♥çã♦ ❛✜♠f(x) = ax+b✱ ♦ ♥ú♠❡r♦ bq✉❡ r❡♣r❡s❡♥t❛f(0) ❝❤❛♠❛✲s❡ ♦ ✈❛❧♦r ✐♥✐❝✐❛❧✱ ❡ q✉❡ ♥♦ ❡①❡♠♣❧♦ r❡♣r❡s❡♥t❛ ❛ ✧❜❛♥❞❡✐✲ r❛❞❛✧✱ ♦✉ s❡❥❛✱ ♦ ✈❛❧♦r ✜①♦✳ ❖ ❝♦❡✜❝✐❡♥t❡ a q✉❡ r❡♣r❡s❡♥t❛ f(1)₋f(0) é ❝❤❛♠❛❞♦ ❞❡ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡f ❡ ♥♦ ❡①❡♠♣❧♦ r❡♣r❡s❡♥t❛ ♦ q✉❡ ❡stá ✈❛r✐❛♥❞♦ ❝♦♥❢♦r♠❡ ❛ q✉✐❧♦♠❡✲ tr❛❣❡♠x r♦❞❛❞❛✳ ❙❡❣✉♥❞♦ ❬✷❪✱ ♦ ♠♦t✐✈♦ ♣❛r❛ ❡st❛ ❞❡♥♦♠✐♥❛çã♦ é q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r

x, h_∈R✱ ❝♦♠ _{h6= 0}✱ t❡♠✲s❡ _{a= [f(x+h)−f(x)]/h}✳ P♦rt❛♥t♦✱_a é ❛ ✈❛r✐❛çã♦ ❞❡_f(x) ♣♦r ✉♥✐❞❛❞❡ ❞❡ ✈❛r✐❛çã♦ ❞❡x✳

■st♦ ❧❡✈❛ à ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s ❢✉♥çõ❡s ❛✜♥s✱ ❞❛❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ❛ s❡❣✉✐r✳

❚❡♦r❡♠❛ ✶✳✷✳✸✳ ✭❈❛r❛❝t❡r✐③❛çã♦ ❞❛s ❋✉♥çõ❡s ❆✜♥s✮ ❙❡❥❛ f : R _7→ R ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ ♦✉ ❞❡❝r❡s❝❡♥t❡✳ ❙❡ ❛ ❞✐❢❡r❡♥ç❛ f(x+h)₋f(x) = φ(h) ❞❡♣❡♥❞❡r ❛♣❡♥❛s ❞❡ h✱ ❡♥tã♦ f é ✉♠❛ ❢✉♥çã♦ ❛✜♠✳

❆ ❞❡♠♦♥str❛çã♦ ❞❡ t❛❧ ❚❡♦r❡♠❛ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♠♦s q✉❡ ❛ ❢✉♥çã♦ f s❡❥❛ ❝r❡s❝❡♥t❡✳ ▲♦❣♦✱ φ : R _7→ R t❛♠❜é♠ é ❝r❡s❝❡♥t❡✱ ❝♦♠φ(0) = 0✳ P❡❧♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✱ t❡♠♦s q✉❡ f(nx) = n.f(x) ♣❛r❛ t♦❞♦ n _∈ N ❡ _{x ∈} R+✳ P♦rt❛♥t♦✱ _{φ(nh) = n.φ(h)} ♣❛r❛ _{n ∈} Z ❡ h_∈R✳

❊♥tã♦✱ ♠❛✐s ✉♠❛ ✈❡③ ♣❡❧♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✱ ♣♦♥❞♦a= φ(1)✱ t❡♠✲s❡ φ(h) = ah ♣❛r❛ t♦❞♦ h _∈R✳ ❖✉ s❡❥❛✱ _{f(x+h)−f(x) = ah}✳ ❈❤❛♠❛♥❞♦ f(0) =b✱ r❡s✉❧t❛f(h) = ah+b✱ ♦✉ s❡❥❛✱ f(x) = ax+b ♣❛r❛ t♦❞♦x_∈R✳

❆ ❞❡♠♦♥str❛çã♦ ♥♦ ❝❛s♦ ❡♠ q✉❡ f é ❞❡❝r❡s❝❡♥t❡ é ❛♥á❧♦❣❛✳

❖ ❝♦❡✜❝✐❡♥t❡at❛♠❜é♠ ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞♦ ❛ ♣❛rt✐r ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ✈❛❧♦r❡s f(x1) ❡f(x2)q✉❡ ❛ ❢✉♥çã♦ f ❛ss✉♠❡ ❡♠ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ✭q✉❛✐sq✉❡r✮x1 ❡ x2✳ ❉❡

❢❛t♦✱ ✈❡r✐✜❝❛✲s❡ q✉❡

f(x1) = ax1+b ✭✶✳✷✮

❡

f(x2) =ax2+b. ✭✶✳✸✮

❙✉❜tr❛✐♥❞♦ ✭✶✳✷✮ ❞❡ ✭✶✳✸✮ ♦❜t❡♠♦s

f(x2)−f(x1) = a(x2−x1)

❡ ♣♦rt❛♥t♦✱

a= f(x2)−f(x1) x2−x1

.

❖ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❛✜♠ é ✉♠❛ r❡t❛✳ P❛r❛ ♣r♦✈❛r♠♦s ❡st❛ ❛✜r♠❛çã♦✱ ✉s❛♠♦s ❛ ❢ór♠✉❧❛ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s P(x1, y1) ❡ Q(x2, y2)✱ s❡❣✉♥❞♦ ❛ q✉❛❧ s❡ t❡♠

d(P, Q) =p(x2−x1)2+ (y2−y1)2✳

❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ❛✜♠ f :R_→R ❞❡✜♥✐❞❛ ♣♦r _{f(x) = ax+b} ❡ s❡❥❛

Gf =_{(x, y); (x, ax+b)_{} ⊂}R2✳ ❙❡❥❛♠ ▼❂_{(x1, ax1+b)}✱ ◆❂_{(x2, ax2+b)}❡ P❂_{(x3, ax3+b)} três ♣♦♥t♦s q✉❛✐sq✉❡r ❞♦ ❣rá✜❝♦✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ s✉♣♦♥❤❛ q✉❡ x1 < x2 <

x3✳ ▼♦str❛♠♦s ❛ s❡❣✉✐r q✉❡ d(M, N) +d(N, P) = d(M, P)✳

❉❡ ❢❛t♦✱ t❡♠♦s q✉❡

d(M, N) =p(x2−x1)2+a2(x2−x1)2 = (x2−x1)

√

1 +a2_.

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱

d(N, P) = (x3−x2)

√

1 +a2_.

▲♦❣♦✱

❞✭▼✱◆✮✰❞✭◆✱P✮❂(x2−x1+x3−x2)

√

1 +a2 ❂_(x

3−x1)

√

1 +a2 ❂❞✭▼✱P✮✳

P♦rt❛♥t♦✱ ✈❡r✐✜❝❛♠♦s q✉❡ três ♣♦♥t♦s q✉❛✐sq✉❡r M✱ N ❡ P s♦❜r❡ ♦ ❣rá✜❝♦ ❞❡f sã♦ ❝♦❧✐♥❡❛r❡s✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ ♦ ❣rá✜❝♦Gf r❡♣r❡s❡♥t❛ ✉♠❛ r❡t❛✳

❋✐❣✉r❛ ✶✳✻✿ ●rá✜❝♦ ❞❛ ❋✉♥çã♦ ❆✜♠

◗✉❛♥❞♦ a > 0✱ ❛ ❢✉♥çã♦ ❛✜♠ f(x) = ax+b é ❝r❡s❝❡♥t❡✱ ✐st♦ é✱ s❡ x1 < x2 ❡♥tã♦

f(x1)< f(x2)✳ ❉❡ ❢❛t♦✱ s❡ x1 < x2✱ ❡♥tã♦ x2−x1 >0✳ ❉❛í

f(x2)−f(x1) =ax2+b−(ax1+b) =a(x2−x1)>0✱

♦✉ s❡❥❛✱f(x1)< f(x2)✳

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ s❡ a < 0✱ f é ❞❡❝r❡s❝❡♥t❡✱ ♦✉ s❡❥❛✱ s❡ x1 < x2✱ ❡♥tã♦ f(x1) >

f(x2)✳

❱❛❧❡ r❡ss❛❧t❛r ♦s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❞❛ ❢✉♥çã♦ ❛✜♠✱ q✉❡ sã♦✿

✶✳ ❆ ❢✉♥çã♦ ❧✐♥❡❛r r❡♣r❡s❡♥t❛❞❛ ♥❛ ❢♦r♠❛ f(x) = ax❀ ❡

✷✳ ❆ ❢✉♥çã♦ ❝♦♥st❛♥t❡ r❡♣r❡s❡♥t❛❞❛ ♥❛ ❢♦r♠❛ f(x) = b✳

✶✳✸ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛

◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ❞♦ tr✐♥ô♠✐♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡ ♦ ❡st✉❞♦ ❞♦ ❣rá✜❝♦ ❞❡st❛ ❢✉♥çã♦ ❝♦♠ s❡✉s ♣r✐♥❝✐♣❛✐s ❡❧❡♠❡♥t♦s✳

❈♦♠❡ç❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✳

❊①❡♠♣❧♦ ✶✳✸✳✶✳ ❆ s♦♠❛ ❞❛s ♠❡❞✐❞❛s ❞❛s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ❧♦s❛♥❣♦ é ✽ ❝♠✳ ◗✉❛❧ ♦ ♠❛✐♦r ✈❛❧♦r ♣♦ssí✈❡❧ ❞❛ ár❡❛ ❞❡ss❡ ❧♦s❛♥❣♦❄

Pr♦❜❧❡♠❛s ❝♦♠♦ ❡st❡ sã♦ s♦❧✉❝✐♦♥❛❞♦s ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳

❉❡✜♥✐çã♦ ✶✳✸✳✷✳ ❬✶❪ ❯♠❛ ❢✉♥çã♦ ❢ : R _7→ R ❝❤❛♠❛✲s❡ q✉❛❞rát✐❝❛ q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦ x_∈R✱ t❡♠✲s❡ _{f(x) = ax}2_{+bx+c✱ ♦♥❞❡ a, b, c}

∈R sã♦ ❝♦♥st❛♥t❡s ❝♦♠ _{a6= 0}✳

❆ ♦r✐❣❡♠ ❞❛s ❢✉♥çõ❡s q✉❛❞rát✐❝❛s ❡stá ❧✐❣❛❞❛ à r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ s❡❣✉♥❞♦ ❣r❛✉✳ ❯♠ ❞♦s ♠❛✐s ❛♥t✐❣♦s ♣r♦❜❧❡♠❛s✱ q✉❡ r❡❝❛❡ ♥✉♠❛ ❡q✉❛çã♦ ❞❡ss❡ t✐♣♦✱ ❝♦♥s✐st❡ ❡♠ ❛❝❤❛r ❞♦✐s ♥ú♠❡r♦s ❝♦♥❤❡❝❡♥❞♦ s✉❛ s♦♠❛ s ❡ s❡✉ ♣r♦❞✉t♦ p✳

❖✉ s❡❥❛✱ s❡ ✉♠ ❞❡ss❡s ♥ú♠❡r♦s ❝❤❛♠❛r♠♦s ❞❡x✱ ♦ ♦✉tr♦ é ❝❤❛♠❛❞♦ ❞❡s₋x✳ ▲♦❣♦✱ ♣❡❧♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛ t❡♠♦s ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦

x.(s₋x) =p. ✭✶✳✹✮

▼❛♥✐♣✉❧❛♥❞♦ ❛ ❊q✉❛çã♦ ✭✶✳✹✮ ❡♥❝♦♥tr❛♠♦s

x2

−sx+p= 0.

❊♥❝♦♥tr❛r ♦ ♥ú♠❡r♦ x ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ ♥ú♠❡r♦ s₋x✱ é r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉x2

−sx+p= 0✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ é ❛❝❤❛r ♦s ✈❛❧♦r❡s ❞❡x ♣❛r❛ ♦s

q✉❛✐s ❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛f(x) =x2

−sx+p s❡ ❛♥✉❧❛✳ ❊ss❡s ✈❛❧♦r❡s sã♦ ❝❤❛♠❛❞♦s ❞❡ ③❡r♦s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ♦✉ ❞❡ r❛í③❡s ❞❛ ❡q✉❛çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡✳

➱ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r q✉❡ s❡ x ❢♦r ✉♠❛ r❛✐③ ❞❛ ❡q✉❛çã♦ x2

−sx+p = 0✱ ❡♥tã♦ s₋x t❛♠❜é♠ ♦ é✱ ♣♦✐s

(s₋x)2

−s(s₋x) +p=s2

−2sx+x2

−s2

+sx+p=x2

−sx+p= 0✳

❉❡✈❡✲s❡ ♦❜s❡r✈❛r ❡♥tr❡t❛♥t♦ q✉❡ ❛♣❡s❛r ❞❡ s❡r ✉♠ ♣r♦❜❧❡♠❛ ❝❧áss✐❝♦✱ ♥❡♠ s❡♠♣r❡ ❡①✐st❡♠ ❞♦✐s ♥ú♠❡r♦s ❝✉❥❛ s♦♠❛ é s ❡ ❝✉❥♦ ♣r♦❞✉t♦ é p✳ P♦r ❡①❡♠♣❧♦✱ ♥ã♦ ❡①✐st❡♠ ❞♦✐s ♥ú♠❡r♦s r❡❛✐s ❝✉❥❛ ❛ s♦♠❛ s❡❥❛ ✻ ❡ ♦ ♣r♦❞✉t♦ s❡❥❛ ✶✵✳ P♦✐s s❡ ♣❡♥s❛r♠♦s ❡♠ ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡♠ q✉❡ ♦ ♣r♦❞✉t♦ s❡❥❛ ✶✵✱ t❡♠♦s ❝♦♠♦ ♣♦ssí✈❡✐s r❡s✉❧t❛❞♦s ✷ ❡ ✺ ♦✉ ✶ ❡ ✶✵✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ t❡♠♦s q✉❡ ❛ s♦♠❛ sã♦ ✼ ♦✉ ✶✶✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❯♠ ❞♦s ♣r♦❝❡❞✐♠❡♥t♦s ♠❛✐s ✐♠♣♦rt❛♥t❡ ❡ út✐❧ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❡st✉❞♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ é ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞♦ q✉❛❞r❛❞♦✳ ❉❡ ✉♠ ♠♦❞♦ ❣❡r❛❧✱ ❞❛❞❛ ❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛f(x) =ax2

+bx+c✱ ♣♦❞❡♠♦s✱ ♠❛♥✐♣✉❧❛♥❞♦✲❛✱ ❡♥❝♦♥tr❛r ❛ ❡①♣r❡ssã♦ ❛❜❛✐①♦✿

f(x) =a(x2+ b

ax) +c=a(x+ b 2a)

2

− b

2

4a +c=a(x+ b 2a)

2

+4ac−b

2

4a . ✭✶✳✺✮

❈❤❛♠❛♥❞♦ − b

2a =m ❡

4ac₋b2

4a =k✱ ✈❡r✐✜❝❛♠♦s ❢❛❝✐❧♠❡♥t❡ q✉❡ k=f(m)✳ ❈♦♠ ❡st❛ ♥♦t❛çã♦✱ t❡♠♦s✱ ♣❛r❛ t♦❞♦x_∈Rq✉❡_{f(x) =a(x−m)}2_+k✱ ♦♥❞❡_m=− b

2a ❡k =f(m)✳

❊st❛ r❡♣r❡s❡♥t❛çã♦ é ❝❤❛♠❛❞❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ❞♦ tr✐♥ô♠✐♦ f(x) = ax2

+bx+c✳ ❆❧❣✉♠❛s ♦❜s❡r✈❛çõ❡s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rá✲ t✐❝❛✿

❖❜s❡r✈❛çã♦ ✶✳✸✳✸✳ ❆ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ♥♦s ❢♦r♥❡❝❡✱ q✉❛♥❞♦b2

−4ac_≥0✱ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ax2_{+bx+c= 0✱ ♣♦✐s ❡st❛ ✐❣✉❛❧❞❛❞❡ é ❡q✉✐✈❛❧❡♥t❡ ❛}

x= −b±

√

b2_−4ac

2a . ✭✶✳✻✮

❆ ✐❣✉❛❧❞❛❞❡ r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❊q✉❛çã♦ ✭✶✳✻✮ é ❛ ❝❤❛♠❛❞❛ ❋ór♠✉❧❛ ❞❡ ❇❤ás❦❛r❛✳ ❉❡ ❢❛t♦✱ ♣❡❧❛ ■❣✉❛❧❞❛❞❡ ✭✶✳✺✮ ❢❛③❡♥❞♦f(x)✐❣✉❛❧ ❛ ③❡r♦ ♦❜t❡♠♦s✿

a(x+ b 2a)

2

+4ac−b

2

4a = 0

a(x+ b 2a)

2

=₋4ac−b

2

4a

(x+ b 2a) 2 = b 2 −4ac 4a2

(x+ b

2a) =±

√

4ac₋b2

2a

x=₋ b 2a ±

√

4ac₋b2

2a

x= −b±

√

b2_−4ac

2a .

◆❛ ❢ór♠✉❧❛x= −b±

√

b2

−4ac

2a ✱ ♦ ♥ú♠❡r♦∆ = b

2

−4acé ❞❡♥♦♠✐♥❛❞♦ ♦ ❞✐s❝r✐♠✐✲ ♥❛♥t❡ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛f(x) =ax2

+bx+c✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡✱ q✉❛♥❞♦ ∆>0✱ ❛ ❡q✉❛çã♦f(x) = 0 t❡♠ ❞✉❛s r❛í③❡s r❡❛✐s ❡ ❞✐❢❡r❡♥t❡s✳ ◗✉❛♥❞♦ ∆ = 0✱ ❛ ♠❡s♠❛ ❡q✉❛çã♦ ♣♦ss✉✐ ❞✉❛s r❛í③❡s r❡❛✐s ❡ ✐❣✉❛✐s ❡ q✉❛♥❞♦∆ <0✱ ❛ ❡q✉❛çã♦ f(x) = 0 ♥ã♦ ♣♦ss✉✐ r❛✐③ r❡❛❧ ✭✈❡❥❛ ❊q✉❛çã♦ ✶✳✻✮✳

❖❜s❡r✈❛çã♦ ✶✳✸✳✹✳ ❆s ❡①♣r❡ssõ❡s−_2ab =m❡ 4ac−b

2

4a =k ❝♦✐♥❝✐❞❡♠ ❝♦♠ ❛s ❝♦♦r❞❡✲ ♥❛❞❛s ❞♦ ✈ért✐❝❡ ❞❛ ♣❛rá❜♦❧❛✱ q✉❡ sã♦ ❞❛❞♦s ♠❛✐s ❛❞✐❛♥t❡✳ ❊st❛s ❝♦♦r❞❡♥❛❞❛s t❛♠❜é♠ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ♣♦r ♠❡✐♦ ❞❛s r❛í③❡s ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳ ❙✉♣♦♥❤❛ q✉❡ x1 = −

b₋√b2

−4ac

2a ❡ x2 =

−b+√b2

−4ac

2a s❡❥❛♠ ❡ss❛s r❛í③❡s✳ ❆ ♠é❞✐❛ ❛r✐t♠ét✐❝❛ ❞❡❧❛s ♥♦s ❞á ❛ ❛❜s❝✐ss❛ ❞♦ ✈ért✐❝❡✱ ♦✉ s❡❥❛✱

xv =₋ b

2a. ✭✶✳✼✮

❙✉❜st✐t✉✐♥❞♦ ♦ ✈❛❧♦r xv ❞❛ ❊q✉❛çã♦ ✶✳✼ ♥❛ ❊q✉❛çã♦ ✶✳✺✱ ♦❜t❡♠♦s

yv =₋∆

4a. ✭✶✳✽✮

❆s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ ✭xv, yv✮ r❡♣r❡s❡♥t❛♠ ♦ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ♦✉ ❞❡ ♠í♥✐♠♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛f(x) =ax2_+bx+c✳

❖ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ é ✉♠❛ ♣❛rá❜♦❧❛✳

❉❡✜♥✐çã♦ ✶✳✸✳✺✳ ❬✶❪ ❉❛❞♦s ✉♠ ♣♦♥t♦ ❋ ❡ ✉♠❛ r❡t❛ ❞ q✉❡ ♥ã♦ ♦ ❝♦♥té♠✱ ❛ ♣❛rá❜♦❧❛ ❞❡ ❢♦❝♦ ❋ ❡ ❞✐r❡tr✐③ ❞ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ ❞✐st❛♠ ✐❣✉❛❧♠❡♥t❡ ❞❡ ❋ ❡

❞❡ ❞✱ ❝♦♥❢♦r♠❡ ❛ ❋✐❣✉r❛ ✶✳✼ ❛❜❛✐①♦✳

❋✐❣✉r❛ ✶✳✼✿ P❛rá❜♦❧❛

❆ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r à ❞✐r❡tr✐③✱ ❜❛✐①❛❞❛ ❛ ♣❛rt✐r ❞♦ ❢♦❝♦✱ ❝❤❛♠❛✲s❡ ♦ ❡✐①♦ ❞❛ ♣❛rá✲ ❜♦❧❛✳ ❖ ♣♦♥t♦ ❞❛ ♣❛rá❜♦❧❛ ♠❛✐s ♣ró①✐♠♦ ❞❛ ❞✐r❡tr✐③ ❝❤❛♠❛✲s❡ ♦ ✈ért✐❝❡ ❞❡ss❛ ♣❛rá❜♦❧❛✳ ❊❧❡ é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ ❝✉❥❛s ❡①tr❡♠✐❞❛❞❡s sã♦ ♦ ❢♦❝♦ ❡ ♦ ♣♦♥t♦ ❉ ✲ ✐♥t❡rs❡çã♦ ❞♦ ❡✐①♦ ❝♦♠ ❛ ❞✐r❡tr✐③✳

Pr♦♣♦s✐çã♦ ✶✳✸✳✻✳ ❈♦♥s✐❞❡r❡ ❛ ♣❛rá❜♦❧❛ ❝✉❥♦ ❢♦❝♦ é ♦ ♣♦♥t♦ ❋❂

0, 1 4a

❡ ❝✉❥❛ ❞✐r❡tr✐③

é ❛ r❡t❛ ❤♦r✐③♦♥t❛❧ y=₋ 1

4a✳ ❊♥tã♦✱ ❛ ❡q✉❛çã♦ ❞❡ t❛❧ ♣❛rá❜♦❧❛ é f(x) =y =ax

2_✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ ❝♦♥✜r♠❛r♠♦s ✐ss♦✱ ✈❛♠♦s ✉t✐❧✐③❛r ❛ ❞❡✜♥✐çã♦ ❞❡ ♣❛rá❜♦❧❛✱ ♦✉ s❡❥❛✱ q✉❡d(P, F) =d(P, Q)✱ ♦♥❞❡ P(x, y) é ✉♠ ♣♦♥t♦ ❞♦ ❣rá✜❝♦ ❡ ◗

x,₋ 1 4a

é ✉♠ ♣♦♥t♦ s♦❜r❡ ❛ ❞✐r❡tr✐③✳

❈♦♠ ❡❢❡✐t♦✱ s❡ d(P, F) = d(P, Q)✱ ❡♥tã♦ t❡♠♦s ♣❡❧❛ ❢ór♠✉❧❛ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s q✉❡

x2

+

ax2

− 1

4a

2

=

ax2

+ 1 4a

2

❉❡s❡♥✈♦❧✈❡♥❞♦ ❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦s

x2+

ax2₋ 1 4a

2

=

ax2+ 1 4a

2

x2

+y2

−2y 1 4a +

1 16a2 =y

2

+ 2y 1 4a + 1 16a2 x2 = y a

y=ax2.

❆s ✜❣✉r❛s ❛❜❛✐①♦ r❡♣r❡s❡♥t❛♠ ♦s ❣rá✜❝♦s ❞❛ ❢✉♥çã♦y=ax2_✳

✭❛✮ ✭❜✮

❋✐❣✉r❛ ✶✳✽✿ ●rá✜❝♦s ❞❛ ❋✉♥çã♦ y=ax2

❆ ♣❛rt✐r ❞♦ ❣rá✜❝♦ ❞❡ f(x) = y=ax2 _{❝❤❡❣❛♠♦s ❛♦ ❣rá✜❝♦ ❞❛s ❢✉♥çõ❡s}

g(x) = a(x₋m)2 _❡

h(x) = a(x₋m)2

+k✳ ❖ ❣rá✜❝♦ ❞❡ g(x) =a(x₋m)2 _{r❡s✉❧t❛ ❞❡}

f(x) =ax2 _{♣❡❧❛ tr❛♥s❧❛çã♦ ❤♦r✐③♦♥t❛❧ ❞❡}

(x, y)_7→(x+m, y)✱ q✉❡ ❧❡✈❛ ♦ ❡✐①♦ ✈❡rt✐❝❛❧x= 0 ♥❛ r❡t❛ ✈❡rt✐❝❛❧ x=m✳

❉♦ ♠❡s♠♦ ♠♦❞♦✱ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ h(x) = a(x₋m)2_{+k r❡s✉❧t❛ ❞❡}

g(x) = a(x₋m)2 _{♣❡❧❛ tr❛♥s❧❛çã♦ ✈❡rt✐❝❛❧ (x, y)}

7→ (x, y +k)✱ q✉❡ ❧❡✈❛ ♦ ❡✐①♦ Ox ♥❛ r❡t❛ y=k ❡ ❛ r❡t❛y =₋ 1

4a ♥❛ r❡t❛y=k− 1 4a✳

❙❡ a > 0✱ ❛ ♣❛rá❜♦❧❛ y = ax2 _{t❡♠ ❛ ❝♦♥❝❛✈✐❞❛❞❡ ✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛ ❡ s❡✉ ✈ért✐❝❡}

(0,0) é ♦ ♣♦♥t♦ ❞❡ ♠❡♥♦r ♦r❞❡♥❛❞❛✳ ❙❡ a < 0✱ ❛ ❝♦♥❝❛✈✐❞❛❞❡ ❞❛ ♣❛rá❜♦❧❛ y = ax2 _é

✈♦❧t❛❞❛ ♣❛r❛ ❜❛✐①♦ ❡ s❡✉ ✈ért✐❝❡ (0,0) é ♦ ♣♦♥t♦ ❞❡ ♠❛✐♦r ♦r❞❡♥❛❞❛✳

➱ ✐♠♣♦rt❛♥t❡ t❛♠❜é♠ ❝♦♠♣r❡❡♥❞❡r ♦ s✐❣♥✐✜❝❛❞♦ ❣rá✜❝♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s a, b, c ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛f(x) = ax2

+bx+c✳

❖ ❝♦❡✜❝✐❡♥t❡a ♠❡❞❡ ❛ ❛❜❡rt✉r❛ ❞❛ ♣❛rá❜♦❧❛✳ ◗✉❛♥t♦ ♠❛✐♦r ❢♦ra ♠❛✐s ❢❡❝❤❛❞❛ s❡rá ❛ ♣❛rá❜♦❧❛ ❡ q✉❛♥t♦ ♠❡♥♦r éa ♠❛✐s ❛❜❡rt❛ é ❛ ♣❛rá❜♦❧❛✳

❖ ❝♦❡✜❝✐❡♥t❡ b é ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ à ♣❛rá❜♦❧❛ ♥♦ ♣♦♥t♦ P❂(0, c)✱ ✐♥t❡r✲ s❡çã♦ ❞❛ ♣❛rá❜♦❧❛ ❝♦♠ ♦ ❡✐①♦y✳ ◆♦ ❈❛♣ít✉❧♦ ✷✱ ❡①♣♦♠♦s s♦❜r❡ ❞❡r✐✈❛❞❛ ❡♠ ✉♠ ♣♦♥t♦ ❡ ❛♣r♦❢✉♥❞❛♠♦s ♠❛✐s s♦❜r❡ ♦ ❝♦❡✜❝✐❡♥t❡bs❡r ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ à ♣❛rá❜♦❧❛✳ ❖ ✈❛❧♦r c q✉❡ r❡♣r❡s❡♥t❛ f(0) é ❛ ♦r❞❡♥❛❞❛ ❞♦ ♣♦♥t♦ ❡♠ q✉❡ ❛ ♣❛rá❜♦❧❛ ❝♦rt❛ ♦ ❡✐①♦Oy✳

P❛r❛ ✜♥❛❧✐③❛r ❛ s❡çã♦✱ r❡s♦❧✈❡♠♦s ♦ ❊①❡♠♣❧♦ ✶✳✸✳✶✳

❘❡♣r❡s❡♥t❛♥❞♦ ❛s ❞✐❛❣♦♥❛✐s ♠❛✐♦r ❡ ♠❡♥♦r ❞♦ ❧♦s❛♥❣♦ ♣♦rD ❡ d✱ t❡♠♦s q✉❡ D+d= 8✳ ❙❛❜❡♠♦s q✉❡ ❛ ár❡❛ ❞♦ ❧♦s❛♥❣♦A é ❞❛❞♦ ♣♦r

A= D.d

2 . ✭✶✳✾✮

❉❛ ✐❣✉❛❧❞❛❞❡ D+d= 8✱ t❡♠♦s

d= 8₋D. ✭✶✳✶✵✮

❙✉❜st✐t✉✐♥❞♦ ❛ ❊q✉❛çã♦ ✭✶✳✶✵✮ ♥❛ ❊q✉❛çã♦ ✭✶✳✾✮✱ ♦❜t❡♠♦s

A= D.(8−D)

2 .

❖✉ s❡❥❛✱

A= −D

2

+ 8D

2 . ✭✶✳✶✶✮

❆ ❡q✉❛çã♦ ❛❝✐♠❛ r❡♣r❡s❡♥t❛ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡♠ q✉❡ ❛ ár❡❛A❡stá ❡♠ ❢✉♥çã♦ ❞❛ ❞✐❛❣♦♥❛❧D✳ ▲♦❣♦✱ ❛ ár❡❛ ♠á①✐♠❛ é ❞❛❞❛ ♣❡❧❛ ♦r❞❡♥❛❞❛ ❞♦ ✈ért✐❝❡ ❞❛ ♣❛rá❜♦❧❛✱ ♦✉ s❡❥❛✱ ♣❡❧❛ ❊q✉❛çã♦ ✭✶✳✽✮✳ ▲♦❣♦✱

yv =₋∆ 4a

A=₋ 42

−4.

−1

2

.0

4.

−1

2

❥á q✉❡ ♦s ❝♦❡✜❝✐❡♥t❡s a✱b ❡c sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱₋1

2✱4 ❡ 0✳ P♦rt❛♥t♦✱

A=₋ 16 (₋2)

A= 8 cm2.

▲♦❣♦✱ ❛ ♠❛✐♦r ár❡❛ ❞♦ ❧♦s❛♥❣♦ ❝✉❥❛ ❛ s♦♠❛ ❞❛s ♠❡❞✐❞❛s ❞❛s ❞✐❛❣♦♥❛✐s é 8 cm é 8cm2_✳

❈❛♣ít✉❧♦ ✷

❱❡❧♦❝✐❞❛❞❡ ❡ ❆❝❡❧❡r❛çã♦ ♣♦r ▼❡✐♦ ❞❡

❉❡r✐✈❛❞❛s

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ ❝❛♣ít✉❧♦ é r❡❧❡♠❜r❛r ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❢ís✐❝♦s✱ ✉s❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦✱ ♣♦r ♠❡✐♦ ❞❡ ❞❡r✐✈❛❞❛s✳

✷✳✶ ❘❡t❛ ❚❛♥❣❡♥t❡ ❡ ❉❡❝❧✐✈❡ ❆♥❣✉❧❛r

◆❡st❛ s❡çã♦ ❞❡✜♥✐♠♦s r❡t❛ t❛♥❣❡♥t❡ q✉❡ ♣❛ss❛ ♣♦r ✉♠ ♣♦♥t♦ ❡♠ ✉♠❛ ❝✉r✈❛ ❡ t❛♠✲ ❜é♠ ❛ ❞❡r✐✈❛❞❛ ♣❡❧❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛✳

■♥✐❝✐❛❧♠❡♥t❡✱ tr❛ç❛♠♦s ❛ r❡t❛ t❛♥❣❡♥t❡ à ✉♠❛ ❝✉r✈❛ ❞❛❞❛ ♥✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣♦♥t♦ ❞❡ss❛ ❝✉r✈❛✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ❛ ❝✉r✈❛ s❡❥❛ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❢✉♥çã♦ f✳

❙❡❥❛ a ❡ f(a) ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ P✱ ♦♥❞❡ ❞❡s❡❥❛✲s❡ tr❛ç❛r ❛ r❡t❛ t❛♥❣❡♥t❡✳ ❈♦♥s✐❞❡r❡ ✉♠ ♣♦♥t♦ ◗ ❞♦ ❣rá✜❝♦ ❞❡ f✱ ❝✉❥❛ ❛❜s❝✐ss❛ é a +h ❡ ♦r❞❡♥❛❞❛ f(a+h)✳ ❈♦♥❢♦r♠❡ ♦ ❣rá✜❝♦ ❞❛ ❋✐❣✉r❛ ✷✳✶ ✱ ♦ ❞❡❝❧✐✈❡ ❞❛ r❡t❛ s❡❝❛♥t❡P Qé ❞❛❞♦ ♣❡❧♦ q✉♦❝✐❡♥t❡

f(a+h)₋f(a)

h ✱

❝❤❛♠❛❞♦ r❛③ã♦ ✐♥❝r❡♠❡♥t❛❧✳

❋✐❣✉r❛ ✷✳✶✿ ❘❡t❛ ❚❛♥❣❡♥t❡ ♥♦ P♦♥t♦ a

❈♦♥s✐❞❡r❡ ♦ ♣♦♥t♦ P ✜①♦✳ ❊♥q✉❛♥t♦ ♦ ♣♦♥t♦ P ❡stá ✜①♦✱ ♦ ♣♦♥t♦ ◗ s❡ ❛♣r♦①✐♠❛ ❞❡ P✱ ♣❛ss❛♥❞♦ ♣♦r s✉❝❡ss✐✈❛s ♣♦s✐çõ❡s Q1✱ Q2✱ Q3✱ ❡t❝✳ ❆ss✐♠✱ ❛ s❡❝❛♥t❡ P Q ❛ss✉♠❡ ❛s

♣♦s✐çõ❡s ❞❛s s❡❝❛♥t❡sP Q1✱P Q2✱ P Q3✱ ❡t❝✳ ❖ q✉❡ ♦❜s❡r✈❛♠♦s ❝♦♠ ❡ss❡ ♣r♦❝❡❞✐♠❡♥t♦

é q✉❡ ❛ r❛③ã♦ ✐♥❝r❡♠❡♥t❛❧✱ q✉❡ é ♦ ❞❡❝❧✐✈❡ ❞❛ s❡❝❛♥t❡✱ s❡ ❛♣r♦①✐♠❛ ❞❡ ✉♠ ✈❛❧♦r m✱ à ♠❡❞✐❞❛ q✉❡ ♦ ♣♦♥t♦ ◗ s❡ ❛♣r♦①✐♠❛ ❞❡ P✳ ❈❛s♦ ✐ss♦ ♦❝♦rr❛✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ q✉❡ é r❡t❛ t❛♥❣❡♥t❡✳

❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❬✹❪ ❆ r❡t❛ t❛♥❣❡♥t❡ à ✉♠❛ ❝✉r✈❛ ♥♦ ♣♦♥t♦ P é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r ❡st❡ ♣♦♥t♦ P ❡ ❝✉❥♦ ❞❡❝❧✐✈❡ ♦✉ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ♦✉ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ é m✳

➱ ✐♠♣♦rt❛♥t❡ s❛❧✐❡♥t❛r q✉❡ ❢❛③❡r ◗ s❡ ❛♣r♦①✐♠❛r ❞❡ P ❝♦♥s✐st❡ ❡♠ ❢❛③❡r ♦ ♥ú♠❡r♦ h❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠♦ ❞❡ ③❡r♦ ♥❛ r❛③ã♦ ✐♥❝r❡♠❡♥t❛❧✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡h❡stá t❡♥❞❡♥❞♦ ❛ ③❡r♦ ❡ r❡♣r❡s❡♥t❛♠♦s ❝♦♠♦ ✧h_→0✧✳

➱ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r q✉❡ q✉❛♥❞♦ h_→0✱ ❛ r❛③ã♦ ✐♥❝r❡♠❡♥t❛❧ s❡ ❛♣r♦①✐♠❛ ❞❡ ✉♠ ✈❛❧♦r ✜♥✐t♦m✳ ❖✉ s❡❥❛✱ m é ♦ ❧✐♠✐t❡ ❞❛ r❛③ã♦ ✐♥❝r❡♠❡♥t❛❧ ❝♦♠ ❤ t❡♥❞❡♥❞♦ ❛ ③❡r♦ ❝❛s♦ ❡st❡ ❧✐♠✐t❡ ❡①✐st❛✳ ▲♦❣♦✱

m = lim h→0

f(a+h)₋f(a)

h . ✭✷✳✶✮

❖ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r ♥♦s ♠♦str❛ ❝♦♠♦ ❞❡t❡r♠✐♥❛r ❛ r❡t❛ t❛♥❣❡♥t❡ ✉t✐❧✐③❛♥❞♦ ❛ s✉❛ ❞❡✜♥✐çã♦✳

❊①❡♠♣❧♦ ✷✳✶✳✷✳ ❊♥❝♦♥tr❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ à ♣❛rá❜♦❧❛ f(x) =x2 _{♥♦ ♣♦♥t♦}

P❂(2,4)✳

❚❡♠♦s q✉❡ f(2) = 22

= 4 ❡ f(2 +h) = (2 +h)2

= 4 + 4h+h2_{✳ ▲♦❣♦✱}

f(2 +h)₋f(2)

h =

4h+h2

h = 4 +h.

P♦rt❛♥t♦✱ s❛❜❡♠♦s ♣♦r ✭✷✳✶✮ q✉❡

m= lim h→0

f(2 +h)₋f(2)

h = 4.

❉❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✱ s❛❜❡✲s❡ q✉❡ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦P(x0, y0)✱ t❡♠ ❝♦♠♦

❡q✉❛çã♦

y₋y0 =m(x−x0).

❊♥tã♦✱

y₋4 = 4(x₋2).

▼❛♥✐♣✉❧❛♥❞♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ ♦❜t❡♠♦s

y= 4x₋4

q✉❡ é ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ♣r♦❝✉r❛❞❛✳

✷✳✷ ■♥t❡r♣r❡t❛çã♦ ●❡♦♠étr✐❝❛ ❡ ❈✐♥❡♠át✐❝❛ ❞❛ ❉❡r✐✲

✈❛❞❛

◆❡st❛ s❡çã♦ ❞❡✜♥✐♠♦s ❛ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ❡♠ ✉♠ ♣♦♥t♦✱ t❛♥t♦ ♣❡❧❛ ✐♥t❡r✲ ♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ q✉❛♥t♦ ♣♦r ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❝✐♥❡♠át✐❝❛ ✭✉t✐❧✐③❛♥❞♦ ♦ ❣rá✜❝♦ ❞♦ ❡s♣❛ç♦ ♣❡❧♦ t❡♠♣♦✮✱ ❛❧é♠ ❞❡ ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ❡①❡♠♣❧♦s ❛♣❧✐❝❛♥❞♦ ❛♠❜❛s ❛s ❞❡✜♥✐çõ❡s✳

◆❛ s❡çã♦ ❛♥t❡r✐♦r ❢♦✐ ❞❡✜♥✐❞♦ ♦ ❞❡❝❧✐✈❡ ❞❡ ✉♠❛ ❝✉r✈❛ y =f(x)✱ ♥✉♠ ♣♦♥t♦ x=a✱ ❝♦♠♦ ♦ ❧✐♠✐t❡ ❞❛ r❛③ã♦ ✐♥❝r❡♠❡♥t❛❧ q✉❛♥❞♦h_→0 ❞❛❞❛ ♣❡❧❛ ❊q✉❛çã♦ ✭✷✳✶✮✳

P❡❧❛ ❊q✉❛çã♦ ✭✷✳✶✮ ♦❜s❡r✈❛✲s❡ q✉❡ ♦ ✈❛❧♦r ❞❡ m ❞❡♣❡♥❞❡ ❞♦ ✈❛❧♦r ❞❡ x = a✳ ❖✉ s❡❥❛✱ ♦ ✈❛❧♦r ❞❡m ❡stá ❡♠ ❢✉♥çã♦ ❞❡a✳ ■st♦ ♥♦s ❞á ❛ ✐❞❡✐❛ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ f ♣❡❧❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛✳

❉❡✜♥✐çã♦ ✷✳✷✳✶✳ ❬✺❪ ❆ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦f ♥♦ ♣♦♥t♦x=a é ✐❣✉❛❧ ❛♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❡ f ♥♦ ♣♦♥t♦ ❞❡ ❛❜s❝✐ss❛ a✳

❉❡♥♦t❛♠♦s ❛ ❞❡r✐✈❛❞❛ ❞❡ f ♥♦ ♣♦♥t♦ a ✭s❡ ❡st❛ ❡①✐st✐r✮ ♣♦r f′

(a)✳ ◆❡st❡ ❝❛s♦ ❡s❝r❡✈❡♠♦s✱

f′

(a) = lim h→0

f(a+h)₋f(a)

h . ✭✷✳✷✮

❖❜s❡r✈❡ q✉❡✱

f′

(x) = lim h→0

f(x+h)₋f(x)

h ✭✷✳✸✮

➱ ✐♠♣♦rt❛♥t❡ s❛❧✐❡♥t❛r♠♦s q✉❡ ❤á ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❝✐♥❡♠át✐❝❛ ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ❞❡r✐✈❛❞❛✳ ❉❛ ❋ís✐❝❛✱ s❛❜❡♠♦s q✉❡ ❛ ♣♦s✐çã♦ ❞❡ ✉♠ ♣♦♥t♦ ♠❛t❡r✐❛❧ ❡♠ ♠♦✈✐♠❡♥t♦✱ s♦❜r❡ ✉♠❛ ❝✉r✈❛ r ✲ tr❛❥❡tór✐❛ ✲ ❝♦♥❤❡❝✐❞❛✱ ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞❛ ❡♠ ❝❛❞❛ ✐♥st❛♥t❡t✱ ♣♦r ♠❡✐♦ ❞❡ s✉❛ ❛❜s❝✐ss❛ s✱ ♠❡❞✐❞❛ s♦❜r❡ ❛ ❝✉r✈❛ r✳ P♦rt❛♥t♦✱ ❛ ❡①♣r❡ssã♦ q✉❡ ❞á ❛ ♣♦s✐çã♦ s ❡♠ ❢✉♥çã♦ ❞♦ ✐♥st❛♥t❡t é

s=s(t)

❊st❛ ❡①♣r❡ssã♦ é ❝❤❛♠❛❞❛ ❡q✉❛çã♦ ❤♦rár✐❛✳

❋✐❣✉r❛ ✷✳✷✿ ❊q✉❛çã♦ ❍♦rár✐❛ ❞♦ ❊s♣❛ç♦ ❡♠ ❋✉♥çã♦ ❞♦ ❚❡♠♣♦

❖❜s❡r✈❛♥❞♦ ❛ ❋✐❣✉r❛ ✷✳✷✱ ❞❛❞♦ ✉♠ ✐♥st❛♥t❡ t0 ❡ s❡♥❞♦ t ✉♠ ✐♥st❛♥t❡ ❞✐❢❡r❡♥t❡ ❞❡

t0✱ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ ❡♥tr❡ ❡st❡s ✐♥st❛♥t❡s✱ ❞❡✜♥✐♥❞♦✲❛ ❝♦♠♦ ♦

q✉♦❝✐❡♥t❡

Vm = s(t)−s(t0) t₋t0

= ∆s

∆t. ✭✷✳✹✮

❈❤❛♠❛✲s❡ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ❡s❝❛❧❛r ❞♦ ♣♦♥t♦ ♥♦ ✐♥st❛♥t❡ t0 ♦✉ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛

♦ ❧✐♠✐t❡

V(t0) = lim

t→t0

Vm= lim t→t0

s(t)₋s(t0)

t₋t0

= lim

∆t→0

∆s ∆t =s

′

(t0). ✭✷✳✺✮

▲♦❣♦✱ s❡❣✉♥❞♦ ❬✺❪ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ s = s(t) ♥♦ ♣♦♥t♦ t=t0 é ✐❣✉❛❧ à ✈❡❧♦❝✐❞❛❞❡ ❡s❝❛❧❛r ❞♦ ♠ó✈❡❧ ♥♦ ✐♥st❛♥t❡ t0✳

❙❛❜❡✲s❡ t❛♠❜é♠ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ v ❞❡ ✉♠ ♣♦♥t♦ ♠❛t❡r✐❛❧ ❡♠ ♠♦✈✐♠❡♥t♦ ♣♦❞❡ ✈❛r✐❛r ❞❡ ✐♥st❛♥t❡ ♣❛r❛ ✐♥st❛♥t❡✳ ❆ ❡q✉❛çã♦ q✉❡ ❞á ❛ ✈❡❧♦❝✐❞❛❞❡v ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦ t é

v =v(t).

❆ ❡①♣r❡ssã♦ ❛❝✐♠❛ é ❝❤❛♠❛❞❛ ❡q✉❛çã♦ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ♥♦ ♣♦♥t♦ t✳

❉❛❞♦ ✉♠ ✐♥st❛♥t❡ t0 ❡ ✉♠ ✐♥st❛♥t❡ t ❞✐❢❡r❡♥t❡ ❞❡ t0✱ ❝❤❛♠❛✲s❡ ❛❝❡❧❡r❛çã♦ ❡s❝❛❧❛r