❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ❚❡rr❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚
❋❯◆➬Õ❊❙ ❊ ❆▲●❯▼❆❙ ❉❊ ❙❯❆❙
P❘❖P❘■❊❉❆❉❊❙
♣♦r▼ár❝✐♦ ❆♥❞ré ❈♦st❛ ❇❛r❜♦s❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❆♥❞ré ●✉st❛✈♦ ❈❛♠♣♦s P❡r❡✐r❛
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋✲ ▼❆❚✴❯❋❘◆✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜✲ t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ❙❡t❡♠❜r♦✴✷✵✶✺
❋❯◆➬Õ❊❙ ❊ ❆▲●❯▼❆❙ ❉❊ ❙❯❆❙
P❘❖P❘■❊❉❆❉❊❙
♣♦r
▼ár❝✐♦ ❆♥❞ré ❈♦st❛ ❇❛r❜♦s❛
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜s✲ s✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚✴❯❋❘◆✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ❋✉♥çõ❡s✳ ❆♣r♦✈❛❞♦ ♣♦r✿
Pr♦❢✳ ❉r✳ ❆♥❞ré ●✉st❛✈♦ ❈❛♠♣♦s P❡r❡✐r❛ ✲ ❯❋❘◆ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❱✐✈✐❛♥❡ ❙✐♠✐♦❧✐ ▼❡❞❡✐r♦s ❈❛♠♣♦s ✲ ❯❋❘◆ ✭▼❡♠❜r♦ ■♥t❡r♥♦✮
Pr♦❢✳ ❉r✳ ❏❡✛❡rs♦♥ ❆❜r❛♥t❡s ❞♦s ❙❛♥t♦s ✲ ❯❋❈● ✭▼❡♠❜r♦ ❊①t❡r♥♦✮
Catalogação da Publicação na Fonte. UFRN / SISBI / Biblioteca Setorial Centro de Ciências Exatas e da Terra – CCET.
Barbosa, Márcio André Costa.
Funções e algumas de suas propriedades / Márcio André Costa Barbosa. - Natal, 2015.
xiv, 84 f. : il.
Orientador: Prof. Dr. André Gustavo Campos Pereira.
Dissertação (Mestrado) – Universidade Federal do Rio Grande do Norte. Centro de Ciências Exatas e da Terra. Programa de Pós-Graduação em Matemática em Rede Nacional.
1. Definição de Função – Dissertação. 2. Algumas propriedades de funções – Dissertação. 3. Modelagem – Dissertação. I. Pereira, André Gustavo Campos. II. Título.
❉❡❞✐❝❛tór✐❛
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ q✉❡♠ ❡✉ ♣♦❞❡r✐❛ ❛❣r❛❞❡❝❡r ♣r✐♠❡✐r❛♠❡♥t❡✱ s❡♥ã♦ àq✉❡❧❡ q✉❡ ❡st❡✈❡ ❡ ♣❡r♠❛✲ ♥❡❝❡ ♣r❡s❡♥t❡ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ q✉❡ ♠❡ ❧❡✈❛♥t❛ q✉❛♥❞♦ tr♦♣❡ç♦ ♦✉ ♠❡s♠♦ q✉❛♥❞♦ ❞❡s❛♥✐♠♦ ❝♦♠ ❛ ✐♥t♦❧❡râ♥❝✐❛ ❞♦s ❤♦♠❡♥s✱ ❡ q✉❡ ♠❡ ❝♦♥❞✉③ ❡♠ s❡✉ ❝❛♠✐♥❤♦ ❞❡ ❧✉③ ❡ s❛❜❡❞♦r✐❛✳ ❱❡r❞❛❞❡✐r♦ ❛♠✐❣♦✱ ❞❡ ❛♠♦r ✐♥✜♥✐t♦ q✉❡ ♥❡♠ ❛ ♠❛t❡♠át✐❝❛ ♠❛✐s ❛✈❛♥ç❛❞❛ ❝♦♥s❡❣✉❡ ❡①♣❧✐❝❛r✳ ❆ss✐♠ é ♦ ♥♦ss♦ ❉❡✉s✱ q✉❡ ❡♥✈✐♦✉ s❡✉ ú♥✐❝♦ ✜❧❤♦ ♣❛r❛ ♥♦s s❛❧✈❛r ❡ ♣❛r❛ ♥♦s ♠♦str❛r ♦ q✉❛♥t♦ ❡❧❡ ♥♦s ❛♠❛ ❞❡ ✈❡r❞❛❞❡✳ ❆♠♦ ❛♦ ❙❡♥❤♦r ❡ ♦ ❛❣r❛❞❡ç♦ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ tr✐st❡③❛✱ ♣♦✐s ♠❡ ✜③❡r❛♠ ❝r❡s❝❡r✱ ❡ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❢❡❧✐❝✐❞❛❞❡ q✉❡ ♦ ❙❡♥❤♦r ♣r♦♣♦r❝✐♦♥♦✉ ♥❛ ♠✐♥❤❛ ✈✐❞❛ ❡ ♥❛ ✈✐❞❛ ❞❡ ♠✐♥❤❛ ❢❛♠í❧✐❛✳
❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦ ♣r♦❢❡ss♦r ❡ ♦r✐❡♥t❛❞♦r ❉r✳ ❆♥❞ré ●✉st❛✈♦✱ ❞♦✉t♦r ♥♦ tít✉❧♦ ❡ ♥❛ ❛rt❡ ❞❡ ❡♥s✐♥❛r✱ q✉❡ ❝♦♠ s✉❛ ♣❛❝✐ê♥❝✐❛ ❡ ❡①tr❡♠❛ ❞❡❞✐❝❛çã♦✱ s❡♠♣r❡ ♠❡ ✐♥❝❡t✐✈♦✉ ❡ ❛♣♦✐♦✉ ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✱ t♦r♥❛♥❞♦ ♣♦ssí✈❡❧ ❝r✐❛r ✉♠❛ ♣r♦♣♦st❛ ❞❡ tr❛❜❛❧❤♦ ❞✐❢❡r❡♥t❡ ❝♦♠ ✐♥✐❝✐❛t✐✈❛ ❞❡ t❡♥t❛r ❛❥✉❞❛r àq✉❡❧❡s q✉❡ ♦ ❧❡❡♠✱ ♠♦str❛♥❞♦ ♦ q✉❛♥t♦ ✈❛❧❡✉ ❛ ♣❡♥❛ ♦ t❡♠♣♦ q✉❡ ♣❛ss❛♠♦s ♣❛r❛ ❞❡s❡♥✈♦❧✈ê✲❧♦✳ ❙❡♠ ♠❡ ❡sq✉❡❝❡r ❞❡ ❛❣r❛❞❡❝❡r ❛ s✉❛ ❡s♣♦s❛✱ ♣r♦❢❛✳ ❉r❛✳ ❱✐✈✐❛♥❡ ❙✐♠✐♦❧✐✱ q✉❡♠ ♠❡ ♦r✐❡♥t♦✉ ♥♦ ✐♥í❝✐♦
❞❡ss❛ ❥♦r♥❛❞❛✱ ♠♦str❛♥❞♦✲s❡ ✉♠❛ ♣❡ss♦❛ ❞❡ ❣r❛♥❞❡ ❝♦r❛çã♦ ❡ ♣r♦❢✉♥❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♥❛ ár❡❛ ❡♠ q✉❡ ❛t✉❛✳
▼❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❡ ❛♠✐❣♦s ❞♦ P❘❖❋▼❆❚✱ s❡♠♣r❡ ❝❛♠✐♥❤❛♥❞♦ ❥✉♥t♦s ❡ ♣❛ss❛♥❞♦ ♣❡❧❛s ♠❡s♠❛s ❞✐✜❝✉❧❞❛❞❡s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ r♦♠♣❡♥❞♦ ❜❛rr❡✐r❛s q✉❡ ♣❛r❡❝✐❛♠ ✐♥q✉❡❜rá✈❡✐s✱ ♠❛s ♠❛♥t❡♥❞♦ s❡♠♣r❡ ✉♠ ❛♠❜✐❡♥t❡ ❛❧❡❣r❡ ❡
❞❡s❝♦♥tr❛í❞♦✱ ❝♦♠ ♣❡q✉❡♥❛s ♣❛✉s❛s ❡♥tr❡ ❛s ❛✉❧❛s ♣❛r❛ ✉♠ ✏❧❛♥❝❤✐♥❤♦✑ ♣r♦✈✐❞❡♥❝✐❛❧✱ q✉❡ ♥♦s ✐♥❞✉③✐❛ ❛ ✉♠ ❞❡❜❛t❡ ❞❡ ❞ú✈✐❞❛s s♦❜r❡ ♦s ❝♦♥t❡ú❞♦s ❛♣r❡s❡♥t❛❞♦s✳
▲❡♠❜r♦ ❞❡ ❛❣r❛❞❡❝❡r ♠✐♥❤❛ ❢❛♠í❧✐❛✿ ❛♠✐❣♦s ✈❡r❞❛❞❡✐r♦s✱ s♦❣r♦✱ s♦❣r❛✱ ❝✉♥❤❛❞♦s✱ ✐r♠ã♦s✱ s♦❜r✐♥❤♦s ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♠❡✉s ♣❛✐s✱ ▼❛♥♦❡❧ ❋é❧✐① ❡ ▼❛r✐❛ ❞❛s ●r❛ç❛s✱ q✉❡ t❛♥t♦ ♠❡ ❛♣♦✐❛r❛♠ ❡ ❛❝r❡❞✐t❛r❛♠ q✉❡ ❡ss❡ ❞✐❛ ✐r✐❛ ❝❤❡❣❛r✳ ❙❡♠ ❝♦♥t❛r ♦s ♠♦♠❡♥t♦s ❞❡ ♦r❛çõ❡s ❡ ❧á❣r✐♠❛s ❞❡ ♠✐♥❤❛ ♠ã❡ ❡ ❞❡ ♠❡✉ ♣❛✐✱ q✉❡ t♦♠❛r❛♠ ♣❛r❛ s✐ ♦ ❢❛r❞♦ ❞❡ ♠✐♥❤❛s ♥♦✐t❡s ❡♠ ❝❧❛r♦✱ ❝♦♠♦ t❛♠❜é♠ ♦s sá❜✐♦s ❝♦♥s❡❧❤♦s ❞❡ ♠❡✉ ✐r♠ã♦ ▼❛✐❦♦♥✱ q✉❡ ❤á ♣♦✉❝♦ t❡♠♣♦ ♣❛ss♦✉ ♣♦r ❡①♣❡r✐ê♥❝✐❛ s❡♠❡❧❤❛♥t❡ ❡✱ q✉❡ ❣r❛ç❛s à ❉❡✉s✱ ❢♦✐ ❜❡♠ s✉❝❡❞✐❞❛✳
❘❡s✉♠♦
◆❡st❡ ❚❈❈ tr❛t❛♠♦s ❞❡ ✉♠ tó♣✐❝♦ ♠✉✐t♦ ❜ás✐❝♦✱ ♣♦ré♠ ❡ss❡♥❝✐❛❧ ❡♠ ▼❛t❡♠át✐❝❛✱ ❛ s❛❜❡r✿ ❋✉♥çõ❡s✳ ▼✉✐t♦s ❛❧✉♥♦s ❡♥tr❛♠ ♥❛ ✉♥✐✈❡rs✐❞❛❞❡ s❡ ♣❡r❣✉♥t❛♥❞♦✿ P❛r❛ q✉❡ s❡r✈❡ ❢✉♥çã♦❄ P❡♥s❛♥❞♦ ♥✐ss♦✱ ♣r♦❝✉r❛♠♦s ♥♦s ❛❞✐❛♥t❛r ❛ ❡ss❛ ♣❡r❣✉♥t❛ ❡ t❡♥t❛♠♦s ♠♦str❛r q✉❡ ❢✉♥çõ❡s ❛♣❛r❡❝❡♠ ❞❡ ♠❛♥❡✐r❛ tã♦ ♥❛t✉r❛❧ q✉❡ ❢♦✐ ♣r❡❝✐s♦ ✉♠ ♥♦♠❡ ♣❛r❛ ❝❛r❛❝t❡r✐③❛r ❡ss❛ ❛ss♦❝✐❛çã♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s✳ ❉❡♣♦✐s q✉❡ ❡ss❛ ❞❡✜♥✐çã♦ ❢♦✐ ❡st❛❜❡❧❡❝✐❞❛✱ ❛♣r♦✈❡✐t❛♠♦s ♣❛r❛ ❛♣r❡s❡♥t❛r ❡ ❢♦r♠❛❧✐③❛r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s q✉❡ ❛s ❢✉♥çõ❡s ♣♦ss✉❡♠✱ ❝♦♠♦✿ ❝r❡s❝✐♠❡♥t♦✴❞❡❝r❡s❝✐♠❡♥t♦✱ ✐♥❥❡t✐✈✐❞❛❞❡✱ ❡t❝✳
P❛❧❛✈r❛s ❝❤❛✈❡✿ ❉❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ ✳ ❆❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❢✉♥çõ❡s ✳ ▼♦❞❡❧❛❣❡♠
❆❜str❛❝t
■♥ t❤✐s ❚❈❈ t❤❡ ❞❡❛❧ ✇✐t❤ ❛ ✉♥❞❡r❧②✐♥❣ ❜✉t ❡ss❡♥t✐❛❧ t♦♣✐❝ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ♥❛♠❡❧②✿ ❋✉♥❝t✐♦♥s✳ ▼❛♥② st✉❞❡♥ts st❛rt t❤❡✐r ❝♦✉rs❡ ✐♥ t❤❡ ❯♥✐✈❡rs✐t② ❛s❦✐♥❣✿ ❲❤② ❞♦ ✇❡ ♥❡❡❞ ❢✉♥❝t✐♦♥s❄ ❲✐t❤ t❤❛t ✐♥ ♠✐♥❞ ✇❡ tr② t♦ ❛♥t❡❝✐♣❛t❡ t♦ t❤✐s q✉❡st✐♦♥ ❛♥❞ ✇❡ tr② t♦ s❤♦✇ t❤❛t ❢✉♥❝t✐♦♥s ❝♦♠❡ ✉♣ s♦ ♥❛t✉r❛❧❧② t❤❛t ❛ ♥❛♠❡ ✇❛s ♥❡❡❞❡❞ t♦ ❡①♣r❡ss t❤❛t ❛ss♦❝✐❛t✐♦♥ ✇❤✐❝❤ ❡①✐sts ❜❡t✇❡❡♥ t❤❡ ❡❧❡♠❡♥ts ♦❢ t✇♦ s❡ts✳ ❆❢t❡r t❤✐s ❞❡✜♥✐t✐♦♥ ✇❛s ❡st❛❜❧✐s❤❡❞✱ ✇❡ ♣r❡s❡♥t ❛♥❞ ❢♦r♠❛❧✐③❡ s♦♠❡ ♦t❤❡r ❝♦♥❝❡♣ts t❤❛t ❢✉♥❝t✐♦♥s ♠✐❣❤t ❤❛✈❡✱ ❛s✿ ❛♥ ✐♥t❡r✈✐❡✇ ♦❢ ❛s❝❡♥❞❡♥❝❡✴❞❡s❝❡♥❞❡♥❝❡✱ ✶✲✶✱ ❡t❝✳
❑❡②✇♦r❞s✿ ❋✉♥❝t✐♦♥✬s ❞❡✜♥✐t✐♦♥✳ ❙♦♠❡ ❢✉♥❝t✐♦♥✬s ♣r♦♣❡rt✐❡s✳ ▼♦❞❡❧❧✐♥❣
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ P❡ss♦❛s ❡ ❛❧❣✉♠❛s ❛ss♦❝✐❛çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✷ ❋❛t✉r❛ ❞♦ ❝❛rtã♦ ❞❡ ❝ré❞✐t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ ❚❛❜❡❧❛ ❞♦s ❈♦rr❡✐♦s ❝♦♠ ✈❛❧♦r❡s ❞❡ ❡♥✈✐♦ ❞❡ ❝❛rt❛s ❝♦♠❡r❝✐❛✐s✳ ✳ ✳ ✳ ✳ ✾ ✷✳✹ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ❛ss♦❝✐❛çã♦ ❡♠ ❢♦r♠❛ ❞❡ ❞✐❛❣r❛♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✸✳✶ ◆♦ ❝ír❝✉❧♦ à ❡sq✉❡r❞❛✱ sen(−α) = −sen(α) = −a✱ ❡ ♥♦ ❝ír❝✉❧♦ à
❞✐r❡✐t❛✱cos(−α) =cos(α) =a ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✸✳✷ ❉✐❛❣r❛♠❛ ❞❡ Pr♦❞✉çã♦ ❞❡ P❡ç❛s ❞❡ ❆✉t♦♠ó✈❡✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✸ ❉✐❛❣r❛♠❛ ❞❡ ❘❡❛❥✉st❡ ❙❛❧❛r✐❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✹✳✶ ❈♦t❛çã♦ ❞♦ ❞ó❧❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✷ ◆ú♠❡r♦ ❞❡ ❤♦♠✐❝í❞✐♦s ♥❛s ❝❛♣✐t❛✐s ❜r❛s✐❧❡✐r❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✳✸ P❡sq✉✐s❛ ❞❡ s❛t✐s❢❛çã♦ ❞♦ ❣♦✈❡r♥♦ ❉✐❧♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✳✹ P❧❛♥♦ ❝♦♠ ❡✐①♦s ♥ã♦✲♦rt♦❣♦♥❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✺ ❉❡s❡♥❤♦ ❞♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ✷ ♥✉♠ ♣❧❛♥♦ ❝♦♠ ❡✐①♦s ♦rt♦❣♦♥❛✐s✳ ✳ ✳ ✳ ✻✸ ✹✳✻ ❉❡s❡♥❤♦ q✉❡ ♠♦str❛ ❝♦♠♦ ✜❝❛r✐❛ ♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ✷ ❡♠ ✉♠ ♣❧❛♥♦✱
❝✉❥♦ â♥❣✉❧♦ ❞❡ ✐♥t❡rs❡çã♦ ❡♥tr❡ ♦s ❡✐①♦s é 45◦
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✹✳✼ ❆❧❣✉♠❛s ✐♥❢♦r♠❛çõ❡s ❞❛❞❛s ❞♦ t❡rr❡♥♦ ♥♦ ❡✐①♦x ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✳ ✻✸
✹✳✽ ■♥❢♦♠❛çõ❡s ❣❡r❛❞❛s ❞♦ t❡rr❡♥♦ ♥♦ ❡✐①♦ y ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✹✳✾ ❋✐❣✉r❛ q✉❡ ♠♦str❛ ♦ ♣♦♥t♦ ✭✶✱✻✻✵✵✵✮ ♦❜t✐❞♦ ♣❡❧❛ ✐♥t❡rs❡çã♦ ❞❛s ♣❡r✲ ♣❡♥❞✐❝✉❧❛r❡s tr❛ç❛❞❛s ♣♦r ✶ ❡ ✻✻✵✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✹✳✶✵ ❋✐❣✉r❛ q✉❡ ♠♦str❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ♦❜t✐❞♦s ♣❡❧❛ ✐♥t❡rs❡çã♦ ❞❡ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✶✶ ❈♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❝♦♠ ✈❛r✐❛çã♦ ❞❡x ❞❡ ✵✱✺ ❡♠ ✵✱✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
✹✳✶✷ ❈♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❡♠ q✉❡ ✐♥s❡r✐♠♦s ❛ ✈❛r✐❛çã♦ ❞❡x ❞❡ ✵✱✶ ❡♠ ✵✱✶✳ ✳ ✻✾
✹✳✶✸ ●rá✜❝♦ ❞❛ ❣❛s♦❧✐♥❛✿ ❙❡♠✐rr❡t❛ ❝♦♠ ♦r✐❣❡♠ ♥♦ ♣♦♥t♦ ✭✵✱✵✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✹✳✶✹ P♦♥t♦s ✭✵✱ ✶✳✸✵✮ ❡ ✭✷✵✱ ✶✳✸✵✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✹✳✶✺ P♦♥t♦s ✭✵✱ ✶✳✸✵✮✱ ✭✷✵✱ ✶✳✸✵✮✱ ✭✷✵✱ ✶✳✽✵✮ ❡ ✭✺✵✱ ✶✳✽✵✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✹✳✶✻ ✈ár✐♦s ♣♦♥t♦s✱ ✉♥s ♣❡rt❡♥❝❡♠ ❛♦ ❣rá✜❝♦ ❞❡f ✭♣♦♥t♦s ❛③✉✐s✮✱ ❡ ♦✉tr♦s
♥ã♦ ♣❡rt❡♥❝❡♠ ✭♣♦♥t♦s ❜r❛♥❝♦s✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✹✳✶✼ ●rá✜❝♦ ❞♦s ❝♦rr❡✐♦s✿ ✈ár✐♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛s ❡♠ ❢♦r♠❛ ❞❡ ❡s❝❛❞❛✳ ✳ ✼✸ ✹✳✶✽ ❋✐❣✉r❛ q✉❡ ♠♦str❛ ♦ ❣rá✜❝♦ ❞❡f ✭❧✐♥❤❛ ❛③✉❧✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹
✹✳✶✾ ❖ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞♦ ❧❛❞♦ ❡sq✉❡r❞♦ ♣♦❞❡ s❡r ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ♥ã♦ é ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✹✳✷✵ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ ✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✷✶ ❋✐❣✉r❛ q✉❡ ♠♦str❛ ♦ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ✭❧✐♥❤❛ ✈❡r❞❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✹✳✷✷ ❋✐❣✉r❛ q✉❡ ♠♦str❛ ❛ ✐♠❛❣❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦ ✭❧✐♥❤❛ ✈❡r❞❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✹✳✷✸ ❋✐❣✉r❛ q✉❡ ♠♦str❛ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦f : [−2,2]−→
R ❝♦♠ ❧❡✐ ❞❡ ❢♦r♠❛çã♦f(x) =x2−1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✹✳✷✹ ❋✐❣✉r❛ ♠♦str❛♥❞♦ ♦ ❞♦♠í♥✐♦ ❞❡ f✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾
✹✳✷✺ ❋✐❣✉r❛ ♠♦str❛♥❞♦ ❛ ✐♠❛❣❡♠ ❞❡ f✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾
✹✳✷✻ ❋✐❣✉r❛ ❝♦♠ ♦ ❣rá✜❝♦ ✭❧✐♥❤❛ ❛③✉❧✮ ❞❡ ✉♠❛ ❢✉♥çã♦ ✐♥❥❡t♦r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✹✳✷✼ ●rá✜❝♦ q✉❡ ♠♦str❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❧✐tr♦s ❞❡ ✉♠❛ ❝❛✐①❛ ❞✬á❣✉❛ ❡♠
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✷✳✶ ◗✉❛♥t✐❞❛❞❡s ❞❡ ❧✐tr♦s× Pr❡ç♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✷✳✷ ▼❡s❡s ❡ ❛♥♦s× P❡s♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✷✳✸ ▼❡s❡s✱ ❛♥♦s✱ ♣❡s♦s ❡ ❛❧t✉r❛s× ■▼❈✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✷✳✹ ❆❧✉♥♦s✱ ♣❡s♦s ❡ ❛❧t✉r❛s × ■▼❈✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✷✳✺ ❉✐❛s ❡♠ ❛tr❛s♦ × ❏✉r♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✷✳✻ ▼❡tr♦s ❛❝r❡s❝✐❞♦s ❛ ✶✵♠ × Pr❡ç♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✸✳✶ ❚❡♠♣♦ ❞❡ ✉s♦× ✈❛❧♦r❡s ❞♦ ❝❛rr♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✸✳✷ ■❞❛❞❡× ❙❛❧ár✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✸ ❆❝rés❝✐♠♦s ❞❡ ✵✱✺ ♠❡tr♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✹✳✶ ❚❛❜❡❧❛ ♣❛r❛ x✈❛r✐❛♥❞♦ ❞❡ ✶ ❡♠ ✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✹✳✷ ❚❛❜❡❧❛ ♣❛r❛ x✈❛r✐❛♥❞♦ ❞❡ ✵✱✺ ❡♠ ✵✱✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❘❡❝♦♥❤❡❝❡♥❞♦ ✉♠❛ ❋✉♥çã♦ ♥♦ ❈♦t✐❞✐❛♥♦ ✸
✷✳✶ ❉♦♠í♥✐♦ ❞❡ ✉♠❛ ❆ss♦❝✐❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ▲❡✐ ❞❡ ❋♦r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸✳✶ ❘❡❧❛❝✐♦♥❛♥❞♦ ❉♦♠í♥✐♦ ❡ ▲❡✐ ❞❡ ❋♦r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✷ ■❣✉❛❧❞❛❞❡ ❞❡ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✹ ■♠❛❣❡♠ ❞❡ ✉♠❛ ❋✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸ ❆❧❣✉♠❛s Pr♦♣r✐❡❞❛❞❡s ❞❡ ❋✉♥çõ❡s ✸✶
✸✳✶ ❋✉♥çõ❡s ■♥❥❡t♦r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷ ❋✉♥çõ❡s ❙♦❜r❡❥❡t♦r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✸ ❋✉♥çõ❡s ❈r❡s❝❡♥t❡s✱ ❉❡❝r❡s❝❡♥t❡s✱ ◆ã♦✲❈r❡s❝❡♥t❡s ❡ ◆ã♦✲❉❡❝r❡s❝❡♥t❡s
✲ ❋✉♥çõ❡s ▼♦♥ót♦♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✹ ❘❡str✐çã♦ ❡ ❊①t❡♥sã♦ ❞❡ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✺ ❋✉♥çõ❡s P❛r❡s ❡ ❋✉♥çõ❡s ❮♠♣❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✻ ❋✉♥çõ❡s ❈♦♠♣♦st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✼ ❋✉♥çõ❡s ■♥✈❡rs❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✹ ❖ ●rá✜❝♦ ❞❡ ✉♠❛ ❋✉♥çã♦ ✻✵
✹✳✶ ❈♦♥str✉✐♥❞♦ ❆❧❣✉♥s ●rá✜❝♦s ❞❡ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✷ ❉♦♠í♥✐♦ ❞❡ ✉♠❛ ❋✉♥çã♦ ❆tr❛✈és ❞♦ ●rá✜❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✸ ■♠❛❣❡♠ ❞❡ ✉♠❛ ❋✉♥çã♦ ❆tr❛✈és ❞♦ ●rá✜❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✹✳✹ ❋✉♥çõ❡s ■♥❥❡t♦r❛s ❆tr❛✈és ❞♦ ●rá✜❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✹✳✺ ●rá✜❝♦s ❞❡ ❋✉♥çõ❡s ❈r❡s❝❡♥t❡s✱ ❉❡❝r❡s❝❡♥t❡s✱ ◆ã♦✲❉❡❝r❡s❝❡♥t❡s ❡
◆ã♦✲❈r❡s❝❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✽✹
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❱❡♠♦s q✉❡ ❢✉♥çõ❡s é ✉♠ ❛ss✉♥t♦ ♠✉✐t♦ t❡♠✐❞♦ ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ♠é✲ ❞✐♦✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♥st❛t❛♠♦s q✉❡ ♦s ❛❧✉♥♦s ❝❤❡❣❛♠ ❛♦ ❡♥s✐♥♦ s✉♣❡r✐♦r s❡♠ ❡♥t❡♥✲ ❞❡r r❡❛❧♠❡♥t❡ t❛❧ ❝♦♥❝❡✐t♦✳ ■ss♦ ❝♦♠♣r♦♠❡t❡ ♠✉✐t♦ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ♣♦st❡r✐♦r ❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❛❧❣✉♠❛s ❢✉♥çõ❡s ♣♦ss✉❡♠✱ ❛ s❛❜❡r✿ ■♥❥❡t✐✈✐❞❛❞❡✱ s♦❜r❡❥❡t✐✈✐❞❛❞❡✱ ❜✐❥❡t✐✈✐❞❛❞❡✱ ❝r❡s❝✐♠❡♥t♦✴❞❡❝r❡s❝✐♠❡♥t♦✱ ♣❛r✐❞❛❞❡✱ ❡t❝✳
❆❝r❡❞✐t❛♠♦s q✉❡ ✐ss♦ ❛❝♦♥t❡❝❡ ❞❡✈✐❞♦ ❛♦ ♠♦❞♦ q✉❡ ♦ ❛ss✉♥t♦ é ❛♣r❡s❡♥t❛❞♦ ❛♦s ❛❧✉♥♦s✳ P❛r❡❝❡✲♥♦s q✉❡ ❛ ❢♦r♠❛ ❞❡ ❝♦♠❡ç❛r ♦ ❡st✉❞♦ ❞❡ ❢✉♥çã♦ ♥ã♦ ❞❡✈❡r✐❛ s❡r ♣❡❧❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ ❞❡✜♥✐çã♦ ❡ ❡♠ s❡❣✉✐❞❛ ❡①❡♠♣❧♦s ❛♣r❡s❡♥t❛♥❞♦ s✐t✉❛çõ❡s q✉❡ s❛t✐s❢❛③❡♠ ♦ ❝♦♥❝❡✐t♦ q✉❡ ❛❝❛❜♦✉ ❞❡ s❡r ❛♣r❡s❡♥t❛❞♦✳ ❆❝r❡❞✐t❛♠♦s q✉❡ ❛ ♠❛♥❡✐r❛ ❞❡ ❛♣r❡s❡♥t❛r t❛❧ ❝♦♥❝❡✐t♦ ❞❡✈❡r✐❛ s❡r ❡♠ ✉♠❛ ♦r❞❡♠ ✐♥✈❡rs❛✳ Pr✐♠❡✐r♦✱ ❞❡✈❡rí❛♠♦s ❛♣r❡s❡♥t❛r s✐t✉❛çõ❡s ❞♦ ❝♦t✐❞✐❛♥♦ ❛♦s ❛❧✉♥♦s✳ ❙✐t✉❛çõ❡s q✉❡ ♣♦ss❛♠ s❡r ♠♦❞❡❧❛❞❛s ❞❡ ♠❛♥❡✐r❛ ♥❛t✉r❛❧✱ ❡ ❡st❛s ♥♦s ❧❡✈❡♠ ♣❛r❛ ✉♠❛ ❢♦r♠❛❧✐③❛çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡s❡❥❛❞♦✳ ❚❡♥t❛♥❞♦ ❡①❡♠♣❧✐✜❝❛r ❝♦♠♦ ❞❡✈❡r✐❛ s❡r ❢❡✐t❛ ❡ss❛ ❛❜♦r❞❛❣❡♠✱ ❞❡s❡♥✈♦❧✈❡♠♦s ❡st❡ tr❛❜❛❧❤♦✳ ❆q✉✐✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❢♦r♠❛ ❛❧t❡r♥❛t✐✈❛✱ q✉❡ ❛❝r❡❞✐t❛♠♦s s❡r ♠❛✐s ❡✜❝❛③ ❡ ♠❛✐s ♥❛t✉r❛❧ ❞❡ ❛♣r❡s❡♥t❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦✱ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❈r❡✲ ♠♦s q✉❡ ❡ss❛ ❛❜♦r❞❛❣❡♠ t♦♠❡ ✉♠ ♣♦✉❝♦ ♠❛✐s ❞❡ t❡♠♣♦✱ ❡ss❡ t❡♠♣♦ ❛ ♠❛✐s s❡rá ❝♦♠♣❡♥s❛❞♦ ♠❛✐s ❛ ❢r❡♥t❡ q✉❛♥❞♦ ♥❡❝❡ss✐t❛r♠♦s ♥♦✈❛♠❡♥t❡ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çõ❡s
❡♠ ❝♦♥t❡①t♦s ♠❛✐s ❡s♣❡❝í✜❝♦s✳ ❆ r❡✈✐sã♦ ❞♦ ❝♦♥❝❡✐t♦ s❡ ❞❛rá ❞❡ ❢♦r♠❛ ♠❛✐s rá♣✐❞❛✳ ❊st❛ ❞✐ss❡rt❛çã♦ é ❞✐✈✐❞✐❞❛ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛✲ ♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦✱ ❛♣r❡s❡♥t❛♥❞♦ ❛ ✐❞❡✐❛ ❞❡ ❞♦♠í♥✐♦✱ ❝♦♥tr❛❞♦♠í♥✐♦✱ ✐♠❛❣❡♠ ❡ ❧❡✐ ❞❡ ❢♦r♠❛çã♦✱ ❜❡♠ ❝♦♠♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ❞❡ ❢✉♥çõ❡s✳ ◆♦ t❡r❝❡✐r♦ ❝❛✲ ♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❝♦♠♦✿ ■♥❥❡t✐✈✐❞❛❞❡✱ s♦❜r❡❥❡t✐✈✐❞❛❞❡ ❡ ❜✐❥❡t✐✈✐❞❛❞❡✱ ❡①t❡♥sã♦✴r❡str✐çã♦✱ ❝♦♠♣♦st❛ ❡ ✐♥✈❡rs❛ ❞❡ ❢✉♥çõ❡s✱ ❝♦♥❝❡✐t♦ ❞❡ ❝r❡s❝✐✲ ♠❡♥t♦✴❞❡❝r❡s❝✐♠❡♥t♦ ❡ ♣❛r✐❞❛❞❡✳ ❆❧é♠ ❞✐ss♦✱ ♥♦ q✉❛rt♦ ❝❛♣ít✉❧♦✱ ❛♣r♦✈❡✐t❛♠♦s ♣❛r❛ ❞❛r ❛ ✐❞❡✐❛ ❞❡ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ❡ ❝♦♠♦ ✈❡r✐✜❝❛r ❛tr❛✈és ❞♦ ❣rá✜❝♦✱ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❛♣r❡s❡♥t❛❞❛s ♥♦ ❈❛♣ít✉❧♦ ✸✳
❈❛♣ít✉❧♦ ✷
❘❡❝♦♥❤❡❝❡♥❞♦ ✉♠❛ ❋✉♥çã♦ ♥♦
❈♦t✐❞✐❛♥♦
❱♦❝ê ❥á ♣❛r♦✉ ♣❛r❛ ♣❡♥s❛r ❝♦♠♦ ❛s ♣❡ss♦❛s tê♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❛ss♦❝✐❛r ❛s ❝♦✐s❛s❄ P♦r ❡①❡♠♣❧♦✱ ❛ss♦❝✐❛❞♦ ❛♦ ♥♦♠❡ ❞❡ ✉♠❛ ♣❡ss♦❛✱ t❡♠♦s ♦ ❈P❋ ✭❈❛❞❛str♦ ❞❡ P❡ss♦❛ ❋ís✐❝❛✮ ❡ ♦ ❘● ✭❘❡❣✐str♦ ●❡r❛❧✮✳ ❏á ♥❛ ❡s❝♦❧❛✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♥❛s ❛✉❧❛s ❞❡ ❡❞✉❝❛çã♦ ❢ís✐❝❛✱ t❡♠♦s ❧✐❣❛❞♦s ❛♦ ♥♦♠❡✱ ♦ ♣❡s♦ ❡ ❛ ❛❧t✉r❛✳ ◆❛ ✉♥✐✈❡rs✐❞❛❞❡✱ t❡♠♦s ♦ ♥ú♠❡r♦ ❞❛ ♠❛trí❝✉❧❛✳ ❊ss❡s sã♦ ❡①❡♠♣❧♦s ❞❡ ❛❧❣✉♠❛s ❞❛s ✐♥ú♠❡r❛s ❛ss♦❝✐❛çõ❡s q✉❡ ♣♦❞❡♠ s❡r ❝r✐❛❞❛s ❡♥✈♦❧✈❡♥❞♦ ♦ ♥♦♠❡ ❞❡ ✉♠❛ ♣❡ss♦❛✳ ❊sq✉❡♠❛t✐❝❛♠❡♥t❡✱ ♣♦❞❡♠♦s ✐♠❛❣✐♥❛r ❛❧❣♦ ❞❛ ❢♦r♠❛ ❞❛ ❋✐❣✉r❛ ✷✳✶✳
❋✐❣✉r❛ ✷✳✶✿ P❡ss♦❛s ❡ ❛❧❣✉♠❛s ❛ss♦❝✐❛çõ❡s✳
❱❡❥❛♠♦s ♦✉tr❛s s✐t✉❛çõ❡s ❞♦ ♥♦ss♦ ❞✐❛ ❛ ❞✐❛ ❡♠ q✉❡ ❛ss♦❝✐❛çõ❡s ❡♥tr❡ ♦❜❥❡t♦s ❞❡ ✈ár✐❛s ♥❛t✉r❡③❛s ❛♣❛r❡❝❡♠ ♥❛t✉r❛❧♠❡♥t❡✳
❊①❡♠♣❧♦ ✷✳✵✳✶ ❯♠❛ ♣❡ss♦❛ s❛✐ ❞❡ ❝❛s❛ ♣❛r❛ ♦ tr❛❜❛❧❤♦ ❞❡ ❝❛rr♦ ❡✱ ❛♦ ♣❛ss❛r ♣♦r ✉♠ ♣♦st♦ ❞❡ ❝♦♠❜✉stí✈❡❧✱ ❡❧❛ ❞❡❝✐❞❡ ♣❛r❛r ♦ ❝❛rr♦ ♣❛r❛ ❛❜❛st❡❝❡r ❝♦♠ ❣❛s♦❧✐♥❛✱ ❝✉❥♦ ♣r❡ç♦ ❞♦ ❧✐tr♦ ❡stá três r❡❛✐s ❡ tr✐♥t❛ ❡ q✉❛tr♦ ❝❡♥t❛✈♦s ✭❘✩ ✸✱✸✹✮✳ ❉❛í✱ ❡♥q✉❛♥t♦ ♦ ❢r❡♥t✐st❛ ❛❜❛st❡❝❡ ♦ t❛♥q✉❡✱ ❛ ♣❡ss♦❛ ✜❝❛ ♦❜s❡r✈❛♥❞♦ ❛ ♠❛r❝❛çã♦ ❞❛ ❜♦♠❜❛ ❛tr❛✈és ❞♦ ✈✐s♦r✱ ♥♦ q✉❛❧ ❛♣❛r❡❝❡✱ ❞❡ ❢♦r♠❛ ✐♥st❛♥tâ♥❡❛✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❣❛s♦❧✐♥❛ ❛❜❛st❡❝✐❞❛ ❡ ♦ ♣r❡ç♦ ❛ s❡r ♣❛❣♦✳ ❈❧❛r♦ q✉❡ ❡st❡s ✈❛❧♦r❡s ♠✉❞❛♠ r❛♣✐❞❛♠❡♥t❡ à ♠❡❞✐❞❛ q✉❡ ♦ t❛♥q✉❡ é ❛❜❛st❡❝✐❞♦✱ ♠❛s✱ ❛✐♥❞❛ ❛ss✐♠✱ ❡❧❛ ✈❡r✐✜❝❛ ❡ ❛♥♦t❛ ❛❧❣✉♥s ✈❛❧♦r❡s✱ ❝♦♠♦ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✷✳✶✳
▲✐tr♦s Pr❡ç♦ ✶✱✹ ✹✱✻✽ ✻✱✾ ✷✸✱✵✺ ✶✶✱✺ ✸✽✱✹✶ ✶✼✱✷ ✺✼✱✹✺ ✷✷✱✽ ✼✻✱✶✺ ✷✾✱✹ ✾✽✱✷✵ ✹✵ ✶✸✸✱✻✵
❚❛❜❡❧❛ ✷✳✶✿ ◗✉❛♥t✐❞❛❞❡s ❞❡ ❧✐tr♦s×Pr❡ç♦s✳
❱❡♠♦s✱ ♥❡st❡ ❡①❡♠♣❧♦✱ q✉❡ ❡①✐st❡ ❛❧❣✉♠❛ ❧✐❣❛çã♦ ❡♥tr❡ ♦s ✈❛❧♦r❡s ❞❛ ❝♦❧✉♥❛ ❞❛ ❡sq✉❡r❞❛ ❝♦♠ ❛q✉❡❧❡s ❞❛ ❝♦❧✉♥❛ ❞❛ ❞✐r❡✐t❛✳ ◆ã♦ sã♦ ♥ú♠❡r♦s ❡s❝♦❧❤✐❞♦s ❛♦ ❛❝❛s♦ q✉❡ ❛♣❛r❡❝❡♠ ❧❛❞♦ ❛ ❧❛❞♦✳
❊①❡♠♣❧♦ ✷✳✵✳✷ ◆❛ ♠❛✐♦r✐❛ ❞❛s ❛❝❛❞❡♠✐❛s é ❝♦♠✉♠✱ ♥♦ ♣r✐♠❡✐r♦ ❞✐❛ ❞❡ ❛✉❧❛✱ ♦ ✐♥str✉t♦r ❢❛③❡r ❛♥♦t❛çõ❡s s♦❜r❡ ❛s ♠❡❞✐❞❛s ❞♦s ❛❧✉♥♦s✿ ❝✐♥t✉r❛✱ ❜í❝❡♣s✱ ♣❡s♦✱ ❛❧t✉r❛✱ ❡t❝✳ ❉❡♣❡♥❞❡♥❞♦ ❞♦ q✉❡ ❡❧❡ q✉❡✐r❛ ❛❝♦♠♣❛♥❤❛r✱ é ♣♦ssí✈❡❧ ❢❛③❡r ✈ár✐❛s ❛ss♦❝✐❛çõ❡s✳
P♦r ❡①❡♠♣❧♦✱ s❡ ❡❧❡ q✉❡r ❛❝♦♠♣❛♥❤❛r ♦ ♣❡s♦ ❞♦ ❛❧✉♥♦ ♠ês ❛ ♠ês✱ ❡❧❡ ♣♦❞❡rá ❝♦♥str✉✐r ✉♠❛ t❛❜❡❧❛✱ ❝♦♠♦ ❛ ❚❛❜❡❧❛ ✷✳✷✳
▼ês ✲ ❆♥♦ P❡s♦ ❖✉t✉❜r♦ ✲ ✷✵✶✹ ✽✽ ❦❣ ◆♦✈❡♠❜r♦ ✲ ✷✵✶✹ ✾✵ ❦❣ ❉❡③❡♠❜r♦ ✲ ✷✵✶✹ ✽✺ ❦❣ ❏❛♥❡✐r♦ ✲ ✷✵✶✺ ✽✺ ❦❣ ❋❡✈❡r❡✐r♦ ✲ ✷✵✶✺ ✽✵ ❦❣
❚❛❜❡❧❛ ✷✳✷✿ ▼❡s❡s ❡ ❛♥♦s×P❡s♦s✳
P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ❡❧❡ q✉❡r ❛❝♦♠♣❛♥❤❛r ♦ ■▼❈ ❞♦ ❛❧✉♥♦ ♠ês ❛ ♠ês✱ s❡rá ♣r❡❝✐s♦✱ ❛❧é♠ ❞♦ ♠ês ❡ ❞♦ ❛♥♦✱ ♦ ♣❡s♦ ❡ ❛ ❛❧t✉r❛ ♣❛r❛ ❝❛❞❛ ♠ês✱ ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ❚❛❜❡❧❛ ✷✳✸✳
▼ês ✲ ❆♥♦ ✲ P❡s♦ ✲ ❆❧t✉r❛ ■▼❈ ❏❛♥❡✐r♦ ✲ ✷✵✶✹ ✲ ✽✺ ❦❣ ✲ ✶✱✼✽ ♠ ✷✻✱✷✸ ❋❡✈❡r❡✐r♦ ✲ ✷✵✶✹ ✲ ✾✺ ❦❣ ✲✶✱✼✽ ♠ ✷✾✱✾✽ ▼❛rç♦ ✲ ✷✵✶✹ ✲ ✾✸ ❦❣ ✲ ✶✱✼✽ ♠ ✷✾✱✸✺ ❆❜r✐❧ ✲ ✷✵✶✹ ✲ ✾✵ ❦❣ ✲ ✶✱✼✽ ♠ ✷✽✱✹✶ ▼❛✐♦ ✲ ✷✵✶✹ ✲ ✾✷ ❦❣ ✲ ✶✱✼✽ ♠ ✷✾✱✵✹
❚❛❜❡❧❛ ✷✳✸✿ ▼❡s❡s✱ ❛♥♦s✱ ♣❡s♦s ❡ ❛❧t✉r❛s×■▼❈✳
❈❛s♦ ♦ ✐♥str✉t♦r q✉❡✐r❛ ❛♣❡♥❛s ❢❛③❡r ✉♠ ❧❡✈❛♥t❛♠❡♥t♦ s♦❜r❡ ♦ ■▼❈ ❛t✉❛❧ ❞♦s ❛❧✉♥♦s✱ ❡❧❡ ♠♦♥t❛rá ✉♠❛ t❛❜❡❧❛ ❝♦♠♦ ❛ ❚❛❜❡❧❛ ✷✳✹✳
❆❧✉♥♦s ❞❛ ❛❝❛❞❡♠✐❛ P❡s♦ ✲ ❆❧t✉r❛ ■▼❈ ❏♦ã♦ ✽✵ ❦❣ ✲ ✶✱✼✽ ♠ ✷✺✱✷✺ P❛trí❝✐❛ ✺✵ ❦❣ ✲ ✶✱✻✹ ♠ ✶✽✱✺✾ ▼❛r❝♦s ✼✵ ❦❣ ✲ ✶✱✼✵ ♠ ✷✹✱✷✷ ❆♥❞r❡③❛ ✻✺ ❦❣ ✲ ✶✱✻✽ ♠ ✷✸✱✵✸ ❙❛♠✉❡❧ ✽✺ ❦❣ ✲ ✶✱✼✺ ♠ ✷✼✱✼✻
❚❛❜❡❧❛ ✷✳✹✿ ❆❧✉♥♦s✱ ♣❡s♦s ❡ ❛❧t✉r❛s×■▼❈✳
❊①❡♠♣❧♦ ✷✳✵✳✸ ◆❛ ❝✐❞❛❞❡ ❞❡ ◆❛t❛❧✱ ❝❛♣✐t❛❧ ❞♦ ❡st❛❞♦ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡✱ ❛s t❛r✐❢❛s ♣r❛t✐❝❛❞❛s ♣❡❧♦s t❛①✐st❛s✱ s❡❣✉♥❞♦ ❛ ❙❊▼❖❇ ✭❙❡❝r❡t❛r✐❛ ❞❡ ▼♦❜✐❧✐❞❛❞❡ ❯r❜❛♥❛✮✱ ♣❛ss❛r❛♠ ❛ s❡r✱ ❛ ♣❛rt✐r ❞❡ ♦✐t♦ ❞❡ ❥❛♥❡✐r♦ ❞❡ ✷✵✶✹✱ ❘✩ ✹✱✺✺ ♣❛r❛ ❜❛♥❞❡✐✲ r❛❞❛✱ ❘✩ ✷✱✸✺ ♣❛r❛ ❜❛♥❞❡✐r❛ ■ ❡✱ ❘✩ ✸✱✸✵ ♣❛r❛ ❜❛♥❞❡✐r❛ ■■✳ ◆❡ss❡ s✐st❡♠❛ ❞❡ t❛r✐❢❛s✱ ♦ ✈❛❧♦r ❞❛ ❜❛♥❞❡✐r❛❞❛ é ✉♠ ✈❛❧♦r ✜①♦ ❛❝r❡s❝✐❞♦ ❛♦ ❝✉st♦ ❞❛ ❝♦rr✐❞❛✱ q✉❡ é ♦❜t✐❞♦ ♠✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ♦ ✈❛❧♦r ❞❛ ❜❛♥❞❡✐r❛ ■ ♦✉ ■■✱ ♣❡❧❛ q✉❛♥t✐❞❛❞❡ ❞❡ q✉✐❧ô♠❡tr♦s r♦❞❛❞♦s✳ ❉❡ss❛ ❢♦r♠❛✱ ♥♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ✈♦❝ê ❡♥tr❛ ♥✉♠ tá①✐ q✉❡ ❡st❡❥❛ ❢✉♥❝✐♦♥❛♥❞♦ ❝♦♠ ❜❛♥❞❡✐r❛ ■✱ ✈♦❝ê ✈❡rá ♦ t❛①í♠❡tr♦ ♠❛r❝❛♥❞♦ ✹✱✺✺ ❡✱ à ♠❡❞✐❞❛ q✉❡ ♦ tá①✐ s❡ ❞❡s❧♦❝❛✱ ✈♦❝ê ✈❛✐ ♦❜s❡r✈❛♥❞♦ ❛❧❣✉♠❛s ♠✉❞❛♥ç❛s ❞❡ ✈❛❧♦r❡s ♥♦ t❛①í♠❡tr♦✿ ✹✱✺✺ ✲ ✻✱✾✵ ✲ ✾✱✷✺ ✲ ✶✶✱✻✵ ✲ ✶✸✱✾✺ ✲ ✶✻✱✸✵ ✲ · · ·✳ ❙❡ ✈♦❝ê ♣❡❣❛ ♦ ♠❡s♠♦ tá①✐✱ ♥✉♠ ❤♦rár✐♦ ❞❡ ❜❛♥❞❡✐r❛
■■✱ ✈♦❝ê ♦❜s❡r✈❛ ♦✉tr♦s ✈❛❧♦r❡s✿ ✹✱✺✺ ✲ ✼✱✽✺ ✲ ✶✶✱✶✺ ✲ ✶✹✱✹✺ ✲ ✶✼✱✼✺ ✲ ✷✶✱✵✺ ✲ · · ·✳
❊①❡♠♣❧♦ ✷✳✵✳✹ ❆❧❣✉♠❛s ✐♥st✐t✉✐çõ❡s ✜♥❛♥❝❡✐r❛s ❞❡ ❝❛rtõ❡s ❞❡ ❝ré❞✐t♦ ✐♥❢♦r♠❛♠ ❡♠ s✉❛s ❢❛t✉r❛s ♦s ♣❡r❝❡♥t✉❛✐s ❞❡ ❥✉r♦s q✉❡ ✐♥❝✐❞✐rã♦ s♦❜r❡ ♦ ♠♦♥t❛♥t❡ ❞❡✈✐❞♦ ❡♠ ❝❛s♦ ❞❡ ❛tr❛s♦ ♥♦ ♣❛❣❛♠❡♥t♦ ♣♦r ♣❛rt❡ ❞♦s ❝❧✐❡♥t❡s✳
❆ ❋✐❣✉r❛ ✷✳✷✱ ❛ s❡❣✉✐r✱ ♠♦str❛ ♣❛rt❡ ❞❛ ❢❛t✉r❛ ❞❡ ✉♠❛ ❞❡st❛s ✐♥st✐t✉✐çõ❡s✱ ♥❛ q✉❛❧ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❡♥❝❛r❣♦s ♣♦r ✜♥❛♥❝✐❛♠❡♥t♦ ❞❡ ❢❛t✉r❛ é ✾✱✾✵✪✱ ❡ ❞❡ ❥✉r♦ ❞❡ ♠♦r❛ ❡ ❞❡ ♠✉❧t❛ sã♦ ✶✱✵✵✪ ❡ ✷✱✵✵✪✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡ ❛ ❢❛t✉r❛ ❢♦r ♣❛❣❛ ❝♦♠ ❛tr❛s♦✱ ✐♥❝✐❞❡ s♦❜r❡ ♦ t♦t❛❧ ❞❛ ❢❛t✉r❛ ♦s ❥✉r♦s✱ q✉❡ sã♦ ❛ s♦♠❛ ❞♦s ❡♥❝❛r❣♦s ♣♦r ✜♥❛♥❝✐❛♠❡♥t♦ ❞❡ ❢❛t✉r❛ ❝♦♠ ♦ ❥✉r♦s ❞❡ ♠♦r❛ ❡ ❛ ♠✉❧t❛✳
❋✐❣✉r❛ ✷✳✷✿ ❋❛t✉r❛ ❞♦ ❝❛rtã♦ ❞❡ ❝ré❞✐t♦✳
❙✉♣♦♥❤❛ q✉❡ ♦ ❝❧✐❡♥t❡ ♥ã♦ ❝♦s❡❣✉✐rá ♣❛❣❛r s✉❛ ❢❛t✉r❛ ✭❛ ❞❛ ❋✐❣✉r❛ ✷✳✷✮ ❛té ♦ ✈❡♥❝✐♠❡♥t♦✱ ❡ t❡♥❤❛ ♣❡❞✐❞♦ ❛♦ s❡✉ ❣❡r❡♥t❡ q✉❡ ❧❤❡ ❡♥✈✐❛ss❡ ♦ q✉❛♥t♦ ♣❛❣❛rá ❞❡ ❥✉r♦s✱ ❝❛s♦ ♣❛❣❛ss❡ ❡♠ ❛té ✺ ❞✐❛s ❛♣ós ♦ ✈❡♥❝✐♠❡♥t♦✳ ❆❧❣✉♥s ❞✐❛s ❞❡♣♦✐s✱ s❡✉ ❣❡r❡♥t❡ ❧❤❡ ❡♥✈✐❛ ✉♠❛ t❛❜❡❧❛✱ ❝♦♠♦ ❛ ❚❛❜❡❧❛ ✷✳✺✳
❉✐❛s ❡♠ ❛tr❛s♦ ❏✉r♦s
✶ ❘✩ ✶✶✱✷✻
✷ ❘✩ ✶✷✱✾✾
✸ ❘✩ ✶✹✱✼✷
✹ ❘✩ ✶✻✱✹✺
✺ ❘✩ ✶✽✱✶✽
❚❛❜❡❧❛ ✷✳✺✿ ❉✐❛s ❡♠ ❛tr❛s♦×❏✉r♦s✳
❊①❡♠♣❧♦ ✷✳✵✳✺ ❯♠ ❜❛♥❝♦ r❡❛❧✐③❛ ❡♠♣rést✐♠♦s ❛ ✉♠❛ t❛①❛ ❞❡ ❥✉r♦s ❞❡ ✸✱✼✽✪ ❛♦ ♠ês✳ ❯♠ ❝❧✐❡♥t❡ q✉❡ ✈❡♥❤❛ ❛ r❡❛❧✐③❛r ✉♠ ❡♠♣rést✐♠♦ ❞❡ ❘✩ ✶✵✵✵✱✵✵ ♥❡st❡ ❜❛♥❝♦✱
❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ q✉✐tá✲❧♦ ❡♠ ✉♠❛ só ♣❛r❝❡❧❛✱ s❡rá ✐♥❢♦r♠❛❞♦ ♣❡❧♦ ❣❡r❡♥t❡ q✉❡✿ ❞❡♣♦✐s ❞❡ ✉♠ ♠ês✱ ♦ ♠♦♥t❛♥t❡ ❞❡✈✐❞♦ é ❞❡ ❘✩ ✶✵✸✼✱✽✵❀ ❞❡♣♦✐s ❞❡ ❞♦✐s ♠❡s❡s✱ é ❞❡ ❘✩ ✶✵✼✼✱✵✸❀ ❞❡♣♦✐s ❞❡ três ♠❡s❡s✱ é ❞❡ ❘✩ ✶✶✶✼✱✼✹❀ ❞❡♣♦✐s ❞❡ q✉❛tr♦ ♠❡s❡s✱ é ❞❡ ❘✩ ✶✶✺✾✱✾✾❀ ❡t❝✳ ❱♦❝ê ♣♦❞❡ ♣❡r❝❡❜❡r q✉❡✱ q✉❛♥t♦ ♠❛✐s t❡♠♣♦ ❡❧❡ ❧❡✈❛r ♣❛r❛ q✉✐t❛r ❛ ❞í✈✐❞❛✱ ♠❛✐♦r s❡rá ♦ ✈❛❧♦r q✉❡ ❡❧❡ t❡rá q✉❡ ♣❛❣❛r✳
❊①❡♠♣❧♦ ✷✳✵✳✻ ❊♠ P❛r♥❛♠✐r✐♠✱ ♠✉♥í❝✐♣✐♦ ❞♦ ❡st❛❞♦ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡✱ r❡❣✐ã♦ ❞❡ ❣r❛♥❞❡ ❝r❡s❝✐♠❡♥t♦ ♥❛ ❝♦♥str✉çã♦ ❝✐✈✐❧✱ ❛❧❣✉♠❛s ✐♠♦❜✐❧✐ár✐❛s ♦♣t❛r❛♠ ♣♦r ✈❡♥❞❡r t❡rr❡♥♦s ✉s❛♥❞♦ ♦ ♣r❡ç♦ ❞♦ ♠❡tr♦ q✉❛❞r❛❞♦✱ ❛❝r❡❞✐t❛♥❞♦ s❡r ✉♠❛ ♠❛♥❡✐r❛ ❧✉✲ ❝r❛t✐✈❛ ❡ rá♣✐❞❛ ❞❡ ✈❡♥❞ê✲❧♦s✳ ❙✉♣♦♥❤❛ q✉❡ ✉♠❛ ❞❡❧❛s só ✈❡♥❞❛ t❡rr❡♥♦s r❡t❛♥❣✉❧❛r❡s ❝♦♠ ✉♠ ❞♦s ❧❛❞♦s ✜①♦ ❡♠ ✶✺ ♠❡tr♦s✱ ❛♦ ♣r❡ç♦ ❞❡ ✹✵✵ r❡❛✐s ♦ ♠❡tr♦ q✉❛❞r❛❞♦✳ ❊♥tã♦✱ ♦ ❝♦♠♣r❛❞♦r ♦❜s❡r✈❛ q✉❡ ♦ ✈❛❧♦r ❞❡ ✉♠ t❡rr❡♥♦ ❡♠ q✉❡ ♦ ♦✉tr♦ ❧❛❞♦ ♠❡❞❡ ✶✵ ♠❡tr♦s é ✻✵ ♠✐❧ r❡❛✐s✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❡ ♣❡r❝❡❜❡ q✉❡ ❛ ❝❛❞❛ ♠❡tr♦ q✉❡ ✈❡♥❤❛ ❛❝r❡s❝❡♥t❛r ❛ ❡st❡ ❧❛❞♦✱ ♦ ✈❛❧♦r ✈❛✐ ❛✉♠❡♥t❛♥❞♦ ❝♦♥❢♦r♠❡ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✷✳✻✳
❆❝rés❝✐♠♦ ❛♦s ✶✵ ♠❡tr♦s Pr❡ç♦s
✵ ✻✵✵✵✵
✶ ✻✻✵✵✵
✷ ✼✷✵✵✵
✸ ✼✽✵✵✵
✹ ✽✹✵✵✵
✺ ✾✵✵✵✵
✳✳✳ ✳✳✳
❚❛❜❡❧❛ ✷✳✻✿ ▼❡tr♦s ❛❝r❡s❝✐❞♦s ❛ ✶✵♠×Pr❡ç♦s✳
❊①❡♠♣❧♦ ✷✳✵✳✼ ❆♦ ❝❤❡❣❛r ♥♦s ❝♦rr❡✐♦s ♣❛r❛ ❡♥✈✐❛r ✉♠❛ ❝❛rt❛✱ ❞❡♣❛r❛♠♦✲♥♦s ❝♦♠ ✉♠❛ t❛❜❡❧❛ ❞❡ ♣r❡ç♦s ♣❛r❛ ❡♥✈✐♦ ❞❡ ❝❛rt❛s ❝♦♠❡r❝✐❛✐s ❞❡ ❞✐✈❡rs♦s ♣❡s♦s✱ ❝♦♥❢♦r♠❡ ♠♦str❛ ❛ ❋✐❣✉r❛ ✷✳✸✳
❋✐❣✉r❛ ✷✳✸✿ ❚❛❜❡❧❛ ❞♦s ❈♦rr❡✐♦s ❝♦♠ ✈❛❧♦r❡s ❞❡ ❡♥✈✐♦ ❞❡ ❝❛rt❛s ❝♦♠❡r❝✐❛✐s✳
❉❡st❛ ❢♦r♠❛✱ ♣❛r❛ ❝❛rt❛s q✉❡ ♥ã♦ ❡①✐❣❡♠ s❡r✈✐ç♦s ❛❞✐❝✐♦♥❛✐s✱ t❡rí❛♠♦s✱ ♣♦r ❡①❡♠✲ ♣❧♦✿ ♣❛r❛ ❡♥✈✐❛r ✉♠❛ ❝❛rt❛ ❝✉❥♦ ♣❡s♦ é ✶✺ ❣r❛♠❛s✱ ♦ ✈❛❧♦r ❛ s❡r ♣❛❣♦ é ❘✩ ✶✱✸✵❀ ♣❛r❛ ♦✉tr❛ ❝✉❥♦ ♣❡s♦ é ✷✸ ❣r❛♠❛s✱ ♦ ✈❛❧♦r é ❘✩ ✶✱✽✵❀ ✉♠❛ ❝✉❥♦ ♣❡s♦ é ✼✽ ❣r❛♠❛s✱ ♦
✈❛❧♦r ❛ s❡r ♣❛❣♦ é ❘✩ ✷✱✹✺❀ ✉♠❛ ❝✉❥♦ ♣❡s♦ é ✶✸✵ ❣r❛♠❛s✱ ♦ ✈❛❧♦r ❛ s❡r ♣❛❣♦ é ❘✩ ✸✱✵✵✳ ❏á ♣❛r❛ ♦❜❥❡t♦s q✉❡ ✉❧tr❛♣❛ss❛♠ ✺✵✵ ❣r❛♠❛s✱ ♦ ✈❛❧♦r t♦♠❛❞♦ ❝♦♠♦ ❜❛s❡ é ♦ ❞❛ ♣r❡st❛çã♦ ❞♦ ❙❊❉❊❳✳
❱❡♠♦s✱ ♣❡❧♦s ❡①❡♠♣❧♦s ❛❝✐♠❛✱ q✉❡ ❡①✐st❡♠ ✈ár✐❛s ❢♦r♠❛s ❞❡ ❛ss♦❝✐❛r ❣r❛♥❞❡③❛s✱ ✭q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♠❜✉stí✈❡❧ ❝♦♠ ♦ ♣r❡ç♦ ♣❛❣♦✱ ❞✐❛s ❡♠ ❛tr❛s♦ ❝♦♠ ❥✉r♦s ♣❛❣♦s✱ ❡t❝✮✳ ❊♠ t♦❞❛s ❡ss❛s ❛ss♦❝✐❛çõ❡s s❡♠♣r❡ ❡①✐st❡♠ ❞♦✐s ❝♦♥❥✉♥t♦s ❡♥✈♦❧✈✐❞♦s✱ ❣❡r❛❧♠❡♥t❡✱ ✉♠ q✉❡ ❝♦♥té♠ ❛s ✐♥❢♦r♠❛çõ❡s q✉❡ ✈♦❝ê ❢♦r♥❡❝❡✱ ❡ ♦✉tr♦✱ q✉❡ ❝♦♥té♠ ❛s r❡s♣♦st❛s q✉❡ ❡stã♦ ❛ss♦❝✐❛❞❛s às ✐♥❢♦r♠❛çõ❡s ❞❛❞❛s✳ ◆♦ ❊①❡♠♣❧♦ ✷✳✵✳✶ ❞❛ ❣❛s♦❧✐♥❛✱ s❡ ✈♦❝ê ❞❡s❡❥❛ ❛❜❛st❡❝❡r ✷✵ ❧✐tr♦s✱ ✈♦❝ê ✐♥❢♦r♠❛ ✐ss♦ ♣❛r❛ ♦ ❢r❡♥t✐st❛ ❡✱ ❛♦ ✜♥❛❧ ❞♦ ❛❜❛st❡❝✐♠❡♥t♦✱ t❡r❡♠♦s ❡♠ ✉♠ ✈✐s♦r ✷✵✱ ❡ ♥♦ ♦✉tr♦ ✈✐s♦r✱ ♦ ✈❛❧♦r ❛ s❡r ♣❛❣♦ ♣♦r ❛q✉❡❧❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♠❜✉stí✈❡❧✳ ◆♦ ❊①❡♠♣❧♦ ✷✳✵✳✼ ❞♦s ❝♦rr❡✐♦s✱ ✈♦❝ê ❡♥tr❡❣❛ s✉❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❛ q✉❛❧ é ♣❡s❛❞❛✱ ❡ss❡ ♣❡s♦ é ❛ ✐♥❢♦r♠❛çã♦ ❢♦r♥❡❝✐❞❛✱ ❡ ❛ s❡❣✉✐r✱ ❛ ❛t❡♥❞❡♥t❡ t❡ ❞❛rá ♦ ♣r❡ç♦ ❛ s❡r ♣❛❣♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ t❛❜❡❧❛ ✈✐❣❡♥t❡✳ P♦❞❡♠♦s ✐♠❛❣✐♥❛r q✉❡ ❛s ❛ss♦❝✐❛çõ❡s q✉❡ ✈✐♠♦s s❡♠♣r❡ ♣❡❣❛♠ ✉♠❛ ✐♥❢♦r♠❛çã♦ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ✭q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♠❜✉stí✈❡❧✱ ♣❡s♦ ❞❛ ❝❛rt❛✱ ❡t❝✮ ❡ ♦s tr❛♥s❢♦r♠❛♠ ❡♠ ♦✉tr❛ ✐♥❢♦r♠❛çã♦✱ só q✉❡ ❡♠ ♦✉tr♦ ❝♦♥❥✉♥t♦ ✭♣r❡ç♦ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♠❜✉stí✈❡❧ ❛❜❛st❡❝✐❞❛✱ ♣r❡ç♦ ♣❛r❛ ❡♥✈✐❛r ❛ ❝❛rt❛ ❝♦♠ ♦ ♣❡s♦ ❢♦r♥❡❝✐❞♦✱ ❡t❝✮✳
❆ss✐♠✱ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❛ ❧✐❣❛çã♦ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❞✐❛✲ ❣r❛♠❛✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✷✳✹✳
❋✐❣✉r❛ ✷✳✹✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ❛ss♦❝✐❛çã♦ ❡♠ ❢♦r♠❛ ❞❡ ❞✐❛❣r❛♠❛✳
P❛r❛ ❞✐❢❡r❡♥❝✐❛r t❛✐s ❝♦♥❥✉♥t♦s✱ ♣♦❞❡rí❛♠♦s ❝❤❛♠❛r ♦ ♣r✐♠❡✐r♦ ❞❡ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛ ✭❛q✉❡❧❡ q✉❡ ❢♦r♥❡❝❡ ❛ ✐♥❢♦r♠❛çã♦ ✐♥✐❝✐❛❧✮✱ ❡ ♦ s❡❣✉♥❞♦✱ ❞❡ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡✲ ❣❛❞❛ ✭❛q✉❡❧❡ q✉❡ ❢♦r♥❡❝❡ ❛ ✐♥❢♦r♠❛çã♦ ❛ss♦❝✐❛❞❛ à ✐♥❢♦r♠❛çã♦ ✐♥✐❝✐❛❧ ❞❛❞❛✮✳
◆♦t❡ q✉❡ ❛s ❛ss♦❝✐❛çõ❡s ❞♦s ❡①❡♠♣❧♦s ❛♥t❡r✐♦r❡s ♣♦ss✉❡♠ ♦✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡✱ ❛ s❛❜❡r✿ ❆ ❝❛❞❛ ✐♥❢♦r♠❛çã♦ ❞❛❞❛✱ ♦❜t❡♠♦s ❛♣❡♥❛s ✉♠❛ ú♥✐❝❛ ✐♥❢♦r♠❛çã♦ ❛ss♦❝✐❛❞❛✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❝❛❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♠❜✉stí✈❡❧ ❛❜❛st❡❝✐❞❛✱ ❡①✐st❡ ❛♣❡♥❛s ✉♠ ú♥✐❝♦ ✈❛❧♦r ❛ s❡r ♣❛❣♦❀ ❛ ❝❛❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❞✐❛s ❡♠ ❛tr❛s♦ ♥♦ ❝❛rtã♦ ❡①✐st❡ ❛♣❡♥❛s ✉♠ ✈❛❧♦r ❞❡ ❥✉r♦s ❛ s❡r ♣❛❣♦✱ ❡t❝✳
P♦r q✉❡ ✐ss♦ é ✐♠♣♦rt❛♥t❡❄ ■♠❛❣✐♥❛ q✉❡ ✈♦❝ê ✈❛✐ ❛❜❛st❡❝❡r ✷✵ ❧✐tr♦s ❞❡ ❝♦♠✲ ❜✉stí✈❡❧ ❡ ❛♣❛r❡❝❡ss❡ ♥♦ ✈✐s♦r ❞❡ ♣r❡ç♦s ❘✩ ✶✵✱✵✵ ❡ ❘✩ ✷✵✱✵✵✳ ◗✉❛❧ ❞♦s ❞♦✐s ✈♦❝ê ♣❛❣❛r✐❛❄ ◗✉❛❧ ❞♦s ❞♦✐s ♦ ❞♦♥♦ ❞♦ ♣♦st♦ ❝♦❜r❛r✐❛❄ ■ss♦ ❣❡r❛r✐❛ ✉♠❛ ❞✉❜✐❡❞❛❞❡✱ ♦✉ s❡❥❛✱ ❣❡r❛r✐❛ ✉♠❛ ❝♦♥❢✉sã♦ q✉❡ ♥ã♦ ♥♦s ♣❡r♠✐t✐r✐❛ s❛❜❡r ❡①❛t❛♠❡♥t❡ ♦ ✈❛❧♦r ❝♦rr❡t♦✳
❖❜s❡r✈❛çã♦ ✷✳✵✳✶ ❖❧❤❡♠♦s ♥♦✈❛♠❡♥t❡ ❛ ❋✐❣✉r❛ ✷✳✶✳ ❆❧❣✉♠❛s r❡❧❛çõ❡s ❞❛q✉❡❧❛s ♣♦❞❡♠ ♥ã♦ ❝✉♠♣r✐r ❡ss❛ ❡①✐❣ê♥❝✐❛✱ ✐st♦ é✱ ❛ q✉❡ ❝❛❞❛ ✐♥❢♦r♠❛çã♦ ❞❛❞❛✱ s❡ ♦❜té♠ ✉♠❛ ú♥✐❝❛ ✐♥❢♦r♠❛çã♦ ❛ss♦❝✐❛❞❛✳ P♦r ❡①❡♠♣❧♦✱ ♥✉♠❛ ❡s❝♦❧❛ ♣♦❞❡ t❡r ❝r✐❛♥ç❛s✱ ❝✉❥♦s ♣❛✐s sã♦ s❡♣❛r❛❞♦s✱ ❡ q✉❡ ❡❧❡s ♦♣t❛r❛♠ ♣❡❧❛ ❣✉❛r❞❛ ❝♦♠♣❛rt✐❧❤❛❞❛ ❞❛ ❝r✐❛♥ç❛✱ ❞❡ ♠♦❞♦ q✉❡ ❛❧❣✉♥s ❞✐❛s ♣♦r s❡♠❛♥❛ ♦ ✜❧❤♦ ✜❝❛ ❝♦♠ ♦ ♣❛✐✱ ❡ ♥♦s ♦✉tr♦s ❞✐❛s✱ ❡❧❡ ✜❝❛ ❝♦♠ ❛ ♠ã❡✳ ◆❡st❛ s✐t✉❛çã♦ ✈❡♠♦s q✉❡ ♦ ♥♦♠❡ ❞❛ ❝r✐❛♥ç❛ ♣♦❞❡ ❡st❛r ❛ss♦❝✐❛❞❛ ❝♦♠ ❞♦✐s ❡♥❞❡r❡ç♦s ❞✐st✐♥t♦s✳ ❯♠ ♦✉tr♦ ❡①❡♠♣❧♦ é ❛ ❛ss♦❝✐❛çã♦ ❡♥tr❡ ♦ ♥♦♠❡ ❞❡ ✉♠❛ ♣❡ss♦❛
❡ ♦ ♥ú♠❡r♦ ❞❡ s❡✉ ❝❡❧✉❧❛r✱ ♣♦✐s ❤♦❥❡ ❡♠ ❞✐❛ q✉❛s❡ t♦❞♦ ♠✉♥❞♦ t❡♠ ❝❤✐♣s ❞❡ ♠❛✐s ❞❡ ✉♠❛ ♦♣❡r❛❞♦r❛ ✐♥st❛❧❛❞♦s ❡♠ s❡✉s t❡❧❡❢♦♥❡s ❝❡❧✉❧❛r❡s✱ ❞❡ ♠♦❞♦ q✉❡ ❛ss♦❝✐❛❞♦ ❛♦ ♥♦♠❡ ❞❡ ✉♠❛ ♣❡ss♦❛✱ ♣♦❞❡♠♦s t❡r ✈ár✐♦s ♥ú♠❡r♦s ❞❡ t❡❧❡❢♦♥❡s ❝❡❧✉❧❛r✳
❊♠ ▼❛t❡♠át✐❝❛✱ t❛✐s s✐t✉❛çõ❡s ♥ã♦ ♦❝♦rr❡♠✱ ❛ ♠❡s♠❛ ♣r❡③❛ ♣❡❧❛ ♣r❡❝✐sã♦✳ ❉❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s ❡❧❡❣❛♥t❡✱ ❋■▲❍❖ ❬✸❪ ♥♦s ❛✜r♠❛ s♦❜r❡ ❛ ❧✐♥❣✉❛❣❡♠ ▼❛t❡♠át✐❝❛✿ ✏◆❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ♥ã♦ ❤á ❧✉❣❛r ♣❛r❛ ❛♠❜✐❣✉✐❞❛❞❡s✱ ♣❛r❛ ✜❣✉r❛s ❞❡ ❧✐♥❣✉❛❣❡♠ ♦✉ ♣❛r❛ ♠❡tá❢♦r❛s✱ q✉❡ sã♦ tã♦ ❝♦♠✉♥s ❡ ❛té ♠❡s♠♦ ❛♣r❡❝✐❛❞❛s ♥❛ ▲✐♥❣✉❛❣❡♠ ❈♦❧♦q✉✐❛❧ ♦✉ ▲✐t❡rár✐❛✑✭❋■▲❍❖✱ ✷✵✵✻✱ ❬✸❪✮✳
❉❡ss❛ ❢♦r♠❛✱ s❡ ✜③❡r♠♦s ✉♠❛ ❛ss♦❝✐❛çã♦ q✉❡ ♥♦s ❧❡✈❡ ❛ ❞✉❜✐❡❞❛❞❡✱ t❛❧ ❛ss♦❝✐❛çã♦ ♥ã♦ s❡r✈✐rá ♣❛r❛ ♦ q✉❡ s❡ ❞❡st✐♥❛✱ ♣♦✐s ❝❛✉s❛rá ♠❛✐s ❝♦♥❢✉sã♦ ❞♦ q✉❡ ❛❥✉❞❛✳ P♦r ❡①❡♠♣❧♦✱ ✐♠❛❣✐♥❛ q✉❡ ✈♦❝ê ✐♥❢♦r♠❛ ❛♦ s❡✉ ❣❡r❡♥t❡ q✉❡ ❡stá ✺ ❞✐❛s ❡♠ ❛tr❛s♦ ❞♦ ♣❛❣❛♠❡♥t♦ ❞♦ s❡✉ ❝❛rtã♦ ❞❡ ❝ré❞✐t♦ ❡ ❡❧❡ ❞✐③ q✉❡ ❛ ♠✉❧t❛ é ❘✩ ✺✵✱✵✵ ♦✉ ❘✩ ✶✵✵✱✵✵✳ Pr✐♠❡✐r♦✱ ✈♦❝ê ✈❛✐ ❛❝❤❛r q✉❡ ❡❧❡ ❡stá ❜r✐♥❝❛♥❞♦✱ ❡ ✈❡♥❞♦ q✉❡ ❡❧❡ ❡s♣❡r❛ ✉♠❛ r❡s♣♦st❛ s✉❛✱ ✈♦❝ê ❞✐③ q✉❡ ✈❛✐ ♣❛❣❛r ♦s ❘✩ ✺✵✱✵✵✳ ❉❡ss❛ ❢♦r♠❛✱ ❡❧❡ ❞✐③ q✉❡ ❣♦st❛r✐❛ q✉❡ ✈♦❝ê ♣❛❣❛ss❡ ♦s ❘✩ ✶✵✵✱✵✵✳ ❏á ✐♠❛❣✐♥♦✉ ❛ ❝♦♥❢✉sã♦❄
❊ ♦ q✉❡ ♣♦❞❡♠♦s ❢❛❧❛r s♦❜r❡ ♦s ❝♦♥❥✉♥t♦s ❞❡ ♣❛rt✐❞❛ ❡ ❝❤❡❣❛❞❛❄
❊ss❡s ❝♦♥❥✉♥t♦s sã♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡s ❡ sã♦ ❞❡t❡r♠✐♥❛❞♦s ♣❡❧♦ ♣r♦❜❧❡♠❛ q✉❡ ❡st❛♠♦s ❞❡s❝r❡✈❡♥❞♦✳ ◆♦ ❝❛s♦ ❞♦ ❝❛rr♦ q✉❡ ✈❛♠♦s ❛❜❛st❡❝❡r✱ s❡ ♦ t❛♥q✉❡ t❡♠ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ✺✺ ❧✐tr♦s✱ ❡♥tã♦ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♠❜✉stí✈❡❧ ❛ s❡r ✐♥❢♦r♠❛❞❛ é ❞❡ ✵ ❛ ✺✺ ❧✐tr♦s ❡✱ ❝♦♠♦ q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ❡♥tr❡ ❡ss❡s ✈❛❧♦r❡s ♣♦❞❡ s❡r ❛❜❛st❡❝✐❞❛✱ ♦ ❝♦♥❥✉♥t♦ q✉❡ r❡♣r❡s❡♥t❛ t❛❧ ❝♦♥❥✉♥t♦ ❞❡ s❛í❞❛ é ❬✵✱✺✺❪✳
❊ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛❄
◆❡st❡ ❝❛s♦ t❡♠♦s q✉❡ ✈❡r q✉❛✐s ♦s ♣r❡ç♦s ❛ss♦❝✐❛❞♦s ❛ ❡st❡s ✈❛❧♦r❡s ❡①tr❡♠♦s✱ s❛❜❡♥❞♦ q✉❡ ♣❛r❛ q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ❛❜❛st❡❝✐❞❛✱ ♦ ✈❛❧♦r ❝♦❜r❛❞♦ ❡stá ❡♥tr❡ ♦s ✈❛❧♦r❡s ❝♦❜r❛❞♦s ♣❡❧❛ q✉❛♥t✐❞❛❞❡ ♠í♥✐♠❛ ❡ ♠á①✐♠❛ ❛ s❡r❡♠ ❛❜❛st❡❝✐❞❛s✳ ◆❡st❡ ❝❛s♦✱ ♣❛r❛ ❛❜❛st❡❝❡r ✵ ❧✐tr♦ ❡❧❡ ♣❛❣❛rá 3,34× 0 = 0✱ ❡ ♣❛r❛ ❛❜❛st❡❝❡r ✺✺ ❧✐tr♦s✱
❡❧❡ ♣❛❣❛rá 3,34×55 = 183,70 ❡✱ ♣♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛ s❡rá [0,183.70]✳
❊♥tr❡t❛♥t♦✱ é ♠✉✐t♦ ❝♦♠✉♠ ✉t✐❧✐③❛r♠♦s ♣❛r❛ ❡st❡s ❝♦♥❥✉♥t♦s✱ ❝♦♥❥✉♥t♦s q✉❡ s✐r✈❛♠ ♥ã♦ ❛♣❡♥❛s ♣❛r❛ s✐t✉❛çõ❡s ♣❛rt✐❝✉❧❛r❡s✱ ❝♦♠♦ ❛ ❞♦ ❝❛rr♦ ❝✐t❛❞♦✱ ♠❛s q✉❡ s✐r✈❛ ♣❛r❛ q✉❛❧q✉❡r ❝❛rr♦✱ ❛ss✐♠ ✉t✐❧✐③❛✲s❡ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛✱ ❝♦♥❥✉♥t♦s ❞♦ t✐♣♦[0,∞)
✭♣♦✐s ♥ã♦ ♣♦❞❡♠♦s ❛❜❛st❡❝❡r ✉♠❛ q✉❛♥t✐❞❛❞❡ ♥❡❣❛t✐✈❛ ❞❡ ❝♦♠❜✉stí✈❡❧✮✱ ❡ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛✱ ❝♦♥❥✉♥t♦s [0,∞) ✭♣♦✐s ♥✉♥❝❛ ♥♦s ❝♦❜r❛r✐❛♠ ❡♠ ✉♠ ♣♦st♦ ✉♠
✈❛❧♦r ♥❡❣❛t✐✈♦ ♣♦r ✉♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♠❜✉stí✈❡❧ ❛❜❛st❡❝✐❞❛✮✳
❱❡❥❛♠♦s ❝♦♠♦ ✜❝❛r✐❛♠ ♦s ❝♦♥❥✉♥t♦s ❞❡ ♣❛rt✐❞❛ ❡ ❞❡ ❝❤❡❣❛❞❛ ♣❛r❛ ♦s ♦✉tr♦s ❡①❡♠♣❧♦s✱ ❡ ❝♦♠♦ ❣❡r❛❧♠❡♥t❡ sã♦ r❡♣r❡s❡♥t❛❞♦s✳
P❡♥s❡ ♥♦ ❊①❡♠♣❧♦ ✷✳✵✳✷ ❞❛ ❛❝❛❞❡♠✐❛✱ s❡ ✈♦❝ê t♦❞♦ ♠ês ✉s❛ ❛ ❜❛❧❛♥ç❛ ♣❛r❛ s❡ ♣❡s❛r✱ ❛s ✐♥❢♦r♠❛çõ❡s q✉❡ ✈♦❝ê ❢♦r♥❡❝❡ sã♦ ♦s ♠❡s❡s ❡ ❛♥♦s ❡♠ q✉❡ ✈♦❝ê s❡ ♣❡s♦✉✳ P♦r ❡①❡♠♣❧♦✱ ♥♦ ❝❛s♦ ❡♠ q✉❡ ♦ ❛❧✉♥♦ s❡ ♣❡s♦✉ ♥♦s ♠❡s❡s ❞❡ ♦✉t✉❜r♦ ❞❡ ✷✵✶✹ ❛té ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✺✱ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❡st❛s ✐♥❢♦r♠❛çõ❡s ♥✉♠ ❝♦♥❥✉♥t♦ ❞♦ t✐♣♦ ④✭♦✉t✉❜r♦✱ ✷✵✶✹✮✱ ✭♥♦✈❡♠❜r♦✱ ✷✵✶✹✮✱ ✭❞❡③❡♠❜r♦✱ ✷✵✶✹✮✱ ✭❥❛♥❡✐r♦✱ ✷✵✶✺✮✱ ✭❢❡✈❡r❡✐r♦✱ ✷✵✶✺✮⑥✱ q✉❡ é ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛✳ ❙✉♣♦♥❞♦ q✉❡ ❝❛❞❛ ♠ês ❛ ❜❛❧❛♥ç❛ ♠♦str♦✉ ✽✽ ❦❣✱ ✾✵ ❦❣✱ ✽✺ ❦❣✱ ✽✺ ❦❣ ❡ ✽✵ ❦❣✱ ❡♥tã♦ ❡ss❡s ♣❡s♦s sã♦ ❛s r❡s♣♦st❛s ♣❛r❛ ❛s ✐♥❢♦r♠❛çõ❡s ❢♦r♥❡❝✐❞❛s ✭♠❡s❡s ❡ ❛♥♦s✮ ❡ ❢❛③❡♠ ♣❛rt❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛ ④✽✽ ❦❣✱ ✾✵ ❦❣✱ ✽✺ ❦❣✱ ✽✵ ❦❣⑥✳
❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ✈♦❝ê ♣♦❞❡ ✐♠❛❣✐♥❛r ❞♦✐s ❝♦♥❥✉♥t♦s ♥♦ ❊①❡♠♣❧♦ ✷✳✵✳✸ ❞❛ ❝♦rr✐❞❛ ❞❡ tá①✐✱ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ ❛s q✉❛♥t✐❞❛❞❡s ❞❡ q✉✐❧ô♠❡tr♦s r♦❞❛❞♦s✱ ❡ ♦✉tr♦ ❝♦♥❥✉♥t♦✱ ❝♦♠ ♦s ✈❛❧♦r❡s ❛ ♣❛❣❛r✱ ♦♥❞❡ ❛ ❝❛❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ q✉✐❧ô♠❡tr♦s r♦❞❛❞♦s✱ ❛ss♦❝✐❛♠♦s ✉♠ ú♥✐❝♦ ✈❛❧♦r ❛ ♣❛❣❛r✳ P♦✐s✱ s❡ ✈♦❝ê ❡♥tr❛ ♥♦ tá①✐ ❡ ❛♥❞❛ ✵ ❦♠✱ ✈♦❝ê ♣❛❣❛ ✉♠ ✈❛❧♦r✱ s❡ ✈♦❝ê ❛♥❞❛ ✶ ❦♠✱ ✈♦❝ê ♣❛❣❛ ♦✉tr♦ ✈❛❧♦r✱ s❡ ❛♥❞❛ ✷ ❦♠s✱ ✈♦❝ê ♣❛❣❛ ♦✉tr♦ ✈❛❧♦r✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ P❡r❝❡❜❛ q✉❡ ❢♦r♠❛♠♦s ✉♠ ❝♦♥❥✉♥t♦ ④✵ ❦♠✱ ✶ ❦♠✱ ✷ ❦♠s✱ ✳✳✳⑥✱ ❝♦♠ ♦s ❞❛❞♦s ✐♥✐❝✐❛✐s q✉❡ s❡rã♦ ✉s❛❞♦s ♣❛r❛ ❡♥❝♦♥tr❛r ♦s ✈❛❧♦r❡s ❛ ♣❛❣❛r✱ ❡ q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛✳ ❊st❡ é ✉♠ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r ✈❛❧♦r❡s q✉❡ ♥ã♦ sã♦ ♥❡❣❛t✐✈♦s ✭♣♦✐s ✈♦❝ê ♥ã♦ ♣❡r❝♦rr❡ ❞✐stâ♥❝✐❛ ♥❡❣❛t✐✈❛✮✱ ❡ ♥❡♠ ❞❡ ✈❛❧♦r❡s ❞❡❝✐♠❛✐s✱ ❝♦♠♦ ✵✱✺ ❦♠ ♦✉ ✸✱✻✻ ❦♠s✳
❊ ❝♦♠♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛❄
❆ r❡s♣♦st❛ ❡stá ♥♦s ✈❛❧♦r❡s q✉❡ ❛♣❛r❡❝❡♠ ♥♦ t❛①í♠❡tr♦ ❞❡♣♦✐s ❞♦ tá①✐ ❛♥❞❛r ❛q✉❡❧❛ q✉❛♥t✐❞❛❞❡ ❞❡ q✉✐❧ô♠❡tr♦s✳ P♦r ❡①❡♠♣❧♦✱ s❡ ♦ t❛①í♠❡tr♦ ♠♦str❛r ♦s ♥ú♠❡r♦s ✹✳✺✺✱ ✻✳✾✵✱ ✾✳✷✺✱ ✶✶✳✻✵✱ ❡t❝✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛ s❡rá ④❘✩ ✹✱✺✺✱ ❘✩ ✻✱✾✵✱ ❘✩ ✾✱✷✺✱✳✳✳⑥ ❝♦♠♣♦st♦ ♣♦r ✈❛❧♦r❡s ❛ ♣❛❣❛r ♥ã♦✲♥❡❣❛t✐✈♦s ✭♣♦✐s ♦ ♣❛ss❛❣❡✐r♦ ♥ã♦ r❡❝❡❜❡ ❞✐♥❤❡✐r♦ ❞♦ t❛①✐st❛✮✳
❊ ♥♦ ❝❛s♦ ❞♦ ❡♠♣rést✐♠♦ ✭❊①❡♠♣❧♦ ✷✳✵✳✺✮ q✉❡ ✈♦❝ê ♣❡❞❡ ❛♦ ❜❛♥❝♦❄
❱♦❝ê ❥á s❛❜❡ ❞✐③❡r q✉❛❧ ✐♥❢♦r♠❛çã♦ é ❞❛❞❛❄ P❛r❛ ✜❝❛r ♠❛✐s ❢á❝✐❧ ❡♥t❡♥❞❡r✱ ✐♠❛❣✐♥❡ q✉❡ ✈♦❝ê ✈❛✐ ❛♦ ❜❛♥❝♦ ❡ ♣❡❞❡ ✉♠ ❡♠♣rést✐♠♦ ❞❡ ✶✵✵✵ r❡❛✐s✳ ❖ ❣❡r❡♥t❡ ✈❡r✐✜❝❛ ❛ t❛①❛ ❞❡ ❥✉r♦s ✈✐❣❡♥t❡ ❡ ♣❡r❣✉♥t❛ q✉❛♥t♦ t❡♠♣♦ ✈♦❝ê ❧❡✈❛rá ♣❛r❛ ♣❛❣❛r✳ ❆ss✐♠✱ s❡ ✈♦❝ê ❞✐③ q✉❡ ♣❛❣❛rá ❞❛q✉✐ ❛ q✉❛tr♦ ♠❡s❡s✱ ♦ ❣❡r❡♥t❡ ❞✐③ q✉❡ ♦ ♠♦♥t❛♥t❡ é ❘✩ ✶✶✺✾✱✾✾✳ P❡r❝❡❜❛✱ ♥❡st❡ ❡①❡♠♣❧♦✱ q✉❡ ❛ ✐♥❢♦r♠❛çã♦ ❞❛❞❛ é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠❡s❡s ✭q✉❛tr♦ ♠❡s❡s✮✱ ❡ ❛ ✐♥❢♦r♠❛çã♦ ❣❡r❛❞❛ ❞❡ss❛ ✐♥❢♦r♠❛çã♦✱ é ♦ ♠♦♥t❛♥t❡ ❛ s❡r ♣❛❣♦ ✭❘✩ ✶✶✺✾✱✾✾✮✳ ❉❡ ♠❛♥❡✐r❛ ♠❛✐s ❣❡r❛❧✱ s❡ ❛ ✐♥❢♦r♠❛çã♦ ❢♦r♥❡❝✐❞❛ ❢♦r q✉❛❧q✉❡r q✉❛♥t✐❞❛❞❡ ✐♥t❡✐r❛ ❞❡ ♠❡s❡s✱ ❛ r❡s♣♦st❛ ❞❡ss❛ ✐♥❢♦r♠❛çã♦ é ♦ ♠♦♥t❛♥t❡ ❛ss♦❝✐❛❞♦ ❛ ❡ss❛ q✉❛♥t✐❞❛❞❡ ❞❛❞❛✱ ❡♥tã♦ ♥♦ss♦ ❝♦♥❥✉♥t♦ ❞❡ s❛í❞❛ é ④✶✱ ✷✱ ✸✱ ✹✱✳✳✳⑥✱ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛ é ④✶✵✸✼✳✽✵✱ ✶✵✼✼✳✵✸✱ ✶✶✶✼✳✼✹✱ ✶✶✺✾✳✾✾✱✳✳✳⑥✳
❱❡♠♦s q✉❡ ✉♠❛ ❛ss♦❝✐❛çã♦ ♣❛r❛ s❡r✈✐r ❛♦s ♣r♦♣ós✐t♦s ❛ q✉❡ ❝♦♠❡♥t❛♠♦s ♣r❡❝✐s❛ s❡r ❝♦❡r❡♥t❡✱ ♦✉ s❡❥❛✱ ✉♠ ❡❧❡♠❡♥t♦ ♥ã♦ ♣♦❞❡ ❡st❛r ❛ss♦❝✐❛❞♦ ❛ ❞♦✐s ♦✉ ♠❛✐s ❡❧❡♠❡♥t♦s✳ ❆❧é♠ ❞✐ss♦✱ só ❝♦♥s✐❞❡r❛♠♦s ♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛ q✉❡ ❡st❡❥❛♠✱ ❞❡ ❢❛t♦✱ ❛ss♦❝✐❛❞♦s ❛ ❛❧❣✉é♠ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛✳ ◆♦ ❊①❡♠♣❧♦ ✷✳✵✳✷ ❞❛ ❛❝❛❞❡♠✐❛✱ s❡ ✉♠❛ ❞❛❞❛ ♣❡ss♦❛ ♥ã♦ ❢❡③ ❛s ♠❡❞✐❞❛s ♥✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠ês ❡ ❛♥♦✱ ❡♥tã♦ ♠❡s♠♦ ❡❧❛ ❡st❛♥❞♦ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛✱ ❝♦♠♦ ❛❧✉♥♦ ❞❛ ❛❝❛❞❡♠✐❛✱ ❡❧❛ ♥ã♦ t❡rá ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦ ❛ss♦❝✐❛❞♦ àq✉❡❧❡ ❛❧✉♥♦ ♥❛q✉❡❧❡ ♠ês ❡ ❛♥♦✱ ♦✉ s❡❥❛✱ s♦♠♦s ✐♥❝❛♣❛③❡s ❞❡ ❞✐③❡r q✉❛❧q✉❡r ❝♦✐s❛ s♦❜r❡ s❡✉ ♣❡s♦✱ ♣♦✐s ♥ã♦ ❡stá ❛ss♦❝✐❛❞♦ ❛ ♥❡♥❤✉♠ ✈❛❧♦r✳
✷✳✶✳ ❉❖▼❮◆■❖ ❉❊ ❯▼❆ ❆❙❙❖❈■❆➬➹❖
✷✳✶ ❉♦♠í♥✐♦ ❞❡ ✉♠❛ ❆ss♦❝✐❛çã♦
◆♦ ❝❛s♦ ❞♦s ❝♦rr❡✐♦s ✭❊①❡♠♣❧♦ ✷✳✵✳✼✮✱ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ✈❛❧♦r ❛ss♦❝✐❛❞♦ ❛ ✈❛✲ ❧♦r❡s ♥❡❣❛t✐✈♦s ♥❡♠ ❛ss♦❝✐❛❞♦ ❛♦ ♣❡s♦ ❞❡ ✵ ❣r❛♠❛ ✭♥❡st❡ ❡①❡♠♣❧♦✱ ♣❡s♦s ♥❡❣❛t✐✈♦s ♥ã♦ ❡①✐st❡♠✮✱ ❧♦❣♦ ♦s ♥ú♠❡r♦s q✉❡ ❝♦♠♣♦rã♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛ sã♦ ❛♣❡♥❛s ♦s ♥ú♠❡r♦s ♠❛✐♦r❡s q✉❡ ③❡r♦✳ ❆ ❡st❡ s✉❜❝♦♥❥✉♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛✱ ❛q✉❡❧❡s q✉❡ r❡❛❧♠❡♥t❡ ♣♦ss✉❡♠ ❛❧❣✉♠ ✭❡ ú♥✐❝♦✮ ❡❧❡♠❡♥t♦ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛ ❛ss♦❝✐❛❞♦ ❛ ❡❧❡✱ ❝❤❛♠❛♠♦s ❞❡ ❞♦♠í♥✐♦ ❞❛ ❛ss♦❝✐❛çã♦✳ ◆♦t❡ q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞♦ ❞♦♠í♥✐♦ sã♦✱ ❞❡ ❢❛t♦✱ ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛ q✉❡ ❡stã♦ ❛ss♦❝✐❛❞♦s ❛ ❛❧❣✉é♠✳
P♦rt❛♥t♦✱ ❛ ♣❛rt✐r ❞❡st❡ ♠♦♠❡♥t♦✱ t♦❞❛ ❛ss♦❝✐❛çã♦ q✉❡ ♠♦♥t❛r♠♦s t❡rá ❝♦♠♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rt✐❞❛ ♦ ❝♦♥❥✉♥t♦ ❞♦♠í♥✐♦ ❞❛ ❛ss♦❝✐❛çã♦✳ ❱❡♠♦s q✉❡✱ ♥♦ ❊①❡♠♣❧♦ ✷✳✵✳✶ ❞❛ ❣❛s♦❧✐♥❛✱ ♦ ❞♦♠í♥✐♦ s❡rá R+ = [0,∞), ♣♦✐s ❛ ❝❛❞❛ q✉❛♥t✐❞❛❞❡ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ③❡r♦✱ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ✉♠ ♥ú♠❡r♦ q✉❡ r❡♣r❡s❡♥t❛ ❛ q✉❛♥t✐❛ ❛ s❡r ♣❛❣❛ ♣♦r ❛q✉❡❧❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♠❜✉stí✈❡❧✳
◆♦ ❊①❡♠♣❧♦ ✷✳✵✳✻ ❞♦ t❡rr❡♥♦✱ ❛s ✐♥❢♦r♠❛çõ❡s q✉❡ ✈♦❝ê ❢♦r♥❡❝❡ sã♦ ❛s q✉❛♥t✐✲ ❞❛❞❡s ❞❡ ♠❡tr♦s q✉❡ ❛❝r❡s❝❡♥t❛♠♦s ❛ ✉♠ ❞♦s ❧❛❞♦s✱ ❡ ❛s ✐♥❢♦r♠❛çõ❡s ❣❡r❛❞❛s sã♦ ♦s ♣r❡ç♦s ❞♦ t❡rr❡♥♦ ❝♦♠ ♦ ❛❝rés❝✐♠♦ ✐♥❢♦r♠❛❞♦✳ P♦r ❡①❡♠♣❧♦✱ s❡ ✈♦❝ê ❞✐③ ❛♦ s❡✉ ❝♦rr❡t♦r q✉❡ q✉❡r ✉♠ t❡rr❡♥♦ ❝♦♠ ❛❝rés❝✐♠♦ ❞❡ ✸ ♠❡tr♦s✱ ❡♥tã♦ ♦ ✈❛❧♦r q✉❡ ❡❧❡ ✈❛✐ t❡ ❞❛r é ✼✽✵✵✵ r❡❛✐s✱ ❡ ♦ t❛♠❛♥❤♦ ❞♦ s❡✉ t❡rr❡♥♦ s❡rá 13m×15m ✭❧ê✲s❡✿ ✶✸ ♠❡✲
tr♦s ❞❡ ❧❛r❣✉r❛ ♣♦r ✶✺ ♠❡tr♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦✮✳ ❆ss✐♠✱ s❡ ✈♦❝ê ❝♦♥s✐❞❡r❛r t♦❞♦s ♦s ❛❝rés❝✐♠♦s ♣♦ssí✈❡✐s✱ ♦ ❞♦♠í♥✐♦ ❞❛ ❛ss♦❝✐❛çã♦Aé ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s
♥ã♦✲♥❡❣❛t✐✈♦s ✭♦s ❛❝rés❝✐♠♦s ❛♦s ✶✵ ♠❡tr♦s✮✱ ♦✉ s❡❥❛✱ ❆ ❂ ④✵✱ ✶✱ ✷✱ ✸✱ ✹✱ ✺✱ ✻✱ ✳✳✳⑥✱ ❡ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ ❇ ♣♦❞❡ s❡r ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s R ✭❝♦♥té♠ t♦❞♦s ♦s ♣♦ssí✈❡✐s ♣r❡ç♦s ❞❡ t❡rr❡♥♦✮✳ ❖❜s❡r✈❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s t❛♠❜é♠ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❛ f✳ P❡r❝❡❜❛ t❛♠❜é♠ q✉❡ ♦s ♥ú♠❡r♦s
r❡❛✐s ♥❡❣❛t✐✈♦s ♥ã♦ ♣♦❞❡♠ ❢❛③❡r ♣❛rt❡ ❞♦ ❞♦♠í♥✐♦✱ ♣♦✐s ♦ ♠❡♥♦r t❡rr❡♥♦ q✉❡ ❡stá s❡♥❞♦ ✈❡♥❞✐❞♦ é ❞❡ 10m×15m ❡✱❞ s❡ ✈♦❝ê ❞❡r ✉♠ ♥ú♠❡r♦ ♥❡❣❛t✐✈♦✱ ✐ss♦ ✈❛✐ ❣❡r❛r
✉♠ t❡rr❡♥♦ ♠❡♥♦r q✉❡ ❡ss❡ ♦✉ ❛té t❡rr❡♥♦s ❝♦♠ ♠❡tr❛❣❡♠ ♥❡❣❛t✐✈❛✳ ◆♦t❡ t❛♠❜é♠
✷✳✷✳ ▲❊■ ❉❊ ❋❖❘▼❆➬➹❖
q✉❡ ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s ♥ã♦✲✐♥t❡✐r♦s t❛♠❜é♠ ♥ã♦ sã♦ ♣❡r♠✐t✐❞♦s✱ ♣♦✐s ❛ ❝♦♥s✲ tr✉t♦r❛ só ❛❞♠✐t❡ ❛❝rés❝✐♠♦s ♠ú❧t✐♣❧♦s ❞❡ ✶ ♠❡tr♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♦s ú♥✐❝♦s ♣♦ssí✈❡✐s ❛❝rés❝✐♠♦s sã♦ ❛q✉❡❧❡s ❛♣r❡s❡♥t❛❞♦s ❡♠ ❆✳ P♦r ❡①❡♠♣❧♦✱ ✲✸ ✭♥ú♠❡r♦ ♥❡❣❛t✐✈♦✮ ❡ ✵✱✺ ✭♥ú♠❡r♦ ❢r❛❝✐♦♥❛❞♦✮ ♥ã♦ ❡stã♦ ♥♦ ❞♦♠í♥✐♦✳
❙❡ ♦❧❤❛r♠♦s ♦ ❊①❡♠♣❧♦ ✷✳✵✳✼ ❞♦s ❝♦rr❡✐♦s✱ ♦ ❞♦♠í♥✐♦ ❞❛ ❛ss♦❝✐❛çã♦ q✉❡ ❛ss♦❝✐❛ ♦ ♣❡s♦ ❞❛ ❝❛rt❛ à t❛r✐❢❛ ❝♦❜r❛❞❛ ♣❛r❛ ❡♥✈✐♦ é ♦ ✐♥t❡r✈❛❧♦ ✭✵✱✺✵✵❪✱ ❡ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♦✉ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s ✭♣♦✐s t❛✐s ❝♦♥❥✉♥t♦s ❝♦♥t❡rã♦ t♦❞♦s ♦s ♣♦ssí✈❡✐s ♣r❡ç♦s✮✳ ❉❡ss❛ ❢♦r♠❛✱ s❡ ✈♦❝ê ♣❡r❝❡❜❡r✱ ♦s ✈❛❧♦r❡s ♥❡❣❛t✐✈♦s ❡ ♦ ③❡r♦ ♥ã♦ ♣♦❞❡♠ ❛♣❛r❡❝❡r ♥♦ ❞♦♠í♥✐♦✱ ♣♦✐s ❛s ❝❛rt❛s ♥ã♦ ♣♦❞❡♠ t❡r ♣❡s♦ ♥❡❣❛t✐✈♦ ♥❡♠ ♣❡s❛r ③❡r♦ ❣r❛♠❛ ❡✱ ♣♦r ❝♦♥t❛ ❞✐ss♦✱ ❡st❡s ✈❛❧♦r❡s ♥ã♦ ❡stã♦ ❛ss♦❝✐❛❞♦s ❛ ♥❡♥❤✉♠ ♣r❡ç♦ ❞♦ ❝♦♥tr❛❞♦♠í♥✐♦✳ ❏á ❛s ❝❛rt❛s q✉❡ ♣❡s❛♠ ♠❛✐s ❞❡ ✺✵✵ ❣r❛♠❛s✱ ♥ã♦ ♣♦❞❡♠ s❡r ❡♥✈✐❛❞❛s ❝♦♠♦ ❝❛rt❛s ❝♦♠❡r❝✐❛✐s ❡✱ ♣♦rt❛♥t♦✱ t❛✐s ♣❡s♦s ♥ã♦ ❡stã♦ ❛ss♦❝✐❛❞♦s ❛ ♥❡♥❤✉♠ ♣r❡ç♦✱ ♥ã♦ ♣♦❞❡♥❞♦ ❢❛③❡r ♣❛rt❡ ❞♦ ❞♦♠í♥✐♦✳
❊♥tã♦✱ ♥❡st❛ s❡çã♦✱ ❞✐s❝✉t✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❛ss♦❝✐❛çã♦ q✉❡ r❡s✉♠✐♠♦s ♥❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳
❉❡✜♥✐çã♦ ✷✳✶✳✶ ✭❉♦♠í♥✐♦ ❞❡ ✉♠❛ ❛ss♦❝✐❛çã♦✮ ❉❛❞❛ ✉♠❛ ❛ss♦❝✐❛çã♦ ❡♥tr❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s❛í❞❛ A ❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛ B✱ ♦ s✉❜❝♦♥❥✉♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡
♣❛rt✐❞❛ A ❢♦r♠❛❞♦ ♣♦r ❛q✉❡❧❡s ❡❧❡♠❡♥t♦s q✉❡ r❡❛❧♠❡♥t❡ ♣♦ss✉❡♠ ❛❧❣✉♠ ✭❡ ú♥✐❝♦✮
❡❧❡♠❡♥t♦ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❡❣❛❞❛ ❛ss♦❝✐❛❞♦ ❛ ❡❧❡✱ é ❝❤❛♠❛❞♦ ❞❡ ❞♦♠í♥✐♦ ❞❛ ❛ss♦✲ ❝✐❛çã♦✳
✷✳✷ ▲❡✐ ❞❡ ❋♦r♠❛çã♦
◆♦t❡ q✉❡✱ ♣❛r❛ ❛❧❣✉♠❛s ❛ss♦❝✐❛çõ❡s✱ ♣♦❞❡♠♦s ❛té ❢♦r♠❛❧✐③❛r ❝♦♠♦ ❛ ❛ss♦❝✐❛çã♦ ❛❝♦♥t❡❝❡✳ ◆♦ ❊①❡♠♣❧♦ ✷✳✵✳✶ ❞❛ ❣❛s♦❧✐♥❛✱ ❝♦♠♦ ♦ ♣r❡ç♦ ❞♦ ❧✐tr♦ é ❘✩ ✸✱✸✹✱ ❡♥tã♦✱ s❡ ❛❜❛st❡❝❡♠♦s ✶ ❧✐tr♦ ♣❛❣❛♠♦s ❘✩ 3,34 = 3,34×1✱ s❡ ❛❜❛st❡❝❡♠♦s ✷ ❧✐tr♦s ♣❛❣❛♠♦s
