❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚
❖ ❯s♦ ❞❛ P❧❛♥✐❧❤❛ ❊❧❡trô♥✐❝❛ ♥♦
❊st✉❞♦ ❞❛s ❋✉♥çõ❡s ❡ ❊q✉❛çõ❡s
P♦❧✐♥♦♠✐❛✐s
†♣♦r
▼❛r❝❡❧♦ ▼♦❣✐❧❦❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡✲ q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡s✲ tr❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❏✉♥❤♦✴✷✵✶✻ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
M696u Mogilka, Marcelo.
O uso da planilha eletrônica no estudo das funções e equações polinomiais / Marcelo Mogilka.- João Pessoa, 2016. 54f. : il.
Orientador: Bruno Henrique Carvalho Ribeiro Dissertação (Mestrado) - UFPB/CCEN
1. Matemática. 2. Planilha eletrônica. 3. Função quadrática. 4. Cálculo de volumes. 5. Funções polinomiais.
❘❡s✉♠♦
❆ ✉t✐❧✐③❛çã♦ ❞❛s t❡❝♥♦❧♦❣✐❛s ❝♦♠♦ ❢❡rr❛♠❡♥t❛s ❞✐❞át✐❝❛s ❥á é ✉♠❛ r❡❛❧✐❞❛❞❡ ✈✐✈✐❞❛ ♣♦r ✉♠❛ ♣❛rt❡ s✐❣♥✐✜❝❛t✐✈❛ ❞♦ ❛❧✉♥❛❞♦ ♥♦ ❇r❛s✐❧✳ ◆♦ ❡♥t❛♥t♦✱ ♦ r✐t♠♦ ✈❡rt✐❣✐♥♦s♦ ❞♦ ❛✈❛♥ç♦ ❞❡ss❛s t❡❝♥♦❧♦❣✐❛s ♥ã♦ é ❛❝♦♠♣❛♥❤❛❞♦ ♣❡❧❛ ❞❡✈✐❞❛ ❛❞❛♣t❛çã♦ ❡ ❛♣❧✐❝❛çã♦ à r❡❛❧✐❞❛❞❡ ❞❛s ❡s❝♦❧❛s ❡✱ s♦❜r❡t✉❞♦✱ à r❡❛❧✐❞❛❞❡ ❞♦s ❛❧✉♥♦s✳ ❉✐❛♥t❡ ❞❡st❡ ❞❡s❝♦♠✲ ♣❛ss♦ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❛♠♣❧✐❛r ❛s ♦❢❡rt❛s ❞❡ tr❛❜❛❧❤♦s ❛❝❛❞ê♠✐❝♦s q✉❡ ✉t✐❧✐③❡♠ ❛s t❡❝♥♦❧♦❣✐❛s ♦❢❡r❡❝✐❞❛s ♥❛ ❢♦r♠❛ ❞❡ ❛♣❧✐❝❛t✐✈♦s✱ s♦❢t✇❛r❡s ♦✉ ♣r♦❣r❛♠❛s ❡s♣❡❝í✜❝♦s ❞❛ ár❡❛ ❞❡ ❡❞✉❝❛çã♦✱ ♣❛r❛ ❛✉①✐❧✐❛r ♦ ♣r♦❢❡ss♦r ♥❛ ❞✐❢í❝✐❧ t❛r❡❢❛ ❞❡ ❡♥s✐♥❛r ❛ ♠❛t❡♠á✲ t✐❝❛✳ ❊ss❛ ❞✐ss❡rt❛çã♦ ♣r❡t❡♥❞❡ ❝♦♥tr✐❜✉✐r ❡①❛t❛♠❡♥t❡ ♥❡ss❡ s❡♥t✐❞♦✳ ❆tr❛✈és ❞❡ ✉♠ ♣r♦❥❡t♦ ❞✐❞át✐❝♦ ❛♣❧✐❝❛❞♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ✕ ♦ Pr♦❥❡t♦ ❉❡❧t❛ ✕ q✉❡ t❡♠ ❝♦♠♦ ♣r✐♥✲ ❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❛♠♣❧✐❛r ❡ ❛♣r♦❢✉♥❞❛r ♦s ❝♦♥❝❡✐t♦s ❞❡ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✱ ❞✐✈✐sã♦ ❞❡ ♣♦❧✐♥ô♠✐♦s ❡ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡s ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s ✉t✐❧✐③❛♥❞♦ ♣❛r❛ t❛♥t♦ ❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❞✐❞át✐❝❛ ❡ ❧✐♥❦ ❡♥tr❡ ♦s s❛❜❡r❡s ❞❛s ár❡❛s ❞❡ ♠❛t❡♠át✐❝❛ ❡ ♣r♦❣r❛♠❛çã♦✳ P♦r ♠❡✐♦ ❞❡ ❡①❡♠♣❧♦s ♣rát✐❝♦s ❞❛ ✉t✐❧✐③❛çã♦ ❞❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ♥❛ ❝r✐❛çã♦ ❞❡ ♣r♦❣r❛♠❛s ♣❡❧♦s ❛❧✉♥♦s q✉❡ ♣❛rt✐❝✐♣❛r❛♠ ❞♦ Pr♦❥❡t♦ ❉❡❧t❛✱ ❡ ❞❛ ❞❡✈✐❞❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✱ ❡ss❡ tr❛✲ ❜❛❧❤♦ ❛❝❛❞ê♠✐❝♦ ♣r❡t❡♥❞❡ ♠♦str❛r q✉❡ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❝r✐❛r ♣r♦❣r❛♠❛s ♥❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ♣❛r❛ ❛✉①✐❧✐❛r ❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s ♦ ❛❧✉♥♦ t❡♠ ❝♦♠♦ s✉❜♣r♦❞✉t♦ ♣♦s✐t✐✈♦ ✉♠ ❛♣r♦❢✉♥❞❛♠❡♥t♦ ❞♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ss❡s s❛❜❡r❡s ♠❛t❡♠á✲ t✐❝♦s ❡✱ ♠❛✐s ❛✐♥❞❛✱ ✉♠❛ ♣♦st✉r❛ ❛♥❛❧ít✐❝❛ ❡ ♣r♦t❛❣♦♥✐st❛ ❞✐❛♥t❡ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❞❡ss❛ ♠❛tér✐❛ ❡♠ ♦♣♦s✐çã♦ à ♣♦st✉r❛ ♣❛ss✐✈❛ ❡ ♣♦✉❝♦ r❡✢❡①✐✈❛ q✉❡✱ ✐♥❢❡❧✐③♠❡♥t❡✱ ❛✐♥❞❛ é ❛ss✉♠✐❞❛ ♣♦r ❣r❛♥❞❡ ♣❛rt❡ ❞♦s ♥♦ss♦s ❛❧✉♥♦s✳
P❛❧❛✈r❛s ❝❤❛✈❡✿ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛✱ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡s✱ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳
❆❜str❛❝t
❙✉♠♠❛r②
❚❤❡ ✉s❡ ♦❢ t❡❝❤♥♦❧♦❣✐❡s ❛s t❡❛❝❤✐♥❣ t♦♦❧ ✐s ❛ r❡❛❧✐t② ❡①♣❡r✐❡♥❝❡❞ ❜② ❛ s✐❣♥✐✜❝❛♥t ♣♦rt✐♦♥ ♦❢ t❤❡ st✉❞❡♥t ❜♦❞② ✐♥ ❇r❛③✐❧✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✐s st✐❧❧ ❝♦♥s✐❞❡r❛❜❧❡ ❞❡♠❛♥❞ ❢♦r ❡❞✉❝❛t✐♦♥❛❧ ♣r♦❥❡❝ts ✐♥✈♦❧✈✐♥❣ t❤❡ t❡❛❝❤✐♥❣ ♦❢ ♠❛t❤❡♠❛t✐❝s ✇✐t❤ t❤❡ t❡❝❤♥♦❧♦❣✐❡s ♦✛❡r❡❞ ❜② ❝♦♠♣✉t❡r ♣r♦❣r❛♠s✳ ❚❤✐s ✇♦r❦ ✇✐❧❧ ❛❞❞r❡ss ✕ t❤r♦✉❣❤ t❤❡ ❉❡❧t❛ Pr♦❥❡❝t ✕ t❤❡ ✉s❛❣❡ ♦❢ ❡❧❡❝tr♦♥✐❝ s♣r❡❛❞s❤❡❡ts ❛s ❛ t♦♦❧ ❢♦r ❛♥❛❧②s✐s ❛♥❞ r❡s♦❧✉t✐♦♥ ♦❢ ❢✉♥❝✲ t✐♦♥❛❧ ❛♥❞ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥s✱ ❡st❛❜❧✐s❤✐♥❣ ❧✐♥❦s ❜❡t✇❡❡♥ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ❦♥♦✇❧❡❞❣❡✳
❑❡②✇♦r❞s✿ ❡❧❡❝tr♦♥✐❝ s♣r❡❛❞s❤❡❡ts✱ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✱ ✈♦❧✉♠❡ ❝❛❧❝✉❧❛t✐♦♥✱ ❢✉♥❝t✐♦♥ ♣♦❧②♥♦♠✐❛❧✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ❚❡❝♥♦❧♦❣✐❛s ❝♦♠♦ ❢❡rr❛♠❡♥t❛s ❞✐❞át✐❝❛s ✹ ✶✳✶ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ▼❛①✐♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ P♦❧② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✹ ▼✉P❛❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✺ ❙✉♣❡r ▲♦❣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✻ ❲✐♥♣❧♦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✼ ❑♠P❧♦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✽ ❲✐♥♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✾ ❯♠ ❜r❡✈❡ ❡st✉❞♦ ❞❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷ ❖ Pr♦❥❡t♦ ❉❡❧t❛ ❡♠ ❡①❡❝✉çã♦ ♥❛ ♣❧❛♥✐❧❤❛ ❝❛❧❝ ✶✸ ✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❋✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ Pr♦❥❡t♦ ❉❡❧t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸ ❆t✐✈✐❞❛❞❡ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✹ ❆t✐✈✐❞❛❞❡ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✺ ❖ Pr♦❥❡t♦ ❉❡❧t❛ ❡①❡❝✉t❛❞♦ ♣❡❧♦s ❛❧✉♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✻ ❆t✐✈✐❞❛❞❡s ❞♦ ❛❧✉♥♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✼ ❆t✐✈✐❞❛❞❡s ❞♦ ❛❧✉♥♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✽ ❆t✐✈✐❞❛❞❡s ❞♦ ❛❧✉♥♦ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✾ ❈♦♥❝❧✉sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸ ◆♦✈❛s ♣r♦♣♦st❛s ♣❛r❛ ♦ ✉s♦ ❞❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ✸✹ ✸✳✶ Pr♦♣♦st❛s ♣❛r❛ ♦ ✷♦ ❛♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸✳✶✳✶ ❆t✐✈✐❞❛❞❡ ✶ ❞♦ ✷♦ ❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸✳✶✳✷ ❆t✐✈✐❞❛❞❡ ✷ ❞♦ ✷♦ ❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✷ Pr♦♣♦st❛ ♣❛r❛ ♦ ✸♦ ❛♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✸✳✷✳✶ ➪r❡❛ ❡①t❡r♥❛ ❡ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✷✳✷ ➪r❡❛ ❡①t❡r♥❛ ❡ ✈♦❧✉♠❡ ❞♦ ❤❡①❛❡❞r♦ r❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✸✳✷✳✸ ➪r❡❛ ❡①t❡r♥❛ ❡ ✈♦❧✉♠❡ ❞❛ ♣✐râ♠✐❞❡ q✉❛❞r❛♥❣✉❧❛r r❡❣✉❧❛r ✳ ✳ ✳ ✹✷ ✸✳✸ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ t❡❧❛ ❞♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ t❡❧❛ ❞♦ s♦❢t✇❛r❡ ▼❛①✐♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ ❡①❡♠♣❧♦ ❞❡ ❢ór♠✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✺ ✉s♦ ❞♦ ❝♦♥❞✐❝✐♦♥❛❧ ❙❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✻ ❙❊ ❛♥✐♥❤❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✼ ❜❛rr❛s ❞❡ ❝♦♠❛♥❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✶ Pr♦❥❡t♦ ❉❡❧t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ③❡r♦s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✸ ③❡r♦s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❝♦♠ ❛ ❝♦♥❞✐❝✐♦♥❛❧ ❙❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✹ ❡①❡❝✉çã♦ ❞❛s ❢ór♠✉❧❛s ❝♦♠ ♦ ❝♦♥❞✐❝✐♦♥❛❧ ❙❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✺ ③❡r♦s ❞❛ ❢✉♥çã♦f(x) = 2x2
−10x+ 5 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✻ ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✼ ✐♠❛❣❡♠ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✽ ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ ❞❛ ❢✉♥çã♦f(x) = 2x2
−4x+ 7 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷✳✾ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ ❞❛ ❢✉♥çã♦f(x) = 2x2
−4x+ 7 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✷✳✶✵ ❛t✐✈✐❞❛❞❡ ✶ ❞♦ ❛❧✉♥♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✶ ❛t✐✈✐❞❛❞❡ ✷ ❞♦ ❛❧✉♥♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✷ ❢ór♠✉❧❛s ❞❛s ❛t✐✈✐❞❛❞❡s ✶ ❡ ✷ ❞♦ ❛❧✉♥♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✸ ❛t✐✈✐❞❛❞❡ ✶ ❞♦ ❛❧✉♥♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✹ ❛t✐✈✐❞❛❞❡ ✷ ❞♦ ❛❧✉♥♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✶✺ ❢ór♠✉❧❛s ❞❛s ❛t✐✈✐❞❛❞❡s ✶ ❡ ✷ ❞♦ ❛❧✉♥♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✶✻ ❛t✐✈✐❞❛❞❡ ✶ ❞♦ ❛❧✉♥♦ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✶✼ ❛t✐✈✐❞❛❞❡ ✷ ❞♦ ❛❧✉♥♦ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✶✽ ❢ór♠✉❧❛s ❞❛s ❛t✐✈✐❞❛❞❡s ✶ ❡ ✷ ❞♦ ❛❧✉♥♦ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✶✾ ❛♥á❧✐s❡ ❞❛ ❢✉♥çã♦f(x) = x2
+x+ 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✸✳✶ ❡♥✉♥❝✐❛❞♦ ❞❛ ❛t✐✈✐❞❛❞❡ ✶ ❞♦ ✷♦ ❛♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✸✳✷ ❣❛❜❛r✐t♦ ❞❛ ❛t✐✈✐❞❛❞❡ ✶ ❞♦ ✷♦ ❛♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✸ ❣❛❜❛r✐t♦ ❞❛ ❛t✐✈✐❞❛❞❡ ✶ ❞♦ ✷♦ ❛♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✹ ❢ór♠✉❧❛s ❞❛ ❛t✐✈✐❞❛❞❡ ✶ ❞♦ ✷♦ ❛♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✸✳✺ ❡♥✉♥❝✐❛❞♦ ❞❛ ❛t✐✈✐❞❛❞❡ ✷ ❞♦ ✷♦ ❛♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✻ ❣❛❜❛r✐t♦ ❞❛ ❛t✐✈✐❞❛❞❡ ✷ ❞♦ ✷♦ ❛♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸✳✼ ❡♥✉♥❝✐❛❞♦ ❞❛ ❛t✐✈✐❞❛❞❡ ✸♦ ❛♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
■♥tr♦❞✉çã♦
❯♠ ❞♦s ❣r❛♥❞❡s ❞❡s❛✜♦s ❞♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ♥♦ ❇r❛s✐❧ é ❛♣r❡s❡♥t❛r ❡st❛ ❞✐s❝✐♣❧✐♥❛ ❛♦ ❛❧✉♥❛❞♦ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❛❝❡ssí✈❡❧ ❡ ❛tr❛❡♥t❡ ✲ ♠❛✐s ❧✐❣❛❞❛ ❛♦ s❡✉ ♠✉♥❞♦ ✲ s❡♠ ❛❜r✐r ♠ã♦ ❞❡ s❡✉ r✐❣♦r ❝✐❡♥tí✜❝♦ q✉❡✱ ❡♥tr❡ ♦✉tr♦s ❛tr✐✲ ❜✉t♦s✱ ❛ ❞❡✜♥❡✳ ▼✉✐t♦ ❥á s❡ ❢❡③ ✲ ❡ s❡ ❢❛③ ✲ ♥❡ss❡ s❡♥t✐❞♦✱ s❡❥❛ ♥♦ ❝❛♠♣♦ t❡ór✐❝♦ ❞❛ ♣❡❞❛❣♦❣✐❛ ❝❧áss✐❝❛✱ s❡❥❛ ♥♦ â♠❜✐t♦ ❞❛s t❡❝♥♦❧♦❣✐❛s ❛♣❧✐❝❛❞❛s à ❡❞✉❝❛çã♦✳ ➱ ♥❡ss❡ s❡✲ ❣✉♥❞♦ s❡❛r❛ q✉❡ ❡st❡ tr❛❜❛❧❤♦ ❢♦❝❛rá s✉❛ ❧✐♥❤❛ ❞❡ ♣❡sq✉✐s❛✱ ❡♠ ♣❛rt✐❝✉❧❛r ♥♦ ✉s♦ ❞❛s ♣❧❛♥✐❧❤❛s ❡❧❡trô♥✐❝❛s ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❛✉①✐❧✐❛r ♦ ❛❧✉♥♦ ♥❛ ❛♥á❧✐s❡ ❡ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s✱ ♥♦ ❝á❧❝✉❧♦ ❡ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ♣♦♥t♦s ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❞❡ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❡ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ ✐♠❛❣❡♠ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳
◆ã♦ ♦❜st❛♥t❡ ♦ ❢❛t♦ ❞❡ q✉❡ ❛s t❡❝♥♦❧♦❣✐❛s ❞❛ ✐♥❢♦r♠❛çã♦ ❡st❛r❡♠ ❞✐ss❡♠✐♥❛❞❛s ♣❡❧❛s ♠❛✐s ❞✐✈❡rs❛s ❝❛♠❛❞❛s ❞❛ ♣♦♣✉❧❛çã♦ ❜r❛s✐❧❡✐r❛ ❡ q✉❡ s❡✉ ✉s♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ s❡❥❛ ❡st✐♠✉❧❛❞♦ ❡ ❡♥❝♦r❛❥❛❞♦ ♣❡❧♦s ❣❡st♦r❡s ❞❛s ❡s❝♦❧❛s ❞❡ ♥í✈❡❧ ❜ás✐❝♦✱ s✉❛ ✉t✐❧✐③❛çã♦ ❡❢❡t✐✈❛ ❛✐♥❞❛ é r❡str✐t❛ ❛ ♣❡q✉❡♥♦s ❣r✉♣♦s ❞❡ ❡st✉❞❛♥t❡s ❞❡♥tr♦ ❞❛s ❡s❝♦❧❛s ❡ ❡♠ ♣❡rí♦❞♦s ♣ré✲❞❡t❡r♠✐♥❛❞♦s✿
✏❊♠ ❡❞✉❝❛çã♦ ❡s❝♦❧❛r✱ ♣♦r ♠✉✐t♦ t❡♠♣♦ ✲ ❡ ❡✉ ❞✐r✐❛ ♠❡s♠♦✱ ❛té ❤♦❥❡ ✲✱ ♥ã♦ s❡ ✈❛❧♦r✐③♦✉ ❛❞❡q✉❛❞❛♠❡♥t❡ ♦ ✉s♦ ❞❡ t❡❝♥♦❧♦❣✐❛ ✈✐s❛♥❞♦ ❛ t♦r♥❛r ♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦✲ ❛♣r❡♥❞✐③❛❣❡♠ ♠❛✐s ❡✜❝✐❡♥t❡ ❡ ♠❛✐s ❡✜❝❛③✑✳ ✭❬✻❪✱ ♣✳✶✸✸✮
❊ss❛ ❢♦r♠❛ ❛✐♥❞❛ ✐♥❝✐♣✐❡♥t❡ ❞❡ ✉t✐❧✐③❛r ❛ t❡❝♥♦❧♦❣✐❛ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛ ✕ ❡♠ ♣❛rt✐❝✉❧❛r ♥♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ✕ ❝♦♠ t♦❞♦ s❡✉ ♣♦❞❡r ❞❡ ❛❣r❡❣❛r s❛❜❡r❡s ❝♦♥❝✐t❛ ❛ ✐♥✐❝✐❛t✐✈❛ ❞❡ ❞❡s❡♥✈♦❧✈❡r ♥♦✈❛s ❢♦r♠❛s ❞❡ ✉t✐❧✐③❛çã♦ ❞❡ss❛s ❢❡rr❛♠❡♥t❛s✱ ♥♦ ❝❛s♦ ❞❡ss❡ tr❛❜❛❧❤♦✱ ❛s ♣❧❛♥✐❧❤❛s ❡❧❡trô♥✐❝❛s✱ q✉❡ ♣r♦♣♦r❝✐♦♥❛♠ ❛♦ ❛❧✉♥♦ ❞❡s❡♥✈♦❧✈❡r ♣r♦❣r❛♠❛s q✉❡ ♦ ❛✉①✐❧✐❡ ❛ ✐♥t❡r♣r❡t❛r ♦s ❝♦♥❝❡✐t♦s✱ ❢ór♠✉❧❛s ❡ ❝á❧❝✉❧♦s ❞❡ ❞❡t❡r♠✐♥❛❞♦s tó♣✐❝♦s ❞❛ ♠❛t❡♠át✐❝❛ ❜ás✐❝❛✿
✏❆❧é♠ ❞❛s ✈❛♥t❛❣❡♥s ❡ ♦s ❜❡♥❡❢í❝✐♦s ❞❛ ✐♥tr♦❞✉çã♦ ❞❡ ♠✐❝r♦❝♦♠♣✉t❛❞♦r❡s ♥❛ ❊❞✉❝❛çã♦✱ é ♥❡❝❡ssár✐♦ ✐♥❞✐❝❛r ❛❧❣✉♠❛s ❞❛s ♣♦ssí✈❡✐s ♠❛♥❡✐r❛s ❞♦ ♠✐❝r♦❝♦♠♣✉t❛❞♦r ❛✉①✐❧✐❛r ♦ ♣r♦❝❡ss♦ ♣❡❞❛❣ó❣✐❝♦✑ ✭❬✷❪✱ ♣✳✸✸✮
❖ ♣r♦❝❡ss♦ ❞❡ ♣r♦❣r❛♠❛r ❛ ♣❧❛♥✐❧❤❛ ♣❛r❛ ❡①❡❝✉t❛r ❞❡t❡r♠✐♥❛❞♦ ❝á❧❝✉❧♦ ♠❛t❡♠á✲ t✐❝♦ ♦✉ ✉t✐❧✐③❛r ❛s ❢❡rr❛♠❡♥t❛s ❞❡ ❛♥á❧✐s❡ ❝♦♥❞✐❝✐♦♥❛❧ q✉❡ ❛ ♣❧❛♥✐❧❤❛ ♦❢❡r❡❝❡✱ r❡q✉❡r ❞♦ ❛❧✉♥♦ ✕ ❛ ♣rát✐❝❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ❝♦♥❢♦r♠❡ ✈❡r❡♠♦s ❛♦ ❧♦♥❣♦ ❞❡ss❡ tr❛❜❛❧❤♦✱
❝♦♠♣r♦✈♦✉ ✕ ✉♠❛ ❝♦♠♣r❡❡♥sã♦ ♠❛✐s ♣r♦❢✉♥❞❛ ❞♦s ❝♦♥❝❡✐t♦s ❛❜♦r❞❛❞♦s ✜①❛♥❞♦ ❛s✲ s✐♠ ♦ q✉❡ ❡stá s❡♥❞♦ ❛♣r❡♥❞✐❞♦ ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❞❡s❧♦❝❛ ♦ ❛❧✉♥♦ ❞❛ ♣♦s✐çã♦ ❞❡ ♠❡r♦ ♦✉✈✐♥t❡ ♣❛r❛ ♣r♦t❛❣♦♥✐st❛ ❞❡ s❡✉ ❛♣r❡♥❞✐③❛❞♦✳ Pr♦t❛❣♦♥✐s♠♦ ❡ss❡ q✉❡ ♣♦ss✐✲ ❜✐❧✐t❛ ❛♦ ❛❧✉♥♦ ❝♦♥❤❡❝❡r✱ ❞❡ ❢❛t♦✱ ♦ q✉❡ ❡stá ❡①❡❝✉t❛♥❞♦ ❜❡♠ ❝♦♠♦ ♦s ♦❜❥❡t✐✈♦s q✉❡ q✉❡r ❛❧❝❛♥ç❛r✿
✏❆ ❛t✐✈✐❞❛❞❡ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ❝♦♥st✐t✉✐✲s❡ ♥✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❆çõ❡s ❡ ❖♣❡r❛çõ❡s✱ ❞✐r❡❝✐♦♥❛❞❛s ♣♦r ✉♠ ▼♦t✐✈♦ ♣❛r❛ ❛t✐♥❣✐r ❞❡t❡r♠✐♥❛❞❛ ❋✐♥❛❧✐❞❛❞❡✑✳ ✭❬✶❪✱ ♣✳✼✶✮✳
❖ ▼♦t✐✈♦ ❡ ❛ ❋✐♥❛❧✐❞❛❞❡ ❝✐t❛❞♦s ❛❝✐♠❛ ♣♦r ❆❜r❡✉ sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♥❡ss❡ ❝♦♥t❡①t♦ ❞❡ ❡❞✉❝❛çã♦ ♠❛t❡♠át✐❝❛✱ ♦ ❞♦♠í♥✐♦ ❡❢❡t✐✈♦ ❞♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s q✉❡ s❡ ♣r♦♣õ❡ ❡st✉❞❛r ❡ ❛ ❛♣❧✐❝❛çã♦ ❞❡ss❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❛❞q✉✐r✐❞♦ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❡ ❛♥á❧✐s❡ ❞❡ s✐t✉❛çõ❡s ❡ ♣r♦❜❧❡♠❛s ♣r♦♣♦st♦s✳ ❆s ❆çõ❡s ❡ ❖♣❡r❛çõ❡s✱ ♥♦ â♠❜✐t♦ ❞❛ ♣r♦♣♦st❛ ♣❡❞❛❣ó❣✐❝❛ s✉❣❡r✐❞❛ ♥❡ss❡ tr❛❜❛❧❤♦✱ s❡ ✉t✐❧✐③❛♠ ❞❛s ♣❧❛♥✐❧❤❛s ❡❧❡trô♥✐❝❛s ❝♦♠♦ ❧✐♥❦ ❡♥tr❡ ♦s s❛❜❡r❡s ♠❛t❡♠át✐❝♦s q✉❡ s❡ q✉❡r ❛♣r♦♣r✐❛r ❡ ❛ ❡❢❡t✐✈❛ ❛♣r♦♣r✐✲ ❛çã♦ ❞❡ss❡s ❝♦♥❤❡❝✐♠❡♥t♦s ♣♦r ♣❛rt❡ ❞♦ ❛❧✉♥♦✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r ❛✐♥❞❛ q✉❡ ✉t✐❧✐③❛r ❛s ♣❧❛♥✐❧❤❛s ❡❧❡trô♥✐❝❛s ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ❞✐❞át✐❝❛ ♥♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠á✲ t✐❝❛ ❡①✐❣❡ q✉❡ ♦ ❛❧✉♥♦ t❡♥❤❛ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ré✈✐♦ ❞❛ ♣❧❛♥✐❧❤❛✱ s❡✉s ❝♦♠❛♥❞♦s ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❛ ❧ó❣✐❝❛ ❞❡ ♣r♦❣r❛♠❛çã♦ ❞❛ ♣❧❛♥✐❧❤❛ q✉❡✱ ❝♦♠ t♦❞♦s s❡✉s ❛❧❣♦✲ r✐t♠♦s ❡ s❡q✉ê♥❝✐❛s ❞❡ ❝♦♠❛♥❞♦s✱ r❡❢♦rç❛ ❛✐♥❞❛ ♠❛✐s ♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ ❞♦ ❛❧✉♥♦✱ ❝r✐❛♥❞♦ ✉♠❛ r❡❛❧✐♠❡♥t❛çã♦ ♣♦s✐t✐✈❛ ♥❛ ❝♦♥str✉çã♦ ❞❡ s❡✉s s❛❜❡r❡s ♥✉♠ ❛♠❜✐❡♥t❡ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ s✐❣♥✐✜❝❛t✐✈❛✿
✏❬✳✳✳❪ ❛♣r❡♥❞✐③❛❣❡♠ s✐❣♥✐✜❝❛t✐✈❛ é ✉♠ ♣r♦❝❡ss♦ ♣♦r ♠❡✐♦ ❞♦ q✉❛❧ ✉♠❛ ♥♦✈❛ ✐♥✲ ❢♦r♠❛çã♦ s❡ r❡❧❛❝✐♦♥❛✱ ❞❡ ♠❛♥❡✐r❛ s✉❜st❛♥t✐✈❛ ✭♥ã♦ ❧✐t❡r❛❧✮ ❡ ♥ã♦ ❛r❜✐trár✐❛✱ ❛ ✉♠ ❛s♣❡❝t♦ r❡❧❡✈❛♥t❡ ❞❛ ❡str✉t✉r❛ ❝♦❣♥✐t✐✈❛ ❞♦ ✐♥❞✐✈í❞✉♦✳ ■st♦ é✱ ♥❡ss❡ ♣r♦❝❡ss♦✱ ❛ ♥♦✈❛ ✐♥❢♦r♠❛çã♦ ✐♥t❡r❛❣❡ ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❡s♣❡❝í✜❝❛ ❬✳✳✳❪✑ ✭❬✼❪ ♣✳✶✶✮✳
❉❡ss❛ ❢♦r♠❛✱ ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♣ré✈✐♦s ❞♦ ❛❧✉♥♦ ✕ ❞❡ ♠❛t❡♠át✐❝❛ ❡ ✐♥❢♦r♠át✐❝❛ ✕ ✐♥t❡r❛❣❡♠ ♣❛r❛ ❢♦rt❛❧❡❝❡r ❛♠❜♦s ❡ ❞❡ ❢♦r♠❛ s✉❜❥❛❝❡♥t❡ ♦s r❡❧❛❝✐♦♥❛ ♥♦ q✉❡ tê♠ ❞❡ ♠❛✐s r❡❧❡✈❛♥t❡ ❡ ❝♦♠✉♠✿ ❝♦♥❝❡✐t♦s✱ r❡❣r❛s✱ s❡q✉❡♥❝✐❛ ❧ó❣✐❝❛ ❞❡ r❛❝✐♦❝í♥✐♦ ❡ ❡①❡❝✉çã♦ ❞❡ ❛❧❣♦r✐t♠♦s ❛tr❛✈és ❞❡ ❝♦♠❛♥❞♦s ♣r❡✈✐❛♠❡♥t❡ ❝♦♥❤❡❝✐❞♦s✳
❊ss❡ ❚❈❈ ❢♦✐ ❞✐✈✐❞✐❞♦ ✕ ♥♦ q✉❡ t❛♥❣❡ s❡✉ ♥ú❝❧❡♦ ❞❡ ♣❡sq✉✐s❛ ✕ ❡♠ ❞✉❛s ♣❛rt❡s✿ ♥❛ ♣r✐♠❡✐r❛ ♣❛rt❡ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❛t✐✈✐❞❛❞❡s ❡♥✈♦❧✈❡♥❞♦ ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ ✷♦ ❣r❛✉ ❡ ❞✐s❝✉ssã♦ ❞♦ ✈❛❧♦r r❡❛❧ ❞♦ ❞✐s❝r✐♠✐♥❛♥t❡✱ ❝á❧❝✉❧♦ ❡ ❛♥á❧✐s❡ ❞♦s ♠á①✐♠♦s ❡
♠í♥✐♠♦s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡ ❞❡t❡r♠✐♥❛çã♦ ❞❛ ✐♠❛❣❡♠ ❞❡ss❛s ❢✉♥çõ❡s ❡①❡❝✉t❛❞❛s ♥❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ♣♦r ❞❡ três ❛❧✉♥♦s ❞❡ ✉♠❛ t✉r♠❛ ❞♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❞❡ ✉♠❛ ❡s❝♦❧❛ ❞❛ ❝✐❞❛❞❡ ❞❡ ❏♦ã♦ P❡ss♦❛✴P❇✳ ◆❡st❛ ❡t❛♣❛ s❡rã♦ ❞❡t❛❧❤❛❞♦s ♦s ♣❛ss♦s ❞❛ ❡①❡❝✉çã♦ ❞❡ss❛s ❛t✐✈✐❞❛❞❡s ❞❛♥❞♦ ê♥❢❛s❡ ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥str✉çã♦ ❞❛ tr❛♥s✐çã♦ ❞❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ✕ ❥á ❛♣r❡s❡♥t❛❞❛ ❛♦s ❛❧✉♥♦s ♥❛s ❛✉❧❛s r❡❣✉❧❛r❡s✱ ❝♦♠ ❛s ❞❡✈✐❞❛s ❞✐s❝✉ssõ❡s ❡ ❡①❡❝✉çã♦ ❞❡ t❛r❡❢❛s ♣r♦♣♦st❛s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✕ ❜❡♠ ❝♦♠♦ ❛s ❞✐✜❝✉❧❞❛❞❡s ❡♥❝♦♥tr❛❞❛s ♣❡❧♦s ❛❧✉♥♦s ♣❛r❛ s❡ ❛♣r♦♣r✐❛r ❞❛s ❢❡rr❛♠❡♥t❛s ❝♦❣♥✐t✐✈❛s ♥❡❝❡ssár✐❛s ♣❛r❛ ❡❢❡t✐✈❛r ❡ss❛ tr❛♥s✐çã♦✳ ❙❡rã♦ ♠♦str❛❞♦s t❛♠❜é♠ ♦s
❞✐❢❡r❡♥t❡s ❝❛♠✐♥❤♦s t♦♠❛❞♦s ♣❡❧♦s ❛❧✉♥♦s ♣❛r❛ ♣r♦❣r❛♠❛r ❛ ♣❧❛♥✐❧❤❛ ♣❛r❛ ❡①❡❝✉t❛r ❢ór♠✉❧❛s ♠❛t❡♠át✐❝❛s ❡ t❛♠❜é♠ ❛♥❛❧✐s❛r s❡✉s r❡s✉❧t❛❞♦s✳ ◆❛ s❡❣✉♥❞❛ ♣❛rt❡ s❡rã♦ ♣r♦♣♦st❛s ♥♦✈❛s ❛t✐✈✐❞❛❞❡s ♥❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ❡♥✈♦❧✈❡♥❞♦ ❛❣♦r❛ ♦s ❛❧✉♥♦s ❞♦ ✷♦
❡ ✸♦ ❛♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ P❛r❛ ♦ s❡❣✉♥❞♦ ❛♥♦ ❛❜♦r❞❛r❡♠♦s ♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s
♣♦❧✐♥♦♠✐❛✐s✳ P❛r❛ ♦ t❡r❝❡✐r♦ ❛♥♦ ♣r♦♣♦r❡♠♦s ♦ ❡st✉❞♦ ❞❛s ár❡❛s ❡ ❞♦s ✈♦❧✉♠❡s ❞♦s ♣r✐♥❝✐♣❛✐s só❧✐❞♦s ❝♦♥✈❡①♦s ❡st✉❞❛❞♦s ♥❡st❛ sér✐❡ t❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ é ❛ss✉♥t♦ r❡❝♦rr❡♥t❡ ♥♦ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❉♦ ❊♥s✐♥♦ ▼é❞✐♦ ✭❊◆❊▼✮✳
❈❛♣ít✉❧♦ ✶
❚❡❝♥♦❧♦❣✐❛s ❝♦♠♦ ❢❡rr❛♠❡♥t❛s
❞✐❞át✐❝❛s
❆ ✉t✐❧✐③❛çã♦ ❞❛s ♠í❞✐❛s ❡❧❡trô♥✐❝❛s ♥❛ ❡❞✉❝❛çã♦ ♠❛t❡♠át✐❝❛ ❝♦♥st✐t✉✐ ✉♠❛ ♣♦✲ ❞❡r♦s❛ ✲ ❛✐♥❞❛ q✉❡ ♣♦t❡♥❝✐❛❧♠❡♥t❡ s✉❜✉t✐❧✐③❛❞❛ ✲ ❢❡rr❛♠❡♥t❛ ❞❡ ❛♣♦✐♦ ❞✐❞át✐❝♦ ♥❛ ❞✐❢í❝✐❧ t❛r❡❢❛ ❞❡ t♦r♥❛r ♦ ❡st✉❞♦ ❞❛ ♠❛t❡♠át✐❝❛ ♠❛✐s ❛❝❡ssí✈❡❧ ❡ ❝♦♥t❡①t✉❛❧✐③❛❞♦ ❡♠ r❡❧❛çã♦ à r❡❛❧✐❞❛❞❡ ❞♦ ❛❧✉♥❛❞♦✳ ❆♥t❡s ❞❡ ❛♣r❡s❡♥t❛r ❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛✱ q✉❡ é ❛ ❢❡rr❛♠❡♥t❛ ✉t✐❧✐③❛❞❛ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❛♣❧✐❝❛çã♦ ❞♦ Pr♦❥❡t♦ ❉❡❧t❛ ✖ ♣r♦❥❡t♦ ♣❡❞❛❣ó❣✐❝♦ q✉❡ é ♦ t❡♠❛ ❝❡♥tr❛❧ ❞❡ ❡st✉❞♦ ❞❡ss❛ ❞✐ss❡rt❛çã♦ ✕ s❡rá ❢❡✐t❛ ✉♠❛ ❜r❡✈❡ ❛♣r❡s❡♥t❛çã♦ ❞♦s ♣r✐♥❝✐♣❛✐s s♦❢t✇❛r❡s✕ t♦❞♦s ❣r❛t✉✐t♦s ❡ ❡♠ ♣♦rt✉❣✉ês ✭❡①❝❡t♦ ♦ s♦❢t✇❛r❡ P♦❧②✮ ✕ ❛♣❧✐❝❛❞♦s à ❡❞✉❝❛çã♦ ♠❛t❡♠át✐❝❛✳
✶✳✶ ●❡♦❣❡❜r❛
❖ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛ é ✉♠❛ ♣♦❞❡r♦s❛ ❢❡rr❛♠❡♥t❛ ❞✐❞át✐❝❛ ❛♣❧✐❝❛❞❛ ♥♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛✳ Pr♦❣r❛♠❛❞♦ ❡♠ ❏❛✈❛✱ ❢✉♥❝✐♦♥❛ ♥♦s s✐st❡♠❛s ♦♣❡r❛❝✐♦♥❛✐s ♠❛✐s ✉t✐❧✐③❛❞♦s ♥❛s ❡s❝♦❧❛s ✕ ▲✐♥✉① ♦✉ ❲✐♥❞♦✇s ✕ ❡ s✉❛ ❛♣❧✐❝❛çã♦ s❡ ❡st❡♥❞❡ ❞❡s❞❡ ❛s sér✐❡s ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ■■ ❛té ❛s ❣r❛❞✉❛çõ❡s ♥❛s ár❡❛s ❞❡ ❝✐ê♥❝✐❛s ❡①❛t❛s✳ ❈♦♠ ❡st❡ s♦❢t✇❛r❡ é ♣♦ssí✈❡❧ ❝♦♥str✉✐r ❡ ❛♥❛❧✐s❛r ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ♣❧❛♥❛s ❡ ❡s♣❛❝✐❛✐s✱ tr❛ç❛r â♥❣✉❧♦s ❡ ♦❜t❡r s✉❛s ♠❡❞✐❞❛s ❡ ♣r♦♣r✐❡❞❛❞❡s✱ tr❛ç❛r ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s✱ ❝❛❧❝✉❧❛r ❡ tr❛ç❛r ❣rá✜❝♦s ❞❛s ❞❡r✐✈❛❞❛s ❞❛s ♣r✐♥❝✐♣❛✐s ❢✉♥çõ❡s r❡❛✐s ❡ ♠❛✐s ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ♦✉tr❛s ❛♣❧✐❝❛çõ❡s ♥♦ ❝❛♠♣♦ ❞❛ á❧❣❡❜r❛✱ tr✐❣♦♥♦♠❡tr✐❛ ❡ ❣❡♦♠❡tr✐❛✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦ é ♠♦str❛❞❛ ✉♠❛ t❡❧❛ ❞♦ ●❡♦❣❡❜r❛ ❝♦♠ ♦s ❣rá✜❝♦ ❞❛s ❢✉♥çõ❡s f(x) =x2
−7x+ 6❡
g(x) = senx✳
✶✳✶✳ ●❊❖●❊❇❘❆
❋✐❣✉r❛ ✶✳✶✿ t❡❧❛ ❞♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛
❖ ●❡♦❣❡❜r❛ t❛♠❜é♠ ♦❢❡r❡❝❡ ♦ r❡❝✉rs♦ ❞❡ ❛♥✐♠❛çã♦ ❛♣❧✐❝❛❞♦ às ❢✉♥çõ❡s✿ ✈❛r✐❛♥❞♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ❢✉♥çõ❡s ♣♦❧✐♥✐♠✐❛✐s ♦✉ tr✐❣♦♥♦♠ér✐❝❛s é ♣♦ssí✈❡❧ ❢❛③❡r ✈❛r✐❛r ❛s ♣ró♣r✐❛s ❝✉r✈❛s ❞❡ss❛s ❢✉♥çõ❡s✱ ♣♦ss✐❜✐❧✐t❛♥❞♦ ❛ss✐♠✱ ❡♥①❡r❣❛r ♦ ❡❢❡✐t♦ ❞♦s ❝♦❡✜❝✐✲ ❡♥t❡s ♥♦s tr❛ç❛❞♦s ❞❡ss❛s ❝✉r✈❛s✳ ❈♦♠♦ ♦ ●❡♦❣❡❜r❛ ❡stá s❡♥❞♦ s❡♠♣r❡ ❛t✉❛❧✐③❛❞♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ r❡❝✉rs♦s ♦❢❡r❡❝✐❞♦s ❛♦s ✉s✉ár✐♦s ❞❡ss❡ ❛♣❧✐❝❛t✐✈♦✱ ❡stá s❡♠♣r❡ ❡♠ ❡①♣❛♥sã♦ ❢❛③❡♥❞♦ ❞❡ss❛ s♦❢t✇❛r❡ ✉♠❛ ✐♥❞✐s♣❡♥sá✈❡❧ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❛❧✉♥♦s ❡ ♣r♦✲ ❢❡ss♦r❡s ❡st✉❞❛r❡♠ ❡ ❡♥s✐♥❛r❡♠ ❛ ♠❛t❡♠át✐❝❛✳
✶✳✷✳ ▼❆❳■▼❆
✶✳✷ ▼❛①✐♠❛
❖ s♦❢t✇❛r❡ ▼❛①✐♠❛ é ✉♠ ♣r♦❣r❛♠❛ q✉❡ ♣♦ss✐❜✐❧✐t❛ ❛♦ ✉s✉ár✐♦ ❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ❡①♣r❡ssõ❡s s✐♠❜ó❧✐❝❛s ❡ ♥✉♠ér✐❝❛s✱ ✐♥❝❧✉✐♥❞♦ ❛ ❞✐❢❡r❡♥❝✐❛çã♦ ❡ ✐♥t❡❣r❛çã♦ ❞❡ ❢✉♥çõ❡s r❡❛✐s✱ ❛♣❧✐❝❛çã♦ ❞❛ sér✐❡ ❞❡ ❚❛②❧♦r ❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▲❛♣❧❛❝❡✱ ❡q✉❛çõ❡s ❞✐❢❡r❡♥✲ ❝✐❛s ♦r❞✐♥ár✐❛s✱ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✱ ♣♦❧✐♥ô♠✐♦s✱ ❝♦♥❥✉♥t♦s✱ ❧✐st❛s✱ ✈❡t♦r❡s✱ ♠❛tr✐③❡s ❡ t❡♥s♦r❡s✳ ❖s r❡s✉❧t❛❞♦s ♥✉♠ér✐❝♦s r❡t♦r♥❛❞♦s ♣❡❧❛ ▼á①✐♠❛ ♦❢❡r❡❝❡♠ ❛❧t♦ ❣r❛✉ ❞❡ ♣r❡❝✐sã♦ ✉t✐❧✐③❛♥❞♦ ❢r❛çõ❡s ❡①❛t❛s✳ ❆s ❢✉♥çõ❡s ❛♥❛❧✐s❛❞❛s ♥♦ ▼❛①✐♠❛ ♣♦❞❡♠ s❡r ❛♣r❡s❡♥t❛❞❛s ❡♠ três ❞✐♠❡♥sõ❡s✳ ❖s ✉s✉ár✐♦s ❞♦ s♦❢t✇❛r❡ ▼❛①✐♠❛ sã♦✱ ♣r✐♥❝✐♣❛❧✲ ♠❡♥t❡✱ ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ s✉♣❡r✐♦r✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦ é ♠♦str❛❞❛ ✉♠❛ t❡❧❛ ❞♦ ▼❛①✐♠❛ ❝♦♠ r❡s♦❧✉çõ❡s ❞❡ ✐♥❡q✉❛çõ❡s ❞♦ ✷♦ ❣r❛✉✱ ❝á❧❝✉❧♦ ❞♦ ❧✐♠✐t❡ ❞❛
❡①♣r❡ssã♦ (1 + 1/x)x
q✉❛♥❞♦ ① t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦ ❡ ♦ ❝á❧❝✉❧♦ ❞♦ ❧✐♠✐t❡ ❞❛ ❡①♣r❡ssã♦
sen(5x
3x) q✉❛♥❞♦ ① t❡♥❞❡ à ③❡r♦✳
❋✐❣✉r❛ ✶✳✷✿ t❡❧❛ ❞♦ s♦❢t✇❛r❡ ▼❛①✐♠❛
✶✳✸✳ P❖▲❨
✶✳✸ P♦❧②
❖ s♦❢t✇❛r❡ P♦❧② é ✉♠ ♣r♦❣r❛♠❛ q✉❡ ♣❡r♠✐t❡ ❛♦ ✉s✉ár✐♦ ❛ ✐♥✈❡st✐❣❛çã♦ ❞❡ só❧✐✲ ❞♦s tr✐❞✐♠❡♥s✐♦♥❛❧♠❡♥t❡ ❝♦♠ ♦ r❡❝✉rs♦ ❞❡ ♠♦✈✐♠❡♥t♦✱ q✉❡ ♣♦ss✐❜✐❧✐t❛ ❛♥❛❧✐s❛r ❡ss❡s ♦❜❥❡t♦s ❞❡ ✈ár✐♦s â♥❣✉❧♦s ❢❛❝✐❧✐t❛♥❞♦ ❛ss✐♠ s✉❛ ❝♦♠♣r❡❡♥sã♦✳ ❖ P♦❧② ♣♦ss✉✐ ✉♠❛ ❡①t❡♥s❛ ❝♦❧❡çã♦ ❞❡ só❧✐❞♦s ♣❧❛tô♥✐❝♦s ❛ ❛rq✉✐♠❡❞✐❛♥♦s ❡♥tr❡ ♦✉tr♦s✳ ❊st❡ ❛♣❧✐❝❛t✐✈♦ r♦❞❛ ♥♦ s✐st❡♠❛ ♦♣❡r❛❝✐♦♥❛❧ ❲✐♥❞♦✇s ❡✱ ❝♦♠♦ ❡stá ❞✐s♣♦♥í✈❡❧ ❛♣❡♥❛s ❡♠ ✐♥❣❧ês✱ é ✐♥❞✐❝❛❞♦ ♣❛r❛ ❛✉①✐❧✐❛r ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s ❛ ♣❛rt✐r ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳
✶✳✹ ▼✉P❛❞
❖ ▼✉P❛❞ é ✉♠ s♦❢t✇❛r❡ ❞❡ ❝♦♠♣✉t❛çã♦ ❛❧❣é❜r✐❝❛ ❝♦♠ ✉♠❛ ✈❛st❛ ár❡❛ ❞❡ ❛♣❧✐❝❛✲ çã♦ ♥❛ á❧❣❡❜r❛ ❡ ♥♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s r❡❛✐s✿ ♣❡r♠✐t❡ ❛♦ ✉s✉ár✐♦ r❡s♦❧✈❡r ❡q✉❛çõ❡s✱ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s✱ ✐♥❡q✉❛çõ❡s✱ ♦♣❡r❛r ❝♦♠ ♠❛tr✐③❡s✱ ❝❛❧❝✉❧❛r ❞❡t❡r♠✐♥❛♥t❡s✱ ♦♣❡✲ r❛r ❝♦♠ ♣♦❧✐♥ô♠✐♦s✱ s✐♠♣❧✐✜❝❛r ❡ ❞❡s❡♥✈♦❧✈❡r ❡①♣r❡ssõ❡s ❛❧❣é❜r✐❝❛s✱ ❝❛❧❝✉❧❛r ❧✐♠✐t❡s✱ ❞❡r✐✈❛❞❛s ❡ ✐♥t❡❣r❛✐s✳ è ✉t✐❧✐③❛❞♦ ♠❛❥♦r✐t❛r✐❛♠❡♥t❡ ♣♦r ❡st✉❞❛♥t❡s ❞♦s ❡♥s✐♥♦s ♠é✲ ❞✐♦ ❡ s✉♣❡r✐♦r✱ ❝♦♥t✉❞♦ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❡❧♦s ❡st✉❞❛♥t❡s ❞❛s ú❧t✐♠❛s sér✐❡s ❞♦ ❢✉♥❞❛♠❡♥t❛❧ ■■✳
✶✳✺ ❙✉♣❡r ▲♦❣♦
❉✐❢❡r❡♥t❡♠❡♥t❡ ❞♦s ♦✉tr♦s s♦❢t✇❛r❡s ❞❛ ár❡❛ ❞❛ ❡❞✉❝❛çã♦ ♠❛t❡♠át✐❝❛✱ ♦ ❙✉♣❡r ▲♦❣♦ ♥ã♦ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ✈♦❧t❛❞♦ ❛♣❡♥❛s ♣❛r❛ ❛✉①✐❧✐❛r ♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦ ❡ ❛♣r❡♥✲ ❞✐③❛❣❡♠ ❞❛ ♠❛t❡♠át✐❝❛✳ ❙✉❛ ár❡❛ ❞❡ ❛t✉❛çã♦ é ❛♠♣❧❛✱ ❡♥✈♦❧✈❡♥❞♦ ✈ár✐❛s ♦✉tr❛s ❞✐s❝✐♣❧✐♥❛s✳ ◆♦ ❝♦♥t❡①t♦ ❞❛ ❡❞✉❝❛çã♦ ♠❛t❡♠át✐❝❛ ♦ s♦❢t✇❛r❡ ❙✉♣❡r ▲♦❣♦ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♥♦ ❡st✉❞♦ ❞❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛✱ ❛❧é♠ ❞❡ ❝♦♥tr✐❜✉✐r ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ ❞♦ ❡st✉❞❛♥t❡ ❡ ♣♦ss✐❜✐❧✐t❛r ❛ ❛q✉✐s✐çã♦ ❞❡ ♥♦çõ❡s ❞❡ ♣r♦❣r❛♠❛çã♦✳
✶✳✻ ❲✐♥♣❧♦t
❖ s♦❢t✇❛r❡ ❲✐♥♣❧♦t é ✉♠ ♣r♦❣r❛♠❛ ❝♦♥❝❡❜✐❞♦ ♣❛r❛ ❣❡r❛r ❣rá✜❝♦s ✷❉ ❡ ✸❉ ❞❡ ❢✉♥çõ❡s r❡❛✐s✳ ❉❡ ❢á❝✐❧ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ♠❛♥✐♣✉❧❛çã♦ ♣♦r ♣❛rt❡ ❞♦s ✉s✉ár✐♦s✱ ♣♦ss✉✐ ♠❡♥✉s ✐♥t❡r❛t✐✈♦s ❝♦♠ ♦♣çã♦ ❞❡ ❛❥✉❞❛ ♣❛r❛ ❛ ✉t✐❧✐③❛çã♦ ❞❡ s❡✉s r❡❝✉rs♦s✳ ➱ ✐♥❞✐❝❛❞♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❛r❛ ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ s✉♣❡r✐♦r✳ ❘♦❞❛ ♥♦ s✐st❡♠❛ ♦♣❡r❛❝✐♦♥❛❧ ❲✐♥❞♦✇s✳
✶✳✼✳ ❑▼P▲❖❚
✶✳✼ ❑♠P❧♦t
❖ ❑♠P❧♦t ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ tr❛ç❛r ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s r❡❛✐s ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐✲ ❛♥♦✱ ♣❛r❛♠étr✐❝❛s ❡ ❢✉♥çõ❡s ♥❛s ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s✳ ❆❝❡✐t❛ ✈ár✐♦s ♠♦❞♦s ❞❡ ❣r❛❞❡ ❡ ♦s ❣rá✜❝♦s ♣♦❞❡ s❡r ✐♠♣r❡ss♦s ❝♦♠ ❛❧t❛ ♣r❡❝✐sã♦ ♥❛ ❡s❝❛❧❛ ❡s❝♦❧❤✐❞❛✳ P♦❞❡✲s❡ tr❛✲ ç❛r ❣rá✜❝♦s ❞❡ ✈ár✐❛s ❢✉♥çõ❡s ❝♦♥❝♦♠✐t❛♥t❡♠❡♥t❡ ❡ ❝♦♠❜✐♥á✲❧❛s ♣❛r❛ ❝r✐❛r ❢✉♥çõ❡s ♥♦✈❛s✳ ❙❡✉ s✐st❡♠❛ ♦♣❡r❛❝✐♦♥❛❧ ♣❛r❛ ❢✉♥❝✐♦♥❛♠❡♥t♦ é ♦ ●◆❯✴▲✐♥✉①✳
✶✳✽ ❲✐♥♠❛t
❊ss❡ ♣♦❞❡r♦s♦ s♦❢t✇❛r❡ ♣❡r♠✐t❡ ❛♦s ✉s✉ár✐♦s ❝♦♥str✉✐r ❡ ♦♣❡r❛r ❝♦♠ ♠❛tr✐③❡s✱ ❝❛❧❝✉❧❛r ♠❛tr✐③ ✐♥✈❡rs❛✱ ❝❛❧❝✉❧❛r ❞❡t❡r♠✐♥❛♥t❡s✱ tr❛ç♦ ❞❛ ♠❛tr✐③✱ ♠❛tr✐③ tr❛♥s♣♦st❛ ❡ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ss❛s ♠❛tr✐③❡s ♣♦❞❡♠ s❡r ♥ú♠❡r♦s r❡❛✐s ♦✉ ❝♦♠♣❧❡①♦s✳ ❙❡✉ ♣ú❜❧✐❝♦ ❛❧✈♦ sã♦ ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❛té ♦ ❡♥s✐♥♦ s✉♣❡r✐♦r✳ ❙❡✉ s✐st❡♠❛ ♦♣❡r❛❝✐♦♥❛❧ ♣❛r❛ ❢✉♥❝✐♦♥❛♠❡♥t♦ é ♦ ❲✐♥❞♦✇s✳
✶✳✾ ❯♠ ❜r❡✈❡ ❡st✉❞♦ ❞❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛
◆❡st❡ ❝❛♣ít✉❧♦ s❡rá ❢❡✐t❛ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s✱ ❡st✉❞♦ ❞♦s ❝♦♠❛♥✲ ❞♦s ❡ ♣❛ss♦s ❞❡ ♣r♦❣r❛♠❛çã♦ ❞❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ❈❛❧❝ ❞♦ ❛♣❧✐❝❛t✐✈♦ ❣r❛t✉✐t♦ ▲✐✲ ❜r❡❖✣❝❡ ✺✳✵ ♠❛✐s ✉t✐❧✐③❛❞♦s ♣❡❧♦s ❛❧✉♥♦s ♥❛s t❛r❡❢❛s q✉❡ s❡r✈✐r❛♠ ❝♦♠♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❞❡ss❛ ❞✐ss❡rt❛çã♦✳
❆s ♣❧❛♥✐❧❤❛s ❡❧❡trô♥✐❝❛s✱ ❡♠ ❣❡r❛❧✱ sã♦ ❛♣❧✐❝❛t✐✈♦s ❝✉❥❛ t❡❧❛ ♣r✐♥❝✐♣❛❧ é ❝♦♠♣♦st❛ ♣♦r ❧✐♥❤❛s✱ ❝♦❧✉♥❛s ❡ ❝é❧✉❧❛s ♦♥❞❡ ♣♦❞❡♠ s❡r ✐♥s❡r✐❞♦s t❡①t♦s✱ ❛❧❣❛r✐s♠♦s✱ sí♠❜♦❧♦s ♠❛t❡♠át✐❝♦s ♦✉ ❢ór♠✉❧❛s✳ ❖s r❡❝✉rs♦s ❞❛s ♣❧❛♥✐❧❤❛s ❡❧❡trô♥✐❝❛s sã♦ ❞✐✈❡rs♦s ❡ ✈ã♦ ❞❡s❞❡ ❛ ❡❞✐çã♦ ❞❡ t❡①t♦s s✐♠♣❧❡s ❝♦♠ ❢ór♠✉❧❛s ❜ás✐❝❛s ❛té ❛ ❡①❡❝✉çã♦ ❞❡ ❝♦♠♣❧❡①❛s ❢ór♠✉❧❛s ♠❛t❡♠át✐❝❛s ❛❝♦♠♣❛♥❤❛❞❛s ❞❡ ❣rá✜❝♦s ❡♠ ✸❉✳
❆ ♣❧❛♥✐❧❤❛ ❞♦ ❈❛❧❝ é ❞✐✈✐❞✐❞❛ ❡♠ ❝é❧✉❧❛s✳ ❈❛❞❛ ❝é❧✉❧❛ t❡♠ ✉♠ ú♥✐❝♦ ❡♥❞❡r❡ç♦✿ ❈♦❧✉♥❛ ❡ ❧✐♥❤❛✱ ♥❡ss❛ ♦r❞❡♠✳ ◆❛ ✜❣✉r❛ ✶ ✭♣ró①✐♠❛ ♣á❣✐♥❛✮ tê♠✲s❡ ✉♠❛ ♣❧❛♥✐❧❤❛ ❞♦ ❛♣❧✐❝❛t✐✈♦ ❈❛❧❝ ✶ ❡ ❛ ❝é❧✉❧❛ ❇✷ ❡stá ❤❛❜✐❧✐t❛❞❛ ♣❛r❛ r❡❝❡❜❡r ❞❛❞♦s✳
❆♣ós ❛ ✐♥s❡rçã♦ ❞❡ ❞❛❞♦s ❛tr❛✈és ❞❛ t❡❝❧❛ ❡♥t❡r ❛ ♣❧❛♥✐❧❤❛ ✈❛✐ ❡♥t❡♥❞❡r ❝♦♠♦ ❢ór♠✉❧❛ ♠❛t❡♠át✐❝❛ s❡ ♦ s✐♥❛❧=❢♦r ❞✐❣✐t❛❞♦ ❛♥t❡s ❞❛ ❢ór♠✉❧❛ q✉❡ s❡ q✉❡r ❡①❡❝✉t❛r✳
❊ss❛ ❢ór♠✉❧❛s sã♦ ♣r♦❣r❛♠❛s q✉❡ ❛ ♣❧❛♥✐❧❤❛ ❡stá ♣r❡♣❛r❛❞❛ ♣❛r❛ ❡①❡❝✉t❛r✳ ◆❛ ✜❣✉r❛ ✷ ✭♣ró①✐♠❛ ♣á❣✐♥❛✮ é ♠♦str❛❞♦ ✉♠ ❡①❡♠♣❧♦ ❞❡ ❢ór♠✉❧❛ ❞✐❣✐t❛❞❛ ♥❛ ❝é❧✉❧❛ ❇✸✳ ❊st❛ ❢ór♠✉❧❛ ❡①❡❝✉t❛ ❛ s♦♠❛ ❞♦ ❝♦♥t❡ú❞♦ ❞❛ ❝é❧✉❧❛ ❆✷ ❝♦♠ ♦ ❝♦♥t❡ú❞♦ ❞❛ ❝é❧✉❧❛ ❇✷✳❈❛s♦ q✉❛✐sq✉❡r ❞❡ss❡s ❝♦♥t❡ú❞♦s ❢♦r ❛❧t❡r❛❞♦ ✕♥❛s s✉❛s r❡s♣❡❝t✐✈❛s ❝é❧✉❧❛s✕ ♦ r❡s✉❧t❛❞♦ ♥❛ ❝é❧✉❧❛ ❇✸ s❡rá ❛t✉❛❧✐③❛❞♦ ❛✉t♦♠❛t✐❝❛♠❡♥t❡✳ ❖❜s❡r✈❛✲s❡ t❛♠❜é♠ q✉❡ ❛s ❝é❧✉❧❛s ❡♥✈♦❧✈✐❞❛s ♥❛ s♦♠❛ ♥❛ ❝é❧✉❧❛ ❇✸ ✜❝❛♠ ❝♦♠ ❛ ♠❡s♠❛ ❝♦r ❞❡ s❡✉s ❡♥❞❡r❡ç♦s✱ ❢❛❝✐❧✐t❛♥t♦ ❛ss✐♠ ❛ ✈✐s✉❛❧✐③❛çã♦ ❞♦ ❛❧✉♥♦ ♥♦ ♠♦♠❡♥t♦ ❞❛ ♣r❛❣r❛♠❛çã♦ ❞❛ ❢ór♠✉❧❛ ❞❡s❡❥❛❞❛✳
✶✳✾✳ ❯▼ ❇❘❊❱❊ ❊❙❚❯❉❖ ❉❆ P▲❆◆■▲❍❆ ❊▲❊❚❘Ô◆■❈❆
❋✐❣✉r❛ ✶✳✸✿ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛
❋✐❣✉r❛ ✶✳✹✿ ❡①❡♠♣❧♦ ❞❡ ❢ór♠✉❧❛
✶✳✾✳ ❯▼ ❇❘❊❱❊ ❊❙❚❯❉❖ ❉❆ P▲❆◆■▲❍❆ ❊▲❊❚❘Ô◆■❈❆
❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ ♣r♦❣r❛♠❛çã♦ ❡♠ ♣❧❛♥✐❧❤❛s ✕ ❡ ❡♠ t♦❞❛s ❛s ♣r♦❣r❛♠❛çõ❡s ✕ é ❛ ♦❜r✐❣❛t♦r✐❡❞❛❞❡ ❞❛ ❡①❡❝✉çã♦ ❝♦rr❡t❛ ❞❡ s❡✉s ❝♦♠❛♥❞♦s✱ ♥ã♦ ♣♦❞❡♥❞♦ ❤❛✈❡r ❛♠✲ ❜✐❣✉✐❞❛❞❡s ❡♥tr❡ ❡❧❡s ♥❡♠ t❛♠♣♦✉❝♦ ❞✐❣✐t❛çã♦ ❡rr❛❞❛ ❞♦s ♠❡s♠♦s✳ ❊ss❛ ❡①✐❣ê♥❝✐❛ r❡❢♦rç❛ ♥♦ ❛❧✉♥♦ ♦ r✐❣♦r ♠❛t❡♠át✐❝♦ q✉❡ ✕ ♣❛r❛ ♠✉✐t♦s ✕ é ✉♠ ❡❧❡♠❡♥t♦ q✉❡ ❞✐✜✲ ❝✉❧t❛ ♦ ❛♣r❡♥❞✐③❛❞♦ ❞❛ ♠❛t❡♠át✐❝❛✱ ♣♦✐s ♥ã♦ ♣❡r♠✐t❡ ♣r♦❝❡❞✐♠❡♥t♦s q✉❡ ❢✉❥❛♠ ❞♦ s❡✉ ❝♦♥❥✉♥t♦ ❞❡ r❡❣r❛s ❡ q✉❛❧q✉❡r ❡rr♦✱ ♣♦r ♣❡q✉❡♥♦ q✉❡ s❡❥❛✱ ❝♦♠♣r♦♠❡t❡ t♦❞♦ ❝á❧✲ ❝✉❧♦ ❝♦rr❡t♦ ❛♥t❡r✐♦r✳ ◆❛ ♣r♦❣r❛♠❛çã♦ ❡♠ ♣❧❛♥✐❧❤❛s ❡❧❡trô♥✐❝❛s ♦❝♦rr❡ ♦ ♠❡s♠♦✿ ✉♠ ♣❡q✉❡♥♦ ❡rr♦ ❞❡ sí♥t❡s❡✱ s❡q✉ê♥❝✐❛ ♦✉ ❞✐❣✐t❛çã♦ ❞❡ s❡✉s ❝♦♠❛♥❞♦s ✐♥✈✐❛❜✐❧✐③❛ ❛ ❡①❡❝✉✲ çã♦ ❞❛ ♣r♦❣r❛♠❛çã♦ ❞❡s❡❥❛❞❛✳ ❯♠ ❡①❡♠♣❧♦ ❞✐ss♦ ❡ ✉♠ ❡rr♦ r❡❝♦rr❡♥t❡ ♦❜s❡r✈❛❞♦ ♥❛ ♣r♦❣r❛♠❛çã♦ ❞❛s ❢ór♠✉❧❛s ♠❛t❡♠át✐❝❛s ♥❛s ♣❧❛♥✐❧❤❛s ♣❡❧♦s ❛❧✉♥♦s ✕ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛s ♣r✐♠❡✐r❛s ✕ é ❡sq✉❡❝❡r ❞❡ ❝♦♠❡ç❛r ✉♠❛ ❢ór♠✉❧❛ ❝♦♠ ♦ s✐♥❛❧ =✭sí♠❜♦❧♦ ♠❛t❡♠á✲
t✐❝♦ q✉❡ r❡♣r❡s❡♥t❛ ✉♠❛ ✐❣✉❛❧❞❛❞❡✮✳ ◆❡ss❡ ❝❛s♦ ❛ ♣❧❛♥✐❧❤❛ ♥ã♦ ❡①❡❝✉t❛ ♦ ♣r♦❣r❛♠❛ ❡ ❡♥t❡♥❞❡ t♦❞❛ ❛ ❢ór♠✉❧❛ ❝♦♠♦ ✉♠ t❡①t♦✳ ❯♠ ❝♦♠❛♥❞♦ ❞❡♠❛♥❞❛❞♦ ♣❡❧❛s t❛r❡❢❛s ❡①❡❝✉t❛❞❛s ♣❡❧♦s ❛❧✉♥♦s ♥❛s ♣❧❛♥✐❧❤❛s q✉❡ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦ é ♦ ❝♦♥❞✐❝✐♦♥❛❧ ❙❊✳ ◆❛ ✜❣✉r❛ ✸ é ♠♦str❛❞♦ ✉♠ ❡①❡♠♣❧♦ ❞♦ ✉s♦ ❞❡ss❡ ❝♦♠❛♥❞♦✳
❋✐❣✉r❛ ✶✳✺✿ ✉s♦ ❞♦ ❝♦♥❞✐❝✐♦♥❛❧ ❙❊
❯♠ ♣r♦❣r❛♠❛ ✉t✐❧✐③❛♥❞♦ ♦ ❝♦♠❛♥❞♦ ❙❊✱ ♠♦str❛❞♦ ❛❝✐♠❛✱ ❝♦♠♣❛r❛ ♦s ✈❛❧♦r❡s ❞❛s ❝é❧✉❧❛s ❆✶ ❡ ❇✶ ❡ ❡①❡❝✉t❛ ❛ ♣r✐♠❡✐r❛ ♦r❞❡♠✿ r❡t♦r♥❛r ♦ t❡①t♦ ✏♠❛✐♦r q✉❡✑ s❡ ♦ ❝♦♥t❡ú❞♦ ❞❡ ❆✶ ❢♦r ♠❛✐♦r q✉❡ ♦ ❝♦♥t❡ú❞♦ ❞❡ ❇✷ ♦✉ ❡①❡❝✉t❛ ❛ s❡❣✉♥❞❛ ♦r❞❡♠✿
✶✳✾✳ ❯▼ ❇❘❊❱❊ ❊❙❚❯❉❖ ❉❆ P▲❆◆■▲❍❆ ❊▲❊❚❘Ô◆■❈❆
r❡t♦r♥❛r ♦ t❡①t♦ ✏♠❡♥♦r q✉❡✑ s❡ ♥ã♦ ❢♦r✳ ➱ ❝❧❛r♦ q✉❡ ♦ t❡①t♦ ❞✐❣✐t❛❞♦ ♣♦❞❡ s❡r q✉❛❧q✉❡r ✉♠✱ ❞❡s❞❡ q✉❡ ✈❡♥❤❛ ❡♥tr❡ ❛s♣❛s✳❖ r❡t♦r♥♦ s❡ ❞á s❡♠♣r❡ ♥❛ ❝é❧✉❧❛ ❡s❝♦❧❤✐❞❛ ♣❛r❛ ❛ ❞✐❣✐t❛çã♦ ❞♦ ♣r♦❣r❛♠❛ ✉t✐❧✐③❛♥❞♦ ♦ ❝♦♠❛♥❞♦ ❙❊✳ ❊♠ ❧✐♥❤❛s ❣❡r❛✐s ♦ ❝♦♠❛♥❞♦ ❙❊ ❝♦♠♣❛r❛ ♦s ✈❛❧♦r❡s ♥✉♠ér✐❝♦s ❞❡ ❞✉❛s ❝é❧✉❧❛s ❛tr❛✈és ❞♦ q✉❡ ❡stá ❞✐❣✐t❛❞♦ ❧♦❣♦ ❡♠ s❡❣✉✐❞❛ à ❞✐❣✐t❛çã♦ ❞❛ ♣❛❧❛✈r❛ ❙❊ ❡ ❡①❡❝✉t❛ ♦ ♣r✐♠❡✐r♦ ❝♦♠❛♥❞♦ s❡ ♦ r❡s✉❧t❛❞♦ ❞❛ ❝♦♠♣❛r❛çã♦ ❢♦r ✈❡r❞❛❞❡✐r♦ ♦✉ ❡①❡❝✉t❛ ♦ s❡❣✉♥❞♦ ❝♦♠❛♥❞♦ s❡ ❢♦r ❢❛❧s♦✳ ❊ss❛ ❢❡rr❛♠❡♥t❛ ❞♦ ❈❛❧ ♣♦ss✐❜✐❧✐t❛ ❛♦ ❛❧✉♥♦ ❝♦♥❞✐♦♥❛r ❛ ❡①❡❝✉çã♦ ❞❡ ✉♠ ♣r♦❣r❛♠❛ ♦✉ ❛♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ t❡①t♦ ❛♦s ❝♦♥t❡ú❞♦s ❞❡ ❞✉❛s ❝é❧✉❧❛s✱ r❡❝✉rs♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ❛♥❛❧✐s❛r ♦ ✈❛❧♦r ❞♦ ❞✐s❝r✐♠✐♥❛♥t❡ ❞❡ ❡q✉❛çõ❡s ❞♦ ✷♦ ❣r❛✉✳ ❖ ❝♦♠❛♥❞♦ ❙❊ ♣♦❞❡ s❡r
❛♥✐♥❤❛❞♦✱ ✐st♦ é✱ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♠❛✐s ❞❡ ✉♠❛ ✈❡③ ♣❛r❛ ❛✉♠❡♥t❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♦♣çõ❡s ❞❡ ❡s❝♦❧❤❛ ♣❛r❛ ❛ ❡①❡❝✉çã♦ ❞❡ ♣r♦❣r❛♠❛s ♦✉ ❛♣r❡s❡♥t❛çã♦ ❞❡ t❡①t♦s✳ ◆❛ ♣rát✐❝❛ ✉♠ ❝♦♠❛♥❞♦ ❙❊ ❡①❡❝✉t❛ ✉♠ ♦✉tr♦ ❝♦♠❛♥❞♦ ❙❊ ❡♠ s❡q✉ê♥❝✐❛ ❛té ❡♥❝♦♥tr❛r ✉♠ ✈❛❧♦r ❧ó❣✐❝♦ ✈❡r❞❛❞❡✐r♦ ♣❛r❛ ♦ ♣ró①✐♠♦ ❝♦♠❛♥❞♦ ❙❊✳ ◆❛ ✜❣✉r❛ ✹ é ♠♦str❛❞♦ ✉♠❛ s❡q✉❡♥❝✐❛ ❞❡ ❞♦✐s ❙❊ ❛♥✐♥❤❛❞♦s✳
❋✐❣✉r❛ ✶✳✻✿ ❙❊ ❛♥✐♥❤❛❞♦
◆♦ ❡①❡♠♣❧♦ ❞❛ ✜❣✉r❛ ✶✳✹ ♦ ❈❛❧❝ ❡①❡❝✉t❛rá ♦ ♣r♦❞✉❞♦ ❞♦s ❝♦♥t❡ú❞♦s ❞❛s ❝é❧✉❧❛s ❆✶ ❡ ❇✶ s❡ ♦ ♥ú♠❡r♦ ❞✐❣✐t❛❞♦ ❡♠ ❆✶ ❢♦r ♠❛✐♦r q✉❡ ♦ ♥ú♠❡r♦ ❞✐❣✐t❛❞♦ ❡♠ ❇✶ ❡✱ s❡ ♥ã♦ ❢♦r✱ ❡①❡❝✉t❛rá ♦ ♣ró①✐♠♦ ❝♦♠❛♥❞♦ ❙❊ q✉❡ ❢❛rá ❛ ❞✐✈✐sã♦ ❞♦s ❝♦♥t❡ú❞♦s ❞❛s ❝é❧✉❧❛s ❆✶ ❡ ❇✶ ♦✉ ❛ s♦♠❛ ❞❡ss❡s ❝♦♥t❡ú❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❞❛ ❡①♣r❡ssã♦ ✏❆✶❁❇✶✑✳
✶✳✾✳ ❯▼ ❇❘❊❱❊ ❊❙❚❯❉❖ ❉❆ P▲❆◆■▲❍❆ ❊▲❊❚❘Ô◆■❈❆
❆ ♣❧❛♥✐❧❤❛ ❈❛❧❝ ♣♦ss✉✐ ❞✐✈❡rs❛s ❜❛rr❛s ♣❛r❛ ❛❝❡ss♦ ❛♦s s❡✉s ❝♦♠❛♥❞♦s✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦ sã♦ ♠♦str❛❞❛s ❛❧❣✉♠❛s ❞❡❧❛s✳
❋✐❣✉r❛ ✶✳✼✿ ❜❛rr❛s ❞❡ ❝♦♠❛♥❞♦s
❆s ❜❛rr❛s q✉❡ ❢♦r❛♠ ♠❛✐s ✉t✐❧✐③❛❞❛s ♥❛s ❛t✐✈✐❞❛❞❡s ❞♦s ❛❧✉♥♦s ❢♦r❛♠ ❛s ❜❛rr❛s ❞❡ ❢❡rr❛♠❡♥t❛s ❞❡ ❢♦r♠❛t❛çã♦✱ ❞❡ ❢❡rr❛♠❡♥t❛s ♣❛❞rã♦ ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❛ ❜❛rr❛ ❞❡ ❢ór♠✉❧❛s✳ ❚♦❞❛ ❢ór♠✉❧❛ ❞✐❣✐t❛❞❛ ❡♠ q✉❛❧q✉❡r ❝é❧✉❧❛ ❞❛ ♣❧❛♥✐❧❤❛ ❛♣❛r❡❝❡ s✐♠✉❧t❛✲ ♥❡❛♠❡♥t❡ ♥❛ ❜❛rr❛ ❞❡ ❢ór♠✉❧❛s✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ ❛❧✉♥♦ ♣♦❞❡ ❝♦♥❢❡r✐r ❛s ❢ór♠✉❧❛s q✉❡ ❞✐❣✐t♦✉ s❡♠ ♣r❡❝✐s❛r ✈❡r✐✜❝❛r ♥❛ ♣ró♣r✐❛ ♣❧❛♥✐❧❤❛✳ ❆ ❢♦r♠❛t❛çã♦ ❞❛s ♣❧❛♥✐❧❤❛s ❞❛s ❛t✐✈✐❞❛❞❡s ❞♦s ❛❧✉♥♦s ❢♦✐ ❞❡✐①❛❞❛ ❛ ❝❛r❣♦ ❞♦s ♠❡s♠♦s✱ ❝♦♠ ❛ r❡ss❛❧✈❛ q✉❡ t❛❧ ❢♦r♠❛t❛çã♦ ❢❛❝✐❧✐t❛ss❡ ❛ ♦♣❡r❛çã♦ ❞❡ s❡✉s ♣r♦❣r❛♠❛s ♣♦r q✉❛❧q✉❡r ♣❡ss♦❛ ❝♦♠ ♦ ♠í♥✐♠♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ ♣❧❛♥✐❧❤❛✱ ♦✉ s❡❥❛✱ ❛ ♣❧❛♥✐❧❤❛ t✐♥❤❛ q✉❡ s❡r ❛❝❡ssí✈❡❧ ❡ ❛✉t♦❡①♣❧✐❝❛t✐✈❛✳
❆❧❣✉♥s ❝♦♠❛♥❞♦s ❢♦r❛♠ ❞❡❧✐❜❡r❛❞❛♠❡♥t❡ ♥ã♦ ✐♥❢♦r♠❛❞♦s ❛♦s ❛❧✉♥♦s✳ ❖ ♦❜❥❡t✐✈♦ ❡r❛ ❢❛③❡r ❝♦♠ q✉❡ ♦s ♠❡s♠♦s ❡♥❝♦♥tr❛ss❡♠ ✉♠❛ ❢♦r♠❛ ❛❧t❡r♥❛t✐✈❛ ❞❡ ❡①❡❝✉t❛r ❞❡t❡r✲ ♠✐♥❛❞❛ ♦♣❡r❛çã♦✳ ❆ ♣♦t❡♥❝✐❛çã♦✱ ♣♦r ❡①❡♠♣❧♦✱ t❡♠ ✉♠ ❝♦♠❛♥❞♦ ♣ró♣r✐♦ ♥♦ ❈❛❧❝✱ ♥♦ ❡♥t❛♥t♦ ❡ss❡ ❝♦♠❛♥❞♦ ♥ã♦ ❢♦✐ ❡♥s✐♥❛❞♦ ❛♦s ❛❧✉♥♦s✱ q✉❡ t✐✈❡r❛♠ q✉❡ ❛♣❧✐❝❛r ❛ ❞❡✜✲ ♥✐çã♦ ❞❡ ♣♦tê♥❝✐❛ ♣❛r❛ ❡①❡❝✉t❛r ❛ ♦♣❡r❛çã♦✳ ❆ r✐❣♦r ♠✉✐t♦s ❞♦s r❡❝✉rs♦s ❞❛ ♣❧❛♥✐❧❤❛ ❈❛❧❝ ♥ã♦ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞♦s ❛♦s ❛❧✉♥♦s ❝♦♠♦ ✉♠❛ ❢♦r♠❛ ❞❡ ❡st✐♠✉❧á✲❧♦s ❛ ✐♥✈❡s✲ t✐❣❛r ❢♦r♠❛s ❞❡ ❞❡s❡♥✈♦❧✈❡r ❢ór♠✉❧❛s ♠❛t❡♠át✐❝❛s ❛♣❡♥❛s ❝♦♠ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❞♦s ❝♦♥❝❡✐t♦s q✉❡ ❛s ❞❡✜♥❡♠✳
❈❛♣ít✉❧♦ ✷
❖ Pr♦❥❡t♦ ❉❡❧t❛ ❡♠ ❡①❡❝✉çã♦ ♥❛
♣❧❛♥✐❧❤❛ ❝❛❧❝
✷✳✶ ■♥tr♦❞✉çã♦
❆ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ❈❛❧❝ ❥á t✐♥❤❛ s✐❞♦ s✐❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦s ❛❧✉♥♦s ❞♦ ✶♦ ❛♥♦ ❇✱
❡♥s✐♥♦ ♠é❞✐♦✱ ❞❡ ✉♠❛ ❡s❝♦❧❛ ❡st❛❞✉❛❧✱ s✐t✉❛❞❛ ♥❛ ❝✐❞❛❞❡ ❞❡ ❏♦ã♦ P❡sss♦❛✱ ❊st❛❞♦ ❞❛ P❛r❛í❜❛✳ ◆♦ ❡♥t❛♥t♦✱ s❡✉ ♣♦t❡♥❝✐❛❧ ❝♦♠♦ ❢❛❝✐❧✐t❛❞♦r❛ ❞♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛✱ ♥ã♦ t✐♥❤❛ s✐❞♦ ❛✐♥❞❛ ❡①♣❧♦r❛❞♦✳ ❆ ✐❞❡✐❛ ❝❡♥tr❛❧ ❞❛ ✉t✐❧✐③❛çã♦ ❞❛ ♣❧❛♥✐❧❤❛ ❝♦♠♦ ❝❛t❛❧✐③❛✲ ❞♦r❛ ❞♦ ❛♣r❡♥❞✐③❛❞♦ ❞❛ ♠❛t❡♠át✐❝❛✱ ✐st♦ é✱ s✉❛ s✉st❡♥t❛çã♦ ❡ ❥✉st✐✜❝❛t✐✈❛✱ r❡s✐❞❡ ♥♦ ❢❛t♦ ❞❡ q✉❡ ❛♦ ♣r♦❣r❛♠❛r ❛ ♣❧❛♥✐❧❤❛ ♣❛r❛ ❡①❡❝✉t❛r ✉♠❛ ❢ór♠✉❧❛ ♠❛t❡♠át✐❝❛ ♦ ♣r♦✲ ❣r❛♠❛❞♦r ✭❛❧✉♥♦✮❡stá ✏❡♥s✐♥❛♥❞♦✑ ❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ❈❛❧❝ ❛ ❡①❡❝✉t❛r ❞❡t❡r♠✐♥❛❞♦ ❛❧❣♦r✐t♠♦ ❡✱ ♣❛r❛ t❛♥t♦✱ ❛❧é♠ ❞❡ t❡r q✉❡ s❛❜❡r ❡♠ ♣r♦❢✉♥❞✐❞❛❞❡ ♦ q✉❡ ❡stá ♣r♦❣r❛✲ ♠❛♥❞♦ ♥❛ ♣❧❛♥✐❧❤❛✱ ❛❝❛❜❛ ♣♦r ✜①❛r ❛✐♥❞❛ ♠❛✐s ♦ ❝♦♥t❡ú❞♦ q✉❡ ❡stá ✏❡♥s✐♥❛♥❞♦✑ à ♣❧❛♥✐❧❤❛ ❛ ❡①❡❝✉t❛r✳ ◆❛s ♣❛❧❛✈r❛s ❞♦ ❡❞✉❝❛❞♦r P❛✉❧♦ ❋r❡✐r❡✿
✏◗✉❡♠ ❡♥s✐♥❛ ❛♣r❡♥❞❡ ❛♦ ❡♥s✐♥❛r✳ ❊ q✉❡♠ ❛♣r❡♥❞❡ ❡♥s✐♥❛ ❛♦ ❛♣r❡♥❞❡r✑✳ ✭❬✹❪✱ ♣✳✷✸✮
➱ ❝❧❛r♦ q✉❡ ♦s tó♣✐❝♦s ❞❛ ♠❛t❡♠át✐❝❛ ✉t✐❧✐③❛❞♦s ♥❛s t❛r❡❢❛s ❡①❡❝✉t❛❞❛s ♣❡❧♦s ❛❧✉♥♦s ✕ q✉❡ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ❝♦♠♦ ♦❜❥❡t♦ ❞❡ ❛♥á❧✐s❡ ♥❡ss❛ ❞✐ss❡rt❛çã♦ ✕ ❥á t✐♥❤❛♠ s✐❞♦ ❛❜♦r❞❛❞♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ♥❛s ❛✉❧❛s r❡❣✉❧❛r❡s✱ ♦ q✉❡ s❡ ♦❜❥❡t✐✈❛✈❛ ❝♦♠ ❛ ✉t✐❧✐③❛çã♦ ❞❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ❡r❛ ❧❛♥ç❛r ♠ã♦ ❞❡ ✉♠ r❡❝✉rs♦ ❞✐❞át✐❝♦ q✉❡ ❛✉①✐❧✐❛ss❡ ❛ ❛♣r❡❡♥sã♦ ❞♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ✈✐st♦s ♥❛s ❛✉❧❛s r❡❣✉❧❛r❡s ❡ ♦s ❡♥❢♦❝❛ss❡♠ s♦❜ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ♥♦ q✉❛❧ ♦s ❛❧✉♥♦s s❡r✐❛♠ ❞❡t❡♥t♦r❡s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ♦s ✏❡♥s✐♥❛ss❡♠✑ à ♣❧❛♥✐❧❤❛ ♥❛ ❢♦r♠❛ ❞❡ ♣r♦❣r❛♠❛s q✉❡ ❡①❡❝✉t❛ss❡♠ ❛s ❢ór♠✉❧❛s ♣♦r ❡❧❡s ❥á ✉t✐❧✐③❛❞❛s ♥♦s ❡①❡r❝í❝✐♦s r❡❣✉❧❛r❡s✳ ❆ ♣rát✐❝❛✱ ♣♦ré♠✱ ♠♦str♦✉ q✉❡ ♥ã♦ ❛♣❡♥❛s ♦s
✷✳✷✳ ❋❯◆❈■❖◆❆▼❊◆❚❖ ❉❖ P❘❖❏❊❚❖ ❉❊▲❚❆
❝♦♥t❡ú❞♦s ❢♦r❛♠ ♠❛✐s ❜❡♠ ✜①❛❞♦s ❡ ✈✐st♦s s♦❜ ♥♦✈♦ ✈✐és✿ ♦s ❛❧✉♥♦s ❛❞q✉✐r✐r❛♠ ✉♠❛ ♣♦st✉r❛ ♠❛✐s ♠❛❞✉r❛ ❡ ❝♦♥s❝✐❡♥t❡ ❡♠ r❡❧❛çã♦ ❛♦s ❝♦♥t❡ú❞♦s ✉t✐❧✐③❛❞♦s ♥❛s t❛r❡❢❛s✱ ✉♠❛ ♣♦st✉r❛ ♣r♦t❛❣♦♥✐st❛ ❡♠ r❡❧❛çã♦ ❛♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❛❜s♦r✈✐❞♦s ❡♠ ♦♣♦s✐çã♦ à ❛♥t✐❣❛ ♣♦st✉r❛ ❞❡s✐♥t❡r❡ss❛❛❞❛ ❡ ♣❛ss✐✈❛ ♦❜s❡r✈❛❞❛ ♥❛s ❛✉❧❛s r❡❣✉❧❛r❡s✳
❖ ❝♦♥❥✉♥t♦ ❞❛s ❛t✐✈✐❞❛❞❡s ❡①❡❝✉t❛❞❛s ♣❡❧♦s ❛❧✉♥♦s ❞♦ ✶♦ ❛♥♦ ❞❡ ✉♠❛ ❡s❝♦❧❛
❡st❛❞✉❛❧ ❞❛ ❝✐❞❛❞❡ ❞❡ ❏♦ã♦ P❡sss♦❛✲P❇ ♥❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ❈❛❧❝ ✕ q✉❡ ❢♦✐ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❞❡ss❡ ❚❈❈ ✕ r❡❝❡❜❡✉ ♦ ♥♦♠❡ ❞❡ Pr♦❥❡t♦ ❉❡❧t❛✱ ❡♠ r❡❢❡rê♥❝✐❛ à ❧❡tr❛ ❣r❡❣❛ ❞❡ ♠❡s♠♦ ♥♦♠❡ q✉❡ r❡♣r❡s❡♥t❛ ♦ ❞✐s❝r✐♠✐♥❛♥t❡ ❞❛s ❡q✉❛çõ❡s ❞♦ ✷♦ ❣r❛✉✳ ❖
♣❡rí♦❞♦ ❡①❡❝✉çã♦ ❞❡ss❡ ♣r♦❥❡t♦ ❢♦✐ ❞❡ ✶✵ ❞❡ s❡t❡♠❜r♦ ❛ ✷✵ ❞❡ ♥♦✈❡♠❜r♦ ❞❡ ✷✵✶✺ ♥✉♠ t♦t❛❧ ❞❡ ✷✵ ❛✉❧❛s ✭✶✺ ❤♦r❛s✮✳ ❉♦s ✶✽ ❛❧✉♥♦s ❞❡ss❛ t✉r♠❛ ❛s t❛r❡❢❛s ❞❡ ✸ ❛❧✉♥♦s ❢♦r❛♠ ❡s❝♦❧❤✐❞❛s ♣❛r❛ s❡r❡♠ ❛♣r❡s❡♥t❛❞❛s ❡ ❞✐s❝✉t✐❞❛s ♥❡ss❡ tr❛❜❛❧❤♦✳ ❊sss❛ ❡s❝♦❧❤❛ t♦♠♦✉ ❝♦♠♦ ❜❛s❡ ♦s ❞✐❢❡r❡♥t❡s ❝❛♠✐♥❤♦s ❡s❝♦❧❤✐❞♦s ♣❡❧♦s ❛❧✉♥♦s ♣❛r❛ ❡①❡❝✉t❛r ❛s t❛r❡❢❛s ♥♦ q✉❡ t❛♥❣❡ q✉❛tr♦ ❛s♣❡❝t♦s ❡ss❡♥❝✐❛✐s✿
✶✳ P❧❛♥❡❥❛♠❡♥t♦✳ ❈♦♠♦ ♦ ❛❧✉♥♦ ❞❡❝✐❞✐✉ ♦ q✉❡ ❢❛③❡r ♣❛r❛ ❡①❡❝✉t❛r ❛s t❛r❡❢❛s ♣r♦♣♦st❛s✱ ❡♠ q✉❡ s❡q✉ê♥❝✐❛ ❡ ❞❡ q✉❡ ❢♦r♠❛✳
✷✳ ❯t✐❧✐③❛çã♦ ❞♦ ❡s♣❛ç♦ ♦❢❡r❡❝✐❞♦ ♣❡❧❛ ♣❧❛♥✐❧❤❛✱ ✐st♦ é✱ ❝♦♠♦ ♦ ❛❧✉♥♦ ♦r❣❛♥✐③♦✉ ❛q✉✐❧♦ q✉❡ ♣r❡t❡♥❞✐❛ ✐♥s❡r✐r ♥❛ ♣❧❛♥✐❧❤❛✳
✸✳ ❊①❡❝✉çã♦ ❞❛s ❢ór♠✉❧❛s✳ ❛ ♣❛rt❡ ❝❡♥tr❛❧ ❡ ♠❛✐s ❝♦♠♣❧❡①❛ ❞❛s t❛r❡❢❛s✿ q✉❛✐s ❢♦r❛♠ ♦s ❝❛♠✐♥❤♦s ✉t✐❧✐③❛❞♦s ♣❡❧♦ ❛❧✉♥♦ ♣❛r❛ ❢❛③❡r ❛ tr❛♥s✐çã♦ ❞❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡✲ ♠át✐❝❛ ♣❛r❛ ❛ ❧✐♥❣✉❡❣❡♠ ❞❡ ♣r♦❣r❛♠❛çã♦ ❞♦ ❈❛❧❝✱ ♦✉ s❡❥❛ ❛ ♣r♦❣r❛♠❛çã♦ ❞❛ ♣❧❛♥✐❧❤❛ ❡♠ s✐✳
✹✳ ❋♦r♠❛t❛çã♦ ❞❛ ♣❧❛♥✐❧❤❛✳ ❉✐③ r❡s♣❡✐t♦ ❛♦ ❛s♣❡❝t♦ ✜♥❛❧ q✉❡ ❛ ♣❧❛♥✐❧❤❛ ✈❛✐ ❛♣r❡s❡♥t❛r✳ ❆ ✐♥t❡r❛çã♦ ❞♦s ❢✉t✉r♦s ✉s✉ár✐♦s ❞♦ ♣r♦❥❡t♦ ❉❡❧t❛ ✭❛❧✉♥♦s ❞❡ ♦✉tr❛s sér✐❡s✮ ❝♦♠ ❛ ♣❧❛♥✐❧❤❛ ✈❛✐ ❞❡♣❡♥❞❡r ❞❛ ✐♥t❡r❢❛❝❡ ❝♦♠ ♦ ✉s✉ár✐♦ q✉❡ ❛ ♣❧❛♥✐❧❤❛ ✈❛✐ ♦❢❡r❡❝❡r✿ ♦ q✉❡ ♦ ♣r♦❣r❛♠❛ ✏Pr♦❥❡t♦ ❉❡❧t❛✑ ❡①❡❝✉t❛✱ ♦s ♣❛ss♦s ♣❛r❛ s❡✉ ❝❛rr❡❣❛♠❡♥t♦ ♣❡❧♦ ✉s✉ár✐♦ ❡ ♦ q✉❡ ♦ ♣r♦❣r❛♠❛ r❡t♦r♥❛✱ ♦✉ s❡❥❛ ♦ ♣r♦❞✉t♦ ✜♥❛❧ ♦❢❡r❡❝✐❞♦ ❛♦ ✏❝❧✐❡♥t❡✑✳
✷✳✷ ❋✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ Pr♦❥❡t♦ ❉❡❧t❛
❖ Pr♦❥❡t♦ ❉❡❧t❛ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ t❛r❡❢❛s ♣r♦♣♦st❛s ❛♦s ❛❧✉♥♦s ❞♦ ✶♦ ❛♥♦ ♠é❞✐♦
❞❛ ❡s❝♦❧❛ ❛❝✐♠❛ r❡❢❡r✐❞❛❀ ❡ss❛s t❛r❡❢❛s ❡♥❢♦❝❛♠ ♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ❣r❛✉ ✷✿ ❝á❧❝✉❧♦ ❞♦s ③❡r♦s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❜❡♠ ❝♦♠♦ s✉❛ ✐♠❛❣❡♠ ❡ ❝♦♦r❞❡♥❛❞❛s ❞♦ s❡✉ ✈ért✐❝❡✳ ❚♦❞❛s ❡st❛s t❛r❡❢❛s tê♠ q✉❡ s❡r ❢❡✐t❛s ♥❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ✕ ♥❡ss❡ ❝❛s♦ ✕ ❛ ♣❧❛♥✐❧❤❛ ❈❛❧❝ ❞♦ ▲✐❜r❡❖✣❝❡ ✺✳✵ ✉t✐❧✐③❛♥❞♦ s❡✉s r❡❝✉rs♦s ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ♣r♦❣r❛♠❛s q✉❡ ❡①❡❝✉t❡♠ ♦s ❝á❧❝✉❧♦s ❞♦s tó♣✐❝♦s ❞❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛s ❞♦ ✷♦ ❣r❛✉
❥á ❝✐t❛❞♦s✳ ◆❛ ✜❣✉r❛ ✷✳✶ sã♦ ♠♦str❛❞❛s ❛s ❛t✐✈✐❞❛❞❡ ❞♦ ♣r♦❥❡t♦ ❉❡❧t❛ q✉❡ ❢♦r❛♠ ♣r♦♣♦st❛s ❛♦s ❛❧✉♥♦s ❞♦ ✶♦ ❛♥♦ ♠é❞✐♦ ❞❡st❛ ❡s❝♦❧❛ ❊st❛❞✉❛❧ ✿
✷✳✸✳ ❆❚■❱■❉❆❉❊ ✶
❋✐❣✉r❛ ✷✳✶✿ Pr♦❥❡t♦ ❉❡❧t❛
✷✳✸ ❆t✐✈✐❞❛❞❡ ✶
❆ ❛t✐✈✐❞❛❞❡ ✶ ♣❡❞✐❛ q✉❡ ♦s ❛❧✉♥♦s ❞❡s❡♥✈♦❧✈❡ss❡♠✱ ✉t✐❧✐③❛♥❞♦ ♦s r❡❝✉rs♦s ❞❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ❈❛❧❝✱ ❢ór♠✉❧❛s q✉❡ ❝❛❧❝✉❧❛ss❡♠ ♦s ③❡r♦s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛
f(x) = ax2
+bx+c ✭✷✳✶✮
✷✳✸✳ ❆❚■❱■❉❆❉❊ ✶
❞❛❞♦s ♣❡❧❛s ❡①♣r❡ssõ❡s✿
x1 = −
b+√b2
−4ac
2a ✭✷✳✷✮
❡
x2 = −
b−√b2
−4ac
2a ✭✷✳✸✮
♦♥❞❡ ♦ ❜✐♥ô♠✐♦
b2
−4ac ✭✷✳✹✮
r❡♣r❡s❡♥t❛ ♦ ❞✐s❝r✐♠✐♥❛♥t❡ ❞❛ ❡q✉❛çã♦ ❞♦ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ ❞♦✐s✱ t❛♠❜é♠ r❡♣r❡s❡♥✲ t❛❞♦ ♣❡❧❛ ❧❡tr❛ ❣r❛❣❛ ❞❡❧t❛✿
∆ =b2−4ac ✭✷✳✺✮
❉❡ss❛ ❢♦r♠❛ ❛s ❡q✉❛çõ❡s ✷✳✶ ❡ ✷✳✷ ✜❝❛♠ ♥❛ ❢♦r♠❛
x= −b±
√
∆
2a ✭✷✳✻✮
P❛r❛ t❛♥t♦ ♦s ❛❧✉♥♦s t❡r✐❛♠ q✉❡✱ ❝♦♠♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛✱ ❡♥❝♦♥tr❛r ✉♠❛ ❢♦r♠❛ ❣❡r❛❧ ❞❡ r❡♣r❡s❡♥t❛r ✕ ♥❛ ❧✐♥❣✉❛❣❡♠ ❞❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛ ✕ ❛s ❢✉♥çõ❡s q✉❛❞rát✐❝❛s✱ ✐st♦ é✱ t❡r✐❛♠ q✉❡ ❡♥❝♦♥tr❛r ❛❧❣♦ q✉❡ ❝❛r❛❝t❡r✐③❛ss❡ ❡ ❞✐❢❡r❡♥❝✐❛ss❡ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❞❡ ♦✉tr❛ ✐♥s❡r✐❞❛ ♥❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛✱ ❥á q✉❡ ❡ss❛ t❛r❡❢❛ ♣❡❞✐❛ ♣❛r❛ ❝❛❧❝✉❧❛r ♦s ③❡r♦s ❞❡ q✉❛❧q✉❡r ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡ ♥ã♦ ❞❡ ✉♠❛ ❡♠ ♣❛rt✐❝✉✲ ❧❛r✳ ❖ ❞❡s❛✜♦ ♣❛r❛ ♦s ❛❧✉♥♦s s❡r✐❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❢❛③❡r ❝♦♠ q✉❡ ❛ ♣❧❛♥✐❧❤❛ ❡♥t❡♥❞❡ss❡ ✕ ❡ ❞✐st✐♥❣✉✐ss❡ ✕ ❛s ❢✉♥çõ❡s
f(x) = 2x2
−x+ 9
❡
f(x) =−x2
+ 6x+ 3
✉♠❛ ❞❛ ♦✉tr❛ s❡♠ ❛♠❜✐❣✉✐❞❛❞❡s✳ ●r❛♥❞❡ ♣❛rt❡ ❞♦s ❛❧✉♥♦s ❝♦♠❡t❡✉ ♦ ♠❡s♠♦ ❡rr♦ ❞❡ ♣r❡s✉♠✐r q✉❡ ❛ ♣❧❛♥✐❧❤❛ ❡♥t❡♥❞❡r✐❛ ❛s ❢✉♥çõ❡s ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ❡st❛✈❛♠ ❛♣r❡✲ s❡♥t❛❞❛s ♥♦ ❧✐✈r♦ ❞✐❞át✐❝♦✳ ❊ss❡ ❡r❛ ✉♠ ❡rr♦ ❡s♣❡r❛❞♦ ♥❛ ♠❡❞✐❞❛ q✉❡ ❛ tr❛❞✉çã♦ ❞❛ ❧✐♥❣✉❡❣❡♠ ♠❛t❡♠át✐❝❛ ✉s✉❛❧ ♣❛r❛ ❛ ❧✐♥❣✉❛❣❡♠ ♣ró♣r✐❛ ❞❛ ♣❧❛♥✐❧❤❛ ❡r❛ ❛ ❣r❛♥❞❡ ❞✐✜❝✉❧❞❛❞❡ q✉❡ ♦s ❛❧✉♥♦s t❡r✐❛♠ q✉❡ ❡♥❢r❡♥t❛r✳ ❆♣ós ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ t❡♥t❛t✐✈❛s✱ ❡rr♦s ❡ r❡✢❡①õ❡s s♦❜r❡ ❡ss❡s ❡rr♦s ✕ s♦❜ s✉♣❡r✈✐sã♦ ❞✐❞át✐❝❛ ❞♦ ♣r♦❢❡ss♦r ✕ ♦s ❛❧✉♥♦s ❝❤❡❣❛r❛♠ à ❝♦♥❝❧✉sã♦ q✉❡ ♦s ❝♦❡✜❝✐❡♥t❡s ♥✉♠ér✐❝♦s ❞❡ ❝❛❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ✕ r❡✲ ♣r❡s❡♥t❛❞♦s ✉s✉❛❧♠❡♥t❡ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ♣❡❧❛s ❧❡tr❛s ❛✱ ❜ ❡ ❝ ✕ ❛s r❡♣r❡s❡♥t❛r✐❛♠✱ s❡♠ ❛♠❜✐❣✉✐❞❛❞❡s✱ ♥❛ ♣❧❛♥✐❧❤❛ ❡❧❡trô♥✐❝❛✳
❱❡♥❝✐❞♦ ❡ss❡ ♦❜stá❝✉❧♦✱ ♦s ❛❧✉♥♦s t❡r✐❛♠ q✉❡ ❞❡s❡♥✈♦❧✈❡r ❛s ❢ór♠✉❧❛s✱ s❡♠♣r❡ ♥❛ ❧✐♥❣✉❛❣❡♠ ❞❛ ♣❧❛♥✐❧❤❛✱ q✉❡ ❡①❡❝✉t❛ss❡♠ ♦ q✉❡ s❡ ♣❡❞✐❛ ♥❛s t❛r❡❢❛s✳ ◆❛ ✜❣✉r❛ ✷✳✷ sã♦ ♠♦str❛❞❛s ❛s ❢♦r♠❛s ❝♦rr❡t❛s ✭❣❛❜❛r✐t♦✮ ❞❡ ❞✐❣✐t❛r ❛s ❢ór♠✉❧❛s ✷✳✷✱ ✷✳✸ ❡ ✷✳✹✳ ❋♦r❛♠ ❞❡✜♥✐❞♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♦s ❡♥❞❡r❡ç♦s ❞♦s ❝♦❡✜❝✐❡♥t❡ ❛✱ ❜ ❡ ❝ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡
