▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
❋❯◆➬Õ❊❙ ❆❋■▼ ❊ ◗❯❆❉❘➪❚■❈❆
❈❖▼ ❆❯❳❮▲■❖ ❉❖ ▼❯▲❚■P▲❆◆❖
❊ ❇▲❖❈❖❙ ❈Ú❇■❈❖❙
♣♦r
❏♦ã♦ P❛✉❧♦ ❆rr✉❞❛ ❞❛ ❙✐❧✈❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥✲ t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐✲ ♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡✲ q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛
❇✐❜❧✐♦t❡❝❛ ❙❡t♦r✐❛❧ ❞♦ ❈❈❊◆
❙✺✽✻❢ ❙✐❧✈❛✱ ❏♦ã♦ P❛✉❧♦ ❆rr✉❞❛ ❙✐❧✈❛✳
❋✉♥çõ❡s ❛✜♠ ❡ q✉❛❞rát✐❝❛ ❝♦♠ ❛✉①í❧✐♦ ❞♦ ♠✉❧t✐♣❧❛♥♦ ❡ ❜❧♦❝♦s ❝ú❜✐❝♦s✴ ❏♦ã♦ P❛✉❧♦ ❆rr✉❞❛ ❞❛ ❙✐❧✈❛✳✕ ❏♦ã♦ P❡ss♦❛✱ ✷✵✶✹✳
✺✺❢✳ ✿ ✐❧✳
❖r✐❡♥t❛❞♦r✿ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛ ❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮ ✲ ❯❋P❇✴❈❈❊◆✳
✶✳ ▼❛t❡♠át✐❝❛✳ ✷✳ ❋✉♥çã♦ ❛✜♠✳ ✸✳ ❋✉♥çã♦ q✉❛❞rát✐❝❛✳ ✹✳❇❧♦❝♦ ❞❡ ❝✉❜♦s✳ ✺✳ ▼✉❧t✐♣❧❛♥♦✳ ✻✳ ❋♦r♠❛ ❝❛♥ô♥✐❝❛✳
❈❖▼ ❆❯❳❮▲■❖ ❉❖ ▼❯▲❚■P▲❆◆❖
❊ ❇▲❖❈❖❙ ❈Ú❇■❈❖❙
♣♦r
❏♦ã♦ P❛✉❧♦ ❆rr✉❞❛ ❞❛ ❙✐❧✈❛
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
ár❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛
❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❆❧❜❡rt♦ ▼❛s❛②♦s❤✐ ❋❛r✐❛ ❖❤❛s❤✐ ✲ ❯❋P❇
Pr♦❢❛✳ ❉r❛✳ ❚❛r❝✐❛♥❛ ▼❛r✐❛ ❙❛♥t♦s ❞❛ ❙✐❧✈❛ ✲ ❯❋❘P❊
●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ♣♦r ♠❡ ❞❛r ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ♣♦❞❡r r❡❛❧✐③❛r ♠❛✐s ✉♠ s♦♥❤♦ ❡♠ ♠✐♥❤❛ ✈✐❞❛✳
❆♦s ♠❡✉s ♣❛✐s ♣♦r ♠❡ ❛♣♦✐❛r❡♠ ❡ ♠❡ ✐♥❝❡♥t✐✈❛r❡♠ ❛♦s ❡st✉❞♦s✳ ➚ ♠✐♥❤❛ q✉❡r✐❞❛ ❡s♣♦s❛ ❘✐❧♠❛ ❆rr✉❞❛ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❡ ♣❛❝✐ê♥❝✐❛✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛ q✉❡ t❛♥t♦ ♠❡ ❛❥✉❞♦✉ ♥❛ ❝♦♥❢❡❝çã♦ ❡ ❝♦rr❡çã♦ ❞❛s ✐❞❡✐❛s ✉t✐❧✐③❛❞❛s ♥❡ss❡ tr❛❜❛❧❤♦✳
❆ t♦❞♦s ♦s Pr♦❢❡ss♦r❡s ❞♦ ❈✉rs♦ P❘❖❋▼❆❚ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ❡♠ ❏♦ã♦ P❡ss♦❛ q✉❡ t❛♥t♦ ❛✉①✐❧✐❛r❛♠ ♥♦ss❛ t✉r♠❛ ♥♦s ❞✐✈❡rs♦s ♠♦♠❡♥t♦s ❞❡ ❞✐✜❝✉❧❞❛❞❡ ❡ ❞❡sâ♥✐♠♦✱ ❝♦♥tr✐❜✉✐♥❞♦ ❜❛st❛♥t❡ ❝♦♥♦s❝♦ ❝♦♠ s❛❜❡❞♦r✐❛ ❡ ♣❛❝✐ê♥❝✐❛✳
❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ t✉r♠❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❆♥tô♥✐♦✱ ❉❡♠✐❧s♦♥✱ ❊❞❥❛♥❡✱ ❊❧✐ P❛✉❧♦✱ ❏♦sé ▼❛r❝♦♥❞❡s✱ ❏♦s✐❧❞♦ ❡ ❘❡♥❛t♦ ❇❡s❡rr❛✱ ❣r❛♥❞❡s ❛♠✐❣♦s q✉❡ ✜③ ♥♦ ❝✉rs♦ ❡ q✉❡ ❝♦♠♣❛rt✐❧❤❛✲ r❛♠ ♠✉✐t❛s ♥♦✐t❡s ❡ ♠❛❞r✉❣❛❞❛s ❞❡ ❡st✉❞♦s ❝♦♠✐❣♦✳
➚ ❇❛♥❝❛ ❝♦♠♣♦st❛ ♣❡❧♦s Pr♦❢❡ss♦r❡s ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛ ✱ ❉r✳ ❆❧❜❡rt♦ ▼❛s❛②♦s❤✐ ❋❛r✐❛ ❖❤❛s❤✐ ❡ ❉r❛✳ ❚❛r❝✐❛♥❛ ▼❛r✐❛ ❙❛♥t♦s ❞❛ ❙✐❧✈❛✳
➚s ♠✐♥❤❛s ❛♠❛❞❛s ✜❧❤❛s P❛♦❧❛ ❆rr✉❞❛✱ P❛✉❧❛ ❆rr✉❞❛ ❡ ❙❛r❛❤ ❆rr✉❞❛ ♣❡❧❛ ❢❡❧✐❝✐❞❛❞❡ q✉❡ ♠❡ ♣r♦♣♦r❝✐♦♥❛♠ t♦❞♦s ♦s ❞✐❛s✳
❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠❛ ♥♦✈❛ ♠❛♥❡✐r❛ ❞❡ s❡ ❞✐s❝✉t✐r ❝♦♥❝❡✐t♦s ❡♥✈♦❧✈❡♥❞♦ ❢✉♥çã♦✱ ❢✉♥çã♦ ❛✜♠ ❡ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳ ❆tr❛✈és ❞❡ ❛t✐✈✐❞❛❞❡s s✐♠♣❧❡s ❝♦♠ ✉s♦ ❞♦s ❜❧♦❝♦s ❞❡ ❝✉❜♦s✱ ❞♦ ♠✉❧t✐♣❧❛♥♦ ❡ ❞❡ s♦❢t✇❛r❡s ❝♦♠♦ ♦ ●❡♦❣❡❜r❛✱ ♦ ❛❧✉♥♦ ♣❛ss❛rá ❞❡ ✉♠❛ ♣♦s✐çã♦ ♣❛ss✐✈❛✱ ❝♦♠♦ s✐♠♣❧❡s ♦✉✈✐♥t❡✱ ❛ ✐♥t❡❣r❛♥t❡ ❛t✐✈♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠✱ ♣♦ss✐❜✐❧✐t❛♥❞♦ ❛♦ ♠❡s♠♦ ❛❞q✉✐r✐r ✉♠❛ ♥♦çã♦ ♠❛✐s ❛♠♣❧❛ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦ ❡ ❛♣❧✐❝á✲ ❧♦ ♣♦st❡r✐♦r♠❡♥t❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ♥❛ ❡s❝♦❧❛ ❡ ❡♠ s✉❛ ✈✐❞❛ ❝♦t✐❞✐❛♥❛✳ ❆❧é♠ ❞❡ss❛s ❛t✐✈✐❞❛❞❡s ❞✐✈❡rs✐✜❝❛❞❛s✱ tó♣✐❝♦s q✉❡ ❣❡r❛❧♠❡♥t❡ ♥ã♦ sã♦ ❝♦❧♦❝❛❞♦s ❡♠ ❞❡st❛q✉❡ ♥❛ ♠❛✐♦r✐❛ ❞♦s ❧✐✈r♦s ❞❡ ❡♥s✐♥♦ ♠é❞✐♦ t❛✐s q✉❛✐s ❛ ❛♥á❧✐s❡ ❞❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❛✜♠ t❡rã♦ ✉♠ ✐♠♣♦rt❛♥t❡ ♣❛♣❡❧ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ q✉❡ s❡ ❡s♣❡r❛ ❞♦s ❡st✉❞❛♥t❡s ❛ r❡s♣❡✐t♦ ❞❡ss❛ ✐♠♣♦rt❛♥t❡ ár❡❛ ❞❛ ▼❛t❡♠át✐❝❛✳
❚♦❞❛s ❛s ❛t✐✈✐❞❛❞❡s ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s ❝♦♠ ❛❧✉♥♦s ❞❛1asér✐❡ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❞❛ ❊s❝♦❧❛ ❊st❛❞✉❛❧ ◆❛t❛❧í❝✐❛ ▼❛r✐❛ ❋✐❣✉❡✐r♦❛ ❞❛ ❙✐❧✈❛✱ ❧♦❝❛❧✐③❛❞❛ ♥❛ ❝✐❞❛❞❡ ❞❡ ❙✉r✉❜✐♠ ♥♦ ❊st❛❞♦ ❞❡ P❡r♥❛♠❜✉❝♦✳
P❛❧❛✈r❛s ❈❤❛✈❡✿ ❋✉♥çã♦ ❛✜♠✱ ❋✉♥çã♦ q✉❛❞rát✐❝❛✱ ❇❧♦❝♦ ❞❡ ❝✉❜♦s✱ ▼✉❧t✐♣❧❛♥♦✱ ❋♦r♠❛ ❝❛♥ô♥✐❝❛✱ ❈❛r❛❝t❡r✐③❛çã♦✳
❚❤✐s ♣❛♣❡r ♣r❡s❡♥ts ❛ ♥❡✇ ✇❛② ♦❢ ❞✐s❝✉ss✐♥❣ ❝♦♥❝❡♣ts ✐♥✈♦❧✈✐♥❣ ❢✉♥❝t✐♦♥✱ ❛✣♥❡ ❢✉♥❝t✐♦♥ ❛♥❞ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✳ ❚❤r♦✉❣❤ s✐♠♣❧❡ ❛❝t✐✈✐t✐❡s ✇✐t❤ t❤❡ ✉s❡ ♦❢ ❜❧♦❝❦s ♦❢ ❝✉❜❡s✱ ❛♥❞ t❤❡ ♠✉❧t✐♣❧❛♥❡ s♦❢t✇❛r❡ s✉❝❤ ❛s ●❡♦●❡❜r❛✱ st✉❞❡♥ts ✇✐❧❧ ♠♦✈❡ ❢r♦♠ ❛ ♣❛ss✐✈❡ ♣♦s✐t✐♦♥✱ ❛s s✐♠♣❧❡ ❧✐st❡♥❡r✱ t❤❡ ❛❝t✐✈❡ ♠❡♠❜❡r ♦❢ t❤❡ t❡❛❝❤✐♥❣✲❧❡❛r♥✐♥❣ ♣r♦❝❡ss✱ ❡♥❛❜❧✐♥❣ t❤❡♠ t♦ ❛❝q✉✐r❡ ❛ ❜r♦❛❞❡r ♥♦t✐♦♥ t❤❡ ❝♦♥❝❡♣t ♦❢ ❢✉♥❝t✐♦♥ ❛♥❞ ❛♣♣❧② ✐t ❧❛t❡r ✐♥ s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ✐♥ s❝❤♦♦❧ ❛♥❞ ✐♥ t❤❡✐r ❞❛✐❧② ❧✐✈❡s✳ ❇❡s✐❞❡s t❤❡s❡ ❞✐✈❡rs✐✜❡❞ ❛❝t✐✈✐t✐❡s✱ t♦♣✐❝s t❤❛t ❛r❡ ✉s✉❛❧❧② ♥♦t ❣✐✈❡♥ ♣r♦♠✐♥❡♥❝❡ ✐♥ ♠♦st ❜♦♦❦s ♦❢ ❤✐❣❤ s❝❤♦♦❧ s✉❝❤ t❤❛t t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❝❛♥♦♥✐❝❛❧ ❢♦r♠ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ t❤❡♦r❡♠ ♦❢ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❛✣♥❡ ❢✉♥❝t✐♦♥ ✇✐❧❧ ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ ✐♥ ❞❡✈❡❧♦♣✐♥❣ ✇❤❛t ✐s ❡①♣❡❝t❡❞ ♦❢ st✉❞❡♥ts ❛❜♦✉t t❤✐s ✐♠♣♦rt❛♥t ❛r❡❛ ♦❢ ♠❛t❤❡♠❛t✐❝s✳
❆❧❧ ❛❝t✐✈✐t✐❡s ✇❡r❡ ❝♦♥❞✉❝t❡❞ ✇✐t❤ ✶st ❣r❛❞❡ st✉❞❡♥ts ♦❢ ❤✐❣❤ s❝❤♦♦❧ ◆❛t❛❧✐❝✐❛ ▼❛r✐❛ ❋✐❣✉❡✐r♦❛ ❞❛ ❙✐❧✈❛✱ ❧♦❝❛t❡❞ ✐♥ t❤❡ ❝✐t② ♦❢ ❙✉r✉❜✐♠✱ ✐♥ t❤❡ st❛t❡ ♦❢ P❡r♥❛♠❜✉❝♦✳
❑❡② ✇♦r❞s✿ ❆✣♥❡ ❢✉♥❝t✐♦♥✱ ◗✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✱ ❇❧♦❝❦ ♦❢ ❝✉❜❡s✱ ▼✉❧t✐♣❧❛♥❡✱ ❈❛♥♦♥✐❝❛❧ ❢♦r♠✱ ❈❤❛r❛❝t❡r✐③❛t✐♦♥✳
■♥tr♦❞✉çã♦ ✶
✶ ◗✉❡stõ❡s ♣r♦♣♦st❛s ♥♦ ❡①❛♠❡ ❞✐❛❣♥óst✐❝♦ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ✷ ✶✳✶ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✶ P♦r❝❡♥t❛❣❡♠ ❞❡ ❛❝❡rt♦s ♣♦r q✉❡stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✷ ●rá✜❝♦ ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡ ❛❝❡rt♦s ♣♦r q✉❡stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✸ ❚❛❜❡❧❛ ❝♦♠ q✉❡stõ❡s ❡ ❛❧t❡r♥❛t✐✈❛s ❛ss✐♥❛❧❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✹ ◆ú♠❡r♦ ❞❡ ❛❧✉♥♦s ♣♦r ❣❛❜❛r✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✺ ▲✐st❛ ❞♦s ♣♦♥t♦s ❞❡ ♠❛✐♦r❡s ❞✐✜❝✉❧❞❛❞❡s ❛♣r❡s❡♥t❛❞♦s ♣❡❧♦s ❡st✉❞❛♥t❡s ✶✶
✷ ◆♦çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❋✉♥çã♦ ✶✷
✷✳✶ ❈♦♦r❞❡♥❛❞❛s ❈❛rt❡s✐❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✶✳✶ ❙✐st❡♠❛ ❈❛rt❡s✐❛♥♦ ❖rt♦❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✳✷ Pr♦❞✉t♦ ❈❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ❘❡❧❛çã♦ ❇✐♥ár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷✳✶ ❉♦♠í♥✐♦ ❡ ■♠❛❣❡♠ ❞❡ ✉♠❛ r❡❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✸ ❋✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸✳✶ ❋✉♥çã♦✲◆♦çã♦ ■♥t✉✐t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸✳✷ ❉❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✸✳✸ ❉♦♠í♥✐♦✱ ❈♦♥tr❛❞♦♠í♥✐♦ ❡ ❈♦♥❥✉♥t♦ ■♠❛❣❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✶✾ ✷✳✹ ❋✉♥çã♦ ✐♥❥❡t✐✈❛ ❡ ❢✉♥çã♦ s♦❜r❡❥❡t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✺ ❋✉♥çã♦ ❈r❡s❝❡♥t❡ ❡ ❢✉♥çã♦ ❞❡❝r❡s❝❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✻ ❋✉♥çã♦ ❆✜♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✼ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ Pr♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✷✳✾✳✶ ❋♦r♠❛ ❝❛♥ô♥✐❝❛ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡ ❞❡t❡r♠✐♥❛çã♦ ❞❛s ❝♦♦r❞❡✲ ♥❛❞❛s ❞♦ ✈ért✐❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✾✳✷ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸ ❆t✐✈✐❞❛❞❡s ❞❡s❡♥✈♦❧✈✐❞❛s ❝♦♠ ❛✉①í❧✐♦ ❞♦ ●❡♦●❡❜r❛✱ ▼✉❧t✐♣❧❛♥♦ ❡ ❇❧♦❝♦s
❈ú❜✐❝♦s ✸✾
✸✳✶ ❆t✐✈✐❞❛❞❡ ❝♦♠ ♦ ▼✉❧t✐♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✶✳✶ ❋✉♥çã♦ ❆✜♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✶✳✷ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✷ ❆✉❧❛ ❝♦♠ ✉s♦ ❞♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✷✳✶ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✸ ❆✉❧❛✴❛t✐✈✐❞❛❞❡ ❝♦♠ ✉s♦ ❞♦s ❜❧♦❝♦s ❝ú❜✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✸✳✶ ❋✉♥çã♦✴●❡♥❡r❛❧✐③❛çã♦ ❞❡ ▼♦❞❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✹ ❊①❛♠❡ ❉✐❛❣♥óst✐❝♦ ❋✐♥❛❧ ✹✺
✹✳✶ ❏✉st✐✜❝❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✷ ❊①❛♠❡ ❉✐❛❣♥óst✐❝♦ ❋✐♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✸ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ❞❛ ❛t✐✈✐❞❛❞❡ ♣♦r tó♣✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✸✳✶ ●rá✜❝♦ ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡ ❛❝❡rt♦s ♣♦r q✉❡stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✸✳✷ ❚❛❜❡❧❛ ❞❡ q✉❡stõ❡s ❝♦♠ ❛❧t❡r♥❛t✐✈❛s ♠❛r❝❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
❋✐❣✉r❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ❋✐❣✉r❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ❋✐❣✉r❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐
❋✐❣✉r❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❋✐❣✉r❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❋✐❣✉r❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ❋✐❣✉r❛ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❋✐❣✉r❛ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ❋✐❣✉r❛ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
❋✐❣✉r❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶ ❙✐st❡♠❛ ❈❛rt❡s✐❛♥♦ ❖rt♦❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ❋✐❣✉r❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ P♦♥t♦s ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ❋✐❣✉r❛ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ❋✐❣✉r❛ ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ❋✐❣✉r❛ ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ❋✐❣✉r❛ ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ❋✐❣✉r❛ ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ❋✐❣✉r❛ ✶✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ❋✐❣✉r❛ ✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
❋✐❣✉r❛ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✹ ❈❛s♦a >0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✷✳✺ ❈❛s♦a <0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
❋✐❣✉r❛ ✶✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ❋✐❣✉r❛ ✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ❋✐❣✉r❛ ✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ❋✐❣✉r❛ ✶✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ❋✐❣✉r❛ ✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✻ ❈❛s♦ ❛❃✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✼ ❈❛s♦ ❛❁✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✸✳✶ ▼✉❧t✐♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✷ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✸ ❇❧♦❝♦s ❈ú❜✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
❋✐❣✉r❛ ✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ❋✐❣✉r❛ ✶✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ P♦r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦s ♣♦r q✉❡stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✶ ❆✈❛❧✐❛çã♦ ❞✐❛❣♥óst✐❝❛ ✐♥✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✷ ❆✈❛❧✐❛çã♦ ❞✐❛❣♥óst✐❝❛ ✜♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✸ ❇❧♦❝♦s ❈ú❜✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✹ ❇❧♦❝♦s ❈ú❜✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✺ ▼✉❧t✐♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✻ ▼✉❧t✐♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✼ ▼✉❧t✐♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✽ ▼✉❧t✐♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✾ ▼✉❧t✐♣❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
❖ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦ é ✉♠ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❛ ▼❛t❡♠át✐❝❛✱ ♦❝✉♣❛♥❞♦ ❧✉❣❛r ❞❡ ❞❡st❛q✉❡ ❡♠ ✈ár✐♦s ❞❡ s❡✉s s❡❣♠❡♥t♦s ❡ ❡♠ ❞✐✈❡rs❛s ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳ ➱ ❜❛st❛♥t❡ ❝♦♠✉♠ ❡①♣r❡ss❛r ❢❡♥ô♠❡♥♦s ❢ís✐❝♦s✱ ❜✐♦❧ó❣✐❝♦s✱ s♦❝✐❛✐s✱ ❡t❝✳ ♣♦r ♠❡✐♦ ❞❡ ❢✉♥çõ❡s✳ ❚♦❞♦s ❡ss❡s ❛s♣❡❝t♦s ❥✉st✐✜❝❛♠ ✉♠ ❡st✉❞♦ ♠❛✐s ❞❡t❛❧❤❛❞♦ ❞❡ss❡ ✐♠♣♦rt❛♥t❡ tó♣✐❝♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❖ tó♣✐❝♦ ❢✉♥çõ❡s é ❛❜♦r❞❛❞♦ ♥❛s ❡s❝♦❧❛s ♥❛ ú❧t✐♠❛ sér✐❡ ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ II ♣♦ré♠✱ ❛♦ ❝❤❡❣❛r❡♠ ♥❛ 1❛ sér✐❡ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞♦s ❡st✉❞❛♥t❡s ♥ã♦
❝♦♥s❡❣✉❡♠ ❝♦♠♣r❡❡♥❞❡r s❡q✉❡r ❛s ♥♦çõ❡s ♠❛✐s ❜ás✐❝❛s ❡♥✈♦❧✈❡♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çõ❡s✱ ❣❡r❛♥❞♦ ❣r❛♥❞❡s ❞✐✜❝✉❧❞❛❞❡s ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ❡ ❝♦♥str✉çã♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦✳ ❆♣❡s❛r ❞♦ ❝♦♥t❡ú❞♦ ❢✉♥çã♦ ❡st❛r ♣r❡s❡♥t❡ ❡♠ t♦❞♦s ♦s ❧✐✈r♦s ❞❛ 1❛ sér✐❡ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❡st❡
✈❡♠✱ ♠✉✐t❛s ✈❡③❡s✱ ❛♣r❡s❡♥t❛❞♦ ❞❡ ♠❛♥❡✐r❛ ♠❡❝â♥✐❝❛✱ ♣❛r❡❝❡♥❞♦ s❡r ❛❧❣♦ ❞✐st❛♥t❡✱ s❡♠ ❝♦♥❡①ã♦ ❛❧❣✉♠❛ ❝♦♠ ♣r♦❜❧❡♠❛s ❞♦ ❝♦t✐❞✐❛♥♦ ❞♦s ❡st✉❞❛♥t❡s✳ ❉❡ss❡ ♠♦❞♦✱ ❛❧é♠ ❞❡ ❣❡r❛r ❝❡rt♦ ❞❡s✐♥t❡r❡ss❡ ♣♦r ♣❛rt❡ ❞♦s ❡st✉❞❛♥t❡s✱ ♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ❡♥✈♦❧✈❡♥❞♦ ❢✉♥çã♦ ♥ã♦ sã♦ ❛ss✐♠✐❧❛❞♦s ❞❡ ❢♦r♠❛ s❛t✐s❢❛tór✐❛✱ ❧❡✈❛♥❞♦ ♦s ♠❡s♠♦s ❛ ❝♦♠❡t❡r❡♠ ❞✐✈❡rs♦s ❡rr♦s q✉❛♥❞♦ s❡ ❞❡♣❛r❛♠ ❝♦♠ s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ ❡♥✈♦❧✈❡♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦✳
◆❡st❛ ♣r♦♣♦st❛ ❞❡ ❛t✐✈✐❞❛❞❡✱ ♣r✐♠❡✐r❛♠❡♥t❡ s❡rá ❡❧❛❜♦r❛❞❛ ✉♠❛ ❛✈❛❧✐❛çã♦ ❞✐❛❣♥óst✐❝❛ ❝♦♥t❡♥❞♦ q✉✐♥③❡ q✉❡stõ❡s ❞❡ ♠ú❧t✐♣❧❛ ❡s❝♦❧❤❛ ❡♥✈♦❧✈❡♥❞♦ ♥♦çõ❡s ❜ás✐❝❛s s♦❜r❡ ❢✉♥çã♦✱ ❢✉♥çã♦ ❛✜♠ ❡ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ q✉❡ s❡rá ❛♣❧✐❝❛❞❛ ❡♠ ❞✉❛s t✉r♠❛s ❞❛1❛ sér✐❡ ❞♦ ❡♥s✐♥♦
♠é❞✐♦ ❞❛ ❊s❝♦❧❛ ❊st❛❞✉❛❧ ◆❛t❛❧í❝✐❛ ❋✐❣✉❡✐r♦❛✱ ❧♦❝❛❧✐③❛❞❛ ♥❛ ❈✐❞❛❞❡ ❞❡ ❙✉r✉❜✐♠ ♥♦ ❊st❛❞♦ ❞❡ P❡r♥❛♠❜✉❝♦✳ ❊ss❛ ❛✈❛❧✐❛çã♦ s❡rá r❡s♦❧✈✐❞❛ ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡✱ ❞✉r❛♥t❡ ❞✉❛s ❤♦r❛s ❡ ♥ã♦ ❤❛✈❡rá ❝♦♥s✉❧t❛ ❞❡ ♠❛t❡r✐❛❧ ❞✐❞át✐❝♦ ♥❡♠ ❝❛❧❝✉❧❛❞♦r❛✳ ❆♣ós ❛ ❛♣❧✐❝❛çã♦ ❞❡ss❡ ❡①❛♠❡ ❞✐❛❣♥óst✐❝♦✱ s❡rá ❢❡✐t❛ ✉♠❛ ❛♥á❧✐s❡ ❞❡t❛❧❤❛❞❛ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣❡❧♦s ❡st✉❞❛♥t❡s✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦s ♣♦♥t♦s ❢♦rt❡s ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❛s ♣r✐♥❝✐♣❛✐s ❞✐✜❝✉❧❞❛❞❡s ❡♥❝♦♥tr❛❞❛s ♣♦r ❡❧❡s ♥❛ r❡s♦❧✉çã♦ ❞❛s q✉❡stõ❡s ❡✱ ❝♦♠ ❜❛s❡ ♥❡ss❡s ❞❛❞♦s✱ s❡rá ❡❧❛❜♦r❛❞♦ ♦ ❈❛♣ít✉❧♦2❞❡st❛ ♣r♦♣♦st❛✱ ❝♦♥t❡♥❞♦ ✉♠ ♠❛t❡r✐❛❧ ❞✐❞át✐❝♦ ❛❜♦r❞❛♥❞♦ tó♣✐❝♦s ❛ r❡s♣❡✐t♦
❞❡ ❢✉♥çã♦✱ ❢✉♥çã♦ ❛✜♠ ❡ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ t❡♥❞♦ ❝♦♠♦ ❢♦♥t❡s ♦s ❧✐✈r♦s ❞❛ 1❛ sér✐❡ ❞♦
❡♥s✐♥♦ ♠é❞✐♦✱ ▼❛t❡♠át✐❝❛✱ ❈♦♥t❡①t♦s ❡ ❆♣❧✐❝❛çõ❡s✱ ❞♦ ♣r♦❢❡ss♦r ▲✉✐③ ❘♦❜❡rt♦ ❉❛♥t❡✱ ❋✉♥❞❛♠❡♥t♦s ❞❛ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r✱ ✈♦❧✉♠❡ 1✱ ❞♦ ♣r♦❢❡ss♦r ●❡❧s♦♥ ■❡③③✐✱ ❛❧é♠ ❞♦s
tr❛❜❛❧❤♦s ❞♦ ♣r♦❢❡ss♦r ❊❧♦♥ ▲❛❣❡s ▲✐♠❛✳
❆❧é♠ ❞❡ss❡ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ ♣❛r❛ ❛s ❛✉❧❛s s❡rã♦ r❡❛❧✐③❛❞❛s ♦✜❝✐♥❛s ❝♦♠ ❞✐✈❡rs❛s ❛t✐✈✐❞❛❞❡s ❝♦♠ ✉s♦ ❞❡ ❛❧❣✉♥s ♠❛t❡r✐❛✐s✱ ❡♠ ❡s♣❡❝✐❛❧✱ ♦ ♠✉❧t✐♣❧❛♥♦ ❡ ♦s ❜❧♦❝♦s ❝ú❜✐❝♦s✱ ♥♦ ❧❛❜♦r❛tór✐♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❡s❝♦❧❛✱ ♣❛r❛ q✉❡ ❤❛❥❛ ✉♠ ♠❛✐♦r ✐♥t❡r❡ss❡✱ ♣❛rt✐❝✐♣❛çã♦ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦s ❡st✉❞❛♥t❡s ❝♦♥s✐❣❛♠ t❡r ✉♠❛ ❜♦❛ ❛ss✐♠✐❧❛çã♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ❢✉♥çã♦✳
❋✐♥❛❧♠❡♥t❡✱ ❛♣ós ♦ ❡st✉❞♦ ❞❡ t♦❞♦ ♠❛t❡r✐❛❧ ❡ r❡❛❧✐③❛çã♦ ❞❛s ♦✜❝✐♥❛s✱ s❡rá ❡❧❛❜♦r❛❞♦
❡①❛♠❡ ✐♥✐❝✐❛❧✳
◗✉❡stõ❡s ♣r♦♣♦st❛s ♥♦ ❡①❛♠❡
❞✐❛❣♥óst✐❝♦ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s
❖ ❡①❛♠❡ ❞✐❛❣♥óst✐❝♦ ❛ s❡❣✉✐r ❢♦✐ ❛♣❧✐❝❛❞♦ ❝♦♠ ❛❧✉♥♦s ❞❡ ❞✉❛s t✉r♠❛s ❞❛ 1❛ sér✐❡ ❞♦
❡♥s✐♥♦ ♠é❞✐♦ ❞❛ ❊s❝♦❧❛ ❊st❛❞✉❛❧ ◆❛t❛❧í❝✐❛ ▼❛r✐❛ ❋✐❣✉❡✐r♦❛ ❞❛ ❙✐❧✈❛✱ ❧♦❝❛❧✐③❛❞❛ ♥❛ ❝✐❞❛❞❡ ❞❡ ❙✉r✉❜✐♠✲P❊✳
❖ ♦❜❥❡t✐✈♦ ❞♦ t❡st❡ ❡r❛ ✈❡r✐✜❝❛r ❛s ♣r✐♥❝✐♣❛✐s ❞✐✜❝✉❧❞❛❞❡s ❡♥❝♦♥tr❛❞❛s ♣❡❧♦s ❛❧✉♥♦s ❛♦ s❡ ❞❡♣❛r❛r❡♠ ❝♦♠ s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ ❡♥✈♦❧✈❡♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦✱ ❢✉♥çã♦ ❛✜♠ ❡ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳
❖ ❡①❛♠❡ ♣♦ss✉✐ q✉✐♥③❡ q✉❡stõ❡s✱ ♦♥❞❡ sã♦ ❛❜♦r❞❛❞❛s ♥♦çõ❡s ❜ás✐❝❛s s♦❜r❡ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦✱ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❝♦♠ ❢✉♥çã♦ ❛✜♠ ❡ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳ Pr✐♠❡✐r❛♠❡♥t❡ s❡rã♦ ❛♥❛❧✐s❛❞❛s ❛s r❡s♣♦st❛s ❞♦s ❡st✉❞❛♥t❡s ❡♠ ❝❛❞❛ q✉❡stã♦ ❡ s❡rã♦ ❢❡✐t❛s ♦❜s❡r✈❛çõ❡s s♦❜r❡ ❛s ♣♦ssí✈❡✐s ❝❛✉s❛s ❞❡ ❛❧❣✉♥s ❡rr♦s✳ ❆ s❡❣✉✐r ♣♦❞❡♠♦s ♦❜s❡r✈❛r ❛s q✉❡stõ❡s ❞♦ t❡st❡ ❡✱ ❛♦ ❧❛❞♦ ❞❛s ❛❧t❡r♥❛t✐✈❛s ❞❡ ❝❛❞❛ q✉❡stã♦✱ t❡♠♦s ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ❛ ♠❛r❝❛r❛♠✳
◗✉❡stã♦ ✵✶✿ ❯t✐❧✐③❛♥❞♦ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❛ ❢✉♥çã♦ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦✱ q✉❡ r❡♣r❡s❡♥t❛ ✉♠❛ ❢✉♥çã♦ ❞❡ ❆ ❡♠ ❇✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❛ ✐♠❛✲ ❣❡♠ ❞❛ ❢✉♥çã♦ é ✐❣✉❛❧ ❛✿
✭❛✮{1,0,4} ✸✵✪ ✭❜✮{4,7} ✷✵✪ ✭❝✮{3,5,7} ✹✻✱✼✪
✭❞✮{3,7} ✸✱✸✪ ✭❡✮{5,7} ✵✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❈
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ❛❧✉♥♦s ❛ r❡s♣❡✐t♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦✱ ❞♦♠í♥✐♦✱ ❝♦♥tr❛❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦ ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞♦ ❉✐❛❣r❛♠❛ ❞❡ ✏✢❡❝❤❛s✑✳
❆♣ós ❛ ❛♥á❧✐s❡ ❝✉✐❞❛❞♦s❛ ❞♦s r❡s✉❧t❛❞♦s✱ ✈❡r✐✜❝♦✉✲s❡ q✉❡ ✹✻✱✼✪ ❞♦s ❡st✉❞❛♥t❡s r❡s✲ ♣♦♥❞❡✉ ❝♦rr❡t❛♠❡♥t❡ ❛ q✉❡stã♦ ✭❛❧t❡r♥❛t✐✈❛✭❈✮{3,5,7}✮ ♣♦ré♠✱ ✸✵✪ ❞♦s ❡st✉❞❛♥t❡s
❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ✭❆✮{1,0,4}♦ q✉❡ ❞❡♠♦♥str❛ ❝❧❛r❛♠❡♥t❡ q✉❡ ❤á ✉♠❛ ✐♥✈❡rsã♦
❡♥tr❡ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❞♦♠í♥✐♦ ❡ ✐♠❛❣❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦✳
◗✉❡stã♦ ✵✷✿ ❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦s A = {2,4,6,8} ❡ B = {1,3,5,7}✳ ❙❡❥❛ R ✉♠❛ r❡✲ ❧❛çã♦ ❡♠ A×B✱ t❛❧ q✉❡ R = {(x, y) ∈ A×B, y = x−1} ❛♣r❡s❡♥t❛❞❛ ♣❡❧♦ s❡✉ ❣rá✜❝♦
❝❛rt❡s✐❛♥♦✳ ■❞❡♥t✐✜q✉❡ ❛ ú♥✐❝❛ ❛✜r♠❛çã♦ ❢❛❧s❛✿
✭❛✮ ✭✷✱✶✮ ♣❡rt❡♥❝❡ à r❡❧❛çã♦ ✺✪ ✭❜✮ ✭✽✱✼✮ ♣❡rt❡♥❝❡ à r❡❧❛çã♦ ✷✺✪ ✭❝✮ ✭✹✱✸✮♣❡rt❡♥❝❡ à r❡❧❛çã♦ ✻✱✼✪ ✭❞✮ ✭✻✱✺✮ ♣❡rt❡♥❝❡ à r❡❧❛çã♦ ✶✱✼✪ ✭❡✮ ✭✽✱✾✮ ♣❡rt❡♥❝❡ à r❡❧❛çã♦ ✻✶✱✼✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❊
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❛❧✉♥♦s tê♠ ♥♦çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ r❡❧❛çã♦ ❜✐♥ár✐❛ ❡ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ♣♦♥t♦s ❞❡ ✉♠ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❡♠ ✉♠ s✐st❡♠❛ ❝❛rt❡s✐❛♥♦ ♦rt♦❣♦♥❛❧✳ ❆♣ós ❛ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s✱ ✈❡r✐✜❝♦✉✲s❡ q✉❡ ✻✶✱✼✪ ❞♦s ❡st✉❞❛♥t❡s r❡s♣♦♥❞❡✉ ❝♦rr❡t❛♠❡♥t❡ ❛ q✉❡stã♦ ✭❛❧t❡r♥❛t✐✈❛ ✭❊✮ ✭✽✱✾✮ ♣❡rt❡♥❝❡ à r❡❧❛çã♦✮ ♣♦ré♠ ✷✺✪ ❞♦s ❡st✉❞❛♥t❡s ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ✭❇✮ ✭✽✱✼✮ ♣❡rt❡♥❝❡ à r❡❧❛çã♦✱ q✉❡ é ✉♠❛ ❛✜r♠❛✲ çã♦ ✈❡r❞❛❞❡✐r❛✱ ♣♦ré♠ ❡ss❡ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ♥ã♦ ♣❡r❝❡❜❡✉ ❛ ❧❡✐ ❞❛ r❡❧❛çã♦R ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞♦ ❣rá✜❝♦✳
◗✉❡stã♦ ✵✸✿ ❩❡r♦ ♦✉ r❛✐③ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ♣♦r ❞❡✜♥✐çã♦✱ é ♦ ✈❛❧♦r ❞❡ x q✉❡ ❛♥✉❧❛ ❛ ❢✉♥çã♦✳ ❉❛❞❛ ❛ ❢✉♥çã♦ f(x) = −2x+ 6✳ ◗✉❛❧ ❞♦s ❝♦♥❥✉♥t♦s ❝♦♥té♠ ♦ ③❡r♦ ♦✉ r❛✐③ ❞❛
❢✉♥çã♦ f ❄
✭❛✮{−3} ✷✻✱✼✪ ✭❜✮ {−2} ✶✽✱✸✪ ✭❝✮ {−1} ✶✺✪
✭❞✮{0} ✶✸✱✸✪ ✭❡✮ {3} ✷✻✱✼✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❊
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ♣♦ss✉❡♠ ♥♦çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ③❡r♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❛✜♠ ❡✱ ❛tr❛✈és ❞❛ r❡s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ 1♦ ❣r❛✉✱ ♣✉❞❡ss❡♠
❝❛❧❝✉❧❛r ❡ss❡ ✈❛❧♦r✳
❱❡r✐✜❝♦✉✲s❡ ♥♦s r❡s✉❧t❛❞♦s q✉❡ ✷✻✱✼✪ ❞♦s ❡st✉❞❛♥t❡s r❡s♣♦♥❞❡✉ ❝♦rr❡t❛♠❡♥t❡ ❛ q✉❡stã♦ ✭❛❧t❡r♥❛t✐✈❛ ✭❊✮{3}✮ ♣♦ré♠✱ ✷✻✱✻✪ ❞♦s ❡st✉❞❛♥t❡s ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ✭❆✮{−3}✳
❱❡r✐✜❝❛✲s❡ q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ❡st✉❞❛♥t❡s ♥ã♦ ❡♥t❡♥❞❡✉ ❡♥✉♥❝✐❛❞♦ ❞❛ q✉❡stã♦✱ ❥á q✉❡ ♥♦ ♠❡s♠♦ ❥á ❛♣❛r❡❝❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ③❡r♦ ❞❛ ❢✉♥çã♦✳ ❖❜s❡r✈❛✲s❡ t❛♠❜é♠ q✉❡ ♣❛rt❡ ❞♦s ❡st✉✲ ❞❛♥t❡s ❝♦♥❢✉♥❞✐✉ ♦s s✐♥❛✐s ❛♦ ❝❛❧❝✉❧❛r❡♠ ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦f ♥♦ ♣♦♥t♦ ❞❡ ❛❜s❝✐ss❛x=−3✳
◗✉❡stã♦ ✵✹✿ ❖ s❛❧ár✐♦ ❞❡ ✉♠ ✈❡♥❞❡❞♦r é ❝♦♠♣♦st♦ ❞❡ ✉♠❛ ♣❛rt❡ ✜①❛ ♥♦ ✈❛❧♦r ❞❡ ❘✩ ✽✵✵✱✵✵✱ ♠❛✐s ✉♠❛ ♣❛rt❡ ✈❛r✐á✈❡❧ ❞❡ ✶✵✪ s♦❜r❡ ♦ ✈❛❧♦r ❞❡ s✉❛s ✈❡♥❞❛s ♥♦ ♠ês✳ ❈❛s♦ ❡❧❡ ❝♦♥s✐❣❛ ✈❡♥❞❡r ❘✩✹✺✵✵✱✵✵✱ q✉❛❧ s❡rá ♦ ✈❛❧♦r ❞❡ s❡✉ s❛❧ár✐♦❄
✭❛✮ ❘✩✽✹✺✱✵✵ ✷✸✱✸✪ ✭❜✮ ❘✩✶✵✵✵✱✵✵ ✺✪ ✭❝✮ ❘✩✶✶✵✵✱✵✵ ✽✱✸✪ ✭❞✮ ❘✩✶✷✺✵✱✵✵ ✺✵✪ ✭❡✮ ❘✩✶✺✵✵✱✵✵ ✽✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❉
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥s❡❣✉✐r✐❛♠ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ❡①♣r❡ssã♦ q✉❡ r❡❧❛❝✐♦♥❛ss❡ ❛s ✈❛r✐á✈❡✐s✱ s❛❧ár✐♦ ❡ t♦t❛❧ ❞❡ ✈❡♥❞❛s✱ ❛ ✜♠ ❞❡ ❡♥❝♦♥tr❛r ✉♠❛ ❢✉♥çã♦ ❡♥tr❡ ❡ss❛s ✈❛r✐á✈❡✐s ❡✱ ✜♥❛❧♠❡♥t❡✱ ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❡ss❛ ❢✉♥çã♦ ❡♠ ❞❡t❡r♠✐✲ ♥❛❞♦ ♣♦♥t♦✳ ❱❡r✐✜❝♦✉✲s❡ ♥♦s r❡s✉❧t❛❞♦s q✉❡ ✺✵✪ ❞♦s ❡st✉❞❛♥t❡s r❡s♣♦♥❞❡✉ ❝♦rr❡t❛♠❡♥t❡ ❛ q✉❡stã♦ ✭❛❧t❡r♥❛t✐✈❛ ✭❉✮ ❘✩✶✷✺✵✱✵✵✮ ♣♦ré♠✱ ✷✸✱✸✪ ❞♦s ❡st✉❞❛♥t❡s ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ✭❆✮ ❘✩✽✹✺✱✵✵✳ P❡r❝❡❜❡✲s❡ q✉❡ ❡ss❡s ❛❧✉♥♦s ❝❛❧❝✉❧❛r❛♠ ❞❡ ❢♦r♠❛ ✐♥❝♦r✲ r❡t❛ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ✶✵✪ s♦❜r❡ ♦ t♦t❛❧ ❞❡ ✈❡♥❞❛s ✭❘✩ ✹✺✵✵✱✵✵✮ ❡♥❝♦♥tr❛♥❞♦ ❘✩✹✺✱✵✵✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ✶✪ ❞❡ ❘✩✹✺✵✵✱✵✵ ✱ ❧❡✈❛♥❞♦ ❡ss❛ ♣❛r❝❡❧❛ ❞❡ ❡st✉❞❛♥t❡s ❛ ❝♦♠❡t❡r ❡ss❡ ❡q✉í✈♦❝♦✳
❯♠❛ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛çã♦ ❛ s❡r ❢❡✐t❛ é q✉❡✱ ❝♦♠♦ ♦ ♣r♦❜❧❡♠❛ s❡ tr❛t❛ ❞❡ ✉♠❛ s✐t✉✲ ❛çã♦ ❞♦ ❝♦t✐❞✐❛♥♦ ❞♦s ❡st✉❞❛♥t❡s✱ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦s ❢♦✐ ❜❡♠ ♠❛✐♦r q✉❡ ❡♠ ♦✉tr❛s q✉❡stõ❡s✳
◗✉❡stã♦ ✵✺✿ ❉❛❞❛ ❛ ❢✉♥çã♦ ❞♦ ✶♦ ❣r❛✉ f(x) = 2x−5✳ ❉❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞❛ ❡①♣r❡ssã♦
f(−3) +f(3)
f(0) .
✭❛✮1 ✷✶✱✼✪ ✭❜✮2 ✷✸✱✸✪ ✭❝✮3 ✹✵✪
✭❞✮4 ✺✪ ✭❡✮5 ✶✵✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❇
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥s❡❣✉✐r✐❛♠ ❝❛❧❝✉❧❛r ♦ ✈❛✲ ❧♦r ❞❛ ❢✉♥çã♦ ❛♦ r❡s♦❧✈❡r ❛ ❡①♣r❡ssã♦ ❞❛❞❛✳ ❱❡r✐✜❝♦✉✲s❡ ♥♦s r❡s✉❧t❛❞♦s q✉❡ ✷✸✱✸✪ ❞♦s ❡st✉❞❛♥t❡s r❡s♣♦♥❞❡r❛♠ ❝♦rr❡t❛♠❡♥t❡ ❛ q✉❡stã♦ ✭❛❧t❡r♥❛t✐✈❛ ✭❇✮ ✷✮ ♣♦ré♠✱ ✹✵✪ ❞♦s
❡st✉❞❛♥t❡s✱ ❛ ♠❛✐♦r✐❛✱ ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ✭❈✮ ✸ ♦ q✉❡ ❝♦♠♣r♦✈❛ ❛ ❞✐✜❝✉❧❞❛❞❡ q✉❡ ❡ss❡s ❛❧✉♥♦s ♣♦ss✉❡♠ ❡♠ ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❡ ❡①♣r❡ssõ❡s ♥✉♠ér✐❝❛s ❝♦rr❡t❛♠❡♥t❡✳
◗✉❡stã♦ ✵✻✿ P❛r❛ tr❛♥s❢♦r♠❛r ❣r❛✉s ❋❛❤r❡♥❤❡✐t ❡♠ ❣r❛✉s ❈❡❧s✐✉s ✉s❛✲s❡ ❛ ❢ór♠✉❧❛✿
C = 5(F −32) 9 ,
♦♥❞❡F é ♦ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❋❛❤r❡♥❤❡✐t ❡ C é ♦ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❈❡❧s✐✉s✳ ◗✉❛❧ ✈❛❧♦r✱ ❡♠ ❣r❛✉s ❈❡❧s✐✉s✱ ❝♦rr❡s♣♦♥❞❡rá ❛95 ❣r❛✉s ❋❛❤r❡♥❤❡✐t ❄
✭❛✮0◦C ✶✱✼✪ ✭❜✮10◦C ✶✶✱✼✪ ✭❝✮ 15◦C ✶✸✱✸✪
✭❞✮25◦C ✶✺✪ ✭❡✮35◦C ✺✽✱✸✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❊
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥s❡❣✉✐r✐❛♠ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ ❞❛❞❛ ❡♠ ❝❡rt♦ ♥ú♠❡r♦✳ ❱❡r✐✜❝♦✉✲s❡ ♥♦s r❡s✉❧t❛❞♦s q✉❡ ✺✽✱✸✪ ❞♦s ❡s✲ t✉❞❛♥t❡s r❡s♣♦♥❞❡r❛♠ ❝♦rr❡t❛♠❡♥t❡ ❛ q✉❡stã♦ ✭❛❧t❡r♥❛t✐✈❛ ✭❊✮ 35♦C)✱ ♠♦str❛♥❞♦ q✉❡
❤♦✉✈❡ ✉♠❛ ♠❡❧❤♦r❛ ♥♦s í♥❞✐❝❡s ❞❡ ❛❝❡rt♦s✱ ♣♦ré♠✱ ✶✺✪ ❞♦s ❡st✉❞❛♥t❡s ❛ss✐♥❛❧♦✉ ❛ ❛❧t❡r✲ ♥❛t✐✈❛ ✭❉✮ 25♦C✱ ♦ q✉❡ ♥♦s ❧❡✈❛ ❛ ❝r❡r q✉❡ r❡❛❧✐③❛r❛♠ ✐♥❝♦rr❡t❛♠❡♥t❡ ❛ s✉❜tr❛çã♦ ❡♥tr❡ ♣❛rê♥t❡s❡s ✭❋✲✸✷✮✱ ❝❤❡❣❛♥❞♦ ❛♦ r❡s✉❧t❛❞♦ ✹✺✱ ❧❡✈❛♥❞♦ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛♦ ❡rr♦✳
◗✉❡stã♦ ✵✼✿ ❆❞♠✐t✐♥❞♦✲s❡ q✉❡ f(x) = mx+n ♣♦ss✉✐ 3❝♦♠♦ ③❡r♦ ❡ f(1) = −8✳ ◗✉❛✐s
♦s ✈❛❧♦r❡s ❞❡ m ❡ n ❄
✭❛✮m= 4 ❡ n =−12 ✸✻✱✼✪ ✭❜✮ m=−4 ❡n = 10 ✶✻✱✼✪
✭❝✮m= 3 ❡n = 4 ✷✻✱✼✪ ✭❞✮ m= 14 ❡ n = 1 ✻✱✼✪
✭❡✮m = 4 ❡ n=−10 ✶✸✱✸✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❆
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥s❡❣✉✐r✐❛♠ ❞❡t❡r♠✐♥❛r ✉♠❛ ❢✉♥çã♦ ❛✜♠ ❛tr❛✈és ❞❡ ❞♦✐s ❞❡ s❡✉s ♣♦♥t♦s✳❱❡r✐✜❝♦✉✲s❡ ♥♦s r❡s✉❧t❛❞♦s q✉❡ ✸✻✱✼✪ ❞♦s ❡s✲ t✉❞❛♥t❡s r❡s♣♦♥❞❡r❛♠ ❝♦rr❡t❛♠❡♥t❡ ✭❛❧t❡r♥❛t✐✈❛ ✭❆✮ m = 4 ❡ n =−12✮ ♣♦ré♠✱ ✷✻✱✼✪
❞♦s ❡st✉❞❛♥t❡s ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ✭✭❈✮ m = 3 ❡ n = 4✮ ♦ q✉❡ ❞❡♠♦♥str♦✉ ❛
❡♥♦r♠❡ ❞✐✜❝✉❧❞❛❞❡ q✉❡ ❡ss❡s ❛❧✉♥♦s tê♠ ❡♠ r❡s♦❧✈❡r ✉♠ s✐♠♣❧❡s s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞♦
1♦ ❣r❛✉✳
◗✉❡stã♦ ✵✽✿ ❆ r❡♣r❡s❡♥t❛çã♦ ❝❛rt❡s✐❛♥❛ ❞❛ ❢✉♥çã♦ y = ax2 + bx + c é ❛ ♣❛rá❜♦❧❛
❛❜❛✐①♦✳ ❚❡♥❞♦ ❡♠ ✈✐st❛ ❡ss❡ ❣rá✜❝♦✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡✿
✭❛✮a <0✱ b <0 ❡ c >0 ✶✵✪ ✭❜✮ a >0✱ b >0 ❡ c <0 ✸✸✱✸✪
✭❝✮a >0✱b >0 ❡ c >0 ✶✸✱✸✪ ✭❞✮ a <0✱ b >0 ❡ c <0 ✷✶✱✼✪
✭❡✮a <0✱ b >0❡ c >0 ✷✶✱✼✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❊
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥s❡❣✉✐r✐❛♠ ✐❞❡♥t✐✜❝❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞❡ s❡✉s ♣❛râ♠❡tr♦sa✱ b ❡ c ✳ ❱❡r✐✜❝♦✉✲s❡ ♥♦s r❡s✉❧t❛❞♦s q✉❡ ✷✶✱✼✪ ❞♦s ❡st✉❞❛♥t❡s r❡s♣♦♥❞❡r❛♠ ❝♦rr❡t❛♠❡♥t❡ ✭❛❧t❡r♥❛t✐✈❛ ✭❊✮ a <0✱ b > 0 ❡ c > 0✮✱ ♣♦ré♠✱ ✸✸✱✸✪✭♠❛✐♦r✐❛✮ ❞♦s ❡st✉❞❛♥t❡s ❛ss✐♥❛✲
❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ✭❇✮ a > 0✱ b >0 ❡ c < 0✱ ❞❡♠♦♥str❛♥❞♦ q✉❡ ❡ss❡s ❡st✉❞❛♥t❡s ♥ã♦
♣♦ss✉❡♠ s❡q✉❡r ❛ ♥♦çã♦ ♠❛✐s ❜ás✐❝❛ ♥❛ ❝♦♥str✉çã♦ ❞♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ q✉❡ é ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ ❝♦♥❝❛✈✐❞❛❞❡ ❞❛ ♣❛rá❜♦❧❛ ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞♦ ♣❛râ♠❡tr♦a✳
◗✉❡stã♦ ✵✾✿ ❖ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠ ♣r♦❥ét✐❧✱ ❧❛♥ç❛❞♦ ♣❛r❛ ❝✐♠❛ ✈❡rt✐❝❛❧♠❡♥t❡✱ é ❞❡s✲ ❝r✐t♦ ♣❡❧❛ ❡q✉❛çã♦ y = −40x2 + 200x✳ ❖♥❞❡ y r❡♣r❡s❡♥t❛ ❛ ❛❧t✉r❛✱ ❡♠ ♠❡tr♦s✱ ❛t✐♥❣✐❞❛
♣❡❧♦ ♣r♦❥ét✐❧ ❡x r❡♣r❡s❡♥t❛ ♦ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦✱ ❡♠ s❡❣✉♥❞♦s✳ ❆ ❛❧t✉r❛ ♠á①✐♠❛ ❛t✐♥❣✐❞❛ ❡ ♦ t❡♠♣♦ q✉❡ ❡ss❡ ♣r♦❥ét✐❧ ♣❡r♠❛♥❡❝❡ ♥♦ ❛r ❝♦rr❡s♣♦♥❞❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛✿
✭❛✮6,25m✱ 5s ✶✶✱✼✪ ✭❜✮ 250m✱ 0s ✺✪ ✭❝✮250m✱5s ✻✵✪ ✭❞✮250m✱ 200s ✷✵✪ ✭❡✮ 10000m✱ 5s ✸✱✸✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❈
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥s❡❣✉✐r✐❛♠ ❞❡t❡r♠✐♥❛r ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s❡✉ ✈❛❧♦r ♠á✲ ①✐♠♦✳ ❱❡r✐✜❝♦✉✲s❡ ♥♦s r❡s✉❧t❛❞♦s q✉❡ ✻✵✪ ❞♦s ❡st✉❞❛♥t❡s r❡s♣♦♥❞❡r❛♠ ❝♦rr❡t❛♠❡♥t❡ ✭❛❧t❡r♥❛t✐✈❛ ✭❈✮ ✷✺✵♠✱✺s✮✱ ♣♦ré♠✱ ✷✵✪ ❞♦s ❡st✉❞❛♥t❡s ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ✭❉✮✷✺✵♠✱✷✵✵s✱ ❞❡♠♦♥str❛♥❞♦ q✉❡ ❡ss❡s ❡st✉❞❛♥t❡s ❝♦♥❢✉♥❞✐r❛♠✲s❡ ❛♦ ❝❛❧❝✉❧❛r ❛s ❝♦♦r✲ ❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ ❞❛ ❢✉♥çã♦ ❞❛❞❛ ❡ ✴♦✉ t✐✈❡r❛♠ ❞✐✜❝✉❧❞❛❞❡s q✉❛♥t♦ ❛ ✐♥t❡r♣r❡t❛çã♦ ❞♦ ❡♥✉♥❝✐❛❞♦ ❞♦ ♣r♦❜❧❡♠❛✳
◗✉❡stã♦ ✶✵✿ ◗✉❛✐s ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ ❞❛ ♣❛rá❜♦❧❛ ❝♦rr❡s♣♦♥❞❡♥t❡ à ❢✉♥çã♦ y= (x−2)2+ 2 ❄
✭❛✮(−2,−2) ✶✵✪ ✭❜✮ (−2,0) ✶✶✱✼✪ ✭❝✮(−2,2) ✹✶✱✼✪
✭❞✮(2,−2) ✶✽✱✸✪ ✭❡✮ (2,2) ✶✽✱✸✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❊
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥❤❡❝❡♠ ❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳ ❱❡r✐✜❝♦✉✲s❡ ♥♦s r❡s✉❧t❛❞♦s q✉❡ ✶✽✱✸✪ ❞♦s ❡st✉❞❛♥t❡s r❡s✲ ♣♦♥❞❡r❛♠ ❝♦rr❡t❛♠❡♥t❡ ✭❛❧t❡r♥❛t✐✈❛ ✭❊✮✭✷✱✷✮✮✱ ♣♦ré♠✱ ✹✶✱✼✪ ❞♦s ❡st✉❞❛♥t❡s✭♠❛✐♦r✐❛✮ ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ✭❈✮ ✭✲✷✱✷✮✱ ♣♦✐s ❢♦r❛♠ ✏❣✉✐❛❞♦s✑ ❛♣❡♥❛s ♣❡❧♦s ✈❛❧♦r❡s ❞♦s ❝♦❡✲ ✜❝✐❡♥t❡s ✈✐s✉❛❧✐③❛❞♦s ♥❛ ❢✉♥çã♦✳
◗✉❡stã♦ ✶✶✿ ◗✉❛✐s sã♦ ♦s ③❡r♦s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❛ s❡❣✉✐rf(x) = 2x2−5x+ 2❄
✭❛✮ 1
2 ❡2 ✸✸✱✸✪ ✭❜✮ 1
2 ❡ 3 ✸✵✪ ✭❝✮
1
2 ✶✺✪
✭❞✮0 ❡ 3 ✶✵✪ ✭❡✮ 0❡ 2 ✺✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❆
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥s❡❣✉✐r✐❛♠ ❝❛❧❝✉❧❛r ♦s ③❡r♦s ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳ ❱❡r✐✜❝♦✉✲s❡ ♥♦s r❡s✉❧t❛❞♦s q✉❡ ✸✸✱✸✪ ❞♦s ❡st✉❞❛♥t❡s r❡s♣♦♥✲ ❞❡r❛♠ ❝♦rr❡t❛♠❡♥t❡ ✭❛❧t❡r♥❛t✐✈❛ ✭❆✮1
2 ❡ ✷✮✱ ♣♦ré♠✱ ✸✵✪ ❞♦s ❡st✉❞❛♥t❡s ❛ss✐♥❛❧❛r❛♠ ❛
✭❛❧t❡r♥❛t✐✈❛ ✭❇✮1
2 ❡ ✸✮✱ ♣♦✐s s❡ ❝♦♥❢✉♥❞✐r❛♠ ❛♦ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ❞♦ 2♦ ❣r❛✉✳
◗✉❡stã♦ ✶✷✿ ◆❛ ✜❣✉r❛ ❛ s❡❣✉✐r t❡♠✲s❡ ✉♠ q✉❛❞r❛❞♦ ✐♥s❝r✐t♦ ❡♠ ♦✉tr♦ q✉❛❞r❛❞♦✳ P♦❞❡✲s❡ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ✐♥t❡r♥♦✱ s✉❜tr❛✐♥❞♦✲s❡ ❞❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ❡①t❡r♥♦ ❛s ár❡❛s ❞♦s ✹ tr✐â♥❣✉❧♦s✳ ❆♣ós ❡ss❡ ❝á❧❝✉❧♦✱ ✈❡r✐✜❝❛✲s❡ q✉❡ ❡ss❛ ár❡❛ r❡s✉❧t❛♥t❡ A é ❢✉♥çã♦ ❞❛ ♠❡❞✐❞❛x✳ ❖ ✈❛❧♦r ♠í♥✐♠♦ ❞❡A é✿
✭❛✮16cm ✸✻✱✼✪ ✭❜✮ 24cm ✶✻✱✼✪ ✭❝✮ 28cm ✶✵✪ ✭❞✮32cm ✸✶✱✼✪ ✭❡✮ 48cm ✺✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❉
❆ q✉❡stã♦ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥s❡❣✉✐r✐❛♠ ❞❡t❡r♠✐♥❛r ✉♠❛ ❡①♣r❡ssã♦ ❛❧❣é❜r✐❝❛ ❛♦ t❡♥t❛r ❡♥❝♦♥tr❛r ♦ ✈❛❧♦r ❞❛ ár❡❛ ❤❛❝❤✉r❛❞❛ ❞❛ ✜❣✉r❛ ❞❛❞❛✳ ❱❡r✐✜❝♦✉✲s❡ ♥♦s r❡s✉❧t❛❞♦s q✉❡ ✸✶✱✼✪ ❞♦s ❡st✉❞❛♥t❡s r❡s♣♦♥❞❡r❛♠ ❝♦rr❡t❛♠❡♥t❡ ✭❛❧t❡r✲ ♥❛t✐✈❛ ✭❉✮✸✷❝♠✮✱ ♣♦ré♠✱ ✸✻✱✼✪✭♠❛✐♦r✐❛✮ ❞♦s ❡st✉❞❛♥t❡s ❛ss✐♥❛❧❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ✭❆✮✶✻ ❝♠ ♣♦✐s s❡ ❝♦♥❢✉♥❞✐r❛♠ ❛♦ ❝❛❧❝✉❧❛r ❛ ár❡❛ s♦❧✐❝✐t❛❞❛ ❡ ♣❡❧♦ ♥ã♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ❝♦♠♦ ❞❡t❡r♠✐♥❛r ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳
❚❡①t♦ ❛ s❡r ✉t✐❧✐③❛❞♦ ♥❛s ♣ró①✐♠❛s ❞✉❛s q✉❡stõ❡s✿
❯♠ ❝♦r♣♦ ❧❛♥ç❛❞♦ ❞♦ s♦❧♦ ✈❡rt✐❝❛❧♠❡♥t❡ ♣❛r❛ ❝✐♠❛ t❡♠ ♣♦s✐çã♦ ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦ ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦ f(t) = 40t−5t2 ♦♥❞❡ ❛ ❛❧t✉r❛ ❢✭t✮ é ❞❛❞❛ ❡♠ ♠❡tr♦s
❡ ♦ t❡♠♣♦ t é ❞❛❞♦ ❡♠ s❡❣✉♥❞♦s✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❡ss❛s ✐♥❢♦r♠❛çõ❡s r❡s♣♦♥❞❛ ❛s q✉❡stõ❡s ✶✸ ❡ ✶✹ ✳
◗✉❡stã♦ ✶✸✿ ◗✉❛❧ ♦ ✐♥st❛♥t❡ ❞❡ t❡♠♣♦✱ ❡♠ s❡❣✉♥❞♦s✱ ❡♠ q✉❡ ♦ ❝♦r♣♦ ❛t✐♥❣✐✉ ❛ ❛❧t✉r❛ ♠á①✐♠❛❄
✭❛✮2 ✸✱✸✪ ✭❜✮ 3 ✶✺✪ ✭❝✮ 8 ✹✸✱✸✪
✭❞✮4 ✶✵✪ ✭❡✮ 28,3 ✷✽✱✸✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❉
◗✉❡stã♦ ✶✹✿ ❆ ❛❧t✉r❛ ♠á①✐♠❛ ❛t✐♥❣✐❞❛ ♣❡❧♦ ❝♦r♣♦✱ ❡♠ ♠❡tr♦s✱ ❢♦✐ ❞❡✿ ✭❛✮80 ✷✺✪ ✭❜✮ 40 ✸✵✪ ✭❝✮ 60 ✶✺✪
✭❞✮30 ✶✸✱✸✪ ✭❡✮ 20 ✶✻✱✼✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❆
❆ q✉❡stõ❡s ✶✸ ❡ ✶✹ t✐♥❤❛♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥s❡❣✉✐r✐❛♠ ❞❡t❡r♠✐♥❛r ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ ❞❡ ✉♠❛ ♣❛rá❜♦❧❛✳ ❖ ♣❡r❝❡♥t✉❛❧ ❞❡ ❡rr♦s ❥á ❢♦✐ ❝♦♠❡♥t❛❞❛ ❡♠ q✉❡stõ❡s ❛♥t❡r✐♦r❡s ❡♥✈♦❧✈❡♥❞♦ ♦ ♠❡s♠♦ tó♣✐❝♦✳
◗✉❡stã♦ ✶✺✿ ❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦f(x) = −x2−6x−9 ✐♥t❡rs❡❝t❛ ♦ ❡✐①♦ x ❡♠✿
✭❛✮x′
= 1 ❡ x′′
= 1 ✺✪ ✭❜✮ x′
=−3 ❡x′′
=−3 ✸✻✱✼✪
✭❝✮x′
= 1 ❡ x′′
=−3 ✶✽✱✸✪ ✭❞✮ x′
=−1 ❡x′′
= 3 ✷✺✪
✭❡✮x′
= 1 ❡ x′′
=−1 ✶✺✪
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛ ✿ ❇
❆ q✉❡stã♦ ✶✺ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✈❡r✐✜❝❛r s❡ ♦s ❡st✉❞❛♥t❡s ❝♦♥s❡❣✉❡♠ ❞❡t❡r♠✐♥❛r ♦s ③❡r♦s ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❡rr♦s ❢♦✐ s❡♠❡❧❤❛♥t❡ à q✉❡stã♦ ❛♥t❡r✐♦r s♦❜r❡ ♦ ♠❡s♠♦ tó♣✐❝♦ ❡ ♦ r❡s✉❧t❛❞♦ ❥á ❢♦✐ ❝♦♠❡♥t❛❞♦ ♥❡ss❛s q✉❡stõ❡s✳
✶✳✶ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s
✶✳✶✳✶ P♦r❝❡♥t❛❣❡♠ ❞❡ ❛❝❡rt♦s ♣♦r q✉❡stã♦
❆ t❛❜❡❧❛ ❡ ♦ ❣rá✜❝♦ ❛❜❛✐①♦ ♠♦str❛♠ ♦ ♣♦r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ✭❞❡ ✉♠ t♦t❛❧ ❞❡ ✻✵ ♣❛rt✐✲ ❝✐♣❛♥t❡s✮ q✉❡ ❛❝❡rt❛r❛♠ ❝❛❞❛ q✉❡stã♦✳ ❖ ♣♦r❝❡♥t✉❛❧ ♠é❞✐♦ ❞❡ ❛❝❡rt♦s ♣♦r q✉❡stã♦ ❢♦✐ ❞❡ ✸✻✪ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✳
◗✉❡stã♦
P♦r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦s✵✶ ✹✻✱✼✪
✵✷ ✻✶✱✼✪
✵✸ ✷✻✱✼✪
✵✹ ✺✵✪
✵✺ ✷✸✱✸✪
✵✻ ✺✽✱✸✪
✵✼ ✸✻✱✼✪
✵✽ ✷✶✱✼✪
✵✾ ✻✵✪
✶✵ ✶✽✱✸✪
✶✶ ✸✸✱✸✪
✶✷ ✸✶✱✼✪
✶✸ ✶✵✪
✶✹ ✷✺✪
✶✺ ✸✻✱✼✪
✶✳✶✳✷ ●rá✜❝♦ ❝♦♠ ♦ ♥ú♠❡r♦ ❞❡ ❛❝❡rt♦s ♣♦r q✉❡stã♦
❖ ❣rá✜❝♦ ❞❡ ❜❛rr❛s ❛ s❡❣✉✐r ❡①♣r❡ss❛ ♦ ♣♦r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ❛❝❡rt♦✉ ❝❛❞❛ q✉❡stã♦ ❞♦ ❡①❛♠❡ ❞✐❛❣♥óst✐❝♦✳ ❆tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞❡ss❡ ❣rá✜❝♦ s❡rá ❢❡✐t♦ ✉♠ ❧❡✈❛♥t❛♠❡♥t♦ ❞♦s ♣♦♥t♦s ♦♥❞❡ ♦s ❡st✉❞❛♥t❡s ♣♦ss✉❡♠ ♠❛✐♦r❡s ❞✐✜❝✉❧❞❛❞❡s ❛ r❡s♣❡✐t♦ ❞♦s ❝♦♥❝❡✐t♦s ❛❜♦r❞❛✲ ❞♦s✳
❆♣ós ❝r✐t❡r✐♦s❛ ❛♥á❧✐s❡ ❞♦ ❣rá✜❝♦✱ ♣♦❞❡✲s❡ ❝♦♥❝❧✉✐r q✉❡ ❛ q✉❡stã♦ ❝♦♠ ♠❛✐♦r í♥❞✐❝❡ ❞❡ ❛❝❡rt♦s ❢♦✐ ❛ ♥ú♠❡r♦ ✵✷✭✸✼ ❛❝❡rt♦s✮✱ ❡♥q✉❛♥t♦ ✐ss♦✱ ❛ q✉❡stã♦ q✉❡ ♦s ❛❧✉♥♦s ♠❛✐s s❡ ❡q✉✐✈♦❝❛r❛♠ ❢♦✐ ❛ ❞❡ ♥ú♠❡r♦ ✶✸✭✻ ❛❝❡rt♦s✮✳
✶✳✶✳✸ ❚❛❜❡❧❛ ❝♦♠ q✉❡stõ❡s ❡ ❛❧t❡r♥❛t✐✈❛s ❛ss✐♥❛❧❛❞❛s
❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ♦ ♣♦r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ♠❛r❝❛r❛♠ ❝❛❞❛ ❛❧t❡r♥❛t✐✈❛ ♣♦r q✉❡stã♦✳ ❆s ❈é❧✉❧❛s ❡♠ ♥❡❣r✐t♦ r❡♣r❡s❡♥t❛♠ ♦s ❣❛❜❛r✐t♦s ❝♦rr❡t♦s✳
◗✉❡stã♦✴●❛❜❛r✐t♦
❆ ❇ ❈ ❉ ❊✶ ✶✽ ✶✷ ✷✽ ✷ ✵
✷ ✸ ✶✺ ✹ ✶ ✸✼
✸ ✶✻ ✶✶ ✾ ✽ ✶✻
✹ ✶✹ ✸ ✺ ✸✵ ✽
✺ ✶✸ ✶✹ ✷✹ ✸ ✻
✻ ✶ ✼ ✽ ✾ ✸✺
✼ ✷✷ ✶✵ ✶✻ ✹ ✽
✽ ✻ ✷✵ ✽ ✶✸ ✶✸
✾ ✼ ✸ ✸✻ ✶✷ ✷
✶✵ ✻ ✼ ✷✺ ✶✶ ✶✶
✶✶ ✷✵ ✶✽ ✽ ✻ ✼
✶✷ ✷✷ ✶✵ ✻ ✶✾ ✸
✶✸ ✷ ✾ ✷✻ ✻ ✶✼
✶✹ ✶✺ ✶✽ ✾ ✽ ✶✵
✶✺ ✸ ✷✷ ✶✶ ✶✺ ✾
✶✳✶✳✹ ◆ú♠❡r♦ ❞❡ ❛❧✉♥♦s ♣♦r ❣❛❜❛r✐t♦
❖ ❣rá✜❝♦ ❛❜❛✐①♦ ❛♣r❡s❡♥t❛ ✉♠❛ ✈✐sã♦ ❛ r❡s♣❡✐t♦ ❞❛s ❛❧t❡r♥❛t✐✈❛s ❛ss✐♥❛❧❛❞❛s ♣❡❧♦s ❡st✉❞❛♥t❡s ❡♠ ❝❛❞❛ q✉❡stã♦✱ ❞❛♥❞♦ ✉♠❛ ✈✐sã♦ ♠❛✐s ❛♠♣❧❛ s♦❜r❡ ♦s r❡s✉❧t❛❞♦s ❞❛ ❛✈❛❧✐❛çã♦✳
✶✳✶✳✺ ▲✐st❛ ❞♦s ♣♦♥t♦s ❞❡ ♠❛✐♦r❡s ❞✐✜❝✉❧❞❛❞❡s ❛♣r❡s❡♥t❛❞♦s ♣❡❧♦s
❡st✉❞❛♥t❡s
❙❡❣✉❡ ❛❜❛✐①♦ ✉♠❛ t❛❜❡❧❛ ♥❛ q✉❛❧ ❝♦♥st❛♠ ♦s tó♣✐❝♦s q✉❡ ♦s ❡st✉❞❛♥t❡s ❛♣r❡s❡♥t❛r❛♠ ♠❛✐s ❞✐✜❝✉❧❞❛❞❡s ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳ ❊ss❛ t❛❜❡❧❛ ✐rá s❡r✈✐r ❝♦♠♦ ♣❛râ♠❡tr♦ ❛✉①✐❧✐❛r ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❞♦ ❝❛♣ít✉❧♦ ✷✳
◗✉❡stã♦
❮♥❞✐❝❡ ❞❡ ❛❝❡rt♦s ✈❡rs✉s ♦❜❥❡t✐✈♦s ✵✸ ✷✻✱✼✪ X ❈❛❧❝✉❧❛r ♦ ③❡r♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❛✜♠✵✺ ✺✪ X ❉❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞❛ ✐♠❛❣❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦ ❡♠ ✉♠ ♣♦♥t♦
✵✽ ✷✶✱✼✪ X ❆♥❛❧✐s❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ♣❛rá❜♦❧❛ ♣❡❧♦s ♣❛râ♠❡tr♦sa✱b ❡c
✶✵ ✶✽✱✸✪ X ❉❡t❡r♠✐♥❛r ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛
✶✸ ✶✵✪ X ❉❡t❡r♠✐♥❛r ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛
◆♦çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❋✉♥çã♦
◆❡st❛ ♣❛rt❡ ❞❛ ♣r♦♣♦st❛ ❞❡ tr❛❜❛❧❤♦ s❡rá ❛♣r❡s❡♥t❛❞♦ ✉♠ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ q✉❡✱ ♣♦s✲ t❡r✐♦r♠❡♥t❡✱ s❡rá ✉t✐❧✐③❛❞♦ ❡♠ s❛❧❛ ❝♦♠ ♦s ❡st✉❞❛♥t❡s ❞❛s t✉r♠❛s q✉❡ r❡❛❧✐③❛r❛♠ ♦ ❡①❛♠❡ ❞✐❛❣♥óst✐❝♦ ❡♥✈♦❧✈❡♥❞♦ ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❞❡ ❢✉♥çã♦✱ ❢✉♥çã♦ ❛✜♠ ❡ ❢✉♥çã♦ q✉❛❞rá✲ t✐❝❛✱ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❝❛♣ít✉❧♦1✳ ❙❡rá ❞❛❞♦ ê♥❢❛s❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❛♦s tó♣✐❝♦s ❡♥✈♦❧✈❡♥❞♦
✐♥t❡r♣r❡t❛çã♦ ❞❡ ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s✱ ❝r❡s❝✐♠❡♥t♦ ❡ ❞❡❝r❡s❝✐♠❡♥t♦✱ ❢✉♥çã♦ ✐♥❥❡t✐✈❛ ❡ s♦✲ ❜r❡❥❡t✐✈❛✱ t❡♦r❡♠❛ ❞❡ ❝❛r❛❝t❡r✐③❛çã♦ ❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ❢✉♥çã♦ ❛✜♠✱ ❡♥q✉❛♥t♦ ♥♦ ✐t❡♠ ❡♥✈♦❧✈❡♥❞♦ ❢✉♥çã♦ q✉❛❞rát✐❝❛ s❡rã♦ ❡①♣❧♦r❛❞♦s tó♣✐❝♦s ❡♥✈♦❧✈❡♥❞♦ ♣r♦❜❧❡♠❛s ❝♦♠ ❢✉♥çã♦ q✉❛❞rát✐❝❛ q✉❡ ❛❜♦r❞❡♠ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ❣rá✜❝♦s✱ ❝á❧❝✉❧♦ ❞❡ ③❡r♦s ❞❛ ❢✉♥çã♦ ❡ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ✈ért✐❝❡✱ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ♠á①✐♠♦s✴♠í♥✐♠♦s ❡ ❛♥á❧✐s❡ ❞♦ ❣rá✜❝♦ ♣♦r ✐♥t❡r♠é❞✐♦ ❞♦s ♣❛râ♠❡tr♦s r❡❛✐s ❛✱ ❜ ❡ ❝✳
Pr✐♠❡✐r❛♠❡♥t❡ s❡rã♦ ❡①♣♦st❛s ❛❧❣✉♠❛s ✐❞❡✐❛s ❝♦♠♦ ♦ ♣❛r ♦r❞❡♥❛❞♦✱ ♦ s✐st❡♠❛ ❝❛rt❡✲ s✐❛♥♦ ♦rt♦❣♦♥❛❧ ❡ ♦ ❝♦♥❝❡✐t♦ ❜ás✐❝♦ ❞❡ r❡❧❛çã♦ ❜✐♥ár✐❛✳ ❊♠ s❡❣✉✐❞❛✱ ♦s ❡st✉❞❛♥t❡s s❡rã♦ ❡♥❝❛♠✐♥❤❛❞♦s ❛♦ ❧❛❜♦r❛tór✐♦ ❞❡ ▼❛t❡♠át✐❝❛ ♦♥❞❡ ✐rã♦ ❞✐s❝✉t✐r s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ ❛ r❡s✲ ♣❡✐t♦ ❞❛s ❢✉♥çõ❡s✱ tr❛❜❛❧❤❛❞❛s ♥♦ ❝❛♣ít✉❧♦✱ ❡ r❡❛❧✐③❛r ❛t✐✈✐❞❛❞❡s ❞✐✈❡rs❛s ❝♦♠ ♦ ♠✉❧t✐♣❧❛♥♦ ❡ ♦s ❜❧♦❝♦s ❝ú❜✐❝♦s✱ ❢❛❝✐❧✐t❛♥❞♦ ❛ss✐♠ ❛ ✈✐s✉❛❧✐③❛çã♦ ❞❡ ❣rá✜❝♦s ❡ ❛ ❛ss✐♠✐❧❛çã♦ ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s✳
✷✳✶ ❈♦♦r❞❡♥❛❞❛s ❈❛rt❡s✐❛♥❛s
❖ ❛❞❥❡t✐✈♦ ❈❛rt❡s✐❛♥♦ s❡ r❡❢❡r❡ ❛♦ ♠❛t❡♠át✐❝♦ ❡ ✜❧ós♦❢♦ ❢r❛♥❝ês ❘❡♥é ❉❡s❝❛rt❡s q✉❡✱ ❞❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✱ ❞❡s❡♥✈♦❧✈❡✉ ✉♠❛ sí♥t❡s❡ ❞❛ á❧❣❡❜r❛ ❝♦♠ ❛ ❣❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛✳ ❖s s❡✉s tr❛❜❛❧❤♦s ♣❡r♠✐t✐r❛♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ár❡❛s ❝✐❡♥tí✜❝❛s ❝♦♠♦ ❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✱ ♦ ❝á❧❝✉❧♦ ❡ ❛ ❝❛rt♦❣r❛✜❛✳ ❆ ✐❞❡✐❛ ♣❛r❛ ♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ❡♠ ✶✻✸✼ ❡♠ ❞✉❛s ♦❜r❛s ❞❡ ❉❡s❝❛rt❡s✳ ❊♠ ✉♠❛ ❞❡ss❛s ♦❜r❛s✱ ❉✐s❝✉rs♦ s♦❜r❡ ♦ ♠ét♦❞♦✱ ❉❡s❝❛rt❡s ❛♣r❡s❡♥t❛ ❛ ✐❞❡✐❛ ❞❡ ❡s♣❡❝✐✜❝❛r ❛ ♣♦s✐çã♦ ❞❡ ✉♠ ♣♦♥t♦ ♦✉ ♦❜❥❡t♦ ♥✉♠❛ s✉♣❡r❢í❝✐❡✱ ✉s❛♥❞♦ ❞♦✐s ❡✐①♦s q✉❡ s❡ ✐♥t❡rs❡❝t❛♠✳
❯♠ s✐st❡♠❛ ❞❡ r❡❢❡rê♥❝✐❛ ❝♦♥s✐st❡ ❡♠ ✉♠ ♣♦♥t♦ ❞❡ ♦r✐❣❡♠✱ ❞✐r❡çã♦ ❡ s❡♥t✐❞♦✱ q✉❡ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❞❡ ✈ár✐❛s ♠❛♥❡✐r❛s✱ ♣♦ré♠✱ ♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s é ♦ ♠❛✐s ♣ró①✐♠♦ ❞♦ ♠✉♥❞♦ r❡❛❧✱ ♣♦✐s ♥♦s ♣❡r♠✐t❡ ♦❜s❡r✈❛r ❛s ❢♦r♠❛s ❞♦ ♠♦❞♦ ♠❛✐s ❛♣r♦①✐♠❛❞♦ ♣♦ssí✈❡❧ ❞♦ ♥♦ss♦ ❞❡ ❡♥①❡r❣❛r ♦ ✉♥✐✈❡rs♦✳
✷✳✶✳✶ ❙✐st❡♠❛ ❈❛rt❡s✐❛♥♦ ❖rt♦❣♦♥❛❧
❖ ❙✐st❡♠❛ ❢♦r♠❛❞♦ ♣❡❧♦s ❡✐①♦s ❖① ❡ ❖②✱ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❡ ❞❡ ♠❡s♠❛ ♦r✐❣❡♠ ❖ é ❝❤❛♠❛❞♦ ❙✐st❡♠❛ ❈❛rt❡s✐❛♥♦ ❖rt♦❣♦♥❛❧✳ ❆ ✜❣✉r❛ ❛❜❛✐①♦ ✐❧✉str❛ ❡ss❡ s✐st❡♠❛ ❡ ❛ r❡♣r❡✲ s❡♥t❛çã♦ ❞♦s q✉❛tr♦ q✉❛❞r❛♥t❡s ❡♠ q✉❡ ♦ ♠❡s♠♦ ✜❝❛ ❞✐✈✐❞✐❞♦✳
❋✐❣✉r❛ ✷✳✶✿ ❙✐st❡♠❛ ❈❛rt❡s✐❛♥♦ ❖rt♦❣♦♥❛❧
❖ P❧❛♥♦ ❈❛rt❡s✐❛♥♦ é ❞✐✈✐❞✐❞♦ ♣❡❧♦s ❡✐①♦s ♦rt♦❣♦♥❛✐s ❡♠ q✉❛tr♦ r❡❣✐õ❡s ❝❤❛♠❛❞❛s q✉❛❞r❛♥t❡s✳ ❆tr❛✈és ❞❡ss❡ s✐st❡♠❛ ♣♦❞❡♠♦s ❧♦❝❛❧✐③❛r ♣♦♥t♦s ♥♦ ♣❧❛♥♦✳ ◆♦ ❙✐st❡♠❛ ❈❛rt❡✲ s✐❛♥♦ ❝❛❞❛ ♣♦♥t♦ ❞♦ P❧❛♥♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ ♣❛r ♦r❞❡♥❛❞♦(x, y)❡ ✈✐❝❡✲✈❡rs❛✳
❊ss❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❝❤❛♠❛❞❛ ❜✐✉♥í✈♦❝❛ ❡ ♣❡r♠✐t❡ r❡♣r❡s❡♥t❛r ❝♦♥❝❡✐t♦s ❣❡♦♠étr✐❝♦ ❡♠ ❧✐♥❣✉❛❣❡♠ ❛❧❣é❜r✐❝❛ ❡ ✐♥t❡r♣r❡t❛r ❣❡♦♠❡tr✐❝❛♠❡♥t❡ r❡❧❛çõ❡s ❡♥tr❡ ♥ú♠❡r♦s r❡❛✐s✳
❯♠ ♣❛r ♦r❞❡♥❛❞♦ p = (x, y) é ❢♦r♠❛❞♦ ♣♦r ✉♠ ♦❜❥❡t♦ x✱ ❝❤❛♠❛❞♦ ❛ ♣r✐♠❡✐r❛ ❝♦♦r✲ ❞❡♥❛❞❛ ❞❡ p ❡ ✉♠ ♦❜❥❡t♦ y✱ ❝❤❛♠❛❞♦ ❛ s❡❣✉♥❞❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ p✳ ❉♦✐s ♣❛r❡s ♦r❞❡♥❛❞♦s p= (x, y)❡q= (u, v)s❡rã♦ ❝❤❛♠❛❞♦s ✐❣✉❛✐s q✉❛♥❞♦x=u❡y=v✱ ✐st♦ é✱ q✉❛♥❞♦ t✐✈❡r❡♠ ❛ ♠❡s♠❛ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛ ❡ ❛ ♠❡s♠❛ s❡❣✉♥❞❛ ❝♦♦r❞❡♥❛❞❛✳
◆♦ ❣rá✜❝♦ ❛ s❡❣✉✐r✱ t❡♠♦s r❡♣r❡s❡♥t❛❞♦s ♦s ♣♦♥t♦sA= (4,3), B = (1,2), C = (−2,4)✱
D= (−3,−4)❡ E = (3,−3)✳
❋✐❣✉r❛ ✷✳✷✿ P♦♥t♦s ♥♦ P❧❛♥♦
✷✳✶✳✷ Pr♦❞✉t♦ ❈❛rt❡s✐❛♥♦
❖ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ A×B ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦sA ❡B é ♦ ❝♦♥❥✉♥t♦A×B ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s(x, y) ❝✉❥❛ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛ x ♣❡rt❡♥❝❡ ❛ A ❡ ❝✉❥❛ s❡❣✉♥❞❛ ❝♦♦r❞❡♥❛❞❛ y ♣❡rt❡♥❝❡ ❛ B✳ ❙✐♠❜♦❧✐❝❛♠❡♥t❡✿
A×B ={(x, y)/ x∈A ❡ y∈B}. A×B ✭❧ê✲s❡ A ❝❛rt❡s✐❛♥♦ B✮✳
❊①❡♠♣❧♦✿ ❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦sA={1,3} ❡B ={1,3,5}✳ ❉❡t❡r♠✐♥❛r✿
❛✮A×B
P❛r❛ q✉❡ ❞❡t❡r♠✐♥❡♠♦s ♦ ❝♦♥❥✉♥t♦ A×B ❜❛st❛ q✉❡ t♦♠❡♠♦s t♦❞♦s ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s
(x, y) ♦♥❞❡ x ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ A ❡ y ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦B✳ ❉❡ss❡ ♠♦❞♦✱t❡r❡♠♦s✿ A×B ={(1,1),(1,3),(1,5),(3,1),(3,3),(3,5)}.
❜✮B×A
P❛r❛ q✉❡ ❞❡t❡r♠✐♥❡♠♦s ♦ ❝♦♥❥✉♥t♦ B ×A ❜❛st❛ q✉❡ t♦♠❡♠♦s t♦❞♦s ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s
(x, y) ♦♥❞❡ x ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ B ❡ y ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ A✳ ❉❡ss❡ ♠♦❞♦✱ t❡r❡♠♦s✿ B×A={(1,1),(1,3),(3,1),(3,3),(5,1),(5,3)}.
