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APLICAÇÕES DO CÁLCULO DIFERENCIAL E INTEGRAL NO ENSINO MÉDIO

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ P✐❛✉í

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛

Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦❢✐ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚

❆♣❧✐❝❛çõ❡s ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ♥♦ ❊♥s✐♥♦

▼é❞✐♦

❆❧❞❡♥♦r ▲♦♣❡s ▲❡♠♦s ❋✐❧❤♦

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❆❧❞❡♥♦r ▲♦♣❡s ▲❡♠♦s ❋✐❧❤♦

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✿

❆♣❧✐❝❛çõ❡s ❞♦ ❈á❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦

❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ à ❈♦♦r❞❡♥❛çã♦ ■♥st✐t✉❝✐♦✲ ♥❛❧ ❆❝❛❞ê♠✐❝❛ ❞♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐✲ ♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ P✐❛✉í✱ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ♦❢❡r❡❝✐❞♦ ❡♠ ❛ss♦❝✐❛çã♦ ❝♦♠ ❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❖r✐❡♥t❛❞♦r✿

Pr♦❢✳ ❉r✳ P❛✉❧♦ ❆❧❡①❛♥❞r❡ ❆r❛ú❥♦ ❙♦✉s❛

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▲❡♠♦s✱ ❆❧❞❡♥♦r ▲♦♣❡s ❋✐❧❤♦✳

❆♣❧✐❝❛çõ❡s ❞♦ ❝á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦

❆❧❞❡♥♦r ▲♦♣❡s ▲❡♠♦s ❋✐❧❤♦ ✕ ❚❡r❡s✐♥❛✿ ✷✵✶✸✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ P❛✉❧♦ ❆❧❡①❛♥❞r❡ ❆r❛ú❥♦ ❙♦✉s❛✳

✶✳ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s✱ ✉♠ s❡r s✉♣r❡♠♦ ❡ ❛❝✐♠❛ ❞❡ q✉❛❧q✉❡r ✈ã r❡❧✐❣✐ã♦✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s ❆❧❞❡♥♦r ▲❡♠♦s ❡ ▼❛r✐❛ ❋❡r♥❛♥❞❡s✱ ♣❡❧♦ ♦ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❡♠ t✉❞♦ q✉❡ r❡❛❧✐③♦ ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❡ ♥❡ss❛ ❡t❛♣❛ ♥ã♦ ❢♦✐ ❞✐❢❡r❡♥t❡✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ❛♠❛❞❛✱ ❈r✉✐③✐♥❤❛✱ q✉❡ ❡st❡✈❡ s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦ ♠❡ ❞❛♥❞♦ ❢♦rç❛✱ s♦❜r❡ t✉❞♦ ♥❡ss❛ ❢❛s❡ ✜♥❛❧ ❞❛ ❞✐ss❡rt❛çã♦ ❡ ❛♦ ♠❡✉ ❛♠❛❞♦ ✜❧❤♦ ❆❧❞❡♥♦r ❚❡r❝❡✐r♦✱ ♣♦r s❡r ❛ r❛③ã♦ ❞♦ ♠❡✉ ❡①✐st✐r ❡ q✉❡ ♥ã♦ s❛❜❡ ♦ t❛♥t♦ q✉❡ ♠❡ ❞❛r ❢♦rç❛s ♣❛r❛ ❧✉t❛r t♦❞♦s ♦s ❞✐❛s ❡ ✈❡♥❝❡r ❝♦t✐❞✐❛♥❛♠❡♥t❡ ❛ ❜❛t❛❧❤❛ ❞❛ ✈✐❞❛✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ✐r♠ã♦s q✉❡ s❡♠♣r❡ ❡stã♦ ♠❡ ❛♣♦✐❛♥❞♦✱ ❡st❛♥❞♦ ❡❧❡s ♣❡rt♦ ♦✉ ❧♦♥❣❡✱ ❛♦s ♠❡✉s ❛♠✐❣♦s ♣r♦❢❡ss♦r❡s ❞❡ ♠✐♥❤❛ q✉❡r✐❞❛ ❝✐❞❛❞❡ ❞❡ ❆❧t♦s ✲ P✐ q✉❡ t❛♥t♦ ♠❡ ❛❥✉❞❛r❛♠ ❛ ❝♦♥❝❧✉✐r ❡st❡ ♠❡str❛❞♦✱ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞❛ ✶a ❚✉r♠❛ ❞♦ P❘❖❋▼❆❚ ✲ ❯❋P■✱ s❡♠ ❡❧❡s

❞✐✜❝✐❧♠❡♥t❡ ❡st❛r✐❛ ❞✐❣✐t❛♥❞♦ ❡st❛s ❧✐♥❤❛s✳

❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ P❛✉❧♦ ❆❧❡①❛♥❞❡✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ♦ ❛♣♦✐♦ ❞❡❞✐❝❛❞♦s ❛ ♠✐♠✱ s❡♠♣r❡ ❝♦♠ ✉♠❛ ♣❛❧❛✈r❛ ❛♠✐❣❛✱ ✉♠ s♦r✐ss♦ ♥♦ r♦st♦ ❡ ✉♠❛ ✐❞❡✐❛ ♣❛r❛ ❛ ❞✐ss❡rt❛çã♦✱ ❛♣r♦✈❡✐t♦ ♣r❛ ❡st❡♥❞❡r ❛ t♦❞♦ ♦ ❝♦r♣♦ ❞♦❝❡♥t❡ ❞♦ P❘❖❋▼❆❚✲❯❋P■ q✉❡ ♠✐♥✐str❛r❛♠ ❞✐s❝✐♣❧✐♥❛s ♥❛ ♠✐♥❤❛ t✉r♠❛✱ ❛ s❛❜❡r✿ Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❍✉♠❜❡rt♦ ❙♦❛r❡s✱ Pr♦❢✳ ❉r✳ ❏❡✛❡rs♦♥ ▲❡✐t❡✱ Pr♦❢✳ ❉r✳ ◆❡✇t♦♥ ❙❛♥t♦s✱ Pr♦❢✳ ❉r✳ ❏✉r❛♥❞✐r ▲♦♣❡s✱ Pr♦❢✳ ❉r✳ P❛✉❧♦ ❆❧❡①❛♥❞r❡✱ Pr♦❢❛✳ ❉r❛✳ ▲✐❛♥❡ ❙♦❛r❡s✱ Pr♦❢✳ ❉r✳ ❘♦❣❡r P❡r❡s ❡ ✉♠ ❛❣r❛❞❡❝✐♠❡♥t♦ t♦❞♦ ❡s♣❡❝✐❛❧ ❛♦ Pr♦❢✳ ❉r✳ ❏✉s❝❡❧✐♥♦ ❙✐❧✈❛ ♣♦r t♦❞♦ ❛ ❢♦rç❛ q✉❡ ♠❡ ❞❡✉ ❞✉r❛♥t❡ ♦ ❝✉rs♦ ❡ ❛❣♦r❛ ♥❛ ❞✐ss❡rt❛çã♦ ❣❡♥t✐❧♠❡♥t❡ ♠❡ ❝♦✲♦r✐❡♥t❛♥❞♦✳

❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ Pr♦❢✳ ❉r✳ P❛✉❧♦ ❆❧❡①❛♥❞❡✱ Pr♦❢✳ ❉r✳ ❏✉s❝❡❧✐♥♦ ❙✐❧✈❛ ❡ Pr♦❢✳ ❉r✳ ❆❢♦♥s♦ ◆♦r❜❡rt♦ ♣♦r t❡r❡♠ ❝♦❧❛❜♦r❛❞♦ ❝♦♠ ♠✐♥❤❛ ❞✐ss❡rt❛çã♦✳ ❆❣r❛❞❡ç♦ ❛♦ Pr♦❢✳ ▼s❝✳ ❘❛✐♠✉♥❞♦ ◆♦♥❛t♦ ❙♦✉s❛ ♣❡❧❛ ❛❥✉❞❛ ❝♦♠ ♦ ❆❜str❛❝t✳

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✐✐✐ ❆❣r❛❞❡ç♦ ❛ ❙❇▼ ✲ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❛♦ ▼❊❈ ✲ ▼✐♥✐stér✐♦ ❞❛ ❊❞✉❝❛✲ çã♦ ♣♦r ✐❞❡❛❧✐③❛r ❡ r❡❛❧✐③❛r ❡ss❡ ♣r♦❥❡t♦ ❣✐❣❛♥t❡s❝♦ ❡ ❞❡ r❡❧❡✈❛♥t❡s ♣rést✐♠♦s à ❡❞✉❝❛çã♦ ❜r❛s✐❧❡✐r❛✳

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✐✈

✏❖ q✉❡ s❛❜❡♠♦s é ✉♠❛ ❣♦t❛✱ ♦ q✉❡ ♥ã♦ s❛❜❡♠♦s é ✉♠ ♦❝❡❛♥♦✧✳

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❘❡s✉♠♦

❖ ❞♦♠í♥✐♦ ❞♦ ❈á❧❝✉❧♦✱ ❡♠ q✉❛❧q✉❡r ♠♦❞❛❧✐❞❛❞❡ ❞❡ ❡♥s✐♥♦✱ s❡♠ ❞ú✈✐❞❛ ♥❡♥❤✉♠❛ é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡✳ ❊st❛ ❢❡rr❛♠❡♥t❛ ♠❛t❡♠át✐❝❛ q✉❡ ✈✐❛❜✐❧✐③❛ ❛ r❡s♦❧✉çã♦ ❞❡ ♠✉✐t♦s ♣r♦❜❧❡♠❛s é ♦ t❡♠❛ ❞❡ss❡ tr❛❜❛❧❤♦✳ P❛rt✐♥❞♦ ❞❛ ♣r❡♠✐ss❛ q✉❡ ❛ ▼❛t❡♠át✐❝❛ é ✉♠❛ ❧✐♥❣✉❛❣❡♠✱ ♦ ❈á❧❝✉❧♦✱ é ✉♠ ❞♦s ♣✐❧❛r❡s ❢✉♥❞❛♠❡♥t❛✐s ❞❡st❛ ❧✐♥❣✉❛❣❡♠ ❡ ❝♦♠♦ t♦❞♦ r❛♠♦ ❞❛ ♠❛t❡♠át✐❝❛ ❢♦✐ ❝♦♥str✉í❞♦ ❛♦ ❧♦♥❣♦ ❞♦s sé❝✉❧♦s✱ t❡♥❞♦ ✉♠ ♠❛✐♦r ❛✈❛♥ç♦ ❝♦♠ ♦s tr❛❜❛❧❤♦s ❞❡ ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③✳ ❊st❡ tr❛❜❛❧❤♦ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ♠♦str❛r ❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❞♦ ❈á❧❝✉❧♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ tr❛③❡♥❞♦ ❛❧❣✉♠❛s s✉❣❡stõ❡s ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ♣♦❞❡♠ s❡r ❛❜♦r❞❛❞♦s ❡♠ ❝❛❞❛ ✉♠❛ ❞❛s três sér✐❡s ❡ ✉♠ ♦✉tr♦ ♦❜❥❡t✐✈♦ t❛♠❜é♠ é ♦ ❞❡ ♠♦str❛r q✉❡ ❛❧❣✉♥s tó♣✐❝♦s ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ sã♦ t♦t❛❧♠❡♥t❡ ❛❝❡ssí✈❡✐s ❛ ❛❧✉♥♦s q✉❡ ❡st❡❥❛♠ ❝✉rs❛♥❞♦ ♦ ❊♥s✐♥♦ ▼é❞✐♦✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ◆❡✇t♦♥✱ ▲❡✐❜♥✐③✱ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ ❊♥s✐♥♦ ▼é❞✐♦✱ ❈á❧❝✉❧♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳

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❆❜str❛❝t

❚❤❡ ♠❛st❡r② ♦❢ ❈❛❧❝✉❧✉s✱ ✐♥ ❛♥② ❧❡✈❡❧ ♦❢ ❡❞✉❝❛t✐♦♥ ✐s ✉♥❞♦✉❜t❡❞❧② ✈❡r② ✐♠♣♦rt❛♥t✳ ❚❤✐s ♠❛t❤❡♠❛t✐❝❛❧ t♦♦❧ t❤❛t ❡♥❛❜❧❡s t❤❡ r❡s♦❧✉t✐♦♥ ♦❢ ♠❛♥② ♣r♦❜❧❡♠s ✐s t❤❡ s✉❜❥❡❝t ♦❢ t❤✐s ✇♦r❦✳ ❙t❛rt✐♥❣ ❢r♦♠ t❤❡ ♣r❡♠✐s❡ t❤❛t ♠❛t❤❡♠❛t✐❝s ✐s ❛ ❦✐♥❞ ♦❢ ❧❛♥❣✉❛❣❡✱ ❈❛❧❝✉❧✉s✱ ✐s ♦♥❡ ♦❢ t❤❡ ❝♦r♥❡rst♦♥❡s ♦❢ t❤✐s ❧❛♥❣✉❛❣❡ ❛♥❞ ❛s ❛ ❜r❛♥❝❤ ♦❢ ♠❛t❤❡♠❛t✐❝s ✐t ❤❛s ❜❡❡♥ ❜✉✐❧t ♦✈❡r t❤❡ ❝❡♥t✉r✐❡s✱ ❤❛✈✐♥❣ ❛ ♠❛❥♦r ❛❞✈❛♥❝❡ ✐♥ t❤❡ ✇♦r❦ ♦❢ ◆❡✇t♦♥ ❛♥❞ ▲❡✐❜♥✐③✳ ❚❤✐s ✇♦r❦ ❛✐♠s t♦ s❤♦✇ t❤❡ ❛♣♣❧✐❝❛❜✐❧✐t② ♦❢ ❈❛❧❝✉❧✉s ✐♥ ❤✐❣❤ s❝❤♦♦❧✱ ❜r✐♥❣✐♥❣ s♦♠❡ s✉❣❣❡st✐♦♥s ❢♦r ♣r♦❜❧❡♠s t❤❛t ❝❛♥ ❜❡ ❛❞❞r❡ss❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ s❡r✐❡s ♦❢ t❤✐s ❧❡✈❡❧ ♦❢ t❡❛❝❤✐♥❣✳ ❆♥♦t❤❡r ❣♦❛❧ ✐s ❛❧s♦ t♦ s❤♦✇ t❤❛t s♦♠❡ t♦♣✐❝s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❛♥❞ ✐♥t❡❣r❛❧ ❈❛❧❝✉❧✉s ❛r❡ ❢✉❧❧② ❛❝❝❡ss✐❜❧❡ t♦ st✉❞❡♥ts ✇❤♦ ❛r❡ ❡♥r♦❧❧❡❞ ✐♥ ❤✐❣❤ s❝❤♦♦❧✳

❑❡②✇♦r❞s✿ ◆❡✇t♦♥✱ ▲❡✐❜♥✐③✱ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s✱ ❍✐❣❤ ❙❝❤♦♦❧✱ ❈❛❧❝✉❧✉s ✐♥ ❤✐❣❤ s❝❤♦♦❧✳

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❙✉♠ár✐♦

❘❡s✉♠♦ ✈

❆❜str❛❝t ✈✐

✶ ❇r❡✈❡ ❍✐stór✐❝♦ ❞♦ ❈á❧❝✉❧♦ ✹

✶✳✶ ◆❡✇t♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ▲❡✐❜♥✐③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s s♦❜r❡ ❈♦♥t✐♥✉✐❞❛❞❡ ❡ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ✶✵ ✷✳✶ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s ❙♦❜r❡ ❈♦♥t✐♥✉✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✶✳✶ ❈♦♥t✐♥✉✐❞❛❞❡ ❞❡ ✉♠❛ ❋✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✶✳✷ ❆ Pr♦♣r✐❡❞❛❞❡ ❞❛ P❡r♠❛♥ê♥❝✐❛ ❞❡ ❙✐♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✶✳✸ Pr♦♣r✐❡❞❛❞❡s ❞❛s ❋✉♥çõ❡s ❈♦♥tí♥✉❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✶✳✹ ❖ ❚❡♦r❡♠❛ ❞❡ ❲❡✐❡rstr❛ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✳✺ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ■♥t❡r♠❡❞✐ár✐♦ ✭❚❱■✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s ❙♦❜r❡ ❉❡r✐✈❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷✳✶ ❆ ❉❡✜♥✐çã♦ ❞❡ ❉❡r✐✈❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷✳✷ ❉❡r✐✈❛❞❛ ❈♦♥❝❡✐t♦ ●❡♦♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷✳✸ ❱❛❧♦r❡s ❊①tr❡♠♦s ❞❡ ✉♠❛ ❋✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷✳✹ ❖ ❚❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷✳✺ ❖ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✻ ❖ ❚❡st❡ ❞❛ ❉❡r✐✈❛❞❛ Pr✐♠❡✐r❛ ❡ ❞❛ ❉❡r✐✈❛❞❛ ❙❡❣✉♥❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷✳✼ ❖ ▼ét♦❞♦ ❞❛s ❆♣r♦①✐♠❛çõ❡s ❙✉❝❡ss✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✸ ❆♣❧✐❝❛çõ❡s ❞♦ ❈á❧❝✉❧♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ✷✾

✸✳✶ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

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❙✉♠ár✐♦ ✈✐✐✐ ✸✳✷ ❆ ❊q✉❛çã♦x✷ =✷x ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸✳✸ ❖ Pr♦❜❧❡♠❛ ❞♦ ❈♦♥❡ ■♥s❝r✐t♦ ❡♠ ✉♠❛ ❊s❢❡r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✹ ❖ Pr♦❜❧❡♠❛ ❞♦ ❚r✐â♥❣✉❧♦ ❈✐r❝✉♥s❝r✐t♦ ❛ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✹ ❯♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❡q✉❛çã♦ x✷ =✷x ✸✽

✹✳✶ ❊q✉❛çõ❡s ❞♦ t✐♣♦ px =xp✱ ❝♦♠ pPr✐♠♦ ❡ ❮♠♣❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

✹✳✶✳✶ ◆ú♠❡r♦s ❆❧❣é❜r✐❝♦s ❡ ◆ú♠❡r♦s ❚r❛♥s❝❡♥❞❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

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■♥tr♦❞✉çã♦

✧❆ ▼❛t❡♠át✐❝❛ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ é ✉♠❛ ✈❛st❛ ❢♦♥t❡ ❞❡ ❢✉♥❞❛♠❡♥t♦s ❜ás✐❝♦s✱ ♦r❣❛♥✐③❛❞♦s ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❢❛❝✐❧✐t❡ ♦ ❛♣r❡♥❞✐③❛❞♦ ❞♦ ❛❧✉♥♦ t♦r♥❛❞♦✲♦ ❝♦♥s❝✐❡♥t❡ ❞❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ▼❛t❡♠át✐❝❛ ♥♦ ♠✉♥❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♥♦ q✉❛❧ ✈✐✈❡♠♦s✳ ❚❛✐s ❢✉♥❞❛♠❡♥t♦s ♥♦s ❞ã♦ s✉♣♦rt❡ ♣❛r❛ s♦❧✉❝✐♦♥❛r ❜❡❧♦s ❡ út❡✐s ♣r♦❜❧❡♠❛s ❞❡ ♥♦ss♦ ❝♦t✐❞✐❛♥♦✧✳ ✭Pr♦❢❡ss♦r ❏✉s❝❡❧✐♥♦ ❙✐❧✈❛ ❡♠ ♣❛❧❡str❛ ♠✐♥✐str❛❞❛ ❡♠ ✷✺ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✵✼ ♥♦ ❈❡♥tr♦ ❋❡❞❡r❛❧ ❞❡ ❊❞✉❝❛çã♦ ❚❡❝♥♦❧ó❣✐❝❛ ❞♦ ❈❡❛rá ✲ ❈❊❋❊❚❈❊ ✲ ❯♥✐❞❛❞❡ ❉❡s❝❡♥tr❛❧✐③❛❞❛ ❞❡ ❏✉❛③❡✐r♦ ❞♦ ◆♦rt❡✮

❯♠ ❢❛t♦ q✉❡ ❞❡✈❡ s❡r r❡♣❡♥s❛❞♦ é ❛ r❡t♦♠❛❞❛ ❞♦ ❡♥s✐♥♦ ❞♦ ❈á❧❝✉❧♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❤❛❥❛ ✈✐st❛ q✉❡ ❛❧❣✉♥s ♣❡sq✉✐s❛❞♦r❡s ❡♠ ❊♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ tê♠ ❧❡✈❛♥t❛❞♦ ❛ q✉❡stã♦ ❞♦s ❋✉♥❞❛♠❡♥t♦s ❞♦ ❈á❧❝✉❧♦ ♥❡ss❛ ♠♦❞❛❧✐❞❛❞❡ ❞❡ ❡♥s✐♥♦✱ ❞❛❞❛ ❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛s ❝✐ê♥❝✐❛s ❡ t❡❝♥♦❧♦❣✐❛s ♠♦❞❡r♥❛s✳ ➪✈✐❧❛ ✭✶✾✾✶✱ ♣✳✷✮ ❢❛③ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❝♦❧♦❝❛çã♦ ❛ ❡ss❡ r❡s♣❡✐t♦✱ q✉❛♥❞♦ ❛✜r♠❛ q✉❡✿

❖ ❈á❧❝✉❧♦ ✈❡♠ ❞❡s❡♠♣❡♥❤❛♥❞♦ ✉♠ ♣❛♣❡❧ ❞❡ ❣r❛♥❞❡ r❡❧❡✈â♥❝✐❛ ❡♠ t♦❞♦ ♦ ❞❡✲ s❡♥✈♦❧✈✐♠❡♥t♦ ❝✐❡♥tí✜❝♦✲t❡❝♥♦❧ó❣✐❝♦✳ P♦rt❛♥t♦✱ ❞❡s❝❛rtá✲❧♦ ♥♦ ❡♥s✐♥♦ é ❣r❛✈❡✱ ♣♦rq✉❡ ❞❡✐①❛ ❞❡ ❧❛❞♦ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ s✐❣♥✐✜❝❛t✐✈❛ ❡ ❝❡rt❛♠❡♥t❡ ❛ ♠❛✐s r❡✲ ❧❡✈❛♥t❡ ❞❛ ▼❛t❡♠át✐❝❛ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞♦ ❛❧✉♥♦ ♥✉♠ ❝♦♥t❡①t♦ ❞❡ ❡♥s✐♥♦ ♠♦❞❡r♥♦ ❡ ❛t✉❛❧✳

❖s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✭P❈◆s✮✱ r❡❝♦♠❡♥❞❛♠ q✉❡ ♦ ❝✉rrí❝✉❧♦ ❞♦ ❊♥s✐♥♦

▼é❞✐♦ ❞❡✈❡ s❡r ❡str✉t✉r❛❞♦ ❞❡ ♠♦❞♦ ❛ ❛ss❡❣✉r❛r ❛♦ ❛❧✉♥♦ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❛♠♣❧✐❛r ❡ ❛♣r♦❢✉♥❞❛r ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❛❞q✉✐r✐❞♦s ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❢♦r♠❛ ✐♥t❡❣r❛❞❛ ❝♦♠ ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ♦r✐❡♥t❛❞❛ ♣❡❧❛ ♣❡rs♣❡❝t✐✈❛ ❤✐stór✐❝♦✲❝✉❧t✉r❛❧ ♥❛ q✉❛❧ ❡stã♦ ❧✐❣❛❞♦s ♦s t❡♠❛s ❡♠ ❡st✉❞♦✳ ■st♦ é ♣r♦♣♦st♦ ✈✐s❛♥❞♦ ❛ ♣r❡♣❛r❛çã♦ ❞♦ ❛❧✉♥♦ ♣❛r❛ ♦ tr❛❜❛❧❤♦ ❡ ❡①❡r❝í❝✐♦ ❞❛ ❝✐❞❛❞❛♥✐❛ ❡ t❛♠❜é♠ ❛ ❝♦♥t✐♥✉❛çã♦ ❞❡ s❡✉s ❡st✉❞♦s ❡♠ ♥í✈❡✐s s✉♣❡r✐♦r❡s✳

◆❛s ❖r✐❡♥t❛çõ❡s ❈✉rr✐❝✉❧❛r❡s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ✭❇❘❆❙■▲✱ ✷✵✵✽✱ ♣✳ ✻✾✮✱ ♣r❡❝♦♥✐③❛✲ s❡ q✉❡ ♦s ❛❧✉♥♦s ❝♦♥❝❧✉✐♥t❡s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ s❛✐❜❛♠

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❙✉♠ár✐♦ ✷ ✉s❛r ❛ ▼❛t❡♠át✐❝❛ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♣rát✐❝♦s ❞♦ q✉♦t✐❞✐❛♥♦❀ ♣❛r❛ ♠♦❞❡✲ ❧❛r ❢❡♥ô♠❡♥♦s ❡♠ ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦❀ ❝♦♠♣r❡❡♥❞❛♠ q✉❡ ❛ ▼❛t❡♠á✲ t✐❝❛ é ✉♠❛ ❝✐ê♥❝✐❛ ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛s ♣ró♣r✐❛s✱ q✉❡ s❡ ♦r❣❛♥✐③❛ ✈✐❛ t❡♦r❡♠❛s ❡ ❞❡♠♦♥str❛çõ❡s❀ ♣❡r❝❡❜❛♠ ❛ ▼❛t❡♠át✐❝❛ ❝♦♠♦ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ s♦❝✐❛❧ ❡ ❤✐st♦r✐❝❛♠❡♥t❡ ❝♦♥str✉í❞♦❀ s❛✐❜❛♠ ❛♣r❡❝✐❛r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ▼❛t❡♠át✐❝❛ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❝✐❡♥tí✜❝♦ ❡ t❡❝♥♦❧ó❣✐❝♦✳

❊ é ❢❛t♦ q✉❡ ♦ ❝á❧❝✉❧♦ ❥á tr❛③ ❡♠ s✉❛ ❡ssê♥❝✐❛ ✉♠ ❢❛t♦r ❞❡ ❝♦♥t❡①t✉❛❧✐③❛çã♦ ❝♦♠ ♦✉tr❛s ❝✐ê♥❝✐❛s✱ ◆❡✇t♦♥✱ ✉♠ ❞♦s ✐❞❡❛❧✐③❛❞♦r❡s ❞❡ss❡ r❛♠♦ ❞❛ ♠❛t❡♠át✐❝❛✱ ✐♥✐❝✐♦✉ s❡✉ ▼ét♦❞♦ ❞❛s ❋❧✉①õ❡s tr❛t❛♥❞♦ ❞❡ ✉♠❛ ❝✉r✈❛ q✉❡ r❡♣r❡s❡♥t❛✈❛ ❛ tr❛❥❡tór✐❛ ❞❡ ♣❛rtí❝✉❧❛s ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦✱ ✉♠ ♣r♦❜❧❡♠❛ q✉❡ r❡♠❡t❡ ❛ ❋ís✐❝❛ ♥♦ ❡♥t❛♥t♦ ❝♦♠ ✉♠❛ ❛❜♦r❞❛❣❡♠ ♣✉r❛♠❡♥t❡ ♠❛t❡♠át✐❝❛✱ ❞❡s❡♥✈♦❧✈✐❛✲s❡ ❛ss✐♠ ❛ ✐❞❡✐❛ ❞❡ ❞❡r✐✈❛❞❛ ❡ t❛♠❜é♠ ♦ ❢❛t♦r ❤✐stór✐❝♦ ❞❛ ❝♦♥✲ tr♦✈❡rs✐❛ ✭◆❡✇t♦♥ ① ▲❡✐❜♥✐③✮ s♦❜r❡ q✉❡♠ ❢♦✐ r❡❛❧♠❡♥t❡ ♦ ✧❣r❛♥❞❡ ❝r✐❛❞♦r✧❞♦ ❝á❧❝✉❧♦✳ ▲♦❣♦ ♣❡r❝❡❜❡✲s❡ q✉❡ ♦ ❝á❧❝✉❧♦ ♣♦ss✉✐ ♦s r❡q✉✐s✐t♦s ❛♣r❡s❡♥t❛❞♦s ♥❛s ❖r✐❡♥t❛çõ❡s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s✱ ❡♠❜♦r❛ ❝♦♥st❛t❡✲s❡ q✉❡ ♥ã♦ é t❡♠❛ ❛❜♦r❞❛❞♦ ♥❛s ❡s❝♦❧❛s ❛t✉❛❧♠❡♥t❡ ❡ ♥ã♦ s❡ ♣♦❞❡ ❝♦♥t❡st❛r q✉❡ ♦ ❞♦♠í♥✐♦ ❞♦s tó♣✐❝♦s✱ ❝♦♠♦✿ ▲✐♠✐t❡✱ ❉❡r✐✈❛❞❛ ❡ ■♥t❡❣r❛❧✱ ♣♦ss✐❜✐❧✐t❛♠ ❛♦ ❛❧✉♥♦ ❞❡ ❡♥s✐♥♦ ♠é❞✐♦ ❛ ♠❛t❡♠❛t✐③❛çã♦ ❞❡ ♣r♦❜❧❡♠❛s q✉♦t✐❞✐❛♥♦s ❡ ♣♦r ❝♦♥s❡❣✉✐♥t❡ ❛ s✉❛ r❡s♦❧✉çã♦✳

▼✉✐t♦s sã♦ ♦s q✉❡st✐♦♥❛♠❡♥t♦s q✉❡ s✉r❣❡♠ ♣♦r ♠♦t✐✈♦ ❞❛ ♥ã♦ ❛❜♦r❞❛❣❡♠ ❞❡ss❡ t❡♠❛✿ ✲ ❙❡rá s❡ ♦s ❛❧✉♥♦s ♥ã♦ ❝♦♥s❡❣✉❡♠ ❛♣r❡♥❞❡r❄ ✲ ❙❡rá s❡ ❡ss❡s tó♣✐❝♦s ❞❡ ❈á❧❝✉❧♦ só s❡✲ rã♦ ✐♠♣♦rt❛♥t❡s ♥♦ ❊♥s✐♥♦ ❙✉♣❡r✐♦r❄ ✲ ❙❡rá q✉❡ ♦s ♣r♦❢❡ss♦r❡s ❡stã♦ ❛♣t♦s ❛ ❡♥s✐♥❛r ♦ ❝á❧❝✉❧♦❄✳ ❊ss❡ ú❧t✐♠♦ q✉❡st✐♦♥❛♠❡♥t♦ ✈❡♠ t❡♥t❛♥❞♦ s❡r s❛♥❛❞♦✱ ♦ ♣ró♣r✐♦ ▼✐♥✐stér✐♦ ❞❛ ❊❞✉❝❛çã♦ ✭▼❊❈✮ ❡♠ ❝♦♥❥✉♥t♦ ❝♦♠ ❛ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r ✭❈❆P❊❙✮ ❡ ❛ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭❙❇▼✮ ❧❛♥ç❛r❛♠ ❡♠ ✷✵✶✶ ♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✭P❘❖❋▼❆❚✮ ♣❛r❛ ❛ ❝❛♣❛❝✐t❛çã♦ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞❡ ♣r♦❢❡ss♦r❡s ❞❡ ♠❛t❡♠át✐❝❛✱ ♣r✐♦r✐t❛r✐❛♠❡♥t❡ ❛♦s q✉❡ ❛t✉❛♠ ♥❛ r❡❞❡ ♣ú❜❧✐❝❛ ❞❡ ❡♥s✐♥♦✱ ♣❛r❛ ❡①❛t❡♠❡♥t❡ tr❛❜❛❧❤❛r t❡♠❛s q✉❡ ♣♦r ✈❡♥t✉r❛ ♥ã♦ ❢♦r❛♠ ❜❡♠ ❛❜♦r❞❛❞♦s ♥❛ ❣r❛❞✉❛çã♦✱ ❝♦♠♦ ❡stá ♥❛ ♣á❣✐♥❛ ❞♦ P❘❖❋▼❆❚ ♥❛ ✐♥t❡r♥❡t✿

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❙✉♠ár✐♦ ✸ ❡♠ ♥í✈❡❧ ❞❡ ♣ós✲❣r❛❞✉❛çã♦ ❧❛t♦ ❡ str✐❝t♦ s❡♥s✉ ❡ ❣❛r❛♥t✐r ❛ t♦❞♦s ❢♦r♠❛çã♦ ❝♦♥t✐♥✉❛❞❛ ❡♠ s✉❛ ár❡❛ ❞❡ ❛t✉❛çã♦✧✳

❊ ♦ ❈á❧❝✉❧♦ é ✉♠ ❞♦s ❛ss✉♥t♦s ❛❜♦r❞❛❞♦s ♥❛ ❣r❛❞❡ ❞❡ ❞✐s❝✐♣❧✐♥❛s ❞♦ P❘❖❋▼❆❚✱ t❡♥❞♦ ✉♠❛ ❞✐s❝✐♣❧✐♥❛✱ ❛ s❛❜❡r✱ ▼❆ ✷✷ ✲ ❋✉♥❞❛♠❡♥t♦s ❞❡ ❈á❧❝✉❧♦✱ ❝♦♠ t♦❞♦ ♦ ❡♠❡♥tár✐♦ ❝♦♠✉♠ ♣❛r❛ ✉♠ ♣r✐♠❡✐r♦ ❝✉rs♦ ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✳

❊❞✉❛r❞♦ ❈♦❧❧✐ ♥♦ ♠❛♥✉❛❧ q✉❡ é ♣❛rt❡ ✐♥t❡❣r❛♥t❡ ❞♦ ❦✐t ❞❡ ♠❛t❡♠át✐❝❛ ❞♦ ♣r♦❥❡t♦ ❆✈❡♥t✉r❛s ♥❛ ❈✐ê♥❝✐❛ ✐♥t✐t✉❧❛❞♦ ✧❆s ❝❡rt❡③❛s ❞♦ ❛❝❛s♦✧❛✜r♠❛✿

❯♠❛ ❞❛s ♠❛✐♦r❡s ❝♦♥tr✐❜✉✐çõ❡s ❞❡ ●❛❧✐❧❡✉ ❢♦✐ ❞❛r ❝❛rát❡r q✉❛♥t✐t❛t✐✈♦ às ❝✐✲ ê♥❝✐❛s ❞❛ ♥❛t✉r❡③❛✱ à ✜❧♦s♦✜❛ ♥❛t✉r❛❧✱ ❝♦♠♦ ❡r❛♠ ❝❤❛♠❛❞❛s ♥❛ s✉❛ é♣♦❝❛✳ ❊❧❡ ❛✜r♠♦✉✿ ✧❆ ✜❧♦s♦✜❛ ❡stá ❡s❝r✐t❛ ♥❡st❡ ❣r❛♥❞❡ ❧✐✈r♦ ✲ ♦ ✉♥✐✈❡rs♦ ✲ q✉❡ ♣❡r♠❛♥❡❝❡ s❡♠♣r❡ ❛❜❡rt♦ ❛ ♥♦ss❛ ✈✐st❛✱ ♠❛s ♥ã♦ ♣♦❞❡♠♦s ❝♦♠♣r❡❡♥❞ê✲❧❛ s❡♠ ❛♣r❡♥❞❡r ♣r✐♠❡✐r♦ ❛ ❧✐♥❣✉❛❣❡♠ ❡♠ q✉❡ ❡stá ❡s❝r✐t♦✳ ❊❧❡ ❡stá ❡s❝r✐t♦ ♥❛ ❧✐♥✲ ❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛✳✳✳✧✳ ➚ ♣❛rt❡ ❛ ♠♦t✐✈❛çã♦ ❝✐❡♥tí✜❝❛✱ ❛ ▼❛t❡♠át✐❝❛ ❡stá ♣r❡s❡♥t❡ ♥❛ ✈✐❞❛ ❞❛s ♣❡ss♦❛s✱ ♥♦ tr❛❜❛❧❤♦ ♦✉ ♥♦ â♠❜✐t♦ ♣❡ss♦❛❧✳ ◆❛s ✜♥❛♥ç❛s✱ ♥♦s ❛s♣❡❝t♦s ❣❡♦♠étr✐❝♦s q✉❡ s✉r❣❡♠ ❡♠ ✈❛r✐❛❞❛s s✐t✉❛çõ❡s ❞♦ ❝♦t✐❞✐❛♥♦✱ ♥❛s r❡♣r❡s❡♥t❛çõ❡s ❣rá✜❝❛s✱ ♥❛s ❡st❛tíst✐❝❛s✳ ❆ ▼❛t❡♠át✐❝❛ é t❛♠❜é♠ ❧✐♥❣✉❛❣❡♠ ❡✱ ♣♦r ✐ss♦✱ é tr❛t❛❞❛ ❝♦♠ t❛♥t❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ❡s❝♦❧❛ ❜ás✐❝❛✳

❖ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ✐♥s❡r✐r ♦ ❈á❧❝✉❧♦ ❝♦♠♦ ❢❡r❛♠❡♥t❛ ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♣r♦❜❧❡♠❛s ❡st❡s ❞♦s ♠❛✐s ❞✐✈❡rs♦s✱ ❞❡s❞❡ ♠❛①✐♠✐③❛çã♦ ✭♠✐♥✐♠✐③❛çã♦✮✱ ♦t✲ ♠✐③❛çã♦ ❡ ✜♥❛❧♠❡♥t❡ ❛s ❡q✉❛çõ❡s ❞♦ t✐♣♦ xp = px ❝♦♠ ♣ ♣r✐♠♦✱ ♣❛r❛ t❛♥t♦ ✐♥✐❝✐❛r❡✲

♠♦s ❝♦♠ ✉♠ ❜r❡✈❡ ❤✐stór✐❝♦ ❞♦ ❈á❧❝✉❧♦✱ ❡♠ s❡❣✉✐❞❛ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s✱ ♣r♦♣♦s✐çõ❡s ❡ t❡♦r❡♠❛s ❞♦ ❝á❧❝✉❧♦ ❡♠ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✱ ❝♦♠♦ ❝♦♥t✐♥✉✐❞❛❞❡ ❡ ❞❡r✐✈❛❞❛❀ s❡❣✉✐♥❞♦✱ t❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣r♦❜❧❡♠❛s ❜ás✐❝♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ q✉❡ sã♦ r❡s♦❧✈✐❞♦s ❝♦♠ té❝♥✐❝❛s ❞❡ ❞❡r✐✈❛❞❛✱ ♣❛ss❛♥❞♦ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛s ❡q✉❛çõ❡s ❞♦ t✐♣♦xp=px ❝♦♠ ♣

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❈❛♣ít✉❧♦ ✶

❇r❡✈❡ ❍✐stór✐❝♦ ❞♦ ❈á❧❝✉❧♦

❆❜♦r❞❛r❡♠♦s ♥❡ss❡ ❝❛♣ít✉❧♦ ✉♠ ❜r❡✈❡ ❤✐stór✐❝♦ ❞♦ ❝á❧❝✉❧♦ ❞❡s❞❡ ♦s ♣r✐♠ór❞✐♦s ❝♦♠ ■s❛❛❝ ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③✱ ❛té ❛ ✐♥❝♦♥❝❧✉sã♦ ❞❛ ❛✉t♦r✐❛ ❞❡ss❡ ✐♠♣♦rt❛♥t❡ r❛♠♦ ❞❛ ♠❛t❡♠át✐❝❛✳

❆s ♣r✐♠❡✐r❛s ✐❞❡✐❛s r❡❧❛❝✐♦♥❛❞❛s ❛♦ ❈á❧❝✉❧♦ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s ♣♦r ❞♦✐s ❞♦s ♠❛✐♦r❡s ♠❛t❡♠át✐❝♦s ❞❡ t♦❞❛ ❤✐stór✐❛✱ q✉❡ sã♦ ■s❛❛❝ ◆❡✇t♦♥ ✭✶✻✹✸ ✲ ✶✼✷✼✮ ❡ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ✈♦♥ ▲❡✐❜♥✐③ ✭✶✻✹✻ ✲ ✶✼✶✻✮✳ ❋❛❧❛r❡♠♦s ❞❡ ❢♦r♠❛ ❜r❡✈❡ s♦❜r❡ ❛ ✈✐❞❛ ❞❡ ❝❛❞❛ ✉♠ ❞❡❧❡s✶

◆♦ ❧✐✈r♦ ✧❆ ❞❡r✐✈❛❞❛ ❡ ❛❧❣✉♠❛s ❛♣❧✐❝❛çã♦✧✱ ❞♦ ♣r♦❢❡ss♦r ❏✉s❝❡❧✐♥♦ ❙✐❧✈❛ ✭✷✵✶✷✮ ❛♣r❡s❡♥✲ t❛❞♦ ♥♦ ✷✵ ❈♦❧óq✉✐♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ r❡❣✐ã♦ ♥♦r❞❡st❡ r❡❛❧✐③❛❞♦ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧

❞♦ P✐❛✉í✱ ♦ ❛✉t♦r ❛♣r❡s❡♥t❛ ✉♠ ❤✐stór✐❝♦ ❛ ❝❡r❝❛ ❞❡ss❡s ❞♦✐s ♠❛t❡♠át✐❝♦s❀ ✈❡❥❛♠♦s ❛❧❣✉♥s ❢❛t♦s r❡❧❡✈❛♥t❡s✳

✶✳✶ ◆❡✇t♦♥

■s❛❛❝ ◆❡✇t♦♥ ♥❛s❝❡✉ ❡♠ ❲♦♦❧st❤♦r♣❡✱ ♥♦ ❛♥♦ ❞❡ ✶✻✹✷✱ ■♥❣❧❛t❡rr❛✱ ❡r❛ ❞❡ ❢❛♠í❧✐❛ ❛❜❛st❛❞❛✱ ◆❡✇t♦♥ t❡✈❡ ✉♠❛ ✐♥❢â♥❝✐❛ ✐♥❢❡❧✐③ ♥ã♦ t❡♥❞♦ ❝♦♥❤❡❝✐❞♦ s❡✉ ♣❛✐ q✉❡ ♠♦rr❡r❛ ❛♥t❡s q✉❡ ❡❧❡ ❝♦♠♣❧❡t❛ss❡ ✉♠ ❛♥♦ ❞❡ ✐❞❛❞❡✱ ❡ t❡♥❞♦ s✉❛ ♠ã❡ s❡ ❝❛s❛❞♦ ♥♦✈❛♠❡♥t❡ ❡❧❡ ♣❛ss❛ ❛ ♠♦r❛r ❝♦♠ s❡✉s ❛✈ós✳ ❈♦♠ ❛ ♠♦rt❡ ❞❡ s❡✉ ♣❛❞r❛st♦ ❡♠ ✶✻✺✸✱ ◆❡✇t♦♥ ♣❛ss♦✉ ❛ ✈✐✈❡r ❝♦♠ ❛ ♠ã❡✱ ❛✈ó✱ ✉♠ ♠❡✐♦✲✐r♠ã♦ ❡ ❞✉❛s ♠❡✐❛✲✐r♠ãs✳ ❊r❛ ✉♠ ❡st✉❞❛♥t❡ ❞❡ ♣♦✉❝❛s ❤❛❜✐❧✐❞❛❞❡s ❡ ❞❡

❚♦❞♦ ♦ t❡①t♦ ❞♦ ❤✐stór✐❝♦ ❞♦ ❈á❧❝✉❧♦ ❢♦✐ r❡t✐r❛❞♦ ❞♦ ❧✐✈r♦ ❞♦ ♣r♦❢❡ss♦r ❏✉s❝❡❧✐♥♦ ❙✐❧✈❛✱ ❆ ❞❡r✐✈❛❞❛ ❡

❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s

(17)

❈❛♣ít✉❧♦ ✶✳ ❇r❡✈❡ ❍✐stór✐❝♦ ❞♦ ❈á❧❝✉❧♦ ✺ ✐♥t❡r❡ss❡ ❞✐♠✐♥✉t♦ ❡✱ ♣♦r ✐ss♦✱ ❢♦✐ ❧♦❣♦ ❝❡❞♦ r❡t✐r❛❞♦ ❞❛ ❡s❝♦❧❛ ♣❛r❛ ❝✉✐❞❛r ❞♦s ✐♥t❡r❡ss❡s ✜♥❛♥❝❡✐r♦s ❞❛ ❢❛♠í❧✐❛✳ ◆❡✇t♦♥ t❛♠❜é♠ ♥ã♦ ❞❡♠♦♥str♦✉ ✐♥t❡r❡ss❡ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❡st❛ ❛t✐✈✐❞❛❞❡✳ ❊♠ ✶✻✾✵✱ ♣♦r ✐♥✢✉ê♥❝✐❛ ❞❡ ❲✐❧❧✐❛♠ ❆②s❝♦✉❣❤✱ t✐♦ ❞❡ ◆❡✇t♦♥✱ ❛ ❢❛♠í❧✐❛ ❞❡❝✐❞✐✉ ♣r❡♣❛rá✲❧♦ ♣❛r❛ ❛ ❡❞✉❝❛çã♦ s✉♣❡r✐♦r✳ ◆❡st❛ é♣♦❝❛ ❡❧❡ s❡ ❛❧♦❥❛✈❛ ❝♦♠ ❛ ❢❛♠í❧✐❛ ❞❡ ❙t♦❦❡s✱ ♦ ❝❤❡❢❡ ❞❛ ❡s❝♦❧❛✱ q✉❡ ♣❡r❝❡❜❡♥❞♦ ♦s t❛❧❡♥t♦s ❞❡ ◆❡✇t♦♥✱ ❢♦✐ ✉♠ ❞♦s r❡s♣♦♥sá✈❡✐s ♣♦r ❝♦♥✈❡♥❝❡r ❛ ❢❛♠í❧✐❛ ❞❡ q✉❡ ❡❧❡ ♣♦ss✉í❛ ❤❛❜✐❧✐❞❛❞❡s ♣❛r❛ ♦ tr❛❜❛❧❤♦ ❛❝❛❞ê♠✐❝♦✳

❋♦✐ só ❡♠ ✶✻✻✶✱ q✉❛♥❞♦ ❥á ❡r❛ ♠❛✐s ✈❡❧❤♦ q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ❝♦❧❡❣❛s✱ q✉❡ ◆❡✇t♦♥ ✐♥❣r❡ss♦✉ ♥♦ ❚r✐♥✐t② ❈♦❧❧❡❣❡ ❞❡ ❈❛♠❜r✐❞❣❡✳ ❆ ♣r❡t❡♥çã♦ ❞❡ ◆❡✇t♦♥ ❡r❛ s❡ ❢♦r♠❛r ❡♠ ❉✐r❡✐t♦✳ ◆❡✇t♦♥ t❡✈❡ ❡♠ ❈❛♠❜r✐❞❣❡ ❛❝❡ss♦ ❛♦s t❡①t♦s ❞❡ ❉❡s❝❛rt❡s✱ ●❛ss❡♥❞✐✱ ❍♦❜❜❡ ❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❞❡ ❇♦②❧❡✳ ❊❧❡ t❛♠❜é♠ ♠♦str♦✉ ✐♥t❡r❡ss❡ ♣❡❧❛s t❡♦r✐❛s r❡✈♦❧✉❝✐♦♥ár✐❛s ❞❡ ❈♦♣ér♥✐❝♦✱ ❑❡♣❧❡r ❡ ●❛❧✐❧❡✉ ❡♠ ❛str♦♥♦♠✐❛ ❡ ót✐❝❛✳ ❊ ❢♦✐ ❣r❛❞✉❛❧♠❡♥t❡ s❡ ❡♥✈♦❧✈❡♥❞♦ ❝♦♠ ♦s ❡st✉❞♦s ❞❡ ♠❛t❡♠át✐❝❛ ❡ ❢ís✐❝❛✳ ❊①✐st❡ ✉♠ r❡❧❛t♦ ♣♦r ♣❛rt❡ ❞❡ ▼♦✐✈r❡✱ ❞❡ q✉❡ s❡✉ ✐♥t❡r❡ss❡ ♣♦r ♠❛t❡♠át✐❝❛ ❢♦✐ ✐♥❝r❡♠❡♥t❛❞♦ ❡♠ ✶✻✻✸ q✉❛♥❞♦ ❝♦♠♣r♦✉ ✉♠ ❧✐✈r♦ ❞❡ ❛str♦✲ ♥♦♠✐❛ ❡ ♥ã♦ ♣ô❞❡ ❡♥t❡♥❞❡r ❛ ♠❛t❡♠át✐❝❛ ❡♥✈♦❧✈✐❞❛✳ ❋♦✐ ❡♥tã♦ q✉❡ ♣❛ss♦✉ ❛ ❛♣r✐♠♦r❛r✲s❡ ❡♠ ♠❛t❡♠át✐❝❛✳ P❛ss♦✉ ❛ ❡st✉❞❛r ❛ r❡❝é♠ ❞❡s❡♥✈♦❧✈✐❞❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ❛tr❛✈és ❞♦s t❡①t♦s ❞❡ ❱✐ét❡ ❡ ❘❡♥é ❉❡s❝❛rt❡s✳ ◆❡st❡ ♣❡rí♦❞♦ ❡❧❡ ❛♣r❡♥❞❡✉ s♦❜r❡ ♦ ♠ét♦❞♦ ❞❡ ❲❛❧❧✐s ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠ q✉❛❞r❛❞♦ ❝♦♠ ár❡❛ s♦❜ s❡❣♠❡♥t♦s ❞❛ ♣❛rá❜♦❧❛ ❡ ❞❛ ❤✐♣ér❜♦❧❡✱ ✉s❛♥❞♦ ♦s ✐♥❞✐✈✐sí✈❡✐s✳ ❈♦♠ ♠❡♥♦s ❞❡ ✷✺ ❛♥♦s ❞❡ ✐❞❛❞❡ ❡❧❡ ✐♥✐❝✐♦✉ s✉❛ ❝❛rr❡✐r❛ ❢❛③❡♥❞♦ ❝♦♥tr✐✲ ❜✉✐çõ❡s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ♠❛t❡♠át✐❝❛✱ ♠❡❝â♥✐❝❛✱ ót✐❝❛ ❡ ❛str♦♥♦♠✐❛✳ ◆❡st❡ ♣❡rí♦❞♦✱ ❝♦♠♦ ❡❧❡ ♣ró♣r✐♦ r❡❧❛t♦✉ ❞❡♣♦✐s✱ ❡❧❡ ❢❡③ q✉❛tr♦ ❞❡ s✉❛s ❣r❛♥❞❡s ❞❡s❝♦❜❡rt❛s✿ ♦ t❡♦r❡♠❛ ❜✐♥♦♠✐❛❧✱ ♦ ❝á❧❝✉❧♦✱ ❛ ❧❡✐ ❞❛ ❣r❛✈✐t❛çã♦ ✉♥✐✈❡rs❛❧ ❡ ❛ ♥❛t✉r❡③❛ ❞❛s ❝♦r❡s✳

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❈❛♣ít✉❧♦ ✶✳ ❇r❡✈❡ ❍✐stór✐❝♦ ❞♦ ❈á❧❝✉❧♦ ✻

❊♠ ♦✉tr✉❜r♦ ❞❡ ✶✻✻✻ ◆❡✇t♦♥ ❡s❝r❡✈❡✉ ✉♠ tr❛t❛❞♦ s♦❜r❡ ❛s ✢✉①õ❡s✳ ❙❡✉ tr❛❜❛❧❤♦ ♥ã♦ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ✐♠❡❞✐❛t❛♠❡♥t❡ ♠❛s ♠✉✐t♦s ♠❛t❡♠át✐❝♦s ❞❛ é♣♦❝❛ ♦ ❝♦♥❤❡❝✐❛♠ ❞❡ ❢♦r♠❛ q✉❡ ♦ tr❛t❛❞♦ ❡①❡r❝❡✉ ❣r❛♥❞❡ ✐♥✢✉ê♥❝✐❛ s♦❜r❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❝á❧❝✉❧♦ ♥❛q✉❡❧❡ ♣❡✲ rí♦❞♦✳ ❊❧❡ ✐♥✐❝✐♦✉ s✉❛s ❝♦♥s✐❞❡r❛çõ❡s tr❛t❛♥❞♦ ✉♠❛ ❝✉r✈❛ ❝♦♠♦ s❡♥❞♦ tr❛❥❡tór✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛✱ ❞❛❞❛ ♣♦r s✉❛s ❝♦♦r❞❡♥❛❞❛s ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦✱ ❡♠ ❧✐♥❣✉❛❣❡♠ ♠♦❞❡r♥❛ ❡❧❡ r❡♣r❡s❡♥t❛✈❛ ❛ ❝✉r✈❛ ❞❡ ✉♠❛ ❢✉♥çã♦ s♦❜ ❛ s✉❛ ❢♦r♠❛ ♣❛r❛♠étr✐❝❛✱ ✉s❛♥❞♦ ♦ t❡♠♣♦ ❝♦♠♦ ♣❛râ♠❡tr♦ ✲ ✉♠❛ ❢♦r♠❛ ❜❛st❛♥t❡ ❛♣r♦♣r✐❛❞❛ ♣❛r❛ q✉❡♠ ❞❡s❡❥❛ ❡st✉❞❛r ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛✳

❊♠ ✶✻✼✶ ◆❡✇t♦♥ ❡s❝r❡✈❡✉ ❉❡ ▼❡t❤♦❞✐s ❡t ❋❧✉①✐♦♥✉♠ ♠❛s ♥ã♦ ♦ ❝♦♥s❡❣✉✐✉ ♣✉❜❧✐❝❛r ❛té q✉❡ ❏♦❤♥ ❈♦❧s♦♥ ♣r❡♣❛r♦✉ ✉♠❛ tr❛❞✉❛çã♦ ♣❛r❛ ♦ ✐♥❣❧ês ❡♠ ✶✼✸✻✳ ◆❡✇t♦♥ ❡♥❝♦♥✲ tr♦✉ ❞✐✈❡rs❛s ❞✐✜❝✉❧❞❛❞❡s ♣❛r❛ ♣✉❜❧✐❝❛r s❡✉s t❡①t♦s ❡♠ ♠❛t❡♠át✐❝❛ ❡✱ ❞❡ ❝❡rt❛ ❢♦r♠❛✱ ❇❛rr♦✇ ✭q✉❡ ❢♦✐ ✉♠ ❞♦s ♣r♦❢❡ss♦r❡s ❞❡ ■s❛❛❝ ◆❡✇t♦♥ ♥♦ ❚r✐♥✐t② ❈♦❧❧❡❣❡✮ t✐♥❤❛ ♣❛rt❡ ❞❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ♣♦r ✐st♦✱ ♣♦✐s ❡♠ t❡♠♣♦s r❡❝❡♥t❡s ♦s ❡❞✐t♦r❡s ❞❡ ❇❛rr♦✇ t✐♥❤❛♠ ✐❞♦ ❛ ❢❛❧ê♥❝✐❛ ❡ ♦s ❞❡♠❛✐s ❡❞✐t♦r❡s s❡ ❡♥❝♦♥tr❛✈❛♠ r❡❝❡♦s♦s ❡♠ ❝♦♥t✐♥✉❛r ❛ ♣✉❜❧✐❝❛çã♦ ❞❡ t❡①t♦s s♦❜r❡ ♠❛t❡♠át✐❝❛✳ ❖ ❧✐✈r♦ ❞❡ ◆❡✇t♦♥ s♦❜r❡ ❆♥á❧✐s❡ ❝♦♥t❡♥❞♦ ✉♠ tr❛t❛♠❡♥t♦ s♦❜r❡ sér✐❡s ✐♥✜♥✐t❛s ❢♦✐ ❡s❝r✐t♦ ❡♠ ✶✻✻✾ ❡ ❝✐r❝✉❧♦✉ ♣❡❧♦s ♠❡✐♦s ❛❝❛❞ê♠✐❝♦s ❡♠ ❢♦r♠❛ ♠❛♥✉s❝r✐t❛✱ só s❡♥❞♦ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✼✼✶✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ s❡✉ ▼❡t❤♦❞ ♦❢ ✢✉①✐♦♥s ❛♥❞ ✐♥✜♥✐t❡ s❡r✐❡s ❢♦✐ ❡s❝r✐t♦ ❡♠ ✶✻✼✶ ❡ s♦♠❡♥t❡ tr❛❞✉③✐❞♦ ♣❛r❛ ♦ ✐♥❣❧ês ❡ ♣✉❜❧✐❝❛❞♦ ♥♦ ❛♥♦ ❞❡ ✶✼✸✻✳ ❖ ♦r✐❣✐♥❛❧ ❡♠ ❧❛t✐♠ só ❢♦✐ ♣✉❜❧✐❝❛❞♦ ♠✉✐t♦s ❛♥♦s ♠❛✐s t❛r❞❡✳ ❆ ♦❜r❛ s❡❣✉✐♥t❡ ❞❡ ◆❡✇t♦♥✱ ❚r❛❝t✉s ❞❡ ◗✉❛❞r❛t✉r❛ ❈✉r✈❛r✉♠✱ ❢♦✐ ❡s❝r✐t♦ ❡♠ ✶✻✾✸ ❡ só ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✼✵✹ s♦❜ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❛♣ê♥❞✐❝❡ ❡♠ s❡✉ ❧✐✈r♦ ❖♣t✐❦s✳

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❈❛♣ít✉❧♦ ✶✳ ❇r❡✈❡ ❍✐stór✐❝♦ ❞♦ ❈á❧❝✉❧♦ ✼ s❡r ♠❛✐s r❡t✐❝❡♥t❡ ❝♦♠ r❡❧❛çã♦ ❛ s✉❛s ❞❡s❝♦❜❡rt❛s✳ ❊♠❜♦r❛ ❛♣r❡❝✐❛ss❡ ❛ ♥♦t♦r✐❡❞❛❞❡ ❡❧❡ r❡ss❡♥t✐❛ ❡♥♦r♠❡♠❡♥t❡ ❛s ❝rít✐❝❛s r❡❝❡❜✐❞❛s ❡ ❥✉❧❣❛✈❛ q✉❡ ✉♠❛ ❢♦r♠❛ ❞❡ ❡✈✐tá✲❧❛s s❡r✐❛ ❛ ♥ã♦ ♣✉❜❧✐❝❛çã♦ ❞❡ s✉❛s ✐❞❡✐❛s✳

❆s r❡❧❛çõ❡s ❞❡ ◆❡✇t♦♥ ❝♦♠ ❍♦♦❦❡ s❡ ❞❡t❡r✐♦r❛r❛♠ ❡♠ ✶✻✼✺✱ q✉❛♥❞♦ ❍♦♦❦❡ ♦ ❛❝✉s♦✉ ❞❡ ♣❧á❣✐♦ ❡♠ ❛❧❣✉♠❛s ❞❡ s✉❛s ❝♦♥❝❧✉sõ❡s ❡♠ ót✐❝❛✳ ■st♦ ❢❡③ ❝♦♠ q✉❡ ◆❡✇t♦♥ ❛❞✐❛ss❡ ❛ ♣✉❜❧✐❝❛çã♦ ❞❡ s✉❛s ❝♦♥❝❧✉sõ❡s ❡♠ ót✐❝❛ ❛té ❛ ♠♦rt❡ ❞♦ ❛❞✈❡rsár✐♦✱ ❡♠ ✶✼✵✸✳

❊♠ ✶✻✽✻ ❍❛❧❧❡②✱ ❛strô♥♦♠♦ ❡ ❛♠✐❣♦ ❞❡ ◆❡✇t♦♥✱ ♦ ❝♦♥✈❡♥❝❡✉ ❛ ♣✉❜❧✐❝❛r ✉♠❛ ❞❡s❝r✐çã♦ ❝♦♠♣❧❡t❛ ❞❡ s✉❛s ❞❡s❝♦❜❡rt❛s s♦❜r❡ ❢ís✐❝❛ ❡ ❛str♦♥♦♠✐❛✳ ❯♠ ❛♥♦ ♠❛✐s t❛r❞❡ ❡❧❡ ♣✉❜❧✐✲ ❝♦✉ ♦ P❤✐❧♦s♦♣❤✐❛❡ ◆❛t✉r❛❧✐s Pr✐♥❝✐♣✐❛ ▼❛t❤❡♠❛t✐❝❛✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ Pr✐♥❝✐♣✐❛✱ ❝♦♠♦ s❡ t♦r♥♦✉ ❝♦♥❤❡❝✐❞♦ ♦ ❧✐✈r♦✳ ❊st❡ é ❝♦♥s✐❞❡r❛❞♦ ♣♦r ♠✉✐t♦s ❤✐st♦r✐❛❞♦r❡s ❞❛ ❝✐ê♥❝✐❛ ❝♦♠♦ ♦ ♠❛✐♦r ❧✐✈r♦ ❝✐❡♥tí✜❝♦ ❥á ❡s❝r✐t♦✳ ◆❡❧❡✱ ◆❡✇t♦♥ tr❛t❛ ❞♦ ♠♦✈✐♠❡♥t♦ ❞♦s ❝♦r♣♦s ❡♠ ♠❡✐♦s r❡s✐st❡♥t❡s ❡ ♥ã♦ r❡s✐st❡♥t❡s ❡ s♦❜ ❛ ❛çã♦ ❞❡ ❢♦rç❛s ❝❡♥trí♣❡t❛s✳ ❖s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❛♣❧✐❝❛❞♦s ❛ ór❜✐t❛s ❝❡❧❡st❡s✱ ♣r♦❥ét❡✐s✱ ♣ê♥❞✉❧♦s ❡ ❝♦r♣♦s ❡♠ q✉❡❞❛ ❧✐✈r❡ ♥❛s ♣r♦①✐♠✐❞❛❞❡s ❞❛ s✉♣❡r❢í❝✐❡ ❞❛ ❚❡rr❛✳ ❊♠ s❡❣✉✐❞❛ ◆❡✇t♦♥ ❢♦r♥❡❝❡✉ ❡①♣❧✐❝❛çõ❡s s♦❜r❡ ❛s ór❜✐t❛s ❡①❝ê♥tr✐✲ ❝❛s ❞♦s ❝♦♠❡t❛s✱ ❛s ♠❛rés ❡ s✉❛s ✈❛r✐❛çõ❡s✱ ❛ ♣r❡❝❡ssã♦ ❞♦ ❡✐①♦ ❞❛ ❚❡rr❛ ❡ ❛ ♣❡rt✉r❜❛çã♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❞❛ ▲✉❛ ♣❡❧❛ ❛tr❛çã♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❞♦ ❙♦❧✳ ❊st❡ tr❛❜❛❧❤♦ ♦ tr❛♥s❢♦r♠♦✉ ♥♦ ♣r✐♥❝✐♣❛❧ ❝✐❡♥t✐st❛ ❞❡ s✉❛ é♣♦❝❛ ❡♠❜♦r❛ ❞✐✈❡rs♦s ♣❡♥s❛❞♦r❡s ❞❛ ❊✉r♦♣❛ ❝♦♥t✐♥❡♥t❛❧ ❡ ❞❛ ♣ró♣r✐❛ ■♥❣❧❛t❡rr❛ r❡❧✉t❛ss❡♠ ❡♠ ❛❝❡✐t❛r s✉❛s ✈✐sõ❡s s♦❜r❡ ❣r❛✈✐t❛çã♦ ❝♦♠♦ ❛çã♦ à ❞✐stâ♥❝✐❛✱ s✉❛ t❡♦r✐❛ ❝♦r♣✉s❝✉❧❛r ❞❛ ❧✉③✱ s✉❛ ✈❡rsã♦ ❞♦ ❝á❧❝✉❧♦ ❡♠ ❝♦♥tr❛♣♦s✐çã♦ à ✈❡rsã♦ ❞❡ ▲❡✐❜♥✐③ ❡ ♦✉tr♦s tó♣✐❝♦s ❧♦❝❛❧✐③❛❞♦s✳

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❈❛♣ít✉❧♦ ✶✳ ❇r❡✈❡ ❍✐stór✐❝♦ ❞♦ ❈á❧❝✉❧♦ ✽

✶✳✷ ▲❡✐❜♥✐③

●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ✈♦♥ ▲❡✐❜♥✐③ ✭✶✻✹✻✲✶✼✶✻✮ ♥❛s❝❡✉ ❡♠ ▲❡✐♣③✐❣✱ ❆❧❡♠❛♥❤❛✱ ✜❧❤♦ ❞❡ ❋r✐❡✲ ❞r✐❝❤ ▲❡✐❜♥✐③ ❡ ❞❡ ❈❛t❤❛r✐♥❛ ❙❝❤✉❝❦✳ ▲❡✐❜♥✐③ ♣❡r❞❡✉ ♦ ♣❛✐ ❝♦♠ ❛♣❡♥❛s s❡✐s ❛♥♦s ❡ ❛♣r❡♥❞❡✉ ❞❛ ♠ã❡ ♦s ✈❛❧♦r❡s r❡❧✐❣✐♦s♦s ❡ ét✐❝♦s q✉❡ ♥♦rt❡❛r❛♠ s✉❛ ✈✐❞❛ ❡ ✜❧♦s♦✜❛✳ ◆❛ ❡s❝♦❧❛ ❡❧❡ ❛♣r❡♥✲ ❞❡✉ ❛ ❧ó❣✐❝❛ ❞❡ ❆r✐stót❡❧❡s✳ ■♥s❛t✐s❢❡✐t♦ ❝♦♠ ❛ ✜❧♦s♦✜❛ ❛r✐st♦té❧✐❝❛✱ ❞♦♠✐♥❛♥t❡ ♥❛ é♣♦❝❛✱ ❡❧❡ ✐♥✐❝✐♦✉ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ s✉❛s ♣ró♣r✐❛s ✐❞❡✐❛s s♦❜r❡ ❝♦♠♦ ❛♣❡r❢❡✐ç♦á✲❧❛✳ ❆✐♥❞❛ ❝r✐❛♥ç❛ ❡❧❡ ❛♣r❡❝✐❛✈❛ ❧❡r ♦s ❧✐✈r♦s ❞❡ s❡✉ ♣❛✐ s♦❜r❡ ♠❡t❛❢ís✐❝❛ ❡ t❡♦❧♦❣✐❛ ♥❛s ✈✐sõ❡s ❞❡ ❡s❝r✐t♦r❡s ❝❛tó❧✐❝♦s ❡ ♣r♦t❡st❛♥t❡s✳ ❖ ♣ró♣r✐♦ ▲❡✐❜♥✐③ r❡❝♦♥❤❡❝❡✉ ♠❛✐s t❛r❞❡ ❝♦♠♦ ✉♠❛ ❝♦♥st❛♥t❡ ❡♠ s✉❛ ✈✐❞❛ ❛ ♣r❡♦❝✉♣❛çã♦ ❡♠ ❡st❛❜❡❧❡❝❡r ✉♠ ♦r❞❡♥❛♠❡♥t♦ ♣♦r trás ❞♦ ♣❡♥s❛♠❡♥t♦ ❧ó❣✐❝♦ ❡ ❞❛ ❞❡❞✉çã♦ ♠❛t❡♠át✐❝❛✱ ❛ss✐♠ ❝♦♠♦ ✉♠❛ t❡♥t❛t✐✈❛ s❡♠♣r❡ ♣r❡s❡♥t❡ ❞❡ ❡st❛❜❡❧❡❝❡r ❝♦♥✲ t❛t♦s ❡♥tr❡ ♣♦♥t♦s ❞❡ ✈✐st❛ ❝♦♥✢✐t❛♥t❡s ❡ ♦ ❞❡ ✉♥✐✜❝❛r ♦s ❞✐✈❡rs♦s s✐st❡♠❛s ❞❡ ♣❡♥s❛♠❡♥t♦✳

❆♦ ❝♦♠♣❧❡t❛r ✶✻ ❛♥♦s✱ ❛❧❣♦ ♥ã♦ ♠✉✐t♦ ✐♥❝♦♠✉♠ ♥❛ é♣♦❝❛✱ ▲❡✐❜♥✐③ ❡♥tr♦✉ ♣❛r❛ ❛ ❯♥✐✈❡r✐❞❛❞❡ ❞❡ ▲❡✐♣③✐❣ ♦♥❞❡ ❡st✉❞♦✉ ✜❧♦s♦✜❛ ❡ ♠❛t❡♠át✐❝❛✳ ◆❡♥❤✉♠❛ ❞❡st❛s ❞✐s❝✐♣❧✐♥❛s ❡r❛✱ ♥❛ é♣♦❝❛✱ ❡♥s✐♥❛❞❛ ❝♦♠ ❣r❛♥❞❡ ♥í✈❡❧ ❞❡ ❛♣r♦❢✉♥❞❛♠❡♥t♦✳ ❆♣ós s✉❛ ❣r❛❞✉❛çã♦✱ ♦❜t✐❞❛ ❡♠ ✶✻✻✸✱ ❡❧❡ ♣❛ss♦✉ ✉♠ ♣❡rí♦❞♦ ❡♠ ❏❡♥❛ ❡st✉❞❛♥❞♦ ❝♦♠ ❊r❤❛r❞ ❲❡✐❣❡❧✱ ✉♠ ✜❧ós♦❢♦ ❡ ♠❛t❡♠át✐❝♦ ❝♦♠ q✉❡♠ ❛♣r❡♥❞❡✉ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ♠ét♦❞♦ ❞❡ ♣r♦✈❛s ♠❛t❡♠át✐❝❛s ❡♠ ❛ss✉♥t♦s t❛✐s ❝♦♠♦ ❧ó❣✐❝❛ ❡ ✜❧♦s♦✜❛✳ ❊♠ ✶✻✻✼✱ ▲❡✐❜♥✐③ r❡❝❡❜❡ ♦ ❣r❛✉ ❞❡ ❞♦✉t♦r ❡♠ ❉✐r❡✐t♦ ♣❡❧❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❆❧t❞♦r❢✳

❊♠ ✶✻✼✶ ❡❧❡ ♣✉❜❧✐❝♦✉ ❍②♣♦t❤❡s✐s P❤②s✐❝❛ ◆♦✈❛ ♦♥❞❡ ❛✜r♠❛✈❛✱ ❡♠ ❛❝♦r❞♦ ❝♦♠ ❑❡♣❧❡r✱ q✉❡ ♦ ♠♦✈✐♠❡♥t♦ é ❞❡❝♦rr❡♥t❡ ❞❛ ❛çã♦ ❞♦ ❡s♣ír✐t♦ s♦❜r❡ ❛ ♠❛tér✐❛✳ ◆❡st❡ ♣❡rí♦❞♦✱ ♣❛ss♦✉ ❛ s❡ ❝♦♠✉♥✐❝❛r ❝♦♠ ❖❧❞❡♥❜✉r❣✱ ♦ s❡❝r❡tár✐♦ ❞❛ ❘♦②❛❧ ❙♦❝✐❡t② ❞❡ ▲♦♥❞r❡s✱ ✉♠ ❞♦s r❡s♣♦♥✲ sá✈❡✐s ♣♦r s❡✉ ❝♦♥t❛t♦ ❝♦♠ ■s❛❛❝ ◆❡✇t♦♥✳

(21)

❈❛♣ít✉❧♦ ✶✳ ❇r❡✈❡ ❍✐stór✐❝♦ ❞♦ ❈á❧❝✉❧♦ ✾ ❞❡ ▲♦♥❞r❡s✱ ▲❡✐❜♥✐③ r❡❝❡❜❡✉ ❝rít✐❝❛s ♥❛ ❘♦②❛❧ ❙♦❝✐❡t②✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♥♦ q✉❡ s❡ r❡❢❡r✐❛ à ♠áq✉✐♥❛ ❞❡ ❝❛❧❝✉❧❛r✳ ❊st❛s ❝rít✐❝❛s t✐✈❡r❛♠ ✉♠ ❡❢❡✐t♦ ✐♥t❡r❡ss❛♥t❡ s♦❜r❡ ▲❡✐❜♥✐③✳ ❊❧❡ ♣❡r❝❡❜❡✉ q✉❡ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❡♠ ♠❛t❡♠át✐❝❛ ❡r❛♠✱ ❞❡ ❢❛t♦✱ ✐♥❝♦♠♣❧❡t♦s ❡ q✉❡ ♥❡❝❡s✲ s✐t❛✈❛♠ ❞❡ ❛♣r✐♠♦r❛♠❡♥t♦✳ ❙❡♠ s❡ ❛❜❛t❡r ❡❧❡ r❡❞♦❜r♦✉ s❡✉s ❡s❢♦rç♦s ❡♠ s❡ ❛♣r♦❢✉♥❞❛r ♥❡st❛ ❞✐s❝✐♣❧✐♥❛✳

▲❡✐❜♥✐③ ❢♦✐ ❛❝❡✐t♦ ♠❡♠❜r♦ ❞❛ ❘♦②❛❧ ❙♦❝✐❡t② ❞❡ ▲♦♥❞r❡s ❡♠ ✶✻✼✸ ❡ ✐♥✐❝✐♦✉ ❡st✉❞♦ s♦❜r❡ ❛ ❣❡♦♠❡tr✐❛ ❞♦s ✐♥✜♥✐t❡s✐♠❛✐s✱ tr♦❝❛♥❞♦ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ s♦❜r❡ ❡st❡s ❡s❢♦rç♦s ❝♦♠ ❖❧❞❡♥✲ ❜✉r❣ q✉❡✱ ♣♦r s✉❛ ✈❡③✱ ♦ ✐♥❢♦r♠♦✉ s♦❜r❡ ♦s ❛✈❛♥ç♦s ❞❡ ◆❡✇t♦♥ ♥❡st❛ ár❡❛✳ ❉❡✈❡✲s❡ ♥♦t❛r q✉❡✱ ♥❡st❡ ♣❡rí♦❞♦✱ ▲❡✐❜♥✐③ ♥ã♦ ❣♦③❛✈❛ ❞❡ ❣r❛♥❞❡ r❡♣✉t❛çã♦ ❝♦♠ ♦s ♠❡♠❜r♦s ❞❛ ❘♦②❛❧ ❙♦❝✐❡t② ❞❡✈✐❞♦ à s✉❛ ✐♥❝❛♣❛❝✐❞❛❞❡ ❞❡ ❝♦♥❝❧✉✐r ❛ ♠áq✉✐♥❛ ❝❛❧❝✉❧❛❞♦r❛✳ ❙✐♠✉❧t❛♥❡❛♠❡♥t❡ ❖❧❞❡❜✉r❣ ❞❡s❝♦♥❤❡❝✐❛ q✉❡ ❡❧❡ ❤❛✈✐❛✱ ❞❡✈✐❞♦ ❛ s❡✉s ❡s❢♦rç♦s ♣❛r❛ s❡ s✉♣❡r❛r✱ tr❛♥s❢♦r♠❛❞♦✲ s❡ ❡♠ ✉♠ ❣ê♥✐♦ ❝r✐❛t✐✈♦ ❞❛ ♠❛t❡♠át✐❝❛✳

❊♠ P❛r✐s✱ ♥❛ ♠❡s♠❛ é♣♦❝❛✱ ▲❡✐❜♥✐③ ❝♦♠❡ç♦✉ ❛ ❞❡s❡♥✈♦❧✈❡r ♦s ♣r✐♥❝í♣✐♦s ❞❡ s✉❛ ✈❡rsã♦ ❞♦ ❝á❧❝✉❧♦✳ ❈♦♥s❝✐❡♥t❡ ❞❡ q✉❡✱ ♣❛r❛ ♦ ♣❧❡♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♠❛✲ t❡♠át✐❝❛✱ ❡r❛ ♥❡❝❡ssár✐❛ ❛ ❛❞♦çã♦ ❞❡ ✉♠❛ ♥♦t❛çã♦ ❝♦♥s❝✐st❡♥t❡ ❡ ❞❡ ❢á❝✐❧ ♠❛♥✐♣✉❧❛çã♦ ❡❧❡ ❞❡❞✐❝♦✉ ✉♠ ❜♦♠ t❡♠♣♦ ♣❛r❛ ♦ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞❡ s✉❛ ♥♦t❛çã♦ q✉❡ é ❜❛s✐❝❛♠❡♥t❡ ❛ ♠❡s♠❛ q✉❡ ✉s❛♠♦s ❛té ❤♦❥❡✳ ➱ s❛❜✐❞♦ q✉❡ s✉❛s ♣r✐♠❡✐r❛s ❛♥♦t❛çõ❡s ❡r❛♠ ❝♦♥❢✉s❛s ❡ ❞❡ ❞✐❢í❝✐❧ ❧❡✐t✉r❛✳ ❏á ❡♠ ✶✻✼✺ ❡❧❡ ❡s❝r❡✈❡✉ ✉♠ ❛rt✐❣♦ ♠❛♥✉s❝r✐t♦ ♦♥❞❡ ✉s❛✈❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❛ ♥♦t❛çã♦f(x)dx✳ ◆♦ ♠❡s♠♦ ❛rt✐❣♦ ❡❧❡ ❛♣r❡s❡♥t♦✉ ❛ r❡❣r❛ ♣❛r❛ ❞✐❢❡r❡♥❝✐❛çã♦ ❞❡ ✉♠

♣r♦❞✉t♦✳ ❊♠ ✶✻✼✻ ▲❡✐❜♥✐③ ❛♣r❡s❡♥t♦✉ ❛ ❞✐❢❡r❡♥❝✐❛çã♦ ❞❡✿

dxn =nxn−✶ dx

♣❛r❛ n✐♥t❡✐r♦ ♦✉ ❢r❛❝✐♦♥ár✐♦✳

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❈❛♣ít✉❧♦ ✷

❆❧❣✉♥s ❘❡s✉❧t❛❞♦s s♦❜r❡ ❈♦♥t✐♥✉✐❞❛❞❡

❡ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡

❖s r❡s✉❧t❛❞♦s q✉❡ ✐r❡♠♦s ❛♣r❡s❡♥t❛r ♥❡ss❡ ❝❛♣ít✉❧♦✱ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦✲ ❜❧❡♠❛s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ sã♦ ♣❛rt❡ ✐♥t❡❣r❛♥t❡ ❞♦ ❡♠❡♥tár✐♦ ❞❡ ✉♠ ♣r✐♠❡✐r♦ ❝✉rs♦ ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✳ ❙❡♥❞♦ q✉❡ t❛✐s r❡s✉❧t❛❞♦s s❡rã♦ t❛♠❜é♠ ✉t✐❧✐③❛❞♦s ❡♠ ❛❧❣✉♠ ♠♦♠❡♥t♦ ❡♠ ♥♦ss♦ tr❛❜❛❧❤♦ ♥❛ ❢♦r♠❛ ❞❡ ❛♣❧✐❝❛çã♦ ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ❛❧❣✉♥s ❞♦s ♣r♦❜❧❡♠❛s ❛♣r❡s❡♥t❛❞♦s✳

P❛r❛ ✉♠ ❛♣r♦❢✉♥❞❛♠❡♥t❡ ♥♦s tó♣✐❝♦s ❞❡ss❡ ❝❛♣ít✉❧♦ r❡❝♦♠❡♥❞❛♠♦s[✻]✱[✼]✱ [✶✵]❡[✶✶]✳

✷✳✶ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s ❙♦❜r❡ ❈♦♥t✐♥✉✐❞❛❞❡

◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ ❝♦♥t✐♥✉✐❞❛❞❡ q✉❡ ✉t✐❧✐③❛r❡♠♦s ♣♦s✲ t❡r✐♦r♠❡♥t❡ ♥♦s ♣r♦❜❧❡♠❛s q✉❡ ✐r❡♠♦s r❡s♦❧✈❡r ❝♦♠ ✉t✐❧✐③❛çã♦ ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✳

✷✳✶✳✶ ❈♦♥t✐♥✉✐❞❛❞❡ ❞❡ ✉♠❛ ❋✉♥çã♦

❆♣❡s❛r ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ s❡r ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❣❧♦❜❛❧ ❞❛s ❢✉♥çõ❡s✱ ❛ ❞❡✜♥✐çã♦ é ❢❡✐t❛ ♣♦♥t♦ ❛ ♣♦♥t♦✳ ❖✉ s❡❥❛✱ ❞❡✜♥✐♠♦s ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ✉♠❛ ❢✉♥çã♦ ❡♠ ✉♠ ❞❛❞♦ ♣♦♥t♦ ✭❞❡ s❡✉ ❞♦♠í♥✐♦✮✳

❉❡✜♥✐çã♦ ✶✳ ✭❈♦♥t✐♥✉✐❞❛❞❡ ❞❡ ✉♠❛ ❋✉♥çã♦ ❡♠ ✉♠ P♦♥t♦✮ ❙❡❥❛♠f:DR✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♥♦ ❞♦♠í♥✐♦ D⊂R ❡ a D✱ ✉♠ ♣♦♥t♦ t❛❧ q✉❡ t♦❞♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ ❝♦♥t❡♥❞♦ a

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❈❛♣ít✉❧♦ ✷✳ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s s♦❜r❡ ❈♦♥t✐♥✉✐❞❛❞❡ ❡ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ✶✶ ✐♥t❡rs❡❝t❛ D\ {a}✳ ❉✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ f é contínua ❡♠ a s❡

❧✐♠

x→af(x) =f(a)✳

❊①❡♠♣❧♦ ✶✳ ❈♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ✉♠❛ ❢✉♥çã♦ ❡♠ ✉♠ ♣♦♥t♦ ❡ ♦ ❢❛t♦ q✉❡ ♣❛r❛ t♦❞♦ ♣♦❧✐♥ô♠✐♦ ♣ ❝♦♠ a R✱ ❧✐♠

x→ap(x) = p(a)❀ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ s❡♥❞♦

f:RR ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❡❧❛ s❡rá ❝♦♥tí♥✉❛ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ s❡✉ ❞♦♠í♥✐♦✳ ❉❡✜♥✐çã♦ ✷✳ ✭❈♦♥t✐♥✉✐❞❛❞❡ ❞❡ ✉♠❛ ❢✉♥çã♦✮ ❙❡❥❛ f:D→R✳ ❉✐③❡♠♦s q✉❡ f é ❝♦♥tí♥✉❛ s❡ f ❢♦r ❝♦♥tí♥✉❛ ❡♠ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ D✳

❖❜s ✶✳ P❛r❛ ♣r♦✈❛r q✉❡ ✉♠❛ ❞❡t❡♠✐♥❛❞❛ ❢✉♥çã♦ é ❝♦♥tí♥✉❛✱ t❡♠♦s q✉❡ ✈❡r✐✜❝❛r ❛ ❞❡✜✲ ♥✐çã♦ ❡♠ ❝❛❞❛ ♣♦♥t♦ ❞❡ s❡✉ ❞♦♠í♥✐♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ ♠♦str❛r q✉❡ ✉♠❛ ❝❡rt❛ ❢✉♥çã♦ ◆➹❖ é ❝♦♥tí♥✉❛✱ ❜❛st❛ ❞❡s❝♦❜r✐r ✉♠ ♣♦♥t♦ ❞❡ s❡✉ ❞♦♠í♥✐♦ ♥♦ q✉❛❧ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥t✐✲ ♥✉✐❞❛❞❡ ❢❛❧❤❡✳

✷✳✶✳✷ ❆ Pr♦♣r✐❡❞❛❞❡ ❞❛ P❡r♠❛♥ê♥❝✐❛ ❞❡ ❙✐♥❛❧

❱❡❥❛♠♦s ♥❡ss❛ s✉❜s❡çã♦ ✉♠❛ ♣r♦♣♦s✐çã♦ q✉❡ tr❛t❛ s♦❜r❡ ♦ s✐♥❛❧ ❞❡ ❢✉♥çõ❡s ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♣♦♥t♦✳

Pr♦♣♦s✐çã♦ ✶✳ ❙❡❥❛♠ f : D R ✉♠❛ ❢✉♥çã♦ ❡ a D t❛❧ q✉❡ t♦❞♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ ❝♦♥t❡♥❞♦ a ✐♥t❡rs❡❝t❛ D\ {a}✳ ❙✉♣♦♥❤❛ q✉❡ f s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ a ❡ f(a) > ✵✳ ❊♥tã♦✱

❡①✐st❡ ✉♠ ♥ú♠❡r♦ r >✵ t❛❧ q✉❡✱

∀x(r−a✱a+r)Df(x)>✵✳

❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s s✉♣♦r✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ r > ✵✱ ❡①✐st❡ x (a−r✱a+r)D t❛❧ q✉❡ f(x)6 ✵✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ ❝❛❞❛ n N ♣♦❞❡♠♦s ❡s❝♦✲ ❧❤❡r an∈

a− ✶ n✱a+

n

∩D t❛❧ q✉❡ f(an)6✵✳

❆ss✐♠ ❝♦♥str✉í♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s (an) t❛✐s q✉❡ |an−a|<

n✳ ■ss♦ q✉❡r

❞✐③❡r q✉❡ ❧✐♠(an) = a✳ ◆♦ ❡♥t❛♥t♦✱ f(an) 6 ✵✱ ♣❛r❛ t♦❞♦ n ∈ N✳ ❊ss❛ s❡q✉ê♥❝✐❛ ♥ã♦

♣♦❞❡ ❝♦♥✈❡r❣✐r ♣❛r❛f(a)>✵✳ ■ss♦ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡f s❡r ❝♦♥tí♥✉❛ ❡♠ a✳

(24)

❈❛♣ít✉❧♦ ✷✳ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s s♦❜r❡ ❈♦♥t✐♥✉✐❞❛❞❡ ❡ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ✶✷ ♠✉❞❛♥ç❛s ❞❡ s✐♥❛✐s ❡♠ t♦r♥♦ ❞❡ss❡s ♣♦♥t♦s✳ ■ss♦ ♦❝♦rr❡✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦ ❝❛s♦ ❞❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s✳

✷✳✶✳✸ Pr♦♣r✐❡❞❛❞❡s ❞❛s ❋✉♥çõ❡s ❈♦♥tí♥✉❛s

▼♦str❛r❡♠♦s ❝♦♠ ❛ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡ q✉❡ ❛ ❛❞✐çã♦ ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s r❡s✉❧t❛ ❡♠ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✱ ♦ ♣r♦❞✉t♦ ❡♥tr❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡ ❛ ❞✐✈✐sã♦ f

g

❝♦♠ g6=✵ ♦♥❞❡ f✱g sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s t❛♠❜é♠ é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳

Pr♦♣♦s✐çã♦ ✷✳ ❙❡❥❛♠ f✱g : D R ❢✉♥çõ❡s✱ D R t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ a D✱ t♦❞♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ ❝♦♥t❡♥❞♦a ✐♥t❡rs❡❝t❛ D\a✳ ❙❡ f ❡ g sã♦ ❝♦♥tí♥✉❛s✱ ❡♥tã♦

✭✐✮ f+g:DR é ❝♦♥tí♥✉❛❀ ✭✐✐✮ f✳g:DR é ❝♦♥tí♥✉❛❀ ✭✐✐✐✮ f

g :D ∗

→R✱ ❡♠ q✉❡ D={xD|g(x)6=}✱ é ❝♦♥tí♥✉❛✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛aD✉♠ ❡❧❡♠❡♥t♦ q✉❛❧q✉❡r ❞♦ ❞♦♠í♥✐♦✳ ❈♦♠♦f❡gsã♦ ❝♦♥tí♥✉❛s✱

❧✐♠

x→af(x) =f(a) e x❧✐♠→ag(x) =g(a)✳

❊♥tã♦✱

❧✐♠

x→a(f+g)(x) = x❧✐♠→a(f(x) +g(x)) =x❧✐♠→af(x) +x❧✐♠→ag(x) = f(a) +g(a) = (f+g)(a)

❧✐♠

x→a(f✳g)(x) = x❧✐♠→a(f(x)✳g(x)) = x❧✐♠→af(x)✳ ❧✐♠x→ag(x)

= f(a)✳g(a) = (f✳g)(a)✳

❖❜s❡r✈❡ q✉❡✱ s❡g(a)6=✵✱ ❛ Pr♦♣✐❡❞❛❞❡ ❞❛ P❡r♠❛♥ê♥❝✐❛ ❞♦ ❙✐♥❛❧ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛

❞❡ ❛❧❣✉♠ r >✵ t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ x(a−r✱a+r)D✱ g(x)6=✵✳ ▼❛✐s ✉♠❛ ✈❡③✱

❧✐♠

x→a

f

g(x) =x❧✐♠→a

f(x) g(x) =

❧✐♠

x→af(x)

❧✐♠

x→ag(x)

= f(a) g(a) =

(25)

❈❛♣ít✉❧♦ ✷✳ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s s♦❜r❡ ❈♦♥t✐♥✉✐❞❛❞❡ ❡ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ✶✸

✷✳✶✳✹ ❖ ❚❡♦r❡♠❛ ❞❡ ❲❡✐❡rstr❛ss

❚♦❞❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛f:XR ❞❡✜♥✐❞❛ ♥✉♠ ❝♦♠♣❛❝t♦ ❳ é ❧✐♠✐t❛❞❛ ❡ ❛t✐♥❣❡ s❡✉s ❡①tr❡✲ ♠♦s✱ ✐st♦ é✱ ❡①✐st❡♠x✱x X t❛✐s q✉❡ f(x)6f(x)6f(x)✳

❉❡♠♦♥str❛çã♦✳ ❈♦♠ ❡❢❡✐t♦✱f(X)s❡♥❞♦ ❝♦♠♣❛❝t♦✱ ♣♦✐s é✱ ❛ ✐♠❛❣❡♠ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠✲

♣❛❝t♦ ♣♦r ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✱ é ❧✐♠✐t❛❞♦ ❡ ❢❡❝❤❛❞♦✳ ▲♦❣♦ s✉♣f(X) f(X) ❡ ✐♥❢f(X) f(X)✳ P♦rt❛♥t♦ ❡①✐st❡♠ x✱x X t❛✐s q✉❡ ✐♥❢f(X) =f(x) ❡ rmsupf(X) =f(x)

✷✳✶✳✺ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ■♥t❡r♠❡❞✐ár✐♦ ✭❚❱■✮

❙❡❥❛f: [a✱b]R❝♦♥tí♥✉❛✳ ❙❡ f(a)< d < f(b) ❡♥tã♦ ❡①✐st❡ c(ab) t❛❧ q✉❡ f(c) =d✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ A= {x [a✱b]❀f(x)< d}✳ ❖ ❝♦❥✉♥t♦ A ♥ã♦ é ✈❛③✐♦ ♣♦✐s f(a) < d✳

❆✜r♠❛♠♦s q✉❡ ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦ ❞❡A é ♠❛✐♦r ❞♦ q✉❡ t♦❞♦s ♦s ♦✉tr♦s✳ ❈♦♠ ❡❢❡✐t♦ s❡❥❛ α A✳ ❈♦♠♦ f(α) < d✱ ✈❡♠♦s q✉❡ α6= b ❡✱ ♣♦rt❛♥t♦✱ α < b✳ ❚♦♠❛♥❞♦ ε = d−f(α)✱

❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ f ♥♦ ♣♦♥t♦ α ♥♦s ❞á ✉♠ δ > ✵ ✭q✉❡ t♦♠❛r❡♠♦s ♣❡q✉❡♥♦✱ ❞❡ ♠♦❞♦ ❛

t❡r [α✱α+δ][a✱b]✮ t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ x [α✱α+δ) t❡♠✲s❡ f(x)< f(α) +ε✱ ♦✉ s❡❥❛✱ f(x)< d✳ ❆ss✐♠✱ t♦❞♦s ♦s ♣♦♥t♦s ❞♦ ✐♥t❡r✈❛❧♦[α✱α+δ)♣❡rt❡♥❝❡♠ ❛A✳ ❆❣♦r❛ ♣♦♥❤❛♠♦s c=supA✳ ❈♦♠♦c é ❧✐♠✐t❡ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦sxn∈A✱ t❡♠♦s✿

f(c) =❧✐♠f(xn)6d✳

❈♦♠♦ A ♥ã♦ ♣♦ss✉✐ ♠❛✐♦r ❡❧❡♠❡♥t♦✱ ♥ã♦ s❡ t❡♠ c A✳ ▲♦❣♦ ♥ã♦ ✈❛❧❡ f(c) < d✱ ♦ q✉❡

♥♦s ♦❜r✐❣❛ ❛ ❝♦♥❝❧✉✐r q✉❡f(c) =d✳

✷✳✷ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s ❙♦❜r❡ ❉❡r✐✈❛❞❛s

(26)

❈❛♣ít✉❧♦ ✷✳ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s s♦❜r❡ ❈♦♥t✐♥✉✐❞❛❞❡ ❡ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ✶✹

✷✳✷✳✶ ❆ ❉❡✜♥✐çã♦ ❞❡ ❉❡r✐✈❛❞❛

❆ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ y = f(x) ❞❡✜♥✐❞❛ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I✱ ❡♠ ✉♠ ♣♦♥t♦ x I✱ é ❞❛❞❛ ♣♦r

f′(x) = ❧✐♠

x→a

f(x+h) −f(x)

h ✱

❝❛s♦ ❡st❡ ❧✐♠✐t❡ ❡①✐st❛✳

❙❡ ♦ ❧✐♠✐t❡ ❡①✐st✐r ❛ ❢✉♥çã♦ fé ❞✐t❛ ❞❡r✐✈á✈❡❧ ❡♠ x✳ ❙❡ f é ❞❡r✐✈á✈❡❧ ♣❛r❛ t♦❞♦ ♣♦♥t♦

❞❡ s❡✉ ❞♦♠í♥✐♦✱ ❞✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ é ❞❡r✐✈á✈❡❧ ❡ q✉❡ ❛ ❢✉♥çã♦f′ :

IRq✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ x∈I ♦ ✈❛❧♦r f′(x) é ❛ ❢✉♥çã♦ ❞❡r✐✈❛❞❛ ❞❡f✳

✷✳✷✳✷ ❉❡r✐✈❛❞❛ ❈♦♥❝❡✐t♦ ●❡♦♠étr✐❝♦

●❡♦♠❡tr✐❝❛♠❡♥t❡ ❛ ❞❡r✐✈❛❞❛ é ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛ ✉♠ ❝❡rt♦ ♣♦♥t♦ ❞❛ ❝✉r✈❛ q✉❡ é ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦y=f(x)✳ ❊♥❝♦♥tr❛r ❛ ❡q✉❛çã♦ ❞❡ss❛ r❡t❛ ❡stá r❡✲

❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♥❝♦♥tr❛r ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛✱ ♣r♦❜❧❡♠❛ q✉❡ ♠♦t✐✈♦✉ ◆❡✇t♦♥ ♥❛ ❝r✐❛çã♦ ❞♦ ❈á❧❝✉❧♦✳

❙❡❥❛f(x) ✉♠❛ ❢✉♥çã♦ ❡ s❡❥❛ x✵ ✉♠ ♣♦♥t♦ ❞❡ s❡✉ ❞♦♠í♥✐♦✱ ❝♦♠ x✶ =x✵+h✳ ◆❛ ✜❣✉r❛

✷✳✶✱ ✈❡❥❛ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ f(x)✱ ♦♥❞❡ tr❛ç❛♠♦s ❛ r❡t❛ s❡❝❛♥t❡ q✉❡ ♣❛ss❛ ♣❡❧♦s

♣♦♥t♦s ❆ ❂ (x✱f(x)) ❡ ❇ ❂ (x✱f(x))✳ ◆♦t❡ q✉❡ ♦ ❣rá✜❝♦ ❢♦✐ tr❛ç❛❞♦ s✉♣♦♥❞♦ h > ✵✳

◆♦ ❡♥t❛♥t♦✱ ❛ s✐t✉❛çã♦ ❤ ❁ ✵ t❛♠❜é♠ ❞❡✈❡ s❡r ❝♦♥s✐❞❡r❛❞❛✳

❖ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ s❡❝❛♥t❡ à ❝✉r✈❛ q✉❡ ♣❛ss❛ ♣♦r ❆ ❂ (x✱f(x)) ❡ ❇ ❂ (x✱f(x)) é ❞❛❞♦ ♣♦r✿

f(x) −f(x) x−x =

f(x +h) −f(x)

h ✳

❚♦♠❛♥❞♦ ❤ ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠♦ ❞❡ ③❡r♦✱ ♦❜t❡♠♦s r❡t❛s q✉❡ ❝♦rt❛♠ ❛ ❝✉r✈❛ ❡♠ ❞♦✐s ♣♦♥t♦sA❡ Bi ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠♦s✳ ■♥t✉✐t✐✈❛♠❡♥t❡ ♣❡r❝❡❜❡♠♦s q✉❡ q✉❛♥❞♦ x✵+h s❡

❛♣r♦①✐♠❛ ❞❡x ❡♥tã♦ ♦s ♣♦♥t♦s f(x+h)❡f(x)♦♥❞❡ ❛ r❡t❛ ❝♦rt❛ ❛ ❝✉r✈❛ ✜❝❛♠ ❝❛❞❛ ✈❡③

♠❛✐s ♣ró①✐♠♦s ❡ ❛ss✐♠ ❡st❛s s❡ ❛♣r♦①✐♠❛♠ ❝❛❞❛ ✈❡③ ♠❛✐s ❞❛ t❛♥❣❡♥t❡ ❡♠x✳ ◗✉❛♥❞♦ ❤

s❡ ❛♣r♦①✐♠❛ ❞❡ ③❡r♦✱ s❡ ♦ q✉♦❝✐❡♥t❡

f(x+h) −f(x)

h ✱

(27)

❈❛♣ít✉❧♦ ✷✳ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s s♦❜r❡ ❈♦♥t✐♥✉✐❞❛❞❡ ❡ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ✶✺ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ t❛♥❣❡♥t❡✳

❋✐❣✉r❛ ✷✳✶✿

◆❛ ✈❡r❞❛❞❡✱ ♦ q✉❡ ❢❛③❡♠♦s é ❞❡✜♥✐r r❡t❛ t❛♥❣❡♥t❡ ❞❛ ❝✉r✈❛ ❡♠ P = (x✱f(x)) ❝♦♠♦ ❛

r❡t❛ q✉❡ ♣❛ss❛ ♣♦rP ❡ ❝✉❥♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r é ❞❛❞♦ ♣♦r

f′(x) = ❧✐♠

h→✵

f(x+h) −f(x) h

❈❛s♦ ♦ ❧✐♠✐t❡ ♥ã♦ ❡①✐st❛✱ ♥ã♦ ❤á r❡t❛ t❛♥❣❡♥t❡ ❛♦ ♣♦♥t♦✱ ❡♠ r❡s✉♠♦✿ ❆ r❡t❛ t❛♥❣❡t❡ ❛ ✉♠❛ ❝✉r✈❛ q✉❡ é ❣rá✜❝♦ ❞❡ y=f(x) ❡♠ ✉♠ ♣♦♥t♦ P = (x✱f(x)) é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r P ❡ ❝✉❥♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r é ❞❛❞♦ ♣♦r

f′(x) = ❧✐♠

h→✵

f(x+h) −f(x)

h ✱

(28)

❈❛♣ít✉❧♦ ✷✳ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s s♦❜r❡ ❈♦♥t✐♥✉✐❞❛❞❡ ❡ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ✶✻ ❊①❡♠♣❧♦ ✷✳ ❱❛♠♦s ♠♦str❛r q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ f(x) =ax✷+bx+c✱ ❝♦♠ a6=✵

é f′(x) =✷ax+b✳ ❉❡ ❢❛t♦✱ ✈❡❥❛✿

f′(x) = ❧✐♠

h→✵

f(x +h) −f(x) h

= ❧✐♠

h→✵

(a(x+h)✷+b(x+h) +c) − ((a(x)✷+bx+c)) h

= ❧✐♠

h→✵

(a(x)✷+✷ax✵h+ah✷ +bx✵+bh+c) − (a(x✵)✷+bx✵+c))

h

= ❧✐♠

h→

a(x)✷+✷axh+ah✷ +bx+bh+c−a(x)✷−bx−c h

= ❧✐♠

h→✵

✷axh+ah✷ +bh h

= ❧✐♠

h→✵

✷axh+bh

h +h❧✐♠→✵ ah✷

h

= ❧✐♠

h→✵(✷ax✵+b) +h❧✐♠→✵ah

= ✷ax+b

❖ q✉❡ ♠♦str❛ q✉❡ s❡ f(x) =ax✷+bx+c✱ ❝♦♠ a6=✵ ❡♥tã♦ f′(x) =✷ax+b✳

✷✳✷✳✸ ❱❛❧♦r❡s ❊①tr❡♠♦s ❞❡ ✉♠❛ ❋✉♥çã♦

❱❡❥❛♠♦s ❛❣♦r❛ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s q✉❡ ✐rã♦ ❞❛r s✉st❡♥t❛çã♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❞♦✐s t❡♦✲ r❡♠❛s ✐♠♣♦rt❛♥t❡♥s ❞♦ ❝á❧❝✉❧♦✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ ❡ ♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✳

❉❡✜♥✐çã♦ ✸✳ ❯♠❛ ❢✉♥çã♦f:DR t❡♠ ♠á①✐♠♦ ❛❜s♦❧✉t♦ ❡♠ cs❡ f(x)6f(c)♣❛r❛ t♦❞♦ x ♥♦ ❞♦♠í♥✐♦ ❉ ❞❡ f✳ ◆❡st❡ ❝❛s♦✱ ♦ ✈❛❧♦r f(c) é ❝❤❛♠❛❞♦ ✈❛❧♦r ♠á①✐♠♦ ❞❡ f ❡♠ D✳

❉❡✜♥✐çã♦ ✹✳ ❯♠❛ ❢✉♥çã♦f:DR t❡♠ ♠í♥✐♠♦ ❛❜s♦❧✉t♦ ❡♠cs❡ f(x)>f(c) ♣❛r❛ t♦❞♦ x ♥♦ ❞♦♠í♥✐♦ ❉ ❞❡ f✳ ◆❡st❡ ❝❛s♦✱ ♦ ✈❛❧♦r f(c) é ❝❤❛♠❛❞♦ ✈❛❧♦r ♠í♥✐♠♦ ❞❡f ❡♠ D✳

❖s ✈❛❧♦r❡s ❞❡ ♠á①✐♠♦ ❡ ♠í♥✐♠♦ ❛❜s♦❧✉t♦ ❞❡ ✉♠❛ ❢✉♥çã♦ sã♦ ❝❤❛♠❛❞♦s ✈❛❧♦r❡s ❡①tr❡♠♦s ❞❛ ❢✉♥çã♦✳ ➱ ❝❧❛r♦ q✉❡ ♥❡♠ t♦❞❛s ❛s ❢✉♥çõ❡s tê♠ t❛❧ ♣r♦♣r✐❡❞❛❞❡✱ ❡❧❛s r❡str✐❣❡♠✲s❡ ❛ t❡r❡♠ ✈❛❧♦r❡s ❞❡ ♠á①✐♠♦ ❡ ♠í♥✐♠♦ ❧♦❝❛❧ ♦ q✉❛❧ ✐r❡♠♦s ❧♦❣♦ ❛ ❢r❡♥t❡ ♠♦str❛r ✉♠ t❡♦r❡♠❛ ✐♠♣♦rt❛♥t❡ ♣❛r❛ t❛✐s ♣♦♥t♦s ❞♦ ❣rá✜❝♦ ❞❡ss❛s ❢✉♥çõ❡s✳

❊①❡♠♣❧♦ ✸✳ ◆❛ ✜❣✉r❛ ✷✳✷ f(x) t❡♠ ♠í♥✐♠♦ ❛❜s♦❧✉t♦ ✐❣✉❛❧ ❛ ✲ ✶✻ ❡ ♥❛ ✜❣✉r❛ ✷✳✸ g(x)

(29)

❈❛♣ít✉❧♦ ✷✳ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s s♦❜r❡ ❈♦♥t✐♥✉✐❞❛❞❡ ❡ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ✶✼

❋✐❣✉r❛ ✷✳✷✿ ❋✐❣✉r❛ ✷✳✸✿

❋✐❣✉r❛ ✷✳✹✿

❉❡✜♥✐çã♦ ✺✳ ❯♠❛ ❢✉♥çã♦ t❡♠ ♠á①✐♠♦ ❧♦❝❛❧ ✭♦✉ ♠á①✐♠♦ r❡❧❛t✐✈♦✮ ❡♠ ✉♠ ♣♦♥t♦c❞❡ s❡✉

❞♦♠í♥✐♦✱ s❡ ❡①✐st✐r ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I✱ t❛❧ q✉❡ c I ❡ f(x) 6f(c) ♣❛r❛ t♦❞♦ x I✳ ◆❡st❡

❝❛s♦✱ ❞✐③❡♠♦s q✉❡ f(c) é ✈❛❧♦r ♠á①✐♠♦ ❧♦❝❛❧ ❞❡ ❢✳

❉❡✜♥✐çã♦ ✻✳ ❯♠❛ ❢✉♥çã♦ t❡♠ ♠í♥✐♠♦ ❧♦❝❛❧ ✭♦✉ ♠í♥✐♠♦ r❡❧❛t✐✈♦✮ ❡♠ ✉♠ ♣♦♥t♦ c❞❡ s❡✉

❞♦♠í♥✐♦✱ s❡ ❡①✐st✐r ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I✱ t❛❧ q✉❡ c I ❡ f(x) >f(c) ♣❛r❛ t♦❞♦ x I✳ ◆❡st❡

❝❛s♦✱ ❞✐③❡♠♦s q✉❡ f(c) é ✈❛❧♦r ♠í♥✐♠♦ ❧♦❝❛❧ ❞❡ ❢✳

❚❡♦r❡♠❛ ✶✳ ❙❡❥❛ f:I →R ✉♠❛ ❢✉♥çã♦ f ❝♦♥tí♥✉❛ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I✳ ❙❡ f t❡♠ ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦ ❧♦❝❛❧ ❡♠ x =c✱ cI ❡ f é ❞❡r✐✈á✈❡❧ ❡♠ c ❡♥tã♦ f′

(c) =✵✳

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