O CÁLCULO DIFERENCIAL E INTEGRAL COMO FERRAMENTA INTERDISCIPLINAR 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❘❖❋▼❆❚

❖ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ❝♦♠♦ ❢❡rr❛♠❡♥t❛

✐♥t❡r❞✐s❝✐♣❧✐♥❛r ♥♦ ❊♥s✐♥♦ ▼é❞✐♦

❏❛♥✐❧s♦♥ ❈❧❛②❞s♦♥ ❙✐❧✈❛ ❇r✐t♦

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❏❛♥✐❧s♦♥ ❈❧❛②❞s♦♥ ❙✐❧✈❛ ❇r✐t♦

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✿

❖ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r

♥♦ ❊♥s✐♥♦ ▼é❞✐♦

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

①①①① ❖ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳

◆♦♠❡ ❞♦ ❆❧✉♥♦ ✕ ❚❡r❡s✐♥❛✿ ❆◆❖✳

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

✶✳ ▼❛t❡♠át✐❝❛

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✐

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❆❣r❛❞❡❝✐♠❡♥t♦s

❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r t♦❞❛s ❛s ♦♣♦rt✉♥✐❞❛❞❡s q✉❡ ♠❡ ❝♦♥❢❡r✐✉ ❡ ♣❡❧❛ ❜♦❛ s❛ú❞❡✱ ♠❡ ♣❡r♠✐t✐♥❞♦ ♦✉s❛r ♥❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❜✉s❝❛r s❡♠♣r❡ ♦ ♠❡❧❤♦r✳

❆❣r❛❞❡ç♦ ❛ t♦❞❛ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛ q✉❡ s❡♠♣r❡ ♠❡ ✐♥s♣✐r❛r❛♠ ❡ ✐♥❝❡♥t✐✈❛r❛♠ ♥❛ ❜✉s❝❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ♠❡s♠♦ s❡♥❞♦ ♠✉✐t❛s ❛s ❞✐✜❝✉❧❞❛❞❡s ♣♦r t♦❞♦s ❡ss❡s ❛♥♦s✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♠✉❧❤❡r P❛trí❝✐❛✱ q✉❡ ❛♦ ❧♦♥❣♦ ❞❡ss❡s ❞♦✐s ❛♥♦s ❞❡ ❝✉rs♦ t❡✈❡ ♣❛❝✐ê♥❝✐❛ q✉❛♥❞♦ ❡✉ ❡st❛✈❛ ❞✐st❛♥t❡ ❛ tr❛❜❛❧❤♦✱ ♣❡❧❛ s✉❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❛♣♦✐♦ ❡♥q✉❛♥t♦ ♠❡ ❞❡❞✐❝❛✈❛ ❛♦s ❡st✉❞♦s✳

❆❣r❛❞❡ç♦ ❛ ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ ♣❡❧❛ ❝♦❧❛❜♦r❛çã♦ ♥♦ ❞❡❝♦rr❡r ❞❡ t♦❞♦ ♦ ❝✉rs♦ ♥❛ ❡①❡❝✉çã♦ ❞❡ tr❛❜❛❧❤♦s ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦s ♠✉✐t♦s ♠♦♠❡♥t♦s ❞❡ ❡st✉❞♦ ❡♠ q✉❡ t♦❞♦s s❡ ❛❥✉❞❛✈❛♠ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❡①❡r❝í❝✐♦s ❡ ♣❡sq✉✐s❛s✱ ♦ q✉❡ ♠❡ ❢❡③ ❛♣r✐♠♦r❛r ❝❛❞❛ ✈❡③ ♠❛✐s ❡♠ ❝❛❞❛ ❞✐s❝✐♣❧✐♥❛✳

❆❣r❛❞❡ç♦ ❛ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ q✉❡ ♠❡ ❢♦✐ ♠✉✐t♦ út✐❧ ♥♦ ❞❡❝♦rr❡r ❞❡ss❡s ❞♦✐s ❛♥♦s✳

❊♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♥♦ss♦s ♣r♦❢❡ss♦r❡s✱ ♣♦✐s ♠❡ ✜③❡r❛♠ ❛♣❛✐①♦♥❛r✲♠❡ ❛✐♥❞❛ ♠❛✐s ♣❡❧❛ ▼❛t❡♠át✐❝❛ ❡ ❛♦ ♣r♦❢❡ss♦r P❛✉❧♦ ❆❧❡①❛♥❞r❡✱ q✉❡ ♠✉✐t♦ ♠❡ ❛❥✉❞♦✉ ♥❛ ❝♦♥str✉çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

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✐✐✐

✏◆✐♥❣✉é♠ ❝❛♠✐♥❤❛ s❡♠ ❛♣r❡♥❞❡r ❛ ❝❛♠✐✲ ♥❤❛r✱ s❡♠ ❛♣r❡♥❞❡r ❛ ❢❛③❡r ♦ ❝❛♠✐♥❤♦ ❝❛✲ ♠✐♥❤❛♥❞♦✱ r❡❢❛③❡♥❞♦ ❡ r❡t♦❝❛♥❞♦ ♦ s♦♥❤♦ ♣❡❧♦ q✉❛❧ s❡ ♣ôs ❛ ❝❛♠✐♥❤❛r✧✳

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❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ♥❛ ♣r✐♠❡✐r❛ ♣❛rt❡✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❜r❡✈❡ ❤✐stór✐❝♦ ❞♦s ♣r✐♥❝✐♣❛✐s ❞❡s❡♥✈♦❧✈❡❞♦r❡s ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ❛♦ ❧♦♥❣♦ ❞❛ ❤✐stór✐❛❀ ❡♠ s❡❣✉✐❞❛✱ é ❢❡✐t❛ ✉♠❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ s♦❜r❡ ❛❧❣✉♥s tó♣✐❝♦s ❞♦ ❝á❧❝✉❧♦✱ ♣♦r ❡①❡♠♣❧♦✿ t❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✱ t❡st❡ ❞❛ ♣r✐♠❡✐r❛ ❡ ❞❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛✱ t❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦❀ ❋✐♥❛❧♠❡♥t❡✱ ✈✐s❛♥❞♦ ♠♦str❛r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ❡st✉❞♦ ❞♦ ❝á❧❝✉❧♦ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❛♣r❡s❡♥t❛♠♦s ❡①❡♠♣❧♦s s✐♠♣❧❡s ❡ ❛♣❧✐❝❛çõ❡s ♠❛✐s ❡❧❛❜♦r❛❞❛s ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ❡♠ ♦✉tr❛s ❈✐ê♥❝✐❛s✳

P❛❧❛✈r❛s ❝❤❛✈❡✿ ❝á❧❝✉❧♦✱ ❞❡r✐✈❛❞❛✱ ✐♥t❡❣r❛❧✱ ❈✐ê♥❝✐❛s✳

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❆❜str❛❝t

❚❤✐s ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✿ t❤❡ ✜rst ♣❛rt ✐s ❛ ❜r✐❡❢ ❤✐st♦r② ♦❢ t❤❡ ♠❛✐♥ ❞❡✈❡❧♦♣❡rs ♦❢ t❤❡ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s t❤r♦✉❣❤♦✉t ❤✐st♦r②✱ t❤❡♥✱ t❤❡r❡ ✐s ❛ t❤❡♦r❡t✐❝❛❧ ❝❛❧❝✉❧❛t✐♦♥ ♦♥ s♦♠❡ t♦♣✐❝s✱ ❢♦r ❡①❛♠♣❧❡✿ ❱❛❧✉❡ ❚❤❡♦r❡♠ ❛✈❡r❛❣❡ t❡st t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✱ ❋✉♥❞❛♠❡♥t❛❧ t❤❡♦r❡♠ ♦❢ ❈❛❧❝✉❧✉s❀ ❋✐♥❛❧❧②✱ ✐♥ ♦r❞❡r t♦ s❤♦✇ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ st✉❞② ♦❢ ❝❛❧❝✉❧✉s ✐♥ ❤✐❣❤ s❝❤♦♦❧✱ ✇❡ ♣r❡s❡♥t s✐♠♣❧❡ ❡①❛♠♣❧❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♠♦r❡ ❡❧❛❜♦r❛t❡ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s ✐♥ ♦t❤❡r s❝✐❡♥❝❡s✳

❑❊❨❲❖❘❉❙✿ ❝❛❧❝✉❧✉s✱ ❞❡r✐✈❛t✐✈❡✱ ✐♥t❡❣r❛❧✱ ❙❝✐❡♥❝❡s✳

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❙✉♠ár✐♦

❘❡s✉♠♦ ✐✈

❆❜str❛❝t ✈

✶ ❯♠ ♣♦✉❝♦ ❞❛ ❍✐stór✐❛ ❞♦ ❈á❧❝✉❧♦ ✸

✶✳✶ ❆♥t✐❣✉✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ■❞❛❞❡ ▼é❞✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ■❞❛❞❡ ▼♦❞❡r♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✹ ■❞❛❞❡ ❈♦♥t❡♠♣♦râ♥❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✷ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✼

✷✳✶ ❆❧❣✉♠❛s ❞❡r✐✈❛❞❛s ❜ás✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✷ ❖ ❚❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✸ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✹ ❈r❡s❝✐♠❡♥t♦ ❡ ❞❡❝r❡s❝✐♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✺ ❉❡r✐✈❛❞❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✺✳✶ ❯s♦ ❞❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ♣❛r❛ ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✸ ❯♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❛♦ ❈á❧❝✉❧♦ ■♥t❡❣r❛❧ ✶✾

✸✳✶ ■♥t❡❣r❛❧ ❞❡✜♥✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✶✳✶ ❖ q✉❡ é ár❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❞❛ ✐♥t❡❣r❛❧ ❞❡✜♥✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✷ ❖ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ✲ ❚❋❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✷✳✶ ❚❡♦r❡♠❛ ❞♦ ✈❛❧♦r ♠é❞✐♦ ♣❛r❛ ✐♥t❡❣r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✷✳✷ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ✭❚❋❈✮ ✲ P❛rt❡ ■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷✳✸ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ✭❚❋❈✮ ✲ P❛rt❡ ■■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

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❙✉♠ár✐♦ ✈✐✐

✸✳✸ ■♥t❡❣r❛❧ ✐♥❞❡✜♥✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ ✐♥t❡❣r❛❧ ✐♥❞❡✜♥✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✹ Pr♦❜❧❡♠❛s ❛♣❧✐❝❛❞♦s ❛ ♦✉tr❛s ❝✐ê♥❝✐❛s ✷✻

✺ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✹✶

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■♥tr♦❞✉çã♦

❙❡❣✉♥❞♦ ❛ ♣r♦♣♦st❛ ❞♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✭P❈◆✬s✮ ❬✹❪✱ ♦ ❝✉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✳

■♥❢❡❧✐③♠❡♥t❡✱ r❡s✉❧t❛❞♦s ❞❡ ❛✈❛❧✐❛çõ❡s ✐♥st✐t✉❝✐♦♥❛✐s ❝♦♠♦ ♦ ❙❆❊❇ ✭❙✐st❡♠❛ ◆❛❝✐♦♥❛❧ ❞❡ ❆✈❛❧✐❛çã♦ ❊s❝♦❧❛r ❞❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛✮ ❡ ♦ ❊◆❊▼ ✭❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✮✱ ♣r♦♠♦✈✐❞♦s ♣❡❧♦ ●♦✈❡r♥♦ ❋❡❞❡r❛❧✱ r❡✈❡❧❛♠ q✉❡ ♠✉✐t♦s ❛❧✉♥♦s t❡r♠✐♥❛♠ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ❝♦♠ ❞✐✜❝✉❧❞❛❞❡s ❡♠ ❝♦♥❝❡✐t♦s ❡ ♣r♦❝❡❞✐♠❡♥t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ▼❛t❡♠át✐❝❛✱ t❛✐s ❝♦♠♦ ♦♣❡r❛r ❝♦♠ ♥ú♠❡r♦s r❡❛✐s✱ ✐♥t❡r♣r❡t❛r ❣rá✜❝♦s ❡ t❛❜❡❧❛s✱ ❞❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✳

❆♣❡s❛r ❞❡ ❛❧❣✉♥s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❛♣r❡s❡♥t❛r❡♠ tó♣✐❝♦s r❡❧❛t✐✈♦s ❛♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ ❝♦♠♦ ❧✐♠✐t❡✱ ❞❡r✐✈❛❞❛ ❡ ✐♥t❡❣r❛❧✱ ❡ss❡s t❡♠❛s✱ ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s✱ ♥ã♦ sã♦ ❡♥s✐♥❛❞♦s s♦❜ ♦ ♣r❡t❡①t♦ ❞❡ s❡r❡♠ ❞✐❢í❝❡✐s ❡ ✐♠♣ró♣r✐♦s ❛ ❡ss❡ s❡❣♠❡♥t♦ ❞❛ ❡❞✉❝❛çã♦ ❡ ❛❝❛❜❛♠ ✜❝❛♥❞♦ r❡str✐t♦s ❛♦ ❡♥s✐♥♦ s✉♣❡r✐♦r✱ ♦ q✉❡ ❧❡✈❛ ♦ ❈á❧❝✉❧♦ ❛ ❢❛③❡r ♣❛rt❡ ❞♦ ❧✐✈r♦ ❞✐❞át✐❝♦✱ ♠❛s ♥ã♦ ❞♦ ❝✉rrí❝✉❧♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳

❙❡❣✉♥❞♦ ●❡r❛❧❞♦ ➪✈✐❧❛✱ ✏♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛ ♣♦❞❡ s❡r ❡♥s✐♥❛❞♦✱ ❝♦♠ ❣r❛♥❞❡ ✈❛♥t❛✲ ❣❡♠✱ ❧♦❣♦ ♥❛ ♣r✐♠❡✐r❛ sér✐❡ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ❛♦ ❧❛❞♦ ❞♦ ❡♥s✐♥♦ ❞❡ ❢✉♥çõ❡s✑ ❬✷❪ ❡ ❬✸❪✳ P❛r❛ ❡❧❡ ♦ ❡♥s✐♥♦ ❞♦ ❝á❧❝✉❧♦ é ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛✱ ♣♦✐s ❛❧é♠ ❞❡ ❛❥✉❞❛r ♥♦ tr❛t❛♠❡♥t♦ ❞❡ ✐♥ú♠❡r❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❢✉♥çõ❡s ❡ ❞❡ t❡r ❛♣❧✐❝❛çõ❡s ✐♥t❡r❡ss❛♥t❡s ❡♠ ♣r♦❜❧❡♠❛s ❞❡ ♠á✲ ①✐♠♦ ❡ ♠í♥✐♠♦✱ ❝r❡s❝✐♠❡♥t♦ ❡ ❞❡❝r❡s❝✐♠❡♥t♦✱ ❞❡♥tr❡ ♦✉tr♦s✱ ✐♥t❡❣r❛✲s❡ ❤❛r♠♦♥✐♦s❛♠❡♥t❡ ❝♦♠ ♠✉✐t❛s ❞❛s ❝✐ê♥❝✐❛s ❝♦♥❤❡❝✐❞❛s✱ ♣♦✐s ♦ ❝á❧❝✉❧♦ ♣♦❞❡ t♦r♥❛r ♦ ❡st✉❞♦ ❞❡ ❛❧❣✉♥s ❞❡st❡s tó♣✐❝♦s ♠❛✐s s✐♠♣❧❡s ❡ ❝♦♠♣r❡❡♥sí✈❡✐s ♣❛r❛ ♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳

❊♠ ❋ís✐❝❛✱ ♦ ❝á❧❝✉❧♦ é ❛♣❧✐❝❛❞♦ ♥♦ ❡st✉❞♦ ❞♦ ♠♦✈✐♠❡♥t♦✱ ♣r❡ssã♦✱ ❞❡♥s✐❞❛❞❡ ❡ ♦✉tr❛s

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❙✉♠ár✐♦ ✷

❛♣❧✐❝❛çõ❡s✳ P♦❞❡ s❡r ✉s❛❞♦✱ ❡♠ ❝á❧❝✉❧♦ ♥✉♠ér✐❝♦✱ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ r❡t❛ q✉❡ ♠❡❧❤♦r r❡♣r❡✲ s❡♥t❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❡♠ ✉♠ ❞♦♠í♥✐♦✳ ◆❛ ❡s❢❡r❛ ❞❛ ♠❡❞✐❝✐♥❛✱ ♦ ❝á❧❝✉❧♦ ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ❡♥❝♦♥tr❛r ♦ â♥❣✉❧♦ ót✐♠♦ ♥❛ r❛♠✐✜❝❛çã♦ ❞♦s ✈❛s♦s s❛♥❣✉í♥❡♦s ♣❛r❛ ♠❛①✐♠✐③❛r ❛ ❝✐r❝✉❧❛çã♦✱ ❡ ❛té ♠❡s♠♦ ❞❡t❡r♠✐♥❛r ♦ t❛♠❛♥❤♦ ♠á①✐♠♦ ❞❡ ♠♦❧é❝✉❧❛s q✉❡ sã♦ ❝❛♣❛③❡s ❞❡ ❛tr❛✈❡ss❛r ❛ ♠❡♠❜r❛♥❛ ♣❧❛s♠át✐❝❛ ❡♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ s✐t✉❛çã♦✱ ♥♦r♠❛❧ ♦✉ ✐♥❞✉③✐❞❛✱ ❡♠ ❝é❧✉❧❛s✳ ◆❛ ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛✱ ♥♦ ❡st✉❞♦ ❞♦s ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s✱ ♦ ❝á❧❝✉❧♦ é ✉s❛❞♦ ♣❛r❛ ❡♥❝♦♥tr❛r ♣♦♥t♦s ♠á①✐♠♦s ❡ ♠í♥✐♠♦s✱ ❛ ✐♥❝❧✐♥❛çã♦✱ ❝♦♥❝❛✈✐❞❛❞❡ ❡ ♣♦♥t♦s ❞❡ ✐♥✢❡①ã♦✳ ◆❛ ❡❝♦♥♦♠✐❛ ♦ ❝á❧❝✉❧♦ ♣❡r♠✐t❡ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦ ❧✉❝r♦ ♠á①✐♠♦ ❢♦r♥❡❝❡♥❞♦ ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❝❛❧❝✉❧❛r ❢❛❝✐❧♠❡♥t❡ t❛♥t♦ ♦ ❝✉st♦ ♠❛r❣✐♥❛❧ q✉❛♥t♦ ❛ r❡♥❞❛ ♠❛r❣✐♥❛❧✳ ❊❧❡ t❛♠❜é♠ ❛❥✉❞❛ ❛ ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ❛♣r♦①✐♠❛❞❛s ❞❡ ❡q✉❛çõ❡s✱ ✉t✐❧✐③❛♥❞♦ ♠ét♦❞♦s ❝♦♠♦ ♦ ♠ét♦❞♦ ❞❡ ◆❡✇t♦♥✱ ✐t❡r❛çã♦ ❞❡ ♣♦♥t♦ ✜①♦ ❡ ❛♣r♦①✐♠❛çã♦ ❧✐♥❡❛r✳

◆❡ss❡ tr❛❜❛❧❤♦✱ ♠♦str❛r❡♠♦s ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞♦ ❝á❧❝✉❧♦✱ ❛♣r❡s❡♥t❛♥❞♦ ❛❧❣✉♥s ❞♦s s❡✉s ♣r✐♥❝✐♣❛✐s ❝♦❧❛❜♦r❛❞♦r❡s✱ ✐❞❡♥t✐✜❝❛♥❞♦✲♦s ❞❡s❞❡ ❛ ❛♥t✐❣✉✐❞❛❞❡✱ ♣❛ss❛♥❞♦ ♣❡❧❛ ■❞❛❞❡ ▼é❞✐❛✱ ❛té ❝❤❡❣❛r ♥❛ ■❞❛❞❡ ▼♦❞❡r♥❛✱ q✉❛♥❞♦ s✉r❣❡♠ ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③✱ ♦s ♣r✐♥❝✐♣❛✐s ❝♦❧❛❜♦r❛❞♦r❡s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✳ ❋❛❧❛r❡♠♦s ✉♠ ♣♦✉❝♦ t❛♠❜é♠ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❝á❧❝✉❧♦ ♥❛ ■❞❛❞❡ ❈♦♥t❡♠♣♦râ♥❡❛✳

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❈❛♣ít✉❧♦ ✶

❯♠ ♣♦✉❝♦ ❞❛ ❍✐stór✐❛ ❞♦ ❈á❧❝✉❧♦

❆ ❤✐stór✐❛ ❞♦ ❝á❧❝✉❧♦ ❡♥❝❛✐①❛✲s❡ ❡♠ ✈ár✐♦s ♣❡rí♦❞♦s ❞✐st✐♥t♦s✱ ❞❡ ❢♦r♠❛ ♥♦tá✈❡❧ ♥❛s ❡r❛s ❛♥t✐❣❛✱ ♠❡❞✐❡✈❛❧ ❡ ♠♦❞❡r♥❛✿ ❬✾❪✱❬✶✵❪ ❡ ❬✶✶❪✳ ❆s ✜❣✉r❛s ♦❜s❡r✈❛❞❛s ♥❡st❡ ❝❛♣ít✉❧♦ ❢♦r❛♠ t✐r❛❞❛s ❞❡ ❬✶✵❪✳

✶✳✶ ❆♥t✐❣✉✐❞❛❞❡

❋✐❣✉r❛ ✵✶✿ ❆rq✉✐♠❡❞❡s

❉❡ ❛❝♦r❞♦ ❝♦♠ ●❛✉ss✱ ❆rq✉✐♠❡❞❡s ✭❋✐❣✉r❛ ✵✶✮✱ ♦ ♠❛✐♦r ♠❛t❡♠át✐❝♦ ❞❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❥á ❛♣r❡s❡♥t❛✈❛ ✐❞é✐❛s r❡❧❛❝✐♦♥❛❞❛s ❛♦ ❈á❧❝✉❧♦ ❞♦✐s sé❝✉❧♦s ❛♥t❡s ❞❡ ❈r✐st♦✳

◆❛ ❆♥t✐❣✉✐❞❛❞❡✱ ❢♦r❛♠ ✐♥tr♦❞✉③✐❞❛s ❛❧❣✉♠❛s ✐❞é✐❛s ❞♦ ❝á❧❝✉❧♦ ✐♥t❡❣r❛❧✱ ❡♠❜♦r❛ ♥ã♦ t❡♥❤❛ ❤❛✈✐❞♦ ✉♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❛s ✐❞é✐❛s ❞❡ ❢♦r♠❛ r✐❣♦r♦s❛ ❡ s✐st❡♠át✐❝❛✳ ❆ ❢✉♥çã♦ ❜ás✐❝❛ ❞♦ ❝á❧❝✉❧♦ ✐♥t❡❣r❛❧✱ ❝❛❧❝✉❧❛r ✈♦❧✉♠❡s ❡ ár❡❛s✱ ♣♦❞❡ s❡r r❡♠♦♥t❛❞❛ ❛♦ P❛♣✐r♦ ❊❣í♣❝✐♦ ❞❡ ▼♦s❝♦✇ ✭✶✽✵✵ ❛✳❈✳✮✱ ♥♦ q✉❛❧ ✉♠ ❡❣í♣❝✐♦ tr❛❜❛❧❤♦✉ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❢r✉st✉♠ ♣✐r❛♠✐❞❛❧✳

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❈❛♣ít✉❧♦ ✶✳ ❯♠ ♣♦✉❝♦ ❞❛ ❍✐stór✐❛ ❞♦ ❈á❧❝✉❧♦ ✹

❊✉❞♦①✉s ✭✹✵✽✕✸✺✺ ❛✳❈✳✮ ✉s♦✉ ♦ ♠ét♦❞♦ ❞❛ ❡①❛✉stã♦ ♣❛r❛ ❝❛❧❝✉❧❛r ár❡❛s ❡ ✈♦❧✉♠❡s✳ ❆rq✉✐✲ ♠❡❞❡s ✭✷✽✼✕✷✶✷ ❛✳❈✳✮ ❧❡✈♦✉ ❡ss❛ ✐❞é✐❛ ❛❧é♠✱ ✐♥✈❡♥t❛♥❞♦ ❛ ❤❡✉ríst✐❝❛✱ q✉❡ s❡ ❛♣r♦①✐♠❛ ❞♦ ❝á❧❝✉❧♦ ✐♥t❡❣r❛❧✳ ❖ ♠ét♦❞♦ ❞❛ ❡①❛✉stã♦ ❢♦✐ r❡❞❡s❝♦❜❡rt♦ ♥❛ ❈❤✐♥❛ ♣♦r ▲✐✉ ❍✉✐ ♥♦ sé❝✉❧♦ ■■■✱ q✉❡ ♦ ✉s♦✉ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦✳ ❖ ♠ét♦❞♦ t❛♠❜é♠ ❢♦✐ ✉s❛❞♦ ♣♦r ❩✉ ❈❤♦♥❣③❤✐ ♥♦ sé❝✉❧♦ ❱✱ ♣❛r❛ ❛❝❤❛r ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛✳

✶✳✷ ■❞❛❞❡ ▼é❞✐❛

◆❛ ■❞❛❞❡ ▼é❞✐❛✱ ♦ ♠❛t❡♠át✐❝♦ ✐♥❞✐❛♥♦ ❆r②❛❜❤❛t❛ ✉s♦✉ ❛ ♥♦çã♦ ✐♥✜♥✐t❡s✐♠❛❧ ❡♠ ✹✾✾ ❞✳❈✳ ❡①♣r❡ss❛♥❞♦✲❛ ❡♠ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❛str♦♥♦♠✐❛ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❜ás✐❝❛✳ ❊ss❛ ❡q✉❛çã♦ ❧❡✈♦✉ ❇❤ás❦❛r❛ ■■✱ ♥♦ sé❝✉❧♦ ❳■■✱ ❛ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ❞❡r✐✈❛❞❛ ♣r❡♠❛t✉r❛ r❡♣r❡s❡♥t❛♥❞♦ ✉♠❛ ♠✉❞❛♥ç❛ ✐♥✜♥✐t❡s✐♠❛❧✱ ❡ ❡❧❡ ❞❡s❡♥✈♦❧✈❡✉ t❛♠❜é♠ ♦ q✉❡ s❡r✐❛ ✉♠❛ ❢♦r♠❛ ♣r✐♠✐t✐✈❛ ❞♦ ✏❚❡♦r❡♠❛ ❞❡ ❘♦❧❧❡✧✳

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❈❛♣ít✉❧♦ ✶✳ ❯♠ ♣♦✉❝♦ ❞❛ ❍✐stór✐❛ ❞♦ ❈á❧❝✉❧♦ ✺

✶✳✸ ■❞❛❞❡ ▼♦❞❡r♥❛

❋✐❣✉r❛ ✵✷✿ ❙✐r ■s❛❛❝ ◆❡✇t♦♥

❛♣❧✐❝♦✉ ♦ ❝á❧❝✉❧♦ às s✉❛s ❧❡✐s ❞♦

♠♦✈✐♠❡♥t♦ ❡ ❛ ♦✉tr♦s ❝♦♥❝❡✐t♦s

♠❛t❡♠át✐❝♦s✲❢ís✐❝♦s✳

❋✐❣✉r❛ ✵✸✿ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠

▲❡✐❜♥✐③✿ ♦ ✐♥✈❡♥t♦r ❞♦ ❝á❧❝✉❧♦✱ ❥✉♥✲

t❛♠❡♥t❡ ❝♦♠ ◆❡✇t♦♥✳

◆❛ ■❞❛❞❡ ▼♦❞❡r♥❛✱ ❢♦r❛♠ ❢❡✐t❛s ❞❡s❝♦❜❡rt❛s ✐♥❞❡♣❡♥❞❡♥t❡s ♥♦ ❝á❧❝✉❧♦✳ ◆♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❱■■ ♥♦ ❏❛♣ã♦✱ ♦ ♠❛t❡♠át✐❝♦ ❙❡❦✐ ❑♦✇❛ ❡①♣❛♥❞✐✉ ♦ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦✳ ◆❛ ❊✉r♦♣❛✱ ❛ s❡❣✉♥❞❛ ♠❡t❛❞❡ ❞♦ sé❝✉❧♦ ❳❱■■ ❢♦✐ ✉♠ ♣❡rí♦❞♦ ❞❡ ❣r❛♥❞❡s ✐♥♦✈❛çõ❡s✳ ❖ ❈á❧❝✉❧♦ ❛❜r✐✉ ♥♦✈❛s ♦♣♦rt✉♥✐❞❛❞❡s ♥❛ ❢ís✐❝❛✲♠❛t❡♠át✐❝❛ ❞❡ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♠✉✐t♦ ❛♥t✐❣♦s q✉❡ ❛té ❡♥tã♦ ♥ã♦ ❤❛✈✐❛♠ s✐❞♦ s♦❧✉❝✐♦♥❛❞♦s✳ ❖✉tr♦s ♠❛t❡♠át✐❝♦s ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❡ss❛s ❞❡s❝♦❜❡rt❛s✱ ❞❡ ✉♠❛ ❢♦r♠❛ ♥♦tá✈❡❧✱ ❝♦♠♦ ❏♦❤♥ ❲❛❧❧✐s ❡ ■s❛❛❝ ❇❛rr♦✇✳ ❏❛♠❡s ●r❡❣♦r② ❞❡s❡♥✈♦❧✈❡✉ ✉♠ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞♦ s❡❣✉♥❞♦ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ❝á❧❝✉❧♦ ❡♠ ✶✻✻✽✳

(16)

❈❛♣ít✉❧♦ ✶✳ ❯♠ ♣♦✉❝♦ ❞❛ ❍✐stór✐❛ ❞♦ ❈á❧❝✉❧♦ ✻

▲❡✐❜♥✐③ ❞❡ t❡r r♦✉❜❛❞♦ ❛s ✐❞é✐❛s ❞❡ s❡✉s ❡s❝r✐t♦s ♥ã♦ ♣✉❜❧✐❝❛❞♦s✳ ◆❡✇t♦♥ t✐♥❤❛ ✉♠ á❧✐❜❡✱ ♣♦✐s à é♣♦❝❛ ❝♦♠♣❛rt✐❧❤❛r❛ s❡✉s ❡s❝r✐t♦s ❝♦♠ ❛❧❣✉♥s ♣♦✉❝♦s ♠❡♠❜r♦s ❞❛ ❙♦❝✐❡❞❛❞❡ ❘❡❛❧✳ ❊st❛ ❝♦♥tr♦✈érs✐❛ ❞✐✈✐❞✐✉ ♦s ♠❛t❡♠át✐❝♦s ✐♥❣❧❡s❡s ❞♦s ♠❛t❡♠át✐❝♦s ❛❧❡♠ã❡s ♣♦r ♠✉✐t♦s ❛♥♦s✳ ❯♠ ❡st✉❞♦ ❝✉✐❞❛❞♦s♦ ❞♦s ❡s❝r✐t♦s ❞❡ ▲❡✐❜♥✐③ ❡ ◆❡✇t♦♥ ♠♦str♦✉ q✉❡ ❛♠❜♦s ❝❤❡❣❛r❛♠ ❛ s❡✉s r❡s✉❧t❛❞♦s ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ▲❡✐❜♥✐③ ✐♥✐❝✐❛♥❞♦ ❝♦♠ ✐♥t❡❣r❛çã♦ ❡ ◆❡✇t♦♥ ❝♦♠ ❞✐❢❡r❡♥❝✐❛çã♦✳ ◆♦s ❞✐❛s ❛t✉❛✐s ❛❞♠✐t❡✲s❡ q✉❡ ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③ ❞❡s❝♦❜r✐r❛♠ ♦ ❝á❧❝✉❧♦ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✳ ▲❡✐❜♥✐③✱ ♣♦ré♠✱ ❢♦✐ q✉❡♠ ❞❡✉ ♦ ♥♦♠❡ ❝á❧❝✉❧♦ à ♥♦✈❛ ❞✐s❝✐♣❧✐♥❛✱ ◆❡✇t♦♥ ❛ ❝❤❛♠❛r❛ ❞❡ ✏❆ ❝✐ê♥❝✐❛ ❞♦s ✢✉①♦s✧✳

❆ ♣❛rt✐r ❞❡ ▲❡✐❜♥✐③ ❡ ◆❡✇t♦♥✱ ♠✉✐t♦s ♠❛t❡♠át✐❝♦s ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♦ ❝♦♥tí♥✉♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❝á❧❝✉❧♦✳

✶✳✹ ■❞❛❞❡ ❈♦♥t❡♠♣♦râ♥❡❛

❋✐❣✉r❛ ✵✹✿ ▼❛r✐❛ ●❛❡t❛♥❛ ❆❣♥❡s✐

(17)

❈❛♣ít✉❧♦ ✷

❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧

❈♦♥s✐❞❡r❡ ✉♠❛ ❝✉r✈❛C q✉❡ ♣♦ss✉❛ ✉♠❛ ❡q✉❛çã♦ ♥❛ ❢♦r♠❛y=f(x)✳ ◗✉❡r❡♠♦s ❡♥❝♦♥tr❛

❛ r❡t❛ t❛♥❣❡♥t❡ ❛C❡♠ ✉♠ ♣♦♥t♦P(a✱f(a))✳ P❛r❛ ✐ss♦ ❝♦♥s✐❞❡r❡♠♦s ✉♠ ♣♦♥t♦Q(x✱f(x))

♣ró①✐♠♦ ❞❡P✱ ♦♥❞❡ x₆=a❡ ❝♦♥s✐❞❡r❡♠♦s ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ s❡❝❛♥t❡ PQ✿ mPQ=

f(x) −f(a)

x−a ✳

❊♥tã♦ ❢❛ç❛♠♦s ❛❣♦r❛Q s❡ ❛♣r♦①✐♠❛r ❞❡P ❛♦ ❧♦♥❣♦ ❞❛ ❝✉r✈❛ C✱ ♦❜r✐❣❛♥❞♦ x t❡♥❞❡r ❛ a✳ ❙❡ mPQ t❡♥❞❡r ❛ ✉♠ ♥ú♠❡r♦ m✱ ❡♥tã♦ ❞❡✜♥✐♠♦s ❛ t❛♥❣❡♥t❡ t ❝♦♠♦ s❡♥❞♦ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦rP ❡ t❡♠ ✐♥❝❧✐♥❛çã♦ m✳ ✭❱❡❥❛ ❛ ✜❣✉r❛ ✵✺✮

❋✐❣✉r❛ ✵✺

❉❡✜♥✐çã♦ ✶✳ ❆ r❡t❛ t❛♥❣❡♥t❡ ❛ ✉♠❛ ❝✉r✈❛y=f(x) ❡♠ ✉♠ ♣♦♥t♦P(a✱f(a)) é ❛ r❡t❛ q✉❡

♣❛ss❛ ♣♦r P q✉❡ t❡♠ ✐♥❝❧✐♥❛çã♦

m = ❧✐♠

x→a

f(x) −f(a)

x−a ✱

❞❡s❞❡ q✉❡ ❡①✐st❛ ♦ ❧✐♠✐t❡✳ ❱✐❞❡ ❬✶❪✳

(18)

❈❛♣ít✉❧♦ ✷✳ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✽

❈♦♥s✐❞❡r❛♥❞♦ h ♦ ✐♥❝r❡♠❡♥t♦ ❞❡ x ❝♦♠ r❡❧❛çã♦ ❛ a✱ ♦✉ s❡❥❛✱ h = x−a✱ t❡♠♦s q✉❡

q✉❛♥❞♦x t❡♥❞❡ ❛ a✱h t❡♥❞❡ ❛ ✵✳ ❆ss✐♠ t❡♠♦s ♦✉tr❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛

t❛♥❣❡♥t❡

m= ❧✐♠

h→✵

f(a+h) −f(a)

h ✳

❉❡✜♥✐çã♦ ✷✳ ❆ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ f❡♠ ✉♠ ♣♦♥t♦ a✱ ❞❡♥♦t❛❞❛ ♣♦r f′

(a)✱ é

f′

(a) = ❧✐♠

x→_a

f(x) −f(a)

x−a ✳

❊①❡♠♣❧♦ ✶✳ ❊♥❝♦♥tr❡ ❛ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ f(x) =x✷−✽x+✾ ❡♠ ✉♠ ♥ú♠❡r♦ a✳

❙♦❧✉çã♦✿ ❉♦ ❧✐♠✐t❡ ❛♣r❡s❡♥t❛❞♦ ❛❝✐♠❛ t❡♠♦s

f′

(a) = ❧✐♠

h→✵

f(a+h) −f(a)

h = h❧✐♠→✵

[(a+h)✷−✽(a+h) +✾] − [a✷−✽a+✾]

h

= ❧✐♠

h→✵

a✷+✷ah+h✷−✽a−✽h+✾−a✷+✽a−✾

h

= ❧✐♠

h→✵

✷ah+h✷−✽h

h =h❧✐♠→✵✷

a+h−✽=✷a−✽✳

❊①❡♠♣❧♦ ✷✳ ❊♥❝♦♥tr❡ ✉♠❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ à ♣❛rá❜♦❧❛y=x✷−✽x+✾ ♥♦ ♣♦♥t♦ (✸✱−✻)✳

❙♦❧✉çã♦✿ P❡❧♦ ❊①❡♠♣❧♦ ✶✱ s❛❜❡♠♦s q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❡f(x) =x✷−✽x+✾ ♥♦ ♥ú♠❡r♦aé f′

(a) =✷a−✽✳ P♦rt❛♥t♦ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❡♠(✸✱−✻)éf′

(✸) =✷(✸) −✽= −✷✳

❆ss✐♠✱ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ éy− (−✻) = (−✷)_·(x−✸)✱ ♦✉ s❡❥❛✱ y= −✷x✳

✷✳✶ ❆❧❣✉♠❛s ❞❡r✐✈❛❞❛s ❜ás✐❝❛s

❙❡❥❛♠ f ❡g ❢✉♥çõ❡s ❞❡r✐✈á✈❡✐s ❡♠ x ❡ ✉♠❛ ❝♦♥st❛♥t❡c✳ ❱❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣♦s✐çõ❡s✿

❬✺❪

Pr♦♣♦s✐çã♦ ✷✳✶✳✶✳ ❙❡❥❛♠ I _⊂ ❘ ✉♠ ✐♥t❡r✈❛❧♦ ❡ f :I _→ ❘ ❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ f(x) = c

∀① ∈❘✱ ❡♥tã♦ f é ❞❡r✐✈á✈❡❧ ❡

f′

(c) =✵✳

❉❡♠♦♥str❛çã♦✳

f(x+h) −f(x)

h =c−c=✵

❡ ❛ss✐♠

f′

(19)

❈❛♣ít✉❧♦ ✷✳ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✾

Pr♦♣♦s✐çã♦ ✷✳✶✳✷✳ ❙❡❥❛♠ I_⊂❘ ✉♠ ✐♥t❡r✈❛❧♦ ❡ f✱g:I_→❘ ❢✉♥çõ❡s ❞❡r✐✈á✈❡✐s✱ ❡♥tã♦ ❛

❢✉♥çã♦f+g é ❞❡r✐✈á✈❡❧ ❡

(f(x) +g(x))′

=f′

(x) +g′

(x)✳

❉❡♠♦♥str❛çã♦✳ ❆♣❧✐❝❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❡ r❡❛rr❛♥❥❛♥❞♦ ♦s t❡r♠♦s✱

(f+g)′

(x) = ❧✐♠

h→✵

f(x+h) +g(x+h)

− f(x) +g(x)

h

= ❧✐♠

h→✵

f(x+h) −f(x)

h +

g(x+h) −g(x)

h

= ❧✐♠

h→✵

f(x+h) −f(x)

h +h❧✐♠→✵

g(x+h) −g(x)

h

= f′

(x) +g′

(x)✱

Pr♦♣♦s✐çã♦ ✷✳✶✳✸✳ ❙❡❥❛♠ I _⊂❘ ✉♠ ✐♥t❡r✈❛❧♦ ❡ f:I _→❘ ✉♠❛ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧✱ ❡♥tã♦

❛ ❢✉♥çã♦ c_·f✱ ♦♥❞❡ c_∈❘✱ é ❞❡r✐✈á✈❡❧ ❡ cf(x)′

=cf′

(x)✳

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

cf(x)′

= ❧✐♠

h→✵

cf(x+h) −cf(x)

h =c·h❧✐♠→✵

f(x+h) −f(x)

h =c·f

′

(x)✳

Pr♦♣♦s✐çã♦ ✷✳✶✳✹✳ ❙❡❥❛♠ I_⊂❘ ✉♠ ✐♥t❡r✈❛❧♦ ❡ f✱g:I_→❘ ❢✉♥çõ❡s ❞❡r✐✈á✈❡✐s✱ ❡♥tã♦ ❛

❢✉♥çã♦f_·g é ❞❡r✐✈á✈❡❧ ❡

(f(x)g(x))′

=f(x)′

g(x) +f(x)g(x)′

✳

❉❡♠♦♥str❛çã♦✳ P♦r ❞❡✜♥✐çã♦✱

f(x)g(x)′

= ❧✐♠

h→✵

f(x+h)g(x+h) −f(x)g(x)

h ✳

P❛r❛ ❢❛③❡r s✉r❣✐r ❛s ❞❡r✐✈❛❞❛s r❡s♣❡❝t✐✈❛s ❞❡f❡ g✱ ❡s❝r❡✈❛♠♦s ♦ q✉♦❝✐❡♥t❡ ❝♦♠♦ f(x+h)g(x+h) −f(x)g(x)

h =

f(x+h) −f(x)

h ·g(x+h) +f(x)·

g(x+h) −g(x)

h

◗✉❛♥❞♦h_→✵✱ t❡♠♦s q✉❡f(x+h) −f(x) h →f

′₍

x)❡g(x+h) −g(x)

h →g

′₍

x)✱ ❡♥❝♦♥tr❛♥❞♦✲

(20)

❈❛♣ít✉❧♦ ✷✳ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✶✵

❊①❡♠♣❧♦ ✸✳ ❈❛❧❝✉❧❡♠♦s ❛ ❞❡r✐✈❛❞❛ ❞♦ ♣r♦❞✉t♦ ❞❛s ❢✉♥çõ❡s

f(x) =x✷+✷x+✶ ❡ g(x) =✷x−✶✳

P❡❧❛ ❞❡r✐✈❛❞❛ ❞♦ ♣r♦❞✉t♦✱

f(x)g(x)′

= (x✷+✷x+✶)′

·(✷x−✶) + (x✷+✷x+✶)_·(✷x−✶)′

= (✷x+✷)_·(✷x−✶) + (x✷+✷x+✶)_·✷ =✻x✷+✻x✳

Pr♦♣♦s✐çã♦ ✷✳✶✳✺✳ ❙❡❥❛♠ I _⊂❘ ✉♠ ✐♥t❡r✈❛❧♦ ❡ f✱g:I_→ ❘ ❢✉♥çõ❡s ❞❡r✐✈á✈❡✐s s❡♥❞♦ g

✉♠❛ ❢✉♥çã♦ ♥ã♦ ♥✉❧❛✱ ❡♥tã♦ ❛ ❢✉♥çã♦ f

g é ❞❡r✐✈á✈❡❧ ❡

f g

′

(x) = f

′₍

x)g(x) −f(x)g′₍

x)

g(x)✷ ❉❡♠♦♥str❛çã♦✳ ❆♣❧✐❝❛♥❞♦ ❛ ❉❡r✐✈❛❞❛ ❞♦ ♣r♦❞✉t♦

f g

′

(x) =

f(x)_· ✶

g(x)

′

=f′

(x)_· ✶

g(x)+f(x)·

−g

′

(x)

g(x)✷

= f

′

(x)g(x) −f(x)g′

(x)

g(x)✷ ❊①❡♠♣❧♦ ✹✳ ❈❛❧❝✉❧❡♠♦s ❛ ❞❡r✐✈❛❞❛ ❞♦ q✉♦❝✐❡♥t❡ ❞❡ l(x) =√x ♣♦r h(x) = −x+✸✳

❙♦❧✉çã♦✿

√

x

−x+✸

′

= (

√

x)′

(−x+✸) −√x(−x+✸)′

(−x+✸)✷ =

✶ ✷·

✶

√

x ·(−x+✸) −

√

x_·(−✶)

(−x+✸)✷ =

= −✶ ✷ · x √ x + ✸ ✷ · ✶ √ x + √ x

(−x+✸)✷ =

✶ ✷ ·

√

x+ ✸

✷· ✶

√

x

(−x+✸)✷ ✳

Pr♦♣♦s✐çã♦ ✷✳✶✳✻✳ ❙❡❥❛ f:❘_→❘ ❞❛❞❛ ❝♦♠♦ f(x) =xn_{✱ ❡♥tã♦}

f′

(x) =n_·xn−✶

❉❡♠♦♥str❛çã♦✳ ❯s❛r❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❡ ❛ ❢ór♠✉❧❛ ❞❛ ❞❡r✐✈❛❞❛ ❞♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s ❢✉♥çõ❡s✱ ♣❛r❛ ♦❜t❡r ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳ ❚♦♠❛♥❞♦f(x) =x ❡g(x) =✶✱ t❡♠♦s

(x_·✶)′

=x′

·✶+x_·✵✳

▼❛s

x′

= ❧✐♠

h→✵

x+h−x

h =h❧✐♠→✵

h

h =h❧✐♠→✵✶

(21)

❈❛♣ít✉❧♦ ✷✳ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✶✶

❞❡ ♠♦❞♦ q✉❡ ❛ ❢ór♠✉❧❛ é ✈á❧✐❞❛ ♣❛r❛n = ✶✳ ❆❣♦r❛ t♦♠❛♥❞♦ ♣♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ❛

✈❛❧✐❞❛❞❡ ♣❛r❛n✱ ✐st♦ é✱

f′

(x) =n_·xn−✶✳

❱❛♠♦s ♠♦str❛r q✉❡ ✈❛❧❡ ♣❛r❛n+✶✱ ✐st♦ é✱ s❡ f(x) =xn+✶ ❡♥tã♦

f′

(x) = (n+✶)_·xn✳

❙❛❜❡♠♦s q✉❡xn+✶ =x_·xn_{✱ ✉s❛♥❞♦ ❡♥tã♦ ❛ ❞❡r✐✈❛❞❛ ❞♦ ♣r♦❞✉t♦✱ t❡♠♦s}

(xn+✶)′

= (x_·xn₎′

=x′

·xn₊_x

·(xn₎′

=xn₊_x

·n✳xn−✶ _⇒

(xn+✶)′

=xn+n_·xn = (n+✶)xn✳

Pr♦♣♦s✐çã♦ ✷✳✶✳✼✳ ❉❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ❝♦♠♣♦st❛ ✭❘❡❣r❛ ❞❛ ❈❛❞❡✐❛✮✳ ❙❡❥❛♠ I_⊂❘

✉♠ ✐♥t❡r✈❛❧♦ ❡f✱g :I _→❘ ❢✉♥çõ❡s ❞❡r✐✈á✈❡✐s✱ ❡♥tã♦ ❛ ❢✉♥çã♦ ❝♦♠♣♦st❛f(g(x))é ❞❡r✐✈á✈❡❧

❡

f(g(x))′

=f′

(g(x))_·g′

(x)✳

❉❡♠♦♥str❛çã♦✳ ❋✐①❡♠♦s ✉♠ ♣♦♥t♦x✳ ❙✉♣♦r❡♠♦s✱ ♣❛r❛ s✐♠♣❧✐✜❝❛r✱ q✉❡g(x+h)−g(x)₆=✵

♣❛r❛ t♦❞♦h s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✳ P♦❞❡♠♦s ❡s❝r❡✈❡r

(f(g(x)))′

= ❧✐♠

h→✵

f(g(x+h)) −f(g(x))

h =h❧✐♠→✵

f(g(x+h)) −f(g(x))

g(x+h) −g(x) ·

g(x+h) −g(x)

h

❙❛❜❡♠♦s q✉❡ ♦ s❡❣✉♥❞♦ t❡r♠♦

g(x+h) −g(x)

h →g

′

(x)

q✉❛♥❞♦ h _→ ✵✳ P❛r❛ ♦ ♣r✐♠❡✐r♦ t❡r♠♦ ❝❤❛♠❡♠♦s a = g(x) ❡ z = g(x +h)✳ ◗✉❛♥❞♦

h_→✵✱z _→a✱ ❧♦❣♦

❧✐♠ h→✵

f(g(x+h)) −f(g(x))

g(x+h) −g(x) =z❧✐♠→a

f(z) −f(a)

z−a =f

′

(a) =f′

(g(x))✳

P❛r❛ ❛♣❧✐❝❛r ❛ ❘❡❣r❛ ❞❛ ❈❛❞❡✐❛✱ é ✐♠♣♦rt❛♥t❡ s❛❜❡r ✐❞❡♥t✐✜❝❛r q✉❛✐s sã♦ ❛s ❢✉♥çõ❡s ❡♥✲ ✈♦❧✈✐❞❛s✱ ❡ ❡♠ q✉❛❧ ♦r❞❡♠ ❡❧❛s sã♦ ❛♣❧✐❝❛❞❛s✳

❊①❡♠♣❧♦ ✺✳ ❈❛❧❝✉❧❡♠♦s ❛ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦

f(x) = _√ ✶

(22)

❈❛♣ít✉❧♦ ✷✳ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✶✷

❖❜s❡r✈❛♠♦s q✉❡

✶

√

x✹+x✷ = (x

✹₊_x✷₎−✶_✷_✳

❆ss✐♠✱ t❡♠♦s ✉♠❛ s✐t✉❛çã♦ ❞❡ ❢✉♥çã♦ ❝♦♠♣♦st❛ ❞♦ t✐♣♦uα_{✱ ❝♦♠}_α_{= −}✶

✷❡u=x✹+x✷✳ ❆ss✐♠✱

f′

(x) = −✶

✷ ·(x✹+x✷)

′

·(x✹+x✷)−✶✷−✶

= −✶

✷ ·(✹x✸+✷x)·(x✹+x✷)−

✸ ✷

= (−✷x✸−x)_·(x✹+x)−✸✷

= −✷x

✸₋_x

p

(x✹+x)✸✳

✷✳✷ ❖ ❚❡♦r❡♠❛ ❞❡ ❘♦❧❧❡

❉❡✜♥✐çã♦ ✸✳ ❙❡❥❛f: [a✱b]_→_R❡ c_∈(a✱b)✳ ❖ ♣♦♥t♦cé ❞✐t♦ ❝rít✐❝♦ ♣❛r❛fs❡ f′

(c) =✵✳

Pr♦♣♦s✐çã♦ ✷✳✷✳✶ ✭❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✮✳ ❙❡ ❛ ❢✉♥çã♦f(x)✱ ❞❡r✐✈á✈❡❧ ♥♦ ✐♥t❡r✈❛❧♦(a✱b)✱

t❡♠ ✉♠ ♠á①✐♠♦ ♦✉ ✉♠ ♠í♥✐♠♦ ♥♦ ♣♦♥t♦ x = x_✶✱ ❡♥tã♦ ❛ ❞❡r✐✈❛❞❛ ❞❡ f(x) é ♥✉❧❛ ❡♠

x=x_✶✱ ✐st♦ é✱ f′

(x_✶) =✵✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛x_✶ ✉♠ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❧♦❝❛❧ ❞❡ f❡ h _∈R t❛❧ q✉❡ x_✶+h_∈ (a✱b)✳

❉❛í t❡r❡♠♦s q✉❡ f(x_✶+h) −f(x_✶)6✵ ❡ ♣♦rt❛♥t♦

f(x_✶+h) −f(x_✶)

h 6✵✱ s❡ h >✵ ❡

f(x_✶ +h) −f(x_✶)

h >✵✱ s❡ h <✵✳

❆ss✐♠✱ t❡r❡♠♦s q✉❡

❧✐♠ h→✵+

f(x_✶+h) −f(x_✶)

h 6✵ ❡ (h❧✐♠→✵−

f(x_✶+h) −f(x_✶)

h >✵✱

❞❛í✱ ❝♦♠♦ ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s ❡①✐st❡♠ ❡ ❝♦✐♥❝✐❞❡♠✱ s❡❣✉❡ q✉❡f′

(x_✶) =✵✳ ◆♦ ❝❛s♦ ❞❡x_✶ s❡r

✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦✱ ❛ ❞❡♠♦♥str❛çã♦ é ❛♥á❧♦❣❛✳

◆♦t❡ q✉❡ ❛ ❝♦♥❞✐çã♦ é ♥❡❝❡ssár✐❛✱ ♠❛s ♥ã♦ s✉✜❝✐❡♥t❡✳ P♦rq✉❡ ♣♦❞❡ ❤❛✈❡r ✉♠ ♣♦♥t♦ ♥♦ ✐♥t❡r✈❛❧♦✱ ♥♦ q✉❛❧ ❛ ❞❡r✐✈❛❞❛ é ♥✉❧❛✱ ♠❛s ♦ ♣♦♥t♦ ♥ã♦ é ♥❡♠ ✉♠ ♠á①✐♠♦ ♥❡♠ ✉♠ ♠í♥✐♠♦✱ ♦✉ ❛ ❢✉♥çã♦ ♣♦ss✉✐ ✉♠ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦ ♥♦ q✉❛❧ ♥ã♦ é ❞❡r✐✈á✈❡❧✳ ■ss♦ ♣♦❞❡ s❡r ❝♦♥st❛t❛❞♦ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❛❧❣✉♠❛s ❢✉♥çõ❡s✱ ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ✵✻✳

(23)

❈❛♣ít✉❧♦ ✷✳ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✶✸

❋✐❣✉r❛ ✵✻

Pr♦♣♦s✐çã♦ ✷✳✷✳✷ ✭❚❡♦r❡♠❛ ❞❡ ❘♦❧❧❡✮✳ ❙❡❥❛ f : [a✱b] _→ _R ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡

❞❡r✐✈á✈❡❧ ❡♠ (a✱b)✳ ❙❡ f(a) =f(b) ❡♥tã♦ ❡①✐st❡ c_∈(a✱b) t❛❧ q✉❡

f′

(c) =✵✱

♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ❡♠(a✱b)✳

❉❡♠♦♥str❛çã♦✳ ❙❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❲❡✐❡rstr❛ss ✭✈✐❞❡ ❬✻❪ ♣á❣✐♥❛ ✷✼✾✮ q✉❡ f ❛❞♠✐t❡

♠á①✐♠♦ ❡ ♠í♥✐♠♦ ❡♠[a✱b]✳ ❙❡ ❛♠❜♦s ❛❝♦♥t❡❝❡♠ ♥♦s ❡①tr❡♠♦s✱ ❞✐❣❛♠♦s✱ f(a)6 f(x) ❡

f(x)6f(b) ♣❛r❛ t♦❞♦ x_∈[a✱b]✱ t❡rí❛♠♦s q✉❡

f(a)6f(x)6f(b) =f(a)✱ _∀x _∈[a✱b]✱

❞❡ss❛ ❢♦r♠❛ f s❡r✐❛ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛ f(a) ❡ ❞❛í q✉❛❧q✉❡r c _∈ (a✱b) s❛t✐s❢❛③ q✉❡ f′

(c) =

✵✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❛♦ ♠❡♥♦s ✉♠ ❞♦s ✈❛❧♦r❡s ❡①tr❡♠♦s ❛❝♦♥t❡❝❡ ❡♠ (a✱b) ❡ ❞❡✈✐❞♦ ❛

Pr♦♣♦s✐çã♦ ✷✳✷✳✶ t❛❧ ♣♦♥t♦ é ❝rít✐❝♦✳

✷✳✸ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦

(24)

❈❛♣ít✉❧♦ ✷✳ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✶✹

❊♠ ♠❛t❡♠át✐❝❛✱ ♦ t❡♦r❡♠❛ ❞♦ ✈❛❧♦r ♠é❞✐♦ ✭✜❣✉r❛ ✵✼✮ ❛✜r♠❛ q✉❡ ❞❛❞❛ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛f❞❡✜♥✐❞❛ ♥✉♠ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ [a✱b] ❡ ❞❡r✐✈á✈❡❧ ❡♠(a✱b)✱ ❡①✐st❡ ❛❧❣✉♠ ♣♦♥t♦

c❡♠ (a✱b) t❛❧ q✉❡✿

f′

(c) = f(b) −f(a)

b−a ✳

●❡♦♠❡tr✐❝❛♠❡♥t❡✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ ❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❡ f ♥♦ ♣♦♥t♦ ❞❡ ❛❜❝✐ss❛ c

é ♣❛r❛❧❡❧❛ à s❡❝❛♥t❡ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ❞❡ ❛❜❝✐ss❛s a❡ b✳

❖ t❡♦r❡♠❛ ❞♦ ✈❛❧♦r ♠é❞✐♦ t❛♠❜é♠ t❡♠ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❡♠ t❡r♠♦s ❢ís✐❝♦s✿ s❡ ✉♠ ♦❜❥❡t♦ ❡stá ❡♠ ♠♦✈✐♠❡♥t♦ ❡ s❡ ❛ s✉❛ ✈❡❧♦❝✐❞❛❞❡ ♠é❞✐❛ é v✱ ❡♥tã♦✱ ❞✉r❛♥t❡ ❡ss❡ ♣❡r❝✉rs♦

✭✐♥t❡r✈❛❧♦[a✱b]✮✱ ❤á ✉♠ ✐♥st❛♥t❡ ✭♣♦♥t♦ c✮ ❡♠ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ✐♥st❛♥tâ♥❡❛ t❛♠❜é♠ év✳

❈♦♥s✐❞❡r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡✱ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s (a✱f(a)) ❡ (b✱f(b))✱ ✐st♦

é✿

y−f(a) = f(b) −f(a)

b−a ·(x−a)✳

❊ss❛ r❡t❛ é ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦

T(x) = f(b) −f(a)

b−a ·(x−a) +f(a)✳

❙❡❥❛g ❛ ❢✉♥çã♦ q✉❡ é ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ f❡ T✱ ✐st♦ ég(x) =f(x) −T(x)✳ ❆ss✐♠✱

g(x) =f(x) −

f(b) −f(a)

b−a ·(x−a) +f(a)

✳

◗✉❛♥❞♦ x=a✱ t❡♠♦s✿

g(a) =f(a) −

f(b) −f(a)

b−a ·(a−a) +f(a)

=f(a) −f(a) =✵

❡✱ q✉❛♥❞♦x=b✱ t❡♠♦s✿

g(b) =f(b) −

f(b) −f(a)

b−a ·(b−a) +f(a)

=f(b) − [f(b) −f(a) +f(a)] =✵✳

❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ g é ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❞✉❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♠ [a✱b] ❡ ❞❡r✐✈á✈❡✐s

❡♠ (a✱b)✱ ❡❧❛ ♣ró♣r✐❛ é ❝♦♥tí♥✉❛ ❡♠ [a✱b] ❡ ❞❡r✐✈á✈❡❧ ❡♠ (a✱b)✳ ▲♦❣♦ ♣♦❞❡♠♦s ✉s❛r ♦

❚❡♦r❡♠❛ ❞❡ ❘♦❧❧❡ ♣❛r❛g✱ ❝♦♥❝❧✉✐♥❞♦ q✉❡ ❡①✐st❡ ✉♠ ♥ú♠❡r♦c♥♦ ✐♥t❡r✈❛❧♦ (a✱b)✱ t❛❧ q✉❡✿

g′

(c) =✵✱

s❡♥❞♦

g′

(x) =f′

(x) −

f(b) −f(a)

b−a

(25)

❈❛♣ít✉❧♦ ✷✳ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✶✺

t❡♠♦s

g′

(c) =f′

(c) −

f(b) −f(a)

b−a

❡✱ ♣♦rt❛♥t♦✱

f′

(c) −

f(b) −f(a)

b−a

=✵✱

❞♦♥❞❡✱

f′

(c) =

f(b) −f(a)

b−a

✳

✷✳✹ ❈r❡s❝✐♠❡♥t♦ ❡ ❞❡❝r❡s❝✐♠❡♥t♦

Pr♦♣♦s✐çã♦ ✷✳✹✳✶✳ ❙❡❥❛♠ I _⊂ ❘ ✉♠ ✐♥t❡r✈❛❧♦ ❡ f : I _→ ❘ ✉♠❛ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ ♥❡ss❡

✐♥t❡r✈❛❧♦✳ ❊♥tã♦ t❡♠♦s q✉❡✿

✐✳ ❙❡ f′

(x)>✵ s♦❜r❡ I✱ ❡♥tã♦ f é ❝r❡s❝❡♥t❡ ♥❡❧❡✳

✐✐✳ ❙❡f′

(x)<✵ s♦❜r❡ I✱ ❡♥tã♦ fé ❞❡❝r❡s❝❡♥t❡ ♥❡❧❡✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ x_✶ ❡ x_✷ ❞♦✐s ♥ú♠❡r♦s q✉❛✐sq✉❡r ♥♦ ✐♥t❡r✈❛❧♦ ❡♠ I ❝♦♠ x_✶ < x_✷✳

❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ ❝r❡s❝❡♥t❡✱ t❡♠♦s q✉❡ ♠♦str❛r q✉❡ f(x_✶) < f(x_✷)✳

❙❛❜❡♠♦s q✉❡ f é ❞❡r✐✈á✈❡❧ ❡♠ (x_✶✱x_✷)✳ ▲♦❣♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✱ ❡①✐st❡ ✉♠

♥ú♠❡r♦c ❡♥tr❡x_✶ ❡x_✷ t❛❧ q✉❡

f(x_✷) −f(x_✶) =f′

(c)(x_✷−x_✶)✳

❆❣♦r❛ f′₍

c) > ✵ ♣♦r ❤✐♣ót❡s❡ ❡ x_✷ −x_✶ > ✵✱ ♣♦✐s x_✶ < x_✷✳ ❆ss✐♠✱ f(x_✷) −f(x_✶) > ✵ ♦✉ f(x_✶)< f(x_✷)✱ ♦ q✉❡ ♠♦str❛ q✉❡f é ❝r❡s❝❡♥t❡✳

❆ ♣r♦♣♦s✐çã♦ ✷✳✹✳✷ é ♣r♦✈❛❞❛ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✳

✷✳✺ ❉❡r✐✈❛❞❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠

❆ ❞❡r✐✈❛❞❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ♦✉ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛✱ r❡♣r❡s❡♥t❛ ❛ ❞❡r✐✈❛❞❛ ❞❛ ❞❡r✐✈❛❞❛ ❞❡st❛ ❢✉♥çã♦✳ ❆ ♥♦t❛çã♦ ❝♦♠✉♠❡♥t❡ ✉t✐❧✐③❛❞❛ ♣❛r❛ ❞❡♥♦t❛r ❛ ❞❡r✐✈❛❞❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ é

y′′

♦✉ d✷y

dx✷✱

(26)

❈❛♣ít✉❧♦ ✷✳ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✶✻

✷✳✺✳✶ ❯s♦ ❞❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ♣❛r❛ ♠á①✐♠♦s ❡ ♠í♥✐♠♦s

❖s ❧❡♠❛s s❡❣✉✐♥t❡s ♥♦s ❛❥✉❞❛r❛♠ ❛ ❞❡♠♦♥str❛r ♦ ✉s♦ ❞❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ♣❛r❛ ❡♥❝♦♥tr❛r ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❧♦❝❛✐s✳

▲❡♠❛ ✶✳ ❙❡❥❛♠ I _⊂ ❘ ✉♠ ✐♥t❡r✈❛❧♦ ❡ f : I _→ ❘ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❙❡ f(x_✵) > ✵✱

❡♥tã♦ ❡①✐st❡δ >✵ t❛❧ q✉❡ s❡ x_∈(x_✵−δ✱x_✵ +δ) ✱ ❡♥tã♦ f(x)>✵✳

❉❡♠♦♥str❛çã♦✳ ❚♦♠❛♥❞♦ ǫ = f(x✵)

✷ > ✵ ❡ ✉s❛♥❞♦ ♦ ❢❛t♦ q✉❡ f é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✱ ❡①✐st❡ δ >✵ t❛❧ q✉❡ s❡ |x−x_✵|< δ✱ ❡♥tã♦ |f(x) −f(x_✵)|< ǫ✱ ♦✉ s❡❥❛✱

x_✵−δ < x < x_✵+δ_⇒−f(x✵)

✷ < f(x) −f(x✵)<

f(x_✵)

✷ ❞❡ ♦♥❞❡ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

▲❡♠❛ ✷✳ ❙❡ fé ✉♠❛ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ ❡♠ (a✱b)✱ ❡♥tã♦ f é ❝♦♥tí♥✉❛ ♥❡st❡ ✐♥t❡r✈❛❧♦✳

❉❡♠♦♥str❛çã♦✳ ▼♦str❛r❡♠♦s q✉❡ f é ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ x_✵ _∈ (a✱b)✳ P❛r❛ ✐st♦✱ ❜❛st❛

♣r♦✈❛r q✉❡

❧✐♠

x→x_✵f(x) =f(x✵) ♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱

❧✐♠

x→x_✵(f(x) −f(x✵)) = ✵✳ ❉❡ ❢❛t♦✱

❧✐♠ x→x_✵

f(x) −f(x_✵)

x−x_✵ ·(x−x✵) =x❧✐♠→x_✵

f(x) −f(x_✵)

x−x_✵ ·x❧✐♠→x_✵(x−x✵) =f

′

(x_✵)_·✵=✵✳

Pr♦♣♦s✐çã♦ ✷✳✺✳✶✳ ❙❡❥❛♠f✉♠❛ ❢✉♥çã♦ ❞❡r✐✈á✈❡❧ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦(a✱b)❝♦♥t❡♥❞♦

♦ ♣♦♥t♦ ❝rít✐❝♦ x_✵ t❛❧ q✉❡ f′

(x_✵) =✵✳ ❙❡ f ❛❞♠✐t❡ ❞❡r✐✈❛❞❛ s❡❣✉♥❞❛ f′′ ❡♠

(a✱b) ❡ s❡

✐✳ f′′

(x_✵)<✵✱ ❡♥tã♦ x =x_✵ é ✉♠ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❧♦❝❛❧✳

✐✐✳ f′′

(x_✵)>✵✱ ❡♥tã♦ x=x_✵ é ✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧✳

❉❡♠♦♥str❛çã♦✳ Pr♦✈❛r❡♠♦s ♦ ✐t❡♠ ✐✱ ♣♦✐s ♦ ♦✉tr♦ ❝❛s♦ é ❛♥á❧♦❣♦✳ ❈♦♠♦f❛❞♠✐t❡ ❞❡r✐✈❛❞❛

❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ❡♥tã♦ ♣❡❧♦ ▲❡♠❛ ✷✱ f′ é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ P♦r ❤✐♣ót❡s❡✱

f′′

(x_✵)

❡①✐st❡ ❞❡ ♠♦❞♦ q✉❡

✵< f′′

(x_✵) = ❧✐♠

x→_x−

✵

f′₍

x) −f′₍

x_✵)

x−x_✵ ⇒x❧✐♠→_x−

✵

f′₍

x)

(27)

❈❛♣ít✉❧♦ ✷✳ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✶✼

❙❡♥❞♦f′

(x) ❝♦♥tí♥✉❛✱ ♣❡❧♦ ▲❡♠❛ ✶✱ ❡①✐st❡ ǫ_✶ >✵ t❛❧ q✉❡ s❡ x _∈(x_✵−ǫ_✶✱x_✵)✱ ❡♥tã♦

f′

(x)

x−x_✵ >✵✳

❙❡♥❞♦x−x_✵ <✵✱ s❡❣✉❡ q✉❡ f′

(x)<✵ ♣❛r❛ t♦❞♦ x_∈(x_✵−ǫ_✶✱x_✵)✳ ❯s❛♥❞♦ ♦ ❧✐♠✐t❡ ❧❛t❡r❛❧

à ❞✐r❡✐t❛✱ ❡①✐st❡ǫ_✷ >✵ t❛❧ q✉❡ s❡ x_∈(x_✵✱x_✵+ǫ_✷)✱ ❡♥tã♦

f′

(x)

x−x_✵ >✵✳

❙❡♥❞♦ x−x_✵ > ✵✱ s❡❣✉❡ q✉❡ f′

(x) > ✵ ♣❛r❛ t♦❞♦ x _∈ (x_✵✱x_✵ +ǫ_✷)✳ ❆ss✐♠✱ t❡♠♦s ✉♠

✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (ǫ_✶✱ǫ_✷) ❝♦♥t❡♥❞♦ x_✵ t❛❧ q✉❡ f′

(x) ♠✉❞❛ ❞❡ s✐♥❛❧✳ ▲♦❣♦✱ ♣❡❧♦ t❡st❡ ❞❛

♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛✱ s❡❣✉❡ q✉❡x=x_✵ é ✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧✳

❊①❡♠♣❧♦ ✻✳ ❆s ❢✉♥çõ❡s f(x) = ✶−x✷ ❡ g(x) = x✷✱ ❞❡✜♥✐❞❛s s♦❜r❡ S = [−✶✱ ✷] ♣♦ss✉❡♠

♣♦♥t♦s ❝rít✐❝♦s ❡♠ x = ✵✳ f′′

(✵) = −✷ < ✵ ❡ g′′

(✵) = ✷ > ✵✳ P❡❧♦ ❝r✐tér✐♦ ❞❛ s❡❣✉♥❞❛

❞❡r✐✈❛❞❛✱ x = ✵ é ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❧♦❝❛❧ ♣❛r❛ f ❡ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧ ♣❛r❛ g ✭✜❣✉r❛

✵✽✮✳

f(x) =✶−x✷ f(x) =x✷

❋✐❣✉r❛ ✵✽

❉❡✜♥✐çã♦ ✹✳

✭✐✮ ❖ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦f t❡♠ ❝♦♥❝❛✈✐❞❛❞❡ ✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛ ♥♦ ♣♦♥t♦ (x_✵✱f(x_✵)) s❡

❡①✐st✐r f′

(x_✵) ❡ s❡ ❡①✐st✐r ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I ❝♦♥t❡♥❞♦ x_✵✱ t❛❧ q✉❡ ♣❛r❛ t♦❞♦s ♦s

✈❛❧♦r❡s ❞❡ x₆=x_✵ ❡♠ I✱ ♦ ♣♦♥t♦ (x✱f(x)) ❞♦ ❣rá✜❝♦ ❡stá ❛❝✐♠❛ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦

❣rá✜❝♦ ❡♠ (x_✵✱f(x_✵))✳

✭✐✐✮ ❖ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ f t❡♠ ❝♦♥❝❛✈✐❞❛❞❡ ✈♦❧t❛❞❛ ♣❛r❛ ❜❛✐①♦ ♥♦ ♣♦♥t♦ (x_✵✱f(x_✵))

s❡ ❡①✐st✐r f′₍

x_✵)❡ s❡ ❡①✐st✐r ✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I ❝♦♥t❡♥❞♦x_✵✱ t❛❧ q✉❡ ♣❛r❛ t♦❞♦s ♦s

✈❛❧♦r❡s ❞❡ x ₆=x_✵ ❡♠ I✱ ♦ ♣♦♥t♦ (x✱f(x)) ❞♦ ❣rá✜❝♦ ❡stá ❛❜❛✐①♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦

(28)

❈❛♣ít✉❧♦ ✷✳ ❯♠ ♣♦✉❝♦ s♦❜r❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ✶✽

Pr♦♣♦s✐çã♦ ✷✳✺✳✷✳ ❙❡ fé ✉♠❛ ❢✉♥çã♦ q✉❡ ♣♦ss✉✐ ❛s ❞✉❛s ♣r✐♠❡✐r❛s ❞❡r✐✈❛❞❛s ❝♦♥tí♥✉❛s

s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦S✱ t❡r❡♠♦s ❛s s✐t✉❛çõ❡s ❛❜❛✐①♦✿

✐✳ ❙❡ f′′

(x) > ✵ ❡♠ ❛❧❣✉♠ ♣♦♥t♦ x ❞❡ S✱ ❡♥tã♦ ♦ ❣rá✜❝♦ ❞❡ f t❡♠ ❛ ❝♦♥❝❛✈✐❞❛❞❡ ✈♦❧t❛❞❛

♣❛r❛ ❝✐♠❛ ♥❛s ✈✐③✐♥❤❛♥ç❛s ❞❡ x✳

✐✐✳ ❙❡ f′′

(x) <✵ ❡♠ ❛❧❣✉♠ ♣♦♥t♦ x ❞❡ S✱ ❡♥tã♦ ♦ ❣rá✜❝♦ ❞❡ f t❡♠ ❛ ❝♦♥❝❛✈✐❞❛❞❡ ✈♦❧t❛❞❛

♣❛r❛ ❜❛✐①♦ ♥❛s ✈✐③✐♥❤❛♥ç❛s ❞❡ x✳

(29)

❈❛♣ít✉❧♦ ✸

❯♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❛♦ ❈á❧❝✉❧♦

■♥t❡❣r❛❧

✸✳✶ ■♥t❡❣r❛❧ ❞❡✜♥✐❞❛

◆♦ ❈❛♣ít✉❧♦ ✷✱ ❡st✉❞❛♠♦s ❛ ❞❡r✐✈❛❞❛ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s✳ ❆ss✐♠ ❝♦♠♦ ❛ ❞❡r✐✈❛❞❛✱ ❛ ✐♥t❡❣r❛❧ t❛♠❜é♠ é ✉♠ ❞♦s ❝♦♥❝❡✐t♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞♦ ❝á❧❝✉❧♦✳ ❏á ✈✐♠♦s q✉❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛ ❡stá ✐♥t✐♠❛♠❡♥t❡ ❧✐❣❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♥❝♦♥tr❛r ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛ ✉♠❛ ❝✉r✈❛ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣♦♥t♦✳ ❆❣♦r❛ ✈❡r❡♠♦s q✉❡ ❛ ✐♥t❡❣r❛❧ ❡stá ❧✐❣❛❞❛ ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡t❡r♠✐♥❛r ❛ ár❡❛ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ q✉❛❧q✉❡r✳

✸✳✶✳✶ ❖ q✉❡ é ár❡❛

❈♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿ ❡♥❝♦♥tr❛r ❛ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦Sq✉❡ ❡stá s♦❜ ❛ ❝✉r✈❛ y =f(x) ❞❡ a ❛té b✳ ■ss♦ q✉❡r ❞✐③❡r q✉❡ S ✭✈❡r ✜❣✉r❛ ✵✾✮ ❡stá ❧✐♠✐t❛❞❛ ♣❡❧♦ ❣rá✜❝♦ ❞❡

✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛f✭♦♥❞❡ f(x)>✵✮✱ ❛s r❡t❛s ✈❡rt✐❝❛✐s x=a ❡x =b✱ ❡ ♦ ❡✐①♦ x✳

❋✐❣✉r❛ ✵✾

(30)

❈❛♣ít✉❧♦ ✸✳ ❯♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❛♦ ❈á❧❝✉❧♦ ■♥t❡❣r❛❧ ✷✵

❯♠ ❝♦♥❝❡✐t♦ ♣r✐♠✐t✐✈♦ ❞❡ ár❡❛ é ♦ ❞❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦✳ ❈❛❧❝✉❧❛r ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ é r❡❧❛t✐✈❛♠❡♥t❡ ❢á❝✐❧✱ ❛ss✐♠ ❝♦♠♦ ❛ ❞❡ ♦✉tr❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❡❧❡♠❡♥t❛r❡s ❝♦♠♦ tr✐❛♥❣✉❧♦ ❡ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ❆ss✐♠✱ ❛ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦Sq✉❛❧q✉❡r ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ❛♣r♦①✐♠❛♥❞♦

❛ r❡❣✐ã♦ ❛tr❛✈és ❞❡ ♣♦❧í❣♦♥♦s✱ ❝✉❥❛s ár❡❛s ♣♦❞❡♠ s❡r ❝❛❧❝✉❧❛❞❛s ♣❡❧♦s ♠ét♦❞♦s ❞❛ ❣❡♦♠❡tr✐❛ ❡❧❡♠❡♥t❛r✳

P❛r❛ ✐ss♦✱ ✈❛♠♦s ❢❛③❡r ✉♠❛ ♣❛rt✐çã♦ P ❞♦ ✐♥t❡r✈❛❧♦ [a✱b]✱ ✐st♦ é✱ ✈❛♠♦s ❞✐✈✐❞✐r ♦

✐♥t❡r✈❛❧♦ [a✱b] ❡♠ n s✉❜✐♥t❡r✈❛❧♦s ✭✈❡❥❛ ❬✽❪✮✱ ♣♦r ♠❡✐♦ ❞♦s ♣♦♥t♦s

x_✵✱x_✶✱x_✷✱ ✳ ✳ ✳ ✱xi−✶✱xi✱ ✳ ✳ ✳ ✱xn✱ ❡s❝♦❧❤✐❞♦s ❛r❜✐tr❛r✐❛♠❡♥t❡✱ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛

a=x_✵ < x_✶ < x_✷ <✳ ✳ ✳< xi−✶ < xi <✳ ✳ ✳< xn=b✳ ❉❡t❡r♠✐♥❡♠♦s ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✐✲és✐♠♦ s✉❜✐♥t❡r✈❛❧♦✱[xi−✶✱xi]❝♦♠♦ s❡♥❞♦

∆xi=xi−xi−✶✳

❱❛♠♦s ❝♦♥str✉✐r r❡tâ♥❣✉❧♦s ❞❡ ❜❛s❡xi−xi−✶❡ ❛❧t✉r❛f(ci)♦♥❞❡cié ✉♠ ♣♦♥t♦ ❞♦ ✐♥t❡r✈❛❧♦

[xi−_✶✱xi]✳ ❆ss✐♠ ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s n r❡tâ♥❣✉❧♦s✱ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦rSn✱ s❡rá✿

Sn=f(c✶)×∆x✶+f(c✷)×∆x✷ +✳ ✳ ✳+f(cn)×∆xn

=

n X

i=✶

f(ci)×∆xi✳

❊ss❛ s♦♠❛ é ❝❤❛♠❛❞❛ ❞❡ ❙♦♠❛ ❞❡ ❘✐❡♠❛♥♥ ❞❛ ❢✉♥çã♦fr❡❧❛t✐✈❛ à ♣❛rt✐çã♦P✳ ◗✉❛♥❞♦ n❝r❡s❝❡✱ é ✏♥❛t✉r❛❧✧❡s♣❡r❛r q✉❡ ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s r❡tâ♥❣✉❧♦s ❛♣r♦①✐♠❡ ❞❛ ár❡❛ Ss♦❜

❛ ❝✉r✈❛✳

❈❤❛♠❛♠♦s ♥♦r♠❛ ❞❛ ♣❛rt✐çã♦ P ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡✉ s✉❜✐♥t❡r✈❛❧♦ ♠❛✐s ❧♦♥❣♦✿

||P||=♠❛①{∆xi❀i=✶✱ ✷✱ ✸✱ ✳ ✳ ✳ ✱n}✳

❉❡✜♥✐çã♦ ✺✳ ❆ ♠❡❞✐❞❛ ❞❛ ár❡❛ A ❞❛ r❡❣✐ã♦ S q✉❡ ❡stá s♦❜ ✉♠ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦

❝♦♥tí♥✉❛ f é

A= ❧✐♠

||P||→✵

n X

i=_✶

f(ci)×∆xi✱ s❡ ❡ss❡ ❧✐♠✐t❡ ❡①✐st✐r✳

(31)

❈❛♣ít✉❧♦ ✸✳ ❯♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❛♦ ❈á❧❝✉❧♦ ■♥t❡❣r❛❧ ✷✶

❉❡✜♥✐çã♦ ✻✳ ❙❡❥❛ f(x) ✉♠❛ ❢✉♥çã♦ ❧✐♠✐t❛❞❛ ❞❡✜♥✐❞❛ ♥♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ [a✱b] ❡ s❡❥❛

P ✉♠❛ ♣❛rt✐çã♦ q✉❛❧q✉❡r ❞❡ [a✱b]✳ ❆ ✐♥t❡❣r❛❧ ❞❡ f(x) ♥♦ ✐♥t❡r✈❛❧♦ [a✱b]✱ ❞❡♥♦t❛❞❛ ♣♦r

Zb

a

f(x) dx✱ é ❞❛❞❛ ♣♦r

Zb

a

f(x) dx= ❧✐♠

||P||→✵

n X

i=✶

f(ci)×∆xi✱ ❞❡s❞❡ q✉❡ ❡①✐st❛ ♦ ❧✐♠✐t❡✳ ❆ss✐♠✱ t❡♠♦s q✉❡

✭✐✮ Z

é ♦ s✐♥❛❧ ❞❡ ✐♥t❡❣r❛çã♦❀

✭✐✐✮ f(x) é ❛ ❢✉♥çã♦ ✐♥t❡❣r❛♥❞♦❀

✭✐✐✐✮ d(x) é ❛ ❞✐❢❡r❡♥❝✐❛❧ q✉❡ ✐❞❡♥t✐✜❝❛ ❛ ✈❛r✐á✈❡❧ ❞❡ ✐♥t❡❣r❛çã♦✳

✸✳✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❞❛ ✐♥t❡❣r❛❧ ❞❡✜♥✐❞❛

❆s ❞❡♠♦♥str❛çõ❡s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ✐♥t❡❣r❛❧ ❞❡✜♥✐❞❛ ♥ã♦ s❡rã♦ ❞❡♠♦♥str❛❞❛s✳ ❱❡❥❛ ❛s ❞❡♠♦♥str❛çõ❡s ❡♠ ❬✻❪ ♣á❣✐♥❛ ✸✽✺✳

Pr♦♣♦s✐çã♦ ✸✳✶✳✶✳ ❙❡❥❛♠f(x)❡g(x)❢✉♥çõ❡s ✐♥t❡❣rá✈❡✐s ♥♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦[a✱b]❡ s❡❥❛

k ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧ q✉❛❧q✉❡r✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✭✐✮

Zb

a

kf(x) dx=k

Zb

a

f(x) dx✳

✭✐✐✮

Zb

a

(f(x)_±g(x)) dx=

Zb

a

f(x) dx_±

Zb

a

g(x) dx

✭✐✐✐✮ ❙❡ ❛❁❝❁❜✱ ❡♥tã♦

Zb

a

f(x) dx=

Zc

a

f(x) dx+

Zb

c

f(x) dx✳

✭✐✈✮ ❙❡f(x)>✵ ♣❛r❛ t♦❞♦ x _∈[a✱b]✱ ❡♥tã♦

Zb

a

f(x) dx>✵✳

✭✈✮ ❙❡ f(x)>g(x) ♣❛r❛ t♦❞♦x _∈[a✱b]✱ ❡♥tã♦✱

Zb

a

f(x) dx> Zb

a

(32)

❈❛♣ít✉❧♦ ✸✳ ❯♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❛♦ ❈á❧❝✉❧♦ ■♥t❡❣r❛❧ ✷✷

✭✈✐✮

Zb

a

f(x) dx

6 Zb

a

|f(x)| dx✳

❈♦♥s✐❞❡r❛çõ❡s✿ ❈❛❧❝✉❧❛r ✉♠❛ ✐♥t❡❣r❛❧ ❛tr❛✈és ❞♦ ❧✐♠✐t❡ ❞❛s ❙♦♠❛s ❞❡ ❘✐❡♠❛♥♥ é ❣❡r❛❧♠❡♥t❡ ✉♠❛ t❛r❡❢❛ tr❛❜❛❧❤♦s❛✳ ❉❡ss❛ ❢♦r♠❛ ❡st❛❜❡❧❡❝❡r❡♠♦s ♦ ❝❤❛♠❛❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ q✉❡ ♥♦s ♣❡r♠✐t✐rá ❝❛❧❝✉❧❛r ✐♥t❡❣r❛✐s ❞❡ ♠❛♥❡✐r❛ ♠✉✐t❛ ♠❛✐s ❢á❝✐❧✳

✸✳✷ ❖ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ✲ ❚❋❈

❈♦♥s✐❞❡r❛❞♦ ✉♠ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s t❡♦r❡♠❛s ❞♦ ❡st✉❞♦ ❞♦ ❝á❧❝✉❧♦✱ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛✲ ♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ♥♦s ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r ❛ ✐♥t❡❣r❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ ✉t✐❧✐③❛♥❞♦ ✉♠❛ ♣r✐♠✐t✐✈❛ ❞❛ ♠❡s♠❛✳

❯s❛r❡♠♦s ♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❝á❧❝✉❧♦✳

✸✳✷✳✶ ❚❡♦r❡♠❛ ❞♦ ✈❛❧♦r ♠é❞✐♦ ♣❛r❛ ✐♥t❡❣r❛✐s

❙❡fé ❝♦♥tí♥✉❛ ❡♠ [a✱b]✱ ❡♥tã♦ ❡①✐st❡x_✵ _∈[a✱b] t❛❧ q✉❡

f(x_✵) = ✶

b−a

Zb

a

f(x) dx

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ fé ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ [a✱b]✱ ❡♥tã♦ _∃x_✶ _∈[a✱b] t❛❧ q✉❡

f(x_✶) é ♦ ✈❛❧♦r ♠í♥✐♠♦ ❞❡ f ❡♠ [a✱b] ❡ _∃x_✷ _∈ [a✱b] t❛❧ q✉❡ f(x_✷) é ♦ ✈❛❧♦r ♠á①✐♠♦ ❞❡ f

❡♠ [a✱b]✳ P♦rt❛♥t♦✱ t❡♠♦s f(x_✶) 6 f(t) 6 f(x_✷)✱ _∀t _∈ [a✱b]✳ ❊♥tã♦✱ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s

❞❡ ✐♥t❡❣r❛✐s ❞❡✜♥✐❞❛s✱ t❡♠♦s

f(x_✶)(b−a)6 Zb

a

f(t) dt6f(x_✷)(b−a)✳

▲♦❣♦✱

f(x_✶)6 Rb

af(t) dt

b−a 6f(x✷)✳

❈♦♠♦ f é ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ ❞❡ ❡str❡♠♦s x_✶ ❡ x_✷✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r

■♥t❡r♠é❞✐ár✐♦✱ ∃x_✵ _∈[a✱b] t❛❧ q✉❡

f(x_✵)6 Rb

af(t) dt

(33)

❈❛♣ít✉❧♦ ✸✳ ❯♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❛♦ ❈á❧❝✉❧♦ ■♥t❡❣r❛❧ ✷✸

✸✳✷✳✷ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ✭❚❋❈✮ ✲ P❛rt❡ ■

❙❡❥❛ ❛ ❢✉♥çã♦ f(x) ❝♦♥tí♥✉❛✳ ❙❡

F(x) =

Zx

a

f(t) dt✱

❡♥tã♦ F′

(x) =f(x)♣❛r❛ t♦❞♦ x_∈[a✱b]✳

❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❛♥❞♦h >✵✱ t❡♠♦s✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✐♥t❡❣r❛❧ ❡ ♣❡❧❛s ♣r♦♣r✐❡❞❛✲

❞❡s ❞❛ ✐♥t❡❣r❛❧ ❞❡✜♥✐❞❛✱ q✉❡

F(x+h) −F(x)

h =

Rx+h

a f(t) dt− Rx

af(t) dt

h =

Rx

af(t)dt+ Rx+h

x f(t) dt− Rx

af(t) dt

h =

Rx+h

x f(t) dt

h ✳

P❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ♣❛r❛ ✐♥t❡❣r❛✐s✱ ❡①✐st❡ th ♥♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ ❞❡ ❡①tr❡♠♦s

x ❡x+h✱ t❛❧ q✉❡

Rx+h

x f(t) dt

h =f(th)✳

P♦rt❛♥t♦✱

F(x+h) −F(x)

h =

Rx+h

x f(t) dt

h =f(th)✳

❈♦♠♦ ❧✐♠ h→✵

f(th) = f(x)✱ ❥á q✉❡ th ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ ❞❡ ❡①tr❡♠♦ x ❡ x +h✱ t❡♠♦s✿

❧✐♠ h→✵+

F(x+h) −F(x)

h =h❧✐♠→✵+

Rx+h

x f(t) dt

h =h❧✐♠→✵+

f(th) =f(x)✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ♠♦str❛✲s❡ ♦ ♠❡s♠♦ r❡s✉❧t❛❞♦ ♣❛r❛h_→✵−_{✳ P♦rt❛♥t♦✱}_F′

(x) =f(x)✳

✸✳✷✳✸ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ✭❚❋❈✮ ✲ P❛rt❡ ■■

❙❡G é t❛❧ q✉❡G′

(x) =f(x) ♣❛r❛ x_∈[a✱b]✱ ❡♥tã♦✱

Zb

a

f(x) dx = G(b) −G(a)✳

❉❡♠♦♥str❛çã♦✳ P❡❧♦ ❚❋❈ ✲ P❛rt❡ ■✱ F′

(x) = f(x)✳ P♦rt❛♥t♦✱ ❝♦♠♦ G′

(x) = f(x)✱ ♣♦r

❤✐♣ót❡s❡✱ t❡♠♦sG′

(x) =F′

(x)_⇒G(x) =F(x) +c✳ ▲♦❣♦✱ G(x) =F(x) +c=

Zx

a