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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈ ▼❡str❛❞♦ Pr♦❢✐ss✐♦♥❛❧✐③❛♥t❡ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ▲✉❝✐❛♥♦ ❑✐✇❛♠❡♥ P❛r❜❡❧♦s ❙❛♥t♦ ❆♥❞ré ✲ ❙P ✷✵✶✹✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈ ❈❡♥tr♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❈♦♠♣✉t❛çã♦ ❡ ❈♦❣♥✐çã♦ P❛r❜❡❧♦s ▲✉❝✐❛♥♦ ❑✐✇❛♠❡♥ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛ ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❥✉♥t♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧✐③❛♥t❡ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ❙❛♥t♦ ❆♥❞ré ✲ ❙P ❆❣♦st♦ ❞❡ ✷✵✶✹✳ ✐✐✐ P❛r❜❡❧♦s ❊st❡ ❡①❡♠♣❧❛r ❝♦rr❡s♣♦♥❞❡ à r❡❞❛çã♦ ✜♥❛❧ ❞❛ ❞✐ss❡rt❛çã♦ ❞❡✈✐❞❛♠❡♥t❡ ❝♦rr✐✲ ❣✐❞❛ ❡ ❞❡❢❡♥❞✐❞❛ ♣♦r ▲✉❝✐❛♥♦ ❑✐✇❛♠❡♥ ❡ ❛♣r♦✈❛❞❛ ♣❡❧❛ ❝♦♠✐ssã♦ ❥✉❧❣❛❞♦r❛✳ ❙❛♥t♦ ❆♥❞ré✱ ✷✻ ❞❡ ❆❣♦st♦ ❞❡ ✷✵✶✹✳ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛ ❖r✐❡♥t❛❞♦r ❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿ ✶✳ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛ ✭❖r✐❡♥t❛❞♦r✮ ✲ ❯❋❆❇❈ ✷✳ Pr♦❢✳ ❉r✳ ❘♦❣ér✐♦ ●❛❧❛♥t❡ ◆❡❣r✐ ✸✳ Pr♦❢✳ ❉r✳ ❙✐♥✉ê ❉❛②❛♥ ❇❛r❜❡r♦ ▲♦❞♦✈✐❝✐ ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❥✉♥t♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❆❇❈✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥✲ çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ à ♠✐♥❤❛ ❡s♣♦s❛ ❊❧❡✐s❛ ❚❤❛❧✐❛ ❚✉r♦❧❛ ◆❛s❝✐♠❡♥t♦ ❑✐✇❛♠❡♥✱ ♠❡✉s ♣❛✐s✱ ❡ ❛♠✐❣♦s❀ ❡ t♦❞♦s ❛q✉❡❧❡s q✉❡ ♠❡ ❛♣♦✐❛r❛♠ ❞✉r❛♥t❡ ❛ ♠✐♥❤❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛✳ ✈ ❆❣r❛❞❡❝✐♠❡♥t♦s Pr✐♠❡✐r❛♠❡♥t❡ à ❉❡✉s ♣♦r t✉❞♦✳ ❆♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✭P❘❖❋▼❆❚✮✱ à ❈❆P❊❙ ♣❡❧♦ ❛✉①í❧✐♦ ❝♦♥❝❡❞✐❞♦✱ à ❯❋❆❇❈ ❡ s❡✉s ♣r♦❢❡ss♦r❡s✱ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛✱ ❛♦s ❝♦❧❡❣❛s ❞❡ t✉r♠❛ ❞❡ ♠❡str❛❞♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ ❛♠✐❣♦ ❋❧❛✈✐♦ ❋❡r♥❛♥❞♦ ❞❛ ❙✐❧✈❛✱ ♣❡❧❛ ❝♦♠♣❛♥❤✐❛ ❡ ❛❥✉❞❛ ❡♠ t♦❞♦ ♦ ❝✉rs♦✳ ✈✐ ❘❡s✉♠♦ ❇❛s❡❛❞♦s ♥♦ tr❛❜❛❧❤♦ ❞❡ ❏♦♥❛t❤❛♥ ❙♦♥❞♦✇ ❬✼❪ ❡ ❆♥t♦♥✐♦ ▼✳ ❖❧❧❡r✲ ▼❛r❝é♥ ❬✻❪✱ ♥❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ♦s ♣❛r❜❡❧♦s✱ ✉♠❛ ❛❞❛♣t❛çã♦ ❞♦s ❛r❜❡❧♦s ♣❛r❛ ♣❛rá❜♦❧❛s✳ ▼♦str❛♠♦s ♠✉✐t❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ♣♦r ♠❡✐♦ ❞❡ ❢♦r♠✉❧❛çã♦ ❛♥❛❧ít✐❝❛✳ P❛❧❛✈r❛s✲❈❤❛✈❡ ❆r❜❡❧♦s✱ ●❡♦♠❡tr✐❛✱ P❛r❜❡❧♦s✱ P❛rá❜♦❧❛s✳ ✈✐✐ ❆❜str❛❝t ❇❛s❡❞ ♦♥ t❤❡ ♣❛♣❡rs ♦❢ ❏♦♥❛t❤❛♥ ❙♦♥❞♦✇ ❬✼❪ ❛♥❞ ❆♥t♦♥✐♦ ▼✳ ❖❧❧❡r✲ ▼❛r❝é♥ ❬✻❪✱ ✐♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② t❤❡ P❛r❜❡❧♦s✱ ❛♥ ❛❞❛♣t✐♦♥ ♦❢ ❛r❜❡❧♦s t♦ ♣❛r❛❜♦❧❛s✳ ❲❡ s❤♦✇ ♠❛♥② ♦❢ t❤❡✐r ♣r♦♣❡rt✐❡s ❢r♦♠ ❛♥ ❛♥❛❧②t✐❝ ❢♦r♠✉❧❛t✐♦♥✳ ❑❡②✇♦r❞s ❆r❜❡❧♦s✱ ●❡♦♠❡tr②✱ P❛r❜❡❧♦s✱ P❛r❛❜♦❧❛s✳ ✈✐✐✐ ❙✉♠ár✐♦ ✶ P❘❊▲■▼■◆❆❘❊❙ ✶✳✶ P❛rá❜♦❧❛s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✶ ❊q✉❛çã♦ ●❡r❛❧ ❞❛ P❛rá❜♦❧❛ ✳ ✳ ✳ ✶✳✶✳✷ ▲❛t✉s ❘❡❝t✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✸ ❈♦♥st❛♥t❡ ❯♥✐✈❡rs❛❧ ❞❛ P❛rá❜♦❧❛ ✶✳✷ ❖ ❚❡♦r❡♠❛ ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ▲✐♠✐t❡s✱ ❉❡r✐✈❛❞❛s ❡ ■♥t❡❣r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ▲✐♠✐t❡ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ ❉❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✸ ■♥t❡❣r❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❖ ♣❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❙❡♠❡❧❤❛♥ç❛ ❡♥tr❡ ✜❣✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✸ ✶✹ ✶✺ ✶✼ ✷✵ ✷✷ ✷✷ ✷✺ ✷✻ ✸✸ ✸✹ ✸✹ ✷ P❆❘❇❊▲❖❙ ✷✳✶ ❯♠ ❡st✉❞♦ ❢✉♥❝✐♦♥❛❧ ❞♦s P❛r❜❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞♦s P❛r❜❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ❖ ♣❛r❛❧❡❧♦❣r❛♠♦ ❛ss♦❝✐❛❞♦ ❛ ✉♠ ♣♦♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✶ P❛r❛❧❡❧♦❣r❛♠♦ ❈ús♣✐❞❡ ❱ért✐❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ❖ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❞♦s P❛r❜❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶ ❆ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ❝✐r❝✉♥s❝r❡✈❡ ♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❞♦ P❛r❜❡❧♦ ✸✾ ✹✵ ✹✸ ✹✺ ✹✽ ✺✶ ✺✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❆❚■❱■❉❆❉❊❙ ✺✼ ✸✳✶ ❈♦♥str✉çã♦ ❞❡ ✉♠ P❛r❜❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✷ ➪r❡❛ s♦❜ ❞❛ ❝✉r✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✐① ① ❙❯▼➪❘■❖ ▲✐st❛ ❞❡ ❋✐❣✉r❛s ✶✳✶ P❛rá❜♦❧❛ P ❝♦♠ ❢♦❝♦ F ✶✳✷ P❛rá❜♦❧❛ P ❝♦♠ ❢♦❝♦ F✱ ✶✳✸ P❛rá❜♦❧❛ ❝♦♠ ❝♦r❞❛ ❢♦❝❛❧ ✶✳✹ P❛rá❜♦❧❛ ❝♦♠ ✶✳✺ P❛rt✐çã♦ ❞❡ ❡ ✈ért✐❝❡ ✈ért✐❝❡ V V ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ C ✳ ✳ ✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ❬✲✷❛✱✷❛❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✻ P❛rá❜♦❧❛ ❡ ♦ tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✼ ❋✉♥çã♦ ❝♦♥tí♥✉❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✽ ❋✉♥çã♦ ♥ã♦ ❝♦♥tí♥✉❛✱ ♦♥❞❡ ✈❡r✐✜❝❛✲s❡ ✉♠ ✑s❛❧t♦✑ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✾ ❋✉♥çã♦ ❝♦♥tí♥✉❛ ❡ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❡♠ C1 C2 ▲❛t✉s ❘❡❝t✉♠ [a, b] ❡ ♦ ♣♦♥t♦ ♣❡rt❡♥❝❡♥t❡ à ♣❛rá❜♦❧❛ ✶✹ h ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✶✵ ❊①❡♠♣❧♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ✳ ✳ ✷✼ ✶✳✶✶ ❊①❡♠♣❧♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ✷✽ ✶✳✶✷ ❆♣r♦①✐♠❛çõ❡s ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✶✸ ❆♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❡ R1 (x − x )dx R1 (x − x2 )dx 0 ✶✳✶✹ ❆♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❡ ✶✳✶✺ ❆♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❡ 0 ✶✳✶✻ ❆♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❡ ✶✳✶✼ P❛r❛❧❡❧♦❣r❛♠♦ ABCD ✶✳✶✽ P❛r❛❧❡❧♦❣r❛♠♦ ABCD 2 ❝♦♠ ♥❂✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ R1 (x − x )dx R1 (x − x2 )dx 0 2 ✷✾ ❝♦♠ ♥❂✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ❝♦♠ ♥❂✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❝♦♠ n = 10 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ 0 △CDA ✸✸ ✶✳✶✾ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✶✳✷✵ ❍❡①á❣♦♥♦s s❡♠❡❧❤❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ P ❡ P′ ❝♦♠ s❡✉s r❡s♣❡❝t✐✈♦s ❢♦❝♦s ✶✳✷✷ ❘♦t❛çã♦ ❡ tr❛♥s❧❛çã♦ ❞❛ P❛rá❜♦❧❛ ✶✳✷✸ ❚r❛♥s❧❛çã♦ ❞❛ P❛rá❜♦❧❛ P P F1 ①✐ F2 ❡ L2 ✸✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ y ❡ ❞✐r❡tr✐③❡s L1 ✳ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ ❡ ❡ △CBA ✳ ✳ ✳ ✳ ✶✳✷✶ P❛rá❜♦❧❛s ❡ ♦s tr✐â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ①✐✐ ✷✳✶ ❆r❜❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✷ P❛r❜❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✸ P❛rá❜♦❧❛ ❝♦♠ r❛í③❡s ❡♠ ✷✳✹ P❛rá❜♦❧❛ ❝♦♠ r❛í③❡s ❡♠ ✵ ❡ ✶ ✷✳✺ P❛r❜❡❧♦ ❞❡ ❙♦♥❞♦✇ ✷✳✻ P❛r❜❡❧♦s s❡♠❡❧❤❛♥t❡s ✷✳✼ P❛r❛❧❡❧♦❣r❛♠♦ ❛ss♦❝✐❛❞♦ ❛♦ ♣♦♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✽ ❖ P❛r❛❧❡❧♦❣r❛♠♦ ❈ús♣✐❞❡ ❱ért✐❝❡ ❞♦ P❛r❜❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✾ ▲❛❞♦s ♣❛r❛❧❡❧♦s ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❝ús♣✐❞❡ ✈ért✐❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ x1 ❡ x2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ x0 ✷✳✶✵ P❛r❛❧❡❧♦❣r❛♠♦ ❈ús♣✐❞❡ ❱ért✐❝❡ ❞♦s P❛r❜❡❧♦s ❡ ♦s ❚r✐â♥❣✉❧♦s ■♥s❝r✐t♦s ✹✹ ✳ ✳ ✳ ✺✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✶✷ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✶✸ ❘❡❧❛çã♦ ❡♥tr❡ ❛s ❛❧t✉r❛s ❞♦s tr✐â♥❣✉❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✷✳✶✹ ❖ P❛r❛❧❡❧♦❣r❛♠♦ ❈ús♣✐❞❡ ❱ért✐❝❡ ❡ ♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ✺✸ ✷✳✶✶ ❘❡t❛ t❛♥❣❡♥t❡ à ♣❛rá❜♦❧❛ ♥♦ ♣♦♥t♦ C1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✺ ❖ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❞♦s P❛r❜❡❧♦s✱ ✉♠❛ ❞✐❛❣♦♥❛❧✱ ❡ ✉♠ ➶♥❣✉❧♦ ❇✐ss❡❝t♦r ✺✹ ✷✳✶✻ ❖ ❈ír❝✉❧♦ ❞♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❡ ♦ ❋♦❝♦ ❞❛ P❛rá❜♦❧❛ ❙✉♣❡r✐♦r✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✶ P❛r❜❡❧♦ ❝♦♥str✉í❞♦ ♣❛r❛ ♦ ❡①❡r❝í❝✐♦ ✹✳✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✷ P❛r❜❡❧♦ ❢♦r♠❛❞♦ ♥♦ ❡①❡r❝í❝✐♦ ✹✳✶✳ ✺✽ ✸✳✸ ❈á❧❝✉❧♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ár❡❛ ♣♦r ❢❛❧t❛ ❝♦♠ ♥❂✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✹ ❈á❧❝✉❧♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ár❡❛ ♣♦r ❡①❝❡ss♦ ❝♦♠ ♥❂✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✺ ❈á❧❝✉❧♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ár❡❛ ♣♦r ❡①❝❡ss♦ ❝♦♠ ♥❂✶✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✻ ❈á❧❝✉❧♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ár❡❛ ♣♦r ❢❛❧t❛ ❝♦♠ ♥❂✶✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✷✳✶✼ ❉✐stâ♥❝✐❛s ❞♦ ❈❡♥tr♦ ❞❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❛♦ ♣♦♥t♦ T1 ❡ ❛♦ ❋♦❝♦ F✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ■◆❚❘❖❉❯➬➹❖ ◆❡st❛ ❞✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦ ❞♦ Pr♦❣r❛♠❛ Pr♦✜ss✐♦♥❛❧ ❞❡ ▼❛t❡♠át✐❝❛✱ ❡st✉❞❛♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ❞♦✐s ✐♠♣♦rt❛♥t❡s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✿ ♣❛rá❜♦❧❛s ❡ ❢✉♥✲ çõ❡s✳ ❚❛✐s ❝♦♥❝❡✐t♦s s❡rã♦ ❛❜♦r❞❛❞♦s ♥♦s P❛r❜❡❧♦s✱ q✉❡ ❞❡r✐✈❛♠ ❞❡ ✉♠❛ ❝♦♥str✉çã♦ ❝❧áss✐❝❛ ❞❛ ●❡♦♠❡tr✐❛ P❧❛♥❛✱ ♦s ❆r❜❡❧♦s✳ P❛r❛ t❛♥t♦✱ ♦ ❞✐✈✐❞✐♠♦s ❡♠ três ❝❛♣ít✉❧♦s✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s q✉❡ ♥♦s ❞❛rã♦ ❜❛s❡ ♣❛r❛ ❞❡♠♦♥str❛r ❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❡♥✈♦❧✈❡♠ ♦s P❛r❜❡❧♦s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛s ♣❛rá❜♦❧❛s ❡ ❛ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❆rq✉✐♠❡❞❡s✱ q✉❡ r❡❧❛❝✐♦♥❛ ❛ ár❡❛ ❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ♣❛r❛❜ó❧✐❝❛ ❝♦♠ ♦ tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ♥❛ ♠❡s♠❛✱ ❡ ♦s ❝♦♥❝❡✐t♦s ❞❡ ▲✐♠✐t❡s✱ ❉❡r✐✈❛❞❛s ❡ ■♥t❡❣r❛✐s✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛♠♦s ♦s P❛r❜❡❧♦s ❞❡ ✉♠❛ ❢♦r♠❛ ❝❧áss✐❝❛ ❡ ❛❜♦r❞❛♠♦s t❛♠❜é♠ ✉♠❛ ❢♦r♠❛ ❛❧t❡r♥❛t✐✈❛ ❞❡ ❡①♣❧♦r❛r t❛❧ ❝♦♥❝❡✐t♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ✐❞❡✐❛ ❞♦s f − belos✱ ✈✐st❛s ❡♠ ❬✻❪✳ ❊✱ ✉t✐❧✐③❛♥❞♦ ❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛ ❡ ♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ ❛♣r❡s❡♥t❛♠♦s ❡ ❞❡♠♦♥str❛♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s r❡❧❛❝✐♦♥❛❞❛s ❛♦s ♣❛r❜❡❧♦s✳ P♦r ✜♠✱ ❛❜♦r❞❛♠♦s ❛❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s r❡❧❛❝✐♦♥❛❞❛s à ✐❞❡✐❛ ❞❡ P❛r❜❡❧♦s q✉❡ ♣♦❞❡♠ s❡r tr❛❜❛❧❤❛❞❛s ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦✱ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ①✐ ✶✷ ▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ❈❛♣ít✉❧♦ ✶ P❘❊▲■▼■◆❆❘❊❙ ❆♣r❡s❡♥t❛r❡♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s q✉❡ ♥♦s ❛❥✉❞❛rã♦ ❡♥t❡♥❞❡r ♦ ❝♦♥❝❡✐t♦ ❞♦s P❛r❜❡❧♦s✱ ❜❡♠ ❝♦♠♦ ❥✉st✐✜❝❛r ❛❧❣✉♠❛s ♣r♦♣♦s✐çõ❡s ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✳ ❖ ♣r✐♠❡✐r♦ ❝♦♥❝❡✐t♦ ❛ s❡r tr❛❜❛❧❤❛❞♦✱ é ♦ ❞❡ P❛rá❜♦❧❛s✱ ✉♠❛ ✈❡③ q✉❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ P❛r❜❡❧♦s ❡stá ❞✐r❡t❛♠❡♥t❡ ❧✐❣❛❞♦ às P❛rá❜♦❧❛s✳ ❚r❛t❛r❡♠♦s ❛q✉✐ ❞❛ s✉❛ ❞❡✜♥✐çã♦ ❡ ❡①♣❧✐❝✐t❛r❡♠♦s ❛❧❣✉♠❛s ♦❜s❡r✈❛çõ❡s r❡❧❡✈❛♥t❡s✱ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ à ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❞❛s ♣❛rá❜♦❧❛s✱ ❞❡♥♦♠✐♥❛❞❛ ❈♦♥st❛♥t❡ ❯♥✐✈❡rs❛❧ ❞❛ P❛r❛❜ó❧❛✱ q✉❡ é à r❡❧❛çã♦ ❡♥tr❡ ❛ ár❡❛ ❞♦ s❡❣♠❡♥t♦ ♣❛r❛❜ó❧✐❝♦ ❡ ♦ tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ♥❡❧❛✱ ❞❡♠♦♥str❛❞❛ ♣♦r ❆rq✉✐♠❡❞❡s✱ ❡ ❛ ✐♥❝❧✐✲ ♥❛çã♦ ❞❛s r❡t❛s q✉❡ ♣❛ss❛♠ ♣♦r s✉❛s r❛í③❡s ❡ ♣❡❧♦ ✈ért✐❝❡✳ P♦r ✜♠✱ ❛♣r❡s❡♥t❛r❡♠♦s t❛♠❜é♠✱ ❛s ❞❡✜♥✐çõ❡s ❞❡ Limites✱ Derivadas ❡ Integrais ❞❡ ❢✉♥çõ❡s✱ ✉♠❛ ✈❡③ q✉❡ ✉t✐❧✐③❛r❡♠♦s t❛✐s ❝♦♥❝❡✐t♦s ♣❛r❛ ♣r♦✈❛r ❛❧❣✉♠❛s ♣r♦♣♦s✐çõ❡s ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✳ ✶✳✶ P❛rá❜♦❧❛s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❉❡✜♥✐çã♦ ✶✳✶✳ ✭P❛rá❜♦❧❛✮ ❙❡❥❛♠ F ✉♠ ♣♦♥t♦ ✜①♦ ♥♦ ♣❧❛♥♦ ❡ L ✉♠❛ r❡t❛ q✉❡ ♥ã♦ ♣❛ss❛ ♣♦r F ✳ ❈❤❛♠❛♠♦s ❞❡ P❛rá❜♦❧❛ P ❛♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ❡q✉✐❞✐st❛♥t❡s ❞❡ F ❡ ❞❛ r❡t❛ L✳ ❖ ♣♦♥t♦ F é ❝❤❛♠❛❞♦ ❞❡ ❋♦❝♦ ❡ ❛ r❡t❛ L é ❝❤❛♠❛❞❛ ❞❡ ❉✐r❡tr✐③✳ ❆ ❞✐stâ♥❝✐❛ p > 0 ❞❡ F ❛té L = 2a é ❞❡♥♦♠✐♥❛❞♦ ♣❛râ♠❡tr♦ ❋♦❝❛❧✳ ❖ ♣♦♥t♦ V q✉❡ ❡stá ❛ ✉♠❛ ♠❡s♠❛ ❞✐stâ♥❝✐❛ a = p2 ❞❡ F ❡ ❞❡ L é ♦ ❱ért✐❝❡ ❞❡ P ✳ ❆ ✜❣✉r❛ ✶✳✶ ❛♣r❡s❡♥t❛ ✉♠❛ ♣❛rá❜♦❧❛ ❝♦♠ F = (0, 0) ❡ V = (0, a) ❡ ❛ ✉s❛r❡♠♦s ♥❛s ❞❡✜♥✐✲ ✲çõ❡s ❡ ❝♦♥str✉çõ❡s ❛ s❡❣✉✐r✳ ✶✸ ✶✹ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ❋✐❣✉r❛ ✶✳✶✿ P❛rá❜♦❧❛ P ❝♦♠ ❢♦❝♦ F ❡ ✈ért✐❝❡ V ✶✳✶✳✶ ❊q✉❛çã♦ ●❡r❛❧ ❞❛ P❛rá❜♦❧❛ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❉❡✜♥✐çã♦ ✶✳✶✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❛ ❡q✉❛çã♦ ❞❛ P❛rá❜♦❧❛ P ✱ ✉s❛♥❞♦ q✉❡✱ ❛ ❞✐stâ♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ C q✉❛❧q✉❡r ❞❛ P❛rá❜♦❧❛ ❛♦ ❋♦❝♦ F = (0, 0) é ✐❣✉❛❧ à ❞✐stâ♥❝✐❛ ❞♦ ♠❡s♠♦ ♣♦♥t♦ à r❡t❛ ❞✐r❡tr✐③ L = 2a✱ ❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ F ❛té L = 2a é ✐❣✉❛❧ ❛ 2a✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✶✳✶✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ t♦♠❡♠♦s ✉♠❛ P❛rá❜♦❧❛ P t❛❧ q✉❡ ♦ ❋♦❝♦ F s❡ ❡♥❝♦♥tr❡ ♥❛ ♦r✐❣❡♠ ❞❡ R2 ❡ ❛ r❡t❛ ❞✐r❡tr✐③ L s❡ ❡♥❝♦♥tr❡ ♣❛r❛❧❡❧❛ ❛♦ ❡✐①♦ x✳ ❙❡♥❞♦ d(C, F ) ❡ d(C, L) ❛ ❞✐stâ♥❝✐❛ ❊✉❝❧✐❞✐❛♥❛ ❡♥tr❡ C ❡ F ❡ C ❡ L✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ s❛❜❡♥❞♦ q✉❡ P é ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ t❛❧ q✉❡ d(C, F ) é ✐❣✉❛❧ ❛ d(C, L)✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡✿ ❋✐❣✉r❛ ✶✳✷✿ P❛rá❜♦❧❛ P ❝♦♠ ❢♦❝♦ F ✱ ✈ért✐❝❡ V ❡ ♦ ♣♦♥t♦ C ♣❡rt❡♥❝❡♥t❡ à ♣❛rá❜♦❧❛ ✶✳✶✳ ✶✺ P❆❘➪❇❖▲❆❙ ❊ ❙❯❆❙ P❘❖P❘■❊❉❆❉❊❙ d(C, F ) = d(C, L) ⇐⇒ » » (x − 0)2 + (y − 0)2 = » » ⇐⇒ x2 + y 2 = (2a − y)2 ⇐⇒ x2 + y 2 = 4a2 − 4ay + y 2 ⇐⇒ x2 = 4a2 − 4ay ⇐⇒ x2 − 4a2 = −4ay x2 − 4a2 = y. ⇐⇒ −4a (x − x)2 + (2a − y)2 e 2a − y ≥ 0 P♦rt❛♥t♦✱ ❛ ❡q✉❛çã♦ ❞❛ ♣❛rá❜♦❧❛ P é ❞❛❞❛ ♣♦r✿ y =a− ✶✳✶✳✷ x2 . 4a ✭✶✳✶✮ ▲❛t✉s ❘❡❝t✉♠ ◗✉❛❧q✉❡r s❡❣♠❡♥t♦ ♣❛ss❛♥❞♦ ♣♦r F ❝♦♠ q✉❛✐sq✉❡r ❡①tr❡♠✐❞❛❞❡s C1 ✱ C2 ♣❡rt❡♥❝❡♥t❡s à P❛rá❜♦❧❛ P é ❝❤❛♠❛❞♦ ❞❡ ❝♦r❞❛ ❢♦❝❛❧✳ ❆ ✜❣✉r❛ ✶✳✶✳✷ ✐❧✉str❛ ❛ ❝♦r❞❛ ❢♦❝❛❧ ❞❡ ❡①tr❡♠✐❞❛❞❡s C1 ❡ C2 ✳ ❋✐❣✉r❛ ✶✳✸✿ P❛rá❜♦❧❛ ❝♦♠ ❝♦r❞❛ ❢♦❝❛❧ C1 C2 ❖ ▲❛t✉s ❘❡❝t✉♠ ❞❡ ✉♠❛ ♣❛rá❜♦❧❛✱ é ❛ ❝♦r❞❛ ❢♦❝❛❧ ❝✉❥♦ ❝♦♠♣r✐♠❡♥t♦ é ♠í♥✐♠♦✳ ❚❛❧ ❝♦r❞❛ ❢♦❝❛❧ é ♦❜t✐❞❛ q✉❛♥❞♦ ♦ s❡❣♠❡♥t♦ C1 C2 é ♣❛r❛❧❡❧♦ à r❡t❛ L✳ ◆❡st❡ ❝❛s♦✱ t♦♠❛♥❞♦✲s❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❡♠ q✉❡ ♦ ❡✐①♦ x é ♣❛r❛❧❡❧♦ à L ❡ F é ❛ ♦r✐❣❡♠✱ t❡♠♦s C1 = (−2a, 0)✱ ❡ C2 = (2a, 0)✳ ✶✻ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ❉❡ ❢❛t♦✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ t♦♠❛♥❞♦✲s❡ C1 = (α, a − α4a ) ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ♣❡rt❡♥❝❡♥t❡ à ♣❛rá❜♦❧❛ P ✱ ❝♦♠ α 6= 0✱ ❡ F = (0, 0) ♦ ❢♦❝♦ ❞❡ P ✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ r q✉❡ ♣❛ss❛ ♣♦r C1 ❡ F ✿ 2 Å a − α4a a αã y= x= − x. α α 4a 2 ❆ ✐♥t❡rs❡❝çã♦ ❡♥tr❡ ❛ r❡t❛ r ❡ ❛ ♣❛rá❜♦❧❛ P ❞á ♦s ♣♦♥t♦s C1 = (α, a− α4a ) ❡ C2 = (β, a− β4a )✳ ▼❛s✱ ❝♦♠♦ C2 t❛♠❜é♠ ♣❡rt❡♥❝❡ à r✱ ❡♥tã♦✿ 2 a− 2 β2 Å a αã ·β = − 4a α 4a 4a2 − β 2 4a2 β − α2 β = 4a 4aα 4a2 α − αβ 2 = 4a2 β − α2 β 4a2 (α − β) = −αβ(α − β). ❉❛❞❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ♣❛rá❜♦❧❛✱ ❡ ❛ ❡s❝♦❧❤❛ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ♣♦❞❡♠♦s ❛❞♠✐t✐r q✉❡ C1 ❡ C2 t❡♥❤❛♠ ❛❜s❝✐ss❛s ❞✐st✐♥t❛s✱ ♦✉ s❡❥❛ α 6= β ✳ ❆ss✐♠ β = ▲♦❣♦✱ −4a2 α ✳ Ñ C2 = 4 16a −4a2 2 ,a − α α 4a é å Ç 16a4 −4a2 C2 = ,a − α 4aα2 Ç å −4a2 4a3 C2 = ,a − 2 . α α ❈❛❧❝✉❧❡♠♦s ❛ ❞✐stâ♥❝✐❛ ❞❡ C1 ❛ C2 ✱ ❧❡♠❜r❛♥❞♦ q✉❡ ❛ ♠❡s♠❛ ❞❡✈❡ s❡r ♠í♥✐♠❛ ✭♣♦r ❞❡✜♥✐çã♦ ❞❡ ▲❛t✉s r❡❝t✉♠ ✮✿ Ã d(C1 , C2 ) = Ã = Ç å2 Ç å2 4a2 α+ α 4a2 α+ α Ç 4a3 α2 −a+ 2 + a− 4a α Ç α2 4a3 + − + 2 4a α å2 . å2 ✶✳✶✳ ✶✼ P❆❘➪❇❖▲❆❙ ❊ ❙❯❆❙ P❘❖P❘■❊❉❆❉❊❙ Ä 2 ä2 é s❡♠♣r❡ α + 4aα 4a2 2 α = − α ⇐⇒ α = −4a2 ❈♦♠♦ ♠❛✐♦r q✉❡ ③❡r♦✱ ❥á q✉❡ ♣❛r❛ s❡r ✐❣✉❛❧ ❛ ③❡r♦ ❞❡✈❡rí❛♠♦s t❡r ✭♦ q✉❡ é ✐♠♣♦ssí✈❡❧✮✱ ♣❛r❛ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ C1 ❡ C2 s❡❥❛ ♠í♥✐♠❛✱ ❞❡✈❡♠♦s t❡r✿ α2 4a3 + 2 = 0 ⇐⇒ 4a α 4a3 α2 = 2 ⇐⇒ ⇐⇒ 4a α ⇐⇒ α4 = 16a4 ⇐⇒ ⇐⇒ α = ±2a. − ❋✐❣✉r❛ ✶✳✹✿ P❛rá❜♦❧❛ ❝♦♠ ▲❛t✉s ❘❡❝t✉♠ ❬✲✷❛✱✷❛❪ P♦rt❛♥t♦✱ ♦ ▲❛t✉s ❘❡❝t✉♠ ♦❝♦rr❡ ♣❛r❛ C1 = (−2a, 0) ❡ C2 = (2a, 0) ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ❋✐❣✉r❛ ✶✳✹✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ▲❛t✉s ❘❡❝t✉♠ ❡q✉✐✈❛❧❡ ❛ ✹ ✈❡③❡s ❛ ❞✐stâ♥❝✐❛ ❞♦ ❢♦❝♦ V✱ F ❛té ♦ ✈ért✐❝❡ ❝♦♠♦ ✐❧✉str❛❞♦ ♥❛ ✶✳✹✳ ❊♥tr❡ C1 ❡ C2 ❡♥❝♦♥tr❛✲s❡ ❛ ♣❛rá❜♦❧❛ P ❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ C1 C2 é 2p = 4a✳ ▼❡t❛❞❡ ❞❡st❡ ❝♦♠♣r✐♠❡♥t♦ p = 2a é ❝❤❛♠❛❞♦ ❞❡ ❙❡♠✐✲▲❛t✉s ❘❡❝t✉♠✳ ¸ C 1 V C2 ❞❡ P é ♦ ❆r❝♦ ▲❛t✉s ❘❡❝t✉♠✳ ✶✳✶✳✸ ✐❣✉❛❧ ❛ ❖ ❛r❝♦ ❈♦♥st❛♥t❡ ❯♥✐✈❡rs❛❧ ❞❛ P❛rá❜♦❧❛ ❯♠ r❡s✉❧t❛❞♦ ✐♠♣♦rt❛♥t❡ q✉❡ ✉t✐❧✐③❛r❡♠♦s ❢✉t✉r❛♠❡♥t❡ ♣❛r❛ ❞❡♠♦♥str❛r ✉♠❛ ❞❛s ♣r♦♣♦s✐✲ çõ❡s é ❛ ❈♦♥st❛♥t❡ ❯♥✐✈❡rs❛❧ ❞❛ P❛r❛❜ó❧❛ q✉❡ ♥♦s ♠♦str❛ q✉❡ ❛ r❛③ã♦ ❡♥tr❡ ♦ ❝♦♠♣r✐♠❡♥t♦ s ❞♦ ❆r❝♦ ▲❛t✉s ❘❡❝t✉♠ ❞❡ q✉❛❧q✉❡r ♣❛rá❜♦❧❛ ❡ ❛ s✉❛ ❙❡♠✐✲▲❛t✉s ❘❡❝t✉♠ é ✉♠❛ ❝♦♥st❛♥t❡ K ✱ ❝❤❛♠❛❞❛ ❞❡ ❈♦♥st❛♥t❡ ❯♥✐✈❡rs❛❧ ❞❛ P❛r❛❜ó❧❛ K = ps ✳ ✶✽ Pr♦♣♦s✐çã♦ ✶✳✶✳ ❈❆P❮❚❯▲❖ ✶✳ ♦ ▲❛t✉s ❘❡❝✲ ¸ ❡♥tr❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❛r❝♦ C 1 V C2 √ ✭❈♦♥st❛♥t❡ ❯♥✐✈❡rs❛❧ P❛r❛❜ó❧✐❝❛✮✿ ❉❡♥♦t❛♥❞♦✲s❡ ♣♦r t✉♠ ❞❡ ✉♠❛ P❛rá❜♦❧❛ P ❝♦♠ ✈ért✐❝❡ V✱ ❛ r❛③ã♦√K ❡ s❡✉ ❙❡♠✐✲▲❛t✉s ❘❡❝t✉♠ é ❝♦♥st❛♥t❡ ❡ ✐❣✉❛❧ ❛ P❘❊▲■▼■◆❆❘❊❙ 2 + ln(1 + C1 C2 2)✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ✈❛♠♦s ❝❛❧❝✉❧❛r✱ ♣♦r ♠❡✐♦ ❞❡ ❈á❧❝✉❧♦ ■♥t❡❣r❛❧ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ ♣❛r❛❜ó❧✐❝♦✳ ❊♠❜♦r❛ ❡st❡ ♥ã♦ s❡❥❛ ✉♠ ❛ss✉♥t♦ ❡st✉❞❛❞♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❛♣r❡s❡♥t❛♠♦s ❛q✉✐ ✉♠❛ ❛r❣✉♠❡♥t❛çã♦ ❛ r❡s♣❡✐t♦ ❞♦ ❝á❧❝✉❧♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝✉r✈❛s✱ ❛ ♣❛rt✐r ❞❛ ❛♣r♦①✐♠❛çã♦ ❞❡st❡ ❝♦♠ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ♣♦❧✐❣♦♥❛❧✳ P❛r❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞❛ ❛r❣✉♠❡♥t❛çã♦✱ r❡❝♦♠❡♥❞❛♠♦s q✉❡ ♦ ❧❡✐t♦r ❡st✉❞❡ ❛❣♦r❛ ❛ ❙❡çã♦ ✷✳✸✱ q✉❡ tr❛t❛ r❛♣✐❞❛♠❡♥t❡ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ▲✐♠✐t❡s✱ ❉❡r✐✈❛❞❛s ❡ ■♥t❡❣r❛✐s ❞❡ ✉♠❛ ❢✉♥çã♦ r❡❛❧✳ ❉❡ q✉❛❧q✉❡r ❢♦r♠❛✱ ❞❛♠♦s ❛♦ ❧❡✐t♦r ✉♠❛ ✐❞❡✐❛ ❞❡ ❝♦♠♦ ❡st❛s ❢❡rr❛♠❡♥t❛s sã♦ ❛♣❧✐❝❛❞❛s às ❢✉♥çõ❡s r❡❛✐s✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ f (x) ✉♠ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ r❡❛❧ [a, b] ❡ ❞❡r✐✈á✈❡❧ ❡♠ (a, b)✳ ❱❛♠♦s ❞✐✈✐❞✐r ♦ ✐♥t❡r✈❛❧♦ [a, b] ❡♠ n ♣❛rt❡s t❛❧ q✉❡ xo = a < x1 < x2 < x3 .... < x(k−1) < xk < ... < xn = b ❡ s❡❥❛ Pk ♦ ♣♦♥t♦ (xk , yk ) ❝♦♠ yk ❂f (xk ) ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ✜❣✉r❛ ✳ ❋✐❣✉r❛ ✶✳✺✿ P❛rt✐çã♦ ❞❡ [a, b] ❖ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ♠❡♥♦r s❡❣♠❡♥t♦ q✉❡ ❧✐❣❛ ♦s ♣♦♥t♦s Pk−1 ❛ Pk é ❞❛❞♦ ♣❡❧❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡❧❡s✳ » S= (xk−1 − xk )2 − (yk−1 − yk )2 . ❆ ♠❡❞✐❞❛ q✉❡ ❛✉♠❡♥t❛♠♦s ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ♥❛ ♣❛rt✐çã♦ ❞❡ [a, b] ✭❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ♣♦♥t♦s x0 , x1 , x2 ..., xn ❡♠ R t❛❧ q✉❡ a = x0 < x1 < x2 ... < xn = b✮✱ ❛♣r♦①✐♠❛♠♦s ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♣♦❧✐❣♦♥❛❧ ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❛r❝♦✳ ❆ss✐♠✱ ❛✉♠❡♥t❛♥❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ ♦ ✈❛❧♦r ❞❡ n✱ t❡♠♦s q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❛r❝♦ é ❞❛❞♦ ♣♦r✿ ✶✳✶✳ P❆❘➪❇❖▲❆❙ ❊ ❙❯❆❙ P❘❖P❘■❊❉❆❉❊❙ n X L = lim n→∞ ◆♦ ❡♥t❛♥t♦❀ ❝♦♥s✐❞❡r❛♥❞♦ ∆ xi ❡ ∆ yi |Pi−1 − Pi |, i=1 (xi−1 − xi ) ✐❣✉❛✐s ❛ t❡♠♦s✿ |Pk−1 − Pk | = = » » Ã = P♦r ❤✐♣ót❡s❡✱ f é ❝♦♥tí♥✉❛ ❡♠ ✶✾ » ∆2xk − ∆2yk = (yi−1 − yi )✱ ❡ (xk−1 − xk )2 − (yk−1 − yi )2 (xk−1 − xi )2 − (f (xk−1 ) − f (xk ))2 ô ñ (xk−1 − xk )2 . r❡s♣❡❝t✐✈❛♠❡♥t❡✱ (f (xk−1 ) − f (xk ))2 . 1− (xk−1 − xk )2 [a, b] ❡ ❞❡r✐✈á✈❡❧ ❡♠ (a, b)✱ ❧♦❣♦ ❛ ❡①♣r❡ssã♦ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❝♦♠ ♦ ✉s♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✱ ♣♦✐s✿ f (xi ) − f (xi−1 ) = f ′ (x̄i ).(xi − xi−1 ), ♣❛r❛ ❛❧❣✉♠ x̄✱ t❛❧ q✉❡ xi−1 < x̄ < xi ✳ Ã |Pi−1 − Pi | = ♦♥❞❡ ∆xi = xi − xi−1 ❖✉ s❡❥❛✱ ô ñ » (f (xi−1 ) − f (xi ))2 = (xi−1 − xi )2 . 1 − [1 + f ′ (x̄i )2 ].∆xi , (xi−1 − xi )2 ♣❛r❛ 1 < i < n✳ ❉❡st❛ ♠❛♥❡✐r❛✱ L = n→∞ lim n X i=1 |Pi−1 − Pi | = n→∞ lim ✉♠❛ ✈❡③ q✉❡ n » X [1 + f ′ (x̄i )2 ].∆xi i=1 n » X ✐♥t❡r✈❛❧♦ [a, b]✱ ♦ ❝♦♠♣r✐♠❡♥t♦ s x✱ t❡♠♦s q✉❡✿ 2 g(x) = Z b» a f (x) = a − x4a ✱ [1 + f ′ (x̄i )2 ]dx, » [1 + f ′ (xk )2 ] r❡❧❛t✐✈❛ à ♣❛rt✐çã♦ P ❞♦ ❞❡ ✉♠❛ ❝✉r✈❛ é ❞❛❞♦ ♣♦r✿ s= P❛r❛ ❛ ❢✉♥çã♦ b [1 + f [(x̄i )2 ] i=1 é ✉♠❛ ❙♦♠❛ ❞❡ ❘✐❡♠❛♥♥ ♣❛r❛ ❛ ❢✉♥çã♦ = Z a» ❝♦♠ 1 + f ′ (x)2 dx. −a ≤ x ≤ a✱ t❛❧ q✉❡ ♦ ✈ért✐❝❡ ❡stá ❡✐①♦ f ′ (x) = − x , 2a y ❡ ❢♦❝♦ ♥♦ ❡✐①♦ ✷✵ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ❞❡ ♠♦❞♦ q✉❡ s❡✉ ❝♦♠♣r✐♠❡♥t♦ é ❞❛❞♦ ♣♦r✿ s= Z 2a   −2a ❋❛③❡♥❞♦✲s❡ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ s = 2a Z 1 √ Å x ã2 1+ − dx. 2a x = 2at✱ t❡♠♦s q✉❡✿ 1 + t2 dt −1 √ #1 2) ln (t + t√ 1 + t 1 + t2 + = 2a 2 2 −1  »  » √ # "√ 2 2) 2 2 1 + (−1) 1 + (−1) ln (−1 + 1+1 ln (1 + 1 + 1 )  + − 2a − + = 2a 2 2 2 2 √ # √ # "√ " √ 2 ln (1 + 2) 2 ln (−1 + 2) = 2a − 2a − + + 2 2 2 2 √ √ = 2a( 2 + ln (1 + 2)). " ▲♦❣♦✱ √ √ ä Ä s = 2a 2 + 2a ln 1 + 2 . P♦rt❛♥t♦✱ ❛ ❈♦♥st❛♥t❡ ❯♥✐✈❡rs❛❧ ❞❛ P❛rá❜♦❧❛ é ❝♦♥st❛♥t❡ ❡ ✐❣✉❛❧ ❛✿ K= ✶✳✷ √ ää √ √ Ä s 1 Ä √ = 2a 2 + 2a ln 1 + 2 = 2 + ln (1 + 2) 2a 2a ❖ ❚❡♦r❡♠❛ ❞❡ ❆rq✉✐♠❡❞❡s ❊st❡ ✐♠♣♦rt❛♥t❡ ❚❡♦r❡♠❛ ♥♦s ♠♦str❛ q✉❡ ❛ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ♣❛r❛❜ó❧✐❝❛ é ✐❣✉❛❧ ❛ 4 ❞❛ 3 ár❡❛ ❞♦ ♠❛✐♦r tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ♥❡❧❛✳ ❚❡♦r❡♠❛ ✶✳✶✳ ❆ ár❡❛ ❞♦ ♠❛✐♦r tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ♥✉♠ s❡❣♠❡♥t♦ ❞❡ ✉♠❛ ♣❛rá❜♦❧❛ é 4 ❞❛ 3 ár❡❛ ❞♦ r❡s♣❡❝t✐✈♦ s❡❣♠❡♥t♦ ♣❛r❛❜ó❧✐❝♦✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ P ✉♠❛ ♣❛rá❜♦❧❛ ❞❡ ✈ért✐❝❡ V ❡ △C1 C2 V ✉♠ tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ♥❡❧❛✱ ✐st♦ é✱ ✉♠ tr✐â♥❣✉❧♦ ❝✉❥♦s ✈ért✐❝❡s ♣❡rt❡♥❝❡♠ à ♣❛rá❜♦❧❛✱ s❡♥❞♦ ❡st❡ ❞❡ t❛❧ ❢♦♠r❛ q✉❡ ♦ s❡❣♠❡♥t♦ ❈❤❛♠❛♥❞♦ ❞❡ ❡♠ B C1 C2 ❡st❡❥❛ ❝♦♥t✐❞♦ ♥♦ ❡✐①♦ x✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✻✳ B ♦ ♣♦♥t♦ ♠é❞✐♦ ❡♥tr❡ ❛ ♦r✐❣❡♠ F C1 ✱ s❡❥❛ A ♦ ♣♦♥t♦ ❝♦♠ ❛❜s❝✐ss❛ ❝❤❛♠❛♠♦s ❞❡ D ♦ ♣♦♥t♦ ♠é❞✐♦ ❡♥tr❡ ❛ ❡ ♦ ♣♦♥t♦ ♣❡rt❡♥❝❡♥t❡ à ♣❛rá❜♦❧❛✳ ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ✶✳✷✳ ❖ ❚❊❖❘❊▼❆ ❉❊ ❆❘◗❯■▼❊❉❊❙ ✷✶ ❋✐❣✉r❛ ✶✳✻✿ P❛rá❜♦❧❛ ❡ ♦ tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ♦r✐❣❡♠ F ❡ ♦ ♣♦♥t♦ C2 ✱ ❡ t♦♠❛♠♦s ♦ ♣♦♥t♦ E ❝♦♠ ❛❜s❝✐ss❛ ❡♠ D t❛♠❜é♠ ♣❡rt❡♥❝❡♥t❡ à ♣❛rá❜♦❧❛✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ ❛ ár❡❛ ❞♦s tr✐â♥❣✉❧♦s tr✐â♥❣✉❧♦s △C1 F V ❡ △C2 F V ✱ C1 V A ❡ C2 V E ❝♦rr❡s♣♦♥❞❡♠ ❛ 1 ❞❛ ár❡❛ ❞♦s 4 r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ←→ F V ✱ ❜❛st❛ ❛♥❛❧✐s❛r s♦♠❡♥t❡ ✉♠❛ ❞❛s 2a·a = a2 ✳ ♠❡t❛❞❡s✳ ❆ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ △C1 F V = T0 é ✐❣✉❛❧ à T0 = 2 ❆❣♦r❛✱ ✈❛♠♦s ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ △C1 V A = T1 ✳ ❚♦♠❡♠♦s ❝♦♠♦ ❜❛s❡ ❞❡st❡ tr✐â♥❣✉❧♦ ♦ s❡❣♠❡♥t♦ C1 V ✳ ❙❡❣✉❡ q✉❡✿ » √ √ C1 V = d(C1 , V ) = (−2a2 ) + (−a2 ) = 4a2 + a2 = a 5. ❈♦♠♦ ❛ P❛rá❜♦❧❛ é s✐♠étr✐❝❛ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ △C1 V A✱ ✈❛♠♦s ❡♥❝♦♥tr❛r ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ A ❞❡ A ❛té à r❡t❛ q✉❡ ❝♦♥té♠ ♦ s❡❣♠❡♥t♦ C1 V ✳ P❛r❛ ❝❛❧❝✉❧❛r ❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ ❡ ❡♠ s❡❣✉✐❞❛ ❝❛❧❝✉❧❛r ❛ ❞✐stâ♥❝✐❛ ❈♦♠♦ B C1 F ✱ t❡♠♦s q✉❡ B = (−a, 0)✳ = (−a, a − a4 ) = (−a, 3a )✳ 4 é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ s❡rá ❞❛❞♦ ♣♦r A = (−a, a − (−a)2 ) 4a P❛r❛ ❡♥❝♦♥tr❛r♠♦s ❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ ♣❛ss❛ ♣♦r C1 ❡ ❆ ❞✐stâ♥❝✐❛ ❞❡ V✿ A ❛té ▲♦❣♦✱ ♦ ♣♦♥t♦ △C1 V A✱ ✈❛♠♦s ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❞❡ r❡t❛ r 1 y = x + a. 2 r é ❞❛❞❛ ♣♦r✿ T1 = d(A, r) = | 12 (−a) − qÄ ä 2 1 2 −3 a 4 + a| + (−1)2 a 4 a = »1 = 4 +1 4   √ a 5 4 = . 5 10 A q✉❡ ✷✷ ❈❆P❮❚❯▲❖ ✶✳ √ √ P♦rt❛♥t♦ T1 ❂a 5 · a105 · 21 = a2 4 P❘❊▲■▼■◆❆❘❊❙ = 14 T0 ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣r♦✈❛✲s❡ q✉❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ △C2 V E é ✐❣✉❛❧ ❛ △C2 F V ✱ q✉❡ é ✐❣✉❛❧ ❛ T0 ✳ 1 4 ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ❙❡❣✉✐♥❞♦ ❡st❡ r❛❝✐♦❝í♥✐♦✱ ✐st♦ é✱ t♦♠❛♥❞♦✲s❡ ♦ ♣♦♥t♦ ♠é❞✐♦ M1 ❡♥tr❡ B ❡ C1 ✱ ❡ ❞❡♣♦✐s ♦ ♣♦♥t♦ ♠é❞✐♦ ❡♥tr❡ M1 ❡ C1 ✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✱ ♣♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ❛ ♠❡t❛❞❡ ❞❛ ár❡❛ ❞♦ s❡❣♠❡♥t♦ ♣❛r❛❜ó❧✐❝♦ s❡ ❛♣r♦①✐♠❛ ❞❛ ár❡❛ ❞❛ sér✐❡ ❣❡♦♠étr✐❝❛ ❞❡ r❛③ã♦ ✐❣✉❛❧ ❛ 41 ❡ ♣r✐♠❡✐r♦ t❡r♠♦ T0 > 0✿ n X Ti = T0 + T1 + T2 + T3 ... + Tn = T0 + i=0 T0 T0 T0 1 + 2 + ... + n = 4 4 4 1− P♦rt❛♥t♦✱ ❛ ár❡❛ ❞❡ ✉♠ s❡❣♠❡♥t♦ ♣❛r❛❜ó❧✐❝♦ é ✐❣✉❛❧ ❛ ♥❡❧❡✳ ✶✳✸ 4 3 1 4 4 = T0 . 3 ✈❡③❡s ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ▲✐♠✐t❡s✱ ❉❡r✐✈❛❞❛s ❡ ■♥t❡❣r❛✐s P❛r❛ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ ❝❡rt♦s ♣♦♥t♦s ❞♦ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛r❡♠♦s s✉❝✐♥t❛♠❡♥t❡ ❛s ❞❡✜♥✐çõ❡s ❞❡ ▲✐♠✐t❡s✱ ❉❡r✐✈❛❞❛s ❡ ■♥t❡❣r❛✐s✳ ✶✳✸✳✶ ▲✐♠✐t❡ ❞❡ ✉♠❛ ❢✉♥çã♦ ❖❜s❡r✈❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡✱ ❝♦♠♦ s❡ ❝♦♠♣♦rt❛ ❛ ❢✉♥çã♦ f (x) = x − 4 q✉❛♥❞♦ x s❡ ❛♣r♦①✐♠❛ ✭❝♦♠ ✈❛❧♦r❡s à ❞✐r❡✐t❛ ❡ à ❡sq✉❡r❞❛✮ ❞❡ ✷✳ ❈❛❧❝✉❧❛♥❞♦✲s❡ ♦s ✈❛❧♦r❡s ❞❡ x − 4 q✉❛♥❞♦ x s❡ ❛♣r♦①✐♠❛ ❞❡ 2 ♣❡❧❛ ❡sq✉❡r❞❛ ✭♣♦r ✈❛❧♦r❡s ♠❡♥♦r❡s q✉❡ 2✮ t❡♠♦s q✉❡✿ x x−4 ✶ ✲✸ ✶✱✺ ✲✷✱✺ ✶✱✻ ✲✷✱✹ ✶✱✽ ✲✷✱✷ ✶✱✾ ✲✷✱✶ ✶✱✾✾ ✲✷✱✵✶ ✶✱✾✾✾ ✲✷✱✵✵✶ ✷ ✲✷ ❊ ❝❛❧❝✉❧❛♥❞♦✲s❡ ♦s ✈❛❧♦r❡s ❞❡ x − 4 q✉❛♥❞♦ x s❡ ❛♣r♦①✐♠❛ ❞❡ 2 ♣❡❧❛ ❞✐r❡✐t❛ ✭♣♦r ✈❛❧♦r❡s ♠❛✐♦r❡s q✉❡ 2✮ t❡♠♦s q✉❡✿ ✶✳✸✳ x ✸ ◆♦t❡ q✉❡ f (x) ✷✸ ▲■▼■❚❊❙✱ ❉❊❘■❱❆❉❆❙ ❊ ■◆❚❊●❘❆■❙ f (2) = −2✱ s❡ ❛♣r♦①✐♠❛ ❞❡ x−4 ✲✶ ✷✱✺ ✲✶✱✺ ✷✱✹ ✲✶✱✻ ✷✱✷ ✲✶✱✽ ✷✱✶ ✲✶✱✾ ✷✱✵✶ ✲✶✱✾✾ ✷✱✵✵✶ ✲✶✱✾✾✾ ✷ ✲✷ ♦q✉❡ ❢♦r♥❡❝❡ ✉♠❛ ✐❞❡✐❛ ❞❡ q✉❡ q✉❛♥❞♦ x s❡ ❛♣r♦①✐♠❛ ❞❡ 2 ❛ ❢✉♥çã♦ −2✳ ❆ s❡❣✉✐r ❡♥tã♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ✐♥t✉✐t✐✈♦ ❞❡ ❢✉♥çã♦✳ I ⊂ R✳ ❉✐③❡♠♦s q✉❡ f é ❝♦♥tí♥✉❛ ❡♠ h ∈ R s❡ ❡①✐st❡ ♦ ❧✐♠✐t❡ ❞❡ f (x) q✉❛♥❞♦ x s❡ ❛♣r♦①✐♠❛ ❞❡ h ❡ ❡st❡ ❧✐♠✐t❡ é ✐❣✉❛❧ ❛ f (h)✳ ❖❧❤❛♥❞♦✲ s❡ ♣❛r❛ ♦ ❣rá✜❝♦ ❞❡ f (x)✱ ♣♦❞❡♠♦s ❞✐③❡r ✐♥t✉✐t✐✈❛♠❡♥t❡ q✉❡ é ❝♦♥tí♥✉❛ ❡♠ h s❡ ♥ã♦ ❤á ✑s❛❧t♦s✑ ♣❛r❛ ♦ ❣rá✜❝♦ ❞❡ f ❡♠ h✱ q✉❡ ♣❡rt❡♥❝❡ ❛♦ ❞♦♠í♥✐♦ I ✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♥✉♠ ✐♥t❡r✈❛❧♦ ❋✐❣✉r❛ ✶✳✼✿ ❋✉♥çã♦ ❝♦♥tí♥✉❛ ■♥t✉✐t✐✈❛♠❡♥t❡✱ ❞✐③❡♠♦s q✉❡ ♦ q✉❛♥❞♦ x t❡♥❞❡ ❛ h✱ limite ❞❡ f (x) ✱ q✉❛♥❞♦ x t❡♥❞❡ ❛ h✱ é ✐❣✉❛❧ ❛ M ✱ ♦✉ s❡❥❛✱ f (x) ❛♣r♦①✐♠❛♠✲s❡ ❝❛❞❛ ✈❡③ ♠❛✐s ❞❡ M ✳ ❙✐♠❜♦❧✐❝❛✲ ♦s ✈❛❧♦r❡❞❡ ❞❡ ♠❡♥t❡✱ ❡s❝r❡✈❡♠♦s ❛ss✐♠✿ lim f (x) = M. x→h ❆ ❞❡✜♥✐çã♦ ❞❡ ❧✐♠✐t❡ é ✉t✐❧✐③❛❞❛ ♣❛r❛ ❡♥t❡♥❞❡r♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛♥❞♦ ❡❧❛ s❡ ❛♣r♦①✐♠❛ ❞❡ ❞❡t❡r♠✐♥❛❞♦s ✈❛❧♦r❡s✳ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛♥❞♦ ❯t✐❧✐③❛♥❞♦✲s❡ ❡st❛ ✐❞❡✐❛ ✐♥t✉✐t✐✈❛✱ x t❡♥❞❡ ❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✈❛❧♦r h✱ ♦❜s❡r✈❛♥❞♦✲s❡ ♦s s❡✉s ❧✐♠✐t❡s ❧❛t❡r❛✐s✱ ✐st♦ é✱ q✉❛❧ ✈❛❧♦r ❛ ❢✉♥çã♦ s❡ ❛♣r♦①✐♠❛ ♣❛r❛ ✈❛❧♦r❡s ♣ró①✐♠♦s ❞❡ h à ❞✐r❡✐t❛ ❡ à ❡sq✉❡r❞❛✳ ✷✹ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ❋✐❣✉r❛ ✶✳✽✿ ❋✉♥çã♦ ♥ã♦ ❝♦♥tí♥✉❛✱ ♦♥❞❡ ✈❡r✐✜❝❛✲s❡ ✉♠ ✑s❛❧t♦✑ ❊①❡♠♣❧♦ ✶✳✶✳ 4x + 3 ❯t✐❧✐③❛♥❞♦✲s❡ ❛ ✐❞❡✐❛ ✐♥t✉✐t✐✈❛ ❞❡ ❧✐♠✐t❡✱ ❝♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ ❝❛❧❝✉❧❡♠♦s limx→0 f (x)✳ x2 − 4x + 3 ❈❛❧❝✉❧❛♥❞♦✲s❡ ♦s ✈❛❧♦r❡s ❞❡ q✉❡✿ x −1 −0, 5 −0, 4 −0, 3 −0, 2 −0, 1 −0, 01 0 2✮ x s❡ ❛♣r♦①✐♠❛ ❞❡ 0 ♣❡❧❛ ❡sq✉❡r❞❛✱ t❡♠♦s x2 − 4x + 3 (−1)2 − 4.(−1) + 3 = 8 (−0, 5)2 − 4 · (−0, 5) + 3 = 5, 25 (−0, 4)2 − 4 · (−0, 4) + 3 = 4, 76 (−0, 3)2 − 4 · (−0, 3) + 3 = 4, 29 (−0, 2)2 − 4 · (−0, 2) + 3 = 3, 84 (−0, 1)2 − 4 · (−0, 1) + 3 = 3, 41 (−0, 01)2 − 4 · (−0, 01) + 3 = 3, 0401 02 − 4 · 0 + 3 = 3 ❏á ❝❛❧❝✉❧❛♥❞♦✲s❡ ♦s ✈❛❧♦r❡s ❞❡ ✈❛❧♦r❡s ♠❛✐♦r❡s q✉❡ q✉❛♥❞♦ f (x) = x2 − x2 − 4x + 3 q✉❛♥❞♦ x s❡ ❛♣r♦①✐♠❛ ❞❡ t❡♠♦s q✉❡✿ x 1 0, 5 0, 4 0, 3 0, 2 0, 1 0, 01 0 x2 − 4x + 3 1 −4·1+3=0 2 0, 5 − 4 · 0, 5 + 3 = 1, 25 0, 42 − 4 · 0, 4 + 3 = 1, 56 0, 32 − 4 · 0, 3 + 3 = 1, 89 0, 22 − 4 · 0, 2 + 3 = 2, 24 0, 12 − 4 · 0, 1 + 3 = 2, 61 0, 012 − 4 · 0, 01 + 3 = 2, 9601 02 − 4 · 0 + 3 = 3 2 0 ♣❡❧❛ ❞✐r❡✐t❛ ✭♣♦r ✶✳✸✳ ▲■▼■❚❊❙✱ ❉❊❘■❱❆❉❆❙ ❊ ■◆❚❊●❘❆■❙ ✷✺ ❆ss✐♠✱ limx→0 x2 − 4x + 3 = 3✳ ❈♦♠♦ ♦ limx→0 f (x) ❡①✐st❡ ❡ é ✐❣✉❛❧ ❛ f (0) = 3✱ s❡❥❛ ♣❡❧❛ ❡sq✉❡r❞❛ ♦✉ ♣❡❧❛ ❞✐r❡✐t❛✱ ❝♦♥✲ ❝❧✉✐♠♦s t❛♠❜é♠ q✉❡ f (x) = x2 − 4x + 3 é ❝♦♥tí♥✉❛ ❡♠ x = 0✳ ✶✳✸✳✷ ❉❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ❆♣r❡s❡♥t❛♠♦s ❛❣♦r❛ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ r❡❛❧✳ ❆ ♣❛rt✐r ❞❡❧❡✱ é ♣♦ssí✈❡❧ ♠❡❞✐r ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r✱ ♦✉ s❡❥❛✱ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛s r❡t❛s t❛♥❣❡♥t❡s ❛♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ r❡❛❧✳ ◆♦ ❡♥t❛♥t♦✱ ♥ã♦ ♥♦s ❞❡❞✐❝❛r❡♠♦s ❛ ✐st♦ ♥❡st❡ t❡①t♦✱ ♣♦✐s ♥ã♦ ❢❛③ ♣❛rt❡ ❞❡ ♥♦ss♦s ♦❜❥❡t✐✈♦s✳ ❉❡✜♥✐çã♦ ✶✳✷✳ ❙❡❥❛ ❢ ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❞❡✜♥✐❞❛ ♥✉♠ ✐♥t❡r✈❛❧♦ r❡❛❧ I ❡ h ∈ R✳ ❖ ❧✐♠✐t❡ f (x) − f (h) x→h x−h lim è ❝❤❛♠❛❞♦ ❞❡ ❞❡r✐✈❛❞❛ ❞❡ ❢ ❡♠ ♣✱ ❞❡s❞❡ q✉❡ ❡❧❡ ❡①✐st❛ ❡ s❡❥❛ ✜♥✐t♦✱ ❡ ✐♥❞✐❝❛✲s❡ ♣♦r f ′ (h)✳ ❆ss✐♠✱ f (x) − f (h) . x→h x−h f ′ (h) = lim ❙❡ f ❛❞♠✐t❡ ❞❡r✐✈❛❞❛ ❡♠ h✱ ❡♥tã♦ ❞✐r❡♠♦s q✉❡ f é ❞❡r✐✈á✈❡❧ ♦✉ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ h✳ ❆ ❞❡r✐✈❛❞❛ t❛♠❜é♠ ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞❛ ❝♦♠♦ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ ✉♠❛ ❢✉♥çã♦ f (x)✱ ✐st♦ é✱ q✉❛♥t♦ ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ ✈❛r✐❛ ❝♦♥❢♦r♠❡ ✈❛r✐❛♠♦s ♦ ✈❛❧♦r ❞❡ x✳ ❊①❡♠♣❧♦ ✶✳✷✳ ❙❡❥❛ f (x) = ax2 + bx + c✳ ❈❛❧❝✉❧❡ f ′ (p)✳ ❙♦❧✉çã♦✿ (ax2 + bx + c) − (ap2 + bp + c) f (x) − f (p) = lim x→p x→p x−p x−p 2 2 2 ax + bx + c − ap − bp − c a(x2 − p2 ) + b(x − p) = lim = lim x→p x→p x−p x−p a(x − p)(x + p) + b(x − p) = x→p lim (a(x + p) + b) = 2ap + b. = x→p lim x−p lim P♦rt❛♥t♦✱ f ′ (p) = 2ap + b✳ ❊①❡♠♣❧♦ ✶✳✸✳ ❙❡❥❛ f (x) = x2 ✳ ❈❛❧❝✉❧❡ f ′ (1) ❡ f ′ (p) ❙♦❧✉çã♦✿ x2 − 1 f (x) − f (1) = lim = lim x + 1 = 2 x→1 x − 1 x→1 x→1 x−1 lim f (x) − f (p) x2 − p2 = lim = lim x + p = x + p. x→p x→p x − p x→p x−p lim ✷✻ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ P♦rt❛♥t♦✱ f ′ (1) = 2 ❡ f ′ (p) = 2p✳ ❆ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ r❡❛❧ f (x) ♥✉♠ ♣♦♥t♦ h✱ q✉❛♥❞♦ ❡①✐st❡✱ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❝♦♠♦ s❡♥❞♦ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ♥♦ ♣♦♥t♦ h✳ P❛r❛ ✐st♦✱ ❜❛st❛ ❝♦♥s✐❞❡r❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❛♥❣✉❧❛r❡s ❞❛s r❡t❛s q✉❡ ♣❛ss❛♠ ♣❡❧♦s ♣♦♥t♦s (x, f (x)) ❡ (h, f (h))✱ t♦♠❛♥❞♦✲s❡ ✈❛❧♦r❡s ❞❡ x ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠♦s ❞❡ h✳ ❋✐❣✉r❛ ✶✳✾✿ ❋✉♥çã♦ ❝♦♥tí♥✉❛ ❡ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❡♠ h ◆❡st❡ ❝❛s♦✱ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❡ y = f (x) ❡♠ x = h é ❞❛❞❛ ♣♦r✿ y − f (h) = f ′ (h) · (x − h). ❊①❡♠♣❧♦ ✶✳✹✳ ❙❡❥❛ f (x) = xn ✳ ❈❛❧❝✉❧❡ f ′ (p) ❙♦❧✉çã♦✿ f (x) − f (p) xn − xp (x − p)(xn−1 + xn−2 .p + ... + x.pn−2 + pn−1 ) = lim = lim x→p x→1 x − p x→1 x−p x−p f ′ (x) = lim = lim xn−1 + xn−2 .p + ... + x.pn−2 + pn−1 = pn−1 + pn−2 .p + ... + p.pn−2 + pn−1 = n.pn−1 x→1 ✶✳✸✳✸ ■♥t❡❣r❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ ❊♠ ♥♦ss♦ tr❛❜❛❧❤♦✱ ✉t✐❧✐③❛r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ■♥t❡❣r❛❧ r❡❧❛❝✐♦♥❛❞♦ à ár❡❛ s♦❜ ♦ ❣rá✲ ✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ r❡❛❧✱ q✉❡ s❡rá ❝❛❧❝✉❧❛❞❛ ❛ ♣❛rt✐r ❞❛s s♦♠❛s ❞❡ ár❡❛s ❞❡ r❡tâ♥❣✉❧♦s ❝♦♥str✉í❞♦s ❛❜❛✐①♦ ❡ ❛❝✐♠❛ ❞♦ ❣rá✜❝♦✳ ❆♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ✐❞❡✐❛s ✉t✐❧✐③❛♥❞♦ ❛ ❢✉♥çã♦ f (x) = ax2 + bx + c✱ ❝♦♠ a < 0, ❝♦♠♦ ♥❛ ✜❣✉r❛ ✶✳✶✵✳ ✶✳✸✳ ▲■▼■❚❊❙✱ ❉❊❘■❱❆❉❆❙ ❊ ■◆❚❊●❘❆■❙ ✷✼ P♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❛ ár❡❛ ❛❜❛✐①♦ ❞♦ ❣rá✜❝♦ ❞❡ f (x)✱ t♦♠❛♥❞♦✲s❡ ❛ ár❡❛ ❞♦s r❡tâ♥❣✉❧♦s ❞❡ ♠♦❞♦ q✉❡ ❛ ❛❧t✉r❛ ❞❡ ❝❛❞❛ r❡tâ♥❣✉❧♦ é ♦❜t✐❞❛ ❝♦♠♦ s❡♥❞♦ ❛ ♠❡♥♦r ✐♠❛❣❡♠ ❞♦s ♣♦♥t♦s q✉❡ ❡stã♦ ♥❛ ❜❛s❡ ❞❡st❡ r❡tâ♥❣✉❧♦✱ ♥❡st❡ ❝❛s♦ ❡st❛♠♦s ❝❛❧❝✉❧❛♥❞♦ ❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❝✉r✈❛ ♣♦r ❢❛❧t❛✱ ✉♠❛ ✈❡③ q✉❡ ❛ ár❡❛ ♦❜t✐❞❛ ❝♦♠ ♦s r❡tâ♥❣✉❧♦s é ♠❡♥♦r q✉❡ ❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❝✉r✈❛✳ ❋✐❣✉r❛ ✶✳✶✵✿ ❊①❡♠♣❧♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❖ ✐♥t❡r✈❛❧♦ [a, b] ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ ❛s r❛í③❡s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ é ❞✐✈✐❞✐❞♦ ❡♠ n ♣❛rt❡s ✐❣✉❛✐s✱ ✐st♦ é x1 ✱ x2 ✱ x3 ✱ ✳✳✳✱ xn−1 ✱ ❝♦♠ x0 = a ❡ xn = b ❞❡ ♠♦❞♦ q✉❡✿ ∆x = b−a . n ∆x é ♦ t❛♠❛♥❤♦ ❞♦ ✐♥t❡r✈❛❧♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ r❡♣r❡s❡♥t❛r ❛ ❜❛s❡ ❞♦s r❡tâ♥❣✉❧♦s✳ ❆ ár❡❛ ❛❜❛✐①♦ ❞❛ ❝✉r✈❛ s❡ ❛♣r♦①✐♠❛ ❞❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s r❡tâ♥❣✉❧♦s✳ ❖ ✈❛❧♦r ❞❛ ár❡❛ ♣♦r ❢❛❧t❛ AF é ✐❣✉❛❧ ❛ s♦♠❛ ❞❛s ár❡❛s ❞❡st❡s r❡tâ♥❣✉❧♦s✿ AF = f (x0 )∆x + f (x1 )∆x + f (x2 )∆x + ... + f (xn−1 )∆x, ♦♥❞❡ xi ∈ [xi , xi+1 ]✱ i = 0, 1, 2, ..., n − 1✱ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♣♦♥t♦ ♠í♥✐♠♦ ❞❡ f ❡♠ [xi , xi+1 ]✳ ◆♦t❡ t❛♠❜é♠ q✉❡ ♣♦❞❡♠♦s ❢❛③❡r ✉♠❛ ❛♣r♦①✐♠❛çã♦ t♦♠❛♥❞♦ ❛ ❛❧t✉r❛ ❞❡ ❝❛❞❛ r❡tâ♥❣✉❧♦ ❝♦♠♦ ❛ ♠❛✐♦r ✐♠❛❣❡♠ ❞♦s ♣♦♥t♦s q✉❡ ❡stã♦ ♥❛ ❜❛s❡ ❞♦s r❡tâ♥❣✉❧♦s✳ ❚❛❧ ❛♣r♦①✐♠❛çã♦ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r AE ✱ é ❝❤❛♠❛❞❛ ❞❡ ❛♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦✱ ♣♦✐s ❛ ár❡❛ ♦❜t✐❞❛ ❝♦♠ ♦s r❡tâ♥❣✉❧♦s é ♠❛✐♦r q✉❡ ❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❝✉r✈❛✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✶✶✳ ❊ t❛❧ ár❡❛ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ❛ss✐♠✿ AE = f (x˜0 )∆x + f (x˜1 )∆x + f (x˜2 )∆x + ... + f (xn−1 ˜ )∆x, ♦♥❞❡ x̃i ∈ [xi , xi+1 ]✱ i = 0, 1, 2, ..., n − 1✱ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♣♦♥t♦ ♠á①✐♠♦ ❞❡ f ❡♠ [x1 , xi+1 ]✳ ❉❡st❛ ❢♦r♠❛ ❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❝✉r✈❛ ❡stá ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❛s s♦♠❛s ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦✱ ❡ q✉❛♥t♦ ♠❡♥♦r ❢♦r ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡ss❛s ár❡❛s✱ ♠❡❧❤♦r s❡rá ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❝✉r✈❛✳ ✷✽ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ❋✐❣✉r❛ ✶✳✶✶✿ ❊①❡♠♣❧♦ ❞❡ ❛♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❙❡♥❞♦ ❛ss✐♠✱ ❡st❛s ❛♣r♦①✐♠❛çõ❡s ✭t❛♥t♦ ♣♦r ❢❛❧t❛✱ q✉❛♥t♦ ♣♦r ❡①❝❡ss♦✮✱ ♣♦❞❡♠ s❡r ♠❡✲ ❧❤♦r❛❞❛s s❡ ❛✉♠❡♥t❛r♠♦s ❛ q✉❛♥t✐❞❛❞❡ ❞❡ r❡tâ♥❣✉❧♦s ✭♣❛rt✐çõ❡s ❞♦ ✐♥t❡r✈❛❧♦ a = x0 ❡ [a, b]✱ ♦♥❞❡ b = xn ✮✳ ❋✐❣✉r❛ ✶✳✶✷✿ ❆♣r♦①✐♠❛çõ❡s ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❆s ár❡❛s ❛♣r♦①✐♠❛❞❛s✱ ♣♦r ❡①❝❡ss♦ ❡ ♣♦r ❢❛❧t❛✱ ♣♦❞❡♠ s❡r ❡s❝r✐t❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r ♠❡✐♦ ❞❡ s♦♠❛tór✐♦ ❛ss✐♠✿ n X f (x0 )∆x + f (x1 )∆x + f (x2 )∆x + ... + f (xn−1 )∆x, n X f (x˜0 )∆x + f (x˜1 )∆x + f (x˜2 )∆x + ... + f (xn−1 ˜ )∆x, i=0 ❡ i=0 P❛r❛ ♦ ❝❛s♦✱ ♣♦r ❡①❡♠♣❧♦ ❞❡ ❢✉♥çõ❡s q✉❛❞rát✐❝❛s✱ s❡ ❛✉♠❡♥t❛r♠♦s ❛r❜✐tr❛r✐❛♠❡♥t❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ r❡tâ♥❣✉❧♦s✱ ❡♥❝♦♥tr❛♠♦s ❡①❛t❛♠❡♥t❡ ❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❝✉r✈❛✳ ❊♠ ♦✉tr❛s ✶✳✸✳ ✷✾ ▲■▼■❚❊❙✱ ❉❊❘■❱❆❉❆❙ ❊ ■◆❚❊●❘❆■❙ ❋✐❣✉r❛ ✶✳✶✸✿ ❆♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❡ R1 0 (x − x2 )dx ❝♦♠ ♥❂✹ ♣❛❧❛✈r❛s✱ s❡ ♦ ♥ú♠❡r♦ ❞❡ r❡tâ♥❣✉❧♦s t❡♥❞❡r ❛♦ ✐♥✜♥✐t♦✱ t❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡✿ lim n→∞ n−1 X i=0 f (xi ) · ∆x. ❙✉♣♦♥❞♦ q✉❡ f s❡❥❛ ❝♦♥tí♥✉❛ ♣❛r❛ a ≤ x ≤ b✱ ❛ ■♥t❡❣r❛❧ ❉❡✜♥✐❞❛ ❞❡ f ❞❡ a ❛té b ❞❡♥♦t❛❞❛ Rb ♣♦r a f (x)dx✱ é ♦ ❧✐♠✐t❡ ❞❛s s♦♠❛s ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦✱ ❢♦r♠❛❞❛ ❝♦♠ n s✉❜✐♥t❡r✈❛❧♦s ❞✐✈✐❞✐♥❞♦ [a, b]✱ q✉❛♥❞♦ n t♦♠❛❞♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ ✐st♦ é✿ f (xi ) · ∆x, f (xi )dx = lim n X f (x̃i ) · ∆x f (xi )dx = lim Z b a ♣❛r❛ ❛ s♦♠❛ ♣♦r ❢❛❧t❛ ❡ n→∞ n X Z b a ♣❛r❛ ❛ s♦♠❛ ♣♦r ❡①❝❡ss♦✳ ❖❜s❡r✈❡♠♦s ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✳ n→∞ i=1 i=1 f (x) = x − x2 2 a (x − x )dx✳ ❊①❡♠♣❧♦ ✶✳✺✳ ❈❛❧❝✉❧❡ ❛s s♦♠❛s ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦ ❞❡ ♥❂ ✹ ❡ ♥❂✶✵ ♦❜t❡♥❞♦✲s❡ ❛ss✐♠ ♣r♦①✐♠❛çõ❡s ♣❛r❛ ❡♠ [0, 1] ♣❛r❛ Rb ❈♦♠♦ ♥♦s ♠♦str❛ ❛ ❋✐❣✉r❛ ✶✳✶✸✱ a = 0 ❡ b = 1✱ ❞❡ ♠♦❞♦ q✉❡ ♣❛r❛ n = 4✱ ∆t = (1 − 0)/4 = 0, 25✳ ▲♦❣♦ x0 = 0✱ x1 = 0, 25✱ x2 = 0, 5✱ x3 = 0, 75 ❡ x4 = 1✳ ❉❛í✱ ❝♦♠ n = 4✱ t❡♠♦s✿ ❙♦♠❛ ♣♦r ❢❛❧t❛✿ f (0)∆x + f (0, 25)∆x + f (0, 5)∆x + f (1) · ∆x = 0 · 0, 25 + 0, 19 · 0, 25 + ❙♦❧✉çã♦ 0, 19 · 0, 25 + 0 · 0, 25 = 0, 095. ❏á ♣❡❧❛ ❋✐❣✉r❛ ✶✳✶✹✱ t❡♠♦s q✉❡✿ ❙♦♠❛ ♣♦r ❡①❝❡ss♦✿ f (0, 25)∆x + f (0, 5)∆x + f (0, 5)∆x + f (0, 75)∆x = 0, 19 · 0, 25 + 0, 25 · 0, 25 + 0, 25 · 0, 25 + 0, 19 · 0, 25 = 0, 22. ✸✵ ❈❆P❮❚❯▲❖ ✶✳ ❋✐❣✉r❛ ✶✳✶✹✿ ❆♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❡ R1 0 P❘❊▲■▼■◆❆❘❊❙ (x − x2 )dx ❝♦♠ ♥❂✹ P♦rt❛♥t♦✱ ❛ ár❡❛ ❞❡❜❛✐①♦ ❞❛ ❝✉r✈❛ f (x) = x − x2 ❞❡ x = 0 ❛té x = 1 ❡stá ❡♥tr❡ ✵✱✵✾✺ ❡ ✵✱✷✷✿ Z 1 0, 095 ≤ 0 (x − x2 )dx ≤ 0, 22. ❚❛❧ ❛♣r♦①✐♠❛çã♦ ✭n = 4✮ ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦✱ sã♦ ✐❧✉str❛❞❛s ♥❛s ❋✐❣✉r❛s ✶✳✶✸ ❡ ✶✳✶✹✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚♦♠❛♥❞♦ n = 10✱ t❡♠♦s ∆x = (1 − 0)/10 = 0, 1❀ ❝❛❧❝✉❧❛♠♦s✿ ❙♦♠❛ ♣♦r ❢❛❧t❛✿ f (0).∆x + f (0, 1) · ∆x + f (0, 2) · ∆x + f (0, 3) · ∆x + f (0, 4) · ∆x + f (0, 5) · ∆x+ + f (0, 6) · ∆x + f (0, 7) · ∆x + f (0, 8) · ∆x + f (0, 9) · ∆x + f (10) · ∆x = = 0, 1 · (0 + 0, 09 + 0, 16 + 0, 21 + 0, 24 + 0, 24 + 0, 21 + 0, 16 + 0, 09 + 0) = 0, 14 ❙♦♠❛ ♣♦r ❡①❝❡ss♦✿ f (0, 1) · ∆x + f (0, 2) · ∆x + f (0, 3) · ∆x + f (0, 4) · ∆x + f (0, 5) · ∆x+ + f (0, 5) · ∆x + f (0, 6) · ∆x + f (0, 7) · ∆x + f (0, 8) · ∆x + f (0, 9) · ∆x = = 0, 1 · (0, 09 + 0, 16 + 0, 21 + 0, 24 + 0, 25 + 0, 25 + 0, 24 + 0, 21 + 0, 16 + 0, 09) = 0, 19 ❖❜s❡r✈❛♠♦s ♥❛s ❋✐❣✉r❛s ✶✳✶✺ ❡ ✶✳✶✻ q✉❡ ❛ s♦♠❛ ♣♦r ❡①❝❡ss♦ é ♠❛✐♦r q✉❡ ❛ ár❡❛ ❛❜❛✐①♦ ❞❛ ❝✉r✈❛ ❡ q✉❡ ❛ s♦♠❛ ♣♦r ❢❛❧t❛ é ♠❡♥♦r✱ ❞❡ ♠♦❞♦ q✉❡✿ 0, 14 ≤ Z 1 0 x − x2 dx ≤ 0, 19. ❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s s♦♠❛s ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦ ♣❛r❛ ♥❂✶✵ é ♠❡♥♦r q✉❡ ♥♦ ❝❛s♦ ♥❂✹✳ ❈♦♥❢♦r♠❡ ♦s s✉❜✐♥t❡r✈❛❧♦s ✈ã♦ ❞✐♠✐♥✉✐♥❞♦✱ ❛s s♦♠❛s ♣♦r ❢❛❧t❛ ❡ ❡①❝❡ss♦ ✜❝❛♠ ♠❛✐s ♣ró①✐♠❛s✱ ♥♦ ❝❛s♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❞❡ ❢✉♥çõ❡s q✉❛❞rát✐❝❛s✱ q✉❡ sã♦ ❛s q✉❡ ♥♦s ✐♥t❡r❡ss❛♠✳ ✶✳✸✳ ✸✶ ▲■▼■❚❊❙✱ ❉❊❘■❱❆❉❆❙ ❊ ■◆❚❊●❘❆■❙ ❋✐❣✉r❛ ✶✳✶✺✿ ❆♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❡ ❋✐❣✉r❛ ✶✳✶✻✿ ❆♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❡ R1 0 (x − x2 )dx ❝♦♠ ♥❂✶✵ R1 0 (x − x2 )dx ❝♦♠ n = 10 Pr✐♠✐t✐✈❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♥✉♠ ✐♥t❡r✈❛❧♦ I ✳ ❯♠❛ ♣r✐♠✐t✐✈❛ ❞❡ f é ✉♠❛ ❢✉♥ çã♦ F ❞❡✜♥✐❞❛ ❡♠ I t❛❧ q✉❡ F ′ (x) = f (x) ♣❛r❛ t♦❞♦ x ∈ I ✳ ❙❡♥❞♦ F ✉♠❛ ♣r✐♠✐t✐✈❛ ❞❡ f ❡♠ I ✱ ❡♥tã♦✱ ♣❛r❛ t♦❞❛ ❝♦♥st❛♥t❡ k ✱ F (x) + k é✱ t❛♠❜é♠✱ ♣r✐♠✐t✐✈❛ ❞❡ f ✳ P♦rt❛♥t♦✱ ❛s ♣r✐♠✐t✐✈❛s ❞❡ f ❡♠ I sã♦ ❛s ❢✉♥çõ❡s ❞❛ ❢♦r♠❛ F (x) + k ✱ ❝♦♠ k ❝♦♥st❛♥t❡✳ ❉✐③❡♠♦s✱ ❡♥tã♦✱ q✉❡ y = F (x) + k ✱ k ❝♦♥st❛♥t❡✱ é ❛ ❢❛♠í❧✐❛ ❞❛s ♣r✐♠✐t✐✈❛s ❞❡ R f ❡♠ I ✳ ❆ ♥♦t❛çã♦ f (x)dx s❡rá ✉s❛❞❛ ♣❛r❛ r❡♣r❡s❡♥t❛r ❛s ❢❛♠í❧✐❛s ❞❛s ♣r✐♠✐t✐✈❛s ❞❡ f ✿ Z f (x)dx = F (x) + k ✳ R ◆❛ ♥♦t❛çã♦ f (x)dx✱ ❛ ❢✉♥çã♦ ❞❡♥♦♠✐♥❛✲s❡ ✐♥t❡❣r❛♥❞♦✳ ❯♠❛ ♣r✐♠✐t✐✈❛ ❞❡ f s❡rá✱ t❛♠✲ ✸✷ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ❜é♠✱ ❞❡♥♦♠✐♥❛❞❛ ✉♠❛ ✐♥t❡❣r❛❧ ✐♥❞❡✜♥✐❞❛ ❞❡ f ✳ ➱ ❝♦♠✉♠ r❡❢❡r✐r✲s❡ ❛ ✐♥t❡❣r❛❧ ✐♥❞❡✜♥✐❞❛ ❞❡ f ✳ ❊①❡♠♣❧♦ ✶✳✻✳ ❙♦❧✉çã♦✿ R Ç 1 xα+1 α+1 Z ❙♦❧✉çã♦✿ f (x)dx ❝♦♠♦ ❛ ❈❛❧❝✉❧❡ xα dx✱ ♦♥❞❡ α 6= −1 é ✉♠ r❡❛❧ ✜①♦✳ ❧♦❣♦✱ ❊①❡♠♣❧♦ ✶✳✼✳ R å′ = xα , xα+1 + k. x = α+1 α ❈❛❧❝✉❧❡ ax2 + bx + cdx✱ ♦♥❞❡ a✱ b✱ ❡ c sã♦ r❡❛✐s ✜①♦s✳ R Ç 1 3 1 2 ax + bx + cx + k 3 2 ❧♦❣♦✱ Z ax2 + bx + cdx = å′ = ax2 + bx + c, ax3 bx2 + + cx + k. 3 2 ❖ Pr✐♠❡✐r♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ❙❡ f ❢♦r ✐♥t❡❣rá✈❡❧ ❡♠ [a, b] ❡ s❡ F ❢♦r ✉♠❛ ♣r✐♠✐t✐✈❛ ❞❡ f ❡♠ [a, b]✱ ❡♥tã♦✿ Z b a f (x)dx = F (b) − F (a). ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ s❡ ❡♥❝♦♥tr❛ ❡♠ ❬✹❪✳ ❊①❡♠♣❧♦ ✶✳✽✳ ❙♦❧✉çã♦✿ ❈❛❧❝✉❧❡ R1 0 (x − x2 )dx✳ F (x) = x2 x3 − 2 3 é ✉♠❛ ♣r✐♠✐t✐✈❛ ❞❡ f (x) = x − x2 ❡ f é ❝♦♥tí♥✉❛ ❡♠ [0, 1]✱ ❛ss✐♠✿ Z 1 0 2 (x − x )dx = ñ 2 x x3 − 2 3 ô1 = 0 1 1 1 − = . 2 3 6 ✶✳✹✳ ✶✳✹ ❖ P❆❘❆▲❊▲❖●❘❆▼❖ ✸✸ ❖ ♣❛r❛❧❡❧♦❣r❛♠♦ Pr♦♣♦s✐çã♦ ✶✳✷✳ ❆ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ♦ ❞✐✈✐❞❡ ❡♠ ❞♦✐s tr✐â♥❣✉❧♦s ❝♦♥❣r✉❡♥✲ t❡s✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ ♦ P❛r❛❧❡❧♦❣r❛♠♦ ABCD ❛❜❛✐①♦✿ ❋✐❣✉r❛ ✶✳✶✼✿ P❛r❛❧❡❧♦❣r❛♠♦ ABCD AC ❞✐✈✐❞❡ ABCD ❡♠ ❞♦✐s △CDA✳ P❡❧♦ ❝❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ▲❛❞♦✲▲❛❞♦✲▲❛❞♦✱ LLL✱ ♣♦❞❡♠♦s ♥♦t❛r q✉❡ ♦s tr✐â♥❣✉❧♦s △CBD ❡ △CDA sã♦ ❝♦♥❣r✉❡♥t❡s✱ ✉♠❛ ✈❡③ q✉❡✱ ♣♦r s❡ tr❛t❛r ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ t❡♠♦s q✉❡ CB é ❝♦♥❣r✉❡♥t❡ ❛ DA ❡ CD é ❝♦♥❣r✉❡♥t❡ ❛ BA ❡ ❝♦♠♦ AC é ❝♦♠✉♠ ❛♦s ❞♦✐s tr✐â♥❣✉❧♦s✱ t❡♠♦s q✉❡ △CBA ❡ △CDA t❛♠❜é♠ sã♦ ❝♦♥❣r✉❡♥t❡s✳ P♦❞❡♠♦s ♦❜s❡r✈❛r ♥❛s ❋✐❣✉r❛s ✶✳✶✼ ❡ ✶✳✶✽ q✉❡ ❛ ❞✐❛❣♦♥❛❧ tr✐â♥❣✉❧♦s✱ △CBA ❡ ❋✐❣✉r❛ ✶✳✶✽✿ P❛r❛❧❡❧♦❣r❛♠♦ ABCD ❡ ♦s tr✐â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s △CDA ❡ △CBA ✸✹ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ❋✐❣✉r❛ ✶✳✶✾✿ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ✶✳✺ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ❚❡♦r❡♠❛ ✶✳✷✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ f é ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ ✉♠ ♥ú♠❡r♦ c ∈ (a, b)✱ [a, b] f✿ R → R [a, b] ⊂ R✳ ❙❡ (a, b)✱ ❡♥tã♦ ❡①✐st❡ ❡ ❝♦♥s✐❞❡r❡ ♦ ✐♥t❡r✈❛❧♦ ❡ ❞❡r✐✈á✈❡❧ ♥♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ t❛❧ q✉❡ f ′ (c) = f (b) − f (a) . b−a ❊st❡ t❡♦r❡♠❛ ♥♦s ❞✐③ q✉❡ s❡ f ❝♦♥tí♥✉❛ ❡ ♥♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ [a, b] ❡ ❞❡r✐✈á✈❡❧ ♥♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (a, b)✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ c ∈ (a, b) t❛❧ q✉❡ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r c é ❛ ♠❡s♠❛ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦r f (a) ❡ f (b)✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✶✳✷✳ ❆ ❞❡♠♦♥tr❛çã♦ ❞❡st❡ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✹❪✳ ❖ ❚❡♦r❡♠❛ 2.3 é t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞♦❞♦ ❱❛❧♦r ▼é❞✐♦ ♣❛r❛ ■♥t❡❣r❛✐s✳ f ✿ R → R ❡ ❝♦♥s✐❞❡r❡ ♦ ✐♥t❡r✈❛❧♦ [a, b] ∈ R✳ [a, b]✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ z ∈ (a, b)✱ t❛❧ q✉❡ ❚❡♦r❡♠❛ ✶✳✸✳ ❙❡❥❛ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ Z b a ❖✉ s❡❥❛✱ ❡①✐st❡ z ∈ (a, b) t❛❧ q✉❡ ❙❡ f é f (x)dx = f (z)(b − a). f (z) = 1 b−a Rb a f (x)dx. ✶✳✻ ❙❡♠❡❧❤❛♥ç❛ ❡♥tr❡ ✜❣✉r❛s ❈♦♠♦ t♦❞♦s ♦s ❝ír❝✉❧♦s sã♦ s❡♠❡❧❤❛♥t❡s ❡♥tr❡ s✐✱ t♦❞❛s ❛s ♣❛rá❜♦❧❛s t❛♠❜é♠ ♦ sã♦✳ ❖ ♠❡s♠♦ ♥ã♦ é ✈❡r❞❛❞❡ ♣❛r❛ ♦✉tr❛s s❡çõ❡s ❝ô♥✐❝❛s✱ ♣♦r q✉❡ ❛ ❝❧❛ss❡ ❞❡ s❡♠❡❧❤❛♥ç❛ ❞❡ ✉♠❛ ✶✳✻✳ ❙❊▼❊▲❍❆◆➬❆ ❊◆❚❘❊ ❋■●❯❘❆❙ ✸✺ ❋✐❣✉r❛ ✶✳✷✵✿ ❍❡①á❣♦♥♦s s❡♠❡❧❤❛♥t❡s ❡❧✐♣s❡ ♦✉ ✉♠❛ ❤✐♣ér❜♦❧❡ ❞❡♣❡♥❞❡ ❞❛ s✉❛ ❡①❝❡♥tr✐❝✐❞❛❞❡✳ ❯♠❛ ✈❡③ q✉❡ ❛ s❡♠❡❧❤❛♥ç❛ ❡♥tr❡ ✜❣✉r❛s✱ s❡rá ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ❣❡♥❡r❛❧✐③❛çã♦ ✭♣❛r❛♠❡tr✐③❛çã♦✮ ❞♦s P❛r❜❡❧♦s✱ ✈❛♠♦s ❞❡♥✐✜✲❧❛ ❛❣♦r❛✿ ❉❡✜♥✐çã♦ ✶✳✸✳ ✭❙❡♠❡❧❤❛♥ç❛ ❡♥tr❡ ✜❣✉r❛s✮✿ ❉✉❛s ✜❣✉r❛s sã♦ s❡♠❡❧❤❛♥t❡s s❡ ❡①✐st✐r ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❥❡t♦r❛ ♥♦ P❧❛♥♦ ❊✉❝❧✐❞✐❛♥♦ E✱ f : E → E✱ t❛❧ q✉❡ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ k > 0✱ ❞❡ ♠♦❞♦ q✉❡ ♣❛r❛ t♦❞♦ ♣❛r P ✱ Q ❞❡ ♣♦♥t♦s ❞❛ ✜❣✉r❛✱ t❡♠♦s P ′ Q′ = kP Q ♦♥❞❡ P ′ = f (P ) ❡ Q′ = f (Q)✳ ❖ ♥ú♠❡r♦ k é ❝❤❛♠❛❞♦ r❛③ã♦ ❞❛ s❡♠❡❧❤❛♥ç❛ f ✳ ◗✉❛♥❞♦ k = 1 ❛ ❛♣❧✐❝❛çã♦ f é ❝❤❛♠❛❞❛ ✉♠❛ ✐s♦♠❡tr✐❛ ❞♦ P❧❛♥♦ ❊✉❝❧✐❞✐❛♥♦ E✳ ◆❡st❡ ❝❛s♦✱ f ♣r❡s❡r✈❛ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♣♦♥t♦s ❞❡ E ❡ ❝♦♠♦ ❡①❡♠♣❧♦s ❞❡ss❛ ❝❧❛ss❡ ❡s♣❡❝✐❛❧ ❞❡ s❡♠❡❧❤❛♥ç❛s ❝✐t❛♠♦s ❛ tr❛♥s❧❛çã♦✱ ❛ r♦t❛çã♦ ❡ ❛ s✐♠❡tr✐❛ ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ r❡t❛✳ ❙❡♥❞♦ ❛ss✐♠✱ s❡ f : E → E é ✉♠❛ ✐s♦♠❡tr✐❛ ❡ P é ✉♠❛ ♣❛rá❜♦❧❛ ❞❡ ❢♦❝♦ F ❡ ❞✐r❡tr✐③ l✱ ✈❡r✐✜❝❛♠♦s ❢❛❝✐❧♠❡♥t❡ q✉❡ f (P ) é ✉♠❛ ♣❛rá❜♦❧❛ ❞❡ ❢♦❝♦ F ′ = f (F ) ❡ ❞✐r❡tr✐③ l′ = f (l)✳ ❉❡ ❢❛t♦✱ x ∈ P ⇐⇒ d(x, F ) = d(x, l)✳ ❈♦♠♦ f é ✉♠❛ ✐s♦♠❡tr✐❛✱ ❡♥tã♦✿ d(x, F ) = d(f (x), f (F )) = d(x′ , F ′ ) ❡ d(x, l) = d(f (x), f (l)) = d(x′ , l′ ), ✉♠❛ ✈❡③ q✉❡ ✐s♦♠❡tr✐❛s ♣r❡s❡r✈❛♠ ❛ ❞✐stâ♥❝✐❛ ❞❡ ♣♦♥t♦ ❛ r❡t❛✱ ❧❡✈❛♠ r❡t❛s ❡♠ r❡t❛s ❡ ❧❡✈❛♠ ✜❣✉r❛s ❡♠ ✜❣✉r❛s✳ ❆ss✐♠✱ x′ = f (x) sã♦ ♣♦♥t♦s ❞❡ ✉♠❛ ♣❛rá❜♦❧❛ ❞❡ ❢♦❝♦ F ′ ❡ ❞✐r❡tr✐③ l′ ✳ ❆ ✜❣✉r❛ ✶✳✻ ♠♦str❛ ❞♦✐s ❤❡①á❣♦♥♦s s❡♠❡❧❤❛♥t❡s t❛❧ q✉❡ A′ B ′ = kAB ✱ B ′ C ′ = kCB ✱ C ′ D′ = kCD✱ D′ E ′ = kDE ✱ E ′ F ′ = kEF ❡ F ′ A′ = kF A✳ ✸✻ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ▲♦❣♦✱ t♦♠❛♥❞♦ ❞✉❛s ♣❛rá❜♦❧❛s P ❡ P ′ ❝♦♠ ❞✐r❡tr✐③❡s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣❛r❛❧❡❧❛s✱ ✉t✐✲ ❧✐③❛♥❞♦ ✉♠❛ ❛❞❡q✉❛❞❛ r♦t❛çã♦✱ tr❛♥s❢♦r♠❛♠♦s P ❡♠ ✉♠❛ ♥♦✈❛ ♣❛rá❜♦❧❛ ❝✉❥❛ ❞✐r❡tr✐③ é ♣❛r❛❧❡❧❛ à ❞✐r❡tr✐③ ❞❡ P ′ ❡ ❝♦♠ ✉♠❛ tr❛♥s❧❛çã♦ ❛❞❡q✉❛❞❛ ♠♦str❛♠♦s q✉❡ q✉❛✐sq✉❡r ❞✉❛s ♣❛rá❜♦❧❛s sã♦ s❡♠❡❧❤❛♥t❡s ❡♥tr❡ s✐✱ ❝♦♠♦ ♠♦str❛♠ ❛s ❋✐❣✉r❛s ✷✳✷✶✱ ✷✳✷✷ ❡ ✷✳✷✸✳ ❋✐❣✉r❛ ✶✳✷✶✿ P❛rá❜♦❧❛s P ❡ P ′ ❝♦♠ s❡✉s r❡s♣❡❝t✐✈♦s ❢♦❝♦s F1 ❡ F2 ❡ ❞✐r❡tr✐③❡s L1 ❡ L2 ❆ ❋✐❣✉r❛ ✷✳✷✷ ♠♦str❛ ❛ r♦t❛çã♦ ❞❛ P❛rá❜♦❧❛ P ❡♠ r❡❧❛çã♦ ❛♦ ❢♦❝♦ F1 s❡❣✉✐❞❛ ❞❡ ✉♠❛ tr❛♥s❧❛çã♦ ♥♦ ❡✐①♦ x ❞❡ ♠♦❞♦ q✉❡ ❛s r❡t❛s ❞✐r❡tr✐③❡s L1 ❡ L2 ✜q✉❡♠ ♣❛r❛❧❡❧❛s✳ ❋✐❣✉r❛ ✶✳✷✷✿ ❘♦t❛çã♦ ❡ tr❛♥s❧❛çã♦ ❞❛ P❛rá❜♦❧❛ P ❊ tr❛♥s❧❛❞❛♥❞♦ ❛ P❛rá❜♦❧❛ P ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ y ✱ t❡♠♦s ❛s P❛rá❜♦❧❛s s❡♠❡❧❤❛♥t❡s✳ ✶✳✻✳ ❙❊▼❊▲❍❆◆➬❆ ❊◆❚❘❊ ❋■●❯❘❆❙ ❋✐❣✉r❛ ✶✳✷✸✿ ❚r❛♥s❧❛çã♦ ❞❛ P❛rá❜♦❧❛ ✸✼ P ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦ y ✸✽ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ❈❛♣ít✉❧♦ ✷ P❆❘❇❊▲❖❙ ❖ ❝♦♥❝❡✐t♦ ❛❝❡r❝❛ ❞♦s ♣❛r❜❡❧♦s ❞❡r✐✈❛ ❞❛ ✜❣✉r❛ ❝❧áss✐❝❛ ❞♦s ❆r❜❡❧♦s✳ ❚❛❧ ✜❣✉r❛ é ❢♦r♠❛❞❛ ♣♦r três s❡♠✐❝ír❝✉❧♦s t❛♥❣❡♥t❡s ❞♦✐s ❛ ❞♦✐s ❡ ❝♦♠ ♦s s❡✉s ❞✐â♠❡tr♦s ❛❧✐♥❤❛❞♦s✳ ❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ ✐♥s♣✐r❛❞♦ ♥♦s tr❛❜❛❧❤♦s ❞❡ ❏♦♥❛t❤❛♥ ❙♦♥❞♦✇ ❡ ❆♥t♦♥✐♦ ▼✳ ❖❧❧❡r✲ ▼❛r❝é♥✳ ❖ ♣r✐♠❡✐r♦ ❞❡✜♥❡✱ ❛♣r❡s❡♥t❛ ❡ ❞❡♠♦♥str❛ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s r❡❧❛❝✐♦♥❛❞♦s ❛♦s P❛r❜❡❧♦s ❡ ♦ s❡❣✉♥❞♦ ❣❡♥❡r❛❧✐③❛ ♦ ❝❛s♦ ❞♦s ❆r❜❡❧♦s ♣❛r❛ q✉❛✐sq✉❡r t✐♣♦ ❞❡ ❢✉♥çõ❡s✱ ❛♣r❡s❡♥t❛♥❞♦ ❡ ❞❡♠♦♥str❛♥❞♦ t❛♠❜é♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❋✐❣✉r❛ ✷✳✶✿ ❆r❜❡❧♦ ◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐r❡♠♦s ❡ ❞✐s❝✉t✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ P❛r❜❡❧♦s✳ ❆♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ❡st❡ t❡♠❛ ❡ ♣♦st❡r✐♦r♠❡♥t❡ ❡♥✉♥❝✐❛r❡♠♦s ❡ ♣r♦✲ ✈❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s t❛♠❜é♠ r❡❧❡✈❛♥t❡s✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ P❛r❜❡❧♦s✱ ❝♦♠♦ ❥á ♠❡♥❝✐♦♥❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✷✱ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❛♦ ❝♦♥❝❡✐t♦ ❞❡ P❛rá❜♦❧❛s✳ ❆❣♦r❛ ❡♥tã♦ ✈❛♠♦s ❞❡✜♥✐r ♦ q✉❡ sã♦ P❛r❜❡❧♦s✿ ✸✾ ✹✵ ❈❆P❮❚❯▲❖ ✷✳ ❉❡✜♥✐çã♦ ✷✳✶✳ ✭P❛r❜❡❧♦s✮✿ ❉❛❞♦s três ♣♦♥t♦s ❝♦❧✐♥❡❛r❡s C1 ✱ C2 ✱ C3 ✱ P❆❘❇❊▲❖❙ P❛r❜❡❧♦ é ❛ r❡✉♥✐ã♦ ❞❡ três ♣❛rá❜♦❧❛s t❡♥❞♦ s❡♥❞♦ ♦s s❡❣♠❡♥t♦s C1 C2 ✱ C2 C3 ❡ C1 C3 ❞❡ ✉♠ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛ q✉❡ ♦s ❝♦♥té♠ C1 C2 ✱ C2 C3 ❡ C1 C3 ❛s ▲❛t❡r❛ ❘❡❝t✉♠ ❞❛s r❡s♣❡❝t✐✈❛s ♣❛rá❜♦❧❛s✳ ❡ ❋✐❣✉r❛ ✷✳✷✿ P❛r❜❡❧♦ ❖❜s❡r❡♠♦s q✉❡✿ ✶✳ ❖s ♣♦♥t♦s C1 ✱ C2 ✱ C3 sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝ús♣✐❞❡s❀ ✷✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ s✉♣♦♥❤❛ C1 ✲C2 ✲C3 ✭♦✉ s❡❥❛✱ C2 ❡stá ❡♥tr❡ C1 ❡ C3 ✮✳ ◆❡st❡ ❝❛s♦✱ ❝❤❛♠❛♠♦s C2 ❞❡ ❝ús♣✐❞❡ ♠é❞✐❛✳ ❆s ♣❛rá❜♦❧❛s q✉❡ ♣❛ss❛♠ ♣♦r C1 ❡ C3 sã♦ t❛♥❣❡♥t❡s ❡♠ C1 ❡ C3 ❀✳ ✷✳✶ ❯♠ ❡st✉❞♦ ❢✉♥❝✐♦♥❛❧ ❞♦s P❛r❜❡❧♦s P♦❞❡♠♦s ❣❡♥❡r❛❧✐③❛r ❛s ❢✉♥çõ❡s q✉❡ ❞❡✜♥❡♠ ❛s P❛rá❜♦❧❛s q✉❡ ❢♦r♠❛♠ ♦s P❛r❜❡❧♦s ✉t✐❧✐✲ ③❛♥❞♦ ❛ s❡♠❡❧❤❛♥ç❛ ❡♥tr❡ ✜❣✉r❛s✳ ❉❡ ❢❛t♦✱ ✈❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♣❛r❛♠❡tr✐③❛r ✉♠❛ ❢✉♥çã♦ ♣❛r❛❜ó❧✐❝❛ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ s✉❛s r❛í③❡s s❡❥❛♠ ✵ ❡ ✶✳ ❙❛❜❡♠♦s q✉❡ ❛ ❡q✉❛çã♦ ❞❛ P❛rá❜♦❧❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ♣❡❧❛ s♦♠❛ ❡ ♣❡❧♦ ♣r♦❞✉t♦ ❞❡ s✉❛s r❛í③❡s ✭q✉❛♥❞♦ ❡①✐st✐r❡♠✮ ❛ss✐♠✿ f (x) = y = ax2 + bx + c = ax2 − a(x1 + x2 )x + ax1 x2 = a(x − x1 ) · (x − x2 ). ❝♦♠ x1 ❡ x2 s❡♥❞♦ ❛s r❛í③❡s ❞❡ y ✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ❝♦♥s✐❞❡r❛r❡♠♦s ♦ ❝❛s♦ ❡♠ q✉❡ s✉❛s r❛í③❡s ❡①✐st❛♠ ❡ q✉❡ a < 0✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✸✳✸✳ P❛r❛ ♦s ❝❛s♦s ♥♦s q✉❛✐s ❛s r❛í③❡s ♥ã♦ ❡①✐st❛♠ ♦✉ ❡①✐st✐r ✉♠❛ ú♥✐❝❛ r❛✐③✱ ♣❡♥s❛r❡♠♦s ❡♠ ✉♠❛ tr❛♥s❧❛çã♦ ♥♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦ ❞♦ ❡✐①♦ y ✳ ✷✳✶✳ ❯▼ ❊❙❚❯❉❖ ❋❯◆❈■❖◆❆▲ ❉❖❙ P❆❘❇❊▲❖❙ ✹✶ ❋✐❣✉r❛ ✷✳✸✿ P❛rá❜♦❧❛ ❝♦♠ r❛í③❡s ❡♠ x1 ❡ x2 ❚r❛♥s❧❛❞❛♥❞♦ ❛ ❢✉♥çã♦ y = f (x) ❡♠ β ✉♥✐❞❛❞❡s ♥❛ ❞✐r❡çã♦ ❞♦ ❡✐①♦ x t❛❧ q✉❡ x1 − β = 0✱ ♦❜t❡♠♦s✿ g(x) = a · [(x + β − x1 ] · [(x + β) − x2 ] g(x) = a · x · [x − (x2 − β)]. ❈❤❛♠❛♥❞♦ x2 − β = α✱ s❡❣✉❡ q✉❡ g(x) = ax · (x − α)✳ ❆❣♦r❛✱ ✉t✐❧✐③❛♥❞♦ ❛ tr❛♥s❢♦r♠❛çã♦ x 7→ αx✱ ❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ✭♣❛r❛❜ó❧✐❝❛✮ ✜❝❛ h(x) = aαx · (αx − α) = aα2 x2 − aα2 x, ❡ s✉❛s r❛í③❡s ❡♥❝♦♥tr❛♠✲s❡ ❡♠ x = 0 ❡ x = 1✱ ❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✸✳✹✳ ❋✐❣✉r❛ ✷✳✹✿ P❛rá❜♦❧❛ ❝♦♠ r❛í③❡s ❡♠ ✵ ❡ ✶ ❱❛♠♦s ❡♥tã♦ ❞❡✜♥✐r ♦ P❛r❜❡❧♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❢✉♥çã♦ h(x) ❛❝✐♠❛✳ ✹✷ ❈❆P❮❚❯▲❖ ✷✳ P❆❘❇❊▲❖❙ ❙❡❥❛ ❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ h(x) : [0, 1] → R✱ ♣♦s✐t✐✈❛ ♥♦ ✐♥t❡r✈❛❧♦ (0, 1) ❡ ❝♦♠ s✉❛s r❛í③❡s ❡♠ x = 0 ❡ x = 1✳ ❚♦♠❛♥❞♦ ✉♠ ♣♦♥t♦ z ∈ (0, 1) ✈❛♠♦s ❞❡✜♥✐r h1 : [0, z] → R ❡ h2 : [z, 1] → R ❝♦♠♦✿ h1 (x) = zh(x/z), ❡ Å x − zã h2 (x) = (1 − z)h . 1−z ❖❜s❡r✈❛♠♦s q✉❡ ❛♠❜❛s sã♦ s❡♠❡❧❤❛♥t❡s ❛ h(x)✱ ♦♥❞❡ h1 (x) ❢♦✐ ♦❜t✐❞❛ ♣♦r ✉♠❛ ❤♦♠♦t❡t✐❛ ❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠ ❡ h2 (x) ♣♦r ✉♠❛ ❤♦♠♦t❡t✐❛ s❡❣✉✐❞❛ ❞❡ ✉♠❛ tr❛♥s❧❛çã♦✱ ♦ ❝❛s♦ ♣❛rt✐✲ ❝✉❧❛r h(x) = x − x2 ✭α = 1 ❡ a = −1✮ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ P❛r❜❡❧♦ ❞❡ ❙♦♥❞♦✇✱ ✐❧✉str❛❞❛ ♥❛ ❋✐❣✉r❛ ✸✳✺✳ ❋✐❣✉r❛ ✷✳✺✿ P❛r❜❡❧♦ ❞❡ ❙♦♥❞♦✇ ˙ ❆ ♦❜s❡r✈❛çã♦ ✷ ❞❛ ❞❡✜♥✐çã♦ ❞❡ P❛r❜❡❧♦s ♥♦s ❞✐③ q✉❡ ❛s ♣❛rá❜♦❧❛s ❢♦r♠❛❞❛s ♣❡❧♦ ❛r❝♦ C 1 C3 ˙ ˙ é t❛♥❣❡♥t❡ às ♣❛rá❜♦❧❛s ❢♦r♠❛❞❛s ♣❡❧♦s ❛r❝♦s C1 C2 ❡ C2 C3 ❡♠ C1 ❡ C3 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❱❛♠♦s ❞❡♠♦♥str❛r ❡st❡ ❢❛t♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ❢♦r♠❛ ❣❡r❛❧ ❞♦s P❛r❜❡❧♦s✳ P❛r❛ q✉❡ ❛s ♣❛rá❜♦❧❛s s❡❥❛♠ t❛♥❣❡♥t❡s ♥♦ ♣♦♥t♦ C1 ✱ ♣♦r ❞❡✜♥✐çã♦✱ ❛ ❞❡r✐✈❛❞❛ ❞❛s ❢✉♥çõ❡s q✉❡ ❛s ❞❡s❝r❡✈❡♠ ❞❡✈❡♠ ❝♦✐♥❝✐❞✐r✳ ❉❡ ❢❛t♦✿ Å ã x 1 Åxã = h′ e h′1 (x) = z h′ z z z Å Å 1 x − zã x − zã h′2 (x) = (1 − z) = h′ . h′ 1−z 1−z 1−z ❊♠ x = 0✱ t❡♠♦s q✉❡✿ Ç å h′1 (0) =h ❡ ❡♠ x = 1✱ t❡♠♦s q✉❡✿ ′ Ç h′2 (1) =h ′ 0 z 1−z 1−z = h′ (0), å = h′ (1). ✷✳✷✳ ✹✸ ❖ ❈❖▼P❘■▼❊◆❚❖ ❉❖❙ P❆❘❇❊▲❖❙ ◆❛s s❡çõ❡s s❡❣✉✐♥t❡s ❡♥✉♥❝✐❛r❡♠♦s ❡ ❞❡♠♦♥str❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ✭❝❛r❛❝t❡ríst✐❝❛s ♦✉ ♣r♦♣♦s✐çõ❡s✮ r❡❧❡✈❛♥t❡s s♦❜r❡ ♦s ♣❛r❜❡❧♦s✳ ❖ ♣r✐♠❡✐r♦ r❡s✉❧t❛❞♦ ♥♦s ❞✐③ q✉❡ ♦ ❛r❝♦ s✉♣❡r✐♦r ❞❡ ✉♠ ♣❛r❜❡❧♦ t❡♠ ♦ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦ q✉❡ ❛ s♦♠❛ ❞♦s ❛r❝♦s ✐♥❢❡r✐♦r❡s ❞♦ ♠❡s♠♦✳ P❛r❛ ♣r♦✈❛r♠♦s ✐st♦✱ ✉t✐❧✐③❛r❡♠♦s ❛ ❈♦♥st❛♥t❡ ❯♥✐✈❡rs❛❧ ❞❛ P❛rá❜♦❧❛✱ ❝♦♠♦ s❡❣✉❡✳ ✷✳✷ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞♦s P❛r❜❡❧♦s ◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛r❡♠♦s ❞♦✐s r❡s✉❧t❛❞♦s r❡❧❛t✐✈♦s ❛♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦s ❛r❝♦s s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r❡s ❞♦s P❛r❜❡❧♦s✳ ❆ ♣r✐♠❡✐r❛ é ✉♠ ❝❛s♦ ❛♥á❧♦❣♦ ❛ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❢✉♥❞❛♠❡♥t❛❧ ❞♦s ❆r❜❡❧♦s ❡ ❛ s❡❣✉♥❞❛ ❛♣r❡s❡♥t❛ ✉♠❛ ❝♦♥str✉çã♦ ❞❡ P❛r❜❡❧♦s ❞❡♥tr♦ ❞❡ P❛r❜❡❧♦s✳ Pr♦♣♦s✐çã♦ ✷✳✶✳ ❖ ❛r❝♦ s✉♣❡r✐♦r t❡♠ ♦ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ s♦♠❛ ❞♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s ❛r❝♦s ✐♥❢❡r✐♦r❡s ❞♦ P❛r❜❡❧♦✳ ❉❡♠♦♥str❛çã♦✳ ▲❡♠❜r❛♥❞♦ q✉❡ ❛ r❛③ã♦ ❡♥tr❡ ♦ ❆r❝♦ ▲❛t✉s ❘❡❝t✉♠ ❡ ♦ ❙❡♠✐❧❛t✉s ❘❡❝t✉♠ é ❝♦♥st❛♥t❡ ✭❈♦♥st❛♥t❡ ❯♥✐✈❡rs❛❧ ❞❛ P❛rá❜♦❧❛✮✱ ✐st♦ é✱ sã♦ ♣r♦♣♦r❝✐♦♥❛✐s✱ ♠♦str❡♠♦s q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❛r❝♦ s✉♣❡r✐♦r é ✐❣✉❛❧ à s♦♠❛ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦s ❛r❝♦s ✐♥❢❡r✐♦r❡s✳ ▲❡♠❜r❡♠♦s ❞♦ ❝❛s♦ ❞♦s ❆r❜❡❧♦s✱ ♣♦✐s sã♦ ❛♥á❧♦❣♦s✳ ❉❡ ❢❛t♦✱ ❞❡♥♦t❡♠♦s ♣♦r ▲❛t✉s ❘❡❝t✉♠ ❡ ♦♥❞❡ k p = 2a C ♦ ❆r❝♦ C = kp✱ ♦ ❙❡♠✐❧❛t✉s ❘❡❝t✉♠ ❞♦ ❛r❝♦ s✉♣❡r✐♦r❀ ❛ss✐♠✱ t❡♠♦s q✉❡ é ❛ ❈♦♥st❛♥t❡ ❯♥✐✈❡rs❛❧ ❞❛ P❛rá❜♦❧❛✳ C1 ❡ C2 ♦s ❆r❝♦s ▲❛t✉s ❘❡❝t✉♠ ❡ p1 ❡ p2 ♦s ❙❡♠✐❧❛t✉s ❘❡❝t✉♠ ❞♦s ❛r❝♦s ˙ ˙ ✐♥❢❡r✐♦r❡s C1 C3 ❡ C 3 C2 ❞♦ P❛r❜❡❧♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ▲♦❣♦✿ ❙❡❥❛♠ t❛♠❜é♠ p1 + p2 = p = 2a, C1 = kp1 , C2 = kp2 . ❈♦♠ ✐ss♦ s❡❣✉❡ q✉❡✿ C = kp = kp1 + kp2 = C1 + C2 . ❋✐❝❛♥❞♦ ❛ss✐♠ ♣r♦✈❛❞❛ ❛ ♣r♦♣♦s✐çã♦✳ ❖ s❡❣✉♥❞♦ r❡s✉❧t❛❞♦ s♦❜r❡ ♦s ♣❛r❜❡❧♦s é ❛♥á❧♦❣♦ ❛ ✉♠❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞♦s ❆r❜❡❧♦s✱ ♥❛ q✉❛❧ ❝♦♥s✐❞❡r❛♠♦s P❛r❜❡❧♦s ❞❡♥tr♦ ❞❡ P❛r❜❡❧♦s ❡ ♠♦str❛♠♦s ✉♠❛ r❡❧❛çã♦ ❡①✐st❡♥t❡ ❡♥tr❡ ❛s P❛rá❜♦❧❛s ✐♥❢❡r✐♦r❡s q✉❡ ♦s ❢♦r♠❛♠✳ Pr♦♣♦s✐çã♦ ✷✳✷✳ ❆❜❛✐①♦ ❞❡ ❝❛❞❛ ❛r❝♦ ✐♥❢❡r✐♦r ❞❡ ✉♠ ♣❛r❜❡❧♦✱ ❝♦♥str✉❛ ❞♦✐s ♥♦✈♦s ♣❛r❜❡❧♦s s❡♠❡❧❤❛♥t❡s ❛♦ ♦r✐❣✐♥❛❧ ✭✈❡r ❋✐❣✉r❛ ✸✳✻✮✳ ❉♦s q✉❛tr♦ ♥♦✈♦s ❛r❝♦s ✐♥❢❡r✐♦r❡s✱ ♦s ❞♦✐s ❞♦ ♠❡✐♦ sã♦ ❝♦♥❣r✉❡♥t❡s✱ ❡ ♦s s❡✉s ❝♦♠♣r✐♠❡♥t♦s sã♦ ✐❣✉❛✐s à ♠❡t❛❞❡ ❞❛ ♠é❞✐❛ ❤❛r♠ô♥✐❝❛ ❞♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s ❛r❝♦s ✐♥❢❡r✐♦r❡s ♦r✐❣✐♥❛✐s✳ ✹✹ ❈❆P❮❚❯▲❖ ✷✳ P❆❘❇❊▲❖❙ ❋✐❣✉r❛ ✷✳✻✿ P❛r❜❡❧♦s s❡♠❡❧❤❛♥t❡s C1 , C2 ❡ C3 ❛s ❝ús♣✐❞❡s ❞❡ ✉♠ ♣❛r❜❡❧♦ P ✱ s❡♥❞♦ C3 ❛ ❝ús♣✐❞❡ ❞♦ C5 ♣♦♥t♦s t❛✐s q✉❡ C4 s❡❥❛ ❛ ❝ús♣✐❞❡ ♠é❞✐❛ ❞❡ ✉♠ ♥♦✈♦ ♣❛r❜❡❧♦ P1 ❢♦r♠❛❞♦ ❛ ♣❛rt✐r ❞❡ C1 , C4 ❡ C3 ✱ ❡ C5 ❛ ❝ús♣✐❞❡ ♠é❞✐❛ ❞♦ ♥♦✈♦ ❛r❜❡❧♦ P2 ❢♦r♠❛❞♦ ❛ ♣❛rt✐r ❞❡ C3 , C5 ❡ C2 ✳ ❈♦♠♦ ♦s ♥♦✈♦s ♣❛r❜❡❧♦s sã♦ ❝♦♥str✉í❞♦s ❞❡ ♠♦❞♦ ❛ s❡r❡♠ s❡♠❡❧❤❛♥t❡s ❛♦ ♣❛r❜❡❧♦ P ✱ t❡♠♦s q✉❡ C1 C3 C3 C5 C1 C4 = = . ✭✷✳✶✮ C4 C3 C3 C2 C5 C2 ❉❡♠♦♥str❛çã♦✳ ♠❡✐♦✳ ❙❡❥❛♠ ❙❡❥❛♠ C4 ❡ ˙ ˙ ˙ ❙❡❥❛♠ lS ✱ lL ✱ lR ✱ l1 ✱ l2 ✱ l3 ✱ l4 ♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s ❛r❝♦s ❧❛t✉s r❡❝t✉♠ C 1 C2 ✱ C1 C3 ✱ C3 C2 ✱ ˙ ˙ ˙ C 1 C4 ✱ C4 C3 ✱ C3 C5 ✱ ˙ ❡ C 5 C2 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ r❡❧❛çã♦ ✭✷✳✶✮ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ l1 lL l3 = = . l2 lR l4 ✭✷✳✷✮ P❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♣r♦♣♦rçã♦ ❞❡ ✭✷✳✷✮✱ s❡❣✉❡ q✉❡✿ lL + lR l1 + l2 = . l2 lR P♦r s✉❛ ✈❡③✱ lL lS = . l2 lR ❚❛♠❜é♠ ❞❡ ✭✷✳✷✮ s❡❣✉❡ q✉❡✿ l3 + l4 lL + lR = . l3 lL ♦✉ s❡❥❛✱ ✭✷✳✸✮ ✷✳✸✳ ✹✺ ❖ P❆❘❆▲❊▲❖●❘❆▼❖ ❆❙❙❖❈■❆❉❖ ❆ ❯▼ P❖◆❚❖ lS lL = . l3 lR ✭✷✳✹✮ ❈♦♠♣❛r❛♥❞♦ ✭✷✳✸✮ ❡ ✭✷✳✹✮✱ ❝♦♥❝❧✉í♠♦s q✉❡ l2 = l3 ✳ ❆❧é♠ ❞✐st♦✱ ♣♦r ✭✷✳✸✮✱ t❡♠♦s q✉❡ l2 = ❖❜s❡r✈❛çã♦ ✷✳✶✳ lL .lR lL .lR lL .lR = = = lS lL + lR lL + lR ❈♦♥str✉í♠♦s ♦ P❛r❜❡❧♦ ❝♦♥str✉✐r ♦s ♥♦✈♦s P❛r❜❡❧♦s C4 ❡ C5 P1 ❡ P2 P 1 lL +lR lL .lR C1 C4 = q ❡♥tã♦ 1 lL . s♦❜r❡ ♦ ✐♥t❡r✈❛❧♦ [0, 1]✳ ❈♦♠♦ q✉❡r❡♠♦s P ✱ ❞❡✈❡♠♦s t♦♠❛r C1 C3 = p ❡♥tã♦ C3 C2 = 1 − p✱ ❡ ❞❡ ♠♦❞♦ ❛ s❡r❡♠ s❡♠❡❧❤❛♥t❡s ❛ ❞❡ ♠♦❞♦ ❛ s❛t✐s❢❛③❡r❡♠ ✭✷✳✶✮✳ ❈❤❛♠❛♥❞♦✲s❡ ❝❤❛♠❛♥❞♦✲s❡ 2 1 = .1 2 lR + C4 C3 = p − q ✳ P♦r ✭✷✳✶✮✱ t❡♠✲s❡ q✉❡✿ p q = . p−q 1−q ▲♦❣♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ✷✳✸ q = p2 ✳ ❖ ♣❛r❛❧❡❧♦❣r❛♠♦ ❛ss♦❝✐❛❞♦ ❛ ✉♠ ♣♦♥t♦ ❈♦♥s✐❞❡r❛♥❞♦✲s❡ ❛s ❢✉♥çõ❡s q✉❡ ❞❡s❝r❡✈❡♠ ✉♠ ♣❛r❜❡❧♦ h(x)✱ h1 (x) ❡ h2 (x)✱ ❡①♣❧✐❝✐t❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✱ t♦♠❡♠♦s ✉♠ x0 ∈ (0, 1) ❡ ❝♦♥s✐❞❡r❡♠♦s ♦ ♣♦♥t♦ P1 = (x0 , h(x0 )) ♣❡rt❡♥✲ ❝❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❡ h(x)✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣♦r s❡♠❡❧❤❛♥ç❛✱ ❡①✐st❡♠ ♣♦♥t♦s P2 = (zx0 , zh(x0 )) ❡ P3 = ((1 − z) · x0 + z, (1 − z) · h(x0 )) q✉❡ ♣❡rt❡♥❝❡♠ ❛♦s ❣rá✜❝♦s ❞❡ h1 (x) ❡ h2 (x) r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡ ❢❛t♦✿ ï h1 (zx0 ) = z · ãò aαzx0 Å αzx0 · − α = z · [aαx0 · (αx0 − α)] = zaα2 x0 · (x0 − 1) = zh(x0 ). z z Ç å ñ ô ñ Ç (1 − z)x0 (1 − z)x0 (1 − z)x0 = (1 − z)aα · α h2 ((1 − z)x0 + z) = (1 − z)h 1−z 1−z 1−z = (1 − z)aαx0 (αx0 − α) = (1 − z)h(x0 ). å ô −α ❱❡r✐✜q✉❡♠♦s q✉❡ ♦s ♣♦♥t♦s P1 ✱ P2 ✱ P3 ❡ C2 ✭❝ús♣✐❞❡ ♠é❞✐❛✮ ❢♦r♠❛♠ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ q✉❡ s❡rá ❝❤❛♠❛❞♦ ❞❡ ♣❛r❛❧❡❧♦❣r❛♠♦ ❛ss♦❝✐❛❞♦ ❛ x0 ✳ ❚❡♠♦s q✉❡✿ P2 C 2 = » (p − px0 )2 + (p · h(x0 ))2 = p · » (1 − x0 )2 + (h(x0 ))2 , ✹✻ ❈❆P❮❚❯▲❖ ✷✳ ❋✐❣✉r❛ ✷✳✼✿ P❛r❛❧❡❧♦❣r❛♠♦ ❛ss♦❝✐❛❞♦ ❛♦ ♣♦♥t♦ P 1 P3 = P❆❘❇❊▲❖❙ x0 » ((1 − p)x0 + p − x0 )2 + ((1 − p).h(x0 ) − h(x0 ))2 = » » = p. (1 − x0 )2 + (h(x0 ))2 = p. (1 − x0 )2 + (h(x0 ))2 . ▲♦❣♦✱ P 2 C 2 = P1 P3 ✳ ❆❧é♠ ❞✐st♦✱ P1 P2 = » (px0 − x0 )2 + (ph(x0 ) − h(x0 ))2 » = (1 − p). (x0 )2 + (h(x0 ))2 , P3 C 2 = » » ((1 − p)x0 )2 + ((1 − p).h(x0 ))2 = p. (1 − x0 )2 + (h(x0 ))2 = » = (1 − p). (x0 )2 + (h(x0 ))2 . ▲♦❣♦✱ P 1 P 2 = P3 C 2 ✳ P♦rt❛♥t♦✱ P1 P 2 P3 C 2 é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ❆ ♣r♦♣♦s✐çã♦ q✉❡ s❡❣✉❡ r❡❧❛❝✐♦♥❛ ❛ ár❡❛ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❛ss♦❝✐❛❞♦ ❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣♦♥t♦ c ❝♦♠ ❛ ár❡❛ ❞♦ P❛r❜❡❧♦✳ ❚♦♠❡♠♦s ✉♠ P❛r❜❡❧♦ P ❞❡✜♥✐❞♦ ♣❡❧❛s ❢✉♥çõ❡s h(x)✱ h1 (x) ❡ h2 (x)✱ ❡st❛❜❡❧❡❝✐❞❛s ❛♥t❡r✐♦r♠❡♥t❡✳ ❈♦♥s✐❞❡r❡ ✉♠ ♣♦♥t♦ c ∈ (0, 1) t❛❧ q✉❡ h(c) s❡❥❛ ♦ ✈❛❧♦r Pr♦♣♦s✐çã♦ ✷✳✸✳ ✷✳✸✳ ✹✼ ❖ P❆❘❆▲❊▲❖●❘❆▼❖ ❆❙❙❖❈■❆❉❖ ❆ ❯▼ P❖◆❚❖ ♠é❞✐♦ ❞❡ h [0, 1]✳ ❡♠ P (c) ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ P (c) é ❞❛❞❛ ♣♦r ❈❤❛♠❛♥❞♦ ❞❡ ❡♥tr❡ ❛s ár❡❛s ❞♦ P❛r❜❡❧♦ P ❡ ❞❡ ❛ss♦❝✐❛❞♦ ❛♦ ♣♦♥t♦ c✳ ❆ r❡❧❛çã♦ Area(P ) = 2Area(P (c)) ✳ ❱❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡✱ ❝♦♠ ♦ ✉s♦ ❞♦❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ P ✳ ❚❛❧ ár❡❛ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ❝♦♠♦ ❛ ár❡❛ A s♦❜ ♦ ❣rá✜❝♦ ❞❛ ♣❛rá❜♦❧❛ s✉♣❡r✐♦r ♠❡♥♦s ❛s ár❡❛s A1 , A2 s♦❜ ♦s ❣rá✜❝♦s ❞❛s ❞✉❛s ♣❛rá❜♦❧❛s ❞❡✜♥✐❞❛s ♣♦r h1 (x) ❡ h2 (x)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚❡♠♦s q✉❡✿ ❉❡♠♦♥str❛çã♦✳ A= Z 1 0 h(x)dx = Z p A1 = 0 Ç = ❡ A2 = Z 1 p aα2 = 1−p =− 0 Ç 2 aα x(x − 1)dx = Z p h1 (x)dx = 0 å aα2 x3 aα2 x2 1 aα2 − . = − 3 2 6 0 å Ç x − 1 dx aα x p 2 å aα2 p2 aα2 x3 aα2 x2 p 2aα2 p2 − 3aα2 p2 − = − = p2 A, = 3p 2 6 6 0 h2 (x)dx = Ç 3 x Z 1 Z 1 p Ç å x−p aα2 aα (x − p) − 1 dx = 1−p 1−p Z 1 2 å 1 aα2 x2 (1 + p) − + px = 3 2 1−p p ÇÇ p å (x2 − x(p + 1) + px)dx Ç 1 p+1 p3 p2 (1 + p) − +p − − + p2 3 2 3 2 a(1 − p)2 aα2 (1 − p)2 =− = (1 − p)2 A 6 6 P♦rt❛♥t♦ ❛ ár❡❛ ❞♦ ♣❛r❜❡❧♦ P é ❞❛❞❛ ♣♦r Area(P ) = A − p2 A − (1 − p)2 A = 2p(1 − p)A. ✭✷✳✺✮ ❆❣♦r❛✱ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦✱ ❧❡♠❜r❡♠♦s ❞❡ q✉❡ ❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ P❛✲ r❛❧❡❧♦❣r❛♠♦ ♦ ❞✐✈✐❞❡ ❡♠ ❞♦✐s tr✐â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s✳ ❙❡♥❞♦ ❛ss✐♠✱ ❝♦♠♦ ❝♦♥❤❡❝❡♠♦s ♦s åå 1 p ✹✽ ❈❆P❮❚❯▲❖ ✷✳ P❆❘❇❊▲❖❙ ♣♦♥t♦s q✉❡ ❢♦r♠❛♠ ♦s tr✐â♥❣✉❧♦s✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r s✉❛ ár❡❛ ♣♦r✿ A(P (c)) = 2 · Area△P1 P2 C2 î c h(c) 1 1 pc pf (c) 1 =2 2 p 0 1 = cph(c) + ph(c) − p2 h(c) − pch(c) ó = [ph(c)(1 − p)] = p(1 − p)h(c). P❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ ♣❛r❛ ✐♥t❡❣r❛✐s✱ P é ❞❛❞❛ ♣♦r ✷✳✸✳✶ 2p(1 − p)h(c)✱ ❞❡ ♠♦❞♦ q✉❡ h(c) = Z 1 0 h(x)dx✱ ❡♥tã♦ ❛ ár❡❛ ❞♦ P❛r❜❡❧♦ Area(P ) = 2.Area(P (c))✳ P❛r❛❧❡❧♦❣r❛♠♦ ❈ús♣✐❞❡ ❱ért✐❝❡ ❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ♠♦str❛ q✉❡ ❛ ár❡❛ ❞❡ ✉♠ P❛r❜❡❧♦ é ♣r♦♣♦r❝✐♦♥❛❧ à ár❡❛ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❢♦r♠❛❞♦ ♣❡❧❛ s✉❛ ❝ús♣✐❞❡ ♠é❞✐❛ ❡ ♦s ✈ért✐❝❡s ❞♦s três ❛r❝♦s ♣❛r❛❜ó❧✐❝♦s q✉❡ ♦ ❢♦r♠❛♠✱ ❝❤❛♠❛❞♦ ❞❡ P❛r❛❧❡❧♦❣r❛♠♦ ❈ús♣✐❞❡ ❱ért✐❝❡✱ ❞❡♥♦t❛❞♦ ♣♦r P (C2 )✳ ❯t✐❧✐③❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❆rq✉✐♠❡❞❡s ✭❚❡♦r❡♠❛ ✶✳✶✮ ♣❛r❛ ❞❡♠♦♥str❛r ❡st❡ ❢❛t♦ q✉❡ ❡stá ✐❧✉str❛❞♦ ♥❛ ❋✐❣✉r❛ ✸✳✽✳ Pr♦♣♦s✐çã♦ ✷✳✹✳ ❆ ❝ús♣✐❞❡ ♠é❞✐❛ ❞❡ ✉♠ ♣❛r❜❡❧♦ P ❡ ♦s ✈ért✐❝❡s ❞❛s ♣❛rá❜♦❧❛s q✉❡ ♦ ❢♦r♠❛♠ ❞❡t❡r♠✐♥❛♠ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ❆ ár❡❛ ❞♦ ♣❛r❜❡❧♦ P é ✐❣✉❛❧ ❛ 4 ✈❡③❡s ❞❛ ár❡❛ ❞♦ 3 s❡✉ ♣❛r❛❧❡❧♦❣r❛♠♦ ❝ús♣✐❞❡ ✈ért✐❝❡✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ C 1 , C2 , C3 ❛s ❝ús♣✐❞❡s ❞♦ ♣❛r❜❡❧♦ P✱ ❝♦♠ C2 s❡♥❞♦ ❛ ❝ús♣✐❞❡ ❞♦ ˙ ˙ ˙ ♠❡✐♦✱ V1 , V2 , V3 ♦s ✈ért✐❝❡s ❞❛s ♣❛rá❜♦❧❛s ❢♦r♠❛❞❛s ♣♦r C 1 C2 ✱ C1 C3 ❡ C2 C3 ✱ r❡s♣❡❝t✐✈❛✲ P ✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ t♦♠❡♠♦s ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ←−→ ❞❡ ♠♦❞♦ q✉❡ C1 C3 s❡❥❛ ♦ ❡✐①♦ Ox ✱ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ C1 C3 s❡❥❛ ❛ ♦r✐❣❡♠ ❡ q✉❡ ❛ ♣❛rá❜♦❧❛ x2 ˙ ❝♦♥t❡♥❞♦ C 1 C3 t❡♥❤❛ ❡q✉❛çã♦ y = a − 4a ✱ ❝✉❥❛s r❛í③❡s sã♦ −2a ❡ 2a✳ ♠❡♥t❡✱ q✉❡ ❢♦r♠❛♠ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❛❜♦r❞❛❣❡♠ ❛♣r❡s❡♥t❛❞❛ ♥❛ ❙❡çã♦ ✸✳✶✱ t♦♠❛♥❞♦✲s❡ t❡♠♦s q✉❡✿ β = 2a✱ α = −4a✱ (x − 2a)2 x2 − 4ax + 4a x2 1 g(x) = a − =a− =a− + x − a = − x(x + 4a), 4a 4a 4a 4a ❡ h(x) = g(−4ax) = x(−4ax + 4a) ❙❡♥❞♦ ❛ss✐♠✱ s❡❣✉❡ q✉❡✿ ✷✳✸✳ ❖ P❆❘❆▲❊▲❖●❘❆▼❖ ❆❙❙❖❈■❆❉❖ ❆ ❯▼ P❖◆❚❖ ✹✾ ❋✐❣✉r❛ ✷✳✽✿ ❖ P❛r❛❧❡❧♦❣r❛♠♦ ❈ús♣✐❞❡ ❱ért✐❝❡ ❞♦ P❛r❜❡❧♦ • C1 = (0, 0)✱ C2 = (z, 0) ❡ C3 = (1, 0)❀ • h(1/2) = a✱ V2 = (1/2, a); • h1 (x) = zh(x/z) = x(−4ax/z + 4a)✱ h1 (z/2) = az ✱ ❡ V1 = (z/2, az)❀ Ä ä Ä = (x − z) − 4a • h2 (x) = (1 − z)h x−z 1−z V3 = ((1 + z)/2, a(1 − z))✳ Ä x−z 1−z ä ä + 4a ✱ h2 ((1 + z)/2) = a(1 − z) ❡ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❝♦♥❢♦r♠❡ ✐❧✉str❛❞♦ ♥❛ ✜❣✉r❛ ✸✳✾✱ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r V1 ❡ V2 t❡♠ ✐♥❝❧✐♥❛çã♦ ✐❣✉❛❧ ❛ 2a✱ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r C2 ❡ V3 t❡♠ ✐♥❝❧✐♥❛çã♦ ✐❣✉❛❧ ❛ 2a✱ ❡♥q✉❛♥t♦ q✉❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r C2 ❡ V2 t❡♠ ✐♥❝❧✐♥❛çã♦ ✐❣✉❛❧ ❛ −2a ❡ ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r V1 ❡ V3 t❡♠ ✐♥❝❧✐♥❛çã♦ ✐❣✉❛❧ ❛ −2a✳ P♦rt❛♥t♦✱ ♦ q✉❛❞r✐❧át❡r♦ C2 V2 V1 V3 é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ❋✐❣✉r❛ ✷✳✾✿ ▲❛❞♦s ♣❛r❛❧❡❧♦s ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❝ús♣✐❞❡ ✈ért✐❝❡ ✺✵ ❈❆P❮❚❯▲❖ ✷✳ ❆ s❡❣✉♥❞❛ ♣❛rt❡ ❞❡ ♥♦ss❛ ❞❡♠♦♥str❛çã♦ ✈❛✐ ♣r♦✈❛r q✉❡ ❛ ár❡❛ ❞♦ ♣❛r❜❡❧♦ P❆❘❇❊▲❖❙ P é ✐❣✉❛❧ ❛ ✈❡③❡s ❞❛ ár❡❛ ❞♦ s❡✉ ♣❛r❛❧❡❧♦❣r❛♠♦❝ús♣✐❞❡ ✈ért✐❝❡✳ P❛r❛ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦✱ ✈❛♠♦s ✉s❛r ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿ • ➪r❡❛ ❞♦ P❛r❜❡❧♦ C1 V 2 C3 V 3 C2 V 1 C1 • ➪r❡❛ ❞❛ P❛rá❜♦❧❛ C1 V 2 C3 ❂ A1 ✱ • ➪r❡❛ ❞❛ P❛rá❜♦❧❛ C1 V 1 C2 ❂ A2 • ➪r❡❛ ❞❛ P❛rá❜♦❧❛ C2 V 3 C3 ❂ A3 ✱ • ➪r❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦ • ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ C1 V 2 C3 ❂ T1 ✱ • ➪r❡❛ ❞♦ ❚rî❛♥❣✉❧♦ C1 V 1 C2 ❂ T2 ✱ • ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ C2 V 3 C3 ❂ T3 ✳ ❂ P1 ✱ ✱ C2 V 1 V 2 V 3 ❂ Q1 ✱ ❋✐❣✉r❛ ✷✳✶✵✿ P❛r❛❧❡❧♦❣r❛♠♦ ❈ús♣✐❞❡ ❱ért✐❝❡ ❞♦s P❛r❜❡❧♦s ❡ ♦s ❚r✐â♥❣✉❧♦s ■♥s❝r✐t♦s ❙❡♥❞♦ ❛ss✐♠✱ t❡♠♦s q✉❡ ❛ ár❡❛ ❞♦ P❛r❜❡❧♦ é ❞❛❞❛ ♣♦r✿ P1 = A1 − (A2 + A3 ), ❡ ❛ ár❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦ ❈ús♣✐❞❡ ❱ért✐❝❡ é ❞❛❞❛ ♣♦r✿ Q1 = T1 − (T2 + T3 ). 4 3 ✷✳✹✳ ❖ ❘❊❚➶◆●❯▲❖ ❚❆◆●❊◆❚❊ ❉❖❙ P❆❘❇❊▲❖❙ ✺✶ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❆rq✉✐♠❡❞❡s ✭❚❡♦r❡♠❛ ✶✳✶✮ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ♣❛r❛❜ó✲ ❧✐❝❛✱ ❛ ➪r❡❛ ❞❡ ❝❛❞❛ r❡❣✐ã♦ P❛r❛❜ó❧✐❝❛ é ✐❣✉❛❧ ❛ 43 ✈❡③❡s ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ♥❡❧❛✳ ▲♦❣♦✱ t❡♠♦s q✉❡✿ P1 = A1 − (A2 + A3 ) = 4 4 4 4 · T 1 − ( · T 2 + · T 3 ) = · Q1 , 3 3 3 3 ♦ q✉❡ ❞❡♠♦♥str❛ ❛ s❡❣✉♥❞❛ ❛✜r♠❛çã♦✳ ✷✳✹ ❖ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❞♦s P❛r❜❡❧♦s ❆ ♣r♦♣♦s✐çã♦ q✉❡ s❡❣✉❡ ❝♦♥stró✐ ✉♠ r❡tâ♥❣✉❧♦ ❛ ♣❛rt✐r ❞❛s ❝ús♣✐❞❡s ❞♦ P❛r❜❡❧♦ ❡ r❡❧❛❝✐♦✲ ♥❛♠ s✉❛s ár❡❛s✳ ❆s q✉❛tr♦ r❡t❛s t❛♥❣❡♥t❡s ❛ ✉♠ P❛r❜❡❧♦ ♥♦s ♣♦♥t♦s ❞❡ ❝ús♣✐❞❡s ❢♦r♠❛♠ ✉♠ r❡tâ♥❣✉❧♦✱ q✉❡ é ❝❤❛♠❛❞♦ ❞❡ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ✭❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✷✳✹✮✳ ❆ ár❡❛ ❞♦ P❛r❜❡❧♦ é ✐❣✉❛❧ ❛ ❞♦✐s t❡rç♦s ❞❛ ár❡❛ ❞♦ s❡✉ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡✳ Pr♦♣♦s✐çã♦ ✷✳✺✳ ❉❡♠♦♥str❛çã♦✳ ◆♦✈❛♠❡♥t❡✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ❝♦♥s✐❞❡r❡♠♦s ❛ ❡q✉❛çã♦ ❞❛ ♣❛rá✲ 2 ❜♦❧❛ s✉♣❡r✐♦r ❞♦ P❛r❜❡❧♦ ❞❛❞❛ ♣♦r y = a − x4a ✳ ❊♥tã♦ y ′ (−2a) = 1✳ ❈♦♠ ✐st♦✱ ♦ â♥❣✉❧♦ ❡♥tr❡ ❛ r❡t❛ t❛♥❣❡♥t❡ à ♣❛rá❜♦❧❛ s✉♣❡r✐♦r ♥♦ ♣♦♥t♦ C1 ♠❡❞❡ π/4 ❝♦♠ ♦ ▲❛t✉s ❘❡❝t✉♠✳ ❉❡ ❢❛t♦✱ ✉♠❛ ✈❡③ q✉❡ ❛ ❡q✉❛çã♦ ❞❡ t❛❧ r❡t❛ t❛♥❣❡♥t❡ é ❞❛ ❢♦r♠❛ y = 1x + b ❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ C1 = (−2a, 0)✱ t❡♠♦s q✉❡ b = 2a✱ ❡ q✉❛♥❞♦ x = 0✱ y = 2a✳ P♦rt❛♥t♦ ♦s ♣♦♥t♦s C1 = (−2a, 0)✱ D = (0, 2a) ❡ O = (0, 0) ❢♦r♠❛♠ ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s✱ r❡tâ♥❣✉❧♦ ◊ ❡♠ O ❡ ♦ â♥❣✉❧♦ OC 1 D = π/4✳ ❆❣♦r❛✱ t♦♠❛♥❞♦✲s❡ ❛s ❡q✉❛çõ❡s ❞❡ h(x)✱ h1 (x) ❡ h2 (x) ❝♦♠♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦✲ s✐çã♦ ✸✳✹✱ t❡♠♦s q✉❡ h′ (0) = 4a✳ ❈♦♠♣❛r❛♥❞♦✲s❡ h′ (0) ❝♦♠ y ′ (−2a)✱ ♦❜t❡♠♦s q✉❡ 4a = 1✳ ❚❛♠❜é♠ t❡♠♦s q✉❡ h′ (1) = −4a = −1✱ h′1 (p) = −4a = −1 ❡ h′2 (p) = 4a = 1✳ ❋✐❣✉r❛ ✷✳✶✶✿ ❘❡t❛ t❛♥❣❡♥t❡ à ♣❛rá❜♦❧❛ ♥♦ ♣♦♥t♦ C1 ✺✷ ❈❆P❮❚❯▲❖ ✷✳ P❆❘❇❊▲❖❙ ❋✐❣✉r❛ ✷✳✶✷✿ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❆ss✐♠✱ ♦ q✉❛❞r✐❧át❡r♦ ❢♦r♠❛❞♦ ♣❡❧❛ ✐♥t❡rs❡❝çã♦ ❞❛s r❡t❛s t❛♥❣❡♥t❡s às ❝ús♣✐❞❡s é ✉♠ r❡tâ♥❣✉❧♦✱ ♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡✱ ❝♦♠♦ ✐❧✉str❛ ❛ ❋✐❣✉r❛ ✸✳✶✷✳ ❆❣♦r❛ ❞❡♠♦♥str❛r❡♠♦s ❛ s❡❣✉♥❞❛ ♣❛rt❡ ❞❛ ♣r♦♣r✐❡❞❛❞❡✳ ❆ss✐♠ ❝♦♠♦ ♥❛ ♣r♦♣r✐❡❞❛❞❡ ❛♥t❡r✐♦r✱ ♣❛r❛ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦✱ ❞❡♥♦t❛r❡♠♦s✿ • ➪r❡❛ ❞♦ ❘❡tâ♥❣✉❧♦ C 2 T 1 T 2 T 3 ❂ R1 ✱ • ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ C1 T 2 C3 ❂ T1 ✱ • ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ C1 T 1 C2 ❂ T2 ✱ • ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ C2 T 3 C3 ❂ T3 ✱ • ➪r❡❛ ❞♦ P❛r❜❡❧♦ • ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ C 1 V 2 C 3 ❂T 4 ✱ • ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ C 1 V 1 C 2 ❂T 5 ✱ • ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ C 2 V 2 C 3 ❂T 6 ✳ C 1 V 2 C 3 V 3 C 2 V 1 ❂ P1 ✱ ❆ ár❡❛ ❞♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ é ✐❣✉❛❧ ❛✿ R1 = T1 − (T2 + T3 ). ▼❛s✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❛ ár❡❛ ❞♦s tr✐â♥❣✉❧♦s ❞♦s tr✐â♥❣✉❧♦s T1 ✱ T2 ❡ T3 ✱ T4 ✱ T5 ♣♦✐s ❛ ❜❛s❡ é ❛ ♠❡s♠❛ ✭♦ ❡ T6 sã♦ ✐❣✉❛✐s à ♠❡t❛❞❡ ❞❛s ár❡❛s ▲❛t✉s ❘❡❝t✉♠ ❞❡ ❝❛❞❛ P❛rá❜♦❧❛✮ ❡ ❛ ❛❧t✉r❛ ✭❝♦♥❢♦r♠❡ ❋✐❣✉r❛ ✸✳✶✸✮ é ❛ ♠❡t❛❞❡ ❞❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ ✉t✐❧✐③❛❞♦ ❛❣♦r❛✳ ✷✳✹✳ ❖ ❘❊❚➶◆●❯▲❖ ❚❆◆●❊◆❚❊ ❉❖❙ P❆❘❇❊▲❖❙ ✺✸ ❋✐❣✉r❛ ✷✳✶✸✿ ❘❡❧❛çã♦ ❡♥tr❡ ❛s ❛❧t✉r❛s ❞♦s tr✐â♥❣✉❧♦s✳ ❋✐❣✉r❛ ✷✳✶✹✿ ❖ P❛r❛❧❡❧♦❣r❛♠♦ ❈ús♣✐❞❡ ❱ért✐❝❡ ❡ ♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❙❡♥❞♦ ❛ss✐♠✱ t❡♠♦s q✉❡✿ R1 = T1 − (T2 + T3 ), P1 = 4 1 1 1 2 2 4 · (T4 − (T5 + T6 )) = · ( T1 − ( T2 + T3 )) = · T1 − (T2 + T3 ) = R1 . 3 3 2 2 2 3 3 Pr♦♣♦s✐çã♦ ✷✳✻✳ ◆♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❞❡ ✉♠ ♣❛r❜❡❧♦✱ ❛ ❞✐❛❣♦♥❛❧ ❢♦r♠❛❞❛ ♣❡❧♦s ♣♦♥t♦s T1 ❡ T3 ✱ ♦♣♦st❛ à ❝ús♣✐❞❡ C2 ✱ é t❛♥❣❡♥t❡ à ♣❛rá❜♦❧❛ s✉♣❡r✐♦r✳ ❡♥❝♦♥tr❛✲s❡ ♥❛ ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ ❞♦ ✈ért✐❝❡ ❖ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛ C2 ✳ C1 = (0, 0)✱ C2 = (2b, 0) ❡ C3 = (4a, 0)✳ ❆ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ♣❛r❜❡❧♦ ❡♠ x = 0 é y = x ❉❡♠♦♥str❛çã♦✳ ❚♦♠❡♠♦s ❛❞❡q✉❛❞❛♠❡♥t❡ ✉♠ s✐st❡♠❛ ❞❡ ❝❝♦♦r❞❡♥❛❞❛s ♥♦ q✉❛❧ ✺✹ ❈❆P❮❚❯▲❖ ✷✳ P❆❘❇❊▲❖❙ ❋✐❣✉r❛ ✷✳✶✺✿ ❖ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❞♦s P❛r❜❡❧♦s✱ ✉♠❛ ❞✐❛❣♦♥❛❧✱ ❡ ✉♠ ➶♥❣✉❧♦ ❇✐ss❡❝t♦r y = −(x − 2b) = −x + 2b✳ T1 = (b, b)✳ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ à ♣❛rá❜♦❧❛ ❞❛ ❡sq✉❡r❞❛ ❞♦ ♣❛r❜❡❧♦ é ❆ ✐♥t❡rs❡❝çã♦ ❞❡st❛s r❡t❛s ❞á ♦ ♣♦♥t♦ T1 ✱ ❞❡ ❝♦♦r❞❡♥❛❞❛s y = −(x − 4a) = −x + 4a ❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ à ♣❛rá❜♦❧❛ ❞❛ ❞✐r❡✐t❛ ❞♦ P❛r❜❡❧♦ é y = x − 2b✳ ❆ ✐♥t❡rs❡❝çã♦ ❞❡st❛s r❡t❛s ❞á ♦ ♣♦♥t♦ T3 ✱ ❞❡ ❝♦♦r❞❡♥❛❞❛s T1 = (2a + b, 2a − b)✳ ❆ ❡q✉❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ P❛r❜❡❧♦ ❡♠ ❆ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r y−b= T1 ❡ T3 x=1 é é ❞❛❞❛ ♣♦r✿ a−b 2a − 2b (x − b) = (x − b), 2a a y= a−b (x − b) + b. a ❆ ❡q✉❛çã♦ ❞❛ ♣❛rá❜♦❧❛ s✉♣❡r✐♦r é ❞❛❞❛ ♣♦r✿ g(x) = a − 2 2a ❞❛ ♣❛rá❜♦❧❛ ❞❡ ❡q✉❛çã♦ f (x) = a− x4a ✳ P❛r❛ ❡♥❝♦♥tr❛r♠♦s ❡♥tr❡ ❛ ❞✐❛❣♦♥❛❧ ♦♣♦st❛ à C2 ❡ ❛ ♣❛rá❜♦❧❛ s✉♣❡r✐♦r ❞♦ P❛r❜❡❧♦✱ ♦❜t✐❞❛ ♣♦r ✉♠❛ tr❛♥s❧❛çã♦ ❞❡ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ (x − 2a)2 , 4a ✷✳✹✳ ✺✺ ❖ ❘❊❚➶◆●❯▲❖ ❚❆◆●❊◆❚❊ ❉❖❙ P❆❘❇❊▲❖❙ ✐❣✉❛❧❛♠♦s f (x) ❛ g(x)✳ ❆ss✐♠✿ b2 (x − 2a)2 a−b x+ =a− a a 4a (x − 2a)2 ⇔ (a − b)x + b2 = a2 − 4 2 x ⇔ + (a − b)x − ax + b2 = 0 4 x2 − bx + b2 = 0 ⇔ 4 ⇔ x = 2b, f (x) = g(x) ⇔ ♣♦✐s t♦♠❛♠♦s x > 0✳ x−2a g ′ (2b) = a−b ✭✉♠❛ ✈❡③ q✉❡ g ′ (x) = − 2a ✮ ❝♦✐♥❝✐❞❡ a ❝♦♠ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r T1 ❡ T3 ✭❞✐❛❣♦♥❛❧✮✱ ♣r♦✈❛♠♦s q✉❡ ❛ ❞✐❛❣♦♥❛❧ ❞♦ r❡tâ♥❣✉❧♦ t❛♥❣❡♥t❡ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s T1 ❡ T3 é t❛♥❣❡♥t❡ à ♣❛rá❜♦❧❛ s✉♣❡r✐♦r ♥♦ ♣♦♥t♦ G ❞❡ ❛❜s❝✐ss❛ x = 2b ✭✈❡r ❋✐❣✉r❛ ✸✳✶✺✮✳ P❛r❛ ♣r♦✈❛r♠♦s q✉❡ ♦ ♣♦♥t♦ x = 2b ♣❡rt❡♥❝❡ à ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ C2 ✱ ❜❛st❛ ♦❜s❡r✈❛r♠♦s q✉❡✿ ❈♦♠♦ ❛ s♦❧✉çã♦ x = 2b é ú♥✐❝❛✱ ❡ • π ÿ ÿ ♦s â♥❣✉❧♦s C 1 C2 T1 ❡ T3 C2 C3 ♠❡❞❡♠ 4 ✱ • ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ • ❝♦♠♦ ❛ r❡t❛ ←−→ CG2 GC2 é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❡✐①♦ x✱ ♣♦✐s C2 ❡ G tê♠ ❛ ♠❡s♠❛ ❛❜s❝✐ss❛✱ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❡✐①♦ π ♠❡❞❡♠ 4 ✱ ❞❡ ♠♦❞♦ q✉❡ ❛ r❡t❛ ✷✳✹✳✶ ←−→ CG2 x✱ ◊ ◊ t❡♠♦s q✉❡ ♦s â♥❣✉❧♦s T 1 C2 G ❡ GC2 T3 é ❛ ❜✐ss❡tr✐③ ❞♦ â♥❣✉❧♦ ❞❡ ✈ért✐❝❡ C2 ✳ ❆ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ❝✐r❝✉♥s❝r❡✈❡ ♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❞♦ P❛r❜❡❧♦ ❆ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡ ♥♦s ♠♦str❛ ♣♦r ♠❡✐♦ ❞❛ ●❡♦♠❡tr✐❛ ❆♥❛❧ít✐❝❛✱ q✉❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ❝✐r❝✉♥s❝r❡✈❡ ♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❞❡ ✉♠ P❛r❜❡❧♦ ♣❛ss❛ ♣❡❧♦ ❢♦❝♦ ❞❛ ♣❛rá❜♦❧❛ s✉♣❡r✐♦r ❞♦ ♣❛r❜❡❧♦✳ Pr♦♣♦s✐çã♦ ✷✳✼✳ ❆ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ❝✐♥❝✉♥s❝r❡✈❡ ♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❞❡ ✉♠ P❛r❜❡❧♦ ♣❛ss❛ ♣❡❧♦ ❢♦❝♦ ❞❛ ♣❛rá❜♦❧❛ s✉♣❡r✐♦r✳ ❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ❞❡♠♦♥str❛r q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞♦ ♣♦♥t♦ T1 ❛té ♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡✲ rê♥❝✐❛ é ✐❣✉❛❧ á ❞✐stâ♥❝✐❛ ❞♦ ❝❡♥tr♦ ❛♦ ❢♦❝♦ ❞❛ ♣❛rá❜♦❧❛ s✉♣❡r✐♦r✳ P❛r❛ ✐st♦✱ ❝♦♥s✐❞❡r❛r❡♠♦s ❛s ♠❡s♠❛s ❝♦♦r❞❡♥❛❞❛s ✉t✐❧✐③❛❞❛s ♥❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✳ ✺✻ ❈❆P❮❚❯▲❖ ✷✳ P❆❘❇❊▲❖❙ ❋✐❣✉r❛ ✷✳✶✻✿ ❖ ❈ír❝✉❧♦ ❞♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❡ ♦ ❋♦❝♦ ❞❛ P❛rá❜♦❧❛ ❙✉♣❡r✐♦r✳ ❙❛❜❡♠♦s q✉❡ ♦ ❢♦❝♦ F ❞❛ ♣❛rá❜♦❧❛ s✉♣❡r✐♦r ❡♥❝♦♥tr❛✲s❡ ♥♦ ♣♦♥t♦ F = (2a, 0)✱ ❡ ♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞♦ r❡tâ♥❣✉❧♦ t❛♥❣❡♥t❡ ❞❡ ✉♠ ♣❛r❜❡❧♦ ❡♥❝♦♥tr❛✲s❡ ♥♦ ♣♦♥t♦ ♠é❞✐♦ s❡❣♠❡♥t♦ T1 T3 ✳ T1 = (b, b) ▲♦❣♦✱ ❝♦♠♦ T1 + T3 C= = 2 Ç ❡ T3 = (2a + b, 2a − b) b + 2a + b b + 2a − b , 2 2 dC,T1 = dC,F = (a + b − b)2 + (a − b)2 = » » (a + b − 2a)2 + (a + 0)2 = F = (2a, 0) F t❡♠♦s q✉❡✿ = (a + b, a). T1 ❡ ❞♦ ❈❡♥tr♦ ❛♦ ❋♦❝♦ F sã♦✿ a2 + (a − b)2 , » (b − a)2 + a2 . P♦rt❛♥t♦✱ ❝♦♠♦ ❛ ❞✐stâ♥❝✐❛ ❞♦ ❈❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❛♦ ♣♦♥t♦ ❈❡♥tr♦ ❛♦ ❋♦❝♦ ❞♦ å ❙❡♥❞♦ ❛ss✐♠✱ t❡♠♦s q✉❡ ❛s ❞✐stâ♥❝✐❛s ❞♦ ❈❡♥tr♦ ❛♦ ♣♦♥t♦ » ❡ C ❞❛ ♣❛rá❜♦❧❛ s✉♣❡r✐♦r ❞♦ P❛r❜❡❧♦✱ ✭dC,T1 T1 = dC,F ✮✱ í❣✉❛❧ à ❞✐stâ♥❝✐❛ ❞♦ ♦ ❢♦❝♦ F ❞❛ ♣❛rá❜♦❧❛ ♣❡rt❡♥❝❡ à ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ❝✐r❝✉♥s❝r❡✈❡ ♦ ❘❡tâ♥❣✉❧♦ ❚❛♥❣❡♥t❡ ❞♦ P❛r❜❡❧♦✳ ❋✐❣✉r❛ ✷✳✶✼✿ ❉✐stâ♥❝✐❛s ❞♦ ❈❡♥tr♦ ❞❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ❛♦ ♣♦♥t♦ T1 ❡ ❛♦ ❋♦❝♦ F✳ ❈❛♣ít✉❧♦ ✸ ❆❚■❱■❉❆❉❊❙ ◆❡st❛ s❡çã♦ ✈❛♠♦s ♣r♦♣♦r ❛❧❣✉♥s ❡①❡r❝í❝✐♦s ❡♥✈♦❧✈❡♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ P❛r❜❡❧♦s✱ ár❡❛ ❛❜❛✐①♦ ❞❛ ❝✉r✈❛ ✭■♥t❡❣r❛❧ ❞❡✜♥✐❞❛✮ ❡ ❞❡r✐✈❛❞❛s✳ ✸✳✶ ❈♦♥str✉çã♦ ❞❡ ✉♠ P❛r❜❡❧♦ ❖s ❞♦✐s ♣r✐♠❡✐r♦s ❡①❡r❝í❝✐♦s q✉❡ s❡❣✉❡♠ ❛♣r❡s❡♥t❛♠ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ♣❛r❜❡❧♦✳ ❈♦♠ ♦ ❛✉①í❧✐♦ ❞♦ ❙♦❢t✇❛r❡ ❣rá✜❝♦ ●❡♦❣❡❜r❛ ✭♦✉ s✐♠✐❧❛r✮ ❡ ✉t✐❧✐③❛♥❞♦ ❛ ❢♦r♠❛ ❣❡r❛❧ ❞♦s P❛r❜❡❧♦s✱ ❝♦♥str✉❛ ✉♠ ♣❛r❜❡❧♦ ❛ ♣❛rt✐r ❞❛ ❢✉♥çã♦ f (x) = −x2 + 5x − 6 ✭❉✐❝❛✿ ♣❛r❛♠❡tr✐③❡ f (x) ❞❡ ♠♦❞♦ q✉❡ ❡❧❛ t❡♥❤❛ s✉❛s r❛í③❡s ♥❛s ❛❜s❝✐ss❛s x = 0 ❡ x = 1✮✳ ❊①❡r❝í❝✐♦ ✸✳✶✳ ❙♦❧✉çã♦✿ ❆s r❛í③❡s ❞❡ f (x) sã♦ x = 2 ❡ x = 3✱ s❡♥❞♦ ❛ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈ê✲❧❛ ❝♦♠♦✿ f (x) = −(x − 2)(x − 3). ❱❛♠♦s ❡♥tã♦✱ ❡s❝r❡✈❡r g(x) ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡ s✉❛s r❛í③❡s s❡ ❡♥❝♦♥tr❡♠ ❡♠ x = 0 ❡ x = 1✳ ❆ss✐♠✿ g(x) = −(x + 2 − 2)(x + 2 − 3) = −x2 + x, ♦❜s❡r✈❛♥❞♦ q✉❡ g(x) é ✉♠❛ tr❛♥s❧❛çã♦ ❞❡ f (x) ❡♠ ✷ ✉♥✐❞❛❞❡s ✭✈❡r ❋✐❣✉r❛ ✹✳✶✮✳ ❆ ❢✉♥çã♦ g(x) s❡rá ♦ ❛r❝♦ s✉♣❡r✐♦r ❞♦ P❛r❜❡❧♦✳ ❆❣♦r❛✱ ✈❛♠♦s ❡s❝r❡✈❡r ❛s ❢✉♥çõ❡s q✉❡ Ä❞❡✜♥✐rã♦ ♦s ❛r❝♦s ✐♥❢❡r✐♦r❡s ä Ä ä ❞♦ P❛r❜❡❧♦✳ ■st♦ s❡rá x−z x ❢❡✐t♦ ✉t✐❧✐③❛♥❞♦ ❛s ❢✉♥çõ❡s h1 (x) = zf z ❡ h2 (x) = (1 − z)f 1−z ✳ ❚♦♠❡♠♦s ✉♠ ♣♦♥t♦ p q✉❛❧q✉❡r✱ t❛❧ q✉❡ p ∈ (0, 1)✳ ❙❡❥❛ p = 0, 3✱ ♣♦r ❡①❡♠♣❧♦✳ ❙❡♥❞♦ ❛ss✐♠✱ t❡♠♦s q✉❡✿ Ç h1 (x) = 0, 3f x 0, 3 å " Ç x = 0, 3 − 0, 3 ✺✼ å2 x2 x =− + x, + 0, 3 0, 3 # ✺✽ ❈❆P❮❚❯▲❖ ✸✳ ❆❚■❱■❉❆❉❊❙ ❡ Ç h2 (x) = 0, 7f x − 0, 3 0, 7 å " Ç x − 0, 3 = 0, 7 − 0, 7 å2 (x − 0, 3)2 x − 0, 3 =− + x − 0, 3. + 0, 7 0, 7 # ❋✐❣✉r❛ ✸✳✶✿ P❛r❜❡❧♦ ❝♦♥str✉í❞♦ ♣❛r❛ ♦ ❡①❡r❝í❝✐♦ ✹✳✶✳ ❘❡♣r❡s❡♥t❛♥❞♦✲s❡ ❣r❛✜❝❛♠❡♥t❡ ❛s ❢✉♥çõ❡s g(x)✱ h1 (x) ❡ h2 (x)✱ ❝♦♥❢♦r♠❡ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♥❛ ❋✐❣✉r❛ ✹✳✷✱ t❡♠♦s ♦ ♣❛r❜❡❧♦ ❞❡s❡❥❛❞♦✳ ❋✐❣✉r❛ ✸✳✷✿ P❛r❜❡❧♦ ❢♦r♠❛❞♦ ♥♦ ❡①❡r❝í❝✐♦ ✹✳✶✳ ❱❡r✐✜q✉❡ q✉❡ ♦ P❛r❜❡❧♦ ❝♦♥str✉í❞♦ ♥♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r s❛t✐s❢❛③ ❛s ♣r♦✲ ♣r✐❡❞❛❞❡s ❞❡ t❛♥❣ê♥❝✐❛ q✉❡ ♦ ❞❡✜♥❡♠✳ ✭❉✐❝❛✿ ✈❡r✐✜q✉❡ s❡ ❛s P❛rá❜♦❧❛s h1 (x) ❡ h2 (x) sã♦ t❛♥❣❡♥t❡s à P❛rá❜♦❧❛ s✉♣❡r✐♦r g(x) ♥❛s ❝ús♣✐❞❡s C1 ❡ C3 ✱ s❡♥❞♦ C2 ❛ ❝ús♣✐❞❡ ♠é❞✐❛✮✳ ❊①❡r❝í❝✐♦ ✸✳✷✳ ✸✳✷✳ ✺✾ ➪❘❊❆ ❙❖❇ ❉❆ ❈❯❘❱❆ ❙♦❧✉çã♦ P❛r❛ q✉❡ ❛s três P❛rá❜♦❧❛s ❡♥❝♦♥tr❛❞❛s ♥♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r ❢♦r♠❡♠ ✉♠ P❛r❜❡❧♦✱ ❛ ♣❛rá✲ ❜♦❧❛ s✉♣❡r✐♦r✱ ❞❡s❝r✐t❛ ♣❡❧❛ ❢✉♥çã♦ g(x) ❞❡✈❡ s❡r t❛♥❣❡♥t❡ à h1 (x) ♥❛ ❝ús♣✐❞❡ C1 = (0, 0) ❡ t❛♥❣❡♥t❡ à h2 (x) ♥❛ ❝ús♣✐❞❡ C3 (1, 0)✳ ❙❡♥❞♦ ❛ss✐♠✱ s❡❣✉❡ q✉❡✿ g(x) = −x2 + x ⇒ g ′ (x) = −2x + 1, h1 (x) = − ❡ h2 (x) = − x2 2x + x ⇒ h′1 (x) = − + 1, 0, 3 0, 3 (x − 0, 3)2 2(x − 0, 3) + x − 0, 3 ⇒ h′2 (x) − + 1. 0, 7 0, 7 ◆♦ ♣♦♥t♦ C1 = (0, 0)✱ t❡♠♦s q✉❡✿ g ′ (0) = −2 · 0 + 1 = 1 ❡ h′1 (0) = − 2·0 + 1 = 1. 0, 3 ❊ ♥♦ ♣♦♥t♦ C2 = (1, 0)✱ t❡♠♦s q✉❡✿ g ′ (1) = −2 · 1 + 1 = −1 ❡ h′2 (1) = − 2 · (1 − 0, 3) + 1 = −1. 0, 7 P♦rt❛♥t♦✱ t❡♠♦s q✉❡ ❛ ✜❣✉r❛ ❢♦r♠❛❞❛ ♥♦ ❡①❡r❝í❝✐♦ ✶ é ✉♠ P❛r❜❡❧♦✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✹✳✷✳ ✸✳✷ ➪r❡❛ s♦❜ ❞❛ ❝✉r✈❛ ❖ ♣ró①✐♠♦ ❡①❡r❝í❝✐♦ ♥♦s ❛♣r❡s❡♥t❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ■♥t❡❣r❛❧ ❉❡✜♥✐❞❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s✳ ❈❛❧❝✉❧❡ ✉♠❛ ❛♣r♦①♠❛çã♦ ♣❛r❛ ❛ ár❡❛ s♦❜ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f (x) = −x + 5x − 6✳ ❋❛ç❛ ✐ss♦ ❝❛❧❝✉❧❛♥❞♦ ❛s s♦♠❛s ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦✱ ❡♠ ❞♦✐s ❝❛s♦s✿ ❊①❡r❝í❝✐♦ ✸✳✸✳ 2 ✭❛✮ ❝♦♠ ✺ r❡tâ♥❣✉❧♦s❀ ✻✵ ❈❆P❮❚❯▲❖ ✸✳ ❆❚■❱■❉❆❉❊❙ ✭❜✮ ❝♦♠ ✶✵ r❡tâ♥❣✉❧♦s✳ ❈♦♠♣❛r❡ ♦s r❡s✉❧t❛❞♦s ❝♦♠ ♦ ✈❛❧♦r ❡①❛t♦ ✭❈❛❧❝✉❧❛♥❞♦ ❛ ■♥t❡❣r❛❧ ❉❡✜♥✐❞❛✮✳ ❙♦❧✉çã♦✿ ✭❛✮ ❆s s♦❧✉çõ❡s ❞❡ f (x) sã♦ ✷ ❡ ✸✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ✐♥t❡r✈❛❧♦ (2, 3) s❡♥❞♦ ❞✐✈✐❞✐❞♦ ❡♠ ∆x = 0, 2 ❡ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ✶✵ r❡tâ♥❣✉❧♦s ❡ ∆x = 0, 1✳ ✺ ♣❛rt❡s ✐❣✉❛❧s✱ t❡♠♦s q✉❡ ❋✐❣✉r❛ ✸✳✸✿ ❈á❧❝✉❧♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ár❡❛ ♣♦r ❢❛❧t❛ ❝♦♠ ♥❂✺✳ ❆ s♦♠❛ ♣♦r ❢❛❧t❛ é ❞❛❞❛ ♣♦r✿ 5 X f (ti )∆t = f ((t0 )∆t + f (t1 )∆t + f (t2 ∆t + f (t3 )∆t + f (t4 )∆t = i=0 = 0 · 0, 2 + 0, 16 · 0, 2 + 0, 24 · 0, 2 + 0, 16 · 0, 2 + 0 · 0, 2 = 0, 112. ❋✐❣✉r❛ ✸✳✹✿ ❈á❧❝✉❧♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ár❡❛ ♣♦r ❡①❝❡ss♦ ❝♦♠ ♥❂✺✳ ❆❣♦r❛✱ ❝❛❧❝✉❧❛♥❞♦ ❛ s♦♠❛ ♣♦r ❡①❝❡ss♦✱ t❡♠♦s q✉❡✿ 5 X f (ti )∆t = f (t1 )∆t + f (t2 )∆t + f (t3 )∆t + f (t4 )∆t + f (t5 )∆t = i=1 = 0, 16 · 0, 2 + 0, 24 · 0, 2 + 0, 25 · 0, 2 + 0, 24 · 0, 2 + 0, 16 · 0, 2 = 0, 210. ✸✳✷✳ ✻✶ ➪❘❊❆ ❙❖❇ ❉❆ ❈❯❘❱❆ ❈♦♠♦ ♣♦❞❡♠♦s ✈❡r✱ ❛ s♦♠❛ ♣♦r ❡①❝❡ss♦ é ♠❛✐♦r ❞♦ q✉❡ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❝♦♠✲ ♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❛ ❝✉r✈❛ ❡ ♦ ❡✐①♦ x ❡ ❛ s♦♠❛ ♣♦r ❢❛❧t❛ é ♠❡♥♦r✳ P♦rt❛♥t♦✱ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❛ ❝✉r✈❛ ❡♥tr❡ ✵✱✶✶✷ ❡ ✵✱✷✶✵✿ 0, 112 < Z 3 2 f (x) = −x2 + 5x − 6 ❡ ♦ ❡✐①♦ x ❡stá f (x)dx < 0, 210. ✭❜✮ ❈♦♠ ✶✵ r❡tâ♥❣✉❧♦s ✐❣✉❛✐s✱ t❡♠♦s q✉❡ ❛ s♦♠❛ ♣♦r ❡①❝❡ss♦ é ❞❛❞❛ ♣♦r✿ 10 X f (ti )∆t = f (t1 )∆t + f (t2 )∆t + f (t3 )∆t + f (t4 )∆t + f (t5 )∆t + f (t6 )∆t+ i=1 +f (t7 )∆t + f (t8 )∆t + f (t9 )∆t + f (t10 )∆t = 0, 09 · 0, 1 + 0, 16 · 0, 1 + 0, 21 · 0, 1+ +0, 24·0, 1+0, 25·0, 1+0, 25·0, 1+0, 24·0, 1+0, 21·0, 1+0, 16·0, 1+0, 09·0, 1 = 0, 19. ❋✐❣✉r❛ ✸✳✺✿ ❈á❧❝✉❧♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ár❡❛ ♣♦r ❡①❝❡ss♦ ❝♦♠ ♥❂✶✵✳ ❊ ❛ s♦♠❛ ♣♦r ❢❛❧t❛ é ❞❛❞❛ ♣♦r✿ 10 X f (ti )∆t = f (t1 )∆t + f (t2 )∆t + f (t3 )∆t + f (t4 )∆t + f (t5 )∆t + f (t6 )∆t+ i=1 + f (t7 )∆t + f (t8 )∆t + f (t9 )∆t + f (t10 )∆t = 0 · 0, 1 + 0, 09 · 0, 1 + 0, 16 · 0, 1+ + 0, 21 · 0, 1 + 0, 24 · 0, 1 + 0, 24 · 0, 1 + 0, 21 · 0, 1 + 0, 16 · 0, 1 + 0, 09 · 0, 1+ + 0 · 0, 1 = 0, 14. ✻✷ ❈❆P❮❚❯▲❖ ✸✳ ❆❚■❱■❉❆❉❊❙ ❋✐❣✉r❛ ✸✳✻✿ ❈á❧❝✉❧♦ ❛♣r♦①✐♠❛❞♦ ❞❛ ár❡❛ ♣♦r ❢❛❧t❛ ❝♦♠ ♥❂✶✵✳ ❆ss✐♠ ❝♦♠♦ ♥♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ♣♦❞❡♠♦s ✈❡r q✉❡✱ ❛ s♦♠❛ ♣♦r ❡①❝❡ss♦ é ♠❛✐♦r ❞♦ q✉❡ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❛ ❝✉r✈❛ ❡ ♦ ❡✐①♦ x ❡ ❛ s♦♠❛ ♣♦r ❢❛❧t❛ é ♠❡♥♦r✳ P♦rt❛♥t♦✱ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❛ ❝✉r✈❛ f (x) = −x2 + 5x − 6 ❡ ♦ ❡✐①♦ x ❡stá ❡♥tr❡ ✵✱✶✹ ❡ ✵✱✶✾✿ 0, 14 < Z 3 2 f (x)dx < 0, 19. ❈♦♠♣❛r❛♥❞♦ ♦s ❞♦✐s r❡s✉❧t❛❞♦s ❛♥t❡r✐♦r❡s ❝♦♠ ❛ ✐♥t❡❣r❛❧ ❞❡✜♥✐❞❛✿ ñ ô 3 x3 5x2 − 6x (−x + 5x − 6)dx = − + f (x)dx = 3 2 2 2 2 ñ ô ñ ô 3 2 3 2 3 5.3 2 5.2 = − + − 6.3 − − + − 6.2 ≈ −4, 5 − (−4, 67) = 0, 17, 3 2 3 2 Z 3 Z 3 2 ♣♦❞❡♠♦s ♥♦t❛r q✉❡ q✉❛♥t♦ ♠❛✐♦r ❢♦r ♦ ♥ú♠❡r♦ ❞❡ r❡tâ♥❣✉❧♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❛ ár❡❛✱ ♠❛✐s ♣ró①✐♠♦ ❞♦ ✈❛❧♦r ❡①❛t♦ s❡rã♦ ♥♦ss♦s ❝á❧❝✉❧♦s✱ ♦✉ s❡❥❛✱ ♠❛✐s ♣ró①✐♠♦ ❞❡ ✵✱✶✼✳ ❈♦♠♦ ❡♠ ❬✸❪✱ ●❡♦❣❡❜r❛ é ✉♠ s♦❢t✇❛r❡ ❞❡ ▼❛t❡♠át✐❝❛ ❞✐♥â♠✐❝❛ ❣r❛t✉✐t♦ ❡ ♠✉❧t✐✲♣❧❛t❛❢♦r♠❛ ♣❛r❛ t♦❞♦s ♦s ♥í✈❡✐s ❞❡ ❡♥s✐♥♦✱ q✉❡ ❝♦♠❜✐♥❛ ❣❡♦♠❡tr✐❛✱ á❧❣❡❜r❛✱ t❛❜❡❧❛s✱ ❣rá✜❝♦s✱ ❡st❛tís✲ t✐❝❛ ❡ ❝á❧❝✉❧♦ ❡♠ ✉♠ ú♥✐❝♦ s✐st❡♠❛✳ ❚♦❞❛s ❛s ✜❣✉r❛s ✉t✐❧✐③❛❞❛s ♥♦ tr❛❜❛❧❤♦ ❢♦r❛♠ ❝♦♥str✉í❞❛s ✉t✐❧✐③❛♥❞♦ ❡st❡ s♦❢t✇❛r❡✱ q✉❡ s❡ ♠♦str♦✉ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛✳ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ❬✶❪ ❆▲❱❆❘❊◆●❆✱ tr✐❜✉✐çã♦ ▼❛✉r♦ ♣❛r❛ ♦ ▲♦♣❡s✳ ❖ ♠ét♦❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❞♦ ❡①❛✉stã♦ ❡ s✉❛ ❝♦♥❤❡❝✐♠❡♥t♦ ❝♦♥✲ ♠❛t❡♠á✲ ❯♥✐✈❡rs✐❞❛❞❡ ❈❛tó❧✐❝❛ ❞❡ ❇r❛sí❧✐❛✱ ✷✵✵✻✳ ❉✐s♣♦♥í✈❡❧ ❡♠✿ ❁✇✇✇✳✉❝❜✳❜r✴s✐t❡s✴✶✵✵✴✶✵✸✴❚❈❈✴✶✷✵✵✻✴▼❛✉r♦▲♦♣❡s❆❧✈❛r❡♥❣❛✳♣❞❢❃✳ ❆❝❡ss♦ ✵✼ ❞❡ ♠❛✐♦ ❞❡ ✷✵✶✹✳ t✐❝♦✳ ❬✷❪ ❇❖●❖▼❖▲◆❨✱ ❆❧❡①❛♥❞❡r✳ ❆r❜❡❧♦s ✲ t❤❡ ❙❤♦❡♠❛❦❡r✬s ❑♥✐❢❡✳ ❉✐s♣♦♥✁ ✈❡❧ ❡♠✿ ❁❤tt♣✿✴✴✇✇✇✳❝✉t✲t❤❡✲❦♥♦t✳♦r❣✴♣r♦♦❢s✴❛r❜❡❧♦s✳s❤t♠❧❃✳ ❆❝❡ss♦ ❡♠✿ ✵✼ ❞❡ ♠❛✐♦ ❞❡ ✷✵✶✹✳ ❬✸❪ ●❊❖●❊❇❘❆✳ ❖ q✉❡ é ♦ ●❡♦❣❡❜r❛❄✳ ❉✐s♣♦♥í✈❡❧ ❡♠✿ ❁❤tt♣✿✴✴✇✇✇✳❣❡♦❣❡❜r❛✳♦r❣✴❝♠s✴♣t❴❇❘✴✐♥❢♦✴✶✸✲✇❤❛t✲✐s✲❣❡♦❣❡❜r❛❧❃✳ ❆❝❡ss♦ ❡♠✿ ✵✶ ❞❡ s❡t❡♠❜r♦ ❞❡ ✷✵✶✹ ❬✹❪ ●❯■❉❖❘■❩❩■✱ ❍❛♠✐❧t♦♥ ▲✉✐③✳ ❏❛♥❡✐r♦✱ ✶✾✾✼✳ ❬✺❪ ●❯■❉❖❘■❩❩■✱ ❍❛♠✐❧t♦♥ ▲✉✐③✳ ❏❛♥❡✐r♦✱ ✶✾✾✼✳ ❯♠ ❈✉rs♦ ❞❡ ❈á❧❝✉❧♦✳ ❱✳ ✶✱ ❊❞✳ ✸✱ ▲❚❈✿ ❘✐♦ ❞❡ ❯♠ ❈✉rs♦ ❞❡ ❈á❧❝✉❧♦✳ ❱✳ ✷✱ ❊❞✳ ✸✱ ▲❚❈✿ ❘✐♦ ❞❡ ❬✻❪ ❖▲▲❊❘✲▼❆❘❈➱◆✱ ❆♥t♦♥✐♦ ▼✳ ❚❤❡ ❢✲❜❡❧♦s✳✱ ❋♦r✉♠ ●❡♦♠❡tr✐❝♦r✉♠✱ ❈❡♥tr♦ ❯♥✐✈❡r✲ s✐t❛r✐♦ ❞❡ ❧❛ ❉❡❢❡♥s❛✱ ❩❛r❛❣♦③❛✱ ❙♣❛✐♥✱ ✷✵✶✸✳ ❬✼❪ ❙❖◆❉❖❲✱ ❏♦♥❛t❤❛♥✳ ❨♦r❦✱ ✷✵✶✸✳ ✱ ❆♠❡r✱ ◆❡✇ ❚❤❡ ♣❛r❜❡❧♦s✱ ❛ ♣❛r❛❜♦❧✐❝ ❛♥❛❧♦❣ ♦❢ ❞❡ ❛r❜❡❧♦✳ ❬✽❪ ❚❙❯❑❊❘▼❆◆✱ ❊♠♠❛♥✉❡❧✳ ❙♦❧✉t✐♦♥ ♦❢ ❙♦♥❞♦✇✬s ♣r♦❜❧❡♠✿ ❛ s②♥t❤❡t✐❝ ♣r♦♦❢ ♦❢ t❤❡ t❛♥❣❡♥❝② ♣r♦♣❡rt② ♦❢ t❤❡ ♣❛r❜❡❧♦s✳ ❙t❛♥❢♦r❞ ❯♥✐✈❡rs✐t②✱ ✷✵✶✸ ❬✾❪ ❙❨▲❱❊❙❚❊❘✱ ❘❡❡s❡❀ ❙❖◆❉❖❲✱ ❏♦♥❛t❤❛♥✳ ❯♥✐✈❡rs❛❧ P❛r❛❜♦❧✐❝ ❈♦♥st❛♥t ❉✐s✲ ♣♦♥í✈❡❧ ❡♠✿ ❁❤tt♣✿✴✴♠❛t❤✇♦r❧❞✳✇♦❧❢r❛♠✳❝♦♠✴❯♥✐✈❡rs❛❧P❛r❛❜♦❧✐❝❈♦♥st❛♥t✳❤t♠❧❃✳ ❆❝❡ss♦ ❡♠✿ ✵✼ ❞❡ ♠❛✐♦ ❞❡ ✷✵✶✹✳ ✻✸
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