Simões, Alexandre Calligaris
Calculando área sob gráficos de funções / Alexandre Calligaris Simões. - Rio Claro, 2014
47 f. : il., figs., gráfs., tabs.
Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas
Orientador: Renata Zotin Gomes de Oliveira
1. Cálculo. 2. Geometria dinâmica. 3. Soma de Riemann. 4. Integral definida. I.Título.
517 S593c
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
❆❧❡①❛♥❞r❡ ❈❛❧❧✐❣❛r✐s ❙✐♠õ❡s
❈❛❧❝✉❧❛♥❞♦ ár❡❛ s♦❜ ❣rá❢✐❝♦s ❞❡ ❢✉♥çõ❡s
❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿
Pr♦❢❛✳ ❉r❛✳ ❘❡♥❛t❛ ❩♦t✐♥ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛ ❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❙✐❞✐♥❡✐❛ ❇❛rr♦③♦
■♥st✐t✉t♦ ❞❡ ◗✉í♠✐❝❛ ❞❛ ❯◆❊❙P✴❆r❛r❛q✉❛r❛
Pr♦❢❛✳ ❉r❛✳ ▼❛rt❛ ❈✐❧❡♥❡ ●❛❞♦tt✐ ❉▼ ✲ ■●❈❊✴❯◆❊❙P✴❘✐♦ ❈❧❛r♦
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡ à ❉❡✉s✱ ♣❡❧❛ ❧✉③ q✉❡ s❡♠♣r❡ ✐❧✉♠✐♥❛ ♥♦ss♦ ❝❛♠✐♥❤♦ ❡ ♥✉♥❝❛ ♥♦s ❛❜❛♥❞♦♥❛r✳
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ♠✐♥❤❛ ❡s♣♦s❛✱ ♠✐♥❤❛ ♠ã❡ ❡ ♠❡✉ ♣❛✐ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛♠ ❡ t♦r❝❡♠ ♣❛r❛ ♦ ♠❡✉ s✉❝❡ss♦✳
❆♦s ♣r♦❢❡ss♦r❡s ❡ ❝♦♦r❞❡♥❛❞♦r❡s ❞♦ ▼❡str❛❞♦ q✉❡ ♥♦s tr❛♥s♠✐t✐r❛♠ s❡✉s ❝♦♥❤❡❝✐✲ ♠❡♥t♦s ❝♦♠ t❛♥t❛ ❞❡❞✐❝❛çã♦✳
➚s ♣r♦❢❡ss♦r❛s ❞❛ ❜❛♥❝❛ Pr♦❢❛✳ ❉r❛✳ ❙✐❞✐♥❡✐❛ ❇❛rr♦③♦ ❡ Pr♦❢❛✳ ❉r❛✳ ▼❛rt❛ ❈✐❧❡♥❡ ●❛❞♦tt✐ ♣❡❧❛s s✉❣❡stõ❡s ❡♥r✐q✉❡❝❡❞♦r❛s ❛♦ ♠❡✉ tr❛❜❛❧❤♦✳
➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ Pr♦❢❛✳ ❉r❛✳ ❘❡♥❛t❛ ❩♦t✐♥ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛ ♣❡❧❛ ❞❡❞✐❝❛çã♦✱ ♣❛❝✐ê♥❝✐❛ ❡ s❛❜❡❞♦r✐❛ ❝♦♠ q✉❡ ❝♦♥❞✉③✐✉ ❡st❡ tr❛❜❛❧❤♦✳
➚ ❈❆P❊❙✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❛♣r❡s❡♥t❛r ✉♠❛ s✉❣❡stã♦ ❞❡ ❝♦♠♦ ❝❛❧✲ ❝✉❧❛r ár❡❛s ❞❡❧✐♠✐t❛❞❛s ♣♦r ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s ♣❛r❛ ❡st✉❞❛♥t❡s ❞❡ ❊♥s✐♥♦ ▼é❞✐♦✱ ✉t✐✲ ❧✐③❛♥❞♦ ♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛✳ ❙❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❛t✐✈✐❞❛❞❡s q✉❡ ❡①❡♠♣❧✐✜❝❛♠ ♦ ❞❡✲ s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♣r♦♣♦st❛✱ q✉❡ ✈❛♦ ❛❧é♠ ❞♦ q✉❡✱ ♥♦r♠❛❧♠❡♥t❡✱ ♦s ❧✐✈r♦s ❞✐❞át✐❝♦s s✉❣❡r❡♠✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ❤❛s ❛s ♠❛✐♥ ♦❜❥❡❝t✐✈❡ t♦ ♣r❡s❡♥t ❛ s✉❣❣❡st✐♦♥ ❤♦✇ t♦ ❝❛❧❝✉❧❛t❡ ❛r❡❛s ❜♦✉♥❞❡❞ ❜② ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s t♦ ❍✐❣❤ ❙❝❤♦♦❧ st✉❞❡♥ts✱ ✉s✐♥❣ t❤❡ ●❡♦❣❡❜r❛ s♦❢t✇❛r❡✳ ■t ✇✐❧❧ ❜❡ ♣r❡s❡♥t ❛❝t✐✈✐t✐❡s t♦ ❡①❡♠♣❧✐❢② t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ♣r♦♣♦s❛❧✱ t❤❛t ❣♦ ❜❡②♦♥❞ ✇❤❛t ♥♦r♠❛❧❧② t❡①t❜♦♦❦s s✉❣❣❡st✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ❘❡❧❛çã♦ ❧❛❞♦✲❞✐â♠❡tr♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷ P♦❧í❣♦♥♦ ❞✐✈✐❞✐❞♦ ❡♠ tr✐â♥❣✉❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✸ ➪r❡❛ ❞♦ ❝ír❝✉❧♦ ♣♦r ❡①❛✉stã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✹ ❆♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❬✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✺ ❆♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❬✽❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✸✳✶ P♦❧í❣♦♥♦ ✐rr❡❣✉❧❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✷ P♦❧í❣♦♥♦ ✐rr❡❣✉❧❛r ♣❛rt✐❝✐♦♥❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✸ ❆♣r♦①✐♠❛çã♦ ❞❛ ár❡❛ ♣♦r ❢❛❧t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✹ ❆♣r♦①✐♠❛çã♦ ❞❛ ár❡❛ ♣♦r ❡①❝❡ss♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✺ ❊①❡♠♣❧♦ ❞❡ ❢♦rç❛ ❝♦♥st❛♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✻ ❊①❡♠♣❧♦ ❞❡ ❢♦rç❛ ✈❛r✐á✈❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✼ ➪r❡❛ ❞♦ s❡♠✐❝ír❝✉❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✽ ❆♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❛ ár❡❛ ❞♦ s❡♠✐❝ír❝✉❧♦ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✷✸
✸✳✾ ❆♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❛ ár❡❛ ❞♦ s❡♠✐❝ír❝✉❧♦ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✷✹
✸✳✶✵ ❊①❡♠♣❧♦ ❞❡ ❢♦rç❛ ✈❛r✐á✈❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✶✶ ➪r❡❛ ♣♦r ❢❛❧t❛ ♣❛r❛ ❛ r❡❣✐ã♦ ❞❛ ❋✐❣✉r❛ ✸✳✶✵✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✳ ✷✺
✸✳✶✷ ➪r❡❛ ♣♦r ❡①❝❡ss♦ ♣❛r❛ ❛ r❡❣✐ã♦ ❞❛ ❋✐❣✉r❛ ✸✳✶✵✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✷✻
✸✳✶✸ ❘❡❣✐ã♦ ❞♦ ❊①❡♠♣❧♦ ✹✳✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✶✹ ❆♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❛ ❋✐❣✉r❛ ✸✳✶✸✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✶✺ ❆♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❛ ❋✐❣✉r❛ ✸✳✶✸✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✳ ✳ ✳ ✷✽
✸✳✶✻ ❆♣r♦①✐♠❛çã♦✱ ❝♦♠ ♦ ❡①tr❡♠♦ ❡sq✉❡r❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛✱ ❞❛ ❋✐❣✉r❛ ✸✳✶✸✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✶✼ ●rá✜❝♦ ❞❡f(x) = −(x−1)(x−3)(x−5)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸✳✶✽ ➪r❡❛ s♦❜ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❞❛ ❋✐❣✉r❛ ✸✳✶✼✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✳ ✸✵
✸✳✶✾ ●rá✜❝♦ ❞❡f(x) ❡ g(x)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✷✵ ➪r❡❛ ❡♥tr❡ ♦s ❣rá✜❝♦s ❞❛s ❢✉♥çõ❡s ❞❛ ❋✐❣✉r❛ ✸✳✶✾✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✸✳✷✶ ➪r❡❛ ❞❡ tr❛❜❛❧❤♦ ❞♦ ●❡♦❣❡❜r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✹✳✶ ❖ ❝❛s♦ f(x)≥0. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✹✳✹ ❘❡❣✐ã♦ A ❞❡❧✐♠✐t❛❞❛ ♣♦rf(x) =−x2
+ 3x. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✹✳✺ ❖ ❝❛s♦ f(x)≤0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✹✳✻ ❖ ❝❛s♦ ♦♥❞❡ f tr♦❝❛ ❞❡ s✐♥❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✸✳✶ ❱❛❧♦r❡s ❞❛s ❛♣r♦①✐♠❛çõ❡s ♣♦r ❢❛❧t❛✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✸✳✷ ❱❛❧♦r❡s ❞❛s ❛♣r♦①✐♠❛çõ❡s ♣♦r ❡①❝❡ss♦✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✳ ✳ ✳ ✳ ✳ ✷✹
✸✳✸ ❱❛❧♦r❡s ❞❛ ár❡❛ ♣♦r ❢❛❧t❛✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸✳✹ ❱❛❧♦r❡s ❞❛ ár❡❛ ♣♦r ❡①❝❡ss♦ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✸✳✺ ❱❛❧♦r❡s ❞❛ ár❡❛ ♣♦r ❢❛❧t❛✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✻ ❱❛❧♦r❡s ❞❛ ár❡❛ ♣♦r ❡①❝❡ss♦✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✼ ❱❛❧♦r❡s ❞❛ ár❡❛ ❝♦♠ ♦ ❡①tr❡♠♦ ❡sq✉❡r❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✽ ❱❛❧♦r ❞❛ ár❡❛✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✵
✷ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛ ❡ ♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✶✶
✷✳✶ ❖ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♥❛ ❤✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷ ❖ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷✳✶ ▼❛t❡♠át✐❝❛ ✲ ❈✐ê♥❝✐❛ ❡ ❆♣❧✐❝❛çõ❡s ✲ ●❡❧s♦♥ ■❡③③✐ ❡ ♦✉tr♦s ✭✷✵✶✵✮ ✶✹ ✷✳✷✳✷ ▼❛t❡♠át✐❝❛ ✲ ❉❛♥t❡ ✭✷✵✵✽✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷✳✸ ❈♦♥❡①õ❡s ❝♦♠ ❛ ▼❛t❡♠át✐❝❛ ✲ ❏✉❧✐❛♥❡ ▼❛ts✉❜❛r❛ ❇❛rr♦s♦ ✭❡❞✐✲
t♦r❛ r❡s♣♦♥sá✈❡❧✮ ✭✷✵✶✵✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸ ❈á❧❝✉❧♦ ❞❡ ár❡❛ ❞❡ r❡❣✐õ❡s ❞❡❧✐♠✐t❛❞❛s ♣♦r ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s ✲ ✉♠❛
♣r♦♣♦st❛ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ✶✽
✸✳✶ ❈❛❧❝✉❧❛♥❞♦ ár❡❛ ❞❡ ✜❣✉r❛s ♣♦❧✐❣♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✷ ❈❛❧❝✉❧❛♥❞♦ ár❡❛ ❞❡ ✜❣✉r❛s ♥ã♦ ♣♦❧✐❣♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✸ ➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲ ✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛ ✷✶ ✸✳✹ ●❡♦❣❡❜r❛ ✲ ❙♦❢t✇❛r❡ ❉✐♥â♠✐❝♦ ❞❡ ▼❛t❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✹ ❆ ■♥t❡❣r❛❧ ❉❡✜♥✐❞❛ ♥♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ✸✽
✹✳✵✳✶ ❈á❧❝✉❧♦ ❞❡ ➪r❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✺ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✹✹
✶ ■♥tr♦❞✉çã♦
❊st❡ tr❛❜❛❧❤♦ t❡♠ ♦ ♣r♦♣ós✐t♦ ❞❡ ❛♣r❡s❡♥t❛r ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ❞❡ ❝♦♠♦ tr❛❜❛❧❤❛r✱ ❝♦♠ ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛ ❞❡ r❡❣✐õ❡s ❞❡❧✐♠✐t❛❞❛s ♣♦r ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s✱ ✉t✐❧✐③❛♥❞♦ ♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛✱ q✉❡ ♣❡r♠✐t❡ ✉♠ ❡st✉❞♦ ♠❛✐s ❞✐♥â♠✐❝♦ ❡ ❛♠♣❧✐❛ ♦ ♣♦❞❡r ❞❡ ♦❜s❡r✈❛çã♦ ❡ ❝♦♠♣❛r❛çã♦ ❞❡ r❡s✉❧t❛❞♦s✳
◆♦ ❝❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥t❛♠♦s ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ s♦❜r❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ●❡✲ ♦♠❡tr✐❛ ❡ ❞♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❡ ❝♦♠♦ ❛❧❣✉♥s ❧✐✈r♦s ❞✐❞át✐❝♦s ✐♥tr♦❞✉③❡♠ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❞❡ ❞❡t❡r♠✐♥❛❞❛s r❡❣✐õ❡s✳
◆♦ ❝❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛ ❞❡ ♣♦❧í❣♦♥♦s✱ ✉♠❛ ❡str❛té❣✐❛ ❞❡ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ s❡ ❡♥❝♦♥tr❛r ár❡❛ ❞❡ ✜❣✉r❛ ♥ã♦ ♣♦❧✐❣♦♥❛❧ ❡ ❝♦❧♦❝❛♠♦s ❛ s✐t✉❛çã♦ ❞❡ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣❡❧♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ♥✉♠ ❞❡t❡r♠✐♥❛❞♦ ✐♥t❡r✲ ✈❛❧♦✳ ❊st❛ ú❧t✐♠❛ s❡rá ♦ t❡♠❛ ♣r✐♥❝✐♣❛❧ ❞❛ ♣r♦♣♦st❛ ❞❡ ❛t✐✈✐❞❛❞❡ ❞✐❢❡r❡♥❝✐❛❞❛ ♣❛r❛ s❡ tr❛❜❛❧❤❛r ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❖s ❝♦♥❝❡✐t♦s ❞❡ ❙♦♠❛ ❞❡ ❘✐❡♠❛♥♥ ❡ ■♥t❡❣r❛❧ ❉❡✜♥✐❞❛✱ ♥❡❝❡ssár✐♦s ❝♦♠♦ s✉♣♦rt❡ t❡ór✐❝♦ ❛♦ ♣r♦❢❡ss♦r✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ♥♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦✳
✷ ❯♠ ♣♦✉❝♦ ❞❡ ❤✐stór✐❛ ❡ ♦s ❧✐✈r♦s
❞✐❞át✐❝♦s
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ♣❛ss❛❣❡♥s ❤✐stór✐❝❛s s♦❜r❡ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❝♦♠♦ ♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❛❜♦r❞❛♠ ❡st❡ ❝♦♥t❡ú❞♦✳
✷✳✶ ❖ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♥❛ ❤✐stór✐❛
❆ ●❡♦♠❡tr✐❛✱ s❡❣✉♥❞♦ r❡❣✐str♦s ❡♥❝♦♥tr❛❞♦s✱ t❡♠ s✉❛ ♦r✐❣❡♠ ♥❛ ❆♥t✐❣✉✐❞❛❞❡✱ ♣♦r ✈♦❧t❛ ❞❡ ✷✵✵✵ ❛✳❈✳✳ ■♥✐❝✐❛❧♠❡♥t❡ ❞❡ ❢♦r♠❛ ❡♠♣ír✐❝❛✱ ♦♥❞❡ ❢♦✐ s❡ ❞❡s❡♥✈♦❧✈❡♥❞♦ ♣❛r❛ ❛t❡♥❞❡r ♥❡❝❡ss✐❞❛❞❡s ♣rát✐❝❛s ❞♦ s❡r ❤✉♠❛♥♦✱ ❝♦♠♣r❡❡♥❞❡r ❡ ♠❛♥✐♣✉❧❛r ♠❡❧❤♦r s❡✉ ♠❡✐♦✳ ❉❛í ❛ ♦r✐❣❡♠ ❞♦ ♥♦♠❡ ●❡♦♠❡tr✐❛✱ ❞❡r✐✈❛❞❛ ❞❛ ♣❛❧❛✈r❛ ❡♠ ❣r❡❣♦ ❣❡♦♠❡tr❡✐♥ q✉❡ s✐❣♥✐✜❝❛✿ ♠❡❞✐çã♦ ❞❡ t❡rr❛✳ ❈♦♥s✐❞❡r❛❞♦ ♦ ✧♣❛✐ ❞❛ ❤✐stór✐❛✧✱ ❍❡ró❞♦t♦ ✭sé❝✳ ❱ ❛✳❈✳✮ ❞❡s❝♦❜r✐✉ s♦❜r❡ ♦ ❛♥t✐❣♦ ❊❣✐t♦ ♦ s❡❣✉✐♥t❡ r❡❧❛t♦ ❬✷❪✿ ✧❉✐ss❡r❛♠✲♠❡ ❛✐♥❞❛ ♦s s❛❝❡r❞♦t❡s q✉❡ ❙❡sóstr✐s r❡❛❧✐③♦✉ ❛ ♣❛rt✐❧❤❛ ❞❛s t❡rr❛s✱ ❝♦♥❝❡❞❡♥❞♦ ❛ ❝❛❞❛ ❡❣í♣❝✐♦ ✉♠❛ ♣♦rçã♦ ✐❣✉❛❧✱ ❝♦♠ ❛ ❝♦♥❞✐çã♦ ❞❡ ❧❤❡ s❡r ♣❛❣♦ t♦❞♦s ♦s ❛♥♦s ❝❡rt♦ tr✐❜✉t♦✳ ❙❡ ♦ r✐♦ ❝❛rr❡❣❛✈❛ ❛❧❣✉♠❛ ♣❛rt❡ ❞♦ ❧♦t❡ ❞❡ ❛❧❣✉é♠✱ ♦ ♣r❡❥✉❞✐❝❛❞♦ ✐❛ ♣r♦❝✉r❛r ♦ r❡✐ ❡ ❡①♣♦r✲❧❤❡ ♦ ❛❝♦♥t❡❝✐❞♦✳ ❖ s♦❜❡r❛♥♦ ❡♥✈✐❛✈❛ ❛❣r✐♠❡♥s♦r❡s ❛♦ ❧♦❝❛❧ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ r❡❞✉çã♦ s♦❢r✐❞❛ ♣❡❧♦ ❧♦t❡✱ ♣❛ss❛♥❞♦ ♦ ❞♦♥♦ ❛ ♣❛❣❛r ✉♠ tr✐❜✉t♦ ♣r♦♣♦r❝✐♦♥❛❧ à ♣♦s✐çã♦ r❡st❛♥t❡✧✳
◆❛s ❝✐✈✐❧✐③❛çõ❡s ❛♥t✐❣❛s✱ ❢❛③❡r ♠❡❞✐çõ❡s ❞❡ t❡rr❡♥♦s ❞❡❧✐♠✐t❛♥❞♦ ♣r♦♣r✐❡❞❛❞❡s ♣❛r❛ ♦ ♣❧❛♥t✐♦ ❡r❛ t❛r❡❢❛ ❝♦rr✐q✉❡✐r❛ ❡ ❞❡ ♠✉✐t❛ ✐♠♣♦rtâ♥❝✐❛✳ ❊♥tr❡ ♦s ❡❣í♣❝✐♦s q✉❡ ❝♦♥✈✐✈✐❛♠ ❝♦♠ ❛s é♣♦❝❛s ❞❡ ❝❤❡✐❛s ❞♦ r✐♦ ◆✐❧♦✱ ❡r❛ ❝♦♥st❛♥t❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ r❡❢❛③❡r ❛s ♠❛r❝❛çõ❡s✳ ◗✉❛♥❞♦ ❛s á❣✉❛s ❡st❛✈❛♠ ❛❧t❛s✱ ♦ q✉❡ r❡st❛✈❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡✈✐❛ s❡r r❡♠❛r❝❛❞♦ ❡ s✉❛ ár❡❛ ❞❡t❡r♠✐♥❛❞❛✱ ♣♦✐s ♦ ✈❛❧♦r ❞♦ ✐♠♣♦st♦ ❡r❛ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ t❛♠❛♥❤♦ ❞♦ ❧♦t❡✳ ◗✉❛♥❞♦ ❛s á❣✉❛s ❜❛✐①❛✈❛♠ ♥♦✈❛♠❡♥t❡ ❛s ♠❛r❝❛çõ❡s ❡r❛♠ r❡❢❡✐t❛s✱ r❡❝♦♥st✐t✉✐♥❞♦ ♦ ❧♦t❡ ♦r✐❣✐♥❛❧✳
❈♦♠ ❡st❛ ♣rát✐❝❛✱ ♦s ❡❣í♣❝✐♦s ❛❞q✉✐r✐r❛♠ ✉♠❛ ❣r❛♥❞❡ ❤❛❜✐❧✐❞❛❞❡ ❡♠ ❞❡♠❛r❝❛r t❡rr❛s ❡ ❝♦♠ ✐ss♦✱ ❞❡ ❢♦r♠❛ ❡♠♣ír✐❝❛✱ ♠✉✐t♦s ♣r✐♥❝í♣✐♦s r❡❧❛t✐✈♦s às r❡t❛s✱ â♥❣✉❧♦s ❡ ♣♦❧í❣♦♥♦s ❢♦r❛♠ ❞❡s❝♦❜❡rt♦s ❡ ♣r❛t✐❝❛❞♦s✳ ❙❛❜✐❛♠ q✉❡ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡ r❡tâ♥❣✉❧♦ ❝♦♠ ♠❡s♠❛ ❜❛s❡ ❡ ❛❧t✉r❛ tê♠ ♠❡s♠❛ ár❡❛✳ ❈♦♥❤❡❝✐❛♠✱ t❛♠❜é♠✱ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❞♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❝♦♠♦ 32
+ 42
= 52✳
◆❡st❛ ❧✐♥❤❛ ♣r❛❣♠át✐❝❛ ❞♦ ♣♦✈♦ ❡❣í♣❝✐♦✱ ♣♦r ✐♥t❡r♠é❞✐♦ ❞❛ ♦❜s❡r✈❛çã♦ ❡ ❞❛ ❡①♣❡✲
❖ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♥❛ ❤✐stór✐❛ ✶✷
r✐♠❡♥t❛çã♦✱ ♦❜t✐✈❡r❛♠ ♠♦❞❡❧♦s ❣❡♦♠étr✐❝♦s ❛tr❛✈és ❞♦ r❛❝✐♦❝í♥✐♦ ✐♥❞✉t✐✈♦✳ ❉❡ ❢❛t♦✱ ❝♦♠ ❡st❛ ❛❝✉♠✉❧❛çã♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s s❡ t♦r♥❛r❛♠ ❛♣t♦s ❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ tr❛✲ ç❛❞♦ ❞❡ ❧✐♠✐t❡s✱ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡ ár❡❛s✱ ❞❡ ♣r♦❥❡t♦s ❛rq✉✐t❡tô♥✐❝♦s ❡ ❡♥❣❡♥❤❛r✐❛ ❞❡ ❝♦♥str✉çõ❡s✱ ❡t❝✳ P❛r❛ ✐❧✉str❛r ♠❡❧❤♦r✱ ❝✐t❛♠♦s ❛❧❣✉♠❛s ♣❛ss❛❣❡♥s✳ ❙❡❣✉♥❞♦ ❊✈❡s❬✶❪✿
• ❈♦♥s✐❞❡r❛✈❛♠ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❝♦♠♦ s❡♥❞♦ ❡q✉✐✈❛❧❡♥t❡ ❛ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦
❞❡ ❧❛❞♦ ✐❣✉❛❧ ❛ 8/9❞❡ s❡✉ ❞✐â♠❡tr♦ ✭❋✐❣✉r❛ ✷✳✶✮✳
❋✐❣✉r❛ ✷✳✶✿ ❘❡❧❛çã♦ ❧❛❞♦✲❞✐â♠❡tr♦✳
❆ss✐♠✱
A=l2
= 8
9d !2
= 8
92r !2 = 16 9 !2 r2 ,
s❡♥❞♦ ❛ r❛③ã♦
16 9
!2
≈3,16
❛ ❛♣r♦①✐♠❛çã♦ q✉❡ ❝♦♥s❡❣✉✐r❛♠✱ ♥❛ é♣♦❝❛✱ ❞♦ ♥ú♠❡r♦ π✳
• ❙❛❜✐❛♠ q✉❡ ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r é ❛ ♠❡t❛❞❡ ❞♦ ♣r♦❞✉t♦ ❞❛ ❜❛s❡ ♣❡❧❛
❛❧t✉r❛✳
• ❙❛❜✐❛♠ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞♦ tr♦♥❝♦ ❞❡ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ q✉❛❞r❛❞❛✳ ▼❡s♠♦ ♥ã♦
❛♣r❡s❡♥t❛♥❞♦ ✉♠❛ ❢ór♠✉❧❛ ❛♥❛❧ít✐❝❛ ♣❛r❛ ❡st❡ ❝á❧❝✉❧♦✱ ✉♠ ♣r♦❜❧❡♠❛ ❡♥❝♦♥tr❛❞♦ ♥✉♠ ♣❛♣✐r♦ ✭Pr♦❜❧❡♠❛ ✶✹ ❞♦ ♣❛♣✐r♦ ▼♦s❝♦✉ ❞❛t❛❞♦ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ♥♦ ❛♥♦ ✶✽✺✵ ❛✳❈✳✮ ♠♦str❛ ♦ ❞♦♠í♥✐♦ q✉❡ t✐♥❤❛♠ ♣❛r❛ ❡st❛ s✐t✉❛çã♦✳
❖ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♥❛ ❤✐stór✐❛ ✶✸
♣rát✐❝❛s✳ ▼❡s♠♦ ❥á ♣♦ss✉✐♥❞♦ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❛❧❣é❜r✐❝♦✱ ♥ã♦ s❡ ♣♦❞❡ ❝♦♥s✐❞❡r❛r ❝♦♠♦ ✉♠❛ ❝✐ê♥❝✐❛ ♦r❣❛♥✐③❛❞❛✳ ❖s ❣r❡❣♦s✱ ❛♦ ❝♦♥trár✐♦ ❞♦s ❡❣í♣❝✐♦s✱ ❛♣r❡❝✐❛✈❛♠ ❛ ●❡♦♠❡tr✐❛ ♥ã♦ ❛♣❡♥❛s ❡♠ ✈✐rt✉❞❡ ❞❡ s✉❛s ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s✱ ♠❛s ❞❡ s❡✉s ✐♥t❡r❡ss❡s t❡ór✐❝♦s✱ ❞❡s❡❥❛♥❞♦ ❝♦♠♣r❡❡♥❞❡r ❛ ♠❛tér✐❛ ♣♦r ❡❧❛ ♠❡s♠❛✱ ❡ ♥ã♦ ❡♠ t❡r♠♦s ❞❡ s✉❛ ✉t✐❧✐❞❛❞❡✳ ❆♦s ❣r❡❣♦s ♥ã♦ ❜❛st♦✉ ❛♣❡♥❛s ♦ ❝r✐tér✐♦ ❡♠♣ír✐❝♦✱ ♣r♦❝✉r❛r❛♠ ❡♥❝♦♥tr❛r ❞❡♠♦♥str❛çõ❡s ❞❡❞✉t✐✈❛s ❡ r✐❣♦r♦s❛s ❞❛s ❧❡✐s ❛❝❡r❝❛ ❞♦ ❡s♣❛ç♦✱ q✉❡ ❣♦✈❡r♥❛♠ ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s ❞❛ ●❡♦♠❡tr✐❛ ❬✸❪✳
◆❛ ●ré❝✐❛ ❛♥t✐❣❛✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✺✵✵ ❛✳❈✳✱ ❝❛❧❝✉❧❛✈❛♠ ár❡❛s ❞❡ q✉❛❧q✉❡r ♣♦❧í✲ ❣♦♥♦ ❞✐✈✐❞✐♥❞♦✲♦ ❡♠ tr✐â♥❣✉❧♦s ✭❋✐❣✉r❛ ✷✳✷✮ ❡ s♦♠❛♥❞♦ ❛ ár❡❛ ❞❡ ❝❛❞❛ ✉♠ ❞❡❧❡s✳
❋✐❣✉r❛ ✷✳✷✿ P♦❧í❣♦♥♦ ❞✐✈✐❞✐❞♦ ❡♠ tr✐â♥❣✉❧♦s✳
P❛r❛ ❛ ❋✐❣✉r❛ ✷✳✷ t❡♠♦s q✉❡ ❛ ár❡❛ A ❞♦ ♣♦❧í❣♦♥♦ ❞❛❞♦ é
A=A1+A2+A3+A4,
♦♥❞❡ Ai é ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ i✱ i= 1,2,3,4.
❈❛❧❝✉❧❛✈❛♠✱ t❛♠❜é♠✱ ár❡❛s ❞❡ ✜❣✉r❛s ❝✉r✈❛s✳ P❛r❛ ❡st❛s s✐t✉❛çõ❡s ✉s❛✈❛♠ ✉♠ ♠ét♦❞♦ ❝❤❛♠❛❞♦ ✧▼ét♦❞♦ ❞❛ ❊①❛✉stã♦✧✱ q✉❡ ❝♦♥s✐st✐❛ ❡♠ ✐♥s❝r❡✈❡r ❡ ❝✐r❝✉♥s❝r❡✈❡r ❛ ✜❣✉r❛ ❝♦♠ ♣♦❧í❣♦♥♦s ❡ ❡♥tã♦ ❛✉♠❡♥t❛r ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❞❡❧❡s✳ ❆ ❋✐❣✉r❛ ✷✳✸ ✐❧✉str❛ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ❝ír❝✉❧♦ ❝♦♠ ♣♦❧í❣♦♥♦s ✐♥s❝r✐t♦s✳
❖ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ✶✹
❙❡♥❞♦An ❛ ár❡❛ ❞♦ ♣♦❧í❣♦♥♦ ✐♥s❝r✐t♦ ❝♦♠ n ❧❛❞♦s✱ ❛✉♠❡♥t❛♥❞♦ n ♦❜s❡r✈❛♠♦s ✭❋✐✲ ❣✉r❛ ✷✳✸✮ q✉❡An✜❝❛ ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠❛ ❞❛ ár❡❛ ❞♦ ❝ír❝✉❧♦✳ ❖ q✉❡ ❡stá ❛❝♦♥t❡❝❡♥❞♦ ❛♣❡s❛r ❞❡✱ ♥❛ é♣♦❝❛✱ ❛✐♥❞❛ ♥ã♦ ❝♦♥❤❡❝❡r❡♠✱ é q✉❡ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ é ✐❣✉❛❧ ❛♦ ❧✐♠✐t❡ ❞❛ ár❡❛ ❞♦ ♣♦❧í❣♦♥♦ ❞❡ n ❧❛❞♦s✱ q✉❛♥❞♦n t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦ ❬✹❪✳
❉✐❛♥t❡ ❞❡st❡ ❝❡♥ár✐♦✱ ❝♦♥s✉❧t❛♠♦s ❛❧❣✉♥s ❧✐✈r♦s ❞✐❞át✐❝♦s ♣❛r❛ ♦❜s❡r✈❛r♠♦s ❝♦♠♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛ é ✐♥tr♦❞✉③✐❞♦ ❡ ár❡❛s ❞❡ q✉❡ t✐♣♦s ❞❡ r❡❣✐õ❡s sã♦ ❛❜♦r❞❛❞❛s✳ ❆♣r❡s❡♥✲ t❛♠♦s ❛ s❡❣✉✐r✱ ❛❧❣✉♥s ❧✐✈r♦s ❛♥❛❧✐s❛❞♦s✳
✷✳✷ ❖ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦
❉❡s❞❡ ✷✵✵✹ ♦ ❣♦✈❡r♥♦ ❢❡❞❡r❛❧ ❞❡s❡♥✈♦❧✈❡ ♦ Pr♦❣r❛♠❛ ◆❛❝✐♦♥❛❧ ❞♦ ▲✐✈r♦ ❉✐❞át✐❝♦ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ✭P◆▲❊▼✮✱ ❢♦r♥❡❝❡♥❞♦ ❧✐✈r♦s ❞✐❞át✐❝♦s ♣❛r❛ ♦s ❛❧✉♥♦s ❞❛ r❡❞❡ ♣ú❜❧✐❝❛✳ ◆❡st❡ ❝❛♣ít✉❧♦ ♠♦str❛r❡♠♦s ❝♦♠♦ ❛❧❣✉♥s ❧✐✈r♦s ❞❡ ♠❛t❡♠át✐❝❛✱ ♣❡rt❡♥❝❡♥t❡s ❛♦ ♣r♦❣r❛♠❛ ♦✉ ♥ã♦✱ ❛❜♦r❞❛♠ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❖s ❧✐✈r♦s ❛♥❛❧✐s❛❞♦s sã♦✿
✶✳ ▼❛t❡♠át✐❝❛ ✲ ❈✐ê♥❝✐❛ ❡ ❆♣❧✐❝❛çõ❡s ✭P◆▲❊▼✮✭✷✵✶✵✮
❆✉t♦r❡s✿ ●❡❧s♦♥ ■❡③③✐✱ ❖s✈❛❧❞♦ ❉♦❧❝❡✱ ❉❛✈✐❞ ❉❡❣❡♥s③❛❥♥✱ ❘♦❜❡rt♦ Pér✐❣♦ ❡ ◆✐❧③❡ ❞❡ ❆❧♠❡✐❞❛
❊❞✐t♦r❛✿ ❙❛r❛✐✈❛
✷✳ ▼❛t❡♠át✐❝❛ ✭P◆▲❊▼✮✲ ❱♦❧✉♠❡ Ú♥✐❝♦ ✭✷✵✵✽✮ ❆✉t♦r✿ ▲✉✐③ ❘♦❜❡rt♦ ❉❛♥t❡
❊❞✐t♦r❛✿ ➪t✐❝❛
✸✳ ❈♦♥❡①õ❡s ❝♦♠ ❛ ▼❛t❡♠át✐❝❛ ✲ ❱♦❧✉♠❡ ✷ ✭✷✵✶✵✮
❆✉t♦r❡s✿ ❖❜r❛ ❝♦❧❡t✐✈❛ ❝♦♠ ❏✉❧✐❛♥❡ ▼❛ts✉❜❛r❛ ❇❛rr♦s♦ ❝♦♠♦ ❡❞✐t♦r❛ r❡s♣♦♥sá✈❡❧ ❊❞✐t♦r❛✿ ▼♦❞❡r♥❛
✷✳✷✳✶ ▼❛t❡♠át✐❝❛ ✲ ❈✐ê♥❝✐❛ ❡ ❆♣❧✐❝❛çõ❡s ✲ ●❡❧s♦♥ ■❡③③✐ ❡ ♦✉tr♦s
✭✷✵✶✵✮
❊st❡ ❧✐✈r♦ ❞✐❞át✐❝♦ ❝♦♠❡ç❛ ♦ ❛ss✉♥t♦ ❛♣r❡s❡♥t❛♥❞♦ ❛❧❣✉♠❛s s✐t✉❛çõ❡s ❞♦ ❝♦t✐❞✐❛♥♦✳ ❉❡✜♥❡ ár❡❛ ❝♦♠♦✿ ❛ ♠❡❞✐❞❛ ❞❛ ❡①t❡♥sã♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡✱ ❡①♣r❡ss❛ ❡♠ ✉♠❛ ✉♥✐❞❛❞❡ ♣❛❞rã♦ ♣r❡❡st❛❜❡❧❡❝✐❞❛✳ ❆ ✉♥✐❞❛❞❡ ❞❡ ár❡❛ é ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ✶✳ ❙❡❣✉✐♥❞♦✱ ♥✉♠ ❜r❡✈❡ r❡s✉♠♦✱ ❛❜♦r❞❛ ♦ ❙✐st❡♠❛ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ▼❡❞✐❞❛s✱ ❞❡st❛❝❛♥❞♦ ❝♦♠♦ ♦ ♠❡tr♦ ❢♦✐ ❞❡t❡r♠✐♥❛❞♦✿ ❛♥t✐❣❛♠❡♥t❡ ✭✶✼✾✾✮✱ ❝♦♠ ✉♠ ♣❛❞rã♦ ❞❡ ♣❧❛t✐♥❛ ❡✱ ❛t✉❛❧♠❡♥t❡ ✭❛ ♣❛rt✐r ❞❡ ✶✾✽✸✮✱ ❝♦♠♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ tr❛❥❡tór✐❛ ❞❛ ❧✉③✱ ♥♦ ✈á❝✉♦✱ ♥♦ ✐♥t❡r✈❛❧♦ ❞❡ 1
❖ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ✶✺
❆♣r❡s❡♥t❛✱ t❛♠❜é♠✱ ❛ ❢ór♠✉❧❛ ❞❛ ár❡❛ ❞❛s ✜❣✉r❛s ❜ás✐❝❛s✿ r❡tâ♥❣✉❧♦✱ q✉❛❞r❛❞♦✱ ♣❛r❛❧❡❧♦❣r❛♠♦✱ ❧♦s❛♥❣♦ ❡ tr❛♣é③✐♦✱ ✉t✐❧✐③❛♥❞♦ ♦ ❞❡s❡♥❤♦ ❞❛ ✜❣✉r❛ ♥♦ ❛✉①í❧✐♦ ❞❛ r❡♣r❡✲ s❡♥t❛çã♦✳
P❛r❛ ♦ tr✐â♥❣✉❧♦✱ ❛❧é♠ ❞❡ ❛♣r❡s❡♥t❛r ❛ ❢ór♠✉❧❛ ❜ás✐❝❛✿ A = b.h
2 ✱ ♦ ❧✐✈r♦ t❛♠❜é♠ tr❛③ ❛s ❡①♣r❡ssõ❡s✿
• ❊♠ ❢✉♥çã♦ ❞❛s ♠❡❞✐❞❛s ❞❡ ❞♦✐s ❧❛❞♦s b ❡ c❡ ❞♦ s❡♥♦ ❞♦ â♥❣✉❧♦ Θ❝♦♠♣r❡❡♥❞✐❞♦
❡♥tr❡ ❡❧❡s
A= b.c.sen Θ
2 .
• ❊♠ ❢✉♥çã♦ ❞❛s ♠❡❞✐❞❛s ❞❡ s❡✉s ❧❛❞♦s a✱ b ❡ c❡ ❞❛ ♠❡❞✐❞❛ ❞♦ r❛✐♦r ❞❛ ❝✐r❝✉♥❢❡✲
rê♥❝✐❛ ✐♥s❝r✐t❛
A=p.r,
♦♥❞❡ p é ♦ s❡♠✐♣❡rí♠❡tr♦ ❞♦ tr✐â♥❣✉❧♦✳
• ❊♠ ❢✉♥çã♦ ❞❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦sa✱b❡c❡ ❞❛ ♠❡❞✐❞❛ ❞♦ r❛✐♦R❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛
❝✐r❝✉♥s❝r✐t❛
A= a.b.c 4.R.
• ◆♦s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❡ ❡q✉✐❧át❡r♦✳
◆♦s ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s✱ ✜❣✉r❛s ❡ ❡①♣r❡ssõ❡s ❞❡♠♦♥str❛♠ q✉❡ t♦❞♦s tê♠ s✉❛s ár❡❛s ✐❣✉❛✐s ❛♦ ♣r♦❞✉t♦ ❞♦ s❡♠✐♣❡rí♠❡tr♦ p ❝♦♠ ♦ ❛♣ót❡♠❛a✳
❆ ❢ór♠✉❧❛ ❞❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ é r❡♣r❡s❡♥t❛❞❛ ✉s❛♥❞♦ ❛ ❢ór♠✉❧❛A =p.a❞♦s ♣♦❧í❣♦♥♦s
r❡❣✉❧❛r❡s✳ ❈♦♠♦ ♥❡st❡s ♣♦❧í❣♦♥♦s q✉❛♥t♦ ♠❛✐♦r ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ♠❛✐s ♣ró①✐♠♦ s✉❛ ár❡❛ é ❞❛ ár❡❛ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ r✱ t❡♠♦s ✉♠❛ t❡♥❞ê♥❝✐❛ ❞❡ q✉❡ a=r ❡ ♦
s❡♠✐♣❡rí♠❡tr♦ p=π.r✳ ▲♦❣♦✱ A=π.r.r✱ ♦✉ s❡❥❛✱ A=π.r2✳
❆✐♥❞❛ s♦❜r❡ ❝ír❝✉❧♦s✱ ❞❡♠♦♥str❛✲s❡ ❛ ár❡❛ ❞♦ s❡t♦r✱ ❝♦r♦❛ ❡ s❡❣♠❡♥t♦ ❝✐r❝✉❧❛r✳ ❚♦❞♦ ❡st❡ ❝♦♥t❡ú❞♦ ❛♣r❡s❡♥t❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡①❡r❝í❝✐♦s ♣❛r❛ ❝❛❞❛ ✜❣✉r❛ ♣❧❛♥❛ ❛♣r❡s❡♥t❛❞❛✳
✷✳✷✳✷ ▼❛t❡♠át✐❝❛ ✲ ❉❛♥t❡ ✭✷✵✵✽✮
❈♦♠♦ ✐♥tr♦❞✉çã♦✱ ❡st❡ ❧✐✈r♦ ❢❛③ ✉♠ ❜r❡✈❡ ❝♦♠❡♥tár✐♦ q✉❡✱ ❞❡s❞❡ ❛ ❛♥t✐❣✉✐❞❛❞❡✱ ♦ ❤♦♠❡♠ ♠❡❞❡ ❡ ❞❡♠❛r❝❛ t❡rr❛s ❡ ✐st♦ s❡ ❢❛③ ❛té ♦s ❞✐❛s ❞❡ ❤♦❥❡✳ P❛ss❛ ✉♠❛ ✐❞❡✐❛ ✐♥t✉✐t✐✈❛ s♦❜r❡ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛✱ ❝♦♠♣❛r❛♥❞♦ ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ s♦❜r❡ ✉♠❛ ♠❛❧❤❛ q✉❛❞r✐❝✉❧❛❞❛✱ ❞❡✜♥✐♥❞♦ ♦ t❛♠❛♥❤♦ ❞❛ ✉♥✐❞❛❞❡✳
❙❡❣✉❡ ❛♣r❡s❡♥t❛♥❞♦ ❛s ❢ór♠✉❧❛s ❞❛s ✜❣✉r❛s ❝♦♥❤❡❝✐❞❛s✱ t❛♠❜é♠ ✉t✐❧✐③❛♥❞♦ ♦ ❞❡✲ s❡♥❤♦ ♣❛r❛ ❛✉①✐❧✐❛r ❛ ❞❡✜♥✐çã♦✳ ❙♦♠❡♥t❡ ❛♦ ✜♥❛❧ ❛♣r❡s❡♥t❛ ❡①❡r❝í❝✐♦s ❞❡ ❛♣❧✐❝❛çã♦✳ ❉❛♥t❡✱ ❞✐❢❡r❡♥t❡ ❞❡ ■❡③③✐✱ ✉s❛ ❛ ❡①♣r❡ssã♦ ✧ár❡❛ ❞❛ r❡❣✐ã♦ q✉❛❞r❛❞❛✱ r❡t❛♥❣✉❧❛r✱ ❡t❝✳✧
❉❡st❛❝❛♠♦s✱ ♥❡st❡ ❧✐✈r♦✱ ♦ ✐t❡♠ ✧❈á❧❝✉❧♦ ❛♣r♦①✐♠❛❞♦ ❞❡ ár❡❛s✧✳ ❖ ❛✉t♦r t♦♠❛✱ ❝♦♠♦ ❡①❡♠♣❧♦✱ ✉♠❛ r❡❣✐ã♦ R ♥ã♦ ♣♦❧✐❣♦♥❛❧✭❋✐❣✉r❛ ✷✳✹✮✱ s♦❜r❡♣õ❡ à ✉♠❛ ♠❛❧❤❛ q✉❛✲
❖ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ✶✻
✶✳ ◗✉❛♥t♦s q✉❛❞r❛❞♦s ✐♥t❡✐r♦s ❝❛❜❡♠ ❞❡♥tr♦ ❞❛ r❡❣✐ã♦R ❄ ◆❡st❡ ❝❛s♦✱ ❛ ❛♣r♦①✐♠❛✲
çã♦ é ♣♦r ❢❛❧t❛✳
❋✐❣✉r❛ ✷✳✹✿ ❆♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❬✽❪✳
➱ ♦❜s❡r✈❛❞♦ q✉❡ ❝❛❜❡♠ ✸✹ q✉❛❞r❛❞♦s ✐♥t❡✐r♦s ❞❡♥tr♦ ❞❡ R✳
✷✳ ◗✉❛♥t❛s q✉❛❞r❛❞♦s ✐♥t❡✐r♦s ❝♦❜r❡♠ ❛ r❡❣✐ã♦ R ❄ ❆❣♦r❛✱ ❛ ❛♣r♦①✐♠❛çã♦ é ♣♦r
❡①❝❡ss♦ ✭❋✐❣✉r❛ ✷✳✺✮✳
❋✐❣✉r❛ ✷✳✺✿ ❆♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❬✽❪✳
❱❡r✐✜❝❛✲s❡ q✉❡ ✻✼ q✉❛❞r❛❞♦s ✐♥t❡✐r♦s ❝♦❜r❡♠ R✳
❖ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ✶✼
❞♦s ❞♦✐s ✈❛❧♦r❡s ❡♥❝♦♥tr❛❞♦s
A≈ 34 + 67
2 = 50,5.
❖ ❛✉t♦r✱ ✜♥❛❧✐③❛ ♦ ❛ss✉♥t♦✱ ❝♦♠❡♥t❛♥❞♦ s♦❜r❡ r❛③ã♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ♣❛r❛ ár❡❛s✳
✷✳✷✳✸ ❈♦♥❡①õ❡s ❝♦♠ ❛ ▼❛t❡♠át✐❝❛ ✲ ❏✉❧✐❛♥❡ ▼❛ts✉❜❛r❛ ❇❛r✲
r♦s♦ ✭❡❞✐t♦r❛ r❡s♣♦♥sá✈❡❧✮ ✭✷✵✶✵✮
◆❡st❛ ♦❜r❛✱ ♦ ❛ss✉♥t♦ é ✐♥✐❝✐❛❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ár❡❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ♣♦❧✐❣♦♥❛❧✳ ❆ ❛✉t♦r❛✱ s❡♠❡❧❤❛♥t❡ ❛ ❉❛♥t❡✱ ♣r❡❢❡r❡ s❡ r❡❢❡r✐r à ár❡❛ ❞❛s ✜❣✉r❛s ❝♦♠♦✿ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ q✉❛❞r❛❞❛✱ r❡t❛♥❣✉❧❛r✱ ❡t❝✳
❙✉❛ ❞❡✜♥✐çã♦✿ ❆ ♣♦rçã♦ ❞♦ ♣❧❛♥♦ ♦❝✉♣❛❞❛ ♣♦r ✉♠❛ s✉♣❡r❢í❝✐❡ ♣♦❧✐❣♦♥❛❧ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ ❆ r❡❛❧ ♣♦s✐t✐✈♦ ❝❤❛♠❛❞♦ ❞❡ ár❡❛✱ ♦❜t✐❞♦ ♣❡❧❛ ❝♦♠♣❛r❛çã♦ ❞❛ ♣♦rçã♦ ♦❝✉♣❛❞❛ ♣❡❧❛ s✉♣❡r❢í❝✐❡ ♣♦❧✐❣♦♥❛❧ ❝♦♠ ❛ ♣♦rçã♦ ♦❝✉♣❛❞❛ ♣♦r ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ❞❡ ár❡❛
❙❡❣✉❡ ❛♣r❡s❡♥t❛♥❞♦ ❛s ❢ór♠✉❧❛s ❞❛s ✜❣✉r❛s ❝♦♥❤❡❝✐❞❛s✱ ❞❡ ♠❛♥❡✐r❛ ✐❣✉❛❧ ❛♦s ♦✉✲ tr♦s ❛✉t♦r❡s✳ ❱❛❧❡ ❞❡st❛❝❛r q✉❡ ♥❡st❛ ♦❜r❛ é ❛♣r❡s❡♥t❛❞❛✱ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦✱ ❛ ❢ór♠✉❧❛ ❞❡ ❍❡r♦♥✱ ♦♥❞❡ ♥✉♠ tr✐â♥❣✉❧♦ ❞❡ ❧❛❞♦s a✱ b ❡c ❡ s❡♠✐♣❡rí♠❡tr♦ p
t❡♠♦s
A=pp.(p−a).(p−b).(p−c)
◆❛s ♦❜r❛s ❛♥❛❧✐s❛❞❛s✱ ✈✐♠♦s q✉❡ é s❡❣✉✐❞❛✱ ❜❛s✐❝❛♠❡♥t❡✱ ❛ ♠❡s♠❛ ❧✐♥❤❛ ♣❛r❛ ❛❜♦r✲ ❞❛r ♦ ❛ss✉♥t♦✳ ❙♦♠❡♥t❡ ❛ ♦❜r❛ ❞❡ ❉❛♥t❡ ♠♦str♦✉ ✉♠❛ ♦♣çã♦ ♣❛r❛ s❡ ❡♥❝♦♥tr❛r ❛ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ♥ã♦ ♣♦❧✐❣♦♥❛❧✳ ❊st❡ ❢❛t♦ ❡❧❡✈❛✱ ❡♠ ✐♠♣♦rtâ♥❝✐❛✱ ❛ ❜✉s❝❛ ❞❡ ❛❧t❡r♥❛t✐✈❛s ❞✐❞át✐❝❛s ♣❛r❛ ❛♠♣❧✐❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ❛❧✉♥♦s s♦❜r❡ ♦ t❡♠❛✳
✸ ❈á❧❝✉❧♦ ❞❡ ár❡❛ ❞❡ r❡❣✐õ❡s
❞❡❧✐♠✐t❛❞❛s ♣♦r ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s ✲
✉♠❛ ♣r♦♣♦st❛ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❝♦♠♦ ❝❛❧❝✉❧❛r ár❡❛ ❞❡ ✜❣✉r❛s ♣♦❧✐❣♦♥❛✐s ✐rr❡❣✉❧❛r❡s ❡ ✜❣✉r❛s ♥ã♦ ♣♦❧✐❣♦♥❛✐s✱ ❜❡♠ ❝♦♠♦ ✉♠❛ ♣r♦♣♦st❛ ❞❡ ✉t✐❧✐③❛çã♦ ❞♦ s♦❢t✇❛r❡ ♠❛t❡♠át✐❝♦ ●❡♦❣❡❜r❛ ♣❛r❛ ❝á❧❝✉❧♦ ❞❡ ár❡❛ ❞❡ r❡❣✐õ❡s ❞❡❧✐♠✐t❛❞❛s ♣♦r ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s✳ P♦r ú❧✲ t✐♠♦ ❛♣r❡s❡♥t❛r❡♠♦s ♦ ●❡♦❣❡❜r❛✱ ♠♦str❛♥❞♦ ✉♠ r♦t❡✐r♦ ❞❡ ♣r❡♣❛r❛çã♦ ❞❡ ✉♠ ❡①❡♠♣❧♦ ❞❛ ❛t✐✈✐❞❛❞❡ ♣r♦♣♦st❛✳
❈❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❛ ár❡❛ ❞❡ ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❚ é ♠❡❞✐r ❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❡st❛✱ ♥♦ ♣❧❛♥♦✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ❞❡✜♥✐r ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛ ❡ ❝♦♠♣❛rá✲❧❛ ❝♦♠ ❛ r❡❣✐ã♦ ❞❛ ✜❣✉r❛✳ ▲♦❣♦✱ ♦ r❡s✉❧t❛❞♦ s❡rá ✉♠ ♥ú♠❡r♦ q✉❡ r❡♣r❡s❡♥t❡ q✉❛♥t❛s ✈❡③❡s ❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛ ❡stá ❝♦♥t✐❞❛ ♥❛ ✜❣✉r❛ ❚✳ ❊❧♦♥❬✺❪ ❛♣r❡s❡♥t❛ ❞❡ ❢♦r♠❛ ❞❡t❛❧❤❛❞❛✱ ❛❧é♠ ❞❡ ♦✉tr♦s ❝♦♥t❡①t♦s✱ ❝♦♠♦ s❡ ❞❡✜♥❡ ár❡❛ ❡ ❛ ❢ór♠✉❧❛ ❞❛ ár❡❛ ❞❡ ✜❣✉r❛s ❝♦♥❤❡❝✐❞❛s✳
❈♦♥s✐❞❡r❛♥❞♦ ♦ ♣ré✈✐♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ❝♦♠♦ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ tr✐â♥❣✉❧♦s ❡ r❡tâ♥✲ ❣✉❧♦s✱ ✈❡❥❛♠♦s ❝♦♠♦ ❞❡t❡r♠✐♥❛♠♦s ❛ ár❡❛ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❡ ❞❡ r❡❣✐õ❡s ❞❡❧✐♠✐t❛❞❛s ♣♦r ❝✉r✈❛s✳
✸✳✶ ❈❛❧❝✉❧❛♥❞♦ ár❡❛ ❞❡ ✜❣✉r❛s ♣♦❧✐❣♦♥❛✐s
❊♠ ✜❣✉r❛s ♣♦❧✐❣♦♥❛✐s✱ ♦♥❞❡ s❡✉s ❧❛❞♦s sã♦ s❡❣♠❡♥t♦s ❞❡ r❡t❛✱ ♥ã♦ ❤á ❞✐✜❝✉❧❞❛❞❡ ❞❡ ❝❛❧❝✉❧❛r s✉❛ ár❡❛ ♣♦✐s q✉❛❧q✉❡r ✜❣✉r❛ ♣♦❧✐❣♦♥❛❧✱ ♣♦r ♠❛✐s ✐rr❡❣✉❧❛r q✉❡ s❡❥❛✱ ♣♦❞❡ s❡r ❝♦❜❡rt❛ ❝♦♠ r❡tâ♥❣✉❧♦s ❡ tr✐â♥❣✉❧♦s ❥✉st❛♣♦st♦s ❬✺❪✳ ❈♦♠ ✐ss♦✱ s✉❛ ár❡❛ t❡rá ♦ ✈❛❧♦r ❞❛ s♦♠❛tór✐❛ ❞❛s ár❡❛s ❞❡st❛s ✜❣✉r❛s✳ ❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦✳
❈❛❧❝✉❧❛♥❞♦ ár❡❛ ❞❡ ✜❣✉r❛s ♣♦❧✐❣♦♥❛✐s ✶✾
❋✐❣✉r❛ ✸✳✶✿ P♦❧í❣♦♥♦ ✐rr❡❣✉❧❛r✳
❖ ♣♦❧í❣♦♥♦ ❞❛ ❋✐❣✉r❛ ✸✳✶ é ❢❛❝✐❧♠❡♥t❡ ♣❛rt✐❝✐♦♥❛❞♦ ❡♠ r❡tâ♥❣✉❧♦s ❡ tr✐â♥❣✉❧♦s ❡ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ ♥❛ ❋✐❣✉r❛ ✸✳✷✱ ❛tr❛✈és ❞❛s s✉❜✜❣✉r❛s ■✱ ■■✱ ■■■✱ ■❱✱ ❥✉st❛♣♦st❛s✳
❋✐❣✉r❛ ✸✳✷✿ P♦❧í❣♦♥♦ ✐rr❡❣✉❧❛r ♣❛rt✐❝✐♦♥❛❞♦✳
❊♥tã♦✱ ♣❛r❛ ❝❛❧❝✉❧❛r s✉❛ ár❡❛✱ ❜❛st❛ ❡♥❝♦♥tr❛r ❛ ár❡❛ ❞❛s ♣❛rt❡s ❡ s♦♠á✲❧❛s✳
❚❡♠♦s✱ ❝♦♠ ✐ss♦✱ ✉♠❛ ❞❡✜♥✐çã♦ ❣❡r❛❧ ❞❡ ár❡❛ ❞❡ ✉♠ ♣♦❧í❣♦♥♦✳ P♦❞❡♠♦s ❛ss♦❝✐❛r ❛ ❝❛❞❛ ♣♦❧í❣♦♥♦P ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦ ♥❡❣❛t✐✈♦✱ ❝❤❛♠❛♥❞♦ ❛ ár❡❛ ❞❡P✱ ❝♦♠ ❛s s❡❣✉✐♥t❡s
♣r♦♣r✐❡❞❛❞❡s✿
✶✳ P♦❧í❣♦♥♦s ❝♦♥❣r✉❡♥t❡s tê♠ ár❡❛s ❝♦♥❣r✉❡♥t❡s❀
✷✳ ❙❡ P é ✉♠ q✉❛❞r❛❞♦ ❝♦♠ ❧❛❞♦ ✉♥✐tár✐♦✱ ❡♥tã♦ ár❡❛ ❞❡ P é ✶❀
✸✳ ❙❡P ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st♦ ❝♦♠♦ r❡✉♥✐ã♦ ❞❡n ♣♦❧í❣♦♥♦sP1, ..., Pn✱ s❡♥❞♦ q✉❡ ❞♦✐s q✉❛✐sq✉❡r ❞❡❧❡s tê♠ ❡♠ ❝♦♠✉♠✱ ♥♦ ♠á①✐♠♦✱ ❛❧❣✉♥s ❧❛❞♦s ✱❡♥tã♦ ❛ ár❡❛ ❞❡ P é ❛
s♦♠❛ ❞❛s ár❡❛s ❞♦s Pi ✳
❙❡❣✉❡ ❞❡ ✸✮ q✉❡ s❡ ♦ ♣♦❧í❣♦♥♦ P ❡st✐✈❡r ❝♦♥t✐❞♦ ♥♦ ♣♦❧í❣♦♥♦ Q ❡♥tã♦ ❛ ár❡❛ ❞❡P s❡rá
❈❛❧❝✉❧❛♥❞♦ ár❡❛ ❞❡ ✜❣✉r❛s ♥ã♦ ♣♦❧✐❣♦♥❛✐s ✷✵
✸✳✷ ❈❛❧❝✉❧❛♥❞♦ ár❡❛ ❞❡ ✜❣✉r❛s ♥ã♦ ♣♦❧✐❣♦♥❛✐s
❆s ♣r♦♣r✐❡❞❛❞❡s ❝✐t❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡ sã♦ s♦♠❡♥t❡ ♣❛r❛ ♣♦❧í❣♦♥♦s ❡ ❛ss✐♠✱ ♥ã♦ s❡r✈✐r✐❛♠ ♣❛r❛ ❡♥❝♦♥tr❛r✱ ♣♦r ❡①❡♠♣❧♦✱ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❡ ❞❛ ❡❧✐♣s❡✱ ♣♦✐s ♥ã♦ ❝♦♥s❡❣✉✐♠♦s ❝♦♠♣♦r ♣❡r❢❡✐t❛♠❡♥t❡ s✉❛s r❡❣✐õ❡s ❝♦♠ tr✐â♥❣✉❧♦s ❡ r❡tâ♥❣✉❧♦s ❥✉st❛♣♦st♦s✳
❆♣r❡s❡♥t❛r❡♠♦s ❡♥tã♦ ✉♠❛ ❡str❛té❣✐❛ ♣❛r❛ ♦❜t❡♥çã♦ ❞❡ ár❡❛s ♥ã♦ ♣♦❧✐❣♦♥❛✐s✳ ❆ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ F ❛r❜✐trár✐❛ ♣♦❞❡ s❡r ❛♣r♦①✐♠❛❞❛ ♣♦r ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦ ♥❡❣❛t✐✈♦✱
q✉❡ ✐♥❞✐❝❛r❡♠♦s ❝♦♠♦ a(F)✱ q✉❡ s❛t✐s❢❛③
a(P)≤a(F)≤a(P′),
♦♥❞❡ a(P)é ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s ♣♦❧í❣♦♥♦s ❝♦♥t✐❞♦s ❡♠F ❡a(P′) é ❛ s♦♠❛ ❞❛s ár❡❛s
❞♦s ♣♦❧í❣♦♥♦s q✉❡ ❝♦♥té♠ F✳
❱❡❥❛♠♦s ❡①❡♠♣❧♦s q✉❡ ✐❧✉str❛♠ ❡st❛ ❛♣r♦①✐♠❛çã♦✳
P❛r❛ s✐♠♣❧✐✜❝❛r✱ ♠♦str❛r❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❝♦♥s✐❞❡r❛♥❞♦ ❛♣❡♥❛s ♣♦❧í❣♦♥♦s r❡t❛♥✲ ❣✉❧❛r❡s ❝♦♥t✐❞♦s ♥❛ r❡❣✐ã♦ F ✭❋✐❣✉r❛ ✸✳✸✮✳ ❆ss✐♠✱ ❝♦♠ ❛ r❡✉♥✐ã♦ ❞❡ s✉❛s ár❡❛s ✈❛♠♦s
♦❜t❡r ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞❛ ár❡❛ ❞❡ F ❝♦♠♦ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛✳
❋✐❣✉r❛ ✸✳✸✿ ❆♣r♦①✐♠❛çã♦ ❞❛ ár❡❛ ♣♦r ❢❛❧t❛✳
a(F)≈
13 X
j=1
a(j).
◆✉♠ s❡❣✉♥❞♦ ❡①❡♠♣❧♦✱ ❛ r❡❣✐ã♦F ❡stá ❝♦♥t✐❞❛ ♥❛ r❡❣✐ã♦ ❢♦r♠❛❞❛ ♣❡❧❛ r❡✉♥✐ã♦ ❞❡
r❡tâ♥❣✉❧♦s✳ ❆❣♦r❛✱ ❛ ár❡❛ ❞❡ F é ♦❜t✐❞❛ ♣♦r ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ✭❋✐❣✉r❛
➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲ ✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛ ✷✶
❋✐❣✉r❛ ✸✳✹✿ ❆♣r♦①✐♠❛çã♦ ❞❛ ár❡❛ ♣♦r ❡①❝❡ss♦✳
◆♦ ❝❛s♦ ❞❛ ❋✐❣✉r❛ ✸✳✹ t❡♠♦s
a(F)≈
16 X
j=1
a(j).
✸✳✸ ➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲
✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛
❆ ❋ís✐❝❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❛♣r❡s❡♥t❛ s✐t✉❛çõ❡s ♦♥❞❡ ♦s ✈❛❧♦r❡s ❞❡ ❝❡rt❛s ❣r❛♥❞❡③❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s ❛♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❞❡ ❛❧❣✉♥s ♣♦❧í❣♦♥♦ ❝♦♥❤❡❝✐❞♦s✳ ❱❡❥❛♠♦s ❞♦✐s ❡①❡♠♣❧♦s✳
❊①❡♠♣❧♦ ✸✳✶✳ ❯♠ ♠ó✈❡❧ t❡♠ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ (d) ❞❡ 5m q✉❛♥❞♦ s♦❢r❡ ❛ ❛çã♦ ❞❡
✉♠❛ ❢♦rç❛(F) ❝♦♥st❛♥t❡ ❞❡ 3N✳ ❈❛❧❝✉❧❛r ♦ tr❛❜❛❧❤♦✭τ✮ r❡❛❧✐③❛❞♦ ♣♦r ❡st❛ ❢♦rç❛✳
❈♦♠♦ τ = F.d s❡❣✉❡ q✉❡ ♦ ✈❛❧♦r ❞♦ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ é ❞❡ 15J✳ ❊st❡ r❡s✉❧t❛❞♦
❡q✉✐✈❛❧❡ à ár❡❛ s♦❜ ❛ r❡t❛ ❤♦r✐③♦♥t❛❧ ♥♦ ✐♥t❡r✈❛❧♦ ❡♥tr❡0m❡5m❞♦ ❣rá✜❝♦F xd✭❋✐❣✉r❛
✸✳✺✮✳
➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲ ✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛ ✷✷
❊①❡♠♣❧♦ ✸✳✷✳ ◆✉♠ ♠ó✈❡❧✱ ❞✉r❛♥t❡ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ ❞❡4m✱ ❛♣❧✐❝❛✲s❡ ✉♠❛ ❢♦rç❛ q✉❡
✈❛r✐❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❣rá✜❝♦ ❞❛❞♦ ♣❡❧❛ ❋✐❣✉r❛ ✸✳✻✳ ◗✉❛❧ ♦ ✈❛❧♦r ❞♦ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦❄
❋✐❣✉r❛ ✸✳✻✿ ❊①❡♠♣❧♦ ❞❡ ❢♦rç❛ ✈❛r✐á✈❡❧✳
❈♦♠♦
τ ≡A,
t❡♠♦s
τ = (1 + 3)4
2 .
◆❡st❡s ❡①❡♠♣❧♦s t❡♠♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✉♠ r❡tâ♥❣✉❧♦ ❡ ✉♠ tr❛♣é③✐♦ ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛s ❢♦rç❛s ❡ ❛s ár❡❛s ♣♦❞❡♠ s❡r ❝❛❧❝✉❧❛❞❛s ❝♦♠ ❢ór♠✉❧❛s ❜ás✐❝❛s ❥á ❝♦♥❤❡❝✐❞❛s ❞❡s❞❡ ♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✳
◆♦ ❡♥t❛♥t♦✱ s❡ ❛ ❢♦rç❛ F ❢♦r ❞❛❞❛ ♣♦r ❢✉♥çõ❡s ❞✐❢❡r❡♥t❡s ❞❛s ❛♣r❡s❡♥t❛❞❛s ♥♦s
❡①❡♠♣❧♦s ❛♥t❡r✐♦r❡s✱ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦ tr❛❜❛❧❤♦ r❡❛❧✐③❛❞♦ ❡♥✈♦❧✈❡ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦✳ P❛r❛ tr❛t❛r♠♦s ❞❡st❛ s✐t✉❛çã♦✱ ✉t✐❧✐③❛r❡♠♦s ♦ s♦❢t✲ ✇❛r❡ ●❡♦❣❡❜r❛✱ q✉❡ ♣❡r♠✐t❡ ✉♠❛ ✈✐s✉❛❧✐③❛çã♦ ❣rá✜❝❛ ❞✐♥â♠✐❝❛ ❞♦ ♣r♦❜❧❡♠❛✳ P❛r❛ ♦ ❧❡✐t♦r q✉❡ ♥ã♦ ❡stá ❢❛♠✐❧✐❛r✐③❛❞♦ ❝♦♠ ♦ ●❡♦❣❡❜r❛✱ ❛♣r❡s❡♥t❛♠♦s ♥♦ ✜♠ ❞♦ ❝❛♣ít✉❧♦ ✉♠❛ ❜r❡✈❡ ❞❡s❝r✐çã♦ ❞♦ s♦❢t✇❛r❡ ❡ ♦s ❝♦♠❛♥❞♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❡①❡♠♣❧♦s q✉❡ ❛♣r❡s❡♥t❛r❡♠♦s ❛ s❡❣✉✐r✳ ■♥✐❝✐❛r❡♠♦s ❝♦♠ ✉♠ ❡①❡♠♣❧♦ ♦♥❞❡ é ♣♦ssí✈❡❧ ❝♦♠♣❛r❛r ♦ ✈❛❧♦r ❡①❛t♦ ❝♦♠ ♦ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦ ♦❜t✐❞♦✳
❊①❡♠♣❧♦ ✸✳✸✳ ❊♥❝♦♥tr❛r ♦ ✈❛❧♦r ❞❛ ár❡❛ ❞❡ ✉♠ s❡♠✐❝ír❝✉❧♦ ❞❡ r❛✐♦ ✷ ✭❋✐❣✉r❛ ✸✳✼✮✳
➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲ ✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛ ✷✸
❆ ❡str❛té❣✐❛ s❡ ❛ss❡♠❡❧❤❛ à ❢♦r♠❛ ❣❡r❛❧ ♣❛r❛ s❡ ❝❛❧❝✉❧❛r ár❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s✱ ♦✉ s❡❥❛✱ ❝♦♥s✐❞❡r❛r ✜❣✉r❛s ❡❧❡♠❡♥t❛r❡s✱ ♥♦ ❝❛s♦✱ r❡tâ♥❣✉❧♦s ❥✉st❛♣♦st♦s✱ ❛♣r♦①✐♠❛♥❞♦ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ s❡♠✐❝✐r❝✉❧❛r ♣❡❧❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s n r❡tâ♥❣✉❧♦s ❝♦♥t✐❞♦s ♥❡❧❛✳
P❛r❛ ✐ss♦✱ t♦♠❛r❡♠♦s s✉❜✐♥t❡r✈❛❧♦s ❞♦ ✐♥t❡r✈❛❧♦ [−2,2] ❞❡ ❧❛r❣✉r❛ 1/n ❡✱ ♥♦ ❝❛s♦ ❞❡
❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛✱ ♦ ✈❛❧♦r ♠í♥✐♠♦ ❞❛ ❢✉♥çã♦ ❡♠ ❝❛❞❛ s✉❜✐♥t❡r✈❛❧♦ s❡rá ❛ ❛❧t✉r❛ ❞♦ r❡tâ♥❣✉❧♦✳ P❛r❛ ❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦✱ ♦ ✈❛❧♦r ♠á①✐♠♦ ❞❛ ❢✉♥çã♦ s❡rá ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ❛ ❛❧t✉r❛ ❞♦s r❡tâ♥❣✉❧♦s✳
◆❡st❛s ❛♣r♦①✐♠❛çõ❡s✱ ♦ ●❡♦❣❡❜r❛ ♥♦s ❢♦r♥❡❝❡ ❣rá✜❝♦s ❞✐♥â♠✐❝♦s ❡ t❛❜❡❧❛ ❞❡ ✈❛❧♦r❡s ❞❛s ár❡❛s✳
◗✉❛♥❞♦ ❛ ❛♣r♦①✐♠❛çã♦ ❢♦r ♣♦r ❢❛❧t❛✱ ❛ ár❡❛ ♦❜t✐❞❛ s❡rá ❞❡♥♦♠✐♥❛❞❛Ani✱ ✐♥❞✐❝❛❞❛ ♥❛ s❡❣✉♥❞❛ ❝♦❧✉♥❛ ❞❛ t❛❜❡❧❛✳ ❆ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛ n ✐♥❞✐❝❛rá ♦ ♥ú♠❡r♦ ❞❡ s✉❜❞✐✈✐sõ❡s
❞♦ ✐♥t❡r✈❛❧♦ ❡ i s✐❣♥✐✜❝❛ q✉❡ é s♦♠❛ ✐♥❢❡r✐♦r✳
◗✉❛♥❞♦ ❢♦r ♣♦r ❡①❝❡ss♦✱ ❛ ár❡❛ ♦❜t✐❞❛ s❡rá ❞❡♥♦♠✐♥❛❞❛Ans✱ ♦♥❞❡n é ♦ ♥ú♠❡r♦ ❞❡ s✉❜❞✐✈✐sõ❡s ❞♦ ✐♥t❡r✈❛❧♦ ❡ s ✐♥❞✐❝❛ s♦♠❛ s✉♣❡r✐♦r✳
❯♠❛ t❡r❝❡✐r❛ ❝♦❧✉♥❛ ❛♣r❡s❡♥t❛rá✱ ♣❛r❛ ❝❛❞❛ ❡t❛♣❛✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ✈❛❧♦r ❡♥❝♦♥✲ tr❛❞♦ ♣❡❧❛ ❢ór♠✉❧❛ ❞❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ✭A = 6,2832✮ ❡ ♦ ✈❛❧♦r An✳ ❊st❛ ❞✐❢❡r❡♥ç❛ é ♦ ❡rr♦ ♦❝♦rr✐❞♦ ❡♠ ❝❛❞❛ ❡t❛♣❛ ❞❛ ❛♣r♦①✐♠❛çã♦✳
Pr✐♠❡✐r❛♠❡♥t❡✱ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛✳
✭❛✮ ♥❂✶✵ ✭❜✮ ♥❂✷✵ ✭❝✮ ♥❂✹✵
❋✐❣✉r❛ ✸✳✽✿ ❆♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❞❛ ár❡❛ ❞♦ s❡♠✐❝ír❝✉❧♦ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳
♥
A
nierro
✶✵
✺✱✷✼✹✸ ✶✱✵✵✽✾
✺✵
✻✱✶✵✹✺ ✵✱✶✼✽✼
✷✺✵ ✻✱✷✹✾✺ ✵✱✵✸✸✼✼
✶✶✵✵ ✻✱✷✼✺✼ ✵✱✵✵✼✺
✹✺✵✵ ✻✱✷✽✶✹ ✵✱✵✵✶✽
✽✼✵✵ ✻✱✷✽✷✷ ✵✱✵✵✵✶
❚❛❜❡❧❛ ✸✳✶✿ ❱❛❧♦r❡s ❞❛s ❛♣r♦①✐♠❛çõ❡s ♣♦r ❢❛❧t❛✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✳
➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲ ✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛ ✷✹
♥♦ s✉❜✐♥t❡r✈❛❧♦ ❝♦♠♦ ❛❧t✉r❛ ❞♦s r❡tâ♥❣✉❧♦s✳ ❊st❛ ❛♣r♦①✐♠❛çã♦ s❡rá ♣♦r ❡①❝❡ss♦✳
✭❛✮ ♥❂✶✵ ✭❜✮ ♥❂✷✵ ✭❝✮ ♥❂✹✵
❋✐❣✉r❛ ✸✳✾✿ ❆♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❛ ár❡❛ ❞♦ s❡♠✐❝ír❝✉❧♦ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✳
♥
A
nserro
✶✵
✻✱✽✼✹✶ ✵✱✺✾✵✾
✺✵
✻✱✹✷✹✹ ✵✱✶✹✶✷
✷✺✵ ✻✱✸✶✸✺ ✵✱✵✸✵✸
✺✸✵ ✻✱✷✾✼✼ ✵✱✵✶✹✺
✶✵✵✵ ✻✱✷✾✶✵ ✵✱✵✵✼✽
✾✵✵✵ ✻✱✷✽✹✵ ✵✱✵✵✵✽
❚❛❜❡❧❛ ✸✳✷✿ ❱❛❧♦r❡s ❞❛s ❛♣r♦①✐♠❛çõ❡s ♣♦r ❡①❝❡ss♦✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳
P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡✱ ❝♦♥❢♦r♠❡ ❛✉♠❡♥t❛♠♦s ❛ q✉❛♥t✐❞❛❞❡ ❞❡ r❡tâ♥❣✉❧♦s ❝♦♠ ❜❛s❡s ❝❛❞❛ ✈❡③ ♠❡♥♦r❡s✱ ❛ ár❡❛ ❞❛ s♦♠❛ ❞♦s r❡tâ♥❣✉❧♦s ✈❛✐ s❡ ❛♣r♦①✐♠❛♥❞♦ ❞❛ ár❡❛ ❞♦ s❡♠✐❝ír❝✉❧♦✱ s❡❥❛ ♥❛ ❢♦r♠❛ ✐♥❢❡r✐♦r ♦✉ s✉♣❡r✐♦r✳ ❆ ❝♦❧✉♥❛ erro ❢❛❝✐❧✐t❛ ❛ ♦❜s❡r✈❛çã♦
❛♣r❡s❡♥t❛♥❞♦ ✈❛❧♦r❡s t❡♥❞❡♥❞♦ ❛ ③❡r♦✳
◆❡st❡ ❡①❡♠♣❧♦✱ ❝♦♠♦ ❛ ✜❣✉r❛ é ❝♦♥❤❡❝✐❞❛ ❡ ❥á t❡♠♦s ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❝❛❧❝✉❧❛r s✉❛ ár❡❛ ♣♦❞❡♠♦s ✉t✐❧✐③á✲❧♦ ❞❡ ♠♦❞♦ ❝♦♠♣❛r❛t✐✈♦ ♣❛r❛ ❞❛r s❡♥t✐❞♦ ❛♦ ✈❛❧♦r ❡♥❝♦♥tr❛❞♦ ❝♦♠ ❛ ❢ór♠✉❧❛ ♦✉✱ ❝♦♠♦ ❡①❡♠♣❧♦ ✐♥✐❝✐❛❧✱ ❞❡♠♦♥str❛r q✉❡ ❡st❡ r❡❝✉rs♦ ❣rá✜❝♦ ♥♦s ❢♦r♥❡❝❡ r❡s✉❧t❛❞♦s ✈❡r❞❛❞❡✐r♦s ❣❡r❛♥❞♦ ✉♠❛ ❝❡rt❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ♣❛r❛ ❡①❡♠♣❧♦s ♣♦st❡r✐♦r❡s✱ ♦♥❞❡ ❛s ✜❣✉r❛s ♥ã♦ sã♦ ❝♦♥❤❡❝✐❞❛s✳
❊♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ♦ ♣r♦❢❡ss♦r ♣♦❞❡ ❝♦♠❡♥t❛r q✉❡ ❡♠ ♥í✈❡✐s ♠❛✐s ❡❧❡✈❛❞♦s ❞❡ ❡st✉❞♦✱ t❡♠♦s ♦ ❝✉rs♦ ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ♦♥❞❡ é ♣♦ssí✈❡❧ ♠♦str❛r ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ✈❛❧♦r ❝♦rr❡t♦ ❞❛ ár❡❛✳
❊①❡♠♣❧♦ ✸✳✹✳ ❖ ❣rá✜❝♦ ❞❛❞♦ ♣❡❧❛ ❋✐❣✉r❛ ✸✳✶✵ ♠♦str❛ ✉♠ ♠ó✈❡❧ s✉❜♠❡t✐❞♦ à ❛çã♦ ❞❡ ✉♠❛ ❢♦rç❛(F) ❞✉r❛♥t❡ ✉♠ ❞❡s❧♦❝❛♠❡♥t♦ (d) ❞❡ 0 ❛ 3m✱ q✉❡ ✈❛r✐❛ ❝♦♥❢♦r♠❡ ❛ ❢✉♥çã♦ F = d2
➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲ ✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛ ✷✺
❋✐❣✉r❛ ✸✳✶✵✿ ❊①❡♠♣❧♦ ❞❡ ❢♦rç❛ ✈❛r✐á✈❡❧✳
❆ ❡str❛té❣✐❛ é ❛ ♠❡s♠❛ ♠♦str❛❞❛ ♥♦ ❊①❡♠♣❧♦ ✹✳✸✱ ♦✉ s❡❥❛✱ r❡❛❧✐③❛r ❛♣r♦①✐♠❛çõ❡s ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦✳ P♦ré♠✱ ♥❡st❛ s✐t✉❛çã♦✱ ♦ ♦❜❥❡t✐✈♦ é ❝❛❧❝✉❧❛r ♦ tr❛❜❛❧❤♦ ❡♥❝♦♥tr❛♥❞♦ ♦ ✈❛❧♦r ❞❛ ár❡❛✱ ❥á q✉❡✱ ♥♦ ♥í✈❡❧ ♠é❞✐♦ ❞❡ ❡♥s✐♥♦✱ ♥ã♦ s❡ ❝♦♥❤❡❝❡ ✉♠❛ ❢ór♠✉❧❛ ❞✐r❡t❛ ♣❛r❛ ❡st❛ ✜❣✉r❛✳
✭❛✮ ♥❂✶✵ ✭❜✮ ♥❂✷✵ ✭❝✮ ♥❂✺✵
❋✐❣✉r❛ ✸✳✶✶✿ ➪r❡❛ ♣♦r ❢❛❧t❛ ♣❛r❛ ❛ r❡❣✐ã♦ ❞❛ ❋✐❣✉r❛ ✸✳✶✵✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✳
♥
A
ni✶✵
✸✱✽✹✼✺
✷✵
✹✱✶✻✽✶
✺✵
✹✱✸✻✺✾
✶✵✵ ✹✱✹✸✷✼
✷✺✵ ✹✱✹✼✸✵
✺✸✵ ✹✱✹✽✼✸
✼✾✵ ✹✱✹✾✶✺
✶✵✵✵ ✹✱✹✾✸✸
➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲ ✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛ ✷✻
❯t✐❧✐③❛♥❞♦ ♥♦ ●❡♦❣❡❜r❛ ♦ ❝♦♠❛♥❞♦ ✧❙♦♠❛❉❡❘✐❡♠❛♥♥■♥❢❡r✐♦r✧❡♠ ❝♦♥❥✉♥t♦ ❝♦♠ ❛ ❢❡rr❛♠❡♥t❛ ✧❈♦♥tr♦❧❡ ❉❡s❧✐③❛♥t❡✧❢♦✐ ♣♦ssí✈❡❧✱ ❞❡ ❢♦r♠❛ ❞✐♥â♠✐❝❛✱ ❛ ♣❡r❢❡✐t❛ ✈✐s✉❛❧✐③❛✲ çã♦ ❞❛ ❛♣r♦①✐♠❛çã♦✱ ♣r✐♠❡✐r❛♠❡♥t❡ ♣♦r ❢❛❧t❛✱ ❡♥tr❡ ❛ ár❡❛ ❞❛ s♦♠❛ ❞♦s r❡tâ♥❣✉❧♦s ❡ ❛ ár❡❛ s♦❜ ❛ ❝✉r✈❛ ♥❡st❡ ✐♥t❡r✈❛❧♦✳ ❖✉tr❛ ❢❡rr❛♠❡♥t❛ ❞❡ ❡①tr❡♠❛ ✉t✐❧✐❞❛❞❡ ♣❛r❛ ♦ ❞❡s❡♥✲ ✈♦❧✈✐♠❡♥t♦ ❞❡st❛ ❛t✐✈✐❞❛❞❡ é ✧●r❛✈❛r ♣❛r❛ ❛ P❧❛♥✐❧❤❛ ❞❡ ❈á❧❝✉❧♦s✧✳ ❈♦♠ ❡❧❛✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♦s ✈❛❧♦r❡s ❞❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s r❡tâ♥❣✉❧♦s s❡♥❞♦ ❛♣r❡s❡♥t❛❞♦s ♥✉♠❛ t❛❜❡❧❛✱ ❝♦♥❢♦r♠❡ ✈❛r✐❛♠♦s ❛ q✉❛♥t✐❞❛❞❡ ❞❡ s✉❜✐♥t❡r✈❛❧♦s ♥♦ ❈♦♥tr♦❧❡ ❉❡s❧✐③❛♥t❡✳ ❖ ❡①❡♠♣❧♦ ♥♦s ♠♦str❛ ♦ ✈❛❧♦r ❞❛ s♦♠❛ s❡ ❛♣r♦①✐♠❛♥❞♦ ❞❡ ✹✱✺✳ ❯♠ q✉❡st✐♦♥❛♠❡♥t♦ q✉❡ ❞❡✈❡♠♦s ❧❡✈❛♥t❛r é✿ ❡st❡ ú♥✐❝♦ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ♦ ❡①♣❡r✐♠❡♥t♦ ❡ s❡✉ ✈❛❧♦r ❛♣r❡s❡♥t❛❞♦ ♣♦❞❡ s❡r ❝♦♥❝❧✉s✐✈♦❄ ❈♦♠♦ ♦ ✈❛❧♦r 4,5 ♥ã♦ ✜❝♦✉ ❡①♣❧í❝✐t♦✱ ♥ã♦ ♣♦❞❡♠♦s t❡r ❛ ❝❡rt❡③❛ ✐♠❡✲
❞✐❛t❛ q✉❡ ❡st❡ s❡rá ♦ ✈❛❧♦r ❞❛ ár❡❛ s♦❜ ❛ ❝✉r✈❛✳ ❈♦♠ ✐ss♦✱ ❞✐r❡❝✐♦♥❛♠♦s ♦s ❛❧✉♥♦s ❛ r❡❛❧✐③❛r ✉♠❛ s❡❣✉♥❞❛ ❡t❛♣❛ ❞♦ ❡①♣❡r✐♠❡♥t♦✱ ❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦✳
✭❛✮ ♥❂✶✵ ✭❜✮ ♥❂✷✵ ✭❝✮ ♥❂✺✵
❋✐❣✉r❛ ✸✳✶✷✿ ➪r❡❛ ♣♦r ❡①❝❡ss♦ ♣❛r❛ ❛ r❡❣✐ã♦ ❞❛ ❋✐❣✉r❛ ✸✳✶✵✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳
♥
A
ns✶✵ ✹✱✽✹✸✶
✷✵ ✹✱✼✷✼✺
✺✵ ✹✱✻✶✸✶
✶✵✵ ✹✱✺✻✶✺
✷✺✵ ✹✱✺✷✺✵
✺✸✵ ✹✱✺✶✷✶
✼✽✵ ✹✱✺✵✽✶
✾✾✵ ✹✱✺✵✻✼
❚❛❜❡❧❛ ✸✳✹✿ ❱❛❧♦r❡s ❞❛ ár❡❛ ♣♦r ❡①❝❡ss♦ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✳
➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲ ✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛ ✷✼
❢✉♥çã♦ ♥❡st❡ ✐♥t❡r✈❛❧♦✳ ◆♦✈❛♠❡♥t❡✱ ❡st❡ ✈❛❧♦r ♣♦❞❡ s❡r ❥✉st✐✜❝❛❞♦ ❛tr❛✈és ❞♦ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✳
◆♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦✱ ❛❧é♠ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❝♦♠ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦✱ ✈❛♠♦s t❛♠❜é♠ ❡①♣❧♦r❛r✱ ♥❛ ♣rát✐❝❛✱ ♦ ❝♦♥❝❡✐t♦ ❞❛ ❙♦♠❛ ❞❡ ❘✐❡✲ ♠❛♥♥✳ P❛r❛ ❢❛❝✐❧✐t❛r ❛ ♣r♦❣r❛♠❛çã♦ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❞❡st❡ ú❧t✐♠♦ ❝♦♥❝❡✐t♦✱ t♦♠❛♠♦s✱ ❡♠ ❢✉♥çã♦ ❞♦ ♣♦♥t♦ ❞❡ ❡①tr❡♠♦ ❡sq✉❡r❞♦ ❞❡ ❝❛❞❛ s✉❜✐♥t❡r✈❛❧♦✱ ❛ ❛❧t✉r❛ ❞♦s r❡tâ♥❣✉❧♦s q✉❡ ❝♦♠♣õ❡♠ ❛ s♦♠❛✳
❊①❡♠♣❧♦ ✸✳✺✳ ❱❛♠♦s ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣❡❧♦ ❣rá✜❝♦ ✭❋✐❣✉r❛ ✸✳✶✸✮ ❞❛ ❢✉♥çã♦ f(x) = −x2
+ 4x ♥♦ ✐♥t❡r✈❛❧♦[0,4]✳
❋✐❣✉r❛ ✸✳✶✸✿ ❘❡❣✐ã♦ ❞♦ ❊①❡♠♣❧♦ ✹✳✺✳
◆♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❝♦♠ ♦ ●❡♦❣❡❜r❛ t❡♠♦s ❛s s❡❣✉✐♥t❡s r❡♣r❡s❡♥t❛çõ❡s ♥♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❛tr❛✈és ❞❡ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛✿
✭❛✮ ♥❂✶✵ ✭❜✮ ♥❂✷✵ ✭❝✮ ♥❂✺✵
➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲ ✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛ ✷✽
♥
A
ni✶✵
✽✱✾✻✵✵
✻✵
✶✵✱✸✾✼✵
✶✵✵
✶✵✱✺✵✺✻
✺✸✵
✶✵✱✻✸✻✹
✼✺✵
✶✵✱✻✹✺✸
✶✵✵✵✵ ✶✵✱✻✻✺✶
❚❛❜❡❧❛ ✸✳✺✿ ❱❛❧♦r❡s ❞❛ ár❡❛ ♣♦r ❢❛❧t❛✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✳
❱❡❥❛♠♦s ♦s ✈❛❧♦r❡s ❛♣r❡s❡♥t❛❞♦s ♣❡❧♦ s♦❢t✇❛r❡ ♥❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦
✭❛✮ ♥❂✶✵ ✭❜✮ ♥❂✷✵ ✭❝✮ ♥❂✺✵
❋✐❣✉r❛ ✸✳✶✺✿ ❆♣r♦①✐♠❛çã♦ ♣♦r ❡①❝❡ss♦ ❞❛ ❋✐❣✉r❛ ✸✳✶✸✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳
♥
A
ns✶✵
✶✷✱✶✻✵✵
✻✵
✶✵✱✾✸✵✹
✷✺✵
✶✵✱✼✸✵✺
✺✸✵
✶✵✱✻✾✻✽
✼✺✵
✶✵✱✻✽✽✵
✶✵✵✵✵ ✶✵✱✻✻✽✸
❚❛❜❡❧❛ ✸✳✻✿ ❱❛❧♦r❡s ❞❛ ár❡❛ ♣♦r ❡①❝❡ss♦✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳
➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲ ✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛ ✷✾
✭❛✮ ♥❂✶✵ ✭❜✮ ♥❂✷✵ ✭❝✮ ♥❂✺✵
❋✐❣✉r❛ ✸✳✶✻✿ ❆♣r♦①✐♠❛çã♦✱ ❝♦♠ ♦ ❡①tr❡♠♦ ❡sq✉❡r❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛✱ ❞❛ ❋✐❣✉r❛ ✸✳✶✸✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✳
❈♦♠ ❛ r❡❢❡rê♥❝✐❛ ♥✉♠ ❡①tr❡♠♦ ❞♦s s✉❜✐♥t❡r✈❛❧♦s ❡st❡ ❡①❡♠♣❧♦ ❛♣r❡s❡♥t♦✉ ❛ s♦♠❛ ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡✳ ❱❡❥❛♠♦s ♦s ✈❛❧♦r❡s ❞❛s ár❡❛s ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ n✳
♥
A
n✶✵ ✶✵✱✺✻✵✵
✷✵ ✶✵✱✻✹✵✵
✶✵✵ ✶✵✱✻✻✺✻
✺✸✵ ✶✵✱✻✻✻✻
✼✺✵ ✶✵✱✻✻✻✻
✾✻✵ ✶✵✱✻✻✻✼
❚❛❜❡❧❛ ✸✳✼✿ ❱❛❧♦r❡s ❞❛ ár❡❛ ❝♦♠ ♦ ❡①tr❡♠♦ ❡sq✉❡r❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛✳
❖❜s❡r✈❛♠♦s ❛tr❛✈és ❞♦s ❡①❡♠♣❧♦s q✉❡ ❛s três ♦♣çõ❡s ♥♦s ❧❡✈❛♠ ❛♦ ♠❡s♠♦ ❧✉❣❛r✱ ♦✉ s❡❥❛✱ ❛s ❛♣r♦①✐♠❛çõ❡s ❝♦♥✈❡r❣❡♠ ❛♦ ♠❡s♠♦ ✈❛❧♦r✳ P♦ss✐❜✐❧✐t❛♥❞♦✱ ❞❡ ♠❛♥❡✐r❛ ❝❧❛r❛ ❡ ❝♦♥✜á✈❡❧✱ t✐r❛r ❝♦♥❝❧✉sõ❡s s♦❜r❡ ♦ ♣r♦❜❧❡♠❛ ❡st✉❞❛❞♦✳
❊♥t❡♥❞❡♠♦s q✉❡✱ ❛♣r❡s❡♥t❛♥❞♦ ❡st❡ ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s ✐♥tr♦❞✉③✐r ♦ ❝♦♥❝❡✐t♦ ❞❛ ❙♦♠❛ ❞❡ ❘✐❡♠❛♥♥ ❞❡ ❢♦r♠❛ ❛❞❡q✉❛❞❛ ❛♦ ◆í✈❡❧ ▼é❞✐♦✱ ❧❡✈❛♥❞♦ ♦ ❛❧✉♥♦ ❛ ♣❡r❝❡❜❡r q✉❡ ❛ s♦♠❛ ❝♦♥✈❡r❣❡ ♣❛r❛ ♦ ✈❛❧♦r ❞❛ ár❡❛ t♦♠❛♥❞♦ ✉♠ ✈❛❧♦r ❛r❜✐trár✐♦ ci ❞❡♥tr♦ ❞❡ ❝❛❞❛ ✐♥t❡r✈❛❧♦✳ ■st♦ ♣♦❞❡ ❝♦♥tr✐❜✉✐r✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♣❛r❛ ❛q✉❡❧❡s ❛❧✉♥♦s q✉❡ ♦♣t❛r❡♠ ♣♦r ❝✉rs♦s ❞❛ ár❡❛ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ♥♦s ❡stá❣✐♦s s❡❣✉✐♥t❡s ❞❡ s❡✉s ❡st✉❞♦s✳
❆s s✐t✉❛çõ❡s ❥á tr❛❜❛❧❤❛❞❛s ♠♦str❛r❛♠✱ ❞❡ ❢♦r♠❛ ❞❡t❛❧❤❛❞❛✱ q✉❡ ❝♦♠ ♦ ♣r♦❝❡❞✐✲ ♠❡♥t♦ ♣r♦♣♦st♦ ♦s ✈❛❧♦r❡s ❞❛s ❛♣r♦①✐♠❛çõ❡s ❝♦♥✈❡r❣❡♠ ♣❛r❛ ♦ ✈❛❧♦r ❞❛ ár❡❛ ♣r♦❝✉r❛❞❛✳ ❙❡❥❛ ♣♦r ❢❛❧t❛✱ s❡❥❛ ♣♦r ❡①❝❡ss♦ ♦✉ ❛✐♥❞❛✱ ♣♦r ✉♠ ❡①tr❡♠♦✱ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♦❝♦rr❡✳
❖s ♣ró①✐♠♦s ❡①❡♠♣❧♦s s❡rã♦ ❞❡s❡♥✈♦❧✈✐❞♦s ♠♦str❛♥❞♦ ❛♣❡♥❛s ✉♠❛ ♦♣çã♦ ❞❡ ❝♦♥✲ ✈❡r❣ê♥❝✐❛✱ s✐♠♣❧✐✜❝❛♥❞♦ ♦ ♣r♦❝❡ss♦ s❡♠ ♣❡r❞❛ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡✳
➪r❡❛ ❞❡ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣♦r ❣rá✜❝♦ ❞❡ ❢✉♥çã♦ ✲ ✉♠❛ s✐t✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❞❛ ✸✵
❊①❡♠♣❧♦ ✸✳✻✳ ❈❛❧❝✉❧❛r ❛ ár❡❛ AT ❞❛ r❡❣✐ã♦ ❡♥tr❡ ❡♥tr❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f(x) =
−(x−1)(x−3)(x−5)❡ ♦ ❡✐①♦ OX✱ ♥♦ ✐♥t❡r✈❛❧♦ [1,5]✳
❋✐❣✉r❛ ✸✳✶✼✿ ●rá✜❝♦ ❞❡ f(x) =−(x−1)(x−3)(x−5)✳
❊s❝♦❧❤❡♥❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♦ ❡①tr❡♠♦ ❡sq✉❡r❞♦ ❞❡ ❝❛❞❛ s✉❜✐♥t❡r✈❛❧♦✱ t❡♠♦s ❡♥tã♦✿
✭❛✮ ♥❂✺ ✭❜✮ ♥❂✶✵ ✭❝✮ ♥❂✷✵
❋✐❣✉r❛ ✸✳✶✽✿ ➪r❡❛ s♦❜ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❞❛ ❋✐❣✉r❛ ✸✳✶✼✱ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡n✳
❙❡♥❞♦A1❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣❡❧♦ ❣rá✜❝♦ ❞❡f ♥♦ ✐♥t❡r✈❛❧♦ [1,3]✭f(x)<0✮ ❡ A2 ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❞❡❧✐♠✐t❛❞❛ ♣❡❧♦ ❣rá✜❝♦ ❞❡f ♥♦ ✐♥t❡r✈❛❧♦[3,5]✭f(x)>0✮✳ ▲♦❣♦✱ ❛ ár❡❛ AT s❡rá ❛ s♦♠❛ ❞❛s ár❡❛s ❞❡ss❛s ❞✉❛s r❡❣✐õ❡s✳ P❛r❛ ♦ ✐♥t❡r✈❛❧♦[1,3]♦ ●❡♦❣❡❜r❛ ❢♦r♥❡❝❡ ✉♠ ✈❛❧♦r ♥❡❣❛t✐✈♦ ♣❛r❛ ❛s ❛♣r♦①✐♠❛çõ❡s ❡ ❞❡ss❛ ❢♦r♠❛✱ ❝♦♥s✐❞❡r❛♠♦s ❝♦♠♦A1 ♦ ♠ó❞✉❧♦ ❞❡ss❡ ✈❛❧♦r ✭✈❡❥❛ ❋✐❣✉r❛ ✸✳✶✽✮✳ ❆ss✐♠✱
AT =A1+A2.
