OS MODELOS DE CRESCIMENTO POPULACIONAL DE MALTHUS E VERHULST - UMA MOTIVAÇÃO PARA O ENSINO DE LOGARITMOS E EXPONENCIAIS

❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❖s ♠♦❞❡❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❞❡

▼❛❧t❤✉s ❡ ❱❡r❤✉❧st ✲ ✉♠❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦

❡♥s✐♥♦ ❞❡ ❧♦❣❛r✐t♠♦s ❡ ❡①♣♦♥❡♥❝✐❛✐s

❘♦❜✐♥s♦♥ ❚❛✈♦♥✐

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r❛

Pr♦❢❛✳ ❉r❛✳ ❘❡♥❛t❛ ❩♦t✐♥ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛

Tavoni, Robinson

Os modelos de crescimento populacional de Malthus e Verhulst: uma motivação para o ensino de logaritmos e exponenciais / Robinson Tavoni. - Rio Claro, 2013 70 f. : il., figs., gráfs., tabs.

Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas

Orientador: Renata Zotin Gomes de Oliveira

1. Matemática - Estudo e ensino. 2. Modelagem

matemática. 3. Função exponencial. 4. Função logarítmica. I. Título.

510.07 T234m

Ficha Catalográfica elaborada pela STATI - Biblioteca da UNESP Campus de Rio Claro/SP

❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

❘♦❜✐♥s♦♥ ❚❛✈♦♥✐

❖s ♠♦❞❡❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❞❡ ▼❛❧t❤✉s ❡

❱❡r❤✉❧st ✲ ✉♠❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ❧♦❣❛r✐t♠♦s ❡

❡①♣♦♥❡♥❝✐❛✐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❛✳ ▼❛r✐❛ ❇❡❛tr✐③ ❋❡rr❡✐r❛ ▲❡✐t❡ ❈❊❆❚❊❈✲ P❯❈ ✲ ❈❛♠♣✐♥❛s

Pr♦❢❛✳ ❉r❛✳ ▼❛rt❛ ❈✐❧❡♥❡ ●❛❞♦tt✐ ❉▼ ✲ ■●❈❊✴❯◆❊❙P✴❘✐♦ ❈❧❛r♦

❆❣r❛❞❡❝✐♠❡♥t♦s

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ♠✐♥❤❛ ♠ã❡✱ ♠❡✉ ♣❛✐ ❡ ♠✐♥❤❛ ✐r♠ã q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ❡ t✐✈❡r❛♠ ♣❛❝✐ê♥❝✐❛ ❝♦♠✐❣♦✳

❆♦s ♣r♦❢❡ss♦r❡s ❡ ❝♦♦r❞❡♥❛❞♦r❡s ❞♦ ▼❡str❛❞♦ q✉❡ ♥♦s ✐♥❝❡♥t✐✈❛✈❛♠ ❡ ♥♦s ❞❛✈❛♠ ❢♦rç❛ ❡ r❡s✐❧✐ê♥❝✐❛ ♣❛r❛ ❝❤❡❣❛r ❛♦ ✜♠✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ Pr♦❢❛

❉r❛ _{❘❡♥❛t❛ ❩♦t✐♥ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛ ❝♦♠ q✉❡♠ ❛♣r❡♥❞✐ ♠✉✐t♦ ❡ s❡♠♣r❡ ❡st❡✈❡ ❞✐s♣♦st❛}

❛ ♠❡ ❡♥s✐♥❛r ❡ ❛❥✉❞❛r ❡ ❛ q✉❡♠ t❛♠❜é♠ s♦✉ ✐♠❡♥s❛♠❡♥t❡ ❣r❛t♦✳ ➚ ❈❆P❊❙✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

❘❡s✉♠♦

❊st❡ tr❛❜❛❧❤♦ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❛♣r❡s❡♥t❛r ✉♠❛ s✉❣❡stã♦ ❞❡ ❝♦♠♦ ✐♥tr♦❞✉✲ ③✐r ♦s ❝♦♥❝❡✐t♦s ❞❡ ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s✱ ✉t✐❧✐③❛♥❞♦ ❝♦♠♦ ♠♦t✐✈❛çã♦ ✉♠ s♦❢t✇❛r❡ q✉❡ ❛♣r❡s❡♥t❛ ❛t✐✈✐❞❛❞❡s s♦❜r❡ ♠♦❞❡❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❞✐s❝r❡t♦s ✲ ▼❛❧t❤✉s ❡ ❱❡r❤✉❧st✳

❆❜str❛❝t

❚❤✐s ✇♦r❦ ❤❛s ❛s ♠❛✐♥ ♦❜❥❡❝t✐✈❡ t♦ ♣r❡s❡♥t ❛ s✉❣❣❡st✐♦♥ ❤♦✇ t♦ ✐♥tr♦❞✉❝❡ t❤❡ ❝♦♥❝❡♣ts ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ ❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥s ✉s✐♥❣ ❛ s♦❢t✇❛r❡ t❤❛t ♣r♦✈✐❞❡s s♦♠❡ ❛❝t✐✈✐t✐❡s ♦♥ ❞✐s❝r❡t❡ ♠♦❞❡❧s ♦❢ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤ ✲ ▼❛❧t❤✉s ❛♥❞ ❱❡r❤✉❧st✳

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ❈♦❜✇❡❜ ❞❛ ❡q✉❛çã♦ ✭✷✳✻✮ ❝♦♠x0 = 3₄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✷✳✷ ❈♦❜✇❡❜ ❞❛ ❡q✉❛çã♦ ✭✷✳✼✮ ❝♦♠x0 = 3₄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✸✳✶ ●rá✜❝♦ ❞♦ ❡①❡r❝í❝✐♦ ❞♦ ✈❡st✐❜✉❧❛r ❞❛ ❯◆■❈❆▼P✴✷✵✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✶ ■♠❛❣❡♠ ❞❛ ♣á❣✐♥❛ ✐♥✐❝✐❛❧ ❞♦ M3 ✲ ▼❛t❡♠át✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✷ Pá❣✐♥❛ ✐♥✐❝✐❛❧ ❞♦ s♦❢t✇❛r❡ ❈r❡s❝✐♠❡♥t♦ P♦♣✉❧❛❝✐♦♥❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✸ ❚❡❧❛ ❞❛ ♣r✐♠❡✐r❛ ❛t✐✈✐❞❛❞❡ ✲ ▼♦❞❡❧♦ ❞❡ ▼❛❧t❤✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✹ ■♠❛❣❡♠ ❞❛ ♣r✐♠❡✐r❛ q✉❡stã♦ ❞♦ s♦❢t✇❛r❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✺ ●rá✜❝♦ ❞❛ ♣r✐♠❡✐r❛ q✉❡stã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✻ ■♠❛❣❡♠ ❞❛ q✉❡stã♦ ✷ ❞♦ ▼♦❞❡❧♦ ❞❡ ▼❛❧t❤✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✼ ●rá✜❝♦ ❞❛ ♣♦♣✉❧❛çã♦ ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦ ✲ ♠♦❞❡❧♦ ❞❡ ▼❛❧t❤✉s✳ ✳ ✳ ✳ ✳ ✺✻ ✹✳✽ ●rá✜❝♦ ❞♦ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧✭R(n)✮ ❡♠ ❢✉♥çã♦ ❞♦ t❛♠❛♥❤♦ ❞❛

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶✾

✷ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ✷✶

✷✳✶ ❊q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ P♦♥t♦ ❞❡ ❊q✉✐❧í❜r✐♦ ❡ ❊st❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸ ❈♦❜✇❡❜ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✹ ❆❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✺ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✻ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ❞❡ ✶❛ _{❖r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽}

✷✳✼ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ✶❛ _{♦r❞❡♠ ❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾}

✷✳✽ ❊q✉❛çõ❡s s❡♣❛rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✾ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✸ ❋✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s ✸✼

✸✳✶ ❆ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✷ ❆ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✸ ❊①❡r❝í❝✐♦s ❞❡ ❱❡st✐❜✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✹ Pr♦♣♦st❛ ❞❡ ❊♥s✐♥♦ ✹✾

✹✳✶ ❖ s♦❢t✇❛r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✷ ❈r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ✲ ▼♦❞❡❧♦ ❞❡ ▼❛❧t❤✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✸ ❈r❡s❝✐♠❡♥t♦ P♦♣✉❧❛❝✐♦♥❛❧ ✲ ▼♦❞❡❧♦ ❞❡ ❱❡r❤✉❧st ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✹ Pr♦♣♦st❛ ❞❡ ❆t✐✈✐❞❛❞❡ ❊①tr❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✺ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✻✼

✶ ■♥tr♦❞✉çã♦

▲♦❣❛r✐t♠♦s ❡ ❡①♣♦♥❡♥❝✐❛✐s ♣♦❞❡♠ s❡r ✐❞❡♥t✐✜❝❛❞♦s ❡♠ s✐t✉❛çõ❡s ❞♦ ❞✐❛✲❛✲❞✐❛ t❛✐s ❝♦♠♦✿ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧✱ ✜♥❛♥❝✐❛♠❡♥t♦ ❞❡ ✉♠ ❝❛rr♦ ♦✉ ❝❛s❛✱ ❛❜s♦rçã♦ ❞❡ ✉♠ r❡♠é❞✐♦✱ ❞❛t❛çã♦ ♣♦r ❝❛r❜♦♥♦✱ r❡s❢r✐❛♠❡♥t♦ ❞❡ ✉♠ ❝♦r♣♦✱ ❡t❝✳ ◆❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛✱ ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s✱ sã♦ ❡♥s✐♥❛❞♦s ❞❡ ♠❛♥❡✐r❛ té❝♥✐❝❛ ❡ ❛s ❛♣❧✐❝❛çõ❡s sã♦ ✏❝♦♥s❡q✉ê♥❝✐❛s✑✳ ◆❡♠ s❡♠♣r❡ é ♣♦ssí✈❡❧ ♣❡r❝❡❜❡r✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ ✏♦s ❧♦❣❛r✐t♠♦s ❢♦r❛♠ ❝r✐❛❞♦s ❝♦♠♦ ✐♥str✉♠❡♥t♦ ♣❛r❛ t♦r♥❛r ♠❛✐s s✐♠♣❧❡s ❝á❧❝✉❧♦s ❛r✐t♠ét✐❝♦s ❝♦♠♣❧✐❝❛❞♦s✳ P♦st❡r✐♦r♠❡♥t❡ ✈❡r✐✜❝♦✉✲s❡ q✉❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦s ❧♦❣❛r✐t♠♦s ♥❛ ▼❛t❡♠át✐❝❛ ❡ ♥❛s ❈✐ê♥❝✐❛s ❡♠ ❣❡r❛❧ ❡r❛ ❜❡♠ ♠❛✐♦r ❞♦ q✉❡ s❡ ♣❡♥s❛✈❛✑ ✭❬✶✻❪✮✳

❊♠ ▲✐♠❛ ❬✶✻❪✮❛ ❞❡✜♥✐çã♦ ❞❡ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧ é ❞❛❞❛ ❛tr❛✈és ❞❛ ár❡❛ ❞❡ ✉♠❛ ❢❛✐①❛ ❞❡ ❤✐♣ér❜♦❧❡✱ ❜❡♠ ❝♦♠♦ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ é ✐♥tr♦❞✉③✐❞♦ ♣♦st❡r✐♦r♠❡♥t❡✱ ❞✐❢❡r❡♥t❡♠❡♥t❡ ❞♦ q✉❡ ❛❝♦♥t❡❝❡ ♥❛ ♠❛✐♦r✐❛ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳ ◆♦ ❧✐✈r♦ ❞❡ ❉❛♥t❡ ❬✶✶❪ é ❢❡✐t♦ ✉♠ ❝♦♠❡♥tár✐♦ s♦❜r❡ ❡ss❛ ❛❜♦r❞❛❣❡♠ ✈✐❛ ár❡❛ ❞❡ ❢❛✐①❛ ❞❡ ❤✐♣ér❜♦❧❡ s♦♠❡♥t❡ ♥♦ ✜♥❛❧ ❞♦ ❝❛♣ít✉❧♦ s♦❜r❡ ❧♦❣❛r✐t♠♦s✳ ❊♠ ♦✉tr♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❡♠ ❣❡r❛❧ é ❛♣r❡s❡♥t❛❞♦ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ❡①♣♦♥❡♥❝✐❛✐s ❡ ♥❛ s❡q✉ê♥❝✐❛ ♦ ❝♦♥t❡ú❞♦ é ❞❡s❡♥✈♦❧✈✐❞♦✳

❖ ✐♥t✉✐t♦ ❞❡ss❛ ♣❡sq✉✐s❛ é ❞❛r s✉♣♦rt❡ s✉✜❝✐❡♥t❡ ❛♦ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛ ♣❛r❛ q✉❡ tr❛❜❛❧❤❡ ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛r✐t♠♦s ❞❡ ♠❛♥❡✐r❛ ❞✐❢❡r❡♥t❡ ❡♠ r❡❧❛çã♦ ❛♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ✉s❛♥❞♦ ✉♠ s♦❢t✇❛r❡ ❝♦♠♦ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦ ❞♦ ❝♦♥t❡ú❞♦ ♠❛t❡♠át✐❝♦✳

◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ♦ s✉♣♦rt❡ t❡ór✐❝♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ✉♠ ❜♦♠ ❡♥t❡♥✲ ❞✐♠❡♥t♦ ❞♦ t❡①t♦✱ ✉♠❛ ✈❡③ q✉❡ ❛ t❡♦r✐❛ ❛♣r❡s❡♥t❛❞❛ ♥ã♦ é tr❛❜❛❧❤❛❞❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✿ ✉♠❛ ✐♥tr♦❞✉çã♦ ❛♦ ❡st✉❞♦ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❡ ❞✐❢❡r❡♥❝✐❛✐s✱ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ✭té❝♥✐❝❛s ❞❡ r❡s♦❧✉çã♦✱ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ ❛♥á❧✐s❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡✮✳ ❖s ❡①❡♠♣❧♦s ❡ ❛❜♦r❞❛❣❡♥s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♣❡r♠✐t✐rã♦ q✉❡ ❛❧❣✉♥s ❞❡ss❡s ❝♦♥❝❡✐t♦s s❡❥❛♠ ✐♥tr♦✲ ❞✉③✐❞♦s ♠❡s♠♦ q✉❡ ❞❡ ❢♦r♠❛ ✐♥t✉✐t✐✈❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛s ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s✳ ◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ s❡rá ❛♥❛❧✐s❛❞♦ ♦ ❡♥s✐♥♦ ❞❡ ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❛ ♣❛rt✐r ❞❡ ❛❧❣✉♥s ❧✐✈r♦s ❞✐❞át✐❝♦s✳ ◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ♦ s♦❢t✇❛r❡ ✏❈r❡s❝✐♠❡♥t♦ P♦♣✉❧❛❝✐♦♥❛❧✑ ❞❛ ❯◆■❈❆▼P ✲ M3₋▼❛t❡♠át✐❝❛ ❡ ▼✉❧t✐♠í❞✐❛ ❬✶✼❪✱ ❜❡♠ ❝♦♠♦ ✉♠❛ s✉❣❡stã♦ ❞❡ ❛t✐✈✐❞❛❞❡s ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛r✐t✲ ♠♦s✳ ◆♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s s♦❜r❡ ♦ tr❛❜❛❧❤♦ ❞❡s❡♥✈♦❧✈✐❞♦✳

✷ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ❡ ❊q✉❛çõ❡s

❉✐❢❡r❡♥❝✐❛✐s

▼♦❞❡❧♦s ♠❛t❡♠át✐❝♦s q✉❡ r❡❧❛❝✐♦♥❛♠ ❛s ✈❛r✐á✈❡✐s ❛tr❛✈és ❞❡ s✉❛s ✈❛r✐❛çõ❡s ❞✐s❝r❡t❛s sã♦ ❢♦r♠✉❧❛❞❛s ❝♦♠ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s✳ ❊ss❛s ❡q✉❛çõ❡s sã♦ ♠❛✐s ❛♣r♦♣r✐❛❞❛s ♣❛r❛ ♠♦❞❡❧❛r✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❡♥tr❡ ❣❡r❛çõ❡s s✉❝❡ss✐✈❛s✱ q✉❛♥❞♦ ❡ss❡ s❡ ❞á ❡♠ ❡t❛♣❛s ❞✐s❝r❡t❛s ❡ ♥ã♦ ♦❝♦rr❡ ✉♠❛ s♦❜r❡♣♦s✐çã♦ ❞❡ ❣❡r❛çõ❡s ❞❛ ❡s♣é✲ ❝✐❡ ❛♥❛❧✐s❛❞❛✳ ❏á ❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s sã♦ ✉t✐❧✐③❛❞❛s ♣❛r❛ ♠♦❞❡❧❛r s✐t✉❛çõ❡s q✉❡ r❡❧❛❝✐♦♥❛♠ ❛s ✈❛r✐á✈❡✐s ❛tr❛✈és ❞❛s ✈❛r✐❛çõ❡s ✐♥st❛♥tâ♥❡❛s✳

◆❡st❡ ❝❛♣ít✉❧♦ tr❛t❛♠♦s ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ♣r✐✲ ♠❡✐r❛ ♦r❞❡♠ ♣❛r❛ ❞❛r s✉♣♦rt❡ ❛♦ ❡st✉❞♦ ❞❡ ♠♦❞❡❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❝✐t❛❞♦s ♥♦ t❡①t♦✳

✷✳✶ ❊q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s

◆❛ ❋ís✐❝❛✱ ❊♥❣❡♥❤❛r✐❛✱ ❊❝♦♥♦♠✐❛✱ ▼❛t❡♠át✐❝❛ ❡ t❛♥t❛s ♦✉tr❛s ár❡❛s ✉t✐❧✐③❛♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ♥❛ ❞❡s❝r✐çã♦ ❞❡ ❞✐✈❡rs♦s ❢❡♥ô♠❡♥♦s✳ ❯♠ ❞♦s ❝❛s♦s ♠❛✐s tí♣✐❝♦s ❛❜♦r❞❛❞♦s ♥❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ é ♦ ❡st✉❞♦ ❞❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ❥✉r♦s ❝♦♠♣♦st♦s✱ q✉❡ tr❛t❛r❡♠♦s ❡♠ ✉♠ ❞❡ ♥♦ss♦s ❡①❡♠♣❧♦s✳ ❉❡✜♥✐çã♦ ✷✳✶✳ ❯♠❛ ❡q✉❛çã♦ ♥❛ ❢♦r♠❛

yn+k=f(yn+k−1, yn+k−2, ...., yn),

♦♥❞❡ k é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ✜①♦✱ é ❝❤❛♠❛❞❛ ❞❡ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s ❞❡ ♦r❞❡♠k✱ ♣♦✐s ❝❛❞❛ ♥ú♠❡r♦ ❞❡♣❡♥❞❡ ❞❡ k ❛♥t❡r✐♦r❡s✳

❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ ❡❧❛ ♣✉❞❡r s❡r r❡♣r❡s❡♥t❛❞❛ ♥❛ ❢♦r♠❛ yn+k+a1yn+k−1+...+anyn=Rn

♦♥❞❡ a1, a2, a3, ..., an sã♦ ♥ú♠❡r♦s r❡❛✐s✱ ❝♦♠an6= ✵ ❡ Rn é ✉♠❛ ❢✉♥çã♦ ❞❡n✱ ❞✐③❡♠♦s q✉❡ é ✉♠❛ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r ❞❡ ♦r❞❡♠ k✱ ♥ã♦ ❤♦♠♦❣ê♥❡❛ ❡ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s✳

✷✷ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s

❊①❡♠♣❧♦ ✷✳✶✳ ❊♠ s❡❣✉✐❞❛ ❡stã♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s✳

• Pn+1 = 1,8Pn é ✉♠❛ ❡q✉❛çã♦ ❧✐♥❡❛r ❤♦♠♦❣ê♥❡❛ ❞❡ ✶❛ ♦r❞❡♠❀

• an+3 =an é ✉♠❛ ❡q✉❛çã♦ ❧✐♥❡❛r ❤♦♠♦❣ê♥❡❛ ❞❡ ✸❛ ♦r❞❡♠❀

• Hn+2 = 3Hn+ 2 é ✉♠❛ ❡q✉❛çã♦ ❧✐♥❡❛r✱ ♥ã♦ ❤♦♠♦❣ê♥❡❛ ❡ ❞❡ ♦r❞❡♠ ✷✳

❖❜s❡r✈❛çã♦ ✷✳✶✳ ❆s ❡q✉❛çõ❡s ❞♦ ❊①❡♠♣❧♦ ✭✷✳✶✮ ♣♦❞❡♠ s❡r ❡s❝r✐t❛s ❞❡ ❢♦r♠❛ ❞✐❢❡r❡♥t❡ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ✈❛❧♦r ✐♥✐❝✐❛❧ ♣❛r❛n✳ P♦r ❡①❡♠♣❧♦✿ an =an−3 ♣❛r❛ n ≥3✳

❉❡✜♥✐çã♦ ✷✳✷✳ ❆s ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ sã♦ ❞❛ ❢♦r♠❛ yn+1+ayn =Rn.

❙❡Rn = 0t❡♠♦s yn+1+ayn= 0✱ ♦✉ s❡❥❛✱ ❡q✉❛çã♦ ❧✐♥❡❛r ❤♦♠♦❣ê♥❡❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❉❛❞❛ ❛ ❡q✉❛çã♦ ❧✐♥❡❛r ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ yn+1 +ayn = 0✱ ♣♦❞❡♠♦s ❡s❝r❡✈ê✲❧❛ ♥❛ ❢♦r♠❛ yn+1 = −ayn✳ ❈♦♥s✐❞❡r❛♥❞♦ y(0) = y0 ❡ s✉❜st✐t✉✐♥❞♦ n ♣♦r ✈❛❧♦r❡s ❞✐s❝r❡t♦s

t❡♠♦s✿

y1 =−ay0

y2 =−ay1 = (−a)(−a)y0 = (−a)2y0,

y3 =−ay2 = (−a)(−a)2y0 = (−a)3y0,

y4 =−ay3 = (−a)(−a)3y0 = (−a)4y0,

✳✳✳

yn= (−a)ny0. ✭✷✳✶✮

▼♦str❡♠♦s ✭✷✳✶✮ ♣♦r ■♥❞✉çã♦ ❋✐♥✐t❛✳ ❈♦♠♦ y0 é ✈á❧✐❞♦ ♣♦r s❡r ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧✱

s✉♣õ❡✲s❡ q✉❡ ✭✷✳✶✮ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛n =k ✭❤✐♣ót❡s❡ ✐♥❞✉t✐✈❛✮✱ ♦✉ s❡❥❛✱ yk= (−a)ky0.

❆❞♠✐t✐♥❞♦ ❛ r❡❝♦rrê♥❝✐❛ ❡ ❛ ❤✐♣ót❡s❡ ✐♥❞✉t✐✈❛✱ ✈❡r✐✜q✉❡♠♦s ❛ ✈❛❧✐❞❛❞❡ ♣❛r❛n=k+ 1✿ yk+1=−ayk = (−a)(−a)ky0 = (−a)k+1y0.

❆ss✐♠✱ ✭✷✳✶✮ ✈❛❧❡ ♣❛r❛ ∀n _∈ N✱ ♣♦rt❛♥t♦ é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ _{yn+1 = −ayn} ❝♦♠ y(0) =y0✳

❊①❡♠♣❧♦ ✷✳✷✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ✉♠❛ ♣❡ss♦❛ ❞❡♣♦s✐t❡ ❘✩ ✶✵✵✵✱✵✵ ♥✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ ♦❢❡r❡❝❡ ✺✪ ❞❡ ❥✉r♦s ❛♦ ♠ês✳ ❙❡❥❛ Pn ♦ ✈❛❧♦r ❛♣ós n ♠❡s❡s ❞❛ ❛♣❧✐❝❛çã♦✳ ❈♦♠♦ ❡ss❡ ✈❛❧♦r é ✐❣✉❛❧ ❛♦ ✈❛❧♦r ❞♦ ♠ês ❛♥t❡r✐♦r ✭Pn−1✮ ♠❛✐s ♦s ❥✉r♦s t❡♠♦s✿

❊q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ✷✸

❆ss✐♠✱ t❡♠♦s ✉♠❛ ❡q✉❛çã♦ ❧✐♥❡❛r ❤♦♠♦❣ê♥❡❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✿ Pn = 1,05Pn−1✳

❈♦♠♦ ♦ r❡♥❞✐♠❡♥t♦ é ♠❡♥s❛❧✱ t❡♠♦s✿

P0 = 1000,

P1 = 1,05P0 = 1,05.1000 = 1050,

P2 = 1,05P1 = 1,05.1,05P0 = (1,05)2P0 = 1102,50,

P3 = 1,05P2 = 1,05.1,052P0 = (1,05)3P0 = 1157,625,

✳✳✳

Pn= (1,05)nP0 = 1000.(1,05)n. ✭✷✳✷✮

P♦rt❛♥t♦✱ ♣❛r❛ s❛❜❡r♠♦s ♦ ✈❛❧♦r ❞❛ ❛♣❧✐❝❛çã♦ ❛♣ósn ♠❡s❡s ✉t✐❧✐③❛♠♦s ❛ ❡①♣r❡ssã♦ ✭✷✳✷✮✳

❊①❡♠♣❧♦ ✷✳✸✳ ❉❛❞❛ ❛ ❡q✉❛çã♦ yn+1 =ayn+b✱ t❡♠♦s✿

y1 =ay0+b,

y2 =ay1+b=a2y0+ab+b,

y3 =ay2+b =a3y0+a2b+ab+b,

✳✳✳

yn=any0+

n−1

X

i=0

aib. ✭✷✳✸✮

❯t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❛ s♦♠❛ ❞♦sn♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛✱ ❝♦♠ a ₆= 1✱ t❡♠♦s✿

yn =any0+b

an

−1 a₋1

. ✭✷✳✹✮

P♦r ✐♥❞✉çã♦✱ ❞❡♠♦♥str❛✲s❡ ❛ ✈❛❧✐❞❛❞❡ ❞❡ ✭✷✳✹✮✳ ❈♦♠♦ y0 é ✈á❧✐❞♦ ♣❛r❛ ❛ ❝♦♥❞✐çã♦

✐♥✐❝✐❛❧✱ ♦✉ s❡❥❛✱ y(0) = y0✱ s✉♣õ❡✲s❡ q✉❡ ✭✷✳✹✮ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ n = k ✭❤✐♣ót❡s❡

✐♥❞✉t✐✈❛✮ ❡ ❛ss✐♠✱

yk =aky0+b

ak₋₁

a₋1

.

❆❞♠✐t✐♥❞♦ ❛ r❡❧❛çã♦ ❞❡ r❡❝♦rrê♥❝✐❛ ❡ ❛ ❤✐♣ót❡s❡ ✐♥❞✉t✐✈❛✱ ✈❡r✐✜q✉❡♠♦s ❛ ✈❛❧✐❞❛❞❡ ♣❛r❛ n =k+ 1✳

yk+1 =ayk+b yk+1 =a

aky0+b

ak

−1 a₋1

+b,

yk+1 =ak+1y0+

✷✹ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s

yk+1 =ak+1y0+

bak+1_−b

a₋1 ,

yk+1 =ak+1y0+b

ak+1₋₁

a₋1

.

P♦rt❛♥t♦✱ ✭✷✳✹✮ ✈❛❧❡ ♣❛r❛ ∀n _∈N ❡ ♣♦rt❛♥t♦ é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ _{yn+1 =ayn+b}✱ ❝♦♠ y(0) =y0✳

✷✳✷ P♦♥t♦ ❞❡ ❊q✉✐❧í❜r✐♦ ❡ ❊st❛❜✐❧✐❞❛❞❡

❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❡st❛❜✐❧✐❞❛❞❡ ♣❛r❛ ❡q✉❛✲ çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s✱ ❢✉♥❞❛♠❡♥t❛✐s ♥❛ ❛♥á❧✐s❡ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s s♦❧✉çõ❡s✱ q✉❛♥t♦ ♥ã♦ s❡ t❡♠ s♦❧✉çã♦ ❛♥❛❧ít✐❝❛✳

❉❡✜♥✐çã♦ ✷✳✸✳ ❙❡❥❛ f : A _⊂ R _→ R✳ ❯♠ ♣♦♥t♦ _x∗ ♥♦ ❞♦♠í♥✐♦ ❞❡ _f é ❞❡♥♦♠✐♥❛❞♦ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❛ ❡q✉❛çã♦ xn+1 = f(xn) q✉❛♥❞♦ ❛ ♣❛rt✐r ❞❡❧❡ ♥ã♦ ♦❝♦rr❡♠ ✈❛r✐❛çõ❡s ❞♦ ❡stá❣✐♦ n ♣❛r❛ ♦ ❡stá❣✐♦ n+ 1✱ ✐st♦ é✿

xn+1 =xn=x∗,∀n∈N.

❉❡✜♥✐çã♦ ✷✳✹✳ ✭❛✮ ❖ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x∗ ❞❡ xn+1 =f(xn) é ❡stá✈❡❧ s❡ ❞❛❞♦ ε >0✱ ❡①✐st❡ δ > 0 t❛❧ q✉❡ _|x0 −x∗| < δ ✐♠♣❧✐❝❛ |fn(x0)−x∗| < ε ✱ ♣❛r❛ t♦❞♦ n > 0✳ ❙❡ x∗

♥ã♦ é ❡stá✈❡❧✱ ❡♥tã♦ ❡❧❡ é ❝❤❛♠❛❞♦ ❞❡ ✐♥stá✈❡❧✳

✭❜✮ ❖ ♣♦♥t♦ x∗ é ❞✐t♦ ❞❡ ❛tr❛çã♦ s❡ ❡①✐st✐r η > 0 t❛❧ q✉❡ _|x0 −x∗| < η ✐♠♣❧✐❝❛

limn→∞xn = x∗✳ ❙❡ ✐st♦ ❢♦r ✈á❧✐❞♦ ♣❛r❛ t♦❞♦ η > 0✱ x∗ é ❝❤❛♠❛❞♦ ♣♦♥t♦ ❞❡ ❛tr❛t♦r ❣❧♦❜❛❧✳

✭❝✮ ❖ ♣♦♥t♦ x∗ é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧ s❡ é ❡stá✈❡❧ ❡ ❛tr❛✲ t♦r✳ ❙❡ ✐st♦ ❢♦r ✈á❧✐❞♦ ♣❛r❛ t♦❞♦ η >0✱ x∗ é ❝❤❛♠❛❞♦ ❞❡ ❣❧♦❜❛❧♠❡♥t❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✳

❚❡♦r❡♠❛ ✷✳✶✳ ❖ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x∗ = b

1−a ♣❛r❛ ♦ s✐st❡♠❛ ❞✐♥â♠✐❝♦ ❛✜♠ xn+1 = axn+b✱ ❝♦♠a 6= 1✱ é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧ s❡|a|<1✱ ♦✉ s❡❥❛✱limn→∞xn=x∗ ♣❛r❛ t♦❞♦ x0✳ ❙❡ |a| > 1 ❡♥tã♦ x∗ é ✐♥stá✈❡❧ ❡ |xn| → ∞ ♣❛r❛ q✉❛❧q✉❡r ✈❛❧♦r ❞❡ x0 6= x∗✳

◗✉❛♥❞♦ a=₋1✱ t❡♠♦s ♦ q✉❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✷✲❝✐❝❧♦✱ ✐st♦ é✱ xk+2 =xk,∀k ∈N✳ ❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡ q✉❡

|x1−x∗|=|ax0+b−

b 1₋a|,

|x1−x∗|=|ax0−

ab

1₋a|=|a||x0−x

∗

|. ❆♥❛❧♦❣❛♠❡♥t❡✱

P♦♥t♦ ❞❡ ❊q✉✐❧í❜r✐♦ ❡ ❊st❛❜✐❧✐❞❛❞❡ ✷✺

❊✱ ♣♦r ✐♥❞✉çã♦✱ s❡❣✉❡ q✉❡

|xk−x∗|=|a|k|x0−x∗|.

❙✉♣♦♥❤❛ q✉❡|a_|<1✳ ❊♥tã♦✱

lim k→∞|a|

k _{= 0}

❡

lim

k→∞|xk−x ∗

|= 0. ▲♦❣♦✱ x∗ é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✳

❙❡|a_|>1✱ ❡♥tã♦ _|a_|k _{→ ∞✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ x}

k ❡stá s❡ ❞✐st❛♥❝✐❛♥❞♦ ❞❡ x∗✳ ▲♦❣♦✱ x∗ é ✐♥stá✈❡❧✳ ❙❡ a = ₋1✱ ❡♥tã♦ xn+2 = −xn+1+b = −(−xn+b) +b = xn✳ ❆ss✐♠✱ q✉❛❧q✉❡r ♦✉tr♦ ✈❛❧♦r é ✐❣✉❛❧✱ ♦✉ s❡❥❛✱x0 =x2 =...✱ ❡x1 =x3 =...✱ ❝♦♥❝❧✉✐♥❞♦ ❛ ♣r♦✈❛✳

❚❡♦r❡♠❛ ✷✳✷✳ ❆ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x∗ ❞❡ xn+1 =f(xn) ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ ✈❛❧♦r ❞♦ ♠ó❞✉❧♦ ❞❡

λ=

df(xn) dxn

xn=x∗

♦♥❞❡ λ é ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ f(xn) ♥♦ ♣♦♥t♦ x∗✳ ❚❡♠♦s✿

• ❙❡ −1 < λ < 1✱ x∗ _{é ❧♦❝❛❧♠❡♥t❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✱ ✐st♦ é✱ s❡ x}

0 ❡stá

♣ró①✐♠♦ ❞❡ x∗ _{❡♥tã♦ x}

n ❝♦♥✈❡r❣❡ ♣❛r❛ x∗❀

❆✐♥❞❛✱ s❡ 0 < λ < 1 ❡♥tã♦ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ é ♠♦♥ót♦♥❛❀ s❡ ₋1 < λ < 0✱ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ é ♦s❝✐❧❛tór✐❛✳

• ❙❡ |λ_|>1✱ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x∗ é ✐♥stá✈❡❧✳ ❉❡♠♦♥str❛çã♦✳ P♦❞❡ s❡r ✈❡r✐✜❝❛❞❛ ❡♠ ❬✶✵❪✳

❖❜s❡r✈❛çã♦ ✷✳✷✳ ❙❡ |λ_|= 1✱ sã♦ ♥❡❝❡ssár✐♦s ❝r✐tér✐♦s ❛❞✐❝✐♦♥❛✐s✳ ❊①❡♠♣❧♦ ✷✳✹✳ ❈♦♥s✐❞❡r❛♠♦s ❛ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s

xn+1 = 0,5xn(1−xn). ✭✷✳✺✮ ❖s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ ✭✷✳✺✮ sã♦ ❞❛❞♦s ♣♦r ✿

x∗ = 0,5x∗(1₋x∗),

0,5(x∗)2_{+ 0,5x}∗ _{= 0.}

P♦rt❛♥t♦✱ x∗ = 0 ♦✉x∗ =₋1✳ ◆❡st❡ ❝❛s♦✱f(x) = 0,5x(1₋x)❡ ❛ss✐♠✱ f′(x) = 0,5₋x✳ ❈❛❧❝✉❧❛♥❞♦ λ✱ t❡♠♦s✿

λ= 0,5₋x∗

◗✉❛♥❞♦ x∗ _{= 0 ⇒ λ = 0,5✱ ♦✉ s❡❥❛✱ x}∗ _{= 0 é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✳ ◗✉❛♥❞♦}

✷✻ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s

✷✳✸ ❈♦❜✇❡❜

❈♦❜✇❡❜ é ✉♠ ✐♠♣♦rt❛♥t❡ ♠ét♦❞♦ ❣rá✜❝♦ ♣❛r❛ ❛♥á❧✐s❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ♣❛r❛ ❛ ❡q✉❛çã♦ xn+1 =f(xn)✳ ❊ss❡ ♠ét♦❞♦✱ q✉❡ ♥❛ s✉❛ tr❛❞✉çã♦ ❧✐t❡r❛❧ s✐❣♥✐✜❝❛ ✏t❡✐❛ ❞❡ ❛r❛♥❤❛✑ s❡rá ❛♣r❡s❡♥t❛❞♦ ❛ s❡❣✉✐r✳

❯♠❛ ✈❡③ q✉❡ xn+1 = f(xn)✱ ♣♦❞❡♠♦s tr❛ç❛r ♦ ❣rá✜❝♦ ❞❡ f ♥♦ ♣❧❛♥♦ ✭xn, xn+1✮✳

❊♥tã♦✱ ❞❛❞♦x0✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ♦ ✈❛❧♦r ❞❡x1 ❞❡s❡♥❤❛♥❞♦ ✉♠❛ r❡t❛ ✈❡rt✐❝❛❧ ❛tr❛✈és

❞❡ x0 ❞❡ ♠♦❞♦ q✉❡ t❛♠❜é♠ ✐♥t❡r❝❡♣t❡ ♦ ❣rá✜❝♦ ❞❡ f ♥♦ ♣♦♥t♦ ✭x0, x1✮✳ ❊♠ s❡❣✉✐❞❛✱

tr❛❝❡ ✉♠❛ r❡t❛ ❤♦r✐③♦♥t❛❧ ❞❡ ✭x0, x1✮ ❛té ❡♥❝♦♥tr❛r ❛ r❡t❛y=x ✭❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡✮ ♥♦

♣♦♥t♦ ✭x1, x1✮✳ ❯♠❛ r❡t❛ ✈❡rt✐❝❛❧ tr❛ç❛❞❛ ❞♦ ♣♦♥t♦ ✭x1, x1✮ ✐♥t❡r❝❡♣t❛rá ♦ ❣rá✜❝♦ ❞❡f

♥♦ ♣♦♥t♦ ✭x1, x2✮✳ ❈♦♥t✐♥✉❛♥❞♦ ❡ss❡ ♣r♦❝❡ss♦✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛rxn ♣❛r❛ t♦❞♦ n >0✳ ❱❛♠♦s ❛♥❛❧✐s❛r ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❝♦❜✇❡❜✳

❊①❡♠♣❧♦ ✷✳✺✳ ❆ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s xn+1 =−

1 2xn+

1

2, ✭✷✳✻✮

t❡♠ ❝♦♠♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x∗ = 1

3✱ q✉❡ é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧ ♣♦✐s✱ ✉s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✭✷✳✷✮ ❞❛ s❡çã♦ ❛♥t❡r✐♦r✱ t❡♠♦s q✉❡ λ=₋1

2✳ ❱❡❥❛♠♦s ❛tr❛✈és ❞♦ ❝♦❜✇❡❜✳

❋✐❣✉r❛ ✷✳✶✿ ❈♦❜✇❡❜ ❞❛ ❡q✉❛çã♦ ✭✷✳✻✮ ❝♦♠ x0 = 3₄ ✳

❆❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ✷✼

❊①❡♠♣❧♦ ✷✳✻✳ ◆❛ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s xn+1=−

3 2xn+

3

2, ✭✷✳✼✮

t❡♠♦s q✉❡ |λ_| > 1✱ ♦✉ s❡❥❛✱ ✉t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✭✷✳✷✮ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ú♥✐❝♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡①✐st❡♥t❡ é ✐♥stá✈❡❧✳

◆♦✈❛♠❡♥t❡✱ ✐♥✐❝✐❛♥❞♦ ♦ ❝♦❜✇❡❜ ♣♦r x0 = 3₄, ♣❡r❝❡❜❡♠♦s q✉❡ xn s❡ ❛❢❛st❛ ❞♦ ♣♦♥t♦ ✜①♦✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ ✐♥st❛❜✐❧✐❞❛❞❡✳

❋✐❣✉r❛ ✷✳✷✿ ❈♦❜✇❡❜ ❞❛ ❡q✉❛çã♦ ✭✷✳✼✮ ❝♦♠ x0 = 3₄ ✳

✷✳✹

❆❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s

❊①❡♠♣❧♦ ✷✳✼✳ ▼♦❞❡❧♦ ❞✐s❝r❡t♦ ❞❡ ▼❛❧t❤✉s✳ ❯♠ ♠♦❞❡❧♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉✲ ❧❛❝✐♦♥❛❧ ❜❛st❛♥t❡ ❝♦♥❤❡❝✐❞♦ é ❞♦ ❡❝♦♥♦♠✐st❛ ✐♥❣❧ês ❚❤♦♠❛s ▼❛❧t❤✉s✱ ❛♣r❡s❡♥t❛❞♦ ❡♠ ✶✼✾✽✳ ❖ ♠♦❞❡❧♦ ♠❛❧t❤✉s✐❛♥♦ ♣r❡ss✉♣õ❡ q✉❡ ❛ ✈❛r✐❛çã♦ ❞❛ ♣♦♣✉❧❛çã♦ ❡♥tr❡ ♦s ✐♥st❛♥t❡s t ❡ t+ 1 é ♣r♦♣♦r❝✐♦♥❛❧ à ♣♦♣✉❧❛çã♦ ♥♦ ✐♥st❛♥t❡ t✳

P(t+ 1)₋P(t) = αP(t). ✭✷✳✽✮ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ♣♦♣✉❧❛çã♦ ✐♥✐❝✐❛❧ P(0) =P0 t❡♠♦s✿

(

P(t+ 1) = (1 +α)P(t) P(0) =P0,

✷✽ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s

♦✉ s❡❥❛✱

P(t) = (1 +α)tP0.

❊①❡♠♣❧♦ ✷✳✽✳ ▼♦❞❡❧♦ ❞✐s❝r❡t♦ ❞❡ ❱❡r❤✉❧st✳ ❬✶✽❪ ❱❡r❤✉❧st ❢♦✐ ✉♠ ♠❛t❡♠át✐❝♦ ❜❡❧❣❛ q✉❡ ✐♥tr♦❞✉③✐✉ ❛ ❡q✉❛çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❧♦❣íst✐❝♦ ♦♥❞❡ ❛ ♣♦♣✉❧❛çã♦ ❞❡✈❡rá ❝r❡s❝❡r ❛té ✉♠ ❧✐♠✐t❡ ♠á①✐♠♦ s✉st❡♥tá✈❡❧✱ ✐st♦ é✱ ❡❧❛ t❡♥❞❡ ❛ s❡ ❡st❛❜✐❧✐③❛r ♥✉♠ ❞❡t❡r♠✐♥❛❞♦ ✈❛❧♦r✳ ❖ ♠♦❞❡❧♦ ❞❡ ❱❡r❤✉❧st é✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ ♦ ♠♦❞❡❧♦ ❞❡ ▼❛❧t❤✉s ♠♦❞✐✜❝❛❞♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ✈❛r✐❛çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ♣ró♣r✐❛ ♣♦♣✉❧❛çã♦ ❡♠ ❝❛❞❛ ✐♥st❛♥t❡ ❡ s❛t✐s❢❛③❡♥❞♦ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s✳

❙❡❥❛P(n)❛ ♣♦♣✉❧❛çã♦ ♥♦ ✐♥st❛♥t❡n✳ ❊♥tã♦✱ ♦ ❝r❡s❝✐♠❡♥t♦ ❛❜s♦❧✉t♦ ❞❡P(n)é ❞❛❞♦ ♣♦r✿

Pn+∆n−Pn = (α−βPn)Pn∆n.

❈♦♥s✐❞❡r❛♥❞♦ P0 ❞❛❞♦✱ ♣♦❞❡♠♦s ♦❜t❡r Pn ❡♠ ❢✉♥çã♦ ❞❡ P0 ❛tr❛✈és ❞❛ ❢ór♠✉❧❛ ❞❡

r❡❝♦rrê♥❝✐❛✿

(

Pn+∆n = (α∆n+ 1)Pn

h

1₋ β∆n

α∆n+1Pn

i

P0,

✭✷✳✶✵✮

❆ ❡q✉❛çã♦ ✭✷✳✶✵✮ ♣♦❞❡ s❡r ❞❛❞❛ ♥❛ ❢♦r♠❛ ♥♦r♠❛❧✐③❛❞❛ ♣♦r✿

 



Nn+1 =rNn(1−Nn) N0 =

P0

P∞

, ✭✷✳✶✶✮

♦♥❞❡ Nn = _α∆β_n+1Pn ❡ (α∆n+ 1) = r✳ P∞ é ♦ ✈❛❧♦r ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♦✉ ✈❛❧♦r ♠á①✐♠♦

s✉st❡♥tá✈❡❧✳

✷✳✺ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s

◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s✭❊❉❖✮✱ ❛❧❣✉♠❛s ❝❧❛ss✐✜❝❛çõ❡s ❡ ♠ét♦❞♦s ❞❡ r❡s♦❧✉çã♦✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♥♦✈❛♠❡♥t❡ tr❛t❛r ♦s ♠♦❞❡❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❞❡ ▼❛❧t❤✉s ❡ ❱❡r❤✉❧st✱ ❞❡st❛ ✈❡③ ❝♦♥tí♥✉♦s✳ ❉❡✜♥✐çã♦ ✷✳✺✳ ❯♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ é ✉♠❛ ❡q✉❛çã♦ q✉❡ ❡♥✈♦❧✈❡ ❢✉♥çõ❡s ✐♥❝ó❣♥✐t❛s ❡ s✉❛s ❞❡r✐✈❛❞❛s✳ ➱ ❝❤❛♠❛❞❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛✭❊❉❖✮ s❡ ❛ ❢✉♥çã♦ ✐♥❝ó❣♥✐t❛ ❞❡♣❡♥❞❡r ❛♣❡♥❛s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧✳

✷✳✻ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ❞❡ ✶

_{❖r❞❡♠}

❞♦ t✐♣♦ dy

dx = F(x, y) ♦✉ y

❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ✶❛ _{♦r❞❡♠ ❧✐♥❡❛r❡s ✷✾}

✐♥t❡r✈❛❧♦ ❛❜❡rt♦ ❞♦ R2✳ ❙❡ ❛ ❢✉♥çã♦ _{F(x, y)}♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ♦ q✉♦❝✐❡♥t❡ ❞❡ ❞✉❛s ♦✉tr❛s ❢✉♥çõ❡s M(x, y) ❡ N(x, y)✱ ❧❡♠❜r❛♥❞♦ q✉❡y′ = dy

dx✱ t❡♠♦s ❡♥tã♦ ♦✉tr❛ ❢♦r♠❛ ❞❡ ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛ ❞❡ ✶❛ _{♦r❞❡♠✿ M(x, y)dx+N(x, y)dy= 0✳}

❯♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ dy

dx = F(x, y) ♥✉♠ ✐♥t❡r✈❛❧♦ α < x < β é ✉♠❛ ❢✉♥çã♦ φ t❛❧ q✉❡ φ′ _{❡①✐st❡ ❡ s❛t✐s❢❛③ φ}′_{(x) = F(x, φ(x))✱ ∀xǫ(α, β)✳}

❊①❡♠♣❧♦ ✷✳✾✳ ❱❛♠♦s ❞❡t❡r♠✐♥❛r ✉♠❛ s♦❧✉çã♦ y=y(x) ❞❛ ❡q✉❛çã♦ dy

dx = 2x+ 3 q✉❡ s❛t✐s❢❛ç❛ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ y(0) = 5✳

❆ss✐♠✱ q✉❡r❡♠♦s ✉♠❛ ❢✉♥çã♦ y = y(x)✱ ❝♦♠ y(0) = 5✱ ❝✉❥❛ ❞❡r✐✈❛❞❛ s❡❥❛ 2x+ 3✳ ❊♥tã♦✱ y =R

(2x+ 3)dx = x2 + 3x+k✱ k _∈ R✱ sã♦ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦✳ ❙❛❜❡♥❞♦✲s❡ q✉❡ y(0) = 5✱ ♦❜t❡♠♦s k = 5✳ ❙❡❣✉❡ q✉❡ y = x2 + 3x+ 5 é ❛ s♦❧✉çã♦ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❛❞❛✳

❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ❡st❛❜❡❧❡❝❡ s♦❜ q✉❡ ❝♦♥❞✐çõ❡s ♣♦❞❡♠♦s ❣❛r❛♥t✐r q✉❡ ✉♠ ♣r♦✲ ❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ t❡♠ ú♥✐❝❛ s♦❧✉çã♦✳

❚❡♦r❡♠❛ ✷✳✸✳ ❚❡♦r❡♠❛ ❞❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ✉♠ ♣r♦✲ ❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ✭❚❊❯✮

❙✉♣♦♥❤❛ q✉❡ ❛s ❢✉♥çõ❡s f ❡ ∂f_∂y sã♦ ❝♦♥tí♥✉❛s ❡♠ ❛❧❣✉♠ r❡tâ♥❣✉❧♦ α _≤ t _≤ β ✱ γ _≤ y _≤ δ ❝♦♥t❡♥❞♦ ♦ ♣♦♥t♦ (t0, y0)✳ ❊♥tã♦✱ ❡♠ ❛❧❣✉♠ ✐♥t❡r✈❛❧♦ t0 −h < t < t0 +h

❝♦♥t✐❞♦ ❡♠ α < t < β ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ (

y′ _{=f(t, y)}

y(t0) =y0

✭✷✳✶✷✮

P❛r❛ ✉♠ ❡st✉❞♦ ♠❛✐s ❞❡t❛❧❤❛❞♦ ❞❡ss❡ t❡♦r❡♠❛ ♦ ❧❡✐t♦r ♣♦❞❡ ❝♦♥s✉❧t❛r ❬✶❪ ♦✉ ❬✷❪✳

✷✳✼ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ✶

_{♦r❞❡♠ ❧✐♥❡❛r❡s}

❉❡✜♥✐çã♦ ✷✳✼✳ ❯♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛✭❊❉❖✮ ❞❡ ✶❛ _{♦r❞❡♠ é ❧✐♥❡❛r✱ s❡}

♣✉❞❡r s❡r ❡s❝r✐t❛

y′+p(x)y =g(x), ✭✷✳✶✸✮ ♦♥❞❡ p ❡ g sã♦ ❢✉♥çõ❡s r❡❛✐s ❡ ❝♦♥tí♥✉❛s✳

P❛r❛ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ ✭✷✳✶✸✮ ♠✉❧t✐♣❧✐❝❛♠♦s ♦s ❞♦✐s ♠❡♠❜r♦s ❞❛ ♠❡s♠❛ ♣♦r ✉♠ ❢❛t♦r µ(x)✱ ♦❜t❡♥❞♦

✸✵ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s

dµ(x)

dx =p(x)µ(x). ❙✉♣♦♥❞♦ µ(x) ♣♦s✐t✐✈♦✱ ♦❜t❡♠♦s

dµ(x)

dx

µ(x) =p(x)

q✉❡✱ ✐♥t❡❣r❛♥❞♦ ❛♠❜♦s ❧❛❞♦s✱ ♦❜t❡♠♦s✿

lnµ(x) = Z

p(x)dx+k.

❊s❝♦❧❤❡♥❞♦ ❛ ❝♦♥st❛♥t❡ ❛r❜✐trár✐❛ k ♥✉❧❛✱ ❡ ❛♣❧✐❝❛♥❞♦ ❛ ❡①♣♦♥❡♥❝✐❛❧✱ ♦❜t❡♠♦s ❛ ❢✉♥çã♦ ♠❛✐s s✐♠♣❧❡s ♣♦ssí✈❡❧ ♣❛r❛ µ✿

µ(x) = eRp(x)dx. ❘❡❡s❝r❡✈❡♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✹✮✱ t❡♠♦s✿

d

dx[µ(x)y] =µ(x)g(x) ❡ ✐♥t❡❣r❛♥❞♦✱ ♦❜t❡♠♦s✿

µ(x)y= Z

µ(x)g(x)dx+c.

P♦rt❛♥t♦✱

y= 1 µ(x)

Z

µ(x)g(x)dx+c

,

♦♥❞❡ c_∈R é ✉♠❛ ❝♦♥st❛♥t❡ ❛r❜✐trár✐❛✳ ❆ ❢✉♥çã♦ _µ(x)é ❝❤❛♠❛❞❛ ❞❡ ❢❛t♦r ✐♥t❡❣r❛♥t❡✳ ❊①❡♠♣❧♦ ✷✳✶✵✳ P❛r❛ ❛ ❡q✉❛çã♦y′₋2xy=x✱ t❡♠♦sp(x) = ₋2x❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ ❢❛t♦r ✐♥t❡❣r❛♥t❡ é ❞❛❞♦ ♣♦r µ(x) = eR−2xdx _{= e}−x2

✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ♣♦r µ(x) ♦❜t❡♠♦s

e−x2y′₋2xe−x2y=xe−x2 =_⇒ d dx[ye

−x2

] =xe−x2.

■♥t❡❣r❛♥❞♦ ❛♠❜♦s ♠❡♠❜r♦s t❡♠♦s✿

ye−x2 = Z

xe−x2dx,

ye−x2 =₋1 2e

−x2

+c,

y=ce−x2 ₋ 1

❊q✉❛çõ❡s s❡♣❛rá✈❡✐s ✸✶

✷✳✽ ❊q✉❛çõ❡s s❡♣❛rá✈❡✐s

❉❡✜♥✐çã♦ ✷✳✽✳ ❯♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✶❛ _{♦r❞❡♠ ❞❡ ✈❛r✐á✈❡✐s s❡♣❛rá✈❡✐s é ✉♠❛}

❡q✉❛çã♦ ❞❛ ❢♦r♠❛

dy

dx =g(x)h(y), ✭✷✳✶✺✮

♦♥❞❡ g ❡ h sã♦ ❢✉♥çõ❡s r❡❛✐s ❞❡✜♥✐❞❛s ❡♠ ✐♥t❡r✈❛❧♦s ❛❜❡rt♦s ❞❡ R✳

❖❜t❡r ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✷✳✶✺✮ é ❡♥❝♦♥tr❛r ✉♠❛ ❢✉♥çã♦y=y(x)✱ t❛❧ q✉❡

y′(x) =g(x)h(y(x)). ✭✷✳✶✻✮ ❈♦♥s✐❞❡r❛♥❞♦ h(y(x))₆= 0 ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ✭✷✳✶✻✮ ❝♦♠♦ ✿

y′(x)

h(y(x)) =g(x).

■♥t❡❣r❛♥❞♦ ❡♠ r❡❧❛çã♦ ❛ x t❡♠♦s✿ Z

y′(x)dx h(y(x)) =

Z

g(x)dx.

❈♦♠ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ v =y(x)_⇒dv=y′_{(x)dx t❡♠♦s}

Z _dv

h(v) = Z

g(x)dx.

P❛r❛ ♦❜t❡r ❛ s♦❧✉çã♦ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ❞❛❞❛ é só ✐♥t❡❣r❛r ❡ ❡s❝♦❧❤❡r k ❛❞❡q✉❛❞❛♠❡♥t❡✱ ♣♦rt❛♥t♦

H(y) =G(x) +k,

♦♥❞❡ H(y) ❡G(x) sã♦ ♣r✐♠✐t✐✈❛s ❞❡ 1

h(y) ❡ g(x) r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊①❡♠♣❧♦ ✷✳✶✶✳ P❛r❛ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ y′ = x+ 1

y4_{+ 1} ❛ ❡s❝r❡✈❡♠♦s ❝♦♠♦✿

(y4+ 1)dy= (x+ 1)dx. ■♥t❡❣r❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s t❡♠♦s✿

Z

(y4+ 1)dy= Z

(x+ 1)dx,

y5

5 +y= x2

2 +x+k.

P♦rt❛♥t♦✱ x2

2 +x− y5

✸✷ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s

❊①❡♠♣❧♦ ✷✳✶✷✳ ❱❛♠♦s ♦❜t❡r ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ dy

dt =−ay+b✱ ♦♥❞❡ a ❡ b sã♦ ❝♦♥st❛♥t❡s r❡❛✐s ❞❛❞❛s✱ a₆= 0 ❡ y₆= _ab✳

P♦❞❡♠♦s ❡s❝r❡✈❡r ❡st❛ ❡q✉❛çã♦ ♥❛ ❢♦r♠❛✿ 1 y_{− a}b

dy

dt =−a.

❊♥tã♦✱ ✐♥t❡❣r❛♥❞♦✱ ♦❜t❡♠♦s

ln_|y₋ b

a|=−at+k,

♦♥❞❡ k _∈R✳ ❆♣❧✐❝❛♥❞♦ ❡①♣♦♥❡♥❝✐❛❧ ❡♠ ❛♠❜♦s ❧❛❞♦s ❞❛ ❡q✉❛çã♦ t❡♠♦s✿

|y₋ b a|=e

−at+k _=ek_e−at_.

▲❡♠❜r❛♥❞♦✲s❡ q✉❡ k é ❝♦♥st❛♥t❡✱ ❡♥tã♦ ek _{t❛♠❜é♠ é ✉♠❛ ❝♦♥st❛♥t❡ ❡ s❡rá ❞❡♥♦t❛❞❛} ♣♦r c✳ ❚❡♠♦s ❝♦♠♦ s♦❧✉çã♦✿

y= b a +ce

−at_.

❊①❡♠♣❧♦ ✷✳✶✸✳ ❬✶❪ ❈♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦ dp

dt = 0,5p−450,

q✉❡ ❞❡s❝r❡✈❡ ❛ ✐♥t❡r❛çã♦ ❞❡ ❞❡t❡r♠✐♥❛❞❛s ♣♦♣✉❧❛çõ❡s ❞❡ r❛t♦s ❞♦ ❝❛♠♣♦ ❡ ❝♦r✉❥❛s✳ ❊♥❝♦♥tr❡♠♦s ❛ ♣♦♣✉❧❛çã♦ ❞❡ r❛t♦ p(t) ❛♣ós ✉♠ t❡♠♣♦ t✱ q✉❡ s❛t✐s❢❛ç❛ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ❞❡ ✽✺✵ r❛t♦s✳

❊s❝r❡✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❝♦♠♦✿ dp dt =

p₋900

2 ❡✱ s❡ p6= 900✱ t❡♠♦s✿ 1

p₋900 dp

dt = 1 2.

■♥t❡❣r❛♥❞♦✱ ♦❜t❡♠♦s✿

ln_|p₋900_|= t 2 +k, ♦♥❞❡ k _∈R✳ ❆♥❛❧♦❣❛♠❡♥t❡ ❛♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ t❡♠♦s✿

|p₋900_|=e2t+k =ek.e t

2 =⇒p−900 =±ek.e t 2.

❊✱ ❝♦♠♦ c=_±ek _{é ✉♠❛ ❝♦♥st❛♥t❡✱ ❡♥tã♦✿}

p= 900 +ce2t.

❆✐♥❞❛ s❛❜❡♠♦s q✉❡ p(0) = 850 ✱ ♦✉ s❡❥❛✱

❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ✸✸

✷✳✾ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s

❊①❡♠♣❧♦ ✷✳✶✹✳ ▼♦❞❡❧♦ ❞❡ ▼❛❧t❤✉s✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦ ♠♦❞❡❧♦ ❞❡ ▼❛❧t❤✉s ❞♦ ❊①❡♠✲ ♣❧♦ ✭✷✳✼✮ ❛ ✈❡rsã♦ ❝♦♥tí♥✉❛ ♣r❡ss✉♣õ❡ q✉❡ ❛ t❛①❛ s❡❣✉♥❞♦ ❛ q✉❛❧ ❛ ♣♦♣✉❧❛çã♦ ❞❡ ✉♠ ♣❛ís ❝r❡s❝❡ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✐♥st❛♥t❡ é ♣r♦♣♦r❝✐♦♥❛❧ ❛ ♣♦♣✉❧❛çã♦ t♦t❛❧ ❞♦ ♣❛ís ♥❛q✉❡❧❡ ✐♥st❛♥t❡✳ ▼❛t❡♠❛t✐❝❛♠❡♥t❡✱ s❡P(t) é ❛ ♣♦♣✉❧❛çã♦ t♦t❛❧ ♥♦ ✐♥st❛♥t❡ t✱ ❡♥tã♦ ♦ ♠♦❞❡❧♦ ❝♦♥tí♥✉♦ é ❞❛❞♦ ♣♦r✿

dP

dt =kP,

♦♥❞❡ k é ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ✭♥❡ss❡ ❝❛s♦ k > 0✮✳ ❊ss❡ ♠♦❞❡❧♦ é ✉t✐❧✐③❛❞♦ ♥♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ♣❡q✉❡♥❛s ♣♦♣✉❧❛çõ❡s ❡♠ ✉♠ ❝✉rt♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ❜❛❝tér✐❛s✱ ♣♦✐s ♥ã♦ ❧❡✈❛ ❡♠ ❝♦♥t❛ ♠✉✐t♦s ❢❛t♦r❡s q✉❡ ♣♦❞❡♠ ✐♥✢✉❡♥❝✐❛r ❛ ♣♦♣✉❧❛çã♦ t❛♥t♦ ❡♠ s❡✉ ❝r❡s❝✐♠❡♥t♦ q✉❛♥t♦ ❡♠ s❡✉ ❞❡❝❧í♥✐♦✳

❙❛❜❡♥❞♦✲s❡ q✉❡ ✉♠❛ ❝❡rt❛ ♣♦♣✉❧❛çã♦ ❝r❡s❝❡ s❡❣✉♥❞♦ ♦ ♠♦❞❡❧♦ ♠❛❧t❤✉s✐❛♥♦✱ t❡♠♦s✿ dP

P =kdt, k >0,

Z _dP

P = Z

kdt,

lnP =kt+c,

P =ec.ekt.

❙❡ P(0) =P0✱ ❡♥tã♦

P =P0ekt.

❊①❡♠♣❧♦ ✷✳✶✺✳ ❉❡❝❛✐♠❡♥t♦ ❘❛❞✐♦❛t✐✈♦✳ ◆♦s ❡❧❡♠❡♥t♦s r❛❞✐♦❛t✐✈♦s ♦ ♥ú♠❡r♦ ❞❡ ♥ú❝❧❡♦s q✉❡ ❞❡❝❛❡♠ ♣♦r ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ♥ú♠❡r♦ t♦t❛❧ ✭N✮ ❞❡ ♥ú❝❧❡♦s r❛❞✐♦❛t✐✈♦s✳ ❆ss✐♠ t❡♠♦s✿

dN

dt =αN, ♦♥❞❡ α é ❛ ❝♦♥st❛♥t❡ ❞❡ ❞❡s✐♥t❡❣r❛çã♦ ✭α <0✮✳

❖❜s❡r✈❛çã♦ ✷✳✸✳ ❯♠❛ ú♥✐❝❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣♦❞❡ s❡r✈✐r ❝♦♠♦ ✉♠ ♠♦❞❡❧♦ ♠❛t❡✲ ♠át✐❝♦ ♣❛r❛ ❡st✉❞♦ ❞❡ ❢❡♥ô♠❡♥♦s ❞✐❢❡r❡♥t❡s✳ ❖s ❞♦✐s ♣r♦❜❧❡♠❛s ❝✐t❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡ ✉s❛♠ ♦ ♠❡s♠♦ ♠♦❞❡❧♦✱ ❞✐❢❡r✐♥❞♦ ❛♣❡♥❛s ♥❛ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❡♠ q✉❡ ♥✉♠ é ♥❡❣❛t✐✈♦ ❡ ♥♦ ♦✉tr♦ é ♣♦s✐t✐✈♦✳ ❊ss❡ ♠❡s♠♦ ♠♦❞❡❧♦ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♥♦ ❡st✉❞♦ ❞♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ✉♠ ❝❛♣✐t❛❧ ❛ ❥✉r♦s ❝♦♠♣♦st♦s ❝♦♥t✐♥✉❛♠❡♥t❡✱ ❛ ♠❡✐❛ ✈✐❞❛ ❞❡ ✉♠❛ ❞r♦❣❛✱ ❡t❝✳

✸✹ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s

r❡♣r❡s❡♥t❛r ❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✉♠ ❝♦r♣♦ ♥♦ ✐♥st❛♥t❡ t✱ Tm ❛ t❡♠♣❡r❛t✉r❛ ❞♦ ♠❡✐♦ q✉❡ ♦ r♦❞❡✐❛ ❡ dT

dt ❛ t❛①❛ s❡❣✉♥❞♦ ❛ q✉❛❧ ❛ t❡♠♣❡r❛t✉r❛ ❞♦ ❝♦r♣♦ ✈❛r✐❛✱ ❛ ❧❡✐ ❞❡ ◆❡✇t♦♥ ❞♦ ❡s❢r✐❛♠❡♥t♦✴❛q✉❡❝✐♠❡♥t♦ é ❝♦♥✈❡rt✐❞❛ ♥❛ s❡♥t❡♥ç❛ ♠❛t❡♠át✐❝❛

dT

dt =k(T −Tm), ✭✷✳✶✼✮

♦♥❞❡ k é ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✳ ❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ❡s❢r✐❛♠❡♥t♦ ♦✉ ❛q✉❡❝✐♠❡♥t♦✱ s❡ Tm ❢♦r ✉♠❛ ❝♦♥st❛♥t❡✱ ❡♥tã♦ k <0✳

❈♦♥s✐❞❡r❛♥❞♦ T(0) =T0✱ r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✼✮✱ t❡♠♦s✿

Z _dT

T ₋Tm =

Z kdt,

❡♥tã♦

ln_|T ₋Tm|=kt+c,

♦♥❞❡ c_∈R✳ ❆ss✐♠✱ t❡♠♦s✿ _eln|T−Tm|=_ekt+c✳ ❈❤❛♠❛r❡♠♦s _ec ❞❡ _C✱ _C

∈R✱ ♦✉ s❡❥❛ T(t) =Cekt+Tm.

❈♦♠♦ T(0) =T0✱ t❡♠♦s✿ C =T0−Tm✳ P♦rt❛♥t♦✱ ❛ s♦❧✉çã♦ ❞❡ ✭✷✳✶✼✮ é ❞❛❞❛ ♣♦r✿ T(t) = (T0−Tm)ekt+Tm.

❊①❡♠♣❧♦ ✷✳✶✼✳ ▼♦❞❡❧♦ ❞❡ ❱❡r❤✉❧st✳ ❖ ♠♦❞❡❧♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❝♦♥✲ tí♥✉♦ ❞❡ ❱❡r❤✉❧st é ❞❛❞♦ ♣♦r✿

 

 dP

dt =rP

1₋ P P∞

P(0) = P0, r >0,

✭✷✳✶✽✮

♦♥❞❡ P(0) = P0 é ❛ ♣♦♣✉❧❛çã♦ ✐♥✐❝✐❛❧✳ ❘❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✽✮ ♣❡❧♦ ♠ét♦❞♦ ❞❡

s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s t❡♠♦s✿

Z _dP

P(1_{− P}P∞)

= Z rdt, Z 1 P + 1 P∞

1₋ P P∞

! dP =

Z rdt,

ln_|P_{| −}ln_|1₋ P P∞|

= Z

rdt,

ln_| P 1₋ P

P∞

|=rt+c.

P♦❞❡♠♦s ✉s❛r ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ P(0) =P0 ❡ ❞❡t❡r♠✐♥❛r ❛ ❝♦♥st❛♥t❡ ❞❡ ✐♥t❡❣r❛çã♦ c✳

c=ln_| P0 1₋ P0

P∞

|=ln_| P0P∞ P∞−P0|

❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ✸✺

P♦rt❛♥t♦

ln_| P 1₋ P

P∞

|=rt+ln_| P0P∞ P∞−P0|

.

❆ss✐♠✱

P P∞−P

= P0

P∞−P0

ert,

♦♥❞❡ ✐s♦❧❛♥❞♦ P ❡♥❝♦♥tr❛♠♦s✿

P(t) = P0P∞

(P∞−P0)e−rt+P0

.

❖❜s❡r✈❡ q✉❡ q✉❛♥❞♦ t_{−→ ∞}✱P(t)_−→P∞✳

✸ ❖ ❡♥s✐♥♦ ❞❡ ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡

❧♦❣❛rít♠✐❝❛s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦

◆❡st❡ ❝❛♣ít✉❧♦ ♣r♦❝✉r❛♠♦s ❛♥❛❧✐s❛r ❝♦♠♦ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❡ ❧♦❣❛✲ rít♠✐❝❛ ✈❡♠ s❡♥❞♦ ❛❜♦r❞❛❞♦s ❡♠ ❛❧❣✉♥s ❧✐✈r♦s ❞✐❞át✐❝♦s ❡ ❛♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ❛❧❣✉♥s t✐♣♦s ❞❡ q✉❡stõ❡s ❡♥✈♦❧✈❡♥❞♦ ❡ss❡s ❝♦♥t❡ú❞♦s q✉❡ ❛♣❛r❡❝❡♠ ❡♠ ❛❧❣✉♥s ✈❡st✐❜✉❧❛r❡s✳

✸✳✶ ❆ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s

❖ ❡♥s✐♥♦ ❞❡ ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s ❢♦✐ ❛♥❛❧✐s❛❞♦ ❝♦♥s✉❧t❛♥❞♦ ♦s s❡✲ ❣✉✐♥t❡s ❧✐✈r♦s ❞♦ P◆▲❊▼✭Pr♦❣r❛♠❛ ◆❛❝✐♦♥❛❧ ❞♦ ▲✐✈r♦ ❉✐❞át✐❝♦ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦✮ ❞❡ ✷✵✶✶✿

• ✏▼❛t❡♠át✐❝❛ ❈♦♠♣❧❡t❛✑ ❞❡ ●✐♦✈❛♥♥✐ ❡ ❇♦♥❥♦r♥♦ ✭❊❞✐t♦r❛ ❋❚❉✮ ❬✶✸❪ • ✏▼❛t❡♠át✐❝❛✱ ❈♦♥t❡①t♦s ❡ ❆♣❧✐❝❛çõ❡s✑ ❞❡ ❉❛♥t❡ ✭❊❞✐t♦r❛ ➪t✐❝❛✮ ❬✶✶❪ • ✏❈♦❧❡çã♦ ◆♦✈♦ ❖❧❤❛r ▼❛t❡♠át✐❝❛✑ ❞❡ ❏♦❛♠✐r ❙♦✉③❛ ✭❊❞✐t♦r❛ ❋❚❉✮ ❬✶✹❪ • ✏▼❛t❡♠át✐❝❛✑ ❞❡ P❛✐✈❛ ✭❊❞✐t♦r❛ ▼♦❞❡r♥❛✮ ❬✶✺❪

◆❛ ♠❛✐♦r✐❛ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❛♥❛❧✐s❛❞♦s ❛ s❡q✉ê♥❝✐❛ ♣r♦♣♦st❛ é✿

• ❙✐t✉❛çã♦ ♣r♦❜❧❡♠❛ ❝♦♠♦ ♠♦t✐✈❛çã♦ ♣❛r❛ ❡st✉❞♦ ❞❡ ❡①♣♦♥❡♥❝✐❛✐s✱ ♥♦ ✐♥í❝✐♦ ❞♦

❝❛♣ít✉❧♦

• ❉❡✜♥✐çõ❡s

• Pr♦♣r✐❡❞❛❞❡s ❡ ❞❡♠♦♥str❛çõ❡s • ❊①❡♠♣❧♦s

• ❆♥á❧✐s❡ ❞❡ ❣rá✜❝♦✭❝r❡s❝❡♥t❡ ♦✉ ❞❡❝r❡s❝❡♥t❡✱ ❞♦♠í♥✐♦✱ ❝♦♥tr❛❞♦♠í♥✐♦✱ ✐♠❛❣❡♠✱

❡t❝✮

• ❏✉st✐✜❝❛♠ ♦ ❢❛t♦ ❞❡ ❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s s❡r❡♠ ✐♥❥❡t✐✈❛✱ s♦❜r❡✲

❥❡t✐✈❛ ❡ ❜✐❥❡t✐✈❛

✸✽ ❋✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s

• ❖ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧e

• ❋✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❛ ✐♥✈❡rs❛ ❞❛ ❡①♣♦♥❡♥❝✐❛❧ • ❆♣❧✐❝❛çõ❡s

❖ ✏❈❛❞❡r♥♦ ❞♦ ❆❧✉♥♦✑✱ ♠❛t❡r✐❛❧ ❞✐s♣♦♥✐❜✐❧✐③❛❞♦ ♣❡❧❛ ❙❡❝r❡t❛r✐❛ ❞❛ ❊❞✉❝❛çã♦ ❞♦ ❊st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦✱ ❛♣r❡s❡♥t❛ s♦♠❡♥t❡ ❡①❡♠♣❧♦s ❞✐❢❡r❡♥❝✐❛❞♦s ❛ s❡r❡♠ tr❛❜❛❧❤❛❞♦s ❝♦♠ ♦s ❛❧✉♥♦s✳ ❈♦♠♦ é s♦♠❡♥t❡ ✉♠ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦✱ ♥ã♦ ❛♣r❡s❡♥t❛♥❞♦ ❞❡✜♥✐çõ❡s ❞♦s ❝♦♥❝❡✐t♦s ❛ s❡r❡♠ tr❛❜❛❧❤❛❞♦s✱ ♦ ♠❡s♠♦ ♥ã♦ s❡rá ♦❜❥❡t♦ ❞❡ ❛♥á❧✐s❡ ♥❡ss❡ tr❛❜❛❧❤♦✳ ❖ ❧✐✈r♦ ❞❡ ❉❛♥t❡ ❬✶✶❪ t❡♠ ❝♦♠♦ ✐♥tr♦❞✉çã♦ ❛ ♠❡✐❛✲✈✐❞❛ ♥❛ r❛❞✐♦❛t✐✈✐❞❛❞❡ ❡ ❞❡♣♦✐s sã♦ ❝✐t❛❞♦s ❡①❡♠♣❧♦s ❞❡ ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ♥❛ ❊❝♦♥♦♠✐❛✱ ❆rq✉✐t❡t✉r❛ ❡ ❇✐♦❧♦❣✐❛✳ ❏♦❛♠✐r ❙♦✉③❛ ❬✶✹❪ tr❛③ ❝♦♠♦ ♠♦t✐✈❛çã♦ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ✉♠❛ ♣❧❛♥t❛ ❛♦ ❞❡❝♦rr❡r ❞♦ ♠ês ❡ ♦ ❝♦♥tr♦❧❡ ❞❡ ♣r❛❣❛s ♣❡❧❛ ❜❛❝tér✐❛ ❇❛❝✐❧❧✉s t❤✉r✐♥❣✐❡♥s✐s✳ ❊♠ s❡✉ ❧✐✈r♦✱ P❛✐✈❛ ❬✶✺❪ ✉s❛ ❝♦♠♦ ✐♥tr♦❞✉çã♦ ❛ ❜✐♣❛rt✐çã♦ ❞❛ ❜❛❝tér✐❛ ❊✳ ❝♦❧✐ ❡ ❛ ♣r❡ssã♦ ❛t♠♦s❢ér✐❝❛ P ♥❛ s✉♣❡r❢í❝✐❡ ❛ ✉♠❛ ❛❧t✉r❛h ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦P(h) = (0,9)h_{✳ ●✐♦✈❛♥♥✐ ❡ ❇♦♥❥♦r♥♦} ❬✶✸❪✱ ♣á❣✐♥❛ ✷✷✹✱ ❝♦❧♦❝❛♠ ❛ s❡❣✉✐♥t❡ s✐t✉❛çã♦✲♣r♦❜❧❡♠❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ♥❛ ♣r♦❞✉çã♦✱ ❞❡ ❢♦r♠❛ ♠❛✐s ❞❡t❛❧❤❛❞❛✿ ✏❯♠❛ ❡♠♣r❡s❛ ♣r♦❞✉③✐✉✱ ♥✉♠ ❝❡rt♦ ❛♥♦✱ 8000 ✉♥✐❞❛❞❡s ❞❡ ❞❡t❡r♠✐♥❛❞♦ ♣r♦❞✉t♦✳ Pr♦❥❡t❛♥❞♦✲s❡ ✉♠ ❛✉♠❡♥t♦ ❛♥✉❛❧ ❞❡ ♣r♦❞✉çã♦ ❞❡50%✱ q✉❛❧ s❡rá ❛ ♣r♦❞✉çã♦ P ❞❡ss❛ ❡♠♣r❡s❛ t ❛♥♦s ❞❡♣♦✐s❄ ❉❛q✉✐ ❛ q✉❛♥t♦s ❛♥♦s ❛ ♣r♦❞✉çã♦ ❛♥✉❛❧ s❡rá ❞❡ 40500✉♥✐❞❛❞❡s❄

P❛r❛ ❝❛❧❝✉❧❛r ❛ ♣r♦❞✉çã♦ P ❞❛ ❡♠♣r❡s❛ t ❛♥♦s ❞❡♣♦✐s✱ ♣♦❞❡♠♦s ✉s❛r ❛ ❢ór♠✉❧❛✿ P = 8000.(1,50)t

❖❜s❡r✈❡ q✉❡ ❛ ♣r♦❞✉çã♦ P ✈❛r✐❛ ❡♠ ❢✉♥çã♦ ❞♦ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ t ❡♠ ❛♥♦s (t = 0,1,2,3, ...)✳

P❛r❛ ❝❛❧❝✉❧❛r ❞❛q✉✐ ❛ q✉❛♥t♦s ❛♥♦s ❛ ♣r♦❞✉çã♦ ❛♥✉❛❧ s❡rá ❞❡ 40500 ✉♥✐❞❛❞❡s✱ ❞❡✲ ✈❡♠♦s ❢❛③❡r P = 40500✳ ▲♦❣♦✿

40500 = 8000.(1,50)t.

❊st❛ ❡q✉❛çã♦ é ❝❤❛♠❛❞❛ ❞❡ ❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧✳✑ ❆♣❡♥❛s é ❝✐t❛❞♦ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❡ ❛ r❡s♦❧✉çã♦ ♣❛ss♦✲❛✲♣❛ss♦ é r❡❛❧✐③❛❞❛ ♣♦st❡r✐♦r♠❡♥t❡✳

❆ ♠❛✐♦r ♣❛rt❡ ❞♦s ❧✐✈r♦s tr❛③ ❝♦♠♦ ❞❡✜♥✐çã♦✿ ❉❡✜♥✐çã♦ ✸✳✶✳ ❯♠❛ ❢✉♥çã♦ f : R_−→ R∗

+✱ ❞❡✜♥✐❞❛ ♣♦r y= f(x) = ax✱ ❝♦♠ a > 0 ❡

a₆= 1✱ é ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳

❈♦♠ ❡①❝❡çã♦ ❞♦ ❧✐✈r♦ ❞❡ P❛✐✈❛ ❬✶✺❪ t♦❞♦s ❞✐s❝✉t❡♠ ♦ ♣♦rq✉ê ❛ ❜❛s❡ ❞❡✈❡ s❡r ♣♦s✐t✐✈❛ ❡ ❞✐❢❡r❡♥t❡ ❞❡ ✶✱ ♣♦✐s✿

❛✮ P❛r❛ a < 0 ♥ã♦ t❡rí❛♠♦s ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ x r❡❛❧✳ P♦r ❡①❡♠♣❧♦✱ s✉♣♦♥❞♦ a=₋4❡ x= 1

2✱ t❡rí❛♠♦s f(

1

2) = (−4)

1

❆ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✸✾

❜✮ ❙❡ a = 1 ♣❛r❛ q✉❛❧q✉❡r ✈❛❧♦r ❞❡ x r❡❛❧✱ ❡♥tã♦ ax _{= 1 ❡ t❡rí❛♠♦s ✉♠❛ ❢✉♥çã♦} ❝♦♥st❛♥t❡✳

❝✮ ❙❡a= 0 ❡x♥❡❣❛t✐✈♦✱ ♥ã♦ ❡①✐st✐r✐❛ax_{✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❛r❛a= 0 ❡x=}

−2✱ t❡♠♦s f(₋2) = 0−2 _{❡ 0}−2 _{♥ã♦ ❡stá ❞❡✜♥✐❞♦ ❡♠ R✳}

❉❡ ✉♠❛ ♠❛♥❡✐r❛ ❣❡r❛❧✱ ♦s ❧✐✈r♦s ❞♦ P◆▲❊▼ ❛q✉✐ ❛♥❛❧✐s❛❞♦s✱ ❞✐s❝✉t❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✮ ♣❛r❛a >1 ❛ ❢✉♥çã♦ é ❝r❡s❝❡♥t❡ ✭x1 > x2 ⇒ax1 > ax2✮✳

✷✮ ♣❛r❛0< a <1✱ ❛ ❢✉♥çã♦ é ❞❡❝r❡s❝❡♥t❡ ✭x1 > x2 ⇒ax1 < ax2✮✳

✸✮ ❙❡ax1

=ax2

⇒x1 =x2 ✱ ♦✉ s❡❥❛✱ ❛ ❢✉♥çã♦ é ✐♥❥❡t✐✈❛✳

✹✮ ❆ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ é s♦❜r❡❥❡t✐✈❛✱ ♦✉ s❡❥❛✱Im(f) = CD(f)❡ ✐♥❥❡t✐✈❛✳ P♦rt❛♥t♦ é ❜✐❥❡t✐✈❛✱ ❡ ❛ss✐♠✱ ❛❞♠✐t❡ ❢✉♥çã♦ ✐♥✈❡rs❛✳

❆ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❞✐s❝✉ssã♦ ❞❡ss❛s ♣r♦♣r✐❡❞❛❞❡s ❝♦❧❛❜♦r❛ ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ❡♥t❡♥❞❛ ❝♦♠♦ r❡s♦❧✈❡r ✉♠❛ ❡q✉❛çã♦ ♦✉ ✐♥❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧✱ ♣♦✐s é ♠✉✐t♦ ❝♦♠✉♠ ♦✉✈✐r♠♦s ✏❡♠ ✉♠❛ ❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❝♦rt❛✲s❡ ❛ ❜❛s❡ ❡ r❡s♦❧✈❡ ♦s ❡①♣♦❡♥t❡s✑ ♦✉ ❡♠ ✉♠❛ ✐♥❡✲ q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧ ✏s❡ ❛ ❜❛s❡ ❡st✐✈❡r ❡♥tr❡ ✵ ❡ ✶ ❝♦rt❛✲s❡ ❛ ❜❛s❡ ❡ ✐♥✈❡rt❡ ♦ s✐♥❛❧ ❞❛ ✐♥❡q✉❛çã♦ tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ♦s ❡①♣♦❡♥t❡s✑✳ ▼❛s s❡rá q✉❡ ♦ ❛❧✉♥♦ ❡♥t❡♥❞❡✉ ♦ ❝♦♥❝❡✐t♦ ♣♦r trás ❞✐ss♦❄ P❛r❛ r❡s♦❧✈❡r ❡q✉❛çõ❡s✱ ✐♥❡q✉❛çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ♦❜t❡r q✉❡ ❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ é ❛ ✐♥✈❡rs❛ ❞❛ ❡①♣♦♥❡♥❝✐❛❧ é ♥❡❝❡ssár✐♦ ❞✐s❝✉t✐r q✉❛♥❞♦ ❛ ❢✉♥çã♦ é ❝r❡s❝❡♥t❡✱ ❞❡❝r❡s❝❡♥t❡ ❡ q✉❡ ❡❧❛ é ✐♥❥❡t✐✈❛✱ s♦❜r❡❥❡t✐✈❛ ❡ ❜✐❥❡t✐✈❛✳

❚♦❞♦s ♦s ❧✐✈r♦s ❞❡✐①❛♠ ❝❧❛r♦ q✉❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ♥ã♦ t♦❝❛ ♦ ❡✐①♦ x✳ ❙❡r✐❛ ✐♥t❡r❡ss❛♥t❡ ♠♦str❛r ❛♦s ❛❧✉♥♦s ✐ss♦ ❝♦♠♦ ❢♦✐ ❢❡✐t♦ ♥♦ ❧✐✈r♦ ✏❆ ♠❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✑✱ ❱♦❧✉♠❡ ✶✱ ❞❛ ❈♦❧❡çã♦ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❙❇▼ ❬✶✷❪✳ ❈♦♠♦ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ax_.ay _{= a}x+y_{✱ ✐st♦ é✱ f(x+y) = f(x).f(y)✱} ❡♥tã♦ f ♥ã♦ ♣♦❞❡ ❛ss✉♠✐r ♦ ✈❛❧♦r ✵✱ ❛ ♠❡♥♦s q✉❡ s❡❥❛ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ ❡①✐st✐r ❛❧❣✉♠ x0 ∈R t❛❧ q✉❡ f(x0) = 0 ❡♥tã♦✱ ♣❛r❛ t♦❞♦ x∈Rt❡r❡♠♦s

f(x) =f(x0+ (x−x0)) =f(x0).f(x−x0) = 0.f(x−x0) = 0

▲♦❣♦ f s❡rá ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✳

▼❛✐s ❛✐♥❞❛✿ s❡ f :R_−→ Rt❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ _ax_.ay _=ax+y ❡ ♥ã♦ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛ ❡♥tã♦ f(x)>0 ♣❛r❛ t♦❞♦x_∈R✱ ♣♦✐s

f(x) = fx 2 +

x 2

=fx 2

.fx

2

=fx 2

2 >0.

➱ ✐♠♣♦rt❛♥t❡ ❧❡♠❜r❛r q✉❡ ❛s ✐❞❡✐❛s ❞❡s❡♥✈♦❧✈✐❞❛s ♥♦ ❡st✉❞♦ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ f(x) = ax_{, a > 0, a 6= 1✱ ♣♦❞❡♠ s❡r ❛♣❧✐❝❛❞❛s ♥❛s ❢✉♥çõ❡s ❞♦ t✐♣♦ f(x) = c.a}kx_{✱ ❝♦♠} c, k_∈R∗✳

❖s ❧✐✈r♦s q✉❡ tr❛❜❛❧❤❛♠ ♦ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ e ❡ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ex _{sã♦ ♦ ❞❡} ❉❛♥t❡ ❬✶✶❪ ❡ ♦ ❞❡ ●✐♦✈❛♥♥✐ ❡ ❇♦♥❥♦r♥♦ ❬✶✸❪✳

◆♦ ❧✐✈r♦ ✏▼❛t❡♠át✐❝❛✱ ❈♦♥t❡①t♦ ❡ ❆♣❧✐❝❛çõ❡s✑ ❬✶✶❪ ♦ ❛✉t♦r ❝♦♥s✐❞❡r❛ ❛ s❡q✉ê♥❝✐❛ (1 + 1

n)

n _{❝♦♠ n}