PARBELOS

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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



❙❯▼➪❘■❖

▲✐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

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