lOMoARcPSD|46958826 ✣➋ ✶
✶✳ ✣å❝ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ❝→❝ ✤à♥❤ ❧þ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝õ❛ ❤➔♠ ❦❤↔ ✈✐ ✈➔ ❝→❝ ❤➺ q✉↔ ❝õ❛ ❝→❝ ✤à♥❤ ❧þ
♥➔②✭ t❤❛♠ ❦❤↔♦ ♣❤➛♥ ✸✳✷✱ ❙♦♦ ❚✳ ❚❛♥ ❙✐♥❣❧❡ ✈❛r✐❛❜❧❡ ✲ ❈❛❧❝✉❧✉s ❡❛r❧② tr❛♥s❝❡♥❞❡♥t❛❧s✮✳
❨➯✉ ❝➛✉ ❤✐➸✉ rã þ ♥❣❤➽❛ ❝→❝ ✤à♥❤ ❧þ✱ ♥➢♠ ❝❤➢❝ ❝→❝ ❣✐↔ t❤✐➳t ❝õ❛ ✤à♥❤ ❧þ ✤➸ →♣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥
❝ö t❤➸✳ ✣÷❛ ❝→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✈➲ ù♥❣ ❞ö♥❣ ❝õ❛ ❝→❝ ✤à♥❤ ❧þ✱ ❤➺ q✉↔ tr➯♥ ✭tr♦♥❣ ❣✐↔✐ t➼❝❤✮ ✭❦❤æ♥❣
❧➜② ❧↕✐ ❝→❝ ✈➼ ❞ö ✤➣ ❣✐↔✐ tr♦♥❣ s→❝❤✳ ❈→❝ ❤➻♥❤ ✈➩ ♠✐♥❤ ❤å❛ ♣❤↔✐ ✤÷ñ❝ ✈➩ ❜➡♥❣ ❝→❝ ♣❤➛♥ ♠➲♠✳
✷✳ ▲➔♠ ❝→❝ ❜➔✐ t➟♣ tø ✶✼✲✷✵ ❝õ❛ ♣❤➛♥ ✸✳✷✳
✸✳ ❚➻♠ ✺ ù♥❣ ❞ö♥❣ t❤ü❝ t➳ ❝õ❛ ❝→❝ ✤à♥❤ ❧þ tr♦♥❣ ♣❤➛♥ ✶✳ ❨➯✉ ❝➛✉ ♥➯✉ ❝ö t❤➸ ❝→❝❤ sû ❞ö♥❣ ✈➔ t➼♥❤
t❤ü❝ t➳ ❝õ❛ ❝→❝ ù♥❣ ❞ö♥❣✳ ✶
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✶✳ ❚r➻♥❤ ❜➔② ❧↕✐ sü ❤➻♥❤ t❤➔♥❤ ✤÷í♥❣ ❝♦♥❣ ❝②❝❧♦✐❞✱ t❤➔♥❤ ❧➟♣ ♣❤÷ì♥❣ tr➻♥❤ t❤❛♠ sè ❝õ❛ ❝②❝❧♦✐❞✱ ①❡♠
✈➼ ❞ö ✻✱ ♣❤➛♥ ✾✳✷✱❙♦♦ ❚✳ ❚❛♥ ❙✐♥❣❧❡ ✈❛r✐❛❜❧❡ ✲ ❈❛❧❝✉❧✉s ❡❛r❧② tr❛♥s❝❡♥❞❡♥t❛❧s✳ ❉ò♥❣ ❝→❝❤ ❦❤↔♦ s→t
❝ü❝ trà ❤➔♠ sè ❝❤♦ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ t❤❛♠ sè ❝❤➾ r❛ ❣✐→ trà ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè y = y(x) ❝❤♦ ❜ð✐
♣❤÷ì♥❣ tr➻♥❤ ❝②❝❧♦✐❞✳
✷✳ ▲➔♠ ❜➔✐ t➟♣ ✹✵ ❝õ❛ ♣❤➛♥ ✾✳✷ tr♦♥❣ ❝➙✉ tr➯♥✳
✸✳ ❉ò♥❣ ♠ët ♣❤➛♥ ♠➲♠ ❤♦➦❝ ♠ët ù♥❣ ❞ö♥❣ ✤➸ ✈➩ ❧↕✐ ❝❤✉②➸♥ ✤ë♥❣ t↕♦ r❛ ✤÷í♥❣ tr♦❝❤♦✐❞ ❝õ❛ ❜➔✐
t➟♣ ✹✵ ❝❤♦ ❝↔ ✸ tr÷í♥❣ ❤ñ♣ ✭r < d, r = d, r > d✮ ✈î✐ d ✈➔ r ✤÷ñ❝ ♥❤➟♣ tò② þ✳ ✷
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✶✳ ❚➻♠ ❤✐➸✉ ✈➲ ✸ ❝→❝❤ t➼♥❤ t❤➸ t➼❝❤ ✈➟t t❤➸ ❜➡♥❣ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ ✤÷ñ❝ tr➻♥❤ ❜➔② tr♦♥❣ ♣❤➛♥ ✺✳✷✱
❙♦♦ ❚✳ ❚❛♥ ❙✐♥❣❧❡ ✈❛r✐❛❜❧❡ ✲ ❈❛❧❝✉❧✉s ❡❛r❧② tr❛♥s❝❡♥❞❡♥t❛❧s✳
❨➯✉ ❤✐➸✉ ✤÷ñ❝ sü ❦❤→❝ ♥❤❛✉ ✈➲ ❝→❝❤ sû ❞ö♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥➔②✳
✷✳ ❚r➻♥❤ ❜➔② ♥❤ú♥❣ ✈➼ ❞ö ✤➦❝ tr÷♥❣ ❝❤♦ ✸ ❝→❝❤ t➼♥❤ tr♦♥❣ ♣❤➛♥ tr➯♥ ✭❦❤æ♥❣ ❞ò♥❣ ❧↕✐ ❝→❝ ✈➼ ❞ö ✤➣
✤÷ñ❝ tr➻♥❤ ❜➔②✳
✸✳ ❉ò♥❣ ♠ët ♣❤➛♥ ♠➲♠ ❤♦➦❝ ♠ët ù♥❣ ❞ö♥❣ ✈➩ ♠ët ✈➟t t❤➸ trá♥ ①♦❛②✱ ✤÷ñ❝ t↕♦ r❛ tø ♠ët ♠✐➲♥
♣❤➥♥❣ ❣✐î✐ ❤↕♥ ❜ð✐ ✤÷í♥❣ ❝♦♥❣ y = f(x) ≥ 0, y = 0, x = a, x = b✳ ▼æ ♣❤ä♥❣ ❝→❝❤ t➼♥❤ t❤➸ t➼❝❤
tr♦♥❣ ❝➙✉ ✶ tr➯♥ ❤➻♥❤ ♥➔②✳ ✸
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✶✳ ❚➻♠ ❤✐➸✉ ✈➲ ❉✐r❡❝t✐♦♥ ❢✐❡❧❞ ✭✾✳✷✮ ✈➔ ♠æ ❤➻♥❤ q✉➛♥ t❤➸ ✤❛ ❧♦➔✐ ✭✾✳✻ Pr❡❞❛t♦r✲Pr❡② ❙②st❡♠✮ tr♦♥❣
❏❛♠❡s ❙t❡✇❛rt✱ ❈❛❧❝✉❧✉s ✲ ❊❛r❧② tr❛♥s❝❡♥❞❡♥t❛❧s✳
✷✳ ❚➻♠ ❤✐➸✉ ❝→❝❤ sû ❞ö♥❣ ❝æ♥❣ ❝ö ❙❧♦♣❡ ❋✐❡❧❞ P❧♦tt❡r ✤➸ ✈➩ ❞✐r❡❝t✐♦♥ ❢✐❡❧❞✳ ❱➩ ♠✐♥❤ ❤å❛ ❞✐r❡❝t✐♦♥
❢✐❡❧❞s ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ y′ = f(x, y) ✈➔ ❤➔♠ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ y′ = f(x, y), y(x0) = y0 ❞ü❛ tr➯♥
❞✐r❡❝t✐♦♥ ❢✐❡❧❞ ♥➔②✳ ✹
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✶✳ ✣å❝ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ♣❤➛♥ ✻✳✹✱ ❆PP▲■❈❆❚■❖◆✿ ❈❖◆❙❯▼❊❘ ❆◆❉ P❘❖❉❯❈❊❘ ❙❯❘P▲❯❙
✭t❤➦♥❣ ❞÷ ♥❣÷í✐ t✐➯✉ ❞ò♥❣ ✈➔ t❤➦♥❣ ❞÷ ♥❤➔ s↔♥ ①✉➜t✱ tr♦♥❣ ❆♣♣❧✐❡❞ ❈❛❧❝✉❧✉s ✺t❤ ❊❞✐t✐♦♥✳ ❨➯✉
❝➛✉ ❤✐➸✉ rã ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ ♣❤→t s✐♥❤ tr♦♥❣ ♣❤➛♥ ♥➔②✳ ✣÷❛ ❝→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❝❤♦ ù♥❣ ✤➣ ♥➯✉✱
❦❤æ♥❣ ❞ò♥❣ ❧↕✐ ♥❤ú♥❣ ✈➼ ❞ö ✤➣ ✤÷ñ❝ tr➻♥❤ ❜➔② tr♦♥❣ t➔✐ ❧✐➺✉✳
✷✳ ❍➔♠ ❝✉♥❣ ✈➔ ❝➛✉ ❝õ❛ ♠ët s↔♥ ♣❤➞♠ ✤÷ñ❝ ❝❤♦ ♥❤÷ ❤➻♥❤ ❜➯♥ ❞÷î✐✳ ❉ò♥❣ tê♥❣ ❘✐❡♠❛♥♥ ÷î❝ t➼♥❤
✷ ❧♦↕✐ t❤➦♥❣ ❞÷ tr♦♥❣ ❝➙✉ tr➯♥✳
✸✳ ▼ët ❝æ♥❣ t② sð ❤ú✉ ♠ët t❤✐➳t ❜à ♠➔ ❣✐→ trà ❝õ❛ ♥â s➩ ❜à ❣✐↔♠ ❧✐➯♥ tö❝ s❛✉ ❧➛♥ ✤↕✐ t✉ ❝✉è✐ ❝ò♥❣✳
❚è❝ ✤ë ❣✐↔♠ ❣✐→ ❧➔ ❤➔♠ sè f = f(t) ✈î✐ t t➼♥❤ t❤❡♦ t❤→♥❣✳ ❈❤✐ ♣❤➼ ❝❤♦ ♠é✐ ❧➛♥ ✤↕✐ t✉ ❧➔ ♠ët ❣✐→
trà A ❝è ✤à♥❤ ♥➯♥ ❝æ♥❣ t② ♠✉è♥ tè✐ ÷✉ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ❣✐ú❛ ❝→❝ ❧➛♥ ✤↕✐ t✉✳ t
❛✴ ●✐↔✐ t❤➼❝❤ t↕✐ s❛♦ R f(s)❞s ❧➔ ❣✐→ trà ❜à ♠➜t s❛✉ t t❤→♥❣ ❦➸ tø ❧➛♥ ✤↕✐ t✉ s❛✉ ❝ò♥❣✳ 0 t
❜✴ ❍➣② ❝❤♦ ❜✐➳t þ ♥❣❤➽❛ ❝õ❛ 1 C = C(t) =
A + R f (s)❞s ✈➔ t↕✐ s❛♦ ❝æ♥❣ t② ♠✉è♥ C ❝â ❣✐→ trà t 0 ♥❤ä ♥❤➜t✳
❝✴ ●✐↔ sû T t❤ä❛ C(T ) = f(T )✱ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ C ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t t↕✐ t = T ✳ ✺
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✶✳ ❱î✐ x ✤ì♥ ✈à ♥❣✉②➯♥ ❧✐➺✉ t❤æ✱ ♥❤➔ s↔♥ ①✉➜t t↕♦ r❛ f(x) ✤ì♥ ✈à s↔♥ ♣❤➞♠✳ ◆❣✉②➯♥ ❧✐➺✉ t❤æ ✤÷ñ❝
♠✉❛ ✈î✐ ❣✐→ $w✴✤ì♥ ✈à ✈➔ s↔♥ ♣❤➞♠ ✤÷ñ❝ ❜→♥ r❛ ✈î✐ ❣✐→ $p✴✤ì♥ ✈à✳ ❙è ✤ì♥ ✈à ♥❣✉②➯♥ ❧✐➺✉ t❤æ
✤÷ñ❝ sû ❞ö♥❣ ✤➸ ❧ñ✐ ♥❤✉➟♥ ✤↕t ❣✐→ trà ❝❛♦ ♥❤➜t ❦þ ❤✐➺✉ ❧➔ x∗✳
❛✴ ❚❤❡♦ ❡♠✱ ♥❤➔ s↔♥ ①✉➜t ♠✉è♥ f′(x) ❞÷ì♥❣ ❤❛② ➙♠❄ ▲þ ❣✐↔✐ ✤✐➲✉ ♥➔②✳
❜✴ ●✐↔✐ t❤➼❝❤ t↕✐ s❛♦ g(x) = pf(x) − wx ❧➔ ❧ñ✐ ♥❤✉➟♥ t❤✉ ✤÷ñ❝ tø x✤ì♥ ✈à ♥❣✉②➯♥ ❧✐➺✉ t❤æ✳
❝✴ ◆➳✉ f ❝â ✤↕♦ ❤➔♠ t↕✐ x∗✱ t➼♥❤ f′(x∗)✳
❞✴ ◆➳✉ f′′(x∗) ❦❤→❝ ✵✱ ❣✐→ trà ❝õ❛ ♥â ❧➔ ➙♠ ❤❛② ❞÷ì♥❣✳
❡✴ ◆➳✉ ◆❤➔ ❝✉♥❣ ❝➜♣ ♥❣✉②➯♥ ❧✐➺✉ t❤æ t❤❛② ✤ê✐ w✱ t❛ ❝â t❤➸ ①❡♠ x∗ ❧➔ ❤➔♠ t❤❡♦ w✱ ①→❝ ✤à♥❤ ❞↕♥❣
dx∗ ✈➔ ❝❤♦ ❜✐➳t ❣✐→ trà ♥➔② ❞÷ì♥❣ ❤❛② ➙♠❄ dw
❢✴ ◆➳✉ w t➠♥❣✱ ♥❤➔ s↔♥ ①✉➜t ♥➯♥ t➠♥❣ ❤❛② ❣✐↔♠ ✈✐➺❝ ♥❤➟♣ ♥❣✉②➯♥ ❧✐➺✉ t❤æ ♥➔②❄
✷✳ ✣å❝ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ♣❤➛♥ ✶✳✸✱❆✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❛♥❞ ❘❡❧❛t✐✈❡ ❝❤❛♥❣❡ tr♦♥❣ ❆♣♣❧✐❡❞ ❈❛❧❝✉❧✉s✲
✺t❤ ❊❞✐t✐♦♥✳ ❨➯✉ ❝➛✉ ♥➯✉ ✤÷ñ❝ ❦❤→✐ ♥✐➺♠ ✭✤à♥❤ ♥❣❤➽❛✮✱ þ ♥❣❤➽❛✱ ❝→❝❤ t➼♥❤✱ s♦ s→♥❤ ✷ ❣✐→ trà ♥➔②✳
❈❤♦ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✷ ❣✐→ trà ♥➔②✱ ❦❤æ♥❣ ❞ò♥❣ ❧↕✐ ✈➼ ❞ö ❝õ❛ s→❝❤✳
✸✳ ❳➨t ❜➔✐ t♦→♥ s❛✉✿ ❑❤✐ ♠→✉ ❧÷✉ t❤æ♥❣ tr♦♥❣ ♠↕❝❤✱ ❧÷ñ♥❣ ♠→✉ ✤✐ q✉❛ ♠ët ✤✐➸♠ ❝❤♦ tr÷î❝ tr♦♥❣
♠ët ✤ì♥ ✈à t❤í✐ ❣✐❛♥ ✭t❤æ♥❣ ❧÷ñ♥❣✮ ✤÷ñ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝ F = kR4✱ tr♦♥❣ ✤â R ❧➔ ❜→♥ ❦➼♥❤ ❝õ❛ ♠↕❝❤ ♠→✉✳
❛✴ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ sü t❤❛② ✤ê✐ t÷ì♥❣ ✤è✐ ❝õ❛ F ①➜♣ ①➾ ✹ ❧➛♥ sü t❤❛② ✤ê✐ t÷ì♥❣ ✤è✐ ❝õ❛ R✳
❜✴ ◆➳✉ ❜→♥ ❦➼♥❤ ♠↕❝❤ ♠→✉ t➠♥❣ ✺✪ t❤➻ ↔♥❤ ❤÷ð♥❣ ♥❤÷ t❤➳ ♥➔♦ ✤➳♥ ❧÷✉ ❧÷ñ♥❣ ♠→✉✳ ✻
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✶✳ ▼ët ❝❤➜t ✤✐➸♠ ❞✐ ❝❤✉②➸♥ t❤❡♦ ❝✉♥❣ ✤÷í♥❣ ✤÷ñ❝ ♠æ t↔ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ t❤❛♠ sè (x(t) = 3t2 − 12t + 16 y(t) = 2 − 9t + 6t2 − t3
tr♦♥❣ ✤â t ❧➔ ❜✐➳♥ t❤í✐ ❣✐❛♥✱ 0 ≤ t ≤ 5✳
✭❛✮ ❇ä q✉❛ ✤ì♥ ✈à t➼♥❤✱ ❤➣② ❝❤♦ ❜✐➳t ❝❤✐➲✉ ❞➔✐ ✤÷í♥❣ ✤✐ ❝õ❛ ❝❤➜t ✤✐➸♠✳
✭❜✮ ❈â ✤✐➸♠ ♥➔♦ ♠➔ ❝❤➜t ✤✐➸♠ ✤✐ q✉❛ ✷ ❧➛♥ ❤❛② ❦❤æ♥❣✱ ♥➳✉ ❝â ❤➣② ❝❤♦ ❜✐➳t ✈à tr➼ ✤✐➸♠ ♥➔② ✈➔
❝→❝ t❤í✐ ✤✐➸♠ ✤✐ q✉❛✳
✭❝✮ ❉ò♥❣ ♠ët ù♥❣ ❞ö♥❣ ✤➸ ✈➩ ❝✉♥❣ ✤÷í♥❣ ♥➔②✱ ♠æ ♣❤ä♥❣ ❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❝❤➜t ✤✐➸♠ tr➯♥ ❝✉♥❣ ✤÷í♥❣✳
✷✳ ❆ ♣✐st♦♥ ✐s ❛tt❛❝❤❡❞ t♦ ❛ ❝r❛♥❦s❤❛❢t ❜② ♠❡❛♥s ♦❢ ❛ ❝♦♥♥❡❝t✐♥❣ r♦❞ ♦❢ ❧❡♥❣t❤ ✱ ❛s s❤♦✇♥ ✐♥ t❤❡
❢✐❣✉r❡✳ ■❢ t❤❡ ❞✐s❦ ✐s ♦❢ r❛❞✐✉s r✱ ❢✐♥❞ t❤❡ ♣❛r❛♠❡tr✐❝ ❡q✉❛t✐♦♥s ❣✐✈✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♣♦✐♥t P
✉s✐♥❣ t❤❡ ❛♥❣❧❡ θ ❛s ❛ ♣❛r❛♠❡t❡r✳ ✼
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✶✳ ❈❤♦ ❤➔♠ sè y = f(x), a ≤ x ≤ b ❝❤♦ ❞↕♥❣ ❜↔♥❣ sè✳ ❉ò♥❣ tê♥❣ ❘✐❡♠❛♥♥ ✤➸ ÷î❝ t➼♥❤ R f(x)❞x a
❜➡♥❣ ♠ët ♣❤➛♥ ♠➲♠ ❤♦➦❝ ♠ët ù♥❣ ❞ö♥❣ tò② þ✳ ❨➯✉ ❝➛✉ t❤➸ ❤✐➺♥ ✤÷ñ❝ ❝→❝❤ ❝ò♥❣ tê♥❣ tr→✐✴tê♥❣
♣❤↔✐✴tê♥❣ tr✉♥❣ t➙♠✳
✷✳ ●✐↔✐ ♣❤➛♥ ❉✐s❝♦✈❡r② Pr♦♠❜❧❡♠ tr♦♥❣ ♣❤➛♥ ✽✳✸✱ ❈❛❧❝✉❧✉s ❡❛r❧② tr❛♥s❝❡♥❞❡♥❛❧s✱ ✻t❤ ❊❞✐t✐♦♥✱ ❏❛♠❡s ❙t❡✇❛rt✳ ✽
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❚❤❛♠ ❦❤↔♦ ❏❛♠❡s ❙t❡✇❛rt✱ ❚r♦② ❉❛②✱ ❇✐♦❝❛❧❝✉❧✉s✲❈❛❧❝✉❧✉s ❢♦r ❧✐❢❡ s❝✐❡♥❝❡✳
✶✳ ❚➻♠ ❤✐➸✉ ✈➲ ❝❡r❡❜r❛❧ ❜❧♦♦❞ ❢❧♦✇ tr♦♥❣ ✈➼ ❞ö ✹✱ ♣❤➛♥ ✻✳✶✳
✷✳ ❚➻♠ ❤✐➸✉ ✈➲ ❙✉r✈✐✈❛❧ ❛♥❞ ❘❡♥❡✇❛❧❀ ❈❛r❞✐❛❝ ♦✉t♣✉t❀ ❇❧♦♦❞ ❢❧♦✇✱ ♣❤➛♥ ✻✳✸✳
❨➯✉ ❝➛✉✿ ❍✐➸✉ ✤÷ñ❝ ❝→❝ t❤✉➟t ♥❣ú ✈➔ ❝→❝❤ t➼♥❤✳ ◆➯✉ ✈➼ ❞ö ❝❤♦ ♠é✐ ♣❤➛♥✱ ❦❤æ♥❣ ❞ò♥❣ ❧↕✐ ✈➼ ❞ö ❝õ❛ s→❝❤✳ ✾
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✶✳ ●✐↔✐ ♣❤➛♥ Pr♦❥❡❝t ✻✳✶✱ ❉✐s❡❛s❡ Pr♦❣r❡ss✐♦♥ ❛♥❞ ■♠♠✉♥✐t② tr♦♥❣ ❏❛♠❡s ❙t❡✇❛rt✱ ❚r♦② ❉❛②✱ ❇✐♦❝❛❧❝✉❧✉s✲
❈❛❧❝✉❧✉s ❢♦r ❧✐❢❡ s❝✐❡♥❝❡✳
✷✳ ❙û ❞ö♥❣ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ ✤➸ t➼♥❤ ❣✐î✐ ❤↕♥ ❝→❝ ❞➣② sè s❛✉ ✭tr➻♥❤ ❜➔② ❝→❝❤ ❧➔♠✱ ♠✐♥❤ ❤å❛ ❜➡♥❣
✤ç t❤à tr➯♥ ♠ët ù♥❣ ❞ö♥❣ ❤♦➦❝ ♠ët ♣❤➛♥ ♠➲♠✮✿ ❛✳ ! 1 r 1 r 2 r n an = 1 + + 1 + + . . . + 1 + . n n n n ❜✳ n n n an = + + . . . + . n2 + 12 n2 + 22 n2 + n2 ✶✵
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✶✳ ●✐↔✐ ♣❤➛♥ Pr♦❥❡❝t ✻✳✶✱ ●✐♥✐ ■♥❞❡① tr♦♥❣ ❏❛♠❡s ❙t❡✇❛rt✱ ❚r♦② ❉❛②✱ ❇✐♦❝❛❧❝✉❧✉s✲❈❛❧❝✉❧✉s ❢♦r ❧✐❢❡ s❝✐❡♥❝❡✳
❨➯✉ ❝➛✉ ♥➢♠ ✤÷ñ❝ ❝→❝ t❤✉➟t ♥❣ú tr♦♥❣ ♣❤➛♥ ♥➔②✳
✷✳ ❉ò♥❣ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ t➻♠ ❣✐î✐ ❤↕♥ ❝õ❛ ❝→❝ ❞➣② sè s❛✉ ✭tr➻♥❤ ❜➔② ❝→❝❤ ❧➔♠✮✿ ❛✳ 1 1 1 an = + + . . . + . n + 1 n + 2 n + n ❜✳ 1p + 2p + . . . + np an = (p > 0). np+1 ✶✶
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❚➻♠ ❤✐➸✉ ✈➲ tå❛ ✤ë ❝ü❝✱ P♦❧❛r ❙②st❡♠✱ tr♦♥❣ ♣❤➛♥ ✾✳✹✱❙♦♦ ❚✳ ❚❛♥ ❙✐♥❣❧❡ ✈❛r✐❛❜❧❡✲ ❈❛❧❝✉❧✉s ❡❛r❧②
tr❛♥s❝❡♥❞❡♥t❛❧s✳ ❨➯✉ ❝➛✉ ▲➔♠ rã ✤÷ñ❝ ♥❤ú♥❣ ✈➜♥ ✤➲ ❞÷î✐ ✤➙②
✶✳ ❈→❝❤ ①→❝ ✤à♥❤ ♠ët ✤✐➸♠ tr♦♥❣ tå❛ ✤ë ❝ü❝✳
✷✳ ▼è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ tå❛ ✤ë ❝ü❝ ✈➔ tå❛ ✤ë ❉❡s❝❛rt❡s✳
✸✳ ❈→❝❤ ✈➩ ✤÷í♥❣ ❝♦♥❣ tr♦♥❣ tå❛ ✤ë ❝ü❝✳
✹✳ ❈→❝❤ ①→❝ ✤à♥❤ t✐➳♣ t✉②➳♥ ✈î✐ ✤÷í♥❣ ❝♦♥❣ tr♦♥❣ tå❛ ✤ë ❝ü❝✳
▲÷✉ þ✿ ◆➯✉ ➼t ♥❤➜t ✷ ✈➼ ❞ö ❝ö t❤➸ ❝❤♦ ♠é✐ ♣❤➛♥✱ ❦❤æ♥❣ ❞ò♥❣ ❧↕✐ ❝→❝ ✈➼ ❞ö ✤➣ ❝â tr♦♥❣ t➔✐ ❧✐➺✉✳
❈→❝ ❤➻♥❤ ✈➩ ♣❤↔✐ ✤÷ñ❝ ✈➩ ❜➡♥❣ ♣❤➛♥ ♠➲♠✳ ✶✷
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❚➻♠ ❤✐➸✉ ✈➲ tå❛ ✤ë ❝ü❝✱ P♦❧❛r ❙②st❡♠✱ tr♦♥❣ ♣❤➛♥ ✾✳✹✱ ✾✳✺✱ ❙♦♦ ❚✳ ❚❛♥ ❙✐♥❣❧❡ ✈❛r✐❛❜❧❡✲ ❈❛❧❝✉❧✉s ❡❛r❧②
tr❛♥s❝❡♥❞❡♥t❛❧s✳ ❨➯✉ ❝➛✉ ▲➔♠ rã ✤÷ñ❝ ♥❤ú♥❣ ✈➜♥ ✤➲ ❞÷î✐ ✤➙②
✶✳ ❈→❝❤ ①→❝ ✤à♥❤ ✶ ✤✐➸♠ tr♦♥❣ tå❛ ✤ë ❝ü❝✳
✷✳ ▼è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ tå❛ ✤ë ❝ü❝ ✈➔ tå❛ ✤ë ❉❡s❝❛rt❡s✳
✸✳ ❈→❝❤ t➼♥❤ ❞✐➺♥ t➼❝❤ ♠✐➲♥ ♣❤➥♥❣ ❣✐î✐ ❤↕♥ ❜ð✐ ❝→❝ ✤÷í♥❣ ❝♦♥❣ tr♦♥❣ tå❛ ✤ë ❝ü❝✳ ❨➯✉ ❝➛✉✿
❼ ❳➙② ❞ü♥❣ ❧↕✐ ❝æ♥❣ t❤ù❝ t➼♥❤ t➼❝❤ ♣❤➙♥ tø tê♥❣ ❘✐❡♠❛♥♥ ✭tr➻♥❤ ❜➔② ♥❣➢♥ ❣å♥✮✳
❼ ❱➟♥ ❞ö♥❣ ✤÷ñ❝ ❝æ♥❣ t❤ù❝ ✤➸ t➼♥❤ ❞✐➺♥ t➼❝❤ ♠✐➺♥ ♣❤➥♥❣✳
▲÷✉ þ✿ ◆➯✉ ➼t ♥❤➜t ✷ ✈➼ ❞ö ❝ö t❤➸ ❝❤♦ ♠é✐ ♣❤➛♥✱ ❦❤æ♥❣ ❞ò♥❣ ❧↕✐ ❝→❝ ✈➼ ❞ö ✤➣ ❝â tr♦♥❣ t➔✐ ❧✐➺✉✳
❈→❝ ❤➻♥❤ ✈➩ ♣❤↔✐ ✤÷ñ❝ ✈➩ ❜➡♥❣ ♣❤➛♥ ♠➲♠✳ ✶✸
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✶✳ ✣å❝ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ù♥❣ ❞ö♥❣ ❝õ❛ t➼❝❤ ♣❤➙♥ ✤➸ t➼♥❤ ❧÷❝ ✈➔ →♣ s✉➜t t❤õ② t➽♥❤ ✭❍②❞r♦t❛st✐❝ ❢♦r❝❡
❛♥❞ ♣r❡ss✉r❡✮✱ ♠♦♠❡♥t ✈➔ tå❛ ✤ë trå♥❣ t➙♠✱ tr♦♥❣ ♣❤➛♥ ✽✳✸ ❝õ❛ ❏❛♠❡s ❙t❡✇❛rt✱ ❈❛❧❝✉❧✉s ❡❛r❧②
tr❛♥s❝❡♥❞❡♥t❛❧s✱ ✻t❤ ❊✐❞❞t✐♦♥✳
❨➯✉ ❝➛✉✿ ❍✐➸✉ ✤÷ñ❝ ❜↔♥ ❝❤➜t ❝→❝ ❦❤→✐ ♥✐➺♠✱ ✈➔ ❝→❝❤ ❤➻♥❤ t❤➔♥❤ ❝→❝ ❝æ♥❣ t❤ù❝ tø ♠æ ❤➻♥❤ t➼❝❤
♣❤➙♥✱ ✈➟♥ ❞ö♥❣ ✤÷ñ❝ tr♦♥❣ ❝→❝ ✈➼ ❞ö ❝ö t❤➸ ✭tr➻♥❤ ❜➔② ♠é✐ ♣❤➛♥ ➼t ♥❤➜t ✷ ✈➼ ❞ö✱ ❦❤æ♥❣ sû ❞ö♥❣
❧↕✐ ✈➼ ❞ö ❝â tr♦♥❣ s→❝❤✳✮
✷✳ ◆➯✉ tè✐ t❤✐➸✉ ✸ ù♥❣ ❞ö♥❣ t❤ü❝ t➳ ❝õ❛ ♣❤➛♥ ✶✳
✸✳ ❉ò♥❣ ♠ët ♣❤➛♥ ♠➲♠ ❤♦➦❝ ♠ët ù♥❣ ❞ö♥❣✱ ❧➟♣ tê♥❣ ❘✐❡♠❛♥♥ ❝õ❛ ♠ët ❤➔♠ sè f tr➯♥ [a, b]✱ ♠æ t↔ ❜➡♥❣ ✤ç t❤à✳ ✶✹
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✶✳ ❚❤❛♠ ❦❤↔♦ P❤➛♥ ✸✳✻✱ ❙♦♦ ❚✳❚❛♥✱ ❆♣♣❧✐❡❞ ❝❛❧❝✉❧✉s ❢♦r ♠❛♥❛❣❡r✐❛❧✱ ❧✐❢❡ ❛♥❞ s♦❝✐❛❧ s❝✐❡♥❝❡s✱ tr➻♥❤
❜➔② ❧↕✐ ❝→❝❤ t➼♥❤ ✤↕♦ ❤➔♠ ❝➜♣ ✶ ❝õ❛ ❤➔♠ ➞♥ ✭■♠♣❧✐❝✐t ❢✉♥❝t✐♦♥s✮ ✈➔ ù♥❣ ❞ö♥❣ tr♦♥❣ t➼♥❤ tè❝ ✤ë
t❤❛② ✤ê✐ ❝õ❛ ❝→❝ ✤↕✐ ❧÷ñ♥❣ t❤ü❝ t➳✳ ❈❤♦ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✈➲ ❝→❝❤ t➼♥❤ ✈➔ ù♥❣ ❞ö♥❣✳ ❑❤æ♥❣ sû ❞ö♥❣
❧↕✐ ❝→❝ ✈➼ ❞ö tr♦♥❣ s→❝❤✳
✷✳ ❍➣② ✤÷❛ r❛ ❝→❝❤ t➼♥❤ ✤↕♦ ❤➔♠ ❝➜♣ ✷✱ sû ❞ö♥❣ ♣❤➛♥ ♠➲♠ ❤♦➦❝ ❝→❝ ù♥❣ ❞ö♥❣ ♦♥❧✐♥❡ t❤ü❝ ❤✐➺♥ t❤❛♦
t→❝ ♥➔②✳ ◆➯✉ ✈➼ ❞ö ♠✐♥❤ ❤å❛✳ ✶✺
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✶✳ ❚❤❛♠ ❦❤↔♦ ♣❤➛♥ ✶✳✹ ✧❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s t♦ ❡❝♦♥♦♥♠✐❝s✧ ❝õ❛ q✉②➸♥ ❆♣♣❧✐❡❞ ❈❛❧❝✉❧✉s✲✺t❤
❡❞✐t✐♦♥✲❍✉❣❤❡s ❍❛❧❧❡t✱ ✤÷❛ r❛ ♠ët sè ù♥❣ ❞ö♥❣ ❝õ❛ ❤➔♠ sè tr♦♥❣ ❦✐♥❤ t➳ ✭tè✐ t❤✐➸✉ ✺ ù♥❣ ❞ö♥❣✮✳ ❨➯✉ ❝➛✉✿
❼ ▼é✐ ♣❤➛♥ ❝❤♦ ✶ ✈➼ ❞ö ❝ö t❤➸✱ ❦❤æ♥❣ ❧➜② ❧↕✐ ✈➼ ❞ö ✤➣ ✤÷ñ❝ s→❝❤ tr➻♥❤ ❜➔② tr♦♥❣ s→❝❤✳
❼ ❑❤æ♥❣ tr➻♥❤ ❜➔② ♣❤➛♥ t➼♥❤ t♦→♥ r÷í♠ r➔✳
❼ ❈→❝ ❤➻♥❤ ✈➩ ♣❤↔✐ ✤÷ñ❝ ✈➩ ❜➡♥❣ ❝→❝ ♣❤➛♥ ♠➲♠ ❤♦➦❝ ù♥❣ ❞ö♥❣✳
✷✳ ●✐↔ sû ❤➔♠ ❝❤✐ ♣❤➼ ✈➔ ❞♦❛♥❤ t❤✉ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❤➔♠ t✉②➳♥ t➼♥❤✳ ❍➔♠ ❝❤✐ ♣❤➼ ❝➟♥ ❜✐➯♥ ✈➔ ❞♦❛♥❤
t❤✉ ❝➟♥ ❜✐➯♥ ✤÷ñ❝ ❝❤♦ ❞↕♥❣ ❜↔♥❣ sè✳ ❉ò♥❣ ♠ët ♣❤➛♥ ♠➲♠ ❤♦➦❝ ♠ët ù♥❣ ❞ö♥❣ ÷î❝ t➼♥❤ ❧ñ✐ ♥❤✉➟♥
tø ❝→❝ t❤æ♥❣ t✐♥ ❝â ✤÷ñ❝✳ ◆➳✉ ♠æ ♣❤ä♥❣ ❜➡♥❣ ✤ç t❤à ❝➔♥❣ tèt✳ ◆➯✉ ❝❤✐ t✐➳t ❝→❝❤ t❤ü❝ ❤✐➺♥ ♥➳✉
t❤æ♥❣ t✐♥ ♥❤➟♣ ✈➔♦ ❧➔ ❝→❝ ❜↔♥❣ sè✱ ❝❤➾ rã ❝→❝❤ ♥❤➟♣ ❞ú ❧✐➺✉ ✈➔ ✤å❝ t❤æ♥❣ t✐♥ ❦➳t q✉↔✳ ✶✻
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✶✳ ❚❤❛♠ ❦❤↔♦ ♠ët sè ♠æ ❤➻♥❤ ❞ù♥❣ ❞ö♥❣ ❝õ❛ ❤➔♠ ♠ô✳ ❳❡♠ ♣❤➛♥ ✺✳✻ ✭❊①♣✳ ●r♦✇t❤✱ ❊①♣✳ ❉❡❝❛②✱
▲❡❛r♥✐♥❣ ❝✉r✈❡s✱❆ss❡♠❜❧② ❚✐♠❡✱ ▲♦❣✐st✐❝ ●r♦✇t❤ ❋✉♥❝t✐♦♥s✮ tr♦♥❣ ❙♦♦ ❈✳❚❛♥✱ ❆♣♣❧✐❡❞ ❈❛❧❝✉❧✉s
❢♦r ♠❛♥❛❣❡r✐❛❧✳
❨➯✉ ❝➛✉✿ ◆➢♠ ✤÷ñ❝ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ ♠ô✳ ◆➯✉ ✈➼ ❞ö ❝❤♦ ❝→❝ ♠æ ❤➻♥❤ ð tr➯♥✱ ❦❤æ♥❣ ❞ò♥❣ ❧↕✐
✈➼ ❞ö ✤➣ ❝â✳ ❈→❝ ❤➻♥❤ ✈➩ ♥➳✉ ❝â ♣❤↔✐ ✤÷ñ❝ ✈➩ ❜➡♥❣ ❝→❝ ♣❤➛♥ ♠➲♠ ❤♦➦❝ ù♥❣ ❞ö♥❣✳
✷✳ ▲➔♠ ❝→❝ ❜➔✐ t➟♣ ✶✵✱ ✶✶✱ ✶✷ tr♦♥❣ ♣❤➛♥ ❚❡❝❤♥♦❧♦❣② ❊①❝❡r❝✐s❡s ❝õ❛ ♠ö❝ ✺✳✻ ♥➔②✳ ✶✼
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✶✳ ❚➻♠ ❤✐➸✉ ✈➲ ❉✐r❡❝t✐♦♥ ❋✐❡❧❞ tr♦♥❣ ♣❤➛♥ ✼✳✷ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ▲♦❣✐st✐❝ tr♦♥❣ ♣❤➛♥ ✼✳✸✱ ❙♦♦ ❚✳❚❛♥✱
❙✐♥❣❧❡ ✈❛r✐❛❜❧❡s ❝❛❧❝✉❧✉s ❊❛r❧② ❚r❛♥s❝❡♥❞❡♥t❛❧s✳
✷✳ ❚➻♠ ❤✐➸✉ ✈➲ ❝æ♥❣ ❝ö ❙❧♦♣❡ ❋✐❡❧❞ P❧♦tt❡r ✤➸ ✈➩ ❉✐r❡❝t✐♦♥ ❢✐❡❧❞✳
❱➩ ♠✐♥❤ ❤å❛ ❞✐r❡❝t✐♦♥ ❢✐❡❧❞s ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ y′ = 2y2 − x ✈➔ ❤➔♠ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ y′ = 2y2 − x, y(2) = 2✳ ✶✽
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