❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷
❑❍❖❆ ❚❖⑩◆
✖✖✖✖✕♦✵♦✖✖✖✖✖

◆●❯❨➍◆ ❚❍➚ ❍⑨

❇❷◆● ❚➐▼ ❑■➌▼ ✭▲❖❖❑❯P ❚❆❇▲❊✮
❱⑨ Ù◆● ❉Ö◆●

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣

❍⑨ ◆❐■✲✷✵✶✾


❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷
❑❍❖❆ ❚❖⑩◆
✖✖✖✖✕♦✵♦✖✖✖✖✖

◆●❯❨➍◆ ❚❍➚ ❍⑨

❇❷◆● ❚➐▼ ❑■➌▼ ✭▲❖❖❑❯P ❚❆❇▲❊✮
❱⑨ Ù◆● ❉Ö◆●

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝

❚❤✳❙ ❚r➛♥ ❚✉➜♥ ❱✐♥❤
❍⑨ ◆❐■✲✷✵✶✾

ớ ỡ
t q tr ự t õ ớ
t tổ t ỡ tợ t ổ tr
t ổ tr tờ ử ộ tổ t t tr sốt tớ tổ
ồ t t trữớ P ở
ổ ỷ ớ ỡ s s t tợ t



r

ữớ ỳ tự t t t

ú ù ữợ tổ tr sốt tớ tổ tỹ õ
tốt ởt ỳ tổ t ỡ t ú
t ỗ sự ọ
t tổ q ợ ổ ự
ồ ỡ ỳ tớ ỹ ừ t ỏ
ổ t tr ọ ỳ t sõt tổ ữủ
sỹ õ õ ỵ qỵ ừ t ổ s õ
ừ tổ ữủ t ỡ
ổ t ỡ
ở t








▲í✐ ❝❛♠ ✤♦❛♥
❚æ✐ ①✐♥ ❦❤➥♥❣ ✤à♥❤ ✤➙② ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ ❝→ ♥❤➙♥ tæ✐
✈î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ t❤➛② ❣✐→♦

❚❤❙✳ ❚r➛♥ ❚✉➜♥ ❱✐♥❤✳ ✣➲ t➔✐ ♥➔② ❝❤÷❛

tø♥❣ ✤÷ñ❝ ❝æ♥❣ ❜è ð ✤➙✉ ✈➔ ❤♦➔♥ t♦➔♥ ❦❤æ♥❣ trò♥❣ ✈î✐ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛
t→❝ ❣✐↔ ❦❤→❝✳

✷


▼ö❝ ❧ö❝
▲í✐ ❝↔♠ ì♥

✶

▲í✐ ❝❛♠ ✤♦❛♥

✷

▼ð ✤➛✉

✻

✶ ❚✃◆● ◗❯❆◆ ❱➋ ❇❷◆● ❚➐▼ ❑■➌▼

✾


✶✳✶

❇↔♥❣ t➻♠ ❦✐➳♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✷

▲à❝❤ sû ♥❣❤✐➯♥ ❝ù✉

✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆●
❇❷◆● ❚➐▼ ❑■➌▼
✶✶
✷✳✶

❱➼ ❞ö ✤ì♥ ❣✐↔♥ ✈➲ ❜↔♥❣ t➻♠ ❦✐➳♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✷✳✷

P❤÷ì♥❣ ♣❤→♣ ♥ë✐ s✉② sû ❞ö♥❣ tr♦♥❣ ❜↔♥❣ t➻♠ ❦✐➳♠ ✳ ✳ ✳ ✳ ✳

✶✷


✷✳✸

❙û ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr♦♥❣ ❜↔♥❣ t➻♠ ❦✐➳♠ ✶✸
✷✳✸✳✶✳

❈➜✉ tró❝ ❝õ❛ ❜↔♥❣ t➻♠ ❦✐➳♠

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✸✳✷✳

❇↔♥❣ t➻♠ ❦✐➳♠ ♠ët ❝❤✐➲✉ ✈➔ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤
tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠ët ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✸✳✸✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

❚ê♥❣ q✉→t ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥
♥ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸ Ù◆● ❉Ö◆● ❈Õ❆ ❇❷◆● ❚➐▼ ❑■➌▼
✸✳✶

✶✼

❇↔♥❣ t➻♠ ❦✐➳♠ ❜❛ ❝❤✐➲✉ ✈➔ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤
tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❜❛ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳


✷✳✸✳✺✳

✶✺

❇↔♥❣ t➻♠ ❦✐➳♠ ❤❛✐ ❝❤✐➲✉ ✈➔ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤
tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤❛✐ ❝❤✐➲✉

✷✳✸✳✹✳

✶✸

✷✺

✷✻

❙û ❞ö♥❣ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠ët ❝❤✐➲✉
✤➸ t➻♠ t✛ ❧➺ ✈è♥ ❦✐♥❤ t➳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✷✼


✸✳✷

❙û ❞ö♥❣ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤❛✐ ❝❤✐➲✉
✤➸ t➻♠ t✛ ❧➺ ✈è♥ ❦✐♥❤ t➳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✳✸


✷✾

❙û ❞ö♥❣ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❜❛ ❝❤✐➲✉
✤➸ t➻♠ t✛ ❧➺ ✈è♥ ❦✐♥❤ t➳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✶

❑➳t ❧✉➟♥

✸✹

❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖

✸✺

✹


❉❛♥❤ s→❝❤ ❜↔♥❣
y = x3 ✳ ✳ ✳ ✳ ✳
3
trà ❤➔♠ y = x t↕✐

✷✳✶

❇↔♥❣ ❜✐➸✉ ❞✐➵♥ ❤➔♠

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳


✷✳✷

❇↔♥❣ ÷î❝ t➼♥❤ ❣✐→

✤✐➸♠

✳ ✳ ✳ ✳

✶✷

✷✳✸

❇↔♥❣ ✤♦ ♥❤✐➺t ✤ë t↕✐ ♥❤➔ ◆❛♠ ❜✉ê✐ tr÷❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✷✳✹

❇↔♥❣ ♠✐♥❤ ❤å❛ ❝➜✉ tró❝ ❝õ❛ ❜↔♥❣ t➻♠ ❦✐➳♠ ❤❛✐ ❝❤✐➲✉ ✳ ✳ ✳ ✳

✶✹

✷✳✺

❇↔♥❣ t➻♠ ❦✐➳♠ ❤❛✐ ❝❤✐➲✉ ♠æ t↔ ♠ët ❤➔♠ ❜✐➸✉ t❤à t❤❡♦ ❤❛✐
❜✐➳♥ ❝❤✐➲✉ ❝❛♦ ✈➔ ❝➙♥ ♥➦♥❣

sinx

x = −1, 5


✶✶

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

t↕✐ ❝→❝ ✤✐➸♠ ✤➦❝ ❜✐➺t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✷✳✻

●✐→ trà ❤➔♠

✸✳✶

❇↔♥❣ ❝❤ù❛ t✛ ❧➺ P❉(%) ✈➔ t✛ ❧➺ ❊❈(%)

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

✸✳✷

❇↔♥❣ ❝❤ù❛ ❦➻ ❤↕♥ ✭♥➠♠✮✱ t÷ì♥❣ q✉❛♥(%) ✈➔ t✛ ❧➺ ❊❈(%) ✳ ✳

✷✾

✸✳✸


❇↔♥❣ ❝❤ù❛ ❦➻ ❤↕♥ ✭♥➠♠✮✱ t÷ì♥❣ q✉❛♥
❧➺ ❊❈

(%)

(%)✱

▲●❉

(%)

✈➔ t✛

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✸✶


❉❛♥❤ s→❝❤ ❤➻♥❤ ✈➩
✷✳✶

❇✐➸✉ ✤ç ♠✐♥❤ ❤å❛ ♥❤✐➺t ✤ë ♥❤➔ ◆❛♠ t↕✐ t❤í✐ ✤✐➸♠ ✶✷❤ tr÷❛

✶✸

✷✳✷


◆ë✐ s✉② t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✷✳✸

◆ë✐ s✉② s♦♥❣ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

✷✳✹

❍➻♥❤ ✈➩ ♠æ t↔ ❜è♥ ✤✐➸♠ ↔♥❤ ✤➣ ❜✐➳t ✈➔ ✤✐➸♠ ↔♥❤ ❝➛♥ ♥ë✐ s✉②✳ ✷✵

✷✳✺

◆ë✐ s✉② t❛♠ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✻

❍➻♥❤ ♠✐♥❤ ❤å❛ ❣✐→ trà ✤✐➸♠ ♥ë✐ s✉②

✸✳✶

❍➻♥❤ ♠✐♥❤ ❤å❛ t✛ ❧➺ ✈è♥ ❦✐♥❤ t➳

✸✳✷

▼æ ♣❤ä♥❣ t✛ ❧➺ ❊❈ ❝➛♥ ♥ë✐ s✉② tr♦♥❣ ✤♦↕♥ ❆❇


✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

✸✳✸

❚✛ ❧➺ ❊❈ ❝➛♥ ♥ë✐ s✉② tr♦♥❣ ❤➻♥❤ ❝❤ú ♥❤➟t ❆❇❈❉ ✳ ✳ ✳ ✳ ✳ ✳

✸✵

✸✳✹

▼æ ♣❤ä♥❣ t✛ ❧➺ ❊❈ ❝➛♥ ♥ë✐ s✉② tr♦♥❣ ❧➠♥❣ trö ❆❇❈❉❊❋●❍

✸✷

✻

p = f (0, 9; 0, 9; 0, 9)

✷✶

✳ ✳

✷✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼



▼ð ✤➛✉
■✳ ▲þ ❞♦ ❝❤å♥ ✤➲ t➔✐
❇↔♥❣ t➻♠ ❦✐➳♠ ✤➣ ✈➔ ✤❛♥❣ ✤÷ñ❝ ù♥❣ ❞ö♥❣ r➜t ♥❤✐➲✉ ✈➔♦ ❝→❝ ♥❣➔♥❤
❝õ❛ ❦❤♦❛ ❤å❝ ✈➔ ❦ÿ t❤✉➟t ❦❤→❝ ♥❤❛✉✳ ❚❤í✐ ❣✐❛♥ ❣➛♥ ✤➙② sü ♣❤→t tr✐➸♥
❝õ❛ ❦❤♦❛ ❤å❝ ♠→② t➼♥❤ ✤➣ ♠ð r❛ ♠ët ❝♦♥ ✤÷í♥❣ ♠î✐ tr♦♥❣ ❝æ♥❣ ♥❣❤➺ ❦ÿ
t❤✉➟t✳ ❑❤↔ ♥➠♥❣ ✈➔ tè❝ ✤ë ❝õ❛ ♠→② t➼♥❤ ✤÷ñ❝ ❝↔✐ t❤✐➺♥ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦
✈✐➺❝ ①û ❧þ ❝→❝ ✈➜♥ ✤➲ ✈➲ ✈➟t ❧þ ✈➔ ❦ÿ t❤✉➟t ♣❤ù❝ t↕♣ ♠➔ tr÷î❝ ✤➙② ❝❤÷❛
✤÷ñ❝ ❣✐↔✐ q✉②➳t✳ ❈❤➼♥❤ ✈➻ t❤➳ ❜↔♥❣ t➻♠ ❦✐➳♠ ♥❤➟♥ ✤÷ñ❝ ♥❤✐➲✉ sü ❝❤ó þ✳
❚✉② ♥❤✐➯♥ s✐♥❤ ✈✐➯♥ ❙÷ ♣❤↕♠ ❚♦→♥ ❤å❝ ♥â✐ ❝❤✉♥❣ ❝❤÷❛ ❝â ♥❤✐➲✉ ✤✐➲✉
❦✐➺♥ ✤➸ t➻♠ ❤✐➸✉ ✈➲ ❜↔♥❣ t➻♠ ❦✐➳♠ ✈➔ ù♥❣ ❞ö♥❣ ❝õ❛ ❜↔♥❣ t➻♠ ❦✐➳♠✳ ❱➻ ✈➟②
tæ✐ ❝❤å♥ ✤➲ t➔✐ ✏❇↔♥❣ t➻♠ ❦✐➳♠ ✈➔ ù♥❣ ❞ö♥❣✑ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣
♥❤➡♠ ✤÷❛ r❛ ♠ët sè ❧➼ t❤✉②➳t ❝ì ❜↔♥ ✈➲ ❜↔♥❣ t➻♠ ❦✐➳♠✱ ♣❤÷ì♥❣ ♣❤→♣ ♥ë✐
s✉② t✉②➳♥ t➼♥❤ ✤÷ñ❝ sû ❞ö♥❣ tr♦♥❣ ❜↔♥❣ t➻♠ ❦✐➳♠ ✈➔ ♠ët ✈➔✐ ù♥❣ ❞ö♥❣
❝ö t❤➸ ❝õ❛ ❜↔♥❣ t➻♠ ❦✐➳♠ ✈➔♦ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝✳

■■✳ ▼ö❝ t✐➯✉ ♥❣❤✐➯♥ ❝ù✉
▼ö❝ t✐➯✉ ❝õ❛ ❦❤â❛ ❧✉➟♥ ❧➔ t➻♠ ❤✐➸✉ ✈➲ ❜↔♥❣ t➻♠ ❦✐➳♠ ✈➔ tø ✤â ✤÷❛
r❛ ❝→❝ ù♥❣ ❞ö♥❣ ❝ö t❤➸ ❝õ❛ ❜↔♥❣ t➻♠ ❦✐➳♠ tr♦♥❣ t♦→♥ ❤å❝ ✈➔ tr♦♥❣ t❤ü❝ t➳✳

■■■✳ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉
◆❣❤✐➯♥ ❝ù✉ ❝ì sð ❧➼ ❧✉➟♥ ✈➲ ❜↔♥❣ t➻♠ ❦✐➳♠✳
◆❣❤✐➯♥ ❝ù✉ ❝ì sð ❧➼ ❧✉➟♥ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ ✤÷ñ❝ sû
❞ö♥❣ tr♦♥❣ ❜↔♥❣ t➻♠ ❦✐➳♠✳
✣÷❛ r❛ ù♥❣ ❞ö♥❣ ❝ö t❤➸ ❝õ❛ ❜↔♥❣ t➻♠ ❦✐➳♠✳

■❱✳ ❈➜✉ tró❝ ❦❤â❛ ❧✉➟♥
✼



✽
◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❦❤â❛ ❧✉➟♥ ❣ç♠ ✸
❝❤÷ì♥❣✿
❈❤÷ì♥❣ ✶✿ ❚ê♥❣ q✉❛♥ ✈➲ ❜↔♥❣ t➻♠ ❦✐➳♠
❈❤÷ì♥❣ ✷✿ ❇↔♥❣ t➻♠ ❦✐➳♠ ✈➔ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr♦♥❣ ❜↔♥❣ t➻♠ ❦✐➳♠
❈❤÷ì♥❣ ✸✿ ù♥❣ ❞ö♥❣ ❝õ❛ ❜↔♥❣ t➻♠ ❦✐➳♠


ữỡ


t
r ồ t t ởt t t t
t t t ử ỡ ỡ tt
tớ ỷ ỵ ởt tr tứ ở ợ tữớ ỡ
tỹ t t ữủ t t ữủ t trữợ
ữ ởt ừ t ừ ữỡ tr ợ
ữủ ữ trỳ tr ự ừ t ự ử ử t
t ụ ữủ sỷ ử rở r tr
s ợ ợ ởt s ử ủ ổ ủ
tr ởt

sỷ ự
rữợ õ sỹ r ớ ừ t t tr ữủ
sỷ ử t tố ở t t từ ổ ừ ự t
ữ tr ữủ rt t ở tố é ở
ờ rt t r ởt tr ỳ s t
ổ õ tr ởt tố ỳ số ỹ tr t P
trs ự qt t ởt ởt
số t ừ ộ số tứ ởt s

số t ợ ởt tứ tr ởt tr s
õ ử ố ữớ s õ t ởt rỗ




❈❤÷ì♥❣ ✶✳

❚✃◆● ◗❯❆◆ ❱➋ ❇❷◆● ❚➐▼ ❑■➌▼

✶✵

①✉è♥❣ ✤➳♥ ♣❤➙♥ sè✧✳ P❤÷ì♥❣ ♣❤→♣ ❞↕② ❤å❝ ❤✐➺♥ ✤↕✐ ❞↕② ❝❤♦ tr➫ ❡♠ ❣❤✐
♥❤î ✏❜↔♥❣ ❝û✉ ❝❤÷ì♥❣✑ ✤➸ tr→♥❤ ✈✐➺❝ ♣❤↔✐ t➼♥❤ t♦→♥ ❧↕✐ ❝→❝ sè t❤÷í♥❣ ❞ò♥❣

[4]

♥❤➜t ✭❧➯♥ ✤➳♥ ✾ ① ✾ ❤♦➦❝ ✶✷ ① ✶✷✮.

❚❤í✐ ❣✐❛♥ ✤➛✉ tr♦♥❣ ❧à❝❤ sû ♣❤→t tr✐➸♥ ♠→② t➼♥❤✱ ✤➛✉ ✈➔♦ ❤♦➦❝ ✤➛✉ r❛
❤♦↕t ✤ë♥❣ ✤➦❝ ❜✐➺t ❝❤➟♠✳ ✣➸ ❣✐↔♠ ❝→❝ ❤♦↕t ✤ë♥❣ t➼♥❤ t♦→♥ ♥❣÷í✐ t❛ →♣
❞ö♥❣ ♠ët ❤➻♥❤ t❤ù❝ ❧÷✉ ❜ë ♥❤î ✤➺♠ t❤õ ❝æ♥❣ ❜➡♥❣ ❝→❝❤ t↕♦ r❛ ❜↔♥❣ t➻♠
❦✐➳♠ ❝❤ù❛ ❝→❝ ♠ö❝ ❞ú ❧✐➺✉ ♣❤ê ❜✐➳♥ ♥❤➜t✳
❇↔♥❣ t➻♠ ❦✐➳♠ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❝❤ù❝ ♥➠♥❣ sî♠ ♥❤➜t ✤÷ñ❝ tr✐➸♥ ❦❤❛✐
tr♦♥❣ ❜↔♥❣ t➼♥❤ ♠→② t➼♥❤✱ ✈î✐ ♣❤✐➯♥ ❜↔♥ ✤➛✉ t✐➯♥ ❝õ❛ ❱✐s✐❈❛❧❝ ✭✶✾✼✾✮ ❜❛♦
❣ç♠ ♠ët ❤➔♠ ▲❖❖❑❯P tr♦♥❣ sè ✷✵ ❤➔♠ ❜❛♥ ✤➛✉ ❝õ❛ ♥â✳ ✣✐➲✉ ♥➔② ✤➣
✤÷ñ❝ sû ❞ö♥❣ ❜ð✐ ❝→❝ ❜↔♥❣ t➼♥❤ t✐➳♣ t❤❡♦✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ ▼✐❝r♦s♦❢t ❊①❝❡❧
❜ê s✉♥❣ ❝→❝ ❤➔♠ ❱▲❖❖❑❯P ✈➔ ❤➔♠ ❍▲❖❖❑❯P ✤➸ ✤ì♥ ❣✐↔♥ ❤â❛ ✈✐➺❝

[4]


t➻♠ ❦✐➳♠ tr♦♥❣ ♠ët ❜↔♥❣ t❤❡♦ ❝❤✐➲✉ ❞å❝ ❤♦➦❝ ❝❤✐➲✉ ♥❣❛♥❣.


❈❤÷ì♥❣ ✷
❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨
❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆●
❚➐▼ ❑■➌▼
✷✳✶ ❱➼ ❞ö ✤ì♥ ❣✐↔♥ ✈➲ ❜↔♥❣ t➻♠ ❦✐➳♠
❱➼ ❞ö s❛✉ ♠✐♥❤ ❤å❛ ❜↔♥❣ t➻♠ ❦✐➳♠ ♠ët ❝❤✐➲✉ ①➜♣ ①➾ ❤➔♠

y = x3 ✳ ❇↔♥❣

t➻♠ ❦✐➳♠ ①→❝ ✤à♥❤ ❞ú ❧✐➺✉ ✤➛✉ r❛ ✭②✮ tr♦♥❣ ♣❤↕♠ ✈✐ ✤➛✉ ✈➔♦ ✭①✮ t❤✉ë❝
✤♦↕♥

[−3, 3]✳

❇↔♥❣ s❛✉ ✤➙② ♠✐♥❤ ❤å❛ ♠è✐ q✉❛♥ ❤➺ ✤➛✉ ✈➔♦ ✈➔ ✤➛✉ r❛✳

❇↔♥❣ ✷✳✶✿ ❇↔♥❣ ❜✐➸✉ ❞✐➵♥ ❤➔♠ y = x3
① −3 −2 −1 ✵ ✶ ✷ ✸
② −27 −8 −1 ✵ ✶ ✽ ✷✼

x = −2
y = −8✳

❱î✐ ❣✐→ trà ✤➛✉ ✈➔♦
✤➛✉ r❛ t÷ì♥❣ ù♥❣


sû ❞ö♥❣ ❜↔♥❣ t➻♠ ❦✐➳♠ ✈➔ ❧➜② ✤ó♥❣ ❣✐→ trà

❑❤✐ ❜↔♥❣ t➻♠ ❦✐➳♠ ❣➦♣ ♠ët ❣✐→ trà ✤➛✉ ✈➔♦ ❦❤æ♥❣ ❦❤î♣ ✈î✐ ❜➜t ❦ý ❣✐→
trà ① ♥➔♦ ❝õ❛ ❜↔♥❣✱ ♥â ❝â t❤➸ ♥ë✐ s✉② ❤♦➦❝ ♥❣♦↕✐ s✉② ✤➸ t➻♠ ❣✐→ trà ✤➛✉
r❛ ② t÷ì♥❣ ù♥❣✳ ❱➼ ❞ö✱ ❜↔♥❣ t➻♠ ❦✐➳♠ ❦❤æ♥❣ ①→❝ ✤à♥❤ ❣✐→ trà ✤➛✉ ✈➔♦ ❧➔

−1, 5

t✉② ♥❤✐➯♥ ❝â t❤➸ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ ❝→❝ ✤✐➸♠ ❞ú ❧✐➺✉ ❧➙♥ ❝➟♥ ❣➛♥

(xi , yi ) ✈➔ (xi+1 , yi+1 )✳
(xi , yi ) ❧➔ (−2, −8)
(xi+1 , yi+1 ) ❧➔ (−1, −1)
♥❤➜t

❱➼ ❞ö✱ ✈î✐ ❤❛✐ ✤✐➸♠ s❛✉✿

❇↔♥❣ t➻♠ ❦✐➳♠ ÷î❝ t➼♥❤ ✈➔ tr↔ ✈➲ ❣✐→ trà ❧➔ ✲✹✱✺✳

✶✶


❈❤÷ì♥❣ ✷✳

❑■➌▼

❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆● ❚➐▼

✶✷


❇↔♥❣ ✷✳✷✿ ❇↔♥❣ ÷î❝ t➼♥❤ ❣✐→ trà ❤➔♠ y = x3 t↕✐ ✤✐➸♠ x = −1, 5
① −3 −2 −1, 5 −1 ✵ ✶ ✷ ✸
② −27 −8 −4, 5 −1 ✵ ✶ ✽ ✷✼

✷✳✷ P❤÷ì♥❣ ♣❤→♣ ♥ë✐ s✉② sû ❞ö♥❣ tr♦♥❣ ❜↔♥❣ t➻♠
❦✐➳♠
❚r♦♥❣ ❣✐↔✐ t➼❝❤ sè✱ ♣❤➨♣ ♥ë✐ s✉② ❧➔ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ①➙② ❞ü♥❣ ❝→❝ ✤✐➸♠
❞ú ❧✐➺✉ ♠î✐ tr♦♥❣ ♣❤↕♠ ✈✐ ❝õ❛ ♠ët t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ❞ú ❧✐➺✉ ✤➣ ❜✐➳t✳
❚r♦♥❣ ❦ÿ t❤✉➟t ✈➔ ❦❤♦❛ ❤å❝ t❤÷í♥❣ ❝â ♠ët sè ✤✐➸♠ ❞ú ❧✐➺✉ t❤✉ ✤÷ñ❝
❜➡♥❣ ✈✐➺❝ ❧➜② ♠➝✉ ❤❛② t❤➼ ♥❣❤✐➺♠✱ ✈➔ t❤û ①➙② ❞ü♥❣ ♠ët ❤➔♠ ♠➔ ❣➛♥ ♣❤ò
❤ñ♣ ✈î✐ ♥❤ú♥❣ ✤✐➸♠ ❞ú ❧✐➺✉ ✤â✳ ◆â t❤÷í♥❣ ✤÷ñ❝ ②➯✉ ❝➛✉ ✤➸ ♥ë✐ s✉②✱ tù❝ ❧➔
÷î❝ t➼♥❤ ❣✐→ trà ❝õ❛ ❤➔♠ ✤â ❝❤♦ ♠ët ❣✐→ trà tr✉♥❣ ❣✐❛♥ ❝õ❛ ❜✐➳♥ ✤ë❝ ❧➟♣✳
▼ët ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ❝❤➦t ❝❤➩ ❧➔ t➼♥❤ ❣✐→ trà ①➜♣ ①➾ ♠ët ❤➔♠ ♣❤ù❝ t↕♣
❜ð✐ ♠ët ❤➔♠ ✤ì♥ ❣✐↔♥✳ ●✐↔ sû ❝æ♥❣ t❤ù❝ ❝❤♦ ♠ët ❤➔♠ ✤➣ ❜✐➳t✱ ♥❤÷♥❣ q✉→
♣❤ù❝ t↕♣ ✤➸ t➼♥❤ ✤÷ñ❝ ❣✐→ trà ❝õ❛ ❤➔♠ ✤â✳ ▼ët ✈➔✐ ✤✐➸♠ ❞ú ❧✐➺✉ ❝❤♦ tr÷î❝
❝â t❤➸ ✤÷ñ❝ ♥ë✐ s✉② ✤➸ t↕♦ r❛ ❣✐→ trà ❝➛♥ t➻♠ tø ♠ët ✤✐➸♠ ❞ú ❧✐➺✉ ❜➜t ❦➻✳
❑➳t q✉↔ ✤↕t ✤÷ñ❝ tr♦♥❣ q✉→ tr➻♥❤ ♥ë✐ s✉② ❝â t❤➸ ❧î♥ ❤ì♥ ❤♦➦❝ ♥❤ä ❤ì♥ ❣✐→
trà t❤ü❝ ❝õ❛ ♥â tø ❧é✐ ♥ë✐ s✉②✳
❱➼ ❞ö✿ ◆❛♠ ✤♦ ♥❤✐➺t ✤ë t↕✐ ♥❤➔ ❜✉ê✐ tr÷❛ ♥❣➔② ✶✷✴✸✴✷✵✶✼ ✈➔ t❤✉ ✤÷ñ❝
❦➳t q✉↔ tr♦♥❣ ❜↔♥❣ ✷✳✸ ❍➣② ÷î❝ t➼♥❤ ♥❤✐➺t ✤ë ð ♥❤➔ ◆❛♠ ❧ó❝ ✶✷❤❄

❇↔♥❣ ✷✳✸✿ ❇↔♥❣ ✤♦ ♥❤✐➺t ✤ë t↕✐ ♥❤➔ ◆❛♠ ❜✉ê✐ tr÷❛
❚❤í✐ ✤✐➸♠ ✭❤✮ ✶✶ ✶✸ ✶✹ ✶✺
◆❤✐➺t ✤ë ✭◦ C ✮ ✷✵ ✷✷ ✷✸ ✷✷✱✺
◆ë✐ s✉② ❤♦↕t ✤ë♥❣ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❝→❝ ❞ú ❧✐➺✉ ✤➸ t➻♠ ❣✐→ trà ÷î❝ t➼♥❤
ð ✤✐➸♠ ❝❤÷❛ rã✳ ✣➸ ÷î❝ t➼♥❤ ♥❤✐➺t ✤ë ð ♥❤➔ ◆❛♠ ❧ó❝ ✶✷❤ tr÷❛✱ ❝❤ó♥❣ t❛
sû ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥ë✐ s✉②✱ ♥ë✐ s✉② ❝✉♥❣ ❝➜♣ ♠ët ♣❤÷ì♥❣ t✐➺♥ ÷î❝ t➼♥❤
❤➔♠ t↕✐ ❝→❝ ✤✐➸♠ tr✉♥❣ ❣✐❛♥✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷

x = 12✳


❈â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ♥ë✐ s✉② ❦❤→❝ ♥❤❛✉✱ ♥❤÷♥❣ ❝➛♥ sû ❞ö♥❣ ♣❤÷ì♥❣
♣❤→♣ ♥ë✐ s✉② ♥➔♦ ❝❤♦ ♣❤ò ❤ñ♣ ✈➲ ❝↔ tè❝ ✤ë ✈➔ ❝❤✐ ♣❤➼ t❤ü❝ ❤✐➺♥✳ ❱➻ t❤➳
❦❤✐ t➼♥❤ t♦→♥ sû ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥ë✐ s✉② ♥➔♦ ❝➛♥ t➼♥❤ ✤➳♥ ♣❤÷ì♥❣ ♣❤→♣
✤â ❝❤♦ ✤ë ❝❤➼♥❤ ①→❝ ✤➳♥ ❜❛♦ ♥❤✐➯✉✱ ♥ë✐ s✉② ♠à♥ ❜❛♦ ♥❤✐➯✉✱ ♥❤✐➲✉ ✤✐➸♠ ❞ú
❧✐➺✉ ✤÷ñ❝ sû ❞ö♥❣ ♥❤÷ t❤➳ ♥➔♦✱✳ ✳ ✳
▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ♥ë✐ s✉② ♣❤ê ❜✐➳♥ ♥❤➜t ♥❤÷✿
✲ ◆ë✐ s✉② t❛♠ ❣✐→❝


ữỡ







ở s t
ở s t t
ở s s t t
ở s t t t
ở s tr ổ
r ử tr ố t t ở t ờ trữ ú
ữ ữủ õ t ữợ t
tr t tớ tỹ ởt ở s t t

ỗ ồ t ở t tớ trữ
ớ trữ ỳ sỷ ử ở s t t
tr ủ ỵ tớ trữ


21 C

ỷ ử ữỡ ở s t t tr
t

trú ừ t
tỷ ừ t ỗ tr
tr r tữỡ ự
ởt t ữủ trữ số ừ õ số
ữủ ử tt ử ởt tr r
t ỡ t ữủ ử ởt
t õ ữủ ồ ởt


❈❤÷ì♥❣ ✷✳

❑■➌▼

❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆● ❚➐▼

✶✹

❈→❝ ❜↔♥❣ t➻♠ ❦✐➳♠ ❤❛✐ ❝❤✐➲✉✱ ❜❛ ❝❤✐➲✉✱ ♥✲❝❤✐➲✉ ❝✉♥❣ ❝➜♣ ❣✐↔✐ ♣❤→♣ rã
r➔♥❣ ❝❤♦ ❣✐î✐ ❤↕♥ ✈è♥ ❝â ❝õ❛ ▲❯❚ ♠ët ❝❤✐➲✉✱ ❝❤♦ ♣❤➨♣ ❞ú ❧✐➺✉ ❞↕♥❣ ❜↔♥❣
✤÷ñ❝ ❧➟♣ ❝❤➾ ♠ö❝ tr➯♥ ❤❛✐ t❤❛♠ sè✱ ❜❛ t❤❛♠✱ ♥ t❤❛♠ sè sè ✤ë❝ ❧➟♣✳
❇↔♥❣ ❞÷î✐ ✤➙② ♠✐♥❤ ❤å❛ ❝➜✉ tró❝ ❝õ❛ ❜↔♥❣ t➻♠ ❦✐➳♠ ❤❛✐ ❝❤✐➲✉✳ ❈→❝
✈❡❝tì ❤♦➦❝ t➟♣ ❞ú ❧✐➺✉ ✤✐➸♠ ❞ø♥❣ ✈➔ ♠ët ♠↔♥❣✱ ✤÷ñ❝ ❣å✐ ❧➔ ❞ú ❧✐➺✉ ❝õ❛
❜↔♥❣✱ t↕♦ t❤➔♥❤ ❜↔♥❣ t➻♠ ❦✐➳♠✳


❇↔♥❣ ✷✳✹✿ ❇↔♥❣ ♠✐♥❤ ❤å❛ ❝➜✉ tró❝ ❝õ❛ ❜↔♥❣ t➻♠ ❦✐➳♠ ❤❛✐ ❝❤✐➲✉
✶
✷
✸
✹
✶ f (1, 1) f (1, 2) f (1, 3) f (1, 4)
✷ f (2, 1) f (2, 2) f (2, 3) f (2, 4)
✸ f (3, 1) f (3, 2) f (3, 3) f (3, 4)
▼é✐ ❜ë ❞ú ❧✐➺✉ ✤✐➸♠ ❞ø♥❣ ❧➔ ♠ët ❝❤➾ ♠ö❝ ❝õ❛ ❝→❝ ❣✐→ trà ✤➛✉ ✈➔♦ ❝❤♦
♠ët t❤ù ♥❣✉②➯♥ ❝ö t❤➸ ❝õ❛ ❜↔♥❣ t➻♠ ❦✐➳♠✳ ▼↔♥❣ ❞ú ❧✐➺✉ ✤â♥❣ ✈❛✐ trá ✤↕✐
❞✐➺♥ ✤÷ñ❝ ❧➜② ♠➝✉ ❝õ❛ ♠ët ❤➔♠ ✤÷ñ❝ ✤→♥❤ ❣✐→ t↕✐ ❝→❝ ❣✐→ trà ✤✐➸♠ ❞ø♥❣✳
❈→❝ ❜↔♥❣ t➻♠ ❦✐➳♠ sû ❞ö♥❣ ❝→❝ t➟♣ ❞ú ❧✐➺✉ ✤✐➸♠ ❞ø♥❣ ✤➸ ❧✐➯♥ ❦➳t ❝→❝ ❣✐→
trà ✤➛✉ ✈➔♦ ❝õ❛ ❜↔♥❣ ✤➸ tr↔ ❦➳t q✉↔ ❣✐→ trà ✤➛✉ r❛ t÷ì♥❣ ù♥❣✳
❱➼ ❞ö✱ ❤➣② ①❡♠ ①➨t ❜↔♥❣ s❛✉✿

❇↔♥❣ ✷✳✺✿ ❇↔♥❣ t➻♠ ❦✐➳♠ ❤❛✐ ❝❤✐➲✉ ♠æ t↔ ♠ët ❤➔♠ ❜✐➸✉ t❤à t❤❡♦ ❤❛✐ ❜✐➳♥ ❝❤✐➲✉ ❝❛♦ ✈➔
❝➙♥ ♥➦♥❣
✶✵✵✭❦❣✮ ✷✵✵✭❦❣✮ ✺✵✵✭❦❣✮ ✶✵✵✵✭❦❣✮
✵✭♠✮
✸
✺✱✺
✻
✽✱✺
✶✵✭♠✮
✹✱✹
✻✱✼
✼✱✽
✶✵✱✶
✷✵✭♠✮
✺✱✽

✼✱✼
✾✱✸
✶✷✱✺
✹✵✭♠✮
✼✱✸
✽✱✻
✶✶✱✸
✶✹✱✷
✻✵✭♠✮
✽✱✺
✶✵✱✾
✶✹✱✺
✶✼✱✺
✶✵✵✭♠✮
✶✵✱✶
✶✺✱✻
✶✼✱✽
✷✶✱✼
❇↔♥❣ ♥➔② ①→❝ ✤à♥❤ ❣✐→ trà ❝õ❛ ♠ët ❤➔♠ ♣❤ö t❤✉ë❝ t❤❡♦ ❤❛✐ ❜✐➳♥ ✤ë❝
❧➟♣ ① ✈➔ ② tr♦♥❣ ✤â ❜✐➳♥ ① ❜✐➸✉ t❤à ❝❤✐➲✉ ❞➔✐ ✭♠✮ ✈➔ ❜✐➳♥ ② ❜✐➸✉ t❤à ❦❤è✐
❧÷ñ♥❣ ✭❦❣✮✳ ❉♦ ✤â✱ ♥â ❧➔ ♠ët ❜↔♥❣ t➻♠ ❦✐➳♠ ❤❛✐ ❝❤✐➲✉✳
◆➳✉ ❜✐➳♥

x = 40(m)

y = 500(kg)✱ ❤➔♠ ♣❤ö t❤✉ë❝ t❤❡♦ ① ✈➔ ②
x = 60(m) ✈➔ ❜✐➳♥ y = 1000(kg) t❤➻ ❤➔♠ ♣❤ö

✈➔ ❜✐➳♥


s➩ ❝â ❣✐→ trà ✶✶✱✸❀ ♥➳✉ ❜✐➳♥

t❤✉ë❝ t❤❡♦ ① ✈➔ ② s➩ ❝â ❣✐→ trà ✶✼✱✺❀✳ ✳ ✳
❱î✐ ♥❤ú♥❣ ❣✐→ trà ❝õ❛ ❜✐➳♥ ①✱ ❜✐➳♥ ② ❦❤æ♥❣ ①✉➜t ❤✐➺♥ tr➯♥ ❜↔♥❣ ✈➼ ❞ö
♥❤÷ ✈î✐

x = 30(m)✱ y = 150(kg)

② s➩ ❧➔ ❜❛♦ ♥❤✐➯✉❄

t❤➻ ❣✐→ trà ❝õ❛ ❤➔♠ ♣❤ö t❤✉ë❝ t❤❡♦ ① ✈➔


❈❤÷ì♥❣ ✷✳

❑■➌▼

❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆● ❚➐▼

✶✺

❈→❝ t❤✉➟t t♦→♥ ♥ë✐ s✉② ❝❤♦ ♣❤➨♣ ❝→❝ ❜↔♥❣ t➻♠ ❦✐➳♠ t↕♦ r❛ ❦➳t q✉↔ ❦❤✐
✤÷ñ❝ tr✉② ✈➜♥ ❝→❝ ❣✐→ trà ❣✐ú❛ ❝→❝ ✤✐➸♠ ♠➝✉✳ P❤÷ì♥❣ ♣❤→♣ ✤ì♥ ❣✐↔♥ ♥❤➜t✱
♥ë✐ s✉② ❧➙♥ ❝➟♥ ❣➛♥ ♥❤➜t✱ ❧➔ t➻♠ ✈➔ tr↔ ✈➲ ❦➳t q✉↔ ❝õ❛ ✤✐➸♠ ♠➝✉ tr÷î❝
✤✐➸♠ ♥ë✐ s✉②✳ ▼➦❝ ❞ò ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ♥❤❛♥❤✱ ♥❤÷♥❣ ♥â t❤÷í♥❣ ♠❛♥❣
❧↕✐ ❦➳t q✉↔ ❦❤æ♥❣ ❧✐➯♥ tö❝✳
▼ët t❤✉➟t t♦→♥ ♥ë✐ s✉② ♥➙♥❣ ❝❛♦ ❤ì♥ ❧➔ t➼♥❤ tr✉♥❣ ❜➻♥❤ ❝â trå♥❣ sè
❣✐ú❛ ❤❛✐ ♠➝✉ ❣✐î✐ ❤↕♥ ✭tr♦♥❣ tr÷í♥❣ ❤ñ♣ ▲❯❚ ✶❉✮✱ ❞ü❛ tr➯♥ ❦❤♦↔♥❣ ❝→❝❤
t÷ì♥❣ ✤è✐ ❝õ❛ ♠➝✉ ✈î✐ ❝→❝ ❧➙♥ ❝➟♥ ✤÷ñ❝ ❣å✐ ❧➔ ♥ë✐ s✉② t✉②➳♥ t➼♥❤✱ ♣❤÷ì♥❣
♣❤→♣ ♥➔② ❝✉♥❣ ❝➜♣ ❦➳t q✉↔ ♠÷ñt ♠➔ ❤ì♥ ✤→♥❣ ❦➸ s♦ ✈î✐ ♥ë✐ s✉② ❧➙♥ ❝➟♥

❣➛♥ ♥❤➜t✳
✣➸ ❣✐↔✐ q✉②➳t ✈➜♥ ✤➲ tr➯♥✱ ❝❤ó♥❣ tæ✐ ❧ü❛ ❝❤å♥ ♣❤÷ì♥❣ ♣❤→♣ ♥ë✐ s✉②
t✉②➳♥ t➼♥❤ ✤➸ ①→❝ ✤à♥❤ ❣✐→ trà ❝õ❛ ❜✐➳♥ tr♦♥❣ ♠✐➲♥ ❦❤♦↔♥❣ ❣✐→ trà ①→❝
✤à♥❤✳

✷✳✸✳✷✳ ❇↔♥❣ t➻♠ ❦✐➳♠ ♠ët ❝❤✐➲✉ ✈➔ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤
tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠ët ❝❤✐➲✉
❚r♦♥❣ t♦→♥ ❤å❝✱ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ ❧➔ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❦❤î♣
✤÷í♥❣ ❝♦♥❣ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ✤❛ t❤ù❝ t✉②➳♥ t➼♥❤ ✤➸ ①➙② ❞ü♥❣ ❝→❝ ✤✐➸♠
❞ú ❧✐➺✉ ♠î✐ tr♦♥❣ ♣❤↕♠ ✈✐ ❝õ❛ ♠ët t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ❞ú ❧✐➺✉ ✤➣ ❜✐➳t✳
◆ë✐ s✉② t✉②➳♥ t➼♥❤ ✤➣ ✤÷ñ❝ sû ❞ö♥❣ tø t❤í✐ ❝ê ✤↕✐ ✤➸ ❧➜♣ ✤➛② ❝→❝
❦❤♦↔♥❣ trè♥❣ tr♦♥❣ ❜↔♥❣✳ ●✐↔ sû r➡♥❣ ♥❣÷í✐ t❛ ❝â ♠ët ❜↔♥❣ ❧✐➺t ❦➯ ❞➙♥
sè ❝õ❛ ♠ët sè q✉è❝ ❣✐❛ ✈➔♦ ❝→❝ ♥➠♠ ✶✾✼✵✱ ✶✾✽✵✱ ✶✾✾✵ ✈➔ ✷✵✵✵ ✈➔ ♥❣÷í✐ t❛
♠✉è♥ ÷î❝ t➼♥❤ ❞➙♥ sè ✈➔♦ ♥➠♠ ✶✾✾✹✳ ◆ë✐ s✉② t✉②➳♥ t➼♥❤ ❧➔ ♠ët ♣❤÷ì♥❣
♣❤→♣ ❞➵ ❞➔♥❣ ✤➸ ❧➔♠ ✤÷ñ❝ ✤✐➲✉ ♥➔②✳ ❑ÿ t❤✉➟t sû ❞ö♥❣ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥
t➼♥❤ ✤➸ ❧➟♣ ❜↔♥❣ ✤÷ñ❝ ❝❤♦ ❧➔ ✤÷ñ❝ sû ❞ö♥❣ ❜ð✐ ❝→❝ ♥❤➔ t❤✐➯♥ ✈➠♥ ❤å❝ ✈➔
t♦→♥ ❤å❝ ♥❣÷í✐ ❇❛❜②❧♦♥ ð ❙❡❧❡✉❝✐❞ ▼❡s♦♣♦t❛♠✐❛ ✭t❤➳ ❦✛ ❜❛ tr÷î❝ ❈æ♥❣
♥❣✉②➯♥✮✱ ✈➔ ❜ð✐ ♥❤➔ t❤✐➯♥ ✈➠♥ ❤å❝ ✈➔ t♦→♥ ❤å❝ ❍② ▲↕♣✱ ❍✐♣♣❛r❝❤✉s ✭t❤➳
❦✛ t❤ù ✷ tr÷î❝ ❈æ♥❣ ♥❣✉②➯♥✮✳
●✐↔ sû t❛ ❝â ❤➔♠

[5]

y = f (x)

❝❤♦ tr÷î❝ ♥❤÷ tr♦♥❣ ❤➻♥❤ ✷✳✷✳

✈➔ t❛ ❝â ❤❛✐ ✤✐➸♠

A(x0 , y0 )✱


✈➔

B(x1 , y1 )


❈❤÷ì♥❣ ✷✳

❑■➌▼

❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆● ❚➐▼

✶✻

❍➻♥❤ ✷✳✷✿ ◆ë✐ s✉② t✉②➳♥ t➼♥❤
❉♦ ❤❛✐ ✤✐➸♠

A(x0 , y0 )✱

✈➔

B(x1 , y1 )

❝â tå❛ ✤ë ❝❤♦ tr÷î❝ ♥➯♥ ♣❤÷ì♥❣

tr➻♥❤ ✤÷í♥❣ t❤➥♥❣ ❆❇ ✤✐ q✉❛ ✤✐➸♠ ❈ ❝â ❞↕♥❣✿

(y1 − y0 )(xc − x0 ) + (x0 − x1 )(yc − y0 ) = 0
⇔ yc =


y0 (x0 − x1 ) − (xc − x0 )(y1 − y0 )
x0 − x1

⇔ yc = y0 +
⇔ yc =

xc − x0
(y1 − y0 )
x1 − x0

x1 − xc
xc − x0
.y0 +
.y1
x1 − x0
x1 − x0

P❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ ✤÷ñ❝ ♠æ t↔ tr♦♥❣ ❤➻♥❤ ✷✳✷ ✤✐➸♠ ❈ ♥➡♠ tr➯♥
✤÷í♥❣ t❤➥♥❣ ❣✐ú❛ ❝→❝ ✤✐➸♠ ❆ ✈➔ ❇ ❧➔ ✤÷ñ❝ ♥ë✐ s✉②✳ ●✐→ trà ♥ë✐ s✉②

yc

❧➔

x −x0
t✉②➳♥ t➼♥❤ t✛ ❧➺ t❤✉➟♥ ✈î✐ t✛ ❧➺ c
x1 −x0 ✱ tr♦♥❣ ✤â
❝õ❛ ✤♦↕♥ t❤➥♥❣ ♥è✐ ✤✐➸♠ ❆ ✈➔ ❇ tr➯♥ trö❝ ❖①✱

x1 − x0 ❧➔ ✤ë ❞➔✐ ❝❤✐➳✉

✈➔ xc − x0 ❧➔ ❦❤♦↔♥❣ ❝→❝❤

❝❤✐➳✉ ❝õ❛ ✤÷í♥❣ ♥è✐ ✤✐➸♠ ❆ ✈➔ ❈ tr➯♥ trö❝ ❖①✳

❱➼

❞ö

sû

❞ö♥❣

y = f (x) = sinx✳

♣❤÷ì♥❣

♣❤→♣

❚❛ ❝â ❜↔♥❣ ✷✳✻

♥ë✐

s✉②

t✉②➳♥

t➼♥❤

tr♦♥❣


❤➔♠


❈❤÷ì♥❣ ✷✳

❑■➌▼

❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆● ❚➐▼

✶✼

❇↔♥❣ ✷✳✻✿ ●✐→ trà ❤➔♠ sinx t↕✐ ❝→❝ ✤✐➸♠ ✤➦❝ ❜✐➺t
◦
◦
①
30◦ 45
60
90◦
√
√
2
3
1
sinx
✶
2
2
2

f (x) = sinx t↕✐ x = 47◦

x ∈ [30◦ , 60◦ ]✳ ●✐→ trà ❝õ❛ ❤➔♠ f (x) = sinx

❚➼♥❤ ❣✐→ trà ❤➔♠
❳➨t

t↕✐

x = 47◦

❜➡♥❣ ❜❛♦

♥❤✐➯✉ t❛ sû ❞ö♥❣ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠ët ❝❤✐➲✉

f
sin 47 = 0, 7074✳

✈î✐ ❣✐→ trà ❤➔♠

t↕✐ ❤❛✐ ✤➛✉ ♠ót✳ ⑩♣ ❞ö♥❣ ❝æ♥❣ t❤ù❝ ♥ë✐ s✉② t❛ ✤÷ñ❝

◦

❑❤✐ t❛ t➠♥❣ ❝→❝ ✤✐➸♠ ❣➛♥ ♠è❝ ♥ë✐ s✉② ❧➯♥✱ ❝❤♦ ♠è❝ ❣✐→ trà

◦

◦

(45 , 60 )✳ ❑❤✐ ✤â
t✉②➳♥ t➼♥❤✱ sin 47 = 0, 7283✳

◦
◦
❱î✐ ✤✐➸♠ ♥ë✐ s✉② ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ (45 , 50 )
◦
❇➡♥❣ ❦➳t q✉↔ t❤ü❝ ♥❣❤✐➺♠✱ sin 47 = 0, 7313✳
t❤➜②

47

◦

♥➡♠ tr♦♥❣ ❦❤♦↔♥❣

x = 45◦ ✱

t❛

→♣ ❞ö♥❣ ❝æ♥❣ t❤ù❝ ♥ë✐ s✉②

◦

t❤➻

sin 47◦ = 0, 73066✳

❚ø ✤â✱ t❛ t❤➜② ❦❤✐ t❛ t➠♥❣ ❝➔♥❣ ♥❤✐➲✉ ✤✐➸♠ ❣➛♥ ✤✐➸♠ ♥ë✐ s✉② t❤➻ ❣✐→ trà
tr↔ ✈➲ ❝➔♥❣ s→t ✈î✐ ❣✐→ trà t❤ü❝ ❝õ❛ ♥â✳

✷✳✸✳✸✳ ❇↔♥❣ t➻♠ ❦✐➳♠ ❤❛✐ ❝❤✐➲✉ ✈➔ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤
tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤❛✐ ❝❤✐➲✉

❚r♦♥❣ t♦→♥ ❤å❝✱ ♣❤➨♣ ♥ë✐ s✉② s♦♥❣ t✉②➳♥ t➼♥❤ ❧➔ ♠ët ♣❤➛♥ ♠ð rë♥❣
❝õ❛ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ ✤➸ ♥ë✐ s✉② ❝→❝ ❤➔♠ ❤❛✐ ❜✐➳♥ ✭✈➼ ❞ö✿ ❤➔♠
❝❤ù❛ ❤❛✐ ❜✐➳♥ ① ✈➔ ②✮ tr➯♥ ❧÷î✐ ✷❉✳ Þ t÷ð♥❣ ❝❤➼♥❤ ❧➔ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ♥ë✐

[1]

s✉② t✉②➳♥ t➼♥❤ tr÷î❝ t❤❡♦ ♠ët ❤÷î♥❣✱ ✈➔ s❛✉ ✤â ❧↕✐ t❤❡♦ ❤÷î♥❣ ❦❤→❝✳
❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❤❛✐ ❝❤✐➲✉✱ t❛ ①➨t ♠ët ❤➔♠ ❤❛✐ ❜✐➳♥
✤✐➸♠

p00 (x0 , y0 )✱ p01 (x0 , y1 )✱ p10 (x1 , y0 )✱ p11 (x1 , y1 )

p = f (x, y)

✈➔ ❜è♥

♥❤÷ tr♦♥❣ ❤➻♥❤ ✷✳✸


❈❤÷ì♥❣ ✷✳

❑■➌▼

❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆● ❚➐▼

✶✽

❍➻♥❤ ✷✳✸✿ ◆ë✐ s✉② s♦♥❣ t✉②➳♥ t➼♥❤
✣➸ t❤✉ ✤÷ñ❝ ❣✐→ trà ❝❤♦ ✤✐➸♠


p0

✤➛✉ t✐➯♥ t❛ ❣✐ú ♥❣✉②➯♥ ❤➡♥❣ sè

❞ö♥❣ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr➯♥ ✤✐➸♠

p00

p0 = p00 + (p10 − p00 )
✣➸ t❤✉ ✤÷ñ❝ ❣✐→ trà ❝❤♦ ✤✐➸♠

p1

p01

p1 = p01 + (p11 − p01 )
p0

✈➔

p1 ✱

p10

✤➸ ✤↕t ✤÷ñ❝

ù♥❣

y0 ✈➔
p1 ✳


ù♥❣

x − x0
x1 − x0

✤➛✉ t✐➯♥ t❛ ❣✐ú ♥❣✉②➯♥ ❤➡♥❣ sè

❞ö♥❣ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr➯♥ ✤✐➸♠

❙❛✉ ❦❤✐ ✤↕t ✤÷ñ❝

✈➔

y0 ✈➔
p0 ✳

✈➔

p11

✤➸ ✤↕t ✤÷ñ❝

x − x0
x1 − x0

t❛ ❧↕✐ →♣ ❞ö♥❣ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ ✤➸ t➻♠

r❛ ✤✐➸♠ ♣ ❜➡♥❣ ❝→❝❤ ❣✐ú ♥❣✉②➯♥ ❤➡♥❣ sè ① ✈➔ ù♥❣ ❞ö♥❣ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥
t➼♥❤ tr➯♥ ✤✐➸♠


p0

✈➔

p1 ✳

p = f (x, y) = p0 +
❚❤❛② ❝æ♥❣ t❤ù❝

p = p00 +

p0

✈➔

p1

y − y0
(p1 − p0 )
y1 − y0

✈➔♦ ♣ t❛ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✿

x − x0
y − y0
x − x0 y − y0
(p10 −p00 )+
(p01 −p00 )+
(p11 −p01 −p10 +p00 )

x1 − x0
y1 − y 0
x1 − x0 y1 − y0


❈❤÷ì♥❣ ✷✳

❑■➌▼

❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆● ❚➐▼

✶✾

❍❛② ♣ ✤÷ñ❝ ✈✐➳t ❞÷î✐ ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮✿

p = c0 + c1 ∆x + c2 ∆y + c3 ∆x∆y
tr♦♥❣ ✤â
♣❤→t

p00

∆x, ∆y

❧➔ ❦❤♦↔♥❣ ❝→❝❤ t÷ì♥❣ ✤è✐ ❝õ❛ ❝→❝ ✤✐➸♠ ✤è✐ ✈î✐ ✤✐➸♠ ①✉➜t

t❤❡♦ ❝→❝ ❤÷î♥❣ ①✱ ② t÷ì♥❣ ù♥❣✱ ✤÷ñ❝ ❜✐➸✉ t❤à tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤

✭✷✳✸✮

∆x =

✈➔ ❤➺ sè

c0
c1
c2
c3

cj

y − y0
x − x0
, ∆y =
x1 − x0
y1 − y0

✤÷ñ❝ ①→❝ ✤à♥❤ tø ❣✐→ trà ❝õ❛ ❝→❝ ✤➾♥❤

= p00 ❀
= p10 − p00 ❀
= p01 − p00 ❀
= p11 − p01 + p00 − p10

P❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮ ❝â t❤➸ ❜✐➳♥ ✤ê✐ ✈➲ ❞↕♥❣✿

p=
+
❍❛②

(x1 − x) (y − y0 )
(x − x0 ) (y − y0 )

p01 +
p11
(x1 − x0 ) (y1 − y0 )
(x1 − x0 ) (y1 − y0 )

(x − x0 ) (y1 − y)
(x1 − x) (y1 − y)
p00 +
p10
(x1 − x0 ) (y1 − y0 )
(x1 − x0 ) (y1 − y0 )

p = p01 .Na + p11 .Nb + p00 .Nc + p10 .Nd

✭✷✳✹✮ tr♦♥❣ ✤â✿

Na =

(x1 − x) (y − y0 )
;
(x1 − x0 ) (y1 − y0 )

Nb =

(x − x0 ) (y − y0 )
;
(x1 − x0 ) (y1 − y0 )

Nc =


(x1 − x) (y1 − y)
;
(x1 − x0 ) (y1 − y0 )

Nd =

(x − x0 ) (y1 − y)
.
(x1 − x0 ) (y1 − y0 )

❍➻♥❤ ❝❤ú ♥❤➟t ❆❇❈❉ ✤÷ñ❝ ❝❤✐❛ t❤➔♥❤ ❜è♥ ✈ò♥❣ t❤❡♦ ❝→❝ ❞á♥❣ ① ✈➔
②✳ ✣➸ ♥ë✐ s✉② ❣✐→ trà ♣ ❝õ❛ ❤➔♠

p = f (x, y)

t↕✐ ✤✐➸♠ ❊ ✭✤÷ñ❝ ❝❤♦ ð ① ✈➔

②✮✱ ❝❤ó♥❣ t❛ t➻♠ t❤➜② ❝→❝ ✈ò♥❣ ❝❤✉➞♥ ❤â❛ ❝õ❛ ❤➻♥❤ ❝❤ú ♥❤➟t ❆❇❈❉✳ ❇è♥
✈ò♥❣ ✤÷ñ❝ ❝❤✉➞♥ ❤â❛ ❝❤➼♥❤ ❧➔

Na , Nb , Nc

✈➔

Nd ✳

[2]

❱➼ ❞ö ✈➲ ♣❤➨♣ ♥ë✐ s✉② s♦♥❣ t✉②➳♥ t➼♥❤ tr♦♥❣ ❝→❝ ❣✐→ trà t❤❛♥❣
✤ë ①→♠✳ ◆ë✐ s✉② ❤❛✐ ❝❤✐➲✉ ①❡♠ ①➨t ✈ò♥❣ ❧➙♥ ❝➟♥ ✷①✷ ❣➛♥ ♥❤➜t ❝õ❛ ❝→❝ ❣✐→

trà ♣✐①❡❧ ✤➣ ❜✐➳t ①✉♥❣ q✉❛♥❣ ✈à tr➼ t➼♥❤ t♦→♥ ❝õ❛ ♣✐①❡❧ ❝❤÷❛ ❜✐➳t✳ ❚➼♥❤ ❣✐→
trà ❝÷í♥❣ ✤ë ♣✐①❡❧ ð ❤➔♥❣ ✷✵✱✷ ✈➔ ❝ët ✶✹✱✺ ❦❤✐ ❜✐➳t ❜è♥ ❣✐→ trà ♣✐①❡❧ ❧➙♥
❝➟♥ ❣➛♥ ♥❤➜t✳


❈❤÷ì♥❣ ✷✳

❑■➌▼

❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆● ❚➐▼

✷✵

❍➻♥❤ ✷✳✹✿ ❍➻♥❤ ✈➩ ♠æ t↔ ❜è♥ ✤✐➸♠ ↔♥❤ ✤➣ ❜✐➳t ✈➔ ✤✐➸♠ ↔♥❤ ❝➛♥ ♥ë✐ s✉②✳
◆❤÷ ✤➣ t❤➜② tr♦♥❣ ❤➻♥❤ ✷✳✹✱ ❣✐→ trà ❝÷í♥❣ ✤ë t↕✐ ♣✐①❡❧ ✤÷ñ❝ t➼♥❤ ð ❤➔♥❣
✷✵✱✷✱ ❝ët ✶✹✱✺ ❝â t❤➸ ✤÷ñ❝ t➼♥❤ ❜➡♥❣ ❝→❝❤ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ ✤➛✉ t✐➯♥ ❣✐ú❛
❝→❝ ❣✐→ trà ð ❝ët ✶✹ ✈➔ ✶✺ tr➯♥ ♠é✐ ❤➔♥❣ ✷✵ ✈➔ ✷✶✳ ❚❛ t❤✉ ✤÷ñ❝ ❦➳t q✉↔✿

I14,5;21 = 162 +

14, 5 − 14
(95 − 162) = 128.5
15 − 14

14, 5 − 14
(210 − 91) = 150, 5
15 − 14
❣✐ú❛ ❝→❝ ❣✐→ trà I14,5;21 ✱ I14,5;20 t❛

I14,5;20 = 91 +

❙❛✉ ✤â ♥ë✐ s✉② t✉②➳♥ t➼♥❤

I14,5;20,2 = 150, 5 +

✤÷ñ❝

20, 2 − 20
(128, 5 − 150, 5) = 146, 1
21 − 20

❑❤✐ ✤â t❛ t❤✉ ✤÷ñ❝ ❦➳t q✉↔ ❣✐→ trà ❝÷í♥❣ ✤ë ♣✐①❡❧ ð ❤➔♥❣ ✷✵✱✷✱ ❝ët ✶✹✱✺ ❧➔
✶✹✻✱✶✳

✷✳✸✳✹✳ ❇↔♥❣ t➻♠ ❦✐➳♠ ❜❛ ❝❤✐➲✉ ✈➔ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤
tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❜❛ ❝❤✐➲✉
p = f (x, y, z)
✈➔ ✽ ✤✐➸♠ p000 (x0 , y0 , z0 )✱ p001 (x0 , y0 , z1 )✱ p100 (x1 , y0 , z0 )✱ p101 (x1 , y0 , z1 )✱
p011 (x0 , y1 , z1 )✱ p111 (x1 , y1 , z1 )✱ p010 (x0 , y1 , z0 )✱ p110 (x1 , y1 , z0 ) ♥❤÷ tr♦♥❣
❚r♦♥❣

❦❤æ♥❣

❣✐❛♥

❜❛

❝❤✐➲✉✱

❣✐↔


sû

t❛

❝â

❤➔♠

❤➻♥❤ ✷✳✺
P❤÷ì♥❣ tr➻♥❤ t❛♠ t✉②➳♥ ❝â ♥❣✉ç♥ ❣è❝ ❜➡♥❣ ❝→❝❤ →♣ ❞ö♥❣ ♣❤➨♣ ♥ë✐ s✉②
t✉②➳♥ t➼♥❤ ❜↔② ❧➛♥❀ ❜❛ ❧➛♥ ✤➸ ①→❝ ✤à♥❤ ✤✐➸♠

p0

✈➔ ❜❛ ❧➛♥ ✤➸ ①→❝ ✤à♥❤ ✤✐➸♠


❈❤÷ì♥❣ ✷✳

❑■➌▼
p1

❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆● ❚➐▼

✷✶

♥❤÷ tr♦♥❣ ♥ë✐ s✉② s♦♥❣ t✉②➳♥ t➼♥❤ ✷❉✱ s❛✉ ✤â t❤➯♠ ♠ët ❧➛♥ ♥ú❛ ✤➸ t➼♥❤

[1]


t♦→♥ ✤✐➸♠ ♣✳

❍➻♥❤ ✷✳✺✿ ◆ë✐ s✉② t❛♠ t✉②➳♥ t➼♥❤
✣➸ t❤✉ ✤÷ñ❝ ❣✐→ trà ❝❤♦ ✤✐➸♠

p00 ✱

ù♥❣ ❞ö♥❣ ♣❤➨♣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr➯♥ ✤✐➸♠

p00 = p000 +
❚÷ì♥❣ tü✱ t➻♠ ✤✐➸♠

p000

✈➔

p100

✤➸ ✤↕t

x − x0
(p100 − p000 )
x1 − x0

p10 ✱ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr➯♥ ✤✐➸♠ p010
p10 = p010 +

y0 ✈➔
✤÷ñ❝ p00 ✳


✤➛✉ t✐➯♥ t❛ ❣✐ú ♥❣✉②➯♥ ❤➡♥❣ sè

✈➔

p110

t❛ ✤÷ñ❝✿

x − x0
(p110 − p010 )
x1 − x0

◆ë✐ s✉② t✉②➳♥ t➼♥❤ ❧➛♥ t❤ù ✸ ❜➡♥❣ ❝→❝❤ ❣✐ú ♥❣✉②➯♥ ❤➡♥❣ sè ①✱ ù♥❣ ❞ö♥❣
♥ë✐ s✉② t✉②➳♥ t➼♥❤ tr➯♥ ❤❛✐ ✤✐➸♠

p0 = p00 +

p10

✈➔

p00

✤➸ t➻♠ r❛ ✤✐➸♠

y − y0
(p10 − p00 )
y1 − y0

p0 ✳



❈❤÷ì♥❣ ✷✳

❑■➌▼

❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆● ❚➐▼

❇➡♥❣ ❝→❝❤ ❧➔♠ t÷ì♥❣ tü✱ ♥ë✐ s✉② t✉②➳♥ t➼♥❤ ✸ ❧➛♥ t➻♠ r❛ ✤✐➸♠

✷✷

p1 ✳

x − x0
(p101 − p001 )
x1 − x0
x − x0
p11 = p011 +
(p111 − p011 )
x1 − x0
y − y0
p1 = p01 +
(p11 − p01 )
y1 − y0
p01 = p001 +

❚ø ❤❛✐ ✤✐➸♠

p0


✈➔

t➼♥❤ tr➯♥ ❤❛✐ ✤✐➸♠

p1 ✱ ❜➡♥❣ ❝→❝❤ ❣✐ú ♥❣✉②➯♥ ②✱ ù♥❣ ❞ö♥❣ ♥ë✐ s✉② t✉②➳♥
p0 ✈➔ p1 t❛ t➻♠ ✤÷ñ❝ ✤✐➸♠ ♣ ❝➛♥ t➻♠ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥

✸❉

p = p0 +

z − z0
(p1 − p0 )
z1 − z0

❚❤❛② ❣✐→ trà tê♥❣ q✉→t ❝õ❛ ❝→❝ ❜✐➸✉ t❤ù❝

p00 ✱ p01 ✱ p10 ✱ p11 ✱p0 ✱p1

✈➔♦ ♣ t❛

✤÷ñ❝ ❜✐➸✉ t❤ù❝ tê♥❣ q✉→t ❝❤♦ ♣❤➨♣ ♥ë✐ s✉② t❛♠ t✉②➳♥ ✤÷ñ❝ ✤÷❛ r❛ tr♦♥❣
♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮

p = c0 +c1 ∆x+c2 ∆y +c3 ∆z +c4 ∆x∆y +c5 ∆y∆z +c6 ∆z∆x+c7 ∆x∆y∆z
tr♦♥❣ ✤â ∆x❀ ∆y ✱ ∆z ❧➔ ❦❤♦↔♥❣ ❝→❝❤ t÷ì♥❣ ✤è✐ ❝õ❛ ❝→❝ ✤✐➸♠ ✤è✐ ✈î✐ ✤✐➸♠
①✉➜t ♣❤→t p000 t❤❡♦ ❝→❝ ❤÷î♥❣ ①✱ ②✱ ③ t÷ì♥❣ ù♥❣✱ ✤÷ñ❝ ❜✐➸✉ t❤à tr♦♥❣
♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮


∆x =
❍➺ sè

c0
c1
c2
c3
c4
c5
c6
c7

cj

x − x0
y − y0
z − z0
, ∆y =
, ∆z =
x1 − x0
y 1 − y0
z1 − z0

✤÷ñ❝ ①→❝ ✤à♥❤ tø ❣✐→ trà ❝õ❛ ❝→❝ ✤➾♥❤

= p000 ;
= p100 − p000 ;
= p010 − p000 ;
= p001 − p000 ;
= p110 − p010 − p100 + p000 ;

= p011 − p001 − p010 + p000 ;
= p101 − p001 − p100 + p000
= p111 − p011 − p101 − p110 + p100 + p001 + p010 − p000 .

▼ët ❝→❝❤ ♥❤➻♥ ❦❤→❝ ❧➠♥❣ trö ❆❇❈❉❊❋●❍ ✤÷ñ❝ ♣❤➙♥ ❤♦↕❝❤ t❤➔♥❤
t→♠ ❦❤è✐ ❜ð✐ ❝→❝ ♠➦t ♣❤➥♥❣ ①✱ ② ✈➔ ③✳ ✣➸ ♥ë✐ s✉② ❣✐→ trà ♣ ❝❤♦ ❜ð✐
❤➔♠

p = f (x, y, z)

t↕✐ ✤✐➸♠ ■ ✤➛✉ t✐➯♥ ❝❤ó♥❣ t❛ t➻♠ ✈ò♥❣ ❝❤✉➞♥ ❤â❛

❝õ❛ ❧➠♥❣ ❦➼♥❤ ❆❇❈❉❊❋●❍✳ ❚→♠ t➟♣ ✤÷ñ❝ ❝❤✉➞♥ ❤â❛ ❜➡♥❣ ❝→❝❤ ❝❤✐❛


❈❤÷ì♥❣ ✷✳

❑■➌▼

❇❷◆● ❚➐▼ ❑■➌▼ ❱⑨ ◆❐■ ❙❯❨ ❚❯❨➌◆ ❚➑◆❍ ❚❘❖◆● ❇❷◆● ❚➐▼

❝❤ó♥❣ ❝❤♦ ❦❤è✐ ❧÷ñ♥❣ ❧➠♥❣ ❦➼♥❤ ❆❇❈❉❊❋●❍✳

Na , Nb , Nc , Nd , Ne , Nf , Ng , Nh
Na =

[2]

✷✸


❚→♠ ✈ò♥❣ ❝❤✉➞♥ ❤â❛

✤÷ñ❝ t➼♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝✿

(x1 − x)(y1 − y)(z − z0 )
(x1 − x)(y − y0 )(z − z0 )
; Nb =
(x1 − x0 )(y1 − y0 )(z1 − z0 )
(x1 − x0 )(y1 − y0 )(z1 − z0 )

(x − x0 )(y1 − y)(z − z0 )
; Nd =
(x1 − x0 )(y1 − y0 )(z1 − z0 )
(x1 − x)(y1 − y)(z1 − z)
Ne =
; Nf =
(x1 − x0 )(y1 − y0 )(z1 − z0 )
(x − x0 )(y1 − y)(z1 − z)
; Nh =
Ng =
(x1 − x0 )(y1 − y0 )(z1 − z0 )
Nc =

(x − x0 )(y − y0 )(z − z0 )
(x1 − x0 )(y1 − y0 )(z1 − z0 )
(x1 − x)(y − y0 )(z1 − z)
(x1 − x0 )(y1 − y0 )(z1 − z0 )
(x − x0 )(y − y0 )(z1 − z)
(x1 − x0 )(y1 − y0 )(z1 − z0 )


❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮ ❝á♥ ✤÷ñ❝ ✈✐➳t ❞÷î✐ ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✻✮ ♥❤÷
s❛✉✿

p = p001 .Na +p011 .Nb +p101 .Nc +p111 .Nd +p000 .Ne +p010 .Nf +p100 .Ng +p110 Nh
❱➼ ❞ö✿ ❈❤♦ ❤➔♠ sè

p = f (x, y, z)

✈î✐ ♠ët sè ✤✐➸♠ ❞ú ❧✐➺✉ ✤➣ ❜✐➳t tr♦♥❣

❜↔♥❣ ✭✷✳✼✮ ❞÷î✐ ✤➙②✿
①

②

③

p = f (x, y, z)

✵

✵

✵

✵

✵

✵


✶

✷

✵

✶

✵

✹

✵

✶

✶

✻

✶

✵

✵

✶

✶


✵

✶

✸

✶

✶

✵

✺

✶

✶

✶

✼

p = f (0, 9; 0, 9; 0, 9)
❜↔♥❣ ✭✷✳✼✮ t❛ ❝â x0 = 0; x1 = 1; y0 = 0; y1 = 1;

❚➼♥❤ ❣✐→ trà ❤➔♠
❉ü❛ ✈➔♦ sè ❧✐➺✉

z0 = 0; z1 = 1

[0, 1] ♥➯♥ t❛ ❝â t❤➸ →♣ ❞ö♥❣ ♥ë✐ s✉② t✉②➳♥ t➼♥❤
❝õ❛ ❤➔♠ f (0, 9; 0, 9; 0, 9)✳

●✐→ trà ✵✱✾ ♥➡♠ tr♦♥❣ ✤♦↕♥
✤➸ t➻♠ r❛ ❣✐→ trà ①➜♣ ①➾