P Pì PP

ề




ỵ tt

số
số tữỡ ố s số tt ố
ỳ số t ỳ số õ
q t t s số

ú tỹ ừ ởt ữỡ tr
ỵ ừ ữỡ tr f (x) = 0
ữỡ ú ừ f (x) = 0 : ổ t ữỡ
sỹ tỗ t t tứ ữỡ s số tữỡ ự
t tứ ữỡ
Pữỡ ổ
Pữỡ ỡ
Pữỡ t t

ú ừ ởt ữỡ tr số
ổ t ữỡ ỡ
ỹ ở tử ừ ữỡ ỡ
s số ừ ữỡ ỡ



✶✳✹ ◆ë✐ s✉② ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❜➻♥❤ ♣❤÷ì♥❣ ❜➨ ♥❤➜t

✶✳ ◆ë✐ s✉② ◆✐✉tì♥
✭❛✮ ❱✐➺❝ ①➙② ❞ü♥❣ ♥ë✐ s✉② ◆✐✉tì♥
✭❜✮ ✣→♥❤ ❣✐→ s❛✐ sè tr♦♥❣ ♥ë✐ s✉②✳
✷✳ P❤÷ì♥❣ ♣❤→♣ ❜➻♥❤ ♣❤÷ì♥❣ ❜➨ ♥❤➜t
✶✳✺ ❚➼♥❤ ❣➛♥ ✤ó♥❣ ❝õ❛ ✤↕♦ ❤➔♠ ✈➔ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤
✶✳ ❚➼♥❤ ❣➛♥ ✤ó♥❣ ❝õ❛ ✤↕♦ ❤➔♠✿ ⑩♣ ❞ö♥❣ ✤❛ t❤ù❝ ♥ë✐ s✉②❀ ⑩♣ ❞ö♥❣ ❦❤❛✐ tr✐➸♥
❚❛②❧♦r
✷✳ ❚➼♥❤ ❣➛♥ ✤ó♥❣ ❝õ❛ t➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤✿ ❈æ♥❣ t❤ù❝ ❤➻♥❤ t❤❛♥❣❀ ❈æ♥❣ t❤ù❝
❙✐♠♣s♦♥

✶✳✻ ❚➼♥❤ ❣➛♥ ✤ó♥❣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝♦s✐ ✤è✐ ✈î✐ ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣
✶✳ P❤÷ì♥❣ ♣❤→♣ ❝❤✉é✐ ❚❛②❧♦r
✷✳ P❤÷ì♥❣ ♣❤→♣ ❒❧❡

✷

❇➔✐ t➟♣

❈❤♦ ❤➔♠ sè y = ln(x1 + 2x2 + 3x3). ❳→❝ ✤à♥❤ ∆y , δy ❜✐➳t x1 = 1, 012; x2 =
1, 2345; x3 = 1, 56789 ✈➔ ❝→❝ ❝❤ú sè ❜✐➸✉ ❞✐➵♥ ♥➯♥ x1 , x2 , x3 ✈ø❛ ✤→♥❣ t✐♥ ✈➔ ✈ø❛ ❝â
♥❣❤➽❛✳
✷✳✷✳ ❙û ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤æ✐✱ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✿
✷✳✶✳

✶✳
✷✳

x2 log 1


2

✷✳✸✳

✶✳
✷✳
✸✳
✹✳

✈î✐ ✤ë ❝❤➼♥❤ ①→❝ 10−3
(x + 1) = 1 ✈î✐ ✤ë ❝❤➼♥❤ ①→❝ ❧➔ 10−2

x3 + 3x2 − 3 = 0

❇➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣✱ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

x − sin x =

1
4

✈î✐ ❦➳t q✉↔ ❝â ❤❛✐ ❝❤ú sè ❧➫ t❤➟♣ ♣❤➙♥ ✤→♥❣ t✐♥

✈î✐ s❛✐ sè ❦❤æ♥❣ ✈÷ñt q✉→ 10−5
x4 − 4x − 1 = 0 ✈î✐ s❛✐ sè ❦❤æ♥❣ ✈÷ñt q✉→ 10−5
lg x − 3x + 5 = 0 ✈î✐ s❛✐ sè ❦❤æ♥❣ ✈÷ñt q✉→ 10−5
x3 − x − 1000

✷



✺✳
✻✳
✼✳
✽✳

✈î✐ s❛✐ sè ❦❤æ♥❣ ✈÷ñt q✉→ 10−5
x3 − 9x2 + 18x − 1 = 0 ✈î✐ s❛✐ sè ❦❤æ♥❣ ✈÷ñt q✉→ 10−5
x3 + 3x2 − 3 = 0 ✈î✐ s❛✐ sè ❦❤æ♥❣ ✈÷ñt q✉→ 10−5

x − cos x = 0

√
1
x+1− =0
x

✈î✐ s❛✐ sè ❦❤æ♥❣ ✈÷ñt q✉→ 10−5

✷✳✹✳ ❇➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆✐✉tì♥✱ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✉
✈î✐ s❛✐ sè ❦❤æ♥❣ ✈÷ñt q✉→ 10−5

✶✳
✷✳
✸✳
✹✳

18 2
x − sin 10x = 0
10

x2 − sin πx = 0
x2 − cos πx = 0
x
2 lg x − + 1 = 0
2

✺✳

lg x −

✻✳

x lg x −

✼✳
✽✳
✾✳

1
=0
x2
12
=0
10

x3 + 3x + 5 = 0
x4 − 3x + 1 = 0
2x − 4x = 0

✷✳✺✳ ❇➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣✱ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

s❛✉ ✈î✐ s❛✐ sè ❦❤æ♥❣ ✈÷ñt q✉→ 10−4
✶✳ x3 + 3x + 5 = 0
✷✳ x4 − 3x + 1 = 0
✸✳ x − cos2 πx = 0
✷✳✻✳ ❉ò♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✤ì♥ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
✤↕✐ sè s❛✉✿

✶✳
✷✳



2, 75x1 + 1, 78x2 + 1, 11x3 = 13, 62
3, 28x1 + 0, 7x2 + 1, 15x3 = 17, 98


1, 15x1 + 2, 72 + 3, 58x3 = 39, 72


3, 2x1 − 1, 5x2 + 0, 5x3 = 0, 9
❝❤♦
1, 6x1 + 2, 5x2 − x3 = 1, 55


x1 + 4, 1x2 − 1, 5x3 = 2, 08

✸

tî✐ ❜❛ ❝❤ú sè ❧➫ t❤➟♣ ♣❤➙♥✳
tî✐ ❦❤✐


x(m) − x(m−1)

k

≤ 10−3


✸✳
✹✳
✺✳
✻✳
✼✳

✽✳



1, 5x1 − 0, 2x2 + 0, 1x3 = 0, 4
−0, 1x1 + 1, 5x2 − 0, 1x3 = 0, 8 tî✐ ♥➠♠ ❝❤ú sè ❧➫ t❤➟♣ ♣❤➙♥✳


−0, 3x1 + 0, 2x2 − 0, 5x0 = 0, 2


1, 02x1 − 0, 05x2 − 0, 1x3 = 0, 795
−0, 11x1 + 1, 03x2 − 0, 05x3 = 0, 849 ✈î✐ sè ❧➛♥ ❧➦♣ ❧➔ ✺


−0, 11x1 − 0, 12x2 + 1, 04x3 = 1, 398



24, 21x1 + 2, 42x2 + 3, 85x3 = 30, 24
2, 31x1 + 31, 49x2 + 1, 52x3 = 40, 95 ❝❤♦ tî✐ ❦❤✐ x(m) − x(m−1) k ≤ 10−5


3, 49x1 + 4, 85x2 + 28, 72x3 = 42, 81

   
−8 1
1
x1
1
 1 −5 1  x2  = 15 ❝❤♦ tî✐ ❦❤✐ x(m) − x(m−1) k ≤ 10−4
1
1 −4
x3
7

   
12
10 1 1
x1
 2 10 1  x2  = 13 ❝❤♦ tî✐ ❦❤✐ x(m) − x(m−1) k ≤ 10−4 . ❙♦ s→♥❤
14
x3
2 2 10
❦➳t q✉↔ ✈î✐ ♥❣❤✐➺♠ ✤ó♥❣ x1 = x2 = x3 = 1?

   

x1
4
0, 24 −0, 08
8
0, 09
3
−0, 15 x2  =  9  ❝❤♦ tî✐ ❦❤✐ x(m) − x(m−1) k ≤ 10−4 .
x3
0, 04 −0, 08
4
20

✶✳ ❳➙② ❞ü♥❣ ✤❛ t❤ù❝ ♥ë✐ s✉② ▲❛❣r❛♥❣❡ ❝❤♦ ❤➔♠ sè
✭❛✮ y = f (x) ❜✐➳t f (0) = 1, f (2) = 3, f (3) = 2, f (5) = 5. ❚ø ✤â ①→❝ ✤à♥❤ ❣✐→
trà ❣➛♥ ✤ó♥❣ ❝õ❛ ❤➔♠ sè t↕✐ x = 3, 24
✭❜✮ y = f (x) ❜✐➳t f (321) = 2, 50651, f (322, 8) = 2, 50893, f (324, 2) = 2, 51081,
f (325) = 2, 51188. ❚ø ✤â t➼♥❤ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛ ❤➔♠ sè t↕✐ x = 323, 5
✷✳ ❍➣② ✤→♥❤ ❣✐→ s❛✐ sè ♥❤➟♥ ✤÷ñ❝ ❦❤✐ ①➜♣ ①➾ ❤➔♠ sè y = sin x ❜➡♥❣ ✤❛ t❤ù❝ ♥ë✐
s✉② ▲❛❣r❛♥❣❡ ❜➟❝ ✺✿ L5(x), ❜✐➳t r➡♥❣ ✤❛ t❤ù❝ ♥➔② trò♥❣ ✈î✐ ❤➔♠ sè ✤➣ ❝❤♦ t↕✐
❝→❝ ❣✐→ trà ❝õ❛ x ❜➡♥❣✿ 00, 50, 100, 150, 200, 250. ❳→❝ ✤à♥❤ ❣✐→ trà ❝õ❛ s❛✐ sè ❦❤✐

✷✳✼✳

x = 120 30

✶✳ ❈❤♦ ❤➔♠ sè y = f (x) = 2x ❜✐➳t f (3, 5) = 33, 115, f (3, 55) = 34, 813,
f (3, 6) = 36, 598, f (3, 65) = 38, 475, f (3, 7) = 40, 477. ❍➣② ❧➟♣ ✤❛ t❤ù❝ ♥ë✐ s✉②
◆✐✉tì♥ t✐➳♥ ①✉➜t ♣❤→t tø 3, 50 ✈➔ ✤❛ t❤ù❝ ♥ë✐ s✉② ◆✐✉tì♥ ❧ò✐ ①✉➜t ♣❤→t tø 3, 7.
❚ø ✤â ❝❤♦ ❜✐➳t ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛ ❤➔♠ sè t↕✐ x = 3, 52 ✈➔ x = 3, 682. ❍➣②
❝❤♦ ❜✐➳t s❛✐ sè t÷ì♥❣ ù♥❣❄

✷✳ ❈❤♦ ❤➔♠ sè y = f (x) ❜✐➳t f (−1) = 3, f (0) = −6, f (3) = 39, f (6) = 822, f (7) =
1611. ❍➣② ①➙② ❞ü♥❣ ✤❛ t❤ù❝ ♥ë✐ s✉② ◆✐✉tì♥ t✐➳♥ ①✉➜t ♣❤→t tø −1 ❝õ❛ ❤➔♠ sè✳
❚ø ✤â t➼♥❤ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛ ❤➔♠ sè t↕✐ x = −0, 25

✷✳✽✳

✹


✸✳ ❈❤♦ ❤➔♠ sè y

= f (x) = sin x ❝â ❜↔♥❣ ❣✐→ trà ❣➛♥ ✤ó♥❣ f (0, 1) = 0, 09983,
f (0, 2) = 0, 19867, f (0, 3) = 0, 29552, f (0, 4) = 0, 38942.

✭❛✮ ❉ò♥❣ ✤❛ t❤ù❝ ♥ë✐ s✉② t✐➳♥ ①✉➜t ♣❤→t tø ✵✱✶ t➼♥❤ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛
❤➔♠ sè t↕✐ x = 0, 14. ❈❤♦ ❜✐➳t s❛✐ sè ❝õ❛ ❣✐→ trà ❣➛♥ ✤ó♥❣ ♥❤➟♥ ✤÷ñ❝❄
✭❜✮ ❉ò♥❣ ✤❛ t❤ù❝ ♥ë✐ s✉② ❧ò✐ ①✉➜t ♣❤→t tø 0, 4 ✤➸ t➼♥❤ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛
❤➔♠ sè t↕✐ ①❂✵✱✹✻✳ ✣→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ❣✐→ trà ❣➛♥ ✤ó♥❣ ♥❤➟♥ ✤÷ñ❝❄
✹✳ ❈❤♦ ❜↔♥❣ ❣✐→ trà ❝õ❛ ❤➔♠ sè y = f (x) = sin x ❧➔ f (150) = 0, 258819; f (200) =
0, 392202; f (250 ) = 0, 422618; f (300 ) = 0, 5.

✭❛✮ ❉ò♥❣ ✤❛ t❤ù❝ ♥ë✐ s✉② ◆✐✉tì♥ t✐➳♥ ①✉➜t ♣❤→t tø 150 ✤➸ t➼♥❤ ❣✐→ trà ❣➛♥
✤ó♥❣ ❝ó❛ ❤➔♠ sè t↕✐ x = 160. ✣→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ❣✐→ trà ❣➛♥ ✤ó♥❣ ♥❤➟♥
✤÷ñ❝❄
✭❜✮ ❉ò♥❣ ✤❛ t❤ù❝ ♥ë✐ s✉② ◆✐✉tì♥ ❧ò✐ ①✉➜t ♣❤→t tø 300 ✤➸ t➼♥❤ ❣✐→ trà ❣➛♥
✤ó♥❣ ❝õ❛ ❤➔♠ sè t↕✐ x = 310. ✣→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ❣✐→ trà ❣➛♥ ✤ó♥❣ ♥❤➟♥
✤÷ñ❝❄
✭❝✮
✷✳✾✳ ❇➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❜➻♥❤ ♣❤÷ì♥❣ ❜➨ ♥❤➜t ①→❝ ✤à♥❤ ❤➔♠ sè
✶✳


y = ax + b ❜✐➳t f (2) = 7, 32; f (4) = 8, 24; f (6) = 9, 2; f (8) = 10, 19; f (10) =
11, 01; f (12) = 12, 05

✷✳

y = ax2 + bx + c ❜✐➳t f (0, 78) = 2, 5; f (1, 56) = 1, 2; f (2, 34) = 1, 12; f (3, 12) =
2, 25; f (3, 81) = 4, 28

✸✳

y = axb ❜✐➳t f (10) = 1, 06; f (20) = 1, 33; f (30) = 1, 52; f (40) = 1, 68; f (50) =
1, 81; f (60) = 1, 91; f (70) = 2, 01; f (80) = 2, 11

✹✳
✺✳

❜✐➳t f (1) = 0, 1; f (2) = 3, f (3) = 8, 1; f (4) = 14, 9; f (5) = 23, 9
y = ax2 +bx+c ❜✐➳t f (0, 56) = −0, 8; f (0, 84) = −0, 97; f (1, 14) = −0, 98; f (2, 44) =

y = ax2 + b

1, 07; f (3, 16) = 3, 66

✶✳ ❈❤♦ ❤➔♠ sè y = f (x) = lg x ✈î✐ ❝→❝ ❣✐→ trà f (50) = 1, 699; f (55) =
1, 7404; f (60) = 1, 7782; f (65) = 1, 8129. ❚➼♥❤ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛ ✤↕♦ ❤➔♠
❝õ❛ ❤➔♠ sè t↕✐ x = 50 ✈➔ s♦ s→♥❤ ✈î✐ ❦➳t q✉↔ t➼♥❤ trü❝ t✐➳♣❄
✷✳ ❈❤♦ ❤➔♠ sè y = f (x) ❜✐➳t ❝→❝ ❣✐→ trà f (0, 98) = 0, 7739332; f (1) = 0, 7651977; f (1, 02) =
0, 7563321. ❚➼♥❤ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛ f (1)?


✷✳✶✵✳

1

❜ð✐ ❝æ♥❣ t❤ù❝ ❤➻♥❤ t❤❛♥❣ ✈➔ ❝❤✐❛ [0; 1]
✶✳ ❚➼♥❤ ❣✐→ trà ❣➛♥ ✤ó♥❣ 1 dx
+x
0
t❤➔♥❤ ✶✵ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉✳ ✣→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ❣✐→ trà ❣➛♥ ✤ó♥❣ ♥❤➟♥ ✤÷ñ❝❄

✷✳✶✶✳

✷✳ ❈❤♦

2
1

sin x
x

✺


✭❛✮ ❚➼♥❤ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛ t➼❝❤ ♣❤➙♥ ❜➡♥❣ ❝æ♥❣ t❤ù❝ ❤➻♥❤ t❤❛♥❣ ✈➔ ❝❤✐❛
[1; 2] t❤➔♥❤ ✶✵ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉✳ ✣→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ❣✐→ trà ❣➛♥ ✤ó♥❣
♥❤➟♥ ✤÷ñ❝❄
✭❜✮ ❍ä✐ ♣❤↔✐ ❝❤✐❛ [1; 2] t❤➔♥❤ ❜❛♦ ♥❤✐➯✉ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ ✤➸ ❦❤✐ t➼♥❤ t➼❝❤
♣❤➙♥ ❜➡♥❣ ❝æ♥❣ t❤ù❝ ❤➻♥❤ t❤❛♥❣ ❜↔♦ ✤↔♠ ✤÷ñ❝ s❛✐ sè ❦❤æ♥❣ ✈÷ñt q✉→
3.10−4


1

✸✳ ❉ò♥❣ ❝æ♥❣ t❤ù❝ ❤➻♥❤ t❤❛♥❣ ✈➔ ❝æ♥❣ t❤ù❝ ❙✐♠♣s♦♥ t➼♥❤ 1 +dxx2 ❦❤✐ ❝❤✐❛ [0; 1]
0
t❤➔♥❤ ✶✵ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉✳ ❈❤♦ ❜✐➳t s❛✐ sè t➻♠ ✤÷ñ❝ ❜➡♥❣ ✈✐➺❝ t➼♥❤ t➼❝❤ ♣❤➙♥
♥❤í ❝æ♥❣ t❤ù❝ ◆✐✉tì♥✲▲❡♣♥✐t
1,1

✹✳ ❉ò♥❣ ❝æ♥❣ t❤ù❝ ❤➻♥❤ t❤❛♥❣ t➼♥❤ (1 +dx4x)2 ❦❤✐ ❝❤✐❛ [0, 1; 1, 1] t❤➔♥❤ ✶✵ ♣❤➛♥
0,1
❜➡♥❣ ♥❤❛✉✳ ✣→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ❣✐→ trà ❣➛♥ ✤ó♥❣ ✈ø❛ t➻♠ ✤÷ñ❝❄
3,5

+x
dx ❦❤✐ ❝❤✐❛ [2; 3, 5] t❤➔♥❤ ✶✷ ♣❤➛♥ ❜➡♥❣
✺✳ ❉ò♥❣ ❝æ♥❣ t❤ù❝ ❙✐♠♣s♦♥✱ t➼♥❤ 11 −
x
2
♥❤❛✉✳ ✣→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ❣✐→ trà ❣➛♥ ✤ó♥❣ ✈ø❛ t➻♠ ✤÷ñ❝❄
3,1

3

✻✳ ✣➸ t➼♥❤ ❣➛♥ ✤ó♥❣ ❝õ❛ x x− 1 dx ❜➡♥❣ ❝æ♥❣ t❤ù❝ ❙✐♠♣s♦♥✱ ❝➛♥ ❝❤✐❛ [2, 1; 3, 1]
2,1
t❤➔♥❤ ❜❛♦ ♥❤✐➯✉ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ ✤➸ ✤↕t ✤÷ñ❝ s❛✐ sè | R |< 10−4
✷✳✶✷✳ ❉ò♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✉é✐ ❚❛②❧♦r ❣✐↔✐ ❜➔✐ t♦→♥
✶✳

y = x2 + y 2 , y(0) = 0


✷✳

y =

xy
, y(0) = 1
2

✈➔ 0 ≤ x ≤ 1

✻