Bài giảng phương pháp tính
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TR
:
17201
TRÌNH
DÙNG CHO SV NGÀNH :
- 2009
Bài gi
2
CNTT
: 3
60
45
15
0
0
0
n cho các bài toán
TS
LT
TH/Xemina
BT
KT
10
8
2
0
1
2
2
1
2
1
1.5. Sai
1
1
15
10
4
1
1
1
2
1
2
1
2
1
2
1
12
9
3
0
2
2
1
2
1
3
1
12
8
3
1
4
1
4
2
11
7
3
1
Bài gi
TS
LT
TH/Xemina
BT
KT
phân
1
3
2
Runger-Kutta
3
1
60
42
15
3
- Anh,
-
-
-
- Sinh
,
/ /2010
Bài gi
1
Trang
1
2
1. 1.
2
1.
3
1. 3. và sai
4
1.
5
1.
7
10
12
14
14
2. 2
14
2. 3.
17
2. 4.
20
2.
26
2. (Newton)
28
33
3:
34
34
34
3. 3. Newton
35
3. 4.
36
37
4TÍCH PHÂN
38
4.
38
4.
38
40
5:
41
5. 1.
41
5.
41
5. Runge-Kutta
42
43
: 6
44
6.
44
6.
46
6.
54
60
60
62
64
65
Bài gi
2
1.1.
1.
. Cho nên
, .
a.
.
Aa
(
). , nên không
. Do
a
Aa
:
Aa
a
(1.1)
a
.
a
a
.
a
(1.1)
a
A = a
a
(1.2)
( 1.1)
:
a -
a A
a
(1.3)
2.
:
a
Aa
A
Aa
(
).
.
:
a
=
a
a
( 1.4)
.
a
=
a
a
( 1.5)
(1.4) (1.5)
.
a
( 1.4)
a
,
a
( 1.5)
a
.
Do ( 1.5) nên ( 1.2) :
A= a ( 1
a
) (1.6)
a
a
.
Bài gi
3
3.
,
.
:
= 10
a
= 0,05
= 2
b
= 0,05m.
.
,
i:
a
10
05,0
= 0,5% <
b
=
2
05,0
= 2,5%
1.2.
1.
,
.
2,74 3
, 0,0207 .
2.
:
A =
10
s
s
a
(1.7)
:
a
s
0 9,
65,807 :
65,807 = 6.10
1
+ 5.10
0
+ 8.10
-1
+ 0.10
-2
+ 7.10
-3
( 1.7)
:
1
= 6,
o
= 5,
-1
= 8,
-2
= 0,
-3
= 7
a
.
s
.
a
0,5 .10
s
s
,
a
>
0,5 .10
s
s
.
.
: Cho a = 65,827
a
6, 5, 8, 2 ,
7, 4 ghi.
a
= 0,0067 6, 5, 8, 2, 7,
4 .
s
s
nghi.
3.
a
.
.
(1.2)
( 1.6) .
Bài gi
4
: .
không
.
, v v
.
1.3. S
1.
.
,
.
.
.
T
:
,
5
,
,
5 ,
tiên < 5
.
: 62,8274
(
a) 62,827;
h 62,83; (
) 62,8.
2.
.
Gi
a
.
aa '
.
a
:
aa '
( 1.8)
, .
.
:
- - a + a - A
:
Aa '
aa '
+
Aa
a
a
a
+
(1.9)
a
.
3.
: = (
2
- 1 )
10
. (Newton)
:
(
2
- 1)
10
= 3363 - 2378
2
( 1.10)
Bài gi
5
:
2
= 1,41421356
(1.10)
2
c
(
1.1):
Bng 1.1
2
1,4
0,0001048576
33,8
1,41
0,00013422659
10,02
1,414
0,00014791200
0,508
1,41421
0,00014866399
0,00862
1,414213563
0,00014867678
0,0001472
.
(1.10) , (1.10)
.
1.4.
1.
.
:
u = f( x,y) (1.11)
, .
:
, ,
, y, u
Dx, dy,
, y, u
x
y
u
, y, u.
(1.1)
:
x
x
;
y
y
(1.12)
Ta
u
u
u
2.
= x + y
Ta suy ra:
u
x
+
y
( 1.12)
:
u
x
y
:
x+y
x
y
(1.13)
:
u
u
.
Bài gi
6
:
(
) (
)
.
. = x -
.
:
u
=
u
u
=
yx
yx
yx
.
.
3.
= xy
du = ydx + xdy
u
y
x
+
x
y
y
x
+
x
y
u
=
y
x
+
x
y
:
u
=
u
u
=
xy
xy
yx
=
x
x
y
y
:
xy
=
x
+
y
( 1.14)
: (
)
(
)
. :
(x
n
)
= n
x
;
(1.15)
4.
/y ()
:
:
x/y
=
x
+
y
( 1.16)
5.
:
Cho : u = f( x
1
, x
2
, ,x
n
)
u
=
x
i
n
n
f
1
i
( 1.17)
u
(1.4)
: (
) (
) :
V=
6
1
3
= 3,7 0,05
= 3,14.
Bài gi
7
.
, theo (1.14) (1.15)
:
v
=
+ 3
d
d
= 0,05/3,7 =0,0135
Suy ra:
V
= 0,0005 + 3.0,0135 = 0,04
: V=
6
1
3
= 26,5 cm
3
V
= 26,5 .0,04 = 1,06 1,1cm
3
V= 26,5 1,1 cm
3
1.5. S
1.
.
.
.
, ta luôn
.
.
.
2.
a) T:
A =
3
1
1
-
3
2
1
+
3
3
1
-
3
4
1
+
3
5
1
-
3
6
1
.
6 .
. .
:
3
1
1
=
1
1
= 1,000
1
= 0
3
2
1
=
8
1
= 0,125
2
= 0
3
3
1
=
27
1
= 0,037
3
= 4.
4
10
3
4
1
=
64
1
= 0,016
4
= 4.
4
10
Bài gi
8
3
5
1
=
125
1
= 0,008
5
= 0
3
6
1
=
216
1
= 0,005
6
= 4.
4
10
a =1,000 - 0,125 + 0,037 - 0,016 + 0,008 - 0,005 = 0,899
aA
=
1
1
1
3
-
125,0
2
1
3
+
037,0
3
1
3
-
016,0
4
1
3
+
008,0
5
1
3
-
005,0
6
1
3
aA
1
1
1
3
+
125,0
2
1
3
+
037,0
3
1
3
+
016,0
4
1
3
+
008,0
5
1
3
+
005,0
6
1
3
1
+
2
+
3
+
4
+
5
+
6
= 9.
4
10
a = 0,899 9.
4
10
:
= 0,899
9.
4
10
( 1.18 )
b) T
:
B =
3
1
1
-
3
2
1
+
3
3
1
-
1
1
n
3
1
n
5.
3
10
.
.
,
. Do
,
:
n
B
=
3
1
1
-
3
2
1
1
1
n
3
1
n
n
B
.
n
BB
,
5.10
-3
.
:
n
BB
=
333
1
1
2
1
1
1
nnn
(
),
= 6 :
3
3
6
10.3
334
1
7
1
BB
Bài gi
9
6
B
=
(xem 1.18):
6
B
= A = 0,899
4
10.9
.899,0
:
B - 0,889 = B -
6
B
+ A - 0,899
899,0899,0
6
ABBB
343
10.410.910.3899,0
B
,0
899
4.
3
10
:
,
.
Bài gi
10
1
NH
1.
(
) . Ta
ô .
.
.
.
,
,
.
2.
y
1i
=qy
i
, ( 1.19 )
y
0
.
i
i
(
),
i
y
i
~
. :
y
y
i
i
( 1. 20 )
i+1
y
~
i + 1
:
y
~
i + 1
= q
y
~
i
= > 0
( 1.21)
(1.19)
:
y
~
i + 1
- y
i+1
=
q
yy
ii
q
y
~
i + 1
- y
i+1
=
q
(
)
~
yy
ii
:
y
i
~
2
= q
y
i
~
1
;
y
i 2
= q
y
i 2
:
y
i
~
2
-
y
i 2
= q(
y
i
~
1
-
y
i 1
) = q
2
(
y
i
~
-
y
i
)
Bài gi
11
:
y
ni
~
-
y
ni
= q
n
(
y
i
~
-
y
i
)
yy
nini
~
=
q
n
yy
ii
~
`
,
y
y
1
~
=
+
yy
nini
~
=
q
n
;
1.
q
1
q
n
yy
nini
~
(
).
.
2.
q
1 -
q
n
q
n
,
yy
nini
~
khi n
,
,
,
.
Bài gi
12
1.
: a = 21
o
o
. T
2.
:
a = 13267 ;
a
= 0,1%
b = 2,32 ;
b
= 0,7%
3.
:
a = 0,39410;
a
= 0,25 .10
-2
b = 38,2543 ;
b
= 0,25 .10
-2
4. :
a = 1,8921 ;
a
= 0,1.10
-2
b = 22,351;
b
= 0,1.
5. (
)
:
a) 2,1514; b)0,16152;
c)0,01204; d) - 0,0015281.
6.
tin :
a) u = ln ( x + y
2
) ; x = 0,97 ; y = 1,132
b) u = (x + y
2
)/z ; x = 3,28; y= 0,932 ; z= 1,132.
7. :
S =
11
1
+
12
1
+
13
1
+
14
1
+
15
1
+
16
1
+
17
1
8. : e = 1 +
!1
1
+
!2
1
+ +
!
1
n
+
10
-4
1.
a
= 0,13.10
-4
;
b
= 0,28.10
-3
2.
a
= 0,13.10
2
;
b
= 0,16.10
-1
3. a) 2; b) 4.
4. a) 3; b)1.
Bài gi
13
5. a)2,15; = 0,14.10
-2
; = 0,65.10
-3
b) 0,162; = 0,48.10
-3
; = 0,3.10
2
c) 0,0120; = 0,4.10
-4
; = 0,33.10
-2
d) -0,00153; = 0,19.10
-5
; = 125. 10
-2
6. a) u = 0,81;
u
= 0,27. 10
-2
;
u
= 0,33. 10
-2
b) u = 3,665;
u
= 0,7. 10
-2
;
u
= 0,20. 10
-2
7. S = 0,511.
8. e = 2,7183 0,0001.
Bài gi
14
2.2. N
y
1.
:
f(x) = 0 (2.1)
:
.
(2.1) (2.1)
:
f() = 0 (2.2)
2. Ý
T
:
y= f(x) (2.3)
(2-
1). G
= 0
= .
(2.3)
:
0 = f() (2.4)
2-1
M
x
y
Bài gi
15
(2.1)
(2.1)
g(x) = h(x) (2.5)
2 (2-2)
y = g(x), y = h(x) (2.6)
= :
g() = h() (2.7)
2
(2.6)
(2.5),
(2.1).
3.
(2.1)
(2.1)
.
2 trên.
:
2.1 -
2
(a<b) sao cho f(a) (b)
f(a).f(b) < 0 (2.8)
(x)
[a, b] [a, b]
(2.1).
(2 -
3). = f(x) x
,
,
nh,
.
(2.1)
[a, b].
2-3
4. ( )
2.2
x
y
M
f
g
y
a
B
b
x
A
Bài gi
16
2.1 - [a, b] (2.1)
.
:
2.2 - [a, b] (x)
,
f(a) (b) ,
(2.8) [a, b]
(2.1).
( 2 - 4).
= f(x)
[a, b].
[a, b]
(2.1).
(x)
.
:
2-4
2.3 - [a, b] (x)
,
(x)
(a), f(b) [a, b]
(2.1)
(2.1)
= f(x)
2.3.
5.
: f(x) = x
3
- x - 1 = 0 (2.9)
y
.
:
(x). , và
2
- 1 = 0 =
3
1
thiên
x
-
-1/
3
1/
3
+
+
0
-
0
+
f(x)
-
M
m
+
y
a
b
x
A
B
Bài gi
17
: M = f (-
3
1
) = -
33
1
+
3
1
- 1 <0
(h. 2-5),
(2.9)
, .
: f(1) = 1
3
- 1 - 1 < 0
f(2) = 2
3
- 2 - 1 > 0
[1, 2]
(2.9)
2-5
[1, 2].
,
(2. 9)
,
[1, 2].
2.3. P
1.
(2.1)
[a,
b].
x
[a, b]
< b - a.
.
[a, b],
= (a +
b)/2. [a, c] hay [c, b].
(c). (c) = 0
.
f(c) 0. (c)
f(a) . (c) (a)
[a, c]. (c) (a) [c,
[a, b]
[a, c]
hay [c, b], [a
1
, b
1
], [a, b] [a, b]
:
b
1
- a
1
=
2
1
(b - a).
[a
1,
, b
1
]
, [a
2
, b
2
], [a
1
, b
1
]
[a, b]
[a
1,
, b
1
] :
b
2
- a
2
=
2
1
(b
1
- a
1
) =
2
2
1
(b - a)
y
x
31
31
Bài gi
18
,
[a
n
, b
n
], [a, b] 1/2
n
[a, b] :
a
n
b
n
; b
n
- a
n
=
n
ab
2
)(
n
, :
nnn
aba
n
ab
2
)(
(2.10)
n
, :
nnn
abb
n
ab
2
)(
(2.11)
, a
n
hay b
n
.
Khi n
n
, b
n
.
.
:
. chính là
.
2.
(2.9)
[1, 2].
:
[1, 2] (1) = 1 - 1 - 1 < 0
f(2) = 2
3
- 2 - 1 > 0
[1, 2]
3/2.
f
2
3
=
2
3
2
-
2
3
- 1 > 0 (1).
[1, 3/2].
[1, 3/2],
5/4.
(5/4) < 0, (1).
[5/4, 3/2].
[5/4, 3/2],
11/8.
(11/8) > 0, (5/4).
[5/4, 11/8].
[5/4, 11/8],
21/16.
(21/16) < 0,
f(5/4).
[21/16, 11/8].
[21/16, 11/8],
43/32.
(42/32) > 0,
f(21/16).
[21/16, 42/32].
21/16 = 1,3125 hay 43/32 = 1,34375
t1/2
5
= 1/32 = 0,03125.
Bài gi
19
5
[1,2] 2 - 1 = 1, (
(2.10)
(2.11)).
3.
1)
(x) = 0
2)
.
3) [a, b]
4)
= (a+b)/2, (c)
f(c)f(a)< 0
Thay b=c
Thay a=c
= b - a
e <
:
a
b
-
a
<
-
b
<
S
S
Bài gi
20
2.4.
1.
(2.1)
phân ly [a,b];
(2.1)
:
X = (x) (2.12)
(2.1)
0
[a,b]
n
theo quy
:
x
n
= (x
n-1
), n = 1,2 (2.13)
x
0
[a,b] (2.14)
,
.
2.
:
2.2 -
n
khi n (2.13) (2.14)
.
n
.
n
.
n
xa . .
:
2.4 -
(2.13)(2.14)
1) [a,b] y
(2.1)
(2.12):
2)
n
(2.13) (2.14) [a,b]:
3) (x) :
| q <1, a<x<b (2.15)
.
(2.13) (2.14)
x
n
khi n (2.16)
:
(2.12)
= ()
(2.13)
:
- x
n
= () - (x
n-1
) (2.17)
.
Bài gi
21
:
-
(x)
[a,b], (a,b)
(a,b),
= a+ (b-a), 0< <1 sao cho:
F (b) - -a)
Á
(2.17)
:
- x
n
= (c)( - x
n-1
) (2.18)
= a + ( - x
n-1
) (a,b)
(2.15)
| q < 1.
(2.18) cho:
| - x
n
| = | - x
n-1
| q|- x
n-1
|
:
| - x
n
| q | - x
n-1
| (2.19)
.
| - x
n
| q | - x
n-1
|
| - x
n-1
| q | - x
n-2
|
| - x
2
| q| - x
1
|
| - x
1
| q| -x
0
|
| - x
n
| q
n
| - x
0
| (2.20)
0
, q
n
0 khi n do 0 < q < 1,
0
: | - x
n
| 0 khi n
(2.16) 2.4.
3.
3) 2.4 2)
0
:
G| q < 1.
(x) > 0
0
[a,b]
, (x) < 0
0
:
x
0
= a khi a < <
(2.21)
x
0
= b khi
< < b
Bài gi
22
( )
(a).
(2.17)
4.
:
(2.13) (2.14)
n
. Khi sso sai
| x - | (2.20) | - x
0
| < b - a:
| - x
n
| q
n
(b - a) (2.22)
:
:
a) Công thư
́
c đa
́
nh gia
́
sai sô
́
thư
́
nhâ
́
t:
(2.19) ta suy ra:
| - x
n
| q | -x
n-1
| = q {| - x
n
+x
n
- x
n-1
|}
:
| - x
n
| q{| - x
n
| + |x
n
- x
n-1
|}
0 q < 1 nên 1 - q > 0.
(1-q)
:
| - x
n
|
q
q
1
| x
n
- x
n-1
| (2.23)
.
b) Công thư
́
c đa
́
nh gia
́
sai sô
́
thư
́
hai:
,
.
2.5
.
2.5.
F (x) = 0 (2.24)
[c,d]
X
[c,d]
.
:
|
X
- X |
m
XF )(
(2.25)
:
|F'(x)| m > 0, c < x<d (2.26)
:
(X) = 0
:
F (
X
) = F (
X
) - F(X)
Á
(2.18) :
F (
X
) = F' â (
X
- X)
