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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
4TÍCH PHÂN 
38
4. 
38
4. 
38

40
5:  
41
5. 1. 
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5. 
41
5. Runge-Kutta
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
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):
Bng 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

1i
=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 
 t1/2
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)

×