1
2
-2022
( )
: : 50
: ,
-
1. - ?
2. - - x3 - x - 1
-
3. () ? - ?
(b) - :
+ 2y + z = 8
2x + 3y + 4z = 20
4x + 3y + 2z = 16
2
2
4. ? - ?
:
1 2 3
3 -2 1
4 1 1
-
5. () ,
() , f(7.5)
1
2
3
4
5
6
7
8
()
1
8
27
64
125
216
343
512
6. (a) n = 4
(b) n = 10 1/3
-
7. () ? - ?
() :
25
27
35
36
3
2
3
2
4
1
2
8. :
()
()
() .
4
2
-2022
( )
: : 50
: ,
-
1. - ?
: :
,
:
1. -
2.
3.
:
1. :
-
,
, -
: 1/3
0.333333... ( ) , 0.33333333 (8 )
5
2
,
: 0.1 + 0.2
0.3 , -
, 0.30000000000000004
-
,
2. :
,
: :
(x) = x - x^3/3! + x^5/5! - x^7/7! + ...
sin(x) (x - x^3/3!) ,
,
, (
)
3. :
, ,
,
:
, 5.23 ,
5.22
0.1°C ,
6
2
:
, ,
-
:
• ,
:
• , A B, ea eb
C = A * B
• , ec , :
• ≈ +
• ,
: 10 , 1% (ea
= 0.01) , 5 , 2% (eb = 0.02)
• : = * = 10 * 5 = 50 ²
• : ec ≈ ea + eb = 0.01 + 0.02 = 0.03 3%
• 50 , 3%
:
• , C = A / B , C
:
• ≈ +
• ,
0.5% (ea = 0.005) 100
1% (eb = 0.01) 50 cm³
= / = 100 / 50 ³ = 2 /³
: ec ≈ ea + eb = 0.005 + 0.01 = 0.015 1.5%
, 2 /³ , 1.5%
:
,
- ,
7
2
:
1. , -
,
2. ,
,
3. ,
4. ,
:
, :
1.
2.
, ,
3. , -
4. :
- ,
5. :3D , -
:
,
:
1. : ,
-
8
2
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4. : -
,
5. ,
(
) -
6. : ,
, -
:
- -,
- :
,
, , -
, ,
,
, ,
, ,
,
, ,
,
9
2
2. - - x3 - x - 1
: - :
- , , -
( ) , x
:
1.
2.
3. x-
4.
5. 2-4
, :
1. :
, ,
x₀
2. : ,
3. - : x-
4. : x- ,
x₁
5. : - ,
,
:
,
:
10
2
x_{n+1} = x_n - f(x_n) / f'(x_n)
:
• x_n
• x_{n+1}
• f(x_n)
• f'(x_n)
- !
:
-
" " ,
(
)
:
1. , - ,
2. :
3. ,
:
1.
,
2. :
3. ,
4. ,
, x^3 - x - 1 = 0
1:
f(x) = x^3 - x - 1 f'(x) = 3x^2 - 1
11
2
2:
, f(1) = 1 - 1 - 1 = -1 f(2) = 8 - 2 - 1 = 5. 1 2
x₀ = 1.5
3: -
x_{n+1} = x_n - f(x_n) / f'(x_n)
1: x₁= 1.5 - (1.5^3 - 1.5 - 1) / (3(1.5)^2 - 1) = 1.5 - (0.875) / (5.75) = 1.5 -
0.152173913 = 1.347826087
2: x₂= 1.347826087 - (1.347826087^3 - 1.347826087 - 1) / (3(1.347826087)^2 - 1)
= 1.347826087 - 0.022411859 / 4.450075681 = 1.347826087 - 0. 005036087 = 1.342790000
3: x₃= 1.342790000 - (1.342790000^3 - 1.342790000 - 1) / (3(1.342790000)^2 - 1)
= 1.342790000 - 0.000069624 / 4.417480725 = 1.34279000 - 0. 000015760 = 1.324774240
4: x₄= 1.324774240 - (1.324774240^3 - 1.324774240 - 1) / (3(1.324774240)^2 - 1)
= 1.324774240 - 0.0000007 / 4.361543042 = 1.324774240 - 0 0000000002 = 1.324774238
1.325
: :
f(1.325) = 1.325^3 - 1.325 - 1 = 2.325390625 - 1.325 - 1 = 0.000390625
,
:
• x ≈ 1.325 x^3 - x - 1 = 0 x
1.325 ,
• , y = x^3 - x - 1 ,
x- x- x-
•
(1 + √5) / 2 , ( 1.618034)
!
:
- , - :
1. : ,
12
2
2. :
3. : 3D
4. :
5. :
6. :
,
,- -
, :
,
x^3 - x - 1 = 0
, , -
,
, ,
,
,
13
2
-
3. () ? - ?
(b) - :
+ 2y + z = 8
2x + 3y + 4z = 20
4x + 3y + 2z = 16
: ) :
,
, :
:
1. 2 3 7
2. 3 2 8
:
x y
1: 2x + 3y = 7 2: 3x + 2y = 8
:
1. :
,
2. - : , -
1 ,
3. :
: 2x + 3y = 0 4x - y = 0
14
2
4. : ,
( )
5. :
6. :
7. ,
8. :
,
) - :
: x + 2y + z = 8 2x + 3y + 4z = 20 4x + 3y + 2z = 16
-
-- :
1:
(x, y, z) ,
:
| 1 2 1 | 8 | | 2 3 4 | 20 | | 4 3 2 | 16 |
2:
1 0
1 ,
2 1 2 : | 1 2 1 | 8 | | 0 -1 2 | 4 | | 4 3 2 | 16 |
3 1 4 : | 1 2 1 | 8 | | 0 -1 2 | 4 | | 0 -5 -2 | -16 |
3:
1 0
2 -1 : | 1 2 1 | 8 | | 0 1 -2 | -4 | | 0 -5 -2 | -16 |
1 2 2 : | 1 0 5 | 16 | | 0 1 -2 | -4 | | 0 -5 -2 | -16 |
2 3 5 : | 1 0 5 | 16 | | 0 1 -2 | -4 | | 0 0 -12 | -36 |
4:
1 0
15
2
3 -12 : | 1 0 5 | 16 | | 0 1 -2 | -4 | | 0 0 1 | 3 |
1 3 5 : | 1 0 0 | 1 | | 0 1 -2 | -4 | | 0 0 1 | 3 |
3 2 2 : | 1 0 0 | 1 | | 0 1 0 | 2 | | 0 0 1 | 3 |
5:
:
= 1 = 2 = 3
:
1: x + 2y + z = 1 + 2(2) + 3 = 1 + 4 + 3 = 8 ✓ 2: 2x + 3y + 4z = 2(1) + 3(2) +
4(3) = 2 + 6 + 12 = 20 ✓ 3: 4x + 3y + 2z = 4(1) + 3(2) + 2(3) = 4 + 6 + 6 = 16 ✓
,
, -
:
-
:
1.
2.
3.
,
( )
:
1. :
,
2. : , -
3. :
,
16
2
4. :-
,
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2. :
, ,
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3. : , (
) -
4.
:
5. - ,
,
6. :
, -
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-
17
2
4. ? - ?
:
1 2 3
3 -2 1
4 1 1
: ?
,
- 2 * 1/2 = 1,
A , A^(-1) A , :
* ^(-1) = ^(-1) * =
I ( 1 0 )
(
) , , "-" ""
?
:
1. Ax = b ,
A , x , b , A^(-1)
x : x = A^(-1)b.
2. :
3. :- ,
4. : ,
5.
18
2
:
1. :
2. ( ):
3. - :
4. :
5. , -
,
:
1.
2.
3.
:
= [1 2 3] [3 -2 1] [4 1 1]
1:
3x3 :
det(A) = a(ei - fh) - b(di - fg) + c(dh - )
: a = A[0][0], b = A[0][1], c = A[0][2] d = A[1][0], e = A[1][1], f = A[1][2] g = A[2][0], h =
A[2][1], i = A[2][2]
:
det() = 1[(-21) - (11)] - 2[(31) - (14)] + 3[(31) - (4-2)] = 1(-3) - 2(- 1) + 3(11) = -3 + 2 + 33 = 32
32
2:
,
19
2
3x3 , 2x2
, (-1)^(i+j) , i j
:
11 = (-1)^(1+1) * [(-21) - (11)] = 1 * (-3) = -3
C12 = (-1)^(1+2) * [(31) - (14)] = -1 * (-1) = 1
13 = (-1)^(1+3) * [(31) - (4-2)] = 1 * 11 = 11
21 = (-1)^(2+1) * [(21) - (31)] = -1 * (-1) = 1
22 = (-1)^(2+2) * [(11) - (33)] = 1 * (-8) = -8
23 = (-1)^(2+3) * [(14) - (31)] = -1 * 1 = -1
31 = (-1)^(3+1) * [(21) - (3-2)] = 1 * 8 = 8
32 = (-1)^(3+2) * [(11) - (32)] = -1 * (-5) = 5
33 = (-1)^(3+3) * [(1*-2) - (2*3)] = 1 * (-8) = -8
, :
= [-3 1 11] [ 1 -8 -1] [ 8 5 -8]
= [-3 1 8] [1 -8 5] [11 -1 -8]
3:
,
:
A^(-1) = (1/32) * [-3 1 8] [ 1 -8 5] [11 -1 -8]
:
^(-1) = [-3/32 1/32 1/4 ] [ 1/32 -1/4 5/32] [11/32 -1/32 -1/4 ]
:
,
, :
[1 2 3] [-3/32 1/32 1/4 ] [3 -2 1] * [1/32 -1/4 5/32] [4 1 1] [11/32 -1/32 - 1/4 ]
, :
[1 0 0] [0 1 0] [0 0 1]
20
2
,
:
1. ,
, - ( )
2. ,
3. : (AB)^(-1) = B^(-
1)A^(-1)
4. : (A^(-1))^(-1) = A
5. : (A^T)^(-1) = (A^(-1))^T
6. 2x2 , : A = [ab] [cd] A^(-1) = (1 /
(ad-bc)) * [ d -b] [-ca] ,
7. , ,
:
1. :3D ,
, 3D " " ,
,
2. : , -
: θ = (X^TX)^(-1) X^T y
3. - ,
4. : ,
,
21
2
5. :
,
6. ,
7.
( )
,
, LU
, ,
, x Ax = b ,
A^(-1) b
,
- ,
- , ,
- 3x3
:
[1 2 3] [3 -2 1] [4 1 1]
:
[-3/32 1/32 1/4 ] [1/32 -1/4 5/32] [11/32 -1/32 -1/4 ]
, ,
,
22
2
-
5. () ,
() , f(7.5)
1
2
3
4
5
6
7
8
()
1
8
27
64
125
216
343
512
:() :
,
:
,
: 9:00
20°C 11:00 24°C 10:00
, 22°C -
:
1. :
2. :
3. :
4. :
:
: 3 5
, 5 ,
:
1.
23
2
2.
3. :
4. :
:
1. :
o :
o :
2. :
o : ,
o : , ,
3. :
o :
o : ,
4. :
o :
o :
5. :
o :
o :
,
() f(7.5) :
:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 () | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512
24
2
f(7.5) 7.5 : 7 8
-- :
1: - x = 7 x = 8
, 7.5
x = 7 , f(x) = 343 x = 8 , f(x) = 512
2: ,
,
3: :
(x) = (x1) + (x - x1) * (f(x2) - f(x1)) / (x2 - x1)
: x ( 7.5) x1 x- (7) x2
x- (8) f(x1) x1 (343) f(x2) x2 (512)
:
f(7.5) = 343 + (7.5 - 7) * (512 - 343) / (8 - 7) = 343 + 0.5 * (169) / 1 = 343 + 0.5 * 169 = 343 +
84.5 = 427.5
, f(7.5) 427.5
:
, x = 7.5 y- ( f(x) )
427.5
:
: ,
f(x) :
1 8: 7 8 27: 19 27 64: 37 64 125: 61 125 216: 91 216 343: 127 343 512: 169
, :
7, 19, 37, 61, 91, 127, 169
: 12, 18, 24, 30, 36, 42
: 6
f(x) = x^3 (x ) :
1^3 = 1 2^3 = 8 3^3 = 27 4^3 = 64 5^3 = 125 6^3 = 216 7^3 = 343 8^3 = 512
25
2
, f(x) = x^3 !
, f(7.5) :
(7.5) = 7.5^3 = 421.875
427.5 ,
?
,
( , ) ,
,
:
x^3 , (7, 343) (8, 512)
x = 7.5
,
,
(427.5) (421.875)
:
:
1. ( )
2. ( )
3. (, , , )
,
,
:
1.
, ,
2. : ,
,
3. :
, ,
26
2
,
4. : ,
5. ,
:
• ,
• ,
, ,
, f(9) ,
, f(9) 9^3 = 729
, ,
:
1. :
2. : -
3. :
4. ,
,
27
2
,
,
,
, ,
, f(7.5) 427.5
f(x) = x^3 , f(7.5)
421.875 -
,
6. (a) n = 4
: , :
∫_0^5▒x/(1+x^4 )
1/(1+x^4) 0 5
:
1. n
2.
3.
4.
28
2
:
∫_a^bf(x)dx ≈ (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]
:
• a
• b
• n
• h = (ba)/n
• x₀, x₁, ..., xₙ [a,b] n
, :
0 5 , a = 0 b = 5. n = 4
1:hh = (ba)/n = (5-0)/4 = 5/4 = 1.25
:x [0, 5] 4 : x₀ = 0 x₁
= 1.25 x₂ = 2.5 x₃ = 3.75 x₄ = 5
3: x f(x) f(x) = 1/(1+x^4)
f(x₀) = f(0) = 1/(1+0^4) = 1 f(x₁) = f(1.25) = 1/(1+1.25^4) ≈ 0.6097561 f(x₂) = f(2.5) =
1/(1+2.5^4) ≈ 0.0609756 f(x₃) = f(3.75) = 1/(1+3.75^4) ≈ 0.0125313 f(x₄) = f(5) = 1/(1+5^4) ≈
0.0015625
4: ∫_0^5▒x/(1+x^4 ) ≈ (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) +
2f(x₃) + f(x₄)]
:
≈ (1.25/2)[1 + 2(0.6097561) + 2(0.0609756) + 2(0.0125313) + 0.0015625]
≈ 0.625[1 + 1.2195122 + 0.1219512 + 0.0250626 + 0.0015625]
≈ 0.625[2.3680885]
≈ 1.4800553125
, 4
1.4801
29
2
, :
1. ? - ,
2. ? ,
,
3. ?
,
-
4. 2 ?? x ( )
-
2
5. ?
: a) (n): , n b)
: ,
,
, n = 4 , ,
, n 8, 16
6. ? : a)
b)
c)
7. ?
: a) ( ):
b) :
c)
:
8. ? , :
) )
,
)
30
2
9. ?
, 1/(1+x^4) x
x = 0 , x
x = 0
, x
10. ? ( 1.4801) x = 0 x = 5
1/(1+x^4) ,
,
11. ? ,
: a) ( )
b) n c)
12. ??
,
13. ?
: ) :
, ) :
) : - n
,
14. ,
: a)
b)
c)
15.
?
: )
: ) :
) :
) :
,
,
31
2
,
(b) n = 10 1/3
() , 1/3
:
1/3
""
,
1/3
, :
∫_0^5▒x/(4x+5)
x = 0 x = 5 y = 1/(4x+5)
n = 10 1/3 , n
1/3 , n
:
1:
(0 5 ) 10 ,
:
h = (b - a) / n b (5) , a (0) , n (10)
= (5 - 0) / 10 = 0.5
0.5
32
2
: x-
11 ( 10 )
:
x0 = 0 x1 = 0.5 x2 = 1.0 x3 = 1.5 x4 = 2.0 x5 = 2.5 x6 = 3.0 x7 = 3.5 x8 = 4.0 x9 = 4.5 x10 = 5.0
3: y-
x- y = 1/(4x+5) :
y0 = 1/(4(0)+5) = 1/5 = 0.2000000
y1 = 1/(4(0.5)+5) = 1/7 ≈ 0.1428571
y2 = 1/(4(1.0)+5) = 1/9 ≈ 0.1111111
y3 = 1/(4(1.5)+5) = 1/11 ≈ 0.0909091
y4 = 1/(4(2.0)+5) = 1/13 ≈ 0.0769231
y5 = 1/(4(2.5)+5) = 1/15 ≈ 0.0666667
y6 = 1/(4(3.0)+5) = 1/17 ≈ 0.0588235
y7 = 1/(4(3.5)+5) = 1/19 ≈ 0.0526316
y8 = 1/(4(4.0)+5) = 1/21 ≈ 0.0476190
y9 = 1/(4(4.5)+5) = 1/23 ≈ 0.0434783
y10 = 1/(4(5.0)+5) = 1/25 = 0.0400000
4: 1/3
1/3 :
∫_a^bf(x)dx ≈ (h/3) * [y0 + 4(y1 + y3 + y5 + y7 + y9) + 2(y2 + y4 + y6 + y8) + y10]
:
(0.5/3) * [0.200000 + 4(0.1428571 + 0.0909091 + 0.0666667 + 0.0526316 + 0.0434783) +
2(0.111111 + 0.0769231 + 0.0588235 + 0.0476190) + 0.04000000]
5:
:
0.5/3 = 0.1666667
y1, y3, y5, y7, y9 : 0.1428571 + 0.0909091 + 0.0666667 + 0.0526316 + 0.0434783 =
0.3965428 4 : 1.5861712
33
2
y2, y4, y6, y8 : 0.1111111 + 0.0769231 + 0.0588235 + 0.0476190 = 0.2944767 2
: 0.5889534
: 0.2000000 + 1.5861712 + 0.5889534 + 0.0400000 = 2.4151246
, 0.1666667 : 2.4151246 * 0.1666667 ≈ 0.4025208
, 0.4025208
, :
∫_0^5▒x/(4x+5) y = 1/(4x+5) x = 0 x = 5
0.4025208
, y = 1/(4x+5) ,
x- x = 0 x = 5 ,
0.4025208
, , 1/3
, - (
, 10)
,
:
(1/4) * ln((45 + 5)/(40 + 5)) ≈ 0.4054651
(0.4025208) , 0.7%
1/3 - 10 - ,
,
, n = 10 n = 100 ,
1/3
,
, (
) ,
1/3 :
1. ,
2. : ,
34
2
3. :
, :
1. ,
-
2. : ,
,
3. , ,
,
:
1.
,
2. : ,
,
3. :
,
4. : , ,
5. : ,
,
, :
1. ,
2.
3.
4. ,
, , -
,
35
2
, ,
,
∫_0^5▒x/(4x+5) 1/3
, 0.4025208
,
x = 0 x = 5 y = 1/(4x+5)
,
-
7. () ? - ?
() :
25
27
35
36
3
2
4
1
2
: a) ? - ?
,
( )
,
:
: 85, 87, 90, 88, 85 : 60, 75, 90, 100, 85
() 87 ,
:
1. :
2. (IQR): 50% 75
(Q3) 25 (Q1)
36
2
3. :
4. :
5. ():
6. : ,
-
,
() :
:
() | () 25 | 3 27 | 2 31 | 4 35 | 1 36 | 2
(N) = 3 + 2 + 4 + 1 + 2 = 12
1: ()
, , ,
(25 × 3) + (27 × 2) + (31 × 4) + (35 × 1) + (36 × 2) = 75 + 54 + 124 + 35 + 72 = 360
= 360 ÷ 12 = 30
2:
:
1. (30)
2.
3.
4. .
5. .
6.
37
2
-- :
X | F | |X - | | F × |X - | 25 | 3 | |25 - 30| = 5 | 3 × 5 = 15 27 | 2 | |27 - 30| = 3 | 2
× 3 = 6 31 | 4 | |31 - 30| = 1 | 4 × 1 = 4 35 | 1 | |35 - 30| = 5 | 1 × 5 = 5 36 | 2 | |36 - 30| = 6
| 2 × 6 = 12
F × |X - | = 15 + 6 + 4 + 5 + 12 = 42
= 42 ÷ 12 = 3.5
3.5
3:
:
1.
2.
3.
4. .
5. .
6.
-- :
X | F | (X - )² | F × (X - )² 25 | 3 | (25 - 30)² = 25 | 3 × 25 = 75 27 | 2 | (27 - 30)² = 9 |
2 × 9 = 18 31 | 4 | (31 - 30)² = 1 | 4 × 1 = 4 35 | 1 | (35 - 30)² = 25 | 1 × 25 = 25 36 | 2 | (36 -
30)² = 36 | 2 × 36 = 72
F × (X - )² = 75 + 18 + 4 + 25 + 72 = 194
= 194 ÷ 12 = 16.1667
= √16.1667 ≈ 4.0207
4.02 ( )
, :
1. (3.5): , 3.5
38
2
2. (4.02): , , 4.02
( )
:
1.
- ,
2. ,
,
3.
,
4. , ,
, 68% , 95%
99.7%
5. , (
), ( ),
( )
, ( )
, 30
4.02 ,
:
• 30
• 3.5 , 30 3.5
• 4.02 ,
,
,
,
,
,
39
2
,
8. :
()
()
() .
: ()
,
: ,
:
1. : ,
, (
) ( )
2. :
3.
, ,
4. ,
:
,
,
: 10 -
:
40
2
2, 1, 3, 2, 0, 2, 1, 4, 2, 3
, : 0 1 1 2
2 4 3 2 4 1
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