1
2
-2021
( )
: : 75
: ,
-
1. ()
() : x3-x-3=0.
2.(a) ?
(b) 3x² + 6x-45=0
-
3. (a) -
?
() - :
2
2
4. ()
() :
-
5. (a) ?
(b) (X = 0) Y
-1
-2
2
4
-1
-9
11
69
6.(a)
?
() 1/3 :
-
7. ()
3
2
(b) ? ?
:
10
20
30
40
50
60
70
80
32
20
25
35
40
28
38
45
8. (a) ?
() :
1993
1994
1995
1996
1997
1998
7
10
12
14
17
24
4
2
-2021
( )
: : 75
: ,
-
1. ()
() : x3-x-3=0.
:: :
:
()
,
, :
1. : -
, -
: 1/3 0.333333 ...
( ) , 6
, 0.333333
5
2
2. : ,
,
: e^x : e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ...
( , 4 ) , -
, ,
3. : , ,
= | - |
π 3.14159... 3.14 , : |3.14159 -
3.14| ≈ 0.00159
4. : ,
= ( / | |) × %
: π : = (0.00159 / 3.14159) × 100% ≈
0.0506%
5. :
: A = πr^2
π r ,
6. ,
: ,
:
1.
2.
3.
4.
6
2
,
: x^3 - x - 3 = 0
:
1. [a, b] f(a) f(b)
2. c = (a + b) / 2
3. f(c)
4. f(c) = 0 ( 0 )
5. f(c) f(a) , a c
6. f(c) f(b) , b c
7. 2-6
, : x^3 - x - 3 = 0
1: f(x) = x^3 - x - 3 [a, b]
f(a) f(b)
: f(1) = 1^3 - 1 - 3 = -3 () f(2) = 2^3 - 2 - 3 = 3 ()
! [1, 2]
: ( ) 0.0001
1: c = (1 + 2) / 2 = 1.5 f(1.5) = 1.5^3 - 1.5 - 3 = -0.875 ()
f(1.5) f(1) , :
: [1.5, 2]
2: c = (1.5 + 2) / 2 = 1.75 f(1.75) = 1.75^3 - 1.75 - 3 = 0.796875 ()
f(1.75) f(2) , :
: [1.5, 1.75]
3:c = (1.5 + 1.75) / 2 = 1.625 f(1.625) = 1.625^3 - 1.625 - 3 = -0.119141 ()
: : [1.625, 1.75]
,
:
7
2
4:c = 1.6875, f(c) = 0.319824 () : [1.625, 1.6875]
5:c = 1.65625, f(c) = 0.095581 () : [1.625, 1.65625]
6:c = 1.640625, f(c) = -0.013092 () : [1.640625, 1.65625]
7:c = 1.648438, f(c) = 0.041058 () : [1.640625, 1.648438]
0.0001
13 , : ≈ 1.6447265625 f(1.6447265625) ≈ -
0.0000047684
, ,
, x^3 - x - 3 = 0 1.6447
: 1.6447^3 - 1.6447 - 3 ≈ -0.0000047684
, ,
:
1.
2. ( )
3.
, :
1.
2.
3. ,
,
,
:
1. ( ): ≈
1.6447298858494002
8
2
2. : ≈ 1.6447265625
= |1.6447298858494002 - 1.6447265625| ≈ 0.0000033233494002
= (0.0000033233494002 / 1.6447298858494002) × 100% ≈ 0.000202%
, ,
,
,
, ,
-, ,
2.(a) ?
(b) 3x² + 6x-45=0
:.
1. :
,
,
!
:
) :-
b)
,
) :
,
9
2
) :
,
) : ,
2. :
:
-- :
1: (x1, y1) (x2, y2) 2:
3: x1 x2 x-
y-
:
y = y1 + ((x - x1) / (x2 - x1)) * (y2 - y1)
:
• x y
• (x1, y1)
• (x2, y2)
: : (1, 10) (3, 20)
x = 2 y-
: y = 10 + ((2 - 1) / (3 - 1)) * (20 - 10) y = 10 + (1/2) * 10 y = 10 + 5 y = 15
, x = 2 y- 15
: : y ^ 20 | * (3,20) | / 15 | * (2,15) <--
| / 10 | * (1,10) | +---+---+---+---> x 1 2 3
,
:
•
•
10
2
•
:
• ,
• -
• -
:
, ,
:
1. : -
2. :
3. :
4. :
5. :
,
,
.
, :
3x² + 6x - 45 = 0
, , -
x- ,
:
1. (x0 x1)
11
2
2.
3. x-
4. , x0 x1
5.
: 3x² + 6x - 45 = 0
1: x0 = 3 x1 = 4
f(x0) = f(3) = 3(3)² + 6(3) - 45 = 27 + 18 - 45 = 0 f(x1) = f(4) = 3(4)² + 6(4) - 45 = 48 + 24 - 45 =
27
! f(x0) f(x1)
:
x0 = 2: f(2) = 3(2)² + 6(2) - 45 = 12 + 12 - 45 = -21 () x1 = 4: f(4) = 3(4)² + 6(4) - 45 =
48 + 24 - 45 = 27 ()
!
2:
:x2 = x0 - (f(x0) * (x1 - x0)) / (f(x1) - f(x0))
:x2 = 2 - (-21 * (4 - 2)) / (27 - (-21)) = 2 - (-42) / 48 = 2 + 0.875 =
2.875
3:f(x2) f(2.875) = 3(2.875)² + 6(2.875) - 45 ≈ 24.796875 + 17.25 - 45 ≈ -
2.953125 ()
4: f(2.875) , f(2) , x0 2.875
: : [2.875, 4]
5: :
2:x2 = 2.875 - (f(2.875) * (4 - 2.875)) / (f(4) - f(2.875)) ≈ 3.0352
f(3.0352) ≈ 0.3199 ()
: [2.875, 3.0352]
3: x2 ≈ 3.0004
f(3.0004) ≈ 0.0036 ()
: [2.875, 3.0004]
4: x2 ≈ 3.0000
12
2
(3.0000) ≈ 0.0000
! 3x² + 6x - 45 = 0 x ≈ 3.0000
: 3(3)² + 6(3) - 45 = 27 + 18 - 45 = 0
, x = 3
:
,
: ,
:
1.
2.
3. ( )
:
1.
2.
3. -
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,
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13
2
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3. (a) -
?
() - :
: :
,
:
) :
) :
,
) :
2. :
:
) :
: : x + y = 5 2x - y = 1
1: y x : y = 5 - x
: : 2x - (5 - x) = 1
3:x : 2x - 5 + x = 1 3x = 6 x = 2
4:y = 5 - x x = 2 y : y = 5 - 2 = 3
: (x, y) = (2, 3)
14
2
) :
: : x + y = 5 2x - y = 1
1: y : (x + y) + (2x - y) = 5 + 1 3x = 6
: x : x = 2
3:y x = 2 : 2 + y = 5 y = 3
: (x, y) = (2, 3)
) :
,
d) :
) :
,
) - :
3. - :
- , ,
,
- :
1: 2:
3:
-
1:
15
2
:a₁x + b₁y + c₁z = d₁ a₂x + b₂y + c₂z = d₂ a₃x + b₃y + c₃z = d₃
: [a₁ b₁ c₁ | d₁] [a₂ b₂ c₂ | d₂] [a₃ b₃ c₃ | d₃]
2:
:
•
• ()
•
, :
•
• .
•
"" , 1
3: -
, ,
,
, - :
:x + 2y + z = 4 2x + 3y + 2z = 7 3x + 4y + z = 12
1: : [1 2 1 | 4] [2 3 2 | 7] [3 4 1 | 12]
2: :
a) x : R2 = R2 - 2R1 R3 =
R3 - 3R1
[1 2 1 | 4] [0 -1 0 | -1] [0 -2 -2 | 0]
b) y
: R3 = R3 - 2R2
[1 2 1 | 4] [0 -1 0 | -1] [0 0 -2 | 2]
3: x, y z - :
: -2z = 2 z = -1
16
2
: -y + 0(-1) = -1 y = 1
: x + 2(1) + 1(-1) = 4 x + 2 - 1 = 4 x = 3
, : x = 3 y = 1 z = -1
:
1(3) + 2(1) + 1(-1) = 4 ✓ 2(3) + 3(1) + 2(-1) = 7 ✓ 3(3) + 4(1) + 1(-1) ) = 12✓
- :
1. : --
,
2. ,
3.
,
4. :
5.
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4. ()
() :
:; :
:
18
2
1. ,
, 2x + 3y = 7 , x y
2. :
,
: 2x + 3y = 7 4x - y = 1
(x y)
3. :
,
: [2 3] [x] = [7] [4 -1] [y] [1]
, , ,
4. : 1:
( ),
2:
A A^(-1) , A
, ,
, 3:
4:
5. :
: 2x + 3y = 7 4x - y = 1
1: [2 3] [x] = [7] [4 -1] [y] [1]
A = [2 3] [4 -1]
A^(-1) = [-1/10 3/10] [ 4/10 2/10]
3: A^(-1) A^(-1) * A * [x] = A^(-1) * [7] [y] [1]
4: [x] = [-1/10 3/10] * [7] = [1] [y] [ 4/10 2/10] [1] [2]
, x = 1 y = 2.
6. :
•
19
2
•
•
7. :
• (-)
• ,
, -
, ()
: -
-
1. - :-
,
- : 1:
( x1, x2, )
: ,
3: ,
4: 2
5: 4
,
2. - :
•
20
2
•
•
4. :
•
• -
•
, - :
x₁ + x₂ - x₃ = -2 x₁ + 3x₂ + 2x₃ = 11 x₁ + 3x₂ + x₃ = -4
1: ,
x₁ 1 :
x₁ + x₂ - x₃ = -2 ( 1) x₁ + 3x₂ + 2x₃ = 11 ( 2) x₁ + 3x₂ + x₃ = -4 ( 3)
2: 1 : x₁ = -2 - x₂ + x₃ 2 : x₂
= (11 - x₁ - 2x₃) / 3 3 : x₃ = -4 - x₁ - 3x₂
3: x₁ = 0, x₂ = 0, x₃ = 0
4: :
1:x₁ = -2 - 0 + 0 = -2 x₂ = (11 - (-2) - 2(0)) / 3 = 13/3 ≈ 4.33 x₃ = -4 - (-2) - 3(4.33) = -15
2:x₁ = -2 - 4.33 + (-15) = -21.33 x₂ = (11 - (-21.33) - 2(-15)) / 3 = 20.78 x₃ = -4 - (-
21.33) - 3(20.78) = -45.01
3: x₁ = -2 - 20.78 + (-45.01) = -67.79 x₂ = (11 - (-67.79) - 2(-45.01)) / 3 = 56.27 x₃ = -4
- (-67.79) - 3(56.27) = -104.02
, -
, -
5. : - ,
:
) : ω ()
- ()
:
x_new = ωx_calculated + (1-ω)x_old
ω 0 2 ω > 1 - , ω < 1
-
21
2
) -,
) : -
d) -,
ω = 0.5 - :
1: x₁_calc = -2 - 0 + 0 = -2 x₁_new = 0.5(-2) + 0.5(0) = -1
x₂_calc = (11 - (-1) - 2(0)) / 3 = 4 x₂_new = 0.5(4) + 0.5(0) = 2
x₃_calc = -4 - (-1) - 3(2) = -9 x₃_new = 0.5(-9) + 0.5(0) = -4.5
2:x₁_calc = -2 - 2 + (-4.5) = -8.5 x₁_new = 0.5(-8.5) + 0.5(-1) = -4.75
x₂_calc = (11 - (-4.75) - 2(-4.5)) / 3 = 8.25 x₂_new = 0.5(8.25) + 0.5(2) = 5.125
x₃_calc = -4 - (-4.75) - 3(5.125) = -14.625 x₃_new = 0.5(-14.625) + 0.5(-4.5) = -9.5625
- , -
- , ,
6. : -
, ,
:
1: [1 1 -1 | -2] [1 3 2 | 11] [1 3 1 | -4]
2: R2 = R2 - R1
: [1 1 -1 | -2] [0 2 3 | 13] [1 3 1 | -4]
3 = 3 - 1: [1 1 -1 | -2] [0 2 3 | 13] [0 2 2 | -2]
3 = 3 - 2: [1 1 -1 | -2] [0 2 3 | 13] [0 0 -1 | -15]
3: - : x₃ = 15 : 2x₂ + 3(15) =
13 2x₂ = 13 - 45 = -32 x₂ = -16 : x₁ + (-16) - 15 = -2 x₁ = -2 + 16 + 15 = 29
, x₁ = 29, x₂ = -16, x₃ = 15
7. :
:
1: 29 + (-16) - 15 = -2 () 2: 29 + 3(-16) + 2(15) = 11 () 3:
29 + 3(-16) + 15 = -4 ()
22
2
,
8. : : -
,
,
:
• ( 1000 ) ,
• , ( ),
• (
), -
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23
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5. (a) ?
(b) (X = 0) Y
-1
-2
2
4
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-9
11
69
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4. :
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24
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2a.
, :
1: , (
x) x-
: y-
:
• : Δy = y₁ - y₀, y₂ - y₁, y₃ - y₂, ...
• : Δ²y = Δy₁ - Δy₀, Δy₂ - Δy₁, ...
• : Δ³y = Δ²y₁ - Δ²y₀, ...
3: :
y = y₀ + p(Δy₀) +(-1)/2!+(-1)(-2)/3!+ ...
:
• y₀ y-
• p = (x - x₀) / h, x , x₀ x- ,
h x-
• Δy₀, Δ²y₀, Δ³y₀, , 2
4: y
2b.
,
:
25
2
1: (x)
, x-
: :
• : y = yₙ - yₙ₋₁, yₙ₋₁ - yₙ₋₂, ...
• : ²y = yₙ - yₙ₋₁, yₙ₋₁ - yₙ₋₂, ...
• : ³y = ²yₙ - ²yₙ₋₁, ...
3: :
y = yₙ + p(yₙ) +(+1)/2!+(+1)(+2)/3!+ ...
:
• yₙ y-
• p = (x - xₙ) / h, x , xₙ x- ,
h x-
• ²³, , 2
4: y
:
1. :
2. : ,
3.
4. :
:
1. :-
2. : ,
3. : - ,
( )
26
2
4.
,
, ,
-
,
:
1: (xᵢ, yᵢ) ,
Lᵢ(x) :
Lᵢ(x) = ∏(j≠i) (x - xⱼ) / (xᵢ - xⱼ)
x = xᵢ 1 x xⱼ
0
: P(x) y-
:
P(x) = ∑(i=0 n) yᵢ * Lᵢ(x)
3: x-
, :
: X = 0 , Y , :
| -1 | -2 | 2 | 4 | -1 | -9 | 11 | 69
1:
L₁(x) = [(x+2)(x-2)(x-4)] / [(-1+2)(-1-2)(-1-4)] = (x+2)(x-2)(x-4) / 15
L₂(x) = [(x+1)(x-2)(x-4)] / [(-2+1)(-2-2)(-2-4)] = (x+1)(x-2)(x-4) / -24
L₃(x) = [(x+1)(x+2)(x-4)] / [(2+1)(2+2)(2-4)] = (x+1)(x+2)(x-4) / 24
L₄(x) = [(x+1)(x+2)(x-2)] / [(4+1)(4+2)(4-2)] = (x+1)(x+2)(x-2) / 90
2:
(x) = -1 * L₁(x) + (-9) * L₂(x) + 11 * L₃(x) + 69 * L₄(x)
3: x = 0
(0) = -1 * [2(-2)(-4) / 15] + (-9) * [1(-2)(-4) / -24] + 11 * [1(2)(-4) / 24] + 69 * [1(2)(-2) / 90] =
-1 * (16/15) + (-9) * (8/24) + 11 * (-8/24) + 69 * (-4/90) = -16/15 - 3 - 11/3 - 46/15 = -16/15 -
45/15 - 55/15 - 46/15 = -162/15 = -10.8
27
2
, X = 0, Y ≈ -10.8.
, , X 0 ,
Y- -10.8
4.
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:
1. :
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31
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6.(a)
?
() 1/3 :
: :
()
""
:
1. : , ,
a b f(x)
, n
2. :
f(x0), f(x1), f(x2), ..., f(xn) , x0 = a xn = b.
3. : ,
,
4. :
:
∫_a^bf(x) dx ≈ (ba)/2n * [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)]
:
• a b
• n
• x0, x1, ..., xn
:
• (ba)/n
• 1/2
32
2
• , (f(x1) f(xn-1))
-
,
: ,
( ),
4
, 1/3 :
() 1/3
1/3 ,
,
1/3 :
1. : , [a,b]
, 1/3 ,
2n
2. : : f(x0), f(x1),
f(x2), ..., f(x2n)
3. 1/3 :
∫_a^bf(x) dx ≈ (ba)/6n * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + 4f(x2n-1) + f(x2n)]
:
• a b
• n ( 2n )
• x0, x1, ..., x2n
:
• (ba)/2n
• 1/3
33
2
• : 1 ,
4 , ( )
2
4:2:4
1/3 ,
, -
, 1/3 :
∫_0^(π/2) √( x) dx
1: 6 (n = 3)
: h = (π/2 - 0) / 6 = π/12
3: x x0 = 0 x1 = π/12 x2 = π/6 x3 = π/4 x4 = π/3 x5 = 5π/12 x6 = π/2
4: f(x) = √(sin x)
f(x0) = √(sin 0) = 0 f(x1) = √(sin(π/12)) ≈ 0.4555 f(x2) = √(sin(π/6)) ≈ 0.6366 f(x3) = √ (sin(π/4))
≈ 0.7654 f(x4) = √(sin(π/3)) ≈ 0.8660 f(x5) = √(sin(5π/12)) ≈ 0.9415 f(x6) = √(sin (π/2)) = 1
5: 1/3
∫_0^(π/2) √(sin x) dx ≈ (π/2 - 0)/(6*3) * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + 4f(x5) +
f(x6)]
≈ (π/18) * [0 + 4(0.4555) + 2(0.6366) + 4(0.7654) + 2(0.8660) + 4(0.9415) + 1]
≈ (π/18) * [0 + 1.8220 + 1.2732 + 3.0616 + 1.7320 + 3.7660 + 1]
≈ (π/18) * 12.6548
≈ 1.1781
, 1/3 6 ∫_0^(π/2)
√(sin x) dx 1.1781
, 1.1803
,
:
1. x = 0 √(sin x) 0 , sin(0) = 0.
34
2
2. - x 0 π/2 , sin(x) 0 1 , √(sin x) ,
- ( )
3. 0 π/2
4. 0 1 , π/2 ( 1.57) ,
π/2 ,
5. 1 , ,
1
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, 6 100 ,
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35
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:
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7. ()
(b) ? ?
:
10
20
30
40
50
60
70
80
32
20
25
35
40
28
38
45
: :
:
1. )
2.
36
2
3.
:
1. ): ""
,
,
, : 80, 85, 90, 95, 100
: a) : 80 + 85 + 90 + 95 + 100 = 450 b)
: 5 c) : 450 ÷ 5 = 90
90
,
()
2. : ,
,
,
: 80, 85, 90, 95, 100
, 5 () , : 90.
, 98, : 80,
85, 90, 95, 98, 100
: 90 95. ,
: (90 + 95) ÷ 2 = 92.5
,
3. :
,
,
: 2, 3, 3, 4, 4, 4, 5, 5
, 4 , 4
( )
37
2
" " :
• ,
• ,
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:
,
-1 +1 :
• +1 -
,
• -1 - ,
• 0
+1 -1 ,
:
• +0.8 ,
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• +0.7 ,
• -0.6 , -
,
-
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,
38
2
:
,
-- :
1. X Y -
2. (X,Y) : a) (X - X ) (Y - Y ) b)
c)
3. 2b 2c
4. : r = Σ((X - Xmean)(Y - Ymean)) / sqrt(Σ(X - Xmean)² * Σ(Y -
Ymean)²)
: r = X = X Y = Y Xmean = X Ymean = Y
Σ = sqrt =
, :
(X): 10, 20, 30, 40, 50, 60, 70, 80 (Y): 32, 20, 25, 35, 40, 28, 38, 45
1: X ( ) = (10 + 20 + 30 + 40 + 50 + 60 + 70 + 80) / 8 = 45
Y () = (32 + 20 + 25 + 35 + 40 + 28 + 38 + 45) / 8 = 32.875
x-x
y-y
x-x
y-y
x-x
x-
)2
y-
y)2
10
32
-35
-0.875
30.625
1225
0.765625
20
20
-25
-
12.875
321.875
625
165.765625
30
25
-15
-7.875
118.125
225
62.015625
40
35
-5
2.125
-10.625
25
4.515625
50
40
5
7.125
35.625
25
50.765625
60
28
15
-4.875
-73.125
225
23.765625
70
38
25
5.125
128.125
625
26.265625
80
35
35
12.125
424.375
1225
147.015625
2:
3:
39
2
Σ((X - Xmean)(Y - Ymean)) = 975 Σ(X - Xmean)² = 4200 Σ(Y - Ymean)² = 480.875
4:
= 975 / (4200 * 480.875) = 975 / (2019675) = 975 / 1421.15 = 0.686
, 0.686
:
0.686
, , - ,
, ( 1 )
,
, -
- , ( , , , )
, :
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2.
3. : ,
,
4. : ,
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40
2
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,
41
2
8. (a) ?
() :
1993
1994
1995
1996
1997
1998
7
10
12
14
17
24
: :
,
:
1. : :
( X ) ( Y
)
:
(X) (Y)
X ( )
Y ( )
2
65
3
70
4
80
5
85
6
90
,
:
42
2
= +
: Y X a Y- (
) b (
)
2. :
: ,
(X1) (X2)
X1 ( )
X2 ( )
Y ( )
2
10
65
3
15
70
4
20
80
5
25
85
6
30
90
, :
= + 1X1 + 2X2
: Y X1 X2
a Y- b1 b2
:
1. :
o
43
2
o
2. :
o , -
o - , -
3. :
o
o - ,
4. :
o -
o ,
5. :
o
o -
6. :
o : Y = a + bX
o : Y = a + b1X1 + b2X2 + ... + bnXn
7. :
o --
o -
, ,
, -
,
44
2
:
,
:
1993
7
1994
10
1995
12
1996
14
1997
17
1998
24
1: 1993, 1994, 1995, 1996, 1997, 1998 1, 2, 3, 4, 5, 6
()
()
1
7
2
10
3
12
4
14
5
17
6
24
2: :
• ΣX (X )
• ΣY (Y )
45
2
• ΣXY (X Y )
• ΣX² ( X )
• n ( )
ΣX = 1 + 2 + 3 + 4 + 5 + 6 = 21
ΣY = 7 + 10 + 12 + 14 + 17 + 24 = 84
ΣXY = (1×7) + (2×10) + (3×12) + (4×14) + (5×17) + (6×24) = 371 ΣX² = 1² + 2² + 3² + 4² + 5² + 6²
= 91
= 6
3: (b) y- (a) :
= ( × ΣXY - ΣX × ΣY) / ( × ΣX² - (ΣX)²)
= (ΣY - × ΣX) /
:
= (6 × 371 - 21 × 84) / (6 × 91 - 21²) = (2226 - 1764) / (546 - 441) = 462 / 105 = 4.4
= (84 - 4.4 × 21) / 6 = (84 - 92.4) / 6 = -8.4 / 6 = -1.4
4: :
= + = -1.4 + 4.4X
X (1993 1, 1994 2, ) Y
5: :
Y = -1.4 + 4.4X :
1. (b = 4.4) 1993 1998 4.4
2. y- (a = -1.4) , " 0"
( )
3.
:
o 1993 (X = 1): Y = -1.4 + 4.4(1) = 3
o 1998 (X = 6): Y = -1.4 + 4.4(6) = 25
4. ,
:
o 1999 (X = 7): Y = -1.4 + 4.4(7) = 29.4
46
2
5.
,
,
:
1993
7
3.0
1994
10
7.4
1995
12
11.8
1996
14
16.2
1997
17
20.6
1998
24
25.0
, ,
:
1.
2.
3. :
,
:
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2.
,
47
2
3. :
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, :
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2. ,
3. ( )
4.
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