Real-number

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((((((((RRRRRRRReeeeeeeeaaaaaaaallllllll NNNNNNNNuuuuuuuummmmmmmmbbbbbbbbeeeeeeeerrrrrrrr SSSSSSSSyyyyyyyysssssssstttttttteeeeeeeemmmmmmmm))))))))
F
ก
““““ F F”””” F 3
F
F F F F . . 2537
www.thai-mathpaper.net

ก 2
ก F F ˈ 2 F F 1 F
F F F F ก ʿก F F F F ʿก ก F F F ก ก
F กF F ˆ 3 F 3.3 F 3.4 F F ก F
ก ก
ʿก F F ก F ก F F ก
F F ก F F F F F ˈ F F ก
F ˈ Fก F F F
F
17 . . 2552

F F ˈ F 3 15 F F
F ก F F F F ก F F ก ก
F F ก F ก ก F F F
ˈ ก F ก F ก F F ก ก
ก F F F F
F F F F F
ก ก F F F Fก F Fก F
F F ˈ F 1
F ก 2 ก ก ก กก F
ˈ ก ก ก ก ˆ กF ก F ก F
3 ก ก ก ก กF ก ก F F F F 4
ˈ F F ก ก ก กF ก F
ก ก F F F กF
ก F ก ก ˈ F
F
13 ก F . . 2549
ก 1
ก F F ˈ ก F F กF 3
ก F กF ก กF 3.4 F กF ก F 3.1
3.2 F กF ก 3.1 ˈ 3.2 F F กF ก กF
F 4.3 ก F F F ก ก
F ก กF ˈ Fก F F F
F
15 . . 2549

1 F 1 13
1.1 F 1
1.2 3
1.3 F 10
1.4 F F 12
2 กก 15 18
2.1 กก 15
2.2 ก k 16
3 ก ก 19 33
3.1 ก ก F 19
3.2 F 21
3.3 ก ก ก n 23
3.4 ก ก F F 29
4 ก F 35 41
4.1 ก ก ˈ ก ก F 35
4.2 ก F F 37
4.3 ก F F 38
4.4 ก F F ก ก 40
ก 43 47
1. ก F 43
2. ก ก (Power Series) 45
3. F ˆ กF 47
ก 49

1
F
1.1 F
(Real Number System) ˈ F F F ก F
F ʽ (Peano s Postulates) ก ก F ก
F F F
1.1
ก 1.1 F 2 F กF ก ก
ก ก 2 F กF ก
ก F F ก ก F 3 F กF
F ก
1.1.1 ก ก (Rational and irrational numbers)
ก F F F F
F ก ก F ก ก ก
F F F F F
(R)
ก (Q) ก (Q′)
(Z) ก F F (Z′)
(Z )
F
ก (Z+
N)

2
F ก F 2
3 , 0.3ɺ, 5, 1 ˈ F F ก F π, e
6 , 3 36 ˈ F
F ก ก ก F F ก F
1.1.2 ก F F
(integers) ˈ ก ก F 3 F กF
ก ( F F F )
F F ˈ F 6, 3, 0, 2, 4 ˈ F
ก F F F กF 2
3 , 1
2 , 0.67ɺ ˈ F
ก ก F ก F F F
ก F F F 4 F
F F ก กก F กก
ก ( ) F F ʽ F F ˈ F
ก F ˈ F ก
F (cuts) ˈ ก F ก
F F F F F F Fก F
ก F F ก F
ʿก 1.1
1. ก F ก F ก 3
2. F F F ˈ πe
, eπ
, 2e
F F
F F
3. F F F ˈ 2
π , e
5
π
F F F

F F 3
1.2
1.2.1
F
1.1 ก F a, b, c ˈ
1) ∀a, b ∈ R, a + b ∈ R ก F ʽ ก ก
2) ∀a, b ∈ R, a ⋅ b ∈ R ก F ʽ ก
3) ∀a, b, c ∈ R, (a + b) + c = a + (b + c) ก F ก ก F ก ก
4) ∀a, b, c ∈ R, (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) ก F ก ก F ก
5) ∀a, b ∈ R, a ⋅ b = b ⋅ a ก F ก ก
6) ∀a, b ∈ R, a + b = b + a ก F ก ก ก
7) ∀a ∈ R, a ⋅ 1 = 1 ⋅ a = a ก F ก ก F ก
8) ∀a ∈ R, a + 0 = 0 + a = a ก F ก ก F ก ก
9) ∀a ∈ R, ∃x ∈ R, a ⋅ x = x ⋅ a = 1 ก F ก ก ก x F ก
ก
10) ∀a ∈ R, ∃y ∈ R, a + y = y + a = 0 ก F ก ก ก ก y F ก
ก ก

4
10 F 1.1 ˈ F F F F
1.2.2
F 1.1 ก F (Trichotomy property of real numbers)
F 1.1 F F ก F x, y ˈ F F F ˈ F F
F
1) x < y
2) x = y
3) y < x
1.2.3 F F
F 1.1
ก F x ˈ F F F F F ˈ F F F
1) x ∈ R+
2) x = 0
3) x ∈ R
1.2
ก F S ⊆ R a, b, a′, b′ ∈ R ก F F
1) a ˈ (upper bound) S ก F x ≤ a ก x ∈ S
2) b ˈ F (lower bound) S ก F b ≤ x ก x ∈ S
a′, b′ ∈ S ก F F
1) a′ ˈ F (least upper bound) S ( F l.u.b S sup S)
ก F a′ ≤ a a ˈ S
2) b′ ˈ F ก (greatest lower bound) S ( F g.l.b S inf S)
ก F b ≤ b′ b ˈ F S

F F 5
F 1.1 ก F A = {x | |x2
1| ≤ 8} F F F A ( F )
ก |x2
1| ≤ 8
F F 8 ≤ x2
1 ≤ 8
7 ≤ x2
≤ 9
F F 2 ก F
1) 7 ≤ x2
ก (x ∈ R)
2) x2
≤ 9 ก 3 ≤ x ≤ 3
ก 3 ≤ x ≤ 3
A = {x | 3 ≤ x ≤ 3}
F y ≤ 3, y < 4, y < 5, ก y ∈ A F A
ก y y ≥ 3 ก F F F F A
ก z z ≤ 3
F F ก ก ก F F F F 1.5
F 1.2 A F 1.1
F 1.3 ก F B = {x | x > 0} F B ˈ F
ก x > 0 F F B F ( ?) F F
1.3 F F B F ˈ
ก A ก F F ก F F sup A = ∞ ก A ˈ
F ก F F sup A = ∞ ก ก ก A ก F F F ก F
F inf A = ∞ ก A ˈ F ก F F inf A = ∞
F F ˈ ก F ก
F ˈ F F
1.3
ก F S ⊆ R ก F F S ˈ (bounded set) ก F S F
F 1.2
ก F S ⊆ R S ≠ φ F S F S F

6
ก F F S ⊆ R F F S F F ก
F F
F ก F S ⊆ R S ≠ φ
F y ˈ F S F F y ≤ s ก s ∈ S
ก F T = {y | y ˈ F S}
F 1.2 F F t = sup T
F F t = inf S
1) F s ∈ S
F F s ˈ T
t ≤ s ก t ∈ T
2) F y′ ˈ F S
y′ ∈ T
F F y′ ≤ t
ก 1) 2) F F S F ก
F ˈ F ก ก F F F F ก
F (ﬂ)ก F S ⊆ R S ≠ φ
F s ∈ S
ก S ˈ
F 1.2 F F s ≤ sup S
1.1 F F inf S ≤ s
F F inf S ≤ sup S
(›) F inf S ≤ sup S S = φ
ก S = φ
sup S = ∞ inf S = ∞
1.1
ก F S ⊆ R S ≠ φ F S F F S F ก
1.2
ก F S ⊆ R S ˈ F S ≠ φ ก F inf S ≤ sup S

F F 7
F sup S < inf S ก F F
S ≠ φ
1.2 F F ˈ F ˈ
F F F ก F F F ก F F ก F F
F ก F S ⊆ R S ≠ φ S ˈ S = { s | s ∈ S}
ก F s′ = sup ( S) F F inf S = s′
F x ∈ S
ก s′ = sup ( S)
F F x ≤ s′
s′ ≤ x
F s′ ˈ F S
F F s ˈ F S
s ≤ y ก y ∈ S
F F y ≤ s ก y ∈ S
s ˈ S
F F s′ ≤ s s ≤ s′
sup ( S) = inf S
F 1.4 ก F S = {x ∈ R | 3 < x < 1} F F F
1) F F F S
2) sup S inf S ( F )
3) sup ( S)
1) {x | x ≥ 1} F {x | x ≤ 3}
2) sup S = 1 inf S = 3
3) 1.3 F F sup ( S) = ( 3) = 3
1.3
ก F S ⊆ R S ≠ φ S ˈ S = { s | s ∈ S}
F F sup ( S) = inf S

8
F ก F S ⊆ R S ≠ φ S ˈ
a ∈ R aS = {as | s ∈ S}
1) F ก F inf (aS) = a inf S
ก a = 0
F F aS = {0⋅s | s ∈ S} = {0 | s ∈ S}
F F F aS {x | x ≤ 0}
inf (aS) = 0 a inf S = 0 ⋅ inf S = 0 = inf (aS)
ก a ≠ 0
1.1 S inf S
ก s ∈ S F F inf S ≤ s a inf S ≤ as
a inf S ˈ F aS
F F c ∈ R ˈ F aS
c ≤ as ก s ∈ S
ก a ≠ 0 F F c
a ≤ s ก s ∈ S
c
a ˈ F S
F F c
a ≤ inf S c ≤ a inf S
2) F ก F sup (aS) = a sup S
F F F F ˈ ʿก
F ก F S, T ⊆ R S, T ≠ φ S, T ˈ
1) F s ∈ S, t ∈ T
1.5
ก F S, T ⊆ R S, T ≠ φ S, T ˈ
S + T = {s + t | s ∈ S t ∈ T} F F F
1) sup (S + T) = sup S + sup T
2) inf (S + T) = inf S + inf T
1.4
ก F S ⊆ R S ≠ φ S ˈ a ∈ R aS = {as | s ∈ S}
F F
1) inf (aS) = a inf S
2) sup (aS) = a sup S

F F 9
F 1.2 sup S, sup T
s ≤ sup S t ≤ sup T
F s + t ≤ sup S + sup T
sup S + sup T ˈ S + T
F F x ∈ R ˈ S + T
ก s ∈ S t ∈ T F F s + t ≤ x
F s ≤ x t ก s ∈ S x t ˈ S
F F sup S ≤ x t
t ≤ x sup S ก t ∈ T x sup S ˈ T
F F sup T ≤ x sup S
sup S + sup T ≤ x
sup (S + T) = sup S + sup T
2) F ˈ ʿก
F ก F S, T ⊆ R S, T ≠ φ S, T ˈ
ก F M = max{sup S, sup T} F F
1) ∀x ∈ S ∪ T, x ≤ M
2) ∀y ∈ R, (y ˈ S ∪ T ﬂ M ≤ y)
F x ∈ S ∪ T
x ∈ S x ∈ T
ก S, T ˈ F 1.2 F F x ≤ sup S x ≤ sup T
x ≤ max{sup S, sup T} = M
F F y ∈ R y ˈ S ∪ T
F y ˈ S ∪ T y ≥ z ก z ∈ S ∪ T
F z ∈ S F y ≥ sup S ≥ z F z ∈ T F y ≥ sup T ≥ z
y ≥ max{sup S, sup T} = M
sup (S ∪ T) = max{sup S, sup T}
1.6 ก F S, T ⊆ R S, T ≠ φ S, T ˈ F
sup (S ∪ T) = max{sup S, sup T}

10
ʿก 1.2
1. ก F S = {y | 0 < y ≤ 3} F sup S F F F
2. F S F 1 ˈ F
3. F 1.4 (2) 1.5 (2)
4. ก F S, T ⊆ R S, T ≠ φ S, T ˈ F
1) F S ⊆ T F sup S ≤ sup T
2) sup (S ∩ T) ≤ min{sup S, sup T}
1.3 F
ก 1.4 ก < F F ก F F F a a ก > F กก F
F ก F a, b, c ∈ R
1) (fl) F a < b fl a + c < b + c
F a < b 1.5 F F b a ∈ R+
ก b a = (b + c) (a + c) ∈ R+
F a + c < b + c
(›) F a + c < b + c fl a 0 F ac < bc
3) F a bc

F F 11
2) F a 0
1.5 F F b a ∈ R+
c(b a) = cb ca = bc ac ∈ R+
ac < bc
3) F a 0
F F c(b a) = bc + ac = ac bc ∈ R+
ac > bc
ʿก 1.3
ก F a, b, c ∈ R F F F
1. F a 0 F 1
a > 0
4. F a < 0 F 1
a < 0
5. F 0 < a 0

12
1.4 F F (Absolute value of real number)
F ก ก 1.5 ก F F F F F x
ก F ก 0 x F
1.6 F ก กF ก F ก ก n
F F F F F F F
1.5
ก F x ˈ F F F F F x F |x|
|x| =
x x ≥ 0
x x < 0
1.6
a > 0 x ˈ F F
1) |x| < a ก F a < x < a
2) |x| ≤ a ก F a ≤ x ≤ a
3) |x| > a ก F x < a x > a
4) |x| ≥ a ก F x ≤ a x ≥ a

F F 13
F F F ก F
F ก F x, y ˈ
1) ก x ≥ 0 F F |x| = x | x| = ( x) = x |x| = | x|
ก x < 0 F F |x| = x | x| = x |x| = | x|
ก ก F F |x| = | x|
F F F F F F F F ˈ ʿก
ʿก 1.4
1. F 1.8 F ˈ
2. ก F x ˈ F F |x| ≤ x ≤ |x|
3. ก F x, y ˈ F F ก F F F F ˈ F
F F F F
1) 1
x = |x 1
| = |x| 1
2) |x y| = |y x|
1.8 ก F x, y ˈ F F F
1) |x| = | x|
2) |x||y| = |xy|
3) x
y = x
y
4) |x + y| ≤ |x| + |y| ก F ก (Triangle Inequality)
5) ||x| |y|| ≤ |x y| ก F ก F ก (Backward Triangle Inequlaity)

2
กก
2.1 กก
2.1.1 กก
กก F
2.1
กก (exponent) an
a ˈ n ˈ ก
an
F ก a n ˈ ก F
an
= a ⋅ a ⋅ a ⋅ ⋅ a
n
2.1 ก F a, b ˈ m, n ˈ ก F
1) am + n
= am
⋅ an
2) (am
)n
= amn
3) a n
= n
1
a
4) (a ⋅ b)m
= am
⋅ bm

16
2.1 F F F F ˈ ʿก F F ก ˈ ก 2.1
ก 2.1 F F F F F F F F ˈ ʿก
ʿก 2.1
1. F 2.1 ˈ
2. F ก 2.1 ˈ
3. F F 00
F F F
2.2 ก k
F F ก 36 2 F F กF 6 ก 6 F ก
k k = 3, 4, 5, FกF F ก
ก 2.1
1) am n
=
m
n
a
a
2) a0
= 1
2.2
ก F a ˈ m, k ˈ ก F
m
ka = k m
a F k m
a
F F ก k am

F F 17
ก F F ก F ก k ก
F ก ก ก F F F
F 2.1 F
3 4
27 F F
2.2 F F
3 4
27 =
4
327 = ( )
4
33
3 = 34
= 81
3 4
27 = 81
F 2.2 F 3 -343 F F
2.2 F F
3 -343 = ( )
1
3-343 = ( )( )
1
33-7 = 7
3 -343 = 7
F 2.3 F
6 3
3 4
64 + 343
216 - 16
ก
6 3
3 4
64 + 343
216 - 16
=
1 1
6 3
1 1
3 4
64 + 343
216 - 16
=
( ) ( )
( ) ( )
1 1
6 36 3
1 1
3 43 4
2 + 7
6 - 2
= 2 + 7
6 - 2 = 9
4
F F
6 3
3 4
64 + 343
216 - 16
= 9
4
2.2 ก F a, b ˈ ก m, n, k ˈ ก k > 2
F
1)
k m
a ⋅ k n
a = k m + n
a
2)
k m
a ⋅ k m
b = ( )k ma b⋅
3)
k m
k m
a
b
= ( )mak
b b ≠ 0

18
F 2.4 F F a - b
a + b
F F F ก F
ก a - b
a + b
F F a + b
a + b
a - b
a + b
= a - b
a + b
⋅ a + b
a + b
=
( ) ( )
( ) ( )
a - b a + b
a + b a + b
⋅
⋅
= 2
(a - b) (a + b)
(a + b)
⋅
=
2 2
a - b
a + b
F 2.5 ก F a = 6, b = 3 F
1
3a
1
3b
F x3
= a = 6 F y3
= b = 3
F F x =
1
3a , y =
1
3b
ก x3
y3
= (x y)(x2
+ xy + y2
)
6 3 = (
1
3a
1
3b )(
2
3a +
1
3(ab) +
2
3b )
3 = (
1
3a
1
3b )(
2
36 +
1
318 +
2
33 )
1
3a
1
3b = 2 1 2
3 3 3
3
6 + 18 + 3
ʿก 2.2
F 2.2 ˈ

3
ก ก
3.1 ก ก F
3.1.1 ก F
ก 3.1 ก F ax + b = 0 F ก F ก F
b F F ax + b + ( b) = ax + 0 = b => ax = b
ก a ก F x = b
a ˈ ก
F 3.1 ก 3x + 1 = x
2 + 3
ก ก 3x + 1 = x
2 + 3
ก F ก F x
2 F 3x x
2 + 1 = x
2 + 3 x
2 = 3
ก F ก F 1 F 3x x
2 + 1 1 = 3 1 = 2
F F 3x x
2 = 2 => 2(3x) - x
2 = 2
ก F 2 F 2(3x) x = 2 ⋅ 2 = 4 => 6x x = 4
F 5x = 4 => ก F 5 F x = 4
5
: 3( )4
5 + 1 = 1
2 ( )4
5 + 3
12
5 + 1 = 2
5 + 3
17
5 = 17
5
ก x = 4
5
3.1
ก F ก F ax + b = 0 a, b ˈ a ≠ 0

20
F 3.2 ก 1 x = x + 3
ก ก F 1 x = x + 3
ก F ก F x F 1 2x = 3
ก F ก F 1 F 2x = 2
ก F 2 F x = 1
: 1 ( 1) = ( 1) + 3
2 = 2
ก x = 1
3.1.2 ก F
ก กF ก F ก กก F ก ก กF ก F ก F
ก F F F F F F ก F F F ก F ก ก
F Fก F F ก F F F ก F F F F ก F F F
ก ก ˈ ก F ก F F F
F 3.3 ก 3x + 2 ≥ 4x + 5
ก ก 3x + 2 ≥ 4x + 5
ก F ก F 2 F 3x ≥ 4x + 3
ก F ก F 4x F x ≥ 3
F ก F 1 F x ≤ 3
ก x ≤ 3 ˈ F F {x | x ≤ 3}
3.2 ก F ก F
1) ax + b < 0
2) ax + b > 0
3) ax + b ≥ 0
4) ax + b ≤ 0
5) ax + b ≠ 0
a, b ˈ a ≠ 0

F F 21
ก กF ก ≠ ก F ก F ก ≠ ˈ =
ก ก กF ก F ˈ ก F ก ˈ ≠
F F
F 3.4 ก 4x + 1 ≠ x 1
ก ก ก F ก ≠ ˈ = F F
4x + 1 = x 1
4x x = 1 1
3x = 2
x = 2
3
ก x ≠ 2
3
3.2 F (Interval)
F 5 8 ก F F ก F
3.3 ก F a, b ˈ a a (a, ∞)
6) x ≥ a [a, ∞)
7) x < a ( ∞, a)
8) x ≤ a ( ∞, a]
9) ∞ < x < ∞ ( ∞, ∞) ก F F F ก

22
F 3.5 ก F A = {x | 1 ≤ x ≤ 6} A F
ก ก F F 3.3 (3) F ˈ F ʽ
A F F F A = [ 1, 6]
F 3.6 ก F A = [2, 7], B = [0, 7], C = [ 1, 7] F n((A ∩ B) C)
ก Fก F ก F F
F
ก n((A ∩ B) C) = n(A ∩ B) n(A ∩ B ∩ C) -----(1)
ก n(A ∩ B) = n([2, 7]) = 6
n(A ∩ B ∩ C) = n([2, 7]) = 6
ก ก (1) F F n((A ∩ B) C) = 6 6 = 0
F F F ก F
ก ก F
A = {2, 3, 4, 5, 6, 7}, B = {0, 1, 2, 3, 4, 5, 6, 7}, C = { 1, 0, 1, 2, 3, 4, 5, 6, 7}
F F A ∩ B = A ( F A ⊆ B)
(A ∩ B) C F ˈ A C
A C ก F ก F A ⊆ C F n(A) = n(A ∩ C)
F F n(A C) = n(A) n(A ∩ C) = 0
ʿก 3.1
1. F ก F 3.3 F 3.4
1 0 72
A
C
B

F F 23
3.3 ก ก ก n
3.3.1 ก ก n
F F ก 3.4 ก ก n F F Pn(x) = 0
ก กF ก ก ก ก F ก ก Fก Fก
ก ก F F ก F F F a, b ˈ ab = 0 F F F a = 0
b = 0 ก F F F F ก F F
F 3.7 ก x2
5x + 6 = 0
ก ก Fก F ˈ ก Fก ก ก F
F F x2
5x + 6 = (x 3)(x 2) = 0
x = 3 x = 2
ก x = 2 x = 3
F ก F ก F F F F
F F
F 3.8 ก x3
+ 2x2
x 2 = 0
ก x3
+ 2x2
x 2 = (x3
+ 2x2
) (x + 2)
= x2
(x + 2) (x + 2)
= (x + 2)(x2
1)
= (x + 2)(x 1)(x + 1)
F F (x + 2)(x 1)(x + 1) = 0
ก x = 2 x = 1 x = 1
3.4
ก n ก F Pn(x) n ≥ 2 F anxn
+ an 1xn 1
+ an 2xn 2
+ +
a1x + a0 an, an 1, an 2, , a1, a0 ˈ an ≠ 0 ก F

24
ก ก F ก ก F ก F F
ก ก ก F
F F P(x) ˈ ก n x c ˈ
F F P(x) = (x c)Pn 1(x) + Rn 1(x) -----(3.3.1)
Pn 1(x) ก n 1 Rn 1(x) F กก P(x) F x c
F x = c ก (3.3.1) F F P(c) = 0 + Rn 1(c) = Rn 1(c) -----(3.3.2)
กก P(x) F x c F F ก P(c) F ก
F ก 3.1 F 3.2 F
F ก F P(x) ˈ ก n
(ﬂ) F x c ˈ ก P(x)
F F P(x) = (x c)Pn 1(x) + Rn 1(x)
ก x c ˈ ก P(x) F F
P(x) = (x c)Pn 1(x)
P(c) = (c c)Pn 1(x) = 0
(›) F P(c) = 0
ก F F P(x) = (x c)Q(x) + Rn 1(x)
ก 3.1 F P(c) = Rn 1(x)
P(x) = (x c)Q(x)
F x c ˈ ก P(x)
F 3.9 ก F P(x) = 2x3
5x2
x 6 ˈ ก P(x) ก F
กก P(x) F x + 3
ก 3.1 F F กก P(x) F x c F ก P(c)
F x + 3 = 0 => x = 3
F F กก P(x) F x + 3 F ก P( 3)
P( 3) = 2( 3)3
5( 3)2
( 3) 6
3.1 (Remainder Theorem or Residue Theorem)
ก F P(x) ˈ ก n F P(x) F x c F F F กก F ก P(c)
3.2 ก ก (Rational Factor Theorem)
ก F P(x) ˈ ก n F F x c ˈ ก P(x) ก F P(c) = 0

F F 25
= 2( 27) 5(9) + 9 6
= 54 45 + 9 6 = 96
F 3.10 ก F P(x) = 6x3
+ 7x2
7x 6 F x 1 ˈ ก P(x)
ก 3.2 F F x 1 ˈ ก P(x) P(1) = 0
F ก P(x) = 6x3
+ 7x2
7x 6 F F
P(1) = 6(1)3
+ 7(1)2
7(1) 6 = 6 + 7 7 6 = 0
ก P(1) = 0 F x 1 ˈ ก P(x)
ʿก 3.3 ก
1. F ก ก ก ก F 3.6, 3.7 3.8
2. F a ˈ F x a x3
+ 2x2
5x 2 F ก 4 F ก F a
F ก ก F F ก F
3. F px3
15x2
+ 27x + q F x 2 x 5 F F ก 22 F F p + q
4. ก
2
x + 1
x
x - 1
x = 1 F
3.3.2 ก ก n
ก ก n ก F F
3.5
ก F P(x) Q(x) ˈ ก n Q(x) ≠ 0 F ก ก n F
1) P(x) ⋅ Q(x) < 0 P(x) ⋅ Q(x) > 0
2) P(x) ⋅ Q(x) ≤ 0 P(x) ⋅ Q(x) ≥ 0
3) P(x)
Q(x) < 0 P(x)
Q(x) > 0
4) P(x)
Q(x) ≤ 0 P(x)
Q(x) ≥ 0

26
1) ก F P(x) ⋅ Q(x) < 0, P(x) ⋅ Q(x) > 0, P(x) ⋅ Q(x) ≤ 0 P(x) ⋅ Q(x) ≥ 0
ก กF ก F ก ก F F ก ก F F ก F
P(x) ⋅ Q(x) = 0 ก F F x ˈ กกF F กF ก F ก x F
F ก F ก ก F
F F ก F ก F F F ก
ก F ก ก F ก F F F
ˈ < ≤ F F F ˈ
> ≥ ก ˈ ก ก F ก F ก ˈ ก F
F 3.11 ก x3
+ 2x2
x 2 > 0
ก F 3.8 F x3
+ 2x2
x 2 = (x + 2)(x 1)(x + 1)
ก F ˈ (x + 2)(x 1)(x + 1) > 0
F F ก x = 2, x = 1, x = 1
F ก F ก F F F F
ก F ˈ > ก F ˈ ก
F F ก {x | 2 < x < 1 x > 1}
F F F A = ( 2, 1) ∪ (1, ∞)
F 3.12 ก F A ˈ ก 6x3
+ 7x2
7x 6 ≤ 0 ก
{x ∈ A | x ∈ I+
x ≤ 5}
ก 6x3
+ 7x2
7x 6 = (x 1)(2x + 3)(3x + 2)
F F (x 1)(2x + 3)(3x + 2) ≤ 0 F ก F กF x = 3
2 , x = 2
3 , x = 1
F ก F ก F F
ก ก F ˈ ≤ F F
2 11
+ +
3
2
12
3
+ +

F F 27
ก A = {x | ∞ ≤ x ≤ 3
2 x ≥ 1} ˈ F F
( ∞, 3
2 ] ∪ [1, ∞) F F F ก ก
{x ∈ A | x ∈ I+
x ≤ 5} F F n({x ∈ A | x ∈ I+
x ≤ 5}) = 5
F 3.13 ก ( )2
x x - 3
2 <
( )2
3 - x
8 ˈ F F
ก. (1, ∞) . ( 2, 100)
. ( 10, 10) . ( ∞, 2)
ก Fก F ˈ ก ก F
ก ( )2
x x - 3
2 <
( )2
3 - x
8
( )2
x x - 3
2 <
( )2
33 - x
2
( )2
x x - 3
2 < 2 - 3x
2
ก ˈ ˆ กF (2 > 0)
F F x2
(x 3) < 2 3x
x2
(x 3) < 2 3x
x3
3x2
+ 3x 2 < 0 -----(1)
ก ก (1) F ก
F P(x) = x3
3x2
+ 3x 2 ﬂ ก P(2) = 0 F x 2 P(x)
ก F ( ก ก ก) F F
x3
3x2
+ 3x 2 = (x 2)(x2
x + 1)
= (x 2)[(x2
x + 1
4 ) + 1 1
4 ]
= (x 2)[(x 1
2 )2
+ 3
4 ]
ก ก (1) F (x 2)[(x 1
2 )2
+ 3
4 ] < 0
F F F [(x 1
2 )2
+ 3
4 ] > 0
x 2 < 0 => x < 2 F F ( ∞, 2)
2) ก F P( x )
Q( x ) < 0 P( x )
Q( x ) > 0, P( x )
Q( x ) ≤ 0 P( x )
Q( x ) ≥ 0
ก กF ก F ก ก F ก F ก ก F 1) ก F ก F
F F F ก ก กF ก ก F ก ก
ก F 1) F F ก F ก F F ก F F F
F F ก ก F F F ก F

28
F 3.14 ˈ ก 5 ≤
2
x - 6
x ≤ 1
ก. 8 . 9
. 10 . 11
ก F 5 ≤
2
x - 6
x ≤ 1 F ก ˈ ก F F 2
1) 5 ≤
2
x - 6
x
2)
2
x - 6
x ≤ 1
=> ก 5 ≤
2
x - 6
x F F
2
x - 6
x + 5 ≥ 0
2
x + 5x - 6
x ≥ 0
x(x + 6)(x 1) ≥ 0
F F ก x = 6, x = 0, x = 1
ก [ 6, 0) ∪ [1, ∞)
=> ก
2
x - 6
x ≤ 1 F F
2
x - 6
x 1 ≤ 0
2
x - x - 6
x ≤ 0
x(x 3)(x + 2) ≤ 0
F F ก x = 2, x = 0, x = 3
ก ( ∞, 2] ∪ (0, 3]
=> ก 2 F F ก (intersection)
{[ 6, 0) ∪ [1, ∞)} ∩ {( ∞, 2] ∪ (0, 3]} = [ 6, 2] ∪ [1, 3]
= { 6, 5, 4, 3, 2} ∪ {1, 2, 3} = { 6, 5, 4, 3, 2, 1, 2, 3}
F F n([ 6, 2] ∪ [1, 3]) = 8

F F 29
ʿก 3.3
ก ˆ F F F
1. ก (4x
2) log(1 x2
) > 0 ˈ F F
ก. ( 2, 1
2 ) . ( 1
2 , 2)
. (0, 10) . ( 1
2 , 20)
2. ก F I S = {x | x - 1 - 1 x - 1 + 1⋅ < 50}
ก S ∩ I F ก F F
ก. 13 . 14
. 15 . 16
3. F S ˈ ก log (log x) + log (9 log x2
) ≥ 1
F a b ˈ ก S F ก F F F ab F F ก F
ก.
7
210 .
9
210
.
11
210 .
13
210
3.4 ก ก F F
F ก ก ก F F F F 1.5 F
ก กF ก ก F F F
3.4.1 ก F F
ก กF ก F F F 1) F F ก P(x) Q(x) F P(x) = 0 Q(x) = 0
ก F F ก F F F ก F F ก ก กF ก
3.6
ก F P(x) Q(x) ˈ ก n c ˈ F ก F F
F
1) |P(x)| + |Q(x)| = c |P(x)| |Q(x)| = c
2) |P(x) ⋅ Q(x)| = c

30
F 3.15 F A ˈ ก |x 4| + |x 3| = 1 F A F ก F
ก. (3, 4) . {x ∈ R | |x 7
2 | ≤ 1
2 }
. ( ∞, 4] . [3, ∞)
F x 4 = 0 => x = 4
F x 3 = 0 => x = 3
F F ก F ก F F F F F F
ก F F F
=> F x < 3 F F
(x 4) (x 3) = 1
2x = 6
x = 3
F F {}
=> F 3 ≤ x ≤ 4 F F
(x 4) + (x 3) = 1
1 = 1
F ก ˈ F ก F F [3, 4]
=> F x > 4 F F
(x 4) + (x 3) = 1
2x 7 = 1
2x = 8
x = 4
F F {}
ก F F ก |x 4| + |x 3| = 1 [3, 4]
3 4
x < 3 3 ≤ x ≤ 4 x > 4
|x 3| = (x 3)
|x 4| = (x 4)
|x 3| = x 3
|x 4| = (x 4)
|x 3| = x 3
|x 4| = x 4

F F 31
F ก .
ก |x 7
2 | ≤ 1
2
1.6 (2) F F 1
2 ≤ x 7
2 ≤ 1
2
กF ก F 3 ≤ x ≤ 4 F F [3, 4] F .
ก กF ก F F F 2) F F F F x, y
F F |xy| = |x||y| ก ก F ก ก F 1) F F
F 3.16 ก |(x 4)(x 3)| = 1
ก |(x 4)(x 3)| = |x 4|⋅|x 3| = 1
F x 4 = 0 => x = 4
F x 3 = 0 => x = 3
ก F F F
=> F x < 3 F F
( (x 4))( (x 3)) = 1
(x 4)(x 3) = 1
x2
7x + 11 = 0 => x =
2
-(-7) (-7) - 4(1)(11)
2
±
x1 = 7 + 5
2 , x2 = 7 - 5
2
F F { 7 - 5
2 } ( 7 + 5
2 F F F ?)
=> F 3 ≤ x ≤ 4 F F
(x 3)(x 4) = 1
(x2
7x + 12) = 1
x2
7x + 13 = 0 => x =
2
-(-7) (-7) - 4(1)(13)
2
±
x1 = 7 + -3
2 , x2 = 7 - -3
2
3 1
x < 3 3 ≤ x ≤ 4 x > 4
|x 3| = (x 3)
|x 4| = (x 4)
|x 3| = x 3
|x 4| = (x 4)
|x 3| = x 3
|x 4| = x 4

32
ก ก ˈ F F F {}
=> F x > 4 F F (x 4)(x 3) = 1
F F ˈ ก ก ก ก F F ก { 7 + 5
2 }
ก F F ก |(x 4)(x 3)| = 1
{ 7 + 5
2 , 7 - 5
2 }
3.4.2 ก F F
ก กF ก F F 3.7 F F F F
F
F 3.17 ก |x2
16| < 9
ก 1.6 (1) F F
9 < x2
16 < 9 ก F
=> 9 < x2
16
x2
7 > 0
(x 7 )(x + 7 ) > 0
F ( ∞, 7 ) ∪ ( 7 , ∞)
=> x2
4 < 9
x2
13 < 0
(x 13 )(x + 13 ) < 0
F ( 13 , 13 )
ก F ก
(( ∞, 7 ) ∪ ( 7 , ∞)) ∩ ( 13 , 13 ) = ( 13 , 7 ) ∪ ( 7 , 13 )
3.7
ก F P(x) Q(x) ˈ ก n F ก F F F
1) |P(x) ⋅ Q(x)| > 0 |P(x) ⋅ Q(x)| < 0
2) |P(x) ⋅ Q(x)| ≥ 0 |P(x) ⋅ Q(x)| ≤ 0
3) |P(x) + Q(x)| ≤ 0 |P(x) + Q(x)| ≥ 0
4) |P(x) + Q(x)| < 0 |P(x) + Q(x)| > 0

F F 33
F 3.18 ก |2 x| ≤ 2 + x
F F F F (2 + x) ≤ 2 x ≤ 2 + x ก ก ˈ ก F F
ﬂ (2 + x) ≤ 2 x
x 2 ≤ 2 x
2 ≤ 2
F ˈ ก F x
ﬂ 2 x ≤ 2 + x
2x ≤ 2 2 = 0
x ≥ 0
ก ( ∞, ∞) ∩ [0, ∞) = [0, ∞)
F 3.19 ก x - 1
x - 2 ≤ 0
ก x ≥ 0 F F x - 1
x - 2 ≤ 0 [1, 2)
ก x < 0 F F -x - 1
-x - 2 ≤ 0
-(x + 1)
-(x + 2) ≤ 0
x + 1
x + 2 ≤ 0 F ( 2, 1]
ก ( 2, 1] ∪ [1, 2)

4
ก F
ก F ˈ ก F F
F F F F ก F ˈ F F ก ก ก F F F
ก F F ก F ก F F F
ก F F F ก 4 F
ก F F F F กF ก ก ˈ ก F ก
F F ก ก ก F F
4.1 ก ก ˈ ก ก F
ก ก ˈ ก ก F กก ก ก ก F F
ก F F ก ˈ ก ก ก F F F
ก F F F ก F F F F F ˈ F
F F F F ก ˈ ก ก F ก F F F
F F F F ก F F F F กก ˈ ก ก
F ก F F
F 4.1 F 6 F F 4 F ก
ก F 2.00002
< ( )2
6 < 3.00002
F F 6 F F F
2 ก 3 F F 2.0000 ก 3.0000 F 2.0000 + 3.0000
2 = 2.5000
2.50002
= 6.2500 F 2.00002
< ( )2
6 < 2.50002
F 2.0000 ก
2.5000 F 2.0000 + 2.5000
2 = 2.2500
2.25002
= 5.0625 F 2.25002
< ( )2
6 < 2.50002
F 2.2500 ก
2.5000 F 2.2500 + 2.5000
2 = 2.3750
2.37502
= 5.6400 F 2.37502
< ( )2
6 < 2.50002
F 2.375 ก
2.5000 F 2.3750 + 2.5000
2 = 2.4375

36
2.43752
= 5.9400 F 2.43752
< ( )2
6 < 2.50002
F 2.4375 ก
2.5000 F 2.4375 + 2.5000
2 = 2.4688
2.46882
= 6.0949 F 2.43752
< ( )2
6 < 2.46882
F 2.4375
ก 2.4688 F 2.4375 + 2.4688
2 = 2.4532
2.45322
= 6.0182 F 2.43752
< ( )2
6 < 2.45322
F 2.4375
ก 2.4532 F 2.4375 + 2.4532
2 = 2.4454
2.44542
= 5.9799 F 2.44542
< ( )2
6 < 2.45322
F 2.4454
ก 2.4532 F 2.4454 + 2.4532
2 = 2.4493
2.44932
= 5.9991 F 2.44932
< ( )2
6 < 2.45322
F 2.4493
ก 2.4532 F 2.4493 + 2.4532
2 = 2.4513
2.45132
= 6.0089 F 2.44932
< ( )2
6 < 2.45132
F 2.4493
ก 2.4513 F 2.4493 + 2.4513
2 = 2.4503
2.45032
= 6.0040 F 2.44932
< ( )2
6 < 2.45032
F 2.4493
ก 2.4503 F 2.4493 + 2.4503
2 = 2.4498
2.44982
= 6.0015 F 2.44932
< ( )2
6 < 2.44982
F 2.4493
ก 2.4498 F 2.4493 + 2.4498
2 = 2.4496
2.44962
= 6.0005 F 2.44932
< ( )2
6 < 2.44962
F 2.4493
ก 2.4496 F 2.4493 + 2.4496
2 = 2.4495
2.44952
= 6.0000 F 2.44932
< ( )2
6 ≮ 2.44952
ก F
F F 6 ≈ 2.4495
ก F F F F ก ก ˈ ก ก ก F ก F F
(initial value) F F F ก F F ก
F 4.2 F 6 F [2, 2.5]
F 22
< ( )2
6 < 2.52
F 2 ก 2.5 F 2 + 2.5
2 = 2.25
2.252
= 5.0625 F 2.252
< ( )2
6 < 2.52
F 2.25 ก 2.5 F 2.25 + 2.5
2 = 2.375

F F 37
2.3752
= 5.6406 F 2.3752
< ( )2
6 < 2.52
F 2.375 ก 2.5 F 2.375 + 2.5
2 = 2.4375
2.43752
= 5.9414 F 2.43752
< ( )2
6 < 2.52
F 2.4375 ก 2.5 F 2.4375 + 2.5
2 = 2.4688
2.46882
= 6.0949 F 2.43752
< ( )2
6 < 2.46882
F 2.4375
ก 2.4688 F 2.4375 + 2.4688
2 = 2.4532
2.45322
= 6.0182 F 2.43752
< ( )2
6 < 2.45322
F 2.4375
ก 2.4532 F 2.4375 + 2.4532
2 = 2.4454
2.44542
= 5.9799 F 2.44542
< ( )2
6 < 2.45322
F 2.4454
ก 2.4532 F 2.4454 + 2.4532
2 = 2.4493
2.44932
= 5.9991 F 2.44932
< ( )2
6 < 2.45322
F 2.4493
ก 2.4532 F 2.4493 + 2.4532
2 = 2.4513
2.45132
= 6.0089 F 2.44932
< ( )2
6 < 2.45132
F 2.4493
ก 2.4513 F 2.4493 + 2.4513
2 = 2.4503
2.45032
= 6.0040 F 2.44932
< ( )2
6 < 2.45032
F 2.4493
ก 2.4503 F 2.4493 + 2.4503
2 = 2.4498
2.44982
= 6.0015 F 2.44932
< ( )2
6 < 2.44982
F 2.4493
ก 2.4498 F 2.4493 + 2.4498
2 = 2.4496
2.44962
= 6.0005 F 2.44932
< ( )2
6 < 2.44962
F 2.4493
ก 2.4496 F 2.4493 + 2.4496
2 = 2.4495
2.44952
= 6.0000 F 2.44932
< ( )2
6 ≮ 2.44952
ก F
F F 6 ≈ 2.4495
ก F F F F F F F ก F F F Fก ˈ ก ก
F F ก ˈ ก F F Fก ก ก F ก F ก F
F ก F ˈ ก ก ก F F F ก ก ก F F F F ก F ก
F F ก F F F

38
ʿก 4.1
1. F 17 F ก F F ( F 4.1231)
2. (algorithm) ก (source code) F ก ˈ ก
ก ก F ก F F 17
4.2 ก F F
ก F F กก ก F ก
F F ก ก F ก ก F ก ก
F ก F ก F F F F F ก F F ก
ก ก F F ก F F F ก F ก F
F F ก ก F F F
F 4.3 F 2 F
ก 12
+ 12
= ( )2
2 F F F ก F F ก ก
F 1 F
4.1 ก F ก ก F 1 F
ก 4.1 F F F F F F ก (1.41, 0)
F F 2 F 1.41
X
Y
1
1
2

F F 39
ʿก 4.2
ก F F F
1. 8
2. 18
3. 72
4. 98
4.3 ก F F
ก F ˆ กF ( ก F ก F F 11
F ) F F ˆ กF F f x = x0 ˈ ก
f′(x0) = 0 0f(x + h) - f(x )
h
h 0
lim
→
-----(4.3.1)
F h F F ก F F f′(x0) ≈ 0 0f(x + h) - f(x )
h -----(4.3.2)
F (4.3.2) F h F h⋅f′(x0) ≈ f(x0 + h) f(x0) F F F
f(x0 + h) ≈≈≈≈ f(x0) + h⋅⋅⋅⋅f′′′′(x0) -----(4.3.3)
F (4.3.3) ก F f(x0 + h) F f(x0) f′(x0) ก F F
ก F F (4.3.3) F กก ˆ กF F ˆ กF F F ก
F ก F ก F h f′(x0) F F F (4.3.3) ก F
F ก F F F F F F F ก
F ก ก ก f F ˈ ˆ กF F F F ˆ กF f ก F ˈ ˆ กF F F
F 4.4 F 6 F ก F x0 = 4
ก F f(x) = x F F f′(x) = 1
2 x
h = 6 4 = 2 f′(4) = 1
2 4
= 1
4 = 0.25
ก f(x0 + h) ≈ f(x0) + h⋅f′(x0) F F
f(6) = f(4 + 2) ≈ f(4) + (2)(0.25) ≈ 2.50
F ก ก F 6 ≈ 2.449 F F ก F ก ก

40
ก F 4.4 F F F F F ˈ x0 = 2.5 ก F F
F F Fก F F ก F F x0 = 4 F F ก F ก
F F
F 4.5 F (numerical value) sin 0.6000 F
ก F x0 = 0.5000 f(x0) = 0.4794 ( ˈ 3 F )
ก F f(x) = sin x (x F ˈ ) F F f′(x) = cos x
h = 0.6000 0.5000 = 0.1000 f′(0) = cos 0.5000 = 0.8776
ก f(x0 + h) ≈ f(x0) + h⋅f′(x0) F F
f(0.6000) = f(0.5000 + 0.1000) ≈ 0.4794 + (0.1000)(0.8776) ≈ 0.567
ʿก 4.3
1. F F F
1)
2.5 -2.5
e + e
2
2) 10001
3) sin 0.75 cos 0.75 (ก F sin 0.70 = 0.6442 cos 0.70 = 0.7648)
F ˆ กF ʽ ก ก
2. F F F F F 1 F F กก ก F ก F ก F

F F 41
4.4 ก F F ก ก
ก F F ก ก ˈ ก F ก ก F F F
F F ก ก F F ก ก ก F ˆ กF
F ก ก F F ˆ กF
F 4.6 ก F f(x) = ex
F f(1.2500) Fก F ก
6 F ก ( F 4 F )
ก ก ex
f(x) ≈ 1 + x + 21
2!x + 31
3!x + 41
4!x + 51
5!x
f(1.2500) ≈ 1 + (1.2500) + 21
2!(1.2500) + 31
3!(1.2500) + 41
4!(1.2500) + 51
5!(1.2500)
≈ 1 + 1.2500 + 0.7813 + 0.3255 + 0.1017 + 0.0254 ≈ 3.4839
F 4.7 ก F f(x) = ln x F f(2.1) Fก F ก F F
x0 = 1 6 F ก ( F 4 F )
ก f(x) = ln x, f(1) = ln 1 = 0 F F
f′(x) = 1
x , f′(1) = 1
f″(x) = 2
1
x
, f″(1) = 1
f′′′(x) = 3
2
x
, f′′′(1) = 2
f(4)
(x) = 4
6
x
, f(4)
(1) = 6
f(5)
(x) = 5
24
x
, f(5)
(1) = 24
ก ก F F T(x) ≈ ( )
(k)
0f (x ) k
0k!
k 0
x x
∞
∑
=
−
≈ ln 1 + 1
1! (2.1 1) + (-1)
2! (2.1 1)2
+ 2
3! (2.1 1)3
+
(-6)
4! (2.1 1)4
+ 24
5! (2.1 1)5
≈ 0 + 1.1000 0.6050 + 0.4437 0.3660 + 0.3221 ≈ 0.8948

42
ʿก 4.4
1. Fก F ก ก F ʿก 4.3
2. ก F F กก ʿก F 1
3. ก F f(x) = sin x F f(0.75) F ก ก ก ก F
F ก F F f(0.75) = 0.6816 F ก F F ก

ก
1. ก F
ก F ˈ ก ก F ก F
ก ก F
ก F P(x) ˈ ก n P(x) = anxn
+ an 1xn 1
+ an 2xn 2
+ + a2x2
+ a1x + a0
an, an 1, an 2, , a2, a1, a0 ˈ an ≠ 0 ˈ ˈ F Q(x) = x c
c ≠ 0 ˈ ˈ F ก F ˈ F
n n - 1 n - 2
2
n n - 1 n
2
n n - 1 n n - 2 n - 1 n
a a a
x = c ca ca + c a
a a + ca a + ca + c a
↓
⋯
⋯
⋯
.1 ก F
ก F F ˈ F ก ก F ก
ก F ก F F F F ก F F F F
F ˈ F F กก F
F ก F ก F F Fก ˈ ก F ก Fก
ˈ ก กก F 1 F ก ก F ˈ ก กF F F
ก Fก ˈ ก F F ˈ ก ก Fก F
F F F ก ก F F F F F ก F F
F .1.1 F P(x)
Q(x) ก F P(x) = x2
2x + 1 F Q(x) = x + 2
ก ก F ก F ก F F
1 -2 1
x = -2
1
↓
F F ก ก F F ก 2 ก 1 F F F

44
F F F ก 2 F F F F F F 2
1 -2 1
x = -2 -2
1
↓
ก 2 กก 2 ˈ F F ก กF F F F ก 4
1 -2 1
x = -2 -2
1 -4
↓
F 2 ก F FกF F 4 F F ก 8 8 F F F 1
1 -2 1
x = -2 -2 8
1 -4 9
↓
F 1 กก 8 ˈ F F ก กF F F F ก 9
F F P(x)
Q(x) =
2
x - 2x + 1
x + 2 = x 4 + 9
x + 2 ก F F F F กก x 4
F ก 9
F .1.2 ก F P(x) = x3
+ 3x 1 Q(x) = x 1 F ก P(x) F Q(x)
ก ก F F
1 0 3 -1
x = 1
1
↓
1 ˈ ก 1 ˈ F F F ก 1 F F F 0 F 0
กก 1 F F F F F
1 0 3 -1
x = 1 1
1 1
↓
1 ˈ ก 1 ˈ F ก กF F F F ก 1 F F F 3
ก 3 กก 1 F F F F F F

F F 45
1 0 3 -1
x = 1 1 1
1 1 4
↓
F 1 ก 4 ˈ F ก กF F F F ก 4
F F F 1 ก 1 กก 4 F F ก 3 F F F F
1 0 3 -1
x = 1 1 1 4
1 1 4 3
↓
ก F F F F F กก x2
+ x + 4 F ก 3
F .1.3 F กก x4
3x3
+ 2x2
x + 1 F x2
4
ก ˈ ก F ก ก กF F F ก (x 2)(x + 2)
F F F F กก x4
3x3
+ 2x2
x + 1 F (x 2)(x + 2)
ก (x 2) ˈ ก ก F F
1 -3 2 -1 1
x = 2 2 -2 0 -2
1 -1 0 -1 -1
↓
ก F F กก ก F x 2 F
F x + 2 ˈ F
1 -1 0 -1 -1
x = -2 -2 6 -12 26
1 -3 6 -13 25
↓
ก x4
3x3
+ 2x2
x + 1 F x2
4 F F x2
3x + 6
F ก 13x + 25

46
2. ก ก (Power Series)
ก ก F F ก ˆ กF (transcendental function) F F ก
ก F F F ก ก FกF ก ก F
ก F y = f(x) = ( )k
k 0
k 0
c x x
∞
∑
=
−
= c0 + c1(x x0) + c2(x x0)2
+ c3(x x0)3
+ c4(x x0)4
+ ---( .2.1)
F F F y′ = f′(x) = c1 + 2c2(x x0) + 3c2(x x0)2
+ 4c4(x x0)3
+ -----( .2.2)
F F F y″ = f″(x) = 2c2 + 6c3(x x0) + 12c4(x x0)2
+ -----( .2.3)
F F F y′′′ = f′′′(x) = 6c3 + 24c4(x x0) + -----( .2.4)
F k F F y(k)
(x) = f(k)
(x) = k!ck => ck =
(k)
f (x)
k! -----( .2.5)
ก ( .2.5) F k ก ก ก F ก F F
ก F F T(x) = ( )
(k)
0f (x ) k
0k!
k 0
x - x
∞
∑
=
-----( .2.6)
ก ( .2.6) ก F ก F F (Taylor s Series) x = x0 F F x0 = 0
F F F ก ( .2.6) F F M(x) =
(k) kf (0)
k!
k 0
x
∞
∑
=
-----( .2.7)
ก ( .2.7) ก F ก (McClaurin s Series)
ก ก (Power Series) ก F F ( )k
k 0
k 0
c x - x
∞
∑
=
x0
ก F Fก ก ก ck F k ก ก

F F 47
F .2.1 ก F f(x) = ex
f(x) F F ก ก x0 = 0
ก f(x) = f′(x) = f″(x) = = f(k)
(x) = ex
F ก ( .2.7) F F
M(x) = f(0)
0! + f (0)
1! x′
+ 2f (0)
2! x′′
+ 3f (0)
3! x′′′
+
= 1 + x + 21
2!x + 31
3!x + 41
4!x +
ʽ (closed form) F M(x) =
k
x
k!
k 1
∞
∑
=
F .2.2 ก F f(x) = sin 2x f(x) F F ก ก x0 = 0
ก f(x) = sin 2x, f′(x) = 2 cos 2x, f″(x) = 4 sin 2x, f′′′(x) = 8 cos 2x,
f(4)
(x) = 16 sin 2x, f(5)
(x) = 32 cos 2x
F ก ( .2.7) F F
M(x) = f(0)
0! + f (0)
1! x′
+ 2f (0)
2! x′′
+ 3f (0)
3! x′′′
+
(4) 4f (0)
4! x +
(5) 5f (0)
5! x +
= 0 + 2x + 0 38
3!x + 0 + 532
5! x +
3. F ˆ กF
ก F u ˈ ˆ กF x (u(x)) F F F
f(u) f′′′′(u)
eu
eu du
dx
sin u cos u du
dx
cos u sin u du
dx
u du1
dx2 u

F F 49
ก
ก ก ก ก F. ก F. F 2. ก : ก F , 2541.
. F Ent 46. ก : ก , 2546.
. F Ent 47. ก : F F ก F, 2547.
. F Ent 48. ก : F F ก F, 2548.
F F. ก F. F 7. ก : ก ก F, 2545.
ก . ก F .4 ( 011, 012). ก : ʽ ก F F, 2539.

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