2015 2016
R
K
x0
R
C1
⇠ 2 I = [x0
8x 2 I , |f 0 (x)| K
(x(k) )k2N
(x
(k)
⇠
|f (x0 )| 
, x0 + ]
8(x, y) 2 I 2 , |f 0 (x)
K
2
8k 2 N x(k+1) = x(k)
x((0) = x0
f
f (x(k) )
f 0 (z)
f 0 (y)| 
z
I
)k2N
8k 2 N, x(k+1)
x0 = x(k)
x0
f (x(k) ) f (x0 )
f 0 (z)
f (x0 )
.
f 0 (z)
f (x(k) )
x
(k)
I
k
(x(k) )k2N
8k 2 N, |x(k)
⇠|  2
k
⇠
I(f ) =
I1,
a=
I2,
↵
1
R
Rb
a
a) f
[a, b]
[a, b]
C1
f
f (x) dx
(f ; a, b) = (b
✓
a+b
2
◆
+
(b
a)2 0
(f (b)
24
f 0 (a)) .
b=1
(f ; a, b) =
K
2
b
a
2
✓
↵ f (a) +
f
✓
a+b
2
◆
◆
(b
+ ↵ f (b) +
a=
1
a)2
4
(f 0 (b)
b=1
f
f 0 (a)) .
x0
n
n
Tn
8x 2 [ 1, 1], Tn (x) = cos(n arccos(x)),
n
xi = cos(✓i ) i = 0, . . . , n
8x 2 R, cos(⇡ + x) =
cos(x), sin(⇡ + x) =
8x 2]
sin(x), cos
1, 1[, (arccos)0 (x) =
p
1
⇣⇡
✓i =
2i+1
2n ⇡
⌘
x = sin(x), (cos(x))2 + (sin(x))2 = 1,
2
x2
1
1
+1
X
1
⇡2
=
.
j2
6
j=1
,
n
[ 1, 1]
n
xi i = 0, . . . , n
Tn
n
8i 2 {0, . . . , n
8i 2 {0, . . . , n
8i 2 {0, . . . , n
1}, 1 + 2
1}, Tn 0 (xi ) =
n
X
( 1)i n
.
sin (✓i )
Tk (xi )Tk (x) =
( 1)i sin(✓i )
n Tn+1 (xi )
1
Tk (x) dx =
1
(
8i 2 {0, . . . , n
8i 2 {0, . . . , n
n
2
2+2
n
X
Tn (x)Tn+1 (xi )
x xi
Tk (xi )
k=1
Z
1
!
Tk (x) dx .
1
y = arccos(x)
2
1 k2
0
8k 2 N, 8x 2 R, cos(k x) sin(x) =
b n2 c
0
k=1
1}, ↵i =
Z
Tn (x)
.
Tn (xi )(x xi )
1}, 8x 2 R, li (x) =
8i 2 {0, . . . , n
k
k
,
.
1
(sin((k + 1) x)
2
sin((k
1}, Tn+1 (xi ) = ( 1)i+1 sin(✓i ).
0
2 @
1}, ↵i =
1
n
n
2
b2c
X
cos(2j✓i )
j=1
8i 2 {0, . . . , n}, ↵i > 0.
4j 2
1
1
A.
1) x)) .
1