4章の演習問題（全問）

1
1
2
0.75
0.5
0.5
1.5
0.25
1
0
0
-0.25
0.5
-0.5
-0.5
-0.75
0
-1 -0.5 0 0.5 -1
1
-1 -0.5 0 0.5 1
-0.75 -0.5 -0.25 0 0.25 0.5 0.75
1
1
1
0.75
0.5
0.5
0.25
0
0
-0.25
-0.5
-0.5
-0.75
-1
-1 -0.5 0 0.5 1
1
0.5
0
-0.5
-1
-0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 -1 -0.5 0 0.5 1
3 [3]
m
−
( ) ∑ m k −
( ) ( )
m m m k k
d 2 d d d
( x − 1) = ( x+ 1) ( x− 1)
= C x+ 1 x+
1
dx dx dx dx
m m m k k
k = 0
m
∑
m
k = 0
m
k = 0
n n n n n
( )
( 1) ( ( ) 1)( 1) ⎤⎡ ( 1) ( 1)( )
n− m−k n−k
= C ⎡
k
n⋅ n− n− m− k + x+ n⋅ n− n− k+ x−1 ⎤
⎣
⎦⎣
⎦
m
n! n!
= ∑ Ck
x+ x−
n− m+ k !( n−k)!
( )
n m k
( 1) ( 1)
− + n−k
…
m=n
n
1 d 1 n! n!
P ( x) = x − 1 = C x+
1 x 1
2 n! dx 2 n! k!( n−
k)!
n
2
( ) ∑
( ) ( − )
n n n n n k
k = 0
n
1
= ∑ n k
+ −
2
n
k = 0
2 k
( C ) ( x 1) ( x 1)
n k n−k
n−k
n
x [1,1]
1 1
( n−n)
P (1) = C 2 0 = C 2 0 = 1
n
2 k ( n−k)
2 n
n n∑( n k)
n
( n n)
2 k = 0 2
1
1
1 1
n
P ( − 1) = C 0 ( − 2) = C 0 − 2 = ( −1)
n
2 k ( n−k)
2 0
n n∑( n k)
n n 0
2 k = 0
2
1
( ) ( )
n