4章の演習問題（全問）

4
4
x
24xy = x + 48 ⇒ xy = + 2
24
4 3 1 3
dx ⋅ y + x ⋅ dy = x dx = x dx
24 6
⎛1
3 ⎞
x⋅ dy = ⎜ x − y⎟dx
⎝6
⎠
1 3 1 3
x − y x − y
dy
= 6 = 6
dx x x
4
4
x + 48
24xy = x + 48 ⇒ y =
24x
3 4
1 3 x x + 48
x − y −
2 2 2 4
dy 6 6 24x
x x 2 x 2 x −16
= = = − − = − =
2 2 2
dx x x 6 24 x 8 x 8x
⎛dy
⎞
ds = 1+ ⎜ ⎟
⎝dx
⎠
2 4
2
4 8 4
2
⎛ x − 16 ⎞ 64x + x −32x
+ 16
dx = 1+ ⎜ ⎟ dx =
dx
2
⎝ x
2
8 ⎠
8
2
( x )
8 4 2 4
x + 32x + 16 x + 16 1 ⎛ 2 16 ⎞
= dx = dx = x dx
2 2 ⎜ +
2 ⎟
8x 8x 8⎝
x ⎠
x :1→
3
[9]
1
P0 =
0 1 = 1
20!
1 d 2 1
P1 = ( x − 1) = 2x=
x
1
21! dx 2
2
1 d 2 2 1 d 2 1 d 3 1 2
P2 = ( x − 1) =
2 2
( 2( x −1) ⋅ 2 x) = ( x − x)
= (3x −1)
22! dx 8dx 2dx
2
1 3 1 4 2 1 5
P3 = (5x − 3 x), P4 = (35x − 30x + 3), P5
= (63x − 70x
3 + 15 x )
2 8 8
(
0
x y 1
)