Homework_06Ans 問1 つぎの関数はいかなる区間で 1 対 1 か． (ⅰ) f

Homework_06Ans
)B <1&; 1 , 1 A
() f (x) = x 3 3x 2
() f (x) =
x +1
x2 +1
)C 5.4A
() sin1 x = cos1 1 x 2
1
x
() cot 1 x = tan1
(0 x 1)
( x > 0)
)D <1/$A
() sin1 ( x 2 )
( x < 1)
x 2 1
() cos 2 x + 1
1
() tan1 x + tan1
1
x
================
>96#?
)B
/$@+28"0@':+%':2-7*:
A':+%':2-&;!,!A
9(3A
)C
>i? sin 1 x = cos 1 1 x 2 , 0 x 1
y = sin 1 x A@
0 x 1, 0 y 2
x = sin y sin 2 y + cos 2 y = x 2 + cos 2 y = 1
cos y = 1 x 2
y = cos 1 1 x 2
y = sin 1 x = cos 1 1 x 2
f (x) = sin 1 x cos 1 1 x 2 d 1
d
f (x) =
sin x cos 1 1 x 2 dx
dx
1
1
d
1 x2
=
2
2 dx
1 x
1 1 x2
(
=
=
1
1 x
2
1
1 x
2
)
1
(
1 1 x
1
x
x
1 x
2
2
)
=
1 2x
2 1 x2
1
1 x
2
1
x
x 1 x2
=0
f (x) = const.
f (0) = sin 1 0 cos 1 1 0 2 = sin 1 0 cos 1 1 = 0
f (x) = sin 1 x cos 1 1 x 2 = 0
sin 1 x = cos 1 1 x 2
1
ii cot 1 x = tan 1 , x > 0
x
y = cot 1 x x = cot y =
1
tan y
tan y =
1
x
y = tan 1
1
x
y = cot 1 x = tan 1
1
x
f (x) = cot 1 x tan 1
1
x
y = cot 1 x x = cot y =
1
1
1
= tan y =
y = tan 1
tan y
x
x
dy
1
d
cot 1 x =
=
dx
dx
dx
dy
1
1
1
=
=
=
d 1
d cos y ( sin y)sin y cos y(cos y)
dy tan y dx sin y
sin 2 y
1
1
1
1
=
=
=
=
2
2
1
cos y
1 + cot y
1 + x2
1
+
sin 2 y
sin 2 y
d 1
1
d
1
d 1
1
f (x) =
cot x tan 1 = 2
dx dx
x
1 + x 1 + ( 1 )2 dx x
x
1
1
1
=
2
1 + x 1 + ( 1 )2 x 2
x
1
1
=
+ 2
=0
2
1+ x
x +1
+ f (x) = const.
f (1) = cot 1 1 tan 1 1 =
=0
4 4
+ f (x) = cot 1 x tan 1
1
=0
x
cot 1 x = tan 1
1
x
'(
(
"%!,
$&#,
f (x) = sin 1 x
)i* y = f (g(x)) = ( f g)(x) = sin 1 (x 2 ) 2
g(x) = x
dy df dg
1
1
2x
dg
=
=
=
( 2x ) =
2
2
2
dx dg dx
1 g dx
1 (x )
1 x4
f (x) = cos 1 x
x 1
) )ii* y = f (g(x)) = ( f g)(x) = cos 1 ( 2
x2 1
x +1
g(x)
=
x2 + 1
2
dy df dg
1 dg
=
=
=
dx dg dx
1 g 2 dx
2x(x 2 + 1) (x 2 1)2x 2
= 2
2
2
2
(x + 1)
x +1
x 2 1 1 2
x + 1 1
iii y = tan 1 x + tan 1
1
x
dy 1 1
1
1 1 =
+
tan x + tan
=
dx dx x 1 + x2
1
1
1 x2
1 + ( )2
x
1
1
=
+ 2
2
1+ x
x +1
=0
=
1
+
1 + x2
d 1
1
1 dx x
1 + ( )2
x