微積分学 II 演習問題
II
II 1 2
1
II 2
6
II 3
10
II 4
16
II 5
19
II 6 2
23
II 7
28
II 8
33
II 9
40
II 10
42
II 11
47
II 12 3
52
II 13
57
II 14
63
II 15
72
II 1 2
1. , .
cos(xy)
(1) lim
(
x y
)(
2 1
)
1 + 2xy
x3 - 3xy
(4)
(
x y
lim )(
0 0
)
x2
+
y2
x4 + 2y4
(7) lim
(
x y
)(
0 0
)
x2 + y2
xy
(10)
lim
(
x y
)(
0 0
)
x4
+
y2
ex2+y2 - 1
(13) lim
(
x y
)(
0 0
)
x2 + y2
x2y2
(16)
lim
(
x y
)(
0 0
)
x2
+
y4
ey sin(xy)
(2) lim
(
x y
)(
0 0
)
xy
2x3 - y3
(5)
lim
(
x y
)(
0 0
)
4x2
+
y2
x3 + y4
(8)
lim
(
x y
)(
0 0
)
x2
+
4y2
xy3
(11)
(
x y
lim )(
0 0
)
x2
+
y4
sin xy
(14)
(
x y
lim )(
0 0
)
x2
+
y2
(17) lim xy
(
x y
)(
0 0
)
x2 + y2
x2 - y2
(3)
lim
(
x y
)(
0 0
)
x2
+
y2
xy2
(6)
lim
(
x y
)(
0 0
)
x2
+
y4
x2 - 2y2
(9)
lim
(
x y
)(
0 0
)
3x2
+
y2
(12) lim xy sin 1
(
x y
)(
0 0
)
x2 + y2
1 - cos x2 + y2
(15) lim
(
x y
)(
0 0
)
x2 + y2
(18) lim (x2 + y2) log(x2 + y2)
(
x y
)(
0 0
)
2. (n) (n = 3, 4, . . . , 18) , R2 fn . (
,
f3
f3
(
x y
)
=
x2 - y2 x2 + y2
.)
f?n
: R2
R
f?n
(
x y
)
=
fn 0
(
x y
)
(
x y
)
=
(
0 0
)
(
x y
)
=
(
0 0
)
, n = 3, 4, . . . , 18 , f?n .
3. () (1) R2 f?2
ey sin(xy)
f?2
(
x y
)
=
ey
xy
xy = 0 xy = 0
,
{ (
x y
)
R2
xy
=
} 0
f?2
.
(2) R2 f?1
cos(xy)
1 xy = -
f?1
(
x y
)
=
1
+
2xy
2 1 xy = -
2
2
,
{ (
x y
)
R2
xy
=
-
1 2
}
f?1
.
1
1
1.
(1)
f
:
{
(
x y
)
R2
xy
=
}
-
1 2
( -,
)
-
1 2
(
-
1 2
,
) ,
g
:
( -,
)
-
1 2
(
-
1 2
,
)
R
(f
(
x y
)
=
xy,
g()t)
=
cos(t) 1 + 2t
,
f,
g
,
lim
(
x y
)(
2 1
)
cos(xy) 1 + 2xy
=
lim g(f
(
x y
)(
2 1
)
(
x y
))
=
g
(
x y
lim )(
2 1
)
f
(
x y
)
=
g(f
(
2 1
))
=
g(2)
=
1 .
5
sin t
(2)
f
:
R2
R,
g
:
R
R
f
(
x y
)
=
xy,
g(t)
=
t
1
t = 0 , f t=0
, lim g(t) = lim sin t
t0
t0 t
(
)
= 1 = g(0) , g 0 .
lim sin(xy)
(
x y
)(
0 0
)
xy
=
lim g(f
(
x y
)(
0 0
)
(
x y
))
=
g
(
x y
lim )(
0 0
)
f
(
x y
)
= g(0) = 1 ,
lim
ey sin(xy) =
lim
ey
lim
sin(xy) = 1.
(
x y
)(
0 0
)
xy
(
x y
)(
0 0
)
(
x y
)(
0 0
)
xy
(3)
x2 - y2
(
x y
lim )(
0 0
)
x2
+
y2
,
c
,
f
: R2
R
x2-y2
f
(
x y
)
=
x2 +y 2
c
(
x y
)
=
(
0 0
)
(
x y
)
=
(
0 0
)
, f
(
0 0
)
.
,
lim g(t)
t0
=
(
0 0
)
g
:
R
R2
,
lim f (g(t))
t0
=
f
(
0 0
)
=
c
.
,
k
,
g
g(t)
=
(t, kt)
,
t2 - k2t2
lim
t0
f
(g(t))
=
lim
t0
t2
+
k2t2
=
1 - k2 1 + k2
,
lim f (g(t))
t0
lim
t0
g(t)
=
(
0 0
)
g
c
.
,
x2 - y2
(
x y
lim )(
0 0
)
x2
+
y2
.
(4)
x3 - 3xy
lim
(
x y
)(
0 0
)
x2 + y2
,
c
,
f
: R2
R
x3-3xy
f
(
x y
)
=
x2 +y 2
c
(
x y
)
=
(
0 0
)
(
x y
)
=
(
0 0
)
, f
(
0 0
)
.
,
lim g(t)
t0
=
(
0 0
)
g
:
R
R2
,
lim f (g(t))
t0
=
f
(
0 0
)
=
c
.
,
k
,
g
g(t)
=
(t, kt)
,
lim f (g(t))
t0
=
lim
t0
t3 - 3kt2 t2 + k2t2
t - 3k
=
lim
t0
1
+
k2
=
-3k 1 + k2
,
lim f (g(t))
t0
lim
t0
g(t)
=
(
0 0
)
g
c
.
,
(
x y
lim )(
0 0
)
x3 x2
- +
xy y2
.
(5) x2 4x2 + y2, y2 4x2 + y2 , |2x3| 2|x|(4x2 + y2), |y3| |y|(4x2 + y2) . |2x3 - y3|
|2x3|
+
|y3|
(2|x|
+
|y|)(4x2
+
y2)
,
(
x y
)
=
(
0 0
)
2x3 - y3 4x2 + y2
2|x| + |y| . ,
(
x y
)
(
0 0
)
,
2|x|
+ |y|
0
,
,
lim
(
x y
)(
0 0
)
2x3 4x2
- y3 + y2
=
0
.
(6)
xy2
(
x y
lim )(
0 0
)
x2
+
y4
,
c
,
f
: R2
R
xy2
f
(
x y
)
=
x2 +y 4
c
(
x y
)
=
(
0 0
)
(
x y
)
=
(
0 0
)
, f
(
0 0
)
.
,
lim g(t)
t0
=
(
0 0
)
g
:
R
R2
lim f (g(t))
t0
=
f
(
0 0
)
=
c
.
,
k
,
g
g(t)
=
(kt2, t)
,
lim f (g(t))
t0
=
kt4
lim
t0
k2t4
+
t4
=
k k2 + 1
,
lim f (g(t))
t0
lim g(t) =
t0
(
0 0
)
g
c
.
,
xy2
(
x y
lim )(
0 0
)
x2
+
y4
.
2
(7) lim
(
x y
)( 00
)
x4 + 2y4 x2 + y2
,
c
,
f
: R2
R
x4+2y4
f
(
x y
)
=
c
x2 +y 2
(
x y
)
=
(
0 0
)
(
x y
)
=
(
0 0
)
,
f
(
0 0
)
.
,
lim g(t)
t0
=
(
0 0
)
g
:
R
R2
,
lim f (g(t)) t0
=
f
(
0 0
)
=c
.
,
k
,
g
g(t)
=
(t, kt)
, lim f (g(t)) = lim
t0
t0
t4 + 2k4t4 t2 + k2t2 =
1 + 2k4 1 + k2
, lim f (g(t)) t0
lim g(t) =
t0
(
0 0
)
g
c . , lim
(
x y
)(
0 0
)
x4 + 2y4 x2 + y2
.
(8) x2 x2 + 4y2, y2 x2 + 4y2 , |x3| |x|(x2 + 4y2), |y4| y2(x2 + 4y2) . |x3 + y4|
|x3|
+ |y4 |
(|x| + y2 )(x2
+ 4y2 )
,
(
x y
)
=
(
0 0
)
x3 + y4 x2 + 4y2
|x| + y2
.
,
(
x y
)
(
0 0
)
,
|x| + y2
0
,
,
lim
(
x y
)(
0 0
)
x3 + y4 x2 + 4y2
=
0
.
(9)
x2 - 2y2
(
x y
lim )(
0 0
)
3x2
+
y2
,
c
,
f
: R2
R
x2-2y2
f
(
x y
)
=
3x2 +y 2
c
(
x y
)
=
(
0 0
)
(
x y
)
=
(
0 0
)
,
g
(
0 0
)
.
,
lim g(t)
t0
=
(
0 0
)
g
:
R
R2
,
lim f (g(t))
t0
=
f
(
0 0
)
=
c
.
,
k
,
g
g(t)
=
(t, kt)
,
t2 - 2k2t2
lim
t0
f (g(t))
=
lim
t0
3t2
+
k2t2
=
1 - 2k2 3 + k2
,
lim f (g(t))
t0
lim
t0
g(t)
=
(
0 0
)
g
(10)
(cxy)lim( 00)x4x+yy2 .,(,xy)lim( 00)3xx2 2c-+2yy22 , f: R2.
R
xy
f
(
x y
)
=
x4 +y 2
c
(
x y
)
=
(
0 0
)
(
x y
)
=
(
0 0
)
, f
(
0 0
)
.
,
lim g(t)
t0
=
(
0 0
)
g
:
R
R2
,
lim f (g(t))
t0
=
f
(
0 0
)
=
c
.
,
k
,
g
g(t)
=
(t, kt3)
, lim
t0
kt4
k
f (g(t)) =
lim
t0
t4
+ k2t6
=
lim
t0
1 + k2t2
c . ,
= k , lim f (g(t)) lim g(t)
xty0
(
x y
lim )(
0 0
)
x4
+
y2
t0
.
=
(
0 0
)
g
(11)
()
()
x2 + y4 2
x2y4
=
|x|y2.
(
x y
)
=
(
0 0
)
|x|y2 x2 + y4
1 2
,
2xy3 x2 + y4
|y| 2
.
,
(
x y
)
(
0 0
)
,
|y|
0
,
,
lim
(
x y
)(
0 0
)
xy3 x2 + y4
=
0
.
(12)
sin 1 x2 + y2
1 xy sin 1 x2 + y2
|xy|
.
,
(
x y
)
(
0 0
)
,
|xy|
0
,
, lim xy sin 1
= 0 .
(
x y
)(
0 0
)
x2 + y2
et-1
(13)
f
:
R2
R,
g
:
R
R
f
(
x y
)
=
x2
+
y2,
g(t)
=
1
t
t = 0 , f t=0
,
lim g(t)
t0
=
lim
t0
et
-1 t
=
1
=
g(0)
,
g
0
.
lim
(
x y
)(
0 0
)
ex2+y2 - 1 x2 + y2
=
lim g(f
(
x y
)(
0 0
)
(
x y
))
=
3
................
................
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