2015¥‰ŒêÆÛÁK †) ‰
morrismodel January 11, 2015
1. ¦
x →+∞sin x lim 1
x |sin t |dt. 0
) .
1 x x →+∞x t |dt =lim 1 x
lim sin |sin |sin t |dt. 0x →+∞x 0
nπ
2n (n +1) π=1
nπ
(n +1) π|sin t |dt
x
≤
1 x |sin t |dt 0
≤1 (n +1)πnπ|sin t |dt 0=2(n +1) nπ
.
¤±
x →lim +∞sin 1 x x |sin t |dt =2
. 0
π
1
2. O Ž -È
(x 2+y 2) dxdy,
D
Ù¥D =
x 2y 2
(x, y ) :+≤1.
a b
) . 4‹I “†:
x =ar cos θ,y =br sin θ,
K k
D
0≤r ≤1, 0≤θ≤2π.
(x +y ) dxdy =
22
1
abrdr
2π
(a 2r 2cos 2θ+b 2r 2sin 2θ) dθ
π
=ab (a 2+b 2)
. 4
3. ¦
f (x, y ) =x y
3+1+y
x +y ≤1x, y ≥0
þ •ŒŠ.
) . N ´•
g (x ) =
f (x, y ) ≤x
3[0, 1]þ4O , x 1−x
=:h (x ) . +3
3
h (x ) =(1+x 2) −−(1+(1−x ) 2) −.
11
0
112
f (, ) =
. 222
4. ¦
1
(xy 2+x 3) dydz +yz 2dzdx +R 3dxdy,
S
3Ù¥S •x 2+y 2+z 2=R 2, z ≥0, •••þ.
) . E -¡
T •x 2+y 2≤R 2, z =0, •••e. d Gauss úª:
(xy 2+1
x 3) dydz +yz 2dzdx +R 3S 3dxdy
+
(xy 2+1
x 3) dydz +yz 2dzdx +R 3dxdy =
T 3(y 2+x 2+z 2) dxdydz {x 2+y 2+z 2≤R 2,z>0}=
1
2(x 2+y 2+z 2) dxdydz {x 2+y 2+z 2≤R 2}=2π55
R .
(xy 2+1
x 3) dydz +yz 2dzdx +R 3dxdy =−πR5T
3.
¤±
(xy 2+17π5
S
3x 3) dydz +yz 2dzdx +R 3dxdy =5R .
5. ¦
∞
1
1+x dx, n >1. 0
) . ·y =
1,
K k ∞
1 0
111−10
1+x dx =−y (−1) 1dy =1
1
n y y 1
y −1−y ) 1(1−1n dy 0=1n B (1n , 1−1n ) =πn sin . 3
6. y ²:
1
tan n xdx
1
, n >0. 2n +2
2n
y . ·y =tan x , K
tan n
xdx =
1
y n
dy
1
y n 0
1+y
2n
. , ˜•¡,
1
y n
1
y n 0
1+y 2dy >02dy =12n +2
. 7. ? Ø? ê
∞(√−√) α
cos n n =1
^‡ÂñÚýéÂñ5.
) . α≤0ž, ϑتu 0, l ØÂñ;
α>2ž,
|(√−√) αcos n |=|cos n |−α(+)
α≤n .
l ? êýéÂñ;
0
k cos n n =1
sin 1−sin 2k +1 1 =2sin ≤, sin (√−√) α=(1
+) α
üN4~ª•u 0, 4
l ? êÂñ. ´
k k (√−√α|cos n |≥ (√−√) αcos 2n n =1
n =1
k =
1
(√2
−√α
n =1
−1 k
√2
(√
−) αcos 2n. n =1
´•
∞(√−√
) αcos 2n Âñ, ∞(√−√
) αu Ñ, n =1
n =1
l
∞(√−√
) α|cos n |u Ñ. n =1
¤± 0
.
8. α∈(0, 1), {a n }•î‚4O ‘ê , …{a n +1−a n }k . , ¦
n lim →∞
a αn +1−a αn .
) . X J {a n }Ã. , |a n +1−a n |
a αn +1−a αn =α(a n +1−a n ) ξαn −1≤αMaαn
−1
→0. X J {a n }k . , ´•a n +1−a n →0. K
a αn +1−a αn =α(a n +1−a n ) ξαn −1≤αaα1
−1(a n +1−a n ) →0. ¤±o k
lim a αn +1−a α
n n →∞
=0
.
5
9. f ∈C [0, 1],0≤f (x ) ≤x , ¦y :
1
1
2
x 2f (x ) dx ≥
f (x ) dx
.
¿¦Ñ Ò¤áž, ¤k ëY ¼êf (x ). y . ·
g (x ) =
x
t 2 x
2
f (t ) dt −
f (t ) dt .
K
g
(x ) =x 2
f (x ) −2f (x )
x
f (t ) dt
f (x ) x 2−2 x
=f (t ) dt
x
≥f (x ) x 2−2tdt
=0.
l
1
2x 2f (x ) dx −
1
f (x ) dx
00
=g (1)−g (0)=
1
g (x ) dx ≥0.
0 Ò¤á …= é? Ûx ∈ (0, 1), x
f (x ) x 2−2
f (t ) dt
=0.
X J f ≡0, K þªw , ¤á. ÄK , •3x 0∈(0, x 2 1), f (x 0) =0. K
x 0
0−2
f (t ) dt ⇒f (t ) =t, ∀t ∈(0, x 0) . 0
½Â
t 0=inf {s ∈[0, 1]:f (t ) >0, ∀t ∈(0, s ) }.
X J t 0
f (t 0) =0…f (t ) =t, ∀t ∈(0, t 0) ,
ù†f 3t 0? ëY g ñ. t 0=1…f (t ) =t, ∀t ∈(0, 1).
n þ, Ò¤á …= f (x ) =0½öf (x ) =x
.
6
10. f (x ) 3[a, +∞) þëY Œ‡,
lim sup |f (x ) +f (x ) |≤M
x →+∞
¦y :
lim x →+sup ∞
|f (x ) |≤M.
y . ? ¿ε>0, •3A >a , x >A ž,
−M −ε
d Cauchy ¥Š½n , •3ξx ∈[A, x ]¦
f (x ) e x −f (A ) e A (f (ξx ) +f (ξx )) e ξx
e −e =
e x
=f (ξx ) +f (ξx ) ∈[−M −ε,M +ε].
lim f (x →+sup ∞f (x ) =lim x →+sup x ) e x
∞e −e
=lim f (x ) e x −f (A ) e A
x →+sup
∞e x −e A ≤M +ε.
Ón ,
lim x →+inf ∞
f (x ) ≥−M −ε.
=
lim x →+sup ∞
(−f (x )) ≤M +ε.
l
lim x →+sup ∞
|f (x ) |≤M +ε.
2d ε>0 ? ¿5Œ•:
lim x →+sup ∞
|f (x ) |≤
M.
7
2015¥‰ŒêÆÛÁK †) ‰
morrismodel January 11, 2015
1. ¦
x →+∞sin x lim 1
x |sin t |dt. 0
) .
1 x x →+∞x t |dt =lim 1 x
lim sin |sin |sin t |dt. 0x →+∞x 0
nπ
2n (n +1) π=1
nπ
(n +1) π|sin t |dt
x
≤
1 x |sin t |dt 0
≤1 (n +1)πnπ|sin t |dt 0=2(n +1) nπ
.
¤±
x →lim +∞sin 1 x x |sin t |dt =2
. 0
π
1
2. O Ž -È
(x 2+y 2) dxdy,
D
Ù¥D =
x 2y 2
(x, y ) :+≤1.
a b
) . 4‹I “†:
x =ar cos θ,y =br sin θ,
K k
D
0≤r ≤1, 0≤θ≤2π.
(x +y ) dxdy =
22
1
abrdr
2π
(a 2r 2cos 2θ+b 2r 2sin 2θ) dθ
π
=ab (a 2+b 2)
. 4
3. ¦
f (x, y ) =x y
3+1+y
x +y ≤1x, y ≥0
þ •ŒŠ.
) . N ´•
g (x ) =
f (x, y ) ≤x
3[0, 1]þ4O , x 1−x
=:h (x ) . +3
3
h (x ) =(1+x 2) −−(1+(1−x ) 2) −.
11
0
112
f (, ) =
. 222
4. ¦
1
(xy 2+x 3) dydz +yz 2dzdx +R 3dxdy,
S
3Ù¥S •x 2+y 2+z 2=R 2, z ≥0, •••þ.
) . E -¡
T •x 2+y 2≤R 2, z =0, •••e. d Gauss úª:
(xy 2+1
x 3) dydz +yz 2dzdx +R 3S 3dxdy
+
(xy 2+1
x 3) dydz +yz 2dzdx +R 3dxdy =
T 3(y 2+x 2+z 2) dxdydz {x 2+y 2+z 2≤R 2,z>0}=
1
2(x 2+y 2+z 2) dxdydz {x 2+y 2+z 2≤R 2}=2π55
R .
(xy 2+1
x 3) dydz +yz 2dzdx +R 3dxdy =−πR5T
3.
¤±
(xy 2+17π5
S
3x 3) dydz +yz 2dzdx +R 3dxdy =5R .
5. ¦
∞
1
1+x dx, n >1. 0
) . ·y =
1,
K k ∞
1 0
111−10
1+x dx =−y (−1) 1dy =1
1
n y y 1
y −1−y ) 1(1−1n dy 0=1n B (1n , 1−1n ) =πn sin . 3
6. y ²:
1
tan n xdx
1
, n >0. 2n +2
2n
y . ·y =tan x , K
tan n
xdx =
1
y n
dy
1
y n 0
1+y
2n
. , ˜•¡,
1
y n
1
y n 0
1+y 2dy >02dy =12n +2
. 7. ? Ø? ê
∞(√−√) α
cos n n =1
^‡ÂñÚýéÂñ5.
) . α≤0ž, ϑتu 0, l ØÂñ;
α>2ž,
|(√−√) αcos n |=|cos n |−α(+)
α≤n .
l ? êýéÂñ;
0
k cos n n =1
sin 1−sin 2k +1 1 =2sin ≤, sin (√−√) α=(1
+) α
üN4~ª•u 0, 4
l ? êÂñ. ´
k k (√−√α|cos n |≥ (√−√) αcos 2n n =1
n =1
k =
1
(√2
−√α
n =1
−1 k
√2
(√
−) αcos 2n. n =1
´•
∞(√−√
) αcos 2n Âñ, ∞(√−√
) αu Ñ, n =1
n =1
l
∞(√−√
) α|cos n |u Ñ. n =1
¤± 0
.
8. α∈(0, 1), {a n }•î‚4O ‘ê , …{a n +1−a n }k . , ¦
n lim →∞
a αn +1−a αn .
) . X J {a n }Ã. , |a n +1−a n |
a αn +1−a αn =α(a n +1−a n ) ξαn −1≤αMaαn
−1
→0. X J {a n }k . , ´•a n +1−a n →0. K
a αn +1−a αn =α(a n +1−a n ) ξαn −1≤αaα1
−1(a n +1−a n ) →0. ¤±o k
lim a αn +1−a α
n n →∞
=0
.
5
9. f ∈C [0, 1],0≤f (x ) ≤x , ¦y :
1
1
2
x 2f (x ) dx ≥
f (x ) dx
.
¿¦Ñ Ò¤áž, ¤k ëY ¼êf (x ). y . ·
g (x ) =
x
t 2 x
2
f (t ) dt −
f (t ) dt .
K
g
(x ) =x 2
f (x ) −2f (x )
x
f (t ) dt
f (x ) x 2−2 x
=f (t ) dt
x
≥f (x ) x 2−2tdt
=0.
l
1
2x 2f (x ) dx −
1
f (x ) dx
00
=g (1)−g (0)=
1
g (x ) dx ≥0.
0 Ò¤á …= é? Ûx ∈ (0, 1), x
f (x ) x 2−2
f (t ) dt
=0.
X J f ≡0, K þªw , ¤á. ÄK , •3x 0∈(0, x 2 1), f (x 0) =0. K
x 0
0−2
f (t ) dt ⇒f (t ) =t, ∀t ∈(0, x 0) . 0
½Â
t 0=inf {s ∈[0, 1]:f (t ) >0, ∀t ∈(0, s ) }.
X J t 0
f (t 0) =0…f (t ) =t, ∀t ∈(0, t 0) ,
ù†f 3t 0? ëY g ñ. t 0=1…f (t ) =t, ∀t ∈(0, 1).
n þ, Ò¤á …= f (x ) =0½öf (x ) =x
.
6
10. f (x ) 3[a, +∞) þëY Œ‡,
lim sup |f (x ) +f (x ) |≤M
x →+∞
¦y :
lim x →+sup ∞
|f (x ) |≤M.
y . ? ¿ε>0, •3A >a , x >A ž,
−M −ε
d Cauchy ¥Š½n , •3ξx ∈[A, x ]¦
f (x ) e x −f (A ) e A (f (ξx ) +f (ξx )) e ξx
e −e =
e x
=f (ξx ) +f (ξx ) ∈[−M −ε,M +ε].
lim f (x →+sup ∞f (x ) =lim x →+sup x ) e x
∞e −e
=lim f (x ) e x −f (A ) e A
x →+sup
∞e x −e A ≤M +ε.
Ón ,
lim x →+inf ∞
f (x ) ≥−M −ε.
=
lim x →+sup ∞
(−f (x )) ≤M +ε.
l
lim x →+sup ∞
|f (x ) |≤M +ε.
2d ε>0 ? ¿5Œ•:
lim x →+sup ∞
|f (x ) |≤
M.
7