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ɢɥɢ
Ɍɚɤ ɤɚɤ
n 1cos −=π
=
2
b
n
()
. ɬɨ, ɩɨɥɚɝɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ
4
π
,0,
()
= cos1
n
n
π
4
==
3
π
y
1
n
π−
.
...,,3,2,1=n
ɧɚɣɞɟɦ
,0,
4
bbbbb
==
ɢ ɬ. ɞ.
54321
π
5
0
– 1
Ɋɢɫ. 8.7
x
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
()
sin
1
3
4
§
=
¨
π
©
1
3sin
5
·
. Ź
+++
...5sin
xxxxf
¸
¹
ɉɪɢɦɟɪ 8.4.
Ɋɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɸ
()
.
π<<π−= xxxf ,
Ɋɟɲɟɧɢɟ.
Ƚɪɚɮɢɤ ɷɬɨɣ ɮɭɧɤɰɢɢ, ɩɪɨɞɨɥɠɟɧɧɨɣ ɧɚ ɜɫɸ ɱɢɫɥɨɜɭɸ ɨɫɶ, ɫɨɫɬɨɢɬ ɢɡ ɩɚ-
ɪɚɥɥɟɥɶɧɵɯ ɨɬɪɟɡɤɨɜ (ɪɢɫ. 8.8). Ɏɭɧɤɰɢɹ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ ɬɟɨɪɟɦɵ 8.1.
Ɍɚɤ ɤɚɤ ɮɭɧɤɰɢɹ ɧɟɱɟɬɧɚ, ɬɨ ɧɚɯɨɞɢɦ ɬɨɥɶɤɨ
ππ
22coscos
b x nx dx dx
n
sin
==−+=
³³
ππ
00
ªº
xnx nx
«»
nn
«»
¬¼
(ɮɨɪɦɭɥɵ (8.9)). ɂɦɟɟɦ:
b
n
π
0
ªº
−
2cos sin 2cos
=+=−
«»
π
«»
¬¼
nnx n
ππ π
nn
π
2
n
0
.
131
y
x
0
ɉɨɥɚɝɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ
ɂɬɚɤ,
()
Ɋɢɫ. 8.8
...,,3,2,1=n
ɩɨɥɭɱɢɦ
§
¨
©
1
sin2
2
1
2sin
3
,
21=b
·
.
−+−=
...3sin
xxxxf
¸
¹
2
−=b
2
2
2
...,
=b
,
3
3
π<<π−
ɗɬɨ ɪɚɜɟɧɫɬɜɨ ɫɩɪɚɜɟɞɥɢɜɨ ɩɪɢ
ɪɚɜɧɚ ɧɭɥɸ.
Ɉɬɦɟɬɢɦ, ɱɬɨ ɜ ɢɧɬɟɪɜɚɥɟ
ɢ ɬɭ ɠɟ ɮɭɧɤɰɢɸ. Ɉɛɟ ɨɧɢ ɜ ɷɬɨɦ ɢɧɬɟɪɜɚɥɟ ɩɪɨɫɬɨ ɪɚɜɧɵ
ɜ ɢɧɬɟɪɜɚɥ
ɱɟɬɧɨ, ɬɨ ɢ ɢɯ ɪɚɡɥɨɠɟɧɢɹ ɜ ɪɹɞɵ Ɏɭɪɶɟ ɩɨɥɭɱɚɸɬɫɹ ɪɚɡɥɢɱɧɵɦɢ. Ź
()
ɨɧɢ ɩɪɨɞɨɥɠɟɧɵ ɩɨ-ɪɚɡɧɨɦɭ: ɩɟɪɜɚɹ – ɱɟɬɧɨ, ɚ ɜɬɨɪɚɹ – ɧɟ-
0,π−
ɮɭɧɤɰɢɢ ɩɪɢɦɟɪɨɜ 2 ɢ 4 ɩɪɟɞɫɬɚɜɥɹɸɬ ɨɞɧɭ
()
π,0
x
. ȼ ɬɨɱɤɚɯ
π±=x
ɫɭɦɦɚ ɪɹɞɚ
x
. Ɉɞɧɚɤɨ ɩɨɫɤɨɥɶɤɭ
8.3. Ɋɹɞ Ɏɭɪɶɟ ɜ ɩɪɨɢɡɜɨɥɶɧɨɦ ɢɧɬɟɪɜɚɥɟ
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɡɚɞɚɱɭ ɨ ɪɚɡɥɨɠɟɧɢɢ ɜ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ, ɡɚɞɚɧɧɨɣ ɜ
ɢɧɬɟɪɜɚɥɟ
ȿɫɥɢ ɜ ɢɧɬɟɪɜɚɥɟ
ɦɵ 8.1, ɬɨ ɪɚɡɥɨɠɟɧɢɟ ɦɨɠɟɬ ɛɵɬɶ ɥɟɝɤɨ ɩɨɥɭɱɟɧɨ ɩɪɢ ɩɨɦɨɳɢ ɡɚɦɟɧɵ ɧɟɡɚɜɢ-
ɫɢɦɨɣ ɩɟɪɟɦɟɧɧɨɣ ɩɨ ɮɨɪɦɭɥɟ
x
ɩɪɨɛɟɝɚɟɬ ɢɧɬɟɪɜɚɥ
Ɋɚɡɥɨɠɟɧɢɟ ɷɬɨɣ ɮɭɧɤɰɢɢ ɜ ɪɹɞ Ɏɭɪɶɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ:
, ɝɞɟ l – ɩɪɨɢɡɜɨɥɶɧɨɟ ɱɢɫɥɨ.
()
ll,−
()
ll,−
xπ=
, ɩɟɪɟɦɟɧɧɚɹ
()
ll,−
1
·
§
() ()
=
′
xfxf
¸
¨
π
¹
©
′
. Ɍɨɝɞɚ ɮɭɧɤɰɢɹ
x
l
∞
a
0
+=
¦
2
1
n
=
()
xf
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ ɬɟɨɪɟ-
1
§
·
′
¨
©
π
, ɢ ɤɨɝɞɚ
¸
¹
()
= xfxf
′
x
ɢɡɦɟɧɹɟɬɫɹ ɜ ɢɧɬɟɪɜɚɥɟ
′
+
′
sincos
xnbxna
nn
()
ππ− ,
.
.
132
ɢɥɢ
l
x
∞
()
xf
a
2
§
0
+=
a
¨
¦
©
=
1
n
π
xn
+
l
π
xn
·
sincos
b
nn
,
¸
l
¹
ɝɞɟ
π
11
·
§
′
=
n
π−
cos
¸
¨
ππ
¹
©
l
1
′′
xdxnxfa cos
()
=
xf
³³
l
−
l
π
xn
,
dx
l
π
11
·
§
′
=
n
π−
sin
¸
¨
ππ
¹
©
l
1
′′
xdxnxfb sin
()
=
xf
³³
l
−
l
xn
π
.
dx
l
1
(ɜ ɢɧɬɟɝɪɚɥɚɯ ɫɥɟɜɚ ɩɪɨɢɡɜɟɞɟɧɚ ɩɨɞɫɬɚɧɨɜɤɚ
Ɂɞɟɫɶ ɫɭɦɦɚ ɪɹɞɚ Ɏɭɪɶɟ ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɫ ɩɟɪɢɨɞɨɦ
′
).
xx =
π
T 2= .
ɑɚɫɬɨɬɵ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɝɚɪɦɨɧɢɤ, ɫɨɫɬɚɜɥɹɸɳɢɯ ɪɹɞ, ɪɚɜɧɵ
π
=ω
ɧɚɢɦɟɧɶɲɚɹ ɱɚɫɬɨɬɚ
.
1
l
ɉɪɢɦɟɪ 8.5.
2
()
Ɋɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɸ
xxf =
ɩɪɢ 11 <<−
.
Ɋɟɲɟɧɢɟ.
ɉɟɪɢɨɞ ɷɬɨɣ ɮɭɧɤɰɢɢ, ɩɪɨɞɨɥɠɟɧɧɨɣ ɧɚ ɜɫɸ ɱɢɫɥɨɜɭɸ ɨɫɶ, ɪɚɜɟɧ ɞɜɭɦ.
ɍɱɢɬɵɜɚɹ ɱɟɬɧɨɫɬɶ ɮɭɧɤɰɢɢ, ɢɦɟɟɦ
11
2,2cos
axdx axnxdx
== = =
0
³³
00
2
22
3
n
π
2
ªº
sin 2 cos 2sin 4
x nx x nx nx
2cos.
=+−=
πππ
«»
n
π
¬¼
22 22 22
nn n
ππ π
1
n
π
0
Ɉɬɫɸɞɚ
4
1
()
§
cos
−= ...3cos
¨
3
π
©
1
2
1
2cos
222
3
·
. Ź
−π+π−π
xxxxf
¸
¹
133
n
π
=ω
n
;
l
Ɋɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɣ, ɡɚɞɚɧɧɵɯ ɧɚ ɩɨɥɨɜɢɧɟ ɩɟɪɢɨɞɚ.
()
ɉɭɫɬɶ ɬɪɟɛɭɟɬɫɹ ɪɚɡɥɨɠɢɬɶ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɣ ɪɹɞ ɮɭɧɤɰɢɸ
ɬɟɪɜɚɥɟ
ɦɨɠɟɦ ɩɪɨɢɡɜɨɥɶɧɨ ɩɪɨɞɨɥɠɢɬɶ ɮɭɧɤɰɢɸ
ɱɬɨɛɵ ɨɛɪɚɡɨɜɚɜɲɚɹɫɹ ɜ ɷɬɨɦ ɢɧɬɟɪɜɚɥɟ ɮɭɧɤɰɢɹ
ɢɧɬɟɪɜɚɥɟ
ɜ ɪɹɞ Ɏɭɪɶɟ, ɩɨɥɭɱɢɦ ɢɫɤɨɦɵɣ ɪɹɞ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɣ ɮɭɧɤɰɢɸ
()
ɝɭɸ ɮɭɧɤɰɢɸ, ɩɨ ɫɭɳɟɫɬɜɭ ɧɟ ɫɜɹɡɚɧɧɭɸ ɫ ɞɚɧɧɨɣ ɮɭɧɤɰɢɟɣ
ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ
ɦɟɬɪɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ, ɬɨ
ɪɹɞ ɛɭɞɟɬ ɫɨɫɬɨɹɬɶ ɢɡ ɨɞɧɢɯ ɫɢɧɭɫɨɜ.
ɧɨɦɟɬɪɢɱɟɫɤɢɣ ɪɹɞ, ɬɨ ɜ ɢɧɬɟɪɜɚɥɟ
ɱɢɫɥɟɧɧɨɟ ɦɧɨɠɟɫɬɜɨ, Ɂɧɚɱɢɬ, ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ ɫɤɨɥɶɤɨ ɭɝɨɞɧɨ ɫɯɨɞɹɳɢɯɫɹ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɯ ɪɹɞɨɜ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɯ ɜ ɢɧɬɟɪɜɚɥɟ
ɰɢɸ, ɚ ɜ ɢɧɬɟɪɜɚɥɟ
()
π,0
()
xf
ɧɚ ɢɧɬɟɪɜɚɥ
()
xF
, ɫɨɜɩɚɞɚɸɳɚɹ ɫ
, ɭɞɨɜɥɟɬɜɨɪɹɥɚ ɭɫɥɨɜɢɹɦ ɬɟɨɪɟɦɵ 8.1. Ɋɚɡɥɨɠɢɜ ɮɭɧɤɰɢɸ
()
π,0
()
()
; ɧɟ ɢɦɟɟɬ ɡɧɚɱɟɧɢɹ, ɱɬɨ ɨɧ ɜ ɢɧɬɟɪɜɚɥɟ
π,0
()
ȼ ɱɚɫɬɧɨɫɬɢ,
()
xF
ɛɭɞɟɬ ɱɟɬɧɨɣ ɮɭɧɤɰɢɟɣ ɢ ɪɹɞ ɛɭɞɟɬ ɫɨɫɬɨɹɬɶ ɬɨɥɶɤɨ ɢɡ ɤɨɫɢɧɭɫɨɜ. ȿɫɥɢ ɠɟ
()
xf
ɩɪɨɞɨɥɠɢɬɶ ɧɟɱɟɬɧɨ ɧɚ ɢɧɬɟɪɜɚɥ
Ɇɵ ɩɪɢɯɨɞɢɦ ɤ ɜɵɜɨɞɭ, ɱɬɨ ɟɫɥɢ ɮɭɧɤɰɢɸ
xf
ɦɨɠɧɨ ɩɪɨɞɨɥɠɢɬɶ ɱɟɬɧɨ ɧɚ ɢɧɬɟɪɜɚɥ
()
xf
ɩɪɨɞɨɥɠɢɬɶ ɫɢɦɦɟɬɪɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ Oy. ɉɪɢ ɷɬɨɦ
()
()
ɉɪɢɦɟɪ 8.6.
ɉɨɫɬɪɨɢɬɶ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
ɫɚɦɵɟ ɪɚɡɧɨɨɛɪɚɡɧɵɟ ɮɭɧɤɰɢɢ.
0,π−
y
()
()
0,π−
ɬɚɤɢɯ ɟɟ ɪɚɡɥɨɠɟɧɢɣ ɫɭɳɟɫɬɜɭɟɬ ɛɟɫ-
π,0
()
ɩɪɟɞɫɬɚɜɥɹɟɬ ɤɚɤɭɸ-ɬɨ ɞɪɭ-
0,π−
()
xf
()
ɡɧɚɱɢɬ, ɝɪɚɮɢɤ ɩɪɨɞɨɥɠɢɬɶ ɫɢɦ-
()
xF
ɛɭɞɟɬ ɧɟɱɟɬɧɨɣ ɮɭɧɤɰɢɟɣ ɢ
()
xf
ɦɨɠɧɨ ɪɚɡɥɨɠɢɬɶ ɜ ɬɪɢɝɨ-
ɨɞɧɭ ɢ ɬɭ ɠɟ ɮɭɧɤ-
()
π,0
xxf sin=
(ɪɢɫ. 8.9).
xf
()
l,0
, ɧɨ ɬɚɤ,
0,π−
xf
ɜ ɢɧɬɟɪɜɚɥɟ
.
; ɡɧɚɱɢɬ,
0,π−
ɜ ɢɧ-
()
xf
()
xF
ɜ
0
Ɋɢɫ. 8.9
Ɋɟɲɟɧɢɟ.
ɗɬɚ ɮɭɧɤɰɢɹ ɩɨɥɭɱɚɟɬɫɹ ɩɪɢ «ɱɟɬɧɨɦ ɩɪɨɞɨɥɠɟɧɢɢ» ɮɭɧɤɰɢɢ
ɬɟɪɜɚɥɚ
ɜ ɢɧɬɟɪɜɚɥ
()
π,0
()
. ɉɨɷɬɨɦɭ
0,π−
π
2
=
0
³
π
0
sin
134
4
=
dxxa
π
x
.
ɢɡ ɢɧ-
xsin
2
=
n
π
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
Ɏɭɧɤɰɢɹ
ɫɱɟɬɚ:
ɉɨɷɬɨɦɭ
ππ
1
[]
cossin
sin
=
−
= ...2cos
π
()
= ...
cos xxx
π
() ()
³³
π
00
§
142
¨
¨
π
3
©
xxf cos=
ɩɨɥɭɱɚɟɬɫɹ ɢɡ ɩɪɟɞɵɞɭɳɟɣ ɩɟɪɟɧɨɫɨɦ ɧɚɱɚɥɚ ɨɬ-
ª
−
«
π
¬
+
= xx
π
1
2cos
15
142
§
−
2cos
¨
3
©
142
§
2cos
¨
3
π
©
1sin1sin
+++
xxx
...4cos
π
·
§
−= xx
sincos
π
2
·
¸
¹
15
¸
¨
2
¹
©
1
15
1
°
dxxnxndxnxxa
®
°
¯
1
2
()
n
−
.
§
4cos
¨
©
·
+−
¸
¹
ɧɟɱɟɬɧɨɟ, ɟɫɥɢ,0
n
−=−−+
4
2
()
−π
1
n
+
nx
12
π
º
·
−+
+
¸
»
2
¹
¼
Ź
....4cos
ɱɟɬɧɨɟ. ɟɫɥɢ,
n
·
¸
.
¸
¹
=
135
ɁȺȾȺɇɂə ȾɅə ɋȺɆɈɋɌɈəɌȿɅɖɇɈȽɈ Ɋȿɒȿɇɂə
8.1. ɇɚ ɨɬɪɟɡɤɟ
ɪɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ:
[]
ππ− ;
()
1)
()
3)
()
5)
()
7)
8.2. ɇɚ ɨɬɪɟɡɤɟ
()
1)
xf
()
3)
xf
()
xf
5)
8.3. ɇɚ ɨɬɪɟɡɤɟ
()
1)
()
3)
xxf sin=
2
xxf =
; 4)
π+= xxf
=
®
¯
=
®
¯
1
°
°
4
=
®
1
°
()
°
¯
4
xxf =
; 2)
xxf =
; 4)
; 2)
()
()
,0,
6)
;0,1
()
xf
()
xf
=
=
()
()
()
xf
()
xf
()
xf
®
¯
®
¯
; 6)
32 += xxf
; 8)
ɪɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ:
[]
ππ− ;
≤≤π−
,0,0
x
2)
π≤<
;0,1
x
,0,
xx
≤≤π−−
4)
;0,0
x
π≤<
x
≤≤π−
xx
π≤<−π
ɪɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ:
[]
ll;−
xxf =
;
3
xxf =
;
π−= xxf
;
xxf 32 −=
.
≤≤π−
,0,0
=
®
¯
=
®
¯
=
®
¯
x
π≤<
;0,
xx
,0,0
x
≤≤π−
;0,sin
xx
π≤<
≤≤π−
,0,0
x
2
xl
≤≤−
lx
≤<−
,0,0
xl
≤≤−
.,
lxx
≤<
π≤<
.0,
xx
,0,1
;,1
136
ɌȿɋɌɕ ȾɅə ɋȺɆɈɄɈɇɌɊɈɅə
x
x
ɂ ɋɂɋɌȿɆȺɌɂɁȺɐɂɂ ɁɇȺɇɂɃ
ȼɚɪɢɚɧɬ 1
Ɂɚɞɚɧɢɟ 1
ɋɪɟɞɢ ɩɪɢɜɟɞɟɧɧɵɯ ɭɬɜɟɪɠɞɟɧɢɣ ɜɟɪɧɵɦɢ ɹɜɥɹɸɬɫɹ:
Ⱥ: ɥɸɛɚɹ ɮɭɧɤɰɢɨɧɚɥɶɧɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɫɯɨɞɢɬɫɹ;
B: ɟɫɥɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɜ ɬɨɱɤɟ, ɬɨ ɨɧɚ ɫɯɨɞɢɬɫɹ ɢ ɧɚ ɦɧɨɠɟɫɬɜɟ, ɟɟ ɫɨɞɟɪɠɚɳɟɦ;
C: ɩɪɟɞɟɥɶɧɚɹ ɮɭɧɤɰɢɹ ɨɩɪɟɞɟɥɟɧɚ ɞɥɹ ɫɯɨɞɹɳɟɣɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ;
D: ɮɭɧɤɰɢɨɧɚɥɶɧɵɣ ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ, ɟɫɥɢ ɩɪɟɞɟɥ ɤɨɧɟɱɧɵɯ ɫɭɦɦ ɪɹɞɚ ɧɟ ɹɜ-
ɥɹɟɬɫɹ ɤɨɧɟɱɧɵɦ;
E: ɢɡ ɚɛɫɨɥɸɬɧɨɣ ɫɯɨɞɢɦɨɫɬɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ ɫɥɟɞɭɟɬ ɟɝɨ ɫɯɨɞɢ
ɦɨɫɬɶ
1) ɬɨɥɶɤɨ B, C, D, E;
2) ɬɨɥɶɤɨ C, D, E;
3) ɬɨɥɶɤɨ D, E;
4) ɬɨɥɶɤɨ B, C, E;
5) ɬɨɥɶɤɨ A, B, C.
Ɂɚɞɚɧɢɟ 2
ɉɭɫɬɶ ɡɚɞɚɧ ɪɹɞ
1) ɪɹɞ ɫɯɨɞɢɬɫɹ ɧɚ ɦɧɨɠɟɫɬɜɟ
2) ɫɭɦɦɚ ɪɹɞɚ ɨɩɪɟɞɟɥɟɧɚ ɧɚ ɜɫɟɣ ɱɢɫɥɨɜɨɣ ɩɪɹɦɨɣ;
3) ɩɪɟɞɟɥɶɧɚɹ ɮɭɧɤɰɢɹ ɪɚɜɧɚ
4) ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɪɚɜɟɧ ɧɭɥɸ;
5) ɩɪɢ
1=
Ɂɚɞɚɧɢɟ 3
ɉɭɫɬɶ ɡɚɞɚɧ ɪɹɞ
1) ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ ɜ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɟ
2) ɪɹɞ ɫɯɨɞɢɬɫɹ ɚɛɫɨɥɸɬɧɨ ɧɚ ɜɫɟɣ ɱɢɫɥɨɜɨɣ ɨɫɢ;
3) ɪɹɞ ɫɯɨɞɢɬɫɹ ɭɫɥɨɜɧɨ ɧɚ ɜɫɟɣ ɱɢɫɥɨɜɨɣ ɨɫɢ;
4) ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ ɜɫɸɞɭ;
5) ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɪɚɜɟɧ ɟɞɢɧɢɰɟ.
∞+
−
n
. ɍɤɚɠɢɬɟ ɜɟɪɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ:
x
¦
11n
=
ɪɹɞ ɫɯɨɞɢɬɫɹ.
∞+
n
()
sin1
¦
n
2
n
1
=
;
()
1;1−
1
, ɟɫɥɢ 1≠x;
−1
nx
. ɍɤɚɠɢɬɟ ɜɟɪɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ:
;
Rx ∈
0
-
137
Ɂɚɞɚɧɢɟ 4
Cɭɦɦɚ ɪɹɞɚ
∞+
()
¦
=
0
n
+
1
n
2
n
ɜ ɬɨɱɤɟ
⋅−
1
x
1
:
=x
0
2
1) 1;
2) 1− ;
3)
;
6,0
4)
5)
;
8,0−
.
4,0−
Ɂɚɞɚɧɢɟ 5
n
x
Ɋɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ
¦
n
ɪɚɜɟɧ:
()
+⋅ 33 n
1) 5;
2) 4;
3) 1;
4) 2;
5) 3.
Ɂɚɞɚɧɢɟ 6
432
2
xxxx
+−+−
4
Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ
1
1
§
1)
¨
©
ª
2)
«
¬
3)
[
4)
()
5)
º
−
−
;
;
»
2
2
¼
1
1
º
;
;
»
2
2
¼
;
)
2;2−
;
2;2−
.
20=x
() () ()
2
2
1
2
2
3
Ɂɚɞɚɧɢɟ 7
Ⱦɥɹ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ ɞɨɫɬɚɬɨɱɧɨ ɜɵɩɨɥɧɟɧɢɹ
ɭɫɥɨɜɢɹ:
1)
2) ɮɭɧɤɰɢɨɧɚɥɶɧɵɣ ɪɹɞ ɫɯɨɞɢɬɫɹ ɯɨɬɹ ɛɵ ɜ ɨɞɧɨɣ ɬɨɱɤɟ
3) ɪɹɞ
– ɧɟɩɪɟɪɵɜɧɚ ɧɚ
()
xf
n
∞+
′
ɫɯɨɞɢɬɫɹ ɧɟɪɚɜɧɨɦɟɪɧɨ ɧɚ
()
nf
n
¦
1n
=
ɞɥɹ ɥɸɛɨɝɨ n;
[]
ba;
;
[]
ba;
;
[]
ba;
4) ɮɭɧɤɰɢɨɧɚɥɶɧɵɣ ɪɹɞ ɞɨɥɠɟɧ ɫɯɨɞɢɬɶɫɹ ɜɫɸɞɭ;
5) ɮɭɧɤɰɢɨɧɚɥɶɧɵɣ ɪɹɞ ɧɟɥɶɡɹ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ.
:
...
138
Ɂɚɞɚɧɢɟ 8
x
x
x
x
x
ɋɪɟɞɢ ɩɪɢɜɟɞɟɧɧɵɯ ɭɬɜɟɪɠɞɟɧɢɣ, ɜɟɪɧɵɦɢ ɹɜɥɹɸɬɫɹ:
A: ɤɚɠɞɵɣ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɵɦ;
B: ɫɭɳɟɫɬɜɭɸɬ ɫɬɟɩɟɧɧɵɟ ɪɹɞɵ, ɤɨɬɨɪɵɟ ɧɟ ɫɯɨɞɹɬɫɹ ɧɢ ɜ ɨɞɧɨɣ ɬɨɱɤɟ;
C: ɟɫɥɢ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɫɯɨɞɢɬɫɹ ɜ ɬɨɱɤɟ
, ɬɨ ɨɧ ɫɯɨɞɢɬɫɹ ɜ ɬɨɱɤɟ
x
0
D: ɧɚ ɤɨɧɰɚɯ ɢɧɬɟɪɜɚɥɚ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɦɨɠɟɬ ɪɚɫɯɨɞɢɬɶɫɹ;
1) ɬɨɥɶɤɨ A ɢ C;
2) ɬɨɥɶɤɨ A ɢ B;
3) ɬɨɥɶɤɨ A ɢ D;
4) ɬɨɥɶɤɨ B ɢ D;
5) ɬɨɥɶɤɨ C ɢ D.
Ɂɚɞɚɧɢɟ 9
∞+
n
Ⱦɥɹ ɪɹɞɚ
1) ɪɹɞ ɫɯɨɞɢɬɫɹ ɜ
2) ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ ɜ ɬɨɱɤɟ
3) ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ ɩɪɢ
4) ɪɹɞ ɫɯɨɞɢɬɫɹ ɩɪɢ
5) ɪɹɞ ɫɯɨɞɢɬɫɹ ɩɪɢ
¦
n
=
x
ɜɟɪɧɵɦ ɭɬɜɟɪɠɞɟɧɢɟɦ ɹɜɥɹɟɬɫɹ:
()
+⋅11
nn
()
;
1=⋅ x
1=
;
2>
;
1
;
<x
2
2<x
.
Ɂɚɞɚɧɢɟ 10
∞+
Ⱦɥɹ ɪɹɞɚ
¦
n
=
1) ɪɹɞ ɫɯɨɞɢɬɫɹ ɜ ɬɨɱɤɟ
2) ɪɹɞ ɫɯɨɞɢɬɫɹ ɩɪɢ
3) ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ ɩɪɢ
4) ɪɹɞ ɫɯɨɞɢɬɫɹ ɩɪɢ
5) ɪɹɞ ɫɯɨɞɢɬɫɹ ɩɪɢ
1
1
§
¨
©
1
·
n
ɜɟɪɧɵɦ ɭɬɜɟɪɠɞɟɧɢɟɦ ɹɜɥɹɟɬɫɹ:
⋅
+
x
¸
n
n
3
¹
1−=
;
1=
;
2−>
;
1
;
<
x
2
1≤x
.
Ɂɚɞɚɧɢɟ 11
∞+
n
Ɋɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɞɥɹ ɪɹɞɚ
¦
nx
ɪɚɜɟɧ:
n
12n
=
1) 4;
2) 1;
3) 3;
4) 2;
5) 10.
139
xx >
;
0
Ɂɚɞɚɧɢɟ 12
+
12
n
()
+
4
xn
()
+
n
!1
ɪɚɜɟɧ:
Ɋɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɞɥɹ ɪɹɞɚ
3
¦
1) 1;
2) 2;
3) 10;
4) 20;
∞+
5)
Ɂɚɞɚɧɢɟ 13
ȿɫɥɢ ɮɭɧɤɰɢɹ
∞
() ( )
¦
n
10n
=
n
ɹɜɥɹɟɬɫɹ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɩɪɢ
−=
xxaxf
ɝɞɚ ɫɪɟɞɢ ɩɪɢɜɟɞɟɧɧɵɯ ɜɵɫɤɚɡɵɜɚɧɢɣ, ɜɟɪɧɵɦɢ ɹɜɥɹɸɬɫɹ:
′
()
xf
=
10=a
0
2
10
()
xf
!10
;
()
xf
0
!2
4
()
xf
0
!5
;
0
;
1)
a
2
2)
a = ;
10
3)
4)
a′=
1
5)
a = .
5
Ɂɚɞɚɧɢɟ 14
ɋɪɟɞɢ ɩɪɢɜɟɞɟɧɧɵɯ ɭɬɜɟɪɠɞɟɧɢɣ, ɜɟɪɧɵɦɢ ɹɜɥɹɸɬɫɹ:
A: ɫɬɟɩɟɧɧɵɟ ɪɹɞɵ ɦɨɠɧɨ ɛɟɫɤɨɧɟɱɧɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ;
B: ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ ɩɪɢ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɢ ɦɟɧɹɟɬɫɹ;
C: ɫɭɦɦɚ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ ɹɜɥɹɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɮɭɧɤɰɢɟɣ ɧɚ ɜɫɟɦ ɢɧɬɟɪɜɚɥɟ ɫɯɨɞɢɦɨɫɬɢ;
D: ɚɧɚɥɢɬɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɹɜɥɹɟɬɫɹ ɫɭɦɦɨɣ ɫɜɨɟɝɨ ɪɹɞɚ Ɍɟɣɥɨɪɚ.
1) ɬɨɥɶɤɨ A ɢ D;
2) ɬɨɥɶɤɨ A ɢ C;
3) ɬɨɥɶɤɨ C ɢ D;
4) ɬɨɥɶɤɨ B ɢ A;
5) ɬɨɥɶɤɨ A, C, D.
Ɂɚɞɚɧɢɟ 15
Ʉɚɤɢɟ ɭɫɥɨɜɢɹ ɹɜɥɹɸɬɫɹ ɧɟɨɛɯɨɞɢɦɵɦɢ ɢ ɞɨɫɬɚɬɨɱɧɵɦɢ ɞɥɹ ɫɯɨɞɢɦɨɫɬɢ
ɪɹɞɚ Ɍɟɣɥɨɪɚ:
A:
∞→nn
, ɝɞɟ
0lim =
R
– ɨɫɬɚɬɨɱɧɵɣ ɱɥɟɧ ɜ ɮɨɪɦɟ Ʌɚɝɪɚɧɠɚ;
R
n
, ɬɨ-
xx =
0
140
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