Matan_Lectures_2013


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Ìàòåìàòè÷åñêèéàíàëèç.Ëåêöèè.
Ä.Â.ÀëåêñååâÎ.Ï.Âèíîãðàäîâ
23àïðåëÿ2013ã.
2
Îãëàâëåíèå
1Ïðîãðàììàêóðñà7
IIñåìåñòð9
2Ìíîæåñòâà11
2.1Îïåðàöèèíàäìíîæåñòâàìè...................
12
2.2×èñëîâûåìíîæåñòâà.......................
13
3Ìåòîäìàòåìàòè÷åñêîéèíäóêöèè.15
3.1¾Õàíîéñêèåáàøíè¿........................
15
3.2Äðóãèåïðèìåíåíèÿìåòîäàìàòåìàòè÷åñêîéèíäóêöèè...
16
3.2.1ÒðåóãîëüíèêÏàñêàëÿ...................
17
3.3ÁèíîìÍüþòîíà...........................
19
4Ìíîæåñòâà.Ìîùíîñòü.×èñëîâûåìíîæåñòâà21
4.1Ìîùíîñòüìíîæåñòâà.Ñ÷åòíûåèíåñ÷åòíûåìíîæåñòâà...
21
4.2×èñëîâûåìíîæåñòâà......................
25
4.3Îêðåñòíîñòè.............................
25
4.4Àêñèîìàîòäåëèìîñòè.......................
26
4.5Âëîæåííûåèñòÿãèâàþùèåñÿñèñòåìûîòðåçêîâ........
28
4.6Îòêðûòûåèçàìêíóòûåìíîæåñòâà................
28
5Ïîñëåäîâàòåëüíîñòè.31
5.1Îïðåäåëåíèåèïðèìåðû.....................
31
5.2Ñâîéñòâàïîñëåäîâàòåëüíîñòåé.Ëîâóøêèèêîðìóøêè....
33
5.3Ïîäïîñëåäîâàòåëüíîñòè.Ïðåäåëè÷àñòè÷íûéïðåäåëïîñëå-
äîâàòåëüíîñòè.*..........................
35
5.4ÁÌÏèÁÁÏ...........................
36
3
4
ÎÃËÀÂËÅÍÈÅ
5.5Àðèôìåòè÷åñêèåñâîéñòâàïðåäåëîâ..............
38
5.6Íåêîòîðûåâàæíûåïðèìåðûá.ì.ïèïðåäåëîâ.........
40
5.7ÒåîðåìûÂåéåðøòðàññàèÁîëüöàíîÂåéåðøòðàññà......
42
5.8Ôóíäàìåíòàëüíûåïîñëåäîâàòåëüíîñòè.ÊðèòåðèéÊîøè.*.
44
5.9×èñëîÝéëåðà...........................
46
6Ðÿäû49
7Äåéñòâèòåëüíûå÷èñëà55
IIIIñåìåñòð57
8Ôóíêöèÿ.Ïðåäåëôóíêöèè.59
8.1×èñëîâûåôóíêöèè........................
60
8.2Ïðåäåëôóíêöèè.ÝêâèâàëåíòíîñòüîïðåäåëåíèéïîÃåéíåè
ïîÊîøè...............................
61
8.3Ïðåäåëûíàáåñêîíå÷íîñòèèîäíîñòîðîííèåïðåäåëû....
62
8.4Àðèôìåòè÷åñêèåñâîéñòâàïðåäåëîâ..............
63
8.4.1Àñèìïòîòè÷åñêèåîáîçíà÷åíèÿ..............
64
8.5Íåïðåðûâíûåôóíêöèè.Îñíîâíûåñâîéñòâà..........
65
8.6Ïðèìåðûíåïðåðûâíûõèðàçðûâíûõôóíêöèé.........
68
9Òðèãîíîìåòðè÷åñêèåôóíêöèè73
9.1×èñëî

...............................
73
9.2Ðàäèàííàÿìåðàóãëà.......................
76
9.2.1Íåêîòîðûåôàêòûèçãåîìåòðèè.............
76
9.3Îïðåäåëåíèåòðèãîíîìåòðè÷åñêèõôóíêöèé..........
77
9.4Ñâîéñòâàòðèãîíîìåòðè÷åñêèõôóíêöèé............
79
9.5Òðèãîíîìåòðè÷åñêèåòîæäåñòâà.................
81
9.6Îáðàòíûåòðèãîíîìåòðè÷åñêèåôóíêöèè............
85
9.7Ïåðâûéçàìå÷àòåëüíûéïðåäåë.................
86
10Ïîêàçàòåëüíàÿèëîãàðèôìè÷åñêàÿôóíêöèè.89
10.1Ëîãàðèôì..............................
89
10.2Ñâîéñòâàëîãàðèôìîâ.......................
90
10.2.1Ïîêàçàòåëüíàÿôóíêöèÿ.Âòîðîéçàìå÷àòåëüíûéïðå-
äåë..............................
91
ÎÃËÀÂËÅÍÈÅ
5
11Ïðîèçâîäíàÿ97
11.1Ââåäåíèå.Ôèçè÷åñêèéèãåîìåòðè÷åñêèéñìûñëïðîèçâîäíîé.
97
11.2Îïðåäåëåíèå.Ïðàâèëàäèôôåðåíöèðîâàíèÿ..........
99
11.2.1Ïðàâèëàäèôôåðåíöèðîâàíèÿ..............
100
11.2.2Ïðîèçâîäíàÿïîêàçàòåëüíîé,ëîãàðèôìè÷åñêîéèñòå-
ïåííîéôóíêöèè......................
102
11.3Ïðîèçâîäíûåýëåìåíòàðíûõôóíêöèé.............
105
11.4Ñâîéñòâàïðîèçâîäíîé.ÒåîðåìûÔåðìà,Ðîëëÿ,Ëàãðàíæà,
Êîøè.................................
106
11.5ÍåðàâåíñòâàÞíãà,üëüäåðà,ÊîøèÁóíÿêîâñêîãî......
109
11.6ÏðàâèëàËîïèòàëÿ.........................
110
12Êðàòíûåïðîèçâîäíûå.ÔîðìóëàËåéáíèöà.Âûïóêëîñòü113
12.1Êðàòíûåïðîèçâîäíûå.......................
113
12.2Âûïóêëîñòüãðàôèêàôóíêöèè.Òî÷êèïåðåãèáà.Íåðàâåíñòâî
Éåíñåíà...............................
115
12.3Êàñàíèåêðèâûõ.Êðóãêðèâèçíû,ýâîëþòàèýâîëüâåíòà...
123
13Ìíîãî÷ëåíÒåéëîðà127
13.1Ìíîãî÷ëåíÒåéëîðà.ÔîðìóëàÒåéëîðà.............
127
13.2ÐÿäÒåéëîðàäëÿýëåìåíòàðíûõôóíêöèé............
128
13.3Ðàçëè÷íûåñïîñîáûîöåíêèîñòàòî÷íîãî÷ëåíà........
128
13.4ÈñïîëüçîâàíèåôîðìóëûÒåéëîðàäëÿïðèáëèæåííûõâû÷èñ-
ëåíèé................................
130
6
ÎÃËÀÂËÅÍÈÅ
Ãëàâà1
Ïðîãðàììàêóðñà
1.
Ñèìâîëèêà,êâàíòîðû,ëîãè÷åñêèåîïåðàöèè(è,èëè,íå).Ìíîæåñòâà,
ïîäìíîæåñòâà,ïóñòîåìíîæåñòâî,êîíå÷íûåèáåñêîíå÷íûåìíîæå-
ñòâà.Îñíîâíûåîïåðàöèèîáúåäèíåíèå,ïåðåñå÷åíèå,ðàçíîñòü(âòîì
÷èñëåñèììåòðè÷åñêàÿ),äîïîëíåíèå,äåêàðòîâîïðîèçâåäåíèå.Îñíîâ-
íûåñâîéñòâà(12øò.).Ôîðìóëàâêëþ÷åíèéèñêëþ÷åíèé.
2.
ÏðèíöèïÄèðèõëå.Ìåòîäìàòåìàòè÷åñêîéèíäóêöèè.
3.
Àðèôìåòè÷åñêàÿ,ãåîìåòðè÷åñêàÿïðîãðåññèè.ÍåðàâåíñòâîÁåðíóë-
ëè.ÁèíîìÍüþòîíà.
4.
Îñíîâíûå÷èñëîâûåìíîæåñòâà
R
;
Q
;
N
;
Z
.×èñëîâûåìíîæåñòâà:ëó÷,
èíòåðâàë,îòðåçîê,îêðåñòíîñòü,
"

îêðåñòíîñòü.Îòêðûòûåèçàìêíó-
òûå(êàêäîïîëíåíèåêîòêðûòûì)ìíîæåñòâà.Îáúåäèíåíèåèïåðå-
ñå÷åíèåîòêðûòûõèçàìêíóòûõìíîæåñòâ.Ñâÿçíîñòü.Êîìïîíåíòû
ñâÿçíîñòè.
5.
Àêñèîìàîòäåëèìîñòè.Òî÷íàÿâåðõíÿÿèíèæíÿÿãðàíüìíîæåñòâà.
Òåîðåìàîñóùåñòâîâàíèèòî÷íîéâåðõíåé(íèæíåé)ãðàíè.Ïðèìåðû.
Öåëàÿèäðîáíàÿ÷àñòü÷èñëà.ÏðèíöèïÀðõèìåäà.
6.
Ïðåäåëüíàÿòî÷êàìíîæåñòâà.Òåîðåìàîñóùåñòâîâàíèèïðåäåëüíîé
òî÷êè(áåñêîíå÷íîãîîãðàíè÷åííîãîìíîæåñòâà).Âíóòðåííÿÿòî÷êà,
ãðàíè÷íàÿòî÷êà,ãðàíèöà,çàìûêàíèåìíîæåñòâà.Ëåììàîçàìûêà-
íèè(çàìûêàíèå=ñàìîìíîæåñòâî+ãðàíèöà).Ýêâèâàëåíòíîñòüîïðå-
äåëåíèéçàìêíóòîãîìíîæåñòâà(êàêäîïîëíåíèÿêîòêðûòîìóèêàê
ñîäåðæàùåãîâñåñâîèïðåäåëüíûåòî÷êè).
7
8
ÃËÀÂÀ1.ÏÐÎÃÐÀÌÌÀÊÓÐÑÀ
7.
Ïîñëåäîâàòåëüíîñòüâëîæåííûõ/ñòÿãèâàþùèõñÿîòðåçêîâ.Òåîðåìà
îíåïóñòîìïåðåñå÷åíèè.Ïðèíöèïñæèìàþùèõîòîáðàæåíèé.
8.
×èñëîâûåïîñëåäîâàòåëüíîñòè.Ìîíîòîííûå,îãðàíè÷åííûåèíåîãðà-
íè÷åííûåïîñëåäîâàòåëüíîñòè.Ëîâóøêèèêîðìóøêèïîñëåäîâàòåëü-
íîñòåé.
9.
Áåñêîíå÷íîáîëüøèå,áåñêîíå÷íîìàëûåïîñëåäîâàòåëüíîñòè.Àðèô-
ìåòè÷åñêèåñâîéñòâàáåñêîíå÷íîáîëüøèõ/ìàëûõïîñëåäîâàòåëüíî-
ñòåé.
10.
Ïðåäåëïîñëåäîâàòåëüíîñòè.Àðèôìåòè÷åñêèåñâîéñòâàïðåäåëîâ.Ïðè-
ìåðûïðåäåëîâ:
1
n
k
;
P
(
n
)
Q
(
n
)
,
n
k
a
n
.(
a�
1
).Áåñêîíå÷íàÿãåîìåòðè÷åñêàÿ
ïðîãðåññèÿ.
11.
Ïðåäåëüíûéïåðåõîäâíåðàâåíñòâàõ.Ëåììàîäâóõìèëèöèîíåðàõ.
ÒåîðåìàØòîëüöà.
12.
ÒåîðåìàÂåéåðøòðàññà.×èñëîå.Èððàöèîíàëüíîñòüå.Âû÷èñëåíèå
ïðåäåëîâ,ñâÿçàííûõñ÷èñëîìå.
13.
ÊðèòåðèéÊîøè.Ôóíäàìåíòàëüíàÿïîñëåäîâàòåëüíîñòü.
14.
ÒåîðåìàÁîëüöàíîÂåéåðøòðàññà.
15.
Ðàöèîíàëüíûåèèððàöèîíàëüíûå÷èñëà.Ïåðèîäè÷åñêèåäðîáèèðà-
öèîíàëüíûå÷èñëà.Ïëîòíîñòüìíîæåñòâðàöèîíàëüíûõèèððàöèî-
íàëüíûõ÷èñåë.Òåîðåìàîðàöèîíàëüíîñòè
cos
m
n

(áåçä-âà,åñëèíå
óñïåþ).
16.
Ðàçëè÷íûåñïîñîáûçàäàíèÿìíîæåñòâàäåéñòâèòåëüíûõ÷èñåë:à)Áåñ-
êîíå÷íûåäåñÿòè÷íûåäðîáèá)Çàìûêàíèåìíîæåñòâàðàöèîíàëüíûõ
÷èñåë.â)Àêñèîìàòè÷åñêèéñïîñîá(17àêñèîì).
×àñòüI
Iñåìåñòð
9
Ãëàâà2
Ìíîæåñòâà
Åñëèâàìçàêàêóþ-ëèáî
ïîìîùüîáåùàþòîêàçàòü
ìíîæåñòâîóñëóã,íåçàáûâàéòå,
÷òîìíîæåñòâîìîæåòáûòü
ïóñòûì.
bash.org.ru
Îïðåäåëåíèå1.
Ìíîæåñòâîìíàçûâàåòñÿíàáîðîáúåêòîâîïðåäåëåííî-
ãîðîäà.Ìíîæåñòâîñ÷èòàåòñÿçàäàííûì,åñëèäëÿëþáîãîîáúåêòàìîæ-
íîîïðåäåëèòü,ïðèíàäëåæèòëèîíýòîìóìíîæåñòâó.
Îïðåäåëåíèå2.
Ìíîæåñòâî,íåñîäåðæàùååíèêàêèõýëåìåíòîâ,íàçû-
âàþòïóñòûìèîáîçíà÷àþò
?
.
Çàìå÷àíèå.
?
ÿâëÿåòñÿïîäìíîæåñòâîìëþáîãîìíîæåñòâà.
Åñëèýëåìåíò
a
ïðèíàäëåæèòìíîæåñòâó
A
,òîýòîîáîçíà÷àþò
a
2
A
.Åñëè
ýëåìåíò
b
íåïðèíàäëåæèòìíîæåñòâó
A
,òîýòîîáîçíà÷àþò
b=
2
A
.Åñëè
âñåýëåìåíòûìíîæåñòâà
A
ïðèíàäëåæàòòàêæåèìíîæåñòâó
B
,òîãîâîðÿò,
÷òî
A
ÿâëÿåòñÿ
ïîäìíîæåñòâîì
B
èîáîçíà÷àþòòàê:
A

B
èëè
B

A
.
Åñëèäëÿëþáîãîýëåìåíòàìíîæåñòâàâûïîëíåíîíåêîòîðîåñâîéñòâî,
òîýòîîáîçíà÷àþò
8
a
2
A
(
:::
)
.Åñëèñóùåñòâóåòýëåìåíòìíîæåñòâà,äëÿ
êîòîðîãîâûïîëíåíîíåêîòîðîåñâîéñòâî,òîýòîîáîçíà÷àþò
9
a
2
A
(
:::
)
.
Ñèìâîëû
8
("äëÿëþáîãî")è
9
("ñóùåñòâóåò")íàçûâàþò
êâàíòîðàìè
.
Ïðèìåð1.
Íàïðèìåðîïðåäåëåíèåïîäìíîæåñòâà
A

B
ìîæíîçàïè-
ñàòüòàê:
8
a
2
A
(
a
2
B
)
.
11
12
ÃËÀÂÀ2.ÌÍÎÆÅÑÒÂÀ
Åñëèíàäîçàäàòüìíîæåñòâîîáúåêòîâ,îáëàäàþùèõíåêîòîðûìñâîé-
ñòâîì
P
,òîýòîîáîçíà÷àþò
f
a
j
P
(
a
)
g
èëè
f
a
:
P
(
a
)
g
2.1Îïåðàöèèíàäìíîæåñòâàìè
Îïðåäåëåíèå3.
Îáúåäèíåíèåììíîæåñòâ
A
è
B
(îáîçíà÷àåòñÿ
A
[
B
)
íàçûâàþòìíîæåñòâî,ñîñòîÿùååèçâñåõýëåìåíòîâ,âõîäÿùèõâ
A
èëè
B
.
Îïðåäåëåíèå4.
Ïåðåñå÷åíèåììíîæåñòâ
A
è
B
(îáîçíà÷àåòñÿ
A
\
B
)
íàçûâàþòìíîæåñòâî,ñîñòîÿùååèçâñåõýëåìåíòîâ,âõîäÿùèõêàêâ
A
,
òàêèâ
B
.
Îïðåäåëåíèå5.
Ðàçíîñòüþìíîæåñòâ
A
è
B
(îáîçíà÷àåòñÿ
A
n
B
)íà-
çûâàþòìíîæåñòâî,ñîñòîÿùååèçâñåõýëåìåíòîâ,âõîäÿùèõâ
A
,èíå
âõîäÿùèõâ
B
.
Îïðåäåëåíèå6.
Ñèììåòðè÷åñêîéðàçíîñòüþìíîæåñòâ
A
è
B
(îáî-
çíà÷àåòñÿ
A
4
B
)íàçûâàþòìíîæåñòâî,ñîñòîÿùååèçâñåõýëåìåíòîâ,
âõîäÿùèõëèáîâ
A
,ëèáîâ
B
.Áîëååôîðìàëüíî,
A
4
B
=(
A
[
B
)
n
(
A
\
B
)
:
Îïðåäåëåíèå7.
Âîìíîãèõñëó÷àÿõìûèìååìäåëîòîëüêîñïîäìíî-
æåñòâàìèíåêîòîðîãîìíîæåñòâà
E
.Íàïðèìåð,ðåøàÿíåðàâåíñòâî,ìû
èìååìäåëîñïîäìíîæåñòâàìè÷èñëîâîéîñè
(
�1
;
1
)=
E
.Âòàêîìñëó-
÷àåìíîæåñòâî
E
,ñîäåðæàùååâñåèíòåðåñóþùèåíàñìíîæåñòâàíàçû-
âàåòñÿîáúåìëþùèììíîæåñòâîìèìîæíîîïðåäåëèòüîïåðàöèþäîïîë-
íåíèÿ.Äîïîëíåíèåì
A
(îáîçíà÷àåòñÿ
A
íàçûâàåòñÿìíîæåñòâîòî÷åê
îáúåìëþùåãîìíîæåñòâà
E
,íåïðèíàäëåæàùèõ
A
,ò.å.
A
=
E
n
A
.
Ïðèìåð2.
Ïóñòü
A
=[0
;
1)
,àîáúåìëþùååìíîæåñòâîâñÿ÷èñëîâàÿ
îñü.Òîãäà
A
=(
�1
;
0)
[
[1
;
+
1
)
.
Îïðåäåëåíèå8.
Äåêàðòîâûìïðîèçâåäåíèåììíîæåñòâ
A
è
B
(îáîçíà-
÷àþò
A

B
)íàçûâàþòìíîæåñòâîïàð
(
a;b
)
,ãäå
a
2
A
è
b
2
B
.Åñëè
ìíîæåñòâà
A
è
B
ñîâïàäàþò,òîèñïîëüçóþòîáîçíà÷åíèå
A
2
=
A

A
.
Îïåðàöèèíàäìíîæåñòâàìèîáëàäàþòñëåäóþùèìèñâîéñòâàìè:
1.
A

A
.
2.
A

B;B

C
!
A

C
.
3.
A

B;B

B
!
A
=
B
.
2.2.×ÈÑËÎÂÛÅÌÍÎÆÅÑÒÂÀ
13
4.
8
A
(
?

A
)
.
5.
(
A
[
B
)
\
C
=(
A
\
C
)
[
(
B
\
C
)
.
6.
(
A
\
B
)
[
C
=(
A
[
C
)
\
(
B
[
C
)
.
7.
A

B
)
A
[
B
=
B
;
A
\
B
=
A
.
8.
A
[

A
=
E
9.
A
\

A
=
?
.
10.
(
A
[
B
)=

A
\

B
.
11.
(
A
\
B
)=

A
[

B
.
12.
A
4
B
=(
A
n
B
)
[
(
B
n
A
)
.
Äîêàæåìñâîéñòâî10.Ïóñòü
x
2
(
A
[
B
)
,òîãäà
x=
2
(
A
[
B
)
,ñëåäîâà-
òåëüíî
x=
2
A
è
x=
2
B
.Àçíà÷èò
x
2
A
è
x
2
B
,ò.å.
A
\
B
.
ÑäðóãîéñòîðîíûÏóñòü
x
2
A
\
B
.Òîãäà
x
2
A
,
x
2
B
)
x=
2
A;x=
2
B
)
x=
2
A
[
B
)
x
2
(
A
[
B
)
.Òàêèìîáðàçîìëþáîéýëåìåíòîäíîãîìíî-
æåñòâàÿâëÿåòñÿýëåìåíòîìäðóãîãî,àçíà÷èò,ìíîæåñòâà
(
A
[
B
)
è
A
\
B
ñîâïàäàþò.
Äîêàçàòåëüñòâîîñòàëüíûõóòâåðæäåíèé112ïðåäîñòàâëÿåòñÿ÷èòàòå-
ëþâêà÷åñòâåóïðàæíåíèÿ.
Óïðàæíåíèå1.
ÍàÍîâûéãîäêäåòèøêàìïðèøåëÄåäÌîðîçñìåøêîì
êîíôåò.Êîíôåòâìåøêåáåñêîíå÷íîìíîãî,èîíèçàíóìåðîâàíûíàòó-
ðàëüíûìè÷èñëàìè.Íàêàæäîéêîíôåòåíàïèñàíååíîìåð,èäëÿêàæäîãî
íàòóðàëüíîãî÷èñëàåñòüðîâíîîäíàêîíôåòàñýòèìíîìåðîì.Çàîäíó
ìèíóòóäîïîëíî÷èÄåäÌîðîçâçÿëêîíôåòó1èïîäàðèëäåòÿì.×åðåç
ïîëìèíóòûîíäàëäåòÿìêîíôåòû2è3,íîïðèýòîìêîíôåòó1
çàáðàë.Åùå÷åðåç÷åòâåðòüìèíóòûîíäàëäåòÿìêîíôåòû4,5,
6è7,íîçàáðàëêîíôåòó2.Èòàêäàëåå:ùåäðûéÄåäÌîðîçêàæ-
äûéðàçäàåòâäâîåáîëüøåêîíôåò,÷åìíàïðåäûäóùåìøàãå,èçà
1
2
n
ìèí.
äîïîëíî÷èäàåòêîíôåòûñíîìåðàìè
2
n
;
2
n
+1
;:::;
2
n
+1

1
èçàáèðàåò
êîíôåòóñíîìåðîì
n
.Íàéòèìíîæåñòâîêîíôåò,êîòîðûåîñòàíóòñÿó
äåòåéâïîëíî÷ü.
2.2×èñëîâûåìíîæåñòâà
Âìàòåìàòèêåèñïîëüçóþòñÿñëåäóþùèåîáîçíà÷åíèÿäëÿ÷èñëîâûõìíî-
æåñòâ:
14
ÃËÀÂÀ2.ÌÍÎÆÅÑÒÂÀ
N
Ìíîæåñòâîíàòóðàëüíûõ÷èñåë.Ýòî÷èñëà,èñïîëüçóåìûåïðèñ÷å-
òå:1,2,3,...,2009,è.ò.ä.
Z
Ìíîæåñòâîöåëûõ÷èñåë.Ýòîíàòóðàëüíûå÷èñëà,âçÿòûåñîçíàêîì
¾+¿èë辿,àòàêæå0:
0
;

1
;

2
;

3
;:::;

2009
;
è.ò.ä.
Q
Ìíîæåñòâîðàöèîíàëüíûõ÷èñåë.Ýòî÷èñëà,ïîëó÷àþùèåñÿïðè
äåëåíèèöåëîãî÷èñëàíàíàòóðàëüíîå,ò.å.äðîáè.Áîëååôîðìàëüíî,
Q
=

m
n
j
m
2
Z
;n
2
N

.
R
Ìíîæåñòâîäåéñòâèòåëüíûõ÷èñåë.Ìîæíîïîíèìàòüäåéñòâèòåëü-
íûå÷èñëàêàêòî÷êèíà÷èñëîâîéïðÿìîéèëèáåñêîíå÷íûåäåñÿòè÷-
íûåäðîáè.Ñòðîãîåîïðåäåëåíèåäåéñòâèòåëüíûõ÷èñåëáóäåòäàíîâ
êîíöåñåìåñòðà.
Çàìå÷àíèå.
Âûïîëíåíîâêëþ÷åíèå:
N

Z

Q

R
.Âïîâñåäíåâ-
íîéæèçíèìûèñïîëüçóåìòîëüêîðàöèîíàëüíûå÷èñëà.Âîçíèêàåòâîïðîñ,
íóæíûëèäåéñòâèòåëüíûå÷èñëà,äàèâîîáùå,ñóùåñòâóþòëè÷èñëà,íå
ÿâëÿþùèåñÿèððàöèîíàëüíûìè.Íàýòîòâîïðîñîòâå÷àåòñëåäóþùååóòâåð-
æäåíèå:
Óòâåðæäåíèå1.
p
2
=
2
Q
.
Äîêàçàòåëüñòâî.
Äîêàçàòåëüñòâîáóäåìïðîâîäèòüìåòîäîì"îòïðîòèâíîãî".ïðåäïîëîæèì,
÷òî
p
2
2
Q
.Òîãäà
p
2
ïðåäñòàâèìââèäåíåñîêðàòèìîéäðîáè
m
n
,òàêêàê
èçïðîèçâîëüíîéäðîáèâñåãäàìîæíîïîëó÷èòüíåñîêðàòèìóþ,ñîêðàòèâ
íàíàèáîëüøèéîáùèéäåëèòåëü÷èñëèòåëÿèçíàìåíàòåëÿ.Òîãäà
2=
m
2
n
2
,
ñëåäîâàòåëüíî
2
n
2
=
m
2
,àçíà÷èò
m
êðàòíîäâóì.Îáîçíà÷èì
m
=2
m
0
.
Òîãäà
n
2
=2
m
0
2
,ñëåäîâàòåëüíî
n
òîæåêðàòíîäâóì,÷òîèïðèâîäèòê
ïðîòèâîðå÷èþñíåñîêðàòèìîñòüþäðîáè
m
n
.
Çàìå÷àíèå.
Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òî
p
3
=
2
Q
,
p
5
=
2
Q
,...è,
âîîáùå,
p
n=
2
Q
,åñëè
n
íåÿâëÿåòñÿòî÷íûìêâàäðàòîì(êâàäðàòîìöåëîãî
÷èñëà).
Ãëàâà3
Ìåòîäìàòåìàòè÷åñêîé
èíäóêöèè.
Îïðåäåëåíèå9.
Ìåòîäîììàòåìàòè÷åñêîéèíäóêöèèíàçûâàåòñÿñïî-
ñîáäîêàçàòåëüñòâàóòâåðæäåíèé,ñîñòîÿøùèéâñëåäóþùåì:ïóñòüåñòü
íåêîòîðîåóòâåðæäåíèå
U
(
n
)
,çàâèñÿùååîò
n
2
N
èíàäîäîêàçàòüýòî
óòâåðæäåíèåäëÿâñåõ
n
2
N
.Äîêàçàòåëüñòâîñîñòîèòèçäâóõýòàïîâ:
1.
Áàçàèíäóêöèè.
Äîêàæåìóòâåðæäåíèå
U
(1)
.
2.
Øàãèíäóêöèè.
Äîêàæåì,÷òîèçóòâåðæäåíèé
U
(1)
;
U
(2)
;:::;
U
(
n

1)
ñëåäóåòóòâåðæäåíèå
U
(
n
)
.
Äîêàçàòåëüñòâî
(êîððåêòíîñòèìåòîäàìàòåìàòè÷åñêîéèíäóêöèè).Ïðåä-
ïîëîæèìîáðàòíîå,ïóñòü
U
(
n
)
âåðíîíåïðèâñåõ
n
2
N
.Òîãäàíàéäåòñÿ
íàèìåíüøåå
n
0
;
ïðèêîòîðîìóòâåðæäåíèåíåâåðíî.Íîìåð
n
0
íåìîæåò
ðàâíÿòüñÿ1,ýòîñëåäóåòèçáàçûèíäóêöèè.Ñëåäîâàòåëüíî,óòâåðæäåíèÿ
U
(1)
;
U
(2)
;:::;
U
(
n
0

1)
âåðíû,àèçíèõèâûòåêàåò
U
(
n
0
)
.Ïðîòèâîðå÷èå.
3.1¾Õàíîéñêèåáàøíè¿
Ðàññìîòðèìèñïîëüçîâàíèåïðèíöèïàìàòåìàòè÷åñêîéèíäóêöèèíàïðèìå-
ðåèãðû¾Õàíîéñêèåáàøíè¿.Ýòîñòàðèííàÿèãðà,êîòîðàÿçàêëþ÷àåòñÿ
âñëåäóþùåì.Íàïîäñòàâêåóêðåïëåíûòðèñòåðæíÿ.Íàëåâûéñòåðæåíü
íàíèçàíîíåñêîëüêîêîëåöðàçíîãîðàçìåðà,âíèçóñàìîåáîëüøîåêîëüöî,
15
16
ÃËÀÂÀ3.ÌÅÒÎÄÌÀÒÅÌÀÒÈ×ÅÑÊÎÉÈÍÄÓÊÖÈÈ.
íàíåìïîìåíüøå,ñâåðõóåùåìåíüøåèò.ï.Çàîäèíõîäìîæíîïåðåíî-
ñèòüòîëüêîîäíîêîëüöî.Ëþáîåêîëüöîìîæíîóêëàäûâàòüëèáîíàáîëü-
øååêîëüöî,ëèáîíàñâîáîäíûéñòåðæåíü.Êàêîâîíàèìåíüøåå÷èñëîõîäîâ
íåîáõîäèìîäëÿòîãî,÷òîáûïåðåíåñòèâñåêîëüöàñëåâîãîñòåðæíÿíàïðà-
âûé(ñðåäíèéñòåðæåíüèñïîëüçóåòñÿêàêâñïîìîãàòåëüíûé)?
Ïîèãðàâíåìíîãîñðàçëè÷íûì÷èñëîìêîëåö,ìîæíîçàìåòèòü,÷òîñ
îäíèìêîëüöîìãîëîâîëîìêàðåøàåòñÿçà1õîä,ñäâóìÿçà3õîäà,ñòðåìÿ
çà7,ñ÷åòûðüìÿçà15è.ò.ä.Ëîãè÷íîïðåäïîëîæèòü,÷òî÷èñëîõîäîâ
äëÿ
n
êîëåöíàåäèíèöóìåíüøå
n
éñòåïåíèäâîéêè.Íîêàêýòîïðîâåðèòü
äëÿâñåõçíà÷åíèé
n
?Èõæåáåñêîíå÷íîìíîãî!Íàïîìîùüïðèõîäèòìåòîä
ìàòåìàòè÷åñêîéèíäóêöèè:
Óòâåðæäåíèå2.
Ãîëîâîëîìêàñ
n
êîëüöàìèìîæíîðåøèòüçà
2
n

1
õîäîâèíåëüçÿçàìåíüøåå÷èñëîõîäîâ.
Äîêàçàòåëüñòâî.
Áàçàèíäóêöèèî÷åâèäíà,ãîëîâîëîìêàñ1êîëüöîìðåøàåòñÿçà1õîä
ïðîñòîïåðåêëàäûâàåìêîëüöîñïåðâîãîñòåðæíÿíàòðåòèé.
Ñäåëàåìøàãèíäóêöèè.Ïóñòüäëÿ
n

1
êîëüöàçàäà÷àðåøàåòñÿçà
2
n

1

1
õîäîâ.Òîãäà,î÷åâèäíî,ìîæíîïåðåëîæèòüêîëüöàñ1ãîïî
n

1

åçàñòîëüêîæåõîäîâíàâòîðîéñòåðæåíü;çàòåìïåðåëîæèòü
n
åêîëüöî
íàòðåòèéñòåðæåíü,àïîòîì,îïÿòüçà
2
n

1

1
õîäîâïåðåëîæèòüêîëü-
öàñ1ãîïî
n

1
åñîâòîðîãîñòåðæíÿíàòðåòèé.Âñåãîïîëó÷àåòñÿ
(2
n

1

1)+1+(2
n

1

1)=2
n

1
õîäîâ,÷òîèòðåáîâàëîñüäîêàçàòü.
Çàìåíüøåå÷èñëîõîäîâãîëîâîëîìêóðåøèòüíåëüçÿ,ïîñêîëüêó,êàê
áûìûíåïåðåêëàäûâàëèêîëüöà,íàäîâêàêîéòîìîìåíòïåðåëîæèòü
n

åêîëüöîñïåðâîãîñòåðæíÿíàòðåòèé,àýòîâîçìîæíî,òîëüêîåñëèâñå
îñòàëüíûåêîëüöàëåæàòíàâòîðîìñòåðæíå.
Óïðàæíåíèå2.
Ïîëåãåíäå,ãäå-òîâäæóíãëÿõåñòüäðåâíèéõðàì,â
êîòîðîììîíàõèðåøàþòòàêóþãîëîâîëîìêóñ64êîëüöàìè.Àêîãäàîíè
ååðåøàòíàñòóïèòêîíåöñâåòà.Îïðåäåëèòü,ñêîëüêîîñòàëîñüäî
êîíöàñâåòà,åñëèíà1õîäîíèòðàòÿòîäíóñåêóíäó,àïåðåêëàäûâàòü
êîëüöàíà÷àëè1ÿíâàðÿ1ãîäàí.ý.
3.2Äðóãèåïðèìåíåíèÿìåòîäàìàòåìàòè÷åñêîé
èíäóêöèè
Òåîðåìà3
(ÍåðàâåíñòâîÁåðíóëëè)
.
Ïóñòü
x�

1
è
x
6
=0
;
n
2
N
nf
1
g
.
Òîãäà
(1+
x
)
n

1+
nx
.
3.2.ÄÐÓÃÈÅÏÐÈÌÅÍÅÍÈßÌÅÒÎÄÀÌÀÒÅÌÀÒÈ×ÅÑÊÎÉÈÍÄÓÊÖÈÈ
17
Äîêàçàòåëüñòâî.

Áàçàèíäóêöèè.
Î÷åâèäíî
(1+
x
)
2
=1+2
x
+
x
2

1+2
x
.

Øàãèíäóêöèè.
Ïóñòü
n

1
.Ïîïðåäïîëîæåíèþèíäóêöèè
(1+
x
)
n

1

1+(
n

1)
x
.
Óìíîæèìîáå÷àñòèíåðàâåíñòâàíà
1+
x�
0
:
Òîãäà
(1+
x
)
n

(1+(
n

1)
x
)(1+
x
)=1+
nx
+(
n

1)
x
2

1+
nx:
3.2.1ÒðåóãîëüíèêÏàñêàëÿ.
Ðàññìîòðèìòðåóãîëüíèê,ñîñòàâëåííûéèç÷èñåëïîñëåäóþùèìïðàâèëàì:

Âïåðâîìðÿäóîäíî÷èñëî
1
.

Âêàæäîìñëåäóþùåìðÿäó÷èñåëíà1áîëüøå,÷åìâïðåäûäóùåì

Âêàæäîìðÿäóêðàéíèå÷èñëàðàâíû1

Íà÷èíàÿñ3ãîðÿäàêàæäîå÷èñëîðàâíîñóììåäâóõñâîèõñîñåäåé:
ñâåðõóñëåâàèñâåðõóñïðàâà.
ÝòóêîíñòðóêöèþíàçûâàþòòðåóãîëüíèêîìÏàñêàëÿ.
paskal
Ðèñ.3.1:ÒðåóãîëüíèêÏàñêàëÿ.
Óòâåðæäåíèå4.
×èñëî,ñòîÿùååâ
n
+1
ðÿäóíà
k
+1
ìåñòåìîæíî
íàéòèïîôîðìóëå
C
k
n
=
n
!
k
!(
n

k
)!
;
ãäå÷èñëà
C
k
n
÷èñëîñî÷åòàíèéèç
n
ýëåìåíòîâïî
k
øèðîêîèñïîëüçóþòñÿâêîìáèíàòîðèêå.
Äîêàçàòåëüñòâî.
Ïðîâåäåìèíäóêöèþïîíîìåðóðÿäà.
Áàçàèíäóêöèè.Äëÿ
n
=1
ïîëó÷èì
C
0
0
=
0!
0!0!
=1
.
Øàãèíäóêöèè.Ïóñòüïðåäïîëîæåíèåèíäóêöèèâåðíîäëÿ
n
ðÿäà.Äîêà-
æåìåãîäëÿ
n
+1
ãîÄëÿêðàéíèõýëåìåíòîâïîëó÷èì:
C
0
n
=
n
!
0!
n
!
=1
;
C
n
n
=
n
!
n
!0!
=1
.Äëÿîñòàëüíûõïîïðåäïîëîæåíèþèíäóêöèèñóììàñîñåäåé
ñâåðõóðàâíà
18
ÃËÀÂÀ3.ÌÅÒÎÄÌÀÒÅÌÀÒÈ×ÅÑÊÎÉÈÍÄÓÊÖÈÈ.
paskalC
C
0
0
C
0
1
C
1
1
C
0
2
C
1
2
C
2
2
C
0
3
C
1
3
C
2
3
C
3
3
C
0
4
C
1
4
C
2
4
C
3
4
C
4
4
Ðèñ.3.2:Áèíîìèàëüíûåêîýôôèöèåíòû.
C
k

1
n

1
+
C
k
n

1
=
(
n

1)!
(
k

1)!(
n

k
)!
+
(
n

1)!
k
!(
n

1

k
)!
=
=
(
n

1)!

k
k
!(
n

k
)!
+
(
n

1)!

(
n

k
)
k
!(
n

k
)!
=
(
n

1)!

n
k
!(
n

k
)!
=
n
!
k
!(
n

k
)!
=
C
k
n
:
Çàìå÷àíèå.
Ïîïóòíîáûëîäîêàçàíîòîæäåñòâî
C
k

1
n

1
+
C
k
n

1
=
C
k
n
.Îíî
íàìïîçäíååïîíàäîáèòñÿ
3.3.ÁÈÍÎÌÍÜÞÒÎÍÀ.
19
3.3ÁèíîìÍüþòîíà.
Íó,êîíå÷íî,ýòîíåñóììà,
ñíèñõîäèòåëüíîñêàçàë
Âîëàíäñâîåìóãîñòþ,õîòÿ,
âïðî÷åì,èîíà,ñîáñòâåííî,
âàìíåíóæíà.Âûêîãäà
óìðåòå?Òóòóæáóôåò÷èê
âîçìóòèëñÿ.Ýòîíèêîìóíå
èçâåñòíîèíèêîãîíåêàñàåòñÿ,
îòâåòèëîí.Íóäà,
íåèçâåñòíî,ïîñëûøàëñÿâñå
òîòæåäðÿííîéãîëîñèç
êàáèíåòà,ïîäóìàåøü,áèíîì
Íüþòîíà!Óìðåòîí÷åðåç
äåâÿòüìåñÿöåâ,âôåâðàëå
áóäóùåãîãîäà,îòðàêàïå÷åíè
âêëèíèêåÏåðâîãîÌÃÓ,â
÷åòâåðòîéïàëàòå.
Ì.À.Áóëãàêîâ,Ìàñòåðè
Ìàðãàðèòà.
Áèíîì
1
âûðàæåíèåñîñòàâëåííîåèçäâóõîäíî÷ëåíîâ.ÔîðìóëàÍüþòî-
íàïîçâîëÿåòâîçâîäèòüäâó÷ëåíâëþáóþíàòóðàëüíóþñòåïåíü.
Òåîðåìà5.
Ïóñòü
n
2
N
:
Òîãäà
(1+
x
)
n
=
C
0
n
+
C
1
n

x
+
C
2
n

x
2
+
:::
+
C
n

1
n

x
n

1
+
C
n
n

x
n
.
Äîêàçàòåëüñòâî.
Èíäóêöèÿïî
n
.Áàçàèíäóêöèè.
(1+
x
)
1
=1+
x
=
C
0
1
+
C
1
1

x
.
ØàãÈíäóêöèè.Ïóñòüôîðìóëàâåðíàäëÿ
n

1
.Óìíîæèìîáå÷àñòèòîæ-
äåñòâà
(1+
x
)
n

1
=
C
0
n

1
+
C
1
n

1

x
+
:::
+
C
n

1
n

1

x
n

1
íà
(1+
x
)
.Ïîëó÷èì
(1+
x
)
n
=
C
0
n

1
+
C
0
n

1

x
+
C
1
n

1

x
+
C
1
n

1

x
2
+
:::
+
C
n

1
n

1

x
n

1
+
C
n

1
n

1

x
n
=
=
C
0
n

1
+(
C
0
n

1
+
C
1
n

1
)

x
+(
C
1
n

1
+
C
2
n

1
)

x
2
+
:::
+(
C
n

2
n

1
+
C
n

1
n

1
)

x
n

1
+
C
n

1
n

1

x
n
=
=
C
0
n
+
C
1
n

x
+
C
2
n

x
2
+
:::
+
C
n

1
n

x
n

1
+
C
n
n

x
n
;
1
binomäâó÷ëåí,ëàò.
20
ÃËÀÂÀ3.ÌÅÒÎÄÌÀÒÅÌÀÒÈ×ÅÑÊÎÉÈÍÄÓÊÖÈÈ.
÷òîèòðåáîâàëîñüäîêàçàòüÇàìåíà
C
0
n

1
íà
C
0
n
è
C
n

1
n

1
íà
C
n
n
çàêîííà,òàê
êàêâñåýòè÷èñëàðàâíû1.
Óïðàæíåíèå3.
ÂûâåñòèôîðìóëóÍüþòîíàäëÿ
(
a
+
b
)
n
.
Óïðàæíåíèå4.
Íàéòèñóììó
C
0
n
+
C
1
n
+
:::
+
C
n
n
.
Óïðàæíåíèå5.
Íàéòèñóììó
C
0
n

C
1
n
+
:::
+(

1)
n

C
n
n
.
Ãëàâà4
Ìíîæåñòâà.Ìîùíîñòü.
×èñëîâûåìíîæåñòâà
4.1Ìîùíîñòüìíîæåñòâà.Ñ÷åòíûåèíåñ÷åò-
íûåìíîæåñòâà.
Cäðåâíîñòèèçâåñòåíïàðàäîêñ:Êàêèõ÷èñåëáîëüøåíàòóðàëüíûõèëè
òî÷íûõêâàäðàòîâ.Ñîäíîéñòîðîíû,íåâñåíàòóðàëüíûå÷èñëàÿâëÿþòñÿ
òî÷íûìèêâàäðàòàìè,áîëååòîãî,êâàäðàòûèäóòâñåðåæåèðåæå.Íàïðè-
ìåð,ìåæäó÷èñëàìè1000001è1002000íåòíèîäíîãîòî÷íîãîêâàäðàòà.
Óïðàæíåíèå6.
Äîêàæèòåýòî.
Ñäðóãîéñòîðîíû,êàæäîìóíàòóðàëüíîìó÷èñëóñîîòâåòñòâóåòåãîêâàä-
ðàò.
Äëÿðàçðåøåíèÿïîäîáíûõïàðàäîêñîâáûëàïðåäëîæåíàêîíöåïöèÿìîù-
íîñòèìíîæåñòâà.
Îïðåäåëåíèå10.
Ìíîæåñòâà
A
è
B
íàçûâàþòñÿ
ðàâíîìîùíûìè
,åñ-
ëèñóùåñòâóåòâçàèìíîîäíîçíà÷íîåñîîòâåòñòâèå(ò.å.áèåêöèÿ)ìåæ-
äóèõýëåìåíòàìè.Ïðèýòîìêàæäîìóýëåìåíòó
a
2
A
ñòàâèòñÿâñîîò-
âåòñòâèåîäèíýëåìåíò
b
=
f
(
a
)
2
B
èêàæäîìó
b
2
B
ñîîòâåòñòâóåò
ðîâíîîäèíýëåìåíò
a
2
A
,òàêîé,÷òî
f
(
a
)=
b
.
Îáîçíà÷àåòñÿìîùíîñòüìíîæåñòâà
j
A
j
,åñëèìíîæåñòâîêîíå÷íîå,òî
j
A
j
=
n
÷èñëîýëåìåíòîâìíîæåñòâà.Äëÿìîùíîñòèáåñêîíå÷íûõìíî-
æåñòâíåêîòîðûåîáîçíà÷åíèÿáóäóòïðèâåäåíûïîçäíåå.
21
22
ÃËÀÂÀ4.ÌÍÎÆÅÑÒÂÀ.ÌÎÙÍÎÑÒÜ.×ÈÑËÎÂÛÅÌÍÎÆÅÑÒÂÀ
Ïðèìåð.
Ðàññìîòðèììíîæåñòâî
[0
;
1]
èìíîæåñòâî
[0
;
1)
.Êàçàëîñüáû,
âîâòîðîìíàîäíóòî÷êóìåíüøå,íîîíèðàâíîìîùíûå.Ïîñòðîèìñëåäó-
þùóþôóíêöèþ:
f
(
x
)=
x
,åñëè
x
6
=1
;
1
=
2
;
1
=
4
;:::;
1
=
2
k
;:::
,àåñëè
x
=2
k
,
òî
f
(
x
)=2
k
+1
.Îíàïåðåâîäèò1â1/2,1/2â1/4,è.ò.ä.Òàêèìîáðàçîì,
êàæäîìóýëåìåíòó
[0
;
1]
ñòàâèòñÿâñîîòâåòñòâèåýëåìåíò
[0
;
1)
èíàîáîðîò.
Îïðåäåëåíèå11.
Ìíîæåñòâî,ðàâíîìîùíîåìíîæåñòâóíàòóðàëüíûõ
÷èñåëíàçûâàåòñÿ
ñ÷åòíûì.
Åãîìîùíîñòüîáîçíà÷àåòñÿñèìâîëîì
@
0
(÷èòàåòñÿàëåôíîëü).
Ëåììà1.
Ïîäìíîæåñòâîñ÷åòíîãîìíîæåñòâàÿâëÿåòñÿêîíå÷íûìèëè
ñ÷åòíûì.
Äîêàçàòåëüñòâî.
Ïóñòü
B

A
,åñëèîíîêîíå÷íî,òîëåììàäîêàçàíà;ïðåäïîëîæèì,÷òîîíî
áåñêîíå÷íî.Çàíóìåðóåìýëåìåíòûìíîæåñòâà
A
=
f
a
1
;a
2
;:::;a
n
;:::
g
,ïóñòü
f
(1)
íàèìåíüøèéíîìåð
k
1
,òàêîé,÷òî
a
k
1
2
B
.Äàëåå,
f
(2)
íàèìåíü-
øèéíîìåð
k
2
6
=
k
1
,òàêîé,÷òî
a
k
2
2
B
.Èòàêäàëåå,
f
(
n
)
íàèìåíüøèé
íîìåð
k
n
6
=
k
1
;:::;k
n

1
,òàêîé,÷òî
a
k
2
2
B
.Îòîáðàæåíèå
f
:
N
7!
B
ÿâëÿ-
åòñÿáèåêöèåé,ò.ê.ëþáîéýëåìåíò
b
2
B
ÿâëÿåòñÿýëåìåíòîì
A
ñêàêèìòî
íîìåðîì
^
k
,ñëåäîâàòåëüíî,áóäåòðàññìîòðåííàíåêîòîðîìøàãå.
Ñëåäñòâèå1.
Ïóñòüñóùåñòâóåòèíúåêöèÿ
f
:
A
7!
N
,òîãäà
A
êî-
íå÷íîåèëèñ÷åòíîå.
Ëåììà2.
Îáúåäèíåíèåêîíå÷íîãîèñ÷åòíîãîìíîæåñòâÿâëÿåòñÿñ÷åò-
íûì.
Äîêàçàòåëüñòâî.
Ïóñòü
A
=
f
a
1
;:::;a
N
g
êîíå÷íîåìíîæåñòâî,à
B
=
f
b
1
;:::;b
n
;:::
g
ñ÷åò-
íîå.Ðàññìîòðèì
f
(
n
)=
(
a
n
;n
6
N
;
b
n

N
;n�N:
.Î÷åâèäíî,
f
:
N
7!
A
[
B

áèåêöèÿ.
Ëåììà3.
Îáúåäèíåíèåäâóõñ÷åòíûõìíîæåñòâÿâëÿåòñÿñ÷åòíûì.
Äîêàçàòåëüñòâî.
Ïóñòü
A
=
f
a
1
;:::;a
n
;:::
g
,
B
=
f
b
1
;:::;b
n
;:::
g
.Ðàññìîòðèì
f
(
n
)=
(
a
k
;n
=2
k
;
b
k
;n
=2
k
+1
:
.
Î÷åâèäíî,
f
:
N
7!
A
[
B
áèåêöèÿ.
Ñëåäñòâèå2.
Îáúåäèíåíèå
N
2
N
ñ÷åòíûõìíîæåñòâÿâëÿåòñÿñ÷åò-
íûì.
4.1.ÌÎÙÍÎÑÒÜÌÍÎÆÅÑÒÂÀ.Ñ×ÅÒÍÛÅÈÍÅÑ×ÅÒÍÛÅÌÍÎÆÅÑÒÂÀ.
23
dekart_prod
Ëåììà4.
Äåêàðòîâîïðîèçâåäåíèåäâóõñ÷åòíûõìíîæåñòâÿâëÿåòñÿ
ñ÷åòíûì.
Äîêàçàòåëüñòâî.
Ïóñòü
A
=
f
a
1
;:::;a
m
;:::
g
,
B
=
f
b
1
;:::;b
n
;:::
g
.Ðàññìîòðèìîòîáðàæåíèå
f
:
A

B
7!
N
,êîòîðîåçàäàíîòàê:
f
(
a
m
;b
n
)=2
m

3
n
.Ðàçëîæåíèå÷èñ-
ëàíàïðîñòûåìíîæèòåëèåäèíñòâåííî,ñëåäîâàòåëüíî,
f
èíúåêöèÿ.Ïî
ñëåäñòâèþèçëåììû
1
A

B
ñ÷åòíî.
Ñëåäñòâèå3.
Îáúåäèíåíèåñ÷åòíîãîêîëè÷åñòâàñ÷åòíûõìíîæåñòâÿâ-
ëÿåòñÿñ÷åòíûì.
Ñëåäñòâèå4.
Ìíîæåñòâî
Z
ñ÷åòíîå.
Äîêàçàòåëüñòâî.
Z
=
N
[f
0
g[
(

N
)
.
Ëåììà5.
Ìíîæåñòâî
Q
ñ÷åòíîå.
Äîêàçàòåëüñòâî.
Ðàññìîòðèììíîæåñòâîïàð
P
=
f
(
m;n
)
j
m
2
Z
;n
2
N
g
=
Z

N
îíî
ñ÷åòíîåïîëåììå
dekart_prod
4.Î÷åâèäíî
Q

P
,ñëåäîâàòåëüíî(ïîëåììå
1)òîæå
ñ÷åòíîå.
Ëåììà6.
Ìíîæåñòâî
[0;1]
íåñ÷åòíîå.
Äîêàçàòåëüñòâî.
Ïðåäïîëîæèìîáðàòíîå,ò.å.÷òî
[0;1]=
f
a
1
;:::;a
n
;:::
g
ò.å.ìîæíîçàíóìå-
ðîâàòüâñå÷èñëà,ïðèíàäëåæàùèåîòðåçêó.Ðàññìîòðèìäåñÿòè÷íûåçàïèñè
êàæäîãîèçýòèõ÷èñåë:
a
i
=0
;
i
1
;:::;
i
n
;:::
.Åñëèâîçìîæíîäâàâàðèàíòà
çàïèñè,áóäåì(äëÿîïðåäåëåííîñòè)âûáèðàòüâàðèàíòñíóëÿìè,àíåñ
äåâÿòêàìè.Ïîñòðîèìïîñëåäîâàòåëüíîñòü

i
=
(
1
;
i
i
=2;
2
;
i
i
6
=2
:
.Î÷åâèäíî,
÷èñëî
b
=0
;
1

2
:::
n
:::
ïðèíàäëåæèò
[0
;
1]
íîíåðàâíîíèîäíîìóèç÷è-
ñåë
a
i
,ïîñêîëüêóîòëè÷àåòñÿîòêàæäîãîèçíèõïîêðàéíåéìåðåâîäíîì
ðàçðÿäå.Ïðîòèâîðå÷èå.
1
Îïðåäåëåíèå12.
Ìîùíîñòüìíîæåñòâà
[0
;
1]
íàçûâàåòñÿ
êîíòèíóóì
èîáîçíà÷àåòñÿ
j
[0
;
1]
j
=
c
.
1
Âíèìàòåëüíûé÷èòàòåëüìîæåòçàìåòèòü,÷òîçàïèñè
0
;
49
::
9
:::
è
0
;
50
:::
0
:::
õîòÿè
îòëè÷àþòñÿâêàæäîìðàçðÿäå,íîîáîçíà÷àþòîäíîèòîæå÷èñëî.Íîâíàøåìñëó÷àå
÷èñëî
b
ñîñòàâëåíîòîëüêîèçöèôð1è2,ïîýòîìóíåìîæåòèìåòüàëüòåðíàòèâíîé
ôîðìûçàïèñè.
24
ÃËÀÂÀ4.ÌÍÎÆÅÑÒÂÀ.ÌÎÙÍÎÑÒÜ.×ÈÑËÎÂÛÅÌÍÎÆÅÑÒÂÀ
Ñëåäñòâèå5.
Ìíîæåñòâî
R
íåñ÷åòíîå.
Ïîñëåäñòâèþèçëåììû
1äëÿäîêàçàòåëüñòâàñ÷åòíîñòèìíîæåñòâà
A
äîñòàòî÷íîïîñòðîèòüåãîèíüåêöèþâ
N
.Ýòîÿâëÿåòñÿ÷àñòíûìñëó÷àåì
íåêîòîðîãîîáùåãîôàêòà,êîòîðûéïîçâîëÿåòëåãêîäîêàçûâàòüðàâíîìîù-
íîñòüìíîæåñòâ.
Òåîðåìà6
(ÊàíòîðÁåðíøòåéí)
.
Ïóñòüäëÿìíîæåñòâ
A
è
B
ñóùåñòâó-
þòèíúåêöèè
f
:
A
7!
B
è
g
:
B
7!
A
.Òîãäà
j
A
j
=
j
B
j
,ò.å.ìíîæåñòâà
A
è
B
ðàâíîìîùíû.
A
0
A
1
A
2
A
3
:::
A
1
B
0
B
1
B
3
:::
B
1
f
g

1
g

1
B
2
f
f
Ðèñ.4.1:ÒåîðåìàÊàíòîðàÁåðíøòåéíà.
geom_proiz
Äîêàçàòåëüñòâî.
Îáîçíà÷èì
A
0
=
A
n
g
(
B
)
,
B
0
=
B
n
f
(
A
)
.
2
A
1
=
g
(
B
0
)
,
B
1
=
f
(
A
0
)
.
Íåñëîæíîçàìåòèòü,÷òî
A
0
\
A
1
=
;
,
B
0
\
B
1
=
;
èïîñòðîèòüáèåêöèþ
ìåæäó
A
0
[
A
1
è
B
0
[
B
1
.
Äàëååðàññìîòðèììíîæåñòâà
^
A
1
=
A
n
(
A
0
[
A
1
)
è
^
B
1
=
B
n
(
B
0
[
B
1
)
è
ïðîâåäåìäëÿíèõàíàëîãè÷íîåïîñòðîåíèå:
A
2
=
^
A
1
n
g
(
^
B
1
)
,
B
2
=
^
B
1
n
f
(
^
A
1
)
è
A
3
=
g
(
B
2
)
,
B
3
=
f
(
A
2
)
.Ïðîäîëæèìýòîïîñòðîåíèå,íà
k
ìøàãåáó-
äåìðàññìàòðèâàòüìíîæåñòâà
^
A
k
=
A
n
(
2
k

1
S
i
=0
A
i
è
^
B
k
=
B
n
(
2
k

1
S
i
=0
B
i
)
.
2
Íàïîìíèì,÷òî
f
(
A
)
îáîçíà÷àåòîáðàçìíîæåñòâà
A
ïðèîòîáðàæåíèè
f
.
4.2.×ÈÑËÎÂÛÅÌÍÎÆÅÑÒÂÀ
25
Äëÿíèõïîñòðîèì
A
2
k
=
^
A
k
n
g
(
^
B
k
)
,
B
2
k
=
^
B
k
n
f
(
^
A
k
)
è
A
2
k
+1
=
g
(
B
2
k
)
,
B
2
k
+1
=
f
(
A
2
k
)
:
Òàêæåîáîçíà÷èì
A
1
=
A
n
(
1
S
i
=0
A
i
è
B
1
=
B
n
(
1
S
i
=0
B
i
)
.
Èòàê,ïîëó÷èëèðàçáèåíèÿìíîæåñòâ
A
è
B
íàíåïåðåñåêàþùèåñÿêîì-
ïîíåíòû:
A
=

1
[
j
=0
A
k

[
A
1
;
B
=

1
[
j
=0
B
k

[
B
1
;
Ïîñòðîèìòåïåðüáèåêòèâíîåñîîòâåòñòâèåìåæäóíèìè.Ðàññìîòðèìîòîá-
ðàæåíèå
F
(
a
)=
(
f
(
a
)
;a
2
A
2
k
èëè
a
2
A
1
;
g

1
(
a
)
;a
2
A
2
k
+1
:
Îíîÿâëÿåòñÿáèåêöèåé,ò.ê.äëÿêàæäîãî
k
=0
;
1
;
2
;:::
îòîáðàæåíèå
f
:
A
2
k
7!
B
2
k
+1
áóäåòáèåêöèåéè
g

1
:
A
2
k
+1
7!
B
2
k
+2
òîæåáóäåòáèåêöèåé.
4.2×èñëîâûåìíîæåñòâà
Âýòîìðàçäåëåðàññìàòðèâàþòñÿðàçëè÷íûåïîäìíîæåñòâàìíîæåñòâàäåé-
ñòâèòåëüíûõ÷èñåë
R
èèõñâîéñòâà.Ñîîòâåòñòâåííî
R
ÿâëÿåòñÿîáúåìëþ-
ùèììíîæåñòâîìèïîäîïîëíåíèåì
A
ïîíèìàåòñÿ
A
=
R
n
A
.
4.3Îêðåñòíîñòè.
Îïðåäåëåíèå13.
Îêðåñòíîñòüþòî÷êè
a
íàçûâàåòñÿèíòåðâàë
U
(
a
)=(
b;c
)
,
ãäå
bac
.
Îïðåäåëåíèå14.
Ïðîêîëîòîéîêðåñòíîñòüþòî÷êè
a
íàçûâàåòñÿìíî-
æåñòâî

U
(
a
)=(
b;a
)
[
(
a;c
)
,ãäå
bac
.
Îïðåäåëåíèå15.
"
oêðåñòíîñòüþòî÷êè
a
íàçûâàåòñÿèíòåðâàë
u
"
(
a
)=(
a

";a
+
"
)
,
ãäå
"�
0
.
Îïðåäåëåíèå16.
Ïðîêîëîòîé
"
îêðåñòíîñòüþòî÷êè
a
íàçûâàåòñÿìíî-
æåñòâî

U
"
(
a
)=(
a

";a
)
[
(
a;a
+
"
)
,ãäå
"�
0
.
26
ÃËÀÂÀ4.ÌÍÎÆÅÑÒÂÀ.ÌÎÙÍÎÑÒÜ.×ÈÑËÎÂÛÅÌÍÎÆÅÑÒÂÀ
Óòâåðæäåíèå7.
Ïåðåñå÷åíèåäâóõîêðåñòíîñòåé(ïðîêîëîòûõîêðåñò-
íîñòåé)ÿâëÿåòñÿîêðåñòíîñòüþ(ïðîêîëîòîéîêðåñòíîñòüþ)
Óòâåðæäåíèå8.
Îáúåäèíåíèåäâóõîêðåñòíîñòåé(ïðîêîëîòûõîêðåñò-
íîñòåé)ÿâëÿåòñÿîêðåñòíîñòüþ(ïðîêîëîòîéîêðåñòíîñòüþ)
Óòâåðæäåíèå9.
Ïåðåñå÷åíèåäâóõ
"
îêðåñòíîñòåé(ïðîêîëîòûõ
"
îêðåñòíîñòåé)
ÿâëÿåòñÿ
"
îêðåñòíîñòüþ(ïðîêîëîòîé
"
îêðåñòíîñòüþ)
Óòâåðæäåíèå10.
Îáúåäèíåíèåäâóõ
"
îêðåñòíîñòåé(ïðîêîëîòûõ
"

îêðåñòíîñòåé)ÿâëÿåòñÿîêðåñòíîñòüþ(ïðîêîëîòîé
"
îêðåñòíîñòüþ)
Äîêàæåìïîñëåäíååóòâåðæäåíèå:
U
"
0
(
a
)
[
U
"
00
(
a
)=(
a

"
0
;a
+
"
0
)
[
(
a

"
00
;a
+
"
00
)=
=
(
a

max(
"
0
;"
00
)
;a
+max(
"
0
;"
00
)=
U
max(
"
0
;"
00
)
(
a
)
;
÷òîèòðåáîâàëîñüäîêà-
çàòü.
Äîêàçàòåëüñòâîîñòàëüíûõóòâåðæäåíèéïðåäîñòàâëÿåòñÿ÷èòàòåëþâ
êà÷åñòâåóïðàæíåíèÿ.
Îïðåäåëåíèå17.
Ìíîæåñòâî
A
íàçûâàåòñÿ
îòêðûòûì
,åñëèäëÿêàæ-
äîéòî÷êè
a
2
A
íàéäåòñÿ
"�
0
(äëÿêàæäîéòî÷êè,âîçìîæíî,ñâîé
"
)
òàêîå,÷òî
U
"
(
a
)

A
.
Íàïðèìåð,ìíîæåñòâà
(0
;
1)
,
(1
;
+
1
)
,
R
ÿâëÿþòñÿîòêðûòûìè,à
N
,
[

1
;
1]
,
(
�1
;
0]
íåÿâëÿþòñÿ.
4.4Àêñèîìàîòäåëèìîñòè.
Îïðåäåëåíèå18.
Ïóñòüìíîæåñòâî
A
òàêîâî,÷òîâñååãîýëåìåíòû
íåïðåâîñõîäÿòíåêîòîðîãî÷èñëà
a
0
,ò.å.
8
a
2
Aa
6
a
0
.Òîãäàãîâîðÿò,
÷òîìíîæåñòâî
A
îãðàíè÷åíîñâåðõó,à
a
0
åãîâåðõíÿÿãðàíèöà(âåðõíÿÿ
ãðàíü)èîáîçíà÷àþòòàê:
A
6
a
0
.
Îïðåäåëåíèå19.
Ïóñòüìíîæåñòâî
A
òàêîâî,÷òîâñååãîýëåìåíòû
íåìåíüøåíåêîòîðîãî÷èñëà
a
1
,ò.å.
8
a
2
Aa

a
1
.Òîãäàãîâîðÿò,÷òî
ìíîæåñòâî
A
îãðàíè÷åíîñíèçó,à
a
1
åãîíèæíÿÿãðàíèöà(íèæíÿÿ
ãðàíü)èîáîçíà÷àþòòàê:
A

a
1
.
Îïðåäåëåíèå20.
Åñëèìíîæåñòâî
A
îãðàíè÷åíîêàêñâåðõó,òàêèñíè-
çó,òîãîâîðÿò,÷òî
A
îãðàíè÷åííîåìíîæåñòâî.
Îïðåäåëåíèå21.
Ïóñòü÷èñëî
a
ÿâëÿåòñÿâåðõíåéãðàíüþìíîæåñòâà
A
,àíèêàêîåìåíüøåååãîíåÿâëÿåòñÿ.Áîëååôîðìàëüíî
A
6
a
è
(
8
a
0
a
)
A
6
6
a
0
.Òîãäàãîâîðÿò,÷òî
a
åñòü
òî÷íàÿâåðõíÿÿãðàíü
ìíîæåñòâà
A
èîáîçíà÷àþòòàê:
a
=sup
A
.
4.4.ÀÊÑÈÎÌÀÎÒÄÅËÈÌÎÑÒÈ.
27
Îïðåäåëåíèå22.
Ïóñòü÷èñëî
a
ÿâëÿåòñÿíèæíåéãðàíüþìíîæåñòâà
A
,àíèêàêîåáîëüøåååãîíåÿâëÿåòñÿ.Áîëååôîðìàëüíî
A

a
è
(
8
a
0
�a
)
A
6

a
0
.
Òîãäàãîâîðÿò,÷òî
a
åñòü
òî÷íàÿíèæíÿÿãðàíü
ìíîæåñòâà
A
èîáî-
çíà÷àþòòàê:
a
=inf
A
.
Ïðèìåð1.
Ðàññìîòðèììíîæåñòâî
(0
;
1)
.Âýòîììíîæåñòâåíåòíàè-
áîëüøåãîýëåìåíòà.Âíåêîòîðîìñìûñëå÷èñëî1õîòåëîñüáûíàçâàòüìàê-
ñèìóìîì,íîîíîíåïðèíàäëåæèòäàííîìóìíîæåñòâó!
Ïðèìåð2.
Ðàññìîòðèììíîæåñòâî
A
=
f
m
n
j
m
2

2
n
2
;m;n
2
N
g
:
Î÷åâèäíî,÷òîâñåýëåìåíòûýòîãîìíîæåñòâàìåíüøå,÷åì
p
2
,íîäëÿëþáî-
ãî
"�
0
ìîæíîïîäîáðàòü
m=n
2A
òàê,÷òî
m=n�
p
2

"
.Ñëåäîâàòåëüíî
sup
A
=
p
2
.
Îïðåäåëåíèå23.
Ïóñòüìíîæåñòâà
A
è
B
òàêîâû,÷òî
8
a
2
A
8
b
2
Ba
6
b
.
Òîãäàýòîîáîçíà÷àþòòàê:
A
6
B
.
Àêñèîìàîòäåëèìîñòè.
Ïóñòü
A
6
B
,
A;B
6
=
?
.Òîãäàíàéäåòñÿòàêîå
÷èñëî
c
,÷òî
A
6
c
6
B
.
Àêñèîìàîòäåëèìîñòèãîâîðèòîâåñüìàâàæíîìñâîéñòâåìíîæåñòâà
äåéñòâèòåëüíûõ÷èñåëíåïðåðûâíîñòè,òîåñòüîòîì,÷òîâíåìíåò¾äû-
ðîê¿.Åñëèïîñìîòðåòü,íàïðèìåð,íàìíîæåñòâîðàöèîíàëüíûõ÷èñåë,òî
ìîæíîóâèäåòü,÷òîäëÿíåãîýòààêñèîìàíåâåðíà.
suprem
Òåîðåìà11.
Åñëèíåïóñòîåìíîæåñòâî
A

R
îãðàíè÷åíîñâåðõó,òî
ñóùåñòâóåò
sup
A
òî÷íàÿâåðõíÿÿãðàíü,.
Äîêàçàòåëüñòâî.
Ðàññìîòðèì
B
ìíîæåñòâîâñåõâåðõíèõãðàíåéìíîæåñòâà
A
.Îíîíåïóñòî
ïîîïðåäåëåíèþîãðàíè÷åííîãîñâåðõóìíîæåñòâà.Î÷åâèäíî,÷òî
A
6
B
,
òîãäà,ïîàêñèîìåîòäåëèìîñòè,ñóùåñòâóåòðàçäåëÿþùàÿèõòî÷êà
c
,òàêàÿ,
÷òî
A
6
c
6
B
.Äîêàæåì,÷òîýòàòî÷êàèåñòü
sup
A
.Äåéñòâèòåëüíî,
ïîñêîëüêó
A
6
c
,òîîíàåñòüâåðõíÿÿãðàíüìíîæåñòâà
A
.Ðàññìîòðèì
ïðîèçâîëüíóþ
a
0
a
.Ïîñêîëüêó
a
0
a
6
B
,òî
a
0
=
2
B
,àçíà÷èò,
a
0
íå
ÿâëÿåòñÿâåðõíåéãðàíüþ
A
.Òåîðåìàäîêàçàíà.
infim
Òåîðåìà12.
Åñëèíåïóñòîåìíîæåñòâî
A
îãðàíè÷åíîñíèçó,òîñóùå-
ñòâóåòåãîòî÷íàÿíèæíÿÿãðàíü,
inf
A
.
Óïðàæíåíèå7.
Äîêàçàòüòåîðåìó
infim
12.
28
ÃËÀÂÀ4.ÌÍÎÆÅÑÒÂÀ.ÌÎÙÍÎÑÒÜ.×ÈÑËÎÂÛÅÌÍÎÆÅÑÒÂÀ
4.5Âëîæåííûåèñòÿãèâàþùèåñÿñèñòåìûîò-
ðåçêîâ.
Îïðåäåëåíèå24.
Ñèñòåìàîòðåçêîâ
f
[
a
n
;b
n
]
g
1
n
=1
íàçûâàåòñÿ
âëîæåí-
íîé
,åñëè
[
a
1
;b
1
]

[
a
2
;b
2
]

:::

[
a
n
;b
n
]

:::;
ò.å.
a
1
6
a
2
6

6
a
n
6

6
b
n
6

6
b
2
6
b
1
.
Îïðåäåëåíèå25.
Ñèñòåìàîòðåçêîâ
f
[
a
n
;b
n
]
g
1
n
=1
íàçûâàåòñÿ
ñòÿãèâà-
þùåéñÿ
,åñëèîíàâëîæåííàÿè,êðîìåòîãîäëÿëþáîãî
8
"�
0
íàéäåòñÿ
n
2
N
,òàêîå,÷òî
j
b
n

a
n
j
":
Òåîðåìà13.
Ïóñòüñèñòåìàîòðåçêîâ
f
[
a
n
;b
n
]
g
1
n
=1
ÿâëÿåòñÿâëîæåí-
íîé.Òîãäàïðåñå÷åíèåîòðåçêîâíåïóñòî:
1
\
n
=1
[
a
n
;b
n
]
6
=
?
:
Äîêàçàòåëüñòâî.
Ðàññìîòðèììíîæåñòâà
A
=
f
a
n
;n
2
N
g
è
A
=
f
a
n
;n
2
N
g
.Ïîóñëîâèþ
A
6
B
.ÒîãäàèçàêñèîìûîòäåëèìîñòèÒîãäàäëÿâñåõ,ò.å.,àçíà÷èò÷òî
èòðåáîâàëîñüäîêàçàòü
styag
Òåîðåìà14.
Ïóñòüñèñòåìàîòðåçêîâ
f
[
a
n
;b
n
]
g
1
n
=1
ÿâëÿåòñÿñòÿãèâàþ-
ùåéñÿ.Òîãäàïåðåñå÷åíèåîòðåçêîâñîñòîèòðîâíîèçîäíîéòî÷êè:
1
\
n
=1
[
a
n
;b
n
]=
f
c
g
:
Äîêàçàòåëüñòâî.
Ïîïðåäûäóùåéòåîðåìåïåðåñå÷åíèå
1
T
n
=1
[
a
n
;b
n
]
íåïóñòî.Äîïóñòèìîíîñî-
ñòîèòèçäâóõèëèáîëååòî÷åê.Âîçüìåìäâåòî÷êè
c
0
è
c
00
èâîçüìåì
1
2
j
c
0

c
00
j
âêà÷åñòâå
"
.Òîãäà,äëÿíåêîòîðîãî
n
2
N
âûïîëíåíî
j
b
n

a
n
j

1
2
j
c
0

c
00
j
,
÷òîíåâîçìîæíîïðè
c
0
;c
00
2
[
a
n
;b
n
]
.Òåîðåìàäîêàçàíà.
4.6Îòêðûòûåèçàìêíóòûåìíîæåñòâà.
3
Íàïîìíèìîïðåäåëåíèåîòêðûòîãîìíîæåñòâà.
3
Íåîáÿçàòåëüíûéìàòåðèàë
4.6.ÎÒÊÐÛÒÛÅÈÇÀÌÊÍÓÒÛÅÌÍÎÆÅÑÒÂÀ.
29
Îïðåäåëåíèå26.
Ìíîæåñòâîíàçûâàþòîòêðûòûì,åñëèëþáàÿòî÷êà
âõîäèòâíåãîâìåñòåñíåêîòîðîéîêðåñòíîñòüþ,ò.å.
8
a
2
A
9
U
(
a
)

A
.
Çàìå÷àíèå.
Âäàííîìîïðåäåëåíèèñëîâîîêðåñòíîñòüìîæíîçàìåíèòü
íà
"
îêðåñòíîñòü.
Îïðåäåëåíèå27.
Ìíîæåñòâîíàçûâàþòçàìêíóòûì,åñëèåãîäîïîëíå-
íèåîòêðûòî.
Ïðèìåð3.
Ìíîæåñòâî(0,1)îòêðûòîå,à[0,1]çàìêíóòîå(ïîäó-
ìàéòå,ïî÷åìó?).
Îïðåäåëåíèå28.
Ãðàíè÷íîéòî÷êîéìíîæåñòâàíàçûâàåòñÿòî÷êà,â
êàæäîéîêðåñòíîñòèêîòîðîéñîäåðæàòñÿêàêòî÷êè,ïðèíàäëåæàùèå
ìíîæåñòâó,òàêèòî÷êè,åìóíåïðèíàäëåæàùèå,ò.å.
8
U
(
a
)
U
(
a
)
\
A
6
=
?
;
U
(
a
)
\
A
6
=
?
.
Ïðèìåð4.
Ìíîæåñòâà(0,1)è[0,1]èìåþòãðàíè÷íûåòî÷êè0è1.
Îïðåäåëåíèå29.
Ìíîæåñòâîâñåõãðàíè÷íûõòî÷åêíåêîòîðîãîìíî-
æåñòâàíàçûâàþòåãîãðàíèöåéèîáîçíà÷àþò
@A
.
Óïðàæíåíèå8.
Ïóñòü
A
îòêðûòî.Äîêàçàòü,÷òî
A
\
@A
=
?
.
Óïðàæíåíèå9.
Ïóñòü
A
çàìêíóòî.Äîêàçàòü,÷òî
@A

A
.
Îïðåäåëåíèå30.
Òî÷êà
a
íàçûâàåòñÿïðåäåëüíîéòî÷êîéìíîæåñòâà
A
,åñëèâêàæäîéååïðîêîëîòîéîêðåñòíîñòèñîäåðæàòñÿòî÷êè,ïðè-
íàäëåæàùèåìíîæåñòâó
A
,ò.å.
8

U
(
a
)

U
(
a
)
\
A
6
=
?
.Ìíîæåñòâîâñåõ
ïðåäåëüíûõòî÷åêìíîæåñòâà
A
îáîçíà÷àåòñÿ
P
(
A
)
.
mn_pred
Òåîðåìà15.
Êàæäîåáåñêîíå÷íîåîãðàíè÷åííîåìíîæåñòâîèìååòõîòÿ
áûîäíóïðåäåëüíóþòî÷êó.
Äîêàçàòåëüñòâî.
Ïîñòðîèìñòÿãèâàþùóþñÿñèñòåìóîòðåçêîâ
f
[
a
n
;b
n
]
g
1
n
=1
ñëåäóþùèìîá-
ðàçîì:

Ìíîæåñòâî
A
îãðàíè÷åíî,ñëåäîâàòåëüíîëåæèòíàíåêîòîðîìîòðåçêå
m
6
A
6
M
.Âîçüìåì
a
1
=
m
è
b
1
=
M
.

Ïóñòü
c
=
a
1
+
b
1
2
ñåðåäèíàîòðåçêà
[
a
1
;b
1
]
.Ðàññìîòðèìîòðåçêè
[
a
1
;c
]
è
[
c;b
1
]
.Õîòÿáûíàîäíîìèçíèõëåæèòáåñêîíå÷íîå÷èñëîòî÷åêìíî-
æåñòâà
A
(èíà÷åîíîáûëîáûêîíå÷íûì).Äîïóñòèìýòîîòðåçîê
[
a
1
;c
]
òîãäàâûáåðåì
a
2
=
a
1
;b
2
=
c
(âïðîòèâíîìñëó÷àå
a
2
=
c;b
2
=
b
1
)
30
ÃËÀÂÀ4.ÌÍÎÆÅÑÒÂÀ.ÌÎÙÍÎÑÒÜ.×ÈÑËÎÂÛÅÌÍÎÆÅÑÒÂÀ

Íàêàæäîìñëåäóþùåìøàãå.Ïóñòü
c
=
a
n
+
b
n
2
ñåðåäèíàîòðåç-
êà
[
a
n
;b
n
]
.Ðàññìîòðèìîòðåçêè
[
a
n
;c
]
è
[
c;b
n
]
.Õîòÿáûíàîäíîìèç
íèõëåæèòáåñêîíå÷íîå÷èñëîòî÷åêìíîæåñòâà
A
.Äîïóñòèìýòîîò-
ðåçîê
[
a
n
;c
]
òîãäàâûáåðåì
a
n
+1
=
a
n
;b
n
+1
=
c
(âïðîòèâíîìñëó÷àå
a
n
+1
=
c;b
n
+1
=
b
n
)
Î÷åâèäíî,ñèñòåìàîòðåçêîâ
f
[
a
n
;b
n
]
g
1
n
=1
áóäåòâëîæåííîé.Äîêàæåì,÷òî
îíàñòÿãèâàþùàÿñÿ.Äåéñòâèòåëüíî,äëèíà
n

ãîîòðåçêàðàâíà
M

m
2
n

1
.Ïî
íåðàâåíñòâóÁåðíóëëè
2
n

1

1+(
n

1)

1=
n
,ñëåäîâàòåëüíî
M

m
2
n

1

1
n
.
Âûáðàâ
n
=[
M

m
"
]+1
,ãäåêâàäðàòíûåñêîáêèîçíà÷àþòöåëóþ÷àñòü÷èñ-
ëà,ïîëó÷èì
j
b
n

a
n
j
=
M

m
2
n

1

M

m
n
"
,îòêóäàèâûòåêàåò,÷òîñèñòåìà
f
[
a
n
;b
n
]
g
1
n
=1
ñòÿãèâàþùàÿñÿ.Ïîòåîðåìå
styag
14ñóùåñòâóåòåäèíñòâåííàÿ
òî÷êàïåðåñå÷åíèÿ
c
.Äîêàæåì,÷òîîíàÿâëÿåòñÿïðåäåëüíîòî÷êîéìíî-
æåñòâà
A
.Äåéñòâèòåëüíî,ðàññìîòðèìïðîèçâîëüíîå
"�
0
èïðîêîëîòóþ
"

îêðåñòíîòü.Î÷åâèäíî,íàéäåòñÿ
n
2
N
òàêîå,÷òî
j
b
n

a
n
j

"
2
,àçíà÷èò
[
a
n
;b
n
]


U
"
(
c
)
.Òîãäàâ
[
a
n
;b
n
]
ñîäåðæèòñÿáåñêîíå÷íîå÷èñëîòî÷åêìíî-
æåñòâà
A
,àçíà÷èòâ

U
"
(
c
)
òîæå.Ýòîâåðíîäëÿëþáîãî
"�
0
,àçíà÷èò
c
ïðåäåëüíàÿòî÷êàìíîæåñòâà
A
.Òåîðåìàäîêàçàíà.
Óïðàæíåíèå10.
Äîêàçàòü,÷òîçàìêíóòîåìíîæåñòâîñîäåðæèòâñå
ñâîèïðåäåëüíûåòî÷êè,ò.å.åñëè
A
çàìêíóòî,òî
P
(
A
)

A
.
Óïðàæíåíèå11.
Äîêàçàòü,÷òî
P
(
A
)
[
A
çàìêíóòîåìíîæåñòâîïðè
ëþáîì
A
.
Óïðàæíåíèå12.
Äîêàçàòü,÷òî
@A
[
A
çàìêíóòîåìíîæåñòâîïðè
ëþáîì
A
.
Óïðàæíåíèå13.
Äîêàçàòü,÷òî
@A
[
A
=
P
(
A
)
[
A
ïðèëþáîì
A
.
Ãëàâà5
Ïîñëåäîâàòåëüíîñòè.
5.1Îïðåäåëåíèåèïðèìåðû
Îïðåäåëåíèå31.
Ïîñëåäîâàòåëüíîñòüþ
(÷èñëîâîé)íàçûâàåòñÿôóíê-
öèÿ,îïðåäåëåííàÿíàìíîæåñòâåíàòóðàëüíûõ÷èñåë
a
:
N
7!
R
.Ïðèíÿòî
îáîçíà÷åíèå
a
(
n
)=
a
n
,÷èñëî
a
n
íàçûâàåòñÿ
n
ì÷ëåíîìïîñëåäîâàòåëüíî-
ñòè.Ñàìóïîñëåäîâàòåëüíîñòüîáû÷íîçàïèñûâàþòñëåäóþùèìîáðàçîì:
f
a
n
g
1
n
=1
.
Ïðèìåð5.
Àðèôìåòè÷åñêàÿïðîãðåññèÿ.Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
,
êàæäûé÷ëåíêîòîðîé(íà÷èíàÿñî2ãî)ðàâåíñóììåïðåäûäóùåãîñíåêî-
òîðûì÷èñëîì
d
,êîòîðîåíàçûâàþòðàçíîñòüþïðîãðåññèè,ò.å.
8
n
2
N
(
a
n
+1
=
a
+
d
.
Âåðíûôîðìóëûäëÿ
n
ãî÷ëåíà
a
n
=
a
1
+
d
(
n

1)
èäëÿñóììûïåðâûõ
n
÷ëåíîâïðîãðåññèè
S
n
=
a
1
+
a
2
+
:::
+
a
n
=(2
a
1
+
d
(
n

1))

n
2
:
Äîêàçàòåëüñòâîèíäóêöèåéïî
n
2
N
,ïðåäîñòàâëÿåòñÿ÷èòàòåëþâ
êà÷åñòâåóïðàæíåíèÿ.
Ïðèìåð6.
Ãåîìåòðè÷åñêàÿïðîãðåññèÿ.Ïîñëåäîâàòåëüíîñòü
f
b
n
g
1
n
=1
,êàæ-
äûé÷ëåíêîòîðîé(íà÷èíàÿñî2ãî)ðàâåíïðîèçâåäåíèþïðåäûäóùåãîñ
íåêîòîðûì÷èñëîì
q
,êîòîðîåíàçûâàþòçíàìåíàòåëåìïðîãðåññèè,ò.å.
8
n
2
N
(
a
n
+1
=
a
n
+
d
)
.Âåðíûôîðìóëûäëÿ
n
ãî÷ëåíà
b
n
=
b
1

q
n

1
31
32
ÃËÀÂÀ5.ÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.
èäëÿñóììûïåðâûõ
n
÷ëåíîâïðîãðåññèè
S
n
=
b
1
+
b
2
+
:::
+
b
n
=
b
1

q
n

1
q

1
;
åñëè
q
6
=1
Äîêàçàòåëüñòâî.
Èíäóêöèÿïî
n
2
N
,ïðåäîñòàâëÿåòñÿ÷èòàòåëþâêà÷åñòâåóïðàæíåíèÿ.
Ïðèìåð7.
×èñëàÔèáîíà÷÷è.Ïîñëåäîâàòåëüíîñòü
f
'
n
g
1
n
=1
âêîòîðîé
ïåðâûåäâà÷ëåíàðàâíû1,àêàæäûéïîñëåäóþùèéðàâåíñóììåäâóõïðåäû-
äóùèõ.
Óòâåðæäåíèå16.
Âåðíàôîðìóëàäëÿ
n
ãî÷ëåíàïîñëåäîâàòåëüíîñòè
Ôèáîíà÷÷è:
'
n
=
1
p
5
"
1+
p
5
2
!
n


1

p
5
2
!
n
#
(ôîðìóëàÁèíý)
:
Äîêàçàòåëüñòâî.
Îáîçíà÷èì

1
=
1+
p
5
2
,

2
=
1

p
5
2
.Íåñëîæíîçàìåòèòü,÷òî

1
+

2
=1
;

1


2
=

1
,ñëåäîâàòåëüíî,ïîòåîðåìåÂèåòà,ýòè÷èñëàåñòüêîðíèêâàä-
ðàòíîãîóðàâíåíèÿ

2



1=0
,àçíà÷èò

2
i
=

i
+1
,
i
=1
;
2
.Äîêàæåì
òåïåðüôîðìóëóÁèíýèíäóêöèåéïî
n
2
N

Áàçàèíäóêöèè.Ïðîâåðÿåì
'
1
=
1
p
5
(

1
1


1
2
)=
p
5
p
5
=1;
'
2
=
1
p
5
(

2
1


2
2
)=
1
p
5
(

1


2
)(

1
+

2
)=
p
5
p
5
=1
:
Òàêèìîáðàçîì,ôîðìóëàâåðíàïðè
n
=1
;
2
.

ØàãèíäóêöèèÏîñêîëüêó
'
1
;
2
êîðíèóðàâíåíèÿ

2



1=0
,
òîäëÿíèõâûïîëíåíîòîæäåñòâî:

2
=

+1
.Óìíîæèâîáå÷àñòèíà

n

1
,ïîëó÷èì

n
+1
=

n
+

n

1
:
(5.1.1)
sumst
Ïðåäïîëîæèì,÷òîôîðìóëàÁèíýâåðíàäëÿ
1
;
2
;:::;n
èäîêàæåìåå
äëÿ
n
+1
:
'
n
+1
=
'
n
+
'
n

1
=
1
p
5
(

n
1


n
2
)+
1
p
5
(

n

1
1


n

1
2
)=
=
1
p
5

(

n
1
+

n

1
1
)

(

n
2
+

n

1
2
)

:
5.2.ÑÂÎÉÑÒÂÀÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÅÉ.ËÎÂÓØÊÈÈÊÎÐÌÓØÊÈ.
33
Ïðèìåíèâðàâåíñòâî
sumst
5.1.1,ïîëó÷èì
'
n
+1
=
1
p
5


n
+1
1


n
+1
2

.
5.2Ñâîéñòâàïîñëåäîâàòåëüíîñòåé.Ëîâóøêè
èêîðìóøêè.
Îïðåäåëåíèå32.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
íàçûâàþò
îãðàíè÷åí-
íîéñâåðõó
åñëèâñååå÷ëåíûíåïðåâîñõîäÿòíåêîòîðîãî÷èñëà
M
,ò.å.
ïðèâñåõ
n
2
N
âûïîëíåíîíåðàâåíñòâî
a
n
6
M
.
Îïðåäåëåíèå33.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
íàçûâàþò
îãðàíè÷åí-
íîéñíèçó
åñëèâñååå÷ëåíûíåìåíüøåíåêîòîðîãî÷èñëà
m
,ò.å.ïðè
âñåç
n
2
N
âûïîëíåíîíåðàâåíñòâî
a
n

m
.
Îïðåäåëåíèå34.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
íàçûâàþò
îãðàíè÷åí-
íîé
åñëèîíàîãðàíè÷åíàñâåðõóèñíèçó.
Îïðåäåëåíèå35.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
íàçûâàþò
âîçðàñòàþ-
ùåé
åñëèêàæäûéåå÷ëåíñòðîãîáîëüøåïðåäûäóùåãî.Ò.å.
8
n
2
N
a
n
+1
�a
n
.
Îïðåäåëåíèå36.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
íàçûâàþò
óáûâàþùåé
åñëèêàæäûéåå÷ëåíñòðîãîìåíüøåïðåäûäóùåãî.Ò.å.
8
n
2
N
a
n
+1
a
n
.
Îïðåäåëåíèå37.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
íàçûâàþò
íåóáûâàþ-
ùåé
åñëèêàæäûéåå÷ëåííåìåíüøåïðåäûäóùåãî.Ò.å.
8
n
2
N
a
n
+1

a
n
Îïðåäåëåíèå38.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
íàçûâàþò
íåâîçðàñòà-
þùåé
åñëèêàæäûéåå÷ëåííåáîëüøåïðåäûäóùåãî.Ò.å.
8
n
2
N
a
n
+1
6
a
n
.
Îïðåäåëåíèå39.
Âîçðàñòàþùèå,óáûâàþùèå,íåâîçðàñòàþùèåèíåóáû-
âàþùèåïîñëåäîâàòåëüíîñòèíàçûâàþò
ìîíîòîííûìè
.
Îïðåäåëåíèå40.
Ìíîæåñòâî
A
íàçûâàþò
ëîâóøêîé
ïîñëåäîâàòåëü-
íîñòè
f
a
n
g
1
n
=1
åñëèíà÷èíàÿñíåêîòîðîãîíîìåðàâñå÷ëåíûïîñëåäîâà-
òåëüíîñòèïîïàäàþòâýòîìíîæåñòâî,ò.å.
9
N
2
N
:
8
n�N
(
a
n
2
A
)
posledo_ogr
Ëåììà7.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
îãðàíè÷åíàòîãäàèòîëüêîòî-
ãäà,êîãäàñóùåñòâóåò
M�
0
,òàêîå,÷òî
[

M;M
]
ëîâóøêàïîñëåäîâà-
òåëüíîñòè.
Äîêàçàòåëüñòâî.
1)Ïóñòüïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
îãðàíè÷åíà,ò.å.ñóùåñòâóþò
m
1
è
m
2
,
òàêèå,÷òî
8
n
(
a
n
2
[
m
1
;m
2
])
.Âûáåðåì
M
=max(
m
1
;m
2
)
,òîãäà
8
n
(
a
n
2
[

M;M
])
,
ò.å.
[

M;M
]
ëîâóøêà
f
a
n
g
1
n
=1
.
34
ÃËÀÂÀ5.ÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.
2)Âîáðàòíóþñòðîíó.Ïóñòü
[

M;M
]
ëîâóøêàïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.Çíà÷èò,íà÷èíàÿñíåêîòîðîãîíîìåðà
N
âûïîëíåíûíåðàâåíñòâà

M
6
a
n
6
M
.Âûáåðåì
m
1
=min(
a
1
;a
2
;:::;a
N

1
;

M
)
è
m
2
=max(
a
1
;a
2
;:::;a
N

1
;M
)
.
Òîãäà,î÷åâèäíî,íåðàâåíñòâà
m
1
6
a
n
6
m
2
âûïîëíåíûäëÿëþáîãî
n
,÷òî
èòðåáîâàëîñüäîêàçàòü.
Îïðåäåëåíèå41.
Ìíîæåñòâî
A
íàçûâàþò
êîðìóøêîé
ïîñëåäîâàòåëü-
íîñòè
f
a
n
g
1
n
=1
åñëèâýòîìíîæåñòâîïîïàäàþò÷ëåíûïîñëåäîâàòåëüíî-
ñòèñîñêîëüóãîäíîáîëüøèìíîìåðîì.
8
N
2
N
9
n�N
:(
a
n
2
A
)
Ïðèìåð8.
Ðàññìîòðèìïîñëåäîâàòåëüíîñòü
f
(

1)
n
n
g
1
n
=1
.Èíòåðâàë
(

1
;
1]
ÿâëÿåòñÿêîðìóøêîé,íîíåëîâóøêîéýòîéïîñëåäîâàòåëüíîñòè.Àîòðå-
çîê
[

1
;
1]
ÿâëÿåòñÿèëîâóøêîéèêîðìóøêîé.
Óïðàæíåíèå14.
Äîêàçàòü,÷òîëþáàÿëîâóøêàÿâëÿåòñÿêîðìóøêîé
(äëÿòîéæåïîñëåäîâàòåëüíîñòè).
Óïðàæíåíèå15.
Ïóñòü
A
è
B
êîðìóøêèïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.
Áóäåòëèêîðìóøêîéà)
A
[
B
;á)
A
\
B
?
Óïðàæíåíèå16.
Ïóñòü
A
è
B
ëîâóøêèïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.
Áóäåòëèëîâóøêîéà)
A
[
B
;á)
A
\
B
?
Ðåøåíèå.(ïóíêòàá.)
Ïîîïðåäåëåíèþëîâóøêèíàéäóòñÿ
N
1
;N
2
òà-
êèå,÷òî
8
n�N
1
(
a
n
2
A
)
8
n�N
2
(
a
n
2
B
)
.Òîãäàâçÿâ
N
0
=max(
N
1
;N
2
)
;
ïîëó÷èì
8
n�N
0
(
a
n
2
A
[
B
)
,ñëåäîâàòåëüíî
A
\
B
ëîâóøêà.
Óïðàæíåíèå17.
Çàêîí÷èòüôðàçó:¾ìíîæåñòâî
M
íåÿâëÿåòñÿëî-
âóøêîéäëÿïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
âòîìèòîëüêîòîìñëó÷àå,
êîãäà...¿(íåèñïîëüçóÿñëîâ¾íåñóùåñòâóåò¿èëè¾íåäëÿâñåõ¿).
Óïðàæíåíèå18.
Ñóùåñòâóåòëèïîñëåäîâàòåëüíîñòü,äëÿêîòîðîéëþ-
áîéèíòåðâàëÿâëÿåòñÿêîðìóøêîé?äëÿêîòîðîéëþáîéèíòåðâàëÿâëÿ-
åòñÿëîâóøêîé?
Óïðàæíåíèå19.
Äîêàçàòü,÷òîäëÿëþáîéîãðàíè÷åííîéïîñëåäîâàòåëü-
íîñòè
f
a
n
g
1
n
=1
ñóùåñòâóåòîòðåçîêäëèíû
1
,êîòîðûéÿâëÿåòñÿå¼êîð-
ìóøêîé.
Óïðàæíåíèå20.
Äîêàçàòü,÷òîäëÿâñÿêîéîãðàíè÷åííîéìîíîòîííîé
ïîñëåäîâàòåëüíîñòèñóùåñòâóåòîòðåçîêäëèíû
1
,ÿâëÿþùèéñÿå¼ëî-
âóøêîé.
5.3.ÏÎÄÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.ÏÐÅÄÅËÈ×ÀÑÒÈ×ÍÛÉÏÐÅÄÅËÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.*
35
5.3Ïîäïîñëåäîâàòåëüíîñòè.Ïðåäåëè÷àñòè÷-
íûéïðåäåëïîñëåäîâàòåëüíîñòè.*
Îïðåäåëåíèå42.
Ïóñòüïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
=
A
N

R
.Ïîä-
ïîñëåäîâàòåëüíîñòüþíàçûâàþòáåñêîíå÷íîåïîäìíîæåñòâî
A
0
A
.Áó-
äåìñ÷èòàòü,÷òîïàðû
(
n
k
;a
n
k
)
2A
0
óïîðÿäî÷åíûòàê,÷òî
n
1
n
2
:::n
k
:::
.
Òîãäà
a
n
k
íàçûâàþò
k
ì÷ëåíîìïîäïîñëåäîâàòåëüíîñòè.
Îïðåäåëåíèå43.
Òî÷êà
a
0
íàçûâàåòñÿïðåäåëüíîéòî÷êîé(÷àñòè÷íûì
ïðåäåëîì)ïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
åñëèëþáàÿîêðåñòíîñòü
U
(
a
0
)
ÿâ-
ëÿåòñÿêîðìóøêîéïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.
Îïðåäåëåíèå44.
Òî÷êà
a
0
íàçûâàåòñÿïðåäåëîìïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
åñëèëþáàÿîêðåñòíîñòü
U
(
a
0
)
ÿâëÿåòñÿëîâóøêîéïîñëåäîâà-
òåëüíîñòè
f
a
n
g
1
n
=1
.Âýòîìñëó÷àåïîñëåäîâàòåëüíîñòüíàçûâàåòñÿñõî-
äÿùåé,àïðåäåëïîñëåäîâàòåëüíîñòèîáîçíà÷àåòñÿòàê:
a
0
=lim
n
!1
a
n
.
Óòâåðæäåíèå17.
Ïóñòü
a
0
=lim
n
!1
a
n
è
a
00
=lim
n
!1
a
n
.Òîãäà
a
0
=
a
00
.Äðó-
ãèìèñëîâàìèóïîñëåäîâàòåëüíîñòèíåìîæåòáûòüäâàðàçíûõïðåäåëà.
Óïðàæíåíèå21.
Äîêàçàòü,÷òîïðåäåëïîñëåäîâàòåëüíîñòèâñåãäàÿâ-
ëÿåòñÿååïðåäåëüíîéòî÷êîé.
chas_pr
Ëåììà8.
Ìíîæåñòâî
A
ÿâëÿåòñÿêîðìóøêîéïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
òîãäàèòîëüêîòîãäà,êîãäà
A
ÿâëÿåòñÿëîâóøêîéíåêîòîðîéïîäïîñëå-
äîâàòåëüíîñòè
f
a
n
k
g
1
k
=1
.
Äîêàçàòåëüñòâî.
Ïóñòü
A
êîðìóøêàïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.Òîãäàíàéäåòñÿ÷ëåí
a
n
1
2
A
.Ïîîïðåäåëåíèþêîðìóøêè
9
n
2
�n
1
:
a
n
2
2
A
,çàòåì
9
n
3
�n
2
:
a
n
3
2
A
,
è.ò.ä.Âèòîãå,ïîëó÷èìïîñëåäîâàòåëüíîñòüíîìåðîâ
n
1
n
2
:::n
k
:::
,
òàêèõ,÷òî
a
n
k
2
A
.Íîýòîîçíà÷àåò÷òî
A
åñòüëîâóøêàïîäïîñëåäîâàòåëü-
íîñòè
f
a
n
k
g
.
Âäðóãóþñòîðîíó,ïóñòü
A
ëîâóøêàïîäïîñëåäîâàòåëüíîñòè
f
a
n
k
g
.Ýòî
îçíà÷àåò,÷òî
8
k�K
(
a
n
k
2
A
)
.Íîïîñêîëüêó
n
1
n
2
:::n
k
:::
òî,î÷åâèäíî,
n
k

k
.Ïóñòüçàäàíîïðîèçâîëüíîå
N
2
N
.Ðàññìîòðèì
k
0
=max(
N
+1
;K
+1)
.Òîãäà
k
0
�K
è,ñëåäîâàòåëüíî,
a
n
k
0
2
A
,àñ
äðóãîéñòîðîíû
k
0
�N
.Ïîñêîëüêó
N
2
N
ïðîèçâîëüíî,òîýòîîçíà÷àåò,
÷òî
A
êîðìóøêàïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.Ëåììàäîêàçàíà.
krm
Ëåììà9.
Ïóñòü
A
=
A
0
[
A
00
êîðìóøêàïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.
Òîãäàõîòÿáûîäíîèçìíîæåñòâ
A
0
è
A
00
òîæåÿâëÿåòñÿêîðìóøêîé
ýòîéïîñëåäîâàòåëüíîñòè.
36
ÃËÀÂÀ5.ÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.
Äîêàçàòåëüñòâî.
Ïðåäïîëîæèìîáðàòíîå.Ïóñòüíè
A
0
íè
A
00
íåÿâëÿþòñÿêîðìóøêàìèïî-
ñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.Òîãäàíàéäóòñÿíîìåðà
N
0
è
N
00
,òàêèå,÷òî
8
n�N
0
(
a
n
=
2
A
0
)
è
8
n�N
00
(
a
n
=
2
A
00
)
.Íîòîãäà
8
n�
max(
N
0
;N
00
)(
a
n
=
2
A
0
[
A
00
=
A
)
,
÷òîïðîòèâîðå÷èòîïðåäåëåíèþêîðìóøêè.Ëåììàäîêàçàíà
Óïðàæíåíèå22.
Âåðíîëèïîäîáíîåóòâåðæäåíèåäëÿëîâóøêè?Ò.å.
Ïóñòü
A
=
A
0
[
A
00
ëîâóøêàïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.Ñëåäóåò
ëèèçýòîãî÷òîõîòÿáûîäíîèçìíîæåñòâ
A
0
è
A
00
òîæåÿâëÿåòñÿ
ëîâóøêîéýòîéïîñëåäîâàòåëüíîñòè?
5.4Áåñêîíå÷íîìàëûå(á.ì.ï.)èáåñêîíå÷íîáîëü-
øèå(á.á.ï.)ïîñëåäîâàòåëüíîñòè
Îïðåäåëåíèå45.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
íàçûâàåòñÿáåñêîíå÷íî
ìàëîéïîñëåäîâàòåëüíîñòþ(ñîêðàùåííîá.ì.ï.)åñëèäëÿëþáîãî
"�
0
îêðåñòíîñòü
U
"
(0)
ÿâëÿåòñÿååëîâóøêîé.
Îïðåäåëåíèå46.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
íàçûâàåòñÿáåñêîíå÷íî
áîëüøîéïîñëåäîâàòåëüíîñòþ(ñîêðàùåííîá.ì.ï.)åñëèäëÿëþáîãî
"�
0
ìíîæåñòâî
R
n
U
"
(0)
ÿâëÿåòñÿååëîâóøêîé.
bmogran
Óòâåðæäåíèå18.
Ëþáàÿá.ì.ï.îãðàíè÷åíà.
Äîêàçàòåëüñòâî.
Ïóñòü
f
a
n
g
1
n
=1
á.ì.ï.Ïîîïðåäåëåíèþá.ì.íàéäåòñÿ
N
2
N
,òàêîå,÷òî
8
n�N
(
a
n
2
U
1
(0))
.Âûáåðåì
C
=max(
j
a
1
j
;
j
a
2
j
;:::;
j
A
N
j
;
1)
.Î÷åâèäíî
(äëÿâñåõíàòóðàëüíûõ
n
)
j
a
n
j
C
,ñëåäîâàòåëüíî

Ca
n
C
,÷òîè
òðåáîâàëîñüäîêàçàòü.
bmnogran
Óòâåðæäåíèå19.
Ëþáàÿá.á.ï.íåîãðàíè÷åíà.
Äîêàçàòåëüñòâî.
Âûòåêàåòèçëåììû
posledo_ogr
7.
Óòâåðæäåíèå20.
Ïóñòü
f
a
n
g
1
n
=1
á.á.ï.è
a
n
6
=0
ïðèâñåõ
n
2
N
.
Òîãäà
b
n
=
1
a
n
á.ì.ï.
Äîêàçàòåëüñòâî.
Çàôèêñèðóåì
"�
0
.Ïóñòü
M
=
1
"
.Ïîîïðåäåëåíèþ
(
�1
;M
)
[
(
M;
1
)
ÿâ-
ëÿåòñÿëîâóøêîé
f
a
n
g
1
n
=1
,÷òîðàâíîñèëüíîòîìó,÷òî
j
a
n
j
�M
íà÷èíàÿñ
íåêîòîðîãîíîìåðà.Ýòîðàâíîñèëüíî(ïðè
a
n
6
=0
)òîìó,÷òî
j
b
n
j
=
1
j
a
n
j

1
M
=
";
5.4.ÁÌÏÈÁÁÏ
37
ÓïðàæíåíèåÏóñòü
f
a
n
g
1
n
=1
á.ì.ï.è
a
n
6
=0
ïðèâñåõ
n
2
N
.Äîêàçàòü,
÷òî
b
n
=
1
a
n
á.á.ï.
bmpred
Óòâåðæäåíèå21.
a
=lim
n
!1
a
n
òîãäàèòîëüêîòîãäà,êîãäà
f
a

a
n
g
1
n
=1
ÿâëÿåòñÿá.ì.ï.
Äîêàçàòåëüñòâî.
Ïóñòü
a
=lim
n
!1
a
n
.Òîãäàäëÿëþáîãî
"�
0
îêðåñòíîñòü
U
"
(
a
)
ëîâóøêà.
Ñëåäîâàòåëüíî
8
n�N
(
j
a

a
n
j
"
)
,àçíà÷èò
8
n�N
(
a

a
n
)
2
U
"
(0)
,÷òî
èòðåáîâàëîñüäîêàçàòü.
Âäðóãóþñòîðîíó.Ïóñòüïîñëåäîâàòåëüíîñòü
f
a

a
n
g
1
n
=1
ÿâëÿåòñÿá.ì.ï.
Òîãäàäëÿëþáîãî
"�
0
îêðåñòíîñòü
U
"
(0)
ÿâëÿåòñÿååëîâóøêîé.Ñëåäî-
âàòåëüíî
j
a

a
n
j
"
,çíà÷èò
a
n
2
U
"
(
a
)
;
÷òîèòðåáîâàëîñüäîêàçàòü.
bmcon
Óòâåðæäåíèå22.
Ïóñòü
f
a
n
g
1
n
=1
á.ì.ï.,
C
íåêîòîðîå÷èñëî.Òîãäà
f
C

a
n
g
1
n
=1
òîæåá.ì.ï.Äðóãèìèñëîâàìè,ïðîèçâåäåíèåá.ì.ï.íà÷èñëî
åñòüá.ì.ï.
Äîêàçàòåëüñòâî.
Åñëè
C
=0
òî
f
0

a
n
g
1
n
=1
,î÷åâèäíî,á.ì.ï.Ïóñòü
C
6
=0
.Òîãäàäëÿïðî-
èçâîëüíîãî
"�
0
âûáåðåì
"
0
=
"
j
C
j

0
.Ïîîïðåäåëåíèþá.ì.ï.íàéäåòñÿ
N
òàêîå,÷òî
8
n�N
(
a
n
2
U
"
0
(0))
.Ýòîîçíà÷àåò,÷òî
8
n�N
j
a
n
j
"
0
,
ñëåäîâàòåëüíî
8
n�N
j
C

a
n
j
"
0
j
C
j
=
"
,àçíà÷èò
f
C

a
n
g
1
n
=1
ÿâëÿåòñÿ
á.ì.ï.
bmsum
Óòâåðæäåíèå23.
Ïóñòü
f
a
0
n
g
1
n
=1
è
f
a
00
n
g
1
n
=1
á.ì.ï.Òîãäà
f
a
n
g
1
n
=1
,ãäå
a
n
=
a
0
n
+
a
00
n
òîæåá.ì.ï.Äðóãèìèñëîâàìè,ñóììàá.ì.ï.åñòüá.ì.ï.
Äîêàçàòåëüñòâî.
Äëÿïðîèçâîëüíîãî
"�
0
âûáåðåì
"
0
=
"
2

0
.Òîãäàíàéäóòñÿ
N
è
N
0
,
òàêèå,÷òî
8
n�N
(
j
a
0
n
j
"
0
)
è
8
n�N
0
(
j
a
00
n
j
"
0
)
.Ñëåäîâàòåëüíî
8
n�
max(
N;N
0
)(
j
a
n
j
=
j
a
0
n
+
a
00
n
j
6
j
a
0
n
j
+
j
a
00
n
j

2

"
0
=
"
)
.Açíà÷èò
f
a
n
g
1
n
=1
á.ì.ï.
Óïðàæíåíèå23.
Ïóñòü
f
a
0
n
g
1
n
=1
è
f
a
00
n
g
1
n
=1
á.ì.ï.Äîêàçàòü,÷òî
f
a
n
g
1
n
=1
,ãäå
a
n
=
a
0
n

a
00
n
òîæåá.ì.ï.Äðóãèìèñëîâàìè,äîêàçàòü,
÷òîðàçíîñòüá.ì.ï.åñòüá.ì.ï.
bmprodc
Óòâåðæäåíèå24.
Ïóñòü
f
a
n
g
1
n
=1
á.ì.ï.,
f
c
n
g
1
n
=1
îãðàíè÷åííàÿïî-
ñëåäîâàòåëüíîñòü.Òîãäà
f
b
n
g
1
n
=1
,ãäå
b
n
=
a
n

c
n
òîæåá.ì.ï.Äðóãèìè
ñëîâàìè,ïðîèçâåäåíèåá.ì.ï.íàîãðàíè÷åííóþåñòüá.ì.ï.
38
ÃËÀÂÀ5.ÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.
Äîêàçàòåëüñòâî.
f
c
n
g
1
n
=1
îãðàíè÷åííàÿïîñëåäîâàòåëüíîñòü,ñëåäîâàòåëüíîñóùåñòâóþò
m;M
,òàêèå,÷òî
8
n
2
N
(
m
6
c
n
6
M
)
.Âûáåðåì
C
=max(
j
m
j
;
j
M
j
)+1

0
,
òîãäà
8
n
2
N
(
j
c
n
j
C
)
:
Äëÿïðîèçâîëüíîãî
"�
0
âûáåðåì
"
0
=
"
C

0
.Ïî
îïðåäåëåíèþá.ì.ï.íàéäåòñÿ
N
òàêîå,÷òî
8
n�N
j
a
n
j
"
0
,ñëåäîâàòåëüíî
8
n�N
j
a
n

c
n
j

j
a
n

C
j
"
0

C
=
"
,àçíà÷èò
f
a
n

c
n
g
1
n
=1
ÿâëÿåòñÿá.ì.ï.
Ñëåäñòâèå6.
Ïóñòü
f
a
0
n
g
1
n
=1
;
f
a
00
n
g
1
n
=1
á.ì.ï.Òîãäàïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
,ãäå
a
n
=
a
0
n

a
00
n
òîæåá.ì.ï.Äðóãèìèñëîâàìè,ïðîèçâåäåíèå
á.ì.ï.åñòüá.ì.ï.
Äîêàçàòåëüñòâî.
Âûòåêàåòèçóòâåðæäåíèÿ
bmogran
18(îòîì,÷òîëþáàÿá.ì.ï.îãðàíè÷åíà).
bmpdiv
Óòâåðæäåíèå25.
Ïóñòü
f
a
n
g
1
n
=1
á.ì.ï.,
A
6
=0
,
8
n
2
N
(
A
+
a
n
6
=0)
.
Òîãäàïîñëåäîâàòåëüíîñòü
n
b
n
=
1
a
n
+
A
o
1
n
=1
îãðàíè÷åíà.
Äîêàçàòåëüñòâî.
Íåîãðàíè÷èâàÿîáùíîñòèñ÷èòàåì
A�
0
.Âûáåðåì
N
òàê,÷òî
8
n�N
(
a
n
2
U
A=
2
(0))
.
Òîãäà
j
A
+
a
n
j
�A=
2
,ñëåäîâàòåëüíî
j
b
n
j

2
=A
.Ïóñòü
M
=max




1
A
+
a
1

;:::;



1
A
+
a
N



;
2
A

.
Î÷åâèäíî
8
n
2
N
(
j
b
n
j
M
)
,
Ñëåäñòâèå7.
Ïóñòü
f
a
n
g
1
n
=1
;
f
b
n
g
1
n
=1
á.ì.ï.,
A
6
=0
,
8
n
2
N
(
A
+
a
n
6
=0)
.
Òîãäà
n
c
n
=
b
n
a
n
+
A
o
1
n
=1
á.ì.ï.
Äîêàçàòåëüñòâî.
Âûòåêàåòèçóòâåðæäåíèé
bmprodc
24è
bmpdiv
25
5.5Àðèôìåòè÷åñêèåñâîéñòâàïðåäåëîâ
Óòâåðæäåíèå26.
lim
n
!1
a
n
=0
òîãäàèòîëüêîòîãäà,êîãäà
f
a
n
g
1
n
=1

á.ì.ï.
Äîêàçàòåëüñòâî.
Âûòåêàåòèçîïðåäåëåíèÿïðåäåëà.
Óòâåðæäåíèå27.
Ïóñòü
lim
n
!1
a
n
=
a
è
lim
n
!1
b
n
=
b
.Òîãäà
lim
n
!1
(
a
n
+
b
n
)=
a
+
b
.
Äîêàçàòåëüñòâî.
Èçóòâåðæäåíèÿ
bmpred
21ñëåäóåò,÷òî
f
a

a
n
g
1
n
=1
è
f
b

b
n
g
1
n
=1
á.ì.ï.Òîãäà
ïîóòâåðæäåíèþ
bmsum
23èõñóììà
f
(
a

a
n
)+(
b

b
n
)
g
1
n
=1
òîæåá.ì.ï.Ïåðå-
ãðóïïèðîâàâ,ïîëó÷èì,÷òî
f
(
a
+
b

(
a
n
+
b
n
)
g
1
n
=1
á.ì.ï.Ñëåäîâàòåëüíî,
predel
(
a
n
+
b
n
)=
a
+
b
.
5.5.ÀÐÈÔÌÅÒÈ×ÅÑÊÈÅÑÂÎÉÑÒÂÀÏÐÅÄÅËÎÂ
39
Óïðàæíåíèå24.
Ïóñòü
lim
n
!1
a
n
=
a
è
lim
n
!1
b
n
=
b
.Äîêàçàòü,÷òî
lim
n
!1
(
a
n

b
n
)=
a

b
.
limprod
Óòâåðæäåíèå28.
Ïóñòü
lim
n
!1
a
n
=
a
è
lim
n
!1
b
n
=
b
.Òîãäà
lim
n
!1
(
a
n

b
n
)=
a

b
.
Äîêàçàòåëüñòâî.
Èçóòâåðæäåíèÿ
bmpred
21ñëåäóåò,÷òî
f
a

a
n
g
1
n
=1
è
f
b

b
n
g
1
n
=1
á.ì.ï.Ðàñ-
ñìîòðèìïîñëåäîâàòåëüíîñòü
f
a

b

a
n

b
n
g
1
n
=1
=
f
a

(
b

b
n
)+(
a

a
n
)

b
n
g
1
n
=1
;
èçóòâåðæäåíèé
bmcon
22è
bmsum
23âûòåêàåò,÷òîîíàá.ì.ï.Ñëåäîâàòåëüíî
lim
n
!1
(
a
n

b
n
)=
a

b
.
limobr
Óòâåðæäåíèå29.
Ïóñòü
lim
n
!1
b
n
=
b
6
=0
8
n
2
N
(
b
n
6
=0)
:
Òîãäà
lim
n
!1
1
b
n
=
1
b
.
Äîêàçàòåëüñòâî.
Ïóñòü,äëÿîïðåäåëåííîñòè
b�
0
.Òîãäàíàéäåòñÿ
N
òàêîå,÷òî
8
n�N
(
b
n
2
U
b=
2
(
b
))
,
àçíà÷èò
b
n
�b=
2

0
.Ðàññìîòðèì
b
0
=min(
j
b
1
j
;
j
b
2
j
;:::;
j
b
N
j
;b=
2)

0
,
ïîñêîëüêóâñå÷èñëà
j
b
1
j
;
j
b
2
j
;:::;
j
b
N
j
;b=
2
ïîëîæèòåëüíûïîóñëîâèþ.Òî-
ãäà
8
n
2
N
(
j
b
n
j
�b
0
)
.Ñëåäîâàòåëüíî



1
b
n



6
1
b
0
,ïîýòîìóïîñëåäîâàòåëü-
íîñòü
f
b
n
g
1
n
=1
îãðàíè÷åíà.Ðàññìîòðèì
1
b
n

1
b
=
b

b
n
b
n

b
=(
b

b
n
)

1
b
n

1
b
.
Íî
f
b

b
n
g
1
n
=1
á.ì.ï.
f
1
b
n
g
1
n
=1
îãðàíè÷åíà,à
1
b
êîíñòàíòà.Òîãäà
f
1
b
n

1
b
g
1
n
=1
á.ì.ï.,ñëåäîâàòåëüíî
lim
n
!1
1
b
n
=
1
b
,÷òîèòðåáîâàëîñüäîêà-
çàòü.
Ñëåäñòâèå8.
Ïóñòü
lim
n
!1
a
n
=
a
,
lim
n
!1
b
n
=
b
6
=0
è
8
n
2
N
(
b
n
6
=0)
.Òîãäà
lim
n
!1
a
n
b
n
=
a
b
.
Äîêàçàòåëüñòâî.
Î÷åâèäíîâûòåêàåòèçóòâåðæäåíèé
limprod
28è
limobr
29.
Óïðàæíåíèå25.
Äîêàçàòü,÷òîåñëè
lim
n
!1
a
n
=
a
,òî
lim
n
!1
j
a
n
j
=
j
a
j
.
Âåðíîëèîáðàòíîåóòâåðæäåíèå?
Óïðàæíåíèå26.
Äîêàçàòü,÷òîåñëè
lim
n
!1
a
n
=
a
,
8
n
2
N
(
a
n

0)
,
a

0
,
òî
lim
n
!1
p
a
n
=
p
a
.
Óïðàæíåíèå27.
Äîêàçàòü,÷òîåñëè
lim
n
!1
a
n
=
a
,òî
lim
n
!1
S
n
n
=
a
,ãäå
S
n
=
a
1
+
a
2
+
:::
+
a
n
.
bmpbou
Óòâåðæäåíèå30.
Ïóñòü
f
a
n
g
1
n
=1
á.ì.ï.è
8
n
2
N
(
j
b
n
j
6
a
n
)
.Òîãäà
f
b
n
g
1
n
=1
á.ì.ï.
40
ÃËÀÂÀ5.ÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.
Äîêàçàòåëüñòâî.
Äëÿïðîèçâîëüíîãî
"�
0
îêðåñòíîñòü
U
"
(0)
ëîâóøêà
f
a
n
g
1
n
=1
.Çíà÷èò
8
n�N
(
a
n
2
U
"
(0))
.Òîãäà
j
a
n
j
"
,ñëåäîâàòåëüíî
j
b
n
j
6
j
a
n
j
"
,àçíà÷èò
b
n
2
U
"
(0)
.Ïîëó÷àåì,÷òî
U
"
(0)
ÿâëÿåòñÿëîâóøêîé
f
b
n
g
1
n
=1
ïðèâñåõ
"�
0
.
Ñëåäîâàòåëüíî
f
b
n
g
1
n
=1
á.ì.ï.
Òåîðåìà31.
(Îäâóõìèëèöèîíåðàõ)
Ïóñòü
8
n
2
N
(
a
n
6
c
n
6
b
n
)
è
lim
n
!1
a
n
=lim
n
!1
b
n
=
C
.Òîãäàïðåäåë
lim
n
!1
c
n
ñóùåñòâóåòèðàâåíòîìóæå
÷èñëó
C
.
Äîêàçàòåëüñòâî.
Çàìåòèì,÷òî
lim
n
!1
(
b
n

a
n
)=
C

C
=0
,ñëåäîâàòåëüíî
f
b
n

a
n
g
1
n
=1

á.ì.ï.Î÷åâèäíî,
0
6
c
n

a
n
6
b
n

a
n
,àçíà÷èòïîïðåäûäóùåìóóòâåðæäå-
íèþ
f
c
n

a
n
g
1
n
=1
á.ì.ï.Àçíà÷èò
f
c
n

a
g
1
n
=1
=
f
(
c
n

a
n
)+(
a
n

C
)
g
1
n
=1
òîæåá.ì.ï.ñëåäîâàòåëüíî
lim
n
!1
c
n
=
C
,
limner
Òåîðåìà32
(Ïðåäåëüíûéïåðåõîäâíåðàâåíñòâàõ)
.
Ïóñòü
lim
n
!1
a
n
=
a
è
8
n�N
(
a
n
6
C
)
.Òîãäà
a
6
C
.Äðóãèìèñëîâàìè,åñëèâñå÷ëåíûïîñëå-
äîâàòåëüíîñòè,íà÷èíàÿñíåêîòîðîãî
N
íåïðåâîñõîäÿòêàêîãî-òî÷èñëà
C
,òîèïðåäåëïîñëåäîâàòåëüíîñòèíåìîæåòáûòüáîëüøå
C
.
Äîêàçàòåëüñòâî.
Ïðåäïîëîæèìîáðàòíîå,ò.å.÷òî
a
=lim
n
!1
a
n
�C
.Âûáåðåì
"
=
a

C
2

ðàññìîòðèìîêðåñòíîñòü
U
"
(
a
)
êîòîðàÿïîîïðåäåëåíèþïðåäåëàåñòüëî-
âóøêàïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.Òîãäàíàéäåòñÿ
a
n
2
U
"
(
a
)
ñîñêîëü
óãîäíîáîëüøèìíîìåðîì
n
.Òîãäà
a
n
�a

"
=
a

a

C
2
=
a
+
c
2
�C
,÷òî
ïðîòèâîðå÷èòóñëîâèþ.
Òåîðåìà33
(ÒåîðåìàØòîëüöà)
.
Ïóñòü
lim
n
!1
x
n
+1

x
n
y
n
+1

y
n
=
l
,ãäå
f
y
n
g
1
n
=1
ñîñòîèòèçïîëîæèòåëüíûõ÷èñåë,ìîíîòîííîâîçðàñòàåòèíåÿâëÿåòñÿ
îãðàíè÷åííîé.Òîãäà
lim
n
!1
x
n
y
n
ñóùåñòâóåòèðàâåí
l
.
Äîêàçàòåëüñòâî.
Ïðåäîñòàâëÿåòñÿ÷èòàòåëþâêà÷åñòâåóïðàæíåíèÿ.
5.6Íåêîòîðûåâàæíûåïðèìåðûá.ì.ïèïðå-
äåëîâ.
Óòâåðæäåíèå34.
Ïîñëåäîâàòåëüíîñòü
f
1
n
g
1
n
=1
á.ì.ï.
5.6.ÍÅÊÎÒÎÐÛÅÂÀÆÍÛÅÏÐÈÌÅÐÛÁ.Ì.ÏÈÏÐÅÄÅËÎÂ.
41
Äîêàçàòåëüñòâî.
Ðàññìîòðèìïðîèçâîëüíîå
"�
0
.Ïóñòü
N
=

1
"

+1
,ãäåêâàäðàòíûå
ñêîáêèîáîçíà÷àþòöåëóþ÷àñòü÷èñëà.Òîãäàäëÿâñåõ
n�N
âûïîëíå-
íî


1
n





1
N


"
,÷òîèòðåáîâàëîñüäîêàçàòü.
Ñëåäñòâèå9.
Ïóñòü
C
2
R
,
k
2
N
.Òîãäà

C
n
k

1
n
=1
á.ì.ï.
Äîêàçàòåëüñòâî.
Î÷åâèäíî.
Óòâåðæäåíèå35.
Ïóñòü
j
q
j

1
,
a
n
=
n

q
n
.Òîãäà
f
a
n
g
1
n
=1
á.ì.ï.
Äîêàçàòåëüñòâî.
Åñëè
j
q
j

1
,òîìîæíîïðåäñòàâèòü
j
q
j
=
1
1+

,ãäå

0
.Òîãäà
(1+

)
n
=1+
C
1
n


+
C
2
n


2
+
:::�C
2
n


2
.
n
j
q
j
n

n
C
2
n

2

4

2

1
n

1
n
ÁÌÏ,
4

2
êîíñòàíòà,ñëåäîâàòåëüíî,ïî
óòâåðæäåíèþ
bmpbou
30
n
j
q
j
n
òîæåÁÌÏ.
Óòâåðæäåíèå36.
lim
n
!1
n
p
n
=1
Äîêàçàòåëüñòâî.
Ðàññìîòðèì
n
p
n
=1+

n
,î÷åâèäíî

n

0
.Âîçâîäÿîáå÷àñòèâ
n

ñòåïåíüèðàñêðûâñêîáêè,ïîëó÷èì
n
=(1+

n
)
n
=1+
C
1
n

n
+
C
2
n

2
n
+
:::�C
2
n

2
n
:
Ñëåäîâàòåëüíî,
0

2
n

n
C
2
n
=
2
n

1
.Ïóñòüçàäàíî
"�
0
,âûáåðåì
N
=

2
"
2

+1
.
Òîãäàïðèâñåõ
n�N
âûïîëíåíî

n

q
2
N
"
.Òàêèìîáðàçîì,
f

n
g
1
n
=1
ÁÌÏ,àçíà÷èò
lim
n
!1
n
p
n
=1
,
Óòâåðæäåíèå37.
Ïóñòü
C�
1
,
a
n
=
n
p
C

1
.Òîãäà
f
a
n
g
1
n
=1
á.ì.ï.
Äîêàçàòåëüñòâî.
Ïóñòü
C
=1+

,ãäå

0
.Îáîçíà÷èì
n
p
C

1=

n
.Òîãäà
(1+

n
)
n
=1+

è,ïîíåðàâåíñòâóÁåðíóëëè,
1+

=(1+

n
)
n

1+
n


n
.Ñëåäîâàòåëü-
íî
0

n


n
.Âûáåðåì
N
=


"

+1
,òîãäà
8
n�N
(
j

n
j


n
"
)
.
Ñëåäîâàòåëüíî

n
=
n
p
C

1
á.ì.ï.
Óïðàæíåíèå28.
Äîêàçàòü,÷òî
f
a
n
g
1
n
=1
,ãäå
a
n
=
n
!
2
n
2
ÁÌÏ.
42
ÃËÀÂÀ5.ÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.
5.7ÒåîðåìûÂåéåðøòðàññàèÁîëüöàíîÂåéåðøòðàññà
Òåîðåìà38.
(Ê.Âåéåðøòðàññ)
Ïðåäåëìîíîòîííîéîãðàíè÷åííîéïî-
ñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
ñóùåñòâóåòèðàâåí
sup
f
a
n
j
n
2
N
g
(äëÿâîç-
ðàñòàþùåéïîñëåäîâàòåëüíîñòè)èëè
inf
f
a
n
j
n
2
N
g
(äëÿóáûâàþùåéïî-
ñëåäîâàòåëüíîñòè).
Äîêàçàòåëüñòâî.
Ðàññìîòðèìâîçðàñòàþùóþïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
.Ìíîæåñòâî
A
=
f
a
n
j
n
2
N
g
îãðàíè÷åíîñâåðõóèíåïóñòî,ñëåäîâàòåëüíîñóùåñòâóåò
a
0
=sup
A
.Ðàñ-
ñìîòðèìïðîèçâîëüíóþîêðåñòíîñòü
U
(
a
0
)=(
b;c
)
.Ïîñêîëüêó
ba
0
=sup
A
,
òîñóùåñòâóåòíîìåð
N
2
N
,òàêîé,÷òî
a
N
�b
.Èçìîíîòîííîñòèïîñëåäî-
âàòåëüíîñòè
f
a
n
g
1
n
=1
âûòåêàåò,÷òî
8
n�N
(
a
n
�a
N
�b
)
.Ñäðóãîéñòîðî-
íû
8
n
2
N
(
a
n
6
sup
Ac
)
.Òàêèìîáðàçîì,
8
n�N
(
a
n
2
(
b;c
)=
U
(
a
0
))
.
Ñëåäîâàòåëüíî
U
(
a
0
)
åñòüëîâóøêà
f
a
n
g
1
n
=1
,
naim_el
Ëåììà10.
Ïóñòüâïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
íåòíàèìåíüøåãîýëå-
ìåíòà,ò.å.
69
n
2
N
(
a
n
=inf
f
a
n
j
n
2
N
g
)
.Òîãäàñóùåñòâóåòìîíîòîí-
íàÿïîäïîñëåäîâàòåëüíîñòü
a
n
1
�a
n
2
�:::�a
n
k
�:::
(
n
1
n
2
:::n
k
).
Äîêàçàòåëüñòâî.
Âûáåðåì
n
1
=1
,òîãäà
a
n
1
=
a
1
.Ïóñòüòåïåðüóæåâûáðàíû
a
n
1
�a
n
2
�:::�a
n
k
,
âûáåðåì
a
n
k
+1
a
n
k
,òàê,÷òîáû
nk
+1
�n
k
.Äîïóñòèìòàêîãî÷ëåíàïî-
ñëåäîâàòåëüíîñòèíåñóùåñòâóåò.Òîãäà
8
n�n
k
(
a
n

a
n
k
)
.Ñëåäîâàòåëüíî,
íåêîòîðûé
a
n
0
=min(
a
1
;a
2
;:::;a
n
k
)
ÿâëÿåòñÿíàèìåíüøèì÷ëåíîìïîñëå-
äîâàòåëüíîñòè
f
a
n
g
1
n
=1
,÷òîïðîòèâîðå÷èòóñëîâèþ.
Ëåììà11.
Âëþáîéïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
ìîæíîâûäåëèòüìî-
íîòîííóþïîäïîñëåäîâàòåëüíîñòü
a
n
1

a
n
2

:::

a
n
k

:::
èëè
a
n
1
6
a
n
2
6
:::
6
a
n
k
6
:::
(
n
1
n
2
:::n
k
).
Äîêàçàòåëüñòâî.
Ïóñòü
a
n
1
íàèìåíüøèé÷ëåíïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.Åñëèòàêîãî
íåñóùåñòâóåò,òîïîëåììå
naim_el
10â
f
a
n
g
1
n
=1
ìîæíîâûáðàòüìîíîòîííóþïîä-
ïîñëåäîâàòåëüíîñòü.Ðàññìîòðèìïîñëåäîâàòåëüíîñòü
f
a
k
g
1
k
=
n
1
.Ëèáîâíåé
íåòíàèìåíüøåãîýëåìåíòàòîãäàïîëåììå
naim_el
10â
f
a
k
g
1
k
=
n
1
ìîæíîâûáðàòü
ìîíîòîííóþïîäïîñëåäîâàòåëüíîñòü.Ëèáîòàêîé÷ëåíñóùåñòâóåòòîãäà
âîçüìåìåãîâêà÷åñòâå
a
n
2
.Äàëåå,íàêàæäîìøàãåâûáèðàåìâïîñëåäî-
âàòåëüíîñòè
f
a
k
g
1
k
=
n
s
íàèìåíüøèé÷ëåíèáåðåìåãîâêà÷åñòâå
a
n
s
+1
.Ýòîò
ïðîöåññëèáîíåîãðàíè÷åííîïðîäîëæàåòñÿ,ëèáî,âêàêîéòîìîìåíòâîç-
íèêíåòñèòóàöèÿ,îïèñàííàÿâëåììå
naim_el
10.Òàêèìîáðàçîì,âëþáîìñëó÷àå,
ìûïîëó÷èììîíîòîííóþïîäïîñëåäîâàòåëüíîñòü.
5.7.ÒÅÎÐÅÌÛÂÅÉÅÐØÒÐÀÑÑÀÈÁÎËÜÖÀÍÎÂÅÉÅÐØÒÐÀÑÑÀ
43
bolzano
Òåîðåìà39.
(ÁîëüöàíîÂåéåðøòðàññà).
Ëþáàÿîãðàíè÷åííàÿïîñëå-
äîâàòåëüíîñòüèìååòñõîäÿùóþñÿïîäïîñëåäîâàòåëüíîñòü.
Äîêàçàòåëüñòâî.
Ïóñòüïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
îãðàíè÷åíà.Äîêàæåì,÷òîîíàèìååò
ïðåäåëüíóþòî÷êó.Ïîñòðîèìñòÿãèâàþùóþñÿñèñòåìóîòðåçêîâ
f
[
a
n
;b
n
]
g
1
n
=1
ñëåäóþùèìîáðàçîì:

Ïåðâûéøàã.Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
îãðàíè÷åíà,ò.å.
9
m;M
j8
n
2
N
(
m
6
a
n
6
M
)
.
Âîçüìåì
u
1
=
m
è
v
1
=
M
.Î÷åâèäíî
[
u
1
;v
1
]
êîðìóøêàïîñëåäîâà-
òåëüíîñòè
f
a
n
g
1
n
=1
.

Íàêàæäîìñëåäóþùåìøàãå.Ïóñòü
c
=
u
n
+
v
n
2
ñåðåäèíàîòðåçêà
[
u
n
;v
n
]
.Ðàññìîòðèìîòðåçêè
[
u
n
;c
]
è
[
c;v
n
]
.Ïîëåììå
krm
9õîòÿáûîäèí
èçíèõÿâëÿåòñÿêîðìóøêîéïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.Äîïóñòèì
ýòîîòðåçîê
[
u
n
;c
]
,òîãäàâûáåðåì
u
n
+1
=
u
n
;v
n
+1
=
c
(âïðîòèâíîì
ñëó÷àå
u
n
+1
=
c;n
n
+1
=
v
n
)
Î÷åâèäíî,ñèñòåìàîòðåçêîâ
f
[
u
n
;v
n
]
g
1
n
=1
áóäåòñòÿãèâàþùåéñÿ(ñì.äîêà-
çàòåëüñòâîòåîðåìû
mn_pred
15).Òîãäàèõïåðåñå÷åíèå
1
T
n
=1
[
u
n
;v
n
]=
f
a
0
g
.Äîêàæåì,
÷òî
a
0
ïðåäåëüíàÿòî÷êàïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.Äåéñòâèòåëüíî,
ðàññìîòðèìïðîèçâîëüíîå
"�
0
.Î÷åâèäíî,íàéäåòñÿ
n
2
N
òàêîå,÷òî
j
v
n

u
n
j
"
,àçíà÷èò
[
u
n
;v
n
]

U
"
(
a
0
)
.Íî
[
u
n
;v
n
]
êîðìóøêàïîñëå-
äîâàòåëüíîñòè
f
a
n
g
1
n
=1
,ñëåäîâàòåëüíî
U
"
(
a
0
)
òîæåêîðìóøêà(
"�
0

ïðîèçâîëüíîå!).Ñëåäîâàòåëüíî
a
0
ïðåäåëüíàÿòî÷êàïîñëåäîâàòåëüíîñòè
f
a
n
g
1
n
=1
.Òîãäàïîëåììå
chas_pr
8ñóùåñòâóåòïîäïîñëåäîâàòåëüíîñòü
f
a
n
k
g
1
k
=1
,
ïðåäåëêîòîðîéðàâåí
a
0
.Òåîðåìàäîêàçàíà.
ÄðóãîåäîêàçàòåëüñòâîòåîðåìûÁîëüöàíîÂåéåðøòðàññàâûòåêàåòèç
òåîðåìûÂåéåðøòðàññàèëåììû
naim_el
10.Íàýêçàìåíåðàçðåøàåòñÿïðèâîäèòü
ëþáîåèçíèõ,ïðèóñëîâèè,÷òîó÷àùèéñÿóìååòäîêàçûâàòü
âñå
âñïîìîãà-
òåëüíûåóòâåðæäåíèÿ.
Ïðèìåð9.
(èòåðàöèîííàÿôîðìóëàÃåðîíà)
Ïóñòü
a
ïðîèçâîëüíîå
ïîëîæèòåëüíîå÷èñëî.Âûáåðåì
u
0

0
ïðîèçâîëüíî,à÷ëåíûïîñëåäîâà-
òåëüíîñòè
f
u
n
g
1
n
=1
ïîëó÷èìèçðåêóððåíòíîãîñîîòíîøåíèÿ
u
n
+1
=
1
2

u
n
+
a
u
n

;
ãäå
n
=0
;
1
;
2
;:::
Òîãäà
lim
n
!1
u
n
=
p
a
.
Äîêàçàòåëüñòâî.
44
ÃËÀÂÀ5.ÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.

Ïîñëåäîâàòåëüíîñòü
f
u
n
g
1
n
=1
îãðàíè÷åíàñíèçóâåëè÷èíîé
p
a
.Äåé-
ñòâèòåëüíî,
u
n
+1

p
a
=
1
2

u
n
+
a
u
n


p
a
=
(
u
n

p
a
)
2
2
u
n

0
:

Ïîñëåäîâàòåëüíîñòü
f
u
n
g
1
n
=1
óáûâàåò.Äåéñòâèòåëüíî
u
n

u
n
+1
=
u
n

1
2

u
n
+
a
u
n

=
u
2
n

a
2
u
n
;
÷òîáîëüøå0ïîïðåäûäóùåìóïóíêòó.

ÏîòåîðåìåÂåéåðøòðàññàïîñëåäîâàòåëüíîñòü
f
u
n
g
1
n
=1
èìååòïðåäåë.
Ïðåäïîëîæèì,îíðàâåí
u
Òîãäàðàññìîòðèì
lim
n
!1
u
n
+1
=lim
n
!1
1
2

u
n
+
a
u
n

ïðåäåëïîñëåäîâàòåëüíîñòè,"ñäâèíóòîé"íà1÷ëåí.Ïîñâîéñòâàì
ïðåäåëîâîíðàâåí
1
2
(
u
+
a
u
)
.Çíà÷èò
u
óäîâëåòâîðÿåòóðàâíåíèþ
u
=
1
2
(
u
+
a
u
)
:
Ðåøàÿåãî,ïîëó÷èì
u
=

p
a
.Êîðåíü
u
=

p
a
íàäîîòáðîñèòü,ïî-
ñêîëüêóâñå÷ëåíûïîñëåäîâàòåëüíîñòè
f
u
n
g
1
n
=1
ïîëîæèòåëüíû,àçíà-
÷èòïðåäåë
lim
n
!1
u
n
íåìîæåòáûòüîòðèöàòåëüíûì(ñì.óòâåðæäåíèå
limner
32).
Âèòîãåïîëó÷èì
lim
n
!1
u
n
=
p
a
,÷òîèòðåáîâàëîñüäîêàçàòü.
Ïîïðîáóåìíàéòè
p
2=1
;
414213562373095
:::
òàêèìñïîñîáîì.Âûáå-
ðåì
a
=2
,
u
0
=1
.Òîãäà
u
1
=1
:
5
;
u
2
=1
:
416666666666666
;
u
3
=1
:
414215686274509
;
u
4
=1
:
414213562374689
;
u
5
=1
:
414213562373094
.Óæåíà5ìøàãåïî-
ãðåøíîñòüâû÷èñëåíèéíåïðåâîñõîäèò
10

15
!
5.8Ôóíäàìåíòàëüíûåïîñëåäîâàòåëüíîñòè.Êðè-
òåðèéÊîøè.*
mydef
Îïðåäåëåíèå47.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
íàçûâàåòñÿôóíäàìåí-
òàëüíîé,åñëèäëÿëþáîãî
"�
0
ñóùåñòâóåòíåêîòîðîå÷èñëî
c
=
c
e
,
òàêîå,÷òî
U
"
(
c
)
ÿâëÿåòñÿëîâóøêîé
f
a
n
g
1
n
=1
.Äðóãèìèñëîâàìè,ôóí-
äàìåíòàëüíîéíàçûâàåòñÿïîñëåäîâàòåëüíîñòü,èìåþùàÿëîâóøêèñêîëü
óãîäíîìàëîãîðàçìåðà.
Áîëååèçâåñòíûìÿâëÿåòñÿäðóãîåîïðåäåëåíèåôóíäàìåíòàëüíîéïîñëå-
äîâàòåëüíîñòè.Åãîêàêðàçîáû÷íîèïîìåùàþòâó÷åáíèêè
5.8.ÔÓÍÄÀÌÅÍÒÀËÜÍÛÅÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.ÊÐÈÒÅÐÈÉÊÎØÈ.*
45
clasdef
Îïðåäåëåíèå48.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
íàçûâàåòñÿôóíäàìåí-
òàëüíîé,åñëèäëÿëþáîãî
"�
0
ñóùåñòâóåòíåêîòîðîå
N
2
N
,òàêîå,
÷òî
8
n
0
;n
00
�N
(
j
a
n
0

a
n
00
j
"
.
Äîêàæåìýêâèâàëåíòíîñòüîïðåäåëåíèé
mydef
47è
clasdef
48.
Äîêàçàòåëüñòâî.
Ïóñòü
f
a
n
g
1
n
=1
ÿâëÿåòñÿôóíäàìåíòàëüíîéâñìûñëåîïðåäåëåíèÿ
mydef
47.Òîãäà
íàéäåòñÿ
U
"=
2
(
c
)=(
c

"
2
;
c
+
"
2
)
ååëîâóøêà.Ýòîîçíà÷àåò,÷òîíà÷èíàÿ
ñíåêîòîðîãîíîìåðà
N
âñå÷ëåíûïîñëåäîâàòåëüíîñòèïîïàäàþòâèíòåð-
âàë
(
c

"
2
;
c
+
"
2
)
,àçíà÷èòèõðàçíîñòüìåíüøå
"
.Ñëåäîâàòåëüíî
f
a
n
g
1
n
=1
ÿâëÿåòñÿôóíäàìåíòàëüíîéâñìûñëåîïðåäåëåíèÿ
clasdef
48.
Ïóñòü
f
a
n
g
1
n
=1
ÿâëÿåòñÿôóíäàìåíòàëüíîéâñìûñëåîïðåäåëåíèÿ
clasdef
48.Òî-
ãäàíàéäåòñÿ
N
2
N
,òàêîå,÷òî
8
n
0
;n
00
�N
(
j
a
n
0

a
n
00
j
"
.Òîãäà
U
"
(
a
N
+1
)
ëîâóøêà,
Òåîðåìà40
(ÊðèòåðèéÊîøè)
.
Ïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
èìååòïðå-
äåëòîãäàèòîëüêîòîãäà,êîãäàîíàôóíäàìåíòàëüíà.
Äîêàçàòåëüñòâî.
Âîäíóñòîðîíóäîêàçàòåëüñòâîòðèâèàëüíî.Äåéñòâèòåëüíî,ïóñòüñóùå-
ñòâóåò
A
=lim
n
!1
a
n
.Òîãäàäëÿëþáîãî
"�
0
îêðåñòíîñòü
U
"
(
A
)
ÿâëÿåòñÿ
ëîâóøêîé
f
a
n
g
1
n
=1
.
Âîáðàòíóþñòîðîíó,ïóñòü
f
a
n
g
1
n
=1
ôóíäàìåíòàëüíàÿ.Òîãäàîíàîãðà-
íè÷åíà,ñëåäîâàòåëüíîèìååòíåêîòîðóþïðåäåëüíóþòî÷êó
A
.Íàéäåòñÿ
N
2
N
òàêîå,÷òî
8
n
0
;n
00
�N
(
j
a
n
0

a
n
00
j

"
2
)
.Âûáåðåì
n
0
�N
òàêîå,
÷òî
a
n
0
2
U
"=
2
(
A
)
(ýòîìîæíîñäåëàòüò.ê.
U
"=
2
(
A
)
êîðìóøêà).Òîãäàïðè
âñåõ
n�n
0
âûïîëíåíî
j
A

a
n
j
6
j
A

a
n
0
j
+
j
a
n
0

a
n
j

"
2
+
"
2
=
";
àçíà÷èò
a
n
2
U
"
(
A
)
.Íîýòîêàêðàçèîçíà÷àåò,
A
=lim
n
!1
a
n
Çàìåòèì,
÷òîïîíÿòèåôóíäàìåíòàëüíîñòèìîæíîââîäèòüíåòîëüêîäëÿ÷èñëîâûõ
ïîñëåäîâàòåëüíîñòåé,íîäëÿëþáûõïîñëåäîâàòåëüíîñòåéîáúåêòîâ,äëÿ
êîòîðûõìîæíîââåñòèïîíÿòèåîêðåñòíîñòè.Òàê,íàïðèìåð,ìîæíîââåñòè
îïðåäåëåíèåôóíäàìåíòàëüíîéïîñëåäîâàòåëüíîñòèòî÷åêíàïëîñêîñòèèëè
âïðîñòðàíñòâå(ïîäóìàéòå,êàê?).
Ñïîñîáíîñòüëþáîéôóíäàìåíòàëüíîéïîñëåäîâàòåëüíîñòèèìåòüïðå-
äåëÿâëÿåòñÿâåñüìàâàæíûìñâîéñòâîììíîæåñòâàäåéñòâèòåëüíûõ÷èñåë.
Äåéñòâèòåëüíî,äëÿìíîæåñòâàðàöèîíàëüíûõ÷èñåëýòîóæåíåâåðíî
ìîæíîïîäîáðàòüôóíäàìåíòàëüíóþïîñëåäîâàòåëüíîñòüâ
Q
,íåèìåþùóþ
ïðåäåëàâ
Q
.Ìíîæåñòâà,äëÿêîòîðûõòàêîåñâîéñòâîèìååòìåñòî,íàçû-
âàþò
ïîëíûìè.
Òàêèìîáðàçîì
R
ïîëíî,à
Q
íåò.
46
ÃËÀÂÀ5.ÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.
Óïðàæíåíèå29.
Äîêàçàòü,÷òîíà÷èñëîâîéïðÿìîéïîëíûìèÿâëÿþò-
ñÿòåèòîëüêîòåìíîæåñòâà,êîòîðûåçàìêíóòû.
5.9×èñëîÝéëåðà
Ðàññìîòðèìñëåäóþùóþçàäà÷ó:ïóñòüíåêîòîðûéáàíêâûïëà÷èâàåò100%
ãîäîâûõ.Î÷åâèäíî,÷òîíåêòî,ïîëîæèâ100ðóá.÷åðåçãîäáóäåòèìåòü
200ðóá.íàñ÷åòó.Íîýòîïðèóñëîâèè,÷òîïðîöåíòûâûïëà÷èâàþòñÿîäèí
ðàçâãîä.Åñëèæåêàæäûåïîëãîäàâûïëà÷èâàåòñÿïî50%îòòåêóùåéñóì-
ìû,òîêêîíöóãîäàíåêòîáóäåòèìåòü100+50+75=225ðóá.Åñëèïðîöåíòû
âûïëà÷èâàòüêàæäûéêâàðòàëòîñóììàáóäåò
100

(1+0
;
25)
4
=244
:
14
,
åñëèïðîöåíòûâûïëà÷èâàþòñÿêàæäûéäåíü(òàêäåëàåòñÿ,íàïðèìåð,íà
íåêîòîðûõìåæáàíêîâñêèõáèðæàõ)òîñóììàáóäåò
100

(1+
1
365
)
365
=271
:
46
.
Âîçíèêàåòâîïðîñ,ê÷åìóáóäåòñòðåìèòüñÿýòàñóììà,åñëèóñòðåìèòü÷èñ-
ëîîòðåçêîâêáåñêîíå÷íîñòè.
vlog
Òåîðåìà41.
Ðàññìîòðèìïîñëåäîâàòåëüíîñòè
a
n
=

1+
1
n

n
b
n
=

1+
1
n

n
+1
.
Ñèñòåìàîòðåçêîâ
[
a
n
;b
n
]
ÿâëÿåòñÿâëîæåííîé.
Äîêàçàòåëüñòâî.
Î÷åâèäíî,
a
n
b
n
.Äîêàæåì,÷òî
a
1
6
a
2
6
:::
6
a
n
6
:::
.Ðàññìîòðèìîò-
íîøåíèåñîñåäíèõ÷ëåíîâýòîéïîñëåäîâàòåëüíîñòè
a
n
+1
a
n
=
(
1+
1
n
+1
)
n
+1
(
1+
1
n
)
n
=

n
+2
n
+1

n
+1


n
n
+1

n
=

n
+2
n
+1



n
(
n
+2)
(
n
+1)
2

n
=

1+
1
n
+1



n
2
+2
n
n
2
+2
n
+1

n
ÏðèìåíèâíåðàâåíñòâîÁåðíóëëè,ïîëó÷èì
a
n
+1
a
n
=

1+
1
n
+1



1

1
n
2
+2
n
+1

n


1+
1
n
+1



1

n
n
2
+2
n
+1

=
n
3
+3
n
2
+3
n
+2
(
n
+1)
3

1
:
Ñëåäîâàòåëüíî,
a
n
+1
�a
n
,ïðèâñåõ
n

1
àçíà÷èòïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
ÿâëÿåòñÿâîçðàñòàþùåé.Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òîïîñëåäî-
âàòåëüíîñòü
f
b
n
g
1
n
=1
ÿâëÿåòñÿóáûâàþùåé.Äåéñòâèòåëüíî,
b
n

1
b
n
=

1+
1
n

1

n

1+
1
n

n
+1
=

n
n
+1



(1+
1
n
2

1

n


n
n
+1



1+
n
n
2

1

=
n
3
+
n
2

n
n
3
+
n
2

n

1

1
:
Ñëåäîâàòåëüíî
b
n
b
n

1
;
Òåîðåìà42.
Ðàññìîòðèìïîñëåäîâàòåëüíîñòè
a
n
=

1+
1
n

n
b
n
=

1+
1
n

n
+1
.
Ñèñòåìàîòðåçêîâ
[
a
n
;b
n
]
ÿâëÿåòñÿñòÿãèâàþùåéñÿ.
5.9.×ÈÑËÎÝÉËÅÐÀ
47
Äîêàçàòåëüñòâî.
Ïîòåîðåìå
vlog
41ïîñëåäîâàòåëüíîñòü
f
[
a
n
;b
n
]
g
1
n
=1
ÿâëÿåòñÿâëîæåííîé.Èç
ýòîãîâûòåêàåò,÷òîâñå
a
n
è
b
n
íåïðåâîñõîäÿò
b
1
=(1+1)
2
=4
.Òîãäà
äëèíà
n
ãîîòðåçêà
j
a
n

b
n
j
=

1+
1
n

n
+1


1+
1
n

n
=
1
n


1+
1
n

n
6
4
n
.
Ïóñòü
"�
0
,ðàññìîòðèì
N
=

4
"

+1
,òîãäà
8
n�N
(
j
a
n

b
n
j
6
4
N
"
)
,
ñëåäîâàòåëüíîñèñòåìà
f
[
a
n
;b
n
]
g
1
n
=1
ÿâëÿåòñÿñòÿãèâàþùåéñÿ.
48
ÃËÀÂÀ5.ÏÎÑËÅÄÎÂÀÒÅËÜÍÎÑÒÈ.
Ãëàâà6
Ðÿäû
Âìàãàçèíçàõîäèòáåñêîíå÷íîå
÷èñëîìàòåìàòèêîâ.Ïåðâûé
ïðîñèòêèëîãðàììêàðòîøêè,
âòîðîéïîëêèëî,òðåòèé
250ãðàìì...¾Ïîíÿë¿
ãîâîðèòïðîäàâåöèêëàä¼òíà
ïðèëàâîêäâàêèëîãðàììà.
bash.org.ru
Ïóñòüçàäàíàïîñëåäîâàòåëüíîñòü
f
a
n
g
1
n
=1
.Âûðàæåíèå
1
P
n
=1
a
n
íàçûâàþò
ðÿäîì,ñîñòàâëåíûìèçýòîéïîñëåäîâàòåëüíîñòè.
Ñóììó
S
N
=
N
P
n
=1
a
n
íàçûâàþò÷àñòè÷íîéñóììîéðÿäà.
Åñëèñóùåñòâóåòïðåäåë
lim
n
!1
S
n
=
S
,òîãîâîðÿò,÷òîðÿäñõîäèòñÿ,à
÷èñëî
S
íàçûâàþòåãîñóììîéèçàïèñûâàþòòàê:
S
=
1
P
n
=1
a
n
.
Ñóììó
R
N
=
1
P
n
=
N
+1
a
n
íàçûâàþòîñòàòêîìðÿäà
1
P
n
=1
a
n
.
Óòâåðæäåíèå43.
Åñëèðÿä
1
P
n
=1
a
n
ñõîäèòñÿ,òîïðåäåë
lim
N
!1
R
N
=0
.
Äîêàçàòåëüñòâî.
Ïðåäîñòàâëÿåòñÿ÷èòàòåëþâêà÷åñòâåóïðàæíåíèÿ.
49
50
ÃËÀÂÀ6.ÐßÄÛ
Òåîðåìà44
(Íåîáõîäèìûéïðèçíàêñõîäèìîñòèðÿäà.)
.
Åñëèðÿä
1
P
n
=1
a
n
ñõîäèòñÿ,òî
lim
n
!1
a
n
=0
.
Äîêàçàòåëüñòâî.
Ïóñòü
lim
n
!1
n
P
k
=1
a
k
=
S
.Òîãäà,î÷åâèäíî,
lim
n
!1
n
+1
P
k
=1
a
k
=
S
.Ñëåäîâàòåëüíî
lim
n
!1
a
n
=lim
n
!1

n
+1
X
k
=1
a
k

n
X
k
=1
a
k
!
=
S

S
=0
;
Òåîðåìà45
(ÊðèòåðèéÊîøèäëÿðÿäîâ)
.
Ðÿä
1
P
n
=1
a
n
ñõîäèòñÿòîãäà
èòîëüêîòîãäàêîãäàäëÿëþáîãî
"�
0
íàéäåòñÿ
N
2
N
òàêîå,÷òî
8
n;n
0
�N
(
j
S
n

S
n
0
j
"
)
.
Äîêàçàòåëüñòâî.
Î÷åâèäíîñëåäóåòèçêðèòåðèÿÊîøèäëÿïîñëåäîâàòåëüíîñòè÷àñòè÷íûõ
ñóìì.
Òåîðåìà46
(ÏðèçíàêÂåéåðøòðàññà)
.
Ïóñòü
8
n
2
N
(
j
b
n
j
6
a
n
)
èðÿä
1
P
n
=1
a
n
ñõîäèòñÿêíåêîòîðîìó÷èñëó
S
.Òîãäàðÿä
1
P
n
=1
b
n
ñõîäèòñÿêíåêî-
òîðîìó
S
0
è
S
0
6
S
.
Äîêàçàòåëüñòâî.
ÏîêðèòåðèþÊîøèäëÿâñåõ
"�
0
íàéäåòñÿ
N
2
N
òàêîå,÷òî
8
n;n
0
�N
(
j
S
n

S
n
0
j
"
)
.
Áóäåìñ÷èòàòü,÷òî
nn
0
,òîãäà
j
b
n
+1
+
b
n
+2
+
:::
+
b
n
0
j
6
j
b
n
+1
j
+
:::
+
j
b
n
0
j
6
a
n
+1
+
:::
+
a
n
0
":
Çíà÷èòêðèòåðèéÊîøèâûïîëíåíäëÿðÿäà
1
P
n
=1
b
n
,ñëåäîâàòåëüíî,îíñõî-
äèòñÿ.Àíåðàâåíñòâî
S
0
6
S
âûòåêàåòèçàíàëîãè÷íîãîíåðàâåíñòâàäëÿ
÷àñòè÷íûõñóìì.
geo
Òåîðåìà47.
Ïóñòü
f
b
n
g
1
n
=1
ãåîìåòðè÷åñêàÿïðîãðåññèÿñîçíàìåíà-
òåëåì
j
q
j

1
.Òîãäàðÿä
1
P
n
=1
b
n
ñõîäèòñÿèåãîñóììàðàâíà
b
1
1

q
.
Äîêàçàòåëüñòâî.
Ðàññìîòðèì÷àñòè÷íûåñóììû
N
P
n
=1
b
n
=
b
1

q
N

1
q

1
ïîôîðìóëåäëÿñóììû
51
êîíå÷íîéãåîìåòðè÷åñêîéïðîãðåññèè.Âçÿâ
lim
N
!1
îòîáåèõ÷àñòåéðàâåí-
ñòâà,ïîëó÷èì
1
P
n
=1
b
n
=
b
1

lim
N
!1
q
N

1
q

1
.Ó÷èòûâàÿóñëîâèå
j
q
j

1
,ïîëó÷èì
1
P
n
=1
b
n
=
b
1


1
q

1
=
b
1
1

q
;
eee
Òåîðåìà48.
Ðÿä
1+
1
P
n
=1
1
n
!
ñõîäèòñÿèåãîñóììàðàâíà
e
.
Äîêàçàòåëüñòâî.
Î÷åâèäíî,÷òîïîñëåäîâàòåëüíîñòü
S
n
ìîíîòîííîâîçðàñòàåò.Àèçíåðàâåí-
ñòâà
S
n
=1+1+
1
2!
+
:::
+
1
n
!

1+1+
1
2
+
1
4
+
:::
+
1
2
n

1
=3

1
2
n

1

3
ñëåäóåòååîãðàíè÷åííîñòü.Ñëåäîâàòåëüíî,ïîòåîðåìåÂåéåðøòðàññàîíà
èìååòïðåäåë
1
P
n
=1
1
n
!
=
e
0
.Ðàññìîòðèìïîñëåäîâàòåëüíîñòü
a
n
=

1+
1
n

n
=1+
1
n

C
1
n
+
:::
+
1
n
n

C
n
n
:
Íåñëîæíîçàìåòèòü,÷òî
C
k
n

1
n
k
6
1
n
!
ïðèâñåõ
k
=1
;
2
;:::;n
.ñëåäîâàòåëüíî
a
n
6
S
n
,àçíà÷èò,ïîëåììå
limner
32(îïðåäåëüíîìïåðåõîäåâíåðàâåíñòâàõ)
e
6
e
0
.
Ñäðóãîéñòîðîíû,âûáåðåìíåêîòîðîå
s
6
n
èðàññìîòðèì
d
s
(
n
)=1+
1
n

C
1
n
+
:::
+
1
n
s

C
s
n
=2+
1
2!


1

1
n


:::


1

s

1
n

:
Î÷åâèäíî,
d
s
(
n
)
6
a
n
.Ñëåäîâàòåëüíî,
e

lim
n
!1
d
s
(
n
)=
c
s
.Íîïîñêîëüêó
lim
s
!1
c
s
=sup
f
c
s
g
=
e
1
,òî
e
1

e
.Òàêèìîáðàçîì,ïîëó÷èì
e
1
=
e
,
Äîêàçàòåëüñòâî.
a
n
=
1
n
!
.Î÷åâèäíî,÷òîïîñëåäîâàòåëüíîñòü
S
n
=
a
1
+
:::
+
a
n
ìîíîòîííî
âîçðàñòàåò.Àèçíåðàâåíñòâà
S
n
=1+1+
1
2!
+
:::
+
1
n
!

1+1+
1
2
+
1
4
+
:::
+
1
2
n

1
=3

1
2
n

1

3
ñëåäóåòååîãðàíè÷åííîñòü.Ñëåäîâàòåëüíî,ïîòåîðåìåÂåéåðøòðàññàîíà
èìååòïðåäåë
1
P
n
=1
a
n
=
e
0
.Äîêàæåì,÷òîîíðàâåí
e
.
52
ÃËÀÂÀ6.ÐßÄÛ
Ðàññìîòðèì
Q
n
=

1+
1
n

n
=1+
1
n

C
1
n
+
1
n
2

C
2
n
+
:::
+
C
n
n
1
n
n
=
1
P
k
=1
b
n;k
,
ãäå
b
n;k
=
(
1
n
k

C
k
n
;k
6
n
0
;k�n
.Äëÿäîêàçàòåëüñòâàòåîðåìûíàìïîòðåáóþòñÿ
ñëåäóþùèåâñïîìîãàòåëüíûåóòâåðæäåíèÿ:
bmen
Ëåììà12.
Ïðèâñåõ
n;k
2
N
âûïîëíÿåòñÿíåðàâåíñòâî
b
n;k
6
a
k
:
Äîêàçàòåëüñòâî(ëåììû
bmen
12)
Ïðè
k�n
íåðàâåíñòâîî÷åâèäíî.Äîêàæåìåãîïðè
k
6
n
.Ïðåîáðàçóåì:
b
n;k
=
C
k
n
n
k
=
n

(
n

1)

:::

(
n

k
+1)
k
!

n
k
=
1
k
!


n
n



n

1
n


:::


n

k
+1
n

6
1
k
!
=
a
n
,
blim
Ëåììà13.
Ïðèâñåõ
k
2
N
ñóùåñòâóåòïðåäåë:
lim
n
!1
b
n;k
=
a
k
:
Äîêàçàòåëüñòâî.
(ëåììû
blim
13)Î÷åâèäíî,
lim
n
!1
b
n;k
=lim
n
!1
1
k
!


n
n



n

1
n


:::


n

k
+1
n

=
=
1
k
!


lim
n
!1
n
n



lim
n
!1
n

1
n



lim
n
!1
n

k
+1
n

=
1
k
!
:
Äîêàæåìòåïåðüòåîðåìó
eee
48.Âûáåðåì
N
2
N
ïðèêîòîðîìîñòàòîêðÿäà
1
P
k
=
N
+1
a
k

"
3
.Ýòîìîæíîñäåëàòü,ïîñêîëüêóðÿä
1
P
k
=1
a
k
ñõîäèòñÿ.Ïîëåììå
bmen
12
1
P
k
=
N
+1
b
n;k
6
1
P
k
=
N
+1
a
k

"
3
.Àïîëåììå
blim
13ïðåäåë
lim
n
!1
N
P
k
=1
b
n;k
=
N
P
k
=1
a
k
,
ñëåäîâàòåëüíî,íàéäåòñÿ
N
1
2
N
,òàêîå,÷òîïðèâñåõ
n�N
1
âûïîëíå-
íîíåðàâåíñòâî


N
P
k
=1
b
n;k

N
P
k
=1
a
k



"
3
.Òîãäàïðèâñåõ
n�N
1
âûïîëíåíî
j
1
P
k
=1
a
k

1
P
k
=1
b
n;k
j
6


N
P
k
=1
b
n;k

N
P
k
=1
a
k


+


1
P
k
=
N
+1
a
k


+
j
1
P
k
=
N
+1
b
n;k
j
=
"
3
+
"
3
+
"
3
=
":
Èçïðîèçâîëüíîñòè
"
âûòåêàåò,÷òî
f
Q
n

S
n
g
1
n
=1
ÁÌÏ,àçíà÷èò
lim
n
!1
S
n
=lim
n
!1
Q
n
=
e;
Òåîðåìà49.
×èñëî
e
èððàöèîíàëüíî.
53
Äîêàçàòåëüñòâî.
Ïðåäïîëîæèìîáðàòíîå,ïóñòü
e
=
m
n
2
Q
.Ðàññìîòðèì
e

S
n
=
1
X
k
=
n
+1
1
k
!

0
:
Î÷åâèäíî,÷òî
1
k
!
6
1
n
!

(
n
+1)
k

n
,ñëåäîâàòåëüíî,
0
e

S
n
6
1
X
k
=
n
+1
1
n
!

(
n
+1)
k

n
=
1
(
n
+1)!

n
n
+1

1
(
n
+1)!
:
Çàìåòèì,÷òî
n
!(
e

S
n
)=
n
!(
m
n

1

1
1!

:::

1
n
!
ïîëîæèòåëüíîåöåëîå
÷èñëî,íî
n
!(
e

S
n
)
n
!

1
(
n
+1)!

1
.Ïðîòèâîðå÷èå.
54
ÃËÀÂÀ6.ÐßÄÛ
Ãëàâà7
Äåéñòâèòåëüíûå÷èñëà
Ðàíååðàññìàòðèâàëèñüäåéñòâèòåëüíûå÷èñëàêàêòî÷êèíà÷èñëîâîéïðÿ-
ìîé.Òåïåðüïðèøëîâðåìÿîïðåäåëèòü,÷òîæåòàêîåäåéñòâèòåëüíî÷èñëî.
Îïðåäåëåíèå49.
Ïóñòü
f
a
n
g
1
n
=

k
,ïîñëåäîâàòåëüíîñòü,÷ëåíûêîòîðîé
a
n
2f
0
;
1
;
2
;
3
;
4
;
5
;
6
;
7
;
8
;
9
g
.Òîãäàäåñÿòè÷íàÿçàïèñü
a

k
a

k
+1
:::a

1
a
0
;a
1
a
2
:::a
n
:::
çàäàåòïîëîæèòåëüíîåäåéñòâèòåëüíîå÷èñëî
a

k

10
k
+
a

k
+1

10
k

1
+
:::
+
a

1

10
1
+
a
0

10
0
+
1
P
n
=1
a
n

10

n
;
àäåñÿòè÷íàÿçàïèñü

a

k
a

k
+1
:::a

1
a
0
;a
1
a
2
:::a
n
:::
çàäàåòîòðèöàòåëü-
íîåäåéñòâèòåëüíîå÷èñëî

a

k

10
k

a

k
+1

10
k

1

:::

a

1

10
1

a
0

10
0

1
P
n
=1
a
n

10

n
.
Äîêàæåìêîððåêòíîñòüâûøåïðèâåäåííîãîîïðåäåëåíèÿ.Äëÿýòîãîäî-
êàæåì,÷òîðÿäñõîäèòñÿè÷òîëþáîå÷èñëîìîæåòáûòüïðåäñòàâëåíîâ
âèäåäåñÿòè÷íîéçàïèñè.
Äðóãîåîïðåäåëåíèåäåñÿòè÷íûõ÷èñåë.
Ïóñòü
R
åñòüìíîæåñòâîîáúåêòîâäëÿêîòîðûõçàäàíûîòíîøåíèå
6
è
èîïåðàöèè
+
è

,óäîâëåòâîðÿþùèõàêñèîìàì:
1.
8
a;b
(
a
+
b
=
b
+
a
)
(Êîììóòàòèâíîñòüñëîæåíèÿ);
2.
8
a;b;c
(
a
+(
b
+
c
)=(
a
+
b
)+
c
)
(Àññîöèàòèâíîñòüñëîæåíèÿ);
3.
9
!0(
8
a
(
a
+0=0+
a
=
a
))
(Ñóùåñòâîâàíèåíóëÿ);
4.
8
a
(
9
!

a
j
a
+(

a
)=(

a
)+
a
=0)
(Îáðàòèìîñòüñëîæåíèÿ);
5.
8
a;b
(
a

b
=
b

a
)
(Êîììóòàòèâíîñòüóìíîæåíèÿ);
6.
8
a;b;c
(
a

(
b

c
)=(
a

b
)

c
)
(Àññîöèàòèâíîñòüóìíîæåíèÿ);
7.
9
!1
6
=0
j8
a
(1

a
=
a

1=
a
)
(Ñóùåñòâîâàíèååäèíèöû);
8.
8
a
6
=0
9
!
a

1
j
a

a

1
=
a

1

a
=1
(Îáðàòèìîñòüóìíîæåíèÿ);
55
56
ÃËÀÂÀ7.ÄÅÉÑÒÂÈÒÅËÜÍÛÅ×ÈÑËÀ
9.
8
a;b;c
(
a

(
b
+
c
)=
a

b
+
a

c
(Äèñòðèáóòèâíîñòüóìíîæåíèÿîòíîñè-
òåëüíîñëîæåíèÿ);
10.
8
a
(
a
6
a
)
(Ñèììåòðè÷íîñòüñðàâíåíèÿ);
11.
Åñëè
a
6
b
è
b
6
c
,òî
a
6
c
(Òðàíçèòèâíîñòüñðàâíåíèÿ);
12.
8
a;b
âûïîëíåíî
a
6
b
èëè
b
6
a
,ïðè÷åìîáàñðàâíåíèÿâûïîëíåíû
òîëüêîåñëè
a
=
b
.(Ñðàâíèìîñòü÷èñåë);
13.
Åñëè
a
6
b
è
0
6
c
,òî
a

c
6
b

c
(Ìîíîòîíííîñòüîïåðàöèèóìíîæåíèÿ);
14.
8

0(
9
n
2
N
j
n


1)
(ÀêñèîìàÀðõèìåäà);
15.
Ïóñòüìíîæåñòâà
A;B
òàêîâû,÷òî
8
a
2
A
8
b
2
Ba
6
b
.Òîãäàñó-
ùåñòâóåò÷èñëî
c
òàêîå,÷òî
8
a
2
A
8
b
2
Ba
6
c
6
b
(Àêñèîìà
îòäåëèìîñòè).
Àêñèîìû14îçíà÷àþò,÷òî
R
ãðóïïàïîñëîæåíèþ,àêñèîìû5-8,÷òî
R
nf
0
g
ãðóïïàïîóìíîæåíèþ.Àêñèîìû19îïðåäåëÿþòïîëå.Àêñèîìû
1012ââîäÿòíà
R
îòíîøåíèåëèíåéíîãîïîðÿäêà.Àïîñëåäíèå2àêñèîìû
êàêðàçèîïðåäåëÿþòñòðóêòóðóìíîæåñòâàäåéñòâèòåëüíûõ÷èñåë.Òàê,
íàïðèìåð
Q
óäîâëåòâîðÿåòâñåìàêcèîìàìêðîìåïîñëåäíåé.Ìîæíîïðè-
âåñòèïðèìåðìíîæåñòâà,íåóäîâëåòâîðÿþùåãîàêñèîìåÀðõèìåäà(ïîäó-
ìàéòå,êàê?).ÌîæíîáûëîáûïîïðèìåðóÅâêëèäàñíà÷àëàâûïèñàòüàê-
ñèîìû,àïîòîìóæåäîêàçàòü,÷òîïîëó÷àþùååìíîæåñòâîåñòüìíîæåñòâî
âñåõäåéñòâèòåëüíûõ÷èñåë.Ðåçþìèðóÿâñåâûøåñêàçàííîå,ìîæíîñêàçàòü,
÷òîñóùåñòâóåòïîêðàéíåéìåðåòðèñïîñîáàîïðåäåëåíèÿäåéñòâèòåëüíûõ
÷èñåë:1)Ãåîìåòðè÷åñêèéñïîñîá(òî÷êèíà÷èñëîâîéïðÿìîé);2)Äåñÿòè÷-
íàÿçàïèñü(ïîñëåäîâàòåëüíîñòüöåëûõ÷èñåë);3)Àêñèîìàòè÷åñêèéñïîñîá
(âûøåïðèâåäåííàÿñèñòåìààêñèîì).
Î÷åâèäíî,òóòïåðå÷èñëåíûäàëåêîíåâñåâîçìîæíûåâàðèàíòûîïðåäå-
ëåíèÿìíîæåñòâàäåéñòâèòåëüíûõ÷èñåë.Íàïðèìåð,ìîæíîáûëîðàññìîò-
ðåòüôóíäàìåíòàëüíûåïîñëåäîâàòåëüíîñòèðàöèîíàëüíûõ÷èñåëèñ÷èòàòü
èõäåéñòâèòåëüíûìè÷èñëàìè.Çàìåòèìòîëüêî,÷òîêàæäûéïîäõîäèìååò
êàêñâîèäîñòîèíñòâà,òàêèíåäîñòàòêè.
×àñòüII
IIñåìåñòð
57
Ãëàâà8
Ôóíêöèÿ.Ïðåäåëôóíêöèè.
Ïåðâûéêóðñ.Ïåðâàÿïàðàïî
ìàò.àíàëèçóâòåõíè÷åñêîì
âóçå.
Ïðåïîäàâàòåëü:
Çàïèñûâàåìòåìó:
Äåéñòâèòåëüíàÿôóíêöèÿ
äåéñòâèòåëüíîéïåðåìåííîé.
Ñþðúåêòèâíûå,èíúåêòèâíûåè
áèåêòèâíûåôóíêöèè.Ñëîæíàÿ
èîáðàòíàÿôóíêöèÿ.
Ãîëîññçàäíåéïàðòû:
ßïåðåäóìàë.Çàáåðèòåìåíÿ
âàðìèþ...
bash.org.ru
Îïðåäåëåíèå50.
Ôóíêöèåé
F
:
A
7!
B
íàçûâàåòñÿíåêîòîðûéçàêîí,
ñòàâÿùèéâñîîòâåòñòâèåíåêîòîðûìýëåìåíòàìíîæåñòâà
A
îäèíèëè
íåñêîëüêîýëåìåíòîâìíîæåñòâà
B
.Áîëååôîðìàëüíî,ôóíêöèåéíàçûâà-
þòìíîæåñòâî
F
A

B
,ñîîòâåòñòâåííî,åñëè
x
2
A
,òîìíîæåñòâî
çíà÷åíèéôóíêöèèíà
x
åñòü
F
(
x
)=
f
y
2
B
j
(
x;y
)
2Fg
.
Îïðåäåëåíèå51.
Îäíîçíà÷íîéôóíêöèåé
F
:
A
7!
B
íàçûâàåòñÿ
íåêîòîðûéçàêîí,ñòàâÿùèéâñîîòâåòñòâèåêàæäîìóýëåìåíòóìíî-
æåñòâà
A
íåáîëååîäíîãîýëåìåíòàìíîæåñòâà
B
.Áîëååôîðìàëüíî,îä-
íîçíà÷íîéôóíêöèåéíàçûâàþòìíîæåñòâî
F
A

B
,òàêîå,÷òîåñëè
(
x
0
;y
0
)
;
(
x
00
;y
00
)
2F
è
y
0
=
y
00
,òî
x
0
=
x
00
.
59
60
ÃËÀÂÀ8.ÔÓÍÊÖÈß.ÏÐÅÄÅËÔÓÍÊÖÈÈ.
Âäàííîìêóðñåðàññìàòðèâàþòñÿòîëüêîîäíîçíà÷íûåôóíêöèè,õîòÿ
ìíîãîçíà÷íûåíååñòüêàêàÿòîýêçîòèêà.Íàïðèìåðôóíêöèÿêîðíåé
êâàäðàòíîãîóðàâíåíèÿ
f
(
a;b;c
)=

b

p
b
2

4
ac
2
a
ïðèíèìàåòíîëü,îäíîèëè
äâàçíà÷åíèÿâçàâèñèìîñòèîòçíàêà
D
=
b
2

4
ac
.
Îïðåäåëåíèå52.
Îáðàçîì
ìíîæåñòâà
M
ïðèîòîáðàæåíèè
F
íàçûâà-
åòñÿìíîæåñòâî
F
(
M
)=
f
y
=
F
(
x
):
x
2
M
g
.
Ïðîîáðàçîì
ìíîæåñòâà
S
ïðèîòîáðàæåíèè
F
íàçûâàåòñÿìíîæåñòâî
F

1
(
S
)=
f
x
:
F
(
x
)
2
S
g
.
Çàìå÷àíèå:
Åñëèìíîæåñòâî
M
èëè
S
ñîñòîèòèçîäíîéòî÷êè,òîôè-
ãóðíûåñêîáêèîïóñêàþòèçàïèñûâàþòòàê:
F
(
f
x
g
)=
F
(
x
)
,
F

1
(
f
y
g
)=
F

1
(
y
)
.
Îïðåäåëåíèå53.
Èíúåêöèåé
íàçûâàþòôóíêöèþ
F
,òàêóþ,÷òîäëÿ
ëþáûõ
x
0
;x
00
2
D
F
èç
x
0
6
=
x
00
ñëåäóåò
F
(
x
0
)
6
=
F
(
x
00
)
.Äðóãèìèñëîâàìè,
èíúåêöèÿôóíêöèÿ,êîòîðàÿíàðàçíûõàðãóìåíòàõïðèíèìàåòðàçíûå
çíà÷åíèÿ.
Îïðåäåëåíèå54.
Ñþðúåêöèåé
íàçûâàþòôóíêöèþ
F
:
A
!
B
,òàêóþ,
÷òîäëÿëþáîãî
y
2
B
ñóùåñòâóåò
x
2
A
òàêîé,÷òî
F
(
x
)=
y
.Äðóãèìè
ñëîâàìè,ñþðúåêöèÿôóíêöèÿ,êîòîðàÿïðèíèìàåòêàæäîåçíà÷åíèåèç
B
.
Îïðåäåëåíèå55.
Áèåêöèåé
(èëè
âçàèìíîîäíîçíà÷íûìîòîáðàæå-
íèåì
)íàçûâàþòôóíêöèþ
F
:
A
!
B
,ÿâëÿþùóþñÿèíúåêöèåéèñþðúåê-
öèåéîäíîâðåìåííî.êàæäîåçíà÷åíèåèç
B
.
8.1×èñëîâûåôóíêöèè
Îïðåäåëåíèå56.
×èñëîâîéôóíêöèåé
íàçûâàþòôóíêöèè
f
(
x
)
,îá-
ëàñòüîïðåäåëåíèÿèîáëàñòüçíà÷åíèéêîòîðîéÿâëÿþòñÿïîäìíîæå-
ñòâàìè÷èñëîâîéïðÿìîé,ò.å.
D
f
;E
f

R
.
Âäàëüíåéøåìâäàííîìêóðñåïîäñëîâîìôóíêöèÿáóäåìïîíèìàòü
÷èñëîâàÿôóíêöèÿ.
Îïðåäåëåíèå57.
×èñëîâàÿôóíêöèÿíàçûâàåòñÿ
÷åòíîé
,åñëèäëÿâñåõ
x
2
D
f
âûïîëíåíî

x
2
D
f
è
f
(

x
)=
f
(
x
)
.×èñëîâàÿôóíêöèÿíàçûâàåòñÿ
íå÷åòíîé
,åñëèäëÿâñåõ
x
2
D
f
âûïîëíåíî

x
2
D
f
è
f
(

x
)=

f
(
x
)
.
Óïðàæíåíèå30.
Äîêàæèòå,÷òîëþáàÿôóíêöèÿ,îïðåäåëåííàÿíàâñåé
÷èñëîâîéïðÿìîéïðåäñòàâèìàââèäåñóììû÷åòíîéèíå÷åòíîéôóíêöèè.
8.2.ÏÐÅÄÅËÔÓÍÊÖÈÈ.ÝÊÂÈÂÀËÅÍÒÍÎÑÒÜÎÏÐÅÄÅËÅÍÈÉÏÎÃÅÉÍÅÈÏÎÊÎØÈ.
61
Îïðåäåëåíèå58.
Ôóíêöèÿ
f
(
x
)
íàçûâàåòñÿ
ïåðèîäè÷åñêîé
åñëèñóùå-
ñòâóåò
T�
0
,òàêîå,÷òî:
1)
8
x
2
D
f
(
x

T
2
D
f
)
;
2)
8
x
2
D
f
(
f
(
x
+
T
)=
f
(
x
))
:
×èñëî
T
íàçûâàþò
ïåðèîäîì
ôóíêöèè
f
(
x
)
.Åñëè
T
min
ïåðèîäèäëÿ
ëþáîãîïåðèîäà
T
âûïîëíåíî
T
min
6
T
,òî
T
min
íàçûâàþò
íàèìåíüøèì
ïåðèîäîì
f
(
x
)
.
8.2Ïðåäåëôóíêöèè.Ýêâèâàëåíòíîñòüîïðå-
äåëåíèéïîÃåéíåèïîÊîøè.
geine
Îïðåäåëåíèå59.
Ïóñòüòî÷êà
x
0
ÿâëÿåòñÿïðåäåëüíîéòî÷êîéìíîæå-
ñòâà
D
f
èäëÿëþáîéïîñëåäîâàòåëüíîñòè
f
x
n
g
1
n
=1

D
f
nf
x
0
g
,òàêîé,
÷òî
lim
n
!1
x
n
=
x
0
ñóùåñòâóåòïðåäåë
lim
n
!1
f
(
x
n
)=
X
.Òîãäàãîâîðÿò,
÷òîôóíêöèÿ
f
èìååòâòî÷êå
x
0
ïðåäåë
,ðàâíûé
X
(èëè÷òîôóíê-
öèÿ
f
(
x
)
ñòðåìèòñÿê
X
ïðè
x
ñòðåìÿùåìñÿê
x
0
)èîáîçíà÷àþòýòî
òàê:
lim
x
!
x
0
f
(
x
)=
X
.
Âêíèãàõïîìàòåìàòè÷åñêîìóàíàëèçó(íàïðèìåð,âøêîëüíîìó÷åáíè-
êå)÷àñòîäàåòñÿäðóãîåîïðåäåëåíèåïðåäåëà:
coshi
Îïðåäåëåíèå60.
Ïóñòü
x
0
2P
(
D
f
)
èäëÿëþáîãî
"�
0
ñóùåñòâóåò

=

(
"
)

0
,òàêîå,÷òî
f
(

U

(
x
0
))

U
"
(
X
)
.Òîãäàãîâîðÿò,÷òîôóíêöèÿ
f
èìååòâòî÷êå
x
0
ïðåäåë,ðàâíûé
X
(èëè÷òîôóíêöèÿ
f
(
x
)
ñòðåìèòñÿ
ê
X
ïðè
x
ñòðåìÿùåìñÿê
x
0
)èîáîçíà÷àþòýòîòàê:
lim
x
!
x
0
f
(
x
)=
X
.
Îïðåäåëåíèå
geine
59íàçûâàþòîïðåäåëåíèåìïðåäåëà
ïîÃåéíå
,îïðåäåëå-
íèå
coshi
60îïðåäåëåíèåìïðåäåëà
ïîÊîøè.
Äîêàæåìðàâíîñèëüíîñòüýòèõ
îïðåäåëåíèé:
Äîêàçàòåëüñòâî.
Ïóñòüïðåäåë
lim
x
!
x
0
f
(
x
)=
X
âñìûñëåîïðåäåëåíèÿ
geine
59(ïîÃåéíå).Äîêàæåì,
÷òîòàêîåæåðàâåíñòâîâåðíîâñìûñëåîïðåäåëåíèÿ
coshi
60(ïîÊîøè).
Áóäåìäîêàçûâàòüìåòîäîìîòïðîòèâíîãî.Äîïóñòèì,÷òîäëÿíåêîòî-
ðîãî
"�
0
âûïîëíåíî
8
�
0

f
(

U

(
x
0
))
6
U
"
(
X
)

.Âîçüìåì

m
=
1
m
è
âûáåðåì
x
n
2

U

m
(
x
0
)
òàê,÷òî
f
(
x
n
)
=
2
U
"
(
X
)
.Òîãäàïðèâñåõ
m
îêðåñò-
íîñòü

U
1
=m
(
x
0
)
ÿâëÿåòñÿëîâóøêîé
f
x
n
g
1
n
=1
,ñëåäîâàòåëüíî,
lim
n
!1
x
n
=0
.
Ðàññìîòðèìîêðåñòíîñòü
U
"
(
X
)
,î÷åâèäíî,îíàíåÿâëÿåòñÿëîâóøêîé
f
(
x
n
)
62
ÃËÀÂÀ8.ÔÓÍÊÖÈß.ÏÐÅÄÅËÔÓÍÊÖÈÈ.
(íèîäèí÷ëåíïîñëåäîâàòåëüíîñòè
f
(
x
n
)
íåïîïàäàåòâíåå),ñëåäîâàòåëü-
íî
lim
n
!1
f
(
x
n
)
6
=
X
ïîëó÷åíîïðîòèâîðå÷èå,÷òîèäîêàçûâàåòòðåáóåìîå
óòâåðæäåíèå.
Äîêàæåìâäðóãóþñòîðîíó.Ïóñòü
lim
x
!
x
0
f
(
x
)=
X
âñìûñëåîïðåäå-
ëåíèÿ
coshi
60(ïîÊîøè).Âûáåðåìïðîèçâîëüíóþïîñëåäîâàòåëüíîñòü
f
x
n
g
1
n
=1
òàêóþ,÷òî
lim
n
!1
x
n
=
x
0
è
8
n
2
N
(
x
n
6
=
x
0
)
.Äëÿëþáîãî
"�
0
ñóùå-
ñòâóåò

=

(
"
)

0
.Ïîîïðåäåëåíèþïðåäåëà

U

(
x
0
)
áóäåòëîâóøêîé
f
x
n
g
1
n
=1
,ñëåäîâàòåëüíî
f
(

U

(
x
0
))

U
"
(
X
)
ëîâóøêà
f
f
(
x
n
)
g
1
n
=1
.Çíà-
÷èò,
lim
n
!1
f
(
x
n
)=
X
(ïîÃåéíå),
8.3Ïðåäåëûíàáåñêîíå÷íîñòèèîäíîñòîðîí-
íèåïðåäåëû.
Îïðåäåëåíèå61.
Áóäåìãîâîðèòü,÷òîïðåäåëïîñëåäîâàòåëüíîñòèðà-
âåíïëþñáåñêîíå÷íîñòèèîáîçíà÷àòü
lim
n
!1
a
n
=+
1
,åñëèäëÿëþáîãî
c
2
R
ìíîæåñòâî
(
c;
+
1
)
ÿâëÿåòñÿååëîâóøêîé.Àíàëîãè÷íî,áóäåìãî-
âîðèòü,÷òîïðåäåëïîñëåäîâàòåëüíîñòèðàâåíìèíóñáåñêîíå÷íîñòèè
îáîçíà÷àòü
lim
n
!1
a
n
=
�1
åñëèäëÿëþáîãî
c
2
R
ìíîæåñòâî
(
�1
;c
)
ÿâëÿåòñÿååëîâóøêîé;áóäåìãîâîðèòü,÷òîïðåäåëïîñëåäîâàòåëüíîñòè
ðàâåíáåñêîíå÷íîñòèèîáîçíà÷àòü
lim
n
!1
a
n
=
1
,åñëèäëÿëþáîãî
c
2
R
ìíîæåñòâî
(
�1
;

c
)
[
(
c;
+
1
)
ÿâëÿåòñÿååëîâóøêîé.
Îïðåäåëåíèå62.
Áóäåìãîâîðèòü,÷òîïðåäåëïîñëåäîâàòåëüíîñòèðà-
âåí
a
+0
èîáîçíà÷àòü
lim
n
!1
a
n
=
a
+0
,åñëèäëÿëþáîãî
"�
0
ìíîæåñòâî
[
a;a
+
"
)
ÿâëÿåòñÿååëîâóøêîé.Àíàëîãè÷íî,áóäåìãîâîðèòü,÷òîïðåäåë
ïîñëåäîâàòåëüíîñòèðàâåí
a

0
èîáîçíà÷àòü
lim
n
!1
a
n
=
a

0
åñëèäëÿ
ëþáîãî
"�
0
ìíîæåñòâî
(
a

";a
]
ÿâëÿåòñÿååëîâóøêîé.
Çàìå÷àíèå.
Âàæíîïîíèìàòü,÷òîíèáåñêîíå÷íîñòü(
1
),íè
a

0
íå
ÿâëÿþòñÿ
äåéñòâèòåëüíûìè÷èñëàìè,ñëåäîâàòåëüíîñíèìèíåëüçÿïðîèç-
âîäèòüàðèôìåòè÷åñêèåîïåðàöèè.Ýòîïðîñòîîáùåïðèíÿòîåôîðìàëüíîå
îáîçíà÷åíèå,êîòîðîåèñïîëüçóåòñÿäëÿñîêðàùåíèÿçàïèñè.
Îïðåäåëåíèå63.
Áóäåìîáîçíà÷àòü
^
R
=
R
[f
a

0
j
a
2
R
g[f1
;
1g

ðàñøèðåííîåìíîæåñòâîäåéñòâèòåëüíûõ÷èñåë.Ðàñøèðèìîïðåäåëåíèå
ïðåäåëàïîÃåéíå.×èñëàâèäà
f
a

0
j
a
2
R
g[f1
;
1g
áóäåìíàçûâàòü
ïñåâäî÷èñëàìè
.Ïóñòü
a;A
2
^
R
.Áóäåìîáîçíà÷àòü
lim
x
!
a
f
(
x
)=
A
,åñëè
8.4.ÀÐÈÔÌÅÒÈ×ÅÑÊÈÅÑÂÎÉÑÒÂÀÏÐÅÄÅËÎÂ
63
äëÿëþáîéïîñëåäîâàòåëüíîñòè
lim
n
!1
x
n
=
a
,ãäå
8
n
2
N
(
x
n
6
=
a
)
âûïîëíåíî
lim
n
!1
f
(
x
n
)=
A
.
Çàìå÷àíèå.
Àíàëîãè÷íîìîæíîðàñøèðèòüîïðåäåëåíèåïðåäåëàïî
Êîøè,åñëèîïðåäåëèòü
"
îêðåñòíîñòèäëÿïñåâäî÷èñåëñëåäóþùèìîáðà-
çîì:

U
"
(
1
)=

U
"
(
1
)=(
�1
;
1
"
)
[
(
1
"
;
+
1
)
;

U
"
(+
1
)=

U
"
(+
1
)=(
1
"
;
+
1
)
;

U
"
(
�1
)=

U
"
(
�1
)=(
�1
;
1
"
)
;

U
"
(
a
+0)=[
a;a
+
"
)
,

U
"
(
a
+0)=(
a;a
+
"
)
;

U
"
(
a

0)=(
a

";a
]
,

U
"
(
a

0)=(
a

";a
)
;
Óïðàæíåíèå31.
Äîêàçàòüýêâèâàëåíòíîñòüîïðåäåëåíèÿñîîòâåòñòâó-
þùèõïðåäåëîâïîÃåéíåèïîÊîøè.
8.4Àðèôìåòè÷åñêèåñâîéñòâàïðåäåëîâ
Óòâåðæäåíèå50.
Ïóñòü
lim
x
!
x
0
f
(
x
)=
F
è
lim
x
!
x
0
g
(
x
)=
G
,òîãäà
lim
x
!
x
0
(
f
(
x
)+
g
(
x
)=
F
+
G
,
Óòâåðæäåíèå51.
Ïóñòü
lim
x
!
x
0
f
(
x
)=
F
òîãäà
lim
x
!
x
0
c

f
(
x
)=
c

F
,
Óòâåðæäåíèå52.
Ïóñòü
lim
x
!
x
0
f
(
x
)=
F
è
lim
x
!
x
0
g
(
x
)=
G
,òîãäà
lim
x
!
x
0
(
f
(
x
)

g
(
x
)=
F

G
,
Óòâåðæäåíèå53.
Ïóñòü
lim
x
!
x
0
f
(
x
)=
F
è
lim
x
!
x
0
g
(
x
)=
G
6
=0
,òîãäà
lim
x
!
x
0
f
(
x
)
g
(
x
)
=
F
G
,
Âñåýòèñâîéñòâàâûòåêàþòèçñîîòâåòñòâóþùèõñâîéñòâïðåäåëîâïî-
ñëåäîâàòåëüíîñòè.Íàäîòîëüêîñäåëàòüçàìå÷àíèåâïîñëåäíåìñâîéñòâå,
÷òîåñëè
lim
x
!
x
0
g
(
x
)=
G
6
=0
,òîâçÿâ
"
=
j
G
j
=
2
,èñîîòâåòñòâóþùåå
�
0
,
ïîëó÷èì
0
=
2
f
(
U

(
x
0
))
,èëè,äðóãèìèñëîâàìè,âíåêîòîðîéîêðåñòíîñòè
òî÷êè
x
0
ôóíêöèÿ
f
(
x
)
íåîáðàùàåòñÿâíîëü.
pred_per_ner_fun
Óòâåðæäåíèå54
(Ïðåäåëüíûéïåðåõîäâíåðàâåíñòâàõäëÿôóíêöèé)
.
Ïóñòü
ñóùåñòâóåòïðåäåë
lim
x
!
x
0
f
(
x
)=
A
,ïðè÷åìôóíêöèÿ
f
(
x
)
òàêîâà,÷òî
8
x
2

U
"
(
x
0
)
(
f
(
x
)
6
C
).Òîãäà
A
6
C
.
64
ÃËÀÂÀ8.ÔÓÍÊÖÈß.ÏÐÅÄÅËÔÓÍÊÖÈÈ.
Äîêàçàòåëüñòâî.
Âûòåêàåòèçòåîðåìûîïðåäåëüíîìïåðåõîäåäëÿïîñëåäîâàòåëüíîñòåé.
dvamenta
Óòâåðæäåíèå55
(Òåîðåìàîäâóõìèëèöèîíåðàõäëÿôóíêöèé)
.
Ïóñòüñó-
ùåñòâóþòïðåäåëû
lim
x
!
x
0
f
1
(
x
)=lim
x
!
x
0
f
2
(
x
)=
F
èâíåêîòîðîéïðîêîëîòîé
îêðåñòíîñòè

U
"
(
x
0
)
òî÷êè
x
0
âûïîëíÿþòñÿíåðàâåíñòâà
f
1
(
x
)
6
g
(
x
)
6
f
2
(
x
)
;
òîãäàïðåäåë
lim
x
!
x
0
g
(
x
)
ñóùåñòâóåòèòîæåðàâåí
F
.
Äîêàçàòåëüñòâî.
Âûòåêàåòèçòåîðåìûîäâóõìèëèöèîíåðàõäëÿïîñëåäîâàòåëüíîñòåé.
8.4.1Àñèìïòîòè÷åñêèåîáîçíà÷åíèÿ
Îïðåäåëåíèå64
(Àñèìïòîòè÷åñêèåîáîçíà÷åíèÿ)
.
Ïóñòüôóíêöèè
f
(
x
)
è
g
(
x
)
îïðåäåëåíûâíåêîòîðîéïðîêîëîòîéîêðåñòíîñòè

U

(
x
0
)
è
g
(
x
)
6
=0
âýòîéîêðåñòíîñòè.Ãîâîðÿò,÷òî
f
(
x
)
åñòü
îáîëüøîå
îò
g
(
x
)
ïðè
x
ñòðåìÿùåìñÿê
x
0
èîáîçíà÷àþò
f
(
x
)=
O
(
g
(
x
))(
x
!
x
0
)
,åñëèôóíêöèÿ
f
(
x
)
g
(
x
)
îãðàíè÷åíàâíåêîòîðîé

U
"
(
x
0
)
.Ãîâîðÿò,÷òî
f
(
x
)
åñòü
îìàëîå
îò
g
(
x
)
ïðè
x
ñòðåìÿùåìñÿê
x
0
èîáîçíà÷àþò
f
(
x
)=
o
(
g
(
x
))(
x
!
x
0
)
,åñëè
lim
x
!
x
0
f
(
x
)
g
(
x
)
=0
.Ãîâîðÿò,÷òî
f
(
x
)
è
g
(
x
)
ýêâèâàëåíòíû
ïðè
x
ñòðåìÿ-
ùåìñÿê
x
0
èîáîçíà÷àþò
f
(
x
)

g
(
x
)(
x
!
x
0
)
,åñëè
lim
x
!
x
0
f
(
x
)
g
(
x
)
=1
.
Îñíîâíûåñâîéñòâà:
1.
Åñëè
f
(
x
)=
o
(
g
(
x
))(
x
!
x
0
)
,òî
f
(
x
)=
O
(
g
(
x
))(
x
!
x
0
)
.
2.
Åñëè
f
(
x
)

g
(
x
)(
x
!
x
0
)
,òî
f
(
x
)=
O
(
g
(
x
))(
x
!
x
0
)
.
3.
Åñëè
f
(
x
)=
o
(
g
(
x
))(
x
!
x
0
)
,òî
g
(
x
)

f
(
x
)

g
(
x
)(
x
!
x
0
)
.
4.
Åñëè
f
(
x
)=
O
(
g
(
x
))(
x
!
x
0
)
,òî
f
(
x
)

g
(
x
)=
O
(
g
(
x
))(
x
!
x
0
)
.
5.
Åñëè
f
(
x
)=
O
(
g
(
x
))(
x
!
x
0
)
è
f
0
(
x
)=
O
(
g
0
(
x
))(
x
!
x
0
)
,òî
f
(
x
)

f
0
(
x
)=
O
(
g
(
x
)

g
0
(
x
))(
x
!
x
0
)
.
6.
Åñëè
f
(
x
)=
o
(
g
(
x
))(
x
!
x
0
)
è
f
0
(
x
)=
o
(
g
0
(
x
))(
x
!
x
0
)
,òî
f
(
x
)

f
0
(
x
)=
o
(
g
(
x
)

g
0
(
x
))(
x
!
x
0
)
.
7.
Åñëè
f
(
x
)=
o
(
g
(
x
))(
x
!
x
0
)
è
f
0
(
x
)=
O
(
g
0
(
x
))(
x
!
x
0
)
,òî
f
(
x
)

f
0
(
x
)=
o
(
g
(
x
)

g
0
(
x
))(
x
!
x
0
)
.
8.5.ÍÅÏÐÅÐÛÂÍÛÅÔÓÍÊÖÈÈ.ÎÑÍÎÂÍÛÅÑÂÎÉÑÒÂÀ.
65
8.
Åñëè
f
(
x
)

g
(
x
)(
x
!
x
0
)
è
f
0
(
x
)

g
0
(
x
)(
x
!
x
0
)
,òî
f
(
x
)

f
0
(
x
)

g
(
x
)

g
0
(
x
)(
x
!
x
0
)
.
9.
Åñëè
f
(
x
)

g
(
x
)(
x
!
x
0
)
è
f
0
(
x
)

g
0
(
x
)(
x
!
x
0
)
,
f
0
(
x
)
;g
0
(
x
)
6
=0
,òî
f
(
x
)
f
0
(
x
)

g
(
x
)
g
0
(
x
)
(
x
!
x
0
)
.
8.5Íåïðåðûâíûåôóíêöèè.Îñíîâíûåñâîéñòâà.
Íåïðåðûâíîéíàçûâàåòñÿ
ôóíêöèÿ,ãðàôèêêîòîðîé
ìîæíîíàðèñîâàòüîäíèì
ïëàâíûìäâèæåíèåìðóêè
ÏðèïèñûâàåòñÿÈ.Íüþòîíó
Îïðåäåëåíèå65.
Ôóíêöèÿ
f
(
x
)
íàçûâàåòñÿ
íåïðåðûâíîé
âòî÷êå
x
0
2
D
f
åñëèååïðåäåë
lim
x
!
x
0
f
(
x
)
ñóùåñòâóåòèðàâåí
f
(
x
0
)
.
Îïðåäåëåíèå66.
Ôóíêöèÿ,íåÿâëÿþùàÿñÿíåïðåðûâíîéâíåêîòîðîé
òî÷êåíàçûâàåòñÿ
ðàçðûâíîé
âýòîéòî÷êå.
Îïðåäåëåíèå67.
Ôóíêöèÿ
f
(
x
)
íàçûâàåòñÿ
íåïðåðûâíîéñëåâà
âòî÷-
êå
x
0
2
D
f
åñëèååïðåäåë
lim
x
!
x
0

0
f
(
x
)
ñóùåñòâóåòèðàâåí
f
(
x
0
)
.Ôóíêöèÿ
f
(
x
)
íàçûâàåòñÿ
íåïðåðûâíîéñïðàâà
âòî÷êå
x
0
2
D
f
åñëèååïðåäåë
lim
x
!
x
0
+0
f
(
x
)
ñóùåñòâóåòèðàâåí
f
(
x
0
)
.
Îïðåäåëåíèå68.
Ôóíêöèÿ
f
(
x
)
íàçûâàåòñÿ
íåïðåðûâíîéíàîòêðû-
òîììíîæåñòâå
M
,åñëèîíàíåïðåðûâíàâîâñåõåãîòî÷êàõ.Ìíîæåñòâî
ôóíêöèé,íåïðåðûâíûõíàìíîæåñòâå
M
îáîçíà÷àþò
C
(
M
)
.
Îïðåäåëåíèå69.
Ôóíêöèÿ
f
(
x
)
íàçûâàåòñÿ
íåïðåðûâíîéíàîòêðû-
òîìîòðåçêå
[
a;b
]
,åñëèîíàíåïðåðûâíàòî÷êàõèíòåðâàëà
(
a;b
)
;
íåïðå-
ðûâíàñïðàâàâòî÷êå
a
èíåïðåðûâíàñëåâàâòî÷êå
b
.Ìíîæåñòâîôóíê-
öèé,íåïðåðûâíûõíàîòðåçêå
[
a;b
]
îáîçíà÷àþò
C
([
a;b
])
.
Çàìå÷àíèå.
Ñâîéñòâîíåïðåðûâíîñòèèìååòâïîëíåïîíÿòíûéãðàôè-
÷åñêèéñìûñëãðàôèêíåïðåðûâíîéôóíêöèèåñòüíåïðåðûâíàÿëèíèÿ,
ò.å.åãîìîæíîíàðèñîâàòüíåîòðûâàÿêàðàíäàøàîòáóìàãè.Áîëüøèíñòâî
ôóíêöèé,ñêîòîðûìèìûáóäåìèìåòüäåëîíåïðåðûâíûå,õîòÿáûâàþò
èñêëþ÷åíèÿ.
farif
Òåîðåìà56.
Ïóñòüôóíêöèè
f
,
g
íåïðåðûâíûíàìíîæåñòâå
M
,òîãäà:
66
ÃËÀÂÀ8.ÔÓÍÊÖÈß.ÏÐÅÄÅËÔÓÍÊÖÈÈ.
1.
Èõñóììàèðàçíîñòü
f
(
x
)+
g
(
x
)
,
f
(
x
)

g
(
x
)
òîæåíåïðåðûâíûíà
ìíîæåñòâå
M
2.
Èõïðîèçâåäåíèå
f
(
x
)
g
(
x
)
íåïðåðûâíîíàìíîæåñòâå
M
3.
Åñëè
g
(
x
)
6
=0
ïðèâñåõ
x
2
M
,òî
f
g
2C
(
M
)
.
Äîêàçàòåëüñòâî.
Ñëåäóåòèçñîîòâåòñòâóþùèõàðèôìåòè÷åñêèõñâîéñòâïðåäåëîâ.
Òåîðåìà57
(Íåïðåðûâíîñòüñëîæíîéôóíêöèè)
.
Åñëè
f
2C
(
M
)
è
g
2C
(
M
1
)
,
ãäå
M
1
=
f
(
M
)
îáðàçìíîæåñòâà
M
,òî
g

f
(
x
)=
g
(
f
(
x
))
2C
(
M
)
.
Äîêàçàòåëüñòâî.
Ðàññìîòðèìïðîèçâîëüíóþ
f
x
n
g
1
n
=1

D
g

f
òàê,÷òî
lim
n
!1
x
n
=
x
0
2
M
.Èç
íåïðåðûâíîñòè
f
(
x
)
ñëåäóåò,÷òî
lim
n
!1
y
n
=lim
n
!1
f
(
x
n
)=
y
0
=
f
(
x
0
)
.Àèç
íåïðåðûâíîñòè
g
(
x
)
âûòåêàåò
lim
n
!1
g
(
y
n
)=
g
(
y
0
)=
z
0
,
logr
Ëåììà14.
Ïóñòü
f
2C
(
f
x
0
g
)
,òîãäàíàéäóòñÿ
M
è
�
0
,òàêèå÷òî
8
x
2
U

(
x
0
)(
j
f
(
x
)
j
6
M
)
.Äðóãèìèñëîâàìè,ôóíêöèÿ,íåïðåðûâíàÿâòî÷-
êå
x
0
,îãðàíè÷åíàâíåêîòîðîéîêðåñòíîñòèýòîéòî÷êè.
Äîêàçàòåëüñòâî.
Âîçüìåì
"
=1
,òîãäàñóùåñòâóåò
�
0
,òàêîå,÷òî
f
(

U

(
x
0
))

U
1
(
f
(
x
0
))
.
Âûáðàâ
M
=
j
f
(
x
0
)
j
+1
ïîëó÷èì
8
x
2
U

(
x
0
)(
j
f
(
x
)
j
6
M
)
,
f_ogran
Òåîðåìà58.
[Îáîãðàíè÷åííîñòèôóíêöèè,íåïðåðûâíîéíàîòðåçêå]Ïóñòü
f
2C
([
a;b
])
.Òîãäàñóùåñòâóåò
M
è
m
,òàêèå,÷òîäëÿëþáîãî
x
2
[
a;b
]
âûïîëíåíûíåðàâåíñòâà
m
6
f
(
x
)
6
M
.Äðóãèìèñëîâàìè,ôóíêöèÿ,íåïðå-
ðûâíàÿíàîòðåçêåÿâëÿåòñÿîãðàíè÷åííîéíàýòîìîòðåçêå.
Äîêàçàòåëüñòâî.
Äîêàæåìîãðàíè÷åííîñòüñâåðõóìåòîäîìîòïðîòèâíîãî.Ïðåäïîëî-
æèìîáðàòíîå,ò.å.òî,÷òîäëÿëþáîãî
M
íàéäåòñÿ
x
2
[
a;b
]
òàêîå,÷òî
f
(
x
)
�M
.Âûáåðåì
f
x
n
g
1
n
=1
,òàê,÷òî
8
m
2
N
(
f
(
x
m
)
�m
)
.Ïîòåîðåìå
bolzano
39
(ÁîëüöàíîÂåéåðøòðàññà)ñóùåñòâóåòñõîäÿùàÿñÿïîäïîñëåäîâàòåëüíîñòü
f
x
m
n
g
1
n
=1
.Îáîçíà÷èì
lim
n
!1
x
m
n
=
x
0
.Ïîëåììå
logr
14íàéäóòñÿ
M
è
�
0
,
òàêèå÷òî
8
x
2
U

(
x
0
)(
j
f
(
x
)
j
6
M
)
,íî
U

(
x
0
)
ëîâóøêàïîñëåäîâàòåëü-
íîñòè
f
x
m
n
g
1
n
=1
,àçíà÷èòíàéäåòñÿ
x
m
n
2
U

(
x
0
)
,òàêîå,÷òî
f
(
x
m
n
)
�M
.
Ïðîòèâîðå÷èå.
8.5.ÍÅÏÐÅÐÛÂÍÛÅÔÓÍÊÖÈÈ.ÎÑÍÎÂÍÛÅÑÂÎÉÑÒÂÀ.
67
Óïðàæíåíèå32.
Äîêàæèòåîãðàíè÷åííîñòüñíèçó.
extrem
Òåîðåìà59
(Îáýêñòðåìàëüíûõçíà÷åíèÿõíåïðåðûâíîéôóíêöèè)
.
Ïóñòü
f
2C
([
a;b
])
.Òîãäàñóùåñòâóåò
x
max
2
[
a;b
]
è
x
min
2
[
a;b
]
,òàêèå,÷òîäëÿ
ëþáîãî
x
2
[
a;b
]
âûïîëíåíûíåðàâåíñòâà
f
(
x
min
)
6
f
(
x
)
6
f
(
x
max
)
.Äðóãè-
ìèñëîâàìè,ôóíêöèÿ,íåïðåðûâíàÿíàîòðåçêå,äîñòèãàåòíàíåìñâîåãî
ìàêñèìóìà(ìèíèìóìà).
Èçîãðàíè÷åííîñòè
f
íà
[
a;b
]
;
î÷åâèäíî,ñëåäóåòîãðàíè÷åííîñòüìíî-
æåñòâà
f
([
a;b
])
.Ñëåäîâàòåëüíî,ñóùåñòâóåò
M
=sup
f
([
a;b
])
.Âûáåðåìïî-
ñëåäîâàòåëüíîñòü
x
n
2
[
a;b
]
òàê,÷òîáû
f
(
x
n
)
2
U
1
=n
(
M
)
.Òàêèå
x
n
ñóùå-
ñòâóþò,ïîñêîëüêó
f
([
a;b
])
\
U
1
=n
(
M
)
6
=
?
ïîîïðåäåëåíèþòî÷íîéâåðõíåé
ãðàíè.Íåñëîæíîçàìåòèòü,÷òî
lim
n
!1
f
(
x
n
)=
M
.Âñå÷ëåíûïîñëåäîâàòåëü-
íîñòè
x
n
2
[
a;b
]
,ñëåäîâàòåëüíî,ïîòåîðåìåÁîëüöàíîÂåéåðøòðàññàèç
f
x
n
g
1
n
=1
ìîæíîâûäåëèòüñõîäÿùóþñÿïîäïîñëåäîâàòåëüíîñòü,
lim
k
!1
x
n
k
=
x
0
.
Ðàññìîòðèì
f
(
x
0
)=lim
k
!1
f
(
x
n
k
)=lim
n
!1
f
(
x
n
)=
M
,
fnzero
Òåîðåìà60
(ÁîëüöàíîÊîøè)
.
Ïóñòü
f
(
x
)
2C
([
a;b
])
,ïðè÷åì
f
(
a
)

0
è
f
(
b
)

0
.Òîãäàñóùåñòâóåò
x
2
[
a;b
]
(âîçìîæíî,íååäèíñòâåííîå),
òàêîå,÷òî
f
(
x
)=0
.
Äîêàçàòåëüñòâî.
Ïîñòðîèìñòÿãèâàþùóþñÿñèñòåìóîòðåçêîâ
f
[
a
n
;b
n
]
g
1
n
=1
ñëåäóþùèìîá-
ðàçîì:

Âûáèðàåìíàïåðâîìøàãå
a
1
=
a
è
b
1
=
b
.

Íà
n
ìøàãåðàññìîòðèì
c
n
=
a
n
+
b
n
2
.Åñòüòðèâàðèàíòà.à)Åñëè
f
(
c
n
)=0
òîòåîðåìàäîêàçàíà.á)Åñëè
f
(
c
n
)

0
,òîâûáåðåì
a
n
+1
=
a
n
,
b
n
+1
=
c
n
;â)Åñëè
f
(
c
n
)

0
,òî
a
n
+1
=
c
n
,
b
n
+1
=
b
n
.Òàêèìîáðàçîì,
íàêàæäîìøàãå
f
(
a
n
)

0
è
f
(
b
n
)

0
.
Î÷åâèäíî,ïîñòðîåííàÿòàêèìîáðàçîìñèñòåìàîòðåçêîâÿâëÿåòñÿâëîæåí-
íîéèñòÿãèâàþùåéñÿ.
Ïóñòü
x
0
=
T
n
[
a
n
;b
n
]
,äîêàæåì,÷òî
f
(
x
0
)=0
.Ïðåäïîëîæèì,÷òî
f
(
x
0
)=
d�
0
.
Ôóíêöèÿ
f
íåïðåðûâíà,ñëåäîâàòåëüíî
f
(
x
0
)=lim
x
!
x
0
f
(
x
)

0
.Î÷åâèäíî,
lim
n
!1
a
n
=
x
0
,ñëåäîâàòåëüíî
lim
n
!1
f
(
a
n
)=lim
x
!
x
0
f
(
x
)=
d�
0
;
÷òîïðîòèâîðå÷èòòåîðåìå
limner
32(îïðåäåëüíîìïåðåõîäåâíåðàâåíñòâàõ)òàê,
êàê
8
n
2
N
(
f
(
a
n
)

0)
.Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òîíåâîçìîæåíñëó÷àé
f
(
x
0
)

0
.Ñëåäîâàòåëüíî,
f
(
x
0
)=0
,
68
ÃËÀÂÀ8.ÔÓÍÊÖÈß.ÏÐÅÄÅËÔÓÍÊÖÈÈ.
fnotr
Ñëåäñòâèå10.
Ïóñòü
f
íåïðåðûâíàÿíàîòðåçêå
[
a;b
]
ôóíêöèÿ,
f
([
a;b
])=
f
y
=
f
(
x
)
j
x
2
[
a;b
]
g
ìíîæåñòâîååçíà÷åíèéíàîòðåçêå.Òîãäà
f
([
a;b
])=[min
x
2
[
a;b
]
f
(
x
)
;
max
x
2
[
a;b
]
f
(
x
)]
.
Ò.å.ìíîæåñòâîçíà÷åíèéíåïðåðûâíîéíàîòðåçêåôóíêöèèîáðàçóåòîò-
ðåçîê.
Äîêàçàòåëüñòâî.
Î÷åâèäíî,
f
([
a;b
])

[min
x
2
[
a;b
]
f
(
x
)
;
max
x
2
[
a;b
]
f
(
x
)]
(ñóùåñòâîâàíèå
min
x
2
[
a;b
]
f
(
x
)
è
max
x
2
[
a;b
]
f
(
x
)
ãàðàíòèðóåòòåîðåìà
extrem
59).Ðàññìîòðèìïðîèçâîëüíîå
y
0
2
[min
x
2
[
a;b
]
f
(
x
)
;
max
x
2
[
a;b
]
f
(
x
)]
.
Ôóíêöèÿ
g
(
x
)=
f
(
x
)

y
0
òàêîâà,÷òî
g
(
x
min
)
6
0
è
g
(
x
max
)

0
,ñëåäîâà-
òåëüíî,ïîòåîðåìå
fnzero
60íàéäåòñÿ
x
0
2
[
a;b
]
òàêîé,÷òî
g
(
x
0
)=0
,àçíà÷èò
f
(
x
0
)=
y
0
,ò.å.
y
0
2
f
([
a;b
])
,
f_inv
Òåîðåìà61
(Îñóùåñòâîâàíèèîáðàòíîéôóíêöèè)
.
Ïóñòü
f
(
x
)
2C
([
a;b
])
èÿâëÿåòñÿñòðîãîìîíîòîííîâîçðàñòàþùåé(óáûâàþùåé).Òîãäàîáðàò-
íàÿôóíêöèÿ
f

1
(
y
)
ñóùåñòâóåòèíåïðåðûâíàíàîòðåçêå
[
f
(
a
)
;f
(
b
)]
(ñî-
îòâåòñòâåííî
[
f
(
b
)
;f
(
a
)]
)
Äîêàçàòåëüñòâî.
Èçñëåäñòâèÿ
fnotr
10èìîíîòîííîñòèâûòåêàåò,÷òî
f
ÿâëÿåòñÿáèåêöèåé,è,
ñëåäîâàòåëüíî,ñóùåñòâóåòîáðàòíàÿêíåéôóíêöèÿ.
Äîêàæåìíåïðåðûâíîñòüîáðàòíîéôóíêöèè.Íåîãðàíè÷èâàÿîáùíîñòè
ðàññóæäåíèé,ñ÷èòàåìôóíêöèþ
f
ìîíîòîííîâîçðàñòàþùåéÂûáåðåìïðî-
èçâîëüíîå
x
0
2
(
a;b
)
è
"�
0
.Ðàññìîòðèìçíà÷åíèÿôóíêöèèâîêðåñòíîñòè
U
"
(
x
0
)
.Èçìîíîòîííîñòèèíåïðåðûâíîñòèñëåäóåò,÷òî
f
(
U
"
(
x
0
))=(
f
(
x
0

"
);
f
(
x
0
+
"
))
.
Îáîçíà÷èì
y
0
=
f
(
x
0
)
èâûáåðåì

=min(
y
0

f
(
x
0

"
)
;f
(
x
0
+
"
)

y
0
)
.
Î÷åâèäíî,
f

1
(
y
0
+

)
6
x
0
+
"
è
f

1
(
y
0


)

x
0

"
,ñëåäîâàòåëüíî,
f

1


U

(
y
0
)


U
"
(
x
0
)
.Òàêèìîáðàçîì,
lim
y
!
y
0
f

1
(
y
)=
x
0
,
Åñëèæå
x
0
=
a
èëè
b
,òîñëåäóåòáðàòüîäíîñòîðîííèåîêðåñòíîñòè
U
+

(
a
)=[
a;a
+

)
è
U


(
b
)=(
b

;b
]
,âîñòàëüíîìäîêàçàòåëüñòâîîñòàåòñÿ
ïðåæíèì.
Óïðàæíåíèå33.
Äîêàçàòüòåîðåìóäëÿìîíîòîííîóáûâàþùåéôóíê-
öèè.
8.6Ïðèìåðûíåïðåðûâíûõèðàçðûâíûõôóíê-
öèé.
Ïðèìåð10.
Ôóíêöèÿ
f
(
x
)=
c
(êîíñòàíòà)íåïðåðûâíàíà
R
.Äåéñòâè-
òåëüíî,
lim
x
!
x
0
f
(
x
)=
c
=
f
(
x
0
)
.
8.6.ÏÐÈÌÅÐÛÍÅÏÐÅÐÛÂÍÛÕÈÐÀÇÐÛÂÍÛÕÔÓÍÊÖÈÉ.
69
Ïðèìåð11.
Ôóíêöèÿ
f
(
x
)=
x
íåïðåðûâíàíà
R
.Äåéñòâèòåëüíî,ïóñòü
lim
n
!1
x
n
=
x
0
,òîãäà
lim
n
!1
f
(
x
n
)=lim
n
!1
x
n
=
x
0
=
f
(
x
0
)
.
Ïðèìåð12.
Ôóíêöèÿ
f
(
x
)=
P
n
(
x
)=
a
n
x
n
+
a
n

1
x
n

1
+
:::
+
a
1
x
+
a
0

íåïðåðûâíàíà
R
.Ýòîâûòåêàåòèçïðåäûäóùèõäâóõïðèìåðîâèòåîðå-
ìû
farif
56.
Ïðèìåð13.
Ôóíêöèè
f
(
x
)=sin
x
è
f
(
x
)=cos
x
íåïðåðûâíûíà
R
.
Äîêàçàòåëüñòâî.
Äîêàæåìäëÿ
f
(
x
)=sin
x
.Ðàññìîòðèìïðîèçâîëüíóþïîñëåäîâàòåëüíîñòü
lim
n
!1
x
n
=
x
0
.Äëÿíåå
j
sin
x
n

sin
x
0
j
=
j
2sin
1
2
(
x
n

x
0
)

cos
1
2
(
x
n
+
x
0
)
j
6
2
j
sin
1
2
(
x
n

x
0
)
j
:
Ïóñòüçàäàíîïðîèçâîëüíîå
"�
0
,âûáåðåì
N
,òàê,÷òîïðè
8
n�N

j
x
n

x
0
j

arcsin

1
2
"

;
òîãäà
sin
j
x
n

x
0
j

1
2
"
(èçìîíîòîííîñòèñèíóñà),à,ñëåäîâàòåëüíî,
j
sin
x
n

sin
x
0
j
"
,
Óïðàæíåíèå34.
Äîêàçàòüòîæåñàìîåäëÿ
f
(
x
)=cos
x
.
Ïðèìåð14.
Ôóíêöèÿ
f
(
x
)=tg
x
íåïðåðûâíàíàêàæäîìèçèíòåðâàëîâ



2
+
n;

2
+
n

,ãäå
n
2
Z
.Ýòîñëåäóåòèçíåïðåðûâíîñòèôóíêöèé
f
(
x
)=sin
x
,
f
(
x
)=cos
x
èòåîðåìû
farif
56.
Ïðèìåð15.
Ôóíêöèÿ
f
(
x
)=ctg
x
íåïðåðûâíàíàêàæäîìèçèíòåð-
âàëîâ
(
n;
+
n
)
,ãäå
n
2
Z
.Ýòîñëåäóåòèçíåïðåðûâíîñòèôóíêöèé
f
(
x
)=sin
x
,
f
(
x
)=cos
x
èòåîðåìû
farif
56.
Ïðèìåð16.
ÔóíêöèÿÄèðèõëå
D
(
x
)=
(
1
;x
2
Q
0
;x=
2
Q
ôóíêöèÿ,ïðè-
íèìàþùàÿçíà÷åíèå1,åñëèàðãóìåíòðàöèîíàëåí,è0,åñëèàðãóìåíò
èððàöèîíàëåí.Òàêêàêâëþáîéîêðåñòíîñòèëþáîéòî÷êèâåùåñòâåííîé
ïðÿìîéñîäåðæàòñÿêàêðàöèîíàëüíûå,òàêèèððàöèîíàëüíûå÷èñëà(à
çíà÷èò,êàêíóëè,òàêèåäèíèöûôóíêöèèÄèðèõëå),íèâîäíîéòî÷-
êåïðåäåë
D
(
x
)
íåñóùåñòâóåò,àçíà÷èò,îíàðàçðûâíàíàâñåé÷èñëîâîé
ïðÿìîé.
70
ÃËÀÂÀ8.ÔÓÍÊÖÈß.ÏÐÅÄÅËÔÓÍÊÖÈÈ.
Ïðèìåð17.
ÔóíêöèÿÐèìàíà
R
(
x
)=
(
1
n
;x
=
m
n
;
(
m;n
)=1
0
;x=
2
Q
ôóíê-
öèÿ,ïðèíèìàþùàÿçíà÷åíèå
1
=n
,åñëèàðãóìåíòïðåäñòàâèìíåñîêðàòè-
ìîéäðîáüþñîçíàìåíàòåëåì
n
,è0,åñëèàðãóìåíòèððàöèîíàëåí.
Óïðàæíåíèå35.
Äîêàçàòü,÷òîôóíêöèÿÐèìàíà
R
(
x
)
íåïðåðûâíàâ
èððàöèîíàëüíûõèðàçðûâíàâðàöèîíàëüíûõòî÷êàõ.
Óïðàæíåíèå36.
Ïðèâåñòèïðèìåðôóíêöèè
f
(
x
)
,ðàçðûâíîéâîâñåõ
òî÷êàõ÷èñëîâîéïðÿìîé,òàêîé,÷òî
j
f
(
x
)
j2C
(
R
)
.
Óïðàæíåíèå37.
à)Ïðèâåñòèïðèìåðôóíêöèè,ðàçðûâíîéâîâñåõòî÷-
êàõ÷èñëîâîéïðÿìîéêðîìåîäíîé,ò.å.íåïðåðûâíîéòîëüêîâîäíîéòî÷êå;
á)ïðèâåñòèïðèìåðôóíêöèè,íåïðåðûâíîéðîâíîâäâóõòî÷êàõ;â)*ðîâíî
â
n
òî÷êàõ,
n
2
N
.
Óïðàæíåíèå38.
Ïóñòü
f
(
x
)
íåêîòîðûéìíîãî÷ëåí,ïðîêîòîðûéèç-
âåñòíî,÷òîóðàâíåíèå
f
(
x
)=
x
íåèìååòêîðíåé.Äîêàæèòå,÷òîòîãäà
èóðàâíåíèå
f
(
f
(
x
))=
x
íåèìååòêîðíåé.
Óïðàæíåíèå39.
Äàíàâûïóêëàÿôèãóðàèòî÷êà
A
âíóòðèíåå.Äîêà-
æèòå,÷òîíàéäåòñÿõîðäà(ò.å.îòðåçîê,ñîåäèíÿþùèéäâåãðàíè÷íûå
òî÷êèâûïóêëîéôèãóðû),ïðîõîäÿùàÿ÷åðåçòî÷êó
A
èäåëÿùàÿñÿòî÷-
êîé
A
ïîïîëàì.
Óïðàæíåíèå40.
Ïóñòü
f
2C
([0
;
1])
òàêàÿ,÷òî
f
(0)=
f
(1)=0
.Äî-
êàæèòå,÷òîíàîòðåçêå
[0;1]
íàéäóòñÿ2òî÷êèíàðàññòîÿíèè
1
10

êîòîðûõôóíêöèÿ
f
(
x
)
ïðèíèìàåòðàâíûåçíà÷åíèÿ.
Óïðàæíåíèå41.
Îôóíêöèè
f
(
x
)
,çàäàííîéíàâñåéâåùåñòâåííîéïðÿ-
ìîé,èçâåñòíî,÷òîïðèëþáîì
a�
1
ôóíêöèÿ
f
(
x
)+
f
(
ax
)
íåïðåðûâíàíà
âñåéïðÿìîé.Äîêàæèòå,÷òî
f
(
x
)
òàêæåíåïðåðûâíàíàâñåéïðÿìîé.
Óïðàæíåíèå42.
Èçâåñòíî,÷òî
D
f
=
R
,èäëÿëþáîãî
x
2
R
âûïîëíåíî
ðàâåíñòâî:
f
(
x
+1)

f
(
x
)+
f
(
x
+1)+1=0
.Äîêàæèòå,÷òî
f=
2C
(
R
)
.
Ñëåäñòâèå11
(Ìåòîäèíòåðâàëîâ)
.
Ðàññìàòðèâàåòñÿíåðàâåíñòâî
F
1
(
x
)

F
2
(
x
)

:::

F
n
(
x
)
G
1
(
x
)

G
2
(
x
)

:::

G
m
(
x
)

0
,ãäå
F
1
;F
2
;:::F
n
;G
1
;G
2
(
x
)
;:::;G
m
íåïðåðûâíûíàíåêîòîðììíîæåñòâå
M
.Òî÷êè
x
1
;x
2
;:::;x
s
,âêîòîðûõõîòÿáûîäíàèçýòèõôóíêöèéðàâ-
íà0ðàçáèâàþò
M
íàíåïåðåñåêàþùèåñÿèíòåðâàëû,âêàæäîìèçêî-
òîðûõçíàêêàæäîéôóíêöèèñîõðàíÿåòñÿýòîãàðàíòèðóåòòåîðå-
ìà
fnzero
60.Ñëåäîâàòåëüíî,îïðåäåëèâêàêèìëèáîîáðàçîìçíàêâûðàæåíèÿ
8.6.ÏÐÈÌÅÐÛÍÅÏÐÅÐÛÂÍÛÕÈÐÀÇÐÛÂÍÛÕÔÓÍÊÖÈÉ.
71
F
1
(
x
)

F
2
(
x
)

:::

F
n
(
x
)
G
1
(
x
)

G
2
(
x
)

:::

G
m
(
x
)
âîäíîéòî÷êåèíòåðâàëà,ìûçíàåìçíàêâîâñåõòî÷-
êàõ.Ñëåäîâàòåëüíî,ìîæíîïîëó÷èòüîòâåò,âûïèñàâèíòåðâàëûñíóæ-
íûìíàìçíàêîì.
Õîòåëîñüáûîáðàòèòüâíèìàíèå÷èòàòåëåéíàòîòôàêò,÷òîôóíê-
öèèìîãóòáûòüëþáûìè,ãëàâíîå,÷òîáûîíèáûëèíåïðåðûâíûíàèíòå-
ðåñóþùåìíàñìíîæåñòâå.Âøêîëüíîéïðîãðàììåîáû÷íîðàññìàòðèâà-
åòñÿ÷àñòíûéñëó÷àé,êîãäàìíîæèòåëèèìåþòâèä
(
x

a
)
k
.
72
ÃËÀÂÀ8.ÔÓÍÊÖÈß.ÏÐÅÄÅËÔÓÍÊÖÈÈ.
Ãëàâà9
Òðèãîíîìåòðè÷åñêèå
ôóíêöèè
9.1×èñëî

Âñïîìíèìíåêîòîðûåïîíÿòèÿèçãåîìåòðèè.Ëîìàíîéèç
n
çâåíüåâíàïëîñ-
êîñòèíàçûâàåòñÿ
n
+1
ðàçëè÷íûõòî÷åê
A
0
;A
1
;:::;A
n
âåðøèíëîìàíîé

n
îòðåçêîâ,ïîñëåäîâàòåëüíîñîåäèíÿþùèõýòèòî÷êè.Ëîìàíàÿáåçñà-
ìîïåðåñå÷åíèéíàçûâàåòñÿïðîñòîé(ñì.ðèñ.
lom1
9.1).Òî÷êè
A
0
è
A
n
íàçûâàþò-
ñÿêîíöàìèëîìàíîé.Åñëèêîíöûëîìàíîéñîâïàäàþò,ëîìàíàÿíàçûâàåò-
ñÿçàìêíóòîé.Ïðîñòàÿçàìêíóòàÿëîìàíàÿíàçûâàåòñÿìíîãîóãîëüíèêîì,
çâåíüÿýòîéëîìàíîéñòîðîíàìèìíîãîóãîëüíèêà,âåðøèíûëîìàíîé
âåðøèíàìèìíîãîóãîëüíèêà.Åñëèëîìàíàÿèìååò
n
çâåíüåâ,òîáóäåò
n

óãîëüíèê.Ìíîãîóãîëüíèêíàçûâàåòñÿâûïóêëûì,åñëèäëÿâñÿêîéåãîñòî-
ðîíûâåðíî,÷òîâñåîñòàëüíûåñòîðîíûëåæàòâîäíîéïîëóïëîñêîñòèîò-
íîñèòåëüíîïðÿìîé,ñîäåðæàùåéâûáðàííóþñòîðîíó.Äëèíàëîìàíîéñóòü
ñóììàäëèíå¼çâåíüåâ,ïåðèìåòðìíîãîóãîëüíèêàñóììàäëèíåãîñòîðîí.
Òåîðåìà62.
Äëèíàîòðåçêà,ñîåäèíÿþùåãîêîíöûëîìàíîé,íåïðåâîñõî-
äèòäëèíûëîìàíîé.
Äîêàçàòåëüñòâî
ïðîâåä¼ìèíäóêöèåéïî÷èñëóçâåíüåâëîìàíîé.

Äëÿ
n
=1
óòâåðæäåíèå,î÷åâèäíî,âåðíî.

Øàãèíäóêöèè:ðàññìîòðèìëîìàíóþ
A
0
A
1
:::A
n
A
n
+1
;
ñîñòîÿùóþèç
(
n
+1)
-ãîçâåíà.Îáîçíà÷èì
l
n
äëèíóëîìàíîé,
A
0
A
1
:::A
n
;
ñîñòîÿùåé
73
74
ÃËÀÂÀ9.ÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÅÔÓÍÊÖÈÈ
lom1
A
0
A
1
:::
:::
A
n
Ðèñ.9.1:Ïðîñòàÿíåçàìêíóòàÿëîìàíàÿ
èç
n
çâåíüåâ.Òîãäà
j
A
0
A
n
+1
j
6
j
A
0
A
n
j
+
j
A
n
A
n
+1
j
6
l
n
+
j
A
n
A
n
+1
j
=
l
n
+1
:
Ïåðâîåíåðàâåíñòâîñóòüíåðàâåíñòâîòðåóãîëüíèêà,àâòîðîåñïðà-
âåäëèâîïîïðåäïîëîæåíèþèíäóêöèè.
Òåîðåìà63.
Åñëèâûïóêëûéìíîãîóãîëüíèê
Q
0
ñîäåðæèòâûïóêëûéìíî-
ãîóãîëüíèê
R;
òîïåðèìåòð
R
íåïðåâîñõîäèòïåðèìåòðà
Q
0
:
P
(
R
)
6
P
(
Q
0
)
:
Ðàññìîòðèìïðîèçâîëüíóþîêðóæíîñòü.Ïóñòü
M
ìíîæåñòâîïåðè-
ìåòðîââïèñàííûõâîêðóæíîñòüâûïóêëûõìíîãîóãîëüíèêîâ.Ìíîæåñòâî
M
íåïóñòî,òàêêàêâîêðóæíîñòüìîæíîâïèñàòüòðåóãîëüíèê,ïåðèìåòð
êîòîðîãîáóäåòýëåìåíòîì
M:
Ïîòåîðåìå2ìíîæåñòâî
M
îãðàíè÷åíîñâåð-
õóïåðèìåòðîìîïèñàííîãîîêîëîîêðóæíîñòèòðåóãîëüíèêà.Ïîòåîðåìå
Âåéåðøòðàññàñóùåñòâóåò
L
=sup
M:
×èñëî
L
íàçûâàåòñÿäëèíîéîêðóæ-
íîñòè.
Îïðåäåëåíèå70.
Äëèíîéîêðóæíîñòèíàçûâàåòñÿòî÷íàÿâåðõíÿÿãðàíü
ìíîæåñòâàïåðèìåòðîââïèñàííûõâîêðóæíîñòüâûïóêëûõìíîãîóãîëü-
íèêîâ.
9.1.×ÈÑËÎ

75
lom1
A
0
A
1
:::
:::
A
n
Ðèñ.9.2:Ïðîñòàÿçàìêíóòàÿëîìàíàÿ
Óòâåðæäåíèå64.
Îòíîøåíèåäëèíûîêðóæíîñòèêå¼äèàìåòðóîäíî
èòîæåäëÿëþáîéîêðóæíîñòè.
Äîêàçàòåëüñòâî.
Âñàìîìäåëå,ïóñòü
D
è
D
0
-äèàìåòðûêàêèõëèáîîêðóæíîñòåéñîáùèì
öåíòðîì
O:
Ãîìîòåòèÿñöåíòðîì
O
èêîýôôèöèåíòîì
k
=
D
0
D
ïåðåâîäèò
îêðóæíîñòü
D
âîêðóæíîñòü
D
0
:
Òàêêàêãîìîòåòèÿåñòüïðåîáðàçîâàíèå
ïîäîáèÿ,òî
D
0
=
kD
èäëèíàîêðóæíîñòè
L
0
=
kL;
âåäüïåðèìåòðûâñåõ
âïèñàííûõâûïóêëûõìíîãîóãîëüíèêîâïðèãîìîòåòèèèçìåíÿòñÿâ
k
ðàç,à
çíà÷èò,èâåðõíÿÿãðàíüïåðèìåòðîâèçìåíèòñÿâîñòîëüêîæåðàç
1
.Òàêèì
îáðàçîì
L
D
=
L
0
D
0
:
Îïðåäåëåíèå71.
×èñëî

åñòüîòíîøåíèåäëèíûîêðóæíîñòèêå¼äèà-
ìåòðó:

=
L
D
:
1
Ïîïðîáóéòåäîêàçàòüýòîñòðîãî.
76
ÃËÀÂÀ9.ÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÅÔÓÍÊÖÈÈ
9.2Ðàäèàííàÿìåðàóãëà
9.2.1Íåêîòîðûåôàêòûèçãåîìåòðèè
Íàïîìíèì,÷òîóãîëãåîìåòðè÷åñêàÿôèãóðàíàïëîñêîñòè,ñîñòîÿùàÿèç
òî÷êè-âåðøèíûóãëà,èäâóõðàçëè÷íûõëó÷åé,èñõîäÿùèõèçýòîéòî÷êè
ñòîðîíóãëà.
Ñâîéñòâàóãëîâ
2
:

Êàæäûéóãîëèìååòîïðåäåë¼ííóþãðàäóñíóþìåðó,áîëüøóþíóëÿ.

Ðàçâ¼ðíóòûéóãîëðàâåí
180

:

Ãðàäóñíàÿìåðàóãëàðàâíàñóììåãðàäóñíûõìåðóãëîâ,íàêîòîðûå
îíðàçáèâàåòñÿëþáûìëó÷îì,ëåæàùèììåæäóñòîðîíàìèóãëà.Ëó÷
ëåæèòìåæäóñòîðîíàìèóãëà,åñëèîíèñõîäèòèçâåðøèíûóãëàè
ïåðåñåêàåòêàêîé-ëèáîîòðåçîêñêîíöàìèíàñòîðîíàõóãëà.

Óãîëðàçáèâàåòïëîñêîñòüíàäâå÷àñòè,êàæäàÿèçêîòîðûõíàçûâà-
åòñÿïëîñêèìóãëîì.

Öåíòðàëüíûìóãëîìíàçûâàåòñÿïëîñêèéóãîëñâåðøèíîéâöåíòðå
íåêîòîðîéîêðóæíîñòè.

Óãëûññîâïàäàþùèìèñòîðîíàìèèìåþòãðàäóñíóþìåðó
0

:

Ïëîñêèåèöåíòðàëüíûåóãëûèìåþòìåðóîò
0

äî
360

:
Óïðàæíåíèå43.
Ëó÷èñõîäèòèçâåðøèíûóãëàèïåðåñåêàåòêàêîé-ëèáî
îòðåçîêñêîíöàìèíàñòîðîíàõóãëà.Äîêàæèòå,îïèðàÿñüíàòåîðåìó
Ïàøà
3
,÷òîâòàêîìñëó÷àåëó÷ïåðåñåêàåòëþáîéäðóãîéîòðåçîêñêîí-
öàìèíàñòîðîíàõóãëà.ÒåîðåìàÏàøàñîñòîèòâòîì,÷òîåñëèïðÿìàÿ
ïåðåñåêàåòîäíóèçñòîðîíòðåóãîëüíèêàèíåïðîõîäèòíè÷åðåçîäíóèç
åãîâåðøèí,òîîíàïåðåñåêàåòîäíóèçäâóõäðóãèõñòîðîíòðåóãîëüíèêà.
ÄîêàæèòåòàêæåòåîðåìóÏàøà.
radians
Îïðåäåëåíèå72.
Ïóñòüïëîñêèéóãîëèìååòìåðó

ãðàäóñîâ.Òîãäàåãî
ðàäèàííàÿìåðàåñòü
'
=

180

:
2
Óïîòðåáëÿÿñëîâîóãîë,áóäåìèìåòüââèäóëèáîïðîñòîóãîë,ëèáîïëîñêèéóãîë,
ëèáîöåíòðàëüíûéóãîë.
3
ÂàêñèîìàòèêåÃèëüáåðòàãåîìåòðèèÅâêëèäàòåîðåìàÏàøàâçÿòàçààêñèîìó.
9.3.ÎÏÐÅÄÅËÅÍÈÅÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÕÔÓÍÊÖÈÉ
77
Òàêèìîáðàçîì,ðàäèàííàÿìåðàåñòüëèíåéíàÿçàìåíàãðàäóñíîéìå-
ðû.Èñõîäÿèçîïðåäåëåíèÿ
radians
72
ñîñòàâèìòàáëèöóñîîòâåòñòâèÿãðàäóñíîéè
ðàäèàííîéìåðûäëÿíåêîòîðûõóãëîâ.


0

30

45

60

90

180

270

360

'
0

6

4

3

2

3

2
2

9.3Îïðåäåëåíèåòðèãîíîìåòðè÷åñêèõôóíêöèé
Îïðåäåëåíèå73.
Ïóñòü
x
2
R
:
Öåëîå÷èñëî
n
òàêîå,÷òî
n
6
xn
+1
íàçûâàåòñÿöåëîé÷àñòüþ
x
èîáîçíà÷àåòñÿ
[
x
]
:
Ñóùåñòâîâàíèåèåäèíñòâåííîñòüöåëîé÷àñòèóñòàíàâëèâàåòñëåäóþ-
ùàÿ
th3
Òåîðåìà65.
8
x
2
R
9
!
n
2
Z
:
n
6
xn
+1
:
Äîêàçàòåëüñòâî.

Ñóùåñòâîâàíèå.
Äëÿ
x
2
[0
;
1)
ïîëàãàåì
n
=0
:
Äëÿ
x

1
ìíîæå-
ñòâî
M
=
f
n
2
N
:
n
6
x
g
íåïóñòî,òàêêàê
1
2
M;
èîãðàíè÷åíî
ñâåðõó÷èñëîì
x:
ÏîòåîðåìåÂåéåðøòðàññàñóùåñòâóåò
m
=sup
M:
Åñëè
m
2
M;
òî
m
6
xm
+1
:
Ñëó÷àé
m=
2
M
íåâîçìîæåí.Äåé-
ñòâèòåëüíî,òàêêàê
m
åñòüòî÷íàÿâåðõíÿÿãðàíü
M;
òîïðèëþáîì
"�
0
íàéäåòñÿ
n
2
M
,òàêîå,÷òî
m

"n
6
m:
Òîãäà,åñëè
m=
2
M;
òîäëÿ
"
=1
ñóùåñòâóåò
n
1
2
M
äëÿêîòîðîãîâûïîëíåíî
m

1
n
1
m
èäëÿ
"
=
m

n
1
ñóùåñòâóåò
n
2
2
M
,äëÿêîòîðîãî
m

(
m

n
1
)=
n
1
n
2
m:
Òàêèìîáðàçîì,
m

1
n
1
n
2
m
èíàòóðàëüíîå÷èñëî
n
0
=
n
2

n
1
òàêîâî,÷òî
0
n
0

1
;
÷òîíåâîç-
ìîæíî.Åñëè
x
0
;
òî

x�
0
èïîäîêàçàííîìóíàéä¼òñÿ
m
0
2
Z
òàêîå,÷òî
m
0
6

xm
0
+1
:
Òîãäà,

(
m
0
+1)
x
6

m
0
:
Òàêèì
îáðàçîì,ëèáî
n
=
x
=

m
0
;
ëèáî
n
=

m
0

1
x

m
0
:

Åäèíñòâåííîñòü.
Äâîéíîåíåðàâåíñòâî
n
6
xn
+1
ðàâíîñèëüíî
òîìó,÷òî
x
=
n
+
;
ãäå

2
[0
;
1)
:
Åñëè,êðîìåòîãî,
x
=
m
+
;
ãäå

2
[0
;
1)
;
òî
0
6
j
m

n
j
=
j



j

1
:
Çíà÷èò,åñëè
m
è
n
öåëûå,òî
m
=
n:
th4
Òåîðåìà66.
Äëÿëþáîãî
x
2
R
ñóùåñòâóåòåäèíñòâåííàÿïàðà
(
;n
)
,
ãäå

2
[0;2

)
n
2
Z
òàêàÿ,÷òî
x
=

+2
n:
Äîêàçàòåëüñòâî.
Ïóñòü
n
=[
x
2

]
öåëàÿ÷àñòü
x
2

:
Òîãäà
n
6
x
2

n
+1
;
÷òîðàâíîñèëüíî
0
6
x

2
n
2
:
Ïîëàãàÿ

=
x

2
n;
èìååìóòâåðæäåíèå
78
ÃËÀÂÀ9.ÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÅÔÓÍÊÖÈÈ
trig_krug
tg
cos
sin
sin

cos

tg

=
sin

cos


1

1
2
1

1

1
2
1
2
1
Ðèñ.9.3:Îïðåäåëåíèåòðèãîíîìåòðè÷åñêèõôóíêöèé.
òåîðåìû.Ñóùåñòâîâàíèåèåäèíñòâåííîñòüñëåäóþòèçñóùåñòâîâàíèÿè
åäèíñòâåííîñòèöåëîé÷àñòèïîòåîðåìå
th3
65.Òåîðåìàäîêàçàíà.
opr5
Îïðåäåëåíèå74.
ÐàññìîòðèìâïðÿìîóãîëüíîéÄåêàðòîâîéñèñòåìåêî-
îðäèíàòíàïëîñêîñòè
Oxy
¾åäèíè÷íóþîêðóæíîñòü¿
x
2
+
y
2
=1
(ñì.Ðèñ.
trig_krug
9.3).
Îòîñèàáñöèññ
Ox
¾âïîëîæèòåëüíîìíàïðàâëåíèè¿,òîåñòüïðîòèâõî-
äà÷àñîâîéñòðåëêè,îòëîæèìóãîë
xOz
ìåðû

ðàäèàí.Ñòîðîíà
Oz
ýòîãî
óãëàïåðåñå÷¼òåäèíè÷íóþîêðóæíîñòüâòî÷êå
M
(
x
0
;y
0
)
:
Ïîîïðåäåëå-
íèþïîëàãàåì
cos

=
x
0
;
sin

=
y
0
:
Åñëèòåïåðü
x
ïðîèçâîëüíîåäåé-
ñòâèòåëüíîå÷èñëî,òîïîòåîðåìå
th4
66äëÿ
x
ñóùåñòâóåòåäèíñòâåííîå
ïðåäñòàâëåíèå
x
=

+2
n:
Ïîëàãàåì
cos
x
=cos
;
sin
x
=sin
;
tg
x
=
sin
x
cos
x
;
ctg
x
=
cos
x
sin
x
;
sec
x
=
1
cos
x
;
cosec
x
=
1
sin
x
:
Îïðåäåëåíèå
opr5
74ïîçâîëÿåòðàñïðîñòðàíèòüïîíÿòèåðàäèàííîéìåðûóã-
ëà(àòàêæåãðàäóñíîéìåðûââèäóëèíåéíîñòèçàìåíûãðàäóñíîéìåðûíà
ðàäèàííóþ)äëÿïðîèçâîëüíîãî
x
2
R
:
Ïóñòü
x

0
è
x
=

+2
n

ïðåäñòàâëåíèå
x;
ïîëó÷åííîåïîòåîðåìå
th4
66.Ïîëàãàåì,÷òîóãîëìåðû
x
ðàäèàíïîëó÷àåòñÿîòêëàäûâàíèåìîòîñèàáñöèññ
Oxn
îáîðîòîââïîëî-
æèòåëüíîìíàïðàâëåíèè(ïðîòèâõîäà÷àñîâîéñòðåëêè),èçàòåìîòêëà-
äûâàíèåìâïîëîæèòåëüíîìíàïðàâëåíèèóãëàìåðû

ðàäèàí.Òàêêàê

x
=



2
n
=(2



)

2

(
n
+1)
èóãëû

è
(2



)
äîïîëíèòåëüíûå,
9.4.ÑÂÎÉÑÒÂÀÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÕÔÓÍÊÖÈÉ
79
òîìîæíîñ÷èòàòü,÷òîóãîë
(2



)
ïîëó÷àåòñÿîòêëàäûâàíèåìóãëà

â
îòðèöàòåëüíîìíàïðàâëåíèè(ïîõîäó÷àñîâîéñòðåëêè),è,òàêèìîáðàçîì,
óãîë
(

x
)
6
0
ïîëó÷àåòñÿîòêëàäûâàíèåìîòîñèàáñöèññ
Oxn
îáîðîòîââ
îòðèöàòåëüíîìíàïðàâëåíèè(ïîõîäó÷àñîâîéñòðåëêè),èçàòåìîòêëàäû-
âàíèåìâòîìæåîòðèöàòåëüíîìíàïðàâëåíèèóãëàìåðû

ðàäèàí.
Çàìåòèì,÷òî
sin
x
è
cos
x
ñóùåñòâóþòäëÿâñåõ
x
2
R
;
à
tg
x;
ctg
x;
sec
x;
cosec
x
ñóùåñòâóþòíåäëÿâñåõ
x
2
R
:
Òðèãîíîìåòðè÷åñêèåôóíêöèèîïðåäåëèìñëåäóþùèìîáðàçîì.
def6
Îïðåäåëåíèå75.
sin:
R
�!
R
;x
7!
sin
x
;tg:
R
nf
x
:cos
x
=0
g�!
R
;x
7!
tg
x
;
cos:
R
�!
R
;x
7!
cos
x
;ctg:
R
nf
x
:sin
x
=0
g�!
R
;x
7!
ctg
x:
9.4Ñâîéñòâàòðèãîíîìåòðè÷åñêèõôóíêöèé
Îïðåäåëåíèå76.
Ïóñòü
M

R
:
Ôóíêöèÿ
f
:
M
�!
R
íàçûâàåòñÿïå-
ðèîäè÷åñêîé,åñëèñóùåñòâóåò

6
=0
òàêîå,÷òîäëÿëþáîãî
x
2
M
âûïîë-
íåíî
f
(
x
+

)=
f
(
x
)
:
×èñëî

íàçûâàåòñÿïåðèîäîìôóíêöèè
f:
Ïîëàãàåì
íîëüïåðèîäîìëþáîéôóíêöèè.Íàèìåíüøèéïîëîæèòåëüíûéïåðèîä
T
íàçûâàåòñÿãëàâíûìïåðèîäîì.
lem1
Ëåììà15.
Åñëè

ïåðèîäôóíêöèè
f
:
R

M
�!
R
;
òîäëÿëþáîãî
m
2
Z
m
òîæåïåðèîä
f:
Äîêàçàòåëüñòâî.
Åñëè

=0
èëè
m
=0
;
òîóòâåðæäåíèå,î÷åâèäíî,
âåðíî.Ïóñòü

6
=0
:
Âñëó÷àå
m�
0
ïðèìåíèìïðèíöèïìàòåìàòè÷åñêîé
èíäóêöèè.Áàçèñèíäóêöèè:
f
(
x
+

)=
f
(
x
)
;
÷òîâåðíî,òàêêàê

-ïåðèîä
f:
Øàãèíäóêöèè:
f
(
x
+(
m
+1)

)=
f
(
x
+
m
+

)=
f
(
x
+
m
)=
f
(
x
)
ïîïðåä-
ïîëîæåíèþèíäóêöèè.Äëÿ
m
0
èìååì
f
(
x
+
m
)=
f
(
x
+
m

m
)=
f
(
x
)
;
òàêêàê
(

m
)

0
èïîäîêàçàííîìó
(

m
)
ïåðèîä
f:
Ëåììàäîêàçàíà.
th5
Òåîðåìà67.
Åñëè
T
ãëàâíûéïåðèîäôóíêöèè
f
:
R

M
�!
R
;
òî
f
mT
:
m
2
Z
g
ìíîæåñòâîâñåõïåðèîäîâôóíêöèè
f:
Äîêàçàòåëüñòâî.
Ïóñòü
M
ìíîæåñòâîâñåõïåðèîäîâôóíêöèè
f:
Ïî
ëåììå
lem1
15,
f
mT
:
m
2
Z
g
M
:
Ïóñòü

2
M
ïðîèçâîëüíûéýëåìåíòèç
M
;
à
n
=[

T
]
öåëàÿ÷àñòü
[

T
]
:
Òîãäà
n
6

T
n
+1
;
÷òîðàâíîñèëüíî
0
6


nTT:
Òàêêàê

è
(

nT
)
ïåðèîäû
f;
òî
f
(
x
+


nT
)=
f
(
x
+

)=
f
(
x
)
;
èòàêèìîáðàçîì
(


nT
)
òîæåïåðèîä
f:
Íî
T
ïîóñëîâèþíàèìåíüøèé
ïîëîæèòåëüíûéïåðèîä
f:
Ïîýòîìóèç
0
6


nTT
ñëåäóåò,÷òî

=
nT:
80
ÃËÀÂÀ9.ÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÅÔÓÍÊÖÈÈ
Çíà÷èò,

2
M
è
M
f
mT
:
m
2
Z
g
:
Òàêêàê
f
mT
:
m
2
Z
g
M
è
M
f
mT
:
m
2
Z
g
;
òî
M
=
f
mT
:
m
2
Z
g
:
Òåîðåìàäîêàçàíà.
svo2
Ñâîéñòâî1.
Ñèíóñèêîñèíóñïåðèîäè÷åñêèåôóíêöèèñãëàâíûìïå-
ðèîäîì
T
=2
:
Ìíîæåñòâîâñåõïåðèîäîâýòèõôóíêöèéåñòü
f
2
n
:
n
2
Z
g
.
Äîêàçàòåëüñòâî.
Ïóñòü
x
=

+2
n
ïðåäñòàâëåíèåïðîèçâîëüíîãî
÷èñëà
x
2
R
;
ïîëó÷åííîåïîòåîðåìå
th4
66.Òîãäà
x
+2

=

+2

(
n
+1)
ïðåä-
ñòàâëåíèå
x
+2
:
Ìûâèäèìèçîïðåäåëåíèÿ
opr5
74,÷òî
cos
x
=cos(
x
+2

)=cos
;
sin
x
=sin(
x
+2

)=sin
:
Çíà÷èò,
T
=2

ïåðèîäôóíêöèéñèíóñèêî-
ñèíóñ.Íàïðîìåæóòêå
[0;2

)
òîëüêîäëÿíóëÿ
cos0=1
:
Ïîýòîìó
T
=2

åñòüíàèìåíüøèéïîëîæèòåëüíûéïåðèîäôóíêöèèêîñèíóñ,âåäüåñëè
T
0
;
0
6
T
0
T;
ïåðèîäêîñèíóñà,òîäîëæíîâûïîëíÿòüñÿ
cos0=cos
T
0
=1
;
îòêóäàñëåäóåò,÷òî
T
0
=0
:
Íàóêàçàííîìïðîìåæóòêåòîëüêîäëÿäâóõòî-
÷åê,íóëÿè
;
ñïðàâåäëèâî
sin

=sin0=0
:
Ïîýòîìó,åñëè
T
0
;
0
6
T
0
T;
ïåðèîäñèíóñà,òîäîëæíîâûïîëíÿòüñÿ
sin0=sin
T
0
=0
;
ñëåäîâàòåëüíî,
T
0
=
:
Íîòîãäà
1=sin

2
=sin(

2
+

)=sin
3

2
=

1
:
Ïðîòèâîðå÷èå.
Çíà÷èò,
T
0
6
=
:
Çíà÷èò,
T
0
=0
:
Èòàê,
T
=2

ãëàâíûéïåðèîäôóíê-
öèèñèíóñ.Òàêêàê
2

åñòüãëàâíûéïåðèîä,òîïîòåîðåìå
th5
67ìíîæåñòâî
f
mT
:
m
2
Z
g
åñòüìíîæåñòâîâñåõïåðèîäîâ.
svo1
Ñâîéñòâî2.
Êîñèíóñ÷¼òíàÿôóíêöèÿ,ñèíóñíå÷¼òíàÿôóíêöèÿ.
Äîêàçàòåëüñòâî.
Ïóñòü
x
=

+2
n
ïðåäñòàâëåíèåïðîèçâîëüíîãî
÷èñëà
x
2
R
;
ïîëó÷åííîåïîòåîðåìå
th4
66.Òîãäà
(

x
)=(2



)+2

(

n

1)
:
Ââèäó
2

-ïåðèîäè÷íîñòèñèíóñàèêîñèíóñàäîñòàòî÷íîäîêàçàòü
cos

=cos(2



)
;
sin

=

sin(2



)
:
Äëÿ

2f
0
;;
3

2
;
g
ýòîóòâåðæäåíèå,î÷åâèäíî,
âåðíî.Ïóñòü
=
2f
0
;;
3

2
;
g
:
Óãëû

è
2



äîïîëíèòåëüíûå.Ïîýòî-
ìó,èçðàâåíñòâàïîãèïîòåíóçåèîñòðîìóóãëóñîîòâåòñòâóþùèõïðÿìî-
óãîëüíûõòðåóãîëüíèêîâ,íàåäèíè÷íîé(¾òðèãîíîìåòðè÷åñêîé¿)îêðóæ-
íîñòè÷èñëó

ñîîòâåòñòâóåòíåêîòîðàÿòî÷êà
A
(
x
0
;y
0
)
;
à÷èñëó
(2



)
ñîîòâåòñòâóåòòî÷êà
B
(
x
0
;

y
0
)
;
âçÿòûåèçîïðåäåëåíèÿ
opr5
74.Ýòîîçíà÷àåò
cos

=cos(2



)
;
sin

=

sin(2



)
;
÷òîèòðåáîâàëîñüäîêàçàòü.
Ñâîéñòâî3.
Îáëàñòüîïðåäåëåíèÿôóíêöèéñèíóñèêîñèíóñ
D
(sin)=
D
(cos)=
R
;
îáëàñòüçíà÷åíèéôóíêöèéñèíóñèêîñèíóñ
E
(sin)=
E
(cos)=[

1;1]

R
:
Äîêàçàòåëüñòâî.
Ñîãëàñíîîïðåäåëåíèÿì
opr5
74è
def6
75,
D
(sin)=
D
(cos)=
R
;
E
(sin)

[

1;1]
;E
(cos)

[

1;1]
:
Òàêèìîáðàçîì,äîñòàòî÷íîäîêàçàòü,
÷òî
[

1;1]

E
(sin)
;
[

1;1]

E
(cos)
:
Ïóñòü
d
2
[

1;1]
:
Òîãäàòî÷êè
A
(
d;
p
1

d
2
)
è
B
(
p
1

d
2
;d
)
ïðèíàäëåæàòåäèíè÷íîéîêðóæíîñòè
x
2
+
y
2
=1
;
òàêêàêèõêîîðäèíàòûóäîâëåòâîðÿþòóðàâíåíèþýòîéîêðóæíîñòè.Ïóñòü
]
xOA
=
;
]
xOB
=
:
Òîãäà
cos

=
d;
sin

=
d:
Ñâîéñòâîäîêàçàíî.
9.5.ÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÅÒÎÆÄÅÑÒÂÀ
81
Ñâîéñòâî4.
Òàíãåíñèêîòàíãåíñíå÷¼òíûåïåðèîäè÷åñêèåôóíêöèè
ñãëàâíûìïåðèîäîì
T
=
:
Îáëàñòüîïðåäåëåíèÿôóíêöèèòàíãåíñåñòü
ìíîæåñòâî
R
nf

2
+
n
:
n
2
Z
g
:
Îáëàñòüîïðåäåëåíèÿôóíêöèèêîòàíãåíñ
åñòüìíîæåñòâî
R
nf
n
:
n
2
Z
g
:
Îáëàñòüçíà÷åíèéýòèõôóíêöèéåñòü
âñÿ÷èñëîâàÿïðÿìàÿ.
Äîêàçàòåëüñòâî.
Èçñâîéñòâ
svo2

svo1
2äëÿñèíóñàèêîñèíóñàñëåäóåò,÷òî
òàíãåíñèêîòàíãåíñíå÷¼òíûå
2

-ïåðèîäè÷åñêèåôóíêöèè.Âäàëüíåéøåì
ìûïîêàæåì,÷òîèõãëàâíûéïåðèîäðàâåí
:
Òàêêàêäëÿïðîèçâîëüíî-
ãî
d

0
òî÷êà
(1;
d
)
ïðèíàäëåæèòïðÿìîé
y
=tg
'
äëÿíåêîòîðîãî
';
0
6
'

2
;
àòî÷êà
(1;

d
)
ïðèíàäëåæèòïðÿìîé
y
=

tg
'
=tg(

'
)
;
òî
ââèäóïðîèçâîëüíîñòè
d
èìååì
E
(tg)=
R
:
Àíàëîãè÷íîóñòàíàâëèâàåòñÿ,
÷òî
E
(ctg)=
R
:
Òàêêàêíàïðîìåæóòêå
(


2
;

2
)
òàíãåíñâñþäóîïðåäåë¼í,
àíàêîíöàõýòîãîïðîìåæóòêàíåîïðåäåë¼í,òîââèäó

-ïåðèîäè÷íîñòè
òàíãåíñàçàêëþ÷àåì,÷òî
D
(tg)=
R
nf

2
+
n
:
n
2
Z
g
:
Àíàëîãè÷íîóñòà-
íàâëèâàåòñÿ,÷òî
D
(ctg)=
R
nf
n
:
n
2
Z
g
:
9.5Òðèãîíîìåòðè÷åñêèåòîæäåñòâà
th8
Òåîðåìà68.
8
x;y
2
R
cos(
x

y
)=cos
x
cos
y
+sin
x
sin
y:
Äîêàçàòåëüñòâî.
1)
x
=

+2
n;y
=

+2
k; ;
2
[0
;
2

)
;n;k
2
Z
;

ïðåäñòàâëåíèÿ
x
è
y:
Äàëåå,

7!
(

)
A
(
x
1
;y
1
)
;
7!
(

)
B
(
x
2
;y
2
)
íàòðèãîíî-
ìåòðè÷åñêîéîêðóæíîñòè
x
2
+
y
2
=1
âïðÿìîóãîëüíîéäåêàðòîâîéñèñòåìå
êîîðäèíàòíàïëîñêîñòè
Oxy
,
(

)
O
(0
;
0)
-íà÷àëîêîîðäèíàò,
x
1
=cos
x;
y
1
=sin
x;x
2
=cos
y;y
2
=sin
y:
Ââèäó÷¼òíîñòèèïåðèîäè÷íîñòèêîñèíóñà
èìååì:
cos(
x

y
)=cos(



+2

(
n

k
))=cos
j



j
=cos(2

�j



j
)=cos
'
,
ãäå
'
=min
fj



j
;
2

�j



jg
=
]
AOB
,
0
6
'
6

.Íàäîäîêàçàòü:
x
1
x
2
+
y
1
y
2
=cos
':
2)Ïóñòü
(

)
C
(
x
1
+
x
2
2
;
y
1
+
y
2
2
)
-ñåðåäèíàîòðåçêà
[
AB
]
;
áûòüìîæåò,â
ñëó÷àå
A
=
B;
âûðîæäåííîãîâòî÷êó.Èìååì:
j
OC
j
2
=(
x
1
+
x
2
2
)
2
+(
y
1
+
y
2
2
)
2
=
=
(
x
2
1
+
y
2
1
)+(
x
2
2
+
y
2
2
)+2(
x
1
x
2
+
y
1
y
2
)
4
=
1+1+2(
x
1
x
2
+
y
1
y
2
)
4
:
Ñëåäîâàòåëüíî,
x
1
x
2
+
y
1
y
2
=2
j
OC
j
2

1
.Ìûâèäèì,÷òîçíà÷åíèå
x
1
x
2
+
y
1
y
2
çàâèñèòòîëüêîîòìåðûóãëà
]
AOB
èíåçàâèñèòîòâçàèìíîãîðàñïîëîæå-
íèÿóãëàèñèñòåìûêîîðäèíàò,ëèøüáûâåðøèíàóãëàñîâïàäàëàñíà÷àëîì
êîîðäèíàò.Ðàññìîòðèìóãîë
]
A
1
OB
1
òàêîé,÷òî
]
A
1
OB
1
=
]
AOB;
=
';
82
ÃËÀÂÀ9.ÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÅÔÓÍÊÖÈÈ
A
1
(cos
';
sin
'
)
,
B
1
(1
;
0)
:
Òîãäà
x
1
x
2
+
y
1
y
2
=1

cos
'
+0

sin
'
=cos
'
.
Òåîðåìàäîêàçàíà.
lem8
Ëåììà16.
Ïðèëþáîì
x
2
R
âûïîëíåíûòîæäåñòâà:
sin(

2

x
)=cos
x;
cos(

2

x
)=sin
x:
Äîêàçàòåëüñòâî.
Ïîòåîðåìå
th8
68ïðèâñåõ
x
2
R
cos(

2

x
)=cos
x
cos

2
+sin
x
sin

2
=cos
x

0+sin
x

1=sin
x:
Ñëåäîâàòåëüíî,
sin(

2

x
)=cos(

2

(

2

x
))=cos
x:
th9
Òåîðåìà69.
8
x;y
2
R
ñïðàâåäëèâûñëåäóþùèåôîðìóëû:
Ôîðìóëûñóììû.
cos(
x

y
)=cos
x
cos
y
+sin
x
sin
y;
cos(
x
+
y
)=cos
x
cos
y

sin
x
sin
y;
sin(
x

y
)=sin
x
cos
y

cos
x
sin
y;
sin(
x
+
y
)=sin
x
cos
y
+cos
x
sin
y:
Ôîðìóëûñóììèðîâàíèÿ.
sin
x
+sin
y
=2sin(
x
+
y
2
)cos(
x

y
2
)
;
sin
x

sin
y
=2sin(
x

y
2
)cos(
x
+
y
2
)
;
cos
x
+cos
y
=2cos(
x
+
y
2
)cos(
x

y
2
)
;
cos
x

cos
y
=

2sin(
x

y
2
)sin(
x
+
y
2
)
:
Ôîðìóëûðàçëîæåíèÿ.
sin
x
sin
y
=
1
2
(cos(
x

y
)

cos(
x
+
y
))
;
cos
x
cos
y
=
1
2
(cos(
x

y
)+cos(
x
+
y
))
;
sin
x
cos
y
=
1
2
(sin(
x

y
)+sin(
x
+
y
))
:
9.5.ÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÅÒÎÆÄÅÑÒÂÀ
83
Ôîðìóëûäâîéíîãîèòðîéíîãîàðãóìåíòà.
Îñíîâíîåòðèãîíîìåòðè÷åñêîåòîæäåñòâî.
cos2
x
=2cos
2
x

1=1

2sin
2
x
=cos
2
x

sin
2
x;
sin2
x
=2sin
x
cos
x;
cos
2
x
+sin
2
x
=1
;
sin3
x
=3sin
x

4sin
3
x;
cos3
x
=4cos
3
x

3cos
x:
Ôîðìóëûïðèâåäåíèÿ.
sin(

2

x
)=cos
x;
cos(

2

x
)=sin
x;
sin(

2
+
x
)=cos
x;
cos(

2
+
x
)=

sin
x;
sin(


x
)=sin
x;
cos(


x
)=

cos
x;
sin(

+
x
)=

sin
x;
cos(

+
x
)=

cos
x;
sin(
3

2

x
)=

cos
x;
cos(
3

2

x
)=

sin
x;
sin(
3

2
+
x
)=

cos
x;
cos(
3

2
+
x
)=sin
x;
sin(2


x
)=

sin
x;
cos(2


x
)=cos
x;
sin(2

+
x
)=sin
x;
cos(2

+
x
)=cos
x:
Ñïðàâåäëèâûòàêæåñëåäóþùèåòîæäåñòâà
:
Ôîðìóëûïðèâåäåíèÿ.
tg(

2

x
)=ctg
x;
ctg(

2

x
)=tg
x;
tg(

2
+
x
)=

ctg
x;
ctg(

2
+
x
)=

tg
x;
tg(


x
)=

tg
x;
ctg(


x
)=

ctg
x;
tg(

+
x
)=tg
x;
ctg(

+
x
)=ctg
x;
tg(
3

2

x
)=ctg
x;
ctg(
3

2

x
)=tg
x;
tg(
3

2
+
x
)=

ctg
x;
ctg(
3

2
+
x
)=

tg
x;
tg(2


x
)=

tg
x;
ctg(2


x
)=

ctg
x;
tg(2

+
x
)=tg
x;
ctg(2

+
x
)=ctg
x:
Äîêàçàòåëüñòâî.
Ïîòåîðåìå
th8
68,
8
x;y
2
R
cos(
x
+
y
)=cos(
x

(

y
))=cos
x
cos(

y
)+sin
x
sin(

y
)=cos
x
cos
y

sin
x
sin
y
ââèäó÷¼òíîñòèêîñèíóñàèíå÷¼òíîñòèñèíóñà.Ïîäîêàçàííîìóèïîëåììå
lem8
16èìååì
sin(
x
+
y
)=cos(

2

(
x
+
y
))=cos((

2

x
)

y
)=cos(

2

x
)cos
y
+sin(

2

x
)sin
y
=sin
x
cos
y
+cos
x
sin
y:
Äàëåå,
sin(
x

y
)=sin(
x
+(

y
))=sin
x
cos(

y
)+cos
x
sin(

y
)=sin(
x

y
)=sin
x
cos
y

cos
x
sin
y:
Ôîðìóëûñóììûäîêàçàíû.Ïîäîêàçàííîìó
cos(
x

y
)+cos(
x
+
y
)=2cos
x
cos
y;
sin(
x

y
)+sin(
x
+
y
)=2sin
x
cos
y;
cos(
x

y
)

cos(
x
+
y
)=2sin
x
sin
y;
÷òîäî-
êàçûâàåòôîðìóëûðàçëîæåíèÿ.Åñëèâôîðìóëàõðàçëîæåíèÿïðîèçâåñòè
çàìåíó

=
x

y;
=
x
+
y;
òîïîëó÷àòñÿôîðìóëûñóììèðîâàíèÿ.
Ñîãëàñíîôîðìóëåêîñèíóñàðàçíîñòè
1=cos0=cos(
x

x
)=cos
2
x
+sin
2
x;
cos2
x
=cos(
x
+
x
)=cos
2
x

sin
2
x
=2cos
2
x

1=1

2sin
2
x:
Ïîôîðìóëåñè-
íóñàñóììû
sin2
x
=sin(
x
+
x
)=2sin
x
cos
x;
sin3
x
=sin(2
x
+
x
)=sin2
x
cos
x
+cos2
x
sin
x
=2sin
x
cos
2
x
+sin
x

2sin
3
x
=3sin
x

4sin
3
x:
Àíàëîãè÷íîïðèïîìîùèôîðìóëûêîñèíóñàñóììûäîêàçûâàåòñÿôîðìóëà
êîñèíóñàòðîéíîãîóãëà.
Ôîðìóëûïðèâåäåíèÿäëÿñèíóñàèêîñèíóñàäîêàçûâàþòñÿåäèíîîáðàç-
íîïðèïîìîùèôîðìóëñóììû,ïîäîáíîòîìóêàêýòîáûëîïðîäåëàíîäëÿ
84
ÃËÀÂÀ9.ÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÅÔÓÍÊÖÈÈ
ôîðìóëû
cos(

2

x
)=sin
x
âëåììå
lem8
16.
Ôîðìóëûïðèâåäåíèÿäëÿòàíãåíñàèêîòàíãåíñàäîêàçûâàþòñÿåäèíî-
îáðàçíîïðèïîìîùèôîðìóëïðèâåäåíèÿäëÿñèíóñàèêîñèíóñà.Íàïðèìåð,
tg(

+
x
)=
sin(

+
x
)
cos(

+
x
)
=

sin
x

cos
x
=tg
x:
Òåîðåìàäîêàçàíà.
Çàìå÷àíèå.
Ôîðìóëûïðèâåäåíèÿóñòàíàâëèâàþò

-ïåðèîäè÷íîñòüòàí-
ãåíñàèêîòàíãåíñà.Íåòðóäíîâèäåòü,÷òî

åñòüãëàâíûéïåðèîäýòèõôóíê-
öèé.Íàïðèìåð,åñëè
T
0
-ïåðèîäòàíãåíñàè
0
6
T
0
;
òî
tg0=0=tg
T
0
:
Ñëåäîâàòåëüíî
T
0
=0
:
Óïðàæíåíèå44.
Äîêàæèòåñëåäóþùèåòîæäåñòâà.

sin
x
=
2tg
x
2
1+tg
2
x
2
;

cos
x
=
1

tg
2
x
2
1+tg
2
x
2
;

tg
x
=
2tg
x
2
1

tg
2
x
2
;

tg2
x
=
2tg
x
1

tg
2
x

tg(
x
+
y
)=
tg
x
+tg
y
1

tg
x
tg
y
;

tg(
x

y
)=
tg
x

tg
y
1+tg
x
tg
y
;

ctg(
x
+
y
)=
1

tg
x
tg
y
tg
x
+tg
y
;

ctg(
x

y
)=
1+tg
x
tg
y
tg
x

tg
y
;

tg
x
+tg
y
=
sin(
x
+
y
)
cos
x
cos
y
;

tg
x

tg
y
=
sin(
x

y
)
cos
x
cos
y
;

a
sin
x
+
b
cos
x
=
p
a
2
+
b
2
sin(
x
+
'
)
;
ãäå
cos
'
=
a
p
a
2
+
b
2
;
sin
'
=
b
p
a
2
+
b
2
;

1+tg
2
x
=
1
cos
2
x
;

1+ctg
2
x
=
1
sin
2
x
;
tg
x

ctg
x
=1;

tg
x
2
=
sin
x
1+cos
x
=
1

cos
x
sin
x
;

j
sin
x
2
j
=
q
1

cos
x
2
;

j
cos
x
2
j
=
q
1+cos
x
2
:
Çàìå÷àíèå.
Òîæäåñòâîåñòüðàâåíñòâî,âåðíîåïðèâñåõäîïóñòèìûõçíà-
÷åíèÿõâõîäÿùèõâíåãîïåðåìåííûõ.
9.6.ÎÁÐÀÒÍÛÅÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÅÔÓÍÊÖÈÈ
85
9.6Îáðàòíûåòðèãîíîìåòðè÷åñêèåôóíêöèè
Îïðåäåëåíèå77.
Ïóñòüîòîáðàæåíèÿ
f
:
X
�!
Y
è
g
:
Y
�!
X
òàêîâû,÷òî
8
x
2
X
9
g
(
f
(
x
))
è
8
y
2
Y
9
f
(
g
(
y
))
(òîåñòüñóùåñòâóþò
êîìïîçèöèè
f

g
è
g

f
),ïðè÷¼ì
8
x
2
X
8
y
2
Yx
=
g
(
f
(
x
))
è
y
=
f
(
g
(
y
))
:
Òîãäà
g
íàçûâàåòñÿîáðàòíûìäëÿ
f
èîáîçíà÷àåòñÿ
f

1
:
Çàìåòèì,÷òîåñëè
g
îáðàòíîåäëÿ
f;
òî
f
îáðàòíîåäëÿ
g:
Òåîðåìà70.
Åñëèîáðàòíîåîòîáðàæåíèåñóùåñòâóåò,òîîíîåäèí-
ñòâåííî.Îáðàòíîåäëÿ
f
îòîáðàæåíèåñóùåñòâóåòòîãäàèòîëüêîòî-
ãäà,êîãäà
f
åñòüáèåêöèÿ,ïðè÷¼ìòîãäà
f

1
òîæåáèåêöèÿè
(
f

1
)

1
=
f:
Äîêàçàòåëüñòâî.
Î÷åâèäíîáèåêòèâíîåîòîáðàæåíèåèìååòîáðàòíîå.
Äåéñòâèòåëüíî,åñëèïðèáèåêòèâíîìîòîáðàæåíèè
fx
1
7!
y
1
;x
2
7!
y
2
;:::;
òî
f

1
ìîæíîîïðåäåëèòüêàêîòîáðàæåíèå,ïðèêîòîðîì
y
1
7!
x
1
;y
2
7!
x
2
èòàêäàëåå.Ïóñòü
g
:
Y
�!
X
îáðàòíîåäëÿ
f
:
X
�!
Y:
Òîãäàäëÿ
âñåõ
y
2
Y
ñïðàâåäëèâî
y
=
f
(
g
(
y
))
:
Çíà÷èò,
f
ñþðúåêòèâíî(êàæäûéýëå-
ìåíò
y
2
Y
èìååòïðîîáðàçòàêîå
x
2
X;
÷òî
y
=
f
(
x
)
:
)Äàëåå,åñëè
f
(
x
)=
f
(
x
0
)
;
òî
x
=
g
(
f
(
x
))=
g
(
f
(
x
0
))=
x
0
:
Çíà÷èò,
f
èíúåêòèâíî(åñëè
x
6
=
x
0
;
òî
f
(
x
)
6
=
f
(
x
0
)
¾ðàçíûåòî÷êèïåðåõîäÿòâðàçíûå¿).Òàêêàê
f
èíúåêòèâíîèñþðúåêòèâíî,òî
f
áèåêòèâíî.Èòàê,åñëè
f
èìååòîáðàò-
íîåäëÿ
g;
òî
f
áèåêöèÿ.Íîòîãäà
g
îáðàòíîåäëÿ
f
òîæåáèåêöèÿ.Åñëè,
êðîìå
g
,îòîáðàæåíèå
g
0
îáðàòíîåäëÿ
f;
òîäëÿâñåõ
y
2
Y
ñïðàâåäëèâî
x
=
g
0
(
y
)=
g
0
(
f
(
x
))=
g
(
f
(
x
))=
g
(
y
)
;
òîåñòü
g
0
=
g:
òàêèìîáðàçîì,
g
åäèíñòâåííîåîáðàòíîåäëÿ
f:
Íîòîãäàè
f
åäèíñòâåííîåîáðàòíîåäëÿ
g
è,
òàêèìîáðàçîì,
(
f

1
)

1
=
f:
Òåîðåìàäîêàçàíà.
Òåîðåìà71.
Ïóñòü
X
è
Y
÷èñëîâûåìíîæåñòâàèîòîáðàæåíèå
f
:
X
�!
Y
èìååòîáðàòíîå.Òîãäàãðàôèêèîáðàòíûõîòîáðàæåíèé
f
è
f

1
ñèììåò-
ðè÷íûîòíîñèòåëüíîïðÿìîé
y
=
x:
Ôóíêöèÿ
f
íå÷¼òíàÿòîãäàèòîëüêî
òîãäà,êîãäà
f

1
íå÷¼òíàÿ,
f
ñòðîãîâîçðàñòàåòòîãäàèòîëüêîòîãäà,
êîãäà
f

1
ñòðîãîâîçðàñòàåò,
f
ñòðîãîóáûâàåòòîãäàèòîëüêîòîãäà,
êîãäà
f

1
ñòðîãîóáûâàåò.
Äîêàçàòåëüñòâî.
Òî÷êà
A
(
x
0
;y
0
)
2

f
òîãäàèòîëüêîòîãäà,êîãäà
B
(
y
0
;x
0
)
2

f

1
ñîãëàñíîîïðåäåëåíèþîáðàòíîãîîòîáðàæåíèÿ.Òî÷êà
C
(
x
0
+
y
0
2
;
y
0
+
c
0
2
)
ñåðåäèíàîòðåçêà
[
AB
]
:
Ïóñòü
(

)
O
(0;0)
íà÷àëîêîîðäèíàò.Î÷åâèäíî
C
2

y
=
x
,
j
OA
j
=
j
OB
j
=
p
x
2
0
+
y
2
0
:
Òîãäàåñëèòî÷êè
A;B;C
íåëåæàòíà
îäíîéïðÿìîé,òî
[
OC
]
åñòüìåäèàíàèâûñîòàâðàâíîáåäðåííîìòðåóãîëü-
íèêå
AOB:
Çíà÷èò,òî÷êè
A
è
B
ñèììåòðè÷íûîòíîñèòåëüíîïðÿìîé
y
=
x:
Åñëèæåòî÷êè
A
,
B
,
C
ëåæàòíàîäíîéïðÿìîé,òîîíèëèáîñîâïàäàþò,
86
ÃËÀÂÀ9.ÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÅÔÓÍÊÖÈÈ
ëèáîâñåðàçëè÷íû,èòîãäàèç
y
0
=
kx
0
,
x
0
=
ky
0
,
x
0
6
=
y
0
ñëåäóåò,÷òî
k
=

1
:
Òàêêàêïðÿìûå
y
=
x
è
y
=

x
ïåðïåíäèêóëÿðíû,òîèâýòîì
ñëó÷àåòî÷êè
A
è
B
ñèììåòðè÷íûîòíîñèòåëüíîïðÿìîé
y
=
x:
Äàëåå,åñëè
f
íå÷¼òíàÿôóíêöèÿ,òî
(
x
0
;y
0
)
2

f
()
(

x
0
;

y
0
)
2

f
:
Òîãäàèìååì:
(
y
0
;x
0
)
2

f

1
()
(
x
0
;y
0
)
2

f
()
(

x
0
;

y
0
)
2

f
()
(

y
0
;

x
0
)
2

f

1
;
òîåñòü,
(
y
0
;x
0
)
2

f

1
()
(

y
0
;

x
0
)
2

f

1
:
Òàêèìîáðàçîì,
f

1
íå÷¼ò-
íàÿôóíêöèÿ.Çíà÷èò,
f
íå÷¼òíàÿôóíêöèÿ
()
f

1
íå÷¼òíàÿôóíêöèÿ.
Åñëè
f
ñòðîãîâîçðàñòàåò,òî
x
1
x
2
()
y
1
y
2
:
Òàêêàê
(
y
0
;x
0
)
2

f

1
()
(
x
0
;y
0
)
2

f
;
òîè
f

1
ñòðîãîâîçðàñòàåò.Çíà÷èò,
f
ñòðîãîâîçðàñòàåò
()
f

1
ñòðîãî
âîçðàñòàåò.Àíàëîãè÷íîèññëåäóåòñÿñëó÷àéñòðîãîãîóáûâàíèÿ.Òåîðåìà
äîêàçàíà.
9.7Ïåðâûéçàìå÷àòåëüíûéïðåäåë
sintg
Ëåììà17.
Äëÿâñåõ

2
(0
;

2
)
âûïîëíÿþòñÿíåðàâåíñòâà
sin

tg

.
Äîêàçàòåëüñòâî.
Ðàññìîòðèìòðåóãîëüíèêè
4
OAB
,
4
OAC
èñåêòîð
OAB
(ñì.ðèñ.
trigas
9.4).
Î÷åâèäíî,÷òî
S
(
4
OAB
)
S
(
OAB
)
S
(
4
OAC
)
.Íî
S
(
4
OAB
)=
1
2
sin

,
S
(
OAB
)=
x
2
è
S
(
4
OAC
)=
1
2
tg
:
Óìíîæàÿíà2,ïîëó÷èìòðåáóåìîå
íåðàâåíñòâî.
-
6
A
B

C
x
y
Ðèñ.9.4:
trigas
Òåîðåìà72
(Ïåðâûéçàìå÷àòåëüíûéïðåäåë)
.
lim
x
!
0
sin
x
x
=1
.
Äîêàçàòåëüñòâî.
Èçëåììû
sintg
17âûòåêàåò,÷òî
x

cos
x
sin
xx
.Ðàçäåëèââñåíà
x
6
=0
,
ïîëó÷èì
cos
x
sin
x
x

1
.Ýòèíåðàâåíñòâàâûïîëíåíûèïðè
x
2
(


2
;
0)
ñëåäóåòèç÷åòíîñòèôóíêöèé
cos
x
è
sin
x
x
.Ïîñêîëüêó,
lim
x
!
0
cos
x
=1
,ïî
òåîðåìå
dvamenta
55(îäâóõìèëèöèîíåðàõäëÿôóíêöèé)ïîëó÷àåì,÷òî
lim
x
!
0
sin
x
x
=1
,
9.7.ÏÅÐÂÛÉÇÀÌÅ×ÀÒÅËÜÍÛÉÏÐÅÄÅË
87
Óïðàæíåíèå45.
Äîêàçàòü,÷òî
lim
x
!
0
1

cos
x
x
2
=
1
2
.
88
ÃËÀÂÀ9.ÒÐÈÃÎÍÎÌÅÒÐÈ×ÅÑÊÈÅÔÓÍÊÖÈÈ
Ãëàâà10
Ïîêàçàòåëüíàÿè
ëîãàðèôìè÷åñêàÿôóíêöèè.
10.1Ëîãàðèôì.
logopr
Òåîðåìà73.
Ïóñòü
a�
1
.Òîãäàñóùåñòâóåòåäèíñòâåííàÿôóíêöèÿ
f
:(0
;
+
1
)
7!
R
,óäîâëåòâîðÿþùàÿóñëîâèÿì:1)
8
x;x
0

0(
f
(
x

x
0
)=
f
(
x
)+
f
(
x
0
))
;
2)
8
x�x
0

0(
f
(
x
)
�f
(
x
0
))
;
3)
f
(
a
)=1
.
Äîêàçàòåëüñòâî.
Âäàííîìäîêàçàòåëüñòâåèñïîëüçóåòñÿîäèíâàæíûéïðèåì:
Ìûïðåä-
ïîëàãàåì,÷òîíåêîòîðûéîáúåêòñóùåñòâóåò,íà÷èíàåìèçó÷àòü
åãî(ïîòåíöèàëüíûå!)ñâîéñòâà.Íàîñíîâåýòèõñâîéñòâñòðîèòñÿ
óêàçàííûéîáúåêò,îòêóäàèâûòåêàåòåãîñóùåñòâîâàíèå.
Ïðåäïîëîæèì,÷òîòàêàÿôóíêöèÿñóùåñòâóåòèïîäóìàåì,êàêèìèñâîé-
ñòâàìèîíàäîëæíàîáëàäàòü.
Ñâîéñòâî1
:Ïðèìåíèâ
n

1
ðàçóñëîâèå1),ïîëó÷èì
f
(
x
n
)=
nf
(
x
)
.
Ñâîéñòâî2
:Ïîäñòàâèââïðåäûäóùååñâîéñòâî
n
p
x
,ïîëó÷èì
f
(
n
p
x
)=
1
n
f
(
x
)
.
Ñâîéñòâî3
:
f
(1)=0
,ïîñêîëüêó
f
(1

1)=
f
(1)+
f
(1)
.
Ñâîéñòâî4
:
f
(
1
x
)=

f
(
x
)
,ïîñêîëüêó
f
(
x

1
x
)=
f
(1)=0
.
Ñâîéñòâî5
:Èçïðåäûäóùèõñâîéñòâèóñëîâèÿ3)âûòåêàåò
8
q
2
Q
(
f
(
a
q
)=
q
)
.
Ñâîéñòâî6
:Èçñâîéñòâà5èóñëîâèÿ2)âûòåêàåò,÷òîåñëè
a
q
0
xa
q
00
,
òî
q
0
f
(
x
)
q
00
.
89
90
ÃËÀÂÀ10.ÏÎÊÀÇÀÒÅËÜÍÀßÈËÎÃÀÐÈÔÌÈ×ÅÑÊÀßÔÓÍÊÖÈÈ.
Ïîñòðîèìòåïåðüôóíêöèþ
f
.Âûáåðåì
n
2
Z
òàêîå,÷òî
a
n
6
xa
n
+1
.
Ïîñòðîèìïîñëåäîâàòåëüíîñòüñòÿãèâàþùèõñÿîòðåçêîâ
f
[
a
k
;b
k
]
g
1
k
=1
ñëåäó-
þùèìîáðàçîì:
a
0
=
n
,
b
0
=
n
+1
.Íà
k

ìøàãåðàññìàòðèâàåì
c
=
a
k

1
+
b
k

1
2
;
åñëè
a
c
6
x
òîâûáèðàåì
a
k
=
cb
k
=
b
k

1
,èíà÷å
a
k
=
a
k

1
,
b
k
=
c
.Òàêèì
îáðàçîì,íàêàæäîìøàãå
a
a
k
6
xa
b
k
,ñëåäîâàòåëüíî
f
(
x
)
2
[
a
k
;b
k
)

[
a
k
;b
k
]
ïðèâñåõ
k
2
N
.Ñëåäîâàòåëüíî,åñëè
f
ñóùåñòâóåò,òîååçíà÷åíèå
f
(
x
)
â
òî÷êåäîëæíîáûòüðàâíî
y
,ãäå
f
y
g
=
1
T
k
=1
[
a
k
;b
k
]
.
Äîêàæåìòåïåðüêîððåêòíîñòüçàäàíèÿôóíêöèè
f
òàêèìîáðàçîì,ò.å.
òî,÷òîïîëó÷åííàÿòàêèìîáðàçîìôóíêöèÿóäîâëåòâîðÿåòóñëîâèÿì1)3).
Äîêàæåìóñëîâèå3).Î÷åâèäíî,íà
k
øàãåïîëó÷èì:
a
1
6
aa
2
k
+1
2
k

\
k
[1
;
2
k
+1
2
k
]=
f
1
g
.Ñëåäîâàòåëüíî
f
(
a
)=1
.
Äîêàæåìóñëîâèå1).Íà
k
øàãåäëÿ
x
,
x
0
a
n
k
2
k
6
xa
n
k
+1
2
k
è
a
n
0
k
2
k
6
xa
n
0
k
+1
2
k
:
Ïåðåìíîæèâýòèíåðàâåíñòâà(âñå÷èñëàâíèõïîëîæèòåëüíû!),ïîëó÷èì
a
n
k
+
n
0
k
2
k
6
x

x
0
a
n
k
+
n
0
k
+2
2
k
;
ñëåäîâàòåëüíî
n
k
+
n
0
k
2
k
6
f
(
x

x
0
)

n
k
+
n
0
k
+2
2
k
:
Ñäðóãîéñòîðîíû,ñëîæèâ
n
k
2
k
6
f
(
x
)

n
k
+1
2
k
è
n
0
k
2
k
6
f
(
x
0
)

n
0
k
+1
2
k
,ïîëó÷èì
n
k
+
n
0
k
2
k
6
f
(
x
)+
f
(
x
0
)

n
k
+
n
0
k
+2
2
k
:
Òàêèìîáðàçîì,÷èñëà
f
(
x

x
0
)
è
f
(
x
)+
f
(
x
0
)
ïðèíàäëåæàòîäíîìóèíòåðâàëó
äëèíû
2
2
k
=2
1

k
,àçíà÷èò
j
f
(
x

x
0
)

(
f
(
x
)+
f
(
x
0
))
j

2
1

k
,èçïðîèçâîëü-
íîñòèâûòåêàåò
f
(
x

x
0
)

(
f
(
x
)+
f
(
x
0
))=0
,
Äîêàæåìòåïåðüñâîéñòâî2).Ðàññìîòðèìïðîèçâîëüíîå
u�
1
èâûáåðåì
n
2
N
òàêîå,÷òî
u
n
�a
.Î÷åâèäíî
f
(
u
)

1
n

0
.Ïóñòüòåïåðü
x�x
0

0
.
Ðàññìîòðèì
f
(
x
)=
f
(
x
0

x
x
0
)=
f
(
x
0
)+
f
(
x
x
0
)
�f
(
x
0
)
,
Èòàê,ïîñòðîåííàÿôóíêöèÿóäîâëåòâîðÿåòóñëîâèÿì1)3)èíèêàêîé
äðóãîéòàêîéôóíêöèèíåñóùåñòâóåò.Ïîñòðîåííàÿôóíêöèÿíàçûâàåòñÿ
ëîãàðèôìïîîñíîâàíèþ
a
èîáîçíà÷àåòñÿ
log
a
x
.Äëÿëîãàðèôìîâïî
îñíîâàíèÿìåè10ïðèíÿòûîáîçíà÷åíèÿ:
log
e
x
=ln
x
log
10
x
=lg
x:
10.2Ñâîéñòâàëîãàðèôìîâ
Óòâåðæäåíèå74.
log
a
x
2C
(0
;
+
1
)
.
10.2.ÑÂÎÉÑÒÂÀËÎÃÀÐÈÔÌÎÂ
91
Äîêàçàòåëüñòâî.
Äîêàæåìñíà÷àëà,÷òî
log
a
2C
(1)
.Âûáåðåìïðîèçâîëüíîå
"�
0
èðàññìîò-
ðèì
1
n
"
.Ïóñòü

=min(
a
1
=n

1
;
1

a

1
=n
)
,òîãäà
log
a
(1+

)
6
log
a
(
a
1
=n
)=
1
n
"
è
log
a
(1


)

log
a
(
a

1
=n
)=

1
n


"
.Ñëåäîâàòåëüíî
lim
x
!
1
log
a
x
=0=log
a
1
,
÷òîèîçíà÷àåòíåïðåðûâíîñòüôóíêöèè
log
a
x
âòî÷êå
x
=1
.
Äîêàæåìòåïåðüíåïðåðûâíîñòü
log
a
x
âòî÷êå
x
=
x
0

0
.Ïðåäñòàâèì
log
a
x
=log
a
x
0
+log
a
x
x
0
.Ôóíêöèÿ
log
a
x
x
0
íåïðåðûâíàâ
x
=
x
0
,ïîñêîëüêó
log
a
x
íåïðåðûâíàâ
x
=1
à
log
a
x
0
ïîñòîÿííàÿ,ñëåäîâàòåëüíî
log
a
x
íåïðåðûâíàâ
x
=
x
0
,
Óòâåðæäåíèå75.
log
b
x
=
log
a
x
log
a
b
.
Äîêàçàòåëüñòâî.
Ðàññìîòðèìôóíêöèþ
f
(
x
)=
log
a
x
log
a
b
.Îíàóäîâëåòâîðÿåòóñëîâèÿì1)è2)èç
òåîðåìû
logopr
73è
f
(
b
)=1
,ñëåäîâàòåëüíî
f
(
x
)=log
b
x
Ïîíÿòèåëîãàðèôìàìîæíîââåñòèèäëÿîñíîâàíèé
0
b
1
.
log
b
x
=

log
1
=b
x
Âýòîìñëó÷àåóñëîâèÿ1)è3)òåîðåìû
logopr
73âûïîëíåíûäëÿ
log
b
,àóñëîâèå
2)ïðåâðàùàåòñÿâ
8
x�x
0

0(log
b
x
log
b
x
0
)
.
10.2.1Ïîêàçàòåëüíàÿôóíêöèÿ.Âòîðîéçàìå÷àòåëüíûé
ïðåäåë.
Ïîñêîëüêóëîãàðèôìåñòüíåïðåðûâíàÿèìîíîòîííàÿôóíêöèÿ,òîñóùå-
ñòâóåòîáðàòíàÿêíåìóïîêàçàòåëüíàÿ,îáîçíà÷àåìàÿ
a
x
.Òàêîåîáîçíà÷å-
íèåèìååòñìûñë,ïîñêîëüêóïðè
x
2
Q
çíà÷åíèÿýòîéôóíêöèèñîâïàäàþò
ññîîòâåòñòâóþùèìèñòåïåíÿìè
a
.Íàèáîëüøååçíà÷åíèåâìàòåìàòè÷åñêîì
àíàëèçåèìååòïîêàçàòåëüíàÿôóíêöèÿ,íàçûâàåìàÿýêñïîíåíòà,ò.å.
e
x
,ãäå
e
÷èñëîÝéëåðà.
Ïóñòü
a�
1
;b
2
(0
;
1)
.Òîãäàâåðíûñîîòíîøåíèÿ:
1.
lim
x
!
+
1
a
x
=+
1
;
2.
lim
x
!
+
1
b
x
=0
;
3.
lim
x
!�1
a
x
=0
;
4.
lim
x
!�1
b
x
=+
1
;
5.
lim
x
!
+
1
log
a
x
=+
1
;
dokaz
6.
lim
x
!
+
1
log
b
x
=
�1
;
92
ÃËÀÂÀ10.ÏÎÊÀÇÀÒÅËÜÍÀßÈËÎÃÀÐÈÔÌÈ×ÅÑÊÀßÔÓÍÊÖÈÈ.
7.
lim
x
!
+0
log
a
x
=
�1
;
8.
lim
x
!
+0
log
b
x
=+
1
;
Äîêàæåì,íàïðèìåðñâîéñòâî
dokaz
6.Âûáåðåìïðîèçâîëüíóþïîñëåäîâàòåëü-
íîñòü
lim
n
!1
x
n
=+
1
:
Ïóñòü
b
1
âûáåðåìïðîèçâîëüíîå
c�
0
,òîãäà
âîçüìåì
x
n

(
1
b
)
[
c
]+1
.Î÷åâèäíî
log
b
x
n


[
c
]

1


c
,ñëåäîâàòåëüíî,
(
�1
;

c
)
ëîâóøêàïîñëåäîâàòåëüíîñòè
log
b
x
n
,
Äîêàçàòåëüñòâî
îñòàëüíûõñâîéñòâïðåäîñòàâëÿåòñÿâêà÷åñòâåóïðàæíåíèÿ.
Òåîðåìà76.
lim
x
!
+
1
ln
x
x
=0
Äîêàçàòåëüñòâî.
Â1-ìñåìåñòðå(óòâåðæäåíèå
sqrtnn
??
)áûëîäîêàçàíî,÷òî
lim
n
!1
n
p
n
=1
.Èç
íåïðåðûâíîñòèëîãàðèôìàâûòåêàåò,÷òî
lim
n
!1
ln
n
p
n
=lim
n
!1
ln
n
n
=0
,ò.å.
ïîñëåäîâàòåëüíîñòü

ln
n
n

1
n
=1
á.ì.ï.Ðàññìîòðèìïðîèçâîëüíóþïîñëåäî-
âàòåëüíîñòü
lim
n
!1
x
n
=+
1
,íåîãðàíè÷èâàÿîáùíîñòèðàññóæäåíèåìîæíî
ñ÷èòàòü,÷òî
x
n

1
.Òîãäà
ln[
x
n
]
[
x
n
]+1

ln
x
n
x
n

ln[
x
n
]+1
[
x
n
]
:
Íîïîñëåäîâàòåëüíîñòü
n
ln[
x
n
]
[
x
n
]
o
1
n
=1
ÿâëÿåòñÿïîäïîñëåäîâàòåëüíîñòüþïî-
ñëåäîâàòåëüíîñòè

ln
n
n

1
n
=1
,àçíà÷èòá.ì.ï.òîæå.Òàêèìîáðàçîì,
lim
n
!1
ln[
x
n
]
[
x
n
]+1
=lim
n
!1
ln[
x
n
]
[
x
n
]

lim
n
!1
[
x
n
]
[
x
n
]+1
=0

1=0
è
lim
n
!1
ln[
x
n
]+1
[
x
n
]
=lim
n
!1
ln[
x
n
]
[
x
n
]
+lim
n
!1
1
[
x
n
]
=0+0=0
.Ñëåäîâàòåëüíî,ïîòåîðåìå
îäâóõìèëèöèîíåðàõ
lim
n
!1
ln
x
n
x
n
=0
,
Ñëåäñòâèå12.
Ïóñòü
; �
0
,òîãäà
lim
x
!
+
1
ln

x
x

=0
:
Äîêàçàòåëüñòâî.
Îáîçíà÷èì
y
=
x
=
,òîãäà
lim
x
!
+
1
ln
x
x
=
=lim
y
!
+
1
ln
y
(
=
y
=lim
y
!
+
1




ln
y
y

=0
.
Ñëåäîâàòåëüíî,
lim
x
!
+
1
ln

x
x

=0
10.2.ÑÂÎÉÑÒÂÀËÎÃÀÐÈÔÌÎÂ
93
Ñëåäñòâèå13.
Ïóñòü

0
,
a�
1
,òîãäà
lim
x
!
+
1
x

a
x
=0
:
Äîêàçàòåëüñòâî.
Ñäåëàåìçàìåíó
a
x
=
y
.Òîãäà
lim
x
!
+
1
x

a
x
=lim
y
!
+
1
ln

y
ln

a

y
=0
:
Ñëåäñòâèå14.
Ïóñòü
k
2
N
,

0
,òîãäà
lim
x
!
+0
ln
k
x

x

=0
:
Äîêàçàòåëüñòâî.
Ñäåëàåìçàìåíó
y
=
1
x
.Òîãäà
lim
x
!
+0
ln
k
x

x

=lim
y
!
+
1
ln
k
1
x
x

=lim
y
!
+
1
(

1)
k
ln
k
x
x

=0
:
vtorpred
Òåîðåìà77
(âòîðîéçàìå÷àòåëüíûéïðåäåë)
.
lim
x
!
0
e
x

1
x
=1
.
Äîêàçàòåëüñòâî.
Äëÿäîêàçàòåëüñòâàíàìïîòðåáóåòñÿíåñêîëüêîâñïîìîãàòåëüíûõëåìì:
Ëåììà18.
lim
x
!
+0
(1+
x
)
1
x
=
e
.
Äîêàçàòåëüñòâî.
Ðàññìîòðèìïðîèçâîëüíóþïîñëåäîâàòåëüíîñòü
lim
n
!1
x
n
=0
,ãäå
x
n

0
.Âû-
áåðåìïîñëåäîâàòåëüíîñòüöåëûõ÷èñåë
y
n
=
h
1
x
n
i
.Î÷åâèäíî,
1
y
n
+1
6
x
n
6
1
y
n
.
Ñëåäîâàòåëüíî

1+
1
y
n
+1

y
n
6
(1+
x
n
)
1
=x
n
6

1+
1
y
n

y
n
+1
.Ïîñêîëüêó
lim
n
!1

1+
1
y
n
+1

y
n
=lim
n
!1

1+
1
y
n

y
n
+1
=
e
,òîïîòåîðåìåîäâóõìèëèöè-
îíåðàõ
lim
n
!1
(1+
x
n
)
1
=x
n
=
e
,
Ëåììà19.
lim
x
!�
0
(1+
x
)
1
x
=
e
.
Äîêàçàòåëüñòâî.
Ðàññìîòðèìïðîèçâîëüíóþïîñëåäîâàòåëüíîñòü
lim
n
!1
x
n
=0
,ãäå
x
n

0
.Âû-
áåðåìïîñëåäîâàòåëüíîñòüöåëûõ÷èñåë
y
n
=
h

1
x
n
i
.Î÷åâèäíî,

1
y
n
6
x
n
6

1
y
n
+1
.
Ñëåäîâàòåëüíî,

1

1
y
n


y
n
6
(1+
x
n
)
1
=x
n
6

1

1
y
n
+1


y
n

1
.Çàìåòèì,
÷òî
lim
n
!1

1

1
n


n
=lim
n
!1

n
n

1

n
=lim
n
!1

1+
1
n

1

n
=
e:
94
ÃËÀÂÀ10.ÏÎÊÀÇÀÒÅËÜÍÀßÈËÎÃÀÐÈÔÌÈ×ÅÑÊÀßÔÓÍÊÖÈÈ.
Ñëåäîâàòåëüíî,
lim
n
!1

1

1
y
n


y
n
=lim
n
!1

1

1
y
n
+1


y
n

1
=
e
,îòêóäàïî
òåîðåìåîäâóõìèëèöèîíåðàõèïîëó÷àåì
lim
n
!1
(1+
x
n
)
1
=x
n
=
e
,
Ëåììà20.
lim
x
!
0
(1+
x
)
1
x
=
e
.
Äîêàçàòåëüñòâî.
Çàôèêñèðóåìïðîèçâîëüíîå
"�
0
.Òîãäàèçïðåäûäóùèõëåììâûòåêàåò,
÷òîñóùåñòâóåò

1

0
òàêîå,÷òîäëÿâñåõ
x
2
(0
;
1
)
âûïîëíåíîíåðàâåíñòâî
j
(1+
x
)
1
x

e
j
"
èñóùåñòâóåò

2

0
,òàêîå,÷òîäëÿâñåõ
x
2
(


2
;
0)
âûïîëíåíîíåðàâåíñòâî
j
(1+
x
)
1
x

e
j
"
.Âçÿâ

=min(

1
;
2
)
,ïîëó÷èì
îïðåäåëåíèåïðåäåëàâòî÷êå.
Ëåììà21.
lim
x
!
0
ln(1+
x
)
x
=1
.
Äîêàçàòåëüñòâî.
1=ln
e
=lnlim
x
!
0
(1+
x
)
1
=x
=lim
x
!
0
ln(1+
x
)
1
=x
=lim
x
!
0
ln(1+
x
)
x
,
Äîêàçàòåëüñòâî(Òåîðåìû
vtorpred
77).
Ñäåëàåìçàìåíó
y
=
e
x

1
èðàññìîòðèì
lim
x
!
0
x
e
x

1
=lim
y
!
0
ln(1+
y
)
y
=1
.Ñëå-
äîâàòåëüíî,
lim
x
!
0
e
x

1
x
=1
,
Ñëåäñòâèå15.
Ïóñòü
a�
0
,
a
6
=1
.Òîãäà
lim
x
!
0
log
a
(1+
x
)
x
=
1
ln
a
.
Ñëåäñòâèå16.
Ïóñòü
a�
0
,
a
6
=1
.Òîãäà
lim
x
!
0
a
x

1
x
=ln
a
.
xprod
Ñëåäñòâèå17.
Ïóñòü
p
2
R
.Òîãäà
lim
x
!
0
(1+
x
)
p

1
x
=
p
.
Âêà÷åñòâåçàêëþ÷åíèÿçàïèøåìâàñèìïòîòè÷åñêîéôîðìåóòâåðæäå-
íèÿ,äîêàçàííûåâýòîìðàçäåëå:
10.2.ÑÂÎÉÑÒÂÀËÎÃÀÐÈÔÌÎÂ
95

log
a
x
=
o
(
x

)
,ïðè
x
!
+
1
,
a�
1
,

0
;

x

=
o
(
a
x
)
,ïðè
x
!
+
1
,
a�
1
,

0
;

log
a
x
=
o
(
1
x

)
,ïðè
x
!
+0
,
a�
1
,

0
;

e
x
=1+
x
+
o
(
x
)
ïðè
x
!
0
;

a
x
=1+ln
a

x
+
o
(
x
)
ïðè
x
!
0
,
a�
0
;

ln(1+
x
)=
x
+
o
(
x
)
ïðè
x
!
0
;

log
a
(1+
x
)=
x
ln
a
+
o
(
x
)
,
x
!
0
,
a�
0
,
a
6
=1
;

(1+
x
)
p
=1+
p

x
+
o
(
x
)
ïðè
x
!
0
.
96
ÃËÀÂÀ10.ÏÎÊÀÇÀÒÅËÜÍÀßÈËÎÃÀÐÈÔÌÈ×ÅÑÊÀßÔÓÍÊÖÈÈ.
Ãëàâà11
Ïðîèçâîäíàÿ
Ñåðåã�à
Âàíÿàïðîèçâîäíàÿîòòîãî
ñàìîãîêàêàÿ?
�-=Grek=-
(òîãîñàìîãî)'=òîãî'*ñàìîãî+
òîãî*ñàìîãî'=ýòîñàìîãîòîãî
ýòîãî
(ïîìàòåðèàëàìbash.org.ru)
11.1Ââåäåíèå.Ôèçè÷åñêèéèãåîìåòðè÷åñêèé
ñìûñëïðîèçâîäíîé.
Cäðåâíèõâðåìåíëþäåéèíòåðåñîâàëàâîçìîæíîñòüèçìåðÿòüìãíîâåííóþ
ñêîðîñòü.
Íàïðèìåð,èçâåñòåíààïîðèÿ(ïàðàäîêñ)ÇåíîíàÝëåéñêîãî:
¾Ëåòÿùàÿ
ñòðåëàíåïîäâèæíà,òàêêàêâêàæäûéìîìåíòâðåìåíèîíàçàíèìà-
åòðàâíîåñåáåïîëîæåíèå,òîåñòüïîêîèòñÿ;ïîñêîëüêóîíàïîêîèòñÿ
âêàæäûéìîìåíòâðåìåíè,òîîíàïîêîèòñÿâîâñåìîìåíòûâðåìåíè,
òîåñòüíåñóùåñòâóåòìîìåíòàâðåìåíè,âêîòîðîìñòðåëàñîâåðøàåò
äâèæåíèå.¿
Äëÿòîãî,÷òîáûðàçîáðàòüñÿñýòèìïàðàäîêñîì,ñëåäóåòêàêèì-òîîá-
ðàçîìèçìåðèòüñêîðîñòüâáåñêîíå÷íîìàëûéïðîìåæóòîêâðåìåíè.Íà
ïîìîùüíàìïðèõîäèòòåîðèÿïðåäåëîâ.
97
98
ÃËÀÂÀ11.ÏÐÎÈÇÂÎÄÍÀß
Ðèñ.11.1:ÇåíîíÝëåéñêèé
fig:zenon
Ïðåäïîëîæèì,÷òîïîëîæåíèåòåëàâ
ìîìåíòâðåìåíè
t
çàäàåòñÿêîîðäèíàòîé
f
(
t
)
.Òîãäàðàññìîòðèìïðîìåæóòîêâðå-
ìåíè
[
t
0
;t
0
+
t
]
.Çàýòîòïðîìåæóòîêâðå-
ìåíèòî÷êàïðîõîäèòðàññòîÿíèå
1
,ðàâíîå
f
(
t
0
+
t
)

f
(
t
0
)
.Òàêèìîáðàçîì,ñðåä-
íÿÿñêîðîñòüíàýòîìîòðåçêåâðåìåíèðàâ-
íà
f
(
t
0
+
t
)

f
(
t
0
)

t
.Ðàññìîòðèìïðåäåëýòî-
ãîâûðàæåíèÿïðè

t
!
0
.Ýòîòïðåäåë
lim

t
!
0
f
(
t
0
+
t
)

f
(
t
0
)

t
èíàçûâàþòìãíîâåííîé
ñêîðîñòüþâìîìåíòâðåìåíè
t
0
.Àíàëîãè÷-
íî,ìîæíîîïðåäåëèòüìãíîâåííîåóñêîðå-
íèå
lim

t
!
0
v
(
t
0
+
t
)

v
(
t
0
)

t
,ãäå
v
(
t
)
ìãíîâåííàÿñêîðîñòüâìîìåíòâðåìåíè
t
.
Çàìåòèì,÷òîòàêèìîáðàçîììîæíîîöåíèâàòüñêîðîñòüèçìåíåíèÿîä-
íîéâåëè÷èíûïðèìàëûõèçìåíåíèÿõäðóãîé.Òàê,íàïðèìåð,âýêîíîìè-
êå,èñïîëüçóåòñÿòåðìèí¾ýëàñòè÷íîñòü¿,êîòîðûéïîêàçûâàåò,íàñêîëüêî
ñèëüíîìåíÿåòñÿîäíàâåëè÷èíàïðèìàëîìèçìåíåíèèäðóãîé.Íàïðèìåð,
Ýëàñòè÷íîñòüñïðîñàïîöåíå
÷óâñòâèòåëüíîñòüñïðîñàêèçìåíåíèþ
öåíû(ïðîöåíòíîåèçìåíåíèåñïðîñàíà1%èçìåíåíèÿöåíû),òîæåìîæåò
áûòüïðåäñòàâëåíàââèäåîòíîøåíèÿèçìåíåíèéñïðîñàèöåíû.
Äðóãàÿèíòåðïðåòàöèÿãåîìåòðè÷åñêàÿ(ñì.ðèñ.
geom_proiz
11.2).Ðàññìîòðèì
òî÷êó
A
(
x
0
;y
0
)
íàãðàôèêåôóíêöèè
y
=
f
(
x
)
.Ïðèìàëîìèçìåíåíèè
x
òî÷êà
B
(
x
0
+
x;y
0
+
y
)
áóäåòðàñïîëàãàòüñÿî÷åíüáëèçêîêòî÷êå
A
,
ñîîòâåòñòâåííî,ïðÿìàÿ
AB
áóäåòâêàêîìòîñìûñëåáëèçêàêêàñàòåëüíîé
,ïðîâåäåíîéâòî÷êå
A
êãðàôèêó.Åñëèæåóñòðåìèòü

x
!
0
,òîâïðåäåëå,
ïîëó÷èìêàñàòåëüíóþ.
Óãëîâîéêîýôôèöèåíòïðÿìîé
AB
ðàâåí

y

x
=
f
(
x
0
+
x
)

f
(
x
0
)

y
,ñëåäî-
âàòåëüíî,åãîïðåäåë
lim

x
!
0
f
(
x
0
+
x
)

f
(
x
0
)

y
áóäåòóãëîâûìêîýôôèöèåíòîì
êàñàòåëüíîéêãðàôèêó.
1
Òóòðàññòîÿíèåáåðåòñÿñîçíàêîì¾+¿,åñëèäâèæåíèåïðîèñõîäèòâïîëîæèòåëüíîì
íàïðàâëåíèèîñè
y
,èñîçíàêîì¾-¿,åñëèîíîïðîèñõîäèòâïðîòèâîïîëîæíîìíàïðàâëå-
íèè.
11.2.ÎÏÐÅÄÅËÅÍÈÅ.ÏÐÀÂÈËÀÄÈÔÔÅÐÅÍÖÈÐÎÂÀÍÈß.
99
x
y
O
A
x
0
f
(
x
0
)
B
x
0
+
x
f
(
x
0
+
x
)
Ðèñ.11.2:Ãåîìåòðè÷åñêèéñìûñëïðîèçâîäíîé.
geom_proiz
11.2Îïðåäåëåíèå.Ïðàâèëàäèôôåðåíöèðîâà-
íèÿ.
proi
Îïðåäåëåíèå78.
Åñëèñóùåñòâóåòïðåäåë
A
=lim

x
!
0
f
(
x
0
+
x
)

f
(
x
0
)
x
;
òîãîâîðÿò,÷òîôóíêöèÿ
f
(
x
)
äèôôåðåíöèðóåìà
âòî÷êå
x
0
,àçíà÷å-
íèåïðåäåëàíàçûâàþò
ïðîèçâîäíîé
ôóíêöèè
f
âòî÷êå
x
0
èîáîçíà÷àþò
A
=
f
0
(
x
0
)
proii
Îïðåäåëåíèå79.
Åñëèñóùåñòâóåò
A
,òàêîå,÷òî
f
(
x
)=
f
(
x
0
)+
A

(
x

x
0
)+
o
(
x

x
0
)(
x
!
x
0
)
;
òîãîâîðÿò,÷òîôóíêöèÿ
f
(
x
)
äèôôåðåíöèðóåìàâòî÷êå
x
0
,àâåëè÷èíó
A
íàçûâàþòïðîèçâîäíîéôóíêöèè
f
âòî÷êå
x
0
èîáîçíà÷àþò
A
=
f
0
(
x
0
)
Òåîðåìà78.
Îïðåäåëåíèÿ
proi
78è
proii
79ýêâèâàëåíòíû.
Äîêàçàòåëüñòâî.
Ïóñòü
A
=lim

x
!
0
f
(
x
0
+
x
)

f
(
x
0
)
x
.Òîãäà
f
(
x
0
+
x
)

f
(
x
0
)
x
=
A
+
o
(1)
,ñëåäîâà-
òåëüíî
f
(
x
0
+
x
)

f
(
x
0
)=
A

x
+
o
(
x
)
,
100
ÃËÀÂÀ11.ÏÐÎÈÇÂÎÄÍÀß
Âîáðàòíóþñòîðîíó,ïóñòü
f
(
x
0
+
x
)=
f
(
x
0
)+
A

x
+
o
(
x
)
.Òîãäà
lim

x
!
0
f
(
x
0
+
x
)

f
(
x
0
)
x
=lim

x
!
0
(
A
+
o
(
x
)
x
)=
A
,
Ëåììà22.
Åñëèôóíêöèÿ
f
(
x
)
äèôôåðåíöèðóåìàâòî÷êå
x
0
,òîîíà
íåïðåðûâíàâýòîéòî÷êå.
Äîêàçàòåëüñòâî.
Î÷åâèäíî,
lim
x
!
x
0
f
(
x
)=lim
x
!
x
0
(
f
(
x
0
)+
f
0
(
x
0
)

(
x

x
0
)+
o
(
x

x
0
))=
f
(
x
0
)
,
Ñëåäñòâèå18.
Åñëèôóíêöèÿ
f
(
x
)
äèôôåðåíöèðóåìàâòî÷êå
x
0
,òîîíà
îãðàíè÷åíàâíåêîòîðîéîêðåñòíîñòè
U
"
(
x
0
)
.
Îïðåäåëåíèå80.
Ìíîæåñòâî
M
íàçûâàþò
îòêðûòûì
,åñëèäëÿëþáî-
ãî
a
2
M
ñóùåñòâóåò
"�
0
,òàêîå,÷òî
U
"
(
a
)

M
.
Îïðåäåëåíèå81.
Ïóñòüìíîæåñòâî
M
îòêðûòîè
f
äèôôåðåíöèðóå-
ìàâîâñåõåãîòî÷êàõÒîãäàãîâîðÿò,÷òî
f
äèôôåðåíöèðóåìàíà
M
è
îáîçíà÷àþò
f
2D
(
M
)
.
Äëÿíàñòàêæåâàæåíñëó÷àé,êîãäàíàäîðàññìàòðèâàòüïðîèçâîäíóþ
ôóíêöèè,îïðåäåëåííîéíàîòðåçêå(êîòîðûé,î÷åâèäíî,íåÿâëÿåòñÿîò-
êðûòûììíîæåñòâîì).Äëÿýòîãîïîíàäîáÿòñÿñëåäóþùèåîïðåäåëåíèÿ:
Îïðåäåëåíèå82.
Åñëèïðåäåë
lim

x
!
+0
f
(
x
0
+
x
)

f
(
x
0
)

x
ñóùåñòâóåò,òîãî-
âîðÿò,÷òîôóíêöèÿ
f
(
x
)
äèôôåðåíöèðóåìàñïðàâà
âòî÷êå
x
0
èíà-
çûâàþòýòîòïðåäåë
ïðàâîéïðîèçâîäíîé
ôóíêöèè
f
(
x
)
âòî÷êå
x
0
.
Àíàëîãè÷íî,åñëèïðåäåë
lim

x
!�
0
f
(
x
0
+
x
)

f
(
x
0
)

x
ñóùåñòâóåò,òîãîâîðÿò,
÷òîôóíêöèÿ
f
(
x
)
äèôôåðåíöèðóåìàñëåâà
âòî÷êå
x
0
èíàçûâàþò
ýòîòïðåäåë
ëåâîéïðîèçâîäíîé
ôóíêöèè
f
(
x
)
âòî÷êå
x
0
.
Îïðåäåëåíèå83.
Åñëèôóíêöèÿ
f
(
x
)
,îïðåäåëåííàÿíàîòðåçêå
[
a;b
]
äèô-
ôåðåíöèðóåìàíà
(
a;b
)
,èìååòïðîèçâîäíóþñïðàâàâòî÷êå
a
èïðîèçâîä-
íóþñëåâàâòî÷êå
b
,òîãîâîðÿò,÷òî
f
äèôôåðåíöèðóåìàíàîòðåçêå
[
a;b
]
èîáîçíà÷àþò
f
2D
[
a;b
]
.
11.2.1Ïðàâèëàäèôôåðåíöèðîâàíèÿ.
Ðàññìîòðèìïðàâèëàíàõîæäåíèÿïðîèçâîäíûõ:
Òåîðåìà79
(Àðèôìåòè÷åñêèåñâîéñòâàïðîèçâîäíîé)
.
Ïóñòüôóíêöèè
f
è
g
äèôôåðåíöèðóåìûâòî÷êå
x
.Òîãäà
11.2.ÎÏÐÅÄÅËÅÍÈÅ.ÏÐÀÂÈËÀÄÈÔÔÅÐÅÍÖÈÐÎÂÀÍÈß.
101
1.
Åñëè
C
êîíñòàíòà,òî
(
Cf
(
x
))
0
=
Cf
0
(
x
)
.
2.
(
f
(
x
)+
g
(
x
))
0
=
f
0
(
x
)+
g
0
(
x
)
;
3.
(
f
(
x
)

g
(
x
))
0
=
f
0
(
x
)

g
0
(
x
)
;
4.
(
f
(
x
)
g
(
x
))
0
=
f
(
x
)
g
0
(
x
)+
f
0
(
x
)
g
(
x
)
;
5.
Åñëè
g
(
x
)
6
=0
,òî

f
(
x
)
g
(
x
)

0
=
f
0
(
x
)
g
(
x
)

f
(
x
)
g
0
(
x
)
g
2
(
x
)
.
Äîêàçàòåëüñòâî.
Ïóñòü
f
(
x
+
x
)=
f
(
x
)+
f
0
(
x
)


x
+
o
(
x
)
;
g
(
x
+
x
)=
g
(
x
)+
g
0
(
x
)


x
+
o
(
x
)
.
Ïðàâèëà1,2è3î÷åâèäíû.Äîêàæåìïðàâèëî4.Ðàññìîòðèì
f
(
x
+
x
)

g
(
x
+
x
)=(
f
(
x
)+
f
0
(
x
)


x
+
o
(
x
))

(
g
(
x
)+
g
0
(
x
)


x
+
o
(
x
))=
=
f
(
x
)

g
(
x
)+
f
(
x
)
g
0
(
x
)


x
+
f
0
(
x
)
g
(
x
)


x
+
f
0
(
x
)
g
0
(
x
)


x
2
+
+(
f
(
x
)+
f
0
(
x
)
x
)

o
(
x
)+
o
(
x
)

(
g
(
x
)+
g
0
(
x
)


x
+
o
(
x
))
:
(11.2.1)
prodprime
Î÷åâèäíî
f
0
(
x
)
g
0
(
x
)


x
2
=
o
(
x
)(
x
!
0)
.Êðîìåòîãî,èçîãðàíè÷åííîñòè
ôóíêöèé
f
(
x
+
x
)
è
g
(
x
+
x
)
âûòåêàåò,÷òî
(
f
(
x
)+
f
0
(
x
)
x
)

o
(
x
)=
o
(
x
)(
ïðè

x
!
0)
è
o
(
x
)

(
g
(
x
)+
g
0
(
x
)


x
+
o
(
x
))=
o
(
x
)(
ïðè

x
!
0)
:
Ñëåäîâàòåëüíî,èçðàâåíñòâà
prodprime
11.2.1âûòåêàåò
f
(
x
+
x
)

g
(
x
+
x
)=
f
(
x
)

g
(
x
)+(
f
(
x
)
g
0
(
x
)+
f
0
(
x
)
g
(
x
))


x
+
o
(
x
)(
x
!
0)
,
Äëÿäîêàçàòåëüñòâàñâîéñòâà5ðàññìîòðèìïðåäåë:
lim

x
!
0
f
(
x
+
x
)
g
(
x
+
x
)

f
(
x
)
g
(
x
)

x
.
Ïðåîáðàçóåìâûðàæåíèå:
f
(
x
+
x
)
g
(
x
+
x
)

f
(
x
)
g
(
x
)
=
f
(
x
+
x
)

g
(
x
)

f
(
x
)

g
(
x
+
x
)
g
(
x
)

g
(
x
+
x
)
=
=

f
(
x
)+
f
0
(
x
)


x
+
o
(
x
)


g
(
x
)

f
(
x
)


g
(
x
)+
g
0
(
x
)


x
+
o
(
x
)

g
(
x
)

g
(
x
+
x
)
=
=
(
f
0
(
x
)
g
(
x
)

f
(
x
)
g
0
(
x
))


x
+
o
(
x
)

g
(
x
)

f
(
x
)

o
(
x
)
g
(
x
)

g
(
x
+
x
)
Òîãäà
lim

x
!
0
f
(
x
+
x
)
g
(
x
+
x
)

f
(
x
)
g
(
x
)

x
=
=lim

x
!
0
(
f
0
(
x
)
g
(
x
)

f
(
x
)
g
0
(
x
))
o
(1)

g
(
x
)

f
(
x
)

o
(1)
g
(
x
)

g
(
x
+
x
)
=
=
f
0
(
x
)
g
(
x
)

f
(
x
)
g
0
(
x
)
g
2
(
x
)
;
102
ÃËÀÂÀ11.ÏÐÎÈÇÂÎÄÍÀß
Òåîðåìà80
(Ïðîèçâîäíàÿñëîæíîéôóíêöèè)
.
Ïóñòü
g
(
x
)
äèôôåðåíöè-
ðóåìàâòî÷êå
x

f
(
y
)
äèôôåðåíöèðóåìàâ
y
=
g
(
x
)
.Òîãäàôóíêöèÿ
h
(
x
)=
f
(
g
(
x
))
èõêîìïîçèöèÿäèôôåðåíöèðóåìàâòî÷êå
x
èååïðî-
èçâîäíàÿðàâíà
h
0
(
x
)=
f
0
(
y
)

g
0
(
x
)
.
Äîêàçàòåëüñòâî.
Îáîçíà÷èì

y
=
g
(
x
+
x
)

g
(
x
)=
g
0
(
x
)


x
+
o
(
x
)
.Òîãäà
f
(
y
+
y
)=
f
(
y
)+
f
0
(
y
)


y
+
o
(
y
)=
f
(
y
)+
f
0
(
y
)

(
g
0
(
x
)


x
+
o
(
x
))+
+
o
(
g
0
(
x
)


x
+
o
(
x
))=
f
(
y
)+
f
0
(
y
)
g
0
(
x
)


x
+
o
(
x
)
;
Ñëåäñòâèå19.
Ïóñòü
f
(
x
)
äèôôåðåíöèðóåìàâòî÷êå
x
,òîãäà

f
(
kx
+
b
)

0
=
k

f
0
(
kx
+
b
)
:
Äîêàçàòåëüñòâî.
Äåéñòâèòåëüíî,
k
(
x
+
x
)+
b
=
kx
+
b
+
k


x
,íî
(
kx
+
b
)
0
=
k
.Ñëåäîâàòåëüíî,

f
(
kx
+
b
)

0
=
f
0
(
kx
+
b
)

(
kx
+
b
)
0
=
k

f
0
(
kx
+
b
)
,
Òåîðåìà81
(Ïðîèçâîäíàÿîáðàòíîéôóíêöèè)
.
Ïóñòüôóíêöèÿ
f
(
x
)
äèô-
ôåðåíöèðóåìàâòî÷êå
x
èïðîèçâîäíàÿíåðàâíà0.Ïóñòü
g
(
y
)=
f

1
(
y
)

ôóíêöèÿ,îáðàòíàÿê
f
(
x
)
.Òîãäà
g
(
y
)
äèôôåðåíöèðóåìàâòî÷êå
y
=
f
(
x
)
,
èååïðîèçâîäíàÿðàâíà
g
0
(
y
)=
1
f
0
(
g
(
y
))
.
Äîêàçàòåëüñòâî.
Î÷åâèäíî,
x
0
=1
.Ïðèìåíèâêòîæäåñòâó
g
(
f
(
x
))=
x
òåîðåìóîïðîèçâîä-
íîéñëîæíîéôóíêöèè,ïîëó÷èì
g
0
(
y
)

f
0
(
x
)=1
,ñëåäîâàòåëüíî
g
0
(
y
)=
1
f
0
(
x
)
=
1
f
0
(
g
(
y
))
;
11.2.2Ïðîèçâîäíàÿïîêàçàòåëüíîé,ëîãàðèôìè÷åñêîé
èñòåïåííîéôóíêöèè.
lem4
Ëåììà23.
lim
x
!
0
(1+
x
)
1
x
=
e
11.2.ÎÏÐÅÄÅËÅÍÈÅ.ÏÐÀÂÈËÀÄÈÔÔÅÐÅÍÖÈÐÎÂÀÍÈß.
103
Äîêàçàòåëüñòâî.
Ðàññìîòðèìïîñëåäîâàòåëüíîñòü
a
n
=(1+
1
m
n
)
m
n
;
ãäå
8
n
2
N
m
n
2
N
è
lim
n
!1
m
n
=+
1
:
Òîãäà
lim
n
!1
a
n
=
e:
Äåéñòâèòåëüíî,ïîñêîëüêóïîñëåäîâà-
òåëüíîñòü
(1+
1
k
)
k
ìîíîòîííîâîçðàñòàåòèñóùåñòâóåòïðåäåë
lim
n
!1
(1+
1
k
)
k
=
e;
òîäëÿïðîèçâîëüíîãî
"�
0
ìîæíîâûáðàòü
k
0
2
N
,äëÿêîòîðîãî
e

(1+
1
k
0
)
k
0
"
.
Òàêèìîáðàçîìäëÿíåêîòîðîãî
n
0
=
n
0
(
"
)
âûïîëíåíî
8
n

n
0
(
m
n
�k
0
)
;
ñëåäîâàòåëüíî,
e

(1+
1
m
n
)
m
n
e

(1+
1
k
0
)
k
0
";
òîåñòü
lim
n
!1
a
n
=
e:
Ðàññìîòðèìïðîèçâîëüíóþïîñëåäîâàòåëüíîñòüïîëîæèòåëüíûõäåéñòâè-
òåëüíûõ÷èñåë,ñõîäÿùóþñÿêíóëþ:
x
n
!
+0
,ïðè
n
!1
.Î÷åâèäíî,
lim
n
!1
1
x
n
=+
1
:
Îáîçíà÷èì
m
n
=[
1
x
n
]
öåëàÿ÷àñòü
1
x
n
:
Áóäåìðàññìàòðè-
âàòüòå
x
n

1
,äëÿíèõâûïîëíåíîíåðàâåíñòâî:
0
m
n
6
1
x
n
m
n
+1
Ñëåäîâàòåëüíî,
(1+
1
m
n
+1
)
m
n

(1+
x
n
)
1
x
n

(1+
1
m
n
)
m
n
+1
:
Òàêêàê
m
n
!
+
1
ïðè
n
!1
;
lim
n
!1
(1+
1
m
n
)
m
n
+1
=lim
n
!1
(1+
1
m
n
)
m
n
(1+
1
m
n
)=
e

1=
e
è
lim
n
!1
(1+
1
m
n
+1
)
m
n
=lim
n
!1

(1+
1
m
n
+1
)
m
n
+1
=
(1+
1
m
n
+1
)

=
e:
Ïîòåîðåìåî¾äâóõìèëèöèîíåðàõ¿
lim
n
!1
(1+
x
n
)
1
x
n
=
e;
àïîîïðåäåëåíèþ
Ãåéíåîäíîñòîðîííåãîïðåäåëàôóíêöèèâòî÷êå
lim
x
!
0
+
(1+
x
)
1
x
=
e:
Ðàññìîòðèìïðîèçâîëüíóþïîñëåäîâàòåëüíîñòüäåéñòâèòåëüíûõ÷èñåë
x
n
2
(0
;
1)
òàêèõ,÷òî
lim
n
!1
x
n
=0
:
Îáîçíà÷èì
y
n
=
j
x
n
j
=

x
n

0
,
î÷åâèäíî
lim
n
!1
y
n
=0+
:
Òîãäà,ïîäîêàçàííîìóðàíåå,
lim
n
!1
(1+
x
n
)
1
x
n
=lim
n
!1
(1

y
n
)

1
y
n
=lim
n
!1
(1+
y
n
1

y
n
)
1
y
n
=
=lim
n
!1
(1+
z
n
)
1
z
n
+1
=lim
n
!1
(1+
z
n
)
1
z
n
(1+
z
n
)=
e

1=
e;
104
ÃËÀÂÀ11.ÏÐÎÈÇÂÎÄÍÀß
òàêêàê
0
z
n
=
y
n
1

y
n
!
0
+
ïðè
n
!1
:
ÏîîïðåäåëåíèþÃåéíåîäíî-
ñòîðîííåãîïðåäåëàôóíêöèèâòî÷êå
lim
x
!
0

(1+
x
)
1
x
=
e:
Èçñóùåñòâîâàíèÿ
îäíîñòîðîííèõïðåäåëîâ
lim
x
!
0

(1+
x
)
1
x
=lim
x
!
0
+
(1+
x
)
1
x
=
e
âûòåêàåòñóùå-
ñòâîâàíèåïðåäåëà
lim
x
!
0
(1+
x
)
1
x
=
e:
Ëåììà24.
Ïðåäåë
lim
x
!
0
ln(1+
x
)
x
ñóùåñòâóåòèðàâåí1.
Äîêàçàòåëüñòâî.
Ðàññìîòðèìïîêàçàòåëüíîñòåïåííóþôóíêöèþ
f
(
x
)=
(
(1+
x
)
1
x
;x
6
=0
e;x
=0
:
.
Ýòàôóíêöèÿîïðåäåëåíàèíåïðåðûâíàíà
(

1
;
+
1
)
;
òàêêàêíàìíîæåñòâå
(

1
;
0)
S
(0
;
+
1
)
ôóíêöèÿ
f
(
x
)=
e
ln(1+
x
)
1
x
=
e
ln(1+
x
)
x
ïðåäñòàâëÿåòñÿââèäå
êîìïîçèöèèíåïðåðûâíûõôóíêöèé,àíåïðåðûâíîñòüâíóëåñëåäóåòèç
ëåììû
lem4
23.Äåéñòâèòåëüíî,
lim
x
!
0
f
(
x
)=lim
x
!
0
(1+
x
)
1
x
=
e
=
f
(0)
:
Ïîýòîìó,
ôóíêöèÿ
ln
f
(
x
)
îïðåäåëåíàèíåïðåðûâíàíà
(

1
;
+
1
)
êàêêîìïîçèöèÿ
íåïðåðûâíûõôóíêöèé,ñëåäîâàòåëüíî,
lim
x
!
0
ln(1+
x
)
x
=lim
x
!
0
ln(1+
x
)
1
x
=ln
f
(0)=ln
e
=1
:
Òåîðåìà82.
Âåðíûñëåäóþùèåïðàâèëà:
1.
(
e
x
)
0
=
e
x
;
2.
(
a
x
)
0
=
a
x
ln
a
;
3.
(ln
x
)
0
=
1
x
;
4.
(log
a
x
)
0
=
1
x
ln
a
5.
(
x

)
0
=
x


1
:
Äîêàçàòåëüñòâî.
(ln
x
)
0
=lim
h
!
0
ln(
x
+
h
)

ln
x
h
=lim
h
!
0
ln(1+
h
x
)
h
=
1
x
lim
h
!
0
ln(1+
h
x
)
h
x
=
1
x
:
(log
a
x
)
0
=(
ln
x
ln
a
)
0
=
1
x
ln
a
:
a
x
=
y
)
x
=log
a
y
)
x
0
=1=(log
a
y
)
0
=
y
0
y
ln
a
)
y
0
=
y
ln
a
)
(
a
x
)
0
=
a
x
ln
a;
â÷àñòíîñòè,
(
e
x
)
0
=
e
x
:
(
x

)
0
=(
e
ln
x

)
0
=(
e

ln
x
)
0
=
e

ln
x

1
x
=
x

x
=
x


1
:
11.3.ÏÐÎÈÇÂÎÄÍÛÅÝËÅÌÅÍÒÀÐÍÛÕÔÓÍÊÖÈÉ
105
11.3Ïðîèçâîäíûåýëåìåíòàðíûõôóíêöèé
Äàëååïðèâîäÿòñÿïðîèçâîäíûåíåêîòîðûõýëåìåíòàðíûõôóíêöèé:
1.
C
0
=0
;
2.
(
e
x
)
0
=
e
x
;
3.
(
a
x
)
0
=ln
a

a
x
;
4.
ln
0
x
=
1
x
;
5.
log
0
a
x
=
1
x
ln
a
;
6.
(
x

)
0
=


x


1
;
7.
sin
0
x
=cos
x
;
8.
cos
0
x
=

sin
x
;
9.
tg
0
x
=
1
cos
2
x
;
10.
ctg
0
x
=

1
sin
2
x
;
11.
arcsin
0
x
=
1
p
1

x
2
;
12.
arccos
0
x
=

1
p
1

x
2
;
13.
arctg
0
x
=
1
1+
x
2
;
14.
arcctg
0
x
=

1
1+
x
2
;
Äîêàçàòåëüñòâî.
1.Î÷åâèäíî.
2.Ïîòåîðåìå
vtorpred
77(îâòîðîìçàìå÷àòåëüíîìïðåäåëå)
e

x
=1+
x
+
o
(
x
)
.
Óìíîæèâîáå÷àñòèíà
e
x
,ïîëó÷èì
e
x
+
x
=
e
x
+
e
X


x
+
o
(
x
)
,
3.Âûòåêàåòèçðàâåíñòâà
a
x
=
e
x
ln
a
èòåîðåìûîïðîèçâîäíîéñëîæíîé
ôóíêöèè;
4.Ôóíêöèÿ
ln
y
ÿâëÿåòñÿîáðàòíîéê
y
=
e
x
.Ïîòåîðåìåîïðîèçâîäíîé
îáðàòíîéôóíêöèè
ln
0
y
=
1
e
x
=
1
y
,
5.Âûòåêàåòèçòîæäåñòâà
log
a
x
=
ln
x
ln
a
;
6.
x

=
e

ln
x
(
x

)
0
=


e

ln
x

ln
0
x
=
x


1
,
7.
sin(
x
+
x
)=sin(
x
)

cos
x
+sin
x

cos
x
=sin
x

(1+
o
(
x
))+cos
x
(
x
+
o
(
x
))=sin
x
+cos
x


x
+
o
(
x
)
,
8.
cos(
x
+
x
)=cos
x

cos
x

sin
x

sin=cos
x

(1+
o
(
x
))

sin
x
(
x
+
o
(
x
))=cos
x

sin
x


x
+
o
(
x
)
,
9.
tg
0
x
=

sin
x
cos
x

0
=
sin
0
x

cos
x

sin
x

cos
0
x
cos
2
x
=
cos
2
x
+sin
2
x
cos
2
x
=
1
cos
2
x
.
10.
ctg
0
x
=

cos
x
sin
x

0
=
cos
0
x

sin
x

cos
x

sin
0
x
sin
2
x
=

sin
2
x

cos
2
x
sin
2
x
=

1
sin
2
x
.
11.Ïîòåîðåìåîïðîèçâîäíîéîáðàòíîéôóíêöèè:
arcsin
0
x
=
1
sin
0
(arcsin
x
)
=
1
cos(arcsin
x
)
=
1
p
1

x
2
:
106
ÃËÀÂÀ11.ÏÐÎÈÇÂÎÄÍÀß
12.Âîñïîëüçóåìñÿòîæäåñòâîì:
arcsin
x
+arccos
x
=

2
,ïîëó÷èì:
arccos
0
x
=


2

arcsin
x

0
=

arcsin
0
x
=

1
p
1

x
2
:
13.Ïîòåîðåìåîïðîèçâîäíîéîáðàòíîéôóíêöèè:
arctg
0
x
=
1
tg
0
(arctg
x
)
=
1
cos

2
(arctg
x
)
:
Âîñïîëüçóåìñÿòåì,÷òî
1+tg
2

=cos

2

.Òîãäà
1
cos

2
(arctg
x
)
=
1
1+tg
2
(arctg)
=
1
1+
x
2
;
14.Âîñïîëüçóåìñÿòîæäåñòâîì:
arctg
x
+arcctg
x
=

2
,ïîëó÷èì:
arcctg
0
x
=


2

arctg
x

0
=

arctg
0
x
=

1
1+
x
2
:
11.4Ñâîéñòâàïðîèçâîäíîé.ÒåîðåìûÔåðìà,
Ðîëëÿ,Ëàãðàíæà,Êîøè.
Ëåììà25.
Åñëèôóíêöèÿ
f
(
x
)
ìîíîòîííîâîçðàñòàåò(íåóáûâàåò)íà
îòðåçêå
[
a;b
]
èäèôôåðåíöèðóåìàíàíåì,òîååïðîèçâîäíàÿ
f
0
(
x
)

0
ïðè
âñåõ
x
2
[
a;b
]
.
Äîêàçàòåëüñòâî.
Ïóñòüôóíêöèÿ
f
(
x
)
íåóáûâàåòíà
[
a;b
]
,òîãäà
(
f
(
x
+
x
)

f
(
x
0
)
;
ïðè

x�
0;
f
(
x
+
x
)
6
f
(
x
0
)
;
ïðè

x
0
:
.
Âëþáîìñëó÷àå,
f
(
x
+
x
)

f
(
x
0
)

x

0
,ñëåäîâàòåëüíî,ïîòåîðåìå
pred_per_ner_fun
54(îïðå-
äåëüíîìïåðåõîäåâíåðàâåíñòâàõ)
f
0
(
x
0
)=lim

x
!
0
f
(
x
+
x
)

f
(
x
0
)

x

0
,
Ëåììà26.
Åñëèôóíêöèÿ
f
(
x
)
ìîíîòîííîóáûâàåò(íåâîçðàñòàåò)íà
îòðåçêå
[
a;b
]
èäèôôåðåíöèðóåìàíàíåì,òîååïðîèçâîäíàÿ
f
0
(
x
)
6
0
ïðè
âñåõ
x
2
[
a;b
]
.
Äîêàçàòåëüñòâî.
Ïðåäîñòàâëÿåòñÿ÷èòàòåëþâêà÷åñòâåóïðàæíåíèÿ.
11.4.ÑÂÎÉÑÒÂÀÏÐÎÈÇÂÎÄÍÎÉ.ÒÅÎÐÅÌÛÔÅÐÌÀ,ÐÎËËß,ËÀÃÐÀÍÆÀ,ÊÎØÈ.
107
ferma
Òåîðåìà83
(Ôåðìà)
.
Ïóñòüôóíêöèÿ
f
(
x
)
ïðèíèìàåòìàêñèìàëüíîå(ìè-
íèìàëüíîå)íàîòðåçêå
[
a;b
]
çíà÷åíèåâòî÷êå
x
0
2
(
a;b
)
,ò.å.
max
x
2
[
a;b
]
f
(
x
)=
f
(
x
0
)
(ñîîòâåòñòâåííî
min
x
2
[
a;b
]
f
(
x
)=
f
(
x
0
)
)èïóñòü
f
(
x
)
äèôôåðåíöèðóåìàâ
ýòîéòî÷êå.Òîãäà
f
0
(
x
0
)=0
.
Ðèñ.11.3:Ï.Ôåðìà(1601
1665)
fig:fermat
Çàìå÷àíèå.
Îáðàòèòåâíèìàíèå,÷òî
x
0
2
(
a;b
)
àíå
[
a;b
]
.Åñëè
x
0
ñîâïàäàåòñîäíèìèçêîíöîâîòðåçêà
[
a;b
]
,òîýòîíåîáÿçàòåëüíîâåðíî.
Íàïðèìåð,ìàêñèìóìôóíêöèè
f
(
x
)=
x
íàîòðåçêå
[0
;
1]
äîñòèãàåòñÿâ
òî÷êå
x
=1
,íîïðîèçâîäíàÿâåçäåðàâíà1èâíîëüíåîáðàùàåòñÿ.
Äîêàçàòåëüñòâî.
Ïóñòü
x
0
òî÷êàìàêñèìóìà,
max
x
2
[
a;b
]
f
(
x
)=
f
(
x
0
)
.Ðàññìîòðèì
A
=lim

x
!
+0
f
(
x
+
x
)

f
(
x
0
)

x
:
Î÷åâèäíî,
f
(
x
0
+
x
)

f
(
x
0
)
,ñëåäîâàòåëüíîïîòåîðåìå
pred_per_ner_fun
54(îïðåäåëüíîì
ïåðåõîäåâíåðàâåíñòâàõ)
A

0
.Ñäðóãîéñòîðîíû
B
=lim

x
!�
0
f
(
x
+
x
)

f
(
x
0
)

x
6
0
:
Aïîñêîëüêóïðåäåë
lim

x
!
0
f
(
x
+
x
)

f
(
x
0
)

x
(ñîáñòâåííîïðîèçâîäíàÿ)ñóùå-
ñòâóåò,òîîíäîëæåíáûòüðàâåí
0
,÷òîèòðåáîâàëîñüäîêàçàòü.
Ãåîìåòðè÷åñêèéñìûñëñîñòîèòâòîì,
÷òîâòî÷êåìèíèìóìàèëèìàêñèìóìàêà-
ñàòåëüíàÿêãðàôèêóôóíêöèè(åñëè,êî-
íå÷íî,îíàñóùåñòâóåò)ãîðèçîíòàëüíà.
Óïðàæíåíèå46.
ÄîêàçàòüòåîðåìóÔåð-
ìàäëÿñëó÷àÿ,êîãäà
x
0
òî÷êàìèíèìó-
ìà.
roll
Òåîðåìà84
(Ðîëëü)
.
Ïóñòüôóíêöèÿ
f
(
x
)
äèôôåðåíöèðóåìàâîâñåõòî÷êàõîòðåçêà
[
a;b
]
èïóñòü
f
(
a
)=
f
(
b
)
.Òîãäàíàéäåòñÿ
òî÷êà
x
0
2
(
a;b
)
,òàêàÿ,÷òî
f
0
(
x
0
)=0
.
Äîêàçàòåëüñòâî.
Åñëè
f
(
x
)
êîíñòàíòà,òîóòâåðæäåíèåî÷å-
âèäíî.Ïóñòüòåïåðüôóíêöèÿ
f
(
x
)
ïðèíè-
ìàåòçíà÷åíèå,îòëè÷íîåîò
f
(
a
)=
f
(
b
)
õî-
òÿáûâîäíîéòî÷êå
x
1
.Äîïóñòèì
f
(
x
1
)
�f
(
a
)=
f
(
b
)
108
ÃËÀÂÀ11.ÏÐÎÈÇÂÎÄÍÀß
(åñëè
f
(
x
1
)
f
(
a
)=
f
(
b
)
,òîäîêàçàòåëü-
ñòâîàíàëîãè÷íî).
x
y
A
B
C
Ðèñ.11.4:ÒåîðåìàÐîëëÿ
fig:th_roll
Ïîëåììå
differ_neprer
??
ôóíêöèÿ
f
(
x
)
íåïðåðûâíà,
ñëåäîâàòåëüíî,ïîòåîðåìå
extrem
59,îíàïðèíè-
ìàåòâíåêîòîðîéòî÷êå
x
0
ìàêñèìàëüíîå
çíà÷åíèå
max
x
2
[
a;b
]
f
(
x
)=
f
(
x
0
)
�f
(
a
)=
f
(
b
)
,
ïðè÷åì
x
0
,î÷åâèäíî,íåñîâïàäàåòñêîíöà-
ìèîòðåçêà
[
a;b
]
.Òîãäà,ïîòåîðåìåÔåðìà
f
0
(
x
0
)=0
,
lagranj
Òåîðåìà85
(Ëàãðàíæà)
.
Ïóñòü
f
(
x
)
äèô-
ôåðåíöèðóåìàíàîòðåçêå
[
a;b
]
.Òîãäà
ñóùåñòâóåò
x
0
2
(
a;b
)
,òàêàÿ,÷òî
f
(
b
)

f
(
a
)=
f
0
(
x
0
)(
b

a
)
.
Äîêàçàòåëüñòâî.
Ðàññìîòðèìôóíêöèþ
g
(
x
)=
x

(
f
(
a
)

f
(
b
))+(
b

a
)
f
(
x
)
.
Î÷åâèäíî,
g
(
a
)=
a

f
(
a
)

a

f
(
b
)+
b

f
(
a
)

a

f
(
a
)=
b

f
(
a
)

a

f
(
b
)
,
g
(
b
)=
b

f
(
a
)

b

f
(
b
)+
b

f
(
b
)

a

f
(
b
)=
b

f
(
a
)

a

f
(
b
)
,
àçíà÷èò,ïîòåîðåìåÐîëëÿ,íàéäåò-
ñÿòî÷êà
x
0
2
(
a;b
)
,òàêàÿ,÷òî
g
0
(
x
0
)=
f
(
a
)

f
(
b
)+(
b

a
)
f
0
(
x
0
)=0
,
ò.å.
f
(
b
)

f
(
a
)=
f
0
(
x
0
)(
b

a
)
,
Ãåîìåòðè÷åñêèéñìûñëýòîéòåîðåìûòàêîâ:Äëÿêàæäîéõîðäû
[
AB
]
íàéäåòñÿïàðàëëåëüíàÿåéêàñàòåëüíàÿ.
Òåîðåìà86
(Êîøè)
.
Ïóñòü
f
(
x
)
;g
(
x
)
2C
[
a;b
]
\D
(
a;b
)
è
8
x
2
(
a;b
)(
g
0
(
x
)
6
=0)
.
Òîãäàñóùåñòâóåò
x
0
2
(
a;b
)
òàêîå,÷òî
f
(
b
)

f
(
a
)
g
(
b
)

g
(
a
)
=
f
0
(
x
0
)
g
0
(
x
0
)
:
Äîêàçàòåëüñòâî.
Îáîçíà÷èì

f
=
f
(
b
)

f
(
a
)
,

g
=
g
(
b
)

g
(
a
)
èðàññìîòðèìôóíêöèþ
h
(
x
)=
f

g
(
x
)


g

f
(
x
)
:
Ïîñêîëüêó
h
(
a
)=
h
(
b
)=
f
(
b
)
g
(
a
)

f
(
a
)
g
(
b
)
(ïðîâåðüòåýòî!),òîïîòåîðåìå
Ðîëëÿíàéäåòñÿ
x
0
2
(
a;b
)
,äëÿêîòîðîãî
h
0
(
x
0
)=
f

g
0
(
x
0
)


g

f
0
(
x
0
)=0
,
à,ñëåäîâàòåëüíî,

f

g
0
(
x
0
)=
g

f
0
(
x
0
)
.Ðàçäåëèâíà
g
0
(
x
0
)
6
=0
,ïîëó÷èì

f

g
=
f
0
(
x
0
)
g
0
(
x
0
)
,
11.5.ÍÅÐÀÂÅÍÑÒÂÀÞÍÃÀ,ÃœËÜÄÅÐÀ,ÊÎØÈÁÓÍßÊÎÂÑÊÎÃÎ.
109
11.5ÍåðàâåíñòâàÞíãà,üëüäåðà,ÊîøèÁóíÿêîâñêîãî.
Ñëåäñòâèå20
(ÍåðàâåíñòâîÞíãà)
.
Ïóñòüçàäàíûïîëîæèòåëüíûå÷èñ-
ëà
a;b�
0
,òàêèå,÷òî
a
+
b
=1
.Òîãäàïðèâñåõ
x�
0
âûïîëíåíîíåðàâåí-
ñòâî
x
a
6
ax
+
b
.
x
y
C
A
B
Ðèñ.11.5:ÒåîðåìàËàãðàíæà
fig:th_lagrange
Ðàññìîòðèìôóíêöèþ
f
(
x
)=
x
a

ax
,
î÷åâèäíî,
f
(1)=1

a
=
b
è
f
0
(
x
)=
ax
a

1

a
=
ax

b

a
=
a

1
x
b

1

.
Âûáåðåìïðîèçâîëüíîå
x�
0
èðàññìîòðèì
äâàâîçìîæíûõñëó÷àÿ:
1.
Ïóñòü
x�
1
,òîãäà,ïîòåîðåìå
Ëàãðàíæà,ñóùåñòâóåò
x
0
2
[1
;x
]
,
òàêîå,÷òî
f
(
x
)

b
=
f
0
(
x
0
)(
x

1)=
a
(
1
x
b
0

1)(
x

1)
.
Ïîñêîëüêó
x
0

1
,òî
(
1
x
b
0

1)

0
,ñëå-
äîâàòåëüíî,
f
(
x
)

b
0
,ò.å.
f
(
x
)
b
.
2.
Ïóñòüòåïåðü
0
x
1
,òî-
ãäàñóùåñòâóåò
x
0
2
[
x;
1]
òà-
êîå,÷òî
b

f
(
x
)=
f
0
(
x
0
)(1

x
)=(
ax
a

1
0

a
)(1

x
)
.
Ïîñêîëüêó
0
x
0

1
,òî
(
1
x
b
0

1)

0
,
ñëåäîâàòåëüíî
b

f
(
x
)

0
è,îïÿòü,
f
(
x
)
b
.
Òàêèìîáðàçîì,ïðèâñåõ
x�
0
,
x
6
=1
çíà-
÷åíèåôóíêöèè
f
(
x
)
ìåíüøå
b
,
Ñëåäñòâèå21
(Íåðàâåíñòâîüëüäåðà)
.
Ïóñòü÷èñëà
p;q�
1
òàêîâû,÷òî
1
p
+
1
q
=1
.Òîãäàäëÿïðîèçâîëüíûõ
a
1
;:::;a
n
;b
1
;:::;b
n

0
âûïîëíåíîíåðàâåíñòâî:
a
1
b
1
+
a
2
b
2
+
:::a
n
b
n
6
(
a
p
1
+
a
p
2
+
:::
+
a
p
n
)
1
=p

(
b
q
1
+
b
q
2
+
:::
+
b
q
n
)
1
=q
Äîêàçàòåëüñòâî.
Îáîçíà÷èì
u
i
=
a
p
i
a
p
1
+
:::
+
a
p
n
è
v
i
=
b
q
i
b
q
1
+
:::
+
b
q
n
,
i
=1
;:::;n
(ñ÷èòàåì,÷òîíå
âñå
a
1
;:::;a
n
;b
1
;:::;b
n
ðàâíûíóëþ,èíà÷åíåðàâåíñòâîòðèâèàëüíî).Âçÿâ
x
=
u
i
v
i
,
a
=
1
p
è
b
=
1
q
çàïèøåìíåðàâåíñòâîÞíãàââèäå:

u
i
v
i

1
=p
6
1
p

u
i
v
i
+
1
q
:
Óìíîæàÿíà
v
i
îáå÷àñòèíåðàâåíñòâàèïðåîáðàçîâûâàÿ(íàïîìíèì,÷òî
1

1
p
=
1
q
),ïîëó÷èì
u
1
=p
i

v
1
=q
i
6
u
i
p
+
v
i
q
:
110
ÃËÀÂÀ11.ÏÐÎÈÇÂÎÄÍÀß
Ïðîñóììèðóåìïîâñåì
i
îò
1
äî
n
èóïðîñòèì:
u
1
=p
1

v
1
=q
1
+
u
1
=p
2

v
1
=q
2
+
:::
+
u
1
=p
n

v
1
=q
n
6
1
(11.5.1)
summa
(Îáðàòèòåâíèìàíèå,÷òî
n
P
i
=1
u
i
=
n
P
i
=1
v
i
=1
).Ïîäñòàâèââ
summa
11.5.1âûðàæåíèÿ
äëÿ
u
i
è
v
i
,èìååì
a
1
b
1
+
:::
+
a
n
b
n
(
a
p
1
+
:::
+
a
p
n
)
1
=p

(
b
q
1
+
:::
+
b
q
n
)
1
=q
6
1
;
îòêóäàèâûòåêàåòòðåáóåìîåíåðàâåíñòâî.
Çàìå÷àíèå.
×àñòíûìñëó÷àåìíåðàâåíñòâàüëüäåðàïðè
p
=
q
=2
ÿâëÿåòñÿíåðàâåíñòâîÊîøèÁóíÿêîâñêîãî:
a
1
b
1
+
a
2
b
2
+
:::a
n
b
n
6
q
a
2
1
+
a
2
2
+
:::
+
a
2
n

q
b
2
1
+
b
2
2
+
:::
+
b
2
n
:
Îíîóòâåðæäàåò,÷òîñêàëÿðíîåïðîèçâåäåíèåâåêòîðîââ
n
ìåðíîìïðî-
ñòðàíñòâåíåïðåâîñõîäèòïðîèçâåäåíèÿèõäëèí.
Óïðàæíåíèå47.
ÏðèêàêèõóñëîâèÿõíàèíåðàâåíñòâîÊîøèÁóíÿêîâñêîãî
îáðàùàåòñÿâðàâåíñòâî?
Ñëåäñòâèå22.
Åñëè
; ;
óãëûòðåóãîëüíèêà,òî
sin

+sin

+sin

6
3
p
3
2
:
Äîêàçàòåëüñòâî.
Çàôèêñèðóåì

èîáîçíà÷èì

=



2
,

=


x
è

=

+
x
.Ðàñ-
ñìîòðèìôóíêöèþ
f
(
x
)=sin

+sin(


x
)+sin(

+
x
)
.Ååïðîèçâîä-
íàÿðàâíà
f
0
(
x
)=

cos(


x
)+cos(

+
x
)=

2sin

sin
x
.Çàìåòèì,÷òî
f
0
(
x
)

0
ïðè
x�
0
,è,íàîáîðîò
f
0
(
x
)

0
ïðè
x
0
.Ñëåäîâàòåëüíî
f
(
x
)
6
f
(0)=sin

+2sin



2
.
Áóäåìòåïåðüâàðüèðîâàòü

.Ðàññìîòðèìôóíêöèþ
g
(
x
)=sin
x
+2sin


x
2
.
Ååïðîèçâîäíàÿðàâíà
g
0
(
x
)=cos
x

cos


x
2
=

2sin
x
+

4
sin
3
x


4
.Çàìå-
òèì,÷òî
g
0
(
x
)

0
ïðè
x�

3
,è,íàîáîðîò,
g
0
(
x
)

0
ïðè
x

3
.Çíà÷èò
g
(
x
)
6
g
(

3
)=
3
p
3
2
,
11.6ÏðàâèëàËîïèòàëÿ.
Òåîðåìà87
(ÏåðâîåïðàâèëîËîïèòàëÿ)
.
Ïóñòü
a
2
R
èäëÿíåêîòîðîãî
"�
0
ôóíêöèè
f
(
x
)
è
g
(
x
)
îïðåäåëåíûâîêðåñòíîñòè

U
"
(
a
)
èíåðàâíû0
11.6.ÏÐÀÂÈËÀËÎÏÈÒÀËß.
111
âîâñåõååòî÷êàõ.Ïóñòü
lim
x
!
a
f
(
x
)=lim
x
!
a
g
(
x
)=0
è
lim
x
!
a
f
0
(
x
)
g
0
(
x
)
=
A
.Òîãäà
ïðåäåë
lim
x
!
a
f
(
x
)
g
(
x
)
ñóùåñòâóåòèðàâåí
A
.
Äîêàçàòåëüñòâî.
Ïîñêîëüêó
lim
x
!
a
f
(
x
)=lim
x
!
a
g
(
x
)=0
,áóäåìñ÷èòàòü
f
(
a
)=
g
(
a
)=0
,ïðè
ýòîìôóíêöèè
f
è
g
áóäóòíåïðåðûâíûâ
a
.Èçîïðåäåëåíèÿäëÿïðåäåëà
lim
x
!
a
f
0
(
x
)
g
0
(
x
)
=
A
âûòåêàåò,÷òîäëÿïðîèçâîëüíîãî
"�
0
íàéäåòñÿ

=

(
"
)
,òà-
êîå,÷òîäëÿêàæäîãî
x
0
2

U

(
a
)
âûïîëíåíîíåðàâåíñòâî
j
f
0
(
x
0
)
g
0
(
x
0
)

A
j
"
.Âû-
áåðåìïðîèçâîëüíîå
x
2

U

(
a
)
,òîãäà
f
(
x
)
g
(
x
)
=
f
(
x
)

f
(
a
)
g
(
x
)

g
(
a
)
f
0
(
x
0
)
g
0
(
x
0
)
,ãäå
x
0
2

U

(
a
)
íåêîòîðîå÷èñëî.Òîãäà
j
f
(
x
)
g
0
(
x
)

A
j
"
.Èçïðîèçâîëüíîñòè
"�
0
âûòåêàåò,
÷òî
lim
x
!
a
f
(
x
)
g
(
x
)
=
A
,
Òåîðåìà88
(ÂòîðîåïðàâèëîËîïèòàëÿ)
.
Ïóñòü
a
2
R
èäëÿíåêîòîðîãî
"�
0
ôóíêöèè
f
(
x
)
è
g
(
x
)
îïðåäåëåíûâîêðåñòíîñòè

U
"
(
a
)
èíåðàâíû
0âîâñåõååòî÷êàõ.Ïóñòü
lim
x
!
a
f
(
x
)=lim
x
!
a
g
(
x
)=
1
è
lim
x
!
a
f
0
(
x
)
g
0
(
x
)
=
A
.
Òîãäàïðåäåë
lim
x
!
a
f
(
x
)
g
(
x
)
ñóùåñòâóåòèðàâåí
A
.
Äîêàçàòåëüñòâî.
(Òî÷íååèäåÿäîêàçàòåëüñòâà)Îñíîâûâàåòñÿíàñëåäóþùåéòåîðåìå:
Òåîðåìà89
(ÒåîðåìàØòîëüöà)
.
Ïóñòüïîñëåäîâàòåëüíîñòü
f
y
n
g
1
n
=1
ìî-
íîòîííîâîçðàñòàåòè
lim
n
!1
y
n
=+
1
.Ïóñòü
lim
n
!1
x
n
+1

x
n
y
n
+1

y
n
=
A:
Òîãäàïðå-
äåë
lim
n
!1
x
n
y
n
ñóùåñòâóåòèðàâåíòîìóæå÷èñëó
A
.
Äîêàçàòåëüñòâî.
Ïîîïðåäåëåíèþïðåäåëàìîæíîçàïèñàòü
x
n
+1

x
n
y
n
+1

y
n
=
A
+

n
,ãäå

n
á.ì.ï.
Ïóñòü
"�
0
ïðîèçâîëüíî.Íàéäåòñÿ
N
=
N
(
"=
2)
,òàêîå÷òîïðèâñåõ
n�N
j

n
j

"
2
.Äîìíîæèâíàçíàìåíàòåëüïîëó÷èì:
x
n
+1

x
n
=
A
(
y
n
+1

y
n
)+

n

(
y
n
+1

y
n
)
.
Çàôèêñèðóåì
n
0
�N
èïðîñóììèðóåìýòèðàâåíñòâàïî
n
îò
n
0
äî
m

1
,
ïîëó÷èì
x
m

x
n
0
=
A
(
y
m

y
n
0
)+

n
0
(
y
n
0
+1

y
n
0
)+
:::
+

m

1
(
y
m

y
m

1
)
:
Ïåðåãðóïïèðóåìñëàãàåìûå
x
m

A

y
m
=
x
n
0

A

y
n
0
+

n
0
(
y
n
0
+1

y
n
0
)+
:::
+

m

1
(
y
m

y
m

1
)
;
112
ÃËÀÂÀ11.ÏÐÎÈÇÂÎÄÍÀß
èïðèìåíèìíåðàâåíñòâîòðåóãîëüíèêàäëÿìîäóëåé:
j
x
m

A

y
m
j
6
j
x
n
0

A

y
n
0
j
+
j

n
0
jj
y
n
0
+1

y
n
0
j
+
:::
:::
+
j

m

1
jj
y
m

y
m

1
j
6
j
x
n
0

A

y
n
0
j
+
"
2

(
y
n
0
+1

y
n
0
)+
:::
:::
+
"
2

(
y
m

y
m

1
)=
j
x
n
0

A

y
n
0
j
+
"
2

(
y
m

y
n
0
)
:
Çäåñüìîäóëè
j
y
k
+1

y
k
j
ðàñêðûâàþòñÿñïîëîæèòåëüíûìçíàêîì,ïîñêîëü-
êó
f
y
n
g
1
n
=1
âîçðàñòàþùàÿïîñëåäîâàòåëüíîñòü.Ðàçäåëèìòåïåðüîáå÷à-
ñòèíåðàâåíñòâàíà
y
m




x
m
y
m

A




6
j
x
n
0

A

y
n
0
j
y
m
+
"
2

(1

y
n
0
y
m
)
:
Ïðåäåëïðàâîé÷àñòèïðè
m
!1
ðàâåí
"
2
,ñëåäîâàòåëüíî,íà÷èíàÿñíåêî-
òîðîãî
M
0
ååâåëè÷èíàìåíüøå
"
,òîåñòü
j
x
m
y
m

A
j
"
,
Äîêàçàòåëüñòâî(2ãîïðàâèëàËîïèòàëÿ).
Âûáåðåìïðîèçâîëüíóþ
ïîñëåäîâàòåëüíîñòü
x
n
!
a
,òàê,÷òî
y
n
=
g
(
x
n
)
ìîíîòîííîâîçðàñòàåòè
lim
n
!1
y
n
=+
1
(ïîñëåäíååñëåäóåòèçóñëîâèÿòåîðåìû).ÏîòåîðåìåÊî-
øèíàèíòåðâàëå
(
x
n
;x
n
+1
)
èëè
(
x
n
+1
;x
n
)
,íàéäåòñÿòî÷êà
c
n
,òàêàÿ,÷òî
f
(
x
n
+1
)

f
(
x
n
)
g
(
x
n
+1
)

g
(
x
n
))
=
f
0
(
c
n
)
g
0
(
c
n
)
Ñëåäîâàòåëüíî,ïîñêîëüêó
c
n
!
a
,òîïðåäåëýòî-
ãîâûðàæåíèÿðàâåí
lim
n
!1
f
(
x
n
+1
)

f
(
x
n
)
g
(
x
n
+1
)

g
(
x
n
))
=
A
.ÏîòåîðåìåØòîëüöàèìååì:
lim
n
!1
f
(
x
n
)
g
(
x
n
)
=
A
,à,ñëåäîâàòåëüíî,
lim
x
!
a
f
(
x
)
g
(
x
)
=
A
,
Ãëàâà12
Êðàòíûåïðîèçâîäíûå.
ÔîðìóëàËåéáíèöà.
Âûïóêëîñòü
12.1Êðàòíûåïðîèçâîäíûå.
Îïðåäåëåíèå84.
Êàêìûóæåâèäåëèïðîèçâîäíàÿíåêîòîðîéôóíêöèè
f
(
x
)
(äèôôåðåíöèðóåìîé)òîæåÿâëÿåòñÿôóíêöèåé,çíà÷èòîòíå¼îïÿòü
ìîæíîâçÿòüïðîèçâîäíóþ.Ýòîíàçûâàåòñÿ
âòîðàÿïðîèçâîäíàÿ
èîáî-
çíà÷àåòñÿ
f
00
(
x
)
èëè
f
(2)
.Àíàëîãè÷íîîïðåäåëÿåòñÿòðåòüÿ(
f
000
(
x
)=
f
(3)
),
÷åòâåðòàÿ(
f
0000
(
x
)=
f
(4)
),...,
n
ÿïðîèçâîäíàÿ(
f
(
n
)
).Ïîäðàçóìåâàåòñÿ,
÷òî
f
èìååò
n

1
þïðîèçâîäíóþâíåêîòîðîéîêðåñòíîñòèòî÷êè
x

ýòà
n

1
ÿïðîèçâîäíàÿäèôôåðåíöèðóåìàâòî÷êå
x
.
Îïðåäåëåíèå85.
Ìíîæåñòâîôóíêöèé,èìåþùèõ
n
þïðîèçâîäíóþâî
âñåõòî÷êàõîòêðûòîãîìíîæåñòâà
M
îáîçíà÷àåòñÿ
D
n
(
M
)
.
Îïðåäåëåíèå86.
Ìíîæåñòâîôóíêöèé,èç
D
n
(
M
)
,èìåþùèõíåïðåðûâ-
íóþ
n
þïðîèçâîäíóþâîâñåõòî÷êàõîòêðûòîãîìíîæåñòâà
M
îáîçíà-
÷àåòñÿ
C
n
(
M
)
.
Çàìå÷àíèå
Êîãäàìíîæåñòâî
M
íåÿâëÿåòñÿîòêðûòûì,íàïðèìåð,â
ñëó÷àå
M
=[
a;b
]
óäîáíîèñïîëüçîâàòüòàêèåæåîáîçíà÷åíèÿ
D
n
[
a;b
]
è
C
n
[
a;b
]
,ïîíèìàÿïîäíåïðåðûâíîñòüþ/äèôôåðåíöèðóåìîñòüþíåïðåðûâ-
íîñòü/äèôôåðåíöèðóåìîñòüñëåâàèëèñïðàâàâñîîòâåñòâóþùèõêîíöàõ
îòðåçêà.
113
114
ÃËÀÂÀ12.ÊÐÀÒÍÛÅÏÐÎÈÇÂÎÄÍÛÅ.ÔÎÐÌÓËÀËÅÉÁÍÈÖÀ.ÂÛÏÓÊËÎÑÒÜ
Äëÿíåêîòîðûõôóíêöèéìîæíîâûïèñàòüÿâíûåçíà÷åíèÿ
n
éïðîèç-
âîäíîé:

(
x

)
(
n
)
=


(


1)

:::

(


n
+1)

x


n
.

(
a
x
)
(
n
)
=(ln
a
)
n

a
x
.

log
a
x
(
n
)
=
(

1)
n
+1
(
n

1)!
ln
n
a

x
n
.

sin
(
n
)
x
=sin

x
+
n
2

=
(
(

1)
k

sin
x;
ïðè
n
=2
k
;
(

1)
k

cos
x;
ïðè
n
=2
k
+1
:

cos
(
n
)
x
=cos

x
+
n
2

=
(
(

1)
k

cos
x;
ïðè
n
=2
k
;
(

1)
k
+1

sin
x;
ïðè
n
=2
k
+1
:
Î÷åâèäíî,êðàòíàÿïðîèçâîäíàÿîòñóììûèëèðàçíîñòèôóíêöèéáó-
äåòñóììîéèëèðàçíîñòüþñîîòâåòñòâóþùèõêðàòíûõïðîèçâîäíûõ.Äëÿ
íàõîæäåíèÿêðàòíûõïðîèçâîäíûõïðîèçâåäåíèÿñóùåñòâóåòñëåäóþùàÿ
ôîðìóëà:
Òåîðåìà90
(ÔîðìóëàËåéáíèöà)
.
Ïóñòü
f
(
x
)
;g
(
x
)
2
D
n
(
M
)
,òîãäà
(
f
(
x
)

g
(
x
))
(
n
)
=
n
X
k
=0
C
k
n
f
(
k
)
(
x
)
g
(
n

k
)
(
x
)
;
ãäåïîäðàçóìåâàåòñÿ,÷òî
f
(0)
(
x
)=
f
(
x
)
,
g
(0)
(
x
)=
g
(
x
)
.
Äîêàçàòåëüñòâî.
Áóäåìâåñòèèíäóêöèåéïî
n
.

Áàçàèíäóêöèè:Äëÿ
n
=1
ïîëó÷àåì
(
f
(
x
)
g
(
x
))
0
=
f
(
x
)
g
0
(
x
)+
f
0
(
x
)
g
(
x
)
ïðàâèëîäèôôåðåíöèðîâàíèÿïðîèçâåäåíèÿ.

Øàãèíäóêöèè:Ïîïðåäïîëîæåíèþèíäóêöèè
(
f
(
x
)

g
(
x
))
(
n

1)
=
n

1
P
k
=0
C
k
n

1
f
(
k
)
(
x
)
g
(
n

k

1)
(
x
)
.
Âîçüìåìïðîèçâîäíóþîòîáåèõ÷àñòåé:
(
f
(
x
)

g
(
x
))
(
n
)
=
n

1
X
k
=0
C
k
n

1

f
(
k
+1)
(
x
)
g
(
n

k

1)
(
x
)+
f
(
k
)
(
x
)
g
(
n

k
)
(
x
)

;
îòêóäàèìååì:
(
f
(
x
)

g
(
x
))
(
n
)
=
n

1
X
k
=0
C
k
n

1
f
(
k
)
(
x
)
g
(
n

k
)
(
x
)+
n
X
k
=1
C
k

1
n

1
f
(
k
)
(
x
)
g
(
n

k
)
(
x
)
;
12.2.ÂÛÏÓÊËÎÑÒÜÃÐÀÔÈÊÀÔÓÍÊÖÈÈ.ÒÎ×ÊÈÏÅÐÅÃÈÁÀ.ÍÅÐÀÂÅÍÑÒÂÎÉÅÍÑÅÍÀ.
115
ñëåäîâàòåëüíî
(
f
(
x
)

g
(
x
))
(
n

1)
=
n
X
k
=0

C
k
n

1
+
C
k

1
n

1

f
(
k
)
(
x
)
g
(
n

k
)
(
x
)=
n
X
k
=0
C
k
n
f
(
k
)
(
x
)
g
(
n

k
)
(
x
)
;
12.2Âûïóêëîñòüãðàôèêàôóíêöèè.Òî÷êèïå-
ðåãèáà.ÍåðàâåíñòâîÉåíñåíà.
Îäíèìèçïðèìåíåíèéêðàòíûõ(òî÷íååâòîðûõ)ïðîèçâîäíûõÿâëÿåòñÿèñ-
ñëåäîâàíèåôóíêöèéíàâûïóêëîñòü.
Îïðåäåëåíèå87.
Ôóíêöèÿ
f
(
x
)
,îïðåäåë¼ííàÿíàïðîìåæóòêå(îòðåç-
êå,èíòåðâàëå,ëó÷å,âñåéïðÿìîé)èïðèíèìàþùàÿäåéñòâèòåëüíûåçíà-
÷åíèÿ,íàçûâàåòñÿ
âûïóêëîéâíèç
,åñëèå¼ãðàôèêíàëþáîìîòðåçêå
[
a;b
]

D
f
,ëåæèòíåâûøåõîðäû
(
a;f
(
a
))

(
b;f
(
b
))
.Àíàëîãè÷íîîïðå-
äåëÿåòñÿïîíÿòèåôóíêöèè
âûïóêëîéââåðõ
ýòîôóíêöèÿ,ãðàôèê
êîòîðîéëåæèòíåíèæåõîðäû.
x
y
A
B
Ðèñ.12.1:Ôóíêöèÿâûïóêëàâíèç
fig:vypukl_vniz
jensenpq
Ëåììà27.
Ïóñòü
f
(
x
)
âûïóêëàâíèçíà
[
a;b
]
.Òîãäàäëÿëþáûõ
p;q

0
,
òàêèõ,÷òî
p
+
q
=1
âûïîëíåíîíåðàâåíñòâî:
f
(
pa
+
qb
)
6
pf
(
a
)+
qf
(
b
)
:
116
ÃËÀÂÀ12.ÊÐÀÒÍÛÅÏÐÎÈÇÂÎÄÍÛÅ.ÔÎÐÌÓËÀËÅÉÁÍÈÖÀ.ÂÛÏÓÊËÎÑÒÜ
x
y
A
B
Ðèñ.12.2:Ôóíêöèÿâûïóêëàââåðõ
fig:vypukl_vverh
Äîêàçàòåëüñòâî.
Ðàññìîòðèìòî÷êè
A
(
a;f
(
a
))
è
B
(
b;f
(
b
))
ãðàôèêàôóíêöèè
y
=
f
(
x
)
.
Î÷åâèäíî,òî÷êà
C
(
pa
+
qb;pf
(
a
)+
qf
(
b
))
ëåæèòíàõîðäå
[
AB
]
,ñëåäîâàòåëü-
íî,ãðàôèêëåæèòíèæåýòîéòî÷êè,
àçíà÷èò
f
(
pa
+
qb
)
6
pf
(
a
)+
qf
(
b
)
;
Ëåììà28.
Ïóñòü
f
(
x
)
âûïóêëà
ââåðõíà
[
a;b
]
.Òîãäàäëÿëþáûõ
p;q

0
,òàêèõ,÷òî
p
+
q
=1
âû-
ïîëíåíîíåðàâåíñòâî:
f
(
pa
+
qb
)

pf
(
a
)+
qf
(
b
)
:
Äîêàçàòåëüñòâî.
Àíàëîãè÷íî.Ïðåäîñòàâëÿåòñÿ÷è-
òàòåëþâêà÷åñòâåóïðàæíåíèÿ.
seredka
Ëåììà29.
Ïóñòü
f
2C
[
a;b
]
èäëÿëþáûõ
a
1
;b
1
2
[
a;b
]
âûïîëíåíîíåðà-
âåíñòâî
f

a
1
+
b
1
2

6
f
(
a
1
)+
f
(
b
1
)
2
:
Òîãäà
f
âûïóêëàâíèçíàîòðåçêå
[
a;b
]
.
Çàìå÷àíèå:
Òàêèìîáðàçîìäëÿïðîâåðêèâûïóêëîñòèíåïðåðûâíîé
ôóíêöèèäîñòàòî÷íîïðîâåðÿòüíåðàâåíñòâîòîëüêîâñåðåäèíåîòðåçêà.
Äëÿðàçðûâíûõôóíêöèéýòî,âîîáùåãîâîðÿ,íåâåðíî,íàïðèìåð,ìîæíî
ïîñìîòðåòüôóíêöèþÄèðèõëå.
Äîêàçàòåëüñòâî.
Âûáåðåìïðîèçâîëüíûå
a
1
;b
1
2
[
a;b
]
èïîêàæåì,÷òîãðàôèêôóíêöèè
y
=
f
(
x
)
ëåæèòíåâûøåõîðäû
[
AB
]
ãäå
A
(
a
1
;f
(
a
1
))
è
B
(
b
1
;f
(
b
1
))
.Äëÿ
óäîáñòâàâûïèøåìóðàâíåíèåïðÿìîé,ïðîõîäÿùåé÷åðåçòî÷êè
A
è
B
:
l
(
x
)=
f
(
a
1
)+
f
(
b
1
)

f
(
a
1
)
b
1

a
1

(
y

a
1
)
:
Âûáåðåìïðîèçâîëüíîå

2
[
a
1
;b
1
]
.Áóäåìîïÿòüñòðîèòüïîñëåäîâàòåëü-
íîñòüñòÿãèâàþùèõñÿîòðåçêîâ:Ïóñòü
c
1
=
a
1
+
b
1
2
.Åñëè


c
1
,òîâûáåðåì
a
2
=
c
1
,
b
2
=
b
1
,âïðîòèâíîìñëó÷àåâûáåðåì
a
2
=
a
1
b
2
=
c
1
.Ïîòîìáåðåì
c
2
=
a
2
+
b
2
2
,è.ò.ä.
12.2.ÂÛÏÓÊËÎÑÒÜÃÐÀÔÈÊÀÔÓÍÊÖÈÈ.ÒÎ×ÊÈÏÅÐÅÃÈÁÀ.ÍÅÐÀÂÅÍÑÒÂÎÉÅÍÑÅÍÀ.
117
Äîêàæåì÷òî
f
(
c
n
)
6
l
(
c
n
)
ìåòîäîììàòåìàòè÷åñêîéèíäóêöèè.Áà-
çàâûòåêàåòèçóñëîâèÿòåîðåìû.Øàãèíäóêöèè.Ïóñòü
f
(
a
n
)
6
l
(
a
n
)
è
f
(
b
n
)
6
l
(
b
n
)
,ñëîæèâýòèíåðàâåíñòâàïîëó÷èì
f

a
n
+
b
n
2

6
f
(
a
n
)+
f
(
b
n
)
2
6
l
(
a
n
)+
l
(
b
n
)
2
=
l
(
c
n
)
:
Òàêèìîáðàçîì
f
(
c
n
)
6
l
(
c
n
)
.
Ïîñëåäîâàòåëüíîñòüîòðåçêîâ
[
a
n
;b
n
]
ñòÿãèâàåòñÿêòî÷êå

,ñëåäîâà-
òåëüíî,âçÿâïðåäåëïðè
n
!1
ïîëó÷èì
lim
n
!1
c
n
=

.Èçíåïðåðûâíîñòè
f
(
x
)
è
l
(
x
)
âûòåêàåò
f
(

)
6
l
(

)
,
Ëåììà30.
Ïóñòü
f
2C
[
a;b
]
èäëÿëþáûõ
a
1
;b
1
2
[
a;b
]
âûïîëíåíîíåðà-
âåíñòâî
f
(
a
1
+
b
1
2
)

f
(
a
1
)+
f
(
b
1
)
2
.Òîãäà
f
âûïóêëàââåðõíàîòðåçêå
[
a;b
]
.
Äîêàçàòåëüñòâî.
Àíàëîãè÷íî.Ïðåäîñòàâëÿåòñÿâêà÷åñòâåóïðàæíåíèÿ.
point3
Ëåììà31.
Ïóñòü
f
(
x
)
âûïóêëàâíèçíà
[
a;b
]
.Òîãäàëþáàÿïðÿìàÿïåðå-
ñåêàåòãðàôèê
y
=
f
(
X
)
ëèáîâäâóõòî÷êàõ,ëèáîïîíåêîòîðîìóîòðåçêó.
Äîêàçàòåëüñòâî.
Ïðåäïîëîæèìîáðàòíîå,÷òîíåêîòîðàÿïðÿìàÿ
l
ïåðåñåêàåòãðàôèêâòðåõ
òî÷êàõ
A
,
B
è
C
,ïðè÷åìíàîòðåçêå
[
AC
]
åñòüòî÷êà,íåëåæàùàÿíàãðàôè-
êåôóíêöèè.Íåîãðàíè÷èâàÿîáùíîñòè,ìîæíîñ÷èòàòü,÷òîîíàëåæèòíà
[
AB
]
.Íîòîãäàòî÷êà
D
(
x
0
;f
(
x
0
))
äîëæíàáûòüðàñïîëîæåíàíèæåõîðäû
AC
(èçâûïóêëîñòè).
Cäðóãîé
[
AB
]
ñòîðîíû,òî÷êà
B
äîëæíàëåæàòüíèæåõîðäû
[
AD
]
.
Ïðîòèâîðå÷èå.
Òåîðåìà91.
Åñëèïðèíåêîòîðîì
"�
0
ôóíêöèÿ
f
(
x
)
âûïóêëàâîêðåñò-
íîñòè
U
"
(
x
0
)
,òîîíàíåïðåðûâíàâòî÷êå
x
0
.
Äîêàçàòåëüñòâî.
Íåîãðàíè÷èâàÿîáùíîñòèðàññóæäåíèéáóäåìñ÷èòàòü,÷òî
f
âûïóêëà
âíèç.Äîêàæåìñíà÷àëà,÷òîôóíêöèÿíåïðåðûâíàñïðàâàâòî÷êå
x
0
.
Ðàññìîòðèìïðîèçâîëüíóþòî÷êó
x
2
(
x
0
;x
0
+
"
2
)
èîáîçíà÷èì

x
=
x

x
0
,

f
=
f
(
x
)

f
(
x
0
)
.Ïóñòü
x
1
=
x
0
+
"
2
,
x
2
=
x
0

"
2
,ðàññìîòðèìòî÷êè,ðàñ-
ïîëîæåííûåíàãðàôèêåôóíêöèè:
X
(
x;f
(
x
))
,
X
0
(
x
0
;f
(
x
0
))
,
X
1
(
x
1
;f
(
x
1
))
è
X
2
(
x
2
;f
(
x
2
))
.
Èçâûïóêëîñòèñëåäóåò,÷òîòî÷êà
X
0
ëåæèòíèæåõîðäû
XX
1
,ñëåäî-
âàòåëüíî,òî÷êà
X
ëåæèòâûøåïðÿìîé
X
0
X
1
(ñì.Ðèñ.
vipuklill
12.2).
118
ÃËÀÂÀ12.ÊÐÀÒÍÛÅÏÐÎÈÇÂÎÄÍÛÅ.ÔÎÐÌÓËÀËÅÉÁÍÈÖÀ.ÂÛÏÓÊËÎÑÒÜ
x
y
O
A
C
B
D
Ðèñ.12.3:Åñëèïðÿìàÿïåðåñåêàåòãðàôèêâ3òî÷êàõ,òîîííåìîæåòáûòü
âûïóêëûì.
point3vyp
x
y
O
X
1
X
2
X
0
X
k
1
k
2
vipuklill
Ðèñ.12.4:Ïðîèçâîëüíàÿòî÷êàíàäóãå
X
0
^X
2
ëåæèòâûøåïðÿìîé
X
1
X
0
.
Cäðóãîéñòîðîíû,òî÷êà
X
ëåæèòíèæåõîðäû
X
0
X
2
(òîæåèçâûïóê-
ëîñòè).Îáîçíà÷èì÷åðåç
k
1
è
k
2
óãëîâûåêîýôôèöèåíòûïðÿìûõ
X
0
X
1
è
X
0
X
2
,ñîîòâåòñòâåííî.Òîãäà,î÷åâèäíî,
k
1

x

fk
2

x:
Óñòðåìëÿÿ

x
!
+0
,ïîëó÷èì(ïîëåììåîäâóõìèëèöèîíåðàõ),÷òî

f
!
0
,
ñëåäîâàòåëüíî,
f
(
x
)
!
f
(
x
0
)
,ò.å.
f
(
x
)
íåïðåðûâíàñïðàâàâòî÷êå
x
0
.
Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òî
f
(
x
)
íåïðåðûâíàñëåâà.Òàêèìîáðàçîì,
f
(
x
)
íåïðåðûâíàâòî÷êå
x
0
.
Òåîðåìà92.
Åñëèïðèíåêîòîðîì
"�
0
ôóíêöèÿ
f
(
x
)
âûïóêëàâîêðåñò-
íîñòè
U
"
(
x
0
)
,òîîíàîáëàäàåòïðàâîéèëåâîéïðîèçâîäíûìèâòî÷êå
x
0
1
.
1
Èçâûïóêëîñòè,âîîáùåãîâîðÿíåñëåäóåòäèôôåðåíöèðóåìîñòü.Òàê,íàïðèìåð,
12.2.ÂÛÏÓÊËÎÑÒÜÃÐÀÔÈÊÀÔÓÍÊÖÈÈ.ÒÎ×ÊÈÏÅÐÅÃÈÁÀ.ÍÅÐÀÂÅÍÑÒÂÎÉÅÍÑÅÍÀ.
119
Äîêàçàòåëüñòâî.
Ðàññìîòðèìïîñëåäîâàòåëüíîñòü
f
"
n
g
1
n
=1
,
"
n
#
+0
.Ïîñòðîèìäëÿêàæäî-
ãî
n
ïðÿìûå
X
0
(
n
)
X
1
(
n
)
è
X
0
(
n
)
X
2
(
n
)
êàêâäîêàçàòåëüñòâåïðåäûäóùåé
òåîðåìû.Íåñëîæíîçàìåòèòü,÷òî,ïîñêîëüêóòî÷êà
X
2
(
n
+1)
ëåæèòíè-
æåõîðäû
X
0
X
2
(
n
)
,òîïîñëåäîâàòåëüíîñòüóãëîâûõêîýôôèöèåíòîâ
k
2
(
n
)
ÿâëÿåòñÿóáûâàþùåé.Ñäðóãîéñòîðîíûîíàîãðàíè÷åíàñíèçóâåëè÷èíîé
k
1
(1)
,ñëåäîâàòåëüíî,ñóùåñòâóåòïðåäåë
lim
n
!1
k
2
(
n
)=
K
.Ýòàâåëè÷èíà,
î÷åâèäíî,ÿâëÿåòñÿïðàâîéïðîèçâîäíîéâòî÷êå
x
0
.
Àíàëîãè÷íîäîêàçûâàåòñÿñóùåñòâîâàíèåëåâîéïðîèçâîäíîéâòî÷êå
x
0
.
Òåîðåìà93.
Ïóñòü
f
(
x
)
2D
2
[
a;b
]
èäëÿëþáîãî
x
2
[
a;b
]
âûïîëíåíî
íåðàâåíñòâî
f
00
(
x
)

0
.Òîãäà
f
(
x
)
ÿâëÿåòñÿâûïóêëîéâíèçíàîòðåçêå
[
a;b
]
.
Äîêàçàòåëüñòâî.
Âûáåðåìïðîèçâîëüíûå
[
a
1
;b
1
]

[
a;b
]
,ïóñòü
c
1
ñåðåäèíàîòðåçêà
[
a
1
;b
1
]
,
d
1
=
b
!

a
1
åãîäëèíà.Ðàññìîòðèìðàçíîñòü
D
=
f
(
a
1
)+
f
(
b
1
)

2
f
(
c
1
)=

f
(
b
1
)

f
(
c
1
)



f
(
c
1
)

f
(
a
1
)

.
ÏðåîáðàçîâûâàÿèïðèìåíÿÿòåîðåìóËàãðàíæà,ïîëó÷èì:
D
=

f
(
b
1
)

f
(
c
1
)



f
(
c
1

f
(
a
1
)

=
f
0
(

2
)(
b
1

c
1
)

f
0
(

1
)(
c
1

a
1
)=
d
2


f
0
(

2
)

f
0
(

1
)

;
ãäåè

1
2
[
a
1
;c
1
]


2
2
[
c
1
;b
1
]
.ÏðèìåíèâòåîðåìóòåîðåìóËàãðàíæà
åù¼ðàç,ïîëó÷èì
D
=
f
00
(

)

(

2


1
)

d
1
2

0
,ïîñêîëüêó

1
6

2
.Òàêèì
îáðàçîì,
f
(
a
1
)+
f
(
b
1
)

2
f
(
a
1
+
b
1
2
)
,ñëåäîâàòåëüíîïîëåììå
seredka
29ãðàôèê
ôóíêöèèÿâëÿåòñÿâûïóêëûìâíèç,
Òåîðåìà94.
Ïóñòü
f
(
x
)
2D
2
[
a;b
]
èäëÿëþáîãî
x
2
[
a;b
]
âûïîëíåíî
íåðàâåíñòâî
f
00
(
x
)
6
0
.Òîãäàãðàôèê
f
(
x
)
ÿâëÿåòñÿâûïóêëûìââåðõ.
Äîêàçàòåëüñòâî.
Óïðàæíåíèå.
Òåîðåìà95.
Ïóñòü
f
(
x
)
2D
2
[
a;b
]
,
f
00
(
x
)
2C
[
a;b
]
èãðàôèê
f
(
x
)
ÿâëÿåòñÿ
âûïóêëûìâíèç.Òîãäàäëÿëþáîãî
x
2
[
a;b
]
f
00
(
x
)

0
.
Äîêàçàòåëüñòâî.
Ïðåäïîëîæèìîáðàòíîå.Ïóñòüñóùåñòâóåò
x
0
2
[
a;b
]
,òàêàÿ,÷òî
f
00
(
x
0
)

0
.
Òîãäà,èçíåïðåðûâíîñòèâòîðîéïðîèçâîäíîéâûòåêàåò,ýòîíåðàâåíñòâî
âûïîëíåíîäëÿâñåõ
x
èçíåêîòîðîé
"
îêðåñòíîñòèòî÷êè
x
0
.Âûáåðåì
a
1
;b
1
2
U
"
(
x
0
)
,
òàê,÷òî
a
1
b
1
,ïóñòü
c
1
=
a
1
+
b
1
2
ñåðåäèíàîòðåçêà
[
a
1
;b
1
]
,
d
1
=
b
1

a
1
åãîäëèíà.
f
(
x
)=
j
x
j
,î÷åâèäíî,âûïóêëàâíèç,íîïðîèçâîäíîéâòî÷êå
x
0
=0
íåñóùåñòâóåò.
120
ÃËÀÂÀ12.ÊÐÀÒÍÛÅÏÐÎÈÇÂÎÄÍÛÅ.ÔÎÐÌÓËÀËÅÉÁÍÈÖÀ.ÂÛÏÓÊËÎÑÒÜ
Èçëåììû
jensenpq
27(c
p
=
q
=
1
2
)âûòåêàåò,÷òîðàçíîñòü
D
=
f
(
a
1
)+
f
(
b
1
)

2
f
(
1
)

0
:
ÏðåîáðàçîâûâàÿèïðèìåíÿÿòåîðåìóËàãðàíæà,ïîëó÷èì:
D
=

f
(
b
1
)

f
(
c
1
)



f
(
c
1

f
(
a
1
)

=
f
0
(

2
)(
b
1

c
1
)

f
0
(

1
)(
c
1

a
1
)=
d
2


f
0
(

2
)

f
0
(

1
)

;
ãäåè

1
2
[
a
1
;c
1
]


2
2
[
c
1
;b
1
]
.ÏðèìåíèâòåîðåìóòåîðåìóËàãðàíæàåù¼
ðàç,ïîëó÷èì
D
=
f
00
(

)

(

2


1
)

d
1
2

0
,ïîñêîëüêó

1


2
.Òàêèìîáðàçîì,
f
(
a
1
)+
f
(
b
1
)
2

f
(
c
1
)
,ñëåäîâàòåëüíî,ïîëåììå
seredka
29ãðàôèêôóíêöèèÿâëÿåòñÿ
âûïóêëûìâíèç,
Òåîðåìà96.
Ïóñòü
f
(
x
)
2D
2
[
a;b
]
,
f
00
(
x
)
2C
[
a;b
]
èãðàôèê
f
(
x
)
ÿâëÿåòñÿ
âûïóêëûìââåðõ.Òîãäàäëÿëþáîãî
x
2
[
a;b
]
f
00
(
x
)
6
0
.
Äîêàçàòåëüñòâî.
Ïðåäîñòàâëÿåòñÿ÷èòàòåëþâêà÷åñòâåóïðàæíåíèÿ.
Ýòèòåîðåìûïîêàçûâàþò,÷òîäëÿäâàæäûíåïðåðûâíîäèôôåðåíöèðó-
åìûõôóíêöèéâûïóêëîñòüãðàôèêàîäíîçíà÷íîñâÿçàíàñîçíàêîìâòîðîé
ïðîèçâîäíîé.Òàêèìîáðàçîìçíàÿçíàêôóíêöèè,ååïåðâîéèâòîðîéïðîèç-
âîäíîé,ìîæíîñóäèòüîôîðìåãðàôèêàôóíêöèèíàäàííîìïðîìåæóòêå.
Òåîðåìà97
(ÍåðàâåíñòâîÉåíñåíà)
.
Ïóñòü
f
(
x
)
2D
2
[
a;b
]
âûïóêëàâíèç
íà
[
a;b
]
Ïóñòüâûáðàíûòî÷êè
x
1
;x
2
;:::;x
n
2
[
a;b
]
è÷èñëà
m
1
;m
2
;:::;m
n

0
òàêèå,÷òî
m
1
+
m
2
+
:::
+
m
n
=1
.Òîãäà
m
1

f
(
x
1
)+
m
2

f
(
x
2
)+
:::
+
m
n

f
(
x
n
)

f
(
m
1
x
1
+
m
2
x
2
+
:::
+
m
n
x
n
)
:
Çàìå÷àíèå1.
Íåñëîæíîçàìåòèòü,÷òîâñëó÷àå
n
=2
òåîðåìàïðîñòî
ïðåâðàùàåòñÿâëåììó
jensenpq
27.Òàêèìîáðàçîì,íåðàâåíñòâîÉåíñåíàÿâëÿåòñÿ
îáîáùåíèåìýòîéëåììû.
Çàìå÷àíèå2.
Òå,êòîóâëåêàåòñÿôèçèêîé,ëåãêîïîéìóòôèçè÷åñêèé
ñìûñëíåðàâåíñòâàÉåíñåíà.Îíîçíà÷àåò,÷òîåñëèâçÿòü
n
òî÷åêñìàññàìè
m
1
,...
m
n
íàãðàôèêåôóíêöèè
y
=
f
(
x
)
,òîöåíòðìàññýòîéñèñòåìûòî÷åê
áóäåòëåæàòüâûøåãðàôèêà.Ñòî÷êèçðåíèÿôèçèêèýòîî÷åâèäíî,àñ
òî÷êèçðåíèÿìàòåìàòèêèïðèäåòñÿäîêàçûâàòü.
Äîêàçàòåëüñòâî.
Áóäåìâåñòèèíäóêöèåéïî
n
.

Áàçàèíäóêöèè.Ïðè
n
=2
ëåììà
jensenpq
27.
12.2.ÂÛÏÓÊËÎÑÒÜÃÐÀÔÈÊÀÔÓÍÊÖÈÈ.ÒÎ×ÊÈÏÅÐÅÃÈÁÀ.ÍÅÐÀÂÅÍÑÒÂÎÉÅÍÑÅÍÀ.
121
x
y
O
X
1
(
m
1
)
X
n
(
m
n
)
X
2
(
m
2
)
:::
X
ö.ì.
m
1
f
(
x
1
)+
:::
+
m
n
f
(
x
n
)
m
1
x
1
+
:::
+
m
n
x
n
f
(
m
1
x
1
+
:::
+
m
n
x
n
)
Ðèñ.12.5:ÍåðàâåíñòâîÉåíñåíà:öåíòðìàññëåæèòâûøåãðàôèêà.

Øàãèíäóêöèè.Äîïóñòèì,÷òîòåîðåìàäîêàçàíàâñëó÷àå
n

1
èäîêà-
æåìäëÿ
n
.Ïóñòü

=
m
1
+
m
2
+
:::
+
m
n

1
è
m
0
i
=
m
i

,
i
=1

n

1
.Î÷å-
âèäíî,
m
0
1
+
:::
+
m
0
n

1
=1
.Îáîçíà÷èì
x
0
1
=
m
0
1
x
1
+
m
2
x
2
+
:::
+
m
0
n

1
x
n

1
.
Òîãäà,ïîïðåäïîëîæåíèþèíäóêöèè:
m
0
1

f
(
x
1
)+
m
0
2

f
(
x
2
)+
:::
+
m
0
n

1

f
(
x
n
)

f
(
x
0
1
)
:
(12.2.1)
jens
Òàêêàê,

+
m
n
=1
,òîïîëåììå
jensenpq
27èìååì:
f
(
x
0
1
)+
m
n
f
(
x
n
)

f
(
x
0
1
+
m
n
x
n
)=
f
(
m
1
x
1
+
m
2
x
+2+
:::
+
m
n
x
n
)
:
Ïîäñòàâèâýòîâíåðàâåíñòâî
jens
12.2.1,ïîëó÷èì:


m
0
1

f
(
x
1
)+
m
0
2

f
(
x
2
)+
:::
+
m
0
n

1

f
(
x
n
)

+
m
n
f
(
x
n
)

f
(
m
1
x
1
+
m
2
x
2
+
:::
+
m
n
x
n
)
:
Ïîñòàâëÿÿ
m
0
i
,
i
=1

n

1
ïîëó÷èì,
m
1

f
(
x
1
)+
m
2

f
(
x
2
)+
:::
+
m
n

f
(
x
n
)

f
(
m
1
x
1
+
m
2
x
2
+
:::
+
m
n
x
n
)
;
Òåîðåìà98
(ÍåðàâåíñòâîÉåíñåíàäëÿâûïóêëîéââåðõôóíêöèè)
.
Ïóñòü
f
(
x
)
2D
2
[
a;b
]
âûïóêëàââåðõíà
[
a;b
]
Ïóñòüâûáðàíûòî÷êè
x
1
;x
2
;:::;x
n
2
[
a;b
]
è÷èñëà
m
1
;m
2
;:::;m
n

0
òàêèå,÷òî
m
1
+
m
2
+
:::
+
m
n
=1
.Òîãäà
m
1

f
(
x
1
)+
m
2

f
(
x
2
)+
:::
+
m
n

f
(
x
n
)
6
f
(
m
1
x
1
+
m
2
x
2
+
:::
+
m
n
x
n
)
:
122
ÃËÀÂÀ12.ÊÐÀÒÍÛÅÏÐÎÈÇÂÎÄÍÛÅ.ÔÎÐÌÓËÀËÅÉÁÍÈÖÀ.ÂÛÏÓÊËÎÑÒÜ
Ñëåäñòâèå23
(ÍåðàâåíñòâîÊîøè)
.
Ñðåäíååãåîìåòðè÷åñêîåíåñêîëüêèõ
ïîëîæèòåëüíûõ÷èñåëíåïðåâîñõîäèòèõñðåäíåãîãåîìåòðè÷åñêîãî.
Äîêàçàòåëüñòâî.
Âûáåðåìâêà÷åñòâå
f
(
x
)=ln
x
.Ïîñêîëüêó
ln
00
x
=

1
x
2

0
,òîãðàôèê
y
=ln
x
áóäåòâûïóêëûìââåðõ.ÏðèìåíèìíåðàâåíñòâîÉåíñåíàñ
m
1
=
m
2
=
:::
=
m
n
=
1
n
,
ïîëó÷èì
1
n
(ln
x
1
+ln
x
2
+
:::
+ln
x
n
)
6
ln

1
n
(
x
1
+
x
2
+
:::
+
x
n
)

:
Ïðåîáðàçóåì
1
n
(ln
x
1
+ln
x
2
+
:::
+ln
x
n
)=ln(
n
p
x
1

:::

x
n
)
èâîñïîëüçóåìñÿ
òåì,÷òî
ln
x
åñòüìîíîòîííîâîçðàñòàþùàÿôóíêöèÿ:
n
p
x
1

:::

x
n
6
1
n
(
x
1
+
x
2
+
:::
+
x
n
)
;
Óïðàæíåíèå48.
Äîêàçàòü,÷òî
(
x
1
+
x
2
+
:::
+
x
n
)
2
6
n
(
x
2
1
+
:::
+
x
2
n
)
Ñëåäñòâèå24
(ÍåðàâåíñòâîÊîøèÁóíÿêîâñêîãî)
.
Äëÿëþáûõ÷èñåë
a
1
,...,
a
n
,
b
1
,...,
b
n
âûïîëíåíîíåðàâåíñòâî
(
a
2
1
+
a
2
2
+
:::
+
a
2
n
)(
b
2
1
+
b
2
2
+
:::
+
b
2
n
)

(
a
1
b
1
+
a
2
b
2
+
:::
+
a
n
b
n
)
2
Äîêàçàòåëüñòâî.
Äîêàæåìñíà÷àëàíåðàâåíñòâîäëÿñëó÷àÿêîãäà
a
1
;:::;a
n
;b
1
;:::;b
n

0
Ôóíêöèÿ
f
(
x
)=
x
2
ÿâëÿåòñÿâûïóêëîéâíèç.Îáîçíà÷èì
M
=
b
2
1
+
:::
+
b
2
n
âûáåðåì
m
i
=
b
2
i
M
è
x
i
=
a
i
b
i
èïðèìåíèìíåðàâåíñòâîÉåíñåíà.Ïîëó÷èì:
m
1

x
2
1
+
m
2

x
2
2
+
:::
+
m
n

x
2
n

(
m
1
x
1
+
m
2
x
2
+
:::
+
m
n
x
n
)
2
:
Ïîëüçóÿñüòåì,÷òî
m
i
x
2
i
=
a
2
i
M
è
m
i
x
i
=
a
i
b
i
M
ïðåîáðàçóåìíåðàâåíñòâîê
âèäó
a
2
1
+
a
2
2
+
:::
+
a
2
n
M


a
1
b
1
+
a
2
b
2
+
:::
+
a
n
b
n
M

2
:
Çàòåìóìíîæèìîáå÷àñòèíà
M
2
èïîäñòàâëÿÿ
M
=
b
2
1
+
:::
+
b
2
n
,ïîëó÷èì
(
a
2
1
+
a
2
2
+
:::
+
a
2
n
)(
b
2
1
+
b
2
2
+
:::
+
b
2
n
)

(
a
1
b
1
+
a
2
b
2
+
:::
+
a
n
b
n
)
2
:
Äëÿòîãî,÷òîáûäîêàçàòüíåðàâåíñòâîäëÿïðîèçâîëüíûõ
a
1
;:::;a
n
;b
1
;:::;b
n
ïîäñòàâèìèõìîäóëèâóêàçàííîåíåðàâåíñòâî,àçàòåìâîñïîëüçóåìñÿòåì,
÷òî
(
a
1
b
1
+
a
2
b
2
+
:::
+
a
n
b
n
)
2
6
(
j
a
1
jj
b
1
j
+
j
a
2
jj
b
2
j
+
:::
+
j
a
n
jj
b
n
j
)
2
:
12.3.ÊÀÑÀÍÈÅÊÐÈÂÛÕ.ÊÐÓÃÊÐÈÂÈÇÍÛ,ÝÂÎËÞÒÀÈÝÂÎËÜÂÅÍÒÀ.
123
kasanie2
x
y
y
=1
y
=cos
x
y
=1

x
2
2
Ðèñ.12.6:Ïðèìåðêàñàíèÿ1ãîè3ãîïîðÿäêà.
12.3Êàñàíèåêðèâûõ.Êðóãêðèâèçíû,ýâîëþ-
òàèýâîëüâåíòà.
Îïðåäåëåíèå88.
Ïóñòüêðèâûå
y
=
f
(
x
)
è
y
=
g
(
x
)
ïåðåñåêàþòñÿâ
íåêîòîðîéòî÷êå
x
=
x
0
èôóíêöèè
f;g
äèôôåðåíöèðóåìûâýòîéòî÷êå.
Òîãäà
óãëîììåæäóêðèâûìè
íàçûâàåòñÿóãîëìåæäóêàñàòåëüíûìè,
ïðîâåäåííûìèâòî÷êå
x
0
êýòèìêðèâûì,ò.å.
j
arctg(
f
0
(
x
0
))

arctg(
g
0
(
x
0
))
j
.
Îïðåäåëåíèå89.
Åñëèêðèâûåïåðåñåêàþòñÿâòî÷êå
x
0
ïîäóãëîì,ðàâ-
íûì0,òîãîâîðÿò,÷òîêðèâûå
êàñàþòñÿ
âýòîéòî÷êå.
Îïðåäåëåíèå90.
Ãîâîðÿò,÷òîêðèâûå
y
=
f
(
x
)
è
y
=
g
(
x
)
èìåþò
êàñàíèå
n

ãîïîðÿäêà
âòî÷êå
x
0
,åñëè
f
(
x
0
)=
g
(
x
0
)
,
f
0
(
x
0
)=
g
0
(
x
0
)
,
f
00
(
x
0
)=
g
00
(
x
0
)
,...
(
n
)
f
(
x
0
)=
g
(
n
)
(
x
0
)
,íî
f
(
n
+1)
(
x
0
)
6
=
g
(
n
+1)
(
x
0
)
.
Ïðèìåð.
Êðèâàÿ
y
=cos
x
èìååòâòî÷êå(0,1)êàñàíèå1ãîïîðÿäêàñ
ïðÿìîé
y
=1
èêàñàíèå3ãîïîðÿäêàñêðèâîé
y
=1

x
2
2
(ñì.ðèñ.
kasanie2
12.3).
Îïðåäåëåíèå91.
Îêðóæíîñòü
(
x

x
c
)
2
+(
y

y
c
)
2
=
R
2
,èìåþùàÿñ
äàííîéêðèâîé
y
=
f
(
x
)
êàñàíèåâòî÷êå
x
0
íåíèæå2ãîïîðÿäêàíàçûâà-
åòñÿ
êðóãêðèâèçíû
âýòîéòî÷êå.Òî÷êà
(
x
c
;y
c
)
íàçûâàåòñÿ
öåíòðîì
124
ÃËÀÂÀ12.ÊÐÀÒÍÛÅÏÐÎÈÇÂÎÄÍÛÅ.ÔÎÐÌÓËÀËÅÉÁÍÈÖÀ.ÂÛÏÓÊËÎÑÒÜ
êðèâèçíû
,
R
íàçûâàåòñÿ
ðàäèóñîìêðèâèçíû
,àâåëè÷èíà
k
=
1
R

êðèâèçíîé
.Ãåîìåòðè÷åñêîåìåñòîöåíòðîâêðèâèçíû
(
x
c
;y
c
)
íàçûâàåò-
ñÿ
ýâîëþòîé
äàííîéêðèâîé.
Òåîðåìà99.
Ïóñòü
f
2D
2
(
x
0
)
è
f
00
(
x
0
)
6
=0
.Òîãäàðàäèóñèöåíòðêðè-
âèçíûíàõîäÿòñÿïîôîðìóëàì(îáîçíà÷åíèÿ
y
0
=
f
0
(
x
0
)
,
y
00
=
f
00
(
x
0
)
,
c
=
p
1+
f
0
(
x
0
)
2
:

R
=
(
p
1+
y
0
2
)
3
j
y
00
j
;

x
c
=
x
0

(1+
y
0
2
)
y
0
y
00
;

y
c
=
y
0
+
1+
y
0
2
y
00
.
Äîêàçàòåëüñòâî.
Íåïîñðåäñòâåííîïðîâåðÿåòñÿðàâåíñòâî1éè2éïðîèçâîäíîé.
Çàïèøåìóðàâíåíèåîêðóæíîñòè
(
x

x
c
)
2
+(
y

y
c
)
2
=
R
2
èâûðàçèìèç
íåãî
y
=
f
(
x
)=
y
c

p
R
2

(
x

x
c
)
2
.Äëÿîïðåäåëåííîñòèâûáåðåìçíàê
¾+¿,äëÿ¾-¿äîêàçûâàåòñÿàíàëîãè÷íî.
Íàéäåìïðîèçâîäíûå:
f
0
(
x
)=

(
x

x
c
)
p
R
2

(
x

x
c
)
2
;
f
00
(
x
)=

p
R
2

(
x

x
c
)
2

(
x

x
c
)
2
p
R
2

(
x

x
c
)
2
R
2

(
x

x
c
)
2
:
Ââåäåìñëåäóþùèåîáîçíà÷åíèÿ:

x
=
x

x
c
,

y
=
y

y
c
=
p
R
2

(
x

x
c
)
2
,
èðåøèìîòíîñèòåëüíîíèõñëåäóþùöþñèñòåìóóðàâíåíèé:
8

:
y
0
=


x

y
;
y
00
=


y


2
x

y

2
y
:
Ýòàñèñòåìàïîëó÷åíàèçóñëîâèÿêàñàíèÿ2ãîïîðÿäêà,ò.å.ðàâåíñòâà
ïåðâîéèâòîðîéïðîèçâîäíûõ.Âûðàçèìèçïåðâîãîóðàâíåíèÿñèñòåìû

x
=

y
0

y
èïîäñòàâèìâîâòîðîåóðàâíåíèå:
y
00
=

1+
y
0
2

y
,îòêóäà

y
=

1+
y
0
2
y
00
:
12.3.ÊÀÑÀÍÈÅÊÐÈÂÛÕ.ÊÐÓÃÊÐÈÂÈÇÍÛ,ÝÂÎËÞÒÀÈÝÂÎËÜÂÅÍÒÀ.
125
kasanie2
x
y
O
(
x
c
;y
c
)
A
x
0

"
X
x
0
B
x
0
+
"
y
=
f
(
x
)
x
y
O
(
x
c
;y
c
)
A
x
0

"
X
x
0
B
x
0
+
"
y
=
f
(
x
)
Ðèñ.12.7:Êðóãêðèâèçíû.
126
ÃËÀÂÀ12.ÊÐÀÒÍÛÅÏÐÎÈÇÂÎÄÍÛÅ.ÔÎÐÌÓËÀËÅÉÁÍÈÖÀ.ÂÛÏÓÊËÎÑÒÜ
Òîãäà

x
=

y
0

y
=
y
0
(1+
y
0
2
)
y
00
:
Ñëåäîâàòåëüíî,
R
=
p

2
x
+
2
y
=
(
p
1+
y
0
2
)
3
j
y
00
j
:
Ïîäñòàâëÿÿâìåñòî

x
è

y
èõçíà÷åíèÿ,ïîëó÷àåìôîðìóëûäëÿöåíòðà
èðàäèóñàêðóãàêðèâèçíû.
Âîçìîæíàñëåäóþùàÿãåîìåòðè÷åñêàÿèíòåðïðåòàöèÿ:ðàññìîòðèìòî÷-
êó
X
(
x
0
;f
(
x
0
))
èáóäåòáðàòüòî÷êè
A
(
x
0

";f
(
x
0

"
))
è
B
(
x
0
+
";f
(
x
0
+
"
))
.
Òîãäàïðåäåëüíîåïîëîæåíèåêðóãà,îïèñàííîãîîêîëîòðåóãîëüíèêà
4
ABX
ïðè
"
!
0
êàêðàçáóäåòêðóãîìêðèâèçíûâóêàçàííîéòî÷êå.
Äëÿäîêàçàòåëüñòâàýòîãîôàêòàïîòðåáóåòñÿñëåäóþùàÿëåììà,êîòî-
ðàÿáóäåòäîêàçàíàâêîíöåñëåäóþùåéãëàâû
Ëåììà32.
Ïóñòü
f
00
(
x
0
)
ñóùåñòâóåò,òîãäà
lim
h
!
0
f
(
x
0
+
h
)+
f
(
x
0

h
)

2
f
(
x
0
)
h
2
=
f
00
(
x
0
)
:
Ãëàâà13
Ìíîãî÷ëåíÒåéëîðà
13.1Ìíîãî÷ëåíÒåéëîðà.ÔîðìóëàÒåéëîðà.
Îïðåäåëåíèå92.
Ïóñòü
f
2D
n
(
a
)
.Ìíîãî÷ëåí
P
n
(
x
)=
f
(
a
)+
f
0
(
a
)

(
x

a
)+
f
00
(
a
)
2!
(
x

a
)
2
+
:::
+
f
(
n
)
(
a
)
n
!
(
x

a
)
n
íàçûâàåòñÿ
ìíîãî÷ëåíîìÒåéëîðà
,àôóíêöèÿ
R
n
(
x
)=
f
(
x
)

P
n
(
x
)

åãî
îñòàòî÷íûì÷ëåíîì
.
Ìíîãî÷ëåíÒåéëîðàîáëàäàåòòåìñâîéñòâîì,÷òî
P
n
(
a
)=
f
(
a
)
,
P
0
n
(
a
)=
f
0
(
a
)
...
P
(
n
)
n
(
a
)=
f
(
n
)
(
a
)
.Ëîãè÷íîïðåäïîëîæèòü,÷òîâîêðåñòíîñòèòî÷êè
a
îí
¾ìàëîîòëè÷àåòñÿ¿îòèñõîäíîéôóíêöèè
f
(
x
)
.Ñëåäóþùàÿòåîðåìàïîêà-
çûâàåò,÷òîçíà÷èò¾ìàëîîòëè÷àåòñÿ¿.
Òåîðåìà100.
Ïóñòü
f
2D
n
(
a
)
è
f
(
n
)
2C
(
a
)
.Òîãäà
R
n
(
x
)=
o

(
x

a
)
n

(
x
!
a
)
,
ò.å.
f
(
x
)=
f
(
a
)+
f
0
(
a
)

(
x

a
)+
f
00
(
a
)
2!
(
x

a
)
2
+
:::
+
f
(
n
)
(
a
)
n
!
(
x

a
)
n
+
o

(
x

a
)
n

(
x
!
a
)
:
(13.1.1)
Peano
Ôîðìóëà
Peano
13.1.1íàçûâàåòñÿôîðìóëàÒåéëîðàñîñòàòî÷íûì÷ëåíîìâ
ôîðìåÏåàíî.
Äîêàçàòåëüñòâî.
127
128
ÃËÀÂÀ13.ÌÍÎÃÎ×ËÅÍÒÅÉËÎÐÀ
Ðàññìîòðèìîòíîøåíèå
R
n
(
x
)
(
x

a
)
n
èïðèìåíèì
n
ðàçïðàâèëîËîïèòàëÿ:
lim
x
!
a
R
n
(
x
)
(
x

a
)
n
=lim
x
!
a
f
(
n
)
(
x
)

f
(
n
)
(
a
)
n
!
=0
;
13.2ÐÿäÒåéëîðàäëÿýëåìåíòàðíûõôóíêöèé.
Âñåôóíêöèèðàññìàòðèâàþòñÿâðàçëîæåíèèïðè
a
=0
(ðÿäÌàêëîðåíà).
1.
Îäíàèçíàèáîëååïðîñòûõôîðìóëïîëó÷àåòñÿäëÿ
e
x
,ïîñêîëüêó
(
e
x
)
(
n
)
=
e
x
.Ïðè
a
=0
èìååì:
e
x
=1+
x
+
x
2
2!
+
x
3
3!
+
:::
+
x
n
n
!
+
o
(
x
n
)
.
2.
Àíàëîãè÷íîäëÿïîêàçàòåëüíîéôóíêöèè:
a
x
=1+
x
ln
a
+
x
2
2!
ln
2
a
+
:::
+
x
n
n
!

ln
n
a
+
o
(
x
n
)
.
3.
Ñòåïåííàÿôóíêöèÿ:
(1+
x
)

=1+


x
+

(


1)
2!

x
2
+
:::
+

(


1)

:::

(


n
+1)
n
!
+
o
(
x
n
)
.
Çàìåòèì,÷òîâñëó÷àåíàòóðàëüíîãî

çäåñüïðè
n


ïîëó÷àåòñÿ
ôîðìóëàáèíîìàÍüþòîíà.
4.
sin
x
=
x

x
3
3!
+
x
5
5!

:::
+
(

1)
n
x
2
n
+1
(2
n
+1)!
+
o
(
x
2
n
+2
)
.
5.
cos
x
=1

x
2
2!
+
x
4
4!

:::
+
(

1)
n
x
2
n
(2
n
)!
+
o
(
x
2
n
+1
)
:
6.
ln(1+
x
)=
x

x
2
2
+
x
3
3

x
4
4
+
:::
+
(

1)
n
+1
x
n
n
+
o
(
x
n
)
.
13.3Ðàçëè÷íûåñïîñîáûîöåíêèîñòàòî÷íîãî
÷ëåíà
Ñóùåñòâóþòèäðóãèåôîðìûçàïèñèîñòàòî÷íîãî÷ëåíà,äëÿòîãî,÷òîáû
ïîëó÷èòüèõäîêàæåìñëåäóþùååóòâåðæäåíèå:
Óòâåðæäåíèå101
(Îáùàÿôîðìàîñòàòî÷íîãî÷ëåíà)
.
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13.3.ÐÀÇËÈ×ÍÛÅÑÏÎÑÎÁÛÎÖÅÍÊÈÎÑÒÀÒÎ×ÍÎÃÎ×ËÅÍÀ
129
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130
ÃËÀÂÀ13.ÌÍÎÃÎ×ËÅÍÒÅÉËÎÐÀ
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13.4.ÈÑÏÎËÜÇÎÂÀÍÈÅÔÎÐÌÓËÛÒÅÉËÎÐÀÄËßÏÐÈÁËÈÆÅÍÍÛÕÂÛ×ÈÑËÅÍÈÉ
131
Òóòâðåñòîðàíâîøåëÿïîíåö.ßóæåðàíüøåâèäåëåãî:îíáðîäèëïîãîðîäó,
ïûòàÿñüïðîäàòüñ÷åòû.Îííà÷àëðàçãîâàðèâàòüñîôèöèàíòàìèèáðîñèëèì
âûçîâ,çàÿâèâ,÷òîìîæåòñêëàäûâàòü÷èñëàáûñòðåå,÷åìëþáîéèçíèõ.
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îäíîãî:÷åìñëîæíååçàäà÷à,òåìóìåíÿáîëüøåøàíñîâïîáåäèòü.
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132
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åãîïîíèìàíèÿ,íåñìîòðÿíàòî,÷òîçà÷àñòóþíåâîçìîæíîíàéòèìåòîäòî÷íîãî
âû÷èñëåíèÿêóáè÷åñêîãîêîðíÿ.Ïîýòîìóìíåòàêèíåóäàëîñüíàó÷èòüåãîáðàòü
êóáè÷åñêèéêîðåíüèëèîáúÿñíèòü,êàêìíåïîâåçëî,÷òîîíâûáðàë÷èñëî1729,03.
Ýòîòïðèìåðïîêàçûâàåò,÷òîñïîìîùüþðàçëîæåíèÿâðÿäÒåéëîðà
ìîæíîâû÷èñëÿòüïðèáëèæåííûåçíà÷åíèÿðàçëè÷íûõôóíêöèé.
2
Ð.Ôåéíìàíàìåðèêàíñêèéôèçèê,àóíèõ,âÑØÀâñåèçìåðÿåòñÿâôóòàõ,äþéìàõ
è.ò.ä.Ä.Â.
Ïðåäìåòíûéóêàçàòåëü
133
Ïðåäìåòíûéóêàçàòåëü
Ýêâèâàëåíòíîñòü,64
Ôóíêöèÿ,59
Äèðèõëå,69
Ðèìàíà,70
áèåêöèÿ,60
÷åòíàÿ,60
÷èñëîâàÿ,60
äèôôåðåíöèðóåìàÿ,99
íàîòðåçêå,100
ñëåâà,100
ñïðàâà,100
èíúåêöèÿ,60
íàèìåíüøèéïåðèîä,61
íå÷åòíàÿ,60
íåïðåðûâíàÿíàìíîæåñòâå
îòêðûòîì,65
íåïðåðûâíàÿíàîòðåçêå,65
íåïðåðûâíàÿâòî÷êå
ñëåâà,65
ñïðàâà,65
íåïðåðûâíàÿâòî÷êå,65
îáðàç,60
îäíîçíà÷íàÿ,59
ïåðèîä,61
ïåðèîäè÷åñêàÿ,61
ïðåäåë,61
ïðîèçâîäíàÿ,99
ýëåìåíòàðíûõôóíêöèé,105
êðàòíàÿ,113
âòîðàÿ,113
ïðîîáðàç,60
ðàçðûâíàÿâòî÷êå,65
ñòðåìèòñÿê,61
ñþðúåêöèÿ,60
âûïóêëàÿâíèç,115
âûïóêëàÿââåðõ,115
âçàèìíîîäíîçíà÷íîåîòîáðà-
æåíèå,60
Ëîãàðèôì,90
Ìíîæåñòâî
ìîùíîñòü
êîíòèíóóì,23
íèæíÿÿãðàíü
òî÷íàÿ,27
îòêðûòîå,26,100
ðàâíîìîùíûåìíîæåñòâà,21
ñ÷åòíîå,22
âåðõíÿÿãðàíü
òî÷íàÿ,26
Îáîëüøîå,64
Îìàëîå,64
Ïîñëåäîâàòåëüíîñòü
êîðìóøêà,34
ëîâóøêà,33
ìîíîòîííàÿ,33
íåóáûâàþùàÿ,33
íåâîçðàñòàþùàÿ,33
îãðàíè÷åííàÿ,33
ñíèçó,33
ñâåðõó,33
óáûâàþùàÿ,33
âîçðàñòàþùàÿ,33
134
ÏÐÅÄÌÅÒÍÛÉÓÊÀÇÀÒÅËÜ
135
Ïðåäåë
ïîÃåéíå,61
ïîÊîøè,61
Ïðîèçâîäíàÿ
ïðàâàÿ,100
ñëåâà,100
Ïñåâäî÷èñëà,62
Ñèñòåìàîòðåçêîâ
ñòÿãèâàþùàÿÿñÿ,28
âëîæåííàÿ,28
Òåéëîðà
ìíîãî÷ëåí,127
îñòàòî÷íûé÷ëåí,127
âôîðìåÊîøè,129
âôîðìåËàãðàíæà,129
âôîðìåÏåàíî,127
âôîðìåØë¼ìèëüõàÐîøà,
130
êðèâûå
öåíòðêðèâèçíû,123
ýâîëþòà,124
êàñàíèå,123
n

ãîïîðÿäêà,123
êðèâèçíà,124
êðóãêðèâèçíû,123
ðàäèóñêðèâèçíû,124
óãîëìåæäó,123
ïîñëåäîâàòåëüíîñòü,31
136
ÏÐÅÄÌÅÒÍÛÉÓÊÀÇÀÒÅËÜ
Ëèòåðàòóðà
nik
[1]
Ñ.Ì.Íèêîëüñêèé,Ì.Ê.Ïîòàïîâ.Àëãåáðàèíà÷àëààíàëèçà.
mak
[2]
È.Ï.Ìàêàðîâ.Äîïîëíèòåëüíûåãëàâûìàòåìàòè÷åñêîãîàíàëèçà.
maj
[3]
Å.Â.Ìàéêîâ.Ìàòåìàòè÷åñêèéàíàëèç.Ââåäåíèå.
bash
[4]
Ì.È.Áàøìàêîâ,Á.Ì.Áåêêåð,Â.Ì.Ãîëüõîâîé.Çàäà÷èïîìàòåìàòèêå.
Àëãåáðàèíà÷àëààíàëèçà(Áèáëèîòå÷êà"Êâàíòâûïóñê22)
vil
[5]
Í.ß.Âèëåíêèí.Ðàññêàçûîìíîæåñòâàõ.
arch
[6]
Ã.È.Àðõèïîâ,Â.À.Ñàäîâíè÷èé,Â.Í.×óáàðèêîâ.Ëåêöèèïîìàòå-
ìàòè÷åñêîìóàíàëèçó.
shi
[7]
Ã.Å.Øèëîâ.Ìàòåìàòè÷åñêèéàíàëèç(ôóíêöèèîäíîãîïåðåìåííîãî).
×àñòü1.
zor
[8]
Â.À.Çîðè÷.Ìàòåìàòè÷åñêèéàíàëèç.×àñòü1.
vav
[9]
Â.Â.Âàâèëîâ,È.È.Ìåëüíèêîâ,Ñ.Í.Îëåõíèê,Ï.È.Ïàñè÷åíêî.
Çàäà÷èïîìàòåìàòèêå.Íà÷àëààíàëèçà:Ñïðàâî÷íîåïîñîáèå.
chainik1
[10]
È.À.Âèîñàãìèð,"Âûñøàÿìàòåìàòèêàäëÿ÷àéíèêîâ.Ïðåäåëôóíê-
öèèhttp://viosagmir.ru/books,2011ã.,98ñò.
chainik2
[11]
È.À.Âèîñàãìèð,"Âûñøàÿìàòåìàòèêàäëÿ÷àéíèêîâ.Ïðîèçâîäíûåè
äèôôåðåíöèàëûhttp://viosagmir.ru/books,2011ã.,36ñò.
137

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