Elementar matematikaning
asosiy formulalari
Kitobcha elementar matematikaning barcha asosiy formulalari va qoidalarini o‘z
ichiga olgan bo‘lib, umumiy o‘rta ta’lim maktablari o‘quvchilari, akademik litsey va
kasb – hunar kollejlari talabalari, Oliy o‘quv yurtlariga kiruvchi abiturientlar va
matematika o‘qituvchilari uchun mo‘ljallangan.
2
Mundarija
ALGEBRA
To‘plamlar nazariyasi elementlari......................................................................................
Tub va murakkab sonlar..............................................................................................
Natural sonlarning kanonik yoyilmasi...........................................……………………
Soning butun va kasr qismi.........................................................................................
Tub sonlar jadvali........................................................................................................
Bo‘linish alomatlari......................................................................................................
Eng katta umumiy bo‘luvchi (EKUB)..........................................................................
Eng kichik umumiy karrali (EKUK) ...........................................................................
O‘lchov birliklari..........................................................................................................
Oddiy kasrlar ustida amallar........................................................................................
Proporsiya ..................................................................................................................
Davriy o‘nli kasrni oddiy kasrga aylantirish ...............................................................
Sonli tengsizliklar..........................................................................................................
O‘rta qiymatlar .................................................................................................................
Protsentlar ....................................................................................................................
Daraja va uning xossalari ...............................................................................................
Haqiqiy sonning moduli.................................................................................................
Arifmetik ildiz va uning xossalari...................................................................................
Qisqa ko‘paytirish formulalari........................................................................................
Kombinatorika elementlari...............................................................................................
Chiziqli tenglama.............................................................................................................
Kvadrat kenglama..........................................................................................................
Kub tenglama......................................................................................................................
Ikki noma’lumli chiziqli tenglamalar sistemasi.................................................................
Arifmetik progressiya...................................................................................................
Geometrik progressiya...................................................................................................
Tengsizliklarning xossalari...........................................................................................
Kasr ratsional tenglama va tengsizliklar...........................................................................
Kvadrat tengsizliklar .....................................................................................................
Irratsional tenglama va tengsizliklar ..........................................................................
Ko‘rsatkichli tenglama va tengsizliklar .........................................................................
Logarifm va uning asosiy xossalari..............................................................................
Logarifmik tenglama va tengsizliklar ...........................................................................
TRIGONOMERTIYA
Asosiy trigonometrik ayniyatlar......................................................................................
Qo‘shish formulalari.......................................................................................................
Ikkilangan va uchlangan burchaklar...............................................................................
Yig‘indini ko‘paytmaga keltirish........................................................................................
Ko‘paytmani yig‘indiga keltirish...................................................................................
Yarim burchak formulalari...........................................................................................
3
Darajani pasaytirish......................................................................................................
Trigonometrik funksiyalarni birini ikkinchisi orqali ifodalash………………………....
Keltirish formulalari .....................................................................................................
Trigonometrik funksiyalarning ayrim burchaklardagi qiymatlari ………………........
Trigonometrik tenglamalar ..........................................................................................
Trigonometrik tengsizliklar...........................................................................................
Ba’zi bir trigonometrik ayniyatlar ..............................................................................
Teskari trigonometrik funksiyalar orasidagi bog‘lanishlar…………………….…………
Teskari trigonometrik funksiyalarning yig‘indisi..........................................................
ALGEBRA VA ANALIZ ASOSLARI
Funksiya va uning asosiy xossalari..............................................................................
Juft va toq funksiyalar..................................................................................................
Davriy funksiya ...........................................................................................................
Funksiyalarning o‘sishi va kamayishi ..........................................................................
Teskari funksiyani topish..............................................................................................
ELEMENTAR FUNKSIYALAR VA ULARNING
ASOSIY XOSSALARI HOSILA
Hosilaning geometrik va fizik ma’nosi...........................................................................
Differensiallashning asosiy qoidalari..............................................................................
Funksiyalarni hosila yordamida tekshirish..................................................................
Funksiyaning eng katta va eng kichik qiymatlari.............................................................
Elementar funksiyalarning hosilalari.............................................................................
AJOYIB LIMITLAR ...................................................................................................
BOSHLANG‘ICH FUNKSIYA .................................................................................
Elementar funksiyalarning boshlang‘ichlari..................................................................
Aniq integralning asosiy xossalari................................................................................
Nyuton-Leybnits formulasi..........................................................................................
Egri chiziqli trapetsiyaning yuzi...................................................................................
Aylanma jismning xajmi .............................................................................................
GEOMETRIYA
PLANIMETRIYA
Burchaklar ....................................................................................................................
Uchburchak ..................................................................................................................
Uchburchaklarning tengligi ..........................................................................................
O‘xshash uchburchaklar ................................................................................................
Gomotetiya .................................................................................................................
Balandlik, bissektrissa, mediana.....................................................................................
Ichki va tashqi chizilgan aylanalar..................................................……………………
Uchburchakning yuzi...................................................................................................
Uchburchakdagi asosiy teoremalar...............................................................................
Teng yonli uchburchak....................................................................................................
4
To‘g‘ri burchakli uchburchak. Pifagor teoremasi..........................................................
Teng tomonli uchburchak.............................................................................................
To‘rtburchaklar ................................................................................................................
Kvadrat .........................................................................................................................
To‘g‘ri to‘rtburchak.......................................................................................................
Parallelogram ................................................................................................................
Romb .............................................................................................................................
Trapetsiya .......................................................................................................................
Aylanaga ichki va tashqi chizilgan to‘rtburchaklar………………………….....................
Ko‘pburchaklar ............................................................................................................
Muntazam beshburchak, oltiburchak va sakkizburchak.................................................
Aylanadagi burchaklar..................................................................................................
Aylanadagi teoremalar.................................................................................................
Doira, sektor, segment ...............................................................................................
STEREOMETRIYA
Prizma ..........................................................................................................................
To‘g‘ri burchakli parallelopiped ....................................................................................
Kub ..............................................................................................................................
Piramida ...........................................................................................................................
Muntazam piramida........................................................................................................
Kesik piramida...............................................................................................................
Muntazam uchburchakli piramida..................................................................................
Muntazam to‘rtburchakli piramida..............................................................................
Silindr ..........................................................................................................................
Konus ...............................................................................................................................
Kesik konus.......................................................................................................................
Sfera va shar......................................................................................................................
Shar segmenti.....................................................................................................................
Shar sektori.....................................................................................................................
Shar halqasi......................................................................................................................
Sharga ichki chizilgan konus..............................................................................................
Tekislikda Dekart koordinatalar sistemasi.................................................................
To‘g‘ri chiziq tenglamasi..................................................................................................
Fazoda Dekart koordinatalar sistemasi.........................................................................
Fazoda vektorlar..........................................................................................................
kollinear va komplanar vektorlar.................................................................................
Vektorlar ustida amallar.................................................................................................
Simmetriya .....................................................................................................................
Muntazam ko‘pyoqlar haqida ma’lumot....................................................................
Ba’zi bir yig‘indilar ....................................................................................................
5
ALGEBRA.
To‘plamlar nazariyasi elementlari
1.
2.
3.
4.
5.
6.
7.
8.
a  A –– a element A to‘plamga tegishli.
a  A –– a element A to‘plamga tegishli emas.
B  A –– B to‘plam A ning qism to‘plami.
B  A –– B va A to‘plamlarning birlashmasi.
B  A –– B va A to‘plamlarning kesishmasi.
A  B –– A tasdiqdan B tasdiq kelib chiqadi.
A  B –– A va B tasdiqlar teng kuchli.
 – bo‘sh to‘plam.
Sonli to‘plamlar
1. N  1,2,3,..., n,... - natural sonlar to‘plami.
N1  1,3,5,...,2n  1,... - toq sonlar to‘plami.
N 2  2,4,6,...,2n,... - juft sonlar to‘plami.
2. Z  ...,  n,..., 2, 1,0,1,2,..., n,... - butun sonlar to‘plami.
m


3. P   x : x  , m  Z , n  N  - ratsional sonlar to‘plami.
n


4. R   x :    x   - haqiqiy sonlar to‘plami.
5. N  Z  P  R .
6. Irratsional sonlar to‘plami: cheksiz, davriy bo‘lmagan o‘nli kasrlar.
Masalan:   3,14... , 2 , 3 9 ,..
Sonli oraliqlar
1. Yopiq oraliq:
 a, b   x : a  x  b, x  R .
2. Ochiq oraliq:
 a, b    x : a  x  b, x  R .
3. Yarim ochiq oraliq:
 a, b    x : a  x  b, x  R .
4. Yarim yopiq oraliq:
 a, b    x : a  x  b, x  R .
6
Tub va murakkab sonlar
1. Faqat o‘ziga va 1 ga bo‘linadigan birdan katta natural sonlar tub sonlar
deyiladi.
2. Uch va undan ortiq natural bo‘luvchiga ega bo‘lgan sonlar murakkab sonlar
deyiladi.
3. Teorema. Agar n sonning
bo‘lmasa, u holda n tub son bo‘ladi.
n dan katta bo‘lmagan tub bo‘luvchisi mavjud
Natural sonlarning kanonik yoyilmasi
1. Har qanday a  N sonini a  p11 p2 2 p33 ... pkk ko‘rinishida ifodalash mumkin,
unga a sonining kanonik yoyilmasi deyiladi. Bunda p1 , p2 , p3 , ..., pk – tub sonlar.
Masalan: 72=2332 ( p1  2, p2  3, 1  3,  2  2 ).
2. a  p11 p2 2 p33 .... pk k sonning bo‘luvchilari soni
n(a)  1  1   2  1  ...   k  1 ,
formula bilan, bo‘luvchilar yig‘indisi esa
p111  1 p2 2 1  1 p33 1  1
pk k 1  1
S (a) 


 ... 
p1  1
p2  1
p3  1
pk  1
formula orqali hisoblanadi.
3. n!  1  2  3  ...  n sonining kanonik yoyilmasida p tub son
n  n   n 
 p      2    3   ... daraja bilan qatnashadi.
 p  p   p 
n  n   n 
4. n! soni k      2    3   ... ta nol bilan tugaydi.
 5  5  5 
Sonning butun va kasr qismi
1. a sonining butun qismi deb, a dan katta bo‘lmagan eng katta butun songa
aytiladi va u  a  orqali belgilanadi. Masalan: [2,4]=2, [–3,52]= –4.
2. Sonning kasr qismi deb ( a -  a  ) ga aytiladi va u
Masalan: {-0,3}=0,7; {2,6}=0,6.
7
a
orqali belgilanadi.
Tub sonlar jadvali (1000 gacha)
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101
103
107
109
113
127
131
137
139
149
151
157
163
167
173
179
181
191
193
197
199
211
223
227
229
233
239
241
251
257
263
269
271
277
281
283
293
307
311
313
317
331
337
347
349
353
359
367
373
379
383
389
397
401
409
419
421
431
433
439
443
449
457
461
463
467
479
487
491
499
503
509
521
523
541
547
557
563
569
571
577
587
593
599
601
607
613
617
619
631
641
643
647
653
659
661
673
677
683
691
701
709
719
727
733
739
743
751
757
761
769
773
787
797
809
811
821
823
827
829
839
853
857
857
863
877
881
883
887
907
911
919
929
937
941
947
953
967
971
977
983
991
997
Bo‘linish alomatlari
1. Agar sonning oxirgi raqami 0 yoki juft bo‘lsa, bu son 2 ga bo‘linadi.
2. Agar sonning raqamlari yig‘indisi 3 (9) ga bo‘linsa, bu son 3 (9) ga bo‘linadi.
3. Agar sonning oxirgi ikki raqamidan tashkil topgan ikki xonali son 4 (25) ga bo‘linsa,
bu son 4 (25) ga bo‘linadi.
4. Agar sonning oxirgi raqami 0 yoki 5 bo‘lsa, bu son 5 ga bo‘linadi.
5. Agar sonning oxirgi raqami 0 bo‘lsa, bu son 10 ga bo‘linadi.
6. Agar sonning toq o‘rindagi raqamlari yig‘indisi bilan juft o‘rindagi raqamlari
yig‘indisining ayirmasi 11 ga bo‘linsa, bu son 11 ga bo‘linadi.
Masalan: 9873424, (9+7+4+4)–(8+3+2)=11.
Eng katta umumiy bo‘luvchi (EKUB)
1 dan boshqa umumiy bo‘luvchilarga ega bo‘lmagan sonlar o‘zaro tub sonlar
deyiladi.
Sonlarning EKUBi deb, shu sonlarning umumiy bo‘luvchilarining eng kattasiga
aytiladi va u quyidagicha topiladi:
1) har bir sonning kanonik yoyilmasida qatnashgan umumiy ko‘paytuvchilar eng kichik
darajasi bilan olinadi;
2) ajratib olingan sonlar ko‘paytiriladi.
8
Eng kichik umumiy karrali (EKUK)
Sonlarning EKUKi deb, shu sonlarga bo‘linadigan sonlarning eng kichigiga
aytiladi va u quyidagicha topiladi:
1) har bir sonning kanonik yoyilmasida qatnashgan ko‘paytuvchilar eng katta darajasi
bilan olinadi;
2) ajratib olingan sonlar ko‘paytiriladi.
Masalan: EKUB(28,144) va EKUK(28,144) ni toping.
28  22  7 , 144  2 4  32  ЭКУБ  28,144   22  4 ,
ЭКУК  28,144   24  32  71  1008 .
Qoldiqli bo‘lish
a  p  q  r (0  r  p ) ,
bu yerda a –bo‘linuvchi, p –bo‘luvchi, q –bo‘linma, r –qoldiq.
O‘lchov birliklari
1. 1 km=1000m.
7.
1m3=1000dm3=1000000sm3.
2. 1 m=10 dm=100sm.
8.
1dm3=1000sm3.
3. 1 sm=10 mm.
9.
1l=1dm3=1000sm3.
4. 1 km2=1000000m2.
10. 1t=10s=1000kg.
5. 1 m2=100dm2=10000sm2.
11. 1kg=1000g.
6. 1 ga=100ar=10000m2.
12. 1g=1000mg.
Oddiy kasrlar ustida amallar
a c ac
 
;
b b
b
a c ad  bc
 
;
b d
bd
a c ac
 
;
b d bd
a c a d ad
:   
.
b d b c bc
Proporsiya
Agar a : b  c : d bo‘lsa, quyidagilar o‘rinli:
ma  nb mc  nd
1) ad  bc ;
2)
;

pa  qb pc  qd
a b b
ac a
3)
 ;
4)
 ;
cd d
bd b
9
Davriy o‘nli kasrlarni oddiy kasrga aylantirish
a0 , a1a2 a3....ak (b1b2b3....bn ) 
a0 a1a2 a3....ak b1b2b3 ....bn  a0a1a2 a3....ak
999...999000...000

 

n
k
Sonli tengsizliklar
1) a  b  a  b  0 ,
2) a  b, b  c  a  c ,
3) a  b, с  0 
4) a  b, с  0 
5) a  b  0  a n
6) a  b  0, c  0
a b  ba;
a b  aс bc;
a b
ac  bc,
 ;
c c
a b
ac  bc,
 ;
c c
 b n , n a  n b (n  ) ;
a b c c

 ,
 .
c c a b
O‘rta qiymatlar
Agar x1 , x2 ,..., xn musbat sonlar bo‘lsa, ularning:
x  x  ...  xn
1) o‘rta arifmetigi: A  1 2
;
n
2) o‘rta geometrigi: G  n x1  x2  ...  xn ;
n
3) o‘rta garmonigi: H  1 1
;
1
x1  x2  ...  xn
x12  x2 2  ...  xn 2
;
n
Koshi teoremasi: H  G  A  K .
4) o‘rta kvadratigi: K 
Protsentlar
aP
;
100
a  100
2) P protsenti a ga teng sonni topish: x 
;
P
a
3) a soni b sonining  100% ini tashkil etadi;
b
a
4) a soni a + b yig‘indining
 100% ini tashkil etadi.
ab
1) a sonining P protsentini topish: x 
10
Daraja va uning xossalari
1) a n  a
 a
 
a  
a;
0
1
2) a  1, a  a ;
n
3) a  p 
1
;
ap
4) a p  a q  a p q ;
 
6) a p
5) a p : a q  a p q ;
q
 a pq ;
p
p
p
ap
a
8)    p .
b
b
p
7)  a  b   a  b ;
Bu yerda n  N , a  0, b  0 ; q , p  .
Haqiqiy sonning moduli
 a,
a 
  a,
a0
agar
bo'lsa;
a0
agar
bo'lsa.
a a
 ;
b b
1) ab  a  b ;
2)
3) a  b  a  b ;
4) a  b  a  b ;
2
5) a  a 2 .
Arifmetik ildiz va uning xossalari

 a  0,

n
a   n a  0,
 n n
 a  a.
 
 
1)
m
3)
m
a 
5)
n
a n p  a n a p ;
7)
m
an b c 
9)
n
 a , n  2k , k  N ,
an  
 a, n  2k  1, k  N .
ab  m a  m b ;
2)
n
am
4)
n
;
p
A B 
mnp
a npb p c ;
Am

2
Am
;
2
m
a ma

;
b mb
mp
a np  m a n ;
6)
m n
8)
n
mn
a;
a n a n a...  n1 a ;
m  A2  B .
11
a
Qisqa ko‘paytirish formulalari
2
3
1)  a  b   a 2  2ab  b 2 ;
3)  a  b   a 3  3a 2b  3ab 2  b 3 ;
2) a 2  b 2   a  b   a  b  ;
4) a 3  b3   a  b  a 2  ab  b 2 ;




5) a 4  b4   a  b   a  b  a 2  b 2 ; 6) (a  b  c) 2  a 2  b 2  c 2  2(ab  ac  bc) .
Kombinatorika elementlari
1. m ta elementdan n tadan barcha o‘rinlashtirishlar soni:
Amn  m(m  1)(m  2)...(m  n  1) 
m!
.
(m  n)!
2. n ta elementdan barcha o‘rin almashtirishlari soni:
Pn  n !  1  2  3  4  ...  n .
3. m ta elementdan n tadan barcha gruppalashlar soni:
Amn m(m  1)(m  2)...(m  n  1)
m!
n
Cm 


.
Pn
1  2  3  ....  n
(m  n)! n !
Chiziqli tenglama
ax  b  0 .
b
1. a  0, b  R bo‘lsa, yagona yechimga ega: x   ;
a
2. a  0, b  0 bo‘lsa, yechimi yo‘q;
3. a  0, b  0 bo‘lsa, yechimi cheksiz ko‘p: x  R .
Kvadrat tenglama
ax  bx  c  0 ( a  0 ).
2
1. D  b 2  4 ac  0 bo‘lsa, 2 ta har xil haqiqiy yechimi bor:
b  b 2  4ac
x1,2 
.
2a
2. D  b 2  4 ac  0 bo‘lsa, 1 ta ikki karrali haqiqiy yechimi bor:
b
x1,2 
.
2a
3. D  b 2  4 ac  0 bo‘lsa, haqiqiy yechimi yo‘q.
12
Yechimlarining xossalari:
Agar x1 va x2 sonlar ax 2  bx  c  0 tenglamaning ildizlari bo‘lsa, u holda:
b
c
1. x1  x2   ,
x1  x2 
(Viet teoremasi);
a
a
2. ax 2  bx  c  a ( x  x1 )( x  x2 ) ;
3.
x12

x22
  x1  x2 
2
b 2  2ac
 2 x1x2 
;
a2
3
 b  3bc
4.    x1  x2   3x1x2 ( x1  x2 )      2 ;
a
a
2
3
1
1 b  2ac
1
1 b  3abc
5. 2  2 
;
 3
;
2
3
x1 x2
c
x1 x2
c3
x13
x23
3
2
b  b 2  4ac

6. to‘la kvadratga ajratish: ax  bx  c  a  x 
.
 
2a 
4a

2
Kub tenglama
x  ax 2  bx  c  0 .
x3 lar bu tenglamaning ildizlari bo‘lsa, u holda
3
Agar x1 , x2,
1. x 3  ax 2  bx  c  ( x  x1 )( x  x2 )( x  x3 ) .
 x1  x2  x3   a,

2. Viet teoremasi:  x1x2  x1x3  x2 x3  b,
 x x x  c.
 1 2 3
Ikki noma’lumli chiziqli tenglamalar sistemasi
 a x  a12 y  b1 ,
Umumiy ko‘rinishi:  11
Geometrik talqini
a
x

a
y

b
.
 21
22
2
a11 a12 b1


bo‘lsa,
a21 a22 b2
sistema yechimga ega emas.
1. Agar
a11 a12

bo‘lsa,
a21 a22
sistema yagona yechimga ega.
2. Agar
a11 a12 b1


bo‘lsa,
a21 a22 b2
sistema cheksiz ko‘p yechimga ega.
3. Agar
13
Arifmetik progressiya an 1  an  d , n  0, 1, 2,... .
d –ayirmasi.
a  am
1. ayirmasini topish: d  an1  an  n
, n m;
nm
2. n - hadini topish: an  a1   n  1 d ;
a  ank
3. o‘rta hadini topish: an  nk
, k  n;
2
4. dastlabki n ta hadining yig‘indisini topish:
2a   n  1 d
a a
Sn  1 n  n  1
n;
2
2
5. n - dan k - gacha bo‘lgan hadlar yig‘indisini topish:
S nk  an  dn(k  1)
k  n ;
6. an  S n  S n1 .
bu yerda a1 – birinchi hadi,
Geometrik progressiya bn1  bn q ,
n  1, 2, 3,... .
q – maxraji.
bu yerda, b1 – birinchi hadi,
b
1. maxrajini topish: q  n1 ;
bn
2. n - hadini topish: bn  b1  q n1 , bn  bnm  q m ;
3. o‘rta hadini topish: bn  bnk  bn k ;
4. dastlabki n ta hadi yig‘indisini topish:


n
bn q  b1 b1 q  1
Sn 

;
q 1
q 1
1. bn  S n  S n1 ;
2. Cheksiz kamayuvchi geometrik progressiya hadlari yig‘indisi:
b
q  1.
S 1 ,
1 q
Tengsizliklarning xossalari
1. Agar f ( x )  g ( x) bo‘lsa, c  0 da cf ( x)  cg ( x) , c  0 da cf ( x)  cg ( x) bo‘ladi.
2. Agar f 2 n ( x)  g 2n ( x)
bo‘ladi.
3. Agar f ( x)  c
o‘rinli.
f
2n

( x)  g 2n ( x) bo‘lsa, u holda f ( x)  g( x)
 f ( x)  c  bo‘lsa, u holda c  f (x)  c  f (x)  c
14
 f (x)  g(x) 
ёки f (x)  c  lar
Kasr ratsional tenglama va tengsizliklar
1.
 f ( x)  0,
f ( x)
0  
g ( x)
 g ( x)  0.
2.
 f ( x)v ( x )  g ( x)h( x)  0,
f ( x ) h( x )

 
g ( x ) v( x)
 g ( x)v( x)  0.
3.
 f ( x ) g ( x)  0,
f ( x)
0  
g ( x)
 g ( x)  0.
4.
 f ( x)  g ( x )h( x )  g ( x)  0,
f ( x)
 h( x)  
g ( x)
 g ( x )  0.
Oraliqlar usuli
Agar a  b  c  d bo‘lsa, u holda
a) ( x  a)( x  b)( x  c)( x  d )  0 tengsizlikning yechimi (; a)  (b; c)  (d ; )
bo‘ladi.
b) ( x  a )( x  b)( x  c )( x  d )  0 tengsizlikning yechimi (a; b)  (c; d ) bo‘ladi.
Bu usul oraliqlar usuli deyiladi.
Kvadrat tengsizlik
ax 2  bx  c  0
ax  bx  c  0
2


1) a  0, D  0 bo‘lsa, x   ; x1    x2 ;  
 x   x1; x2  ;
2) a  0, D  0 bo‘lsa, x   ;  
 x  x1  ;
3) a  0, D  0 bo‘lsa, x   ;  
 x   ;
4) a  0, D  0 bo‘lsa, x   x1; x2 
  ; x1    x2 ;    ;
5) a  0, D  0 bo‘lsa, x  x1
 x   ;    ;
6) a  0, D  0 bo‘lsa, x 
 x   ;    .
15
Irratsianal tenglama va tengsizliklar
1.
2k
f ( x)  0
2.
2k
f ( x)  g  x 
3.
2k
f ( x)  2 k g  x 
 f ( x)  0,
 
 f ( x)  g ( x).
4.
2 k 1
f ( x)  g  x 
 f ( x)  g 2 k 1 ( x ) .
5.
2k
6.
2 k 1
7.
2k
8.
2 k 1
 f ( x)  0 .

 g ( x )  0,

2k
 f ( x)  g ( x).
 f ( x)  0, g ( x )  0,
 
2k
 f ( x)  g ( x).
f ( x)  g  x 
f ( x)  g  x 
 f ( x )  g 2 k 1 ( x) .
 g ( x)  0,
 g ( x)  0,
 
ёки

2k
 f ( x)  0.
 f ( x )  g ( x ).
f ( x)  g  x 
f ( x)  g  x 
 f ( x)  g 2 k 1 ( x ) .
Ko‘rsatkichli tenglama va tengsizliklar
1. a f ( x )  1  f ( x )  0.
2.
 f ( x ) g ( x )  1
 f ( x)  1,
 f ( x )  0,
 
yoki 
 g ( x)  R,
 g ( x)  0.
3. a f ( x )  a g ( x )  f ( x)  g ( x)
(a  0) .
4. a f ( x )  b g ( x )  f ( x)  g ( x )log a b (a, b  0) .
0  a  1,
5. a f ( x )  a g ( x )  
 f ( x)  g ( x),
a  1,
yoki 
 f ( x)  g ( x).
b  0, a  1,
0  a  1, b  0,
6. a f ( x )  b, 0  a  1  
yoki 
 f ( x)  log a b,
 f ( x)  log a b.
b  0, a  1,
7. a f ( x )  b  
 f ( x)  log a b,
b  0, 0  a  1,
yoki 
 f ( x )  log a b.
16
Logarifm va uning asosiy xossalari
b  log a N ( a  0, a  1, N  0 )  N  a b .
1) log a 1  0 , log a a  1 ,
a
2) log a (bc )  log a b  log a c ,
m
log a b ,
n
log a c
4) logba c 
,
1  log a b
3) log a n b m 
5)
6)
7)
8)
log aN
N;
b
log a    log a b  log a c ;
c
1
log a b 
;
logb a
log c b
log a b 
;
log c a
log c
log a
a b c b ;
log a b  log c d  log a d  log c b ,
log10 x  lg x – o‘nli logarifm;
log e x  ln x – natural logarifm;
Agar a  1, 0  b  1 yoki 0  a  1, b  1 bo‘lsa log a b  0 ;
9) Agar a  1, b  1 yoki 0  a  1,0  b  1 bo‘lsa log a b  0 ;
10) Agar b  a  1 bo‘lsa, log b p  log a p bo‘ladi ( p  0 );
11) Agar 0  a  b  1 bo‘lsa, log b p  log a p bo‘ladi ( p  0 );
12) a  b  0 bo‘lsin.
agar 0  p  1 bo‘lsa, log p a  log p b bo‘ladi,
agar p  1 bo‘lsa, log p a  log p b bo‘ladi.
Logarifmik tenglama va tengsizliklar
 f ( x)  0, 0  a  1,
 
b
 f ( x)  a .
1. log a f ( x )  b
2. log f ( x ) a  b
 f ( x)  1, a  0,

 
1
 f ( x)  a b .
3. log a f ( x )  log a g ( x)
4. log f ( x ) g ( x)  b

0  a  1, f ( x)  0,

 f ( x)  g ( x).
0  f ( x)  1,
 
b
 g ( x )  f ( x ).
17
5. log ( x ) f ( x)  log ( x ) g ( x)
6. log ( x )
0   ( x)  1, f ( x)  0,
 
 f ( x )  g ( x).
0   ( x )  1,
 ( x )  1,


f ( x)  log ( x ) g ( x)   f ( x)  0,
yoki  g ( x)  0,
 f ( x)  g ( x ).
 f ( x)  g ( x ).


7. log f ( x ) g ( x)  b
0  f ( x )  1,
 f ( x)  1,

  g ( x)  0,
yoki 
b
 g ( x )  f ( x ).

b
 g ( x)  f ( x).
TRIGONOMETRIYA
 
Sin  y ,
y
tg   ,
x
1
Sec  ,
x
180
  рад ,

 рад 

 0
180
Со s   x ,
x
ctg   ,
y
1
С s с 
.
y
1 rad  5701715 ;
  3,141592 . . .
Trigonometrik funksiyalarning choraklardagi ishoralari
Cosx
Sinx
tgx, ctgx
Asosiy trigonometrik ayniyatlar
1. Sin 2 x  Cos 2 x  1 .
2. tgx  ctgx  1.
3. 1  tg 2 x 
1
.
Cos 2 x
Sinx
.
Cosx
Cosx
5. ctgx 
.
Sinx
4. tgx 
6. 1  ctg 2 x 
18
1
.
Sin 2 x
Qo‘shish formulalari
sin(   )  sin  cos   cos  sin  ;
cos(   )  cos  cos   sin  sin  ;
tg (   ) 
tg  tg 
;
1  tg tg 
сtg (   ) 
сtg сtg   1
.
сtg  сtg 
Ikkilangan va uchlangan burchaklar
sin 2 x  2sin x cos x ;
2tgx
tg 2 x 
;
1  tg 2 x
sin 2 x  2sin x cos x ;
ctg 2 x  1
ctg 2 x 
;
2ctgx
sin 3 x  3sin x cos 2 x  sin 3 x  3sin x  4sin 3 x ;
cos3 x  cos3 x  3cos x sin 2 x  4cos3 x  3cos x ;
tgx  tg 2 x 3tgx  tg 3 x
tg 3 x 

.
1  tgxtg 2 x 1  3tg 2 x
Yig‘indini ko‘paytmaga keltirish
1. Sinx  Siny  2 Sin
x y
x y
Cos
.
2
2
2. Cosx  Cosy  2Cos
x y
x y
Cos
.
2
2
3. Cosx  Cosy  2Sin
x y
x y
Sin
.
2
2




4. Cosx  Sinx  2 Sin   x   2Cos   x  .
4

4





5. Cosx  Sinx  2Cos   x   2 Sin   x  .
4

4

6. pCosx  qSinx  rSin( z  x ) , r 
7. tgx  tgy 
Sin  x  y 
,
CosxCosy
8. tgx  ctgy 
Cos  x  y 
,
CosxSiny
p 2  q 2 , Sinz 
p
q
, Cosz  .
r
r
ctgx  ctgy 
Sin  x  y 
.
SinxSiny
tgx  ctgy  
Cos  x  y 
.
CosxSiny
19
Ko‘paytmani yig‘indiga keltirish
1
 Sin  x  y   Sin  x  y   .
2
1
2. CosxCosy   Cos  x  y   Cos  x  y   .
2
1
3. SinxSiny   Cos  x  y   Cos  x  y   .
2
1. SinxCosy 
Yarim burchak formulalari
1. Sin
2. tg

1  Cos

,
2
2
Cos

1  Cos

.
2
2
 1  Cos
Sin
1  Cos



.
2
Sin
1  Cos
1  Cos
3. ctg
 1  Cos
Sin
1  Cos



.
2
Sin
1  Cos
1  Cos
Darajani pasaytirish
1  Cos 2
,
2
3Cos  Cos3
Cos 3 
,
4
Cos 2 
1  Cos 2
.
2
3sin   sin3
Sin3 
.
4
x
Sinx, Cosx, tgx va ctgx larni tg orqali ifodasi
2
2tg ( x / 2)
2tg ( x / 2)
Sinx 
;
tgx 
;
2
1  tg ( x / 2)
1  tg 2 ( x / 2)
Cosx 
Sin 2 
1  tg 2 ( x / 2)
;
1  tg 2 ( x / 2)
ctgx 
1  tg 2 ( x / 2)
.
2tg ( x / 2)
Trigonometrik funksiyalarni birini ikkinchisi orqali ifodalash
sin x
cos x
tgx
tgx
ctgx
1
sin x
sin x
 1  cos 2 x
 1  tg 2 x
1
 1  ctg 2 x
ctgx
cos x
 1  sin 2 x
cos x
 1  tg 2 x
 1  ctg 2 x
20
sin x
tgx
 1  sin 2 x
ctgx
 1  sin 2 x
sin x
1
sec x
 1  sin 2 x
csc x
1
sin x
tgx
1
ctgx
1
tgx
ctgx
 1  tg 2 x
 1  ctg 2 x
ctgx
 1  cos 2 x
cos x
cos x
 1  cos 2 x
1
cos x
1
 1  tg 2 x
tgx
2
 1  cos x
 1  ctg 2 x
Keltirish formulalari


x
2
x
3
x
2
2  x
Sin
Cosx
 Sinx
Cosx
 Sinx
Cos
 Sinx
Cosx
 Sinx
Cosx
tg
 ctgx
tgx
 ctgx
tgx
ctg
 tgx
 ctgx
 tgx
 ctgx
Trigonometrik funksiyalarning ayrim
burchaklardagi qiymatlari
Gradus
o‘lchovi
0
300
450
600
900
1200
1350
Radian
o‘lchovi
0

6

4

3

2
2
3
3
4
sin x
cos x
tgx
ctgx
sec x
csc x
0
1
2
1
3
2
2
2
1
2
0
3
3
–
1
2
3
–
1
1
2
3
3
2
2
2
3
0
1

2
2

2
–
0
–
2
2
3
2
1
3
2
2
2
3
3
 3
1
21

3
3
1
2
2
1
2
3
 2
2
1500
1800
2100
2250
2400
2700
3600
Gradus
o‘lchovi
150
5
6

7
6
5
4
4
3
3
2
2
1
2
0
1

2
3
2
1
3

2
2

2
1

2

2
2
3

2

1
3
3
0
3
3
2
3
1
2

3

 3
–
1
3
–
0
1
0
Radian
o‘lchovi

12

0
3
2
–
2
1
 2
3
3
2
 2
2

3
–
1
1
–
0
–
sin x
cos x
tgx
ctgx
3 1
2 2
3 1
2 2
2 3
2 3
180

10
5 1
4
5 5
2 2
360

5
5 5
2 2
5 1
4
540
3
10
5 1
4
750
5
12
3 1
2 2
5 5
2 2
3 1
2 2
5 1
10  2 5
10  2 5
5 1
5 1
10  2 5
5 1
5 1
10  2 5
10  2 5
10  2 5
5 1
2 3
2 3
Trigonometrik tenglamalar
n
1) Sinx  a,
a  1,
x   1 arcSina  n , n  Z ;
2) Cosx  a,
a  1,
x   arcCosa  2n , n  Z .
a
Sinx  a
0
x k , k Z
1
x 
2
Cosx  a
x    k , k Z
2
 2 k , k  Z
x  2 k , k  Z
22
1
2
k
x   1 
1
2
x   1
–1
x  

3
2
3
2

2
2
2
2

3) tgx  a,
4) ctgx  a,
k 1
2
k 1
k 1
3
6
 2 k , k  Z
4
x   3
 2 k , k  Z
4
 2 k , k  Z
x  arctga  n , n  Z ;
x  arcctga  n , n  Z .
x k , k Z
1
x 
–1
x  
x 
4
x  
3
x 
x  
x 
4
x  3
k , k Z
6
x  5
k , k Z
x 
23
 k , k Z
k , k Z
 k , k Z
 k , k Z
6
3
x  2
k , k Z
 k , k Z
4
x 
 k , k Z
6
2
x 
 k , k Z
3
6
ctgx  a
 k , k Z
4
3
 3
3
x  
 k , k Z
4
0
 3
 2 k , k  Z
 2 k , k  Z
6
x   5
k , k Z
tgx  a
3
x  
k , k Z
a
3
3
x    2 k , k  Z
 k , k Z
3
4
 2 k , k  Z
3
x   2
 2 k , k  Z
k
x   1 
x   1
x  
k , k Z
 k , k Z
6
k
x   1 
x   1
6
 k , k Z
3
 k , k Z
Trigonometrik tengsizliklar
1) Sinx  a
 a  1 ,
2) Sinx  a
 a  1 , 
3) Cosx  a
 a  1 ,  x   arcCosa  2n ; arcCosa  2n  ;
4) Cosx  a
 a  1 ,  x   arcCosa  2n ;2  arcCosa  2n  ;
 x   arcSina  2n ;  arcSina  2n  ;
x     arcSina  2n ; arcSina  2n  ;



 x  arctga  n ;  n  ;
2


5) tgx  a
a  R ,
6) tgx  a
a  R , 
 

x     n ; arctga  n  ;
 2

7) ctgx  a
a  R , 
x   n ; arcctga  n  ;
8) ctgx  a
a  R , 
x   arcctga  n ; n  . (Bu yerda n  Z .)
Ba’zi trigonometrik ayniyatlar
1
1. sin x sin(60  x)sin(60  x )  sin 3 x .
4
2. 16 Sin100 Sin300 Sin500 Sin700 Sin900  1 .
3. 16Cos800 Cos600 Cos 400 Cos 200 Cos00  1 .
4. 16 Sin 200 Sin 400 Sin 600 Sin800  3 .
5. 16Cos100 Cos300 Cos500 Cos 700  3 .
6. Cos

4
5 1
Cos
Cos
 .
7
7
7 8
7. Cos

2
4
1
Cos
Cos
 .
7
7
7
8
8. Cos

3
5
1
Cos Cos
 .
7
7
7
8
9. Cos
2
4
6 1
Cos
Cos
 .
7
7
7 8
10. Cos

3
1
Cos
 .
5
5
4
24
11. Cos
12. Sin

3 1
 Cos
 .
5
5 2

3
5
(2n  1)
2
Sin Sin
  Sin
 n .
4n
4n
4n
4n
2
Sin 2 n1 x
13. Cosx  Cos 2 x  Cos 4 x  ...  Cos 2 x  n1 
.
Sinx
2
n
Sin
14. Cosx  Cos 2 x  ...  Cosnx 
1
nx
(n  1) x
Cos
2
2
.
x
Sin
2
Teskari trigonometrik funksiyalar orasidagi bog‘lanishlar


arcSinx  arcCosx  ; arctgx  arcctgx 
2
2
1. arcSin ( x)   arcSinx; arcCos( x)    arcCosx .
2. arctg ( x)   arctgx; arcctg ( x)    arcctgx .
3. arcSinx 

x
 arcCosx  arctg
.
2
2
1 x
4. arcCosx 
5. arctgx 

x
 arcSinx  arcctg
.
2
2
1 x

x
 arcctgx  arcSin
2
1  x2
6. arcctgx 

x
 arctgx  arcCos
.
2
2
1 x
Trigonometrik va teskari trigonometrik funksiyalar
orasidagi bog‘lanishlar
1. Sin  arcSinx   x , x   1;1 .
  
2. arcSin(Sinx )  x , x    ;  .
 2 2
3. Cos  arcCosx   x , x   1;1 .
4. arcCos (Cosx)  x , x   0;  .
5. arctg (tgx)  x,
  
x   ;  .
 2 2
6. tg (arctgx)  x,
25
xR.
7. arcctg (ctgx)  x, x   0;   .
8. ctg (arcctgx)  x,
9. Sin  arcCosx   1  x 2 , x   1;1 .
10. Cos  arcSinx   1  x 2 , x   1;1 .
11. Sin  arctgx  
13. Sin  arcctgx  
15. tg  arcSinx  
17. tg  arcSinx  
19. Sin  arcctgx  
x
1 x
1
1 x
2
x
1 x
2
x
1 x
2
16. tg  arcSinx  
, x   1;1 .
18. tg  arcSinx  
1
2
1 x
20. Sin  arcctgx  
.
1 x
x
1 x
1 x
3. arcSinx  arcSiny  arcCosx ( 1  x 2 1  y 2  xy ) .
4. arcSinx  arcSiny  arcCosx ( 1  x 2 1  y 2  xy ) .
 arcCos xy  1  x 2  1  y 2 , x  y,

5. arcCosx  arcCosy  
arcCos xy  1  x 2  1  y 2 , x  y.

6. arctgx  arctgy  arctg
x y
, xy  1 .
1  xy
7. arctgx  arctgy  arctg
x y
, xy  1 .
1  xy
8. arcctgx  arcctgy  arcctg

xy  1
, x  y.
x y
26
2
.
2
, x   1;1 ,
, x   1;1 .
 1 x
2. arcSinx  arcSiny  arcCosx ( 1  x 2 1  y 2  xy ) .

.
1
1. arcSinx  arcSiny  arcCosx ( 1  x 2 1  y 2  xy ) .

2
x
Teskari trigonometrik funksiyalar yig‘indisi

2
x
14. Cos  arcctgx  
.
, x   1;1 .
 1 x
1
12. Cos  arctgx  
.
2
xR .
2
.
Funksiya va uning asosiy xossalari
X sonlar to‘plamidan olingan x ning har bir qiymatiga biror qonuniyat yoki
qoida yordamida Y sonlar to‘plamidan olingan yagona y qiymat mos kelsa, bunday
moslik funksiya deyiladi va y  f ( x) ko‘rinishida belgilanadi.
X to‘plam funksiyaning aniqlanish sohasi deyiladi va D ( f ) ko‘rinishida
belgilanadi.
Y to‘plam esa funksiyaning qiymatlari to‘plami deyiladi va Е ( f ) ko‘rinishida
belgilanadi.
Funksiyaning aniqlanish sohasini (a.s.)
topishga doir misollar:
1) y  2 n f ( x) funksiyaning a.s. f ( x )  0 tengsizlikning yechimi bo‘ladi;
1
2) y 
funksiyaning a.s. f ( x )  0 tengsizlikning yechimi bo‘ladi;
f ( x)
 f ( x )  0,

3) y  log g ( x ) f ( x) funksiyaning a.s.  g ( x)  0, sistemani yechimi bo‘ladi.
 g ( x)  1,

Funksiyaning qiymatlar sohasini
topishga doir misollar:
4ac  b 2
1) y  ax  bx  c va y0 
 0 bo‘lsin.
4a
2

agar a  0 bo‘lsa, E ( y )   y0 ;  ;
agar a  0 bo‘lsa, E ( y )  0; y0  bo‘ladi;
2) y  a cos kx  b sin kx bo‘lsa, E ( y )    a 2  b 2 ; a 2  b 2  .


Juft, toqligi
Agar x  D ( f ) uchun  x  D ( f ) va f ( x)  f ( x) bo‘lsa, y  f ( x) funksiya
juft, f ( x)   f ( x) bo‘lsa, y  f ( x) funksiya toq deyiladi, aks holda juft ham emas,
toq ham emas.
Masalan: y  x 2 –juft funksiya, y  x3 –toq funksiya.
27
Agar f ( x) , g ( x) – juft,  ( x) ,  ( x ) – toq funksiyalar bo‘lsa, u holda f ( x)  g ( x)
– juft,  ( x)   ( x ) – toq, f ( x)  g ( x) – juft,  ( x )   ( x) – juft,
f ( x) : g ( x) – juft, g ( x)   ( x) – toq, g ( x) :  ( x) – toq,  ( x) :  ( x ) – juft.
g ( x)   ( x) – juft ham, toq ham emas.
Juft funksiyaning grafigi Oy o‘qiga nisbatan simmetrik bo‘ladi.
Toq funksiyaning grafigi koordinatalar boshiga nisbatan simmetrik bo‘ladi.
Davriyligi
Agar ixtiyoriy x  D ( f ) uchun ( x  Т )  D( f ) Т  0  va f ( x  T )  f ( x )
bo‘lsa, y  f ( x) davriy funksiya, T soni esa uning davri deyiladi.
Agar T  0 soni y  f ( x) funksiyaning davri bo‘lsa, nT (n  ) soni ham
y  f ( x) funksiyaning davri bo‘ladi.
Agar y  f ( x) funksiyaning eng kichik musbat (e.k.m.) davri T bo‘lsa, u holda
T
y  f (kx  b) funksiyaning e.k.m. davri
bo‘ladi.
k
Bir necha davriy funksiyalarning yig‘indisidan iborat funksiyaning e.k.m. davri
qo‘shiluvchi funksiyalar e.k.m. davrlarining EKUK iga teng.
Funksiyaning o‘sishi va kamayishi (monotonligi)
y  f ( x) funksiya (a;b) oraliqda aniqlangan va shu oraliqdan olingan ixtiyoriy
x1 , x2  x1  x2  lar uchun f ( x1 )  f ( x2 ) bo‘lsa, y  f ( x) funksiya (a;b) oraliqda
o‘suvchi; f ( x1 )  f ( x2 ) bo‘lsa, y  f ( x) funksiya (a;b) oraliqda kamayuvchi deyiladi.
Teskari funksiyani topish
y  f ( x) funksiyaga teskari funksiyani topish uchun
1) y  f ( x) tenglamadan D ( f ) ni hisobga olgan holda x topiladi;
2) hosil bo‘lgan tenglikda x va y larning o‘rni almashtiriladi.
2
Masalan: y 
 3 ( x  1) ga teskari funksiyani topaylik:
x 1
2
2
2
 y  3  x 1 
x
1.
x 1
y 3
y 3
2
Endi x va y larning o‘rni almashtiriladi: y 
 1.
x3
28
Elementar funksiyalar va ularning asosiy xossalari
y  kx  b – chiziqli funksiya
1. Aniqlanish sohasi: D ( y )  R .
2. Qiymatlar sohasi: E ( y )  R .
3. k  0 bo‘lsa, son o‘qida o‘suvchi;
k  0 bo‘lsa, son o‘qida kamayuvchi.
Bu yerda, k - to‘g‘ri chiziqning OX o‘qi bilan hosil qilgan
burchak tangensi.
y  x funksiya
1. Aniqlanish sohasi: D ( y )  R .
2. Qiymatlar sohasi: E ( y )   0;    .
3.  0;  oraliqda o‘suvchi,  ;0 oraliqda kamayuvchi.
4. Juft junksiya.
y  x funksiya
1. Aniqlanish sohasi:
D ( y )  [0, ) .
2. Qiymatlar sohasi:
E ( y )  [0, ) .
3.  0;  oraliqda o‘suvchi,
y
1.
2.
3.
4.
5.
1
funksiya
x
Aniqlanish sohasi: D ( y )  (;0)  (0; ) .
Qiymatlar sohasi: E ( y )  (;0)  (0; ) .
Toq funksiya.
(;0) va (0; ) oraliqda kamayuvchi funksiya.
Asimptotalari: x  0, y  0 .
y  x 2 funksiya
1. Aniqlanish sohasi: D ( y )  (; ) .
2. Qiymatlar sohasi: E ( y )   0;   .
3. Juft funksiya.
4.  0;  oraliqda o‘suvchi,  ;0 oraliqda kamayuvchi.
29
Kvadratik funksiya y  ax 2  bx  c (a  0)
1. Aniqlanish sohasi: D( y )   ;   .
2. Qiymatlar sohasi: a  0 bo‘lsa, E ( y )   y0 ;   ,
a  0 bo‘lsa, E ( y )   ; y0  .
3. Parabola uchining koordinatalari:
b
4ac  b 2
x0   , y0 

2a
4a
y  ax 2  bx  c  a ( x  x0 )2  y0 .
b
4. Simmetriya o‘qi: x   .
2a
5. a  0 bo‘lsa,  x0 ;   oraliqda o‘suvchi,  ; x0  oraliqda kamayuvchi.
a  0 bo‘lsa,  ; x0  oraliqda o‘suvchi,  x0 ;   oraliqda kamayuvchi.
Parabolaning joylashishi:
1. Agar a  0 bo‘lsa:
D0
2. Agar a  0 bo‘lsa:
D0
D0
D0
y  x3 funksiya
1.
2.
3.
4.
Aniqlanish sohasi: D ( y )  (; ) .
Qiymatlar sohasi: E ( y )  (; ) .
Toq funksiya.
(; ) oraliqda o‘suvchi.
30
D0
D0
y   x funksiya
1. Aniqlanish sohasi: D ( y )  R .
2. Qiymatlar sohasi: E ( y )  [0,1) .
3. Davriy, e.k.m. davri 1 ga teng.
y   x  funksiya
1. Aniqlanish sohasi: D ( y )  R .
2. Qiymatlar sohasi: E ( y )  Z .
3. Kamaymaydigan funksiya.
Ko‘rsatkichli funksiya y  a x  a  0, a  1
1. Aniqlanish sohasi: D( y )   ;   .
2. Qiymatlar sohasi: E ( y )   0;   .
3. a  1 bo‘lsa,  ;   oraliqda o‘suvchi;
0  a  1 bo‘lsa,  ;   oraliqda kamayuvchi.
4. Grafigi (0;1) nuqtadan o‘tadi.
5. Asimptotasi: y  0 .
Logarifmik funksiya y  log ax
1. Aniqlanish sohasi:
D( y )   0;   .
2. Qiymatlar sohasi: E ( y )   ;   .
3. a  1 bo‘lsa,  0;  oraliqda o‘suvchi;
0  a  1 bo‘lsa,  0;  oraliqda kamayuvchi.
4. Grafigi (1;0) nuqtadan o‘tadi.
5. Asimptotasi: x  0 .
y  sin x funksiya
31
 0  a  1
1.
2.
3.
4.
5.
6.
Aniqlanish sohasi: D ( y )  R .
Qiymatlar sohasi: E ( y )  [1;1] .
Toq funksiya: sin( x)   sin x .
Eng kichik musbat davri: T  2 .
Nollari: x0   n, n  Z .
Ishora o‘zgarmas oraliqlar:
x   2 n,   2 n   n   b o ‘ ls a , Sinx  0 ;
x     2 n,2 n   n    b o ‘ ls a , Sinx  0 .

 

7 .    2 n;  2 n   n    oraliqlarda o‘suvchi;
2
 2

3



2

n
;

2

n
 2
  n    oraliqlarda kamayuvchi.
2

8 . Eng katta qiymati 1 : Sinx  1  x   2 n, n  Z .
2

9 . Eng kichik qiymati  1 : Sinx  1  x    2 n, n  Z .
2
1 0 .  2 n;  2 n  n    oraliqlarda qavariq;
  2 n;
2  2 n
n  
oraliqlarda botiq.
y  cos x funksiya
1.
2.
3.
4.
5.
6.
Aniqlanish sohasi: D ( y )  R .
Qiymatlar sohasi: E ( y )  [1;1] .
Juft funksiya: cos( x)  cos x .
Eng kichik musbat davri: T  2 .

Nollari: x0    n, n  Z .
2
Ishora o‘zgarmas oraliqlar:

 

x     2 n,  2 n   n    bo‘lsa, Cosx  0 ;
2
 2

3


x    2 n,
 2 n   n    bo‘lsa, Cosx  0 .
2
2

32
7.
   2 n; 2 n   n    oraliqlarda o‘suvchi;
 2 n;   2 n  n   oraliqlarda kamayuvchi.
Eng katta qiymati 1 , Cosx  1  x  2 n, n  Z .
Eng kichik qiymati  1 , Cosx  1  x    2 n, n  Z .


1 0 .   2 n  x   2 n  n  Z  oraliqlarda qavariq.
2
2

3
 2 n  x 
 2 n  n  Z  oraliqlarda botiq.
2
2
8.
9.
y  tgx funksiya

  n, n  Z .
2
Q i y ma t l a r s o ha s i : E ( y )  R .
To q fu n k s i y a : tg ( x)  tgx .
Eng kichik musbat davri: T   .
N o l l a r i : x0   n, n  Z .
Is ho ra o ‘ z g a r ma s o ra l i q l a r :



x    n,   n   n    b o ‘ ls a , tgx  0 ;
2


 

x      n,  n   n    b o ‘ ls a , tgx  0 .
 2


 

  n   n  Z  o ra l i q l a r d a o ‘ s u v c h i.
    n;
2
 2


As i mp t o t a l a r i : x    n, n  Z .
2
1 . A n i q l a n i s h s o ha s i : x 
2.
3.
4.
5.
6.
7.
8.
33
y  ctgx funksiya
1.
2.
3.
4.
5.
6.
7.
8.
A n i q l a n i s h s o ha s i : x   n, n  Z .
Q i y ma t l a r s o ha s i : E ( y )  R .
To q fu n k s i y a : ctg ( x )  ctgx .
Eng kichik musbat davri: T   .

N o l l a r i : x0    n, n  Z .
2
Is ho ra o ‘ z g a r ma s o ra l i q l a r :



x    n,   n   n    b o ‘ ls a , сtgx  0 ;
2


 

x      n,  n   n    b o ‘ ls a , сtgx  0 .
 2

 n;    n, n  Z o ra l i q l a r d a k a ma y u v c h i .
As i mp t o t a l a r i : x   n, n  Z .
y  arcsin x funksiya
A n i q l a n i s h s o ha s i : D ( y )  [1;1] .
  
2 . Q i y ma t l a r s o ha s i : E ( y )    ;  .
 2 2
3 . To q fu n k s i y a : arcSin( x)   arcSinx .
4 . A n i q l a n i s h s o ha s i d a o ‘ s u v c h i fu n k s i y a .
1.
34
y  arccos x funksiya
1.
2.
3.
4.
A n i q l a n i s h s o ha s i : D ( y )  [1;1] .
Q i y ma t l a r s o ha s i : E ( y )   0;  .
To q ha m, j u ft h a m e ma s :
arcCos ( x )    arcCosx .
A n i q l a n i s h s o ha s i d a k a ma y u v c h i fu n k s i y a .
y  arctgx funksiya
1 . A n i q l a n i s h s o ha s i : D ( y )  R .
  
2 . Q i y ma t l a r s o ha s i : E ( y )    ;  .
 2 2
3 . To q fu n k s i y a : arctg ( x)   arctgx .
4 . Aniqlanish sohasida o‘suvchi funksiya.

5 . As i mt o t a la r i y   .
2
y  arcctgx funksiya
1 . Aniqlanish sohasi: D ( y )  R .
2 . Q i y ma t l a r s o ha s i : E ( y )   0;  .
3 . To q h a m e ma s , j u ft ha m e ma s :
arcctg ( x)    arcctgx .
4 . A n i q l a n i s h s o ha s i d a
k a ma y u v c h i fu n k s i y a .
5. As i mp t o t a l a r i y  0, y   .
Hosila
y  f ( x) funksiyaning x  x0 nuqtadagi hosilasi
y
f ( x0  x )  f ( x0 )
y  f ( x0 )  lim
 lim
x  x0 x
x  x0
x
Hosilaning geometrik ma’nosi
y  f  x  funksiya grafigiga x  x0 nuqtada o‘tkazilgan urinmaning burchak
koeffitsienti k  tg  y ( x0 ) .
Urinma tenglamasi:
y  f ( x0 )  f '( x0 )  x  x0  .
35
Hosilaning fizik ma’nosi
Harakatlanayotgan jismning t0 vaqtda bosib o‘tgan yo‘li S  f (t ) , uning t0
vaqtdagi tezligi V (t0 ) , tezlanishi esa a (t0 ) bo‘lsa, u holda
V (t0 )  f (t0 ) , a (t0 )  V (t0 ) .
Differensiallashning asosiy qoidalari

u  v 

u  v 
 u   v ;

 u  uv  uv
;
  
v2
v

 f  g  x 
 uv  uv ;
 f  g  x   g x  .
Funksiyani hosila yordamida tekshirish
y  f  x  funksiya (a, b) oraliqda aniqlangan bo‘lsin.
1) Agar (a, b) oraliqda y  0 bo‘lsa, u holda funksiya shu oraliqda o‘suvchi;
2) Agar (a, b) oraliqda y  0 bo‘lsa, u holda funksiya shu oraliqda kamayuvchi;
3) Agar (a, b) oraliqda y  0 bo‘lsa, u holda funksiya shu oraliqda botiq, agar y  0
bo‘lsa, shu oraliqda qavariq bo‘ladi.
Funksiyaning eng katta va eng kichik qiymatlari
f '  x   0 tenglamaning ildizlari va hosila mavjud bo‘lmagan nuqtalar y  f  x 
funksiyaning statsionar nuqtalari deyiladi.
 a, b kesmada uzluksiz bo‘lgan y  f  x  funksiyaning shu kesmadagi eng katta
va eng kichik qiymatlari f  x1  , f  x2  , . . ., f  xn  , f  a  , f  b  sonlar orasida
bo‘ladi. Bu yerda x1 , x2 ,... xn lar y  f  x  funksiyaning  a, b  oraliqdagi statsionar
nuqtalari.
36
Elementar funksiyalarning hosilalari
Funksiya
Hosilasi
Funksiya
Hosilasi
yС
y  0
yx
y  1
y  xm
y  m  x m 1
y  ctgkx
y  ax
y  a x  ln a
y  arcsin kx
y 
y  arccos kx
y  
y  log a x
y 
1
x ln a
y  
y  Sinkx
y  k  Coskx
y  arctgkx
y  Coskx
y   k  Sinkx
y  arcctgkx
y  tgkx
n
y   g ( x) 
y
1
g ( x)
n
y  n g ( x)
y
n
1
g ( x)
y  Sing ( x )
y  Cosf ( x )
y  log ag ( x )
y 
k
Cos 2 kx
n 1
y  n  g ( x ) 
y  
y 
g ( x)
ng ( x)
g n1 ( x )
g ( x)
n n g n1 ( x)
g ( x)
y  
n n g n1 ( x)
y  Cosg ( x)  g ( x )
y 
Sinx
x
 lim
 1.
x 0 x
x 0 Sinx
Sinpx
3. lim
 p,    p   .
x 0
x
1  (kx) 2
k
y 
1  (kx) 2
k
y  
1  (kx) 2
y  a g ( x ) ln a  g ( x)
g ( x )
Cos 2 f ( x )
g ( x)
y  
Sin 2 g ( x)
y 
y  arcS in f ( x )
y 
y  arcCosf ( x )
y  
y  arctgf ( x)
y  arcctgg ( x)
2. lim x x  1 .
x 0
4. lim(1 
x 0
37
k
y  a g (x)
y  ctgg ( x )
g ( x)
g ( x)ln a
2
y  e x
Ajoyib limitlar
1. lim
1   kx 
y  ex
y  tgf ( x )
y   Sinf ( x)  f '( x)
k
Sin 2 kx
k
1
x) x
 e.
f '( x )
1  f 2 ( x)
f '( x )
1  f 2 ( x)
f '( x )
y 
1  f 2 ( x)
g ( x)
y  
1  g 2 ( x)
C x 1
5. lim
 ln C , C  0 .
x 0
x
n
 1
7. lim 1    e  2,7183... .
n 
n
tgx
 1.
x 0 x
6. lim
Boshlang‘ich funksiya (integral)
Boshlang‘ich funksiya va uni topishning sodda qoidalari
Agar berilgan oraliqdagi barcha x uchun F ( x)  f  x  tenglik bajarilsa, u holda
F ( x) funksiya shu oraliqda f ( x) funksiyaning boshlang‘ich funksiyasi deyiladi.
Agar F ( x) funksiya y  f ( x) funksiyaning boshlang‘ich funksiyasi bo‘lsa:
1) har qanday o‘zgarmas S uchun F ( x)  С ham f ( x) ning boshlang‘ich funksiyasi
bo‘ladi;
1
2)
F (kx  b) funksiya y  f (kx  b) funksiyaning boshlang‘ich funksiyasi bo‘ladi.
k
Elementar funksiyalarning boshlang‘ichlari
Funksiya
Boshlang‘ichi
y  xm
m  1
x m 1
Y
m 1
ax
Y
C
ln a
y  ax
y
1
x
y  Sinkx
y  Coskx
y  tgkx
y
1
x2  a2
y  ex
Funksiya
y  ctgkx
1
Sinx
1
y
Cosx
1
y
Sin 2 x
1
y
Cos 2 x
1
y 2
a  x2
1
y
a2  x2
1
y
x2  a 2
y
Y  ln x  C
1
Y    Coskx  C
k
1
Y   Sinkx  C
k
1
Y   ln Coskx  C
k
1
xa
Y
ln
C
2a x  a
Y  ex  С
38
Boshlang‘ichi
1
Y  ln Sinkx  C
k
x
Y  ln tg  C
2
x 
Y  ln tg     C
2 2
Y  ctgx  C
Y  tgx  C
1
x
Y  arctg  C
a
a
x
Y  arcsin  C
a
Y  ln x  x 2  a 2  C
Aniq integralning asosiy xossalari
b
b
b
с
b
 kf ( x)dx  k  f ( x)dx ;
 f ( x)dx   f ( x)dx   f ( x)dx ;
a
a
a
a
b
b
 f ( x)dx   f ( x)dx ;
a
a
1
 f (kx  p)dx  k  f (t )dt ;
a
b
b
с
kb  p
b
ka  p
b
  f ( x )  g ( x)  dx   f ( x)dx   g ( x)dx .
a
a
a
Nyuton-Leybnits formulasi
Agar F  x  funksiya y  f  x  funksiyaning boshlang‘ich funksiyasi bo‘lsa, u
b
holda  f ( x )dx  F (b)  F (a) .
a
Egri chiziqli trapetsiyaning yuzi
1.
b
b
S   f  x  dx  F  x   F  b   F  a  .
a
a
b
2.
S    f1 ( x )  f 2 ( x)  dx .
a
b
3.
S    f 2 ( y )  f1 ( y ) dy .
a
Aylanma jismning hajmi
Egri chiziqli trapetsiyaning Ox o‘qi atrofida aylanishidan hosil bo‘lgan jism hajmi
b
y  f  x
V    f 2 ( x )dx .
a
a
39
b
GEOMETRIYA
Planimetriya
Tekislikda istalgan uchtasi bir to‘g‘ri chiziqda yotmaydigan n (n  2) ta nuqtalar
n(n  1)
berilgan bo‘lsa, ikkitasini o‘z ichiga oluvchi
ta to‘g‘ri chiziq o‘tkazish
2
n2  n  2
mumkin. Bu to‘g‘ri chiziqlar tekislikni
ta qismga ajratadi.
2
To‘g‘ri chiziqni kichik lotin xarflari a , b, c, d , . . . va nuqtalarni katta lotin xarflari
A, B, C , D, . . . bilan belgilanadi.
Burchaklar
Boshi bir nuqtada bo‘lgan ikkita nurdan tashkil topgan shakl burchak deyiladi,
berilgan nuqta burchakning uchi, nurlar esa burchakning tomonlari deyiladi.
Burchak kattaligi kichik yunon harflari  ,  , ,  ,... bilan belgilanadi.
Burchakning uchidan chiqib, uni teng ikkiga bo‘lgan nur bissektrisa deyiladi.
Turlari:
O‘tkir burchak: 0    90 .
To‘g‘ri burchak:   90 .
O‘tmas burchak: 90    180 .
Yoyiq burchak:   180 .
    180
Qo‘shni burchaklar
1   2 ; 1   2
Vertikal burchaklar.
40
Ikki paralel to‘g‘ri chiziqni uchinchi chiziq kesib o‘tganda
xosil bo‘lgan burchaklar
Mos burchaklar: 1 va 5; 2 va 6; 3 va 7; 4 va 8.
Ichki almashinuvchi burchaklar: 3 va 5; 4 va 6.
Tashqi almashinuvchi burchaklar: 1 va 7; 2 va 8.
Ichki bir tomonli burchaklar: 3 va 6; 4 va 5.
Tashqi bir tomonli burchaklar: 1 va 8; 2 va 7.
1=3=5=7;
2=4=6=8;
0
4+5=3+6=180 .
Uchburchak
ABC uchburchakning mos tomonlari a, b, c va mos burchaklari  ,  ,  orqali
ifodalaymiz.
1)       180 ;
2) uchburchak tengsizligi
a  b  c; b  c  a; a  c  b ;
a  b  c;
b  c  a; a  c  b ;
3) uchburchakning katta burchagi qarshisida katta
tomoni, kichik burchagi qarshisida kichik tomoni
yotadi;
4) tashqi burchaklari: 1 , 1,  1 : 1  1   1  360 ;
1     ; 1     ;
1     .
Uchburchaklarning tengligi
Agar ABC va A1B1C1 larda:
1) a  a1 , b  b1 ,    1 ;
2) a  a1 ,   1 ,    1 ;
3) a  a1 , b  b1 , с  с1 bo‘lsa, u holda
ABC va A1B1C1 lar teng bo‘ladi.
O‘xshash uchburchaklar
1. ABC  A1B1C1
A  A1 , B  B1, C  C1 ,

  AB
AC
BC


 A B AC B C .
 1 1
1 1
1 1
41
2. A  A1 , B  B1  ABC  A1B1C1 .
AB
AC
3.

, A  A1  ABC  A1B1C1 .
A1B1 A1C1
AB
AC
BC
4.


 ABC  A1B1C1 .
A1B1 A1C1 B1C1
2
2
S ABC  AB   PABC 
5. ABC  A1B1C1 

 .
 
S A1B1C1  A1B1   PA1B1C1 
Gomotetiya
ABC  A1B1C1 .
Balandlik, bissektrisa, mediana
Balandlik
Uchburchak uchidan shu uch qarshisidagi
tomonga tushirilgan perpendikulyar balandlik
deyiladi.
ha , hb , hc –uchburchak balandliklari bo‘lsin:
2S
1) ha 
 bSin  cSin ;
a
1 1 1
2) ha : hb : hc  : :  bc : ac : ab ;
a b c
1 1 1 1
3)    , r – ichki chizilgan aylana
r ha hb hc
radiusi.
Mediana
Uchburchak uchi bilan shu uch qarshisidagi tomon o‘rtasini tutashtiruvchi kesma
mediana deyiladi.
42
Uchburchakning uchta medianasi bir nuqtada kesishadi va uchidan boshlab
hisoblaganda shu nuqtada 2:1 nisbatda bo‘linadi.
Medianalar kesishish nuqtasi uchburchakning og‘irlik markazi deyiladi.
ma , mb , mc –uchburchakning medianalari bo‘lsin:
1
1 2
1) AP  ma 
2 b 2  c2  a 2 
b  c 2  2bcCos ;
2
2
2
2) a 
2 mb 2  mc 2  ma 2 ;
3
1
3) OE  BD ;
6
1
4) S EOP  S EOQ  S ABC ;
24
SBQE  SBEP  1 / 8 SABC .




Bissektrisa
Uchburchak uchidan chiqib, shu uchidagi burchakni
teng ikkiga bo‘luvchi va ikkinchi uchi shu burchak
qarshisidagi tomonda yotuvchi kesma bissektrisa
deyiladi.
Bissektrissalar la , lb , lс – bilan belgilanadi.
1) 2 1   ;
a c
S1 b
2)  1 ;
 ;
b c2
S2 a

2
abCos
1
2  ba  с с .
3) lс 
ab  a  b  c  c  b  a  
1 2
ba
ba
Ichki va tashqi chizilgan aylanalar
Ichki chizilgan aylana markazi bissektrisalar kesishgan nuqtada bo‘ladi. Ichki
chizilgan aylana radiusini r bilan belgilaymiz.
Tashqi chizilgan aylana markazi uchburchak tomonlarining o‘rta
perpendikulyarlari kesishgan nuqtada bo‘ladi. Tashqi chizilgan aylana radiusini R
orqali belgilaymiz.
p ( p  a )( p  b)( p  c )
abc
S
1) r  
; p
;
2
p
p
abc
abc
2) R 

;
4 S 4 p ( p  a )( p  b)( p  c)
a
b
c
3) R 
;


2sin  2sin  2sin 
43
4) Ichki va tashqi chizilgan aylanalar markazlari
orasidagi masofa O1O2  R 2  2 Rr ;
cba
5) y 
.
2
1) S
2) S
3) S
4) S
Uchburchak yuzi
a  ha b  hb c  hc
1
a 2 Sin Sin



,
S  abSin 
;
2
2
2
2
2 Sin
 p ( p  a )( p  b)( p  c) (Geron formulasi);
abc
abc

 pr , p 
;
4R
2
4
m  mb  mc

m(m  ma )(m  mb )(m  mc ) , m  a
.
3
2
Uchburchakdagi asosiy teoremalar
1. Sinuslar teoremasi:
a
b
c


 2R ,
Sin Sin Sin
R – tashqi chizilgan aylana radiusi.
2. Kosinuslar teoremasi:
a 2  b 2  c 2  2bcCos .
3.
a  bCos  cCos .
4. Molveyde formulasi:
 
Cos
ab
2 .


c
Sin
2

 p  b  p  c  ;
6. Sin 
2
bc
5. Tangenslar teoremasi
 

tg
ctg
ab
2 
2 .

a  b tg    tg   
2
2
p p  a

.
Cos 
2
bc
Teng yonli uchburchak
4a 2  c 2
c 4a 2  c 2
; S
;
2
4
c  2a  c 
a2
R
;
r
.
2hc
4hc
hc 
44
To‘g‘ri burchakli uchburchak
1) а 2  в 2  с 2 (Pifagor teoremasi);
b
a
b
a
2) Sin  , Cos  ; tg  , ctg  ;
c
c
a
b
2
2
3) a  ca,
b  cb;
ab
h 2  ab,
hс  ;
c
c
abc
ab
4) R   mc , r 
,
rR
;
2
2
2
ab ch a 2tg c 2 Sin 2
5) S 



 r 2  2rR;
2
2
2
4
2
2
d  b
 y b
6)    ,
   ,
a
a
c
x
l – bissektrisa.
Teng tomonli uchburchak
3a
3a
, r
, R  2r ;
3
6
3 2
h  3 r  1,5 R  r  R, S 
a .
4
R
To‘rtburchaklar
Kvadrat
2
d
P  4a ,
S  a2 
,
2
a
d
r ,
R .
d a 2,
2
2
P  2(a  b) ,
d  а 2  b2 ,
To‘g‘ri to‘rtburchak
S  ab ,
d
R .
2
Parallelogramm
1) P  2(a  b) ;
2) 2a 2  2b 2  d12  d 2 2 ;
1
3) S  a  ha  a  b  Sin , S  d1  d 2  Sin .
2
45
Romb
1
2) S  a  h  2a  r  d1  d 2 ,
2
3) h  2r .
1) 4a 2  d12  d 2 2 ;
S  a 2  Sin ;
Trapetsiya
1
1) S  mh  d1  d 2 Sin ;
2
c 2 a  d 2b
2) d1  ab 
;
ab
3) S 2  S 4  S1  S3 ,
S

S1  S3
2
.
4) AE=EC, BF=FD;
ab
MN 
–o‘rta chiziq;
2
ab
b AO OD a
EF 
, ME  FN  ;

 .
2
2 OC OB b
Teng yonli trapetsiya
2
ba
b  a

2
2
x
, c h 
;
2
4
1
ab
S  d 2 Sin 
h;
2r  h  ab .
2
2
To‘g‘ri burchakli trapetsiyada
d12  d 22  a 2  b2 .
Aylanaga tashqi va ichki chizilgan to‘rtburchaklar
1) Aylanaga tashqi chizilgan to‘rtburchak
acbd ;
S  pr  (a  c)r  (b  d )r , bu yerda 2 p  a  b  c  d .
2) Aylanaga ichki chizilgan to‘rtburchak
        180 ;
ad  bc  d1  d 2 ;
S
R
 p  a  p  b  p  c  p  d  ;
1
4S
 ab  cd  ac  bd  ad  bc  .
46
Ko‘pburchaklar
1) Ichki burchaklari yig‘indisi:  n  2   ;
2) Tashqi burchaklari yig‘indisi: 2 ;
n  n  3
3) Diagonallari soni:
ta.
2
Muntazam ko‘pburchaklar
1) Ichki burchagi:
2) Tashqi burchagi:
 n  2 ;
n
2
;
n
1
4R2  a2 ;
2
r  n  a n  a 4R 2  a2
4) S n 

;
2
4


5) a  2 R  Sin  2r  tg .
n
n
3) r 
Muntazam beshburchak, oltiburchak, sakkizburchak
Muntazam beshburchak
Ichki burchaklari yig‘indisi: 540 ;
Ichki burchagi: 108 ;
Tashqi burchagi: 72 ;
R
a
10  2 5  2r 5  2 5 ;
2
a
R
a
R
50  10 5  r 5  1 ; r 
5 1 
25  10 5 ;
10
4
10
5 2
a2
1 5
d
a , d – diagonal; S  R 10  2 5 
25  10 5  5r 2 5  2 5 .
2
8
4




Muntazam oltiburchak
Ichki burchaklari yig‘indisi 7200;
Ichki burchagi: 120 ;
Tashqi burchagi: 60 ;
2
a 3
aR r 3; r
;
d1  a 3 ,
d 2  2 R  2a ;
3
2
3
3
S  R 2 3  a 2 3  2r 2 3 .
2
2
47
Muntazam sakkizburchak
Ichki burchaklari yig‘indisi 1080 ;
Ichki burchagi: 135 ;
Tashqi burchagi: 45 ;
a  R 2  2  2r
R


2 1 ;
a
R
a
42 2  r 42 2 ; r 
2 2 
2
2
2
 
2  2a  2  1  8r 
d1  a 2  2 ;
S  2R2


2 1 ;
d 2  a 1  2 ; d3  2 R  a 4  2 2 ;
2
2

2 1 .
Aylanadagi burchaklar
x
 
2


2
2
a  b  c c
y

 
2
 
2

Aylanadagi teoremalar
PL  PK  PN  PM
AS  BS ,
ASO  BSO
48
 
2
ab  cd
Doira, sektor, segment, halqa
Aylana va doira
1) Aylana uzunligi: l  2 R ;
 D2
2) Doira yuzi: S   R 2 
;
4
 R
3) Yoy uzunligi: lёй 
.
180
Sektor
S сект
yuzi
 R 2

.
360
Segment yuzi
 R 2 1 2
S сегм 
 R Sin .
360 2
Aylananing ikki paralel vatarlari
orasidagi bo‘lagi yuzi
S кес
 R2
1

      R 2  Sin  Sin  .
360
2
Halqa yuzi


Sх   R2  r 2 .
STEREOMETRIYA
Prizma
Ixtiyoriy prizma
Yon sirti: Sён  Pп к  l .
To‘la sirti: SТ  Sён  2 S ас .
Hajmi: V  S п.к .  l  S ac  h .
Diagonallar soni: n (n  3) .
Bu yerda S п.к . – perpendikulyar kesim yuzi, Рп.к. perpendikulyar kesim perimetri.
To‘g‘ri burchakli parallelepiped:
Yon sirti: Sён  2  ac  bc  .
To‘la sirti: Sт  2  ab  ac  bc  .
Hajmi: V  abc .
49
d 2  a 2  b2  c 2 .
3 ta simmetriya tekisligiga ega.
8 ta uchi, 12 ta qirrasi, 6 ta yoqi, 4 ta dioganali bor.
Kub
Yon sirti:
To‘la sirti:
Sён  4а 2 .
SТ  6a 2 .
V  a3 .
a
a 3
d  a 3 ; r  ;R 
.
2
2
Hajmi:
9 ta simmetriya tekisligiga ega.
Piramida
Ixtiyoriy piramida
To‘la sirti: ST  S ac  Sён .
1
1
Hajmi: V  S ас  h  SТ r .
3
3
Muntazam piramida
l –yon qirra, f – apofema.
nar
Pас  n  a , S ac 
.
2
1
S ён  Pас  f . S ас  SёнCos ,  –asosidagi ikki yoqli burchak.
2
2
a
2
R     r 2 ; l 2  R 2  h2 ; f 2  r 2  h2 .
2
Kesik piramida
1
V  h S1  S1S 2  S 2 .
3
Muntazam kesik piramida uchun:
1
S ён   P1  P2   l , l -apofema.
2
S т  S1  S 2  Sён .


Muntazam uchburchakli piramida
l – yon qirra, f – apofema,
 – ikki yoqli burchak.
a2
a2
2
f 
H .
l
 H2 .
12
3
3
1
a2 3
a2 3
S ён  af , S ас 
,
V  Sас  H 
H.
2
4
3
12
50
Muntazam to‘rtburchakli piramida
l – yon qirra,
f – apofema.
a2
f 
 H2 .
4
a2
a 3
a 3
l
 H2 . r 
, R
.
2
6
3
1
a2 3
S ac
2
, Sас  a , V  Sас  H 
H.
S ён  2af 
3
12
Cos
Silindr
R –asosining radiusi, H –balandlik.
Sас   R 2 .
S ён  2 RH .
Sт.с.  2 R  R  H  .
V   R2H .
Yon sirti yoyilmasi:
Konus
L -yasovchi, R -asosinig radiusi, H -balandlik.
L  R2  H 2 .
S ён   RL .
Sт.с.   R  R  L  .
1
1
V   R 2 H  Sён d .
3
3
Yon sirti yoyilmasining uchidagi
burchakni topish:
  2 R / L .
Kesik konus
L -yasovchi, R, r -asosining radiuslari, H -balandlik.
 R  r 2  H 2 .
Sён   L  R  r  .
1
Sт.с.   R 2  r 2  L  R  r  . V   H R 2  Rr  r 2 .
3
L

Sfera va shar
R -radiusi,
S  4 R 2   d 2 .



d -diametr.
4

V   R3  d 3 .
3
6
51
Shar segmenti
R -sharning radiusi, h -segment balandligi.
r  h  2R  h  .
Sён  2 Rh   r 2  h 2 .


Sт.с.   2 Rh  r
2
.
 h2
V
3


 3R  h   h  3r
6
2

 h2 .
Shar sektori
Sт.с.   R  2h  r  .
2 2

V
R h  d 2h .
3
6
Shar halqasi
S ён  2 RH .


Sт.с.   2 RH  R12  R2 2 .
1
V   H 3R12  3R22  H 2 .
6


Sharga ichki chizilgan konus
l -yasovchi, R -sharning radiusi,
H -konusning balandligi, r -radiusi.
r2  H 2
R
.
x  H  R.
2H
Konusga ichki chizilgan shar
RH
r
.
LR
Tekislikda Dekart koordinatalar sistemasi
To‘g‘ri chiziq tenglamasi
1) To‘g‘ri chiziq tenglamasi y  kx  b , bu yerda k –
burchak koeffitsienti k  tg ,  – Ox o‘qi bilan hosil
qilgan burchak;
2) A1  x1; y1  va A2  x2 ; y2  nuqtalardan o‘tuvchi to‘g‘ri
chiziq tenglamasi:
y  y1
x  x1

; y  k ( x  x1 )  y1 ,
y2  y1 x2  x1
y  y2
bunda k  1
;
x1  x2
52
3) A  x1, y1  nuqtadan o‘tuvchi to‘g‘ri chiziq: y  y1  k ( x  x1 ) ;
4) Ikki to‘g‘ri chiziq orasidagi burchak tangensi:
k k
tg  1 2 ;
1  k1  k2
5) Ikki to‘g‘ri chiziqning parallellik alomati: k1  k2 ;
6) Ikki to‘g‘ri chiziqning perpendikulyarlik alomati: k1  k2  1 ;
7) Ikki to‘g‘ri chiziqning kesishish alomati: k1  k2 ;
8) A(x1, y1 ), B(x2 , y2 ) va С(x3 , y3 ) nuqtalarning bir to‘g‘ri chiziqda yotish sharti:
x0  x1 y0  y1

;
x2  x0 y2  y0
9) A( x0 , y0 ) nuqtadan ax  by  c  0 to‘g‘ri chiziqqacha bo‘lgan masofa:
ax  bу0  c
d 0
;
a 2  b2
10) Parallel to‘g‘ri chiziqlar orasidagi masofa:
c c
h 1 2 ;
a 2  b2
11) Tekislikda uchlari A(x1, y1 ) , B(x 2 , y2 ) va C(x 3 , y3 )
nuqtalarda bo‘lgan ABC uchburchakning yuzi
1
S  ( x2  x1 )( y3  y1 )  ( x3  x1 )( y2  y1 ) .
2
Aylana tenglamasi
Markazi  a; b  nuqtada, radiusi R ga teng aylana tenglamasi:
 x  a
2
2
  y  b   R2 .
Fazoda Dekart koordinatalar sistemasi
Fazoda A( x1, y1 , z1 ), B( x2 , y2 , z2 ) va С(x 3 , y3 , z3 ) nuqtalar berilgan bo‘lsin.
1. AB kesma uzunligi AB  (x 2  x1 ) 2  ( y2  y1 ) 2  ( z2  z1 ) 2 .
2. AB kesma o‘rtasining koordinatasi
x x x
y  y2  y3
z z z
x 1 2 3, y 1
, z 1 2 3.
2
2
2

3. AB kesmani
nisbatda bo‘lish

 x   x2
 y   y2
 z   z2
.
x 1
, y 1
, z 1
 
 
 
4. ABC uchburchak medianalari kesishgan O( x; y; z ) nuqta
koordinatasi:
53
x1  x2  x3
y  y2  y3
z z z
, y 1
, z 1 2 3.
3
3
3
5. Markazi  a; b; c  nuqtada bo‘lgan R radiusli sfera tenglamasi:
x
 x  a
2
2
2
  y  b    z  c   R2 .
Fazoda vektorlar

Boshi A  x0 ; y0 ; z0  va oxiri B  x1; y1; z1  nuqtalarda bo‘lgan vektor АВ kabi



belgilanadi. Vektorlar kichik lotin а , b , с . . . harflari bilan belgilanadi.



AB  x1  x0 ; y1  y0 ; z1  z0 .
1. Koordinatalari:

AB  ( x1  x0 )2  ( y1  y0 ) 2  ( z1  z0 )2 .
2. Moduli (uzunligi):
Birlik vektorlar






i  (1,0,0), j  (0,1,0), k  (0,0,1) ; i  1, j  1, k  1 ;
 
 
 




i  j  0, i  k  0, k  j  0 ;
a( x; y; z )  xi  y j  zk .
 
x
y
z
e
;
;
 x2  y 2  z 2 x2  y2  z 2 x2  y 2  z 2


 – birlik vektor.


Kollinear va komplanar vektorlar
Bir to‘g‘ri chiziqqa parallel bo‘lgan vektorlar, kollinear vektorlar deyiladi.
Kollinear vektorlar bir xil yo‘nalgan yoki qarama qarshi yo‘nalgan bo‘ladi.
Bir xil yo‘nalgan vektorlarning uzunliklari teng bo‘lsa, ular teng vektorlar
deyiladi.
Uzunligi nolga teng bo‘lgan vektor, nol vektor deyiladi.
Bir tekislikka parallel bo‘lgan uchta vektor komplanar vektor deyiladi.
a  x1; y1; z1  , b  x2 ; y2 ; z2 

Vektorlar ustida amallar
vektorlar berilgan bo‘lsin.

1) a  b  x1  x2 ; y1  y2 ; z1  z2 ;

2)  a   x1;  y1;  z1

3) a  a  a
2
2
a ,

(   R );
a  aa ;
4) Skalyar ko‘paytma
54
 

a) a  b  x1  x2  y1  y2  z1  z2 ;
5) Parallelik sharti:
b) a  b  a  b  Cos a b ;
x1 y1 z1

 ;
x2 y2 z2
6) Perpendikulyarlik sharti: a  b  0 ;
7) Vektorlar orasidagi burchak kosinusi:
 

Cos a b 
a b

ab
x1x2  y1 y2  z1z2
x12

y12

z12
x22

y22

z22
.
SIMMETRIYA
O‘qqa nisbatan simmetriya
Hususiy xollarda
1) Berilgan oy o‘qiga nisbatan, ox o‘qiga nisbatan simmetriya shakllar
2) y  kx  b va y   kx  b to‘g‘ri chiziqlar oy o‘qiga nisbatan simmetrik; y  kx  b
va y   kx  b lar ox o‘qiga nisbatan simmetrik;
3) O‘zaro teskari funksiyalar grafiklari y  x to‘g‘ri chiziqqa nisbatan simmetrik
bo‘ladi;
4) Juft funksiyaning grafigi oy o‘qiga nisbatan, toq funksiyaning grafigi esa
koordinatalar boshiga nisbatan simmetrik bo‘ladi;
5) To‘g‘ri burchakli parallelepipedda 3 ta, kubda esa 9 ta simmetriya tekisligi mavjud.
55
Nuqtaga nisbatan simmetriya
Berilgan shakl
Koordinatalar boshiga
nisbatan simmetriya
Tekislikka nisbatan simmetriya
ABC D to‘rtburchak va ABCD to‘rtburchak
MNKL tekislikka nisbatan simmetrik.
Bunda
MA  MA; NB  NB;
KC  KC ; LD  LD.
Muntazam ko‘pyoqlar haqida ma’lumotlar
Tetraedr
Kub
Oktaedr
Dodekaedr
Ikosaedr
R
r
Cos
a 6
4
a 3
2
a 2
2
a 3
5 1
a 6
12
a
2
a 6
6
1
3
a
10  2 5
4
0

a 25  11 5
2
10
1
3
5

5
S
a
2
V
3
6a 2
a3
2
a3 2
3
2a
3
3a 2 25  10 5
a (3  5)
5

5a 2 3
5
4 3
Bu yerda  ikki yoqli burchak.
56
a2 2
12
a3
15  7 5
4
5a 3
(3  5)
12
Ba’zi yig‘indilar
n(n  1)
;
2
1)
1  2  3  4  5  6  ...  n 
2)
1  3  5  7  9  ...  (2n  1)  n 2 ;
3)
2  4  6  8  10  ...  2n  n(n  1) ;
4)
12  22  32  ...  n 2 
5)
 n(n  1) 
1  2  3  ...  n  
 ;
 2 
6)
13  33  53  ...  (2n  1)3  n2 (2n 2  1) ;
7)
14  24  34  ...  n 4 
8)
2 2  42  6 2  ...  (2n)2 
9)
20  21  22  23  2 4  ...  2 n1  2 n  1 ;
n(n  1)(2n  1)
;
6
2
3
3
3
3
n(n  1)(2n  1)(3n 2  3n  1)
;
30
2n(n  1)(2n  1)
;
3
10) 2 2  62  10 2  ...  (4n  2)2 
11) 12  32  52  ...  (2n  1)2 
4n(2n  1)(2n  1)
;
3
n(n  1)(n  2)
;
3
12) 1  4  2  7  3  10  ....  n(3n  1)  n(n  1) 2 ;
13) 1  2  3  2  3  4  3  4  5  ....  n( n  1)( n  2) 
n(n  1)(n  2)(n  3)
;
4
n(n 2  1)(3n  2)
14) 1  2  2  3  3  4  ....  ( n  1)n 
;
12
2
2
2
2
15) 2  12  3  2 2  4  32  ....  ( n  1) n 2 
n(n  1)(n  2)(3n  1)
;
12
57
a (a n  1)
16) a  a  a  a  a  ....  a 
,
a 1
0
1
2
3
4
n
a  1;
17) 2  20  3  21  4  2 2  ....  (n  1)2n 1  n  2 n ;
18)
1
1
1
1
n
;


 ... 

1 2 2  3 3  4
n  ( n  1) n  1
19)
1
1
1
n
;

 ... 

1 3 3  5
(2n  1)  (2n  1) 2n  1
20)
1
1
1
1
n
;


 ... 

1  5 5  9 9  13
(4n  3)  (4n  1) 4n  1
21)
1
1
1
1
n
;


 ... 

1  6 6  11 11  16
(5n  4)  (5n  1) 5n  1
3
7
13
n 2  n  1 n (n  2)
22)


 ... 

;
1 2 2  3 3  4
n  (n  1)
n 1
23)
12
22
32
n2
n(n  1)


... 

;
1 3 3  5 5  7
(2n  1)  (2n  1) 2(2n  1)
24)
7
7
7
7
1
;


 ... 
1
1  8 8  15 15  22
(7 n  6)(7 n  1)
7n  1
25)
1 2
3
n
n2
 2  3  ...  n  2  n ;
2 2
2
2
2
26)
1 3 5
2n  1
2n  3
 2  3  ...  n  3 
;
2 2
2
2
2n
27)
0 1 2 3
n 1
1
    ... 
1 ;
1! 2! 3! 4!
n!
n!
28)
1
1
1
1
n
;


 ... 

1  4 4  7 7 10
(3n  2)(3n  1) 3n  1
29)
1
1
1
1
n
;


 ... 

45 5 6 67
( n  3)( n  4) 4( n  4)
58
30)
1
1
1
1
1
1
;


 ... 
 
4  8 8  12 12  16
4 n(4n  4) 16 16(n  1)
31)
1
1
1
1
n
;


 ... 

1  10 10  19 19  28
(9n  8)(9n  1) 9n  1
32)
1
1
1
1
n
;


 ... 

5  11 11  17 17  23
(6n  1)(6n  5) 5(6n  5)
1
22
32
n2
n(n  1)
33)


 ... 

;
1 3 3  5 3  5
(2n  1)  (2 n  1) 2(2n  1)
34)

1
1
1
1
11
1


 ... 
  
;
1 2  3 2  3  4 3  4  5
n  ( n  1)( n  2) 2  2 (n  1)(n  2) 
35)
1
1
1
1
n(n  1)
;


 ... 

1 3  5 3  5  7 5  7  9
(2n  1)(2n  1)(2n  3) 2(2n  1)(2n  3)
36)

1
1
1
11
1

 ... 
  
;
1 2  3  4 2  3  4  5
n  ( n  1)(n  2)(n  3) 3  6 ( n  1)(n  2)(n  3) 
1 
1
 1  1  1  
37) 1  1  1   ....1 
;

 2  3  4   n  1  n  1
1 
1 
1  
1  n2

38) 1  2 1  2 1  2  ....1 

;
2 
 2  3  4   (n  1)  2n  2
39) 1  1! 2  2! 3  3! ....  n  n !  (n  1)! 1 ;
40)
3
4
n2
1
1
.

 ... 
 
1! 2! 3! 2! 3! 4!
n! (n  1)! (n  2)! 2! (n  2)!
59
60