Elementar matematika formulalari

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

Elementar matematika formulalari

Products

Support

Elementar matematika formulalari

Add this document to collection(s)

Add this document to saved

Suggest us how to improve StudyLib