ប្រជុំរូបមន្តគណិតវិទ្យា by Van Vannak

eroberogeday

lwm pl:ún nig Esn Bisidæ briBaØabR&tEpñkKNitviTüa

−b ± aa22

b 2 − 4a c == || aa|| 2a ∆y iθ ∆x β

x ∫ x dx = n + 1 n

i2 = − 1

n+1

z = a + i.b

dy + c dx

2 (X − X) ∑ i i =1

1 2 = 1 + cot φ 2 sin φ x

 1 e = lim  1 +  xx→∞ →∞  x

គណះកមមករនពន្ឋ នង ិ ិ េរៀបេរៀង េ

េ

ក លម ឹ ផលគន ុ

ក ែសន ពសដ្ឋ ិ ិ

គណះកមមករ្រតួតពនតយបេចច កេទស ិ ិ េ េ េ

ក លម ឹ ឆុន

េ

ក នន់ សុខ

េ

ក្រសី ទុយ រ ី

េ

ក អុង ឹ សំ

ក ទតយ ិ េម៉ង

ក ្រពម ឹ សុនតយ ិ

គណះកមមករ្រតួតពនយអកខ ិ េ

ក លម ឹ មគក ិ សរិ

ក អុង ឹ សំ

កញញ លី គុ

វរុិ ទ្ឋ

រចនទំពរ័ នង ិ ្រកប

ករយកុំ ពយទ័ ិ ូ រ

េ

ង

េ

្ណ ក -i-

ក ្រពំ ម៉

matikar

CMBUkTI1 CBUkTI2 CMBUkTI3 CMBUkTI4 CMBUkTI5 CMBUkTI6 CBUkTI7 CMBUkTI8 CMBUkTI9 CMBUkTI10 CMBUkTI11 CBUkTI12 CMBUkTI13 CMBUkTI14 CMBUkTI15 CMBUkTI16 CMBUkTI17 CMBUkTI18

tk;viTüa sMNMu cMnYn BhuFa RbB&næráb´ smIkar nig vismIkar sVIúténcMnYnBit GnuKmn_RtIekaNmaRt GnuKmn_Gics,ÚNg´Esül nig elakarIt lImIt nig PaBCab´énGnuKmn_ edrIevénGnuKmn_ GaMgetRkal smIkarDIeprg´Esül viuTr&kñúglMh cMnYnkMupøic RTwsþIbTkñúgRtIekaN vismPaBcMnYnBit PaBEckdac´ nig viFIEckGWKøIt GnuKmn_GIuEBbUllik viPaKbnßM nig RbUáb‘ÍlIet

TMB&r 001 005 007 012 017 028 035 039 049 057 065 069 082 088 101 111 116 131

RbCMurUbmnþKNitviTüa

CMBUkTI01

tk;viTüa

1-sMeNI niymn&y - sMeNI KWCaGMN¼GMNagTaMgLayNaEdleKGacseRmcfaBit ¦k¾ minBit . - eKtageQµa¼énsMeNIedayGkßr p , q , r , s , .... . - ebI p CasMeNIBitena¼ p mantémøPaBBitesµInwg 1 KW t¿ (p) = 1 - ebI p CasMeNIminBitena¼ p mantémøPaBBitesµInwg 0 KW t¿ (p) = 0 2-Qñab´tk;viTüa k¿Qñab´nig ( ∧ ) - eKkMnt´sresr p ∧ q Ganfa p nig q - sMeNI p ∧ q BitEtkñúgkrNIsMeNI p nig q Bit x¿Qñab´¦ ( ∨ ) - eKkMnt´sresr p ∨ q Ganfa p ¦ q - sMeNI p∀∀q minBitEtkñúgkrNIsMeNI p nig q minBitTaMgBIr

-1-

RbCMurUbmnþKNitviTüa K¿Qñab´mig ( ) - eKkMnt´sresr p Ganfa min p - sMeNI p nig sMeNI p mantémøPaBBitxusKña . X¿QñabńaM[ ( ⇒ ) - eKkMnt´sresr p ⇒ q Ganfa p naM[ q - sMeNI p ⇒ q minBitEtkñúgkrNIsMeNI p Bit nig q minBit eRkABIen¼vaCasMeNIBit . p CalkçxNÐRKb´RKanédIm,I[ q . q CalkçxNÐcaMácédIm,I[ p . g¿Qñab´smmUl ( ⇔ ) - eKkMnt´sresr p ⇔ q Ganfa p smmUl q - sMeNI p ⇔ q BitEtkñúgkrNIEdlsMeNI p nigsMeNI q mantémøPaBbitdUcKña . p CalkçxNÐcaMác´ nigRKb´RKanédIm,I[ q . - CaTUeTA p ⇔ q = (p ⇒ q) ∧ (q ⇒ p)

-2-

RbCMurUbmnþKNitviTüa 3-RbePTénsRmaybBa¢ak´ k¿ sRmaybBa¢ak´edaypÞal´ RbePTénsRmaybBa¢ak´en¼KWCakarRsaybBa¢ak´Rtg´@eTAtamGIVEdl eKcg´án . x¿ sRmaybBa¢ak´tamsMeNIpÞúyBIsmµtikmµ rebobeda¼Rsay «bmafa eKcg´bgHajsMeNI p ⇒ q Bit - CMh‘anTI1 RtUvkMnt´sMeNI p nig sMeNI q [ánRtwmRtUv. - CMh‘anTI2 RtUvkMnt´sMeNI p nigsMeNI q . - CMh‘anTI3 eKepþImBI q bBa©ak´rhUteKánsMeNI p EdlCa sMeNIpÞúyBIsmµtikmµ KWmann&yfaeKánbgHajfa q ⇒ p Bit dUcen¼eKánsMeNI p ⇒ q Bit . K¿ sRmaybBa¢ak´pÞúyBIkarBit rebobeda¼Rsay - CMh‘anTI1 tag p CasMeNIEdlRtUvbgHaj . - CMh‘anTI2 RtUvkMnt´sMeNI p . - CMh‘anTI3 «bmafasMeNI p Bit rYcbkRsaybnþbnÞab´rhUt -3-

RbCMurUbmnþKNitviTüa dl´ánlTæplpÞúyBIRTwsþIKNitviTüa . eKánsMeNI p minBit . dUcen¼sMeNI p Bit (eRBa¼témøPaBBitrvagsMeNI p nig p mantémøpÞúyKña ) . X¿ sRmaybBa¢ak´tameTVlkçxNÐ rebobeda¼Rsay - CMh‘anTI1 bgHajlkçxNÐcaMác´ p ⇒ q - CMh‘anTI2 bgHajlkçxNÐRKb´RKan´ q ⇒ p g¿ sRmaybBa¢ak´tam«TahrN_pÞúj viFIen¼tRmUv[eKrk«TahrN_mYymkbBa¢ak´fasMeNIEdlRtUvbgHaj CasMeNIminBit .

-4-

RbCMurUbmnþKNitviTüa

CMBUkTI02

sMNMu sMNMu KWCabNþúMénvtSú EdlkMntédaylkçxNÐCak´lak´ . cMnYnFatuénsMNMu A tageday n( A ) . sMNMuTeT KWCasMNMuEdlKµanFatuesa¼ ehIytageday φ . karkMnt´sMNMumanBIrrebob ½ kMnt´tamkarerobrabéQµa¼Fatu nig kMnt´tamlkçN¼rYménFatu . sMNMurab´GsĆasMNMuEdlmancMnYnFatuCacMnYnkMnt´ . sMNMu GnnþCasMNMuEdlmancMnYnFatueRcInrab´minGs´ . sMNMuesµIKñakalNasMNMuTaMgBIrmanbBa¢IeQµa¼FatudUcKña . A CasMNMurgén B lu¼RtaEtRKb´ x ∈ A ena¼ x ∈ B ebI A CasMNMurgén B ena¼ n( A ) ≤ n(B ) . ebI A CasMNMurgpÞal´én B ena¼ n( A ) < n(B ) . sMNMusklKWCasMNMuEdlmanRKb´FatuEdleKáneRCIserIs ykmksikßa . sMNMurgbMeBj A = { x / x ∈ A , x ∈ U } sMNMuRbsBV A ∩ B = { x ∈ A nig x ∈ B }

-5-

RbCMurUbmnþKNitviTüa sMNMuRbCMu A ∪ B = { x ∈ A ¦ x ∈ B } sMNMu A nig B CasMNMudac´Kñalu¼RtaEt A ∩ B = φ . cMeBa¼RKb´sMNMu A nigsMNMuskl U eKán ½ A∩A = φ , A∪A = U . ebI A nig B CasMNMurab´Gs´ena¼eKánrUbmnþ ½

n ( A ∪ B ) = n( A ) + n( B ) − n( A ∩ B )

lkçN¼

DeMogan

A∪B= A∩B

;

½ A∩B = A∪B

lkçN¼pþMú 1/

A ∩ ( B ∩ C) = ( A ∩ B ) ∩ C

2 / A ∪ ( B ∪ C) = ( A ∪ B ) ∪ C

lkçN¼bMEbk 1/

A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C)

2 / A ∪ ( B ∩ C) = ( A ∪ B ) ∩ ( A ∪ C )

-6-

RbCMurUbmnþKNitviTüa

CMBUkTI03

cMnYn BhuFa RbBn&æráb´ 1¿ cMnYn bnÞat´cMnYn Kl´ EpñkGviC¢man

EpñkviC¢man O

cMnucÉkta

lkçN¼éntémødac´xat . | a | = a ebI a ≥ 0 . | a | = − a ebI a < 0 sMNMucMnYnKt´FmµCati IN = { 1 , 2 , 3 , ..... } sMNMucMnYnKt´rWLaTIhV Z = { ... , − 2 , − 1 , 0 , 1 , 2 , ... } m cMnYnsniTanmanTRmg´ Edl m nig n CacMnYnKt´rILaTIb n sMNMucMnYnsniTantageday Q

-7-

RbCMurUbmnþKNitviTüa RKb´cMnYnBitviC¢man a nig b eKán cMeBa¼ a > b nig b > 0 eKán ½

ab = a . b a = b

a<b⇔ a <

a 2 = ( a )2

a b

TRmg´BnøaténRbB&næráb´eKal 10 manrag abcd10 = a × 103 + b × 102 + c × 10 + d

TRmg´BnøaténRbB&næráb´eKal 2 manrag abcd 2 = a × 23 + b × 22 + c × 2 + d

2¿ ÉkFa nig BhuFa ÉkFaKWCakenßamEdlRbmaNviFIelIGefrmanEtviFIKuN nig sV&yKuNEdlmanniTsßnþKt´viC¢man ¦ sUnü . ÉkFadUcKña KWCaÉkFaEdlmanEpñkGefrdUcKña . dWeRkénÉkFa CaplbUkniTsßnþrbs´GefrnImYy@énÉkFa . BhuFa CaplbUkéneRcInÉkFaxus@Kña . dWeRkénBhuFa KWCadWeRkrbs´tYEdlmandWeRkx<s´CageK .

-8-

RbCMurUbmnþKNitviTüa 3¿ RbmaNviFIelI BhuFa edIm,IbUk ¦ dkBIrBhuFa eKRtUvbUk ¦ dkÉkFaEdldUcKña . edIm,IKuNBhuFa nig BhuFaeKyktYnImYy@énBhuFaTImYyKuN RKb´tYénBhuFaTIBIr rYceFIVRbmaNviFI . 4¿ rUbmnþ 1¿ (a + b) = a + 2ab + b 2¿ (a − b) = a − 2ab + b 3¿ (a − b)(a + b) = a − b 4¿ (a + b + c) = a + b + c + 2ab + 2bc + 2ca 5¿ (a + b) = a + 3a b + 3ab + b 6¿ (a − b) = a − 3a b + 3ab − b 7¿ (a + b)(a − ab + b ) = a − b 8¿ (a − b)(a + ab + b ) = a + b 9¿ acx + (ad + bc)x + bd = (ax + b)(cx + d) 5¿ RbmaNviFIEckBhuFa «bmafaeKmankenßamBIr A nig B EdlmanGefrdUcKña ehIymandWeRkerogKña m nig n . ebI m ≥ n eKGacrkkenßam 2

-9-

RbCMurUbmnþKNitviTüa BICKNitBIr Q nig R Edl A = B × Q + R . dWeRkén R tUcCagdWeRkén B . Q CaplEck ehIy R Ca sMNl´énkñúgviFIEck . plEck Q mandWeRk m − n . ebI R = 0 eKán A = B × Q ena¼eKfa A Eckdacńwg B 6¿ tYEckrYmFMbMput nig BhuKuNrYmtUcbMput tYEckrYmFMbMputénkenßam A nig B KWCaplKuNktþarYmEdl manniTsßnþtUcCageK . BhuKuNrYmtUcbMput KWCaplKuNRKb´ktþarYmEdlmanniTsßnþ FMCageK . 7¿ viFan edIm,IKNnatYEckrYmFMbMput ½ 1-dakĆplKuNktþa RKb´tYTaMgGs´ 2-eRCIserIsykEtktþarYmEdlmanniTsßnþtUcCageK. 3-tYEckrYmFMbMput CaplKuNénktþarYmTaMgena¼ . edIm,IKNnatYEckrYmtUcbMput ½ 1-dakĆplKuNktþa RKb´tYTaMgGs´ 2-eRCIserIsykEtktþaminrYm nig kþarYmEdlmanniTsßn - 10 -

RbCMurUbmnþKNitviTüa FMCageK. 3-BhuKuNrYmtUcbMput CaplKuNénktþaTaMgena¼ . 8¿ RbmaNviFIbUk ¦ dkkenßamRbPaK A B Α+B A B A−B + = ni g − = C C C C C C Edl C ≠ 0 . 9¿ viFan edIm,IeFIV viFIbUk ¦ dkkenßamRbPaKeKRtUv ½ 1-tRmUvRbPaKnImYy@[manPaKEbgrYmdUcKña 2-eFIVRbmaNviFIbUk ¦ dkEtPaKyk rkßaTukPaKEbgrYm 3-sRmYllTæpl 10¿ RbmaNviFIKuN nig RbmaNviFIEck A C A×C A C A×D × = ni g ÷ = B D B×D B D B×C Edl B , C , D xusBIsUnü . 11¿ viFan dak´PaKyk nig PaKEbgénkenßamTaMgGs´CaplKuNktþa sRmYlkenßamRbPaKnImYy@ eFIVRbmaNviFIKuN ¦ viFIEcktamrUbmnþxagelI . - 11 -

RbCMurUbmnþKNitviTüa

CMBUkTI04

smIkar nig vismIkar 1-smIkardWeRkTIBIrmanmYyGBaØat k-niymn&y smIkarEdlmanragTUeTA ax + bx + c = 0 ehAfasmIkardWeRkTIBIr manmYyGBaØatEdl x CaGBaØat ehIyelxemKuN a , b , c CacMnYnefr nig a ≠ 0 . x-dMeNa¼RsaysmIkardWeRkTIBIr snµtfaeKmansmIkar ax + bx + c = 0 , a ≠ 0 DIsRKImINg;smIkar ∆ = b − 4ac -ebI ∆ > 0 smIkarmanb¤sBIrCacMnYnBitepSgKñaKW ³ 2

x1 =

−b+ ∆ −b− ∆ ; x2 = 2a 2a

-ebI ∆ = 0 smIkarmanb¤sDúb x = x = − 2ba -ebI ∆ < 0 smIkarmanb¤sBIrepSgKñaCacMnYnkMupøicqøas;Kña ³ 1

x1 =

−b+i −∆ −b−i −∆ ; x2 = 2a 2a

- 12 -

RbCMurUbmnþKNitviTüa K-TMnak´TMng…s nig elxemKuN ebI α nig β Cab¤srbs;smIkar ax 2 + bx + c = 0 , a ≠ 0 enaHeKman ³ -plbUkb¤s S = α + β = − ba -plKuNb¤s P = α.β = ac X-muFaKNna…sénsmIkardWeRkTIBIrgay ]bmafaeKmansmIkardWeRkTIBIr ax 2 + bx + c = 0 , a ≠ 0 -ebI a + b + c = 0 smIkarmanb¤s x1 = 1 ; x 2 = ac -ebI b = a + c smIkarmanb¤s x1 = −1 , x 2 = − ac g-rUbmnþdak´CaplKuNktþa ebI α nig β Cab¤srbs;smIkar ax 2 + bx + c = 0 , a ≠ 0 enaHeK)an ³ ax 2 + bx + c = a( x − α )( x − β ) . c-beg›ItsmIkardWeRkTIBIr ebIeKdwgplbUk α + β = S nig plKuN αβ = P enaH α nig β Cab¤ssmIkardWeRkTIBIr x 2 − Sx + P = 0 .

- 13 -

RbCMurUbmnþKNitviTüa 2-vismPaB k-lkçN£vismPaB !> cMeBaHRKb;cMnYnBit a , b , c ebI a > b enaHeK)an a + c > b + c b¤ a − c > b − c . @> cMeBaHRKb;cMnYnBit a , b , c eKman ³ -ebI a > b nig c > 0 enaH ac > bc -ebI a > b nig c < 0 enaH ac < bc x-vismPaBmFümnBVnþ nig mFümFrNImaRt cMeBaHRKb;cMnYnBit a ≥ 0 nig b ≥ 0 eKman ³ a+b ≥ ab . 2 vismPaBenHkøayCasmPaBluHRtaEt a = b . 3-vismIkartémødac´xat ebI α > 0 enaHeK)an ³ !> | ax + b |< α ⇔ ax + b < α nig ax + b > −α @> | ax + b |> α ⇔ ax + b > α b¤ ax + b < −α #> | ax + b |= α ⇔ ax + b = ± α

- 14 -

RbCMurUbmnþKNitviTüa 4-sBaØarbs´eTVFadWeRkTImYy cMeBaHeTVFa f (x) = ax + b man x = − ba Cab¤s eKkMNt;sBaØaeTVFaenH eTAtamsBaØarbs; a dUctaragxageRkam ³ −∞

−

sBaØapÞúyBI a

f ( x ) = ax + b

b a

+∞

sBaØadUc a

5-sBaØarbs´RtIFadWeRkTIBIr cMeBaHRtIFa f (x) = ax + bx + c manb¤sBIr α nig β Edl α < β . 2

x f ( x ) = ax 2 + bx + c

−∞

+∞

sBaØapÞúyBI a sBaØadUc a

sBaØadUc a

6-cemøIyvismIkardWeRkTIBIr -krNI ∆ > 0 nig a > 0 man…s α , β (α < β) k> ax + bx + c > 0 mancemøIy x < α , x > β . x> ax + bx + c < 0 mancemøIy α < x < β . K> ax + bx + c ≥ 0 mancemøIy x ≤ α , x ≥ β . X> ax + bx + c ≤ 0 mancemøIy α ≤ x ≤ β . -krNI ∆ = 0 nig a > 0 man…sDúb 2

- 15 -

RbCMurUbmnþKNitviTüa k> ax + bx + c > 0 mancemøIyRKb;cMnYnBitelIkElgEt x = − 2ba x> ax + bx + c < 0 KµancemøIy . K> ax + bx + c ≥ 0 mancemøIyRKb;cMnYnBitTaMgGs; . X> ax + bx + c ≤ 0 mancemøIy x = − 2ba . -krNI ∆ < 0 nig a > 0 man…sCacMnYnkMupøic k> ax + bx + c > 0 mancemøIyRKb;cMnYnBitTaMgGs; . x> ax + bx + c < 0 KµancemøIy . K> ax + bx + c ≥ 0 mancemøIyRKb;cMnYnBitTaMgGs; . X> ax + bx + c ≤ 0 KµancemøIy . 2

- 16 -

RbCMurUbmnþKNitviTüa

CMBUkTI05

sIVúténcMnYnBit I-

sIVútnBVnþ nig sIVútFrNImaRt

1-sIVútcMnYnBit sIVúténcMnYnBitKWCaGnuKmn_elxEdlkMnt;BIsMNMu IN eTAsMNMu IR . eKkMnt;sresrsIVútmYyeday (Un ) b¤ (U n ) n∈IN Edl Un = f (n) 2>GefrPaBénsIVút k-sIVútekIn eKfasIVút (Un ) CasIVútekInelI IN kalNaRKb; n ∈ IN eKman U > U x-sIVútcuH eKfasIVút (U ) CasIVútcuHelI IN kalNaRKb; n ∈ IN eKman U n+1 < U n K-sIVútm:UNUtUn eKfasIVút (U ) CasIVútm:UNUtUnkalNavaCasIVútekInCanic© b¤ CasIVút cuHCanic© . n +1

- 17 -

RbCMurUbmnþKNitviTüa 3>sIVútTal; k-sIVútTal;elI eKfasIVút (U ) CasIVútTal;elIkalNamancMnYnBit M EdlbMeBj lkç½xN½Ð ∀n ∈ IN : U n ≤ M . x-sIVútTal;eRkam eKfasIVút (U ) CasIVútTal;eRkamkalNamancMnYnBit m Edl cMeBaH ∀n ∈ IN : Un ≥ m . K-sIVútTal; eKfasIVút (U ) CasIVútTal;kalNavaCasIVút Tal;elIpg nigTal; eRkampg . 4>sIVútxYb eKfasIVút (U ) CasIVútxYbEdlmanxYbesµI p kalNa cMeBaH ∀n ∈ IN : Un + p = Un , p ∈ IN * . 2-sIVútnBVnþ -sIVútnBVnþ KWCasIVúténcMnYnBitEdlmantYnImYy² ¬ eRkABItYTImYy ¦ esµInwgtYmunbnÞab;bUkcMnYnefr d mYyehAfaplsgrYm b¤ ersugénsIVút rUbmnþplsgrYm d = un+1 − un . -tYTI n énsIVútnBVnþ un = u1 + (n − 1)d n

- 18 -

RbCMurUbmnþKNitviTüa -plbUk n tYdMbUgénsIVútnBVnþ Sn =

∑ (uk ) = u1 + u2 + u3 + ... + un =

k =1

n ( u1 + u2 ) 2

3-sIVútFrNImaRt -sIVútFrNImaRt KWCasIVúténcMnYnBitEdlmantYnImYy² ¬eRkABItYTImYy¦ esµInwgtYmunbnÞab;KuNnwgcMnYnefr q mYyEdlxusBIsUnü . cMnYnefr q ehAfapleFobrYm b¤ ersugénsIVút . rUbmnþpleFobrYm q = uun+1 . n -tYTI n énsIVútFrNImaRt un = u1 × q n−1 -plbUk n tYdMbUgénsIVútFrNImaRt qn − 1 S n = ∑ (uk ) = u1 + u2 + u3 + ... + un = u1 q−1 k =1 n

4-rUbmnþplbUksIVútnBVnþmanlMdab´x<s´ n

∑ (k ) = 1 + 2 + 3 + ... + n =

k =1 n

2) 2 2 2 2 n( n + 1)( 2n + 1) ( k = 1 + 2 + 3 + ... + n = ∑ 6

k =1 n

n( n + 1) 2

( )

∑k

k =1

n 2 ( n + 1) 2 = 1 + 2 + 3 + ... + n = 4 3

- 19 -

RbCMurUbmnþKNitviTüa -rebobKNnaplbUktYénsIVútepßg@ 1-nimµitsBaØa ∑ sRmab´plbUkénsIVút plbUk n tYdMbUgénsIVút u , u , u ,...., u kMnt;tageday ³

S n = ∑ (uk ) = u1 + u2 + u3 + ..... + un k =1

2-lkçN£plbUktYénsIVút 1> ∑ (λ ) = λ + λ + λ + ..... + λ = nλ n

k =1 n

2> ∑ ( λ u ) = λ ∑ (u ) ¬ λ CacMnYnefr ¦ k

k =1 n

3> ∑ (u

k =1 n

k =1

k =1 n

+ v k − w k ) = ∑ (uk ) + ∑ (v k ) − ∑ (w k )

4> ∑ (u + v ) = ∑ (u ) + 2∑ (u v ) + ∑ (v ) 3-rebobKNnaplbUksIVútEdlmanTRmg´ ½ S = 1 + 2 + 3 + ..... + n Edl p = 1 ; 2 ; 3 ; ... edIm,IKNnaplbUkenHeKRtUvGnuvtþn_tamCMhanxageRkam ³ -KNna (n + 1) − n -[témø n = 1 ; 2 ; 3 ; ....; n -eFIVviFIbUk . n

k =1

k k

p +1

- 20 -

k =1

RbCMurUbmnþKNitviTüa 4-rebobKNnaplbUksIVútEdlmanTRmg´ ½ Sn =

1 1 1 1 + ..... + + + a1a 2 a 2a 3 a 3 a4 an an+1

Edl a − a = d efr ehIy d ≠ 0 . edIm,IKNnaplbUkenHeKRtUv ³ -bEmøgtY a a1 = d1 . aa a− a = d1  a1 − a1 n +1

n+1

n n +1



n n+1

  n +1 

-[témø n = 1 ; 2 ; 3 ; ....; n -eFIVviFIbUk 5-rebobKNnaplbUksIVútEdlmanTRmg´ ½ Sn =

1 1 1 1 + ..... + + + a1a 2a 3 a 2a 3a4 a 3a4 a5 a n a n + 1a n + 2

Edl a − a = d efr ehIy d ≠ 0 . edIm,IKNnaplbUkenHeKRtUv ³ -bEmøgtY a a 1 a = d1 . aa a −aa = d1  a a1  -[témø n = 1 ; 2 ; 3 ; ....; n -eFIVviFIbUk n+ 2

n+ 2

n n +1 n + 2

n n+1 n+ 2

- 21 -

n n +1

−

1   a n+1an+ 2 

RbCMurUbmnþKNitviTüa 6-rebobKNnaplbUksIVútEdlmanTRmg´ ½ S n = a1b1 + a2 b2 + a3 b3 + ...... + a n bn

Edl (a ) CasIVútnBVnþmanplsgrYm d nig (b ) CasIVútFrNImaRmanersug q . edIm,IKNnaplbUkenHeKRtUvKNna S − q S rYcTajrk S . 7-sMKal´ eK[ S = ∑ (u ) = u + u + u + .... + u edIm,IKNnaplbUkxagelIenHeKRtUv ³ -sresrtY u Carag u = t − t b¤ u = t − t ¬ ebIGac ¦ -krNIeKGacsresr u = t − t enaHeK)an ³ n

k =1

k +1

S n = ∑ (uk ) = ∑ (t k +1 − t k ) = t n+1 − t1

-krNIeKGacsresr u

k =1

= t k − t k +1

enaHeK)an ³

S n = ∑ (uk ) = ∑ (t k − t k +1 ) = t1 − t n+1

- 22 -

k +1

RbCMurUbmnþKNitviTüa -rebobkMnt´tYTI n tamplsgtYénsIVút 1-plsgtYlMdab´TImYy ½ -eKmansIVút (a ): a ; a ; a ; ..... ; a ehIy b = a − a ; b = a − a ; b = a − a ; ..... ena¼eKfasIVút (b ) : b ; b ; b ; .... ; b CaplsgtYlMdab´TImYyénsIVút (a ). -rUbmnþKNnatY a eKman b = a − a eKán ∑ (b ) = ∑ (a − a ) III

n +1

n −1

k =1 n −1

eday ∑ (a k =1

n −1

k +1

k =1

k +1

− ak ) = (a 2 − a1 ) + (a 3 − a 2 ) + ... + (a n − a n−1 ) = a n − a1

n −1

eKán ∑ (b ) = a − a dUcen¼ a = a + ∑ (b ) . 2-plsgtYlMdab´TIBIr ½ -eKmansIVút (a ): a ; a ; a ; ..... ; a ehIy k =1

n −1

k =1

b1 = a2 − a1 ; b2 = a 3 − a 2 ; b3 = a4 − a3 ; .....; bn = an+1 − a n

(bn ) : b1 ; b2 ; b3 ; .... ; bn

CaplsgtYlMdab´TImYyénsIVút - 23 -

(an )

RbCMurUbmnþKNitviTüa n −1

-rUbmnþKNnatY a KW a = a + ∑ (b ) . -sIVút ( c ) CaplsglMdab´TIBIrénsIVút (a ) KWCaplsg lMdab´TImYyénsIVút (b ) Edl c = b − b ; n = 1 , 2 , 3 , ... rUbmnþKNnatYTI n KW b = c + ∑ (c ) ; n ≥ 2 . n

k =1

n+1

n −1

-vicarGnumanrYmKNitviTüa niymn&y ½

i =1

CasMeNIEdlTak;TgnwgcMnYnKt; n edIm,IRsaybBa¢ak;fa P(n)BitcMeBaHRKb; n ∈ IN * eKRtUv ³ !> epÞógpÞat;fa P(n) BitcMeBaH n = 1 @> ]bmafa P(n)BitcMeBaHtémø n #> RsaybBa¢ak;fa P(n) BitnaM[)an P(n + 1) Bit P (n)

- 24 -

RbCMurUbmnþKNitviTüa -rebobKNnatYTUeTAénsIVúttamTMnak´TMngkMeNIn 1-krNIs:al´TMnak´TMngkMeNIn un +1 = a un + b ebIeKs:al´fa (un ) CasIVúténcMnYnBitehIyepÞógpÞat´ TMnak´TMngkMeNIn un +1 = a un + b cMeBa¼RKb´ n ∈ IN * nigmantY u1 = α ( | a | ≠ 1 , a ≠ 0 ) . edIm,IkMnt´rktY un eKRtUvBicarNadUcxageRkam ½ rk¦ssmIkar r = a r + b (ehAfasmIkarsMKal´énsVúIt) tagsIVútCMnYy Vn = un − r rYcRtUvbgHajfa (Vn ) CasIVútFrNImaRt . rk[eXIjnUvtY Vn bnÞab´mkeKTaj un = Vn + r . 2-krNIs:al´TMnak´TMngkMeNIn un + 2 = a un +1 + bun ebIeKs:al´fa (un ) CasIVúténcMnYnBitehIyepÞógpÞat´ TMnak´TMngkMeNIn un + 2 = a un +1 + b un cMeBa¼RKb´ n ∈ IN * nigmantY u1 = α , u2 = β edIm,IkMnt´rktY un eKRtUvBicarNasmIkar r 2 = a r + b ¦ ( E ) : r 2 − a .r − b = 0 ( ehAfasmIkarsMKal´énsVúIten¼ ) eKRtUvsikßakrNIepßg@dUcxageRkam ½ ebI ∆ = a 2 + 4b > 0 IV

- 25 -

RbCMurUbmnþKNitviTüa smIkarsMKal´ (E ) man¦sBIrepßgKñaCacMnYnBit r1 nig r2 . kñúgkrNIen¼edIm,IKNna un eyIgRtUvGnuvtþn_dUcxageRkam ½ -tagsIVútCMnYyBIrKW x n = un + 1 − r1 un nig y n = un + 1 − r2 un -rkRbePTénsIVút ( xn ) nig ( yn ) rYcKNna xn nig yn CaGnuKmn_én n . «bmafaeKán xn = f (n) nig yn = g(n)  un + 1 − r1un = f ( n) -eyIgánRbB&næsmIkar u − r u = g(n)  n+1

-eda¼Rsayrk

eKTTYlán

2 n

un =

f ( n) − g( n) r2 − r1

ebI ∆ = a 2 + 4b = 0 smIkarsMKal´ (E ) man¦sDub r1 = r2 = r0 kñúgkrNIen¼edIm,IKNna un eyIgRtUvGnuvtþn_dUcxageRkam ½ -tagsIVútCMnYy Vn = un+1 − r0 un rYcrkRbePTénsIVút (Vn ) nigKNna Vn CaGnuKmn_én n . «bmafa Vn = f (n) . -eKTajánsmIkar un +1 − r0un = f (n) rYcRtUvbMElgCaTRmg´ ½ u n + 1 un f ( n) − = ( EcksmI k arnw g r0 n + 1 ) n+1 n n+1 r0

- 26 -

RbCMurUbmnþKNitviTüa -Taj[án

n −1 f ( k )  n u1 un = r0  + ∑ [  r0 k = 1 r0 k + 1

 ] 

ebI ∆ = a 2 + 4b < 0 smIkarsMKal´ (E) man¦sBIrCacMnYnkMupøicqñas´Kña KW r1 = p + i .q , r2 = p − i .q , p , q ∈ IR . kñúgkrNIen¼edIm,IKNna un eyIgRtUvGnuvtþn_dUcxageRkam ½ -tagsIVútCMnYy Z n = un +1 − ( p + i .q) un rYcRtUvRsayfa ( Z n ) CasIVútFrNImaRténcMnYnkMupøic . rUcKNna Z n CaGnuKmn_én n . -«bmafa Z n = An + i .Bn ; An , Bn ∈ IR , n ∈ IN * . -eKánsmIkar un +1 − ( p + iq) un = An + i .Bn -Taj[ánfa un = − Bqn . 3-krNIs:al´TMnak´TMngkMeNIn un + 2 = a un +1 + bun + c ebIeKs:al´fa (un ) CasIVúténcMnYnBitehIyepÞógpÞat´ TMnak´TMngkMeNIn un + 2 = a un +1 + b un cMeBa¼RKb´ n ∈ IN * nigmantY u1 = α , u2 = β . edIm,IkMnt´rktY un eKRtUvGnuvtþn_dUcxageRkam ½ tagsIVútCMnYy wn = un + λ - 27 -

RbCMurUbmnþKNitviTüa eKán un = wn − λ , un+1 = wn+1 − λ , un+ 2 = wn+ 2 − λ yk un , un +1 , un + 2 CMnYskñúg un + 2 = a un +1 + bun + c eKánsmIkar ½ w n + 2 − λ = a ( w n + 1 − λ ) + b( w n − λ ) + c

w n + 2 = a w n + 1 + b w n + (1 − a − b ) λ + c

RtUv[

eKTaján ( Edl a + b ≠ 1 ) . kñúgkrNIen¼eKánTMnak´TMngkMeNIn (1 − a − b ) λ + c = 0

λ=

c a + b−1

wn + 2 = a wn +1 + b wn

eda¼RsayrktY wn tamviFIsaRsþEdlánsikßarYcehIy xagelI bnÞab´mkTajrktY un = wn − λ = wn − a + cb − 1

CMBUkTI06

GnuKmn_RtIekaNmaRt - 28 -

RbCMurUbmnþKNitviTüa 1¿ TMnak´TMngRKw¼ !> sin 2 θ + cos2 θ = 1 sin θ @> tan θ = cos θ θ #> cot θ = cos sin θ

$> tan θ . cot θ = 1 %> 1 + tan 2 θ = 12

cos θ ^> 1 + cot 2 θ = 12 sin θ

2¿ rUbmnþplbUk nig pldk !> sin(a + b) = sin a cos b − sin b cos a @> cos(a + b) = cos a cos b − sin a sin b a + tan b #> tan(a + b) = 1tan − tan a tan b $> sin(a − b) = sin a cos b − sin b cos a %> cos(a − b) = cos a cos b + sin a sin b a − tan b ^> tan(a − b) = 1tan + tan a tan b 3¿ rUmnþmMuDub !> sin 2a = 2 sin a cos a @> cos 2a = cos2 a − sin 2 a = 2 cos2 a − 1 = 1 − sin 2 a #> tan 2a = 2 tan a2 $> cot 2a = cot2 cota −a 1 2

1 − tan a

- 29 -

RbCMurUbmnþKNitviTüa 4¿ rUbmnþknø¼mMu a !> sin a2 = 1 − cos 2 2

a 1 + cos a = 2 2 a 1 − cos a tan 2 = 2 1 + cos a

@> cos #>

5¿ kenßam

sin x , cos x , tan x

CaGnuKmn_én

t = tan

x 2

2t 1 + t2 1 − t2 @> cos x = 2 1+ t 1 − t2 #> tan x = 2 1+ t 6¿ kenßam sin 3a , cos 3a , tan 3a

!> sin x =

!> sin 3a = 3 sin a − 4 sin 3 a @> cos 3a = 4 cos3 a − 3 cos a #> tan 3a = 3 tan a − tan a 1 − 3 tan a 7¿ rUbmnþbMElgBIplKuNeTAplbUk !> cos a cos b = 12 [ cos(a + b) + cos(a − b) ] 3

@> sin a sin b = 12 [ cos(a − b) − cos(a + b) ] - 30 -

RbCMurUbmnþKNitviTüa #> sin a cos b = 12 [ sin(a + b) + sin(a − b) ] $> sin b cos a = 12 [ sin(a + b) − sin(a − b) ] 6¿ rUbmnþbMElgBIplbUkeTAplKuN !> cos p + cos q = 2 cos p +2 q cos p −2 q @> cos p − cos q = −2 sin p +2 q sin p −2 q #> sin p + sin q = 2 sin p +2 q cos p −2 q $> sin p − sin q = 2 sin p −2 q cos p +2 q sin(p + q ) %> tan p + tan q = cos p cos q sin(p − q ) ^> tan p − tan q = cos p cos q p + q) &> cot p + cot q = sin( sin p sin q q − p) *> cot p − cot q = sin( sin p sin q 7¿ smIkarRtIekaNmaRt !> smIkar sin u = sin v mancemøIy

 u = v + 2kπ  u = π − v + 2kπ , k ∈ Z 

- 31 -

RbCMurUbmnþKNitviTüa @> smIkar cos u = cos v mancemøIy

 u = v + 2kπ  u = − v + 2kπ , k ∈ Z  #> smIkar tan u = tan v mancemøIy u = v + kπ

8¿ rUbmnþbEmøgFñÚEdlKYkt´sMKal´

π  sin ( − θ) = cos θ  2  !>  cos ( π2 − θ) = sin θ @>  π  tan ( − θ) = cot θ  2 π   sin ( 2 + θ) = cos θ  #>  cos ( π2 + θ) = − sin θ  π  tan ( + θ) = − cot θ  2  sin ( π + θ) = − sin θ $>  cos (π + θ) = − cos θ  tan ( π + θ) = tan θ   sin (θ + 2kπ ) = sin θ %>  cos (θ + 2kπ) = cos θ  tan (θ + kπ ) = tan θ 

- 32 -

 sin ( π − θ) = sin θ   cos ( π − θ) = − cos θ  tan ( π − θ) = − tan θ 

, ∀k ∈ Z

RbCMurUbmnþKNitviTüa 9¿ RkahVikGnuKmn_RtIekaNmaRt !> ExßekagGnuKmn_ y = sin x y 4 3 2 1

-4

-3

-2

-1

10x

-1 -2

@> ExßekagGnuKmn_

y = cos x

y 4 3 2 1

-4

-3

-2

-1

-1 -2

- 33 -

RbCMurUbmnþKNitviTüa

#> ExßekagGnuKmn_

y = tan x y 4 3 2 1

-4

-3

-2

-1

-1 -2

$> ExßekagGnuKmn_

y = cot x

y 4 3 2 1

-3

-2

-1

-1 -2 -3 -4

- 34 -

9 x

RbCMurUbmnþKNitviTüa

CMBUkTI07

GnuKmn_Gics,¨ÚNg´Esül nig GnuKmn_elakarIt 1-អនុគមន៍អិចសប៉ូណង់ែសយល

អនុគមន៍អិចសប៉ូ ណង់ែសយល ជអនុគមន៍កំនត់

េដយ y = f ( x ) = a x ែដល x ∈ IR និង a ជចំនួនពិត ិ មន និងខុសពី 1 ។ វជជ

រកបៃនអនុគមន៍អិចសប៉ូណង់ែសយល y

-2

-1

- 35 -

RbCMurUbmnþKNitviTüa y

-2

-1

ចំេពះរគប់ចំនួនពិត a > 0 និង a ≠ 1 េគបន 1/ a x = ak

⇔

x=k

2/ af ( x ) = ag ( x ) ⇔ f ( x ) = g( x )

- 36 -

4 x

RbCMurUbmnþKNitviTüa 2-អនុគមន៍េលករតី

េបេគមន y = ax េនះ x = loga y ែដល y > 0 , a > 0 និង a ≠ 1 ។ េគថ f ( x) = a x មនអនុគមន៍រចស់ f −1 ( x) = log a x ។ ដូចេនះ y = log a x េហថអនុគមន៍េលករ ីតៃន x មនេគល a ។

លកខណះៃនេលករ ីត ិ មន x និង y , a > 0 , a ≠ 1 េគមន រគប់ចំនួនពិតវជជ 1/ log a ( xy ) = log a x + log a y  x y

2/ log a   = log a x − log a y 3/ log a xn = n log a x 4/ log a x =

1 log x a

5/ log a a = 1 6/ log a 1 = 0 7/ alog a b = b

- 37 -

RbCMurUbmnþKNitviTüa រកបៃនអនុគមន៍េលករ ីត y

y = log a x , a > 1

-1

y = log a x , 0 < a < 1

-1

- 38 -

RbCMurUbmnþKNitviTüa

CMBUkTI08

lImIt nig PaBCab´énGnuKmn_ 1-lImIténGnuKmn_Rtg´cMnYnkMNt´ niymn½y ³ GnuKmn_ f manlImItesµI L kalNa x xitCit a ebIRKb; cMnYn ε > 0 mancMnYn δ > 0 Edl 0 <| x − a |< δ naM[ | f ( x ) − L |< ε . eKsresr ³ lim f ( x) = L . niymn½y ³ eKfaGnuKmn_ f xiteTA + ∞ b¤ − ∞ kalNa x xiteTACit a ebIcMeBaHRKb;cMnYn M > 0 man δ > 0 Edl 0 <| x − a |< δ naM[ f ( x ) > M b¤ f ( x ) < − M . eKsresr lim f (x) = +∞ b¤ lim f (x) = −∞ . 2-lImIténGnuKmn_Rtg´Gnnþ eKfaGnuKmn_ f manlImItesµI L kalNa x eTACit + ∞ b¤ − ∞ ebIcMeBaHRKb;cMnYn ε > 0 eKGacrk N > 0 Edl x > N b¤ x < − N naM[ | f (x) − L |< ε . x→a

x→a

- 39 -

RbCMurUbmnþKNitviTüa

eKsresr lim f (x) = L b¤ lim f (x) = L . eKfaGnuKmn_ f manlImIt + ∞ kalNa x eTACit + ∞ ebIcMeBaHRKb;cMnYn M > 0 eKman N > 0 Edl x > N naM[ f ( x) > M .eKsresr lim f ( x ) = +∞ . eKfaGnuKmn_ f manlImIt + ∞ kalNa x eTACit − ∞ ebIcMeBaHRKb;cMnYn M > 0 eKman N > 0 Edl x < − N naM[ f ( x) > M .eKsresr lim f ( x ) = +∞ . 3-RbmaNviFIelIlImIt ebI lim f (x) = L ; lim g(x) = M nig lim h(x) = N Edl L ; M ; N CacMnYnBitenaHeK)an ³ 1 / lim[f ( x ) ± g( x )] = L ± M ¬ a CacMnYnkMNt; b¤Gnnþ ¦ x → +∞

x → −∞

x → +∞

x → −∞

x→a

2 / lim[f ( x ) + g( x) − h( x )] = L + M − N x→a

[k f ( x)] = k lim [f ( x)] 3 / lim x→a x→a →

→

[f ( x).g( x).h( x)] = L.M .N 4 / lim x→a →

 f ( x)  L 5 / lim = ;M≠0  x→a   g( x )  M n [ ] 6 / lim f ( x ) = L x→a n

Edl n CacMnYnKt;FmµCatiminsUnü . - 40 -

RbCMurUbmnþKNitviTüa

4-lImIténGnuKmn_GsniTan 1 / lim ( x ) = a cMeBaH a ≥ 0 nig n ∈ IN 2 / lim ( x ) = a cMeBaH a < 0 nig n CacMnYnKt;ess n

x→a

[

]

3 / lim n f ( x ) = n lim[f ( x )] = n L x→a x→a →

→

ebI L ≥ 0 nig n CacMnYnKt;KU b¤ L < 0 nig n CacMnYnKt;ess. 5-lImIténGnuKmn_bNþak´ ebI f nig g CaGnuKmn_Edlman lim[g(x)] = L nig lim f (x) = f (L) enaH lim f [g ( x )] = f ( L ) . 6-lImIttamkareRbobeFob ebIeKmanGnuKmn_ f ; g nigcMnYnBit A Edl lim g( x) = +∞ nig f (x) ≥ g(x) cMeBaHRKb; x ≥ A enaH lim f (x) = −∞ . ebIeKmanGnuKmn_ f ; g nigcMnYnBit A Edl lim g( x) = +∞ nig f (x) ≤ g(x) cMeBaHRKb; x ≥ A enaH lim f (x) = −∞ . ebIeKmanGnuKmn_ f ; g; h nigcMnYnBit A Edl lim g( x ) = lim h( x ) = λ nig g( x ) ≤ f ( x ) ≤ h( x ) cMeBaHRKb; x ≥ A enaH lim f (x) = λ . x→a

x→L

x→a

x→ +∞

x → +∞

x→ +∞

x → +∞

x→ +∞

x → +∞

- 41 -

RbCMurUbmnþKNitviTüa ebIeKmanGnuKmn_ f ; g nigcMnYnBit A

Edl lim f (x) = λ lim g( x) = λ ' nig f ( x ) ≤ g( x ) cMeBaHRKb; x ≥ A enaH λ ≤ λ' . 7-lImItragminkMnt´ 0 lImItEdlmanragminkMnt;t 0 viFan edIm,IKNnalImItEdlmanragminkMnt; 00 eKRtUvbMEbk PaKyk nig PaKEbgCaplKuNktþa ehIysRmYlktþarYm rYcKNna lImIténkenSamfµI . ∞ lImItEdlmanragminkMnt;t ∞ viFan edIm,IKNnalImItEdlmanragminkMnt; ∞∞ eKRtUvdak;tY EdlmandWeRkFMCageKenAPaKyk nig PaKEbgCaplKuNktþa ehIysRmYlktþarYm rYcKNnalImIténkenSamfµI . lImItEdlmanragminkMnt;t + ∞ − ∞ viFan edIm,IKNnalImItEdlmanragminkMnt; + ∞ − ∞ eKRtUv dak;tYEdlmandWeRkFMCageKenAPaKyk nig PaKEbgCaplKuN x → +∞

- 42 -

x→ +∞

RbCMurUbmnþKNitviTüa

énktþa ehIysRmYlktþarYm rYcKNnalImIténkenSamfµI . 8-lImIténGnuKmn_RtIekaNmaRt -ebI a CacMnYnBitsßienAkñúgEdnkMnt;énGnuKmn_RtIekaNmaRtEdl [enaHeK)an lim sin x = sin a ; lim cos x = cos a nig lim tan x = tan a . -viFan ebI x Cargjvas;mMu b¤FñÚKitCar:adüg;enaHeK)an sin x 1 − cos x = 1 nig lim =0 . lim x x 9-lImIténGnuKmn_Giucs,:ÚNg´Esül x→a

x→a

x→0

1 / xlim e x = +∞ → +∞ 2 / xlim e x = 0 → −∞ ex 3 / xlim = +∞ → +∞ x ex 4 / lim n = +∞ ( n > 0) x → +∞ x xn 5 / xlim = 0 (n > 0) → +∞ x e

- 43 -

RbCMurUbmnþKNitviTüa

10-lImIténGnuKmn_elakarItenEB 1 / lim ln x = +∞ x → +∞

2 / lim ln x = −∞ x→ 0+

ln x =0 x → +∞ x 4 / lim x ln x = 0 x→0 3 / lim →

ln x = 0 (n > 0) → +∞ x n 6 / lim x n ln x = 0 5 / xlim → +∞ x→0+

11-lImItsIVút nig es‘rI RbmaNviFIelIlImIt eKmansVIút (a ) nig (b ) Edlman lim a = M nig lim b = N eK)an k> lim ka = k .M x> lim (a + b ) = M + N , lim (a − b ) = M − N K> lim (a b ) = M.N ebI N ≠ 0 enaH lim ba = MN n

n→ +∞

- 44 -

RbCMurUbmnþKNitviTüa

lImItsVIútFrNImaRtGnnþ k>ebI r > 1 enaH lim r = +∞ ehIy r CasVIútrIkeTArk + ∞ x>ebI r = 1 enaH (r ) CasIVútefrehIy lim r = 1 . K>ebI r = 0 enaH (r ) CasIVútefrehIy lim r = 0 . X>ebI r ≤ −1 enaHsIVú (r ) CasIVútqøas;ehIykalNa n → +∞ eKminGackMNt;lImItén (r ) )aneT . sVIútFrNImaRtGnnþEdlrYm ³ sVIút (r ) smmUl − 1 ≤ r ≤ 1 es‘rIrYm nig es‘rIrIk ³ k-ebIes‘rI ∑ (a ) Caes‘rIrYmenaH lim a = 0

n→ +∞

n → +∞

∞

n =1

n → +∞

∞

x-ebIsIVút (a ) minrYmrk 0 eTenaH ∑ (a ) Caes‘rIrIk . PaBrYmnigrIkénes‘rIFrNImaRtGnnþ ³ RKb;es‘rIFrNImaRtGnnþ a + ar + ar + ar + .... + ar Edl a ≠ 0 Caes‘rIrYm b¤ rIkeTAtamkrNIdUcxageRkam ³ k-ebI | r |< 1 enaHes‘rIrYmeTArk 1 −a r . x-ebI | r |≥ 1 enaHes‘rIrIk . n

n =1

- 45 -

n −1

+ ...

RbCMurUbmnþKNitviTüa

12-PaBCab´énGnuKmn_Rtg´mYycMnuc niymn½y ³ GnuKmn_ y = f (x) Cab;Rtg;cMnuc x = c kalNa f bMeBj lkçxNÐTaMgbIdUcxageRkam !- f kMNt;cMeBaH x = c @- f manlImItkalNa x → c #- lim f (x) = f (c) 13-lkçN³énGnuKmn_Cab´ ebI f nig g CaGnuKmn_Cab;Rtg; x = c enaHeK)an 1> f (x) ± g(x) CaGnuKmn_Cab;Rtg; x = c 2> f (x).g(x) CaGnuKmn_Cab;Rtg; x = c 3> gf ((xx)) CaGnuKmn_Cab;Rtg; x = c Edl g(c) ≠ 0 . 14-PaBCab´elIcenøa¼ niymn½y ³ -GnuKmn_ f Cab;elIcenøaHebIk (a , b ) luHRtaEt f Cab;cMeBaH RKb;témø x éncenøaHebIenaH . x→c

- 46 -

RbCMurUbmnþKNitviTüa

-GnuKmn_ f Cab;elIcenøaHbiT [a , b ] luHRtaEt f Cab; elIcenøaH ebIk (a , b ) nigmanlImIt lim f (x) = f (a) ; lim f (x) = f (b) ¬GnuKmn_ f Cab;Rtg; a xagsþaM Cab;Rtg; b xageqVg ¦ 15-PaBCab´énGnuKmn_bNþak´ ebIGnuKmn_ g Cab;Rtg; x = c nigGnuKmn_ f Cab;Rtg; g(c) enaHGnuKmn_bNþak; (f g)(x) = f [ g(x) ] Cab;Rtg; c . 16-Gnuvtþn_ GnuKmn_bnøaytamPaBCab´ ebI f CaGnuKmn_minkMnt;Rtg; x = a nigmanlImIt lim f (x) = L enaHGnuKmn_bnøayén f tamPaBCab;Rtg; x = a kMnt;eday  f ( x ) ebI x ≠ a g( x ) =  ebI x = a  L x→a +

x→b −

x→a

- 47 -

RbCMurUbmnþKNitviTüa

17-RTwsþIbTtémøkNþal RTwsþIbT ³ ebIGnuKmn_ f Cab;elIcenøaHbiT [ a , b ] nig k CacMnYnmYyenAcenøaH f (a) nig f (b) enaHmancMnYnBit c mYyya:gtickñúgcenøaHbiT [a , b ] Edl f (c) = k . y

(c ) : y = f ( x )

f (b ) k f (a )

- 48 -

RbCMurUbmnþKNitviTüa

CMBUkTI09

edrIevénGnuKmn_ 1-edrIevénGnuKmn_Rtg´ x 0 niymn½y ³ edrIevénGnuKmn_ y = f (x) CalImIt ¬ebIman ¦ énpleFob kMenIn ∆∆yx kalNa ∆x xiteTACit 0 . eKkMnt;sresr

f ( x) − f ( x 0 ) f (x 0 + h) − f (x 0 ) ∆y y'0 = lim = lim = lim ∆x → 0 h→0 ∆x x → x 0 x − x 0 h

2-PaBmanedrIev nig PaBCab´ snµtfaGnuKmn_ f (x) kMnt;elIcenøaH I ehIy x0 CacMnYnBitenAkñúg cenøaH I nig h CacMnYnBitminsUnüEdl x0 + h Carbs; I . -cMnYnedrIeveqVgRtg;cMnYn x0 énGnuKmn_ f (x) kMnt;tageday f (x + h) − f (x ) f ' ( x ) = lim . h -cMnYnedrIevsþaMRtg;cMnYn x0 énGnuKmn_ f (x) kMnt;tageday f (x + h) − f (x ) f ' ( x ) = lim . h -edrIevénGnuKmn_ f (x) Rtg; x0 ebIman kMnt;tageday −

h→0−

h→0+

- 49 -

RbCMurUbmnþKNitviTüa f (x0 + h) − f (x0 ) h→0 h f (x0 + h) − f (x0 ) lim h→0 h f (x0 + h) − f (x0 ) f (x0 + h) − f (x0 ) lim = lim h h h→0− h→0+ f ' ( x 0 ) = lim

mankalNa

ehIy

3-GnuKmn_edrIev

k¿ni k¿niymn& mny& -ebI f CaGnuKmn_mYykMnt´elIcenøa¼ I nigmanedrIevRtg´RKb´cMnuc enAkñúgcenøa¼ I ena¼eKfaGnuKmn_ f manedrIevelIcenøa¼ I . -GnuKmn_EdlRKb´ x ∈ I pßMáncMnYnedrIevén f Rtg´ x ehAfa GnuKmn_edrIevén f EdleKkMnt´sresr f : x ֏ f ' (x) . y

( C ) : y = f (x)

(T)

cMnYnedrIevénGnuKmn_

f ( x)

Rtg´cMnuc - 50 -

KWCaemKuNRáb´Tisén

RbCMurUbmnþKNitviTüa bnÞat´b¨¼nwgExßekag (c) : y = f (x) Rtg´cMnucmanGab´sIus ehIysmIkarbnÞat´b¨¼ena¼kMnt´eday ½ ( T) : y − y = f ' ( x ) ( x − x ) . 0

x = x0

4-edrIevénGnuKmn_bNþak´ ebI y = f (u) nig u = g(x) enaHeK)an dy dy du d f [u( x)] = f ' (u ) × u' ( x ) = × = y'×u' b¤ dx du dx dx 5-rUbmnþedrIevénGnuKmn_ GnuKmn_

edrIev

!> y = k @> y = xn #> y = 1

y' = 0

$> y =

%> y = e x ^> y = a x &> y = ln x *> y = sin x (> y = cos x

y' = n x n − 1 1 y' = − 2 x 1 y' = 2 x y' = e x

y' = a x ln a 1 y' = x y = cos x y' = − sin x - 51 -

RbCMurUbmnþKNitviTüa !0> y = tan x !!> y = cot x !@> y = arcsin x !#> y = arccos x !$> y = arctan x CaTUeTA

GnuKmn_

!> y = un @> y = u #> y = u.v $> y = u v

%> y = ln u ^> y = sin u &> y = cos u *> y = eu (> y = tan u !0> y = arcsin u

1 2 = 1 + tan x 2 cos x 1 y' = − 2 = − (1 + cot 2 x ) sin x 1 y' = 1 − x2 1 y' = − 1 − x2 1 y' = 1 + x2 y' =

edrIev

y' = n.u'.u n − 1 u' y' = 2 u y' = u' v + v' u u' v − v' u y' = v2 u' y' = u y' = u'. cos u y' = − u' sin u

y' = u'.eu y' = u' (1 + tan 2 u ) u' y' = 1 − u2 - 52 -

RbCMurUbmnþKNitviTüa !!> y = arccos u

y' = −

1 − u2 u' y' = 1 + u2 u'   y' = u V  v' ln u + v  u  

!@> y = arctan u !#> y = u V

6-edrIevlMdab´x<s´ ebIGnuKmn_ y = f (x) manedrIevbnþbnÞab;dl;lMdab; n enaH y = f (x) ehAfaedrIevTI n énGnuKmn_ y = f (x) ehIy f (x) = dxd f (x) . 7-el,Ónénclna niymn½y ³ el,ÓnénclnamYyenAxN³ t KW V(t ) = S' (t ) = dS dt Edl S(t ) Cacm¶ayenAxN³ t . 8-sMTu¼énclna sMTuHénclnamYyenAxNH t KW a(t ) = dVdt(t ) = V' (t ) Edl V(t ) Cael,ÓnénclnaenAxN³ t . (n)

(n )

( n −1)

- 53 -

RbCMurUbmnþKNitviTüa

9-GnuKmn_GsniTan >GnuKmn_ y = ax + b Edl a ≠ 0 EdnkMNt; ³ GnuKmn_mann½ykalNa ax + b ≥ 0 -ebI a > 0 enaH x ≥ − ba ehIy D = [− ba , + ∞) -ebI a < 0 enaH x ≤ − ba ehIy D = (−∞ , − ba ] edrIev y' = 2 axa + b -ebI a < 0 enaH y' < 0 naM[GnuKmn_cuHCanic©elIEdnkMNt;. -ebI a > 0 enaH y' > 0 naM[GnuKmn_ekInCanic©elIEdnkMNt; . >GnuKmn_ y = ax + bx + c man ∆ = b − 4ac EdnkMNt; ³ GnuKmn_mann½ykalNa ax + bx + c ≥ 0 -krNI a > 0 Rkabén y = ax + bx + c manGasIumtUteRTtBIrKW k-ebI x → +∞ enaH y = a(x + 2ba ) CaGasIumtUteRTt . k-ebI x → −∞ enaH y = − a (x + 2ba ) CaGasIumtUteRTt -krNI a < 0 2

- 54 -

RbCMurUbmnþKNitviTüa

RkabénGnuKmn_ y = ax + bx + c KµanGasIumtUteT . 2ax + b edrIev y' = mansBaØadUc 2ax + b 2 ax + bx + c -ebI a < 0 GnuKmn_manGtibrmamYyRtg; x = − 2ba . -ebI a > 0 GnuKmn_manGb,brmamYyRtg; x = − 2ba . 10-GnuKmn_RtIekaNmaRtcRmu¼ >cMNucsMxan;²sRmab;sikSaGnuKmn_RtIekaNmaRt -EdnkMNt; -xYbénHnuKmn_ -PaBKUessénGnuKmn_ -TisedAGefrPaBénGnuKmn_ >xYbénGnuKmn_ -xYbénGnuKmn_ y = sin(ax) KW |2πaπ| 2π π -xYbénGnuKmn_ y = cos(ax) KW | a | >PaBKUessénGnuKmn_ -GnuKmn_ f (x) CaGnuKmn_esselI I kalNa ∀x ∈ I , − x ∈ I 2

- 55 -

RbCMurUbmnþKNitviTüa

ehIy f (− x) = −f (x) . -GnuKmn_ f (x) CaGnuKmn_KUelI I kalNa ∀x ∈ I , − x ∈ I ehIy f (− x) = f (x) . 11-RTwsþIbT ebImanBIrcMnYnBit m nig M EdlcMeBaHRKb; x ∈ I : m ≤ f ' ( x ) ≤ M enaHRKb;cMnYnBit a , b ∈ I Edl a < b eK)an m(b − a) ≤ f (b) − f (a) ≤ M(b − a) . eK[GnuKmn_ f manedrIevelIcenøaH [a , b ] . ebImancMnYn M EdlRKb; x ∈ [a , b ] : | f ' (x) |≤ M enaHeK)an ³ | f (b ) − f (a ) |≤ M | b − a | . ebI f CaGnuKmn_Cab;elIcenøaH [a , b ] manedrIevelI cenøaH (a , b ) nig f (a) = f (b) enaHmancMnYnc ∈ (a , b ) mYy y:agticEdl f ' (c) = 0 . ebI f CaGnuKmn_Cab;elIcenøaH [a , b ] manedrIevelI cenøaH (a , b ) enaHmancMnYn c ∈ (a , b ) mYyy:agtic Edl f ' (c) = f (bb) −− fa(a) . - 56 -

RbCMurUbmnþKNitviTüa

GaMgetRkal

CMBUkTI10

1-GaMgetRkalminkMNt´ RBImITIv k> niymn&y snµtfa f (x) CaGnuKmn_kMnt´elIcenøa¼ I . eKfa F(x) CaRBImITIvén f (x) elIcenøa¼ I kalNa F' ( x ) = f ( x ) RKb´ x ∈ I . «TahrN_1 GnuKmn_ F(x) = x CaRBImITIvmYyénGnuKmn_ f ( x ) = 3x elIcenøa¼ ] − ∞ , + ∞ [ BIeRBa¼ F' (x) = 3x = f (x) cMeBa¼RKb´ x ∈ ] − ∞ , + ∞ [ «TahrN_2 GnuKmn_ F(x) = sin x CaRBImITIvmYyénGnuKmn_ f ( x ) = cos x elIcenøa¼ ] − ∞ , + ∞ [ BIeRBa¼ F' (x) = cos x = f (x) cMeBa¼RKb´ x ∈ ] − ∞ , + ∞ [ . «TahrN_3 GnuKmn_ F(x) = ln x CaRBImITIvmYyénGnuKmn_ 1 elIcenøa¼ ] 0 , + ∞ [ f ( x) = x BIeRBa¼ F' (x) = 1x = f (x) cMeBa¼RKb´ x ∈ ] 0 , + ∞ [ . 3

- 57 -

RbCMurUbmnþKNitviTüa x> RTwsþIbT ebIGnuKmn_ F(x) nig G(x) CaRBImITIvénGnuKmn_ f (x) elIcenøa¼ I ena¼eKman F(x) = G(x) + c cMeBa¼RKb´ x ∈ I . Edl c CacMnYnefr . GaMgetRkalminkMnt´ k> niymn&y ebIGnuKmn_ F(x) CaRBImITIvénGnuKmn_ f (x) ena¼GaMgetRkalminkMnt´énGnuKmn_ f (x) kMnt´eday ∫ f ( x).dx = F( x) + c Edl c CacMnYnefr . x> RTwsþIbT snµtfa f (x) nig g(x) CaGnuKmn_BIrmanRBImITIvelIcenøa¼rYm I eKman ½ a/ b/

∫ [ f ( x) + g( x) ]dx = ∫ f ( x).dx + ∫ g( x).dx ∫ [ k . f ( x)]dx = k ∫ f ( x) .dx

rUbmnþGaMgetRkalsMxan´@ 1> ∫ k .dx = kx + c 2> ∫ xn .dx = n 1+ 1 .xn + 1 + c +

, n ≠ −1

3> ∫ dxx = ln | x | + c 4> ∫ dx2 = − 1x + c x

- 58 -

RbCMurUbmnþKNitviTüa 5> ∫ 6> ∫ 7> ∫

dx = 2 x+c x dx 1 = ln | ax + b | + c ax + b a dx 2 = ax + b + c ax + b a

8> ∫ exdx = ex + c 9> ∫ eax .dx = a1 eax + c 10> ∫ a x .dx = ln1a . a x + c 11> ∫ dx2 = tan x + c

cos x dx = − cot x + c 2 sin x sin x.dx = − cos x + c

12> ∫ 13> ∫ 14> ∫ cos x.dx = sin x + c 15> ∫ tan x.dx = − ln | cos x | +c 16> ∫ cot x.dx = ln | sin x | +c 17> ∫ 18> ∫ 19> ∫

dx 1 x−a . ln +c = 2 2 2 a x + a x −a dx 1  x = arctan  +c x2 + a2 a a dx = ln | x + x 2 + a 2 | + c x2 + a2

- 59 -

RbCMurUbmnþKNitviTüa 20> ∫ 21> ∫ 22> ∫ 23> ∫ 24> ∫

dx x2 − a2 dx

= ln | x + x 2 − a 2 | + c

 x = arcsin  + c a a2 − x2 x 2 a2 2 2 2 x + a .dx = x + a + ln | x + x 2 + a 2 | + c 2 2 x 2 a2 2 2 2 x − a .dx = x − a − ln | x + x 2 − a 2 | + c 2 2 x 2 a2  x 2 2 2 a − x .dx = a − x + arcsin  + c 2 2 a

viFIbþÚrGefrkñúgGaMgetRkalminkMnt´ 1-«bmafaeKmanGaMgetRkal I = ∫ f [g(x)].g' (x).dx ebIeKtag u = f (x) ena¼ du = f ' (x).dx eKán I = ∫ f [g(x)].g' (x).dx = ∫ f (u).du = F(u) + c 2-«bmafaeKmanGaMgetRkal I = ∫ f (x).dx tag x = ϕ(t ) ena¼ dx = ϕ' (t ).dt eKán I = ∫ f (x).dx = ∫ f [ϕ(t )].ϕ' (t ).dt . 3-rUbmnþRKw¼ k- ∫ k P' (x).dx = k .P(x) + c - 60 -

RbCMurUbmnþKNitviTüa xKXgc-

1 n [ ] P ( x ) . P ' ( x ). dx = .P n + 1 ( x ) + c ∫ n+1 P' ( x ) ∫ P( x) .dx = ln | P( x) | + c P' ( x) ∫ P( x) .dx = 2 P( x) + c

∫

e P ( x ) .P' ( x ).dx = e P ( x ) + c

∫

P' ( x ) 1 . dx = − +c 2 P( x ) P ( x)

GaMgetRkaledayEpñk ebIeKman u = f (x) nig ∫

, n ≠ −1

u . dv = u .v −

∫

v .du

v = g( x )

eKán ½

2-GaMgetRkalkMNt´

>niymn½y f CaGnuKmn_Cab;elIcenøaH [a , b ] . GaMgetRkalkMNt;BI a eTA b én y = f (x) kMNt;eday ∫ f ( x).dx = F(b) − F(a) Edl F' (x) = f ( x) . b

>épÞRkLaénEpñkbøg; -ebIGnuKmn_ y = f (x) Cab;elIcenøaH [a , b ] enaHépÞRkLa énEpñkbøg;EdlxNÐedayExSekag GkS½Gab;sIus bnÞat;Qr - 61 -

RbCMurUbmnþKNitviTüa x=a , x=b y

kMNt;eday S = ∫ f (x).dx ebI f (x) ≥ 0 a

C : y = f ( x) A S a

B b

-ebIGnuKmn_ y = f (x) Cab;elIcenøaH [a , b ] enaHépÞRkLa énEpñkbøg;EdlxNÐedayExSekag GkS½Gab;sIus bnÞat;Qr b x = a , x = b kMNt;eday S = − ∫ f ( x ).dx ebI f ( x ) ≤ 0 . y

b x

-ebI f nig g CaGnuKmn_Cab;elI [a , b ]enaHeK)anépÞRkLa enAcenøaHExSekagtagGnuKmn_TaMgBIrkMNt;eday ³ - 62 -

RbCMurUbmnþKNitviTüa S=

∫ [f ( x) − g(x)].dx a

Edl f (x) ≥ g(x) RKb; x ∈ [a , b ] . y

3-maDsUlId nig RbEvgFñÚ

ebIGnuKmn_ f viC¢manehIyCab;elIcenøaH [a , b ] enaHmaDénsUlId brivtþn_)anBIrgVilCMuvijGkS½Gab;sIusénépÞEdlxNÐedayRkabtag GnuKmn_ y = f (x) GkS½Gab;sIus bnÞat;Qr x = a nig x = b kMNt;eday V = lim ∑ [πf (x ).∆x] = π ∫ f (x).dx . maDénsUlIdbrivtþkMNt;)anBIrgVilCMuvijGkS½ (ox) énépÞ xNÐeday Rkab y = f (x) nig y = g(x) elIcenøaH [a , b ]

n→ +∞

k =1

- 63 -

RbCMurUbmnþKNitviTüa

Edl f (x) ≥ g(x) kMNt;eday V = π ∫ [f b

]

( x ) − g 2 ( x ) .dx

a x

GnuKmn_ F EdlkMNt;elIcenøaH [a , b ] eday F(x) = ∫ f (t ).dt ehAfaGnuKmn_kMNt;tamGaMgetRkalkMNt; . 1 témømFümén f kMNt;Cab;elI [a , b ] KW y = f ( x ).dx ∫ b−a

RbEvgFñÚénRkabtag f elI[a , b ] KW L = ∫ a

- 64 -

1 + f '2 ( x ) .dx

RbCMurUbmnþKNitviTüa

CMBUkTI11

smIkarDIepr¨g´Esül 1-smIkarDIepr¨g´EsüllMdab´TI1

smIkarDIepr:g;EsüllMdab;TI ! manragTUeTA > dy = f ( x ) mancemøIyTUeTA y = ∫ f ( x ).dx + c dx = f ( x ) mancemøIyTUeTA G ( y ) = F( x ) + C > g(y ). dy dx Edl G(y ) = ∫ g(y ).dy . + ay = 0 mancemøIyTUeTA y = A.e > y'+ay = 0 b¤ dy dx Edl A CacMnYnefr . > y'+ay = p(x) mancemøIyTUeTA y = y + y Edl y CacemøIyénsmIkar y'+ay = 0 nig y CacemøIyBiessmYy énsmIkar y'+ay = p(x) . − ax

- 65 -

RbCMurUbmnþKNitviTüa 2-smIkarDIepr¨g´EsüllMdab´TI2 k-smIkarDIepr¨g´EsüllIenEG‘lMdab´2

niymn½y ³ smIkarDIepr:g;EsüllIenEG‘lMdab;TI2 GUm:UEsn nigman emKuNCacMnYnefrCasmIkarEdlGacsresrCaragTUeTA ³] ay' '+ by'+ cy = 0 Edl a ≠ 0 , a, b, c ∈ IR .

x-dMeNa¼RsaysmIkarDIepr¨g´EsüllMdab´TI2

-smIkarsmÁal; smIkarsmÁal;énsmIkarDIepr:g;EsüllIenEG‘lMdab;TI2 GUmU:Esn nigmanemKuNCacMnYnefr ay' '+by'+cy = 0 CasmIkardWeRkTIBIr aλ + bλ + c = 0 Edl a ≠ 0 , a, b, c ∈ IR -viFIedaHRsaysmIkarDIepr:g;EsüllIenEG‘lMdab;2 ]bmafaeKmansmIkarDIepr:g;EsüllIenEG‘lMdab;@dUcxageRkam³ (E) : y' '+ by'+ cy = 0 Edl b , c ∈ IR . smIkar ( E) mansmIkarsmÁal; λ + bλ + c = 0 (1) KNna ∆ = b − 4c -krNI ∆ > 0 smIkar (1) manb¤sBIrCacMnYnBitepSgKñaKW 2

- 66 -

RbCMurUbmnþKNitviTüa

nig λ = β enaHsmIkar (E) mancemøIyTUeTA CaGnuKmn_rag y = A.e + B.e Edl A , B CacMnYnefrmYyNak¾)an . -krNI ∆ = 0 smIkar (1) manb¤sDúbKW λ = λ = α enaHsmIkar (E) mancemøIyTUeTACaGnuKmn_rag λ1 = α

αx

βx

y = Ax.e αx + B.e αx

Edl A , B CacMnYnefrmYyNak¾)an . -krNI ∆ < 0 smIkar (1) manb¤sBIrepSgKña CacMnYnkMupøicqøas;KñaKW λ = α + i.β nig λ = α − i.β ¬ α , β ∈ IR ¦enaHsmIkar (E) mancemøIyTUeTACaGnuKmn_rag 1

y = ( A cos β x + B sin β x )e αx

Edl A , B CacMnYnefrmYyNak¾)an .

K-dMeNa¼RsaysmIkarDIepr¨g´EsüllMdab´TI2minGUmUEsn

]bmafaeKmansmIkarDIepr:g;EsüllIenEG‘lMdab;TI@minGUmU:Esn y' '+ by'+ cy = P( x ) Edl P( x ) ≠ 0 . edIm,IedaHRsaysmIkarenHeKRtUv ³ EsVgrkcemøIyBiessminGUmU:Esn tageday y P

- 67 -

RbCMurUbmnþKNitviTüa

rbs;smIkar y' '+by'+cy = P(x) Edl y manTRmg;dUc P(x) . rkcemøIyTUeTAtageday y énsmIkarlIenEG‘ lMdab;TI@GUmU:Esn y' '+by'+cy = 0 . eK)ancemøIyTUeTAénsmIkarDIepr:g;EsüllIenEG‘lMdab;TI@ minGUm:UEsnCaplbUkrvag y nig y KW y = y + y . P

- 68 -

RbCMurUbmnþKNitviTüa

viucTr&kñúglMh

CMBUkTI12 ១-វុិចទ័រកនង ុ លំហ ក/ នយមន័ យ ិ

អងកត់មនទិសេដ AB េនកនង ុិ រ ុ លំហេហថវចទ័ កនង ុ ុ លំហែដលមន A ជគល់និង B ជចង។  →

េគកំនត់សរេសរេដយ AB ។ ខ/ កូអរេដេនៃនវុិចទ័រកនង ុ លំហ

→ → →

កនង ុ លំហរបកបេដយតរមុយ (O, i , j , k ) ចំេពះ

រគប់ចំនុច P មនរតីធតុ ( a , b , c ) ែតមួយគត់ែដល  →

 →

→

U = OP = a . i + b . j + c . k ។ រតីធតុ (a , b , c ) េហថ

កូអរេដេនៃនចំនច ុ P ែដលេគសរេសរ P(a , b , c ) ។ ២-ផលគុណសកែលៃនវុិចទ័រកនង ុ លំហ ក/ នយមន័ យ ិ

→

θ O

→

u - 69 -

RbCMurUbmnþKNitviTüa →

→

- ផលគណស ក ែលៃនពីរវចទ័ ួ ពិត ុ ុ ិ រ u និង v គឺជចំនន → →

→

កំនត់េដយ u . v = | u | .| v | cos θ ។ →

→

( θ ជមំុរវងវចទ័ ុ ិ រ u និង v ) →

→

→ →

- េប u = 0 ឬ v = 0 េនះ u . v = 0 ។ ខ/េគល នង ិ តរមុយអរតូណរម៉ល់ z

→

k →

→

េគេហេគលអរតូណរម៉ល់ៃនវចទ័ ុ ិ រ គឺរគប់រតីធតុ → → →

→

( i , j , k ) ែដល | i | = | j | = | k | = 1 → →

→ →

និង i . j = j . k = k . i = 0 ។ - 70 -

RbCMurUbmnþKNitviTüa គ/ រទសត ឹ ប ី ទ

កនង ក ែលរវង ុ ុ េគលអរតូណរម៉ល់ៃនលំហផលគណស →

→

ពីរវចទ័ ុ ិ រ u = ( x1 , y 1 , z1 ) និង v = ( x 2 , y 2 , z 2 ) គឺជចំនួន → →

ពិតកំនត់េដយ u . v = x1x 2 + y 1y 2 + z1z 2 ។ ឃ/ កូរូែល →

→

-ពីរវចទ័ ុ ិ រ u = ( x1 , y 1 , z1 ) និង v = ( x 2 , y 2 , z 2 ) → →

អរតូកូណល់គនលះរតតែត u . v = 0 បនន័យថ ុ →

→

។

u ⊥ v ⇔ x1 x 2 + y 1 y 2 + z 1 z 2

→

-សកែល និង ណមៃនវចទ័ ុ ិ រ u = (a , b , c ) កំនត់េដយ

→

( u ) =a +b +c 2

→

និង | u | = a 2 + b 2 + c 2

។

→

-មំរុ វងពីរវចទ័ ុ ិ រ u = ( x1 , y 1 , z1 ) និង v = ( x 2 , y 2 , z 2 ) → →

→

→ →

→

កំនត់េដយ u . v = | u | .| v | cos θ ⇒ cos θ = ឬ cos θ =

u.v →

| u |.| v |

x1 x 2 + y 1 y 2 + z 1 z 2 x1 + y 1 + z 1 . x 2 + y 2 + z 2 2

- 71 -

→

។

RbCMurUbmnþKNitviTüa ៣-សមករែសវ ៊កុនងលំហ ី z c

P I

(S )

→

k b

→

i a

→

នយមន័ យ ិ

សំណុំ ៃនចំនច ុ P( x , y , z ) ែដលមនចមងយេថរ

ិ I កំ R ។ េសម R ពីចំនុចនឹង I(a ; b ; c ) េហថែសវ៊ផចត សមីកររបស់ែសវេ៊ នះគឺ

(S ) : ( x − a ) 2 + ( y − b ) 2 + ( z − c ) 2 = R 2

- 72 -

។

RbCMurUbmnþKNitviTüa ៤-សមករបន ទ ត់កុនងលំហ ី z →

u (L)

→

k →

→

j x

សមីករបនទត់ (L ) កត់តមចំនច ុ A( x A , y A , z A ) →

េហយមនវចទ័ ុ ិ ររបប់ទិស u (a , b , c ) កំនត់េដយ

 x = x A + at  (L ) :  y = y A + bt ( េហថសមីករប៉រ ៉ែម៉ត)  z = z + ct ; t ∈ IR  A x − xA y − y A z − z A (L ) : = = ( េហថសមីករឆលះុ ) a b c

- 73 -

RbCMurUbmnþKNitviTüa ៥-សមករ=ល ងក់ ង ី ុន តរមុយអរតូណរម៉ល់

ក/សមករ=ល ង ់កត់តមចំនុចមួយនងវុ ួ ី ិ ិ ចទ័រណរម៉ល់មយ

សមីករបលងក ់ ត់តមចំនុច A( x A , y A , z A ) េហយមន →

វចទ័ ុ ិ រណរម៉ល់ n(a ; b ; c ) កំនត់េដយ ( P ) : a( x − x A ) + b( y − y A ) + c( z − z A ) = 0

។

ខ/ចមងយពចំ ុ មួយេទ=លងម់ យ ួ កនុងលំហ ី នច z

d( A ; ( P ))

→

k →

→

j x

ចមងយពីចណ ំ ុ ច A( x A ; y A ; z A ) េទបលង់ ( P ) ែដល - 74 -

RbCMurUbmnþKNitviTüa មនសមីករ ax + by + cz + d = 0 កំនត់េដយ d( A ; ( P )) =

| ax A + by A + cz A + d | a +b +c 2

។

៦-ផលគុណៃនពរវុ ី ិ ចទ័រកនង ុ លំហ

!>niymn½yplKuNénBIrviucTr½ ebI u = u . i + u . j + u . k nig v = v . i + v . j + v . k CaviucTr½kñúglMh . plKuNénviucTr½ u nig v KWCaviuTr½kMNt;eday³ →

→

→ →

→

u× v = (u2 v3 − u3 v2 ) i − (u1 v3 − u3 v1 ) j + (u1v2 − u2 v1 ) k →

→

edIm,IgayRsYlkñúgkarKNnaplKuNénBIrviucTr½ u nig v eKsnµt eRbIedETmINg;lMdab;bIdUcxageRkam ³ →

→

u× v = u1 v1

u2 v2

u3 v3

→

→ →

u× v =

u2 u 3 v2 v3

→

i−

u1 u2 → j+ k v3 v1 v2

u1 u3 v1

→

- 75 -

RbCMurUbmnþKNitviTüa

@>lkçN³énplKuNénBIrviucTr½ ebI u , v nig w CaviucT½rkñúglMh nig c CacMnYnBitenaHeK)an ³ k> u× v = −( v× u ) x> u× ( v + w ) = u× v + u× w K> c ( u× v ) = (c u ) × v = u× (c v ) X> u× o = o × u = o g> u× u = o c> u .( v× w ) = ( u× v ). w #>bMNkRsayénplKuNénBIrviucTr½tamEbbFrNImaRt → →

→

u× v

→

→ →

→

ebI u nig v CaviucTr½minsUnüenAkñúglMh nigtag θ CamMurvagviucTr½ u nig v enaHeK)an ³ →

→

- 76 -

RbCMurUbmnþKNitviTüa →

→

k> u× v GrtUkUNal;eTAnwg u pg nig v pg . x> | u× v |= | u | . | v | . sin θ K> ebI u× v = o enaH u nig v CaviucTr½kUlIenEG‘nwgKña . X> | u× v | ³ CaépÞRkLaénRbelLÚRkamEdlsg;elI u nig v . g> 12 | u× v | ³ CaépÞRkLaénRtIekaNEdlsg;elI u nig v . $>m:Um:g; M énkmøaMg F cMeBaHcMNuc P →

→

 →

P →

Q →

→

ebI Q CacMNuccab;énkmøaMg F enaHm:m:g;énkmøaMg F cMeBaHcMNuc P KW | M | = | PQ × F | . →

 →

→

- 77 -

RbCMurUbmnþKNitviTüa

%>plKuNcRmuHénbIviucTr½kñuglMh k>niymn½y eKmanviucTr½bI u , v nig w enAkñúglMh . plKuNcRmuHén u , v nig w tamlMdab;KWCacMnYnBitkMNt;eday u .( v× w ) = r . x>RTwsþIbTTI1 ebIeKmanviucTr½ u = u . i + u . j + u . k v = v . i + v . j + v . k nig w = w . i + w . j + w . k → →

→

→ →

→

u .( v × w ) = v 1 w1

v2 w2

v3 w3

→

K>RTwsþIbTTI2 maDrbs;RbelBIEb:tEkgEdlsg;elIviucTr½ u , v nig w KW ³ → →

→

nigmaD W rbs;etRtaEG‘tKW ³ W = | u .( v6× w ) | )ann½yfaykmaDrbs;etRtaEG‘tEcknwg 6 . →

→

V = | u .( v × w ) |

- 78 -

RbCMurUbmnþKNitviTüa

^>bnÞat; nig bøg;kñúglMh k-bnÞat´kñúglMh RTwsþIbT ³ eKmanbnÞat; L mYyRsbnwgviucTr½ v = (a, b, c) ehIykat;tam cMnuc P (x , y , z ) enaHsmIkar)a:ra:Em:RténbnÞat; L KW ³ →

x = x 0 + at , y = y 0 + bt , z = z 0 + ct ( t ∈ IR )  x = x 0 + at   y = y 0 + bt ; t ∈ IR  z = z + ct  0

b¤

ebI a , b , c

xusBIsUnüenaHsmIkarqøúHénbnÞat; L KW ³ x−x y−y z−z L: = = . a b c x-bøg´kñúglMh RTwsþIbT ³ ebIbøg;mYykat;tamcMNuc P(x , y , z ) nigmanviucTr½Nrm:al; n = (a, b, c) enaHbøg;mansmIkarsþg;da ³ 0

→

a( x − x 0 ) + b( y − y 0 ) + c( z − z 0 ) = 0

edayBnøatsmIkarenHehIytag d = −(ax + by enaHeK)ansmIkarTUeTA ax + by + cz + d = 0 . 0

- 79 -

+ cz 0 )

RbCMurUbmnþKNitviTüa

K-mMurvagbøg´BIr eKmanbøg;BIr α nig α manviucTr½Nrm:al;erogKña n nig n enaHmMu θ rvagBIrviucTr½enH CamMurvagbøg;TaMgBIrEdlGackMNt;)an →

→

tamTMnak;TMng cos θ =

→

| n1 . n 2 | →

→

| n1 | . | n 2 |

X-smIkarEs‘V smIkarsþg;daénEsV‘Edlmanp©it C(x , y , z ) nigkaM r KW (x − x ) + (y − y ) + (z − z ) = r . BnøatsmIkarsþg;daeK)ansmIkarTUeTAénEsV‘Edlmanrag 0

x 2 + y 2 + z 2 − 2x 0 x − 2 y 0 y − 2 z 0 z + k = 0

Edl k = x + y + z − r . g-cm¶ayBIcMNucmYyeTAbøg´kñúglMh RTwsþIbT cm¶ayBIcMnuc Q eTAbøg; α EdlcMNuc Q minenAkñúgbøg; 2

kMNt;eday D =

 → →

| PQ . n | →

→

Edl P CacMNucenAkñúgbøg;ehIy n

|n|

CaviucTr½Nrm:al;énbøg; . - 80 -

RbCMurUbmnþKNitviTüa

ebI Q(x , y , z ) nig α : ax + by + cz + d = 0 ehIy n = (a, b, c) enaH D = | ax + by + cz + d | . a +b +c c-cm¶ayBIcMNucmYyeTAbnÞat´kñúglMh RTwsþIbT cm¶ayBIcMnuc Q eTAbnÞat; L kñúglMhkMNt;eday ³ 0

→

 →

→

| PQ × u | →

→

Edl P CacMNucenAelIbnÞat;ehIy u CaviucTr½

|u|

R)ab;TisénbnÞat; L .

- 81 -

RbCMurUbmnþKNitviTüa

cMnYnkMupøic

CMBUkTI13

1-niymn½y ³ > cMnYnkMupøicCacMnYnEdlmanTRmg; z = a + i.b Edl a nig b CaBIrcMnYnBit ehIy i ehAfaÉktanimµitEdl i = −1 b¤ i = − 1 . > eKtagsMNMucMnYnkMupøiceday ¢ . > a ehAfaEpñkBitEdleKkMnt;tageday Re(z ) = a . > b ehAfaEpñknimµitEdleKkMnt;tageday Im( z ) = b . 2-RbmaNviFIelIcMnYnkMupøic k> RbmaNviFIbUk nig RbmaNviFIdk snµtfaeKman z1 = a1 + i.b1 nig z 2 = a 2 + i.b 2 Edl a1 , a 2 , b1 , b 2 ∈ IR . eK)anrUbmnþ z + z = (a + a ) + i.(b + b ) 2

nig

z 1 − z 2 = (a1 + a 2 ) − i .(b1 + b 2 )

- 82 -

RbCMurUbmnþKNitviTüa x> RbmaNviFIKuN nig RbmaNviFIEck snµtfaeKman z1 = a1 + i.b1 nig z 2 = a 2 + i.b 2 Edl a1 , a 2 , b1 , b 2 ∈ IR . eK)anrUbmnþ z × z = (a a − b b ) + i.(a b + a b ) 1

nig

1 2

z 1 a 1a 2 + b 1b 2 a 2 b 1 − a 1b 2 = + i . z2 a 22 + b 22 a 22 + b 22

2 1

3-cMnYnkMupøicqøas;nigm:UDúléncMnYnkMupøic > cMnYnkMupøicqøas;éncMnYnkMupøic z = a + i.b KWCacMnYnkMupøicEdltageday z = a − i .b . > kñúgtRmuyGrtUnrm:al; eKGactagcMnYnkMupøic z = a + i.b edaycMnuc P(a , b ) . y b

P | z |= r

0 y'

- 83 -

RbCMurUbmnþKNitviTüa > rgVas; OP = r =| z | ehAfam:UDúléncMnYnkMupøic eKkMnt;sresr ³ | z |= r = a + b . 2

z = a + i .b

5-sV½yKuNén i cMeBaHRKb;cMnYnKt; k ∈ IN eKmansV½yKuNén i dUcxageRkam ³ i =1 , i =i , i = −1 nig i = −i . 4k +1

4k + 2

4k + 3

]TahrN_ TahrN ³ KNna S = 1 + i + i + i + ..... + i . tamrUbmnþplbUksIVútFrNImaRt 1 + q + q + q + ..... + q 2

2006

1 − i 2007 S= i 2007 = i 4×501+ 3 = − i 1− i 1+ i (1 + i ) 2 1 + 2i − 1 2i S= = = = = i 1 − i (1 − i )(1 + i ) 1+1 2

eK)an eday dUcenH 6-Gnuvtþn_kñúgdMeNaHRsaysmIkardWeRkTIBIr eKmansmIkardWeRkTIBIr az 2 + bz + c = 0 Edl a ≠ 0 nig a, b, c ∈ IR . DIsRKImINg;énsmIkarKW ∆ = b 2 − 4ac . -ebI ∆ > 0 smIkarmanb¤sBIrCacMnYnBitKW ³ z1 =

−b+ ∆ −b− ∆ , z2 = 2a 2a

-ebI ∆ = 0 smIkarmanb¤sDúbCacMnYnBitKW³ - 84 -

1 − q n +1 = 1− q

. z1 = z 2 = −

b 2a

RbCMurUbmnþKNitviTüa -ebI ∆ < 0 smIkarmanb¤sBIrCacMnYnkMupøicqøas;KñaKW ³ z1 =

−b+i |∆| −b−i |∆| , z2 = 2a 2a

7-rebobKNnab¤skaeréncMnYnkMupøic ³ edIm,IKNnab¤skaeréncMnYnkMupøic z = a + i.b eKRtUvGnuvtþn_dUcteTA ³ -tag W = x + i.y ; x, y ∈ IR Cab¤skaerén z = a + i.b . -eK)an W = z eday W 2 = (x + i.y )2 = (x 2 − y 2 )+ 2ixy -eK)an (x − y )+ 2ixy = a + i.b x 2 − y 2 = a -eKTaj)anRbBnæ½ 2xy = b 2



¬RtUvedaHRsayRbBn½æenHrkKUcemøIy x; y ¦ 8-TRmg;RtIekaNmaRténcMnYnkMupøic y Q

| z |= r

R a

enAkñúgbøg;kMupøic (xoy) eK[cMnuc P(a ;b) tag[cMnYnkMupøic z = a + i.b . - 85 -

RbCMurUbmnþKNitviTüa tag θ CamMutUcbMputén

 → →   0x ; 0P     

EdlehAfaGaKuym:g;éncMnYnkMupøic z = a + i.b . kñúgRtIekaNEkg OPR eKman OP = OR + RP 2

eday

OP = r =| z |  OR = a  RP = OQ = b 

eK)an r

müa:geTot

= a2 + b2

b¤ r =

OR a  cos θ = =  OP r  sin θ = RP = b  OP r

a2 + b2

naM[

¬ehAfam:UDúlén z = a + i.b ¦.

a = r . cos θ  b = r . sin θ

eK)an z = a + i.b = r. cos θ + i.r. sin θ = r(cos θ + i. sin θ) dUcenH

z = r .(cos θ + i . sin θ )

Edl

a b  θ = θ = cos ; sin  r r   2 2 r = a + b > 0

9-RbmaNviFIelIcMnYnkMupøicRtIekaNmaRt k> rUbmnþviFIKuN ³ snµtfa Z = r (cos θ + i. sin θ ) nig Z = r (cos θ + i. sin θ eK)an Z × Z = r .r [cos(θ + θ ) + i. sin(θ + θ )] . 1

x> rUbmnþviFIEck ³ - 86 -

)

RbCMurUbmnþKNitviTüa snµtfa Z = r (cos θ + i. sin θ ) nig Z = r (cos θ eK)an ZZ1 = rr1 [cos(θ1 − θ 2 ) + i. sin(θ1 − θ 2 )] . 1

+ i . sin θ 2 )

K> sV½yKuNTI n éncMnYnkMupøicRtIekaNmaRt ³ snµtfaeKman Z = r (cos θ + i. sin θ) nigRKb;cMnYnKt; n ∈ Z eK)an Z = [r. (cos θ + i. sin θ)] = r [cos(nθ) + i. sin(nθ)] n

X> rUbmnþdWmr½ ³ cMeBaHRKb; n ∈ Z eKman

(cos θ + i. sin θ)n = cos(nθ) + i. sin(nθ)

10-b¤sTI n éncMnYnkMupøic³ RTwsþIbT ³ snµtfa Z = r (cos θ + i. sin θ)CacMnYnkMupøicminsUnü ehIy n ∈ IN * . ebI Wk Cab¤sTI n éncMnYnkMupøicxagelIenaHeK)an ³   θ + 2kπ   θ + 2kπ   Wk = n r cos  + i . sin   n    n 

Edl k = 0 , 1, 2, 3 ,......,n − 1 . 11-TRmg;Giucs,:ÚNg;EsüléncMnYnkMupøic ³ cMnYnkMupøic Z Edlmanm:UDúl | Z |= r nig GaKuym:g; arg(Z) = θ manTRmg;Giucs,:ÚNg;EsülkMnt;eday Z = r . ei .θ . - 87 -

RbCMurUbmnþKNitviTüa

CMBUkTI14

RTwsþIbTkñúgRtIekaN ១-ទំនក់ទន ំ ងមរតកនង ី ុ រតេកណែកង ឧបមថេគមនរតីេកណ ABC មួយែកងរតង់ កំពូល A និងមនកំពស់ AH ។ េគមនទំនក់ទំនងសំខន់ៗដូចខងេរកម 1/ AB = BH .BC 2 2/ AC = HC.BC 2 3/ AH = BH .HC 2

4/ BC = AB + AC 2

5/ AH .BC = AB.AC

( រទឹសប ីត ទពីតហគរ ័ )

1 1 1 = + AH 2 AB 2 AC 2 AB 7/ sin α = BC AC 8/ cos α = BC B AB 9/ tan α = AC

- 88 -

RbCMurUbmnþKNitviTüa ២-រទសត ូ ឹ ប ី ទសុន ី ស A b

ឧបមថេគមនរតី ឧបមថេគមនរតីេកណ ABC មួយចរ ឹកកនុង

រងវង់កំ R េហយមនរជុង BC = a ; AC = b និង AB = c ។

a b c េគមនទំនក់ទំនង = 2R = = sin A sin B sin C

៣-រទសត ូ ឹ ប ី ទកូសន ីុ ស រតីេកណ ABC មួយមនរជុង BC = a ; AC = b និង AB = c ។ េគមនទំនក់ទំនង 1/ a 2 = b 2 + c 2 − 2bccsoA 2/ b 2 = c 2 + a 2 − 2ca cos B 3/ c 2 = a 2 + b 2 − 2ab cos C

- 89 -

RbCMurUbmnþKNitviTüa ៤-រូបមនតចេំ ណលែកង រតីេកណ ABC មួយមនរជុង BC = a ; AC = b និង AB = c ។ េគមនទំនក់ទំនង 1/ a = b cos C + c cos B

2/ b = c cos A + a cos C 3/ c = a cos B + b cos A

៥-រទសត ឹ ប ី ទតង់ហសង់

រតីេកណ ABC មួយមនរជុង BC = a ; AC = b និង AB = c ។ េគមនទំនក់ទំនង

A−B a−b 2 1/ = A+B a+b tan 2 B−C tan b−c 2 2/ = B+C b+c tan 2 C− A tan c−a 2 3/ = C+ A c+a tan 2 tan

- 90 -

RbCMurUbmnþKNitviTüa ៦-កំរងវង ់ចរកកន យ ឹ ង ី ុ មុៃំ នរតេកណមួ រតីេកណ ABC មួយមនរជុង BC = a ; AC = b និង AB = c េហយ p =

a+b+c ។ 2

តង rA , rB , rC ជកំរងវង់ចរ ឹកកនុងមុំ A , B , C

ៃនរតីេកណ ABC ។េគមនទំ ។េគមនទំនក់ទំនង K C

rA J rA

H A

A p−c p−b = = B C 2 tan tan 2 2 B p−a p−c 2/ rB = p. tan = = C A 2 tan tan 2 2 C p−b p−a 3/ rC = p. tan = = A B 2 tan tan 2 2

1/ rA = p. tan

- 91 -

rA L

(rA )

RbCMurUbmnþKNitviTüa ៧-កេនសមកំរងវង ់ចរកកន យ ឹ ង ី ុ ៃនរតេកណមួ រតីេកណ ABC មួយមនរជុង BC = a ; AC = b និង AB = c េហយ p =

a+b+c ។ 2

តង r ជកំរងវង់ចរ ឹកកនុងៃនរតីេកណេនះ ។

េគមនទំនក់ទំនង

A B C r = (p − a ) tan = (p − b ) tan = (p − c) tan 2 2 2 A L K I r

- 92 -

RbCMurUbmnþKNitviTüa ៨-រូបមនតគណនៃផទរកឡរតេកណ ី រតីេកណ ABC មួយមនរជុង BC = a ; AC = b

និង AB = c េហយ p = a + b + c ជកនលះបរ ិមរត។ 2

តង r និង R េរៀងគនជកំរងវង់ចរ ឹកកនុងនិងេរក

ៃនរតីេកណេនះ , តង rA , rB , rC ជកំរងវង់ ចរ ឹកកនុងមុំ និង h a , h b , h c ជកពស់គូស ពីកំពូល A , B , C ៃនរតីេកណ ។

1 1 1 a.h a = b.h b = c.h c 2 2 2 abc 2/ S = = pr 4R 3/ S = p(p − a )(p − b )(p − c) 1 1 1 4/ S = b.c sin A = c.a sin B = a.b sin C 2 2 2 A B C 5/ S = p(p − a ) tan = p(p − b ) tan = p(p − c ) tan 2 2 2 6/ S = 2R 2 sin A sin B sin C 1/ S =

7/ S = (p − a )rA = (p − b )rB = (p − c )rC 8/ S =

r . rA . rB . rC

- 93 -

RbCMurUbmnþKNitviTüa ៩-រទសត ឹ ប ី ទេមដយនកនង ី ុ រតេកណ រតីេកណ ABC មួយមនរជុង BC = a ; AC = b និង AB = c ។ AL = m a ; AM = m b ; AN = m c ជេមដយនៃនរតីេកណ ABC ។ A

េគមនទំនក់ទំនង

b2 + c2 a2 − 1/ m a = 2 4 2 2 c + a b2 2 − 2/ m b = 2 4 2 2 a + b c2 2 3/ m c = − 2 4 2

- 94 -

RbCMurUbmnþKNitviTüa ១០១០-រទសត ំ ង ឹ ប ី ទបនទត់ពះុ មុក ុន រតីេកណ ABC មួយមនរជុង BC = a ; AC = b និង AB = c ។ AA' = L a ; BB' = L b ; CC' = L c ជរបែវងៃនបនទត់ពុះកនុងៃនមុំ A , B , C ។ A

C' I

េគមនទំនក់ទំនង 2bc A cos b+c 2 2ac B 2/ L b = cos a+c 2 2ab C 3/ L c = cos a+b 2

1/ L a =

- 95 -

RbCMurUbmnþKNitviTüa ១១១១-កេនសម cos A cos = 2 C cos = 2

A B C ; cos ; cos 2 2 2 p(p − a ) B ; cos = bc 2

p(p − b ) ac

p(p − c ) ab A B C ១២១២-កេនសម sin ; sin ; sin 2 2 2 A (p − b )(p − c ) B (p − a )(p − c ) sin = ; sin = 2 bc 2 ac C (p − a)(p − b ) sin = 2 ab A B C ១៣១៣-កេនសម tan ; tan ; tan 2 2 2 A (p − b )(p − c) B (p − a )(p − c ) tan = ; tan = 2 p( p − a ) 2 p(p − b ) tan

C (p − a )(p − b ) = 2 p(p − c )

- 96 -

RbCMurUbmnþKNitviTüa ១៤១៤-ទំនកទំនងេផសងៗេទៀតគួកត់សគ ំ ល់ កនុងរតីេកណ ABC មួយេគមនទំនក់ទំនង ខងេរកម A 2

B 2

1/ sin A + sin B + sin C = 4 cos cos cos

C 2

A B C sin sin 2 2 2 3/ tan A + tan B + tan C = tan A tan B tan C

2/ cos A + cos B + cos C = 1 + 4 sin

4/ cot A cot B + cot B cot C + cot C cot A = 1 A B C A B C + cot + cot = cot cot cot 2 2 2 2 2 2 A B B C C A 6/ tan tan + tan tan + tan tan = 1 2 2 2 2 2 2 7/ sin 2 A + sin 2 B + sin 2 C = 1 − 2 cos A cos B cos C

5/ cot

8/ cos 2 A + cos 2 B + cos 2 C = 2 + 2 cos A cos B cos C A B C A B C + cos 2 + cos 2 = 2 + 2 sin sin sin 2 2 2 2 2 2 A B C A B C 10/ 10/ sin 2 + sin 2 + sin 2 = 1 − 2 sin sin sin 2 2 2 2 2 2

9/ cos 2

- 97 -

RbCMurUbmnþKNitviTüa ១៥១៥-រទសត ឹ ប ី ទ Ceva

F E K B

កនុងរតីេកណ ABC មួយ,បនទត់បី AD ; BE ; CF របសពវគនរតង់ចំនុច K ែតមួយលុះរតែត AF BD CE . . =1 ។ FB DC EA

- 98 -

RbCMurUbmnþKNitviTüa ១៦១៦-រទសត ឹ ប ី ទ Menelaus A F F

ិ េនេល AB , BC , AC េគមនបីចំនុច F , D , E សថត ៃនរតីេកណ ABC ។ បីចំនុច F , D , E រត់រតង់គន លុះរតែត

AF BD CE . . =1 ។ BF CD AE

១៧១៧-រទសត ឹ ប ី ទឡបនច ិ ិ កនុងរតីេកណ ABC ែដលមនរជុង a ; b ; c ចរ ឹក កនុងរងវង់ផិត ច O កំ R េហយ G ជទីរបជុំទមងន់ៃន

- 99 -

RbCMurUbmnþKNitviTüa a2 + b2 + c2 រតីេកណ ABC េគមន OG = R − 9 2

១៧១៧-រទសត ឹ ប ី ទ Stewart េប L ជចំនុចេនេលរជុង BC ៃនរតីេកណ ABC ែដល AL = l ; BL = m ; LC = n , a , b , c ជរជុង េនះេគមន a ( l 2 + mn ) = b 2 m + c 2 n ។

- 100 -

។

RbCMurUbmnþKNitviTüa

vismPaBcMnYnBit

CMBUkTI15

១/ វិ សមភព មធយមនពវនត មធយមធរណមរត ី

(The AM-GM Inequality ) ិ មន a1 ,a 2 ,a 3 ,...,an ចំេពះរគប់ចំនួនពិតមិនអវជជ េគបន

a1 + a 2 + a 3 + ... + an n ≥ a1 .a 2 .a 3 ...an ។ n

ិ ល យជសមភពលុះរតែត និង វសមភពេនះក រគន់ែត a1 = a 2 = a 3 = .... = an ។ សរមយបញ សរមយបញជក់

a1 + a 2 -eKman 2 ≥ a1a2 smmUl ( a1 − a2 )2 ≥ 0 dUcen¼vismPaBBi mPaBBitcMeBa¼ Ba¼ n = 2 .

-«bmafasmPaBen¼ «bmafasmPaBen¼Bitdl´tYTI a1 + a 2 + a 3 + ...... + ak ≥ k

eyIgnwgRsayfavaBitdl´tYTI

n=k k

a1 .a 2 .a 3 ......ak

k +1

a1 + a 2 + a 3 + .... + ak + ak +1 ≥ k +1

- 101 -

k +1

Bit

KW ½ a1 .a 2 .a 3 ...ak .ak +1

RbCMurUbmnþKNitviTüa

tagGnuKmn_ mn_

(x + a1 + a 2 + a 3 + ...... + ak )k +1 f (x) = x Edl x > 0 , ak > 0 , k = 1 , 2 , 3 , ... .

eyIgán

(k + 1)(x + a1 + .... + ak )k x − (x + a1 + ... + ak )k +1 f '(x) = x2 (x + a1 + .... + ak )k [ (k + 1)x − (x + a1 + .... + ak )] = x2

(x + a1 + a2 + .... + ak )k ( kx − a1 − a2 − .... − ak ) = x2

a1 + a 2 + ..... + ak ebI f '(x) = 0 naM[ x = k cMeBa¼ x = a1 + a2 +k..... + ak eKán eKán ½

a1 + a 2 + ... + ak k +1 + a + ... + a ) 1 k a + a 2 + .... + ak k f( 1 )= a1 + a 2 + ... + ak k k (

a + a 2 + ...... + ak  a + a 2 + ... + ak  f( 1 ) = (k + 1)k +1  1  k k  

cMeBa¼ Ba¼

∀x ≥ 0

eyIgTaján ½ - 102 -

RbCMurUbmnþKNitviTüa a1 + a 2 + .... + ak f (x) ≥ f ( ) k k a + a + a + ..... + a  2 3 k  f (x) ≥ (k + 1)k +1  1  k  

tamkar«bma smmUl

a1 + a 2 + a 3 + ..... + ak ≥ k

a1 .a 2 .a 3 ......ak

 a1 + a 2 + .... + ak    ≥ a1 .a 2 ...ak k   k

(x + a1 + a 2 + .... + ak )k +1 ≥ (k + 1)k +1 a1a2 ...ak x x + a1 + a 2 + ..... + ak ≥ k +1 a1 .a 2 .......ak x (*) k +1

x = a k +1 > 0

CYskñúg (*) eKán

a1 + a 2 + a 3 + ... + ak + ak +1 k +1 ≥ a1 .a 2 .a 3 ......ak .ak +1 k +1

dUcen¼

a1 + a 2 + a 3 + ... + an n ≥ a1 .a 2 .a 3 ...an ។ n

២/ វិ សមភព កូស-ីុ សវស ័ (Cauchy-Schwarz’s Inequality)

ក-រទសត ឹ ប ី ទ ចំេពះរគប់ចំនួនពិត a1 ; a 2 ;...; an ; b1 ; b 2 ; ...; b n េគបន - 103 -

RbCMurUbmnþKNitviTüa (a1b1 + a2b2 + ... + anbn )2 ≤ (a12 + a22 + ... + an2 )(b12 + b22 + ... + bn2 ) ឬ 2

( )

n n  n  2 2 a b a b ≤ × ( ) ∑ ∑ ∑  k k  k k  k =1  k =1 k =1

។

ិ ល យជសមភពលុះរតែតនិងរគន់ែត វសមភពេនះក

a1 a 2 a 3 a = ... = n ។ = = b1 b 2 b 3 bn សរមយបញជក់

eyIgeRCIserIsGnuKmn_ Y kMnt´ mnmY_my

∀x ∈ IR eday

f (x) = (a1x + b1 )2 + (a 2 x + b 2 )2 + ..... + (an x + bn )2 n

f (x) = ∑

k =1

eday

( )x ak2

k =1

( )

+ 2 ∑ ( ak bk ) x + ∑ bk2

∀x ∈ IR RtIFa f (x) ≥ 0

Canic©ena¼

af > 0  ∆ ' ≤ 0

( )

n n  n  2 ∆ ' =  ∑ ( ak bk )  − ∑ ak × ∑ bk2 ≤ 0  k =1  k =1 k =1 2 n n  n  2 ឬ  ∑ ( ak bk )  ≤ ∑ ak × ∑ bk2  k =1  k =1 k =1

( )

- 104 -

( )

RbCMurUbmnþKNitviTüa 2/ Cauchy-Schwarz in Engle form ចំេពះរគប់ចំនួនពិត x1 , x 2 ,..., xn , y 1 , y 2 ,..., y n > 0 េគបន

x1 2 x 2 2 xn 2 (x1 + x 2 + ... + xn )2 + ... + + ≥ y1 y 2 yn y1 + y 2 + ... + y n

ិ វសមភពេនះក ល យជសមភពលុះរតែតនិងរគន់ែត

x1 x 2 x 3 xn = ... = ។ = = y1 y 2 y 3 yn សរមយបញជក់ 2

( )

n n  n  2 2 ិ តមវសមភព a b ≤ a × b ( ) ∑ ∑ ∑  k k  k k  k =1  k =1 k =1 xk េបេគយក ak = , bk = y k yk 2

n x 2 n   xk  k េគបន  ∑  . yk   ≤ ∑   × ∑ ( yk )  k =1 y  k =1  y k  k =1    k 2 n   x ( ) ∑  k  n x 2  , េគទញ ∑  k  ≥  k =n1 k =1  y k  ∑ (y ) n

k =1

- 105 -

RbCMurUbmnþKNitviTüa x1 2 x 2 2 xn 2 (x1 + x 2 + ... + xn )2 ដូចេនះ + ... + + ≥ y1 y 2 yn y1 + y 2 + ... + y n ៣/ វិ សមភពហូលឌ័រ (HÖlder’s Inequality) ិ មន a i,j ,1 ≤ i ≤ m , 1 ≤ j ≤ 1 រទសត ឹ ប ី ទ រគប់ចំនួនពិតវជជ

    m េគបន ∏  ∑ aij  ≥  ∑ ∏ a ij  i =1  j=1   j=1 i =1  m

។

រូបមនតេផសងេទៀតៃន HÖlder’s Inequality រគប់ចន ំ ន ួ វជជ ិ មន x1 , x 2 ,...., xn នង ិ y 1 , y 2 ,..., y n

1 1 ចំេពះ p > 0,q > 0 និង + = 1 p q េនះេគបន

 p x y ≤ x ( ) ∑ k k ∑ k  n

k =1

 k =1

1 p

  ×  ∑ yk q    k =1  n

៤/ វិ សមភពមនកូ ិ សីគ (Minkowski’s Inequality) រទសត ួ វជជ ិ មន a1 ,a 2 ,...,an នង ិ ឹ ប ី ទទ១ ី ចំេពះចំនន

b1 ,b 2 ,....,bn ; ∀n ∈ ℕ នងចំ ិ េពះ p ≥ 1 េគបន :

- 106 -

1 q

។

RbCMurUbmnþKNitviTüa 1 p p

1 p p

   (a + b ) a ≤ + ∑ ∑     ∑ bk  k k k   k =1   k =1   k =1  n

។

រទសត ួ វជជ ិ មន a1 ,a 2 ,...,an នង ិ ឹ ប ី ទទ២ ី ចំេពះចំនន

b1 ,b 2 ,....,bn ; ∀n ∈ ℕ ែដល n ≥ 2 េគបន n

k =1

∏ ( ak ) + n ∏ ( bk ) ≤ n ∏ ( ak + bk )

។

៥/ វិ សមភពែបនូល(ី Bernoulli’s Inequality) •

•

a ចំេពះ x > −1 , a ∈ (0,1) េគបន: េគបន: (1 + x) < 1 + ax

ចំេពះ x > −1,n < 1 េគបន: (1 + x)a > 1 + ax

៦/ វសមភព CHEBYSHEV(Chebyshev’s Inequality) ិ េគេអយពរស៊ វ ៃនចំនន ួ ពតវជជ ី ីត ិ ិ មន

a1 ,a 2 ,...,an នង ិ b1 ,b 2 ,....,bn នង ិ n∈ℕ* -ចំេពះ a1 ≤ a 2 ≤ ... ≤ an នង ិ b1 ≤ b 2 ≤ ... ≤ bn

- 107 -

RbCMurUbmnþKNitviTüa n

n 1 n េគបន : ∑ ( ak bk ) ≥ ∑ ( ak ) × ∑ ( bk ) n k =1 k =1 k =1

-ចំេពះ a1 ≤ a 2 ≤ ... ≤ an នង ិ b1 ≥ b 2 ≥ ... ≥ bn n

n 1 n េគបន ∑ ( ak bk ) ≤ ∑ ( ak ) × ∑ ( bk ) n k =1 k =1 k =1

៧/ វិ សមភព JENSEN (Jensen’s Inequality)

វិ សមភព JENSEN ទរមង់ទ១ ី េគឲយ n ចំនន ួ ពត ិ x1 ,x 2 ,...,xn ∈ I •

េប f ''(x) < 0 នង េគបន: ិ ∀xk ∈ I េគបន:

1 n 1 n  f (x ) ≤ f (x ) [ ] ∑ ∑ k  ។ k  n k =1  n k =1  •

េប f ''(x) > 0 នង េគបន: ិ ∀xk ∈ I េគបន:

1 n 1 n  ∑ [ f (xk )] ≥ f  ∑ (xk ) ។ n k =1  n k =1 

- 108 -

RbCMurUbmnþKNitviTüa វិ សមភព JENSEN ទរមង់ទ២ ី េគឲយ n ចំនន ួ ពត ិ x1 ,x 2 ,...,xn ∈ I នងចំ ិ េពះរគប់ ចំនន ួ វជជ ិ មន a1 ,a 2 ,...,an ែដលផលបូក •

∑ ( ak ) = 1

k =1

េប f ''(x) < 0 នង េគបន: ិ ∀xk ∈ I េគបន:

n  a f (x ) ≤ f (a x ) [ ] ∑ k k ∑ k k  ។   k =1  k =1 n

•

េប f ''(x) > 0 នង េគបន: ិ ∀xk ∈ I េគបន:

n  f (a x ) ≥ f (a x ) [ ] ∑ ∑ k k  ។ k k   k =1  k =1 n

វិ សមភព JENSEN ទរមង់ទ៣ ី េគឲយ n ចំនន ួ ពត ិ x1 ,x 2 ,...,xn ∈ I នងចំ ិ េពះរគប់ ចំនន ួ វជជ ិ មន a1 ,a 2 ,...,an ។ •

េប f ''(x) < 0 នង េគបន: ិ ∀xk ∈ I េគបន:

- 109 -

RbCMurUbmnþKNitviTüa  n  a f (x ) (a x ) [ ] ∑ k k ∑ k k   k =1 k =1 ≤ f   ។ n n  ∑ ( ak )  ∑ ( ak )  k =1  k =1 n

•

េប f ''(x) > 0 នង េគបន: ិ ∀xk ∈ I េគបន: n  n  a f (x ) (a x ) ∑[ k k ] ∑ k k  k =1 ≥ f  k =1n  n  ∑ ( ak )  ∑ ( ak )  k =1  k =1 ៨/ វិ សមភព Schur (Schur’s Inequality) ិ មន a ,b ,c និង n > 0 េគបន រគប់ចំនួនពិតវជជ

an (a − b)(a − c) + bn (b − c)(b − a) + cn (c − a)(c − b) ≥ 0

ិ តចំេពះ a = b = c ។ វសមភពេនះពិ ៩/ វិ សមភព (Rearrangement’s Inequality) ិ មន េគឲយ ( an )n≥1 និង ( bn )n≥1 ជសវត ីុ ៃនចំនួនពិតវជជ េកន ឬចុ ឬចុះរពមគន ។ ចំេពះរគប់ចមលស់ (cn ) ៃនចំ ៃនចំនួន n

k =1

( bn ) េគបន ∑ ( ak bk ) ≥ ∑ ( ak ck ) ≥ ∑ ( ak bn−k +1 ) - 110 -

RbCMurUbmnþKNitviTüa

CMBUkTI16

PaBEckdac´ nig viFIEckGWKøIt !>PaBEckdac;kñúg Z k-niymn½y -cMnYnKt;rWuLaTIb a CaBhuKuNéncMnYnKt;rWLaTIb b luHRtaEtman cMnYnrWuLaTIb q mYyEdl a = b.q . kñúgkrNIenH b ehAfatYEckén a . -ebI b ≠ 0 enaHeKfa b CatYEckmYyén a b¤ b Eckdac; a ehIy eKkMnt;sresr b | a Ganfa b Eckdac; a . x-lkçNHEckdac;énplbUknigpldk man a , b , c nig x CacMnYnKt;rWLaTIbEdl x xusBIsUnü . ebI x | a , x | b nig x | c enaH x | (a + b − c) . K-lkçNHEckdac;nwgmYycMnYn -mYycMnYnKt;Eckdac;nwg 2 luHRtaEtelxxÞg;rayEckdac;nwg 2 )ann½yfacMnYnenaHRtUvmanelxxagcugCaelxKU ³ ( 0, 2, 4 , 6,8 ) . - 111 -

RbCMurUbmnþKNitviTüa

-mYycMnYnEckdac;nwg 5 luHRtaEtvamanelxcug 0 b¤ % . -mYycMnYnEckdac;nwg $ luHRtaEtcMnYnBIrxÞg;xagcugEckdac;nwg $ -mYycMnYnEckdac;nwg @% luHRtaEtcMnYnBIrxÞg;xagcugEckdac;nwg @% KW 00 , 25 , 50 , 75 . -mYycMnYnEckdac;nwg # ¬ nwg ( ¦ luHRtaEtplbUkelxRKb;xÞg;én cMnYnenaHEckdac;nwg # ¬ nwg ( ¦ . -mYycMnYnEckdac;nwg !! luHRtaEtpldkrvagplbUkelxxÞg;ess nigplbUkelxxÞg;KUéncMnYnenaH ¬rab;BIsþaMeTAeqVg ¦ Eckdac;nwg !! . @>viFIEckEbbGWKøIt k-niymn½y eFIVviFIEckEbbGWKøIténcMnYnKt;rWLaTIb a nigcMnYnKt;FmµCati b KW kMNt;cMnYnKt;rWuLaTIb q nigcMnYnKt;FmµCati r Edl a = bq + r eday 0 ≤ r < b . a ehAfatMNaMgEck , b ehAfatYEck , q ehAfaplEck nig r ehAfasMNl; .

- 112 -

RbCMurUbmnþKNitviTüa

x-RTwsþIbT ebI a CacMnYnKt;rWLaTIb nig b CacMnYnKt;FmµCati enaHmancMnYnKt; rWuLaTIb q EtmYyKt; nig cMnYnKt;FmµCati r EtmYyKt;Edl a = b.q + r eday 0 ≤ r < b .

cMnYnbzm tYEckrYm nig BhuKuNrYm cMnYnbfm

-eKfacMnYnKt;FmµCati n CacMnYnbfmkalNa n mantYEck EtBIKt;KW 1 nig n xøÜnÉg . -RKb;cMnYnKt;FmµCati n > 1 mantYEckCacMnYnbfmmYyEdlCa tYEcktUcbMputeRkABI 1 . -ebI n ∈ IN ehIy n minEmnCacMnYnbfm enaHmancMnYnbfm b Edl n Eckdac;nwg b nig b2 ≤ n . -edIm,IsÁal;facMnYnKt;FmµCati a NamYyCacMnYnbfm eKRtUvEck a nigcMnYnbfmt²KñaEdltUcCagva . ebIKµanviFIEckNamYypþl; sMNl;sUnüeTenaH nig plEcktUcCagtYEckEdl)anykmkeRbI enaH a CacMnYnbfm . -ebI n ∈ ℕ ehIy n Eckmindac;nwgcMnYnbfmEdlmankaertUc - 113 -

RbCMurUbmnþKNitviTüa

Cagb¤esµI n enaH n CacMnYnbfm . -sV‘IúténcMnYnbfm CasIVútGnnþtY . -RKb;cMnYnKt;FmµCatiminbfm ehIyFMCag 1 GacbMEbkCaplKuN énktþabfm)an ehIy)anEtmYyEbKt; . tYEckrYm nig BhuKuNrYm -eK[ a nig b CacMnYnKt;FmµCati . cMnYnKt;FmµCati d CatYEck rYmén a nig b kalNa d CatYEckén a pg nigCatYEckén b pg. -tYEckrYmFMbMputéncMnYnKt;FmµCati a nig b CacMnYnKt;FMCageK énbNþatYEckrYmén a nig b . nimµitsBaØa δ = PGCD(a,b) b¤ δ = GCD(a,b) CatMNag[tYEckrYmFMbMputéncMnYnKt;FmµCati a nig b . -cMnYnKt;FmµCati a nig b CacMnYnbfmrvagKñakalNa tYEckrYmFMbMput GCD(a,b) = 1 nigRcas;mkvij . -ebI a , b ∈ ℕ Edl a = bq + r , 0 < r < b enaHeK)an GCD(a,b) = GCD(b,r) . -RKb; a , b ∈ ℕ nigRKb;tYEckrYm d én a nig b eK)an ³

- 114 -

RbCMurUbmnþKNitviTüa

k> GCD(na ,nb) = nG(a,b) Edl n ∈ ℕ x> GCD  da , db  = GCD(a,b) d   -RTwsþIbT Bezout cMnYnKt;FmµCati a nig b bfmrvagKñaluHRtaEtmancMnYnKt;rWuLaTIb u nig v Edl au + bv = 1 . -RTwsþIbT Gauss : ebI c | ab nig GCD(a,b) = 1 naM[ c | b . -ebI a | n ; b | n nig GCD(a,b) = 1 naM[ ab | n . -BhuKuNrYmtUcbMputénBIrcMnYnKt; a nig b KWCacMnYnKt;FmµCati EdltUcCageKkñúgbNaþBhuKuNrYmviC¢manxusBIsUnüén a nig b EdlkMNt;eday µ = PPCM(a,b) b¤ µ = LCM(a,b) -RKb;cMnYnKt;FmµCati a ,b nig n nigRKb;tYEckrYm a nig b eK)an k> LCM(na,nb) = n LCM(a,b) a b LCM(a,b) x> LCM( d , d ) = d -RTwsþIbT ³ ebI a nig b CacMnYnKt;FmµCatienaHeK)an GCD(a,b) × LCM(a,b) = a × b . - 115 -

RbCMurUbmnþKNitviTüa

CMBUkTI17

GnuKmnGIuEBbUlik (Hyperbolic Functions)

!>kenSamBICKNitsþg;da GnuKmn_GIuEBbUlikman ³ e −e sIunUsGIuEBbUlik kMNt;edaysmIkar sinh x = 2 e +e kUsIunUsGIuEBbUlik kMNt;edaysmIkar cosh x = 2 e −e tg;sg;GIuEBbUlik kMNt;edaysmIkar tanh x = e +e −x

−x

e x − e −x coth x = x e + e−x

kUtg;sg;GIuEBbUlik kMNt;edaysmIkar @>TMnak;TMngsMxan;² k> cosh x − sinh x = 1 g> 1 − tanh x = cosh1 x sinh x 1 x> tanh x = cosh c> coth x − 1 = x sinh x x K> coth x = cosh sinh x X> tanh x = coth1 x

- 116 -

RbCMurUbmnþKNitviTüa

sRmaybBa¢ak; eday sinh x = e eK)an

e x + e −x cosh x = 2 2 −x −x 2 x x e + e  e −e   −   cosh 2 x − sinh 2 x =  2 2     x

− e−x 2

nig

e 2x + 2 + e −2x e 2x − 2 + e −2x = − 4 4 e 2x + 2 + e −2x − e 2x + 2 − e −2x = 4 =1

dUcenH cosh x − sinh x = 1 . tam cosh x − sinh x = 1 x − sinh x 1 eK)an cosh cosh = x cosh 2

dUcenH

sinh 2 x 1 sinh x 1− tanh x = = cosh x cosh 2 x cosh 2 x 1 1 − tanh 2 x = cosh 2 x cosh 2 x − sinh 2 x 1 = sinh 2 x sinh 2 x cosh 2 x 1 cosh x − 1 = coth x = sinh x sinh 2 x sinh 2 x 1 coth 2 x − 1 = cosh 2 x

eday .

müa:geTot dUcenH

eday .

- 117 -

RbCMurUbmnþKNitviTüa

#>rUbmnþplbUk !> sinh( x + y ) = sinh x cosh y + sinh y cosh x @> cosh(x + y ) = cosh x cosh y + sinh x sinh y x + tanh y #> tanh(x + y ) = 1tanh + tanh x tanh y x coth y + 1 $> coth(x + y ) = coth coth x + coth y sMraybBa¢ak; eKman sinh x = e −2e , cosh x = e +2e x

eK)an ehIy

−x

e y − e −y e y + e −y sinh y = , cosh y = 2 2 (e x − e − x )(e y + e − y ) sinh x cosh y = 4 x+ y −( x+ y ) e −e + e x− y − e −( x + y ) sinh x cosh y = (a ) 4 (e y − e − y )(e x + e − x ) sinh y cosh x = 4 x+ y − ( x+ y ) e +e − e x− y − e −( x + y ) sinh y cosh x = (b ) 4

bUksmIkar (a) nig (b) eK)an ³

e x + y − e − (x − y) sinh x cosh y + sinh y cosh x = = sinh( x + y ) 2

- 118 -

RbCMurUbmnþKNitviTüa (e x + e − x )(e y + e − y ) cosh x cosh y = 4 e x+ y + e x− y + e − x+ y + e −( x+ y ) cosh x cosh y = (c ) 4 x −x y −y (e − e )(e − e ) sinh x sinh y = 4 e x + y − e x − y − e − x+ y + e − ( x+ y ) sinh x sinh y = (d ) 4

mü:ageTot ehIy

bUksmIkar (c) nig (d) eK)an ³

e x+ y + e −( x + y ) cosh x cosh y + sinh x sinh y = = cosh(x + y ) 2

dUcenH cosh(x + y ) = cosh x cosh y + sinh x sinh y . sinh( x + y ) eKman tanh(x + y ) = cosh( x + y) =

sinh x cosh y + sinh ycsohx cosh x cosh y + sinh x sinh y

sinh x sinh y + ) cosh x cosh y = sinh x sinh y cosh x cosh y (1 + ) cosh x cosh y tanh x + tanh y = 1 + tanh x tanh y tanh x + tanh y tanh( x + y ) = 1 + tanh x tanh y cosh x cosh y (

dUcenH

- 119 -

RbCMurUbmnþKNitviTüa

#>GnuKmn_énGaKuym:g;GviC¢man !> sinh( − x) = − sinh x @> cosh(− x) = cosh x #> tanh(− x) = − tanh x $> coth(− x) = − coth x sRmaybBa¢ak; eKman sinh x = e −2e eK)an sinh( − x) = e 2− e = − e −2e = − sinh x ehIy cosh x = e +2e eK)an cosh(− x) = e 2+ e = cosh x . −x

−x

sinh( − x ) sinh x =− = − tanh x cosh( − x ) cosh x cosh( − x ) cosh x coth( − x ) = =− = − coth x sinh( − x ) sinh x tanh( − x ) =

%>rUbmnþpldk !> sinh( x − y ) = sinh x cosh y − sinh y cosh x @> cosh(x − y ) = cosh x cosh y − sinh x sinh y - 120 -

RbCMurUbmnþKNitviTüa x − tanh y #> tanh(x − y ) = 1tanh − tanh x tanh y − coth x coth y $> coth(x − y ) = 1coth x − coth y sRmaybBa¢ak; eKman sinh( x + y ) = sinh x cosh y + sinh y cosh x

(i )

cosh( x + y ) = cosh x cosh y + sinh x sinh y (ii ) tanh x + tanh y tanh( x + y ) = (iii ) 1 + tanh x tanh y coth x coth y + 1 coth( x + y ) = (iv ) coth x + coth y

edayCMnYs y eday − y kñúg (i), (ii ), (iii ) nig (iv) eK)an sinh( x − y ) = sinh x cosh(− y ) + sinh( − y ) cosh x = sinh x cosh y − sinh y cosh x cosh( x − y ) = cosh x cosh( − y ) + sinh x sinh( − y ) = cosh x cosh y − sinh x sinh y tanh x + tanh( − y ) tanh x − tanh y tanh( x − y ) = = 1 + tanh x tanh(− y ) 1 − tanh x tanh y coth x coth( − y ) + 1 1 − coth x coth y coth( x − y ) = = coth x + coth( − y ) coth x − coth y

- 121 -

RbCMurUbmnþKNitviTüa

^>rUbmnþmMuDub

( DOUBLE ANGLE FORMULAS FORMULAS )

!> sinh 2x = 2 sinh x cosh x @> cosh 2x = cosh x + sinh 2

= 2 cosh 2 x − 1 = 1 + 2 sinh 2 x 2 tanh x tanh 2x = 1 + tanh 2 x 1 + coth 2 x coth 2x = 2 coth x

#> $> sRmaybBa¢ak; eKman sinh( x + y ) = sinh x cosh y + sinh y cosh x

(i )

cosh( x + y ) = cosh x cosh y + sinh x sinh y (ii ) tanh x + tanh y tanh( x + y ) = (iii ) 1 + tanh x tanh y coth x coth y + 1 coth( x + y ) = (iv ) coth x + coth y

edayCMnYs y eday x kñúg (i), (ii ), (iii ) nig (iv) eK)an ³ sinh 2x = 2 sinh x cosh x

cosh 2x = cosh 2 x + sinh 2 x 2 tanh x 1 + coth 2 x tanh 2x = coth 2x = 2 2 coth x 1 + tanh x

nig

- 122 -

RbCMurUbmnþKNitviTüa

&>rUbmnþknøHmMu

( HALE ANGLE FORMULAS )

!> sinh x2 = cosh2x − 1 @> cosh x2 = cosh2x + 1 cosh x − 1 #> tanh x2 = cosh x+1 x+1 $> coth x2 = cosh cosh x − 1 2

*>rUbmnþBhumMu

( MULTIPLE ANGLE FORMULAS )

!> sinh 3x = 3 sinh x + 4 sinh x @> cosh 3x = 4 cosh x − 3 cosh x x + tanh x #> tanh 3x = 3 tanh 1 + 3 tanh x $> sinh 4x = 8 sinh x cosh x + 4 sinh x cosh x %> cosh 4x = 8 cosh x − 8 cosh x + 1 x + 4 tanh 4x ^> tanh 4x = 14+tanh 6 tanh x + tanh x 3

- 123 -

RbCMurUbmnþKNitviTüa

(>rUbmnþplbUk-pldk nig plKuN ( SumSum-Difference and Product of Hyperbolic Functions )

!> sinh x + sinh y = 2 sinh x +2 y cosh x −2 y @> cosh x + cosh y = 2 cosh x +2 y cosh x −2 y #> sinh x − sinh y = 2 sin x −2 y cosh x +2 y $> cosh x − cosh y = 2 sinh x +2 y sinh x −2 y sinh( x + y ) %> tanh x + tanh y = cosh x cosh y sinh( x − y ) ^> tanh x − tanh y = cosh x cosh y sinh( x + y ) &> coth x + coth y = sinh x sinh y sinh( y − x ) *> coth x − coth y = sinh x sinh y

(> sinh x sinh y = 12 [cosh(x + y ) − cosh(x − y )] - 124 -

RbCMurUbmnþKNitviTüa

!0> cosh x cosh y = 12 [cosh(x + y ) + cosh(x − y )] !!> sinh x cosh y = 12 [sinh( x + y ) + sinh( x − y )] !@> sinh y cosh x = 12 [sinh( x + y ) − sinh( x − y )] !0>RkabénGnuKmn_GIuEBbUlik ( Graphs of Hyperbolic Functions )

!> y = sinh x y 5

y = sinh x

4 3 2 1

-4

-3

-2

-1

-1 -2 -3

- 125 -

RbCMurUbmnþKNitviTüa

@> y = cosh x y

y = cosh x

5 4 3 2 1

-4

-3

-2

-1

4 x

-1

#> y = tanh x 3y 2

y = tanh x 1

-4

-3

-2

-1

-1 -2

- 126 -

RbCMurUbmnþKNitviTüa

$> y = coth x

y 3

y = coth x

2 1

-4

-3

-2

-1

-1 -2 -3

!!>TMnak;TMngrvagGnuKmn_GIuEBbUliknigGnuKmn_RtIekaNmaRt ( Relationships between Hyperbolic and Trigonometry Functions )

!> sin(ix) = i sinh x

$> sinh(ix) = i sin x

@> cos(ix) = cosh x

%> cosh(ix) = cos x

#> tan(ix) = i tanh x

^> tanh(ix) = i tan x

- 127 -

RbCMurUbmnþKNitviTüa

sRmaybBa¢ak; tamrUbmnþGWEl

e ix + e − ix cos x = 2

nig

e ix − e − ix sin x = 2i

CMnYs x eday ix eK)an ³ e i ( ix ) + e − i ( ix ) e − x + e x cos( ix ) = = cosh x = 2 2 e i ( ix ) − e − i ( ix ) e − x − e x e x − e −x sin( ix ) = =i = i sinh x = 2i 2i 2 sin( ix ) i sinh x ehIy tan(ix) = cos( = = i tanh x . ix ) cosh x

!@>Gnuvtþn_ eKman sinh( x + y ) = sinh x cosh y + sinh x cosh y CMnYs x = ia nig y = ib eK)an ³ sinh i(a + b ) = sinh( ia ) cosh(ib ) + sinh( ib ) cos(ia )

eday sinh i(a + b) = i sin(a + b) , sinh(ia) = i sin a - 128 -

RbCMurUbmnþKNitviTüa sinh( ib ) = i sin b , cosh(ia ) = cos a , cosh(ib )

eK)an i sin(a + b) = i sin a cos b + i sin b cos a dUcenH

sin(a + b ) = sin a cos b + sin b cos a

eKman cosh(x + y ) = cosh x cosh y + sinh x sinh y CMnYs x = ia nig y = ib eK)an ³ cosh i(a + b ) = cosh(ia ) cosh(ib ) + sinh( ia ) sinh( ib )

eday cosh i(a + b) = cos(a + b) , sinh(ia) = i sin a sinh( ib ) = i sin b , cosh(ia ) = cos a , cosh(ib )

eK)an cos(a + b) = cos a cos b + i dUcenH

sin a sin b

cos(a + b ) = cos a cos b − sin a sin b

eday i

!#>lImIténGnuKmn_GIuEBbUlik !> lim sinhx x = 1

@> lim tanhx x = 1

x→0

- 129 -

= −1

RbCMurUbmnþKNitviTüa

sRmaybBa¢ak; eKman

e x − e −x e 2x − 1 sinh x = = 2 2e x

eK)an

sinh x e2x − 1 e 2x − 1 1 lim = lim = lim . x =1 x x→0 x 0 x 0 → → x 2x e 2xe

!$>edrIevénGnuKmn_GIuEBbUlik !> y = sinh x ⇒ y' = cosh x @> y = cosh x ⇒ y' = sinh x #> y = tanh x ⇒ y' = cosh1

$> y = coth x ⇒ y' = − sinh1

x 2

!%>GaMgetRkalénGnuKmn_GIuEBbUlik !> ∫ sinh x.dx = cosh x + c #> ∫ tanh x.dx = ln(cosh x) + c 2. ∫ cosh x.dx = sinh x + c 4. ∫ coth x.dx = ln(sinh x ) + c

- 130 -

RbCMurUbmnþKNitviTüa

CMBUkTI18

viPaKbnßM nig RbUáb‘ÍlIet I-

viPaKbnßM

1>hVak;tUErül (Factorial) ³ EdlehAfahVak;tUErüléncMnYn n CaplKuNén n cMnYnKt;viC¢mandMbUg b¤ CaplKuNcMnYnKt;viC¢mantKña BI 1 rhUtdl; n EdleKkMnt;sresr ³ n! = 1 × 2 × 3 × ....... × n . 2>tMerobminsareLIgvij (Arrangement ) ³ tMerob p FatukñúgcMeNam n FatuénsMnMu E KWCasMnuMrgén E Edlman p Fatuxus²Kña erobtamlMdab;mYykMnt; . eKkMnt;tagcMnYntMerob p FatukñúgcMenam n Fatueday ³ n! A(n , p ) = = n(n − 1)(n − 2 ).....(n − p + 1) . (n − p ) ! 3>cMlas;minsaeLIgvij (Permultation ) ³ cMlas; n Fatuxus²Kña KWCatMerob n Fatu kñúgcMeNam n Fatu . eKkMnt;cMnYncMlas; n Fatueday P = n! = 1 × 2 × 3 × ..... × n .

- 131 -

RbCMurUbmnþKNitviTüa 4>bnSMminsareLIgvij (Combination ) ³ bnSM p FatukñúgcMeNam n Fatu CatMerobminKitlMdab;EdlkMnt;eday ³ A(n , p ) n! C(n, p ) = = (n ≥ p ) . p! p!.(n − p )! 5>tMerobsareLIgvij  Arrangement with Re petition  ³ tMerobsareLIgvij p Fatu kñgcMeNam n FatuKWCatMerobEdlFatunImYy² Gacmanvtþman 1,2,3,..., n dg . eKkMnt;sresr ³ A(n, p) = np . 6>cMlas;sareLIgvij (Permultation with Re petition) ³ eKeGaysMnMu E man n Fatu EdlkñúgenaH n1 CaFatuRbePTTI1 , n2 CaFatuRbePTTI2 , n CaFatuRbePTTI3 ,........., np CaFatuRbePTTI p Edl n + n + n + .... + n = n cMnYncMlas;sareLIgvijén n Fatu KWCacMlas;GacEbgEck)anEdlkMnt;tageday ³ n! P= . n !.n !.n !.....n ! 3

7>bnSMsareLIgvij (Combiation with Re petition) ³ bnSMsareLIgvijén p Fatu kñúgcMenam n FatuKWCabnSM EdlFatunImYy²Gacman vtþmaneRcIndg. - 132 -

RbCMurUbmnþKNitviTüa eKtagbnSMsareLIgvijén p Fatu kñúgcMenam n Fatueday ³ C(n, p ) =

8>eTVFajÚtun

(n + p − 1) ! p!.(n − 1) !

 Binom de Newton   

(a + b )n = Cn0 an + Cn2 an −1 .b + Cn2 an − 2 .b 2 + ....... + Cnnbn

Edl Cpn = C(n, p) = p!(nn−! p)! . sMKal; ³ eTVFasMxan;²KYrkt;sMKal; !> (1 + x)n = Cn0 + C1n x + Cn2 x2 + Cn3 x3 + ........... + Cnnxn @> (x + 1)n = Cn0 xn + C1n xn −1 + Cn2 xn − 2 + .......... + Cn0 II-

RbUáb‘ÍlIet

RbU)ab‘‘ÍlIet mansarHsMxan;kñúgCIvPaBRbcaMéf¶rbs;eyIg EdleyIgeRbIR)as;vasMrab; vas;kMriténPaBmineTogTat; . kalNaeyIgeRKageFVIGIVmYykalNaGñk]tuniym TsSn_TayGakasFatub¤ Rkumh‘unFanara:b;rgeFIVeKalneya)ayrbs;Rkumh‘un caM)ac; RtUveRbI RbU)ab‘ÍlIet edIm,IeFVIesckþIsMerccitþ b¤eFVI kareRCIserIs .

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RbCMurUbmnþKNitviTüa 1-RBwtþikarN_-lMhsMNak ³ k>viBaØasa ³ viBaØasa KWCakarBiesaFn_mYyEdl ³ -GaceGayeKdwg nUvsMNMulT§plEdl)anekIteLIg -BMuGacdwgR)akdfa lT§plNaEdlnwgekItmaneLIg -karBiesaFn_ GacsareLIgvij CaeRcIndg kñúglkç½xNÐdUcKña . x>skl b¤lMhsMNak ³ sMnMuénlT§plTaMgGs;EdlGacman rbs;viBaØsamYy ehAfa skl EdleKtageday S . K>RBwtþikarN_ ³ CasMNMurg rbs;skl b¤lMhsMNak . ]TahrN_ ³ ebIeyIge)aHkak;Edlmanmux H nigxñg T cMnYnmYydgenaHeKGac)anlT§pl H b¤ T . -sMnMu {H, T} ehAfa lMhsMNak tageday S = {H, T} . -ebIeKR)afñae)aH)anmux H enaHsMNMu {H} ehAfaRBwtþikarN_ tageday A = {H} -cMnYnFatuénlMhsMNak ehAfacMnYnkrNIGac eKtageday n(S) = 2 . -cMnYnFatuénRBwtþikarN_ ehAfacMnYnkrNIRsb eKtageday n(A ) = 1 .

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RbCMurUbmnþKNitviTüa 2>rUbmnþeKalénRbU)ab ³ enAkñúgBiesaFn_mYy EdlmanlMhsMNak S RbU)ab‘ÍlIeténRBwtþikarN_ A ekIteLIgkMnt;eday ³ P(A ) =

cMnYnkrNIR sb cMnYnkrNIG ac

n( A ) n(S )

eday A ⊆ S enaH

0 ≤ P(A ) ≤ 1

3>rUbmnþKNnaRbU)ab‘ÍlIet ³ k-rUbmnþ RbU)ab‘ÍlIeténRbCMuRBwtþikarN_BIr ebI A nig B CaRBwtþikarN_BIrmincuHsMrugKñaenaHeK)an ³ P (A ∪ B ) = P (A ) + P (B )

ebI

nig B CaRBwtþikarN_BIrsamBaØenaHeK)an ³

P (A ∪ B ) = P (A ) + P (B ) − P (A ∩ B )

x-rUbmnþ RbU)ab‘ÍlIeténRbCMuRBwtþikarN_bI ³ ebI A , B nig C CaRBwtþikarN_bImincuHsMrugKñaBI²enaHeK)an ³ P (A ∪ B ∪ C ) = P (A ) + P (B ) + P (C )

ebI

nig B CaRBwtþikarN_bIsamBaØenaHeK)an ³

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C)

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RbCMurUbmnþKNitviTüa K-CaTUeTA ³ ebI A , A , A ,......, A CaRBwtþikarN_mincuHsMrugKñaBI²enaHeK)an ³ P (A ∪ A ∪ ... ∪ A ) = P (A ) + P (A ) + ... + P (A ) . 1

X-rUbmnþRbU)abénRBwtþikarN_BIrpÞúyKña ³ ebI A nig A CaRBwtþikarN_BIrpÞúyKñaenaHeK)an ³ P(A ) + P(A ) = 1 ⇔ P(A ) = 1 − P(A ) . g-rUbmnþRbU)ab‘ÍlIetmanlk½çx½NÐ ³ RbU)ab‘ÍlIeténRBwtþikarN_ A edaydwgfa manRBwtþikarN_ B )anekIt eLIgrYcehIy ehAfa RbU)abmanlkç½x½NÐ EdleKtageday P(A / B) GanfaRbU)abén A eday)andwg B . dUcenH cMeBaHRBwtþikarN_ A nig B eday P(B) ≠ 0 eKmanrUbmnþ ³ P (A ∩ B ) P (A / B ) = b¤eKGacTaj P(A ∩ B ) = P(A / B )× P(B ) P (B ) c-rUbmnþRbU)abénRBwtþikarN_BIrminTak;TgKña ³ RBwtþikarN_ A nig B EdlGaRsy½nwgKñakñúgviFIEdlkarekIteLIgénRBwtþikarN_mYy minmanCab;Bak;Bn§½nwgkarekIteLIgénRBwtþikarN_mYyeTot eyIgehAfa RBwtþikarN_minTak;TgKña . P(A / B) = P(A ) ebI A nig B minTak;TgKñasmmUl ³  P(B / A ) = P(B) 

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RbCMurUbmnþKNitviTüa ebI

nig B CaRBwtþikarN_ minTak;TgKñaenaHeK)an ³ P(A ∩ B) = P(A ) × P(B) . A

eroberogeday lwm plÁún Tel : 017 768 246 Email:lim_phalkun@ymail.com Website: www.mathtoday.wordpress.com www.todaymath.blogspot.com

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