វិញ្ញសាត្រៀមប្រលង Bacc II

KNitviTüa fñak;TI 12 viBaØasaTI 1

eRtomRblgsBaØab½Rt Bacc II

I. eKeGayGnuKmn_ f kMNt;elI IR eday 2

2

3

1

x ax bf x

x

ehIy C CaRkabén f kñúgtRmuy

GrtUNrem ; ;o i j . kMNt; a nigb edIm,IeGaybnÞat; T b:HeTAnwgRkab C Rtg;cMnucmanGab;

suIs 0 mansmIkar 4 3y x . sikSaTItaMg C eFobeTAnwg T .

II. KNnalImIt ³

3 3 3

22 2

ln lnln 1lim ; lim ; lim ; lim

2009 1xx x x x

x x gxx xA B C D

e x xx

III. kñúgeyIgmansmIkar : ln3 0E y y .

1/. rksMNuMcemøIyén E . 2/. kMnt; f EdlCacemøIyén E edaydwgfa 1 2f .

3/. bgðajfa 13xg x CacemøIyén E . (smÁal; xU x a enaH lnxU x a a ).

IV. eKeGay)a:ra:bUl 2: 2P y x x nigbnÞat; :D y x .

1/. sg; P nig D kñúgtRmuyGrtUNrma:l; rYcbBa¢ak;eGay)anc,as;nUvkUGredaencMNucRbsBV.

2/. KNnaRklaépÞEdlx½NÐeday P nig D .

V. 1/. sisS10nak;cUlrYmRbLgRbNaMgrt;cm¶ay1000m. etImanbu:nµanrebobEdlsisSmkdl;

TImYy/ TIBIr nigTIbI.

2/. kñúgfg;mYymanbU‘l12EdlcuHelxerogBI1dl;12. eKcab;ykRBmKñanUvbU‘lbI. KNna

RbU)abénRBwtþikarN_EdlelxenAelIbU‘lTaMgbI ³

k> tMerobsmRsb begáIt)anCasVúItnBVnþplsgrYm3.

x. tMerobsmRsb begáIt)anCasVúItFrNImaRtersug1

2.

VI. bMElgsmIkarTUeTAénGuIEBbUl 2 29 2 54 116 0x y x y CaTMrg;sþg;da. rkp©it/ kMBUl/ kMnuM nig

GasuImtUt rYcsg;Rkabrbs;va.

VII. eKeGay f CaGnuKmn_kMnt;elI IR eday 2 xf x x e x nigmanRkab C kñúgtRmuyGrtUNrma:l; 0 ; ;i j .

1/. sg;taragGefrPaBén 1 1xh x x e (min)ac;KNnalImIteT).

2/. KNna limx

f x

nig limx

f x

. sg;taragGefrPaBén f .

3/. bgðajfabnÞat; :D y x CaGasuImtUtén C . rkcMNucRbsBVén C nig D rYcsikSa

TItaMgén C eFobnwg D .

4/. kMNt;bnÞat;b:H T eTAnwg C Rtg;cMnucmanGab;suIs0 . sg; ;C D nig T kñúgtRmuy 0 ; ;i j .

5/. ]bmafa CacMnYnBittUcCag b¤esµI 2 . KNnaépÞRklax½NÐedayExSekag C nigGasuImtUt

rbs;vaelIcenøaH[ ; 2] .

VIII. enAkñúgtRmuyGrtUNrma:l;manTisedAviC¢man 0 ; ; ;i j k eK[cMnuc 0;2;2A nigbnÞat; D Edl

mansmIkar)a:ra:Em:t

1

1

2

x t

y t t IR

z t

1/. bgðajfabøgkat;tam A ehIyEkgnwgbnÞat; D .

2/. bøg; P kat;tamG½kSox Rtg; ;M oy Rtg; N nigoz Rtg; P . rkkUGredaenéncMnuc ;M N nig P

rYcsg;cMNuc ; ; ;A M N P enAkñúgtRmuy 0 ; ; ;i j k .

3/. bgðajfaRtIekaN MNP CaRtIekaNsm½gS.

4/. KNnaMP MN rYcTajrképÞRklaénRtIekaN MNP .

&Y'



I. eKeGaycMnYnkMupøic 3 1 3 1A i nig1

x iyB

i

Edl ;x y CacMnYnBit.

1/. sresr 2A CaTRmg;BiCKNit ehIyCaTRmg;RtIekaNmaRt.

2/. sresr B CaTRmg;BiCKNit. rk x nig y edaydwgfa 22 0B A . ( B CacMnYnkMupøicén B ).

II. KNnalImIt ³

22 2

20 0 0

sin 1 cos ln 2 2 1lim ; lim ; lim ln cos ln ; lim

tan

xx

x x xx

e e x x x xA B C x x D

x x x x

III. 1/. kMNt;cMnYnBit ;a b nig c edIm,IeGay)an

2

2 2

2 2

1 21 2 1

x x a b c

x xx x x

cMeBaHRKb; x

Edl 2 ; 1x x .

2/. KNnaGaMgetRkal2

0

21

2 2[ ]( 1) ( 2)

x xI dx

x x

.

IV. eKeGay 21

0

cos

1 2sin

xI dx

x

nig 2

20

sin 2

1 2sin

xI dx

x

.

1/. KNna 1I nig 2I . 2/. Tajrktémøén 2I .

V. eKmantaragTinñn½ydUcxageRkam ³ x 2 4 6 8 10 y 2 2 6 10 10

1/. rksmIkaréntMErtMrg;lIenEG‘.

2/. cUrKNnatémøén y cMeBaH 3 ; 9x x .

VI. RsaybBa¢ak;fa 2

ln 12

xx x cMeBaHRKb; 0x .

VII. kMnt;cMelIyBiess f mYyénsmIkar : 4 12 9 0E y y y EdlExSekagtagGnuKmn_ f kat;tam

cMNuc 0;5A ehIysmIkarbnÞat;b:HnwgExSekagRtg;cMnucenaHRsbnwgbnÞat; 2y x .

VIII. GnuKmn_ f kMnt;elI ] 0 ; [ eday 1 ln

1x

f x xx

nigmanRkab C kñúgtRmuyGrtUNrma:l;.

1/. sg;taragGefrPaBén 22 lng x x x (min)ac;KNnalImIteT) rYcTajrksBaØaén g x

elI ] 0 ; [ .

2/. KNna 0

limx

f x

nig limx

f x

. Tajfa C manGasuImtUtQr.

3/. sikSaTMnak;TMng

2

g xf x

x . sg;taragGefrPaBén f .

4/. bgðajfa : 1y x CaGasuImtUtén C .

5/. rkcMnucRbsBVén C CamYy rYcbBa¢ak;eGay)anc,as;TItaMgén C eFobCamYy .

sg;ExSekag C .

IX. kñúgtRmuyGrtUNrma:l;manTisedAviC¢man 0 ; ; ;i j k eKmancMnuc 2; 1;1A nigvuicT½r 3;2;3AB .

1/. rkkUGredaencMnuc B .

2/. eKeGaycMnuc 0;1;1C nig 3;4;4D . epÞógpÞat;facMnuc ; ;A B C nig D CakMBUlénRbelLÚ

RkammYy rYcrkRklaépÞva.

3/. rksmIkarbøg; P kat;tamcMnuc A nig B ehIyRsbnwgG½kSGab;suIs.

4/. rkcMgayBIcMnucC eTAbøg; P .

&Y'


eRtomRblg Bacc II

I. eKeGay 2 2 2 2Z i .

1/. KNna cos8

nig sin

8

rYcsresr Z CaTMrg;RtIekaNmaRt.

2/. KNna cos16

rYcrkmU:Dul nigGaKuym:g;én 2W Z .

II. 1/. KNnalImIt 2

ln 1lim ; 0

x

x

e xx

x

.

2/. KNnaedrIevén ln 1 ; 0xy e x x . TajrkRBImITIv 1

ln 11

xx e

f x e xx

eday

dwgfaRBImITIvenHesµI1cMeBaH 0x .

III. eKeGayGnuKmn_ f kMnt;elI[ 0 ; [ eday lnx

xf x

e x

ebI 0x nig 0 0f . sikSaPaBCab;

nigmanedrIevsþaMén f Rtg; 0 .

IV. eKeGay f nig g CaGnuKmn_manedrIevRtg; 0x . ehIyepÞógpÞat; 0 0 0f g nig 0 0g .

RsaybBa¢ak;fa

0

0lim

0x

f x f

g x g

. Gnuvtþn_

sin

0 0 0

1 sin cos sinlim ; lim ; lim

x x

x x x

e x x e x x

x x x

.

V. eKeGay f kMnt;elI[ 1; [ eday ln 2f x ax b x .

1/. kMnt;cMnYnBit a nigb Edl 1 3f nig1

02

f

.

2/. bgðajfa 3

ln 2 32

G x x x x

CaRBImITIvmYyelI[ 1; [ én ln 2 3g x x .

KNna 2

0I f x dx .

VI. sisSmñak;)a:n;sµanfaRbU)abénkarRblgnieTÞs A elImuxviC¢aKNitviTüa b¤rUbviTüaman90%

RbU)abénkarRblg)annieTÞs A elIEpñkrUbviTüaman80%bu:EnþRbU)abénnieTÞs A TaMgBIrmuxEt

60%. rkRbU)abénkar)a:n;sµanrbs;enaH)annieTÞs A elIEpñkKNitviTüa.

VII. kñúgtRmuyGrtUNrma:l;manTisedAviC¢man 0 ; ; ;i j k eKmancMnuc 0;0;3S nigvuicT½r 1;1;1U .

1/. rksmIkar)a:ra:Em:ténbnÞat; D Edlkat;tamcMNuc S ehIyRsbnwgvuicT½rU .

2/. ; ;M x y z CacMNucmYyenAkñúgtRmuy 0 ; ; ;i j k . KNnaplKuNsáaElU SM . kMNt;rk

smIkarmYyEdlCab;Tak;Tgnwg ; ;x y z edIm,IeGayvuicT½r SM GrtUkUNal;nwgvuicT½rU .

etIsMNuMcMNuc M CaGIV?

3/. eKeGaycMnuc 3;0;0A nig 0;3;0B . bgðajfaRtIekaN SABCaRtIekaNsm½gS.

4/. KNnaplKuNvuicT½rSA SB . TajrképÞRkLaénRtIekaN SAB.

5/. rksmIkarEsV‘Edlmanp©itO ehIykat;tamcMNuc ;S Anig B .

VIII. f CaGnuKmn_kMnt;cMeBaH ] ; 0 [ ]1; [x D eday 1

ln2

x xy f x

x

ehIymanExSekag

C enAkñúgtRmuyGrtUNrma:l;mYy.

1/. KNnalImIt 0

lim ; lim ; limx x x

f x f x f x

nig 1

limx

f x

. rksmIkarGasuImtUtQrTaMgBIrén C .

2/. KNna nigsikSasBaØaén f x ebIeKdwgfa 1 0x x cMeBaHRKb; x D .

KNnatémøGb,brma nigGtibrmaén f . sg;taragGefrPaBén f .

3/. bgðajfabnÞat; :2

xL y CaGasuImtUteRTténExSekag C .

4/. sg;GasuImtUtTaMgGs; nigExSekag C enAkñúgtRmuyGrtUNrma:lmYy. (eKyk ln 2 0,7 ;

1

ln ln 2 0,72

) rktémø a edIm,IeGaysmIkar

1ln

2

x xa

x

manb¤sviC¢manBIr

epSgKñaedayeRbIExSekag C .

&Y'



I. eKeGaycMnYnkuMpøic 1 cos sin ;Z i .

1/. cMeBaH3

/ cUrKNnamU:Dul nigGaKuym:g;én Z .

2/. kñúgbøg;kuMpøic rksMnuMéncMnucrUbPaBén Z kalNa ERbRbYl.

3/. bgðajfaEpñkBitén1

ZCacMnYnefrEdlRtUvkMnt;.

II. kMnt; ; ;a b c edIm,IeGay 2 2xF x ax bx c e CaRBImITIvmYyén 2 24 xf x x e . rkRBImITIv

én f x EdlesµI0 Rtg; 0x .

III. 1/. KNna1

0 1

x

x

eA dx

e

nig

1

201

x

x

eB dx

e

.

2/. kMnt; ;a b nig c ebI

2 2

1

11 1

bt cta

tt t

.

3/. TajrktMélén

1

201 x

dxI

e

rYcKNna

1

301

x

x

xe dxJ

e

.

IV. f CaGnuKmn_kMnt;eday

11

1 ln 1

xe ef x x

xx

f e k

ebI . cUrkMNt;tMél k edIm,IeGay f CaGnuKmn_

Cab;Rtg; 1x .

V. kñúgtRmuyGrtUNrma:l; 0 ; ;i j eK[ExSeekag

2 24: 1

25 9

x yE

.

1/. bBa¢ak;RbePTénExSekag E rYcbBa¢ak;kUGredaenénp©it/ kMNuM/ kMBUl nigcMNucRbsBVrvag

E nigG½kStUc.

2/. sg; E .

VI. kñúgtRmuyGrtUNrma:l;manTisedAviC¢man 0 ; ; ;i j k eKmancMnuc 1;1;0 ; 2;0;1 ; 1;2;2 ;A B C

0;1; 2D nig ; ;M x y z .

1/. rkTMnak;TMngrvag ;x y nig z edIm,IeGay AM Ekgnwg BM . etIsMNuMcMnucéncMNuc M CaGIV?

2/. KNna n BC BD . rkcMnYnBitm edIm,IeGayépÞRkLa S én BCD epÞógpÞat;smPaB

3 mS e .

3/. rksmIkarbøg; BCD . rkkUGredaenéncMnucRbsBV N rvagbøg; BCD nigbnÞat;

1 1 2

:1 1 1

x yL

.

4/. KNnacm¶ayBIcMNuc A eTAbøg; BCD rYcTajrkmaDénetRtaEGt ABCD.

VII. f CaGnuKmn_kMnt;eday 1

1 ln1

xf x

x

manExSekag C .

1/. rkEdnkMnt;énGnuKmn_. KNnalImItén f Rtg;cugEdnkMnt;. bBa¢ak;smIkarGasuImtUt

énRkab C .

2/. KNna f x ehIysikSasBaØaén f x . KUstaragGefrPaBénGnuKmn_ f x .

3/. KNna 2f nig 2f . bgðajfabnÞat;b:HeTAnwgRkab C Rtg;cMnucmanGab;suIs 2x

nig 2x RsbKña. sresrsmIkarbnÞat;b:HTaMgBIr.

4/. sg;ExSekag C kñúgtMruyGrtUNrma:l; 0 ; ;i j .

&Y'


eRtomRblgBacc II

I. 1/. KNna2008 2008

1 1

2 2

i iA

CaTRmg;BiCKNit.

2/. kMNt;cMnYnBit a nigb edIm,IeGay 1 2Z A i Cab¤sénsmIkar 2: 10 0E az bz .

3/. rkb¤s 2Z mYyeToténsmIkar E . sresr4

1

2

zB

z

CaTRmg;RtIekaNmaRt.

II. GnuKmn_ g kMnt;eday 2

2cos 1

4sin 3

xg x

x

cMeBaH

3x

.

1/. KNna 3

limx

g x

.

2/. kMnt;GnuKmn_ f EdlCaGnuKmn_bnøaytamPaBCab;énGnuKmn_ g Rtg; 3

x

.

III. GnuKmn_ f x nig g x bMeBjl½kçxNÐxageRkam ³

; ; 0 1 ; 0 0i f x g x ii f x g x iii f iv g

1/. kMnt;GnuKmn_ f x g x nig f x g x .

2/. cUrkMnt;GnuKmn_ f x nig g x .

IV. f CaGnuKmn_kMnt;cMeBaH 0x eday 2 lnf x x x .

1/. RsaybBa¢ak;faGnuKmn_ f manGb,brma rYcKNnatMélGb,brmaenaH.

TajbBa¢ak;fa 0f x cMeBaH 0x .

2/. RsaybBa¢ak;facMeBaH 1x eyIg)anln 2x

xx x

rYcTajbBa¢ak;fa ³ln

lim 0x

x

x .

V. fñaMmYyRKab;RtUv)ancab;ykecjBIRbGb;mYyEdlmanfñaMGas<IrIn 6 RKab; KINal; 4 RKab;

nigvitamIn5 RKab;. rkRbU)abénfñaMEdlRtUvcab;ykecjenaHKW ³

k> Gas<IrIn x> KINal; K> vItamIn X> minEmnGas<IrIn g>Gas<IrIn rWKINal;

VI. GnuKmn_ f kMnt;elI ]1; [ eday 1

ln1

xf x

x x

.

1/. KNna 1

limx

f x

nig limx

f x

.

2/. KNna f x rYckMnt;TisedAGefrPaBén f .

3/. sg;taragGefrPaBén f rYckMnt;sBaØaénGnuKmn_ f x cMeBaH x ]1; [ .

4/. sg;ExSekag C kñúgtMruyGrtUNrma:l; 0 ; ;i j .

5/. g CaGnuKmn_kMnt;elI ]1; [ eday 1 ln1

xg x x

x

. KNna g x rYckMnt;TisedA

GefrPaBén g .

VII. kñúgtRmuyGrtUNrma:l;manTisedAviC¢man 0 ; ; ;i j k eKmancMnuc 1;0;2 ; 2;0;1A B nig 1;2;4C .

1/. KNna BA BC . rksmIkarbøg; ABC .

2/. rksmIkarbøg; P Edlkat;tam C ehIyEkgvuicT½r BA . bgðajfabøg; P Ekgbøg; ABC .

3/. KNnaépÞRkLaRtIekaN ABC rYcmaDénctumuxOABC.

4/. KNnacMgayBIcMnuc C eTAbnÞat; (AB) .

5/. kMnt;kUGredaenéncMnucG EdlepÞógpÞat; 6 3 2OG OA OB OC .

&Y'


eRtomRblg Bacc II

I. KNnalImIt ³ 2009

2lim

1

n

n

nA

n

1lim 1n

nB n

n

II. 1/. kMnt; K edIm,IeGay 2 2 ln 3ln 2 ;xx K x K e x .

2/. kMnt; m edIm,IeGay 3ln 2 1 5y mx m x m Kµann½ycMeBaH x .

III. KNnaedrIevénGnuKmn_ 2

ln 2 1 ln3

F x x x rYcTajyk2

1

ln

1 ln

e xI

x x

.

IV. 1/. bgðajfa *1 ;xe x x .

2/. Tajfa 1 1

ln 2009 ln 2008 ln 20072008 2007

.

V. kñúgfñak;eronmYymansisScMnYn 20 nak; kñúgenaHmanRbus14nak; nigRsI6 nak;. eKcg;begáItRkum

mYyEdlmansisS 4 nak; etImanbu:nµanrebobekIteLIg? kñúgkrNIxageRkam ³

k> rkcMnYnsisSRbus nigsisSRsIesµIKña.

x> ya:gticmansisSRsImñak;.

VI. kñúgtRmuyGrtUNrma:l;manTisedAviC¢man 0 ; ; ;i j k eKmancMnuc 1;0;1A vuicT½r 1;2;1AB .

1/. KNnakUGredaencMnuc B . rksmIkarbøg; P Edlkat;tam A ehIyEkgnwg AB .

2/. eK[cMnuc 2;1;0C nig 1;3;1D . rkkUGredaenvuicT½r AC nigCD. KNna AB AC rYc

bgðajfa ABDC CactuekaNEkg.

3/. KNna AB AC . TajrképÞRkLaépÞénctuekaN ABDC .

4/. rkmaDctumuxOABC. Tajrkcm¶ayBIOeTAbøg; ABC .

VII. eKeGayGnuKmn_ f kMnt;elI eday 1xf x e x nig g kMNt;elIcenøaH ] 0 ; [ eday

1 2lng x x x . C CaRkabén f nig C CaRkabén g kñúgtRmuyGrtUNrma:l; 0 ; ;i j .

1/. k> KNnalImIt limx

f x

nig limx

f x

rYcsg;taragGefrPaBén f .

x> bgðajfabnÞat; 1y x CaGasuImtUteRTtén C kalNa x . bBa¢ak;TItaMgén

C eFobnwgGasuImtUt.

K> kMNt;kUGredaenéncMnucP én C EdlbnÞat;b:Hnwg C Rtg;cMnucenHRsbnwgbnÞat; y x .

2/. k> KNnalImIt limx

g x

nig 0

limx

g x

rYcsg;taragGefrPaBén g x .

x> ehA T CabnÞat;b:Hnwg C Rtg;Q EdlmanGab;suIsesµI1. epÞógpÞat;fabnÞat; T

kat;tam P .

3/. sg;Rkab C nigbnÞat;b:Hrbs;vakñúgRkabxagelI.

&Y'


eRtomRblg Bacc II

I. 1/. eKeGay ln 2 2 ln 2 2 2 ln 2 2 2 : sin cosa b x x Edl

1

sin cos2

x x . cUrsresrCaTMrg;RtIekaNmaRténcMnYnkuMpøic ln 2 8Z a bi .

2/. eKeGay

3

2

2 1

1 3

iZ

i

. cUrsresr Z CaTMrg;BICKNit nigCadTMrg;RtIekaNmaRt rYcTaj

rktMélR)akdén cos2

nig sin

2

.

II. eKeGayGnuKmn_ f kMnt;elI[ 0 ; [ eday

ln 1 ln 0

0 0

f x x x x x x x

f

ebI

1/. etIGnuKmn_ f Cab;Rtg; 0 b¤eT?

2/. kMnt;lImIt

0limx

f x

x. etIeKbkRsaylTæplenHtamFrNImaRtdUcemþc?

III. eKeGaysmIkarDIepr:g;Esül 2 4 20cos2 1y y y x .

1/. edaHRsaysmIkar 2 4 0 2y y y .

2/. kMnt;cMelIyénsmIkar 1 manTMrg; cos2 sin 2f x a x b x Edla nigb CacMnYnBitRtUvkMnt;.

IV. f CaGnuKmn_kMnt;elIcenøaH ] 0 ; [ Edl

ln1; ;

3 2 0 1

xax b x a b

xf x

x x

ebI

ebI

kMnt;tMél a nigb edIm,IeGayGnuKmn_ f x manedrIevRtg; 1x .

V. KNnalImIt ³ 1

lim 1x

xA x e

1

20

1 cos 2 sinlim

x

x

x e e xB

x

VI. kñúgtRmuyGrtUNrma:l; 0 ; ; ;i j k CalMh eKman 4 cMnuc 0;1;0 ; 2;3;1A B .

1/. RsaybBa¢ak;fa ; ; ;A B C D CakMBUlénctumux rYcTajrkmaDénctumux ABCD.

2/. Rsayfactumux ABCDmanmuMEkg3 Rtg;kMBUl A .

3/. rksmIkarEsV‘ S Edlkat;tam 4 cMnuc ; ; ;A B C D .

VII. A/. f CaGnuKmn_kMnt;elI *

eday 23 3 2lnf x x x cUrkMnt;TisedAGefrPaBén f .

KNna 1f rYckMnt;sBaØaén f x .

B/. g CaGnuKmn_kMnt;elI *

eday

22ln 1 3

2

x xg x

x

ehIy C CaRkabén g .

1/. KNna 0

limx

g x

nig limx

g x

rYcbBa¢ak;smIkarGasuImtUtQrén C .

2/. k> bgðajfa 3

:2

L y x CaGasuImtUtén C kalNa x xitCit.

x> rkkUGredaencMnuc M CacMnucRbsBVrvag L nig C . rYcsikSaTItaMgrvagRkab

C eFobnwgbnÞat; L .

3/. KNna g x rYckMnt;sBaØaén g x . bgðajfag manGtibrma rYcKNnatMélGtibrmaenaH.

4/. KUstaragGefrPaBén g . KNna1

2g

nig 2g rYcsg;Rkab C nig L .

ebI 1,64 ; ln 2 0,69e .

&Y'


eRtomRblg Bacc II

I. f CaGnuKmn_kMnt;eday

20091 0 1

ln

0 1

xf x e x xx

f m

ebI ngi

KNna 0

limx

f x

. TajrktMél m edIm,IeGay f CaGnuKmn_Cab;xagsþaMRtg; 0x .

II. GnuKmn_ f manedrIevRtg; x a . RsaybBa¢ak;fa

limx a

x f a a f xf a f a

x a

.

Tajrkcos cos

limx a

x a a x

x a

nig

1

ln 2 ln 1lim

1x

x x

x

.

III. eKeGay 2 lny x x Edl x CacMnYnBit.

1/. sikSasBaØa y rYcTajrktMél x Edl 0y .

2/. KNna2

1y dx .

IV. KNnalImIt ³ k> 1 ln 1

limln

x

x

e x x x

x x

x>

0

sin 2009 2lim

sin 2008x

x x

x x

K>ln

lim1xx

x x

e

X>

3 2

23

3 1 3 3lim

4 3x

x x m x m

x x

V. P CaExSekagtag 2 2f x x x . A nigB CaBIrcMnucén P EdlmanGab;suIserogKña 1 nig1.

1/. sg;ExSekag P .

2/. rksmIkarbnÞat; AB .

3/. KNnaRklaépÞEdlx½NÐedayExSekag P nigGgát; AB .

VI. KeGay 0 ; ; ;i j k CatMruyGrtUNrma:l;manTisedAviC¢manénlMh. eKeGaybIcMnuc 1;1;0 ;A

0;0;1D nig 1;1;0C kñúgtMruyenH.

1/. sg;cMnuc B mYyEdlOABCCakaer. KNnakUGredaen B .

2/. eKeGayRbelBIEb:tEkgOABC DEFG manmuxBIrCakaerKWOABCnig DEFG .

kMnt;kUGredaenén cMnuc ; ;E F G .

3/. KNnamaDetRtaEGt DBCD.

VII. f CaGnuKmn_kMnt;eday

2

1 lnf x

x x

manRkabtMnag C .

1/. cUrrkEdnkMnt;énGnuKmn_ f .

2/. KNnalImItén f kalNa x xiteTAcugEdnkMnt;. bBa¢ak;smIkarGasuImtUténRkab C .

3/. bgðajfaGnuKmn_ f manGb,brmaRtg; 1x rYcKNna 1f . KUstaragGefrPaBén f .

4/. KNna 2f nig 3f . sg;ExSekag C .

5/. KNnaRklaépÞenAcenøaHRkab C / G½kS x ox RtUvnwgcenøaH1 2x .

6/. kMnt;tMél a edIm,IeGaysmIkar

2

1 lna

x x

KµanrwstamRkab.

&Y'


eRtomRblg Bacc II

I. KNnatMél dy nig y rYcrktMélRbEhl f x x énGnuKmn_

1/. ln 1f x x x kalNa 0 1x nig 0,1x .

2/. 0sin ; ;2 4

f x x x x

.

II. TwkexµARtUv)anFøak;eTAelIRkdasbegáIt)anCasñamRbLak;tagCargVg; EdlRklaépÞvaekIneday

GRta 25 /cm s . rkGRtaERbRbYlénkaMeBlRkLaépÞesµInwg 230cm .

III. eKeGaysmIkarDIepr:g;Esül : sinE y y x .

1/. kMnt;cMnYnBit a edIm,IeGayGnuKmn_ g kMnt;elI eday cosg x ax x CacemøIyén E .

2/. bgðajfaRKb;cemøIy f epSgeTotén E epÞógpÞat; 0f g f g .

3/. edaHRsaysmIkar 0y y . TajrkcemøIyTUeTAén E .

IV. 1/. eKeGayGnuKmn_ lny a x b Edl a nigb CacMnYnefrbgðajfa2

20

d y dy

dx dx

.

2/. C CaRkabénGnuKmn_ 3 lng x ax b x . kMnt;tMéla nigb edIm,IeGay1

1;2

I

CacMnucrbt;én C .

V. eKeGayeGlIb 2 2: 4 36E x y .

1/. KNnaRbEvgG½kSFM nigG½kStUc kUGredaenénkMnuM rYcsg;eGlIb E .

2/. kMnt;smIkarbnÞat; L Edlb:HeGlIb E ehIyRsbnwgbnÞat; : 2009y x .

VI. eKeGay f CaGnuKmn_kMnt;elI eday 3

11x

f x xe

ehIy C CaRkabén f kñúgtMruy

GrtUNrma:l; 0 ; ;i j .

1/. KNna limx

f x

nig limx

f x

.

2/. KNna f x rYcsikSasBaØarbs;va. sg;taragGefrPaBén f

3/. bgðajfa 1 : 1D y x CaGasuImtUteRTtén C enAnig 2 : 2D y x CaGasuImtUteRTt

én C xag. bBa¢ak;eGayc,as;TItaMgbnÞat;TaMgBIreFobnwg C .

4/. rksmIkarbnÞat;b:H T eTAnwg C Rtg;cMnucmanGab;suIs0 . sg; 1 2; ;T D D nigExSekag C .

5/. KNnaRklaépÞEdlx½NÐeday C nigG½kSGab;suIselIcenøaH 0 ; 2 .

VII. eKeGay 2;3;5A nigbøg; : 2 3 17 0P x y z .

1/. kMnt;smIkar)a:ra:Em:t nigsmIkarqøúHénbnÞat; L Edlkat;tam A ehIyEkgnwgbøg; P .

2/. bnÞat; L kat;G½kSOZ Rtg; M . rkkUGredaencMnuc M .

3/. rksmIkarEsV‘ S EdlmanGgát;p©it AM .

4/. KNnacMgayBIcMnuc 2;1;3B eTAbnÞat; L .

5/. rkkUGredaen AEdlqøúH A eFobnwgbøg; P . KNnaRbEvgGgát; A A .

&Y'

KNitviTüa fñak;TI 12 viBaØasaTI 10 eRtomRblg Bacc II

I. eKeGaycMnYnkuMpøic 6 2 6 2Z i nig2

x iyW

i

Edl ;x y CacMnYnBit.

1/. sresr Z CaTMrg;BiCKNit rYcCaTMrg;RtIekaNmaRt.

2/. sresrW CaTMrg;BiCKNit. rk x nig y edIm,IeGay 2 4 0 (Z W W CacMnYnkuMpøícqøas;én )W .

II. 1/. eRbIDIepr:g;Esül cUrKNnatMélRbEhléncMnYn 3 9 nig cos58 ; sin88 .

2/. k> KNna

1

1 20096 7

0 0

1 1lim ; 1 ; 1 sin

1 2 cossin

x

x

eA B x x dx C x dx

x

x

.

x> KNna 1

0lim ln 1 x

xx

rYcTajrk

0

sin 2 1lim

ln 1

x

x

e x

x

.

III. fñak;eronmYymansisSBUEkKNitviTüa 2 nak; nigBUEkrUbviTüa3nak; ehIyfñak;eronmYyeTotman

sisSBUEkKNitviTüa 4 nak; nigBUEkrUbviTüa3nak;. sisSmñak;RtUveRCIserIsedayécdnüecjBI

fñak;mYykñúgcMeNamfñak;TaMgBIrenH. rkRbU)abEdleKeRCIserIssisSBUEkKNitviTüa.

IV. 1/. edaHRsaysmIkarDIepr:g;Esül : 3 2 0E y y y .

2/. kMnt;cMelIy g x én E edIm,IeGayRkabtMNag g b:HeTAnwgbnÞat; : 3D y x Rtg; 0;0O .

V. eKeGay xf x e nig ln 1 1g x x .

1/. epÞógpÞat;faExSekag 1 :C y f x nig 2 :C y g x mancMnucrYm 0;1A .

2/. KNna 0f nig 0g rYcTajfa 1C nig 2C b:HKñaRtg; A .

3/. sresrsmIkarbnÞat;b:HrYmrvag 1C nig 2C .

VI. g CaGnuKmn_kMnt;cMeBaH 1x eday 1

1 2ln 1g x xx

nig C CaRkabén f .

1/. cUrsikSaTisedAGefrPaBén g . eKeGay ln 2 0,7 nig ln 3 1,1 .

2/. KNna 1

limx

g x

nig limx

g x

. rksmIkarGasuImtUténRkab C .

3/. KUstaragGefrPaBén g . rYcsg;Rkab C

4/. bgðajfasmIkar 3g x manrwsEtmYyKt;enAelIcenøaH[ 2 ; [ .

5/. KNnaRklaépÞ A enAcenøaHExSekag C nigbnÞat; : 1L y x EdlRtUvnigcenøaH2 4x .

VII. 1/. eKeGaybøg; : 1 0P x y z nigEsV‘ S EdlmansmIkar 2 2 2 2 2 6 0x y z x y .

k> rkkUGredaenp©it I nigkaM R énEsV‘ S .

x> KNnacm¶ayBIp©it I énEsV‘ S eTAbøg; P . etIeKsniñdæanya:gNacMeBaHTItaMgbøg;

P nigEsV‘ S .

2/. enAkñúgtRmuyGrtUNrma:l;manTisedAviC¢man ; ; ;o i j k eKeGaybøg;BIr :3 4 6 12 0Q x y z

nig : 0R x y z .

k> rksmIkar)a:ra:Em:ténbnÞat; D RbsBVrvagbøg; Q nig R edayyk Z t Ca

)a:ra:Em:t. bBa¢ak;kUGredaenénviucT½rR)ab;TisU énbnÞat; D .

x> tag q CavuicT½rn½rma:l;énbøg; Q nig r CavuicT½rn½rma:l;énbøg; R . KNnaplKuN

vuicT½r q r . bgðajfa q r RsbnwgU .

K> bøg; Q kat;G½kS ; ;ox oy oz erogKñaRtg; ;A B nigC . KNnamaDénctumuxOABC.

KNitviTüa fñak;TI 12 viBaØasaTI 11 eRtomRblg Bacc II

I. eKeGay 1Z i .

1/. KNna 2 2A iZ Z i rYucsresr A CaTMrg;RtIekaNmaRt.

2/. KNna3

3

Z iZB

i

rYcsresr B CaTMrg;RtIekaNmaRt.

II. f nig g CaGnuKmn_EdlmanedrIev f nig g nigepÞógpÞat; ff gg . etImanTMnak;TMngdUcemþc

rvag f nig g ?

III. f CaGnuKmn_kMnt;eday 4

; 11

xf x x

x

.

1/. bgðajfacMeBaH 2 ; 3x eK)an 3

34

f x .

2/. edayeRbIvismPaBkMeNInmankMnt;bgðajfaRKb; 2 ; 3x eK)an 3 14

3 84

xf x x

.

bkRsaytamRkaPic.

IV. eKman 1

1:

sin 1

xeC y

x

nig 2 : ln 1C y x x . rksmIkarbnÞat;b:H 1C eTAnwg C Rtg;

0x nigbnÞat; 2T eTAnwg 2C Rtg; 0x .

V. sresrTMrg;sþg;daéneGlIbEdlmanp©it 0 ;0 ; kMBUl 0 ;5 ; nigkMnuM 0 ;4 rYcsg;Rkab.

VI. 1/. h CaGnuKmn_kMnt;elI eday 1x xh x e xe . KNna 0g rYckMnt;sBaØaén g x .

sg;taragGefrPaBén g .

2/. f CaGnuKmn_kMnt;elI eday 11x

xf x

e

cMeBaH 9x nig 0 2f C CaRkabén f

kñúgtRmuyGrtUNrma:l; ; ;o i j edayykmYyÉktaesµI 2cm.

k/ KNna limx

f x

nig limx

f x

. TajrkGasuImtUtedk 2L én C . etIGnuKmn_ f

Cab;Rtg; 0x EdrrWeT?

x/ bgðajfabnÞat; : 1y x CaGasuImtUteRTtén C kalNa x xiteTA.

bBa¢ak;TItaMg C eFob .

K/ KNnaedrIev f x ehIysikSasBaØaén f x rYckMnt;TisedAGefrPaBén f .

sg;taragGefrPaBén f .

X/ sg; ;D nig C enAkúñgtMruyEtmYy.

VII. enAkñúgtRmuyGrtUNrma:l; ; ; ;o i j k énlMh eKeGaycMNuc 3;0;0 ; 3;6;0 ; 0;6;0A B C nig

0;0;3D .

1/. sg;BIra:mIt DOABC .

2/. cUrBnül;faOABCCactuekaNEkg ehIy DO CakMBs;énBIra:mIt DOABC .

KNnamaDBIra:mItenH.

3/. G CaTIRbCuMTMgn;én BCD . rkkUGredaenén H .

4/. rksmIkar)a:ra:Em:ténbnÞat; L kat;tamG ehIyEkgnwgbøg; ABC Rtg; H .

rkkUGredaenén H .

5/. eRbobeFobmaDetRtaEG‘tDOABnig DOBC.

KNitviTüa fñak;TI 12 viBaØasaTI 12 eRtomRblgBacc II

I. KNnaGaMgetRkal ³

3 1 2009

9 10

0 0; 2 2 ; 1

1 ln

dxI J x dx K x x dx

x x

II. eKeGaysmIkarDIepr:g;Esül : 2 0E f x f x .

1/. tag g x f x . bgðajfa g CacMelIyénsmIkar 2 0g x g x .

2/. edaHRsaysmIkar 2 0g x g x . rYcrkcMelIyénsmIkar 2 0f x f x .

3/. epÞógpÞat;cemøIyénsMnYr 2 edaHRsaysmIkar 2 0f x f x tamsmIkarsMKal;.

III. RsayfasmIkar 3 23 5 4 2 0x x x manrwsbIepSgKñaenAkñúgcenøaH ] 3 ;1[ .

IV. KNnalImIt ³ 20

ln 1lnlim lim

2009xx x

xx xA B

e x x

V. f CaGnuKmn_kMnt;cMeBaH 0x kMnt;eday 2 lnx x

g xx

ehIymanRkab C enAkñúgtMruyGrtUNr

ma:l; ; ;o i j EdlmanÉkta 2cm.

1/. KNna 0

limx

g x

nig limx

g x

. kMnt;smIkarGasuImtUtQrénRkab C .

2/. KNna g x nigsikSasBaØaén g x ebIeKdwgfa 2 1 ln 0x x cMeBaHRKb; 0x .

sg;taragGefrPaBén g .

3/. bgðajfabnÞat; :L y x CaGasuImtUteRTtén C sikSaTItaMgrvag L nig C .

4/. bnÞat; T b:HExSekag C Rtg;cMnuc A ehIy T Rsbnwg L . rkkUGredaenéncMnuc H .

5/. KNna 3f ehIysg; ;T L nig C enAkñúgtMruyGrtUNrma:l;EtmYy.

VI. yk g CaGnuKmn_ f kMnt;eday ln 11

xx

x

eg x e

e

manExSekag C kñúgtMruyGrtUNrma:l;

; ;o i j ÉktaRbEvg1cm .

1/. sikSaGefrPaBén g . TajrksBaØaén g x .

2/. bgðajfa : 1D y x CaGasuImtUtén C xag.

3/. KUs C nig D kñúgtMruyEtmYy.

4/. yk f CaGnuKmn_kMnt;elI eday ln 1x xf x e e manRkabtMNag C .

k/ KNnalImIt limx

f x

. etIGñkTaj)anGIVcMeBaH C .

x/ Tajfa ln 1

11

x

x

x

ef x e

e

KNnalImIt limx

f x

.

VII. enAkñúglMhRbkbedaytRmuyGrtUNrma:l;manTisedAviC¢man ; ; ;o i j k eKeGaycMnuc 1;0;0 ;A

1; 1;1B nig 2;0;1D .

1/. kMnt;kUGredaenéncMnucG EdlepÞógpÞat; 2 2 0GA GB GC .

2/. eKP¢ab;eTARKb;cMnuc M énlMhnUvvuicT½r mV EdlkMnt;eday 2 2mV MA MB MC .

dak; mV CaGnuKmn_énvuicT½r MG .

3/. cMnuc M mankUGredaen ; ;x y z .

k/ KNna mAB V x/ KNna mAB V

4/. kMnt;rkcMnuc M EdlepÞógpÞat;RBmKña mAB V OA nig 0mAB V .


I. 1- k- edaHRsaykñúgsMnuM C smIkar 0842 xx .

x-rkm:UDul nigGaKuym:g;énrws1x nig

2x rbs;va .

2-eKman )26()26( iZ . sresr 2Z CaTMrg;BiCKNit rYcCaTMrg;RtIekaNmaRt

II.RbFanviBaØasaKNitviTüaénkarRblgmYymanlMhat; BICKNit 7 nig FrNImaRt 3 ebkçCnmñak;eQµaHsux

RtUveFVIlMhat; ; 3 kñúgcMenamlMhat;TaMg10 eTAtamkareRCIserIsrbs;xøÜn .kMnt;RbU)abénRBitþikarN_³

A sisSeQµaHsuxeFIVlMhat;BICKNItTaMg3 B sisSeQµaHsux eFVIlMhat;FrNImaRt2Kt;

III. 1-rkkMBUl-kMnuM-bnÞab;R)ab;Tisén)a:ra:bUlEdlmansmIkar ³

04384 2 xyy

2-KNnaGaMgetRkal³

2ln

0

4

0

4

3

cos;

1

1

x

dxJdx

e

eI

x

x

IV. eKeGayGnuKmn_

)0()sin(sinsin

)( xx

xxf nig

)6log()6log(0

0)()(

KKg

xxfxg

ebI

cUrkMNt;tMél K edIm,IeGay g CabnøayPaBCab;én f Rtg; 0x

V.GnuKmn_ f kMnt;eday x

xexf

x )( ehIy )(C CaRkabtMNag f .

1- rkEdnkMnt; énGnuKmn_ f rYcrklImItén )(xf Rtg;cugEdnkMnt;enaH .

Tajrk smIkarGasuImtUténRkab )(C .

2- bgðajfa f mantMél Gb,brmamYy . KNnatMélGb,brmaenaH .

3- KUstaragGefrPaBén f .

4- KNnatMél )3(;)2(;)1( fff rYcsg;Rkab )(C kñúgtMruyGrtUNrm:al;. )7,2( e

5- cUrBiPakSatagRkabnUvGtßiPaB nigsBaØarwsénsmIkar xme x )1( eTAtamtMél)a:ra:Em:Rt m .

VI.kñúgtMruyGrtUNrm:al;manTisedAviC¢man );;;(

kjio eKmanbIcMnuc )1;2;1();1;1;1( BA nig )1;2;2(C .

k¼> KNna

ACAB rYcTajfacMnuc CBA ;; minrt;Rtg;CYr .

x¼>rYcbBa¢ak;RbePT RtIekaN ABC .

K¼> rksmIkarbøg; )(ABC rYcbgðajfacMnuc )2;1;0(D )(ABC nig cMnuc )3;2;1(E )(ABC .

X¼> KNnamaDetRtaEGt OABC rYcrkmaDRbelBIEb:tEkgEdlmanvimaRt

OCOBOA ;; .

g¼> rksmIkar)a:ra:Em:RténbnÞat; )(OE rYcrkkUGredaencMnucRbsBVrvagbnÞat; )(OE nigbøg; )(ABC


I-eKeGayRbB½n§smIkar

02

42

21

21

ZZi

ZZ

k¼ edaHRsayRbB½n§smIkarenH

x¼ rkm:UDul nig GaKuym:g;én 1Z nig 2Z

II-1- KNnalImIt ³

tgxxAx

)2

(lim

2

; 12

12lim

2008

2007

1

xx

xxB

x

2-rkEdnkMnt; ³ )1ln(1)1ln(1 xxy

III-a).bgðajfa K IN* Rsayfa³ !)!1(!. KKKK

b). Tajbgðajfa ³ )!1(!.....!3.3!2.2!1.1 nnn

IV-ebIeKmanGnuKmn_ xxxf 22 cossin)(

k-kMnt;rk a nig b edIm,IeGay xbaxf 4cos)(

x-KNna dxxx

0

22 cossin

V-kñúges,agmYymanb‘Ul s 5nigb‘UlexµA 3. eKTajykb‘UlBIr . KNnaRbU)abénRBwtþkarN¾ ³

AmanBN_dUcKña - B manBN_xusKña

. VI-1-kñúgtMruyGrtUnrm:al; );;;(

kjiO eKeGaycMnuc A(0;0;-3); B(2;0;-1) nig bøg; 01783)( zyxP .

k- rkkUGredaen I RbsBVrvagbnÞat;kat;tam A nig B CamYy (P) x- rkkUGredaenén C enAelI (P) edIm,IeGayRtIekaN ABC CaRtIekaNsmgS½

2-eKeGaybøg; 03783: zyxQ

k-RsaybBa¢ak;fa P nig Q RsbKña

x-rkcMgayrvag P nig Q

VII- f CaGnuKmn_kMnt;edayx

xxxf

1ln)(

manRkab )(C kMnt; 0x

a) KNnalImIt én xf kalNa xx ;0 .Taj rkGasuImtUt

b) KNna )(xf . sikSasBaØaén f TajrktMélGtibrma .sg;taragGefrPaB

c) RsayfaExSekag )(C mancMnucrbt;mYy . rYcKNnakUGredaencMnucrbt;

d) sresrsmIkarbnÞat;b:Hnwg )(C Rtg; M EdlmanGab;suIs ex

e) KNna

ef

1 . sg;Rkab )(C kñúgtMruy );;(

jio ebI 6,121

e ; 36,01

;7,2 e

e


I-KNnalImIt³

1). 3)1

3(lim

x

x x

x 2).

x

ee xx

x

2007

0lim

II- f CaGnuKmn_ kMnt;elI IR eday

3)0(

021

)(

f

xe

xxf

xebI )(C CaRkabén f enAkñúgtMruyGrtUnrm:al; .

k¼> RsaybMPøWfa f Cab;Rtg; 0x

x¼> rklImItén f kalNa x xiteTA ; . TajrkGasuImtUtén )(C .

. III- a). kMnt;smIkarbnÞat;b:H )(d nigExSekag xxxyC 2ln:)( Rtg; )2;1(A

b). kMnt;kUGredaencMnucRbsBVrvag )(d nig xeyH x 3:)(

IV-eKcg;sg;FugragsuILaMg)atCafas kaM R mankMBs; h nigmanmaD eKsÁal; 327cm . etIeKRtUveRCIserIs

R nig h y:agNa edIm,IeGayGs;sah‘uytic .

V- f CaGnuKmn_kMnt;eday 0;ln)()( 2 xxmxxfm . )( mC CaExSekagtMNag mf .

k- RsaybBa¢ak;fa )( mC kat;tamcMnucnwg 1 EdlRtUvrkkUGredaen .

x- kMnt;tMél m edIm,IeGay mf mantMélGb,brmaRtg; e

x1

.

VI-1- sg;RkabGuIEBbUl 0393629 22 yxyx

2-kñúgtMruyGrtUnrm:al; );;;(

kjio énlMhmanTisedAviC¢maneKeGayEsV‘r

0642:)( 222 zyxzyxS

k- rkkUGredaenp©it I nigkaM R rbs;EsV‘r )(S

x- Es‘Vr )(S kat;GkS½ )();();( ozoyox erogKñaRtg; BA; nig C EdlepSgBI O .

rkkUGredaenéncMnuc CBA ;; enH

K- bgðajfabøg; )(ABC mansmIkar 01642

zyx

X- )(D CabnÞat;kat;tamp©itEsV‘r )(S ehIyEkgbøg; )(ABC Rtg; H .

sresrsmIkar)a:ra:Em:RténbnÞat; )(D .

g- rkmaDetRtaEG‘t OABC

VII-eKeGayGnuKmn_ 12)(2 xxf manRkabtMNag )(H

k-sg;Rkab )(H enAkñúgtMruyGrtUnrm:al;

x- HBA ; EdlmanGab;suIserogKña ba; .bgðajfa ba; eK)an

ba

bfafbaf

2

K-KNnaépÞRklax½NedayExSekag )(H nigGkS½Gab;suIs

eKeGayGnuKmn_ 32

22)(

2

xx

xxf manRkabtMNag (C)

k- bBa¢ak;EdnkMnt; rYcKNnalImItRtg;cMnucEdnkMnt; . TajrksmIkarGasuImtUténExSekag (C).

sg;taragGefraBén f

x- rkkUGredaenéncMnuc I rvagExSekag (C) nigG½kS oxx' . bgðajfa I Capi©tqøúHén (C) .

KNna )0(f nig )2(f

K- sg;Rkab (C) . edayeRbIExSekag (C) sikSaGtißPaB nigsBaØarwsénsmIkar

0)23()1(22 mxmmx eTAtamtMél):ar:aEm:Rt m VIII. ABCD CactumuxmYyEdlmanRTnugTaMg 6 b:un²Kña . A CacMeNalEkgén A elIbøg; )(BCD ehIy O CacMnucmYyén

][ AA Edl )(0.3 cmxAOBOA .

1- cUrKNna

OBOA. CaGnuKmn_én x

2- cUrKNna

AOBcos rYcTajrk AB CaGnuKmn_én x

KNitviTüa fñak;TI 12

I- 1/. edaHRsaysmIkar 2 3 0 ;y y y E .

2/. kMNt;cMelIy g x mYyén E edIm,IeGayRkabtagGnuKmn_ g b:HnwgbnÞat; :2

xd y Rtg; 0;0O .

II. fg;mYymanbU‘lRkhm 2 nigbU‘lexov 6 . bU‘lBIrRtUvykecjedayécdnü.

1/. rkRbU)aBEdlbU‘lTaMgBIrBN’Rkhm.

2/. rkRbU)abEdlya:gticmanbU‘lmYyBN’Rkhm.

3/. rkRbU)abEdlbU‘lmYyKt;BN’Rkhm.

III. eKeGayGnuKmn_

7 10

2 5

xf x

x x x

Edl 0 ; 2 ; 5x x x .

1/. kMnt;cMnYnBit ;A B nigC edIm,IeGay 2 5

A B Cf x

x x x

RKb; 0 ; 2 ; 5x x x .

2/. KNna 3

4I f x dx .

IV. RtIekaN ABC man ; 3AB x AC x nig 150A .

k.KNnaRklaépÞRtIekaN ABC CaGnuKmn_én x .

x.ebIeKbegáIn x edayGRta0,1 /cm s . KNnaGRtakMenInénépÞRtIekaN ABC xN³ 4x cm

V. kñúgtMruyGrtUnrm:al; );;;(

kjio eKeGay 01862:)( 222 zyxzyxS .

1- rkkUGredaenp©it I nig kaM R rbs;Es‘V )(S

2- )(D CabnÞat;kat;tamcMnuc I ehIyRsbbøg; );;(

jio nig );;(

kjo .

sresrsmIkar)a:ra:Em:RténbnÞat; )(D

3- rkkUGredaencMnuc M nig N EdlCaRbsBVrvagbnÞat; )(D CamYyEsV‘r )(S

4- )(P nig )(Q Cabøg;b:HEsVr )(S Rtg; M nig N . sresrsmIkarbøg; )(P nig )(Q

VI.eKeGayGnuKmn_ f kMnt;elI IR eday 2 1 ;xf x x e C CaRkabén

f kñúgtMruyGrtUNrma:l; 0 ; ;i j EdlmanÉktaelIG½kS 2cm.

1/. KNnalImIténGnuKmn_ f Rtg;cugEdnkMnt;. TajrksmIkarGasuImtUtén C .

2/. KNna f x nigsikSasBaØaén f x rYcbgðajfa f manGtibrmamYy.

3/. sg;taragGefrPaBénGnuKmn_ f .

4/. sresrsmIkarbnÞat; L b:HeTAnwgExSekag C Rtg;cMnuc 0;1A .

5/. KNna 1

1 ;2

f f

rYcsg;bnÞat; L nigExSekag C kñúgtMruyGrtUnrma:l; 0 ; ;i j

6/. KNnaRklaépÞEpñkbøg;Edlx½NÐedayExSekag C G½kS x ox bnÞat; 0 ; 1x x .

rkkUGredaenkMBUl/ kMnuM nigbnÞat;R)ab;Tisén)a:ra:bUlxageRkam ³

21/ . 2 0P y y x 22 / . 4 4 0P x x y 23/ . 2 4 7 0P y y x .

VI. enAkñúgtMruyGrtUnrma:l; 0 ; ; ;i j k EdlmanTisedAviC¢manénlMh eKeGaycMnuc 2;2;1 ;A

0;2;3B nig 1;0;2C .

1/. cUredAcMnuc ;A B nig C .

2/. rkkUGredaenénCA CB . TajrkRkLaépÞénRtIekaN ABC .

3/. bgðajfabIcMnuc ;A B nigC rt;minRtg;CYrKña rYcTajrksmIkarbøg; ABC .

4/. KNnacMgayBIcMnuc 0;2;0S eTAbøg; : 3R x z .

5/. RsaybMPøWfaRtIekaN ABC CaRtIekaNsm)atkMBUlC . 6/. KNnamaDetRtaEGt SABC.

KNitviTüa fñak;TI 12

I-KNna

2

1 2 3

1 ln1I dx

x x xx

2

0

sin;

1 cos

xJ dx

x

II- eKeGayGnuKmn_ f mYykMnt;elIcenøaH ]4

;0[

eday x

xfcos

1)( .

k¼> kMnt; a nig b edIm,IeGay x

xb

x

xa

x sin1

cos

sin1

cos

cos

1

x¼> KNna 4

0 cos

x

dxI

III-eKeGaysmIkarDIepr:g;Esül : 0E f x f x .

1-tag g x f x . bgðajfa g CacMelIyénsmIkar 0y y .

2-edaHRsaysmIkar 0g x g x . rYcrkcMelIyénsmIkar 0f x f x .

3-epÞógpÞat;cemøIyénsMnYr 2 edaHRsaysmIkar 2 0f x f x tamsmIkarsMKal; IV-kñúgRbGb;mYyeKdak;sMbuRtepJIeTAbreTscMnYn 100 sMbuRt kñúgenaHmansMbuRt sMrab;epJIeTAGaemrikcMnYn 10

sMbuRt . eKyksMbuRt 3 edayécdnü .

KNnaRbU)abEdlyk)an ³

k¼> sMbuRtsMrab;epJIeTAGaemrikmYyya:gtic .

x¼> sMbuRtsMrab;epJIeTAGaemrikmYyya:geRcIn

V-enAkñúgtMruy ; ;o i j eKeGayExSekag 2: 6 9C y x x .

1/ bBa¢ak;RbePTénExSekag C rYckMNt;kUGredaen kMBUl/ kMnuM nigbnÞat;R)ab;Tisén C .

2/ sg;ExSekag C kñúgtMruy ; ;o i j .

VI- kñúgtRmuyGrtUnrma:l; ; ; ;o i j k énlMh eKeGaycMnuc 3;0;0 ; 0;2;0A B nig 0;0;1C .

k> KNna N AB AC rYcTajfa ; ;A B C minenAelIbnÞat;EtmYy. KNnaRklaépÞén ABC .

x-rksmIkarbøg; ABC . rkcm¶ayOI BIKl;O eTAbøg; ABC . epÞógpÞat;fa

2 2 2 2

1 1 1 1

OI OA OB OC .

K>KNnamaDetRtaEG‘tOABC. KNnakUGredaenéncMnuc I .

X>rkkUGredaenp©it J nigkaM R rbs;EsV‘ S Edlkat;tamcMnuc ; ; ;O A B C

VII- f CaGnuKmn_kMnt;eday

2

1 lnf x

x x

manRkabtMnag C .

1/ cUrrkEdnkMnt;énGnuKmn_ f .

2/ KNnalImItén f kalNa x xiteTAcugEdnkMnt;. bBa¢ak;smIkarGasuImtUténRkab C .

3/ bgðajfaGnuKmn_ f manGb,brmaRtg; 1x rYcKNna 1f . KUstaragGefrPaBén f .

4/ KNna 2f nig 3f . sg;ExSekag C .

5/ KNnaRklaépÞenAcenøaHRkab C / G½kS x ox RtUvnwgcenøaH1 2x .

6/ kMnt;tMél a edIm,IeGaysmIkar

2

1 lna

x x

KµanrwstamRkab.

VIII eKmanGnuKmn_ f kMnt;elI eday 22 ;xf x e C CaRkabén f kñúgtMruyGrtUNrma:l; ; ;o i j

ÉktaelIG½kS1cm .

1/. KNna lim ; limx x

f x f x

. TajrksmIkarGasuImtUtén C .

2/. KNna f x / sikSasBaØaén f x rYcTajfa f cuHelI .

3/. sg;taragGefrPaBén f . sresrsmIkarbnÞat; T b:HeTAnwgExSekag C Rtg;cMnucEdl

manGab;suIs 2x .

4/. KNna 0 ; 1 ; 2 ; 3f f f f sg;bnÞat; T nigExSekag C kñúgtMruyGrtUNrma:l; ; ;o i j .

5/. KNnaRkLaépÞx½NÐedaybnÞat; T nwgExSekag C nigbnÞat;QrmansmIkar 0x nig 4x .

វិញ្ញសាត្រៀមប្រលង Bacc II

Science

Transcript of វិញ្ញសាត្រៀមប្រលង Bacc II