Page 1


y

1

0

1

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

្ណ ក -i-

ក ្រពំ ម៉


រមភកថ

sYsþImitþGñksikSa CaTIemRtI ! ! esovePA កំែណលំ ត់គណតវិ ិ ទយថនក់ទ1 ី 2 kMritx<s;EdleyIgxJMú)an eroberogenH rYmmanR)aMCMBUkFM² Edl)andkRsg;ecjBIesovePAKNitviTüa fñak;TI12kMritx<s; rbs;RksYgGb;rM yuvCn nigkILa . kñúgCMBUknImYy²eyIgxJMú)ansegçbemeron nig dkRsg;lMhat;kñúgemeronénCMBUk nImYy²mkeFIVdMeNaHRsayy:agek,aHk,ay EdlGacCaCMnYysμartIdl;GñksikSa kñúgRKb;mCÆdæan . eTaHCay:agNak¾eday karbkRsaynUvral;lMhat;kñúgesovePA enHvaminl¥hYseKhYsÉg TaMgRsugenaHeT . kMhusqÁgnana TaMgbec©keTs nig GkçraviruTæR)akdCaekItmanedayGectna BMuxaneLIy GaRs½yehtuenH eyIgxJMúrg;caMCanic©nUvmtiriHKn;EbbsßabnaBIsMNak; GñksikSakñúgRKb;mCÆdæan edIm,IEktRmUvesovePAenH[kan;EtmansuRkitüPaB EfmeTot . CaTIbBa©b; eyIgxMJúGñkeroberog sUmCUnBrdl;GñksikSaTaMgGs; TTYl)aneCaKC½ykñúgCIvit nig mansuxPaBl¥ . )at;dMbgéf¶TI 8 Ex ]sPa qñaM 2010 Gñkeroberog lwm plÁún - ii -


matikaerOg TMB½r CMBUkTI1 lImIténsVIút

01

CMBUkTI2 Gnuvtþn_edrIevénGnuKmn_ CMBUkTI3 emeronTI1 GnuKmn_ GsniTan emeronTI2 GnuKmn_RtIekaNmaRtcRmuH CMBUkTI4 emeronTI1 emeronTI2 CMBUkTI5 emeronTI1 emeronTI2

27 53 65

GaMgetRkalkMNt; maDsUlId nig RbEvgFñÚ

83 96

smIkarDIepr:g;EsüllMdab;TI1 smIkarDIepr:g;EsüúllIenEG‘lMdab;TI2

115 136

- iii -


CMBUkTI1

lImIténsVIút

CMBUkTI1

lImIténsVúIt

emeronsegçb

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 ¡ 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 → +∞ ¡

n

n

n

n→ +∞

n→ +∞

n→ +∞

n

n→ +∞

n→ +∞

n

n

n

n

n→ +∞

n

n

n

n

n → +∞

n

n

n

n→ +∞

n

n

n → +∞

n

n

n → +∞

n

-1-


CMBUkTI1

lImIténsVIút

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 n

n

n =1

n

n → +∞

n

=0

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

2

-2-

n

3

n −1

+ ...


CMBUkTI1

lImIténsVIút

lMhat; !>etIsVIút (U ) EdlmantYTUeTAdUcxageRkam CasVIútrYm b¤ rIk ? k> U = 3n + 5n + 1 x> U = 2nn ++n5 K> U = sin5 2n X> U = n2+n3 + n3n+ 5 g> U = nn sin+ n1 c> U = 2 − n3 + 4n @>KNnalImIténsVIútxageRkam 5n + n − n lim k> lim n8n +−3nn +− 11 x> n + n − 1 n

2

2

n

n

2

3

n

n

n

n

n

2

2

3

2

n→ +∞

2

n→ +∞

5n 3 + ( −1)n

2

2

n 2 + sin n lim n→ +∞ 5n 2 + cos πn

K> lim n + (− 1) X> g> lim (n − cos πn) c> lim [− 5n + (− 1) n ] #> KNnalImIténsVIútxageRkam k> lim ( n + 1 − n ) x> lim n ( n − 3 − n ) K> lim (n n + 1 − 1) X> lim ⎛⎜ (n + n1!)!−n! − n2 + 3 ⎞⎟ . n

n→ +∞

2

2

3

n→ +∞

n → +∞

n→ +∞

n→ +∞

2

n→ +∞

n→ +∞ ⎝

-3-

n

3


CMBUkTI1

lImIténsVIút

$>eKmansVIútEdlkMnt;cMeBaHRKb; n ∈ IN mantYTUeTAdUcxageRkam³ n 2 n! = n(n − 1) × ... × 1 U = Edl ,V = n! 2 k>KNna lim UU nig lim VV 3

n

n

n

n

n +1

n → +∞

n +1

n→ +∞

n

n

2n + n 3 lim n→ +∞ n!+ n 3

x>KNna %>KNnalImIténsVIút (a ) EdleKsÁal;tYdUcxageRkam³ k> a = 2 , a = 12 a + 3 x> a = 3 , a = 2a − 5 K> a = 1 , a = 13 a + 43 . ^>Binitües‘rIxageRkamenH etICaes‘rIrIk rW rYm? x> ∑ 1000(1.055) k> ∑ 3.⎛⎜⎝ 32 ⎞⎟⎠ n

1

n +1

1

n +1

n

1

n +1

n

n

n

n =0

n=0

K> g>

n

n+ n ∑ 3 n =1 2n − 1 3 9 27 81 ... 2+ + + + 2 8 32 128 ∞

-4-

2n + 1 ∑ n −1 n =1 2 ∞

X> c> ∑ ( ∞

n =1

2n + 1 − 2n − 1 )


CMBUkTI1

lImIténsVIút

&>rkplbUkénes‘rIxageRkam³ k>1 + 0.1 + 0.01 + 0.001 + ... x> ∑ 4n 2− 1 ∞

2

n =1

X>

n

c>

K> ∑ (2n − 1)(2n + 1) n =1

5(− 1) ∑ 4n n =1 ∞

g> *>KNna k> lim ⎛⎜⎝ 1 − 21 ⎟⎞⎠...⎛⎜⎝ 1 − n1 ⎞⎟⎠ 2

n→ +∞

π⎞ ⎛ ∑ 2⎜ cos ⎟ 3⎠ n=0 ⎝ 2 3 2 ⎛ 2⎞ ⎛ 2⎞ + ⎜ ⎟ + ⎜ ⎟ + ... 3 ⎝ 3⎠ ⎝ 3⎠

1

n

2

x> lim ⎛⎜⎜ n + n + ... + n ⎞⎟⎟ n +2 n +n⎠ ⎝ n +1 !@>KNna lim [(q − 1) + q (q − q ) + ... + q ( ) (q − q )] Edl q = 2 . 1 ( ) (> eK[sVIút a kMNt;eday a = 2 + q Edl q ≠ −1. sikSalImIténsVIút (a ) kalNa n → +∞ . !0>cMeBaHRKb; n ∈ IN eKman n→ +∞

4

4

2

4

2 n −1

2

n

n −1

n→ +∞

1 2

n

n

n

n

n 2 2 2 2 Sn = + + ... + = ∑ (2n − 1)(2n + 1) p=0 (2p + 1)(2p + 3) 1× 3 3 × 5

-5-


CMBUkTI1

lImIténsVIút

k>KNna S CaGnuKmn_én n edayeRbI (2p + 1)(22p + 3) CaTRmg; (2pa+ 1) + (2pb+ 3) . x>KNna lim S . !!>eKmanes‘rIGnnþmYy r r r r + + + ... + + ... bgðajfaes‘rIGnnþ r +1 n

n

n→ +∞

2

2

2

2

(r

2

2

)

+1

(r

2

2

+1

)

n −1

enaHrYmcMeBaHRKb;témø r . !@>kMNt;es‘rIxageRkam etIes‘rImYyNarYm mYyNarIk? ebICaes‘rIrYm cUrrkplbUk. k> ∑ 2⎛⎜⎝ cos π3 ⎞⎟⎠ x> ∑ ⎛⎜⎝ tan π4 ⎞⎟⎠ ∞

n

n

n =1

n =1

5(− 1) ∑ n 4 n =1 ∞

n

K> X> . !&>eKman ΔABC mYymanRklaépÞesμInwg 6 Ékta. sg; ΔA' B' C'; eday A' , B' nig C' CacMnuckNþalénRCug . cUrkMNt;plbUk S = S + S + S + ... . n ∑ n =1 2n + 5

1

2

-6-

3


CMBUkTI1

lImIténsVIút

dMeNaHRsay !> etIsVIút (U ) EdlmantYTUeTAdUcxageRkam CasVIútrYm b¤ rIk ? k> U = 3n + 5n + 1 x> U = 2nn ++n5 K> U = sin5 2n X> U = n2+n3 + n3n+ 5 g> U = nn sin+ n1 c> U = 2 − n3 + 4n dMeNaHRsay sikSaPaBrYm b¤ rIkénsVIút k> U = 3n + 5n + 1 eKman lim U = lim (3n + 5n + 1) n

2

2

n

n

2

3

n

n

n

n

2

n

2

2

n

2

n → +∞

n

n → +∞

5 1 ⎤ ⎡ = lim ⎢n 2 ( 3 + + 2 )⎥ n → +∞ ⎣ n n ⎦ = lim ( 3n 2 ) = +∞ n → +∞

dUcenH (U ) CasIVúrIkxiteTArk + ∞ . x> U = 2nn ++n5 eKman lim U = lim 2nn ++n5 = lim 2nn n

2

n

2

2

2

n → +∞

n

n → +∞

2

-7-

n → +∞

2

=

1 2


CMBUkTI1

lImIténsVIút

dUcenH (U ) CasIVútrYmxiteTArk 12 . K> U = sin5 2n eKman − 1 ≤ sin 2n ≤ 1 naM[ − 51 ≤ U ≤ 51 eday lim 51 = 0 enaH lim U = 0 . dUcenH (U ) CasIVútrYmxiteTArk 0 . X> U = n2+n3 + n3n+ 5 n

n

n

n

n

n → +∞

n

n → +∞

n

n

n

3

n

eKman

2

⎛ 2n 3n 3 ⎞ ⎟⎟ + 2 lim Un = lim ⎜⎜ n → +∞ n → +∞ n + 3 n + 5⎠ ⎝ ⎛ 2n 3n 3 ⎞ = lim ⎜⎜ + 2 ⎟⎟ n → +∞ n n ⎠ ⎝ = lim ( 2 + 3n ) = +∞ n → +∞

dUcenH (U ) CasIVútrIkxiteTArk + ∞ . g> U = nn sin+ n1 eKman − 1 ≤ sin n ≤ 1 naM[ − n n+ 1 ≤ u ≤ n n+ 1 eday lim n n+ 1 = 0 enaH lim U = 0 . dUcenH (U ) CasIVútrYmxiteTArk 0 . n

n

2

n

2

n → +∞

2

n → +∞

n

-8-

n

2


CMBUkTI1

lImIténsVIút

c> U = 2 − n3 + 4n eKman lim U = lim ⎛⎜⎝ 2 − n3 + 4n ⎞⎟⎠ 3 = 2 ¬eRBaH lim = lim n dUcenH (U ) CasIVútrYmxiteTArk 2 . @>KNnalImIténsVIútxageRkam k> lim n8n +−3nn +− 11 x> lim 5nn n

n → +∞

n

n → +∞

n → +∞

n → +∞

4 =0 n

¦

n

2

2

n→ +∞

5n 3 + ( −1)n

K> lim n + (− 1) g> lim (n − cos πn) dMeNaHRsay KNnalImIténsVIútxageRkam k> lim n8n +−3nn +− 11 n

n→ +∞

2

2

n→ +∞

2

n→ +∞

2

n2 1 = lim = n → +∞ 8n 2 8 5n 3 + n 2 − n lim n→ +∞ n 2 + n − 1 5n 3 = lim 2 = lim 5n = +∞ n → +∞ n n → +∞

x>

-9-

+ n2 − n 2 n→ +∞ +n−1 n 2 + sin n lim n→ +∞ 5n 2 + cos πn 3

X> c> lim [− 5n n → +∞

3

+ ( − 1) n 3 n

]


CMBUkTI1

K> lim

lImIténsVIút 5n 3 + ( −1)n

n + ( − 1) 5n 3 = lim = lim 5n 2 = +∞ n → +∞ n n → +∞ n 2 + sin n lim n→ +∞ 5n 2 + cos πn sin n 1+ 2 n = lim cos πn n → +∞ 5+ n2 1 sin n 1 − 1 ≤ sin n ≤ 1 − 2≤ 2 ≤ 2 n n n 1 sin n lim → 0 n → +∞ =0 2 2 n → +∞ n n 1 cos( πn ) 1 − 1 ≤ cos(πn ) ≤ 1 − 2≤ ≤ 2 n n2 n 1 cos( πn ) lim → 0 n → +∞ =0 2 2 n → +∞ n n 2 n + sin n 1 = lim n → +∞ 5n 2 + cos πn 5 n

n→ +∞

X>

eday ehIy dUcKñaEdr ehIy dUcenH g> lim (n n→ +∞

enaH kalNa

enaH

enaH

kalNa

enaH

.

2

− cos 2 πn )

⎡ 2 cos2 πn ⎤ )⎥ = +∞ = lim ⎢n (1 − 2 n → +∞ n ⎣ ⎦ 2 cos πn 1 cos 2 πn 0≤ ≤ 2 ⇒ lim =0 2 2 n → +∞ n n n

eRBaH

- 10 -

.


CMBUkTI1

lImIténsVIút

#> KNnalImIténsVIútxageRkam k> lim ( n + 1 − n ) x> lim n ( K> lim (n n + 1 − 1) X> lim ⎛⎜ (n + n1!)!−n! − n2 + 3 ⎞⎟ . ⎝ ⎠ dMeNaHRsay KNnalImIténsVIútxageRkam k> lim ( n + 1 − n ) n→ +∞

n→ +∞

2

n→ +∞

n→ +∞

n→ +∞

= lim

n → +∞

= lim

n → +∞

n +1− n n+1+ n 1 =0 n+1+ n

x> lim n ( n→ +∞

n − 3 − n)

n (n − 3 − n ) n → +∞ n−3+ n n = −3 lim n → +∞ n − 3 + n n = −3 lim n → +∞ n + n n 3 = −3 lim =− n → +∞ 2 n 2

= lim

- 11 -

n − 3 − n)


CMBUkTI1

lImIténsVIút

K> lim (n n→ +∞

)

n2 + 1 − 1

1 ⎤ ⎡ = lim ⎢n( n 2 + 1 − )⎥ n → +∞ ⎣ n ⎦ = lim n n 2 + 1 = +∞ n → +∞

X> lim ⎛⎜ (n + n1!)!−n! − n2 + 3 ⎞⎟ n→ +∞ ⎝

⎡ n! 2 ⎤ = lim ⎢ − + 3⎥ n → +∞ n! (n + 1) − n! n ⎣ ⎦ ⎞ ⎛1 2 ⎞ ⎛ 1 = lim ⎜ − + 3 ⎟ = lim ⎜ − + 3 ⎟ = 3 n → +∞ ⎝ n n ⎠ n → +∞ ⎝ n ⎠

$>eKmansVIútEdlkMnt;cMeBaHRKb; n ∈ IN mantYTUeTAdUcxageRkam³ 2 n n! = n(n − 1) × ... × 1 Edl ,V = U = n! 2 k>KNna lim UU nig lim VV n

3

n

n

n

n +1

n → +∞

n

n +1

n→ +∞

n

2n + n 3 lim n→ +∞ n!+ n 3

x>KNna dMeNaHRsay k>KNna lim UU nig n +1

n → +∞

eKman

n

Vn+1 n→ +∞ V n lim

2n n3 U n = n , Vn = n! 2

Edl n! = n(n − 1) × ... × 1

- 12 -


CMBUkTI1

eK)an eK)an ehIy eK)an

lImIténsVIút (n + 1)3 n +1 Un + 1 (n + 1)3 1 1 3 2 = = = ( 1 + ) 3 3 Un 2 n n 2n 2n U 1 lim n + 1 = n → +∞ U 2 n 2n + 1 2 Vn + 1 (n + 1)! = = n+1 Vn 2n n! V 2 lim n + 1 = lim =0 n → +∞ n + 1 n → +∞ V n 2n + n 3 lim n→ +∞ n!+ n 3 n3 n 2 (1 + n ) 2n + n 3 2 lim = lim 3 n → +∞ n!+ n n → +∞ n3 n! (1 + ) n! n 2 = lim = +∞ n → +∞ n!

.

x>KNna eK)an

- 13 -


CMBUkTI1

lImIténsVIút

%>KNnalImIténsVIút (a ) EdleKsÁal;tYdUcxageRkam³ k> a = 2 , a = 12 a + 3 x> a = 3 , a = 2a − 5 K> a = 1 , a = 13 a + 43 . dMeNaHRsay KNnalImIténsIVút (a ) k> a = 2 , a = 12 a + 3 KNnatYTUeTAénsIVút a = 12 a + 3 smIkarsmÁal;énsIVútKW r = 12 r + 3 ⇒ r = 6 tagsIVútCMnYy b = a − 6 eK)an b = a − 6 = ⎛⎜⎝ 12 a + 3 ⎟⎞⎠ − 6 = 12 (a − 6) eKTaj b = 12 b enaH (b ) CasIVútFrNImaRtman q = 12 nigtY b = a − 6 = 2 − 6 = −4 eK)an b = −4 × ⎛⎜⎝ 12 ⎞⎟⎠ naM[ a = 6 − 4⎛⎜⎝ 12 ⎞⎟⎠ dUcenH lim a = 6 . n

1

n +1

1

n +1

n

1

n +1

n

n

n

n +1

1

n

n +1

n

n +1

n

n +1

n +1

1

n

n

n

n

n

1

n −1

n −1

n

n → +∞

n

n

- 14 -


CMBUkTI1

lImIténsVIút

x> a = 3 , a = 2a − 5 edaHRsaydUcxagelIeK)an a = 5 − 2 nig K> a = 1 , a = 13 a + 43 . n +1

1

n

n

n

n +1

1

lim an = −∞

n → +∞

n

⎛1⎞ an = 2 − ⎜ ⎟ ⎝ 3⎠

n −1

edaHRsaydUcxageleK)an nig lim a ^>Binitües‘rIxageRkamenH etICaes‘rIrIk rW rYm? x> ∑ 1000(1.055) k> ∑ 3.⎛⎜⎝ 32 ⎞⎟⎠ ∞

n

n → +∞

n

=2

n

n =0

n=0

n+ n ∑ 3 n =1 2n − 1 3 9 27 81 2+ + + + ... 2 8 32 128

2n + 1 ∑ n −1 n =1 2

K> g> dMeNaHRsay sikSaPaBrYm b¤ rIkénes‘rI k> ∑ 3.⎛⎜⎝ 32 ⎞⎟⎠ plbUkedayEpñkrbs;es‘rIKW ∞

.

X> c> ∑ ( ∞

n =1

2n + 1 − 2n − 1 )

n

n=0

n +1

⎛ 3⎞ 1 − ⎜ ⎟ k n ⎛ 3⎞ ⎝ 2⎠ Sn = ∑ 3⎜ ⎟ = 3 × 3 k =0 ⎝ 2 ⎠ 1− 2

- 15 -

⎡ ⎛ 3 ⎞n + 1 ⎤ = −3 ⎢1 − ⎜ ⎟ ⎥ ⎢⎣ ⎝ 2 ⎠ ⎥⎦


CMBUkTI1

eKman

lImIténsVIút ⎡ ⎛ 3 ⎞n + 1 ⎤ lim Sn = −3 lim ⎢1 − ⎜ ⎟ ⎥ = +∞ n → +∞ n → +∞ ⎢⎣ ⎝ 2 ⎠ ⎥⎦ n ∞ ⎛ 3⎞ ∑ 3.⎜ ⎟ n=0 ⎝ 2⎠

Caes‘rIrIk . dUcenH x> ∑ 1000(1.055) Caes‘rIrIkeRBaH r = 1.055 > 1 . K> ∑ 2nn+ −n1 Caes‘rIrYm . ∞

n

n =0 ∞ n =1

3

2n + 1 ∑ n −1 n =1 2 2n + 1 1 ⎞ ⎛ lim n − 1 = lim ⎜ 2 + n −1 ⎟ = 2 ≠ 0 n → +∞ 2 n → +∞ ⎝ 2 ⎠ ∞ 2n + 1 ∑ n −1 n =1 2 3 9 27 81 2+ + + + ... 2 8 32 128 n −1 3 9 ⎛ 3⎞ Sn = 2 + + + ... + 2 ⎜ ⎟ 2 8 ⎝ 4⎠ n ⎛ 3⎞ 1− ⎜ ⎟ n ⎡ 3 4⎠ ⎛ ⎞ ⎤ ⎝ = 2× = 8 ⎢1 − ⎜ ⎟ ⎥ 3 ⎢⎣ ⎝ 4 ⎠ ⎥⎦ 1− 4 ⎡ ⎛ 3 ⎞n ⎤ lim Sn = 8 lim ⎢1 − ⎜ ⎟ = 8⎥ n → +∞ n → +∞ ⎢⎣ ⎝ 4 ⎠ ⎥⎦ ∞

X> eKman dUcenH g>

Caes‘rIrIk .

plbUkedayEpñk

enaHvaCaes‘rIrYm .

eday

- 16 -


CMBUkTI1

lImIténsVIút

c> ∑ ( 2n + 1 − 2n − 1) manplbUkedayEpñk ∞

n =1

S n = ∑ ( 2k + 1 − 2k − 1 ) = 2n + 1 − 1 n

k =1

eday lim S = lim ( 2n + 1 − 1) = +∞ dUcenH ∑ ( 2n + 1 − 2n − 1) Caes‘rIrIk . &>rkplbUkénes‘rIxageRkam³ k>1 + 0.1 + 0.01 + 0.001 + ... x> ∑ 4n 2− 1 n → +∞ ∞

n

n → +∞

n =1

n =1

π⎞ ⎛ ∑ 2⎜ cos ⎟ 3⎠ n=0 ⎝ 2 3 2 ⎛ 2⎞ ⎛ 2⎞ + ⎜ ⎟ + ⎜ ⎟ + ... 3 ⎝ 3⎠ ⎝ 3⎠

X>

K> ∑ (2n − 1)(2n + 1) 1

n =1

5(− 1) ∑ 4n n =1 ∞

2

n

n

g> c> dMeNaHRsay rkplbUkénes‘rI ³ k>1 + 0.1 + 0.01 + 0.001 + ... eyIgeXIjfa 1 , 0.1 , 0.01 , ... Caes‘rIFrNImaRtGnnþEdl a = 1 nig r = 0.1 . - 17 -


CMBUkTI1

lImIténsVIút

dUcenH 1 + 0.1 + 0.01 + 0.001 + ... = 1 −10.1 = 109 . x> ∑ 4n 2− 1 eKman 4k 2− 1 = (2(2kk++11) )(− 2(2kk−−11) ) = 2k1− 1 − 2k1+ 1 eK)an ∞

n =1

2

2

1 ⎞ 1 2n ⎛ 2 ⎞ n ⎛ 1 1 = − = − = ⎜ ⎟ ⎜ ⎟ ∑ 2 ∑ 2k + 1 ⎠ 2n + 1 2n + 1 k = 1 ⎝ 4k − 1 ⎠ k = 1 ⎝ 2k − 1 2n n → +∞ →1 2n + 1 ∞ 2 ∑ 4n 2 − 1 = 1 n =1 ∞ 1 1 = ∑ (2n − 1)(2n + 1) 2 n =1 n

kalNa dUcenH

enaH

K>

π⎞ 2 ⎛ = 2 cos =4 ∑ ⎜⎝ 3 ⎟⎠ π n=0 1 − cos 3 n ∞ 5(− 1) 1 = 5 × =4 ∑ 4n 1 n =1 1+ 4 2 3 2 ⎛ 2⎞ ⎛ 2⎞ 2 1 + ⎜ ⎟ + ⎜ ⎟ + ... = . =2 3 ⎝ 3⎠ ⎝ 3⎠ 3 1− 2 3

X> g> c>

n

- 18 -


CMBUkTI1

lImIténsVIút

*>KNna k> lim ⎛⎜⎝ 1 − 21 ⎟⎞⎠...⎛⎜⎝ 1 − n1 ⎞⎟⎠ n→ +∞

2

2

1 k2 − 1 k − 1 k + 1 1− 2 = = × k k k k2 n 1 ⎞ n ⎛ k − 1⎞ n ⎛ k + 1⎞ ⎛ ∏ ⎜⎝ 1 − k 2 ⎟⎠ = ∏ ⎜⎝ k ⎟⎠ × ∏ ⎜⎝ k ⎟⎠ k=2 k=2 k=2 1 n+1 n+1 = . = n 2 2n n+1 1 1⎞ ⎛ 1⎞ ⎛ = lim ⎜ 1 − 2 ⎟...⎜ 1 − 2 ⎟ = lim n → +∞ ⎝ 2 2 ⎠ ⎝ n ⎠ n → +∞ 2n ⎛ n ⎞ n n ⎟⎟ lim ⎜⎜ 4 ... + + + 4 4 n→ +∞ n +2 n +n⎠ ⎝ n +1

Binitü eK)an

dUcenH >

x> cMeBaHRKb; k = 1 , 2 , ..., n eKman n4 + 1 ≤ n4 + k ≤ n4 + n

n4 + 1 ≤ n4 + k ≤ n4 + n n n n ≤ ≤ n4 + n n4 + k n4 + 1 n n n n n n ≤ ≤ ∑ n4 + n ∑ n4 + k ∑ n4 + 1 k =1 k =1 k =1 2 n n n n2 ≤∑ 4 ≤ 4 n + n k =1 n + k n4 + 1 n2 n2 lim = lim =1 n → +∞ n 4 + n n → +∞ n 4 + 1

eKTaj eday

- 19 -


CMBUkTI1

lImIténsVIút

eKTaj dUcenH (>KNna

n

lim

n

=1

.

n +k n n ⎞ ⎛ n + + + lim ⎜ 4 ... ⎟=1 4 4 n → +∞ ⎝ n + 1 n +2 n +n⎠ n → +∞

[

4

k =1

(

)

(

)]

(

)

(

)]

lim (q − 1) + q 2 q 2 − q + ... + q 2(n − 1 ) qn − qn − 1

n → +∞

1 22

Edl q = . dMeNaHRsay [

lim (q − 1) + q 2 q 2 − q + ... + q 2(n − 1 ) qn − qn − 1

n → +∞

n

q 2 ( k − 1 ) (q k − q k − 1 ) ∑ n → +∞

= lim

k =1 n

= lim

n → +∞

3k − 3 q (q − 1) ∑

k =1

1 − q 3n = lim (q − 1) n → +∞ 1 − q 3 = lim

1−

n → +∞

eRBaH

3n 22

1−

lim

n → +∞

3 22

3n 22

1 (2 2

− 1) = +∞

= +∞

. - 20 -


CMBUkTI1

lImIténsVIút

!0> eK[sVIút (a ) kMNt;eday a = 2 +1q Edl q ≠ −1. sikSalImIténsVIút (a ) kalNa n → +∞ . dMeNaHRsay sikSalImIténsVIút eK)an lim a = lim 2 +1q -ebI q > 1 enaH lim a = lim 2 +1q = 0 1 13 = = lim a lim -ebI q = 1 enaH 2+q -ebI − 1 < q < 1 enaH lim a = lim 2 +1q = 12 -ebI q < −1 enaH lim a = lim 2 +1q = 12 !!>cMeBaHRKb; n ∈ IN eKman n

n

n

n

n → +∞

n

n

n → +∞

n

n → +∞

n → +∞

n

n → +∞

n

n → +∞

n → +∞

n

n

n → +∞

n

n

n → +∞

n → +∞

n

n 2 2 2 2 Sn = + + ... + = ∑ (2n − 1)(2n + 1) p=0 (2p + 1)(2p + 3) 1× 3 3 × 5

k>KNna S CaGnuKmn_én n edayeRbI (2p + 1)(22p + 3) CaTRmg; (2pa+ 1) + (2pb+ 3) . x>KNna lim S . n

n→ +∞

n

- 21 -


CMBUkTI1

lImIténsVIút

dMeNaHRsay k>KNna S CaGnuKmn_én n n

2

(2p + 1)(2p + 3)

=

a b + (2p + 1) (2p + 3)

.

b¤ 2 = a(2p + 3) + b(2p + 1) b¤ 2 = (2a + 2b)p + 3a + b ⎧ 2a + 2b = 0 eKTaj ⎨3a + b = 2 naM[ a = 1 , b = −1 ⎩

eK)an (2p + 1)(2 2p + 3) = 2p1+ 1 − 2p1+ 3 ehIy S

n

n

2 p = 0 ( 2p + 1)( 2p + 3) n ⎛ 1 1 ⎞ = ∑⎜ − ⎟ 2p + 3 ⎠ p = 0 ⎝ 2p + 1

=∑

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

dUcenH . x>KNna lim S eKman S = 1 − 2n1+ 3 dUcenH lim S = lim (1 − 2n1+ 3 ) = 1 . n

n→ +∞

n

n → +∞

n

n → +∞

- 22 -


CMBUkTI1

lImIténsVIút

!@>eKmanes‘rIGnnþmYy

r2 r2 r2 + + ... + + ... r + 2 n −1 2 r + 1 r2 + 1 2 r +1 2

(

)

bgðajfaes‘rIGnnþenaHrYmcMeBaHRKb;témø r . dMeNaHRsay bgðajfaes‘rIGnnþrYmcMeBaHRKb;témø r eKman r + r r+ 1 + r + ... + r 2

2

2

(r

2

2

2

)

+1

n

(r

2

plbUledayEpñkrbs;es‘rIenHKW b¤ S

(

2

+1

)

n −1

+ ...

r2 Sn = ∑ 2 k −1 k = 1 (r + 1) n

n

1 ∑ (r 2 + 1)k −1 k =1 1 1− 2 ⎡ ⎤ 1 (r + 1)n 2 = (r 2 + 1) ⎢1 − 2 =r × n⎥ 1 + ( r 1 ) ⎣ ⎦ 1− 2 r +1 1 →0 r n → +∞ 2 n (r + 1)

=r

)

2

kalNa enaH cMeBaHRKb; . eK)an lim S = r + 1 naM[es‘rIxagelICaes‘rIbRgYm . !#>kMNt;es‘rIxageRkam etIes‘rImYyNarYm mYyNarIk? ebICaes‘rIrYm cUrrkplbUk. x> ∑ ⎛⎜⎝ tan π4 ⎞⎟⎠ k> ∑ 2⎛⎜⎝ cos π3 ⎞⎟⎠ 2

n → +∞

n

n

n =1

n =1

- 23 -

n


CMBUkTI1

lImIténsVIút

K> dMeNaHRsay sikSaPaBrIk rYmrbs;es‘rI

X>

n ∑ n =1 2n + 5

n =1

eRBaHvaCaes‘rIFrNImaRtGnnþman

n

.

π 3 =2 π 1 − cos 3 π 1 | r |= cos = < 1 3 2

k> ∑ 2⎛⎜⎝ cos π3 ⎞⎟⎠ Caes‘rIrYmxireTArk n

5(− 1) ∑ n 4 n =1 ∞

2 cos

.

π⎞ ⎛ tan ⎟ ∑⎜ 4⎠ n =1 ⎝ n π⎞ ⎛ lim an = 1 ≠ 0 an = ⎜ tan ⎟ = 1n = 1 → +∞ n 4 ⎝ ⎠ n ∞ π⎞ ⎛ ∑ ⎜ tan ⎟ 4⎠ n =1 ⎝ ∞ n n an = ∑ 2n + 5 n =1 2n + 5 ∞ n n 1 lim an = lim = ∑ n → +∞ n → +∞ 2n + 5 2 n =1 2n + 5 ∞ 5(− 1)n −5 = −4 ∑ n 1 4 n =1 1+ 4

x>

eday

n

ehIy

dUcenH K>

Caes‘rIBRgIk . eday manlImIt naM[ Caes‘rIBRgIk .

X>

Caes‘rIrYmxiteTArk

eRBaHvaCaes‘rIFrNImaRtGnnmanþ - 24 -

1 1 | r |=| − |= < 1 4 4

.


CMBUkTI1

lImIténsVIút

!$>eKman ΔABC mYymanRklaépÞesμInwg 6 Ékta. sg; ΔA' B' C'; eday A' , B' nig C' CacMnuckNþalénRCug ΔABC . cUrkMNt;plbUk S = S + S + S + ... . dMeNaHRsay kMNt;plbUk S = S + S + S + ... 1

1

2

2

3

A

B'

A' '

C'

C' '

B' B

A'

C

tag a , b , c CaRCug ΔABC EdlmanépÞRkLa S = 6 nigknøHbrimaRt p = a + b2 + c tag S CaRkLaépÞ ΔA' B' C' S CaRkLaépÞ ΔA' ' B' ' C' ' -------------------------------1

2

3

- 25 -

3


CMBUkTI1

lImIténsVIút

eday A ' , B' , C' CacMNuckNþalén BC , CA , AB enaH a b c p ni g B' C' = ; A' C' = , A' B' = p' = 2 2 2 2 eK)an S = p2 ( p2 − a2 )( p2 − b2 )( p2 − 2c ) = 14 S dUcKñaEdreKTaj S = 41 S ; S = 41 S , ..... eK)an S = S + S + ..... = 1S− q = 6 1 = 8 . 2

1

3

2

1

4

3

1

1

1

2

1−

- 26 -

4


CMBUkTI2

edrIevénGnuKmn_

CMBUkTI2

emeronsegçb

edrIevénGnuKmn_

eK[GnuKmn_ f kMNt;nigCab;ehIymanedrIevelI I . 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 ] manedrIevelIcenøaH (a , b ) nig f (a) = f (b) enaHmancMnYn c ∈ (a , b ) mYyy:agticEdl f ' (c) = 0  ebI f CaGnuKmn_Cab;elIcenøaH [a , b ] manedrIevelIcenøaH (a , b ) enaHmancMnYn c ∈ (a , b ) mYyy:agticEdl f ' (c) = f (bb) −− fa(a) . 

- 27 -


CMBUkTI2

edrIevénGnuKmn_

lMhat; !>rk dy , Δy , dy − Δy nig Δdyy énGnumn_xageRkam k> y = x − 3x + 4 cMeBaH x = 3 , Δx = 0.2 x> y = 12 − 5x cMeBaH x = 2 , Δx = 0.07 . @>eRbIDIepr:g;Esül edm,IrktémøRbEhléncMnYnxageRkam³ k> 37 x> 65 K> 26 X> 126 g> 50.4 c> 79.5 q> 62.3 C> 218.6 #> Rkumh‘unplitsmÖar³eRbIR)as;mYy)anTTYlR)ak;cMnUlsrubBIkar lk; smÖar³ x eRKOgEdl[tamGnuKmn_ x R( x ) = 20x − Ki t CamW u n erolEdl 0 ≤ x ≤ 600 . 30 edayeRbIDIepr:g;)a:n;sμantémøRbEhlénkMeNInR)ak;cMNUl ebI smÖar³ lk;ERbRbYlBI 150 eRKOg eTA 160 eRKOg . $> eragcRkplitsmÖar³eRbIR)as;mYy)ancMNayR)ak;srubkñugkar 2

3

3

3

3

2

- 28 -


CMBUkTI2

edrIevénGnuKmn_

plitsmÖar³ x eRKOgEdl[tamGnuKmn_ C( x ) = 930 + 15x + 0.2x Ban;erol . edayeRbIDIepr:g;)a:n;sμantémøRbEhlénkMeNInR)ak;cMNayebIsmÖar³ Edl)anplitekInBI 60 eRKOg eTA 62 eRKOg . %>shRKasplitsmÖar³eGLicRtUnicmYy )ancMNayR)ak;srubkñúg mYyExsRmab;plitsmÖar³ x eRKOgEdl[tamGnuKmn_ C( x ) = 0.1x + 4x + 200 KitCaBan;erol . ehIyshRKas)an TTYlR)ak;cMNUlmkvij[tamGnuKmn_ R( x ) = 54x − 0.3x KitCaBan;erol . k>sresrGnuKmn_R)ak;cMeNj P(x) x>edayeRbIDIepr:g;)a:n;sμantémøRbEhlénkMeNInR)ak;cMeNj ebIbrimaNsmÖar³Edl)anlk;ekInBI 40 eRKOg eTA 44 eRKOg . ^>tamkarGegátrbs;Gñksßiti)an[dwgfa cMnYnRbCaBlrdæenAkñugTIRkug mYyryHeBl t qñaMeTAmuxeTot mankarekIneLIgEdl[tamGnuKmn_ P(t ) = 10(40 + 2t ) − 160t ¬nak;¦ . edayeRbIDIepr:g;EsüledIm,I)a:n;sμankMeNInRbCaBlrdækñúgTIRkugenaH 2

2

2

2

- 29 -


CMBUkTI2

edrIevénGnuKmn_

ebI t ERbRbYlBI 6 eTA 6.25 qñaM . &>)aLúgmYymanragCaEsV‘ . eRbIDIepr:g;EsüledIm,IKNnatémø RbEhlénkMeNInmaD)aLúg ebIeBlRtUvkMedAéf¶)aLúgrIkmaDEdl kaMrbs;vaERbRbYlBI 2m eTA 2.15m . *>eK[GnuKmn¾ f manedrIevelI (− 2 , ∞ ) Edl f (x) = x + 2 . k>rktMélGmén f (x) cMeBaHRKb; x ∈ [− 1 , 2]. x>bgðajfa cMeBaHRKb; x ∈ [− 1 , 2] eK)an 14 x + 54 ≤ x + 2 ≤ 12 x + 32 . (>eK[GnuKmn¾ f kMNt;elI ⎡⎣⎢ 0 , π2 ⎞⎟⎠ Edl f (x) = tan x . bgðajcMeBaHRKb;cMnYnBit a nig b Edl 0 ≤ a ≤ b ≤ π2 b−a b−a ≤ tan b − tan a ≤ eK)an cos . a cos b !0>eK[GnuKmn¾ f kMNt;elIcenøaH I . eRbIRTwsþIbTr:Ul ¬ebIGac¦ rkRKb;témø c kñúgcenøaH I Edl f ' (c) = 0 ³ k> f (x) = x − 4x , c ∈ (− 2 , 2) x> f (x) = (x − 1)(x − 2)(x − 3) , c ∈ (1 , 3) 2

2

3

- 30 -


CMBUkTI2

edrIevénGnuKmn_

x 2 − 2x − 3 f (x ) = , c ∈ (− 1 , 3 ) x+2 x2 − 1 f (x ) = , c ∈ (− 1 , 1) x ⎛π π⎞ f (x ) = sin 2x , c ∈ ⎜ , ⎟ ⎝ 6 3⎠ x πx f (x ) = − sin , c ∈ (− 1 , 0 ) 2 6

K> X> g> . c> !!>eK[GnuKmn¾ f kMNt;elIcenøaH I . eRbIRTwsþIbTtémømFüm rkRKb;témø c ∈ (a , b) Edl f ' (c) = f (bb) −− fa(a) ³ k> f (x) = x , c ∈ (− 2 , 1) x> f (x) = x(x − x − 2) , c ∈ (− 1 , 1) K> f (x) = x , c ∈ (0 , 1) X> f (x) = x x+ 1 , c ∈ ⎛⎜⎝ − 12 , 2 ⎞⎟⎠ !@>shRKasplitsmÖar³eGLicRtUnicmYy)ancMnayR)ak;srubkñúg mYyéf¶sMrab;plitsmÖar³ x eRKOgEdl[GnuKmn¾ C(x ) = 2400 + 28x + 0.02x KitCaBan;erol. k>kMnt;R)ak;cMNaysrubkñúgkarplitsmÖar³ 10 eRKOg 20 eRKOg nig 30 eRKOg. 2

2

3

2

- 31 -


CMBUkTI2

edrIevénGnuKmn_

x>)a:n;sμantémøRbEhlénR)ak;cMNaykñúgkarplitsmÖar³eRKOgTI 11 eRKOgTI 21 nig eRKOgTI 31 . !#>eragBum<e)aHBum<TsSnavdþImYy)ancMNaysrubkñúgmYyExsRmab; e)aHBum<TsSnavdIþ x c,ab; Edl[tamGnuKmn¾ C(x ) = 0.0001x + x + 465 KitCaBan;erol ehIyeragBum<)anlk; ecjvij TsSnavdIþ 1 c,ab;éfø P = D(x) = 4 − 0.0002x Ban;erol. k>sresrGnuKmn¾R)ak;cMnUlsrub R(x). x> sresrGnuKmn¾R)ak;cMenjsrub P(x). K>KNnaR)ak;cMenjsrubebIkñúg mYyExeragBum<lk;Gs; 3000 c,ab; 3500 c,ab; nig 4000 c,ab;. X>>)a:n;sμantémøRbEhlénR)ak;cMenjEdl)anBIkarlk; TsSnavdIþc,ab;TI 3001 c,ab;TI 3501 nigc,ab;TI 4001. !$> shRKasplitsmÖar³eRbIR)as;mYy)ancMnayR)ak;srubkñúgkar plitsmÖar³ x eRKOg [tamGnuKmn¾ C(x) = 480 + 26x + −0.1x Ban;erol. k>kMNt;GnUKmn¾R)ak;cMNaymFüm C(x). 2

2

- 32 -


CMBUkTI2

edrIevénGnuKmn_

x>KNnaR)ak;cMNaymFümbEnßm kalNa x = 30 , x = 50 , x = 70 . !%> shRKasplitsmÖar³eGLicRtUnicmYy)ancMnayR)ak;srubkñúg karplitsmÖar³ x eRKOg Edl[GnuKmn¾ C(x ) = 1080 + 42x + 0.3x Ban;erol. kMNt;brimaNsmÖar³Edl shRKasRtUvplitedIm,I[R)ak;cMNaymFümmankMritGb,brma ebI 0 ≤ x ≤ 90 . !^>Rkumh‘unplitsmÖar³eRbIR)as;mYy)ancMNaysrubkñúgkarplit smÖar³ x eRKOg[yamGnuKmn¾ C(x) = x + 20x + 1050 Ban;erol ehIyRkumh‘un)anlk;ecjvijTTYl)anR)ak;cMNUlsrub [tamGnuKmn¾ R(x) = 140x − 0.5x Ban;erol. kMNt;tMrit brimaNsmÖar³EdlRkumh‘unRtUvplitniglk;edIm,I[Rkumh‘unTTYl)an R)ak;cMenjGtibrima ebI 0 ≤ x ≤ 70 . 2

2

2

- 33 -


CMBUkTI2

edrIevénGnuKmn_

dMeNaHRsay !>rk dy , Δy , dy − Δy nig Δdyy énGnumn_xageRkam k> y = x − 3x + 4 cMeBaH x = 3 , Δx = 0.2 x> y = 12 − 5x cMeBaH x = 2 , Δx = 0.07 . dMeNaHRsay rk dy , Δy , dy − Δy nig Δdyy énGnumn_xageRkam k> y = x − 3x + 4 cMeBaH x = 3 , Δx = 0.2 eK)an dy = y'.dx = (2x − 3).dx eday x = 3 , Δx = 0.2 ≈ dx eK)an dy = (2 × 2 − 3)(0.2) = 0.2 . ehIy Δy = f (x + Δx) − f (x) 2

2

= f ( 3 + 0.2) − f ( 3) = f ( 3.2) − f ( 3)

[

] [

]

= ( 3.2) 2 − 3( 3.2) + 4 − ( 3) 2 − 3( 3) + 4 = (5.2)(0.2) − 3(0.2) = ( 2.2)(0.2) = 0.044 .2 ehIy dy − Δy = 0.2 − 0.044 = 0.156 nig Δdyy = 0.0044 = 4.54 . - 34 -


CMBUkTI2

edrIevénGnuKmn_

x> y = 12 − 5x cMeBaH x = 2 , Δx = 0.07 . eK)an dy = y'.dx = − 2 125− 5x .dx 5 (0.07 ) 2 12 − 10 0.35 =− = −0.123 2 2

=−

ehIy Δy = f (2 + 0.07) − f (2)

= 12 − 5( 2.07) − 12 − 5( 2) = 1.65 − 2 = 1.284 − 1.414 = −0.13 dy = 0.946 dy − Δy = −0.123 + 0.13 = 0.007 Δy

ehIy nig @>eRbIDIepr:g;Esül edIm,IrktémøRbEhléncMnYnxageRkam³ k> 37 x> 65 K> 26 X> 126 g> 50.4 c> 79.5 q> 62.3 C> 218.6 dMeNaHRsay eRbIDIepr:g;Esül edIm,IrktémøRbEhléncMnYnxageRkam³ k> 37 3

3

3

3

- 35 -

.


CMBUkTI2

edrIevénGnuKmn_

eKman f (x + Δx) = f (x) + f ' (x).dx tagGnuKmn_ f (x) = x ⇒ f ' (x) = 2 1 x yk x = 36 ; Δx = 1 = dx eK)an 37 = 36 + 2 136 × 1 = 6 + 121 = 6.08 . x> 65 eKman f (x + Δx) = f (x) + f ' (x).dx tagGnuKmn_ f (x) = x ⇒ f ' (x) = 2 1 x yk x = 64 ; Δx = 1 = dx 1 1 eK)an 65 = 64 + 2 64 × 1 = 8 + 16 = 8.06 . K> 26 eKman f (x + Δx) = f (x) + f ' (x).dx tagGnuKmn_ f (x) = x = x ⇒ f ' (x) = 1 3

3

1 3

3 3 x2

yk x = 27 ; Δx = −1 = dx eK)an 26 = 27 − 1 = 3 − 271 = 2.96 . 3

3

3 3 27 2

- 36 -


CMBUkTI2

edrIevénGnuKmn_

X> 126 eKman f (x + Δx) = f (x) + f ' (x).dx tagGnuKmn_ f (x) = x = x ⇒ f ' (x) = 3

1 3

3

yk x = 125 ; Δx = 1 = dx eK)an 125 = 125 + 1 3

3

3

3 125

g> c> q> C>

50.4 = 7.099 79.5 = 8.916 3

62.3 = 3.964

3

218.6 = 6.023

- 37 -

2

= 5+

1 3 3 x2

1 = 5.013 75

.


CMBUkTI2

edrIevénGnuKmn_

#> Rkumh‘unplitsmÖar³eRbIR)as;mYy)anTTYlR)ak;cMnUlsrubBIkar lk; smÖar³ x eRKOgEdl[tamGnuKmn_ x R( x ) = 20x − KitCamWunerolEdl 0 ≤ x ≤ 600 . 30 edayeRbIDIepr:g;)a:n;sμantémøRbEhlénkMeNInR)ak;cMNUl ebI smÖar³ lk;ERbRbYlBI 150 eRKOg eTA 160 eRKOg . dMeNaHRsay edayeRbIDIepr:g;)a:n;sμantémøRbEhlénkMeNInR)ak;cMNUl tag y = R(x) = 20x − x30 ebI smÖar³lk;ERbRbYlBI 150 eRKOg eTA 160 eRKOgenaH Δx = 10 eK)an dR(x) = R' (x).dx Et R' (x) = 20 − 15x enaH dR(x) = (20 − 150 ).10 = 100 ¬KitCamWunerol¦ . 15 $> eragcRkplitsmÖar³eRbIR)as;mYy)ancMNayR)ak;srubkñugkar plitsmÖar³ x eRKOgEdl[tamGnuKmn_ C( x ) = 930 + 15x + 0.2x Ban;erol . 2

2

2

- 38 -


CMBUkTI2

edrIevénGnuKmn_

edayeRbIDIepr:g;)a:n;sμantémøRbEhlénkMeNInR)ak;cMNayebIsmÖar³ Edl)anplitekInBI 60 eRKOg eTA 62 eRKOg . dMeNaHRsay edayeRbIDIepr:g;)a:n;sμantémøRbEhlénkMeNInR)ak;cMNay ebIsmÖar³Edl)anplitekInBI 60 eRKOg eTA 62 eRKOgenaH Δx = 2 . eday C(x) = 930 + 15x + 0.2x enaH C' (x) = 15 + 0.4x eK)an dC(x) = C' (x).dx = (15 + 0.4 × 60)(2) = 78 ¬Ban;erol¦ %>shRKasplitsmÖar³eGLicRtUnicmYy )ancMNayR)ak;srubkñúg mYyExsRmab;plitsmÖar³ x eRKOgEdl[tamGnuKmn_ C( x ) = 0.1x + 4x + 200 KitCaBan;erol . ehIyshRKas)an TTYlR)ak;cMNUlmkvij[tamGnuKmn_ R( x ) = 54x − 0.3x KitCaBan;erol . k>sresrGnuKmn_R)ak;cMeNj P(x) x>edayeRbIDIepr:g;)a:n;sμantémøRbEhlénkMeNInR)ak;cMeNj ebIbrimaNsmÖar³Edl)anlk;ekInBI 40 eRKOg eTA 44 eRKOg . 2

2

2

- 39 -


CMBUkTI2

edrIevénGnuKmn_

dMeNaHRsay k>sresrGnuKmn_R)ak;cMeNj P(x) eK)an P(x) = R(x) − C(x) P( x ) = 54x − 0.3x 2 − 0.1x 2 − 4x − 200

dUcenH P(x) = −0.4x + 50x − 200 . x>edayeRbIDIepr:g;)a:n;sμantémøRbEhlénkMeNInR)ak;cMeNj ebIbrimaNsmÖar³Edl)anlk;ekInBI 40 eRKOg eTA 44 eRKOg enaH Δx = 44 − 40 = 4 eRKÓg . eK)an dP(x) = P' (x).dx eday P' (x) = −0.8x + 50 eK)an dP(x) = (−0.8 × 40 + 50)(4) = 72 ¬KitCaBan;erol¦ ^>tamkarGegátrbs;Gñksßiti)an[dwgfa cMnYnRbCaBlrdæenAkñugTIRkug mYyryHeBl t qñaMeTAmuxeTot mankarekIneLIgEdl[tamGnuKmn_ P(t ) = 10(40 + 2t ) − 160t ¬nak;¦ . edayeRbIDIepr:g;EsüledIm,I)a:n;sμankMeNInRbCaBlrdækñúgTIRkugenaH ebI t ERbRbYlBI 6 eTA 6.25 qñaM . 2

2

- 40 -


CMBUkTI2

edrIevénGnuKmn_

dMeNaHRsay edayeRbIDIepr:g;EsüledIm,I)a:n;sμankMeNInRbCaBlrdækñúgTIRkugenaH ebI t ERbRbYlBI 6 eTA 6.25 qñaMenaH Δt = 6.25 − 6 = 0.25 qñaM eKman P(t ) = 10(40 + 2t ) − 160t eK)an P' (t) = 40(40 + 2t ) − 160 ehIy dP(t ) = P' (t ).dt = [40(40 + 2 × 6) − 160] (0.25) = 480 ¬nak;¦ &>)aLúgmYymanragCaEsV‘ . eRbIDIepr:g;EsüledIm,IKNnatémø RbEhlénkMeNInmaD)aLúg ebIeBlRtUvkMedAéf¶)aLúgrIkmaDEdl kaMrbs;vaERbRbYlBI 2m eTA 2.15m . dMeNaHRsay eRbIDIepr:g;EsüledIm,IKNnatémøRbEhlénkMeNInmaD)aLúg tag V(r) CamaDrbs;)aLúg ehIy r CakaMrbs;)aLúg eKman V(r ) = 43π r ⇒ V' (r ) = 4πr eday r ERbRbYlBI 2m eTA 2.15m enaH Δr = 0.15m ehIy dV(r) = V' (r).dr 2

3

2

- 41 -


CMBUkTI2

edrIevénGnuKmn_ = (4 × 3.14 × 2 2 )(0.15) = 7.536m 3

*>eK[GnuKmn¾ f manedrIevelI (− 2 , ∞ ) Edl f (x) = k>rktMélGmén f ' (x) cMeBaHRKb; x ∈ [− 1 , 2]. x>bgðajfa cMeBaHRKb; x ∈ [− 1 , 2] eK)an 14 x + 54 ≤ x + 2 ≤ 12 x + 32 . dMeNaHRsay k>rktMélGmén f (x) cMeBaHRKb; x ∈ [− 1 , 2] f ( x) = x + 2 ⇒ f ' ( x) =

1 2 x+2

x+2

.

eKman − 1 ≤ x ≤ 2 ⇒ 1 ≤ x + 2 ≤ 4 b¤ 1 ≤ x + 2 ≤ 2 eKTaj 14 ≤ 2 x1+ 2 ≤ 12 dUcenH 14 ≤ f ' (x) ≤ 12 . x>bgðajfa 14 x + 54 ≤ x + 2 ≤ 12 x + 32 cMeBaHRKb; x ∈ [− 1 , 2] tamvismPaBkMeNInmankMNt;cMeBaHGnuKmn_ f kñúg x ∈ [− 1 , 2] eK)an 14 (x + 1) ≤ f (x) − f (−1) ≤ 12 (x + 1) Et f (−1) = 1 1 1 1 1 x+ ≤ x+ 2 −1≤ x+ 4 4 2 2 - 42 -


CMBUkTI2

edrIevénGnuKmn_

dUcenH 14 x + 54 ≤ x + 2 ≤ 12 x + 32 . ⎡ π⎞ (>eK[GnuKmn¾ f kMNt;elI ⎣⎢ 0 , 2 ⎟⎠ Edl f (x) = tan x . bgðajcMeBaHRKb;cMnYnBit a nig b Edl 0 ≤ a ≤ b < π2 b−a b−a ≤ tan b − tan a ≤ eK)an cos . a cos b dMeNaHRsay bgðajcMeBaHRKb;cMnYnBit a nig b Edl 0 ≤ a ≤ b ≤ π2 b−a b−a ≤ tan b − tan a ≤ eK)an cos a cos b eKman f (x) = tan x ⇒ f ' (x) = cos1 x cMeBaH 0 ≤ a ≤ b < π2 ebI a ≤ x ≤ b enaH cos b ≤ cos x ≤ cos a b¤ cos1 a ≤ f ' (x) ≤ cos1 b edayGnuvtþn_vismPaBkMeNInmankMNt;eTAnwgGnuKmn_ f cMeBaH x ∈ [a , b ] eK)an 2

2

2

2

2

2

2

1 1 ( b − a ) ≤ f ( b ) − f ( a ) ≤ (b − a ) 2 2 cos a cos b b−a b−a tan b tan a ≤ − ≤ cos 2 a cos 2 b

dUcenH

.

- 43 -


CMBUkTI2

edrIevénGnuKmn_

!0>eK[GnuKmn¾ f kMNt;elIcenøaH I . eRbIRTwsþIbTr:Ul ¬ebIGac¦ rkRKb;témø c kñúgcenøaH I Edl f ' (c) = 0 ³ k> f (x) = x − 4x , c ∈ (− 2 , 2) x> f (x) = (x − 1)(x − 2)(x − 3) , c ∈ (1 , 3) K> f (x) = x −x +2x2− 3 , c ∈ (− 1 , 3) x −1 ( ) X> f x = x , c ∈ (− 1 , 1) g> f (x) = sin 2x , c ∈ ⎛⎜⎝ π6 , π3 ⎞⎟⎠ c> f (x) = x2 − sin π6x , c ∈ (− 1 , 0). dMeNaHRsay rkRKb;témø c kñúgcenøaH I Edl f ' (c) = 0 ³ k> f (x) = x − 4x , c ∈ (− 2 , 2) eKman f (x) CaGnuKmn_BhuFaCab;RKb; x ∈ (−2 , 2) ehIy f ( −2) = f ( 2) = 0 . tamRTwsþIbTr:UlmancMnYn c ∈ ( −2 , 2 ) Edl f ' (c) = 0 . eKman f ' (c) = 3c − 4 3

2

2

3

2

- 44 -


CMBUkTI2

edrIevénGnuKmn_

ebI f ' (c) = 0 ⇒ 3c − 4 = 0 ⇒ c = − 2 3 3 , c = 2 3 3 x> f (x) = (x − 1)(x − 2)(x − 3) , c ∈ (1 , 3) eKman f (x) CaGnuKmn_BhuFaCab;RKb; x ∈ (1 , 3) ehIy f (1) = f ( 3) = 0 . tamRTwsþIbTr:UlmancMnYn c ∈ (1 , 3 ) Edl f ' (c) = 0 . eKman f (x) = (x − 1)(x − 2)(x − 3) = x − 6x + 11x − 6 eK)an f ' (x) = 3x − 12x + 11 ebI f ' (c) = 0 ⇒ 3c − 12c + 11 = 0 6− 3 6+ 3 . Δ ' = 36 − 33 = 3 > 0 ⇒ c = ,c = 3 3 K> f (x) = x −x +2x2− 3 , c ∈ (− 1 , 3) eKman f (x) CaGnuKmn_BhuFaCab;RKb; x ∈ (−1 , 3) ehIy f ( −1) = f ( 3) = 0 . tamRTwsþIbTr:UlmancMnYn c ∈ (1 , 3 ) Edl f ' (c) = 0 . eKman f ' (x) = (2x − 2)(x +(x2+) −2)(x − 2x − 3) 2

1

2

3

2

2

2

1

2

2

2

2

- 45 -


CMBUkTI2

edrIevénGnuKmn_ 2x 2 + 4x − 2x − 4 − x 2 + 2x + 3 = ( x + 2)2 x 2 + 4x − 1 = ( x + 2)2

ebI

c 2 + 4c − 1 f ' (c ) = =0 2 (c + 2 )

naM[ c

2

+ 4c − 1 = 0

Δ' = 4 + 1 = 5 > 0 ⇒ c1 = 2 + 5 , c 2 = 2 − 5

dUcenH c = 2 + 5 , c = 2 − 5 . !!>eK[GnuKmn¾ f kMNt;elIcenøaH I . eRbIRTwsþIbTtémømFüm rkRKb;témø c ∈ (a , b) Edl f ' (c) = f (bb) −− fa(a) ³ k> f (x) = x , c ∈ (− 2 , 1) x> f (x) = x(x − x − 2) , c ∈ (− 1 , 1) K> f (x) = x , c ∈ (0 , 1) X> f (x) = x x+ 1 , c ∈ ⎛⎜⎝ − 12 , 2 ⎞⎟⎠ dMeNaHRsay rkRKb;témø c ∈ (a , b) Edl f ' (c) = f (bb) −− fa(a) ³ k> f (x) = x , c ∈ (− 2 , 1) eKman f ' (x) = 2x 1

2

2

2

3

2

- 46 -


CMBUkTI2

edrIevénGnuKmn_

eK)an f ' (c) = 2c = f (11)−−(f−(2−)2) = 11 +− 42 = −1 eKTaj c = − 12 . x> f (x) = x(x − x − 2) , c ∈ (− 1 , 1) eKman f (x) = x − x − 2x ⇒ f ' (x) = 3x − 2x − 2 eK)an f ' (c) = 3c − 2c − 2 = f (11)−−(f−(1−)1) = − 22− 0 = −1 eKTaj 3c − 2c − 1 = 0 ⇒ c = − 13 , c = 1 eday c ∈ (−1 , 1 ) dUcenH c = − 13 . K> f (x) = x , c ∈ (0 , 1) eKman f ' (x) = 3x eK)an f ' (c) = 3c = f (11) −− f0(0) = 1 ⇒ c = ± 33 eday c ∈ (0 , 1 ) dUcenH c = 33 . X> f (x) = x x+ 1 , c ∈ ⎛⎜⎝ − 12 , 2 ⎞⎟⎠ eKman f ' (x) = (x +11) 2

3

2

2

2

2

1

3

2

2

2

- 47 -

2


CMBUkTI2

eK)an

edrIevénGnuKmn_ 2 1 +1 2 − − ) f ( 2 ) f ( 1 3 2 = = = f ' (c ) = 2 5 1 3 (c + 1) 2 − (− ) 2 2 3 3 c1 = −1 + , c 2 = −1 − 2 2

. eKTaj !@>shRKasplitsmÖar³eGLicRtUnicmYy)ancMnayR)ak;srubkñúg mYyéf¶sMrab;plitsmÖar³ x eRKOgEdl[GnuKmn¾ C(x ) = 2400 + 28x + 0.02x KitCaBan;erol. k>kMnt;R)ak;cMNaysrubkñúgkarplitsmÖar³ 10 eRKOg 20 eRKOg nig 30 eRKOg. x>)a:n;sμantémøRbEhlénR)ak;cMNaykñúgkarplitsmÖar³eRKOg TI 11 eRKOgTI 21 nig eRKOgTI 31 . dMeNaHRsay 2

k>kMnt;R)ak;cMNaysrubkñúgkarplitsmÖar³ 10 eRKOg 20 eRKOg nig 30 eRKOg eKman C(x) = 2400 + 28x + 0.02x 2 ¬KitCaBan;erol¦ -R)ak;cMNaysrubkñúgkarplitsmÖar³ 10 eRKOgKW C(10) = 2400 + 280 + 2 = 2682 Ban;erol . - 48 -


CMBUkTI2

edrIevénGnuKmn_

-R)ak;cMNaysrubkñúgkarplitsmÖar³ 20 eRKOgKW C( 20) = 2400 + 560 + 8 = 2968 Ban;erol . -R)ak;cMNaysrubkñúgkarplitsmÖar³ 30 eRKOgKW C(10) = 2400 + 540 + 18 = 2958 Ban;erol .

x>)a:n;sμantémøRbEhlénR)ak;cMNaykñúgkarplitsmÖar³eRKOg TI 11 eRKOgTI 21 nig eRKOgTI 31 eKman C(x) = 2400 + 28x + 0.02x ¬KitCaBan;erol¦ 2

eK)an C' (x) = 28 + 0.04x

-témøRbEhlénR)ak;cMNaykñúgkarplitsmÖar³eRKOgTI 11 KW C' (11) = 28 + 0.04 × 11 = 28.44 Ban;erol. -témøRbEhlénR)ak;cMNaykñúgkarplitsmÖar³eRKOgTI 21 KW C' ( 21) = 28 + 0.04 × 21 = 28.84 Ban;erol . -témøRbEhlénR)ak;cMNaykñúgkarplitsmÖar³eRKOgTI 31 KW C' ( 31) = 28 + 0.04 × 31 = 29.24 Ban;erol. !#>eragBum<e)aHBum<TsSnavdþImYy)ancMNaysrubkñúgmYyExsRmab; e)aHBum<TsSnavdIþ x c,ab; Edl[tamGnuKmn¾ C(x ) = 0.0001x + x + 465 KitCaBan;erol ehIyeragBum<)anlk; 2

- 49 -


CMBUkTI2

edrIevénGnuKmn_

ecjvij TsSnavdIþ 1 c,ab;éfø P = D(x) = 4 − 0.0002x Ban;erol. k>sresrGnuKmn¾R)ak;cMnUlsrub R(x). x> sresrGnuKmn¾R)ak;cMenjsrub P(x). K>KNnaR)ak;cMenjsrubebIkñúg mYyExeragBum<lk;Gs; 3000 c,ab; 3500 c,ab; nig 4000 c,ab;. X>>)a:n;sμantémøRbEhlénR)ak;cMenjEdl)anBIkarlk; TsSnavdIþc,ab;TI 3001 c,ab;TI 3501 nigc,ab;TI 4001. dMeNaHRsay k>sresrGnuKmn¾R)ak;cMnUlsrub R(x) eK)an R(x) = P × x = (4 − 0.0002x)x = 4x − 0.0002x x> sresrGnuKmn¾R)ak;cMenjsrub P(x) eK)an P(x) = R(x) − C(x) 2

= 4x − 0.0002x 2 − 0.0001x 2 − x − 465 = −0.0003x 2 + 3x − 465

K>KNnaR)ak;cMenjsrubebIkñúg mYyExeragBum<lk;Gs; 3000 c,ab; 3500 c,ab; nig 4000 c,ab; -ebI x = 3000 c,ab; - 50 -


CMBUkTI2

edrIevénGnuKmn_

enaH P(3000) = −0.0003(3000) + 3(3000) − 465 = 5835 Ban;erol . 2

-ebI x = 3500 c,ab; enaH P(3500) = −0.0003(3500) + 3(3500) − 465 = 6360 Ban;erol . -ebI x = 4000 c,ab; enaH P(4000) = −0.0003(4000) + 3(4000) − 465 = 6735 Ban;erol . X>)a:n;sμantémøRbEhlénR)ak;cMenjEdl)anBIkarlk; TsSnavdIþc,ab;TI 3001 c,ab;TI 3501 nigc,ab;TI 4001. eKman P(x) = −0.0003x + 3x − 465 eK)an P' (x) = −0.0006x + 3 -témøRbEhlénR)ak;cMenj)anBIkarlk; TsSnavdIþc,ab;TI 3001 KW P' (3000) = −0.0006(3000) + 3 = 1.2 Ban;erol . -témøRbEhlénR)ak;cMenj)anBIkarlk; TsSnavdIþc,ab;TI 3501 2

2

2

- 51 -


CMBUkTI2

edrIevénGnuKmn_

KW P' (3500) = −0.0006(3500) + 3 = 0.9 Ban;erol . -témøRbEhlénR)ak;cMenj)anBIkarlk; TsSnavdIþc,ab;TI 4001 KW P' (4000) = −0.0006(4000) + 3 = 0.6 Ban;erol .

- 52 -


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

CMBUkTI3 emeronTI1

GnuKmn¾GsniTan >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 Δ = 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 . 2

2

2

2

- 52 -


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

k-ebI x → −∞ enaH y = − a(x + 2ba ) CaGasIumtUteRTt -krNI a < 0 RkabénGnuKmn_ y = ax + bx + c KμanGasIumtUteT . 2ax + b edrIev y' = mansBaØadUc 2ax + b 2

2 ax 2 + bx + c

-ebI a < 0 GnuKmn_manGtibrmamYyRtg; x = − 2ba . -ebI a > 0 GnuKmn_manGb,brmamYyRtg; x = − 2ba .

- 53 -


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

lMhat; !>rkGasIumtUteRTtrbs;GnuKmn¾ y = 2x + 3 − 4x + x + 1 . @>rkedrIevénGnuKmn¾xageRkam³ k> y = (2x − 3) x − 3x + 4 x> y = x + 6x + 5 + x2 + 3 . #>sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = 3x − 2 + x − 1 x> y = x + x − 3x + 2 $>k>sikSaGefrPaB nig sg;Rkab (C) énGnuKmn¾ y = 9 − x . x>rkcMnucnwg EdlbnÞat; d : mx − y + 3 − 4m = 0 kat;tamcMeBaHRKb;tMél m . K>eRbIRkab (C) BiPakSatamtémø m GtßiPaBénb¤srbs;smIkar 9 − x − mx + 4m − 3 = 0 . 2

2

2

2

2

2

m

2

%>eK[GnuKmn¾ y = mx + m

2

+ x2 + 1

- 54 -

.


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

k>RsaybBa¢ak;fa GasIumtUteRTtxagsþaMrbs;RkabxagelI b:Hnwg )a:ra:bUlnwgmYy. x>ebI m = 1 sikSaGefrPaBnigsg;Rkab C rbs;GnuKmn¾xagelI. ^> eK[GnuKmn¾ y = 2x(4 − x) . k>sikSaGefrPaB nig sg;Rkab C rbs;GnuKmn¾. x>eRbIRkab C BiPakSatamtémø m GtßiPaBénb¤srbs;smIkar 2x(4x − x ) = mx + 2 2 − 5m . &>k>kMNt;témø m edIm,I[smIkar x + 2x + 1 = m manb¤s. x>kMNt;témø m edIm,I[vismIkar x + 2x + 1 < m manb¤s. *> eK[GnuKmn¾ y = f (x) = x + 4x + 2x + 1 . k>sikSaGefrPaB nig sg;RkabénGnuKmn¾. x>eRbIRkabrktémø m edIm,I[vismIkar 4x + 2x + 1 ≤ m − x manb¤s. (>eK[GnuKmn¾ y = f (x) = x2 + 12 12 − 3x . k>sikSaGefrPaB nig sg;Rkab C énGnuKmn¾. x>RsaybBa¢ak; − 2 ≤ x 12 − 3x ≤ 4 . 2

2

2

2

2

2

- 55 -


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

K>edaHRsaysmIkar 12 − 3x = 4 − x rYcepÞopÞat;lT§plén smIkaredayRkabénGnuKmn¾xagelI. !0> eK[GnuKmn¾ y = x + 2x + 1 manRkab C . k>rkGasIumtUteRTtrbs;Rkab C . x>tamRkabrktémø m edIm,I[smIkar x + 2x + 1 = m manb¤s. !!>KUbmYy ABCDEFGH manrgVas;RCug a . I CacMnuckNþalén [AB] ehIy J CacMnuckNþalén [EH ]. cMnuc M mYyrt;enAelIRCug énKUb. rkcMgayxøIbMputEdl M rt;BI I eTA J . 2

2

2

- 56 -


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

dMeNaHRsay !>rkGasIumtUteRTtrbs;GnuKmn¾ y = 2x + 3 − 4x + x + 1 . dMeNaHRsay rkGasIumtUteRTt eKman y = 2x + 3 − 4x + x + 1 eK)an y = 2x + 3 − 4 | x + 18 | + ε(x) Edl lim ε(x) = 0 -ebI x → +∞ 1 11 enaH y = 2x + 3 − 2(x + 8 ) = 4 CaGasIumtUtedkrbs;Rkab . -ebI x → −∞ enaH y = 2x + 3 + 2(x + 18 ) = 4x + 134 CaGasIumtUteRTt . @>rkedrIevénGnuKmn¾xageRkam³ k> y = (2x − 3) x − 3x + 4 x> y = x + 6x + 5 + x2 + 3 . 2

2

x→ ±∞

2

2

2

- 57 -


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

dMeNaHRsay rkedrIevénGnuKmn¾xageRkam³ k> y = (2x − 3) x − 3x + 4 eK)an y' = (2x − 3)' x − 3x + 4 + ( 2

2

= 2 x − 3x + 4 +

( 2x − 3) 2

2

2 x 2 − 3x + 4

4x 2 − 12x + 16 + 4x 2 − 12x + 9

=

2 x 2 − 3x + 4 8x 2 − 24x + 25

=

x>

x 2 − 3x + 4 )' ( 2x − 3)

2 x 2 − 3x + 4 x2 2 y = x + 6x + 5 + +3 2 2x + 6 y' = +x 2 2 x + 6x + 5 x+3 = +x 2 x + 6x + 5

eK)an

=

dUcenH y' =

x + 3 + x x 2 + 6x + 5 x 2 + 6x + 5 x + 3 + x x 2 + 6x + 5 x + 6x + 5 2

- 58 -

.


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

#>sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = 3x − 2 + x − 1 x> y = x + x − 3x + 2 dMeNaHRsay sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = 3x − 2 + x − 1 EdnkMNt; D = [1 , + ∞ ) TisedAGefrPaB -edrIev y' = 3 + 2 x1 − 1 -cMeBaHRKb; x > 1 eK)an y' > 0 naM[ y CaGnuKmn_ekIn. KNnalImIt ³ lim (3x − 2 + x − 1) = +∞ . taragGefrPaB 2

x → +∞

+∞

x 1 +

y'

+∞

y

0

- 59 -


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

sMNg;Rkab y 6 5

C : y = 3x − 2 + x − 1

4 3 2 1

-2

-1

0

1

2

3

4

5

6

7

x

-1

x> y = x +

x 2 − 3x + 2 y 4

3

2

1

-2

-1

0

1

2

-1

- 60 -

3

4

5x


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

$>k>sikSaGefrPaB nig sg;Rkab (C) énGnuKmn¾ y = 9 − x . x>rkcMnucnwg EdlbnÞat; d : mx − y + 3 − 4m = 0 kat;tamcMeBaHRKb;tMél m . K>eRbIRkab (C) BiPakSatamtémø m GtßiPaBénb¤srbs;smIkar 2

m

9 − x 2 − mx + 4m − 3 = 0

dMeNaHRsay k>sikSaGefrPaB nig sg;Rkab (C) énGnuKmn¾ y = EdnkMNt; D = [−3 , 3 ] TisedAGefrPaB y' =

− 2x

2 9 − x2

=−

9 − x2

x 9 − x2

ebI y' = 0 ⇒ x = 0 GnuKmn_manGtibrmaeFobRtg; x = 0 KW y = (0) = 3 taragGefrPaB x

−3

0 +

y'

3

y - 61 -

3


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

sMNg;Rkab y 5 4 3

I

2 1

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

x

-1

x>rkcMnucnwg EdlbnÞat; d : mx − y + 3 − 4m = 0 kat;tamcMeBaHRKb;tMél m eKGacsresr y − 3 = m(x − 4) smIkarenHepÞógpÞat;Canic©RKb; m smmUl x = 4 , y = 3 dUcenH I(4 ; 3 ) CacMnucnwg . K>eRbIRkab (C) BiPakSatamtémø m GtßiPaBénb¤srbs;smIkar 9 − x − mx + 4m − 3 = 0 CasmIkarGab;sIuscMNucRbsBV rvag (C) : y = 9 − x nwg d : y = mx − 4m + 3 tamRkahVikeK)an ³ m

2

2

m

- 62 -


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

-cMeBaH m = 0 smIkarmanb¤sDúb x = x = 0 . -cMeBaH m = 73 smIkarmanb¤sEtmYyKt;KW x = −3 . -cMeBaH m = 3 smIkarmanb¤sEtmYyKt;KW x = 3 . 3 -cMeBaH m ∈ ( 7 , 3 ) smIkarmanb¤sEtmYyKt;KW − 3 < x < 3 -cMeBaH m > 3 b¤ m < 0 smIkarKμanb¤s . %>eK[GnuKmn¾ y = mx + m + x + 1 . k>RsaybBa¢ak;fa GasIumtUteRTtxagsþaMrbs;RkabxagelI b:Hnwg )a:ra:bUlngmY w y. eKman y = mx + m + x + 1 y = mx + m + | x | + ε( x ) Edl lim ε( x ) = 0 eKTaj y = mx + m + x = (m + 1)x + m CaGasIumtUtxag sþaMrbs;Rkabtag y = mx + m + x + 1 . yk P : y = ax + bx + c Ca)a:ra:bUlnwgRtUvrk smIkarGab;sIus ax + bx + c = (m + 1)x + m b¤ ax + (b − m − 1)x + c − m = 0 (E) smIkar (E) manb¤sDubRKb; m luHRtaEt Δ = 0 ∀m ∈ IR 1

2

2

2

2

2

2

x→ ±∞

2

2

2

2

2

2

2

2

2

- 63 -


CMBUkTI3

GefrPaBnigRkabénGnuKmn¾

Δ = (b − m − 1) 2 − 4a(c − m 2 ) Δ = b 2 − 2b(m + 1) + (m + 1) 2 − 4ac + 4am 2 = b 2 − 2mb − 2b + m 2 + 2m + 1 − 4ac + 4am 2 = (1 + 4a )m 2 + ( 2 − 2b )m + (b − 1) 2 − 4ac 1 ⎧ = − a ⎪ 4 ⎪ Δ = 0 ∀m ∈ IR ⎨b = 1 ⎪c = 0 ⎪ ⎩ x2 +x P:y = − 4

ebI

smmUl

eK)an . x>ebI m = 1 sikSaGefrPaBnigsg;Rkab C rbs;GnuKmn¾xagelI cMeBaH m = 1 eK)an y = x + 1 + x + 1 2

y

C : y = x + 1 + x2 + 1

1 0

1

- 64 -

x


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

>cMNucsMxan;²sRmab;sikSaGnuKmn_RtIekaNmaRt -EdnkMNt; -xYbénHnuKmn_ -PaBKUessénGnuKmn_ -TisedAGefrPaBénGnuKmn_ >xYbénGnuKmn_ -xYbénGnuKmn_ y = sin(ax) KW |2πa | -xYbénGnuKmn_ y = cos(ax) KW |2πa | >PaBKUessénGnuKmn_ -GnuKmn_ f (x) CaGnuKmn_esselI I kalNa ∀x ∈ I , − x ∈ I ehIy f (− x) = −f (x) . -GnuKmn_ f (x) CaGnuKmn_KUelI I kalNa ∀x ∈ I , − x ∈ I ehIy f (− x) = f (x) . - 65 -


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

lMhat; !>sikSaGefrPaB nig sg;RkabénGnuKmn¾ y = cos . sin 2x . @> sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = 2 sin x − 3 cos x x> y = 5 sin x + cos x . #>sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = cos 2x − cos 3x x> y = 2x cos 2x − sin x . $>sikSaGefrPaB nig sg;RkabénGnuKmn¾ k> y = 2 sin x − sin 2x x> y = cos x + x sin x . %> sikSaGefrPaB nig sg;RkabénGnuKmn¾ k> y = 1 −cossinx x x> y = cossinx x− 1 . ^> rkbrimaénGnuKmn¾ y = 2 +sincosx x elIcenøaH [0 , 2]. &>eRbIRkabén y = sin 2x − 3 sin x edaHRsaysmIkar sin 2x − 3 sin x = 0 ebI − 2π ≤ x ≤ 2π . *> eK[GnuKmn¾ y = a sin 2x + sin x . sikSaGefrPaB nig sg;RkabcMeBaH a = 1. 2

2

- 66 -


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

(>eK[GnuKmn¾ y = a sin xa −coscosx x − 1 . sikSaGefrPaB nig sg;RkabcMeBaH a = 1. lMhat;CMBUk 3 !> sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = 3x − 5 x> y = 7 − 4x . @> sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = 2 + x − 2x − 3 x> y = 2 + 8 − 2x . #> sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = 2x − 1 − 3x − 5 x> y = x + 1 − 4 − x . $> sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ x−1 k> y = f (x) = sin x + 2 sin x2 x> y = f (x) = 2cos . cos x + 1 %> sikSaGefrPaB nig sg;RkabénGnuKmn¾ 2

2

2

y = f (x ) = 3 sin x + cos x

^>eK[GnuKmn¾ y = −2x + m x + 1 k>sikSaGefrPaB nig sg;RkabcMeBaH m = 4 . x>rktémø m edIm,I[GnuKmn¾Kμantémøbrma. 2

- 67 -


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

dMeNaHRsay !>sikSaGefrPaB nig sg;RkabénGnuKmn¾ y = cos dMeNaHRsay sikSaGefrPaB nig sg;Rkab

2

x. sin 2x

y = cos 2 x. sin 2x

>EdnkMNt; D = IR >xYb ³ eKman y = cos x sin 2x = 12 (1 + cos 2x) sin 2x GnuKmn_manxYb p = π eRBaH f (x + π) = 12 [1 + cos(2π + 2x)]sin( 2π + 2x) 2

1 = (1 + cos 2x ) sin 2x = f ( x ) 2

>PaBKUess ³ f ( x ) CaGnuKmn_esseRBaH f ( − x ) = cos ( − x ) sin( − x ) 2

= − cos 2 x sin x = −f ( x)

dUcenHeKsikSaEtkñúgcenøaH [0 , π2 ] - 68 -

.


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

>TisedAGefrPaB edrIev f ' (x) = 12 (1 + cos 2x)' sin 2x + 12 (sin 2x)' (1 + cos 2x) = − sin 2 2x + cos 2x(1 + cos 2x )

= −1 + cos 2 2x + cos 2x + cos 2 2x = 2 cos 2 2x + cos 2x − 1 = ( 2 cos 2x − 1)(cos 2x + 1)

edayRKb; x ∈ IR : − 1 ≤ cos 2x ≤ 1 enaH cos 2x + 1 ≥ 0 naM[ f ' (x) mansBaØadUc 2 cos 2x − 1 -ebI 2 cos 2x − 1 = 0 ⇒ cos 2x = 12 b¤ 2x = π3 ⇒ x = π6 -ebI 2 cos 2x − 1 > 0 ⇒ cos 2x > 12 b¤ 0 < x < π6 -ebI 2 cos 2x − 1 < 0 ⇒ cos 2x < 12 b¤ π6 < x < π2 taragGefrPaB x

π 6

0

π 2

+

f ' (x)

π f( ) 6

f ( x)

0

0 - 69 -


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH π (0 , ) 2 π x= 2 π x= 2

π 3 3 f( ) = 6 8 π f( ) = 0 2

kñúgcenøaH GnuKmn_manGtibrma . eK)an f ' ( π2 ) = 0 nig . cMeBaH Rkab (c) manGkS½ (ox) CabnÞat;b:H . dUcenHRtg; y

1

0

- 70 -

(c) : y = cos 2 x sin 2x 1

x


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

@> sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = 2 sin x − 3 cos x x> y = 5 sin x + cos x . dMeNaHRsay sikSaGefrPaB nig sg;Rkab k> y = 2 sin x − 3 cos x y

1

0

1

x

x> y = 5 sin x + cos x y

1 0

1

x

- 71 -


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

#>sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = cos 2x − cos 3x x> y = 2x cos 2x − sin x . dMeNaHRsay k> y = cos 2x − cos 3x y

1

0

1

x

x> y = 2x cos 2x − sin x y

1 0

1

x

- 72 -


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

$>sikSaGefrPaB nig sg;RkabénGnuKmn¾ k> y = 2 sin x − sin 2x x> y = cos dMeNaHRsay k> y = 2 sin x − sin 2x

2

x + x sin x

.

y

1

0

x> y = cos

2

1

x

x + x sin x y

1 0

1

- 73 -

x


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

%> sikSaGefrPaB nig sg;RkabénGnuKmn¾ x> y = cossinx x− 1 . k> y = 1 −cossinx x dMeNaHRsay k> y = 1 −cossinx x y

1 0

1

x

x> y = cossinx x− 1 y

1 0

1

- 74 -

x


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

^> eK[GnuKmn¾ y = a sin 2x + sin x . sikSaGefrPaB nig sg;RkabcMeBaH a = 1. dMeNaHRsay sg;RkabcMeBaH a = 1 eK)an y = sin 2x + sin x y

1

0

1

x

&>eK[GnuKmn¾ y = a sin xa −coscosx x − 1 . sikSaGefrPaB nig sg;RkabcMeBaH a = 1. dMeNaHRsay sg;RkabcMeBaH a = 1 sin x − 1 − cos x − 1 = −1 + eK)an y = sin x cos x cos x - 75 -


CMBUkTI3

GnuKmn他RtIekaNmaRtcRmuH y

1 0

1

x

- 76 -


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

lMhat;CMBUk3 nigdMeNaHRsay !> sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = 3x − 5 x> y = 7 − 4x . dMeNaHRsay sikSaGefrPaB nig sg;RkabénGnuKmn_ k> y = 3x − 5 >EdnkMNt; D = [ 53 , + ∞) >TisedAGefrPaB edrIev y' = 2 33x − 5 cMeBaH x > 53 eK)an y' > 0 naM[vaCaGnuKmn_ekInelI ( 53 , + ∞ ) lImIt lim 3x − 5 = +∞ taragGefrPaB y

x → +∞

x

5 3

1

+

y' y

+∞

+∞

0 - 77 -

0

1

x


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

x> y = 7 − 4x >EdnkMNt; D = (−∞ , 74 ] >TisedAGefrPaB edrIev y' = 7−−24x cMeBaH x < 74 eK)an y' < 0 naM[vaCaGnuKmn_cuHelI (−∞ , 74 ) lImIt lim 7 − 4x = +∞ taragGefrPaB x → −∞

x

7 4

−∞

y' y

+∞

0 y

1

0

1

- 78 -

x


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

@> sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = 2 + x − 2x − 3 x> y = 2 + 8 − 2x . dMeNaHRsay sikSaGefrPaB nig sg;RkabénGnuKmn_ k> y = 2 + x − 2x − 3 2

2

2

y

y=2 1

0

x> y = 2 +

1

x

8 − 2x 2 y

y=2 1

0

1

x

- 79 -


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

#> sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ k> y = 2x − 1 − 3x − 5 x> y = x + 1 − dMeNaHRsay sikSaGefrPaB nig sg;RkabénGnuKmn_ k> y = 2x − 1 − 3x − 5

4 − x2

.

y

1

0

1

x> y = x + 1 −

x

4 − x2 y

1

0

1

x

- 80 -


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

$> sikSaGefrPaB nig sg;RkabénGnuKmn¾xageRkam³ x−1 . k> y = f (x) = sin x + 2 sin x2 x> y = f (x) = 2cos cos x + 1 dMeNaHRsay sikSaGefrPaB nig sg;RkabénGnuKmn_ k> y = f (x) = sin x + 2 sin x2 y

1 0

1

x

x−1 x> y = f (x) = 2cos cos x + 1 y

1 0

1

x

- 81 -


CMBUkTI3

GnuKmn¾RtIekaNmaRtcRmuH

%> sikSaGefrPaB nig sg;RkabénGnuKmn¾ y = f (x ) = 3 sin x + cos x

dMeNaHRsay sikSaGefrPaB nig sg;RkabénGnuKmn¾ y = f (x ) = 3 sin x + cos x y

1

0

1

x

- 82 -


CMBUkTI4

GaMgetRkalkMNt;

CMBUkTI4 emeronTI1

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

a

>épÞRkLaénEpñkbøg; -ebIGnuKmn_ y = f (x) Cab;elIcenøaH [a , b ] enaHépÞRkLaénEpñkbøg; EdlxNÐedayExSekag GkS½Gab;sIus bnÞat;Qr x = a , x = b kMNt;eday S = ∫ f (x).dx ebI f (x) ≥ 0 b

a

y C : y = f (x)

A

S a

B b

- 83 -

x


CMBUkTI4

GaMgetRkalkMNt;

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

y

a

a

b

x

S

(C) : y = f ( x ) B

A

-ebI f nig g CaGnuKmn_Cab;elI [a , b ] enaHeK)anépÞRkLaenAcenøaH ExSekagtagGnuKmn_TaMgBIrkMNt;eday S = ∫ [f (x) − g(x)].dx Edl f (x) ≥ g(x) RKb; x ∈ [a , b ] . b

a

y

A

B

D

C

a

b - 84 -

x


CMBUkTI4

GaMgetRkalkMNt;

lMhat; !>edayeRbIniymn½yKNnaGaMetRkalxageRkam³ k> ∫ 3xdx x> ∫ 4xdx K> ∫ x dx X> ∫ (x − 5x)dx g> ∫ x dx . @>KNnatémøRbEhlénépÞRklaEdlxN½ÐedayRkabtag y = f (x) nigGk½S x' ox elIcenøaH [a , b] Edl³ k> f (x) = 9 − x Edl [a , b] = [− 3 , 2] , n = 5 x> f (x) = x +1 2 Edl [a , b] = [− 1 , 3] , n = 4 #>KNnaRklaépÞxN½ÐedayRkabnigGk½SGab;sIuselIcenaøHEdl[³ k> f (x) = x Edl x ∈ [1 , 3] x> f (x) = x + 2x − 3 Edl x ∈ [1 , 3] K> f (x) = 2 − x Edl x ∈ [− 3 , − 2] X> f (x) = 2x3+ 1 Edl x ∈ [0 , 2] . 2

4

0

2

2

2

2

0

2

0

2

1

2

2

2

3

- 85 -

2


CMBUkTI4

GaMgetRkalkMNt;

$>KNnaRklaépÞxN½ÐedayRkabtagGnuKmn¾TaMgBIr³ k> f (x) = x + 2 nig g(x) = x , x ∈ [2 , 5] x> f (x) = x nig g(x) = x , x ∈ [0 , 1] K> f (x) = 2x + 1 nig g(x) = 3x + 2 , x ∈ [0 , 2] X> f (x) = e nig g(x) = x , x ∈ [1 , 4] %> KNnaRklaépÞxN½ÐedayExSekagtagGnuKmn¾ x = y nig y = x − 2 . ^> KNnaRklaépÞxN½ÐedayExSekagtagGnuKmn¾ y = f (x ) nig y = g(x ) = 2 − x nig x' ox . &> KNnaRklaépÞxN½ÐedayExSekagtagGnuKmn¾ 1 ni g y = g (x ) = e ni g x ∈ [0 , 4]. y = f (x ) = x+1 2

2

3

x −1

2

0.7 x

- 86 -


CMBUkTI4

GaMgetRkalkMNt;

dMeNaHRsay !>edayeRbIniymn½yKNnaGaMgetRkalxageRkam³ k> ∫ 3xdx x> ∫ 4xdx K> ∫ x dx X> ∫ (x − 5x)dx g> ∫ x dx . dMeNaHRsay ⎡ 3x ⎤ k> ∫ 3xdx = ⎢ 2 ⎥ = 6 − 0 = 6 2

4

0

2

2

2

2

0

2

0

2

2

1

2

2

2

0

0

x> ∫ 4xdx = [2x ] = 32 − 8 = 24 K> ∫ x dx = ⎡⎢⎣ 13 x ⎤⎥⎦ = 83 4

2 4 2

2

2

2

2

3

0

0

X> ∫ ( 2

0

g> ∫

2

1

)

2

5 ⎤ 8 20 22 ⎡1 x 2 − 5x dx = ⎢ x 3 − x 2 ⎥ = ( − ) = − 2 ⎦0 3 2 3 ⎣3 2

8 1 7 ⎡1 ⎤ x dx = ⎢ x 3 ⎥ = − = ⎣3 ⎦ 1 3 3 3 2

- 87 -

.


CMBUkTI4

GaMgetRkalkMNt;

@>KNnaRklaépÞxN½ÐedayRkabnigGk½SGab;sIuselIcenaøHEdl[³ k> f (x) = x Edl x ∈ [1 , 3] x> f (x) = x + 2x − 3 Edl x ∈ [1 , 3] K> f (x) = 2 − x Edl x ∈ [− 3 , − 2] X> f (x) = 2x3+ 1 Edl x ∈ [0 , 2] . dMeNaHRsay KNnaRklaépÞ k> f (x) = x Edl x ∈ [1 , 3] eK)an S = ∫ x dx 2

2

3

2

y

3

2

1

3

⎡1 ⎤ = ⎢ x3 ⎥ ⎣3 ⎦ 1

27 1 26 − = 3 3 3 26 S= 3 =

dUcenH

1 0

¬ÉktaépÞ¦

- 88 -

1

x


CMBUkTI4

GaMgetRkalkMNt;

x> f (x) = x eyIg)an

2

+ 2x − 3

Edl x ∈ [1 , 3]

y

3

S = ∫ ( x 2 + 2x − 3).dx 1 3

⎤ ⎡1 = ⎢ x 3 + x 2 − 3x ⎥ ⎦1 ⎣3 32 5 = 9 − (− ) = 3 3 32 S= 3

1 0

ÉktaépÞ

dUcenH

K> f (x) = 2 − x Edl x ∈ [− 3 , − 2] eK)an S = ∫ (2 − x ).dx 3

−2

3

−3

−2

⎡ x4 ⎤ = ⎢ 2x − ⎥ 4 ⎦ −3 ⎣ = ( − 4 − 4 ) − ( −6 − = −2 +

81 ) 4

81 73 = 4 4

- 89 -

1

x


CMBUkTI4

GaMgetRkalkMNt;

X> f (x) = 2x3+ 1 Edl x ∈ [0 , 2] . eK)an y

S=

2

3dx ∫ 2x + 1 0

⎤ ⎡1 = 3 ⎢ ln | 2x + 1 |⎥ ⎦ ⎣2 =

2

0

1

3 ln 5 2

0

1

dUcenH S = 32 ln 5 ÉktaépÞ #>KNnaRklaépÞxN½ÐedayRkabtagGnuKmn¾TaMgBIr³ k> f (x) = x + 2 nig g(x) = x , x ∈ [2 , 5] x> f (x) = x nig g(x) = x , x ∈ [0 , 1] K> f (x) = 2x + 1 nig g(x) = 3x + 2 , x ∈ [0 , 2] X> f (x) = e nig g(x) = x , x ∈ [1 , 4] 2

2

3

x −1

- 90 -

x


CMBUkTI4

GaMgetRkalkMNt;

dMeNaHRsay KNnaRklaépÞ k> f (x) = x + 2 nig g(x) = x eyIg)an 2

, x ∈ [2 , 5] y

5

S = ∫ ( x 2 + 2 − x ).dx 2 5

1 ⎤ ⎡1 = ⎢ x 3 + 2x − x 2 ⎥ 2 ⎦2 ⎣3 235 14 69 = − = 6 3 2

dUcenH S = 34.5 ÉktaépÞ x> f (x) = x nig g(x) = x eK)an 2

S=

∫ (x 1

2

3

1 0

1

x

, x ∈ [0 , 1] y

)

− x 3 .dx 1

0 1

1 ⎤ ⎡1 = ⎢ x3 − x4 ⎥ 4 ⎦0 ⎣3 =

1 1 1 − = 3 4 12 1 S= 12

dUcenH

0

ÉktaépÞ - 91 -

1

x


CMBUkTI4

GaMgetRkalkMNt;

K> f (x) =

2x + 1

nig g(x) = 3x + 2

, x ∈ [0 , 2]

y

1 0

1

eyIg)an S = ∫ (3x + 2 − 2

x

2x + 1 ) .dx

0

2

3 ⎡3 2 ⎤ 1 2 = ⎢ x + 2x − ( 2x + 1) ⎥ 3 ⎢⎣ 2 ⎥⎦ 0

1 125 ) − (0 − ) 3 3 31 − 5 5 = 3 31 − 5 5 S= 3 = (10 −

dUcenH

.

- 92 -


CMBUkTI4

GaMgetRkalkMNt;

X> f (x) = e nig g(x) = x x −1

, x ∈ [1 , 4]

y

1 0

1

x

eK)an 4

S = ∫ (e x−1 − x ).dx 1 4

1 ⎤ ⎡ = ⎢ e x −1 − x 2 ⎥ 2 ⎦1 ⎣

1 = (e 3 − 8) − (1 − ) 2 17 = e3 − 2 2e 3 − 17 S= 2

dUcenH

. - 93 -


CMBUkTI4

GaMgetRkalkMNt;

$> KNnaRklaépÞxN½ÐedayExSekagtagGnuKmn¾ x = y nig y = x − 2 . dMeNaHRsay KNnaRklaépÞ 2

y

y

1

0

1

1

x

0

1

eKman y = x − 2 naM[ x = y + 2 eK)an S = ∫ (y + 2 − y ).dy 2

2

−1

2

1 ⎤ ⎡1 = ⎢ y 2 + 2y − y 3 ⎥ 3 ⎦ −1 ⎣2 20 − 3 + 12 − 2 9 8 1 1 = (6 − ) − ( − 2 + ) = = 3 2 3 6 2

- 94 -

x


CMBUkTI4

GaMgetRkalkMNt;

%> KNnaRklaépÞxN½ÐedayExSekagtagGnuKmn¾ 1 y = f (x ) = ni g y = g (x ) = e ni g x ∈ [0 , 4]. x+1 dMeNaHRsay KNnaRklaépÞ 0.7 x

y

1 0

4

S = ∫ (e 0.7 x − 0

1

x

1 ).dx x+1 4

⎡ 1 0.7 x ⎤ =⎢ e − ln( x + 1)⎥ = 20.45 ⎣ 0.7 ⎦0

dUcenH S = 20.45 ÉktaépÞ . - 95 -


CMBUkTI4

GaMgetRkalkMNt;

CMBUkTI4 emeronTI2

maRtsUlItnigRbEvgFñÚ 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 ).Δx] = π ∫ f (x).dx . maDénsUlIdbrivtþkMNt;)anBIrgVilCMuvijGkS½ (ox ) énépÞxNÐeday Rkab y = f (x) nig y = g(x) elIcenøaH [a , b ]Edl f (x) ≥ g(x) kMNt;eday V = π ∫ [f (x) − g (x)].dx .



b

n

2

n→ +∞

2

k

k =1

a

b

2

2

a x

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

a

b

m

a

b



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

- 96 -

1 + f '2 ( x ) .dx


CMBUkTI4

GaMgetRkalkMNt;

lMhat; !>k>KNnamaDsUlItbrivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRklaEdlxNÐedayRkab tagGnuKmn¾ y = 2x + 1 Gk½S x' ox bnÞat;Qr x = 1 nig x = 3 . x>KNnamaDsUlItbrivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRklaEdlxNÐedayRkab tagGnuKmn¾ y = x + 1 Gk½S x' ox bnÞat;Qr x = 0 nig x = 3 . K>KNnamaDsUlItbrivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRklaEdlxNÐedayRkab tagGnuKmn¾ y = x − 3 Gk½S x' ox bnÞat;Qr x = 4 eTA x = 9 . @>k>KNnamaDsUlItbrivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRklaEdlxNÐedayRkab tagGnuKmn¾ f (x) = x nig g(x) = 4x − x . x>KNnamaDsUlItbrivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRklaEdlxNÐedayRkab tagGnuKmn¾ f (x) = x − 2x + 3 nig g(x) = 9 − x . 2

2

2

2

- 97 -


CMBUkTI4

GaMgetRkalkMNt;

#>cMeBaHGnuKmn¾ F(x) = ∫ sin πtdt . KNna x> F' (0) k> F(− 1) K> F' ⎛⎜⎝ 12 ⎞⎟⎠ X> F" (x). $>épÞRkla A xNÐedayRkabtagGnuKmn¾ g(t ) = 4 − t4 nig 4⎞ ⎛ ( ) [ ] Gk½SGab;sIuselIcenaøH 1 , x kMNt;eday A x = ∫ ⎜⎝ 4 − t ⎟⎠dt . k>KNna A CaGnuKmn¾én x . etIRkabGnuKmn¾ A mansmIkar GasIumtUtedk rW eT? x>rksmIkarGasIumtUteRTténRkabtagGnuKmn¾ g . %> KNna F' (x) ebI k> F(x) = ∫ (4t + 1)dt x> F(x) = ∫ t dt K> F(x) = ∫ t dt X> F(x) = ∫ t1 dt g> F(x) = ∫ sin t dt c> F(x) = ∫ 2 1+ t dt . ^>k>bgðajfaebI f (t ) CaGnuKmn¾ess enaH F(x) = ∫ f (t ) dt RKb; x ∈ IR CaGnuKmn¾essEdrrWeT? x

1

2

x

2

1

x+ 2

x

x

−x

3

sin x

x2

0

2

2

x3

0

0

sin x

x

a

- 98 -


CMBUkTI4

GaMgetRkalkMNt;

&>eKe)aHdMusársUkULa)anTTYlvik½yb½RtcMnYn 1200 erogral; 30 éf¶mþg. sársUkULaRtUv)anlk;bnþ[Gñklk;rayeday GRtaefrnig x Caéf¶bnÞab;BITTYlvik½yb½Rtmkdl; ehIybBa¢ITUTat; kMNt;eday I(x) = 1200 − 40x . KNnamFümRbcaMéf¶énkarTUTat;. KNnamFümRbcaMéf¶énlk;sársUkULa ebIsársUkULamYyRKb;éfø 300 erol. *>k>KNnaRbEvgFñÚénRkabtagGnuKmn¾ y = x3 + 41x BI x = 1 eTA x = 3 . x>]bmafa f (x) = 12 (e + e ). bnÞat;CYbGk½SGredaenRtg; B ehIyb:HnigRkabtag f Rtg;cMnuc A(a , f (a)) Edl a > 0 . eRbobeFobRbEvgénGgÁt; AB nigRbEvgFñÚrénRkabtag f enAbenøaHbnÞat;Qr x = 0 nig x = a . (>rképÞRkla S énEsV‘rEdlmankaM r . 3

x

−x

- 99 -


CMBUkTI4

GaMgetRkalkMNt;

lMhat;CMBUk4 !>KNnaGaMgetRkalxageRkam³ k> ∫ (4 − x )(2 + x) dx K> ∫ x −xx − 2 dx g> ∫ cos mx cos nxdx q> ∫ 3 +sincos2x x dx @>KNna k> ∫ e cos 2t dt (a ≠ 0) x dx K> ∫ e +e +2 g> ∫ dx 2

n

2

2

2

1

0

2

π

0

π 2 0

π

2

− at

2

0

ln 6

−x

0

2

x

2

1

0

4 − x2

x> ∫ x − 6xx + 8 dx X ∫ x −x1 dx x c> ∫ 1 + 3x dx C> ∫ x cos xdx . 8

2

6

3

1

1

2

0

π 2 0

2

2

x> ∫ sin(πxln x) dx X> ∫ (4 − x − 1 − x )dx c> ∫ 2ax − x dx e

1

1

2

2

0

a

2

0

q> ∫ (x + ) dx C> ∫ x −2xx + 1 dx #>KNnamaDsUlItbivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRkla xNÐedayRkaPictagGnuKmn¾ f (x) = x + 6 nig g(x) = x elIcenøaH x ∈[− 2 , 3] . a

0

2

3 2 2 a

1

0

2

2

- 100 -


CMBUkTI4

GaMgetRkalkMNt;

$> KNnamaDsUlItbivtþkMNt;)anBIrgVilénépÞRklaxNÐedayRkaPic tagGnuKmn¾ y = 3 − x nig Gk½S x' ox , (− 1 ≤ x ≤ 2) CMuvijGk½S x' ox . %> KNnamaDsUlItbivtþkMNt;)anBIrgVilcMnYn 360 CMuvijGk½S Gab;sIusénépÞRklaxNÐedayRkaPictagGnuKmn¾ y = 1 − x nig Gk½S x' ox , x ∈[− 1 , 1] . ^> KNnamaDsUlItbivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRkla xNÐedayRkaPictagGnuKmn¾ f (x) = 2x nig g(x) = 4x − x . &> KNnamaDsUlItbivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRkla xNÐedayRkaPictagGnuKmn¾ f (x) = 2 − x nig g(x) = x elIcenøaH x ∈[0 , 1]. *> KNnamaDsUlItbivtþkMNt;)anBIrgVil cMnYn 180 CMuvijGk½S x' ox énépÞRklaxNÐedayRkaPictagGnuKmn¾ f (x) = 8 − x nig g (x ) = x . (>KNnaépÞRklaEdlxNÐedayRkabtagGnuKmn¾ y = sin π2 x nig y=x . 0

2

2

2

2

0

2

2

4

- 101 -


CMBUkTI4

GaMgetRkalkMNt;

!0> KNnaépÞRklaEdlxNÐedayRkabtagGnuKmn¾ y = sin x nig 5π π y = cos x enAcenøaH x = nig x = . 4 4 !!> KNnaépÞRklaEdlxNÐedayRkab C nig C GnuKmn¾ y = sin x , (0 ≤ x ≤ π ) nig y = sin 3x , (0 ≤ x ≤ π ) !@>BinitüRkab y = sin 2x nig y = cos x Edl 0 ≤ x ≤ π . k>rkkUGredaencMNucRbsBVénRkabTaMgBIr. x>rképÞRkla S EdlxNÐedayRkabTaMgBIr. !#> C CaRkabtag f (x) = ln(x + 1) nig L CabnÞat;b:Hnwg C Edl manemKuNR)ab;TisesμInwg 12 . k>rksmIkarbnÞat;b:H L . x>KNnamaDsUlItbivtþkMNt;)anBIrgVilCMuvijGk½S x' ox én épÞRklaxNÐedayRkab C , L nig Gk½SGredaen. !$>tag C CaRkab y = e nig L CabnÞat;b:Hnwg C Rtg;cMnuc P(1 , e ) . k> rksmIkarbnÞat; L . x>rképÞRkla S EdlxNÐedayRkab C bnÞat; L nigGk½SGredaen. 3

3

1

2 x −1

- 102 -

2


CMBUkTI4

GaMgetRkalkMNt;

!%> tag C CaRkab y = xe , L , L CabnÞat;b:H C EdlKUs ecjBIcMnuc ⎛⎜⎝ 12 , 0 ⎞⎟⎠ . k>rksmIkarbnÞat;b:H L , L . x>rképÞRkla S EdlxNÐedayRkab C nig bnÞat;b:HEdlmankñúg sMnYrTI !. x

1

1

2

- 103 -

2


CMBUkTI4

GaMgetRkalkMNt;

dMeNaHRsay !>k>KNnamaDsUlItbrivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRklaEdlxNÐedayRkab tagGnuKmn¾ y = 2x + 1 Gk½S x' ox bnÞat;Qr x = 1 nig x = 3 . x>KNnamaDsUlItbrivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRklaEdlxNÐedayRkab tagGnuKmn¾ y = x + 1 Gk½S x' ox bnÞat;Qr x = 0 nig x = 3 . K>KNnamaDsUlItbrivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRklaEdlxNÐedayRkab tagGnuKmn¾ y = x − 3 Gk½S x' ox bnÞat;Qr x = 4 eTA x = 9 . dMeNaHRsay k>KNnamaDsUlIt 2

y 7 6

3

V = π ∫ ( 2x + 1)2 .dx 1

[

4

]

3 π ( 2x + 1)3 1 6 158π π = ( 343 − 27 ) = 6 3

=

5

3 2 1

-3

¬ ÉktamaD ¦ .

-2

-1

0 -1

- 104 -

1

2

3

4

5

6 x


CMBUkTI4

GaMgetRkalkMNt;

x>KNnamaDsUlItbrivtþ

y

3

V = π ∫ ( x 2 + 1)2dx 0 3

= π ∫ ( x4 + 2x 2 + 1)dx 0 3

2 ⎡1 ⎤ = π ⎢ x4 + x 3 + x⎥ 3 ⎣4 ⎦0 =

1 0

1

x

165π 4

dUcenH V = 1654 π ¬ÉktamaD ¦ . K>KNnamaDsUlItbrivtþ y 2 1

-2

-1

0

1

2

3

4

5

6

7

8

9

10 x

-1 -2 -3

eyIg)an

9

9

V = π ∫ ( x − 3) dx = π ∫ ( x − 6 x + 9)dx 2

4

4

9

3 ⎡ x2 ⎤ 3π 2 = π ⎢ − 4x + 9x ⎥ = ⎢⎣ 2 ⎥⎦ 4 2

- 105 -

¬ÉktamaD ¦ .


CMBUkTI4

GaMgetRkalkMNt;

@>k>KNnamaDsUlItbrivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRklaEdlxNÐedayRkab tagGnuKmn¾ f (x) = x nig g(x) = 4x − x . x>KNnamaDsUlItbrivtþkMNt;)anBIrgVilCMuvijGk½S x' ox énépÞRklaEdlxNÐedayRkab tagGnuKmn¾ f (x ) = x − 2x + 3 nig g(x ) = 9 − x . dMeNaHRsay k>KNnamaDsUlItbrivtþ 2

2

2

y 4

3

2

1

-2

-1

0

1

2

3

4

5

-1

eyIg)an V = π ∫ [(4x − x 2

]

) − ( x 2 ) 2 . dx

2 2

0

- 106 -

x


CMBUkTI4

GaMgetRkalkMNt; 2

V = π ∫ (16x 2 − 8x 3 )dx 0

⎡ 16 3 ⎤ V = π ⎢ x − 2x 4 ⎥ ⎣3 ⎦ 32π V= 3

2

=

32π 3

0

dUcenH ¬ÉktamaD ¦ . x>KNnamaDsUlItbrivtþ y 13 12 11 10 9 8 7 6 5 4 3 2 1 -5 -4 -3 -2 -1 0 -1

1

2

3

4

5

6

7

8

9 10 11 12x

-2

eK)an V = π ∫ [(9 − x) − (x − 2x + 3) ]dx bnÞab;BIKNnaeK)an V = 250π ¬ÉktamaD ¦ . 3

2

2

−2

- 107 -

2


CMBUkTI4

GaMgetRkalkMNt;

#>cMeBaHGnuKmn¾ F(x) = ∫ sin πtdt . KNna x> F' (0) k> F(− 1) K> F' ⎛⎜⎝ 12 ⎞⎟⎠ X> F" (x). dMeNaHRsay eKman F(x) = ∫ sin πtdt x

1

x

1

x

1 ⎤ cos πt ⎥ π ⎦1

⎡ = ⎢− ⎣ 1 1 = − cos πx − π π

k> F(− 1) eK)an F(−1) = − π1 (−1) − π1 = 0 . x> F' (0) eKman F' (x) = sin πx eK)an F' (0) = 0 K> F' ⎛⎜⎝ 12 ⎞⎟⎠ π 1 eK)an F' ( 2 ) = sin 2 = 1 . X> F" (x) = π cos πx . - 108 -


CMBUkTI4

GaMgetRkalkMNt;

$>épÞRkla A xNÐedayRkabtagGnuKmn¾ g(t ) = 4 − t4 nig Gk½SGab;sIuselIcenaøH[1 , x] kMNt;eday 4⎞ ⎛ A(x ) = ∫ ⎜ 4 − ⎟dt . t ⎠ ⎝ KNna A CaGnuKmn¾én x . etIRkabGnuKmn¾ A mansmIkar GasIumtUtedk rW eT? dMeNaHRsay k>KNna A CaGnuKmn¾én x 2

x

2

1

x⎛

4⎞ − 4 ∫1 ⎜⎝ t 2 ⎟⎠.dt x 4⎤ 4 ⎡ = ⎢4t + ⎥ = 4x + − 8 t⎦1 x ⎣ 4 A( x ) = 4x − 8 + x

A (x ) =

dUcenH . eday lim A(x) = lim (4x − 8 + 4x ) = +∞ . naM[RkabGnuKmn¾ A KμansmIkarGasIumtUtedkeT . x→ +∞

x→ +∞

- 109 -


CMBUkTI4

GaMgetRkalkMNt;

%> KNna F' (x) ebI k> F(x) = ∫ (4t + 1)dt K> F(x) = ∫ t dt g> F(x) = ∫ sin t dt dMeNaHRsay KNna F' (x) ebI k> F(x) = ∫ (4t + 1)dt eK)an F(x) = [2t + t ]

x> F(x) = ∫ t dt X> F(x) = ∫ t1 dt c> F(x) = ∫ 2 1+ t dt .

x+ 2

x

3

−x

x

x2

sin x

2

0

x3

0

sin x

0

x+ 2

x

x+ 2 x

2

[

]

= 2( x + 2) 2 + x + 2 − ( 2x 2 + x ) = 2x 2 + 8x + 8 + x + 2 − 2x 2 − x = 8x + 10

dUcenH F' (x) = 8 . x> F(x) = ∫ t dt eK)an F(x) = ⎡⎢⎣ 14 t ⎤⎥⎦ dUcenH F' (x) = 0 . x

3

−x

x

=0

4

−x

- 110 -

2


CMBUkTI4

GaMgetRkalkMNt;

K> F(x) = ∫

sin x

t dt

0

eK)an

sin x

⎡2 3 ⎤ F( x ) = ⎢ t 2 ⎥ ⎢⎣ 3 ⎥⎦ 0

dUcenH F' (x) = cos x 1 ( ) X> F x = ∫ t dt

3

2 = sin 2 x 3 sin x

x2

2

2

eK)an F(x) = ⎡⎢⎣− 1t ⎤⎥⎦

x2

=− 2

1 1 + 2 2 x

dUcenH F' (x) = x2 . g> F(x) = ∫ sin t dt eK)an F(x) = [− cos t] = − cos x + 1 dUcenH F' (x) = 3x sin x c> F(x) = ∫ 2 1+ t dt . eK)an F(x) = [ln | 2 + t |] = ln 2 − ln | 2 + sin x | x )' cos x =− dUcenH F' (x) = − (22 ++ sin . sin x 2 + sin x 3

x3

0

x3 0

2

3

3

0

sin x

0 sin x

- 111 -


CMBUkTI4

GaMgetRkalkMNt;

^>k>bgðajfaebI f (t ) CaGnuKmn¾ess enaH F(x) = ∫ f (t ) dt RKb; x ∈ IR CaGnuKmn¾essEdrrWeT? dMeNaHRsay sikSaPaBKUessénGnuKmn_ F(x) man F(x) = ∫ f (t ) dt eK)an F(− x) = ∫ f (t ).dt tag u = −t ⇒ du = −dt cMeBaH t = a enaH u = −a ehIy t = − x enaH u = x eK)an F(− x) = ∫ f (−u) (−du) = ∫ f (−u)du x

a

x

a

−x

a

−a

x

−a

−a

x

−a

b¤ F(− x) = ∫ f (−t )dt = ∫ f (t )dt eRBaH f (t ) CaGnuKmn¾ess x

x x

F( − x ) = − ∫ f (t )dt −a

x

x

0

0

-ebI a = 0 enaH F(x) = ∫ f (t )dt nig F(− x) = − ∫ f (t )dt eK)an F(− x) = −F(x) enaH F(x) CaGnuKmn_ess . -ebI a ≠ 0 enaH F(x) minEmnCaGnuKmn_esseT . - 112 -


CMBUkTI4

GaMgetRkalkMNt; x3 1 y= + 3 4x

&>k>KNnaRbEvgFñÚénRkabtagGnuKmn¾ BI x = 1 eTA x = 3 . x>]bmafa f (x) = 12 (e + e ). bnÞat;CYbGk½SGredaenRtg; B ehIyb:HnigRkabtag f Rtg;cMnuc A(a , f (a)) Edl a > 0 . eRbobeFobRbEvgénGgÁt; AB nigRbEvgFñÚrénRkabtag f enAbenøaHbnÞat;Qr x = 0 nig x = a . dMeNaHRsay y k>KNnaRbEvgFñÚ tamrUbmnþ L = ∫ 1 + y' .dx −x

x

3

2

1 3

eday y = x3 + 41x eK)an y' = x − 41x 2

L=

3

∫ 1 3

1 1 + ( x 2 − 2 ) 2 .dx 4x

= ∫ (x 2 + 1

2

1 2 ) .dx 2 4x 3

1⎤ 53 ⎡1 = ⎢ x3 − ⎥ = 4x ⎦ 1 6 ⎣3

- 113 -

1 0

1

x


CMBUkTI4

GaMgetRkalkMNt;

x> AB nigRbEvgFñÚrénRkabtag f enAcenøaHbnÞat; x = 0 nig x = a y

A 1

0

1

x

B

smIkarbnÞat; T b:HRkabtag f (x) = 12 (e + e ) KW 1 T : y − f (a ) = f ' (a )( x − a ) Edl f ' (a ) = (e − e ) 2 ebI x = 0 ⇒ y = −af ' (a) + f (a) enaH B(0 , − af ' (a) + f (a)) eK)an AB = a + a f ' (a) = a 1 + f ' (a) −x

x

a

2

ehIy

2

2

−a

2

1 a = a 1 + ( e a − e − a ) 2 = (e a + e − a ) 4 2 a 1 x 1a x −x 2 L = ∫ 1 + (e − e ) .dx = ∫ (e + e − x ).dx 4 20 0 1 x 1 −x a = e − e 0 = (e a − e − a ) 2 2

[

]

.

- 114 -


CMBUkTI5

smIkarDIepr:g;Esül

CMBUkTI5 emeronTI1

smIkarDIepr:g;EsüllMdab;TI1 smIkarDIepr:g;EsüllMdab;TI ! manragTUeTA > dy = f ( x ) mancemøIyTUeTA y = ∫ f ( x ).dx + c dx > g(y ). dy = f ( x ) mancemøIyTUeTA G ( y ) = F( x ) + C 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

e

p

- 115 -

p

e


CMBUkTI5

smIkarDIepr:g;Esül

lMhat; !>edaHRsaysmIkarDIepr:g;Esül³ k> y' = 2x − x + 1 x> y' = e K> y' = x 2+x 1 X> y' = x x− 1 kMNt;elI (− 1 , 1) g> xy' = 1 kMNt;elI (0 , + ∞ ). @> edaHRsaysmIkarDIepr:g;EsültamlkçxNÐEdl[³ k> yy' = cos x , y⎛⎜⎝ π2 ⎞⎟⎠ = e x> y' = e , y(0) = 5 K> (3x − 2)y' = 6 , y(1) = 4 X> tany' x = 1 , y(0) = 0 #>edaHRsaysmIkarDIepr:g;EsüllIenEG‘lMdab;TI! dy + 2y = 0 3 +y=0 k> dy x> dx dx K> 2y'−3y = 0 X> y'+ y 2 = 0 . −2x

2

2

2

2x

2

- 116 -


CMBUkTI5

smIkarDIepr:g;Esül

$> edaHRsaysmIkarDIepr:g;EsüllIenEG‘lMdab;TI!tamlkçxNÐ Edl[³ k> − y'+2y = 0 , y(3) = −2 x> 2y'+ y = 0 , y(ln 4) = 15 K> 7y'+4y = 0 , y(7) = e X> 2y'−5y = 0 , y(1) = −3 %>cUrbgðajfaGnuKmn¾nimYy²cxageRkamenHCacMelIyénsmIkar DIepr:g;EsülenAxagsþaM³ k> y = x + e , y'− y = 1 − x x> y = e − x − 1 , y'−3y = 3x + 2 K> y = sin x + cos x , y'+ y = 2 cos x X> y = x + ln x , xy'− y = 1 − ln x . ^>eKmansmIkarDIepr:g;Esül (E) : y'+2y = x . k>kMNt;BhuFa g mandWeRkTI@ EdlCacMelIyBiessén (E) . x>tag h CaGnuKmn¾Edl h(x) = f (x) − g(x) . ebI h EdlCa cMelIyBiessénsmIkarDIepr:g;Esül y'+2y = 0 enaHbgðajfa f CacMelIyTUreTAén (E). −4

x

3x

2

- 117 -


CMBUkTI5

smIkarDIepr:g;Esül

&> eKmansmIkarDIepr:g;Esül (E) : y'−2y = 1 +−e2 . k>edaHRsaysmIkarDIepr:g;Esül y'−2y = 0 EdlepÞogpÞat; y (0 ) = 1 . x>tag f CaGnuKmn¾manedrIev IR Edl f (x) = e g(x) . KNna f ' (x) CaGnuKmn¾én g(x) nig g' (x). K>bgðajfaebI f CacemøIyén (E) luHRtaEt g' (x) = 1−+2ee X>TajrkkenSam g(x) rYc f (x) Edl f CacemøIyén (E) . *>edaHRsaysmIkarDIepr:g;Esül dy 1 +y= −y=e x> k> dy dx dx e +1 K> y'+ y = 1 X> y'+ y = sin x (>edaHRsaysmIkarDIepr:g;EsültamlkçxNÐedIm k> y'− y = 1 , y(0) = 1 x> y'+2y = 1 , y(0) = 0 !0>edaHRsaysmIkarDIepr:g;Esül = sin 5x k> dy x> dx + e dy = 0 dx dy = 2x X> ( x + 1) = x+6 K> e dy dx dx −2x

2x

−2x −2x

3x

2x

3x

x

- 118 -


CMBUkTI5

smIkarDIepr:g;Esül

lMhat; nig dMeNaHRsay !>edaHRsaysmIkarDIepr:g;Esül³ k> y' = 2x − x + 1 x> y' = e K> y' = x 2+x 1 X> y' = x x− 1 kMNt;elI (− 1 , 1) g> xy' = 1 kMNt;elI (0 , + ∞ ). dMeNaHRsay edaHRsaysmIkarDIepr:g;Esül³ k> y' = 2x − x + 1 eK)an y = ∫ (2x − x + 1).dx dUcenH y = 23 x − 12 x + x + c x> y' = e eK)an y = ∫ e dx dUcenH y = − 12 e + c 2

2

2

2

2

3

2

−2x

−2x

−2x

- 119 -

−2x


CMBUkTI5

smIkarDIepr:g;Esül

K> y' = x 2+x 1 eK)an y = ∫ x 2+x 1.dx dUcenH = ln( x + 1) + C X> y' = x x− 1 kMNt;elI (− 1 , 1) eK)an y = ∫ x x− 1dx 2

2

2

2

2

1 ⎛ 1 1 ⎞ + ⎜ ⎟.dx ∫ 2 ⎝ x + 1 x − 1⎠ 1 dx 1 dx = ∫ + ∫ 2 x+1 2 x−1 1 1 = ln | x + 1 | + ln | x − 1 | + C 2 2 1 y = ln ( x + 1)( x − 1) + C 2 =

dUcenH g> xy' = 1 kMNt;elI (0 , + ∞ ). eK)an y' = 1x ⇒ y = ∫ dxx = ln | x | +C dUcenH y = ln | x | +C .

- 120 -


CMBUkTI5

smIkarDIepr:g;Esül

@> edaHRsaysmIkarDIepr:g;EsültamlkçxNÐEdl[³ k> yy' = cos x , y⎛⎜⎝ π2 ⎞⎟⎠ = e x> y' = e , y(0) = 5 K> (3x − 2)y' = 6x , y(1) = 4 X> tany' x = 1 , y(0) = 0 dMeNaHRsay edaHRsaysmIkarDIepr:g;Esül k> yy' = cos x , y⎛⎜⎝ π2 ⎞⎟⎠ = e eK)an ∫ yy' dx = ∫ cos xdx 2x

2

ln y = sin x + c ⇒ y = e sin x+c π π x = ⇒ y ( ) = e1+ c = e ⇒ c = 0 2 2

cMeBaH dUcenH y = e . x> y' = e , y(0) = 5 eK)an y = ∫ e dx = 12 e + c cMeBaH x = 0 eK)an y(0) = 12 + c = 5 ⇒ c = 92 dUcenH y = 12 e + 92 sin x

2x

2x

2x

2x

- 121 -


CMBUkTI5

smIkarDIepr:g;Esül

K> (3x − 2)y' = 6x , y(1) = 4 eK)an y = ∫ 36xxdx− 2 = ln | 3x − 2 | +C ebI x = 1 ⇒ y(1) = C = 4 dUcenH y = ln | 3x − 2 | +4 X> tany' x = 1 , y(0) = 0 eK)an y = ∫ tan xdx = − ln | cos x | +C ebI x = 0 ⇒ y = C = 0 dUcenH y = − ln | cos x | . #>edaHRsaysmIkarDIepr:g;EsüllIenEG‘lMdab;TI! dy + 2y = 0 3 +y=0 k> dy x> dx dx K> 2y'−3y = 0 X> y'+ y 2 = 0 . dMeNaHRsay edaHRsaysmIkarDIepr:g;Esül + 2y = 0 eday a = 2 k> dy enaHcemø I y TU e TArbs; s mI k ar dx KW y = A.e Edl A CacMnYnefr . 2

2

2

2

−2x

- 122 -


CMBUkTI5

smIkarDIepr:g;Esül

dy 1 + y = 0 b¤ + y=0 x> 3 dy dx dx 3

eday enaHcemøIyTUeTArbs;smIkarKW y = A.e Edl A CacMnYnefr . K> 2y'−3y = 0 b¤ y'− 32 y = 0 1 a= 3

1 − x 3

3 x 2 A.e

eday enaHcemøIyTUeTArbs;smIkarKW y = Edl A CacMnYnefr . X> y'+ y 2 = 0 eday a = 2 enaHcemøIyTUeTArbs;smIkarKW y = A.e Edl A CacMnYnefr . $> edaHRsaysmIkarDIepr:g;EsüllIenEG‘lMdab;TI!tamlkçxNÐ Edl[³ k> − y'+2y = 0 , y(3) = −2 x> 2y'+ y = 0 , y(ln 4) = 15 K> 7y'+4y = 0 , y(7) = e X> 2y'−5y = 0 , y(1) = −3 dMeNaHRsay edaHRsaysmIkarDIepr:g;Esül 3 a=− 2

2x

−4

- 123 -


CMBUkTI5

smIkarDIepr:g;Esül

k> − y'+2y = 0 , y(3) = −2 smIkarGacsresr y'−2y = 0 eday a = 2 enaHcemøIyTUeTArbs;smIkarKW y = Ae ebI x = 3 ⇒ y(3) = Ae = −2 ⇒ A = −2e dUcenH y = −2e . x> 2y'+ y = 0 , y(ln 4) = 15 smIkarGacsresr y'+ 12 y = 0

2x

−6

6

2 x− 6

eday

1 a= 2

enaHcemøIyTUeTArbs;smIkarKW y = Ae

ebI x = ln 4 ⇒ y(ln 4) = Ae 1

2 − x y= e 2 5

dUcenH K> 7y'+4y = 0

1 ln 4 2

=

1 x 2

1 2 ⇒A= 5 5

.

, y (7 ) = e − 4

.smIkarGacsresr y'+ 74 y = 0

eday enaHcemøIyTUeTArbs;smIkarKW y = ebI x = 7 ⇒ y(7) = Ae = e ⇒ A = 1 dUcenH y = e . 4 a= 7

−4

−4

4 − x 7

- 124 -

4 − x Ae 7


CMBUkTI5

smIkarDIepr:g;Esül

X> 2y'−5y = 0 , y(1) = −3 smIkarGacsresr y'− 52 y = 0 eday

enaHcemøIyTUeTArbs;smIkarKW y =

5 a= 2

5 Ae 2

5 2

5 x Ae 2

ebI x = 1 ⇒ y(1) = = −3 ⇒ A = −3e dUcenH y = −3 e . %>cUrbgðajfaGnuKmn¾nimYy²cxageRkamenHCacMelIyénsmIkar DIepr:g;EsülenAxagsþaM³ k> y = x + e , y'− y = 1 − x x> y = e − x − 1 , y'−3y = 3x + 2 K> y = sin x + cos x , y'+ y = 2 cos x X> y = x + ln x , xy'− y = 1 − ln x . dMeNaHRsay karbgðaj k> y = x + e , y'− y = 1 − x man y = x + e ⇒ y' = 1 + e eK)an y'− y = (1 + e ) − (x + e ) = 1 − x Bit 5 ( x −1 ) 2

x

3x

x

x

x

x

x

- 125 -


CMBUkTI5

smIkarDIepr:g;Esül

dUcenH y = x + e CacemøIyénsmIkar y'− y = 1 − x . x> y = e − x − 1 , y'−3y = 3x + 2 eKman y' = 3e − 1 eK)an y'−3y = 3e − 1 − 3e + 3x + 3 = 3x + 2 Bit dUcenH y = e − x − 1 CacemøIysmIkar y'−3y = 3x + 2 . K> y = sin x + cos x , y'+ y = 2 cos x eKman y' = cos x − sin x eK)an y'+ y = cos x − sin x + cos x + sin x = 2 cos x Bit dUcenH y = sin x + cos x CacemøIysmIkar y'+ y = 2 cos x . X> y = x + ln x , xy'− y = 1 − ln x eKman y' = 1 + 1x ⇒ xy' = x + 1 eK)an xy'− y = x + 1 − x − ln x = 1 − ln x Bit dUcenH y = x + ln x CacemøIyrbs;smIkar xy'− y = 1 − ln x x

3x

3x

3x

3x

3x

- 126 -


CMBUkTI5

smIkarDIepr:g;Esül

^>eKmansmIkarDIepr:g;Esül (E) : y'+2y = x . k>kMNt;BhuFa g mandWeRkTI@ EdlCacMelIyBiessén (E) . x>tag h CaGnuKmn¾Edl h(x) = f (x) − g(x) . ebI h EdlCa cMelIyBiessénsmIkarDIepr:g;Esül y'+2y = 0 enaHbgðajfa f CacMelIyTUreTAén (E). dMeNaHRsay k>kMNt;BhuFa g mandWeRkTI@ EdlCacMelIyBiessén (E) tag g(x) = ax + bx + c CacemøIyrbs;smIkar (E) eK)an g' (x) + 2g(x) = x (1) Et g' (x) = 2ax + b enaHTMnak;TMng ¬!¦Gacsresr 2

2

2

2ax + b + 2(ax 2 + bx + c) = x 2 2ax 2 + ( 2a + 2b )x + b + 2c = x 2 1 1 1 a= , b=− ,c= 2 2 4 1 1 1 g( x ) = x 2 − x + 2 2 4

eKTaj . dUcenH x> bgðajfa f CacMelIyTUreTAén (E). man h CaGnuKmn¾Edl h(x) = f (x) − g(x) ebI h CacMelIyBiessénsmIkarDIepr:g;Esül y'+2y = 0 - 127 -


CMBUkTI5

smIkarDIepr:g;Esül

eK)an h' (x) + 2h(x) = 0 eday h' (x) = f ' (x) − g' (x) enaH f ' (x) − g' (x) + 2f (x) − 2g(x) = 0 [ f ' ( x ) + 2f ( x ) ] − [ g' ( x ) + 2g( x ) ] = 0 ( 2)

yk

(1)

CYskñúg (2) eK)an

. sRmaybBa¢ak;enHmann½yfa ebI h CacMelIyBiessénsmIkar DIepr:g;Esül y'+2y = 0 enaH f CacMelIyTUreTAén (E). &> eKmansmIkarDIepr:g;Esül (E) : y'−2y = 1 +−e2 . k>edaHRsaysmIkarDIepr:g;Esül y'−2y = 0 EdlepÞogpÞat; y (0 ) = 1 . x>tag f CaGnuKmn¾manedrIev IR Edl f (x) = e g(x) . KNna f ' (x) CaGnuKmn¾én g(x) nig g' (x). K>bgðajfaebI f CacemøIyén (E) luHRtaEt g' (x) = 1−+2ee X>TajrkkenSam g(x) rYc f (x) Edl f CacemøIyén (E) . f ' ( x) + 2f ( x) − x 2 = 0 ⇒ f ' ( x ) + 2f ( x ) = x 2

−2x

2x

−2x

−2x

- 128 -


CMBUkTI5

smIkarDIepr:g;Esül

dMeNaHRsay k>edaHRsaysmIkarDIepr:g;Esül y'−2y = 0 EdlepÞogpÞat; y (0 ) = 1 ³ eKman y'−2y = 0 ⇒ y = A.e Edl A CacMnYnefr ebI x = 0 ⇒ y(0) = A = 1 . dUcenH y = e . x>KNna f ' (x) CaGnuKmn¾én g(x) nig g' (x) eKman f (x) = e g(x) eK)an f ' (x) = 2e g(x) + e g' (x) dUcenH f ' (x) = [2g(x) + g' (x)] e . K>bgðajfaebI f CacemøIyén (E) luHRtaEt g' (x) = 1−+2ee ebI f CacemøIyén (E) : y'−2y = 1 +−e2 enaHeK)an −2 eday f (x) = e g(x) f ' ( x) − 2f ( x ) = 1+ e nig f ' (x) = [2g(x) + g' (x)]e enaHeK)an 2x

2x

2x

2x

2x

2x

−2x −2x

−2x

2x

−2x

2x

[2g( x ) + g' ( x)]e 2 x − 2e 2 x g( x ) =

- 129 -

−2 1 + e −2x


CMBUkTI5

smIkarDIepr:g;Esül − 2e − 2 x g' ( x ) = 1 + e −2x − 2e − 2 x g' ( x ) = 1 + e −2x

eKTaj . müa:geTotebI ehIyeday f ' (x) = [2g(x) + g' (x)] e eK)an 2x

− 2e − 2 x 2 x f ' ( x) = [2g( x ) + ]e −2x 1+ e 2 2x f ( x ) = g ( x ). e f ' ( x) = 2g( x )e 2 x − 1 + e −2x 2 −2 f ' ( x) = 2f ( x) − ⇒ f ' ( x ) − 2 f ( x ) = 1 + e −2x 1 + e −2x − 2e − 2 x f g' ( x ) = (E) 1 + e −2x

eday

. dUcenH ebI CacemøIyén luHRtaEt X>TajrkkenSam g(x) rYc f (x) Edl f CacemøIyén (E) − 2e eKman g' (x) = 1 + e eK)an g(x) = ∫ 1−+2ee .dx = ∫ (11++ ee )'.dx dUcenH g(x) = ln | 1 + e | +C . müa:geToteday f (x) = g(x).e dUcenH f (x) = e . ln(1 + e ) + C.e Edl C ∈ IR . −2x

−2x

−2x

−2x

−2x

−2x

−2x

2x

2x

−2x

- 130 -

2x


CMBUkTI5

smIkarDIepr:g;Esül

*>edaHRsaysmIkarDIepr:g;Esül 1 dy k> dy −y=e x> +y= dx dx e +1 K> y'+ y = 1 X> y'+ y = sin x dMeNaHRsay edaHRsaysmIkarDIepr:g;Esül k> dy −y=e dx − y = e nig y Ca tag y = ke CacemøIyBiessén dy dx − y = 0 enaH y = y + y CacmøIyTUeTA cemøIyénsmIkar dy dx −y=e . énsmIkar dy dx − y = 0 naM[ y = A.e Edl A ∈ IR eK)an dy dx − y = e enaHeK)an eday y = ke CacemøIyBiessén dy dx dy dy −y =e Et dx = 3ke dx eK)an 3ke − ke = e ⇒ k = 12 . eK)an y = 12 e . dUcenH y = A.e + 12 e Edl A CacMnYnefr . 3x

2x

3x

3x

3x

e

p

e

p

3x

x

e

3x

3x

p

p

p

3x

3x

p

3x

3x

3x

3x

p

x

3x

- 131 -


CMBUkTI5

smIkarDIepr:g;Esül

1 x> dy +y= dx e +1 b¤ y'+ y = e 1+ 1 KuNGgÁTaMgBIrnwg e eK)an 2x

2x

x

ex y' e + e y = 2 x e +1 ex x ⇒ ye x = ( ye )' = 2 x e +1 x

x

e x dx e 2x + 1

tag t = e ⇒ dt = e dx eK)an ye = ∫ t dt+ 1 = arctan(t ) + C b¤ ye = arctan(e ) + C dUcenH y = e arctan(e ) + C.e . K> y'+ y = 1 b¤ y'+ y − 1 = 0 tag z = y − 1 enaH z' = y' smIkarGacsresr z'+ z = 0 ⇒ z = A.e eday z = y − 1 ⇒ y = z + 1 = A.e + 1 Edl A ∈ IR . X> y'+ y = sin x cemøIy y = A.e + 12 (sin x − cos x) Edl A ∈ IR . x

x

x

2

x

x

−x

x

−x

−x

−x

−x

- 132 -


CMBUkTI5

smIkarDIepr:g;Esül

(>edaHRsaysmIkarDIepr:g;EsültamlkçxNÐedIm k> y'− y = 1 , y(0) = 1 x> y'+2y = 1 , y(0) = 0 dMeNaHRsay edaHRsaysmIkarDIepr:g;EsültamlkçxNÐedIm k> y'− y = 1 , y(0) = 1 eKman y'− y = 1 KuNGgÁTaMgBIrnwg e eK)an −x

y' e − x − e − x y = e − x ( ye − x )' = e − x ⇒ ye − x = ∫ e − x dx ⇒ ye − x = −e − x + k ⇒

y = −1 + ke − x

ebI x = 0 enaH y(0) = −1 + k = 1 ⇒ k = 2 dUcenH y = −1 + 2e . x> y'+2y = 1 , y(0) = 0 edaHRsaydUcxagelIEdreK)ancemøIyrbs;smIkarKW 1 1 y= − e . 2 2 −x

−2x

- 133 -


CMBUkTI5

smIkarDIepr:g;Esül

!0>edaHRsaysmIkarDIepr:g;Esül = sin 5x x> dx + e dy = 0 k> dy dx dy = 2x X> ( x + 1) = x+6 K> e dy dx dx dMeNaHRsay edaHRsaysmIkarDIepr:g;Esül = sin 5x k> dy dx eKTaj y = ∫ sin 5x.dx = − 15 cos 5x + C x> dx + e dy = 0 eKTaj y = ∫ e dx = − 13 e + C = 2x K> e dy dx eKTaj y = ∫ 2xe dx tag f (x) = 2x ⇒ f ' (x) = 2 ehIy g' (x) = e ⇒ g(x) = ∫ e .dx = −e tamrUbmnþ ∫ f (x).g' (x).dx = f (x)g(x) − ∫ g(x)f ' (x).dx eK)an y = −2xe + ∫ 2e dx = −2xe − 2e + C 3x

x

3x

−3x

−3x

x

−x

−x

−x

−x

−x

- 134 -

−x

−x

−x


CMBUkTI5

smIkarDIepr:g;Esül

dUcenH y = −2(x + 1)e = x+6 X> (x + 1) dy dx eK)an y = ∫ xx ++ 16.dx

−x

+C

.

5 )dx x+1 y = x + 5 ln | x + 1 | + C y = ∫ (1 +

- 135 -


CMBUkTI5

smIkarDIepr:g;Esül

CMBUkTI5 emeronTI2

smIkarDIepr:g;EsüllIenEG‘lMdab;TI2 !-smIkarDIepr:g;EsüllIenEG‘lMdab;2

niymn½y

smIkarDIepr:g;EsüllIenEG‘lMdab;TI2 GUm:UEsn nigmanemKuNCa cMnYnefrCasmIkarEdlGacsresrCaragTUeTA ay' '+by'+cy = 0 Edl a ≠ 0 , a, b, c ∈ IR . 2-dMeNaHRsaysmIkarDIepr:g;EsüllIenEG‘lMdab;2Gum:UEsn nigmanemKuNCacMnYnefr

k>smIkarsmÁal;

smIkarsmÁal;énsmIkarDIepr:g;EsüllIenEG‘lMdab;TI2 GUm:UEsn nigmanemKuNCacMnYnefr ay' '+by'+cy = 0 CasmIkardWeRkTIBIr aλ + bλ + c = 0 Edl a ≠ 0 , a, b, c ∈ IR . 2

- 136 -


CMBUkTI5

smIkarDIepr:g;Esül

x>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 Δ = b − 4c -krNI Δ > 0 smIkar (1) manb¤sBIrCacMnYnBitepSgKñaKW λ = α λ = β enaHsmIkar (E) mancemøIyTUeTACaGnuKmn_rag y = A.e + B.e Edl A , B CacMnYnefrmYyNak¾)an . -krNI Δ = 0 smIkar (1) manb¤sDúbKW λ = λ = α enaHsmIkar (E) mancemøIyTUeTACaGnuKmn_rag y = Ax.e + B.e Edl A , B CacMnYnefrmYyNak¾)an . -krNI Δ < 0 smIkar (1) manb¤sBIrepSgKñaCacMnYnkMupøicqøas;KñaKW λ = α + i .β nig λ = α − i .β ¬ α , β ∈ IR ¦ enaHsmIkar (E) mancemøIyTUeTACaGnuKmn_rag 2

2

1

2

αx

βx

1

αx

1

αx

2

y = ( A cos β x + B sin β x )e αx

Edl A , B CacMnYnefrmYyNak¾)an . - 137 -

2


CMBUkTI5

smIkarDIepr:g;Esül

3-dMeNaHRsaysmIkarDIepr:g;EsüllIenEG‘lMdab;2minGum:U EsnnigmanemKuNCacMnYnefr ]bmafaeKmansmIkarDIepr:g;EsüllIenEG‘lMdab;TI@minGUm:UEsn y' '+ by'+ cy = P( x ) Edl P( x ) ≠ 0 . edIm,IedaHRsaysmIkarenHeKRtUv ³ EsVgrkcemøIyBiessminGUm:UEsn tageday y rbs;smIkar y' '+ by'+ cy = P( x ) Edl y manTRmg;dUc P( x ) . rkcemøIyTUeTAtageday y énsmIkarlIenEG‘lMdab;TI@GUm:UEsn y' '+ by'+ cy = 0 . eK)ancemøIyTUeTAénsmIkarDIepr:g;EsüllIenEG‘lMdab;TI@ minGUm:UEsnCaplbUkrvag y nig y KW y = y + y . P

P

c

P

- 138 -

c

P

C


CMBUkTI5

smIkarDIepr:g;Esül

lMhat; !>epÞógpÞat;faGnuKmn_ f CacemøIyénsmIkarDIepr:g;EsülEdl[ k> f (x) = (2x + 1)e , y' '+2y'+ y = 0 x> f (x) = e sin x , y' '+2y'+2y = 0 K> f (x) = Ae + Bxe , y' '−2y'+ y = 0 Edl A nig B CacMnYnefrNamYyk¾)an . @> edaHRsaysmIkar k> 2y' '−3y'+ y = 0 x> − 4y' '+7y'+2y = 0 K> y' '−2y' = 0 X> y' '−3y'+3y = 0 g> 2y' '+3y'−2y = 0 c> y' '−3y'+ y = 0 #>kñúgkrNInImYy²xageRkam eK[ f CaGnuKmn_kMNt;elI IR . rksmIkarDIepr:g;EsüllIenEG‘lMdab;TIBIr GUm:UEsnEdlman GnuKmn_ f CacemøIy k> f (x) = (x + 1)e x> f (x) = 2e + 3e K> f (x) = (2 cos 3x − 3 sin 3x)e −x

−x

x

x

−2x

−x

3x

x

- 139 -


CMBUkTI5

smIkarDIepr:g;Esül

$> edaHRsaysmIkarDIepr:g;EsültamlkçxNÐEdl[³ k> y' '− y = 0 , y(0) = 1 , y' (0) = −2 x> y' '−2y'+3y = 0 , y(0) = 2 , y' (0) = 1 K> y' '+ y = 0 , y( π2 ) = 3 , y' ( π2 ) = 2 X> y' '−3y'+2y = 0 , y(0) = 1 , y' (0) = 3 . %>eKmansmIkarDIepr:g;Esül y' '−4y'+2y = 4 k>rkGnuKmn_efr k EdlCacemøIyBiessén (E) x>edaHRsaysmIkar y' '−4y'+2y = 4 K>rkcMelIyBiessén (E) EdlepÞogpÞat;lkçxNÐedIm y (0 ) = 2 2 , y' (0 ) = 0 .

- 140 -


CMBUkTI5

smIkarDIepr:g;Esül

dMeNaHRsay !>epÞógpÞat;faGnuKmn_ f CacemøIyénsmIkarDIepr:g;EsülEdl[ k> f (x) = (2x + 1)e , y' '+2y'+ y = 0 x> f (x) = e sin x , y' '+2y'+2y = 0 K> f (x) = Ae + Bxe , y' '−2y'+ y = 0 Edl A nig B CacMnYnefrNamYyk¾)an . dMeNaHRsay epÞógpÞat;faGnuKmn_ f CacemøIyénsmIkarDIepr:g;Esül k> f (x) = (2x + 1)e , y' '+2y'+ y = 0 GnuKmn_ f (x) = (2x + 1)e CacemøIysmIkar y' '+2y'+ y = 0 luHRtaEt f (x) , f ' (x) , f ' ' (x) epÞógpÞat;nwgsmIkareBalKW f ' ' ( x ) + 2f ' ( x ) + f ( x ) = 0 cMeBaHRKb; x . eKman f ' (x) = 2e − (2x + 1)e = (−2x + 1)e nig f ' ' (x) = −2e − (−2x + 1)e = (2x − 3)e eK)an −x

−x

x

x

−x

−x

−x

−x

−x

−x

−x

−x

f ' ' ( x ) + 2f ' ( x ) + f ( x ) = ( 2x − 3 − 4x + 2 + 2x + 1)e − x = 0

dUcenHGnuKmn_ f CacemøIyénsmIkarDIepr:g;EsülEdl[ . - 141 -


CMBUkTI5

smIkarDIepr:g;Esül

x> f (x) = e sin x , y' '+2y'+2y = 0 eKman f ' (x) = −e sin x + e cos x b¤ f ' (x) = −f (x) + e cos x (1) ehIy f ' ' (x) = −f ' (x) − e cos x − e sin x b¤ f ' ' (x) = −f ' (x) − e cos x − f (x) (2) bUksmIkar (1) nig (2) eKTTYl)an −x

−x

−x

−x

−x

−x

−x

f ' ' ( x ) + f ' ( x ) = − f ' ( x ) − 2f ( x )

b¤ f ' ' (x) + 2f ' (x) + 2f (x) = 0 . dUcenH f (x) = e sin x CacemøIyrbs; y' '+2y'+2y = 0 . K> f (x) = Ae + Bxe , y' '−2y'+ y = 0 Edl A nig B CacMnYnefrNamYyk¾)an . eKman f ' (x) = Ae + Be + Bxe nig f ' ' (x) = Ae + 2Be + Bxe eK)an f ' ' (x) − 2f ' (x) + f (x) = 0 Bit dUcenH f (x) = Ae + Bxe CacemøIyrbs;smIkarDIepr:g;Esül y' '−2y'+ y = 0 Edl A nig B CacMnYnefrNamYyk¾)an . −x

x

x

x

x

x

x

x

x

x

x

- 142 -


CMBUkTI5

smIkarDIepr:g;Esül

@> edaHRsaysmIkar k> 2y' '−3y'+ y = 0 x> − 4y' '+7y'+2y = 0 K> y' '−2y' = 0 X> y' '−3y'+3y = 0 g> 2y' '+3y'−2y = 0 c> y' '−3y'+ y = 0 dMeNaHRsay k> 2y' '−3y'+ y = 0 mansmIkarsmÁal; 2λ − 3λ + 1 = 0 eday a + b + c = 0 ⇒ λ = 1 ; λ = ac = 12 dUcenHcemøIyTUeTArbs;smIkarDIepr:gEsülenHKWCaGnuKmn_ y = Ae + Be Edl A , B CacMnYnefr . x> − 4y' '+7y'+2y = 0 mansmIkarsmÁal; − 4λ + 7λ + 2 = 0 eday Δ = 49 + 32 = 9 ⇒ λ = 2 ; λ = ac = − 14 dUcenHcemøIyTUeTArbs;smIkarDIepr:gEsülenHKWCaGnuKmn_ y = Ae + Be Edl A , B CacMnYnefr . 2

1

x

2

1 x 2

2

2

1

2x

1 − x 4

- 143 -

2


CMBUkTI5

smIkarDIepr:g;Esül

K> y' '−2y' = 0 mansmIkarsmÁal; λ − 2λ = 0 eKTajb¤s λ = 0 ; λ = 2 dUcenHcemøIyTUeTArbs;smIkarDIepr:gEsülenHKWCaGnuKmn_ y = A + Be Edl A , B CacMnYnefr . X> y' '−3y'+3y = 0 mansmIkarsmÁal; λ − 3λ + 3 = 0 eday Δ = 9 − 12 = −3 ⇒ λ = 32 ± i 23 eKTaj α = 32 , β = 23 . dUcenHcemøIyTUeTArbs;smIkarDIepr:gEsülenHKWCaGnuKmn_ 3 3 y = (C cos x + D sin x) e Edl C , D CacMnYnefr 2 2 g> 2y' '+3y'−2y = 0 cemøIy y = Ae + Be Edl A , B CacMnYnefr . c> y' '−3y'+ y = 0 cemøIy y = Ae + Be Edl A , B CacMnYnefr . 2

1

2

2x

2

1, 2

3x 2

−2x

3− 5 x 2

1 x 2

3+ 5 x 2

- 144 -


CMBUkTI5

smIkarDIepr:g;Esül

#>kñúgkrNInImYy²xageRkam eK[ f CaGnuKmn_kMNt;elI IR . rksmIkarDIepr:g;EsüllIenEG‘lMdab;TIBIr GUm:UEsnEdlman GnuKmn_ f CacemøIy k> f (x) = (x + 1)e x> f (x) = 2e + 3e K> f (x) = (2 cos 3x − 3 sin 3x)e dMeNaHRsay rksmIkarDIepr:g;EsüllIenEG‘lMdab;TIBI k> f (x) = (x + 1)e eKman f ' (x) = e − 2(x + 1)e = (−2x − 1)e ehIy f ' ' (x) = −2e − 2(−2x − 1)e b¤ f ' ' (x) = 4xe . eKman f ' ' (x) + 4f ' (x) + 4f (x) = (4x − 8x − 4 + 4x + 4)e = 0 dUcenH y' '+4y'+4y = 0 CasmIkarEdlRtUvrk . x> f (x) = 2e + 3e cemøIy y' '−2y'−3y = 0 K> f (x) = (2 cos 3x − 3 sin 3x)e cemøIy y' '−2y'+10y = 0 . −2x

−x

3x

x

−2x

−2x

−2x

−2x

−2x

−2x

−2x

−2x

−x

3x

x

- 145 -


CMBUkTI5

smIkarDIepr:g;Esül

$> edaHRsaysmIkarDIepr:g;EsültamlkçxNÐEdl[³ k> y' '− y = 0 , y(0) = 1 , y' (0) = −2 x> y' '−2y'+3y = 0 , y(0) = 2 , y' (0) = 1 K> y' '+ y = 0 , y( π2 ) = 3 , y' ( π2 ) = 2 X> y' '−3y'+2y = 0 , y(0) = 1 , y' (0) = 3 . dMeNaHRsay edaHRsaysmIkarDIepr:g;EsültamlkçxNÐEdl[³ k> y' '− y = 0 , y(0) = 1 , y' (0) = −2 mansmIkarsmÁal; λ − 1 = 0 eKTajb¤s λ = 1 ; λ = −1 dUcenHcemøIyTUeTArbs;smIkarDIepr:gEsülenHKWCaGnuKmn_ y = Ae + Be ehIy y' = Ae − Be eday y(0) = 1 nig y' (0) = −2 1 3 ⎧A + B = 1 A = − , B = − eK)an ⎨A − B = −2 naM[ 2 2 2

1

x

2

−x

x

dUcenH y = − 12 e

x

3 − e −x 2

.

- 146 -

−x


CMBUkTI5

smIkarDIepr:g;Esül

x> y' '−2y'+3y = 0 , y(0) = 2 , y' (0) = 1 cemøIy y = (cos 2x − 22 sin 2x) e K> y' '+ y = 0 , y( π2 ) = 3 , y' ( π2 ) = 2 cemøIy y = −2 cos x + 3 sin x X> y' '−3y'+2y = 0 , y(0) = 1 , y' (0) = 3 cemøIy y = −e + 2e %>eKmansmIkarDIepr:g;Esül y' '−4y'+2y = 4 k>rkGnuKmn_efr k EdlCacemøIyBiessén (E) x>edaHRsaysmIkar y' '−4y'+2y = 4 K>rkcMelIyBiessén (E) EdlepÞogpÞat;lkçxNÐedIm y (0 ) = 2 2 , y' (0 ) = 0 . dMeNaHRsay k>rkGnuKmn_efr k EdlCacemøIyBiessén (E) ebIGnuKmn_ y = k CacemøIy Biessén (E) enaHeK)an x

x

2x

(k )' '−4(k )'+2k = 4

2k = 4 ⇒ k = 2

dUcenH y

p

=2

CacemøIyBiessén (E) . - 147 -


CMBUkTI5

smIkarDIepr:g;Esül

x>edaHRsaysmIkar y' '−4y'+2y = 4 tag y CacemøIyén y' '−4y'+2y = 0 eK)an y = y + y CacemøIyrbs;smIkar y' '−4y'+2y = 4 . smIkar y' '−4y'+2y = 0 mansmIkarsmÁal; λ − 4λ + 2 = 0 c

c

p

2

Δ' = 4 − 2 = 2 ⇒ λ 1 = 2 + 2 , λ 2 = 2 − 2

eKTaj y = Ae + Be . K>rkcMelIyBiessén (E) EdlepÞogpÞat;lkçxNÐedIm eKman y = Ae + Be eK)an y' = (2 + 2 )Ae + ( 2 − 2 )Be eday y(0) = 2 2 , y' (0) = 0 enaHeK)an ( 2+ 2 ) x

( 2− 2 ) x

( 2+ 2 ) x

( 2− 2 ) x

( 2+ 2 ) x

( 2− 2 ) x

⎧A + B = 2 2 ⇒ A = −2 + 2 , B = 2 + 2 ⎨ ⎩( 2 + 2 )A + ( 2 − 2 )B = 0

dUcenH y = (−2 +

2 )e ( 2+

2 )x

+ ( 2 + 2 )e ( 2−

- 148 -

2 )x


CMBUkTI5

smIkarDIepr:g;Esül

lMhat;TI1 1>edaHRsaysmIkar g' ' (x) − 5g' (x) + 6g(x) = 0 (E) 2> kMnt;cMelIy g(x) mYyénsmIkar (E) Edl g(0) = 0 nig g' (0) = 1 . ¬RbFanRbLgqmasTI2qñaM2004¦ dMeNaHRsay 1> edaHRsaysmIkar g' ' (x) − 5g' (x) + 6g(x) = 0 mansmIkarsMKal; r − 5r + 6 = 0 eday Δ = 25 − 24 = 1 nMa[manb¤s

(E)

2

r1 =

5+1 5−1 = 2 , r2 = =3 2 2

tamrUbmnþ g(x) = A.e

r1x

+ B.e r2 x , A , B ∈ IR

dUcenHcMelIysmIkar g(x) = A.e - 149 -

2x

+ B.e 3 x , A, B ∈ IR


CMBUkTI5

smIkarDIepr:g;Esül

2> kMnt;cMelIy g(x) mYyénsmIkar (E) eKman g(x) = A.e + B.e naM[ g' (x) = 2A.e + 3B.e 2x

3x

2x

3x

tambMrab;eKman ⎧⎨gg'(0(0))==01 smmUl ⎧⎨2AA++B3=B0= 1 ⎩

naM[

⎧ A = −1 ⎨ ⎩B = 1

dUcenH g(x) = −e + e . lMhat;TI2 edaHRsaysmIkarDIepr:g;Esül (E) : y' '−3y'+2y = 0 2x

3x

edaydwgfa y(0) = 1 , y' (0) = 0 . dMeNaHRsay edaHRsaysmIkarDIepr:g;Esül³ (E) : y' '−3y'+2y = 0 - 150 -


CMBUkTI5

smIkarDIepr:g;Esül

mansmIkarsMKal; r

2

− 3r + 2 = 0

eday a + b + c = 0 naM[ r

1

tamrUbmnþ y = Ae

r1x

y = A.e x + B.e 2 x

= 1 , r2 =

+ Be r2 x

nig

eK)an

y' = A.e x + 2B.e 2 x , A, B ∈ IR

edaytambMrab;eKman ⎧⎨yy'(0(0))==10 b¤ ⎩

naM[

c =2 a

⎧A + B = 1 ⎨ ⎩ A + 2B = 0

⎧A = 2 ⎨ ⎩ B = −1

dUcenH y = 2e − e CacMelIysmIkar . lMhat;TI3 eK[smIkarDIepr:g;Esül x

2x

(E) : y' '−4y'+4y = 4x 2 − 24x + 34

k-kMnt;cMnYnBit a, b nig c edIm,I[ y (x) = ax CacMelIyedayBiessmYyrbs;smIkar (E) .

2

P

- 151 -

+ bx + c

n


CMBUkTI5

smIkarDIepr:g;Esül

x-bgðajfaGnuKmn_ y = y (x) + y (x) CacMelIyTUeTArbs; (E) enaHGnuKmn_ y (x )CacMelIyrbs;smIkarGUmU:Esn (E') : y' '−4y'+4y = 0 . K-edaHRsaysmIkar (E')rYcrkcMelIyTUeTArbs;smIkar (E). dMeNaHRsay k- kMnt;cMnYnBit a, b nig c P

c

c

(E) : y' '−4y'+4y = 4x 2 − 24x + 34

edIm,I[GnuKmn_ y (x) = ax + bx + c CacMelIyedayBiess mYyrbs;smIkar (E)luHRtEtGnuKmn_ y (x ), y' (x ) nig y' ' (x ) epÞógpÞat;nwgsmIkar (E ) . eK)an (E) : y' ' (x) − 4y' (x) + 4y (x) = 4x − 24x + 34 2

P

p

p

p

2

p

eday

P

⎧ y P (x ) = ax 2 + bx + c ⎪ ⎨ y'p (x ) = 2ax + b ⎪ ⎩ y' 'p (x ) = 2a - 152 -

p


CMBUkTI5

smIkarDIepr:g;Esül

eK)an (2a ) − 4(2ax + b ) + 4(ax 2 + bx + c ) = 4x 2 − 24x + 34

naM[ 4ax 2 + (4b − 8a )x + (2a − 4b + 4c ) = 4x 2 − 24x + 34

eKTaj)an

⎧4a = 4 ⎪ ⎨4b − 8a = −24 ⎪ 2a − 4b + 4c = 34 ⎩

naM[

⎧a = 1 ⎪ ⎨b = − 4 ⎪c = 4 ⎩

dUcenH a = 1 , b = −4 , c = −4 nig y (x) = x − 4x + 4 = (x − 2) . x-karbgðaj GnuKmn_ y = y (x) + y (x) CacMelIyrbs; (E) luHRtaGnuKmn_ y, y' , y' 'epÞógpÞat;smIkar (E) . edayeKman y' = y' (x) + y' (x) nig y' ' = y' ' (x) + y' ' (x) enaHeK)an ³ 2

2

P

P

c

p

p

c

c

- 153 -


CMBUkTI5

[y' ' [y' '

smIkarDIepr:g;Esül

] [

] [

]

2 + y ' ' − 4 y ' + y ' + 4 y + y = 4 x − 24x + 34 p c p c p c

]

2 [ ] − 4 y ' + 4 y + y ' ' − 4 y ' + 4 y = 4 x − 24x + 34 (1) p p p c c c

tamsRmayxagelIeKman y' 'p (x ) − 4y' P (x ) + 4y p (x ) = 4x 2 − 24x + 34 (2 )

¬ eRBaH y (x) CacMelIyrbs;smIkar (E) ¦ . tamTMnak;TMng ¬! ¦ nig ¬@¦ eKTaj)an ³ p

4x 2 − 24x + 34 + [y' ' c − 4y'c +4y c ] = 4x 2 − 24x + 34

naM[eKTaj)an y' ' (x) − 4y' (x) + 4y (x) = 0 TMnak;TMngenHbBa¢ak;faGnuKmn_ y (x) CacMelIyrbs; smIkar (E') : y' '−4y'+4y = 0 . K-edaHRsaysmIkar (E')³ y' '−4y'+4y = 0 smIkarsMKal; r − 4r + 4 = 0 , Δ' = 4 − 4 = 0 naM[smIkarmanb¤sDúb r = r = r = 2 c

c

c

h

2

1

- 154 -

2

0


CMBUkTI5

smIkarDIepr:g;Esül

dUcenHcMelIysmIkar (E') CaGnuKmn_ y c (x ) = (Ax + B ).e 2 x , A, B ∈ IR

.

TajrkcMelIyTUeTArbs;smIkar (E)³ tamsMrayxagelIcMelIysmIkar (E) KWCaGnuKmn_TMrg; y = y p (x ) + y c (x )

edayeKman y

p

(x ) = (x − 2 ) 2

nig y (x) = (Ax + B).e

dUcenH y = (x − 2) + (Ax + B).e CacMelIyrbs;smIkar (E). 2

- 155 -

c

2x

, A, B ∈ IR

2x


CMBUkTI5

smIkarDIepr:g;Esül

lMhat;TI4 eK[smIkarDIepr:g;Esül (E) : y'−4y = −4x + 10x − 6 k-kMnt;cMnYnBit a, b nig c edIm,I[ y (x) = ax + bx + c CacMelIyedayBiessmYyrbs;smIkar (E) . x-bgðajfaGnuKmn_ y = y (x) + y (x) CacMelIyTUeTArbs; (E ) enaHGnuKmn_ y (x ) CacMelIyrbs;smIkarGUmU:Esn (E') : y'−4y = 0 . K-edaHRsaysmIkar (E')rYcrkcMelIyTUeTArbs;smIkar (E). dMeNaHRsay k- kMnt;cMnYnBit a, b nig c 2

2

P

P

h

h

(E ) : y'−4y = −4x 2 + 10x − 6

edIm,I[GnuKmn_ y (x) = ax + bx + c CacMelIyBiess mYyrbs;smIkar (E)luHRtEtGnuKmn_ 2

P

- 156 -


CMBUkTI5

smIkarDIepr:g;Esül

y p (x ), y'p (x )

eK)an (E) : y' eday

P

nig y' '

p

(x ) epÞógpÞat;nwgsmIkar (E ) .

(x ) − 4y p (x ) = −4x 2 + 10x − 6

⎧⎪ y P (x ) = ax 2 + bx + c ⎨ ⎪⎩ y'p (x ) = 2ax + b

eK)an (2ax + b ) − 4(ax + bx + c) = −4x + 10x − 6 naM[ − 4ax − (4b − 2a)x + (b − 4c) = −4x + 10x − 6 2

2

2

eKTaj)an

2

⎧ − 4a = −4 ⎪ ⎨4b − 2a = −10 ⎪b − 4c = −6 ⎩

naM[

⎧a = 1 ⎪ ⎨b = − 2 ⎪c = 1 ⎩

dUcenH a = 1 , b = −2 , c = 1 nig y (x) = x − 2x + 1 = (x − 1) . x-karbgðaj GnuKmn_ y = y (x) + y (x) CacMelIyrbs; (E) luHRtaGnuKmn_ y, y'epÞógpÞat;smIkar (E) . edayeKman y' = y' (x) + y' (x) enaHeK)an ³ 2

2

P

P

h

p

h

- 157 -


CMBUkTI5 [y' [y'

smIkarDIepr:g;Esül p

(x ) + y'h (x )] − 4[y p (x ) + y h (x )] = −4x 2 + 10x − 6

p

(x ) − 4y p (x )] + [y'h (x ) − 4y h (x )] = −4x 2 + 10x − 6 (1)

tamsRmayxagelIeKman y' P (x ) − 4y p (x ) = −4x 2 + 10x − 6 (2 )

¬ eRBaH y (x) CacMelIyrbs;smIkar (E) ¦ . tamTMnak;TMng ¬! ¦ nig ¬@¦ eKTaj)an ³ p

− 4x 2 + 10x − 6 + [y'h (x ) − 4y h (x )] = −4x 2 + 10x − 6

naM[eKTaj)an y' (x) − 4y (x) = 0 . TMnak;TMngenHbBa¢ak;faGnuKmn_ y (x) CacMelIyrbs; smIkar (E') : y'−4y = 0 . K-edaHRsaysmIkar (E')³ y'−4y = 0 eday a = 4 dUcenHcMelIysmIkar (E') CaGnuKmn_ h

h

h

y h (x ) = k .e 4 x , k ∈ IR

.

- 158 -


CMBUkTI5

smIkarDIepr:g;Esül

TajrkcMelIyTUeTArbs;smIkar (E)³ tamsMrayxagelIcMelIysmIkar (E) KWCaGnuKmn_TMrg; y = y p (x ) + y h (x )

edayeKman y

p

(x ) = (x − 1)2 nig y h (x ) = k .e 4x

dUcenH y = (x − 1)

2

+ k .e 4 x , k ∈ IR

- 159 -

.

កំនែលំហាត់គណិតវិទ្យា ១២ កំរិតខ្ពស់  

សៀវភៅគណិតវិទ្យា ដោនឡូត

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