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deilppA amehtaM scit snoitacilbup smeG VTS
1
6102 LIRPA
𝑒𝑚𝑖𝑇
�
𝑇
h
𝑒𝑒𝑟
h
𝑠𝑟𝑢𝑜
�
𝑚𝑢𝑚𝑖𝑥𝑎𝑀
𝑠𝑘𝑟𝑎𝑀
:
57
�
:B.N [ - TRAP fo hcae ni snoitseuq EVIF yna rewsnA )1( - TRAP &A - dna B
snoitseuq hcae fo snoisivid owt yna TRAP ni -C
(2 ) hcaE snoitseuq skram)owt(2 seirrac TRAP ni -A skram)eerht(3,
in TRAP - dna B ni noisivid hcae rof skram)evif(5 TRAP -C ].
TRAP – A
(f fI .1 x si tahw neht noitcnuf ytisned ytilibaborp a si )
fo eulav eht
∫
𝑓
(
𝑥)
𝑥𝑑
∞
−
∞
?
21 era noitubirtsid laimonib a fo ecnairav dna naem eht fI .2
’p‘ dnif , 6 dna
‘ elbairav modnar a fI .3 X hcus noitubirtsid nossioP swollof ’ taht
𝑃(
𝑋
�
1)
�
𝑃
�
𝑋
�
2
� naem eht dnif ,
eht fo noitaived dradnats dna naem eht nwod etirW .4 noitubirtsid lamron dradnats
t5 = s fI .5 2 + t6 + yticolev laitini eht dnif ,5
evruc eht ot lamron fo epols eht dniF .6 y = x3 ,4( ta –2 )
o eht etatS .7 fo eerged dna redr
�
𝑑
𝑦
𝑑
𝑥�
2
�
7
𝑑
2
𝑦
𝑑
𝑥
2
�
2
𝑦
�
0
:evloS .8
(
𝐷
2
�
94
)
𝑦
�
0
TRAP – B
ytilibaborp gniwollof eht sah ’X‘ elbairav modnar a fI .9
)X(E dnif , noitubirtsid
deilppA amehtaM scit snoitacilbup smeG VTS
2
lamron fo seitreporp eerht yna noitneM .01 c evru
.11 fI x ea = t eb + –t lauqe syawla si noitarelecca eht taht wohS .
revo dessap ecnatsid eht ot
fo eulav muminim eht dniF .21 y = x 2 – 4x
:evloS .31
𝑥
𝑥𝑑
�
𝑦
𝑦𝑑
�
0
.41 fo rotcaf gnitargetni eht dniF
𝑦𝑑
𝑥𝑑
�
1
𝑥
𝑦
�
𝑥
.51 :evloS (
𝐷
2
�
5
𝐷
�
6)
𝑦
�
0
fo largetni ralucitrap eht dniF .61 (
𝐷
2
�
01
𝐷
�
1)
𝑦
�
𝑒
−
𝑥
TRAP - C
)a( .71 elbairav modnar A ’X‘ gniwollof eht sah ytilibaborp
noitubirtsid
𝑋
0 1 2 3
𝑃
�
𝑋
�
𝑥
�
1
3
1
6
1
6
1
3
)i( dniF
𝐸(
𝑋) dna
�
𝑖𝑖
�
𝐸(
𝑋
2) )b( elbairav modnar A ’X‘ gniwollof eht sah ytilibaborp
noitubirtsid
X 0 1 2 3 4
)x = X( P a a3 a5 a7 a9
)i( dniF )ii( dna ’a‘
𝑃
�
𝑋
�
2
�
X 1 2 3
P (X)
1
2
0
1
2
deilppA amehtaM scit snoitacilbup smeG VTS
3
)c( dna 51 = n fi ,noitubirtsid laimonib a nI
𝑃(
𝑋
�
1)
�
3
𝑃(
𝑋
�
0)
,
’p‘ fo eulav eht dnif fI )a( .81
%3 ,evitcefed era sblub cirtcele eht fo b 001 fo elpmas a ni taht ytilibaborp eht dnif sblu
( . evitcefed era sblub 5 yltcaxe
𝑒
−
3
�
0
.
8940
�
(b ) si naem noitubirtsid lamron a nI
01
dradnats dna
si noitaived
3
morf lavretni ytilibaborp eht dniF .
X ot 6.8 = X 8.21 =
)c( atad gniwollof eht rof enil thgiarts a tiF
)a( 91 nevig si elcitrap a fo noitauqe dellevart ecnatsid eht fI
yb noitarelecca eht taht wohS . t6 nis b + t6 soc a = s
ecnatsid sti sa seirav
)b( evruc eht ot stnegnat eht ot noitauqe eht dniF
y = x2 + x – ta 6 eht stuc ti erehw tniop eht x – sixa
fo seulav muminim dna mumixam eht dniF )c(
y 2 = x3 – 51 x2 – 63 x 81 +
esab fo enoc ralucric thgir a fo emulov eht dniF )a(.02
suidar
′
𝑟
′ thgieh dna
′
�
′ noitargetni gnisu yb
)b( :evloS
𝑛𝑎𝑡
𝑥
𝑒𝑠
𝑐
2
𝑦
𝑦𝑑
�
𝑛𝑎𝑡
𝑦
𝑒𝑠
𝑐
2
𝑥
𝑥𝑑
�
0
:evloS )c(
𝑦𝑑
𝑥𝑑
�
2
𝑦
𝑥
�
𝑥
2
𝑛𝑖𝑠
𝑥
:evloS )a( .12 (
𝐷
2
�
𝐷
�
2)
𝑦
�
0
:evloS )b( (
𝐷
2
�
8
𝐷
�
61 )
𝑦
�
2
𝑒
𝑥
:evloS )c( (
𝐷
2
�
61 )
𝑦
�
𝑛𝑖𝑠
9
𝑥
𝑥 0 1 2 3 4
𝑦 01 41 91 62 13
deilppA amehtaM scit snoitacilbup smeG VTS
4
TRAP - A
(f fI .1 x si tahw neht noitcnuf ytisned ytilibaborp a si )
fo eulav eht
∫
𝒇
(
𝒙)
𝒙𝒅
∞
−
∞
?
ehT
eulav
fo
∫
𝑓
(
𝑥)
𝑥𝑑
�
1
∞
−
∞
.2 21 era noitubirtsid laimonib a fo ecnairav dna naem eht fI
’p‘ dnif , 6 dna
neviG : naeM =
𝑝𝑛 21 = - - - - - )1(
= ecnairaV
𝑞𝑝𝑛 6 = - - - - - )2(
�
2
�
�
1
�
⇒
𝑞𝑝𝑛
𝑝𝑛
�
6
21
⇒
𝑞
�
1
2
∴
𝒑
�
1
�
𝑞
�
1
�
1
2
�
𝟏
𝟐
3. ‘ elbairav modnar a fI X swollof ’ p hcus noitubirtsid nossio
taht
𝑷(
𝑿
�
𝟏)
�
𝑷
�
𝑿
�
𝟐
� naem dnif ,
:alumroF
𝑃(
𝑋
�
𝑥)
�
𝑒
−
𝜆
𝜆
𝑥
𝑥
!
neviG
𝑃(
𝑋
�
1)
�
𝑃
�
𝑋
�
2
�
𝑒
−
𝜆
𝜆
1
1
!
�
𝑒
−
𝜆
𝜆
2
2
!
𝜆
1
�
𝜆
2
2
2
1
�
𝜆
2
𝜆
⇒
𝜆
�
2
∴ naeM
�
2
SREWSNA
rewsnA
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
5
4. etirW nwod eht naem fo noitaived dradnats dna
dradnats eht n noitubirtsid lamro
naeM
𝜇
�
0
noitaiveD dradnatS
𝜎
�
1
5. t5 = s fI 2 + t6 + yticolev laitini eht dnif ,5
:neviG t5 = s 2 + t6 + 5
v
�
𝑠𝑑
𝑡𝑑 = 5 ( t2 ) )1(6 + + 0 = 6 + t01
yticolev laitinI
v
� �
𝑠𝑑
𝑡𝑑 �
𝑡
=
0 = + )0(01 6 = ces / stinu 6
6 . fo epols eht dniF eht evruc eht ot lamron y = x3 ( ta 4 , –2)
:neviG y = x3
𝑦𝑑
𝑥𝑑
�
3
𝑥
2
�
𝑦𝑑
𝑥𝑑�
�
4
,
−
2
�
�
3(
4)
2
�
84
,tnegnat eht fo epolS m 84 =
,lamron eht fo epolS
−
1
𝑚
�
−
1
84
7 . fo eerged dna redro eht etatS
�
𝒅
𝒚
𝒅
𝒙�
𝟐
�
𝟕
𝒅
𝟐
𝒚
𝒅
𝒙
𝟐
�
𝒚𝟐
�
𝟎
redrO
�
2
dna
eergeD
�
1
.8 :evloS
�
𝑫
𝟐
�
𝟗𝟒
�
𝒚
�
𝟎
neviG (
𝐷
2
�
94
)
𝑦
�
0
si noitauqe yrailixuA
𝑚
2
�
94
�
0
𝑚
2
�
94
rewsnA
rewsnA
rewsnA
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
6
𝑚
�
� √
94
�
�
7
∴ si noitulos ehT
𝑦
�
𝐴
𝑒
𝑚
1
𝑥
�
𝐵
𝑒
𝑚
2
𝑥
𝑦
�
𝐴
𝑒
7
𝑥
�
𝐵
𝑒
−
7
𝑥
TRAP - B
.9 ytilibaborp gniwollof eht sah ’X‘ elbairav modnar a fI
d noitubirtsi f , )X(E dni
:alumroF
𝐸
�
𝑋
�
�
∑
𝑥
i
p
i
n
i
=
1
𝐸
�
𝑋
� =
𝑥
1
𝑝
1
�
𝑥
2
𝑝
2
�
⋯
�
𝑥
𝑛
𝑝
𝑛
= �
1
�
1
2�
�
(
2
�
0)
� �
3
�
1
2�
=
1
2
�
0
�
3
2
=
1
+
3
2
=
4
2
=
2
.01 noitneM yna eerht fo seitreporp n lamro c evru )i( . depahs lleb si evruc lamron ehT
)ii( . enil eht tuoba lacirtemmys si tI
𝑋
�
𝜇
.)iii( = edoM = naideM = naeM
𝜇
.11 fI s ea = t eb + –t . wohS eht taht syawla si noitarelecca
lauqe ecnatsid eht ot revo dessap
s :neviG = ea t eb + –t
X 1 2 3
P (X)
𝟏
𝟐
𝟎
𝟏
𝟐
rewsnA
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
7
v
�
𝑠𝑑
𝑡𝑑 ea = t – eb –t { ecnis
𝑑
𝑡𝑑(
𝑒
−
𝑡)
�
�
𝑒
−
𝑡 }
𝑎
�
𝑑
2
𝑠
𝑑
𝑡
2 ea = t eb + –t
⇒ a s =
∴
𝐴 noitarelecc revo dessap ecnatsid eht ot lauqe syawla si
.21 fo eulav muminim eht dniF y = x 2 – 4x
:neviG y = x 2 – 4x
y1 = 2x – 4
y2 = 2
tuP y 1 0 =
⇒ 2x – 0 = 4
2x 4 =
⇒ x = 2
woN (y2 ) x = 2 = 2 0 >
y si muminim ta x 2 =
ehT muminim fo eulav y )2( = 2 – )2(4
4 = – = 8 – 4
.31 evloS :
𝒙𝒅𝒙
�
𝒚𝒅𝒚
�
𝟎 G nevi
𝑥𝑑𝑥
�
𝑦𝑑𝑦
�
0
𝑥𝑑𝑥
�
�
𝑦𝑑𝑦
�
𝑥
𝑥𝑑
�
� �
𝑦
𝑦𝑑
𝑥
2
2
�
�
𝑦
2
2
�
𝐶
𝑥
2
2
�
𝑦
2
2
�
𝐶
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
8
.41 fo rotcaf gnitargetni eht dniF
𝒚𝒅
𝒙𝒅
�
𝟏
𝒙
𝒚
�
𝒙
neviG
𝑦𝑑
𝑥𝑑
�
1
𝑥
𝑦
�
𝑥
mrof eht fo si sihT
𝑦𝑑
𝑥𝑑
�
𝑦𝑃
�
𝑄
,ereH
𝑃
�
1
𝑥
;
𝑄
�
𝑥
rotcaF gnitargetnI
�
𝑒∫
𝑥𝑑𝑃
�
𝑒∫
1
𝑥
𝑥𝑑
�
𝑒
gol
𝑥
�
𝑥 15. evloS : �
𝑫
𝟐
�
𝑫𝟓
�
𝟔�
𝒚
�
𝟎
neviG (
𝐷
2
�
5
𝐷
�
6)
𝑦
�
0
uqe yrailixuA a si noit
𝑚
2
�
5
𝑚
�
6
�
0
(
𝑚
�
2)(
𝑚
�
3)
�
0
𝑚
�
2
�
0
𝑚
�
3
�
0
𝑚
�
2
𝑚
�
3
∴ si noitulos ehT
𝑦
�
𝐴
𝑒
𝑚
1
𝑥
�
𝐵
𝑒
𝑚
2
𝑥
𝑦
�
𝐴
𝑒
2
𝑥
�
𝐵
𝑒
3
𝑥
16 . eht dniF p ralucitra i fo largetn
�
𝑫
𝟐
�
𝟎𝟏
𝑫
�
𝟏�
𝒚
�
𝒆
−
𝒙
neviG (
𝐷
2
�
01
𝐷
�
1)
𝑦
�
𝑒
−
𝑥
𝑃
.
𝐼
.
�
𝑒
−
𝑥
𝐷
2
−
01
𝐷
+
1
ecalpeR
𝐷
𝑦𝑏
�
1
𝑃
.
𝐼
.
�
𝑒
−
𝑥
�
�
1
�
2
�
01
�
�
1
�
�
1
rewsnA
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
9
𝑃
.
𝐼
.
�
𝑒
−
𝑥
1
�
01
�
1
∴
,
𝑃
.
𝐼
.
�
𝑒
−
𝑥
21
TRAP - C
(.71 a .) elbairav modnar A ’X‘ gniwollof eht sah
ytilibaborp noitubirtsid
𝑿
𝟎 1 2 3
𝑷
�
𝑿
�
𝟏
𝟑
𝟏
𝟔
𝟏
𝟔
𝟏
𝟑
)i( dniF
𝑬(
𝑿) dna
�
𝒊𝒊
�
𝑬�
𝑿
𝟐�
)i( :alumroF
𝐸
�
𝑋
�
�
∑
𝑥
i
p
i
n
i
=
1
𝐸
�
𝑋
� =
𝑥
1
𝑝
1
�
𝑥
2
𝑝
2
�
⋯
�
𝑥
𝑛
𝑝
𝑛
= �
0
�
1
3�
�
�
1
�
1
6�
�
�
2
�
1
6�
� �
3
�
1
3�
=
0
�
1
6
�
2
6
�
1
=
1
+
2
+
6
6
=
9
6
�
3
2
�
𝑖𝑖
�
:alumroF
𝐸
�
𝑋
2
�
� ∑
𝑥
i
2
n
i
=
1
P
i
𝐸
�
𝑋
2
�
�
𝑥
1
2
𝑝
1
�
𝑥
2
2
𝑝
2
�
⋯
�
𝑥
𝑛
2
𝑝
𝑛
= �
0
2
�
1
3�
� �
1
2
�
1
6�
�
�
2
2
�
1
6�
�
�
3
2
�
1
3�
= �
𝟎
�
1
3�
� �
𝟏
�
1
6�
�
�
𝟒
�
1
6�
�
�
𝟗
�
1
3�
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
01
=
0
�
1
6
�
4
6
�
3
=
1
+
4
+
81
6
=
32
6
.)b(.71 elbairav modnar A ’X‘ gniwollof eht sah
p ytilibabor noitubirtsid
X 0 1 2 3 4
P (X) a 3a 5a a7 9a
ii( dna ’a‘ )i( dniF )
𝑷
�
𝑿
�
𝟐
�
)i( taht wonk eW
∑
𝑷
𝒊 1 =
a + 3a + 5a + a7 + 9 a 1 =
52 = a 1
⇒52
1a
)ii(
𝑃
�
𝑋
�
2
� = P (X = 2 + ) P (X = 3) + P (X = 4)
= 7+ a5 + a 9a
= 12 a
= 1252
1
5212
.)c(.71 noitubirtsid laimonib a nI fi , = n 51 dna
𝑷(
𝑿
�
𝟏)
�
𝟑
𝑷(
𝑿
�
𝟎)
,
dnif fo eulav eht ‘p’
neviG
𝑛
�
51
si noitubirtsid laimoniB
𝑷
(
𝑿
�
𝒙)
�
𝒄𝒏
𝒙
𝒑
𝒙
𝒒
𝒏
−
𝒙
𝑃
(
𝑋
�
𝑥)
�
51
𝑐
𝑥
𝑝
𝑥
𝑞
51
−
𝑥
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
11
neviG
𝑷
(
𝑿
�
𝟏)
�
𝟑
𝑷(
𝑿
�
𝟎)
51
𝑐
1
𝑝
1
𝑞
51
−
1
�
3
51
𝑐
0
𝑝
0
𝑞
51
−
0
51
𝑝
𝑞
41
�
3
�
1
�
1
�
𝑞
51
51
𝑝
𝑞
41
�
3
𝑞
51
51
𝑝
�
3
𝑞
51
𝑝
�
3(
1
�
𝑝) [
∴
𝑞
�
1
�
𝑝]
51
𝑝
�
3
�
3
𝑝
51
𝑝
�
3
𝑝
�
3
81
𝑝
�
3
𝑝
�
3
81
𝑝
�
1
6
.)a(.81 fI
𝟑
% ,evitcefed era sblub cirtcele eht fo
sblub 001 fo elpmas a ni taht ytilibaborp eht dnif
5 yltcaxe . evitcefed era sblub (
𝒆
−
𝟑
�
𝟎
.
𝟖𝟗𝟒𝟎
�
:alumroF
𝑃(
𝑋
�
𝑥)
�
𝑒
−
𝜆
𝜆
𝑥
𝑥
!
neviG
𝑃
�
%3
�
3
001
;
𝑛
�
001
taht wonk eW
𝜆
�
𝑃𝑛
�
001 �
3
001�
�
3
𝑃(
𝑋
�
𝑥)
�
𝑒
−
3(
3)
𝑥
𝑥
!
:evitcefed era 5 yltcaxE
𝑃
�
𝑋
�
5
�
𝑃(
𝑋
�
5)
�
𝑒
−
3(
3)
5
5
!
𝑃(
𝑋
�
5)
�
0
.
8940
�
342
�
021
�
0
.
8001
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
21
.)b(.81 si naem noitubirtsid lamron a fI 𝟎𝟏 dradnats dna
si noitaived 𝟑 . morf lavretni ytilibaborp eht dniF
X = ot 6.8 X 8.21 =
,neviG naeM 𝜇 = 01
noitaiveD dradnatS 𝜎 = 3
taht wonk eW 𝑧 = 𝑋−𝜇𝜎
= 𝑋 − 01
3
P morf lavretni ytilibabor X = ot 6.8 X 8.21 =
𝑷( 𝟖. 𝟔 < 𝑋 < 𝟐𝟏 . 𝟖 )
nehW 𝑋 = 8.6 nehW 𝑋 = 21 .8
𝑧 =8.6 − 01
3 𝑧 =
21 .8 − 013
𝑧 =−1.4
3 𝑧 =
2.83
𝑧 = −0. 664 = −0. 74 𝑧 = 0. 339 = 0. 39
∴ 𝑷( 𝟖. 𝟔 < 𝑋 < 𝟐𝟏 . 𝟖 ) = 𝑷(−𝟎. 𝟕𝟒 < 𝑧 < 0. 39 )
−∞ 𝑧 = −0. 74 0 𝑧 = 0. 39 ∞
= 𝑃(−0. 74 < 𝑧 < 0) + 𝑃(0 < 𝑧 < 0. 39 )
= 𝑃(0 < 𝑧 < 0. 74 ) + 𝑃(0 < 𝑧 < 0. 39 )
= 0. 8081 + 0. 8323
= 𝟎. 𝟔𝟒𝟎𝟓
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
31
)c(.81 atad gniwollof eht rof enil thgiarts a tiF
𝒙 0 1 2 3 4
𝒚 01 41 91 62 13
teL
𝑦
�
𝑥𝑎
�
𝑏
…
�
1
� tif tseb fo enil eht eb era snoitauqe lamron eht nehT
𝑎 ∑
𝑥
𝑖
�
𝑏𝑛
� ∑
𝑦
𝑖 .... )2(
𝑎 ∑
𝑥
𝑖
2
�
𝑏 ∑
𝑥
𝑖
� ∑
𝑥
𝑖
𝑦
𝑖 .... )3( etupmoc eW ∑
𝑖𝑥
, ∑
𝑥
𝑖
2
, ∑
𝑖𝑦
dna
∑
𝑖𝑥
𝑖𝑦
.elbat gniwollof eht morf
𝑖𝑥
𝑖𝑦
𝑖𝑥
2
𝑖𝑦𝑖𝑥 0 01 0 0 1 41 1 14 2 91 4 83 3 62 9 87 4 13 61 1 42
∑
𝒙
𝒊
�
01 ∑
𝒚
𝒊
�
001 ∑
𝒙
𝒊
𝟐
�
03 ∑
𝒙
𝒊
𝒚
𝒊
�
452 ,ereH
𝑛
�
5
teg ew ,snoitauqe lamron eht gnisU
)2(
⟹
𝒂 ∑
𝒙
𝒊
�
𝒃𝒏
� ∑
𝒚
𝒊
𝑎(
01 )
�
�
5
�
𝑏
�
001
01
𝑎
�
5
𝑏
�
001
)3(
⟹
𝑎 ∑
𝑥
𝑖
2
�
𝑏 ∑
𝑥
𝑖
� ∑
𝑥
𝑖
𝑦
𝑖
𝑎(
03 )
�
𝑏
�
01
�
�
452
03
𝑎
�
01
𝑏
�
452
: eluR s’remarC yB
01
𝑎
�
5
𝑏
�
001 dna
03
𝑎
�
01
𝑏
�
452
�
�
�
01
5
03
01 �
�
001
�
051
�
�
05
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
41
�
𝑎
�
�
001
5
452
01 �
�
0001
�
0721
�
�
072
�
𝑏
�
�
01
001
03
452 �
�
0452
�
0003
�
�
064
𝑎
�
�
𝑎
�
�
�
072
�
05
�
𝟓
.
𝟒
,
𝑏
�
�
𝑏
�
�
�
064
�
05
�
𝟗
.
𝟐
)1(
⇒
𝑦
�
𝑥𝑎
�
𝑏
tuP
𝑎
�
5
.
4 dna
𝑏
�
9
.
2
𝒚
�
𝟓
.
𝟒
𝒙
�
𝟗
.
𝟐 .tif tseb fo enil eht si hcihw ,
�.91 .)a eht fI dellevart ecnatsid fo noitauqe nevig si elcitrap a
yb a = s soc + t6 b t6 nis . noitarelecca eht taht wohS
sa seirav ecnatsid sti
s t6 nis b + t6 soc a = –––– )1(
v
�
𝑠𝑑
𝑡𝑑
v a = { (– )t6nis 6 } b + { )t6 soc( 6}
v = – t6 soc b6 + t6nis a6
a
�
𝑑
2
𝑠
𝑑
𝑡
2
a = – a6 { )t6soc( 6 } b6 + {(– 6 )t6 nis }
a = – t6soc a63 – t6nis b63
a = – 63 { t6 nisb + t6soca }
a = – 63 {s} [ gnisu ])1(
a = – 63 { ecnatsiD }
∴
.ecnatsid sti sa seirav noitarelecca
.)b(.91 noitauqe eht dniF fo stnegnat eht eht ot evruc
y = x2 + x – 6 ti erehw tniop eht ta eht stuc x – sixa
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
51
y = x2 + x – 6
𝑦𝑑
𝑥𝑑
�
2
𝑥
�
1
evruc ehT y = x2 + x – 6 stuc x – sixa . tuP 0 = y
x2 + x – 6 0 = (x – 2 ( ) x + 3 0 = )
x – 0 = 2 x + 3 0 = x 2 = x = –3
( dna )0 ,2( – )0 ,3
)i( )0 ,2( ta tnegnat eht fo epolS
�
𝑦𝑑
𝑥𝑑�
�
2
,
0
�
�
2(
2)
�
1
�
5
�
𝑚
fo noitauqE eht ( ta tnegnat si )0 ,2 y – y1 = m ( x – x1)
ereH m = 5 , x1 ,2 = y1 0 =
y – = 0 5 (x –2)
y = 5x – 01
5x – y – 0 = 01
�
𝒊𝒊
�
( ta tnegnat eht fo epolS – )0 ,3
�
𝑦𝑑
𝑥𝑑�
�
−
3
,
0
�
�
2(
�
3)
�
1
�
�
5
�
𝑚
fo noitauqE eht ( ta tnegnat –3 si )0 , y – y1 = m (x – x1) ereH m = –5 , x1 = –3 , y1 0 =
y – = 0 –5 (x – (–3))
y = – 5 (x + )3
y = –5x – 51
5x + y 0 = 51 +
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
61
.)c(.91 eht dniF fo seulav muminim dna mumixam
y = 2x3 – 51 x2 – 63 x 81 +
y 2 = x3 – 51 x2 – 63 x 81 +
y 1 2 = (3x2) – 51 (2x) – 63 (1) 0+
y1 6 = x2 – 03 x – 63
y2 6 = (2x) – 03 (1) – 0
y2 21 = x – 03
tuP y 1 0 =
⇒ 6x2 – 03 x – 0 = 63
nO teg ew ,6 yb x2 – 5x – 0 = 6
( x 1 + ( ) x – 6 0 = )
x + 1 0 = x – 6 0 =
x = – 1 x 6 =
esaC )i( : nehW x = – 1
[ woN y2] x =-1 (21 = – 1 ) – 03
= –1 2 – 03 = – 24 0 <
y ta mumixam si x = – 1
fo eulav mumixam ehT y (2 = – 1)3 – (51 – 1)2 – (63 – 1 81 + )
= – 2 – + 51 63 81 +
= 73
)ii( esaC : nehW x = 6
[ woN y2] x = 6 (21 = 6 ) – 03
= 27 – 03 = 24 0 >
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
71
y ta muminim si x = 6
(2 = y fo eulav muminim ehT 6)3 – (51 6) 2 – 6(63 ) 81 +
(2 = 612 ) – (51 63 ) – 612 81 +
4 = 23 – 045 – 612 81 +
= – 603
.)a(.02 esab fo enoc ralucric thgir a fo emulov eht dniF
suidar ′𝒓′ thgieh dna ′𝒉′ noitargetni yb
thgir a gnitator yb demrof si enoc ralucric thgir A
tuoba elgnairt delgna 𝑥- sixa .
Y
𝒚 = 𝒙𝒎 r
𝑿′ x ) 0= 𝜃 ℎ M 𝑿
𝒀′
era stimil ehT 𝑥 = 0 dna 𝑥 = ℎ
∴ 𝑎 = 0 𝑑𝑛𝑎 𝑏 = ℎ
enil eht fo noitauqE 𝑨𝑶 si 𝒚 = 𝒙𝒎 … … …. (𝟏)
nI △ 𝑀𝐴𝑂 , nat 𝜃 =𝑝𝑝𝑜 . 𝑒𝑑𝑖𝑠𝑗𝑑𝑎 . 𝑒𝑑𝑖𝑠
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
81
nat
𝜃
�
𝑟
�
⇒
𝑚
�
𝑟
ℎ ecnis
𝑚
�
nat
𝜃
(
1)
⇒
𝑦
� �
𝑟
��
𝑥
𝑒𝑚𝑢𝑙𝑜𝑉
𝑓𝑜
𝑒𝑛𝑜𝑐
�
𝜋 ∫
𝑦
2
𝑏
𝑎
𝑥𝑑
�
𝜋 � �
𝑥𝑟
� �
2
ℎ
0
𝑥𝑑
�
𝜋 �
𝑟
2
𝑥
2
�
2
ℎ
0
𝑥𝑑
�
𝜋
𝑟
2
�
2 �
𝑥
2
ℎ
0
𝑥𝑑
�
𝜋
𝑟
2
�
2 �
𝑥
3
3�
0
ℎ
�
𝜋
𝑟
2
�
2 �
�
3
3 �
�
𝜋
𝑟
2
ℎ
3
𝑐𝑖𝑏𝑢𝑐
𝑠𝑡𝑖𝑛𝑢
.)b(.02 evloS :
𝒏𝒂𝒕
𝒙
𝒆𝒔
𝒄
𝟐
𝒚
𝒚𝒅
�
𝒏𝒂𝒕
𝒚
𝒆𝒔
𝒄
𝟐
𝒙
𝒙𝒅
�
𝟎
neviG
𝑛𝑎𝑡
𝑥
𝑒𝑠
𝑐
2
𝑦
𝑦𝑑
�
𝑛𝑎𝑡
𝑦
𝑒𝑠
𝑐
2
𝑥
𝑥𝑑
�
0
nat
𝑥
ces
2
𝑦
𝑦𝑑
�
�
nat
𝑦
ces
2
𝑥
𝑥𝑑
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
91
ces
2
𝑦
nat
𝑦
𝑦𝑑
�
�
ces
2
𝑥
nat
𝑥
𝑥𝑑
�
ces
2
𝑦
nat
𝑦
𝑦𝑑
�
� �
ces
2
𝑥
nat
𝑥
𝑥𝑑
gol (
nat
𝑦)
�
�
gol
(
nat
𝑥)
�
gol
𝐶
gol (
nat
𝑦)
�
gol
(
nat
𝑥)
�
gol
𝐶
gol (
nat
𝑦
nat
𝑥)
�
gol
𝐶
nat
𝑦
nat
𝑥
�
𝐶
(.02 .)c evloS :
𝒚𝒅
𝒙𝒅
�
𝒚𝟐
𝒙
�
𝒙
𝟐
𝒏𝒊𝒔
𝒙
neviG
𝑦𝑑
𝑥𝑑
�
2
𝑦
𝑥
�
𝑥
2
nis
𝑥
mrof eht fo si sihT
𝑦𝑑
𝑥𝑑
�
𝑦𝑃
�
𝑄
ereH
𝑃
�
�
2
𝑥
,
𝑄
�
𝑥
2
nis
𝑥
rotcaF gnitargetnI
�
𝑒∫
𝑥𝑑𝑃
�
𝑒∫
−
2
𝑥
𝑥𝑑
�
𝑒
−
2 ∫
1
𝑥
𝑥𝑑
�
𝑒
−
2
gol
𝑥
�
𝑒
gol
𝑥
−
2
�
𝑥
−
2
�
1
𝑥
2
∴
, si noitulos ehT
𝑦
𝑒∫
𝑥𝑑𝑃
� ∫
𝑄
𝑒∫
𝑃
𝑥𝑑
𝑥𝑑
�
C
𝑦
1
𝑥
2
� �
𝑥
2
nis
𝑥 �
1
𝑥
2�
𝑥𝑑
�
C
� �
nis
𝑥
𝑥𝑑
�
C
�
�
soc
𝑥
�
C
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
02
.)a(.12 evloS : �
𝑫
𝟐
�
𝑫
�
𝟐�
𝒚
�
𝟎
neviG (
𝐷
2
�
𝐷
�
2)
𝑦
�
0
uqe yrailixuA a si noit
𝑚
2
�
𝑚
�
2
�
0
(
𝑚
�
1)(
𝑚
�
2)
�
0
(
𝑚
�
1)
�
0
(
𝑚
�
2)
�
0
𝑚
�
1
𝑚
�
�
2
∴
si noitulos ehT
𝑦
�
𝐴
𝑒
𝑚
1
𝑥
�
𝐵
𝑒
𝑚
2
𝑥
𝑦
�
𝐴
𝑒
𝑥
�
𝐵
𝑒
−
2
𝑥
.)b(.12 evloS : �
𝑫
𝟐
�
𝑫𝟖
�
𝟔𝟏 �
𝒚
�
𝒆𝟐
𝒙
,neviG (
𝐷
2
�
8
𝐷
�
61 )
𝑦
�
2
𝑒
𝑥
si noitauqE yrailixuA
𝑚
2
�
8
𝑚
�
61
�
0
(
𝑚
�
4)(
𝑚
�
4)
�
0
𝑚
�
4
�
0
𝑚
�
4
�
0
𝑚
�
4
𝑚
�
4
∴
, noitcnuF yratnemelpmoC
�
�
𝒙𝑨
�
𝑩
�
𝒆
𝒙𝒎
(
𝐶
.
𝐹)
� (
𝑥𝐴
�
𝐵)
𝑒
4
𝑥
largetnI ralucitraP
�
𝑒
𝑥𝑎
𝑓(
𝐷)
𝑃
.
𝐼
.
�
2
𝑒
𝑥
𝐷
2
−
8
𝐷
+
61
ecalpeR
𝐷
𝑦𝑏
1
𝑃
.
𝐼
.
�
2
𝑒
𝑥
�
1
�
2
�
8
�
1
�
�
61
𝑃
.
𝐼
.
�
2
𝑒
𝑥
1
�
8
�
61
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
12
𝑃
.
𝐼
.
�
2
𝑒
𝑥
9
,noituloS lareneG
𝑦
�
𝐶
.
𝐹
�
𝑃
.
𝐼
𝑦
� (
𝑥𝐴
�
𝐵)
𝑒
4
𝑥
�
2
𝑒
𝑥
9
�.12 .)c evloS : �
𝑫
𝟐
�
𝟔𝟏 �
𝒚
�
𝒏𝒊𝒔
𝒙𝟗
neviG
(
𝐷
2
�
61 )
𝑦
�
𝑛𝑖𝑠
9
𝑥
si noitauqe yrailixuA
𝑚
2
�
61
�
0
𝑚
2
�
�
61
𝑚
�
� √
�
61
𝑚
�
�
𝑖
4
,ereH
𝛼
�
0
,
𝛽
�
4
∴ noitcnuf yratnemelpmoC
�
𝒆
𝒙𝜶
�
𝑨
𝒔𝒐𝒄
𝒙𝜷
�
𝒏𝒊𝒔𝑩
𝒙𝜷
�
�
𝐶
.
𝐹
�
𝑒
0
𝑥
�
𝑠𝑜𝑐𝐴
4
𝑥
�
𝑛𝑖𝑠𝐵
4
𝑥
�
�
𝑠𝑜𝑐𝐴
4
𝑥
�
𝑛𝑖𝑠𝐵
4
𝑥
𝑟𝑎𝑙𝑢𝑐𝑖𝑡𝑟𝑎𝑃
𝑙𝑎𝑟𝑔𝑒𝑡𝑛𝐼
�
nis
𝑥𝑎
𝑓
�
𝐷
�
�
nis
9
𝑥
𝐷
2
�
61
ecalpeR
𝐷
2
�
� [
9]
2
�
�
18
�
nis
9
𝑥
�
18
�
61
�
nis
9
𝑥
�
56
si noitulos lareneG
𝑦
�
𝐶
.
𝐹
�
𝑃
.
𝐼
∴
𝑦
�
𝑠𝑜𝑐𝐴
4
𝑥
�
𝑛𝑖𝑠𝐵
4
𝑥
�
nis
9
𝑥
56
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