Applied Mat hemat ics STV Gem s pub lic at ion s A PRIL 2 016 APRIL 2016 aplied maths...Applied Mat...
Transcript of Applied Mat hemat ics STV Gem s pub lic at ion s A PRIL 2 016 APRIL 2016 aplied maths...Applied Mat...
deilppA amehtaM scit snoitacilbup smeG VTS
1
6102 LIRPA
ππππ
οΏ½
π
h
πππ
h
π ππ’π
οΏ½
ππ’πππ₯ππ
π ππππ
:
57
οΏ½
: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 β A
(f fI .1 x si tahw neht noitcnuf ytisned ytilibaborp a si )
fo eulav eht
β«
π
(
π₯)
π₯π
β
β
β
?
21 era noitubirtsid laimonib a fo ecnairav dna naem eht fI .2
βpβ dnif , 6 dna
β elbairav modnar a fI .3 X hcus noitubirtsid nossioP swollof β taht
π(
π
οΏ½
1)
οΏ½
π
οΏ½
π
οΏ½
2
οΏ½ 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 β2 )
o eht etatS .7 fo eerged dna redr
οΏ½
π
π¦
π
π₯οΏ½
2
οΏ½
7
π
2
π¦
π
π₯
2
οΏ½
2
π¦
οΏ½
0
:evloS .8
(
π·
2
οΏ½
94
)
π¦
οΏ½
0
TRAP β B
ytilibaborp gniwollof eht sah βXβ 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 + βt lauqe syawla si noitarelecca eht taht wohS .
revo dessap ecnatsid eht ot
fo eulav muminim eht dniF .21 y = x 2 β 4x
:evloS .31
π₯
π₯π
οΏ½
π¦
π¦π
οΏ½
0
.41 fo rotcaf gnitargetni eht dniF
π¦π
π₯π
οΏ½
1
π₯
π¦
οΏ½
π₯
.51 :evloS (
π·
2
οΏ½
5
π·
οΏ½
6)
π¦
οΏ½
0
fo largetni ralucitrap eht dniF .61 (
π·
2
οΏ½
01
π·
οΏ½
1)
π¦
οΏ½
π
β
π₯
TRAP - C
)a( .71 elbairav modnar A βXβ gniwollof eht sah ytilibaborp
noitubirtsid
π
0 1 2 3
π
οΏ½
π
οΏ½
π₯
οΏ½
1
3
1
6
1
6
1
3
)i( dniF
πΈ(
π) dna
οΏ½
ππ
οΏ½
πΈ(
π
2) )b( elbairav modnar A βXβ gniwollof eht sah ytilibaborp
noitubirtsid
X 0 1 2 3 4
)x = X( P a a3 a5 a7 a9
)i( dniF )ii( dna βaβ
π
οΏ½
π
οΏ½
2
οΏ½
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
π(
π
οΏ½
1)
οΏ½
3
π(
π
οΏ½
0)
,
βpβ 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
π
β
3
οΏ½
0
.
8940
οΏ½
(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 β ta 6 eht stuc ti erehw tniop eht x β sixa
fo seulav muminim dna mumixam eht dniF )c(
y 2 = x3 β 51 x2 β 63 x 81 +
esab fo enoc ralucric thgir a fo emulov eht dniF )a(.02
suidar
β²
π
β² thgieh dna
β²
οΏ½
β² noitargetni gnisu yb
)b( :evloS
πππ‘
π₯
ππ
π
2
π¦
π¦π
οΏ½
πππ‘
π¦
ππ
π
2
π₯
π₯π
οΏ½
0
:evloS )c(
π¦π
π₯π
οΏ½
2
π¦
π₯
οΏ½
π₯
2
πππ
π₯
:evloS )a( .12 (
π·
2
οΏ½
π·
οΏ½
2)
π¦
οΏ½
0
:evloS )b( (
π·
2
οΏ½
8
π·
οΏ½
61 )
π¦
οΏ½
2
π
π₯
:evloS )c( (
π·
2
οΏ½
61 )
π¦
οΏ½
πππ
9
π₯
π₯ 0 1 2 3 4
π¦ 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
β«
π
(
π)
ππ
β
β
β
?
ehT
eulav
fo
β«
π
(
π₯)
π₯π
οΏ½
1
β
β
β
.2 21 era noitubirtsid laimonib a fo ecnairav dna naem eht fI
βpβ dnif , 6 dna
neviG : naeM =
ππ 21 = - - - - - )1(
= ecnairaV
πππ 6 = - - - - - )2(
οΏ½
2
οΏ½
οΏ½
1
οΏ½
β
πππ
ππ
οΏ½
6
21
β
π
οΏ½
1
2
β΄
π
οΏ½
1
οΏ½
π
οΏ½
1
οΏ½
1
2
οΏ½
π
π
3. β elbairav modnar a fI X swollof β p hcus noitubirtsid nossio
taht
π·(
πΏ
οΏ½
π)
οΏ½
π·
οΏ½
πΏ
οΏ½
π
οΏ½ naem dnif ,
:alumroF
π(
π
οΏ½
π₯)
οΏ½
π
β
π
π
π₯
π₯
!
neviG
π(
π
οΏ½
1)
οΏ½
π
οΏ½
π
οΏ½
2
οΏ½
π
β
π
π
1
1
!
οΏ½
π
β
π
π
2
2
!
π
1
οΏ½
π
2
2
2
1
οΏ½
π
2
π
β
π
οΏ½
2
β΄ naeM
οΏ½
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
π
οΏ½
0
noitaiveD dradnatS
π
οΏ½
1
5. t5 = s fI 2 + t6 + yticolev laitini eht dnif ,5
:neviG t5 = s 2 + t6 + 5
v
οΏ½
π π
π‘π = 5 ( t2 ) )1(6 + + 0 = 6 + t01
yticolev laitinI
v
οΏ½ οΏ½
π π
π‘π οΏ½
π‘
=
0 = + )0(01 6 = ces / stinu 6
6 . fo epols eht dniF eht evruc eht ot lamron y = x3 ( ta 4 , β2)
:neviG y = x3
π¦π
π₯π
οΏ½
3
π₯
2
οΏ½
π¦π
π₯ποΏ½
οΏ½
4
,
β
2
οΏ½
οΏ½
3(
4)
2
οΏ½
84
,tnegnat eht fo epolS m 84 =
,lamron eht fo epolS
β
1
π
οΏ½
β
1
84
7 . fo eerged dna redro eht etatS
οΏ½
π
π
π
ποΏ½
π
οΏ½
π
π
π
π
π
π
π
οΏ½
ππ
οΏ½
π
redrO
οΏ½
2
dna
eergeD
οΏ½
1
.8 :evloS
οΏ½
π«
π
οΏ½
ππ
οΏ½
π
οΏ½
π
neviG (
π·
2
οΏ½
94
)
π¦
οΏ½
0
si noitauqe yrailixuA
π
2
οΏ½
94
οΏ½
0
π
2
οΏ½
94
rewsnA
rewsnA
rewsnA
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
6
π
οΏ½
οΏ½ β
94
οΏ½
οΏ½
7
β΄ si noitulos ehT
π¦
οΏ½
π΄
π
π
1
π₯
οΏ½
π΅
π
π
2
π₯
π¦
οΏ½
π΄
π
7
π₯
οΏ½
π΅
π
β
7
π₯
TRAP - B
.9 ytilibaborp gniwollof eht sah βXβ elbairav modnar a fI
d noitubirtsi f , )X(E dni
:alumroF
πΈ
οΏ½
π
οΏ½
οΏ½
β
π₯
i
p
i
n
i
=
1
πΈ
οΏ½
π
οΏ½ =
π₯
1
π
1
οΏ½
π₯
2
π
2
οΏ½
β―
οΏ½
π₯
π
π
π
= οΏ½
1
οΏ½
1
2οΏ½
οΏ½
(
2
οΏ½
0)
οΏ½ οΏ½
3
οΏ½
1
2οΏ½
=
1
2
οΏ½
0
οΏ½
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
π
οΏ½
π
.)iii( = edoM = naideM = naeM
π
.11 fI s ea = t eb + βt . wohS eht taht syawla si noitarelecca
lauqe ecnatsid eht ot revo dessap
s :neviG = ea t eb + βt
X 1 2 3
P (X)
π
π
π
π
π
rewsnA
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
7
v
οΏ½
π π
π‘π ea = t β eb βt { ecnis
π
π‘π(
π
β
π‘)
οΏ½
οΏ½
π
β
π‘ }
π
οΏ½
π
2
π
π
π‘
2 ea = t eb + βt
β a s =
β΄
π΄ noitarelecc revo dessap ecnatsid eht ot lauqe syawla si
.21 fo eulav muminim eht dniF y = x 2 β 4x
:neviG y = x 2 β 4x
y1 = 2x β 4
y2 = 2
tuP y 1 0 =
β 2x β 0 = 4
2x 4 =
β x = 2
woN (y2 ) x = 2 = 2 0 >
y si muminim ta x 2 =
ehT muminim fo eulav y )2( = 2 β )2(4
4 = β = 8 β 4
.31 evloS :
ππ π
οΏ½
ππ π
οΏ½
π G nevi
π₯ππ₯
οΏ½
π¦ππ¦
οΏ½
0
π₯ππ₯
οΏ½
οΏ½
π¦ππ¦
οΏ½
π₯
π₯π
οΏ½
οΏ½ οΏ½
π¦
π¦π
π₯
2
2
οΏ½
οΏ½
π¦
2
2
οΏ½
πΆ
π₯
2
2
οΏ½
π¦
2
2
οΏ½
πΆ
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
8
.41 fo rotcaf gnitargetni eht dniF
ππ
ππ
οΏ½
π
π
π
οΏ½
π
neviG
π¦π
π₯π
οΏ½
1
π₯
π¦
οΏ½
π₯
mrof eht fo si sihT
π¦π
π₯π
οΏ½
π¦π
οΏ½
π
,ereH
π
οΏ½
1
π₯
;
π
οΏ½
π₯
rotcaF gnitargetnI
οΏ½
πβ«
π₯ππ
οΏ½
πβ«
1
π₯
π₯π
οΏ½
π
gol
π₯
οΏ½
π₯ 15. evloS : οΏ½
π«
π
οΏ½
π«π
οΏ½
ποΏ½
π
οΏ½
π
neviG (
π·
2
οΏ½
5
π·
οΏ½
6)
π¦
οΏ½
0
uqe yrailixuA a si noit
π
2
οΏ½
5
π
οΏ½
6
οΏ½
0
(
π
οΏ½
2)(
π
οΏ½
3)
οΏ½
0
π
οΏ½
2
οΏ½
0
π
οΏ½
3
οΏ½
0
π
οΏ½
2
π
οΏ½
3
β΄ si noitulos ehT
π¦
οΏ½
π΄
π
π
1
π₯
οΏ½
π΅
π
π
2
π₯
π¦
οΏ½
π΄
π
2
π₯
οΏ½
π΅
π
3
π₯
16 . eht dniF p ralucitra i fo largetn
οΏ½
π«
π
οΏ½
ππ
π«
οΏ½
ποΏ½
π
οΏ½
π
β
π
neviG (
π·
2
οΏ½
01
π·
οΏ½
1)
π¦
οΏ½
π
β
π₯
π
.
πΌ
.
οΏ½
π
β
π₯
π·
2
β
01
π·
+
1
ecalpeR
π·
π¦π
οΏ½
1
π
.
πΌ
.
οΏ½
π
β
π₯
οΏ½
οΏ½
1
οΏ½
2
οΏ½
01
οΏ½
οΏ½
1
οΏ½
οΏ½
1
rewsnA
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
9
π
.
πΌ
.
οΏ½
π
β
π₯
1
οΏ½
01
οΏ½
1
β΄
,
π
.
πΌ
.
οΏ½
π
β
π₯
21
TRAP - C
(.71 a .) elbairav modnar A βXβ gniwollof eht sah
ytilibaborp noitubirtsid
πΏ
π 1 2 3
π·
οΏ½
πΏ
οΏ½
π
π
π
π
π
π
π
π
)i( dniF
π¬(
πΏ) dna
οΏ½
ππ
οΏ½
π¬οΏ½
πΏ
ποΏ½
)i( :alumroF
πΈ
οΏ½
π
οΏ½
οΏ½
β
π₯
i
p
i
n
i
=
1
πΈ
οΏ½
π
οΏ½ =
π₯
1
π
1
οΏ½
π₯
2
π
2
οΏ½
β―
οΏ½
π₯
π
π
π
= οΏ½
0
οΏ½
1
3οΏ½
οΏ½
οΏ½
1
οΏ½
1
6οΏ½
οΏ½
οΏ½
2
οΏ½
1
6οΏ½
οΏ½ οΏ½
3
οΏ½
1
3οΏ½
=
0
οΏ½
1
6
οΏ½
2
6
οΏ½
1
=
1
+
2
+
6
6
=
9
6
οΏ½
3
2
οΏ½
ππ
οΏ½
:alumroF
πΈ
οΏ½
π
2
οΏ½
οΏ½ β
π₯
i
2
n
i
=
1
P
i
πΈ
οΏ½
π
2
οΏ½
οΏ½
π₯
1
2
π
1
οΏ½
π₯
2
2
π
2
οΏ½
β―
οΏ½
π₯
π
2
π
π
= οΏ½
0
2
οΏ½
1
3οΏ½
οΏ½ οΏ½
1
2
οΏ½
1
6οΏ½
οΏ½
οΏ½
2
2
οΏ½
1
6οΏ½
οΏ½
οΏ½
3
2
οΏ½
1
3οΏ½
= οΏ½
π
οΏ½
1
3οΏ½
οΏ½ οΏ½
π
οΏ½
1
6οΏ½
οΏ½
οΏ½
π
οΏ½
1
6οΏ½
οΏ½
οΏ½
π
οΏ½
1
3οΏ½
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
01
=
0
οΏ½
1
6
οΏ½
4
6
οΏ½
3
=
1
+
4
+
81
6
=
32
6
.)b(.71 elbairav modnar A βXβ gniwollof eht sah
p ytilibabor noitubirtsid
X 0 1 2 3 4
P (X) a 3a 5a a7 9a
ii( dna βaβ )i( dniF )
π·
οΏ½
πΏ
οΏ½
π
οΏ½
)i( taht wonk eW
β
π·
π 1 =
a + 3a + 5a + a7 + 9 a 1 =
52 = a 1
β52
1a
)ii(
π
οΏ½
π
οΏ½
2
οΏ½ = 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
π·(
πΏ
οΏ½
π)
οΏ½
π
π·(
πΏ
οΏ½
π)
,
dnif fo eulav eht βpβ
neviG
π
οΏ½
51
si noitubirtsid laimoniB
π·
(
πΏ
οΏ½
π)
οΏ½
ππ
π
π
π
π
π
β
π
π
(
π
οΏ½
π₯)
οΏ½
51
π
π₯
π
π₯
π
51
β
π₯
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
11
neviG
π·
(
πΏ
οΏ½
π)
οΏ½
π
π·(
πΏ
οΏ½
π)
51
π
1
π
1
π
51
β
1
οΏ½
3
51
π
0
π
0
π
51
β
0
51
π
π
41
οΏ½
3
οΏ½
1
οΏ½
1
οΏ½
π
51
51
π
π
41
οΏ½
3
π
51
51
π
οΏ½
3
π
51
π
οΏ½
3(
1
οΏ½
π) [
β΄
π
οΏ½
1
οΏ½
π]
51
π
οΏ½
3
οΏ½
3
π
51
π
οΏ½
3
π
οΏ½
3
81
π
οΏ½
3
π
οΏ½
3
81
π
οΏ½
1
6
.)a(.81 fI
π
% ,evitcefed era sblub cirtcele eht fo
sblub 001 fo elpmas a ni taht ytilibaborp eht dnif
5 yltcaxe . evitcefed era sblub (
π
β
π
οΏ½
π
.
ππππ
οΏ½
:alumroF
π(
π
οΏ½
π₯)
οΏ½
π
β
π
π
π₯
π₯
!
neviG
π
οΏ½
%3
οΏ½
3
001
;
π
οΏ½
001
taht wonk eW
π
οΏ½
ππ
οΏ½
001 οΏ½
3
001οΏ½
οΏ½
3
π(
π
οΏ½
π₯)
οΏ½
π
β
3(
3)
π₯
π₯
!
:evitcefed era 5 yltcaxE
π
οΏ½
π
οΏ½
5
οΏ½
π(
π
οΏ½
5)
οΏ½
π
β
3(
3)
5
5
!
π(
π
οΏ½
5)
οΏ½
0
.
8940
οΏ½
342
οΏ½
021
οΏ½
0
.
8001
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
21
.)b(.81 si naem noitubirtsid lamron a fI ππ dradnats dna
si noitaived π . morf lavretni ytilibaborp eht dniF
X = ot 6.8 X 8.21 =
,neviG naeM π = 01
noitaiveD dradnatS π = 3
taht wonk eW π§ = πβππ
= π β 01
3
P morf lavretni ytilibabor X = ot 6.8 X 8.21 =
π·( π. π < π < ππ . π )
nehW π = 8.6 nehW π = 21 .8
π§ =8.6 β 01
3 π§ =
21 .8 β 013
π§ =β1.4
3 π§ =
2.83
π§ = β0. 664 = β0. 74 π§ = 0. 339 = 0. 39
β΄ π·( π. π < π < ππ . π ) = π·(βπ. ππ < π§ < 0. 39 )
ββ π§ = β0. 74 0 π§ = 0. 39 β
= π(β0. 74 < π§ < 0) + π(0 < π§ < 0. 39 )
= π(0 < π§ < 0. 74 ) + π(0 < π§ < 0. 39 )
= 0. 8081 + 0. 8323
= π. ππππ
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
31
)c(.81 atad gniwollof eht rof enil thgiarts a tiF
π 0 1 2 3 4
π 01 41 91 62 13
teL
π¦
οΏ½
π₯π
οΏ½
π
β¦
οΏ½
1
οΏ½ tif tseb fo enil eht eb era snoitauqe lamron eht nehT
π β
π₯
π
οΏ½
ππ
οΏ½ β
π¦
π .... )2(
π β
π₯
π
2
οΏ½
π β
π₯
π
οΏ½ β
π₯
π
π¦
π .... )3( etupmoc eW β
ππ₯
, β
π₯
π
2
, β
ππ¦
dna
β
ππ₯
ππ¦
.elbat gniwollof eht morf
ππ₯
ππ¦
ππ₯
2
ππ¦ππ₯ 0 01 0 0 1 41 1 14 2 91 4 83 3 62 9 87 4 13 61 1 42
β
π
π
οΏ½
01 β
π
π
οΏ½
001 β
π
π
π
οΏ½
03 β
π
π
π
π
οΏ½
452 ,ereH
π
οΏ½
5
teg ew ,snoitauqe lamron eht gnisU
)2(
βΉ
π β
π
π
οΏ½
ππ
οΏ½ β
π
π
π(
01 )
οΏ½
οΏ½
5
οΏ½
π
οΏ½
001
01
π
οΏ½
5
π
οΏ½
001
)3(
βΉ
π β
π₯
π
2
οΏ½
π β
π₯
π
οΏ½ β
π₯
π
π¦
π
π(
03 )
οΏ½
π
οΏ½
01
οΏ½
οΏ½
452
03
π
οΏ½
01
π
οΏ½
452
: eluR sβremarC yB
01
π
οΏ½
5
π
οΏ½
001 dna
03
π
οΏ½
01
π
οΏ½
452
οΏ½
οΏ½
οΏ½
01
5
03
01 οΏ½
οΏ½
001
οΏ½
051
οΏ½
οΏ½
05
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
41
οΏ½
π
οΏ½
οΏ½
001
5
452
01 οΏ½
οΏ½
0001
οΏ½
0721
οΏ½
οΏ½
072
οΏ½
π
οΏ½
οΏ½
01
001
03
452 οΏ½
οΏ½
0452
οΏ½
0003
οΏ½
οΏ½
064
π
οΏ½
οΏ½
π
οΏ½
οΏ½
οΏ½
072
οΏ½
05
οΏ½
π
.
π
,
π
οΏ½
οΏ½
π
οΏ½
οΏ½
οΏ½
064
οΏ½
05
οΏ½
π
.
π
)1(
β
π¦
οΏ½
π₯π
οΏ½
π
tuP
π
οΏ½
5
.
4 dna
π
οΏ½
9
.
2
π
οΏ½
π
.
π
π
οΏ½
π
.
π .tif tseb fo enil eht si hcihw ,
οΏ½.91 .)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 = ββββ )1(
v
οΏ½
π π
π‘π
v a = { (β )t6nis 6 } b + { )t6 soc( 6}
v = β t6 soc b6 + t6nis a6
a
οΏ½
π
2
π
π
π‘
2
a = β a6 { )t6soc( 6 } b6 + {(β 6 )t6 nis }
a = β t6soc a63 β t6nis b63
a = β 63 { t6 nisb + t6soca }
a = β 63 {s} [ gnisu ])1(
a = β 63 { ecnatsiD }
β΄
.ecnatsid sti sa seirav noitarelecca
.)b(.91 noitauqe eht dniF fo stnegnat eht eht ot evruc
y = x2 + x β 6 ti erehw tniop eht ta eht stuc x β sixa
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
51
y = x2 + x β 6
π¦π
π₯π
οΏ½
2
π₯
οΏ½
1
evruc ehT y = x2 + x β 6 stuc x β sixa . tuP 0 = y
x2 + x β 6 0 = (x β 2 ( ) x + 3 0 = )
x β 0 = 2 x + 3 0 = x 2 = x = β3
( dna )0 ,2( β )0 ,3
)i( )0 ,2( ta tnegnat eht fo epolS
οΏ½
π¦π
π₯ποΏ½
οΏ½
2
,
0
οΏ½
οΏ½
2(
2)
οΏ½
1
οΏ½
5
οΏ½
π
fo noitauqE eht ( ta tnegnat si )0 ,2 y β y1 = m ( x β x1)
ereH m = 5 , x1 ,2 = y1 0 =
y β = 0 5 (x β2)
y = 5x β 01
5x β y β 0 = 01
οΏ½
ππ
οΏ½
( ta tnegnat eht fo epolS β )0 ,3
οΏ½
π¦π
π₯ποΏ½
οΏ½
β
3
,
0
οΏ½
οΏ½
2(
οΏ½
3)
οΏ½
1
οΏ½
οΏ½
5
οΏ½
π
fo noitauqE eht ( ta tnegnat β3 si )0 , y β y1 = m (x β x1) ereH m = β5 , x1 = β3 , y1 0 =
y β = 0 β5 (x β (β3))
y = β 5 (x + )3
y = β5x β 51
5x + y 0 = 51 +
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
61
.)c(.91 eht dniF fo seulav muminim dna mumixam
y = 2x3 β 51 x2 β 63 x 81 +
y 2 = x3 β 51 x2 β 63 x 81 +
y 1 2 = (3x2) β 51 (2x) β 63 (1) 0+
y1 6 = x2 β 03 x β 63
y2 6 = (2x) β 03 (1) β 0
y2 21 = x β 03
tuP y 1 0 =
β 6x2 β 03 x β 0 = 63
nO teg ew ,6 yb x2 β 5x β 0 = 6
( x 1 + ( ) x β 6 0 = )
x + 1 0 = x β 6 0 =
x = β 1 x 6 =
esaC )i( : nehW x = β 1
[ woN y2] x =-1 (21 = β 1 ) β 03
= β1 2 β 03 = β 24 0 <
y ta mumixam si x = β 1
fo eulav mumixam ehT y (2 = β 1)3 β (51 β 1)2 β (63 β 1 81 + )
= β 2 β + 51 63 81 +
= 73
)ii( esaC : nehW x = 6
[ woN y2] x = 6 (21 = 6 ) β 03
= 27 β 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 β (51 6) 2 β 6(63 ) 81 +
(2 = 612 ) β (51 63 ) β 612 81 +
4 = 23 β 045 β 612 81 +
= β 603
.)a(.02 esab fo enoc ralucric thgir a fo emulov eht dniF
suidar β²πβ² thgieh dna β²πβ² noitargetni yb
thgir a gnitator yb demrof si enoc ralucric thgir A
tuoba elgnairt delgna π₯- sixa .
Y
π = ππ r
πΏβ² x ) 0= π β M πΏ
πβ²
era stimil ehT π₯ = 0 dna π₯ = β
β΄ π = 0 πππ π = β
enil eht fo noitauqE π¨πΆ si π = ππ β¦ β¦ β¦. (π)
nI β³ ππ΄π , nat π =πππ . ππππ πππ . ππππ
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
81
nat
π
οΏ½
π
οΏ½
β
π
οΏ½
π
β ecnis
π
οΏ½
nat
π
(
1)
β
π¦
οΏ½ οΏ½
π
οΏ½οΏ½
π₯
πππ’πππ
ππ
ππππ
οΏ½
π β«
π¦
2
π
π
π₯π
οΏ½
π οΏ½ οΏ½
π₯π
οΏ½ οΏ½
2
β
0
π₯π
οΏ½
π οΏ½
π
2
π₯
2
οΏ½
2
β
0
π₯π
οΏ½
π
π
2
οΏ½
2 οΏ½
π₯
2
β
0
π₯π
οΏ½
π
π
2
οΏ½
2 οΏ½
π₯
3
3οΏ½
0
β
οΏ½
π
π
2
οΏ½
2 οΏ½
οΏ½
3
3 οΏ½
οΏ½
π
π
2
β
3
ππππ’π
π π‘πππ’
.)b(.02 evloS :
πππ
π
ππ
π
π
π
ππ
οΏ½
πππ
π
ππ
π
π
π
ππ
οΏ½
π
neviG
πππ‘
π₯
ππ
π
2
π¦
π¦π
οΏ½
πππ‘
π¦
ππ
π
2
π₯
π₯π
οΏ½
0
nat
π₯
ces
2
π¦
π¦π
οΏ½
οΏ½
nat
π¦
ces
2
π₯
π₯π
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
91
ces
2
π¦
nat
π¦
π¦π
οΏ½
οΏ½
ces
2
π₯
nat
π₯
π₯π
οΏ½
ces
2
π¦
nat
π¦
π¦π
οΏ½
οΏ½ οΏ½
ces
2
π₯
nat
π₯
π₯π
gol (
nat
π¦)
οΏ½
οΏ½
gol
(
nat
π₯)
οΏ½
gol
πΆ
gol (
nat
π¦)
οΏ½
gol
(
nat
π₯)
οΏ½
gol
πΆ
gol (
nat
π¦
nat
π₯)
οΏ½
gol
πΆ
nat
π¦
nat
π₯
οΏ½
πΆ
(.02 .)c evloS :
ππ
ππ
οΏ½
ππ
π
οΏ½
π
π
πππ
π
neviG
π¦π
π₯π
οΏ½
2
π¦
π₯
οΏ½
π₯
2
nis
π₯
mrof eht fo si sihT
π¦π
π₯π
οΏ½
π¦π
οΏ½
π
ereH
π
οΏ½
οΏ½
2
π₯
,
π
οΏ½
π₯
2
nis
π₯
rotcaF gnitargetnI
οΏ½
πβ«
π₯ππ
οΏ½
πβ«
β
2
π₯
π₯π
οΏ½
π
β
2 β«
1
π₯
π₯π
οΏ½
π
β
2
gol
π₯
οΏ½
π
gol
π₯
β
2
οΏ½
π₯
β
2
οΏ½
1
π₯
2
β΄
, si noitulos ehT
π¦
πβ«
π₯ππ
οΏ½ β«
π
πβ«
π
π₯π
π₯π
οΏ½
C
π¦
1
π₯
2
οΏ½ οΏ½
π₯
2
nis
π₯ οΏ½
1
π₯
2οΏ½
π₯π
οΏ½
C
οΏ½ οΏ½
nis
π₯
π₯π
οΏ½
C
οΏ½
οΏ½
soc
π₯
οΏ½
C
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
02
.)a(.12 evloS : οΏ½
π«
π
οΏ½
π«
οΏ½
ποΏ½
π
οΏ½
π
neviG (
π·
2
οΏ½
π·
οΏ½
2)
π¦
οΏ½
0
uqe yrailixuA a si noit
π
2
οΏ½
π
οΏ½
2
οΏ½
0
(
π
οΏ½
1)(
π
οΏ½
2)
οΏ½
0
(
π
οΏ½
1)
οΏ½
0
(
π
οΏ½
2)
οΏ½
0
π
οΏ½
1
π
οΏ½
οΏ½
2
β΄
si noitulos ehT
π¦
οΏ½
π΄
π
π
1
π₯
οΏ½
π΅
π
π
2
π₯
π¦
οΏ½
π΄
π
π₯
οΏ½
π΅
π
β
2
π₯
.)b(.12 evloS : οΏ½
π«
π
οΏ½
π«π
οΏ½
ππ οΏ½
π
οΏ½
ππ
π
,neviG (
π·
2
οΏ½
8
π·
οΏ½
61 )
π¦
οΏ½
2
π
π₯
si noitauqE yrailixuA
π
2
οΏ½
8
π
οΏ½
61
οΏ½
0
(
π
οΏ½
4)(
π
οΏ½
4)
οΏ½
0
π
οΏ½
4
οΏ½
0
π
οΏ½
4
οΏ½
0
π
οΏ½
4
π
οΏ½
4
β΄
, noitcnuF yratnemelpmoC
οΏ½
οΏ½
ππ¨
οΏ½
π©
οΏ½
π
ππ
(
πΆ
.
πΉ)
οΏ½ (
π₯π΄
οΏ½
π΅)
π
4
π₯
largetnI ralucitraP
οΏ½
π
π₯π
π(
π·)
π
.
πΌ
.
οΏ½
2
π
π₯
π·
2
β
8
π·
+
61
ecalpeR
π·
π¦π
1
π
.
πΌ
.
οΏ½
2
π
π₯
οΏ½
1
οΏ½
2
οΏ½
8
οΏ½
1
οΏ½
οΏ½
61
π
.
πΌ
.
οΏ½
2
π
π₯
1
οΏ½
8
οΏ½
61
rewsnA
rewsnA
deilppA amehtaM scit snoitacilbup smeG VTS
12
π
.
πΌ
.
οΏ½
2
π
π₯
9
,noituloS lareneG
π¦
οΏ½
πΆ
.
πΉ
οΏ½
π
.
πΌ
π¦
οΏ½ (
π₯π΄
οΏ½
π΅)
π
4
π₯
οΏ½
2
π
π₯
9
οΏ½.12 .)c evloS : οΏ½
π«
π
οΏ½
ππ οΏ½
π
οΏ½
πππ
ππ
neviG
(
π·
2
οΏ½
61 )
π¦
οΏ½
πππ
9
π₯
si noitauqe yrailixuA
π
2
οΏ½
61
οΏ½
0
π
2
οΏ½
οΏ½
61
π
οΏ½
οΏ½ β
οΏ½
61
π
οΏ½
οΏ½
π
4
,ereH
πΌ
οΏ½
0
,
π½
οΏ½
4
β΄ noitcnuf yratnemelpmoC
οΏ½
π
ππΆ
οΏ½
π¨
πππ
ππ·
οΏ½
ππππ©
ππ·
οΏ½
οΏ½
πΆ
.
πΉ
οΏ½
π
0
π₯
οΏ½
π πππ΄
4
π₯
οΏ½
πππ π΅
4
π₯
οΏ½
οΏ½
π πππ΄
4
π₯
οΏ½
πππ π΅
4
π₯
ππππ’πππ‘πππ
ππππππ‘ππΌ
οΏ½
nis
π₯π
π
οΏ½
π·
οΏ½
οΏ½
nis
9
π₯
π·
2
οΏ½
61
ecalpeR
π·
2
οΏ½
οΏ½ [
9]
2
οΏ½
οΏ½
18
οΏ½
nis
9
π₯
οΏ½
18
οΏ½
61
οΏ½
nis
9
π₯
οΏ½
56
si noitulos lareneG
π¦
οΏ½
πΆ
.
πΉ
οΏ½
π
.
πΌ
β΄
π¦
οΏ½
π πππ΄
4
π₯
οΏ½
πππ π΅
4
π₯
οΏ½
nis
9
π₯
56
rewsnA