 $9$/,$d­2 '$  81,'$'( ±  6e5,( '2 (16,12 0e',2
',6&,3/,1$ 0$7(0È7,&$
3529$ (/$%25$'$ 3(/2 352)(6625 2&7$0$5 04548(6
5(62/8d­2 ( &20(17È5,2 3(/$ 352)$ 0$5,$ $17Ð1,$ *289(,$
4XHVW}HV GH D 6RPH RV YDORUHV GDV SURSRVLo}HV YHUGDGHLUDV H PDUTXH R
YDORU QD IROKD GH UHVSRVWDV
6REUH Q~PHURV UHDLV p YHUGDGH TXH
$ VRPD GH GRLV Q~PHURV LUUDFLRQDLV p XP Q~PHUR LUUDFLRQDO
4 3−4
1− 3
p XP Q~PHUR UDFLRQDO
∀ [ ∈ 5 x 2 ± [
∀ D E F ∈ 5 D E ⇒ DF EF
1
1
⇔ [ RX [ !
x
2
5(62/8d­2
)$/6$
3RUTXH D VRPD GH GRLV Q~PHURV LUUDFLRQDLV SRGH VHU XP Q~PHUR LUUDFLRQDO RX XP Q~PHUR
UDFLRQDO FRPR SRU H[HPSOR (
)
2 + − 2 = 0
9(5'$'(,5$
(4
) (
)
3−4
4 3 −1
=
= −4
1− 3
1− 3
TXH p XP Q~PHUR UDFLRQDO
9(5'$'(,5$
2
2
(0,2333....) =  21  =  7  = 49 =
900
 90   30 
2
6H [
)$/6$
∈5 − *
x2 − x = 2 x
3RU H[HPSOR VH [
)$/6$
3RUTXH FRPR D E H F VmR Q~PHURV UHDLV H F
WHPRV
(−2) 2 − (− 2) = 2 + 2 = 4 = 2 − 2 ≤ TXDQGR D WHUHPRV DF ≥ EF
9(5'$'(,5$
1
1 − 2x
<2⇒
< 0 DV UDL] GD IXQomR 1[
x
x
1
H D GD IXQomR '[
2
1 − 2x
WHPRV
x
[ p
)D]HQGR R HVWXGR GD YDULDomR GRV VLQDLV GH I[
10
[ p G
1D ILJXUD $%&'() p XP KH[iJRQR UHJXODU GH ODGR XF
3RGHVH DILUPDU TXH
2 kQJXOR ( Ĝ ' PHGH ž
E
$ GLVWkQFLD GR SRQWR * j UHWD $% p LJXDO D
$*
9 3 XF
6 5 XF
D
N
M
F
$ iUHD GR SROtJRQR $%&'*() p LJXDO D $ iUHD GR WULkQJXOR $*) p C
3 XD
3 XD
2V WULkQJXORV *(0 *01 H *1' WrP PHVPD iUHD
A
B
5(62/8d­2
)$/6$
2 kQJXOR ( Ĝ ' PHGH ž SRUTXH FRPR RV kQJXORV LQWHUQRV GH XP KH[iJRQR UHJXODU PHGHP
ƒ HQWmR RV kQJXORV
GÊD H GD̂E
PHGHP ƒ
9(5'$'(,5$
1D ILJXUD PRGLILFDGD GH DFRUGR FRP D OHLWXUD GD TXHVWmR YHPRV TXH R WULkQJXOR $%2 p HTLOiWHUR
6 3
= 3 3 6HQGR RV WULkQJXORV
2
GI ED
GI
6
=
⇒
=
⇒ 2GI = GI + 3 3 ⇒ GI = 3 3 ⇒ *+
GO FC
GI + 3 3 12
H
DVVLP
+2
11
*('
H
*)&
3× 3 3 = 9 3 VHPHOKDQWHV
)$/6$
2 WULkQJXOR $+* p UHWkQJXOR ORJR
2
⇒ AG = 252 = 6 7
9(5'$'(,5$
$ iUHD GR SROtJRQR $%&'*() p ×
( )
AG 2 = AH 2 + GH 2 ⇒ AG 2 = 3 2 + 9 3
36 3
= 63 3 4
9(5'$'(,5$
$ iUHD GR WULkQJXOR $*) p
1
1
3
× AF × FG × sen120° = × 6 × 12 ×
= 18 3 2
2
2
9(5'$'(,5$
2V WULkQJXORV *01 H *$% VmR VHPHOKDQWHV H D UD]mR HQWUH VXDV OLQKDV FRUUHVSRQGHQWHV p LJXDO j
UD]mR HQWUH DV DOWXUDV 3 3
9 3
=
1
MN 1
MN 1
= ⇒
= ⇒ MN = 2 (QWmR (0
ORJR
3
AB 3
6
3
1'
&RQFOXVmR RV WULkQJXORV *(0 *01 H *1' WrP EDVHV H DOWXUDV FRP PHGLGDV LJXDLV ORJR VmR
HTXLYDOHQWHV RX VHMD SRVVXHP D PHVPD iUHD
12
6HMDP &$
5 H &%
UHVSHFWLYDPHQWH e YHUGDGH TXH
5 RV SUHoRV GH FXVWR GDV PHUFDGRULDV $ H %
$ PHUFDGRULD % p PDLV FDUD TXH D PHUFDGRULD $
6H D PHUFDGRULD $ IRU YHQGLGD FRP XP OXFUR GH VREUH R SUHoR GH FXVWR HQWmR VREUH
R SUHoR GH YHQGD HVWH OXFUR VHUi GH 6H DV PHUFDGRULDV $ H % IRUHP YHQGLGDV FRP XP OXFUR GH H UHVSHFWLYDPHQWH
VREUH RV SUHoRV GH FXVWR HQWmR QD YHQGD GDV GXDV PHUFDGRULDV R OXFUR VREUH R SUHoR GH
FXVWR VHUi GH 6H QD YHQGD GHVVDV PHUFDGRULDV R OXFUR IRU GH VREUH R SUHoR GH YHQGD HQWmR HVWH
VHUi LJXDO D 5 6H D PHUFDGRULD $ IRVVH YHQGLGD GLDV DSyV D FRPSUD SRU 5 WHULD VLGR
PHOKRU SDUD R FRPHUFLDQWH WHU DSOLFDGR R YDORU GD FRPSUD SDUD UHQGHU MXURV VLPSOHV GH
DR PrV GXUDQWH RV GLDV
6H R YDORU WRWDO GD FRPSUD GDV PHUFDGRULDV $ H % IRVVH DSOLFDGR D MXURV FRPSRVWRV GH
DR PrV DSyV GRLV PHVHV R PRQWDQWH REWLGR VHULD 5 5(62/8d­2
9(5'$'(,5$
600
= 1,50 = 150%
400
)$/6$
× 9(5'$'(,5$
/ × × 9
&
⇒/
L
80 1
=
= = 0,1666...
V 480 6
/RJR
L
70
7
=
=
= 7%
V 1000 100
9(5'$'(,5$
9
& 9
⇒ 9
&
⇒9
C 1000
=
= 1250
0,8
0,8
)$/6$
6H D PHUFDGRULD $ IRVVH YHQGLGD SRU 5 R OXFUR HP GLDV FRUUHVSRQGHULD D XP
UHQGLPHQWR GH
100
= 25% 400
$SOLFDQGR 5 D MXURV VLPSOHV D DR PrV GXUDQWH GLDV R UHQGLPHQWR VHULD GH
0,10 × 72
= 0,24 = 24% 30
0
9(5'$'(,5$
ð
× × 13
6REUH VHTrQFLDV SRGHVH DILUPDU TXH
$ VRPD GRV SULPHLURV WHUPRV GD VHTrQFLD p 2 TXDUWR WHUPR GD 3$ ± log2 [ log2 [ p [
6H WUrV WHUPRV FRQVHFXWLYRV GH XPD 3* VmR [
[
H HQWmR D UD]mR GHVVD 3* p
1
16
x −1
x −1
x −1
+
+
+ ... = p [
3
9
27
$ VROXomR GD HTXDomR
6H D VRPD GR VHJXQGR FRP R TXLQWR WHUPRV GH XPD 3$ p H D VRPD GR WHUFHLUR FRP R
GpFLPR WHUPRV p HQWmR D UD]mR GHVVD 3$ p LJXDO D
2
3
$ VHTrQFLD [ [ [ [ WHP WHUPRV
5(62/8d­2
9(5'$'(,5$
$ VHTrQFLD p XPD 3$ GH UD]mR H D
D
WHUPRV YDOH
× VRPD GRV &RPR 6Q
(1 + 58)× 20 = 590 2 VHX YLJpVLPR WHUPR p
n (a 1 + a 20 )
D
2
VRPD GRV VHXV SULPHLURV
2
9(5'$'(,5$
6H D VHTrQFLD ± log2 [
YDOH
D UHODomR
log2 [ p XPD 3$ SDUD YDORUHV GH [ ! H [! ⇒ [ !
log2 [ log2 [ 5HVROYHQGR HVWD HTXDomR YHP log2 [ [ ⇒
log2 [ log2 [ ⇒ log2 [ log2 log2 [ ⇒ log2 [ð
log2 [ ⇒ [ð
[ð [ log2 [ ± 1± 1+ 8 1± 3
1
=
⇒ x = 1 ou x = - ∉ D( f ) 4
4
2
⇒ [
(QWmR [
H D VHTrQFLD
ID]HQGR D VXEVWLWXLomR GH [ SHOR YDORU p FXMD UD]mR p /RJR R TXDUWR WHUPR
GD p 9(5'$'(,5$
6H D VHTrQFLD [
[[
[
[
H ⇒ [
[
[
p XPD 3* HQWmR ð
⇒[
[
[
4x
2 x +3
− 1 $ UD]mR GD 3* p T
⇒ 2
4x
= 2 x +3.2 6x ⇒
6XEVWLWXLQGR [ SRU ± YHP T
4 −1
2 −2
1
= 2 =
−1+ 3
16
2
2
)$/6$
$ VROXomR GD HTXDomR
x −1
x −1
x −1
+
+
+ ... = p [
3
9
27
2V WHUPRV GR SULPHLUR PHPEUR GD HTXDomR DFLPD VmR WHUPRV GH XPD SURJUHVVmR JHRPpWULFD
LQILQLWD GH UD]mR
a1
1
(QWmR D VRPD GHVVHV WHUPRV WHP FRPR OLPLWH
=
3
1− q
14
x −1
3 = x −1 1
2
1−
3
x −1
= 6 ⇒ x − 1 = 12 ⇒ x = 13 2
9(5'$'(,5$
a 2 + a 5 = 8
2a + 5r = 8
2
⇒ 1
⇒ 6r = 4 ⇒ r =

3
a 3 + a 10 = 12 2a 1 + 11r = 12
)$/6$
$ VHTrQFLD GRV H[SRHQWHV GD 3* [ [ [ [ IRUPDP XPD 3$ QD TXDO
Dn
HQWmR Q
⇒ Q ±
⇒Q
a1 = 1 T
H
C
V
B
A
1D ILJXUD YHPRV XP FXER GH DUHVWD FP H D SLUkPLGH GH YpUWLFH 9 H EDVH $%&
e YHUGDGH TXH
$ SLUkPLGH p WULDQJXODU UHJXODU
6 3 FP
$ DUHVWD GD EDVH GD SLUkPLGH p LJXDO D
2 UDLR GD EDVH GHVVD SLUkPLGH p LJXDO D
2 6 FP
$ DOWXUD GD SLUkPLGH UHODWLYD j EDVH $%& p LJXDO D K
2 3 FP
2 YROXPH GD SLUkPLGH p XY
2 YROXPH GD HVIHUD LQVFULWD QR FXER p LJXDO D π XY
5(62/8d­2
9(5'$'(,5$
&RQVLGHUDQGR D EDVH GD SLUkPLGH FRPR R WULkQJXOR UHWkQJXOR %9& H FRPR DOWXUD
YROXPH VHUi , 9
6H FRQVLGHUDUPRV
AV R
VHX
1 36
× × 6 = 36 3 2
FRPR EDVH GD SLUkPLGH R WULkQJXOR HTLOiWHUR $%& FXMRV ODGRV VmR
GLDJRQDLV GDV IDFHV GR FXER TXH PHGHP
6 2
( )
1 6 2
×
3
4
6 3h = 36 ⇒ h = 2 3 SRGHUi VHU FDOFXODGR GR VHJXLQWH PRGR ,, 9
&RPSDUDQGR , H ,, WHUHPRV
H FRPR DOWXUD
2
VH GH
3
PHGLGD K R VHX YROXPH
× h = 6 3h 3DUD TXH D SLUkPLGH VHMD UHJXODU + WHP TXH VHU R FHQWUR GD EDVH $%& RX VHMD
15
2 6 2 3
×
= 2 6
3
2
$+
&RPSURYHPRV HVWH YDORU DSOLFDQGR R WHRUHPD GH 3LWiJRUDV QR WULkQJXOR UHWkQJXOR 9+$ $+ð
(2 3 )
$+ð
2
± ⇒ $+
6
)$/6$
$R UHVROYHUPRV R LWHP DQWHULRU YLPRV TXH DV DUHVWDV GD EDVH PHGHP
6 2 FP
9(5'$'(,5$
2 UDLR GD EDVH GHVVD SLUkPLGH p $+
2 6 FP
9(5'$'(,5$
$LQGD GD UHVROXomR GR LWHP SRGHPRV FRPSURYDU TXH D DOWXUD GD SLUkPLGH UHODWLYD j EDVH
$%& p LJXDO D K
2 3 FP
)$/6$
3HORV FiOFXORV DFLPD YHPRV TXH R YROXPH GD SLUkPLGH p XY
9(5'$'(,5$
2 UDLR GD HVIHUD LQVFULWD QR FXER p LJXDO D FP (QWmR R VHX YROXPH p
4Œ × 33
= 36ΠXY
3
6REUH H[SRQHQFLDLV H ORJDULWPRV SRGHVH DILUPDU TXH
2 FRQMXQWR VROXomR GR VLVWHPD $ IXQomR \
b
 
a
E D D
SRVVXL VROXomR LQWHLUD
≠ R E ≠ R p FUHVFHQWH
[
$GPLWLQGR TXH ORJ
^[
[±
−x
D VROXomR GH p [
4
3
∈ 5 log1 3 [± ! ± ` ⊂ > @
2V JUiILFRV GH \
log2 [± H \
TXDGUDQWH
$ LQYHUVD GD IXQomR \
±
log2 x VH LQWHUFHSWDP QXP SRQWR GR SULPHLUR
log4 [± p D IXQomR \
16
±[
5(62/8d­2
)$/6$
[±
&RPR D EDVH GR VLVWHPD p /RJR R VLVWHPD QmR SRVVXL VROXomR LQWHLUD
HQWmR [ ± ⇒ [ 9(5'$'(,5$
&RPR QR Q~PHUR SRVLWLYR
IXQomR \
b
 
a
−x
a
 
b
9(5'$'(,5$
$GPLWLQGR TXH ORJ
b
a
WHPVH E D D
≠ R E ≠ R HQWmR
6HQGR
⇒
a
b
! /RJR D
x
p FUHVFHQWH
D VROXomR GH ORJ [
ORJ log5
log10 − log2
1 − 0,3
7
4
⇒x=
−1 =
−1 = −1 = log2
log2
0,3
3
3
[
b
a
log1 3 [± ! ± ⇒ log1 3 [± ! log1 3
1
 
 3
⇒ [ ORJ ORJ ⇒
−1
6HQGR D EDVH GR ORJDUtWPR PHQRU
TXH HVWDPRV GLDQWH GH XPD IXQomR GHFUHVFHQWH H [ ± R VHX ORJDULWPDQGR HQWmR
[ ⇒
[ (QWmR p YHUGDGHLUR TXH ^[
∈ 5 log1 3 [± ! ± ` ⊂ > @
)$/6$
6H KRXYHU LQWHUVHomR HQWUH RV JUiILFRV GDV IXQo}HV DV FRRUGHQDGDV GHVWH SRQWR VHUmR GDGDV
SHOD VROXomR GR VLVWHPD IRUPDGR SHODV HTXDo}HV \
log2 x FRP
log2 [± H \
>[ ! H [ ! ⇒ [ ! @
,JXDODQGR RV GRLV YDORUHV GH \
log2 [±
9(5'$'(,5$
log 2 2 log2 x ⇒
log2 x ⇒ log2 [±
log2 [±
⇒[
YDORU TXH QmR SHUWHQFH DR GRPtQLR GD IXQomR
[
$ LQYHUVD GD IXQomR \
±
log4 [± p D GDGD D SDUWLU GD UHODomR [
log4 \ ⇒
log4 \
log2 [ ⇒ [ ± ± [
⇒\±
41− x ⇒
\
±[
6REUH FRPELQDWyULD SRGHVH DILUPDU TXH
A 2n + 3 = 2 + 4Cn2 ⇒ n ∈ {1, 2, 3, 4}
2 Q~PHUR GH VXEFRQMXQWRV GH ^D E F G H` FRP HOHPHQWRV VHQGR D XP GRV VHXV
HOHPHQWRV p LJXDO D 2 Q~PHUR GH ILODV TXH SRGHP VHU IRUPDGDV FRP UDSD]HV H PRoDV GH PRGR TXH QmR
ILTXHP MXQWRV GRLV UDSD]HV RX GXDV PRoDV p LJXDO D &RQVLGHUDQGRVH RV YpUWLFHV H R FHQWUR GH XP RFWyJRQR UHJXODU R Q~PHUR GH WULkQJXORV
TXH VH SRGH REWHU OLJDQGRVH HVVHV SRQWRV p LJXDO D 2 Q~PHUR GH DQDJUDPDV GD SDODYUD &$&$78$ TXH FRPHoD SRU FRQVRDQWH p SHVVRDV SRGHP VHQWDUVH HP XPD ILOD GH FDGHLUDV GH PRGRV
17
5(62/8d­2
)$/6$
A 2n + 3 = 2 + 4Cn2 ⇒ n ∈ {1, 2, 3, 4}
 n (n − 1) 
2
2
2
A 2n +3 = 2 + 4C 2n ⇒ (n + 3)(n + 2 ) = 2 + 4
 ⇒ n + 5n + 6 = 2 + 2 n − n ⇒ n − 7 n - 4 = 0
 2! 
(
)
$V UDt]HV GHVWD HTXDomR VmR Q~PHURV LUUDFLRQDLV
9(5'$'(,5$
1D UHSUHVHQWDomR GH XP FRQMXQWR QD IRUPD GH OLVWDJHP D RUGHP GRV HOHPHQWRV QmR p
LPSRUWDQWH RX VHMD ^D E F `
^D F E `
2V VXEFRQMXQWRV HP TXHVWmR GHYHP WHU HOHPHQWRV HQWUH RV TXDLV GHYH DSDUHFHU VHPSUH R
HOHPHQWR D DVVLP D TXDQWLGDGH WRWDO GH VXEFRQMXQWRV VHUi GDGR SHOD FRPELQDomR GRV RXWURV
HOHPHQWRV DJUXSDGRV
D C 24 =
4×3
= 6
2
9(5'$'(,5$
&RPR RV UDSD]HV H DV PRoDV GHYHP IRUPDU ILODV GH PRGR TXH GXDV SHVVRDV GR PHVPR
VH[R QmR HVWHMDP MXQWDV WHUHPRV VHPSUH D GLVSRVLomR D VHJXLU 6HQGR TXH RV UDSD]HV
GHYHUmR VHPSUH SHUPXWDU RV OXJDUHV HQWUH VL EHP FRPR DV PRoDV
5
0
/RJR R Q~PHUR WRWDO GH ILODV p × 5
0
×
5
0
5
)$/6$
2 Q~PHUR GH WULkQJXORV VHUi HQFRQWUDGR ID]HQGR D FRPELQDomR GRV SRQWRV D H
GLPLQXLQGR R Q~PHUR GH GLDJRQDLV TXH SDVVDP SHOR FHQWUR
C 39 +
n 9× 8× 7 8
=
− = 84 − 4 = 80
2 3 × 2 ×1 2
9(5'$'(,5$
9DPRV LQLFLDOPHQWH GHWHUPLQDU D TXDQWLGDGH GH DQDJUDPDV TXH FRPHoDP SHOD OHWUD F
3DUD D SULPHLUD FDVD j HVTXHUGD WHPRV SRVVLELOLGDGH H SDUD DV
& $ & $ 7
VHLV
FDVDV
UHVWDQWHV
WHUHPRV
6! 6 × 5 × 4 × 3/ × 2/ × 1
=
= 3!
3/ × 2/ × 1
8 $
SRVVLELOLGDGHV
$JRUD YDPRV GHWHUPLQDU D TXDQWLGDGH GH DQDJUDPDV TXH FRPHoDP SHOD OHWUD
3DUD D SULPHLUD FDVD j HVTXHUGD WHPRV SRVVLELOLGDGH H SDUD DV
VHLV FDVDV
UHVWDQWHV WHUHPRV
6!
6 × 5 × 4 × 3/ × 2/ × 1
=
= 2!×3!
2 × 3/ × 2/ × 1
/RJR R WRWDO GH SRVVLELOLGDGHV p 7
W
& $ & $ 8 $
SRVVLELOLGDGHV
)$/6$
&RPR D RUGHP HP TXH DV SHVVRDV SRGHP VHQWDU p LPSRUWDQWH HODV SRGHUmR VHQWDUVH HP
ILOD GH
× ×× PRGRV GLIHUHQWHV
18
y
D
A
C
0
1D ILJXUD $%&' p XP UHWkQJXOR WDO TXH &
x
B
H '
± e YHUGDGH TXH
$ iUHD GHVVH UHWkQJXOR p LJXDO D XD
$ HTXDomR GD UHWD $& p [ ± \ $ HTXDomR GD UHWD SDUDOHOD j UHWD $& SDVVDQGR SRU % p \ ± [ $ HTXDomR GD UHWD TXH SDVVD QD RULJHP H p SHUSHQGLFXODU j UHWD $& p [ \
$V UHWDV [
H \
VmR HL[RV GH VLPHWULD GR UHWkQJXOR $%&'
2 SRQWR &
± p REWLGR SRU PHLR GH XPD URWDomR GR SRQWR & QR VHQWLGR SRVLWLYR
HP WRUQR GR SRQWR 2
5(62/8d­2
)$/6$
$ iUHD GHVVH UHWkQJXOR p LJXDO $%
> ± @
× ± ×
XD
9(5'$'(,5$
$ UHWD
AC
SDVVD SHORV SRQWRV SDUWLU GD UHODomR
p [ ± \ × %&
3 −1 x 3
=0 ⇒
2 0 y 2
H HQWmR D VXD HTXDomR SRGH VHU HQFRQWUDGD D
±\ [ ±\ ⇒ [ ± \ ⇒ [ ± \ 9(5'$'(,5$
$ HTXDomR UHGX]LGD GD UHWD AC p \
HOD SDUDOHOD
WDPEpP
p
1
2
1
1
x + ORJR R VHX FRHILFLHQWH DQJXODU H GH WRGD UHWD D
2
2
2 FRHILFLHQWH DQJXODU GD UHWD TXH WHP FRPR HTXDomR
1
H SDVVD SRU % SRLV ± 2
\ ± [ p
)$/6$
$ HTXDomR GH XPD UHWD TXH SDVVD QD RULJHP p GD IRUPD \
AC WHP FRPR FRHILFLHQWH DQJXODU ± HQWmR \
19
[
D[ 6H HOD p SHUSHQGLFXODU j UHWD
⇒ [ \
9(5'$'(,5$
2EVHUYDQGR D ILJXUD YHPRV TXH RV SDUHV GH SRQWRV & H ' H % H $ VmR VLPpWULFRV HP
UHODomR j UHWD [
H TXH RV SDUHV % H & H $ H ' VmR VLPpWULFRV HP UHODomR j UHWD \
/RJR DV UHWDV [
H \
VmR HL[RV GH VLPHWULD GR UHWkQJXOR $%&'
9(5'$'(,5$
2 SRQWR &
± p REWLGR SRU PHLR GH XPD URWDomR GR SRQWR & QR
VHQWLGR SRVLWLYR HP WRUQR GR SRQWR 2
4XHVW}HV H (IHWXH RV FiOFXORV QHFHVViULRV H PDUTXH R UHVXOWDGR QD )ROKD
GH 5HVSRVWDV
&RQVLGHUH D HTXDomR PDWULFLDO
1 2 
 0 1
X
A+X
AB
  e B = 
+
=
RQGH A = 
3
2
6
 3 − 1
 3 2
&DOFXOH R YDORU GD H[SUHVVmR GHW; ± GHW$
5(62/8d­2
X
A+X
AB
+
=
⇒ ; $ ;
3
2
6
⇒ ;
$VVLP GHW; ± GHW$
5(63267$ ± $% ± $
⇒ ;
3
−1
⇒ GHW;
− 12 4
 6 5  3 6 
 - 

 ⇒ GHW ; −
3
1
9
−
3

 

;
$%
1 2   0 1 1 2 
 - 3

 

3
−
1
3
2
3
−
1


 

±
⇒ GHW;
&RPSUHL FHUWR Q~PHUR Q GH WHOHYLVRUHV LJXDLV SRU 5 6H R SUHoR GH FDGD XP GHOHV
IRVVH UHGX]LGR HP 5 FRP D PHVPD TXDQWLD HX SRGHULD FRPSUDU Q WHOHYLVRUHV
&DOFXOH R YDORU GH Q
5(62/8d­2
6H
Q
WHOHYLVRUHV LJXDLV IRUDP FRPSUDGRV SRU 5 3600
n
WHOHYLVRU
UHDLV
6H R SUHoR GH WHOHYLVRU WLYHVVH VLGR UHGX]LGR GH 5 SRU
,
IRL DGTXLULGR DR SUHoR GH
 3600

− 90  UHDLV

 n

WHOHYLVRUHV
WHUVHLD SDJR
H SRGHULDP WHU VLGR FRPSUDGRV FRP RV PHVPRV 5 R TXH HTXLYDOH GL]HU TXH R SUHoR GH
,JXDODQGR DV VHQWHQoDV , H ,, WHUHPRV
Q ± Qð Q
WHOHYLVRU
Q
⇒ Qð Q ± WHOHYLVRU
 3600

− 90 

 n

⇒Q
5HVSRVWD 20
RX
YDOHULD ,,
3600
 40 
−1
⇒ 
n+2
 n

n/ = -1/ 0/ 3600
n+2
40
⇒
n+2
PDLV 
Download

9$/,$d2 `$ 81,`$`( ± 6e5,( `2 (16,12 0e`,2 3529$ (/$%25$`$ 3(/2 352