unidad 8 - funciones, límites y continuidad - problemas resueltos
TRANSCRIPT
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8/11/2019 Unidad 8 - Funciones, Lmites y Continuidad - Problemas resueltos
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1 BACHILLERATO - Matemticas - Unidad 8 - Funciones, !mites " continuidad
#$%18& 'I
#$%18& ' II a( =+ 03x ( ) NoExistem == 90=)( = 22xy 2=m 32 = xy '2663c( = 25y 0=m = 0..tgarc 0= d( ( ) += 3132 xy 32=m ( )= 32..tgarc '18146 e( = 02x ( ) NoExistem == 90=*( = 4xy 1=m = 1..tgarc 45=
#$%18& ' III a( ( )( ) 081 + xx 1 8 +1+x + +8x +
( )( )81 + xx + + [ ]8181 xRx )( ( ) 03 22 >+xx +
x + ( ) 22 3+x + + ( ) 22 3+xx + ( )+> 00xRx
#$%18 ' 1 a( ( ) = drticaFuncinCuaxf ( ) RfD =)( ( ) = rsaalidadInveproporcionxf ( ) [ ]2= RfD
#$%18 ' 2 a( ( ) [ )+= 0fD )( ( )( ) + 0111 2 xxx 11: xD
( )( ) + 01112 xxx 11: xxD ( ) [ ]11= RxfDc( ( ) RfD =
#$%18 ' 3 #o. e/em0o 10 x ctaRe
75 x SimpleCurva
#$%18 ' 4 a( ( ) ( )4=fD ( ) ( )3=fR)( ( ) [ ] [ ]5.25.110 =fD ( ) ( )11=fR
#$%182 ' 5 1341.650 13541.860 13642.300 1343.040 13844.150
#$%182 ' 6 La grfica c)% La 7ue incu"e un dec.ecimiento de a distancia, asta ce.o#o. tanto, es a 9nica 7ue .e0.esenta una :;ueta a casa
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1 BACHILLERATO - Matemticas - Unidad 8 - Funciones, !mites " continuidad
La tablaes e>tensa% La frmulaes en$o..osa%
La grficaes a ms con;eniente%
#$%182 ' 8 a( En ( ) ( )1=fD , e ( ) parbolafR =En ( ) ( )= 1fD , e ( ) rectafR =
)(
#a.a 1
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1 BACHILLERATO - Matemticas - Unidad 8 - Funciones, !mites " continuidad
x 1 5 1, 1,2 1, 1,1 1,@ 1,6 1,@& 1,@P ' + ' + ' + ' + ' +
( ) [ ]6036,16035,1101 f
#$%131 ' 13 a( x ',3 '1,1 ',33 '1,1 ',333 '1,1f ',&5@ ',625 ',&5& ',632& ',&@ ',6338 21. =lm
)(
x ,3 2%1 ,33 2,1 ,333 2,1 f 1,33@2 5,5 1,3336 5, 1,3333 5,1 2. =lm c( x @,3 6,1 @,33 6,1 @,333 6,1 f @,3268 6,568 @,332& 6,5& @,3332 6,5 4. =lm d( x ',1 ,1 ',1 ,1 ',1 ,1 f @,3@6 @,3@6 @,3 @,3 @, @, 3. =lm e( x 1,3 5,1 1,33 5,1 1,333 5,1
f 1,583 1,&&56 1,6 1,658@ 1,6158 1,61& 2. =lm*( x 2,3 8,1 2,33 8,1 2,333 8,1 f 11,3633 15,633 11,33& 15,& 11,333& 15,& 12. =lm
#$%131 ' 14 a( ( )=af 2 ( )=bf 0 ( )=cf NoExiste )( ( )= axlm 2 ( )= +axlm 2 ( )= axlm 2 ( )= bxlm 3 ( )= +bxlm 0 ( )= bxlm NoExiste ( )= cxlm + ( )= +cxlm NoExiste ( )= cxlm NoExiste
#$%135 ' 15 a( +
+=
+
+
222
2
2
11
11
1
1
x
xx
xx
x
xx
x
=++
1
0
01
00 ( )=+xlm 0
)(
+=
+
x
x
xx
x
xx
x
23
15
23
15
=+
3
5
03
05 ( )=+xlm
3
5
c( +
=
+ 17
1
7
1
7
7
7
1
7
1
!
!
x
x
x
x
==+
01
0
10
0 ( )=+xlm 0
d( +
+
=+
+
3
3
33
3
33
2
3
3
82
133
82
133
x
xx
xx
xxx
x
x
x
=+ + 23
02003 ( )=+xlm 2
3
e( +++
++
=
+++
++
432
43
444
2
4
3
444
4
1111
111
1
1
xxxx
xx
xx
x
x
x
x
x
xx
x
x
x
+==+++++
0
1
0000
001 =lm +
@=51
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1 BACHILLERATO - Matemticas - Unidad 8 - Funciones, !mites " continuidad
*( +
+=
+
+
32
32
333
3
33
326
13
326
13
xx
xx
xx
x
x
x
xx
x
==+
+0
6
0
006
00 ( )=+xlm 0
#$%135 ' 16 a( =xxx
xx
x
71
2
7
2
==+ 212
01
2
( )=xlm 2
)(
+=
+
7
72
77
7
77
5
13
12
13
12
x
xx
xx
x
xx
x
==+
03
0
03
00 ( )=xlm 0
c( ( )
( ) == xxxx
xx
3232
1
32
33 ( ) < + 01 ( )=xlm 0
d( +=
+
3
2
33
2
33
3
17
2
1
17
2
xx
x
xx
x
x
x
x
x
=
00
01 ( )=xlm
e( ( ) ( )
+x
x
xlmxlm72
1
2
7 0
1
+ ( )=xlm 0
*( ( ) ( )[ ]+ x
xxlmxlm 43
4
3 22
( )=xlm +
#$%13@ ' 17 a( ( ) =
0
12xlm )( ( ) = +
0
97xlm +
c( ( ) = +0
11xlm d( ( ) = +
++
0
33xlm +
e( ( ) = +
+
0
65xlm *( ( ) = +
0
44xlm +
#$%13@ ' 18 a( ( ) axlm + ( ) +axlm +)( ( ) axlm + ( ) +axlm NoExistec( ( ) axlm NoExiste ( ) +axlm +
#$%13& ' 19 a( =+
4
0
22
220 )(
( ) ( )
( ) ( ) +
=+
2
5
22
52
x
x
xx
xx
=+
4
3
22
52
4
3
#$%13& ' 20 a( ( ) ( )= glmflm =03 0 )( ( )
( ) ==0
3
glm
flm ( )0. =gSi"m
c( ( )[ ] ( ) == 03glmflm 1 d( ( )[ ] ( ) == 30flmglm 0
#$%13& ' 21
+
3
3
3
2
3
333
3
22
3
x
x
x
x
x
xx
x
x
x
=+
200
001
2
1
+
xx
x
xx
x
6
222
==+
1
0
01
000
6=51
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#$%132 ' 22 a( #ER$IC%"ES 092 =x 3=x 3=x
( ) = +
0
73xlm ( ) =
+
0
73xlm +
( ) =
0
53xlm ( ) = +
+
0
53xlm +
ES&'RI('N$%"
22
2
22
9
12
xx
xxx
x
( ) 010x 0 0=y
')"IC*%S ( ) 0:9
122
=
xx
xxlm No$iene
)( #ER$IC%"ES =++ 0232 xx ( )( ) =++ 012 xx 2=x 1=x
( ) ( )( )
=
10
132xlm ( ) ( )( )
=
++
10
132xlm +
( ) ( )( )=
01
21xlm ( )
( )( )=
++
01
21xlm +
ES&'RI('N$%" ( )xlm ( ) ++++
001
003 No$iene
')"IC*%S ( )xlm ++
++
23
5273
2 xx
xx 73 = xy
c( #ER$IC%"ES ( )
0
50xlm 0=x
( ) =
0
50xlm ( ) =
+
+
0
50xlm +
ES&'RI('N$%" ( )xlm =+3
02
3
2
3
2=y
')"IC*%S ( ) 0:3
52=
+ x
x
xxlm No$iene
d( #ER$IC%"ES ( ) 0xlm +
0
10 0=x
ES&'RI('N$%" ( )=xlm No$iene
')"IC*%S ( ) +x
xxlm 1
4 xy 4=
#$%132 ' 23 a( #ER$IC%"ES ( ) =xlm No$iene
ES&'RI('N$%" ( )
=xlm
No$iene ')"IC*%S ( ) =+=
x
xxlmm
32
1
( )[ ]= mxxflmn + xx 32 =+
++
xx
xxxx
3
33
2
22 ( ) ( )
+
+
xx
xx
3
3
2
22
0
+= nmxy xy = )( #ER$IC%"ES ( ) 3xlm + 3=x
ES&'RI('N$%" ( ) +=xlm No$iene
')"IC*%S ( ) =+
=
+
=23
33
3
1:
3
1
xx
xlmx
x
xxlmm 1
&=51
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( )( )
=
+
+
=
+
+
=
+
=
xx
xx
xlm
xx
x
xx
x
lmmxx
xxlmn
3
13
13
3
1
3
1
3
1
3
2
3
23
3
2
3
x 23+= xy +x 23+= xy
c( #ER$IC%"ES #a.a yRx 022
+x
No$ieneES&'RI('N$%" ( )=xlm 3 3=y
')"IC*%S ( ) 0:2
132
2
=
+
xx
xxlm No$iene
d( #ER$IC%"ES ( ) +=+ 013xlm =+ 013x 1=x ES&'RI('N$%" ( ) 0=xlm 0=y
')"IC*%S ( ) 0:13
=
+ x
x
xxlm No$iene
#$%133 ' 24 a(
polinmicaf
= continuaf
Rx
)( =1x f = continuaf ( )1 Rx c( = 012x = 1x f = continuaf ( )11= Rx d( =0x f = continuaf ( )0Rx e( Rx 012 +x = continuaf Rx=
*( ( )( )
43
2
xx
xf = 43x =f = continuaf ( )43= Rx
$( aParteEnterf EnteroxNoDefinida = =continuaf (Rx = ( polinmicaf =continuaf Rx , si ( ) ( )+ 11 xlmxlm
[ ] [ ]= xlmfxlm 2)1( 2 ( )[ ] [ ]=+ + 11 xlmfxlm 2
( )1
f 0
( ) ( ) ( ) idaddiscontinufxlmxlm = + 111 ( )1 Rx
#$%133 ' 25 a( ( )( )
+1
11
x
xxlm ( ) 21 =+xlm ( )=1f 2
)( ( ) ( )( ) ( )
++
22
21
xx
xxlm
4
3
2
1=
++
x
xlm ( )=2f
4
3
#$%133 ' 26 ( )( )
( )( )
+21
11
xx
xxlm =
+
=+
21
11
2
1
x
xlm 2 12 + +
2
3=+
#$%51 ' 27 a( ( ) ( )( )22
3253
nnlmnlm
+=+
2
3
)( ( )nnn
nlmnlm
2
21
+
+=+ =+ 01
11
c( ( ) 2222
1
11
nnn
nnlmnlm
++
=+ =++10
000
d( ( )nnnnn
nnnnnlmnlm
1
1
++
+=+ ==+
2
0
11
110
#$%51 ' 28 a( ( ) ( )
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1 BACHILLERATO - Matemticas - Unidad 8 - Funciones, !mites " continuidad
c( ( ) =+
+=+
nnn
nnnlmnlm
31
1
+
+611
11
nn
nlm =
++
00
01 + nteNoConverge
#$%51 ' 29 a( =na ( ) nn 1 =na ( ) ( )nnne 1)( =na n =na n c( =na n =na ( ) en 32
d( =na ( ) ( )71 n
n
=na ( )1
23
++ nnn
#$%51 ' 30 a( ( )
( ) =
++
++++
3
32
21
312
n
n
n
n
( )( )23
3
++ nn 0> 1+< nn aa Creciente
)( ( ) ++
=+n
nlmnlm
21
32=
1
22 2 Acotada superiormente
#$%5@ ' 31 a( ( ( ++
=++
+++
nnnn
nnnn
1
1
1
11
1
0
)( =++
+++nnnn
nnnnnnnn22
2222
3
33
nnnn
n
++
223
4
=++
++
=
2
4
0101
4
3
4
22
2
22
2
n
n
n
n
n
n
n
n 2
c( =++
+=
++
+++
nnn
n
nnn
nnnnnn
51
51
51
5151
2222
2222
++
+=
nn
n
5111
15
2
+=
+++
11
5
0101
05
2
5
#$%5@ ' 32 a( ( )
( )( )
+
+=
++
++
++ 152
215
21
11
1
21
n
nnn
nn
2e
)(
++=
++
+
+++ nn
n
nnn
nnnn
3
1
3
2
1
2
22
3
11
3
11 =0e 1
c( ( )
+
+=
++
+
+ 7
6
6
7
67
11
7
61
n
nn
n
nn
6e
d(
+=
+
32
32
3
11
3
11
n
n
nn
nn
2e
e(
++=
++
++
++ 72
72
7
11
7
11
n
nnn
nn=1e e
*( ( ) ( )
++
+=
+++
++
++
++ 1
5
5
3
2
1
3
2
22
53
11
3
51
n
n
n
n
n
n
nnn
n 1e
#$%5 ' 33 a( 012 +x ( ) =fD R)( = 01x 1=x ( )=fD { }1Rc( =+ 02x 2=x 042 x = 00f 2xf 4=f
En 0.inci0io, ( )=fD { }2R #e.o es admisi)e, =D R d( = 042x 2=x 2=x ( ) = 002f ( ) 21 xf 41=f
2=51
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$(
+
2
2
34
12
x
x ==+
4
2
04
02
2
1(
+
3
32
11
15
x
xx ==+
1
0
01
000
#$%52 ' 46 a( +
+
2
214
51
xx
x
=++
004
01
2
1
)( +
x
xx
12
3383
3
==+
2
2
02
0083
1
c(
+
+=
++
x
x
x
xx
51
3
5
3
=+01 + d(
510
5
10
13
2
61
x
x
x
x
x ==
1010
#$%52 ' 47 a( = 00 0
)( ( )( )
( )
++
=
++
+++
xxxx
xxxx
3
3
3
33 =
30
c( ( ) +
=+
x
xx
x
11
1
1
12
==+ 1
1
01
11
d( ( )( )
++=
+++++
2
11
1111
11 xx
xxxx
xx =
+2
+
#$%52 ' 48 a( ( )=+xlm + ( )=xlm )( ( )=+xlm ( )=xlm +
#$%52 ' 49 a( ( )
2
1
3
x
x =0
2 )( ( ) =
xxx
x 1
1
1 =0
1
c( ( ) =
xxx
x 1
1
1 =+0
1 + d(
( )
=
1
1
1
12 xx
x=+
0
1 +
e( NoExiste ( )"""" gyf$ras *( ( )
2
33
xx
x=
+025
+
$( ( )
1
33
xx
x = 025 (
x
xx1
1
3
1 2 ( ) =1
1
#$%52 ' 50 a( ==+
2
10
24
645 )( ==
+6
18
39
993
c( =
=+
2
2
11
311 d(
( )
( ) =
+
+
=+
10
30
1
3
1
3
x
x
xx
xx3
e == 16952 4 *(( )( ) +=
+1
1
11x
x
xx2
$( == 0399 22x 0 ( ( ) =
22 399 x +0
i( ( )( )
( )( ) +
=+
2
1
21
11
x
x
xx
xx=
+21
112 /( ( )( )
+
=+
1
1
11
1
xxx
x
2
1
1=51
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( ( )( )( )
( )( ) ( )( )++=
+
++310
33
3109xx
xx
xxx( ) ==++ 61939109 114
(( ) ( )
( )
+=
+
x
x
xx
xx 1
2
21==
+1
2
1
112
#$%52 ' 51 ( ) ( ) ( ) ( ) +=+=++ 323233 22
-x
-
-x-
-
xx-x-x32 x
#$%52 ' 52 ( )( ) ( )( )
( )( )
( )( )
+++
=+++
=++
+=
++
2222
21
2
11
12
11
2
11
3
1
1
xx
x
xxx
xx
xxx
xx
xxxx 1
#$%52 ' 53 a( 2xx
= 200
0 )( NoExiste ( )"""" dyc$ras
c(
21
1 =
= 1
1
21
11 d( ( ) + 112 ==+ 1212 1
e( ( ) = 16132 5 *( ++=++ ++++ 142144 7 $( ( ) = 18142 7 ( ++=++ 142144 7
#$%58 ' 54 a( Funcin 0oinmica No$iene )( #ER$IC%"ES y + 012x No$iene
ES&'RI('N$%" ( ) =+
+ 2
01
2
11
22
xxlm 2=y
')"IC*%S Gi 0m ( ) 0:1
22
2
+
= xx
xxlmm No$iene
c( #ER$IC%"ES y =+ 05x 5=xES&'RI('N$%" ( ) 0xlm 0=y
')"IC*%S Gi 0m ( )
+= 0:
5
3x
xxlmm No$iene
d( ( )( )
( )( ) +
=+
3
1
31
11
x
x
xx
xx #a.a 1x Funcin ! "efi!i"aen 1=x
#ER$IC%"ES y = 03x 3=x
ES&'RI('N$%" ( ) +
= 1341
112
2
xx
xxlm 1=y
')"IC*%S Gi 0m ( )
+
= 0:
34
12
2
xxx
xxlmm No$iene
e( #ER$IC%"ES y =+ 01x 1=xES&'RI('N$%" ( ) xlm No$iene
')"IC*%S Gi 0m ( )
+
= xx
xxlmm :
1
2
1
( ) ( ) +
=+
=
+=
xx
xx
x
xxlmn
11
1
11
1
2
1 1= xy
*(
#ER$IC%"ES
y
=++ 012
xx = imaginariox
No$iene
ES&'RI('N$%" x y No$iene
')"IC*%S Gi 0m ( )
++++
= xxx
xxxlmm :
1
722
3
1
11=51
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( ) ( ) +++
=
++++
=1
731
1
722
2
2
3
xx
xxx
xx
xxxlmn 1 1= xy
$( #ER$IC%"ES y +0x 0=x JGo ado +0 (ES&'RI('N$%" ( ) xlm No$iene
')"IC*%S Gi 0m ( )
+= x
x
xxlmm :
13
+
1
11 3
x1
( ) ( ) =
+=
xx
x
xxlmn
11
13
0 xy=
( #ER$IC%"ES y 4x
+4
17112 2
x
xx 4=xES&'RI('N$%" ( ) xlm No$iene
')"IC*%S Gi 0m ( )
+= x
x
xxxlmm :
4
17112 2
2
( ) ( ) +
+
=4
1732
4
17112 2
x
xx
x
xxxlmn 3 32 = xy
#$%58 ' 55 ++=
cx
bay x ay 2y 2=a
y cx =1x 1=c
( )
+=16
263b
f
=16
23b
5=b
#$%58 ' 56
( )( )( )( )xx
xx
+
+31
2121
#ER$IC%"ES y 1=x 3=xES&'RI('N$%" x ylm. 2 2=y
')"IC*%S Gi 0m
( )
+
= 0:
23
212
2
xxx
xxlmm No$iene
#$%58 ' 57 ( )( )
( )( ) =
+
=51
52
xx
xxy
1
2
+
x
x ( ) ==
+
=6
3
15
255xlm
2
1 )( o#alorFinit
En consecuencia, 5=x !0uede se. as!ntota
#$%58 ' 58 ( )( )( )
( )( ) =
++
=11
311
xx
xxxy 3x
#a.a 1=x a" 7ue asi$na. e ;ao. ( ) = 311xlm 2#a.a 1=x a" 7ue asi$na. e ;ao. ( ) = 311xlm 4
#$%58 ' 59 a( ( ) polinmicaxf = dContinuida K Rx= )( 042 >+x dContinuida K Rx= c( = 02x 2=x dContinuida K { }2= Rx
d(
=+ 0522
xx 61=x dContinuida K { }6161 +Rx e( =++ 0452 xx 1=x 4=x dContinuida K { }14 Rx *( Rx 052 ++ xx dContinuida K Rx= $( 049 2x dContinuida K ( ] [ )+ 3232x
15=51
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1 BACHILLERATO - Matemticas - Unidad 8 - Funciones, !mites " continuidad
( + 0832 2 xx =4
733x
+
+
4
733
4
733x
#$%58 ' 60 a( A 0.io.i, ( )xf es continua, e>ce0to 0a.a 1=x , 0o. se. 0oinmica% ( ) 1xlm ( ) +=+ 861262x ( ) +++ =+=+ 87171 xxlm Lue$o, es tam)iDn continua en 1=x ( ) == ContinuaEnxf R
)( ( ) Continuapolinmicaxf = udas en e ;ao. 1=x L!mites ate.aes ( ) ( ) += 86121xlm ( ) ( ) ++ = 6711xlm Gon distintos #o. tanto, ( )xf no es continua en 1=x =dEnContinuida { }1R c( iscontinuidad de sato in*inito en 0=x L!mites ate.aes en 1=x
( ) ( )
( ) =
+
1
6121xlm 8 ( ) ( ) =+ ++ 71lxlm 8 Continuidad en 1=x
En consecuencia, =dEnContinuida { }0R
d(
i$e i$uadad !mites ate.aes
1@=51
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1 BACHILLERATO - Matemticas - Unidad 8 - Funciones, !mites " continuidad
( ) ( ) =+ 100 2xlm 1 ( ) ( ) =+ + baxlm 00 b 1=b
( ) ( ) =+ baxlm 33 13 +a ( ) ( ) == ++ 533xlm 2 1=a
#$%58 ' 64 Los t.oos de*inito.ios son :c!ti!u#< Ha" 7ue i$uaa. !mites ate.aes ( ) 0xlm 3 ( ) ( ) =+ ++ nmxlm 00 n 3=n ( ) ( ) =+ nmxlm 22 32 +m ( ) +2xlm 1 2=m
#$%53 ' 65 a( ( )( )
+
+++
+23
12
213
1121
n
n
n
naa
nn ( )( ) >
++ 02353
7
nn nn aa >+1 $recie!te
+
23
12
n
nan < 1na
( )( ) ( )
ce0to en { }= 12963t
#$%511 ' 84 =
++
x
x 34
21
++
++ 342
2
34
34
21
x
x
x
x =42
e21e
=
+
+
2
24
11
ax
x
( )
++
+
+
2
2
24
1
1
4
24
11
x
ax
x
x
( )
4
1 a
e
( )
=4
1
2
1 a ( )12 = a
#$%511 ' 85 ( )
( )!1
12 11
1
+
+=
++
+
n
na
nn
n
( )
( )
=
+
+=
+++
!12
!12 11
1
nn
nn
a
ann
nn
n
n ( )
( )
=
+
+ +
1
12 1
nn
nn
n n
n
n
+ 1
2
n
n
n
na
a
+=+
1121 ( ) =
+
n
n
a
axlm 1 e2
#$%511 ' 86a( ( ) xf .a m/s estrecha ( ) xg .a m/s ancha )( ( ) ( )xgxf cuando x
c( ( ) ( ) ( )( ) ( )++ 32227 23 xxxxxxgxf ( )28 +x( )[ ]=+ gfxlm 0 ( )[ ]= gfxlm 0
d( ( ) ( ) 28
322 ++ xxxxf 322 xx, ( )xg 8R
x ( ) ( ),lmflm = ( ) ( )glmflm ( ) ( )xg%sntotaxf
#$%511 ' 87 a( La $.*ica .e0.esenta a recta ( ) a 3= xyCoo.denadas en o.i$en ( )03 " ( )30 #endiente 1=m O.denada o.i$en 3=n
)( ( )b ( ) ( )
+
+
= 333
x
xx
y 3= xy#a.a 3=x a *uncin est :i!"etermi!a"a