PQ
R;''
$<
3I( LBA0 QQD4
$C
4 :''
'4 6(
'-
4 "08'/'-
4 "
(
4 " - ' / PQ?$(
, PBBB C $( ' 1 :( ,
(-''$
I9 M N56
 PA A!R BR QQQ
$
$'
(
"'-XX
(
$(,
" I9 M -
1 '- '- - 1
N $
(
N " X
X ( 9 $
4 '4 NN
4 N
4 N
4 PQ
?"''
+)
plantas
vento
luz
parasitas
GAFANHOTOS
predadores
umidade
outros
gafanhotos
,'2'-'-
4 P
'4 AE
4 LD
4 DL
4 EA
PQ?D;M(0
'
.3e4<
)=&K5 ,-
4 '4 4 -
4 -
4 PQ%F?>($5
BB" 1
8 '
B "
BA$1
'1'
BL$'2
(
BD$ ( - '
)PQ?25
'6Y)6"
4 $(
'4 $
'
4 $
4 $
'
4 $-'
++
+PQ??R"
4 '4 '-
4 '
4 4 , P Q%F ? " ' 0 8 ( 60
Luz
CO2 O2
O2 CO2
Moluscos
Zooplâncton
Insetos
Fitoplâncton
Peixes
Bactérias
e
fungos
BB"((
-
B ( 1 '/ BA, '0'
BL$'(
(9
BDV ' 0
09-
3LA4
. P Q
D R $ KKKu KKKu KKKu
$ '
4 +00
'4 +
4 +00
4 +
4 ,
P Q?? R $ / ( 6 4 '4 4 4 4 +,
P Q
D R 6 $1 '6(
4 '4 4 4 4 PQ?D)
Milho
Galinha
Porco
Piolho
Solitária
Homem
,(
4 '4 _
4 A_
4 L_
4 _A_
PQ%F?H'
;" 3!e 4 ' ' $(?0
V'0F
, ( ' 5 '0-@1'
5R M/''<
$+]$lIF)
G
BB"$8/e
B = - BA, -
BL,(1
BD" / '
(
+.
PQ
?)92'
-
4
3
2
1
,
$(
4 +→→→'(
'4 +→→→
4 +→'→→
4 +→→5→'
4 +→→→'
PQ? D"9
$9'
4 -
'4 2 4 4 -
4 '
)PQUD,'+) )A
$9 AL0
4
'4
4
4
4
2
2
2
+ P Q? D " 'C / ' ( $01
4 "/(18
'4 "/(1
4 "'C(18
4 "'C(1
4 "/'C1
, P Q? D " 'C 3)"A4 ( SEB A!E
34 QESL E QSALDB60
ABEB 5 EEB PEB * )"A(/CO
4 *'-.'
'4 *$1C.
'
4 $
4 =.
4 =
1C.
(
.PQ%F?"
'G
BB"/9(9'(
B "(-/
BA$ ' - BL$ 1
1 '9
PQ?D>
-F
$
((
4 ^-
'4 =-''
4 ,/
4 ?'
4 ?--
P QD ?'0 / ( 6
4 '(
'4 4 4 4 ,
PQDD"'1
CO2
3
1
VEGETAIS
2
ANIMAIS
,
4
5
COMBUSTÍVEIS
DECOMPOSITORES
"2 E
4 '
'4 '
4 '
4 '
4 '
PQ?2=' `1
?' "'((
9 ( '
$ ( ('92
$' ' ' '-
N $''(
'
( ( '
N $' -
*O
4 $N
'4 $N
4 $NN
4 $NN
4 NN
PQD
"10'
0(
/ $ (
4
'4
4
4
4
,
$
+
I
'
'
$
PQ?$'/8
'
= ' 1(1
'
4 =
'4 =1
4 =
4 I
(
4 '9
)PQD"/(
(9
(9,1?'
'(
:'(/
:'(/(
:'(9C
N :'(C
6
4 '4 N
4 N
4 4 N
+PQ
??5
,s
Ms
6s
s
,
'9
4 ,r(Mr6
'4 ,r6(Mr
4 ,r6(Mr
4 ,r(Mr6(
4 ,(M61
,
, P QDF ;$ < " ( 5 0 ?'0
$
"
'(1
+
5
N +
5
,
)0
'
4
'4
4
4
4
N
N
N
.PQ?D63434
*(
O
n
n
n
a.
c.
t
I
I
II
II
I
t
n
II
e.
t
n
I
I
II
b.
II
d.
t
PQ?2"'8'/
t
% de sobreviventes
I
0
II
III
IV
Tempo
$ 1 10'-(
4
'4
4
4
4
N
N
PQ?D]?D"
(6
,
B "
$(F.$2
11(
BA "F
.0
BL ,)
'
BD ,=0
'
BE6 6 0 /
'
PQ
?)
8
>
(
$
2
$
V?"M6,6
4
'4
4
4
4
P Q? D $ F N ABBB N5 ' 5 ( ( '- 0 '( 1-' 5 1
-5'
G"I,$H=$$I=6A!KSKS!
" ' F 6 '( / $ 5 / - 6
(
4
'4
4
4
4
PQ
RH11+'
'9'('9
-
'
H5
4 =C'(
'4 X)X
4 ?'-'(
4 ?'-
4 ?'-1
P Q?2 M ' 0(6
'
$ - ' 0 0 4 '4 4 4 4 ) P Q
R 6 ' 0 ' " - 4 '
'4 4 4 0
4 + P Q?D 2 i)-0'd ( ( / ?
'
$(1
4 ?11'
'-
'4 ? (5 1 4 ?(51
4 ?'(
4 ? ( ' (
,)
,PQ??R$
C
C > ' / (
4 '4 C
4 4 C
4 .PQ?D2>(1
-3
4 ' 8 $
4 '4 '
4 4 4 P Q?? R , 0 (
3 4
C
3A4
C
34
34
34'
34
34'
34
34(
4 AA '4 A A A A
4 A A AA
4 AAA A 4 AAA AA P Q? D > 1 " 4 '4 4 4 '-
4 -
,+
PQ?2,-('
;K<6
'
?'0
B " ;0< ' /
'(.
BA,
9('
'
'.
BD$''
.
BS$-0
-.
P$'8
1
.
LA$'-
'.
PD$-(;<-
0
PQ?D2,2/
M7)
N@
346/2
-
34" 8 1 34*
'
34$((
346 '
$/(
4 N@@N@
'4 @@N@@
4 @@NN@
4 N@NN@
4 @N@NN
P Q
? ?' (0 0 , / ,,
4
'4
4
4
4
P QR " ' 5 S =>?@?A@@B ;$
H<8
'
6'-
<0
*
'
*'
)
)
4
$
=
?'015'(
9?'0'('
9
'
$ '
9 4 "/1
'4 H1
4 "/
4 ?
)PQUD"
CC'C
?CC'C
4 '4 1
4 4 4 + P Q? 2 $ ' 5 2(''
4 @
'4 @
4 4 @
4 @
, P QD
N
.
''
./(
51
0'
,.
4
'4
4
4
4
1
. P Q? 2 , ' "1
? ( 1
"'(
$ 4 9'(
'4 '9(
4 '(9
4 (9'
PQ?@
$-34
34)"II6$3?4
$ F
F :'
) ,6-
H(
H(
M$9
N )
N F
N N)
B )KN
BAFK
BD)K
BS$KN
PFKN
LAFKN
PD$K
PQ
?)'
?'
(C
?'
65
.
$
1(?"M6,6
4 '4 4 4 4 P Q? 2 $ '
'9 >9"'
@1
)
I1-
N I1'
F"($
$ )
F M1
) M$9
= )
$/)"II6$(
4 $.).=.NF
'4 =.$.F.N)
4 F.=.$.N)
4 F.$.).N=
4 =.$.).NF
P QDF " 1 - ' F 364 "' '
= C 1
= 1 1 / 9
1
'- $ '
4 '4 4 4 4 P Q
? $ ' 8,'
.
4
'4
4
4
4
(
9(
(
(
9(
PQ?D
$'-
$ $'
N $'
6
4 '4 4 N
4 4 N
)PQ?"'1
F(
''
$3434
B $0!=0L
BA$0D60 BD$0!@0E
BSF0PF0S
P@0A70E
LA:0LF0P
+PQ?"+M07-
ABB'+>
'(1
.
6 ( =
'3434
B "(5(
''
BA$ ' 1 5 ' +
BD$ 32 '
0 4 0 '
BS"+1''
P,+
(
'
, P Q D " F ' I ' @$1C
)
)
N M1
34^''+-
34^1-$/
'
34^(
34^ ' ' ) 1
$'/
4
'4
4
4
4
N
N
N
N
N
.PQ?2$(W=''1,
1 5 ' K )(
B 1
-(8
.
BA ' -0-.
BD
0-'.
BS0
.
P ' - ' 3)"A4 .
LA'('0(
/-.
PD / - 1 .
) P Q?2 + )- @- $
)- @- 1
)
(
B (--1
1.
BA - ' (
1
.
BD / 5
.
BS.
10.
P C - ( 155
)PQ?D2"'
*(1O
4 "1
'4 +
1
4 7-
4 +
(
4 ,2(
)PQ?R?261('
I'
'
$ >(2
'1-
F >
-
) >(
/
/-
F-
A +-
L 7
D $
.
$
(
4 $0 F0A)0L.
'4 $0AF0L)0D.
4 $0AF0D)0L.
4 $0 F0L)0D.
4 $0 F0D)0L
)PQ
?)
'-
N
N -
+ ;#1 9(!(
4 '4 N
4 N
4 N
4 NN
)PQ?2$'
$ ' ? (
BB?
? 9 C ,'--(10'--'0
01-
N$----(
N $ )' - $
4 NN
'4 4 N
4 N
4 ?NN
.
)PQ?D"(
4 (
'4 (
4 (
4 (
4 (
))PQ?
DM1
4 -'-
'4 -'-
4 -'-
4 '--
4 '--
)+ P Q
R > ( ' 1 $- '0 ( 1
$ C '- ' 4 $
1('18(
'4 $
1('18('
4 $
1('18(
4 $
1('18(C
4 " / ' '8(
),PQR"''
6(0XX
" " 1
XX>''5
' ? C
$(
4 '4 4 4 4 .)
).PQ
R"'
.+
$
4
'4
4
4
4
"
2(881
$(
$
-1
?1'
2
?'
2
+PQR"'C1'(9'
$'911(
4
'4
4
4
4
"A
)"A
$+
)P: A"P
,$=+:A
+PQ
?
R"11
1 ( 9 '9 6 '( 1
4 M$
'4 +M
4 +@
4 +@M
4 +@$
+PQ?D2$
4 +-'1
'4 $(
)"A(
4 $
(
4 i* - 1 '-
4 $
(('C
+PQ?,3434/
'( (
1
" ' ) B "
BA "/
BD "'
BS "/(
P "58
'
LA $'5
+PQ
R$
?
34(
A +
342
L ?
34
D )
34-
34
$
(
4 0A0 0L0D
'4 L0 0A0D0L
4 D0 0L0L0A
4 D0A0 0 0A
4 D0A0 0 0L
+PQ?$
$8F
)$
)F
34@
'
34+/
A 346/
L 34Y
D 1
34M
.,
$
('$(
4 L0D0A0L0 '4 L0A0 0L0D
4 D0L0D0 0A
4 D0A0 0L0L
4 A0L0 0D0L
..
+)PQDD)
1
N $
4 $
'4 $
4 "N-
4 $((
4 $
0
++PQR=(
1'
$'1
$
5
'
(2
4 '4 A
4 L
4 D
+,PQ?D"$F
Adulto
Esporos
A
Adulto
Gametas
B
Embrião
Adulto
Embrião
Gametas
4
'4
4
4
4
'('
'
'(
+.PQ?D*-'O
4
'4
4
4
4
$1-
"
$1
6
]
,PQ?DF,C/9
6
$F-3
4
4 $ 2
'4 $ 2
4 $
4 $
2
4 $ 2
,PQ
?)'
%"$
" '176(
?
=
=
6
4
'4
4
4
4
"
(
"(
$(
1
$(1
,
, P QR " " 2 L 2 A D
" A D 4
'4
4
4
4
$((
$(
+
+(
$(
, P Q?2 "' 0- )''0
4
'4
4
4
4
$L(
$P(5!
$
A(1'-'
"- ((
""AD(E
,PQ?DF=
$
$'
1'1
(9
+"I*>6
$
51
0
(0
4
'4
4
4
4
$
(5
$
(5
$
(
$
(
$
,PQ?D2)
'0
4 V1'
'4 )(1
4 )''
2
4 ?'
(
4 )('9
,)PQFF>;(<(1
-'
4
'4
4
4
4
C(
C1
,+PQ?D2$-3-49
9 ' -
4
'4
4
4
4
9
0
1,0
0,02
B
A
0,5
0,01
1
5
10
15
Ovulação
20
25
Concetração de B (ug/ml)
Concetração de A (ug/ml)
,,PQ?D"
C
Dias
Menstruação
*C$FO
4
'4
4
4
4
H10
@01
H1
+-
6-
,. P Q? D I '0 34
CKK
'(
6 1 '
35(4
=
01
4 '4 4 4 4 .PQR"
' = ' /($(9 , ( '9-'
4 BA)"A
'4 4 )"A
4 C,A
4 $+,$=+
.PQR>'(
26
(
=
(
4 '9-
'4 '9-
4 '9
4 '9
.PQ
DR?'(
4 0B
'4 0
4 4 B
4 .PQ?D2
O
O
O
+
4 8(
'4 /
4 8
4 8
4 /
.PQ
D2+('M
)"II6$
4 =88(
'4 =2/
4 =22
4 =28(
.PQR$
($ 0'0
= 8 (
('
4 ='-
'4 =
4 $
-
4 $2
.)PQ?
' A (
7
%
\NA
' 5 )*' # %(
!:
0
!
)€,)$:"G6AD DAP
)
()"II6"
B $ + '( M1 MJ
BA@
BDM+0
BS$
P@
LA+1
.+ P Q? 6 !ES H C ?' 5
BB"]
;<F9;<
B $
(/'(
BA"5/10
C/
BL6''
., P Q? 2 ) e '-(,)"II6"
4
'4
4
4
4
"('
$((
e((2
C(C
D(2
..PQD6
ABB 5'-
^ aJ J $0 aJ J ' $ BBABB
$ ' 1- ) M : : :
4
'4
4
4
4
/(
('(
/(
/(
('/
PQ
?>
=
2 0 '- N 4
'4
4
4
4
(
/
2F
F
2
-
F
)
+
,
.F
F
)
+F
,
.?
)
+
,
.F
F
)
+
,
.
F
???
F
)F
+
,
.
??
F
F
))
+
,
.????
)??
)
)
)
)
)
))
)+
),
).
+
+F
+F
+)
+
+
+)
++
+,
+.
,
,
,
,
,
,
,)
,+
,,
,.
.F
.
.F
.
.F
.F
.)
.+??
.,F
..
)
2
E
PQ
FD]
,
5
$3S A4 F3L P4
DEm
PQDR]
"9'(
($Fs3AD4)s3SD4
+ 9 $ F(5
) 4
§1 ·
¨ , 0¸ ©2 ¹
'4 (1, 0 ) 4
§3 ·
¨ , 0¸ ©2 ¹
4 (2, 0) 4
§5 ·
¨ , 0¸ ©2 ¹
2A)
PQ]
, '0 $ F 3 A4 ) 3A L4 $ '"J"$
(
4
'4
4
4
4
2A6
B
L
K A
$($
4 3EDr 27 4
'4 3PD4
4 SE4
4 3P 27 4
4 3PEr 27 4
2A$
PQDR]
?$s3K B4Fs3 B4)s34e4
(9
9$)(
4 '4 A
4 D
4 2 4 3 2AF
PQ
?2]
) $ 3L 4 F 3 L4 " $ ( ( F
8JsA
" ( $F(M3AA4
+
$ 1 $F(Js
N $9$
F))3LL4(A
V
4 $(
'4 $ 4 $ N 4 $ N 4 2A)
)PQ
D]
,$F)=(
$9$3BB4(
4 30 '4 46 4 26 4 22 4 38 2A)
+PQ?2]
$ 9 ( 8 , 9 L4oDesBL4rDeo B
sB(
4 BE
'4 4 E
4 A
4 AE
2A=
,PQ
FD]
)9
9 Ar
EJsPB$/
9(
4 '4 A
4 L
4 D
4 E
2AF
.PQ
FD]
,
B J G Q J
'?
0'
P
4 's
'4 'sKQ
4 'sKP
4 'srQ
4 'srP
2A6
PQ
?D]
$ Js
rLJsKLrA!
J = ,
4 DEE
'4 DQE
4 EAE
4 EEE
4 ESE
2AF
PQ?]
, +3J4 ( / ? / 4 Dπ
'4 Aπ
4 Pπ
4 Lπ
4 π
2A$
PQ?F]
?53 4
3LKA4
?
J s E '
15
2
10
=−
3
3
= 2
10
=
3
15
=
2
4
x=−
'4
x
4
x
4
x
4
x
2A=
PQ?2]
$ ;@+<
$1
(AB`
4 * ( 9 L r
DJsEBLrDJsAEO
'4 * (
=O
4 * ( O
2A
4 E
'4 =3! 4
4 s BB
PQDD]
, '
$ .
4 A
'4 E
4 Q
4 S
4 P
2A)
PQ?2]
, !3PQ4(8
A r LJ s P '
4 3 LB4
'4
§ 35 ·
¨ , 0¸ © 2 ¹
4 3 SB4
4
§ 39 ·
¨ , 0¸ © 2 ¹
4 3ALB4
2A=
)PQ
?DD].
, $3 4 F3E 4 )3P L4 =3A L4 ( r : 3 x − 5 y − 11 = 0 $ 8 $F)= 4 LKEJKEsB
'4 LKEJsB
4 PK BJK sB
4 QK EJKAsB
4 AKABJK sB
2A)
+PQ?'%9 '"%D]
, / λ 3BB4 rJs $ ' (
4 π − 1 '4
4
4
2
π −1
4
π−2
2
π−2
4
4 πKA
2A=
,PQDD]
$ /
AKDr
3Jr 4AsB
4 3KA 4D
'4 3AK 4A
4 3DK 4A
4 3K A4A
4 3AA4A
2AF
.PQ?2]
, /
)3EL4 LrDJK AsB
$
/(
4 ArJAK BKPJrAEsB
'4 ArJAK BKPJrLPsB
4 ArJAK BKPJrDQsB
4 ArJAr BrPJr PsB
4 ArJAr BrPJrQsB
2A$
PQ
DD]
)$
s3K 4Fs3AA4
4 6 ( $ F
'4 ?5 ) = 9 $F) S
2A
4 LrJsL
'4 sKD
PQ?2]
)/
Ar
JAs!$3J4
/(
4 AE
'4 A 4 D
4 !
4 AS
2AF
PQ
DD]
) 3AK4r
3A r 4J r S r D s B J
(9
4 =9
J6
'4 )rLJr AsB
s =$
" 6'
/ "$(9
2A
4 sA.3BKD4
'4 3rP4ArJAsLP⇔Ar ArJAsB
PQUD]
N-0( ( 1C ? / 6''
-0(5''8
ArJArArJr sB
+'
8
'
5'(
ArJAK
AKLJr sB
$9
(
4 2( 2 − 1) '4 A
4 2 2 4 2 − 2 4 5 2A$
PQ?F]
+ J s r L 5 8
/ A K S r JA s Q 4
'4
4
4
4
3
5
4
3
8
5
5
3
9
4
2AF
PQ?2]
, I ?
8'
JsA
$3BL4
F3DB4
$'I?(
4 KBDE
'4 KBEE
4 KBPE
4 KB!E
4 KBSE
2A=
)PQ?]
> 5 AA S AB P
'
" '
+ 9 -(
4 L
'4 D
4 E
4 P
2A)
+PQ
?2]
$ $ F
(
4 ArAJAKPKSJsQ
'4 ArAJAKPKSJsKQ
4 ArAJAKPrSJsKQ
4 ArAJArPKSJsKQ
4 AArJAKSKPJsQ
2AF
,PQ?2]
? 3KA4ArD3Jr
E4AsLPJ
r
(
4 S
'4 !
4 P
4 D
4 L
2A)
.PQ]
$ 5
(
x 2 y2
+
= 1 ( $ s 'π
a 2 b2
6 PA r AEJA s DBB PArQJAs DD(
4 Aπ
'4 ABπ
4 Sπ
4 AEPπ
4 π
2A)
PQD]
?' C A r DJA s Q
'( 9(
B 0
BA $ C '3LB43KLB4
BD ?$FC
= 6 C 9 $=6 F=6
BS $ / 2 C
P " §¨ 2
©
2,
1·
¸ 8C
2¹
2AAL
PQ%"<'F]
$- + +A
B '-
1
f ( x ) = ( x − 3) 2 + 5 9
) ' 0 0
B $ 9 (
P
BA $9(
B
BL + +A / P B
BD " - !+A
BE " - 10 m 9+ 2ABD
PQD]
$ ( ArJAs AKJAs '( 9(
B $ '
BA $ ('
BD "
BS " C A K JA s '
P $ K J r A s B
ArJAs 2AAP
PQ?]
> '- 5
R 2 @RR'
?'0-
1 1
1 ''-
) ' $
4 NN
'4 NN
4 N
4 4 N
2A6
PQ
FR]
$
34 ArLJKPsBAJKLKAsB
34 (3J4
"JAArPJKLJAsQ(
34 r 'J r s B ' 's
B
34 JsAJs x "J2
@&&&'
) 5 '
x =
y2
12
(
' 5 R '
) ' 1
1(
3BL4
91(
( ' 5
R ( A 3
4
1
1(
3LB4
N 1(0P
N 9 1 ( ( ' 5
R(L 3
4
(1,
2
)
2 (
34 8
/ArJAs (8
/ArJAsA
$ 2
4 B
'4 4 A
4 L
4 D
2A)
PQDF]
?∈Is − 1 (
­x = 5 cos t
®
¯ y = isent
4
'4
4
4
4
2A6
=
>/
>'(
=
>('
^DC
PQ
DD]
" - 2 1 s AB D K
QS!(
4 2 '4 B
4 3 4 2A$
PQ?2]
) C +34sDrLr'Arr
'0+3A4s+3Ar4sBAsK ,
r'rr(
4
'4
4
4
4
KA
K P
S
Q
2A6
PQ]
?J20
- 2 z = x − iy (
x + iy
4 '4 A
4 A_JA
4 J
2A$
PQ
DD]
) 2 AsK 6Br rArLrerAB L
4 B
'4 4 4 r
2A=
?521 sAK!1A
s r E 1L s K r L " 2
31 r1Ar1L4A(O
4
'4
4
4
4
I
?
*
+
PQ%"<5D]
) ' 2
2A6
)PQ?F]
?21 s r L 1A s K K L ( - 2AN-s 40 5 π u.v. +PQUD]
I0 $F)(9
2 1 1A1L12'S
1 s A ) ' 4 3KA B4 ( ( 9
'4 1A(51 4 1AsK1L
4 1Ar1LsKA
4 K1 s1A
2A=
,PQ?F]
?1 1A
1A K A1 r D s B z1 z 2
+
z 2 z1
(
PQ?2]
,$07
$F)= 2
]B] ]A]L]D]E
4 K '4 − i 3 4 B
4 i 3 4 )
2A$
.PQ
2]
) 1 1A 2
1 2
+ x = −2 , ( x
1 r1A
4 '4 A
4 D
4 L
2AF
PQ?22]
)'C344s4DKAE
4 45
2344tB.
'4 344/
C.
4 0 2 5(1344
2A
4 − 5 < x < 5 '4 @0
'(0
2 8 1 ( 52
4 $ (1C( B
? ]B 2
( / $F)= 2
(
4 ] '4 ]A
4 ]L
4 ]D
4 ]E
2AF
PQ?F]
? z ⋅ (1 + i 3 ) 21
(
4 E
'4 !E
4 PE
4 AAE
4 LDE
2A6
PQ?F]
"2111Ss P
r1r1Ar1Lrer1!sKPKL(
4 1s 2 '4 1sK K
4 1sK r
4 1s K
4 1s
2A)
PQD]
,'(
2 3 $ K
7
9 / )0
B r = 3 2 ) ( 2
z c = cos 4π + isen 4π 3
BA
r=2 3
3
) ( 2
z c = cos 7π + isen 7π 6
BL
r=2 3
) ( 2
z c
BD
r= 3
= cos
5π
5π
+ isen
3
3
)(2
z c = cos
BE
6
11π
11π
+ isen
6
6
r= 2
) ( 2
-A
2ABD
+PQD2D]
)211
r z K sL?'1
?9(B(
D 3 5(
4 − 4 2 − 4 2 i '4 − 4 2 + 4 2 i 4 − 4 2 i 4 4 2 i 4 4 2 2A=
PQDD]
(
2]]rL
⋅]s A
2A/
L
)PQ%"<'F]
?'0 2
π
π·
§
z1 = 2¨ cos + isen ¸ 6
6¹
©
3π
3π ·
§
z 2 = 2¨ cos
+ isen
¸ 2
2 ¹
©
( π 3
B 1sD
BA 1AsKSrS 3i BD 1K z sD 3i BS "1L(π
P 1⋅ z (2
2AL ,PQ?F]
?521 sDBmr
⋅DBm1AsDBmK⋅DBm
" 2 1 ⋅1A (
4 Bm
'4 ABm
4 DBm
4 SBm
4 PBm
2A$
.PQ
?DD]
,$0734
$(21s
rJ5-314(
+
OA 5
3θ4(9
OA 0 I 314 " 2 1 s (
;<
4
4
4 =2
1s3rA4D(2
'4 ? 1 2 1 ( 1B
5 ( 3B 4 u B
1
2A$
D
`
HSM2IF
PQ
FD]
$ ' C +34 Em 21
2A
4 "2BA0A
'4 D
PQ?22]
)C34sLKQArAE0
AE?'021s
Ar ( 1 9 5
( 1 4
'4
" 2 1 C
*34
*34sAK+34(
4
4
'4
4
4
4
2A)
A
L
D
E
,
PQDD]
"CArr'(rA
K "K'(
4
'4
4
4
4
B
A
L
D
)PQ
DD]
)C34sAK r#r
A ( # 9'(
4 +9#
34K (
LO
'4 ?#sD'
'134
2A=
§π π·
sen ¨ + ¸ ©a b¹
PQ?D]
"'C34s
LrAr'r?'K r α α u B 1 34sB34
3K 4(S
4 α.
'4 343r 4
(AsK 2A
4 αsA
'4 AKArE
2A
4 #s
'4
1
−
2
+PQ?]
, 34sLr'Arr
?'34(34sAK
Q3D4
PQD'(D("9W'%"']
" C D KLL r
!AKrDAKAr (
4
'4
4
4
4
DA
D
S
Sr SK 2A)
PQ?RD]
"C+ 34s3r 4⋅3Ar 4⋅3Lr 4⋅
e⋅3 BBr 4+A34sA
r r *34 I34?*34
+ 34 r(
4
'4
4
4
4
2A$
EBDQ
E AEAP
EBE B 4
'4
4
4
2A=
, ( 1 E
LA
AS
.
,PQ?D]
)C34sDr 4 $134
'4 6 34 C 2A
4 '4
.PQ?2]
=0'C+34s3r 4E
1 (
4 P
'4 AD
4 LA
4 DB
4 DS
2A)
PQDD]
$
LKLArDKAsB
1?1
4 (1 + 3 ⋅ i ) (1 − 3 ⋅ i ) '4 3 r43 K4
4 3Ar43AK4
4 3K r43K K4
4 (− 1 + 3 ⋅ i ) (− 1 − 3 ⋅ i ) 2AF
PQ
?2]
> C 1 L J L ,
C(
4 34sKLrLAKrL
'4 34sLrLAKrL
4 34sKLrLAKKL
4 34sKLLrLAKrL
4 34sKLrLAKLrL
2A$
PQ?D]
" C 344 s 4D r 4L r 4A r 4 K S
2 2 r 1 '
1(
4 =1344
'4 ?' 1 344 C
1
2A
4 s0A's0AsS134
A0A r 0
'4 34s#⋅30 43rL43Ar 4.#∈
Ig
PQ?2]
>C34sAr'
r12AK
E63'r4(
4 E '4 LL
4 AE
4 AA
4
2A)
PQ
DD]
" C 34 s L K AA K Q r S /1K
4 =
'4 ) 314 1 s r ( 2A
4 sDs±L
'4 314s!K PQDD]
?'0
LrDArKPs
B1(8
"5
3?4
(
4 ?s‚KLKAK ƒ
'4 ?s‚KLKAr ƒ
4 ?s‚r rArLƒ
4 ?s‚K rArLƒ
4 ?s‚KAr rLƒ
2AF
)PQUD]
?αβγ1
LrA
r r s B α s K A ( 1 α⋅β⋅γ(
4 E
'4 Q
4 K E
4 K A
4 KQ
2A)
+PQ?2]
? 1 A K A# r LP s B # ( ( 4 9
1 (
4 4 + 3 7 '4 8 + 2 7 4 D
4 6 + 4 6 4 8 + 4 6 2AF
,PQ?F]
? ( ( 1 C 34 s AA r r E ! 134sLArKAA3
4((
1C34r34
4 D
'4 DE
4 E
4 SE
4 Q
2A$
.PQ?2]
$ AL K LA K L r A s B 5
‚K 'ƒ
+Ar'A(
4 13 '4
4
4
7
2
15
4
4 D
4 17 4
2A6
PQDD]
> C +34 K *34 A *
C ( K A *34L+
*3 4r*3A4(
4 '4 B
4 KA
4 K 4 A
2A6
RI
PQ?D]
$(8
2
, '(O
4 *52
' ( a + b = a + b '4 *52
'AK'AsB(
s'
4 * 5 2 (
a 2 = a 4 *52
' t ' (
R't R
4 * 5 2 Bt (At a 2A6
PQ
?]
G ' > ''
'
,
8 '
4 !R A
'4 ER A
4 !RAD
4 ERAD
4 !RP
2A)
PQ
?]
) 5 E 2
2 3 2 53*K,4)∪]O
4 2 '4 B
4 K 4 π
4 A L
2A6
PQ?F]
?5,]*I52
= 7s ‚ ∈ ] 
90
n
∈ ,ƒ @ s ‚ ∈ * 
A∈]ƒ=s‚∈I x ∈*ƒ(
4 @∩=⊂,
'4 @∩](51
4 2 ∈3=K*4
4 7∪=⊂@
4 7∩@∩=A
2A6
PQD]
$'( 9(
B BA
BD
BS
§1 1· 1
2¨ − ¸ = © 2 3¹ 3
3
> 2
2
1
= 0,01010101 ... 90
15
7 3 80 4
3
TADU
P $ 2
2
2ABL
)PQ?R?2]
=0 T'U U'T U'U T'T 3' K 4 = MsTL BU,sUP DT+sTE AT
3M∩+4∪3+K,4(
4 '4 L
4 E
4 !
4 Q
2A)
+PQ?22]
,
I':q#
(
! :
!
0
$ 50s‚(
! :
ƒ D s ‚!
0
:
ƒ s ‚0
! :
ƒ ‚'( '
- 5 ƒ
=0
5 3$∩F4) K ) 4 '('
'4 '(-
4 '
4 '
-
4 5
2A$
,PQ
?2]
= EB )
,:'
E-AE
, - = 4 Q
'4 B
4 DB
4 AA
4 L 2A$
.PQ?D]
, @
• N/O
•
N/1(
2'O
) ' 60N ,
• >(5.
• $ ( 5 .
• +(51
( 2'
= 2 ;?< 8 2 ;,< 8
4
'4
4
4
4
!S PA
!SDS
QBPB
QBA B
!DA!B
2AF
PQ?F]
EBB ' Y n ] =
DB AABY !Bn
SBYn
) ' 0
4 !B]
'4
B]
4 EB]
4 QB]
4 ( ]
2A)
PQDR]
> *1
AB
AA * O
4 E
'4 L
4 AA
4 AB
4 A
2AF
PQ
?2]
" SBB 3M4 3@4 3*4 $ ' Disciplina M
F
Q
M e F F e Q M e Q M, F e Q
Número de
400 1220 1080 220
800
180
100
estudantes
+ ( '( 9( 2
/
(
4 BB
'4 ABB
4 QBB
4 ABB
4 PEB
2AF
PQ?]
> 2 ' 0AE`
0 !`'
0 E` '
0 S`2
, 0 *
) >YQ
\\Q
*
) '
0 * ' ) PZQ
$'( 9
4 $
'4 $
4 $
4 ,
2A$
PQ?]
?' 5 ' $s‚4∈3gR4tABBƒ
Fs‚4∈0R4(2Sƒ
)s‚4∈0R4(2Lƒ
5
#
D6MN
5
#
d\
5
#
7
Z
R YNN R
N 0 06DRLdM
$'( 9
4
'4
4
4
2A)
$
$
(
$
PQD2D]
?'5$s‚ALEƒFs‚LDƒ
)$∩)s‚AƒF∩)s‚Dƒ$∪F∪
)s‚ ALDEƒ
B $∩F∩)s∅
BA )KFs‚ Aƒ
BD )K$s‚LEƒ
BS L∈)
P 3$∩F4∪)s‚ AEƒ
2ABL
?SM2IF
PQ?F]
>Y'P
( I„DBBB
" n ' I„EBB (
' E
) '
- 5
nY
4
'4
4
4
4
AAB(
AAB(
LAB(
LAB(
DDB(
DB
LAB
ADB
LAB
ADB
2A)
PQ?2]
> ' I„ Q SBBBB I„DEBB)
( I„ PEBB ?5 1 5AB`$
(
4
'4
4
4
4
2A=
A
L
D
E
P
PQ
?D]
" N - $ F N$34sABBrLNF34sEBBBKL
= - 4 sEBB
'4 sPBB
4 s!BB
4 sSBB
4 sQBB
2A=
PQ
FD]
+ 1
9$
J (
) B BB ' (
= $ EB
= F 5 EB
$5''
'
($
F(
4
'4
4
4
2A=
4
PQDD]
$ 34 s r ' ( 0
?'034sA'3'4sA
"3L4(
4 A
'4 D
4 KA
4 B
4 K 2A)
)PQ?2]
> / $ (
0 '
(
sKBDrABB?5# #A 2 (
I„A BBBBB"# r#A(
4 DEB
'4 EBB
4 EEB
4 PBB
4 PEB
2AF
+PQ
DD]
"/
('
+ ( B2'
(
AB + ( AB D 2' B 2' )
34 2' 4 6' 34 BLB
)
+
'4 +D
2' ( 2'O
6 AE
2'O
2A
4 4 634≠3AA4
f ( 2 ) = 2 O
2A
4
f
( 2 ) = 2 22++22 f
( 2 ) = 2 22++22 ⋅
=
2 −2
2 −2
4−4 2 +2 2 −4
−2
=
( 2 ) = − −2 22 f ( 2 )= 2 f
'4 3K B43B 4
4
'4 I„ E 2' I„ LAB 2'
,PQ?D]
=
m
f (x) = 2 −
x+n
≠K
4 ,ssA
f ( 2 ) = 2 '4 , s s A 4 , s s A ' u KA
' 4 '4 >1 '
4 .PQ?'%9 '"%D]
$ / +"A ( /
( $ +"A 1
=
EB +"A B` +"A :
4 SELE
'4 QPAE
4 EQQE
4 !QSE
4 QS E
2AF
PQ?D]
"
8 I @ ( ' ' " ( OA, AB, BC, CD DE G ' I„ DLSBBBB +
'
I„ BBBBB $ PQDD]
?1
AAKEKDsB
1 + 1 (
m
n
4
'4
4
4
4
5
4
3
−
2
3
4
7
4
5
2
−
,
2A$
PQDD]
$ x + 3 3x + 1
=
x −1
x+3
2
4
'4
4
4
4
2A6
PQ?2]
" 2 ( 1 A DKALKLArr's
B"⋅'(
4
'4
4
4
4
KS
KD
KLA
K P
KPD
2A)
4
'4
4
4
4
2A)
I„ BBBB
I„ABBBB
I„AAEBB
I„DEBBB
I„PBBBB
PQ?2]
"5
3AK D4A⋅
3LJKQ4Ls BSJ
3 J4 ? (
4
'4
4
4
4
2A)
B
A
D
S
PQ?2]
6
(I„DBBBB
"?G
!S
(AA"
?G(
4
'4
4
4
4
I„
I„
I„
I„
I„
BAPBBBB
BL!BBBB
BDSBBBB
BEQBBBB
B!BBBBB
2AF
)PQ
?]
, ' ( '
5 ( J s r ' r + ' '
' ' 4
'4
4
4
4
2A=
+PQR]
" ' $F')34
$(F3LB4
" 9 ) ( DE
" $F
4 E
'4 P
4 3 5 4 6 2 2A)
,PQ
D]
6 2 -
-JsKArPBr BBsB(
$- 2 2 - 2 (
"
2 ( 5 2
(
4 LL
'4 L 4 LD
4 LP
4 LS
2AF
.PQ
?D]
$ ''
@ 5-'/
' ( 9
1X
sKB Ar A
r AE = '
.
4
'4
4
4
4 P '4 PL
4 !A
4 !E
4 SL
2A$
PQ?'" "9(!( ('AD]
" 34 s A r ' r B ≤ × ≤ D 9!×
D
1
<m≤
4
1
m< 2
1
m>− 2
1
− ≤m≤
2
−
1
4
1
2
4 ≠B
2A6
PQ]
$1
AK
ErPsKL(
4
'4
4
4
4
B
!
B
L
D
2A6
PQ%"5#]
= ( x 2 + 20) =| x + 1 | +3 ∈I1(
"'(
4 K!
'4 KD
4 A
4 D
4 !
2A6
PQ?D]
6 ( 10
AA K A3 K 4 K s B - 2$
9
0
1 4
'4
4
4
4
KA
KL
KE
KD
KP
2A$
PQ?RD]
)
­ x, se x ≥ 0
| x |= ®
¯− x, se x < 0
(
2
= ( 5
2
§
·
2
¨ | x | + | x | + | x | + | x | + ¸ = 5 5 + 1 ¨
¸
5
5
5 5
©
¹
(
)
4
'4
4
4
4
2A=
UKππAU
UK π πLT
UKπL π T
UKAπAπU
UKπAπT
PQ]
"5
34
sr KK (
4 TKAAU
'4 UK∞AU
4 TBAU
4 TBr∞T
4 UBAU
2A)
)PQ?2]
?5 # 2 JsJsK
r# P
,
#(
4 4 2 '4 6 2 4 S
4 7 2 4 B
2A$
+PQ
DD]
) 34 s A r  r 
4 $'34
=
'4 ?sKL
1 34s A
2A
4 s− '4 sE
,PQ
FD]
$
)344
" 2 5
)344s I(
4 P
'4 E
4 D
4 L
4 A
2AF
.PQ]
= I→I 34 s
AKPrE22JB
/ 2 A
L0
3 4s3A4s3L4s
JB,
rA
rL(
4 P
'4 !
4 S
4 Q
2A=
PQD]
= ' )344sr
AKAKEttE(
4
'4
4
4
4
4
2A)
PQ?]
$ 34 s  r LK  r  1 ' 8
4 tKL
'4 KLttL
4 uL
4 tA
4 uKA
2A6
PQ?2]
)
34s K
$ 4
'4
4
2AF
PQ?].
" 2 4 1 3 x − 3 ≤ 6 5
4 (
'4 (2 3 4 ('
4 (
4 ( ]−∞ , 2[ 2A6
PQF].
?' 50
x − 2 − 2 x − 1 = −1 x ∈ R 0(
5
B 1
BA
BL
BD/
BE
2ABL
PQ? D],
?5 f(x) = | x + 2 | e g(x) = 2 | x − 2 | 4 I
f (x) = g(x) '4 ) h(x) =| x + 2 | −2 | x − 2 | 2A
4
3&&&'4
­2 ½
V = ® , 6¾ ¯3 ¿
'4
)PQDD]
?3D4As P⋅A x (
4 A!
'4 D
4 1 2
4
4
4
−
1
27
2AF
+PQDD]
5'+%
1' ! / C
-
C).
J
5 YNLY ) YYL UT 'LR\UT
6\QLNQ*'YNNR
) ( 9 -'
A≈BPQL≈ BE≈ P Q≈AQD
( 8 '
AB A
4 ABASAB Q
'4 ABPSABDB
4 ABAAAB !
4 ABAEAB S
4 ABE!ABAQ
2AF
,PQDD]
? 3 n +3 − 3.3 n −1
3.3 n + 2
'(0
4
3 n +1 −
1
9
'4 KLrA
4 L
4 26 4
27
16
9
2A=
.PQ?2]
) A s BL V
1 AAKP⋅ArEsB(
"γ'
(
4
7
3
'4 A
4
4
5
3
4
3
4 '4 BAD
4 B!
4 B D
4 D
4 2A$
PQ?D]
> 11 SE m) 6
PE m) ? 11 )
s PB×AKBS×
rAE
*
11 O
4 BAE
'4 BPS
4 AE
4 PPL
4 BB
2A)
PQ?RD]
) ' ( +Nγ s # + ( N(9#γ
" 0 / '- '
2A6
PQ
?2]
$ F
5 AB L ' 2' 2 '8
) 8
! BB
2 8 2 8 A (
4 ABB
'4 A!BB
4 ADLBB
4 S BB
4 ABBB
2A=
PQ
FD]
$ / 3*4 3+4 *s rD⋅3BS4A+
, 6 ( +
* = + '
Q −1
4
4
P = log 0,8
'4
§ Q −1·
P = log 0,8 ¨
¸
© 8 ¹
' 6 BB
'-
Q −1
4
4
P = 0,5 ⋅ 0,8
4
P = 0 ,8
4
§Q ·
P = 0,5 ⋅ log 0,8 ¨ − 1¸ ©4
¹
Q −1
8
2A$
PQUD]
?
I→I(34s
LK '(
4 =34sI34sI
'4 ( 4 (5'5
4 (5('5
4 34sIgr
2A)
PQ?2]
>B#5I„DBBBBBB
9Ns'
A
?EI„AB
BBBBB A 4 I„ QABBBB
'4 I„ !PBBBB
4 I„!PBBBB
4 I„EABBBB
4 I„DSABBB
2A)
)PQ
DD]
$ - ' 2' LABBBB
A ' 6
( ' =
)- @ ( -
^ +
+3$++4
4 ) $++
'4 ? -
N34 s NBAK NB ( ( *(
1 B` O >1 BA≈BLB
2A
4 3ADrπ4× BDA
'4 ≈L B
+PQD'(D("9W'%"']
?' ' 2 1
2
log 1 a = log 1 b = 16
3
5
16
6 log 2 §¨ a 3 ⋅ 4 b 5 ·¸ (
©
¹
4 P
'4 E
4 KE
4 KP
4 K!
2A=
,PQ%"'9(D]
" : ( :_ 6:sK
T:_U>6M(
:sL
(
' 1 C ' C : s 4 :_
C ( BB 1 '4 :_
C ( BBB 1 4 1
:_
( BBB1
C
4 1
:_
( L 1 C
4 :_
C(L1
2A$
.PQUD]
" 5 2 § x 2 + 5x + 4 ·
¸
f ( x ) = log x +5 ¨¨
2
¸
© x −1 ¹
(
4 I
'4 ‚∈IR≤KE≥ ƒ
4 ‚∈IR≤KEu ƒ
4 ‚∈IRKPt≤KE≥ ƒ
4 ‚∈IRKEttKDu ƒ
2A6
PQ%"<'F]
)0 W 2 (
+ s
3AA43DA43SA4e3 log 2n 2 4∈] *+ (
B W
BA AW
BL
BD
BE
1
n!
1
( n − 1)!
1
2 n!
2ABL
PQD'(D("9W'%"']
V
A3r 4rA3KA4sA
4 '4 A 5
(B
4 A 5
(… 4 A 5
( 4 2A6
PQ?D]
?'
3rL4 2 x −9 ArK s
B(
4 1
'4 21(KL
4 1LKL
4 21KLB 4 1
2A6
PQDD]
? r Ar L r D s KAB (
4 B
'4 B 4 BB
4 BB 4 2A=
PQ?F]
$
AAKAQs 7 3log 2 2
7
)
4 sKDS
'4 sKA!
4 sA!
4 sLP
4 sDS
2A6
PQ
FD]
$ 34 s
3 A4K
) ' A3E4KA3L4
(
4 BEB
'4 B!E
4 BB
4 AE
4 EB
2AF
)PQ
FD]
"2
D!K P!sB
(
4 '4 A
4 L
4 D
4 E
2AF
+PQ?'!"@%'"!&!!F 9(
D]
"5:1!'
AB L
?AB LABPB
'
8
M34s S⋅KBA
)M34
s AB LsAABAP
$AsB!Ls Es
P sAD(
MS
4 ABD!
'4 ABEP
4 ABEA
4 ABDE
4 ABEQ
2A)
,PQ?D]
" 2 , -
,34 s ,B Kλ ,B 2-sB
λ$'
B,
'
( QQ 3QQ41-
+
'4
4
,
4
$
4 B,B.
'4 2,B
QQ
.
4 03 RA4QQ
(9!(9
$03 RA4-
( 2
-
BAsBL. BEsB!
2A
4 B,BsP
'4 ,Bs BP
4 RAsP
.PQ
?]
"
34s$3 KK#4≥B
$#(1
1
/ '1 8 / / " 34(
4
4
2A$
)PQ?D]
> ? 0(
(2
4
log 4
3h + 6
2
'4
3h + 4
log 4 2
4
3h + 2
log 4 2
4
3h + 2
log 4 4
4
3 h +1
log 4 2
2A)
)PQFF]
$ ' F ( / FB -
§ B
M = log a ¨¨
© B0
B3A4 ≈ BLB B3L4 ≈ BDS B3E4 ≈
B!B
·
¸ ;
¸
¹
E'
( BB<
, 0 A s
BLB L s BDS 0 HFs A
× BEFB(
B K AQ
BA K A!
BL K AE
BD K AL
BE K A 2ABA
)PQ
DD]
> ' 6 1 ' $
' 13m)4
+34s⋅ B'
'
$ ' ' '
H
Temperatura
Perda anual de
(º C)
0
capacidade (%)
1,6
55
20,0
)'+34
'
4 ''
+34 ' sBsEE.
'4 ' '?
.
2A
4
'4 s P'sBBA
)PQD]
, I '53G4
6
s DD r EM 6 AB G QB I6'
4 T L DU
'4 T ! SU
4 T B L B DU
4 T B ! B SU
4 T BEL BEDU
2A=
)PQ?2]
?'
'2
'( C 3(e[4e
( 2 '( 4 4 2'
?
'(
2
5AB
O>''
x
log x
1
2
3
4
5
0 0,30 0,48 0,60 0,70
4 D '4 DL
4 DE
4 D!
4 DQ
2AF
C-P($1$T0196(-)
+ 1 ' C +34 J s +34
& ,
( 10 ' KE (
A!
)PQF]
"21
+34s TKEA!U(
B 4 P
BA4 E
BL4 D
BD4 L
BE4 A
2ABA
C-P($1$T0196(-))
>
AB`
" 5 1AB`$
2
I„ BBBBBBB - (
2s BBBBB⋅3BS4
"L
))PQ
FD]
+
AE`
) A s
BLB 4 L-
'4 D-
4 P-
4 S-
4 B-
2A)
C-P($1$T0196(-)+
, / ' ' " '
3,434
) 2 / '
EBBB'
)+PQ
FD]
? ,34 ' '
AS / A m
4 L AE
'4 AEBB
4 S!E
4 AEB
4 PAE
2A=
C-P($1$T019=-),L).L+
= K $ F K 1 ' ?
( $ ' F 6
RH $ F C,Cs L!m)CsAL!Dm)CsLL!Sm)
1 BD
m) " $ ( ' s K
S3C K 4 3C K 4 ≤ C ≤ F(
RH
$
4
'4
4
4
BB B
B AB
AB
BB
2A)
).PQ?]
? $ PSRH3t PS4'3:4
m)
4
'4
4
4
L!≤:tLQLQDtC≤D LS≤:tDBP
LSt:tDB
L!≤:tLSPLQStC≤D 2A=
+PQ?]
*
$F
' (
4
'4
4
4
LQAm)
LSPm)
LSm)
DBm)
2A=
D2M
I
2I
PQ
DD]
" 9 9 (
PB
( 3+$4 $ 9(
),PQ?]
$ 0L!Sm)'
4 LBA
'4 ABA
4 EA
4 LEA
2A)
PQ]
?5 34 (
2 5
(D1(
? AE (
4 AD
'4 AS
4 LA
4 LP
2A)
PQ]
? / 2 A Lee ( ( 1 / J JA JLeJe J s ( 4 (51(
'4 (51(
4 (51(
4 (51(
2A$
PQ?D]
)/
1 (
$ 2 QP_
(
4
'4
4
4
4
2A)
DE
D!
DQ
EB
EA
PQ?D9 *("R]
" / ( $ /
/
( / EΩ 40 Ω
33
M0 2 /
'(0
4 DE
'4 SB
4 BE
4 AB
4 AE
2AF
)PQD'(D("9W'%"']
,(3 ALDE4
'0ArEsQLEKLs P
6
4
'4
4
4
4
a5
a2
21
8
15
7
13
5
11
4
10
3
2A=
+PQFF]
6
/ $
'' 9 ` AD` LS` / 3FI6q6I AB L !E4
? 9 (8
2
9
PQDD]
$ ' / , '
(
'
B PL`
BA PE`
BL P!`
BD PQ`
BE !A`
2ABD
,PQ
DD]
=1 / 2
3 A L D e4 ( C / 3 R RA RL RD e4 ( (3+$4
) 1 H 1 1 AD!E
?'
BB
'O ) 1 4 = C 3ARE
DRQ}e4
'4 ?5'
C N
'sAR3r4
2A
4 DRE
'4 + C R R' R
+$
$
.PQDD]
= 5 > I„ EBBB / I„ EBB /
I„ BBB/I„ EBB
I„ EBB /
$ 2 62
4
'4
4
4
4
2A$
4
'4
4
4
4
AL AEA
ALEEBA
ALE!EA
DA EBA
DAEAEA
2A$
PQ?2]
" 2 5 $ / 2 LB_5(
4
'4
4
4
4
2A=
SBB
SP SSD
QBB
QEB
PQ]
?(AB L2
(
4 '4 L
4 E
4 !
2A$
PQ?'%9 '"%D]
" ( 2 3 2 , 4 E
'4 S
4 A
4
4 D
2A6
PQ]
)
34sAA K E?53 A L e4 ( 1L f (a 1 ) = 1 $
8
ELs E!
$ (( DE
3E4sAA N 33 4 3A4 3L4 e4 ( (1PD
$'( 9
4 ? '4 ? N 4 ? 4 ? N 4 2AF
PQ?D]
$ 1
1
1
"(
/ / +7(
> 2 / =- 3 m
4 AP :1 - A '( - =- 3 Lm 4 EAA
:1'
6 2 5 1 / H 3 Bm
4 $ / 11(
4 2614 4 '4 26112 8 4 26110 4 4 2614 8 4 2614 12 2A=
)PQ?2]
? ( ( ( AB ( (
((
4 2 39 '4 2 41 4 2 43 4 2 55 4 3 39 2A=
+PQ?F]
>/ E
( AE 4 SB
4 QP
4 BB
2A)
PQ?F]
6'
!PSBB '( PBB ?
EB`
( 4 D
'4 E
4 P
4 !
4 S
2A=
PQ?2]
4 > ' $ C '
;>'
!E (
6 , D
6 O
'4 " ' 1 8 $
;? ( L
<
G
2A
4 )D
'4 S = a 1 = 3 = 6 + (
4 AEEr 127 2 '4 AEPr 128 2 4 382 2 4
4
32 2
2 −1
127 2 − 1
2 −1
2A$
,PQ22]
"(
( / $ s ‚' 'A A e ' ƒ
Fs‚' 'Ae'ƒ(
( ) s ‚ A e ƒ (?'0$
s ‚KE PD KA LA eƒ AB
(
4 EE
'4 SE
4 LBB
4 A L
2A=
.PQDD]
$ ' 5- - $ F) s
)= =6 s 6@ @7 s 7: : s G ) 9 1
$+
-
1− r
1 − 0,5
? 9P
'
4 PE
'4 !A
PQR]
6 9 DB AEAB
BE L , 0 . . . '02
$ 1 (
1
)A Bs BBB2
5 (
4 E
'4 P
4 !
4 S
2AF
PQ%"'9(D]
$ ' $ 6 ') 1
6A
V I IA IL e
86 6A6Le+
7( 3+74 1 1
2
) 0 1 '( +751(
4
'4
4
2
2
1
2
1
4
4
2
4
4
2
8
2A=
PQD]
G ' 1
6
' 1
, G
' 1
E A!BE
4 ?' G 1
-O
$ 1 '4 , 1
62
)
ABPP! E 1
*O
$ 1 4
'4
4
2A
4 0
?Es A!BE
s!
sL
sE
>@-
4
+ (s E
'4 0
s!
sABPP! E
sL
s
>@-
+(
ABPP! E1
PQ
2]
$ ' '
/9'
' BB 9 9 /
9 DEm ' , 0 5EB9
' 9 ' 9
100 2 50
(2 − 1) 2 49
100 2 49
(2 − 1) 2 50
100 49
(2 − 1) 2 49
100 2 50
(2 − 1) 2 50
2A$
\
?
PQ
DD]
$1+
(ARQ?
(EE+
4 A
'4 L
4 B
4 E
2A)
PQ
DD]
>I„ABBBB/
2 , /
I„ QBB ,
/ 1
I„ !BB , / I„
SBB ?' - (
2
4 I„PBB
'4 51
4 51I„PBB
4 I„PEB
2A$
+
PQ
DD]
) ? ' 1 4 (1EB`
'4 EB`
4 4 (1AE`
2A$
PQ?22]
H
$- 2 2=
$ >?„ P! ' 6
A !`
/AB A
:5BO0 1!06B51'!dYNLZDL
-0.
= AB A
2A
I/
P!ppppppp A !B`
ppppppp BBBB`
"'/0 s L!
$ >?„
L!'AB L
PQ?'%9 '"%D]
M'5
'
AB` , 5 I 5 ' B`+
M
I„ ASSB ' I I„
LPBB , '
I 5 M 8551
4 I„APB
'4 I„A B
4 I„ LB
4 I„ QB
4 I„AQB
2A)
)PQDD]
> 2 '' '
1 AB` ?
' ' ! '' (
4 S!E
'4 QLS
4 !DA
4 PQL
4 EP!
2A$
+PQ?D]
> / I„
BBBBBBBB P`
!E`
$ ' +-
I„
!ABBBBB
4 I„ABBBBBBB
'4 I„ !EBBBBB
4 I„ EBBBBBB
4 I„ AEBBBBB
4 I„ BBBBBBB
2A$
,
,PQ?D]
> 5 ' - $- ( 1
( 1 '
?'0 ' 5S`5
1
(
!(9
BSsBBL
AsBLB
Ms)⋅3 r4
4 B
'4 E
4 AB
4 AE
4 LB
2A$
.PQ?2]
5 #` / PB` E ) (
# ( PQD'(D("9W'%"']
> 5 I„ BBBBBB 1LB
PB I„ EEDLBB I„ BESBBB
G 1
1+
5 ?5
E` / '5
1O
4 I„ABBBBBB
'4 I„A ABBB
4 I„A LABBB
4 I„AAASBBB
4 I„AASABBB
2A6
ZL
PQ
DD]
§a
¨
) 1 A = ¨ − 1
¨ c
©
1
1·
¸
0 b ¸ − 2 0 ¸¹
'2
4 6 ' $sK$
'4 =s 'sK § x · §1·
¨ ¸ ¨ ¸
A¨ y ¸ = ¨ 1 ¸ ¨ z ¸ ¨d¸
© ¹ © ¹
O
4 S
'4 Q
4 B
4 A
4 P
2A)
2A
4 sB'sAsK '4 sBsKD
PQ
FD]
/ $ '
.
" J1=
1
2 '(
ª5 5 3º
«
»
y z ] ⋅ «6 3 3» = [96 105 79] «¬ 4 5 2»¼
ª x º ª5 5 3º ª 96 º
» « »
« » «
« y » ⋅ «6 3 3» = «105» «¬ z »¼ «¬4 5 2»¼ «¬ 79 »¼
ª5 5 3º
«
»
«6 3 3» ⋅ [x y z ] = [96 105 79] «¬4 5 2»¼
ª5 5 3º ª x º ª 96 º
«
» « » « »
«6 3 3» ⋅ « y » = «105» «¬4 5 2»¼ «¬ z »¼ «¬ 79 »¼
ª x º ª 96 º ª5 5 3º
« » « » «
»
« y » ⋅ «105» = «6 3 3» «¬ z »¼ «¬ 79 »¼ «¬ 4 5 2»¼
4 [x
'4
4
4
4
2A=
PQDD]
)
$rFYsYr
A) 5 - ( 1 Y 1 $
2(
4 FK≠"(1
" ( 1 '4 F5
4 F≠""(1
4 F K 5 ( 1
4 $)5
2A=
PQ
2]
) $I→MA3I4 MA3I4(51
A A( t ) =
t2
t −1
t + 1 2t − 1
=
4 0-.[0-oL.
'4 0- . ( 1 4 0-o.[0-.o0-.
4 0-.(1
2AF
PQ22]
§ − 1·
§0·
¨ ¸
¨ ¸
B = ¨ 0 ¸ C = ¨ 4 ¸ ¨1¸
¨ 2¸
© ¹
© ¹
­i + 2 j, se i ≠ j
a ij ®
(
¯x, se i = j
= 1
A = (a ij ) 3 x 3 2"$FK)
5(
4 !
'4 B
4 4 L
2A$
)PQD'(D("9W'%"']
= 1
ª a 1 − aº
A=«
»
0 ¼
¬2a
2
1$AK$(
4 '4 Ar
4 AK
4 KAK
4 B
2A)
+PQUD]
?1
1
x + y + z 3y − z + 2 º
ª
«
»
4
5
−5 »
«
«¬ y − 2z + 3
»¼
z
0
(((
4 B
'4 4 P
4 L
4 KE
2A)
,PQ]
?5 Y 5 1 A×A $
$
1
Y1
1 5Y
$ 1 A×A
5Y
N ? $ F Y $rF '(
Y
$'( 9
4 ?(
'4 ? N 4 ? N 4 ?(
4 2AF
.PQ%"<'F]
$ 0 1 >
' 15
2 B B '
)0 ( 1Fs3'543'54s33PK544
B BA
BL
BD
BE
2ABL
PQ?2]
? M 1 MK M
=M+
M ( 4 =s+
'4 M1
4 MK M 4 M
4 M⋅MK ( +=
2A$
PQ]
> 1 + s 354 ( (
5 s 5 + 1
ª 2 − 3 5º
«
»
«− 3 7 4» ((
«¬ 5
4 1 »¼
ª x + y x − y xy º
?1 M = «« 1 y − x 2 y»» ((
«¬ 6
x + 1 1 »¼
0 M(
4 K '4 KA
4 4 A
2AF
PQDD]
=1
( 8
1 ?
ª2 1 º
A=«
»
¬0 − 1¼
ªx y º
» 1
¬z w¼
ªcos x −senxº
A=«
» ( ¬ senx cos x ¼
4 ,1
'4 4 4 *1
4 ' 2
2A=
PQ?D]
?5
2A Q
)PQ?D]
)1
2a + 1º
ª a
A=«
» ¬a − 1 a + 1 ¼
(2?'
$$K 5(
ª2a − 1º
«
» ¬ −1 ¼
$K (
4
'4
4
4
4
D$K 2A
ª 5 −7º
A −1 = «
»
¬− 2 3 ¼
ª 20 −28º
4.A −1 = «
»
¬− 8 12 ¼
ª 3 1º
«
» ¬11 4¼
r'rr
rJr1r&(
4 B
'4 4 KA
4 L
4 KD
2A$
PQ22]
ª3 7 º
A=«
» ¬2 5¼
1
B = «
= 1
ªa b º
«
»
¬c d ¼
PQ
?]
$ 1 5
(5
+ '1
E
P
!
S
Q
2A$
+PQDR]
ª2 x
$1 M = ««0 x
«¬0 1
0º
»
x » x »¼
4
'4
4
4
4
2A$
B BK K A AK ,PQ
D]
)1
ª 3 0º
ª1 3º
A=«
» B = «
»
−
1
1
¬
¼
¬ b 1¼
=$1$
$A 1 $⋅$ V 4 5 '∈I 0 ª 3 0º
A ⋅ BT = «
» ¬− 3 1¼
b=
3
2
1 $ r AF (
A B C D E F G
H
I
Número
1
2
3
8
9 10 11
Letra
L
M
N
4
O
5
6
P
7
Q
R
S
J
T
K
U
Número 12 13 14 15 16 17 18 19 20 21
Letra
Número
V
W
1
ª47 30 29º
»
¬ 28 21 22¼
4 ' s A# #∈] $ r
AF(
4 5'∈I1$rAF
2A=
.PQ?22]
> ( 1 1 >
1 ( ' 2
8 '
Letra
ª2 1º
» ¬1 1¼
10s «
s «
'4 '∈I03$rF43$K
F4s$ArFA
4 + 1 1A†ADD ×0[;;(
1
-YpY.
$
D '(0D× [D ×
0×([(
> X
Y
Z
_
22 23 24 25 26 27
+ K ª S E Nº
ª19 5 14º
» s «
»
H
A
S
¬
¼
¬ 8 1 19¼
Ms «
+ 1 A†A $ (
1 ( 1 6 s $ × M ( $ ' '('
) 1 D 2A
ª 1 −1º
B=«
»
¬− 1 2 ¼
ª19 9 7 º
ªS I G º
F6s «
» s «
»
9
12
15
¬
¬ I L O¼
¼
$(?7H"
PQ?2]
§ 2 −1·
¸¸ A = ¨¨
©1 1 ¹
) 1
§ x· §1·
¸¸ = ¨¨ ¸¸ © y¹ © 2¹
" A¨¨
2 J 6234
s3$4
$10 BB(
N ?D(10L
D s det( B) = 1 ; ( 1
9
A
) 4 $N
'4 $N
4 $
4 $
2A=
PQ
DD]
4
§ 1 a 1·
¨
¸
1 a ¸ '
¨ 1 b 1¸
©
¹
)1 M = ¨ b
2 +
4 1M(
'4 M(
4 M(AK'A
4 1M(8
2AF
PQDD]
? 1
x º
ª3
«
» 4
x
+ 1¼
¬
0
DQ > (
4 E
'4 KL
4 4 KD
4 A
2A=
PQ
?2]
"4
1
0
0
log x 16 log x 8 log x 4 = 1 (
1
1
2
4 D
'4 P
4 A
4 LA
4 S
2AF
PQ]
?$$K ª2 3º
»
¬ 4 8¼
1 A = «
1Fs$KA
$K (
'4
−111
2
−83
2
4 K PP
4 97 2
4 PA
2AF
PQ]
?2J1&
( 1 ªx
1 M = ««m
«¬ u
y zº
»
n p»
v w »¼
(
4 '4 B
4 &
4 L
2AF
)PQ?2]
?5 MLL ,DD 1 '
52
ª 2a 2b 2i 2c º
ªa b c º
«
»
0 0 2j 0 »
«
»
M = «d e f » N = «
«2d 2e 2a 2f »
«¬g h i »¼
«
»
¬«2g 2h 2c 2i »¼
? M ( 2 # ,
4 K P5#
'4 P5#
4 KA5#
4 A5#
4 B
2A$
PQ?F]
+PQD'(D("9W'%"']
) 1
0 º
ªx 5
«
»
A = «−1 4
1 »
«¬ − 2 1 1 + x »¼
= 1$ s
2 ? ' 1 $sBu' b a
F s
4 K '4 KA
4 KL
4 KD
4 KE
2A6
,PQ
?2]
" 1
ª1
«
«0
«¬ x
0
x
0
-1 º
»
0 »
- 1»¼
(
4 4gB
'4 4g 4 4f 4 Bt4f 4 4fB4g 2A=
.PQDD]
= 1
ª2 1 −3º
A=«
»
¬0 − 1 2 ¼
ª4 1 º
«
»
B = « 2 2 » 3$⋅F4(
«¬ 3 − 1»¼
4
'4
4
4
4
2A)
AP
K S
KLA
AS
A
§0
¨
¨ 3x
¨
¨9
©
2
0
0
0 ·¸
4 ¸
¸
2x ¸
¹
§
¨ sen 3x cos 3x
¨ − cos 3x sen 3x
¨
¨¨ 0
0
©
·
0 ¸
0 ¸
x ¸¸
¸
3− x ¹
5
3$F4 ≤ B 3$F4 1 $F
2A
"5
(UK∞BU
∪TALT‚∈.≤BA≤tLƒ
PQD2D]
?'1'
'
ªa
1 3º
ªa b c º
»
«
»
2 5» Fs «2 4 8»
«¬ c 4 1»¼
«¬3 5 1»¼
$s ««b
1
2
B $s F
BA 3A$4sDF
BD $sF
BS §¨ B ·¸ s$
©2¹
P $sKAF
2ABL
PQ?2]
) 1
A
§ 1 0·
§2 1·
¸¸ B = ¨¨
¸¸ A = ¨¨
2
1
©
¹
©0 2¹
?5Ms$⋅FF(1
F " 1 M(
4 RS
'4 RP
4 RD
4 }
2A)
PQD]
" 1
ª 1
«
«− 2
«
« x
¬
2
x
0
0º
»
0» »
x »¼
(
4 uwD
'4 tB
4 tA
4 twDuB
4 uwAtwP
2A=
PQDF]
?
§m n ·
¸¸ A = ¨¨
© 2 − 10 ¹
1 $K det A −1 = −
1
6
4 ErsKL
'4 EKsL
4 ErsL
4 rs 4 KEsL
2A)
PQD]
? $ ( 1 A † A B?Fs0A‡$)sL‡F0 F0 ( 1 F )(
4 wPB
'4
4
4
4
−3
20
−20
3
9
40
40
9
2A=
)PQ?22]
> 1 I„ !!BB $
I„
BBBB 0
I„ BBB
" I„!LBB/
I„ BBBI„EBB
/I„ BB"
0
(
4 I„ BLBB
'4 I„ B!BB
4 I„ LBB
4 I„ !BB
4 I„ ALBB
2AF
+PQ
DD]
"(3((4
BB (!E#"(
(QB#(PE#
4 * O
'4 ? B ( !E
# !A # * ( (
BO
2A
4 DB
'4 BA#
,PQ%/$A"( 1$A"D]
$
( A
*EB
4 AB
'4 Q
4 S
4 A 4 AA
2AF
.PQ?D]
> '- 1 E B
)
I„ !LEBB +
I„
AAEBB ) (
4 QEBB
'4
LBB
4 SLBB
4 !EBB
4 QABB
2A)
PQ
DD]
, 1 , + a
/
-
J1
4 " 1
1 ­3x + y − z = 0,20
°
2 y + z = 0,55 ®
°
z = 0,25
¯
)J
'4 ? 1
AD` ≤ r J r 1 ≤
ED` ≥ B` J ≥ AB` 1 s B`
' 3J4
1
2A
4 sB Bs B`JsB Es E`1s
BAEsAE`
'4
+
PQ?2]
= -4e$
­ x + 3y − z = 9
°
®2 x − y + z = −4
°− x + 11y − 5z = m
¯
(
4 s B
'4 5
4 ≠LE
4 sLE
4 5
2A$
PQDD]
­ax + 4 y = a 2
°̄x + ay = −2
" °®
J(
4 ≠KA
'4 ≠A
4 s±A
4 sKA
4 sA
2A=
PQD]
)
­x − 2 y + az = 3
°
®bx + 2 y − 2z = 0 °4 x − 2 y + 2 z = 6
¯
' $ '
'( 9(
B ? 3 AL4 ( (
BA ? s ' s B (
BD 6 ' 3 B 4 ( BS ? s A ' s K (
P ? J s 1 ' s B (
2A L
PQD2D]
$
B " 1 ­2 x + y = 3
(A
®
¯x − 2 y = −1
BA $ 1 ­2 x + y = 9
(
®
¯x − 3y = −13
B
BD + s KD ­2 x − ay = 8
®
¯x + 2 y = 4
(
­ax + y = 1
®
¯x + y = 2
(
BS + s P
­2 x + 3 y − z = 0
" °®x + 2y + 4z = 0 (
°x − 14z = 1
¯
2AAB
PQUD]
" [ ­x − y + z = 2
°
®3x + ky + z = 1 °− x + y + kz = 3
¯
A (6
+$ ' 3 ((
4 S
'4 B
4 A
4 D
4 P
2A$
C\
F
PQ]
?Yn5P A
2 5 Y → n (
4 PPEASB
'4 PSESAB
4 PEPSAB
4 PESASB
2A$
PQ?2]
>(
( 6 3'!!'!!4
* ;<
;'<!O
4 BW
'4 AEAB
4 L EB
4 PLBB
4 10! 4!6!
2A)
PQ
FD]
+ C' 1
$ ( 5 )
,
LBB ) 12 4 EABB
'4 Q BB
4 LBBB
4 EPBB
4 SABB
2A=
PQ?2]
6LA'5S
' S S 'S60
'5 1
2 1
(
4 !
'4 Q
4 AE
4 LA
4 LL
2AF
PQ%/$A"( 1$A"D]
6 0 G M " 2 (
4 AB PB
'4 AEAB
4 BBSB
4 EBDB
4 DBLAB
2A)
)PQDD]
) , I 5 S(
'
I ,0'
5O
4 !AB
'4 LPB
4 SB
4 EB
4 !A
2A$
+PQ]
$ A L D E P
' 2 " 2 2 ALQ_
(
4 APEDL '4 APED L
4 APEL D
4 APDEL 2AF
,PQ%"<5D]
$ 5
P
1
.
" 2 5
(
4 !EP
'4 Q!A
4 BAD
4 PLB
4 SSS
2AF
.PQ?2]
" 2 !
5 D(
4 D
'4 AS
4 LE
4 DA
4 DQ
2A=
PQ?2]
" 1 2 A L D E E P
(
4 3DW4D
'4 PDW
4 E PW
4 3 PW4E
4 P P
2A)
PQ
FD]
> ' ) ) 5
/ 0 3> F $4 3
$@
64
$ 5
$ I G F ? + ) 1 5 2
/ 0
(
4 DB
'4 AB
4 !B
4 PB
4 DB
2A=
PQ22]
" # 1 '
)# s LE ) s )#A s L
( 2 4 !L
'4 LA
4 LEP
4 PL
2A=
PQ%"'9(D]
6 0
( "
29
( (
4 A
'4 S
4 P
4 AD
4 AB
2A6
)
PQ?2]
$C ' / ' /
I ' ' ( '' 5O
2A 2 3
PQ?D]
N ' ?( $
)F
)51
$
5 (
4 !`
'4 !` B`
4 B` L`
4 L` P`
4 P`
2AF
)PQ?22]
$ 5 '' ' '
'9 ' / > 5 / )0 '' B5
' A ' AA 0 '' XX
5 '9'0
4
'4
4
4
4
2A$
''''
''''
''''
''''
''''
+PQ?]
> ( !
E
= P / A A
L
/+1@
) 0
D $0 '
- ''
5
( (X"$!$%9
4 AP!`
'4 ELL`
4 LS!`
4 D Q`
2A=
,PQDD]
"
P , 5 2 1 = D
5
5
5
"2
51
4 E
'4 AB
4 S
4 P
4 A 2A)
.PQ]
>AEP
?'0 AS` L
L$2
'(
)
5LBO
4 ASEPB
'4 SE 4 LSBB
4 BAS PB
4 E BP
2A$
PQD]
+52
A 510 ) " 2 ' 52 E(
4 C1530 ⋅ C 620 '4 A1530 ⋅ A 620 4 C1530 + C 620 4 A1530 + A 620 4 C 5021 2A$
PQ?F]
6
D F 'r
2'
'
) ' ( 4 sL
'4 sD
4 sE
4 sP
4 s!
2AF
PQDD]
* 2 2AA B2
C-P($1$T0196(-
F 3I„LBPQ4
' 1 =(
F
# H
: q 3H:q4 " 2
'
=AB BAB H:q PAP`
, - '
' >?„ L 3I„
PPQP 4 3>I?$
AB AFL4
>I?$FI$?H6I"i$$$d(Lm?(/!
D1(
?+ 5AB AM6I)$="$
3)
9' AB A4
PQFF]
? 5/ 1 ( , 2 (
B EA
BA EP
BL PB
BD PD
BE PS
2ABA
)
C-P($1$T0196(-
+ ,
P?
- 2 '
PQ]
+ ' ' ' / "
2 1
(
4 P
'4 AB
4 Q
4 A
4 S
2AF
C-P($1$T0196(-
\
"0
'9 5 : 3
14 5 EE > 1
(
$ 1 ''(.
'9 3FI6q6IAB LPD4
PQFF]
) '- (
' : '9LADA
, '
02
( (
B Q
BA B
BL A
BD D
BE E
2ABL
C^F
PQDD]
" '
§ 11 ·
¨¨ ¸¸ © 4x ¹
§ x + 3y ·
¨¨
¸¸ © y ¹
0
? (
4 PE
'4 LLB
4 EE
4 DPA
4
2A$
PQ]
+ # u #
§n·
n!
¸¸ =
Ws
k
k
!(
n
- k)!
© ¹
¨¨
AL?
§ n · §n·
¸¸ = 3¨¨ ¸¸ © k + 1¹ © k ¹
# 1 ¨¨
0
4
'4
4
4
2A)
sD#r sD#rA
sD#rL
sD#rD
)
PQ?2]
" ( 0
9+
H HA HL A HD LL HE DPD HP E B BE ?5 + 2 9 + 34*
2 ,
P (
Q
4
'4
4
4
4
n 2 − 3n + 2
2(n − 2 )
(
'C
7
$F
,
$rF(
4 4 A
4
1
1
1
1
+
+
+ ... +
1⋅ 2 2 ⋅ 3 3 ⋅ 4
1 999 ⋅ 2 000
70x B
81
4
)
(
4
2A)
PQ?]
?5
21
!W s S3
K 4W K !AB 6 4 (2
'4 (2
4 (
4 (2P
4 (2 B
2AF
PQDD]
) 2 1
1
1
≥ 0
= −
n ⋅ (n + 1) n n + 1
§ x2
−1 ·
¨
¸
¨ 2 + Ax ¸
©
¹
'4
n 2 − 3n + 2
2n − 1
n 2 − 3n + 2
2(2n − 1)
n 2 − 2n + 2
4n − 2
n 2 − 2n + 2
2n − 1
4 QQQE× BKL
'4 QQE× BKA
4 QQQE× BKA
4 QQE× BK 4 QQQE× BK 2A6
)PQ?2]
,
3
3
4
5 3
5
3
4
8 3
3
2
A
3
2A=
+PQD2D]
" ( 'C §¨ x m +
©
1 ·
¸
xm ¹
k
(3#
214
?' # B #(2
BA #(2Q
BD #∈T BABU
BS #t E
P #(E
2AB!
,PQ]
n
? §¨ 2x 2 + 1 ·¸ x¹
©
Q(
!
A 4 S
'4 P
4 AD
4 LA
2AF
.PQ?2]
" §¨ x +
©
1 ·
¸
x3 ¹
PQDD]
$ '
L (''
12
(
4 AP
'4 PQ
4 AAB
4 ASB
4 L B
2A)
PQDR]
60 ''$(
4 1 '4
4
4
4
§
,'C ¨¨ x 3 +
©
1 ·
¸
y 2 ¸¹
24
( LP J (
4 K A
'4 KAD
4 KP
4 K S
4 KD
2AF
C
\DFF
PQ
DD]
>C'
(ABEB,
DBB''
2(
(
4 RD
'4 ARE
4 ARL
4 LRE
2AF
2
1
3
1
4
2
5
2
7
2AF
PQ?2]
=$F
$ '' $(+3$4sBD''
$F(+3$∪F4sBS+0
''F(
4 ERP
'4 DRE
4 LRD
4 ARL
4 }
2A=
PQ?22]
+ C ( =
tRR'RRM>J"=$Ju
$ BAB L
)
+ 5 $ s ‚
ƒ + 5$
=
''
(
4
'4
4
4
4
1
5
2
5
3
5
4
5
4
4
2A6
+PQD'(D("9W'%"']
G LB
P 0 L - (''A
5O
4
'4
4
4
2AF
PQ
DD]
> / 2 12
4 + / 2
(''O
'4 ?/2I„
ABB 2O
1
10
1
7
4
190
203
184
203
92
145
90
140
85
140
2AF
,PQ%/$A"( 1$A"D]
$ 2A
4 ≈BBBDE
'4 AB
)PQ?2]
> 9 1 LB
1
L 1
LB"'0
-
L '' 1(
4
'4
4
1
63
1
21
2
21
>1 π s L 5 '' '
4
2− 3
2
'4
3− 3
2
4
2− 3
3
4
4− 3
3
4
3− 3
3
2A$
))
.PQ?'!"@%'"!&!!F 9(
D]
>F'
($
'2
('
60
?'
''
)'(
4 2 '4
4
4
4
3
5
6
1
6
2
5
1
5
.E
> G
(
2A+'-
1 5 + , G
?'0
5(11
'' 8 -
+ G(
4 1 '4
4
4
4
2AF
PQ%"'9(D]
>((-
D$
2 ( ' 6 D
G + (?
E
+.EG
6
1
5
1
8
1
16
1
4
2A=
PQ? D]
6 DB AP
/ 5 AA / /5
$
''
5 (
4 DE`
'4 DS`
4 E `
4 ED`
4 E!`
2A$
PQUD]
6 L?
2
(
'' 2(
)+
4
'4
4
4
4
1
2
3
4
7
8
2
3
5
8
=)'
11''
(
4 5 '4
2AF
PQ?D]
6
(
' 5 ' > ' E` 1 1 L` 1
1 " '=
PB` DB` 60 1 ' $
'' 1(
4 QED`
'4 QEP`
4 QES`
4 QPB`
4 QPA`
2A)
PQFF]
4
4
4
128
7
128
15
256
17
256
25
512
2ABA
PQ?]
> AB
$ ' '
2 , ( 4 EE`PE`
'4 PE`
4 DE`
4 DE`EE`
2A=
)PQ?2]
6''
?'
'
('?'
1'
),
'
('
"2'
(
4
'4
4
4
4
A LP
D EP
P 2A6
+PQ?2]
> ( ' ADPS>'(
2(
.'(
$''((
2 5 E(
4
'4
4
4
4
BLDE
BLEE
BLPE
BL!E
BLSE
2A=
,PQ?22]
)
(
$F I
' /,''
8
'(
4
'4
4
4
4
2AF
3
4
1
2
3
8
1
4
1
8
.PQ?2]
>
D D
, ' + '0 )''
4 .
'4 / .
4 .
4 /.
4 2A
4 RAD
'4 B
4 RD
4 RL
4 LRS
PQD]
" 5 0 = ' $ = O
4 DDB
'4 QAB
4 AB P
4 DBLA
4 E!PB
2A6
).
PQDD]
6 AB
ALB 6
/ AB`
( B` = ' '
/ ' $ '' ' 5 (
4
'4
4
4
4
24
350
24
47
47
350
23
350
23
47
2A6
PQ?D]
" ( 5 ' '
( +
F 2
5 5 ( 6 2 (
8 ' .('
' ? 5
'' 5
1 5(
4
'4
1
3
5
12
4
4
4
17
36
1
2
19
36
2A)
PQ?D]
=50 ' ? @
' 8 34 @
'
34 ' 34 /
'
4 * @
O
'4 ) , 2 3 / '4 6 ,0'
4 ) + '' ' N + t
REB
?)
I2'@F
K QSS@
(
/
2A
4 ,s!,sQ0s
D?sD?sL
'4 ,s)D.A⋅)Q.A⋅)!. ⋅)D. ⋅)L. ⋅L!
?,sAE⋅L ⋅! 4 8748 13585
C-P($1$T0196(-
, '
BBBBB / ' L (
$F)$''
+
'
embarques do mesmo
passageiro
5
4
números de pessoas
1.000
1.500
3
3.000
2
10.000
1
60.000
G '1L
'
0'6
5 PQ
FD]
? 5 '
PQ
FD]
)
+ BBBB ' $
12'
+ ' F 1'
+1'
V3?434
4 '4 4 4 4 2A$
C-P($1$T0196(-
6 5 -
)
( 1 5 1 (
5 ? 9
'
)
/
$
'' 5
(
(
4 1 '4
4
4
4
2
5
8
2
3
3
4
5
6
2A)
C
E
PQ?D]
)(
' $ 2 ( (
+
4 EE!SQ B
'4 DEP!SS
4 DEP!SQ
4 EEE!!Q
4 EE B B B B
2A=
PQ?2]
$ ( (P
$ '
/ (
('
4 '4 5 ' 4 4 /-'
4 '
2A=
PQ?22]
, 2 I2 /
9 =tR&&&u$ !
AB L
$
4 PA`
'4 PL`
4 PD`
4 PE`
4 PP`
2A6
PQ?22]
,
F ) ( ( /'
((
2A )0 ( ( / '(0
$ S $ASLA
$LS !
" ( ( (A
PQ?22]
$'
F)0"
m 5 P AB L
=0
+%fWWWg
fLTT
T
YNLZTNYT
LMN
YNLZ$
g0%YNYNLZ
-0.
+
g+5
:5BO0 1!06B51'!YLYNLZL
-0.
=
4 ( )0" ( F '4 ?' ( / LBB BB ' )0
" 2A
4 $
APPDL!
'4 = ( / ( LBB BB'
BBBL` )0
)0"
'(0
= AB L
8 C AB L
2A L!'
+PQ?]
+1
'
" '
)
$
M 7 M 7 ? 7
)PQ?22]
" (CF
AB L
34 1
4>>Q
34 5 + # LR
\d
34 0 4 -
. ) # LR
LP
$/'( 9'(
4
'4
4
4
2A$
N0N0N
N0N0@
N0@0@
@0@KN
+
,PQ22]
> 5 ( A L e #
"'(0 5 (MM
2 M M≠M≠M≠#
?
5 ''
4 M
'4 M
4 #
4 M
2A=
.PQD'(D("9W'%"']
= '
'34(''
' ?' µ σ0
4 µsDBB Btσt AB
'4 µsDBB BBtσt B
4 µsD BQBtσt BB
4 µsD BSBtσtQB
4 µsDAB!BtσtSB
2A)
PQD'(D("9W'%"']
> S 5 ' ' ' 8 ' ''"5
$F)=6@7:
?
'
"(
4 QBBB.Q E.QEBB
'4 QB B.Q AB. BBBB
4 QBLB.Q AB. BBBB
4 Q AE.QAEB. BBBB
4 Q AE. BBBB.QSBB
2A=
PQ%"'9(D]
> M ( ' 6 AS (
( !B $- BB DB I (
('0
4 '4 B`
4 AB`
4 B`
4 AB`
2A$
PQD'(D("9W'%"']
> $ F ) = 6 " 2 '
+
"(
(
4 LAB
'4 LLB
4 LEB
4 DBB
4 DEB
2A6
PQ?D]
" ' * ( O
4 LEB
'4 LSB
4 LQB
4 D B
4 DDB
2A=
PQFF]
$5'
)6,>If,M9$;9"''
@=?+ABBP SL
=
( 5 (
B BQL
BA B BL BQ
BD !
BE AE
2ABD
PQD]
$ ( ' 1
2 ' ,
' ' 0 "
(
4 PP
'4 !D
4 SE
4 Q!
4 A A
2AF
C
2
D
PQDD]
,-''0
$F)=F)@69
+
P A B A "
-(
4 SL
'4 BL
4 AL
4 PL
4 ADL
2A)
PQDD]
> 5 ( ' 0
' L ?' LAD L 4 LPB
'4 LES
4 A!D
4 APS
4 AEA
2A6
PQ?22]
" 5 ( ' ' $5(
'''0
$
' (8'0
) ' ( - B`'
' 0 '
2A
$ AD BEPL
PQ
FD]
> ' 05 9
$ 05 P
'( P $ 05 '
$ A ( 4 72(3 + 3 ) '4 36(6 + 5 ) 4 108(2 + 5 ) 4 27 (8 + 7 ) 4 54(4 + 7 ) 2A6
+)
PQ%"5#]
> ( 1 - s DB
DB$'
9 9 ' ?' s BBB L s AB 3 = 1,73 (
'(
4 EP
'4 ED
4 LA!
4 E!
4 LAD
2A)
)PQ?D]
$ 9$F)=6@7:$Fs A$=s
S $@ s Q 6 DE 8 ?'
96@+5-
=*F):6+7 L (
4 DSS
'4 ELA
4 !AB
4 E!P
4 DEP
2A=
+PQD]
$ 1 ( ' 8
'/ ( '
1 ' (
3$)+I'M$
M
KA 6K?+6
H@AB 4
>
L
P
A '
@
RR&&&'RRRRp
=' '
' ? NL ' ND ' NP'
(
3$ 3 s !4
++
4 NLtNPtND
'4 NLtNDtNP
4 NDtNLtNP
4 NPtNLtND
4 NPtNDtNL
2AF
,PQ]
> ' - 9 A B D ( LB '
? 2 ! Q A AB '
4 DSBA
'4 BDA
4 L!PA
4 DSSA
4 D PA
2A$
.PQ?2]
> ' '
' ' ? ' 4 BL#
'4 BA#
4 B#
4 #
4 BB
2A$
PQ?2]
> 1 9 B E
A6(
1 E BBB $-
/ 4 AE
'4 A!E
4 LB
4 LAE
4 LE
2A6
PQ
?D]
> 0 $
' B
'6
' 5 F+
' )'
(F
* 9 5$(FO
+,
4 AB
'4 10 + 10 2 4 LB
4 10 + 2 10 4 10 2 + 2 10 2AF
PQ?D]
/ ' ('(
$ 1 '(
4 1 '4
4
4
4
8
1
6
2
9
1
4
1
3
$
"
"Nα
'
2AF
PQ]
?5Yn]/
+ ( 9 5 ' ( 9Yn]5
( L L " 5 +
( 4 '4 4 4 2A)
PQR]
> $F)= " ' α 6 ( "N F)
)
ˆ $F)= "N .
ˆ "N ( Bt≤ π .
2
ˆ 9 "Nα.
ˆ9$F)=N
2'(J
"
J9L
9
(
4
'4
+.
4
4
,
2A$
PQD]
$95
1
@
-=:-
9
9
3$)+I'M$
M
KA 6K?+6
H@AB 4
) 9 ' 5 A : :- ? ? 9(
4 ?s3r43K4
'4 ?s3r43K4
4 ?s3r4A
4 ?s3K4A
4 ?sA⋅A
2AF
)PQ?]
?'0 ' 9
( $F)= 5
' 24 2 ?'0 '( 9 ( N ' P
$ 9 A (
4 QPB
'4 LAP
4 B!P
4 ELP
4 E!P
2A=
+PQ?F]
6 9
-5'(
A
?(
DABL 9
4 PBL
'4 DBL
4 A BL
4 LLBL
4 DABL
2A$
,PQ?2]
=50 - 9 ' $ ' -
AL
- )
3 = 1,7 -(
4 AEEBB
'4 AAQEB
4 !SEB
4 ELBB
4 !PEB
2A)
.PQR]
> - - A
"(
)1-
2A 1 4
PQ?D]
6 9 ( 5(
4 a 3 '4 a 2 4
a 3
2
4
a 2
2
4
a 2
4
2A=
PQ?D]
$ ' !0D ( 0D " α?'0
5
(!
α α
BC ( AD $(D!
( 9 - ' BC 5
α9 θ
B t θ t πRA ? !D s " 2 / 2 " θ
4 !0D.
'4 9 0!D AB .
4 9 0! AD 2A
4
'4
4
l 3senθ
6
l
sen 2 θ + 1 2
l
⋅ 4 cos θ + 5
2
PQ
?]
> ' @1 '9 '"
DAL LBL )
( A (
,
PQ]
$ $ 'N$
F'AA
NF 5 9?$?F
4
'4
4
4
4
P
!
S
Q
B
2AF
,
PQ22]
> "
$- RL 0 2 $(
S
,( A
SB
4
'4
4
4
'4
4
4
2A$
PQ
?D]
> 1 (
5
' = 2'
1 3$πsL D4
4
'4
4
4
4
2A)
L DB
LLDP
LDED
LE A
LS P
VB
'
/
V
A 4
4
1
4
1
2
1
4
1
2
1
2
1 8
1
6
1
6
1
2
1
4
2A$
)PQD]
, + 2' ( 5 (EL`
5 / 6 5 G(
/ L
? G( DB` = 1,4 = 1,183 4
S`
'4 DB`
4 SL`
4 PBB`
4 ! A`
2A)
+PQ?'%9 '"%D]
$
' B A 9
A
*
91
Pπ A
?' 0
3 4 4 AB`
'4 AE`
4 LE`
4 LB`
4 DB`
2A$
,PQ?2]
" $F)= P 6 LPBm
( AB " - L(
4 DPπ
'4 DSπ
4 EBπ
4 EDπ
4 EPπ
2A=
.PQ
?]
+
( / ?'0
(L(
4 E
'4 P
4 S
4 Q
4 A
2A=
PQUD]
+
" Is P 0 9 θ s AAEm 6 0
s +10'
AC BD " '
$(
,
4
2 39 cm '4
15 39
cm 8
4
55
cm 8
4
2 55 cm
4
15 55
cm 8
,
2A6
PQ?D]
> ( L '
'(
C 6 H'0
"'
(
'
" 1 BB(L
4 E!DBBπ
'4 EPABBπ
4 EDSBBπ
4 DSPBBπ
4 DEDBBπ
2AF
PQ?'!"@%'"!&!!F 9(
D]
>
'S9 P
((
B
)πsL(
L
(
4 D A
'4 PEL
4 !DP
4 QLE
4 S!A
2A=
PQD]
> 1 '
'
C ?'
LP ( DS 9 ( O
=1(
$ 1 2A
V (9
9 ) 1340
=0 ( 8 '0
B RL BP RL
!(
3 3 ≈ 1,44 3 5 ≈ 1,71 + 1 (
PB
?' 1 (
' 1 0
'1
1
)
A1
NA
1
N )(
0
+
303 4 cm (
PQ?22]
> /ABAB
9 ' (
' ,
4 ES P
'4 SA SB
4 SD PS
4 SQ EB
4 QB SB
2A)
PQ?2]
>C(
(:=50
:'
C
$
$-
(
4
'4
4
4
4
H
4
H
6
H
8
H
12
H
24
2A6
)PQ?22]
=50 A ' '
( 9 ' $( ' H'0
?''
πRP 2ADIDIPI
+PQ]
> I 9 N ? ( EB` (
4 BBB`
'4 AEB`
4 A EB`
4 AL!E`
2A=
,PQ?2]
> 3 4 L 34'(
L?
?
SB` )
2AQP
.PQR]
>$L
(α
>10
@ @ $ "'
)(@5'(
'
5 ' ? 8 ( 9 FT, 4 B
'4 Q
4 S
4 !
2A)
PQ?RD]
> / '- C 5 ( 34
?'0C
34
' 3I4 ) C ( ' " -
) '
( h2
R2
(
,)
4 A
'4 D
4 S
4 8 4
3
4
3
2A)
PQ?D]
) P 3 4 P 3A4 'P3L4
-5
'
3D4
BB - / ( ARL AA LL 1
(2‰
DD ( ' 2ANN@@N
PQ%/$A"( 1$A"D]
> E P 9 " 2
9 >1 π s L 2
(
4 S D
'4 !BA
4 LQP
4 EBB
4 P S
2AF
PQFF]
?'(
( EBB $
/
8 . '( (
3 '( (4 3FI6q6I AB L !P4
)0
'
π s L 0 / B AB
BA AE
BL LB
BD LE
BE DB
2ABL
C-P($1$T019=-L
$ ((
,+
" L
" / $ F
8 1 PQ
FD]
=
5 4 L
'4 LE
4 D
4 DE
4 E
2A=
PQ
FD]
>
(
" 4 (ABAEπL
'4 (ABAEπL
4 ABAEπLDBEπL
4 (DBEπL
4 DBEπLPB!EπL
2A$
C
2
PQ
FD]
,'+*I?
/
( /
α β Btαtβtπ
?'αsBS0
β(
4 KBS
'4 BS
4 KBP
4 BP
4 KBA
2A)
PQ]
? I → I ( 34 s
A r (
4 DE
'4 LB
4 E
4 B
2A$
PQUD]
?534sArβrβ?34sB
3π
β=
2
,,
4 '4 K 4 K B
4 K 4 B
2A=
PQDD]
" 5 3?4 A⋅A r 3A4 u A B t t π (
4
π
5π
­
½
S = ® x ∈ (0, π) | 0 < x < ou
< x < π¾ 6
6
¯
¿
'4
π
2π ½
­
S = ®x ∈ (0, π) | < x <
¾
3
3 ¿
¯
4
π
2π
­
½
S = ®x ∈ (0, π) | 0 < x < ou
< x < π¾ 3
3
¯
¿
4
π
5π ½
­
S = ®x ∈ (0, π) | < x <
¾
6
6 ¿
¯
4 ?s‚∈3Bπ4ƒ
+PQ
FD]
$
34s3r4DK3K4D
,.
" '(
4 5π 2A$
'4
PQ
DD]
?5s"
(
4
'4
4
4
2A)
)PQUD]
6I
sen 2 x
senx
4
4
( 3 − 1)/ 2 (1 − 3 )/ 2 ( 5 − 1)/ 2 (1 − 5 )/ 2 f (x) =
4
12
4π
9
3π
8
5π
6
2π
3
2A$
,PQ
?]
> 5 0 '
$J
θ
(
(
4 ‚∈IR≠#π#∈]ƒ
'4 ‚∈IRA#πttπrA#π#∈]ƒ
4 ‚∈IR π rA#π≤≤ 3π rA#π#∈]ƒ
2
2
4 ‚∈IRA#πt≤ rA#π∨ 3π rA#π
π
2
2
≤tAπrA#π#∈]ƒ
4 ‚∈IRA#π≤≤ π rA#π∨ 3π rA#π
2
≤tAπrA#π#∈]ƒ
2A=
2
4
'4
4
4
4
2A=
JsAθ
JsAθrA
JsθrA
JsAθ
JsAθrA
.PQ?]
) '5 1 ' @- + / $
8E3sB4
-
§ πt π ·
T ( t ) = 24 + 3 cos¨ + ¸ © 6 3¹
34
- 34 3m)4
" '
4 PAEEm) B
'4 AA!m) B
4 AA!m) E
4 PAEEm) E
2A)
PQ?2]
, '0 J s KD K > (' (
4 J≤Kπ≤D
'4 J≤Krπ≤KD
4 D≤J≤Kπ
4 KD≤J≤Krπ
4 KD≤J≤KKπ
2A=
PQ
D]
$ - J s
34
?'
(
4 s πRD
∈]
'4 Lπ(1
4 34sAK 4 π
4 34sAK 2A=
PQ?22]
6 5 (
.
?'0 (AA
9θ(
!(
ρs RL
s × BE,RA
s BRA
4 LRE
'4 LRD
4 BR 4 DRL
4 ERL
2AF
PQDD]
> ,
S # 9 +'
9
PBm
9LBm
' $
4 SB#R
'4 ADB#R
4 AB#R
4 EB#R
4 ABB#R
2AF
PQD]
$
$ ' (
2'$
$
F , (
3$F,4 '
C ' 6 5 1E`SLL`>
E` 1 'BBEI
) '0
A
?
$F, E`
4 E
'4 AB
4 2 + 1 20
4
401 − 2 4
4,01 +
3
2A=
PQ
?2]
"
π
7π
π
7π
(
y = sen − cos
+ sec + tg
1
20
4
3
+1 2
'4
3
−1
2
4
4
4
2A6
3 +3
3 −1
3 +1 6
3
4
.
)PQ
DD]
> 2' 1
( ( 3 a 0
4
' 0 9 θ ' '
.
3&&&'$4
4 ? ( 9θ
'4 ) 3θ4s RD B t θ t πRA
( %3Aθ4K%3Aθ4
2A
4 1 '4
2
7
17
+PQDD]
$ ' '
L +(
+**
1(
' '
A 6 Š 1 = α s BS + 9Š(
4 DS
'4 EB
4 LS
4 DD
4 DB
2A)
,PQFF]
$ 1
9 6 '
N "
(
9(2
/$F
, 2 A
$F
? 9 $0
(
B 1 + cos θ
1 − cos θ
BA
1 − cos θ
1 + cos θ
BL
1 − senθ
1 + cos θ
BD
1 + senθ
1 − senθ
BE
1 − senθ
1 + senθ
PQ?22]
"' 99
.
2AB .PQ?]
6 M0GM-
3 SQL0 QSL4 $ " ( PB
B $
'S
$ 9 9 '5
O " I 3 DLP0 D!P4 C ' D! ' /
"'
+ * ' "
9 α /8
(8
) 9 ") ' GM-
"9α
4
4 3 −3
10
'4
4− 3
10
4
4−3 3
10
4
4+3 3
10
4
4 3 +3
10
2A$
PQ?22]
H
" ! % ' 9 + 1 9/
( 9
+%
fWWW)T)T)TNdLLLN)g0
%MYNLY]0b
> 9 / E
(BSS
) ' 2A A
4 9
.
'4 1 0
1 ( AAEBEPAP#R
=
2 ≈ 1,41 3 ≈ 1,73 2A
4
senβ =
PQ
FD]
, 34s
34s
0,26
≈ 0,29 0,88
.
'4 ≈ Q!D !!!#R
PQ]
? ( D−Ds 1 (
2
4
'4
3
2
4
3
3
4
3
$(
cos( 2senx ) =
2A=
PQ?]
> ' SB / 9 AEm 1 ) EB 8 /0'9EBm
'
2
2
4 BA
'4 BL
4 BD
4 BE
4 BP
2A)
PQ?D]
= '0 520
§ p+q·
§ p−q ·
sen p + sen q = 2 ⋅ sen ¨
¸ ⋅ cos¨
¸
© 2 ¹
© 2 ¹
?'AEm(BDA
AEm ( BQ ( 4
'4
4
4
4
DD.
DA.
DB.
LS.
LP
2A)
H⋅Q(/
4 BrS
'4 A⋅3PrA4
4 A⋅3 BrS4
4
4
2A6
1
2
1
2
⋅3PrA4
⋅3 BrS4
\
PQ?
6 1 6 ' 0 - 0 ' '
$ '
8 * ;<O
4 "'
'
'4 $'1
1 '
4 " ' ( 0
4 $ ' 1 ( 8
4 + ' ' '
(
PQD
(C
0 5 1 " 1 ( ( 1 $ 1 1 .
3RR'$4
$ ' 1
+ 1 0 1 8 + 1 0 18
, 1 8 1 - 1 N ,1
6
4
'4
4
4
4
N
N
N
PQ?
$ 1
4 $1
( '4 " 1
( 2 ( 4 ,1
/ ( '
4 $- 1
4 $ 1
1 0 (
PQ
FR
> ' 1
(
1 $ "'0
'
'
$)"II6$?
4 '4 4 4 4 ,
.)
PQD
@ALI'?( Q!BI
34
DE!A
DE!7H?>36>$4
$'( 3@AL41'
C;<
6 I 3@ AD4 6 1 '
''
6 ( $ ( 5- 23@AE4
?'0
@ADMC;<
$1
/'
$ ' '
(
@AE)(2
)'ADAE
' 9 $ ( (
? - ( ( 5- AE ( 2
" I ( N , AE ( 1
(-'
$
4 ? '4 ? N 4 ? N 4 ? 4 ? N )PQR
/6 6A6L
6A
6 6L?
(
9
$ ( /
$
6 6A.
- (
, ( 6 6A '6L
+PQ
= rD r 19
6 1
8rD0
1r
O
4 ARL
'4 LRA
4 ERD
4 RL
4 LRD
,PQ
* ( r rA rL rD ( $
0 )5
B ( ( $ - (
(
4
1 q
4πe 0 d 2
'4
2 q
πe 0 d 2
4
1 q
πe 0 d 2
4
4 q
πe 0 d 2
.PQD
> 1 ( u B
+ .+
BD " ( 0 D(
BS G ' 1 - 0 ' ( 5 9
0 ( ( AB
P , ( - '(3
04 '-
(D
( -68
=9
θ 8 0
3
4 ( + )
, ' 4
'4
4
4
4
3⋅r⋅64⋅θ
3⋅0⋅6⋅ 2 4⋅θ
3⋅r⋅64⋅θ⋅θ
3⋅r⋅64⋅θ
⋅r⋅6⋅θ
PQD
G(
0 ' 1 ' ( 0D. ( 9 SBm$(
( D ' ( 0 ( H0 '( 9(
B " (5(0
BA ? G (
5 0 - ( 0 (
(1
PQ?22
> =,$ + 0 ()
' 9 + 1
1 ' 8
→
→
F res = −b v
$ 1 $-
0 > 1
0
AB N )
DB '0 .,
"(
( 6 s AB‡ BD,R) 9
(s!B)
( 5 ' (- P‡ BK
Q
)Q ‡ BKL #
, (
4 (
.
'4 .
4 1 ( PQUD
> BE A ' 5 8 ( &
) | g |= 10m / s 2 0
1
, - ( R 8 (
4
'4
4
4
4
LP‡ BL
LP‡ BP
EB‡ BP
!B‡ BP
AP‡ B0P
PQ?2
$
(
/$F)
$ $ ) (
8
4 B× B A ( 2-
'4 PD× B A - 2(
4 B× B A - 2(
4 PD× B A ( 2-
4 ( (
PQ?F
V
4 $F/(
)
'4 $F/(
)
4 $)/(
F
4 ($F
)
N$tNFtN)
4 ($F
)
N$uNFuN)
PQ?22
)08
B 18
..
' r
? -(
1/
/(
4 T = 2 π
L
g
'4 T = 2 π
mL
qE
4 T =
1
2π
mL
qE
4 T =
1
2π
mL
qE + mg
4 T = 2 π
mL
qE + mg
)PQ?F
$
H@
(
+PQD
>B # BKP
) ( - BE
,R) =1 / B RA 4 SRA BRA
'4 QRA BRA
4 SRA ARA
4 QRA RA
4 SRAQRA
,PQD
V (
( 6 ( 1 (
* - ( -1
0 1 9 /
(
(
(!s
εB(
(
=10 0-sA
× B0A!#(sA× B0 Q)
' ( s D ×
B Q ,R) 1 1
= 1 A(
4 " ( ( '4 $ ( @ 5- ' 8
4 $
'(
4 $ ((
4 " ' ( ' ( 1 2e 2 a 2
4πİ0 3c 3
0 E
$9(sD× B ( ( 1 - 9(Cs B0 Q
4 ) / ( ! s α A α s
2
1 2e
4πİ0 3c 3
sP× B0EAj⋅$
'4 ) - $
.PQUD
> @ ;,3#s
Q BQ ,AR)A4 *
8 9 = ( LPB ,R) ( SBN
<$
4 ADµ)
'4 Bµ)
4 LB)
4 S)
4 B)
PQD
$'( 9(
B )
(
' ( 0
(
BA )
(
' (0
BD H ( 8 BS $ ( 0 P 6 ( ' 1
5-
PQ
* uA
uLuD
, 5 ' ( $ (
4 '4 A
4 L
4 D
PQ?
) 1 , '
4 ( (
'4 ( 9
8
4 ( ( 8
4 '
( 1
4 - ( 9
8
PQD2D
) 0 D
Z 9 0 (
D
4 " 4 ? ' ( Z $ - ( ) B 0 D 1 BA $0 3 *D
4
9 *
4
BD " ( 0 (
( D
BS " ( 0 D P $ PQD
) (
( I
'( 9(
B + t I ((
BA + = u I (
=(=A
BD $ ( Q 2 4 πR
BS " ( Ns
kQ
tI
2
(# 27Q2 16d
PQ?
6 =6I$,
('
' ( , (
<3 + '
#
'#
'1+
;#
'
=
,
+?9AER?
+$FI$M6KABBE
d
P $ (
(
(
)PQ
"'
) 9 ' / 2?'
( 0
$ '( 9 5
4 $5
'4 $ '
4
'4
4
4
s3≠B4
sA
sRA
ssB
+PQ
> 9 C/?'
( ( 0
4 ( - (
'4 ( ( 4 ( 4 ( - (
,PQ?
, 9) s B@)AsLB@* s ) *A s L ) $
9
( s $- -
LPBBB#R
9 d ?'
4 9 .
'4 * .
4 9 )A - '.
4 9) A`
? tt 1
1
A
≈ 1 + x 2 4 2
2
1− x
.PQ?F
= ' ' > "
' '
;6 '- 9 ) s PBµ@ (' ABN(
ABN
0 ' 0 9 )A s DBµ@ + ? 0
?A M ) ' ) 0 )
9 ( 1 , W<
" 1
'
'- '( ' '
) '- (6s §¨ 1 ·¸ )>A
©2¹
9)s
q
U
+
$@ (
@A
'
, ( U 1 BB PAG6B(
B9
LBm U (
?'0 L Ω B
(DLAG"
(
•
•
•
•
) .
.
.
@
PQD
> C I s BΩ ( 9 E µ@ 6 5 (
' AN1
1/
s AΩ ) ' 5 '( 9(
B $ ( I( $
BA $ ( (
BD $
I( BN
BS $
(
P $(
(AEBµG
PQ
R
4
'4
4
4
4
LEB
DDB
ELB
PAB
! B
3
3
3
3
3
PQ?D
" ( '
( -$
' ( 2 ( '
" '
(
9 8's
S , 4 0 ( 0 3)04 3)Ar4 .
'4 ( '.
4 :) : ( )0 )Ar '.
4 (
' ( 9 A@
,"66$="6
)(s0 P B0 Q)
@s B0 A@
s B0Q
[R2
C-P($1$T0196(-
* (
/ * ( '
(
( ( @ 1 C 0Y
PQ
R
> $ 1 (
4 1 '4 ( 4 4 9
4 1 PQ
DD
+
0sEB× BK
D
A9s
AB ( s 2 2 C = ε 0 A εB≈
d
QB× B
PQ
D
>$
9L
/ $ ) 9 ) 9 )B 4
'4
4
4
4
)s)B
)uD)B
Bt)t)B
)Bt)tA)B
A)Bt)tD)B
K A
C
Vm
, s
DE× BKQ)(
4 A× BAN
'4 EB× BKDN
4 AB× BLN
4 B× BKABN
C-P($1$T019=-)L+L,
!(
$
$]
=2L)#]'$
+(LX]$
) [ a ]π
πε a
.
.LX $ ] )PQ?D
" / ( 9
8 = (=/
.
+PQ?D
µ0a_L
/(0a0)µ
µ 0 a 0 L µ $ 9 ! a L '$ ) - ( 1 A]$
,PQ?D
, I) ' ) ' ( a $
)((
) (
$b 2A=
2A$
)2A
6 ( - 6 6A * r
*A s P µ) *a *aA
5*a s*aAsSµ)
+2A$
,2AF
.2A=
2A A
2A
4 ( sP2
R
$ ( 8
5
'4 N vT(M) = PAE× BwLR
N vT(N) sEB× BwLR
4 $- 0
s F C s B ( ( s2T
+
V [ C → d
2F
\
-
2AF
2A=
2A)
QM vTM ǻZ M 15
=
=
=
QN vTN
ǻZ N 12
QN
QM
2A$
2A=
2A)
N
s vTM
vT
=
Q d ⋅b
=
= cte. vT
V
s AE
ǻZ N 12
= sBS
ǻZ M 15
Ÿ
)
2A6
)2A$
+2A=
,2A
4 $
(
s!C
+ / ( -
( '
:[
:[ 14 × 1019 C ⋅ N
C ⋅ Kg
2 × 10 −27
s eE s 2 × 10
mp
−19
s D× BA!TA
$(
A
s α C s P ×
A A
0 Q
B0EA U ⋅ 3 D × BA!
T 4 B s !P× B0 !@s !P×
B0 D@≅ A× B0 D@
'4 $ - ( '
,
- ' ( ( 8 0 Q
⋅ D× B Q⋅D× B0 E
s sA× B
N
s A× B0 E@s A× B0 L@
C
$(
s
1
Ar 2
2
( E pot − Eirr )
mp
[
s
s
2
(1,12 × 10 −13 − 1,2 × 10 −14 ) m / s 2 × 10 −27
10 27 (1,12 × 10 −13 − 0,12 × 10 −13 ) m / s s 10 27 × 10 −13 (1,12 − 0,12) m / s 7
s B! (1,12 − 0,12) m / s = 1,0 ×10 m / s s B!R
.2A6
2AAQ
2A=
2A)
2AA!
2AF
2AA )2A$
+2A$
,2A
4 )sDB:
'4 > sB 2
4 )Ad≅LBT r
1
18
⋅3 B0D4AU:
4 ≅PBB⋅ B!R
.2A
$() -
ABN(s) >(sP⋅ B0P⋅ As!A⋅ B0
P
)
$-
=
'(0)s) r )A s P⋅ B0P r D⋅ B0P s B⋅ B0P (
)
s q U
>s q
C
=
72 ⋅10 −6
10 ⋅10 −6
s!AN
+
$ ( 1
) 6 s
1
)
2
>A s
1
P⋅
2
B0P⋅ AA s L⋅ DD⋅ B0P s
DLA⋅ B0PsDLA⋅ B0DG
,
'(0
6s6 r6As 1 C1U12 r 1 )A U 22 2
2
M> s>As!AN
$'(0
6 s 6 r 6A s 1 P⋅ B0P⋅!AA r 1 D⋅ B0P⋅!AA s
2
2
E⋅E SD⋅ B0PsAEQ⋅ B0PsAEQ⋅ B0DG
∆6s6K6sK !ASµG
"'06(6
) 0 9
$
(
( ( G '( (
8
2AAL
2A=
2A
4 "(
(
" ((
'4 6sS⋅ BPNR
4 •
+
−
@)s AS⋅ B− A,
•
+
Yo
@)sAEP⋅ B− A,
4 *s!PS⋅ B− L)
2A$
2A6
2A)
)2A*sE)
+2A6sBNR
,2AsNIRIsE$
c
PQD
$ ' 0 C
$
' 1 BBB$3$‹4
?'0((
PB × B0 Q 2 ((
' 4 BPA× B0 S
'4 PB× B0 P
4 E!P× B L
4 LPB× BA 4 AAE× BAA
PQ?D
$ ' ( / ( , / L (3$4QQBq
QSB q 9 H HA PB q ,(
(DDBBq
A+
,
( D ( 8
(O
…M……M
G
A $ Y I*(
( $
)=O
…$s) r=…$s) 0=…
$s=0)
G
L $ C I " ( ( ( O
… M … …
M
4 #q L.
'4 ( '
.
4 ( '
,"66$="6
$ ( B N AABN
"'•
PQ?2
$ 9 / 8 '
)/(
1 ' 9 , F L
(
/$2(
+.
$ C I6?+",=$$
G
PQ
?
60 '
' - ' ' ' '5 ' ' '
6 ' + ' .
?'
-
'
?' ' ' '
) )A
)L
+ ( 1
4
'4
4
4
4
PQ?D
,
(/
' 1 0 $ 5 AE$
$
$
? ) )A
$ / ( 9
BBΩ
" (
N ? ' LB$
V
4
'4
4
4
4
N
N
)PQ
) /
I IA $ / ( I IA ( 1 / $ ( 0
4 I sIA
'4 I sDIA
4 I s 1 IA
4
4 I sAIA
+PQ
) / / 6 ( 9
$ / ( 9(
4 LRA
'4 L
4 A
4 ARL
,PQ?D
$ ' (
, 1ŒI IA
) 1/I s
IAsL?
5 CB =50 5
( C
) / 5 DB` 5 DB` / 0 $ ∆ 4
∆t =
'4
∆t =
4
∆t =
4
∆t =
4
∆t =
40 Rmc∆T
ε2
60Rmc∆T
ε2
45Rmc∆T
16ε 2
30Rmc∆T
4ε 2
15Rmc∆T
8ε 2
.PQ?2
> / Ω BB N " 0 H
3 DA † BL GR# m) 1
AAPB † BL
GR#4 $ 8
AB m) ) 8 1
0 BBm)(
4 S LP
'4 AAPB
4 LLP
4 AEQP
PQ?
> / 0 ' @10
/ 0 ' ( ( ' 1 /
( ( /
I + (
4 IŸŸ+
'4 IŸŸ+Ž
4 IŸŽŸ+Ž
4 IŸŽŸ+34
4 IŸ34Ÿ+
PQ?
H-'
$ $/ 6 36$
K e ;
0
e4 / EL` ABBS ABLE =
(
' - 3- 4 $( ' - 0(
$ - ( ' $RR&&&RJJR
OsL LBAB'AB $)"II6$
4 $ ( 1
( 9 ( ( ' '
'4 $ 1 ( ' ( 0
1
4 $ 1
( ? ( ( 1
4 $ ( 1
( ' '
3 -4 ( 0 4 $ - 1
( PQD2
> ( 1 3 0'4(
8 ' N$F s AN 0 EB $ " / s BA Ω /
' 0 3 G4 1 $ (
9 V
4 $/2(EE&
'4 " (PBG
4 $ / ' (Eq
4 $/5(
EB` / '
PQDDD
:5 ( / 3
N
4 > 01 BB N *
AB N / DB q ,
/ 4
'4
4
4
4
AB
LB
EB
B
AB
PQD
> ( '' ( / EB*(
AAB N ' DB $ $ ' '( 9(
B $ 1 (ABBN
BA$ / (SSBq
BD$ /
(SBq
BS$ / ( !A` /
P?8(AAB
N //
1
PQ?2
" '
1
364 AN/34A
Ω 0
1364SN/
34DΩ
4 ) '4 , '
O
)PQD
> ( ' AAB N (
( B
$ $ / ( 2 ( ABBB q $ 8 1
8 / ( 4 ABB.A.BSB
'4 ABB.A.BQ 4 DBB.D. 4 DBB.D.BSB
4 DBB.D. E
+PQ
$ 1 !EBBBBN
( 9(
4 1(
'4 1'
4 1 G
4 /'1
,PQ
>
∆2s2$K2F(
0D
Cs Ω)'∆2
((
4
'4
4
4
.PQ?
" (
( C / ' 1 6 +
5 ((
' ( 8
, 4 ( (
'4 ( 8
( 34 /(
PQD
= 5 (
>
'
? (
' / (
4 RD
'4 RA
4 4 A
4 D
PQUD
, ( ' 1
( /
( 1 $9/
1
3 q7
B N4 $- 0 a 4
'4
4
4
4
EB$ AEN
AE$ AEN
EB$AEBN
AE$AEBN
!E$EBBN
PQ
D
$ 1
' q ? I
51
/ I s
#r B#sAB× BKDR+
U
> 1 B B B I q 5
'?IAsABILs E

4
'4
4
4
4
=I
=I
=I
=I
=I
sAEΩI sLBΩ
sABΩI sLBΩ
s
BΩI sAEΩ
sQBΩI sALΩ
s!!ΩI sQBΩ
PQD2
"
'
?/
BBB" BBBB"'0
4 '4 4 4 PQ22
>(
' 3F4 9 3H4 )
( ( 3$4
$'( 9
/
'(
$- 1 6 O
4 $ (
. 0 '
'4 $ A 5 ( 0
4 $L(
A1
( 9 4 $ D ( / 8
L
( 9'(
PQ?
+ (
- 5 ' ' 3 4 " N
34Y34
4
'4
4
4
B AE
ABE
A )PQD
"' /'
E / 9 / '(
/
,034
934pppppp
'R'/
8/
4
'4
4
4
4
A
L
D
E
+PQ?
, ( ' /
'(Is Ω
*()"II6$$
$ N N
O
4 $ N
'4 $BEN
4 A$ N
4 B$EN
4 A$PN
,PQ?
$ ;)
/ 9 , )
+0 ' /
4
'4
4
4
4
BΩ
SΩ
PΩ
DΩ
AΩ
)
PQ?22
$
6 ( 9 / IH /I@$/I@
1
' - 9 (
'
$N(AABN
9/ BB
qAABN
* ' /(
4
'4
4
4
4
EB`
EB`
8
AE`
AE`
.PQD
> ' 1 A N (
C/S
Ω$(( A$
4 /(9IH.
'4 (9
I@sLIHs PQ
?
$51
( > ( (
9 5
()
' ( 1 6 (
5
( 0 ,f? $ H a ,- 1 ( ,f ( ,fN5
'
T$
RR&&&''RR'R5QABQ U
+ ( ' / BB N /AB"
+ 6 H '(
4 $H)
M
'4 $H)
6
4 $ H )
M
H
4 $ H )
M
$
4 $H)
)
PQ$'$
"(
1 '
+ (2
2 8 2 "
( 2
/ " F ( ( ( '
h/3PB:14(
/
6
A, i 2
11
8
= A, i 2
11
10
=
A, i 2
11
12
=
A, i 2
11
14
=
A, i 2
11
4
i1 =
'4
i1
4
i1
4
i1
4
i1
24
A i 3
11
27
=
A i 3
11
30
=
A i 3
11
33
=
A i 3
11
36
=
A i 3
11
=
=
=
=
=
=
30
A
11
35
A
11
40
A
11
45
A
11
50
A
11
PQD
>(
6 s6As DN
/I s BIAsLB
I s B $
$ I P$
" IA(
$/IA
Aq
+
N " I('
* s B RA πsL
6
4
'4
4
4
4
N
N
N
N
C-P($1$T0196(-
=
N1LB† BSR
$
BRA
PQ
DD
1
11
C
>
( (
1
= 9 , 0 × 10 9 Nm 2 /C 2 4 πε 0
) BRm)
) EDB
R
PQ?D
" ' '
/ $
) ' B q ( BP $
=
'
C-P($1$T0196(-
6 ' C - 6 @ M -
4 " ( (
'
F '
$ ( 0 [ BAE
A F /
- ) 5 '5 '4 " ( > ' 6 s P N /sA
#Ω ? ( ' 0 ,
C-P($1$T019=-)L+
> ( 9
)'
K(1
( B$.
K1( ABN.
K ' 9 /PBq
)PQR
$ / S9(
4 LB
'4 PB
4 AB
4 ADB
+PQR
" 2 9 4 B
'4 E
4 AB
4 LB
C-P($1$T0196(-,
!(
$
$]
=2L)#]'$
+(LX]$
) [ a ]π
πε a
.
.LX $ ] ,PQ?D
" 3;4 ( 5 <16( 8 ( (
" (;-(
L[R]Q[#
( , [R][# >1 ( d # ; " 1 = 9 1 ? ( ' ; =/ #1%!(
C-P($1$T019=-.L
, SBB$
N 1 1 ( N 6 1'
'' 2 $ N (
1'1
=
RR&#R&#R+p
pN
$BA'AB .PQ
2
+ A N
1
.
'
8(1
(
6
0
4 $ ( '4 j 1 4 j
/
4 $ / PQ
2
> - ( (LNV
/ # ( ' -
$ / -)s B
RA
4 B '4 BL
4 B!
4 BQ
C-)P($1$T0196(-
(176( ! (5L 1 01%!(
%'; "(-
• $
s B R A .
=ρs BRLs BBB
#RL
• N 1 s LB× BS
R
•)+#sPPL× BKLDG×s
D D× BK EN×.
•)πsL D
PQ?R?2
> ' - 1εs AN/
6 ' ( ( ' -,@'(
/ 2 + '
(
4 =0
'
/ 2 G '4 ) / '
4 ) / I /
4 ?' / η ' ( 1 N ' 1 ε
/ ' /
4 @
'G
C-+P($1$T0196(-
+ ( 2 2 1
/ ( ( 2 ; 5
1
(
1
) 1
- $(
5 +6 G 0 / d ' '#
"
SB`
'
I$M$H:"G@.
@6II$I",;.
?"$I6?+$"@@?
+ML QDABBQ
3;>M$Io6?AB 4
;>M$Ic6??+)N)/:5
=t
RR&&&5'R
'R'RAB RBERu$
A!5AB PQ%"< "'IAsIL
s AI $ N $ $F'
PB
4
'4
4
4
4
EN3LRI 4$
LBN3LRI 4$
LBN3LRAI 4$
LBN3ARLI 4$
PBN3ARLI 4$
C-,P($1$T0196(-
) ABBS '- 5 1
) 6
@1'2'=
M?2$((
PQF
> ( / EDBBBq d AB )
(LS)'
? 1 (0
- A! ) ( LS ) EDBBBq '1
/q
B DDBBB
BA LLBBB
BL AABBB
BD BBB
BE BBBB
2F
c
-
2A6
2A
4 ! E#q
'4 , H
sAQ$
,A9
$
AsLB$
4 s $
2A
$
9F
) (
((
(
? F ) ( F ( 8
()
A$s)r=
$$3$4(
( '
6 9 ( )=
=$s)r=
L
) / (
/ (
$)$
$
) ( 5 ( ) (
$
2A-$
2AF
)2A$
+2A=
,2A6
.2A=
2A)
2A6
2A$
2A=
2ABL
2A
4 i = 2 A ≅BP!$
3
'4 ; ηG = 8 ≅BSQ
9
I ηR = 3
4
≅B!E
)2AF
+2A)
,2A)
.2A
4 "(
( ( G
( 8 ' 5 '4 " ( (
/
/
( 3+ s N4
= /
( 3N4 /(
2A$
2A)
2A)
2AF
2A$
2A)
)2AF
+2A)
,2A$
.2A6
2A
4 IHsDSDΩ
'4 ≈BLA0
2A6
2A)
2A6
2A B
2A
4 AB† BD,RA
'4 AN
)2A$
+2A)
,2AQA
.2A=
2AF
2A
4 6 0
/ ( ( / /((
=
'4
4
4
4
[LYN0![!4([
dN0
sB Ω
IsB Ω
ηsEB`
$'(
2 ×'
'
2A)
2ABA
2
PQ?
N(
( ' '2 ' '-
@'
'2 ( $ 1 ( 1 ( ) '2
(
= ' (
1 ' '2
(
PQ
= 1(""
@E$"
"
4
4
'4
4
) '
( 1 " " 1 4
4
PQ?
" 9
( 9(
' (/
82
' j (
5
,1 A (
1 ' j A ( (
N " ( ( 8 LRD 8
A
6 (
0
1;
C4
hTY
3' 4 )
-
.*'
) $
(!
'( 9
4
'4
4
4
0
0
0
00
4 ? '4 ? 4 ?
4 ?
4 ?N
PQ
?D
>''5
5 ) '' '/IsSΩP
B(
'4 " (
4 $ $= (
8 4 $ $F )=
1
8
'"
4 $ $= F)
/ DE B0A,
1 0'
"
/
AΩ
1EBN
(
''
= µ s D π ⋅ BK! R$
3'(4
4
'4
4
4
4
Aπ⋅
Dπ⋅
Pπ⋅
Sπ⋅
Qπ⋅
BKE
BKE
BKE
BKE
BKE
PQ?F
> (
' ( ( -
( ( >10 ' '
$ (
?'0-
( ( BBE (( E$10
(
4 $ $= F)
/1
)PQD2D
6 S Q:)"'
( ( ( $ ' H " ( )
B $ (
( /
/ BA " ( (
' FsA µ 0 i π d
BD " ( 9 8 Fs µ 0 i 4π d
BS " ( 98A
Fs µ 0 i 4π d
P " 5(@s
µ 0 i1 ⋅ i 2
⋅
⋅ L 2π
d
+PQ
R
,PQ?2
$ ' '
( '
' ( $ (
1/0
$ ( - F " (3M 4
3MA ML4 $ ( ?'0 ('
M ( $
( ' (
!(
• )H
• I/I
• MM
• )M #
• )MAML
A#
• M-(F
• $
+ 0
' C
(
4 nY
'4 Y
n
4 Yn
4 n
Y
.PQ?D
6'-
"$
1 '5 ( ( " N'
4
'4
4
4
4
I3M04RHF
I3M0A4RHFA
I3M0A4RDHFA
3M04RAIHF
3M0A4RAIHF
)
4 $ -/5-
4 =
- / 5-
4 , - 5-
$ F ' 1 $ - = 2
(N
4 (
(
'4 ( ( 4 ( (
4 ( (
4 ( ( (
PQ
?
" 2 C + - ?
<H:) 3H :
)4W - e$f e0f ) M( > ( " )
M( 1 8 + (
PQD
>M(
*(
1
( E '
, '( ( H = '( 9(
B "-
'(*3:r64
BA + 5- (
- 5Bs E H
BD , 5- ( '
BS $
(
P $ ( ( 4 M/ - 5-
'4 M/ - 5-
PQD
>''(C
''
,
(
4 C(
8
''
+
'4 '' 0 C ( ((''
4 '' ( C ((''
4 8 '' C (
((
''
4 '' C ( ( ( ''(
PQ?2
= ( ' 1 , . 0 ( 0 + (0
4 5
( 2
'4 5
( 0
4 '
4 PQ?2
N / 5 ' ) '
0 $- 0 1 0
+ 10
0 '
4 '4 4 4 4 '-
PQ
*'
(
O
4 @ '4 = ?
4 :
4 " 4 " )PQ
$5-/
V0 ( B (
4 '4 4 4 '-
,
) +PQ?F
4 " ( 2
'4 " ' 1 ('(
1
4 $ / (
8 4 "-
(
' ( 2D
4 " 5- ((8
= AsA AsD >3NFK
N$4 @ ( B 3
4
9QBm B ,
1
•
•
B 5 66 6?
6? .
5- B ,PQ
$ @ D (
D >
(
2
.PQ22
>(
( e
'
+ (
( D
4 ( J
*(F(
4 5- ('
(
'4 5- ( '
(
.
4 ( 5-5(
[-.T-D.
4 (
0
(
- 5-
[-.T-D.
PQ
?2
, ' ( (
B = 5- $ ( F ( PRA
) ' ' ( (
(
4
'4
4
4
4
8
@J
8
)'
G
$
)
PQD
>/
( (
/ $
- D (
8
?'((Q ⋅
B KL # 5- ( LB ⋅ BL R
-
'
(-
' $F
=[ P⋅ BK Q)
4 DD⋅ BK SGDL!⋅ BKLAG
'4 DD⋅ BKLAGBG
4 BGDL!⋅ BKL G
4 BGBG
4 DD⋅ B SGDL!⋅ BLAG
PQD
6 ( - ( ( ' "
9 9 ( (
,∆s ) 1$
PQD
" ( ( (
$
/ (
, ' 5
8
, A ''58
(
8
( 1 ''
PQ
"'
$ '
, '/
= '( 9
) (
(1
' ' ' ' ((
4
'4
4
4
4
4 $ '
'4 " '
(
4 "(
'
4 6'
1
'
4 "-
'
PQ
$ @ S '
1 >
/ ( I s BB P
Ω 6 '
( 1 H s B 6 /
-
BB > ( F s BS ( ' 3 4
sDG
$ 2 1 B m)
4
'4
4
4
4
LBR
ABR
BR
SBR
BER
C-P($1$T0196(-)
!(
$
$]
=2L)#]'$
+(LX]$
) [ a ]π
πε a
.
.LX $ ] )PQ?D
> (
" &
( B ( , BB 6' 5 ' ( (.
B 6' 5 (
'
(.
BA 6' 5 ' ( 5- .
BL 6' 5 ' ' 1
.
BD $9
8
2F
2
-
2A=
2A=
2A=
2A)
2A6
)2AL +2AF
,2AF
.2A$
2A$
2A L
2AF
2AF
2A)
2A$
)2A$
+2A
• $ 1 (
v1
= 2
v2
•
$1( R 1
,2AF
.2A)
2A)
2A$
R2
=
1
2
2A
εs − ∆Φ .∆Φ=Φ)0Φ.
I
66,
C M0MJ
1 SS! ' $'
6 ' QBE $ '
M0MJ ( < -/W
∆t
ΦsD0.
2
Φ)sD0)sDπ a 4
A
ΦsD0sDπ
∆Φ=Φ)0Φs −3 DπA
4
∆s +
) ' @
ε= 3 DπAR
4
$1
♦ .
♦ 0
/ '7<(W
7 I6
s
ε 3 Bπa 2
=
R 4 R
R
2A$
2AF
2A)
)2A@N@N@
?E
PQ?F
+''
M9
,&
6)
PQD
$ ' '( 9(
B " I I 1 / BA > ( /
' 2
+#
BD $-'(
'2C
(
1
BS " C (
0
(
P $ 0 ( - PQ?2
" A L D ( '
6
DH
(
?' 0 $ H ( 0 $ ( /
0 D '
6)"II6$?
4
'4
4
4
PQR
> / G AB 1 -
/ + 1 '' 8
$ 0 ( 9 - 18
)EL LL1
A
0AB
$- BB 4 PAE
'4 AE
4 AEB
4 EBB
PQ?D
$ 6?$=$")"MFIR 0
E ' AB Cm6;5 0
C
)
*)
4
:U ) + A $ LZP +
LL ' $ 6 + 0 /4Y:U
*' LZL YNN
+7
? P AB ' 0
L FRL
M0 ( 2 $ 0
0 L (S
4 BB
'4 EB
4 AE
4 AE
4 PA
C-P($1$T0196(-)
!(
$
$]
=2L)#]'$
+(LX]$
) [ a ]π
πε a
.LX.$]
)PQ?D
) 8 C
BB $ 1 1
B $ $ 2
C '
BA V
( /
BL , F / ( -'
( -'
( (
2 +#
BD +1/
(
( 2 0 BN
C-P($1$T019=-+L,
$ ' ABBA !e h 1 ' ' ? ' ( ' 6
I 1 '
$ ' 5
$ $ ( 8 8
' -
6I
Q C (
=
tRR&&&'RR DARRBPR
pBP‘BQu
34$BA'AB +PQ
2
$ I
0 $ 5 4 " ( 1 2 8 '4 "(1
2 8 (
4 $ 2 ( - 2
4 , (
2
(
0
( 2 8
,PQ
2
$ '
2
8 0
4 "(
/ '4 " ( -
2 +#
4 $( '(
( /
4 $ C-P($1$T0196(-.
(176( ! (5L 1 01%!(
%'; "(-
• $
s B R A .
=ρs BRLs BBB
#RL
• N 1 s LB× BS
R
•)+#sPPL× BKLDG×s
D D× BK EN×.
•)πsL D
.PQ?R?2
> 1 λ s DBB s DBB× BKQ s BB q R A =
6 QBE
C ( 1 ( - > ' 91
4 -1
'4 '
A B
4 2 - C-P($1$T0196(-
? -s P× BK Q
). - s !× BKA! #.
s BRA. s!P:.1s
L× BSR
PQ
D
)
9 s
BγsBγ
6sBγAB(
γ = 1 / 1 − v 2 / c 2 ( H1
= 6A K AA s 3BA4A ' <
51W
2F
?E
-
2A
♦ , 1 (
- 6 '5
< (W <(W ( 314 $'
( 1
3 4
/
' -60
5 /
♦ ,
5/
)
66
/
( 6 (
<2W
3
- 9 C (4 $
6
2A D
2A=
2A$
2A=
)2AN@N@@
+2A)
,2A=
.2A
4 εsDQ!× BK QG
'4 sBB G
4 sAB × B P-
2A
6 $
6 A
G<
51
W=
+
s
$ s B 6 s s
PQDD]
><C
"
'
W'-6
1
3N4"
?'0 ' 1
$0 ' '( )?"D'0
$0'
(-'
'(/
10
N 02'(
3 4 3) A:AA" 4 0 -
6
O
4 '4 N
4 4 N
4 N
PQ%"'9(D]
>
<W-(
4 '4 (
4 '
4 2
4 5-
PQ?]
,00C
C @C (
GC
'9 * (
CO
4 @
'4 *-
'
4 ?'
,
4 )'
-
1
4 @
PQD]
"
1
>(
'(1
(>
(
(5
(
" ( ( -
$
(
4 V'-
('
'4 $-'
'
4 $/
((/1
'
4 * 5 4 $((∆:
PQ22]
$(
"
'9
C$''
Espécie química
Fenômeno
Dióxido de carbono
Formação do diamante
Ozônio
Água
Queda de uma fruta
Cozimento de um alimento
Nitrogênio
Surgimento de queimada
Cloro
Quebra de grafite
='
4 -('9
'4 C
)PQ?]
$ M‹ G M‹ '/0
'( - 1 / , 4 $'
A34
'4 $
.
4 "
4 "
+PQ
DD]
> ' / 1 > - ('
/ - $ ' ' / 4 H0(C
5O
'4 + 5 ' ( = 5 ,PQ?2D]
" 0 '9 ( 2 1
)/'9
3+q$#!
++6F#L_ABBP4
V'LEm)'('9
0
4
'4
4
4
4
-
-
-
.PQ
?2]
"'
?">]$M1!%7!9!(<>"'(=
tRR&R01R
R00000Ru
$L AB LT$U
V
4 ' '9 9 '4 -'
(
4 ( 9 1 4 (
1/
4 '
PQ]
$ 3QEE` r DE` 4 ( 5 $ 8
+'0SEm) pppppppppppppppp
$
4
'4
4
4
4
-K
K
K
K
K
PQ?F]
"
$ ( 1 > 4 $8'
8
'4 "((
4 $ ( '9
4 $ BB)'
4 "(B)
(('9
PQ?D]
, '- ' $
'0 '9 "
'
1$
BB , '
B ,'
BA $'
BL )'
(
'
(
BD ,'
2
h
PQ
?D]
$
'
4 ('
'4 (
4 5
2
4 2(('10
4 (
PQD]
$'( 9(
B " ( '( BA $ - - - ...
BD $-('
BS " 1
P @1
'
PQUD]
6 )
J
$ *
0
4)J
7
<
=*A
0 5 7 )*A
J
$
- J
. )
$C
$
"
<=7
"
5
<=
"
H)
$ J
"
' 4
+
J
RR&&&A'RRRpp
ppp
)'(
4
'4
4
4
4
('
(-'
(-'0 A
((
0
)PQ]
?<;(A OW)$'?'
0 )/ :5 < ( ' ( '?'''
?'(W
?''2
4 $-'
'4 "('
'1
4 $'
A
4 $2''((
+PQ?22]
" 3 - '4 ( /
8 ' '
1 3A`4 6 '
61$'
)'
4 '1
( =/.
'4 /-'
,PQ
?]
"(,2(1'
, ( 'MV
'$ AAB A
4
'4
4
4
4
1
'
'
.PQD2]
"' N , ^
$
4
'4
4
4
"/
$
"N(
$
PQ?DF]
, ( 1'
"'((
( '( (
$
'55
=
tRR'R&RAB ARBLR#0J00J0u$ EAB A
= ' ( ( 4
'4
4
4
4
"-'
$
'(
"-'
"('9
"'(''
PQD]
$3434'( 934
B $(C'9
BA ?'9-/
BD $()")"A)"LAK'9-
BS " ' ' - ' 'A'L
P "-"A(1C
((9
PQ%"'9(D]
)
'
'
-
?/
4
'4
4
4
4
'
'
'
-
'-
PQ?2D]
6 '- '8''9
, 2 4
'4
4
4
4
A..
L...
L...
D....
D....
PQ?2D]
$ ( '9
$ '9 '9 //
4 ^
'4 ^
4 $
4 $
4 $
PQD2D]
= ' '
)
B " ('9
BA "AD/
BD " A L D A D /L(/
BS "LD
P " A 10 )PQ
DD]
)5O<>/
/ ( ' )0 SBBB
'9-'$:DBDE
DE`3R4W
?'5
4 2 '(DE
'4 -'(
DE
4 2-'
(DE
4 ' (DE BB
+PQD2D]
)'
Vinagre
Acetona
Grafite
Gasolina comum
Água potável
Gás carbônico
Ar atmosférico não poluído Hidrogênio gasoso
B ;'C'9
BA N(
BD :/('9
BS ^'C/
,PQD]
)('''AEm)
'( 9(
B $(/'
-
(/
+
BA " / 0'(/
BD 3$F)=64'9
BS $$=/
5(
P $$)((
/
.PQ?
DD]
>'''
'- / J A L '9
6J '
6JA/
6JL'1
)/
?-J /
A "JAL
L $J A/
63434
4 '4 A
4 L
4 L
4 AL
PQ?D]
$('1'(
1
>
" '9 - $
L)3":4Ar$A3?"D4L→L)?"DrA$3":4L
$(
4 '4 4 4 4 ,
PQ?'%9 '"%D]
" 3$$?4 ( $ '- ( 9 ( " $$? ( ' $ $$? ( 1 $- $$? 1
( -
3&&&'$4
" 1 1
$$?
4
'4
4
4
4
1
'
1
'
PQ?22]
> - 0 +'91/
3(
4-'
'9
)'
4 (
1'9.
'4 (
1'9(
1'9
PQ]
=
'8/
1-
4
'4
4
4
=
(
?
(
=
H
(
.
PQ?'%/$A"( 1$A"D]
$
5
⋅!$*'%
9P\nYN
⋅*'%*')
F
4
#F*'
⋅1*'
⋅(%
)#9
⋅!
%4
⋅1%)*'
)+#
⋅(*'%$
#J
32# BB!AB L$4
6
5
4
'4
4
4
4
1
1
PQ]
6 1 BBHEBH
/ ( BB H ,) $- 3&&&'$4
"(
4
'4
4
4
4
P `
LQ`
AS`
AA`
`
)PQ
?D]
"-((-0
4 '4 4 4 4 +PQ?2]
$ BBLRH BLBRH
> ' 5 2
>'5
>
$RH
4 QBBK BBK BBL
'4 BBBK BBBK BBB
4 BSBK BBK BLB
4
BBKQEBK BAB
,PQ
R]
><$W5
((QE`E`
'( ><FW5
((QE`
E`'(A6AE) ''(
'6
0''"O
4 , 5 ' 8 '4 , 5 ' 8 4 $' ' 8 /
4 " ' ' '( 8 4 " ' ' '( A 8 .PQD]
$ 0 ' " '(
+ BBH BBH
'0 Em) " (
,( QDHABBH
$(BSBRH(
BBRH8'
)'(
4
'4
4
4
4
1( BBRH
(EB`
(/1
BBH
QB
(
PQ
?]
$'-3#434
34G'
3H43H4M/
(V
'9(
(
4 '4 (
4 4 4 PQ?22]
> 2' ' '
)0'AE
1
AE0
4
'4
4
4
4
+
$
+
$
)
PQ?D]
$($
- * BB0
)
^ BBRL
;BQARL
4
'4
4
4
4
QAL
DEL
BQL
QPL
QAL
PQ2]
$6
8'0
'
"-
4
'4
4
4
(
(
PQ]
$2210
('(
'
$ '
( @A"LrL)"→A@rL)"A
!(6
3∆O4AEm)#GR
$
3’O4AEm)(
4
'4
4
4
4
ADS#GR
KADS#GR
ED A#GR
KED A#GR
LASA#GR
PQ
DD]
6 / ' ? @ ? ?
)
"
1 BC
$
'D
( C - /6'AB Lq6>
$( '9 1 4 ) ' ( C
'4 =
#GK '9
C 30LPP4 - / 3SA4 30ADA4 ( ( (O G )PQ?'%9 '"%D]
, '- )'
$'#GR(
4
'4
4
4
4
KE! rA !
K!SQ
KA !
rE! +PQ%"'9(D]
AM34r"A34→AM"34
6 ’:msKPB P#G
+
(
4 ((
'4 M34r 1 "A34→M"34∆:msKPB P#G
2
4 M34r"A34rLBBS#G→M"34
4 AM34r"A34rPB P#G→AM"34
4
,PQ%"<'F]
= ( ' ? ( ( 1 V'
$ ('1
6
8'?$
( 5 " ( 'DE-
3M$)6=".
,";>6I$AB A. AP0 AS4
M$)6="=.,";>6I$9
:?+$'AAS!DELSAB A
)0 8'(
B $1'
BA $
0'(
BL " ( 1 ' BD $1'9
BE " ' '
'
.PQD]
$3434'( 934
B *(8
''
BA "∆:
BD "∆:
BS $ ' ' ' '
P ?
)K)(LDS#GR
)≡)( BDD
#GR
PQD2D]
)
:A"34→:A"34→:A"34
B $(
:A"34t:A"34t:A"34
BA $1
(
BD ,
(
BS ,
'∆:tB
P $1
(3∆:uB4
PQ?]
@'>"H Q'AB L'
1''<]b1
>N / )"
J
)$ *' ]qb=
!()' 3)A:P"344
5K LPS#GR) AR.: R." PR
@RR'R0'R0RR
AB LRBQR QR100000000
''01‘,sDBDR BRAB L
F
(X'9(
4 $ ' E ' ' 3
-4LDAB#G
'4 " 0'(2
4 "'
4 ,('
(S
PQ?'%/$A"( 1$A"D]
$ -'
)P: A"P34rP"A34→P)"A34rP:A"34
B
!(∆: 'sKASBA!#GR
) BB3)P: A"P45
(
4 L #G'
'4 L #G'
4 E#G'
4 E#G'
4 PA#G'
)
PQ?'" "9(!( ('AD]
;+'<
3RR1''RR4
$(
AB L
LE1
EBB#((
?' / ' / ‡ K A P' )P: A"P1
P!B#1
4 AD
'4 LE
4 SB
4 ADB
4 LPB
PQD]
)
'314
/(
13
C 4 H10(g ) + O 2( g ) → 4CO 2( g ) + 5H 2 O (l) 2
=3)s A."s P:s K 4
?'''
N
N
$
((
$
((
?ES')"A1(DPBH3),+4
?'QB`)"A1SBPDH3),+4
$
''
$((
4
'4
4
4
4
N
NN
N
NN
PQ]
=
):D34rA"A34→A:A"34r)"A34.∆:sKAAB#R0
4
'4
4
4
4
'LA'DDB#
'DS'PPB#
'LAA"A34
' PB'AAB#
((
+
)PQ22]
$
'
(9'
+ 0 >(
5
5
H
H
C
H
C
+ H
H
Metal
H
H
C
C
H
H
H
H
H
C
C = +146.0 kcal.mol-1
C
H = +100.0 kcal.mol-1
C
C = +82.9 kcal.mol-1
H
H = +104.2 kcal.mol-1
)0 0 #K (
4 PB
'4 LL
4 DLL
4 P!
+PQ?F]
Ligação química
Energia de ligação
H−I
(kJmol −1 )
300
Cl − Cl
243
H − Cl
433
I−I
152
A:34r)A34→A:)34rA34
$
('
6
;'< ( 0 * (
;'0<$
'9 '(0
'
$
'
(
4 $(:34((A34
'4 $
(DDL#G
4 $ (
LLQ#GK 4 $
' !E#G
4 "
' B S#G
,
,PQ?2]
.
$
'0
)K
)!Q #@$H?$
4
'4
4
4
$
)K:( QQB#K $
:K::A( BDA#K $
'9(KABL#K $'
AB'( ! !#K .PQ?]
)
'∆:s0 ED#GR
)(-
)s)(
'( 9
=+
3(4
)K)ADL#GR)K)LD!#GR)K)LL #GR
4
'4
4
4
!PP#GR
APE#GR
PQD#GR
P A#GR
)PQ?22]
$'
+('
3#R4
(
4
'4
4
4
4
SP DED
PPS LDD
PPS LD
!!S LDD
!!S LD
)PQ]
6'2
" - $ )L:E,L"Q34 ( ,A34 )"A34 :A"34 "A34 $ '
$
(
$
∆:uB
$
(
$'( 9
4 ?(
'4 ?(
4 ?(
4 ?
4 ?
)PQ?2]
$
M
o34(
024(
M
o34r4:A"34→TM3:A"44U
o34
)01
0ar34MAr34)r34
/
(
4 MAr34tar34t)r34
'4 MAr34t)r34tar34
4 ar34t)r34tMAr34
4 )r34tar34tMAr34
)PQ?2D]
" ( ( 6 ( C'(-
"):@L(1
):D34rL@A34→):@L34rL:@34
)
!(
∆H 0f ( kJ ⋅ mol −1 )
− 1437
CHF3 (g )
CH 4 (g )
HF(g )
− 75
− 271
$0
#GK (
4
'4
4
4
4
K PLL
KA !E
KALAE
r PLL
rA !E
)PQ]
, ' ' 6 ( )
4
'4
4
4
:A34r}"A34→:A"34KPSL#
:A"34KPSL#→:A34r}"A34
:A"34→:A34r}"A34rPSL#
:A34r}"A34→:A"34rPSL#
)PQUD]
"' ' ' 1 ) ' (
)
4 )"A34→)34r"A34
'4
4
4
4
1
CO ( g ) + O 2(g ) → CO 2 ( g )
2
1
C (graf) + O 2( g ) → CO ( g )
2
1
CO 2 ( g ) → CO ( g ) + O 2 ( g )
2
C ( graf ) + O 2 ( g ) → CO 2( g )
∆:srLQD:GR
∆H° = −284 KJ / mol ∆H° = +110KJ/mol
∆H° = +284KJ/mol ∆H° = −394KJ / mol )
))PQDD]
"CF0
:'"F0:'
3a)4
$(,)"II6"
4
'4
4
4
4
'
(SQ#GR
)0)(ADD#GR
a)34(K! !#GR
1
a34(D S#GR
((
)+PQ?F]
$91
(
" -234
>
4 $
-234(KQB#GK '4 $:34"A341
B:"34
4 $
-234
(8
'9
4 $'
-234(
'
-
4 $
),PQD2D]
=7710
( + ( 1 - ( /"
(')
B $
(((
BA $
(∆:s−SB#R
BD "Y
BS $
M34:A"34FM3":4A34:A34
P $
$F'
).PQ?22]
$ 3∆:4 ( 1 8 $
. )L:S34rE"A34→L)"A34rD:A"34∆:msKAAAB#G
. )34r"A34→)"A34∆:msKLQD#G
. :A34r}"A34→:A"34∆:msKASP#G
$ ' +PQUD]
"3#4(
(2
('("'
( '0' R
>
(
)
34 ) A:AP34→)P: D34rA)L:P34
? ' 0 /
'
) A:AP34r 37 "A34→ A)"A34r L:A"34∆:m)sK!E LB#GR
2
)P: D34r 19 "A34→P)"A34r!:A"34∆:m)sKD PLB#GR
2
9
)L:P34r "A34→L)"A34rL:A"34∆:m)sKAAABB#GR
2
>1 H : 0 0 4
'4
4
4
4
K LSQPB#GR
K LBB#GR
r BQBB#GR
r LBB#GR
r LSQPB#GR
+PQ]
)
#GR
)"34r:A34→)34r:A"34∆:sK EB#GR
)"34r}"A34→)"A34∆:sKA!L#GR
:A34r}"A34→:A"34∆:sKAL #GR
+0
' )34
)"A34(
4
'4
4
4
4
KPED#GR
KEBD#GR
rEBD#GR
rPED#GR
KLED#GR
+PQ
?2]
" ' 34 ( '9 -V21
'N5
+'34r)A34→+')A34∆:sKLEQD#
∆:sKLAQL#
+'34rA)A34→+')D34
V
∆:B
+'34rA)A34→+')D34(
)
4
'4
4
4
4
KLB #G
rLB #G
KPBA#G
rPSS!#G
KPSS!#G
+PQ
?]
=
∆:sKE! E#G
A:A34r"A34→A:A"34.
,A"E34r:A"34→A:,"L34. ∆:sK!PP#G
},A34rLRA"A34r}:A34→:,"L34.∆:sK !D#G
$
∆:
A,A34rE"A34→A,A"E34
4 KLP!#G
'4 KAS!#G
4 rLP!#G
4 rAS!#G
4 rLSL#G
+PQD]
)
'3AEm) 4'( 9(
?34r"A34→?"A34∆:sKAQ!#GR
)34r"A34→)"A34∆:sKLQD#GR
)34rA?34→)?A34∆:srSS#GR
B $
'(K B!P#GR
)?A34rL"A34→)"A34rA?"A34
BA $
'((
)"A34rA?"A34→)?A34rL"A34
BD $ )"A34 ' ?"A34
BS $ A )"A34 /(!SS#GR
P $
)?A34((
+PQR]
$
'
/
:A"34r)34→)"34r:A34
, AE) )
:A34r 1 "A34→:A"34∆:msKADAB#G
2
)34r"A34→)"A34∆:msKLQLE#G
"A34rA)"34→A)"A34∆:msKD!!B#G
)5
#/
+)PQ?22]
" ( ' 5 6 (
' 2 6'8
(
(
?
2
?
2
)'
4 /(3"4#Ω)
$s BKL$ N $K s Ω
'4 '1
'
++PQ
?2]
6
3'14('( 9(
4 ''1L
'4 "' ALAL
4 ''1A
4 ''1
4 "'(LA
+,PQ?D]
"'/'
LA6'
'C'1
))
2 9 $ 9C
)+
$ '
' C $'''A
A $'''A
L $' 2 ' '
'L
D "2'''L( P
$
4
'4
4
4
4
? L
?AD
?ALD
? AL
$ ALD
+.PQ?]
" - ' 1 ' $ '1
'(
4 A
'4 LA
4 L
4 A
4 L
,PQ?]
$'
?'
4
'4
4
4
4
$
δ)K)(AKL
$
δ)K:(LK
$
δ)K:(LKA
$
π)K)(K
$
π)K:(AK
,PQ]
;*"9(*'<$
(
$'
4 @
'(L
'4 "'('
4 $
4 H0
/
,PQD]
$'( 9(
B )'//9
BA $ '1
( 1 '
BD , ' ( '
BS ,('
P "'C
,PQ?]
$ ' 1 $1C ' O 34 ( ( 5
ou
$((
4
'4
4
4
4
+
+-)E:P
,''1
,/
+''1L
,PQ
?2]
$(
'( 9'C
):L):):L):A)):
),
4
'4
4
4
4
,PQDR]
I 5 F 5 6 > -'1
=0'1
4
'4
4
4
4
-)Q: ,L"A'
-)Q:Q,L"A'
-)Q: L,L"A/'
3π4
3σ4
3π41
3σ4
,)PQ]
$ ( ' 5-
9>(,a
O
O
):$,6$>?:">IGM/R(1 %(< #%"'
/$"9 4PSALABBL34
6,a'O
).
4
'4
4
4
4
M
)'
-
$'C
+(
,+PQ?D9 *("R]
$(
)10
'
9
('6
1
$' 9 4 *'9
O
'4 *(
('O
,,PQ]
> q1 '
' C '(
OH
H
H
H
O
TESTOSTERONA
)
8(0
4
'4
4
4
P'
'C(/
'
(C-) !:AD"A
+
,.PQ]
+ SES SP @ $ a#( $' ? ) $ M F / ' 9 $ ' 0
9)'
9 9 29
4 $/
'4 +
4 )
'
4 6/
4 F'
.PQ]
= AB A '1 L ' ?' ' 1 ( APD ,
.(18'
C(161'(
1,
1(3(4
C3'4'1C3'4
,8
4 "(
(DB`
'4 "(
4 "9'1
4 "'1('1
.PQ]
"(
V1
O
H 3C
O
NH
HO
CAPSAICINA
?
9
4
'4
4
4
'(
(
(
(
+
.PQ]
, ( N ) :- ( '6-(
1
" M( ?( 6 $ " 0 '( '
' 6
(5'
O
OH
O
O
$
9
4
'4
4
4
4
'(
(
(
.PQ]
$-('9
" ' ' 6
' ' > F > '(
3):L)""4A+'rA):L)")→+')ArAY
)
0Y(
4 '4 (
4 (
4 '
.PQD]
$
$'('-
1 9 $ 00
2 F ( ' 0 '(
N 1e
11
6 /
-
+
$(
6
8
(
+
4 ('-):
'4 (-5-
-
4 'C 4 '1 E-' A
4 (''3AEBR4
.PQ%"< ]
" - 5 + 3B)) 4 ( '$A3+H$A4=(/
- , ( 9
4
'4
4
4
4
'(
'(
'(
'(
((
.)PQ
2]
$(((
*90
HO
O
O
CH3
O
4
'4
4
4
'(
'(
.+PQR]
" - ( ( -
O
H
+
" - ( ( -
.,PQ?2D]
+ 1 / 1 ( "'
$-1
'
(
4
'4
4
4
4
A
L
D
E
..PQ?]
)
-/
A,A"E34→D,"A34r"A34
)-/5P× BKL
HKK'( 9(
,"AHKK
4 S× BKL
'4 AD× BKL
4 P× BKL
4 A× BKL
PQ?D]
0
/
M
:)>
:)
- 1 ' ( J - /' - / 1 '
"''''
'
4 $ ' * :A6
6 M ' 0 '9 '4 ,'''1
M
+
PQ?2]
) ( $ F
)=, BPB
$ PB F $- F DB)AB='DB$
$
'
(
4 $rF→)r=
'4 $rLF→A)r=
4 A$rLF→A)r=
4 L$rLF→A)r=
PQ%"'9(D]
)
A,"A34→A,"34r"A34
)
1
-/LA× BKLR,(
/R(
4 P× BKL
'4 DS× BKL
4 LA× BKL
4 PD× BKL
4 BA× BKL
PQD]
+ 1 5 "
15'1
1$
'314
/(
C 4 H10(g ) +
13
O 2( g ) → 4CO 2( g ) + 5H 2 O (l) 2
"'
'RH
[C 4 H10 ](mol/L) 22,4 20,8 18,2 16,6 15,4 14,9
Tempo(horas)
0
1
2
3
4
5
$(1BE L
4 K ERHKA RH
'4 ERHA RH
4 ERHKA RH
4 A RH ERH
4 K ERHA RH
+)
PQ
DD]
*''-QB`('
6 ( '1 ?
(
" ' ( - ALD
4 = D - '1
'
O6
'4 >'C1-
BB Q)
G
=$
34(ALBB 3$q#@
)!U
DE3L4AB!0AAD Q!!4
PQD]
"
-(
1 1 - 1 $ 1
( '
/
$
(
)A:E?:34r:A34→)A:P34r:A?34
4 ? 1
5 1 '- /
Experiências
[Etanotiol] [Hidrogênio] Velocidade inicial
1
(mol/L)
2
(mol/L)
1
(mol/min)
4
2
2
2
8
3
3
6
8
4
6
6
16
)''
'4 ) ( 1
( BR
R
++
)PQ]
> 1 LDD H ),+ , -''
(
4 PB(R
'4 PBA BAL(R
4 SDR
4 AADHR
4 PBR
+PQ?D]
>29
1
"
(
$ArFA→A$F
$
4
'4
4
4
4
(∆:sK B#G
(∆:sr B#G
(LB#G
(LB#G
(DB#G
,PQ22]
= $ 8 "'
(
$F8
)=
+,
)'
0
4
'4
4
4
( E#G
(DB#G
((
(0LB#G
.PQ22]
"/
):D34rA"A34→)"A34rA:A"34
= 'C (
+.
PQ%"$(%92]
$
11
)
/
4
'4
4
4
$1
1
,1
$
≥
$
1
PQ
R]
" ' 1 6 )
'(
4
'4
4
4
$ '
$A
'
$ 6 r6L
$ A '
(6Ar6L
4 $ '
(6 PQ2]
$
,
3∆:4
4
'4
4
4
.
.
'.
'.
,
PQ?D]
>
' + S ' EB H :) 6 ' 1 ' $ '
C -(
,": " ' BB H " '
Volume de
Volume
Volume de
Temperatura
de H 2 O
1
2
HCl (aq)
(mL)
50
50
(mL)
50
45
NaOH (aq)
(mL)
0
5
máxima
( º C)
23,0
24,4
3
4
5
6
50
50
50
50
40
35
30
25
10
15
20
25
25,8
27,2
28,6
30,0
7
8
50
50
20
15
30
35
30,0
30,0
Tubo
4 ) ' ,":
'4 $
'''OG
'
4 )
H0 :)'
,":1ABH0 PQ]
)('@?
:)
@34rA:)34→@)A34r:A34
$@$
(
4 KK
'4 KK
4 KK
4 KK
4 KK
,
PQ
R]
)
+$rF)→$)rF
?$)r=→$r)=
"(
'
=
(
4 3A43L4
F
'4 3 4
4 3D40000000000
4 3E4(2
4 (
)PQ?2D]
" -'
'
?
($rF→
)r=
/
6
8
(
4 '
'4 '
,
4 '
4 '
4 '
+PQUD]
> (0 0 -''-$1
( $- '
C '
( ''
Método
1
2
3
4
Estado do
Temperatura Tempo
Comprimido da água
de reação
Inteiro
10º C
50 s
Triturado
60º C
15 s
Inteiro
60º C
25 s
Triturado
10º C
30 s
= ' ' 1 $ ( 3 4 $
15
$/(/
=
(1
4
'4
4
4
4
,PQ]
" - / ," ,"A - ' 1C"
6 "Lr,"→"Ar,"A
6A,"Ar"L→A"Ar,"
6
"%'( 9
,
4
'4
4
4
4
","(
""A(
1C
",","A
0/
"
C-P($1$T0196(-.
,
(#1$($9!( = 0 5 ' 1 $
8 D BB':5-
(C
( > ( )/ F- > @ M 7 '$(
$,N?$@=$6>
= BB'9''-
511
$ '9 H - ( -
3$I75AB
AB4
.PQD$D]
V'9
4 '4 2
4 1
4 -5
4 -
C-P($1$T0196(-
"/ / ( 1 ' 6 '
PQ?'!"@%'"!&!!F 9(D]
)
RA,A34r RA"A34→,"34∆:srQB#GR,"
$ (
4
'4
4
4
4
,A34r"A34→A,"34rQB#G
RA,A34r RA"A34→,"34r SB#G
,A34r"A34rQB#G→A,"34
,A34r"A34r SB#G→A,"34
,A34r"A34→A,"34r SB#G
C-P($1$T0196(-
$
/ , 1 0
/
$F?+
3&&&'$4
PQ?'!"@%'"!&!!F 9(D]
'
(
4 '4 4 4 4 C-P($1$T0196(-
" / ' +-0 '' ( + /
''5
(
' 1 / ) :A 1
'
:L)0):A0":34rA:A"34r}"A34→E:A34rA)"A34
PQ?2D]
)
∆:')A:E":sK LPS#GK ∆:
:A"sKASP#GK ,
+0 / #G(
4
'4
4
4
4
rPA
r PED
KPA
K BSA
KA!QS
,)
C-P($1$T0196(-
=t&&& 'RR BSALB 000000u
$AQ'AB A
PQ?D9 *("R]
= '
5(
$
4 (
'4 1
-
(
4 -
(
4 '
(
4 (
C-)P($1$T0196(-
0*+
0
)
63;0(! 6:!0 ) / *)5*)
J
+ # )$
//O#
7)/
35"92"1'AB AAD4
PQD$D]
"**+ SSLB#GR#($'
' )A:P" 3 s AQ PDB
#GR#4(
4 LE
'4 !A
4 BB
4 AS
4 EB
C-+P($1$T0196(-
0
/
)$#)
(J
> 5''
'?
# )$ wz ' 5
'
$
$
(
'
)0
$
*7
)
*'
35"92"1'AB A!!4
=I
')
*'
) A:AA" r:A"→D)A:E":rD)"Ar QP#G
PQD$D]
+')
*'
4 '
DQ#G
'4 '
QS#G
4 '
QP#G
4 '
DQ#G
4 '
QP#G
C-,P($1$T0196(-)
"(
'$(
( ' '
$ '0 $
- ( '- 'C $ 0 ( 1 '1
=(
1 '( '
9
$ ' ( - 3)P: B"E434r:A"34→)P: A"P34
- 5
1
$ A Em)'
2BD`/
G-
,+
1
QB`(
' ( ' ' 3?6H67:M.+"Ha$I+"NAB ADB0DE4
?6H67:M+.+"Ha$I+"N=
'"%9"<"'$ "'%F "
?+= BAB A
)PQFF]
" ' '
9'
)0 B!P#RH B!QRLABm)
(
A× BKL)K 1
(B!DDMGRH
?
34 $(AEm)PS!m)
34 $ABm)(
!!× BA#RL
34 >PB'EBB
ABm) ? ( LEm) '
'
34 " BL!DD× BPG1
$/'(
B NNN@
BA NN@N
BL @@@N
BD @NN@
BE @@N@
C-.P($1$T019=-+L,
$'
,,
+PQ%"'9(D]
?
(
4 1
--
'4 1
--
4 1
--
4 1
-1
--
4 -1
,PQ%"'9(D]
$
---3-4
(
4 ∆:EK∆:AK∆: '4 ∆: r∆:Ar∆:Lr∆:D
4 ∆:EK∆:LK∆:D
4 ∆:Er∆:Lr∆:D
4 ∆: r∆:Lr∆:D
C-P($1$T0196(-.
f / ( $ '((
"
'(
'-5
O
H 2C
O
C
C17H31
O
HC
O
C
C17H33
etanol
O
H 2C
O
C
C17H31
O
etanol
glicerina + 2 CH3CH2-O-C-C17H31 +
éster do ácido linoleico
O
+ CH3CH2-O-C-C17H33
éster do ácido oleico
.PQ?2]
H'C8'((
,
H'(2
'C
,.
C-P($1$T0196(-
+ ( K A 1 '9 aA?A"S 1 aA?A"L1
$
('
0176(76(
aA?A"S34rAa34→AaA?"D34rA34
AaA?A"L34rA34→aA?D"P34rAa34
PQ?2]
)0
'0((CD
H
'
C-P($1$T0196(-
)
''-
-'
6
MD† BAR
'
B-YN.*'
*'
%
/NY\
+LB
-
$G
//+).
3&&&''4
PQ?RD]
$'C'-
4 (
'4 '
.
4 (
4 ('
4 C-P($1$T0196(-
= AB ! - ? + I
M- F 3IMF4 8 - '(
F
> - - 0AD ' 6 -20AL!
@0AD '
)?2+3)"?+$4
$ ' ->"A>@P
>"A34rD:@34→>@D34rA:A"34
>@D34r@A34→>@P34
3$&&&''R00RAB ARBAR
0000'000
00AB !:F"=
+6,R),6,ABBQ&&&'4
PQ?2D]
)'
Substância ∆H of (kJ ⋅ mol −1 )
− 1100
UO 2 (s)
UF4 (s)
UF6 (g )
− 1900
− 2150
HF(g )
H 2 O (g )
− 270
− 242
"
'
>@P
#G⋅K (
4
4
4
4
4
KDED
K!PD
K APD
rDED
r APD
C-P($1$T0196(-
" ( 9
$
'1
.
):L0):A0":34rL"A34→A)"A34rL:A"34∆:sK LPPS#G
pppppppppppppppppppppppppppppppppppppppppppppp
CH 3
CH 3
C
CH 3
CH2
CH
CH3(l)
rAERA"A34→S)"A34rQ:A"34∆:sKEDP B#G
CH3
PQ?D]
=
)"A34∆:sKLQL#GR
:A"34∆:sKASP#GR
0
'
(#GR
4 K B!E
'4 r B!E
4 KA!!A
4 rA!!A
4 rPS!!
C-P($1$T0196(-
2" LNN + ) / G 1 * G $ )$
" 5 + ' + 0 *'
*'
*'%
O
CH2
O
C
R
O
CH
O
C
R + 2H2O
lipase
O
CH2
O
C
R
triacilglicerol
CH 2
OH
O
lipase
CH
O
CH 2
OH
C
O
R + 2R
C
OH
monoacilglicerol
3=$>#+"
7 QQA$4
.
PQ?D]
,
0pppppppp
pppppppp
pppppppp
$
4 1KK
'4 KK
4 1KK
4 1KK
4 KK
C-)P($1$T0196(-
5 4"
) !e LPYY 0 LPP\ B 4 )
A
*'3
$
+4"
Ld5 LP5 LR5 3 4 ) 9 4"
) G -5Y. +
1'
7*A4"
3
*'
5Y
3RR'$4
PQ?'%/$A"( 1$A"D]
6'
0"A
8'"
1
3&&&4
" Am)
AAm)
(
4
'4
4
4
4
/ Am)(BBSH"AR
/(5
/AAm) B
/ ( ( B 1 (
.
C-+P($1$T0196(-)
$ 3)P:E,:A4 (0 2 /'
'1
D)P:E,"ArQ@rD:A"→D)P:E,:ArL@L"DrEDB#G
)PQ?'!"@%'"!&!!F 9(D]
$
'∆:#G(
4 K PSB
'4 K LE
4 rA PB
4 KA PB
4 r LE
6Y" S0)8 L!
>1
''-
+PQ%"'9(D]
) ' ,(
4 //'
'4 4 '
4 '
/
4 C-.P($1$T0196(-,
2- 9-3"A4'
( !B` "A - ( ' 8
/
.
-
"A34r)34→34r)"A34∆:
srEEB#G⋅K 9
34rA)A34→)D34∆:
sKSBD#G⋅K 9/
)D34r"A34→"A34rA)A34∆:
sK DB#G⋅K ,PQDD]
)(
)D34"A34(
4
'4
4
4
4
('
LED#G
('
AED#G
('
AED#G
('
LED#G
('
SBD#G
C-P($1$T0196(-.
= ( ( ( + ( 1 2 9 (
2 '' $ ' 33ABBL$4
.PQ?'%9 '"%D]
) 2 # s DA #G (
(1
29
ABBB#
4
'4
4
4
4
`
A`
L`
D`
E`
.
C-P($1$T0196(-
" ' $ ' 2
'
)P: A"P34rP"A34→P)"A34rP:A"34r
$'
H
H
O
C
C
OH
C
O
HO
C
H
H
H
C
OH
HO
C
H
H
C
OH
H
C
OH
H
C
OH
H
C
OH
CH2OH
CH 2OH
d-frutose
d-glicose
$'1
$
'''1
C'0
H
CH 2OH
O
OH
O
C
OH
OH
CH2OH
H
C
OH
HO
C
H
H
C
OH
H
C
OH
O
OH
OH
OH
CH2OH
α-d-glicose
OH
β-d-glicose
d-glicose
PQ?]
6
3’OB)4AEm)
composto ǻH 0 f (kJ/mol)
C 6 H12 O 6 (s)
− 1.271
CO 2 (g)
H 2 O(l)
− 393,5
− 285,8
)'
'
'AEm)
4
'4
4
4
KAABQ#GR
KAPBD#GR
KASBE#GR
K EBS#GR
.)
2F
HE
-
2A6
2AF
2A$
2A$
2A
4 "L s 1C. ,A s / )A s
'4 @
1 )2A=
+2A
4 $ - ( ' 6 / - '4 $ 5 -
d -
,2A$
.2AF
2AF
2A=
2A@NN@@
2A$
2A L
2A)
)2A)
+2A
4
.+
'4 $(
'( -'
,2A$
.2A$
2A)
2A !
2AF
2AF
2A=
2AAL
)2AF
+2AB!
,2AAD
.2A)
2A)
2A)
2A
4 =
'4 =
,). =
0
2AF
2AF
2A=
)2AF
+2A=
,2A6
.2A)
2A)
2A6
2A)
2AF
2AF
2A
4 $
(C(
,:D,"L→,A"rA:A"
'4 $ ( ∆:sΣ:KΣ:
,:D,"L→,A"rA:A"
KLPPrSAA3KADA4.
∆:sKDSDrSArLPPsKLP#GK )
( 0 (
)2A$
+2AF
,2ABA
.2AB!
2AA!
2AF
2A=
2AF
2A6
2A$
)2AF
+2A=
,2A$
.2A=
)2A=
)2A$
)2A=
)2AF
)2A=
)2A)
))2A)
)+2A$
),2AL ).2A
+ ' ' 0
' ( 1
5 2
+0
'
4 )L:S34rE"A34→L)"A34rD:A"34
∆:msKAAAB#G
3
∆:m4
Q# _ KQ → K,Q# _ Q#∆
∆Ka_[R
.,
4 )34 r "A34 → )"A34 ∆:m s K
LQD#G
3
L4
Q# <"9_Q#→
→Q#∆
∆K
aXQK.[RaK,[R
4 :A34r}"A34→:A"34∆:msKASP
#G
3
D4
KQ#_Q#→
→KQ∆
∆Ka
XQK,)[RaK[R
?0 =:
Q# _ KQ → K,Q# _ Q#
∆Ka_[R
∆KaK
Q# <"9_Q#→
→Q#∆
,[R
KQ# _ Q# → KQ ∆K a K
[R
ggggggggggggggggggggggggggggggggggg
→K,Q#∆
∆KaK
Q# <"9_KQ#→
)[R
+2A)
+2A6
+2AF
+2A=
+2AA +2ADLE× BD#G
)'
+)2A
4 Is × BEΩ BB#Ω
" I '( '
1 '4 $ ' 1
/ ' > 1 (
/ 2
'1
(A
++2A6
+,2AF
+.2A)
,2A=
,2A)
,2AAS
,2A)
,2A=
,2A=
,)2A$
,+2A
4 9 A L
. L '4 ( / 'C ,,2A)
,.2A6
.2AF
.2A=
.2A$
.2A$
QD47'-
.2A)
.)2A=
..
.+2A
$
'1
( '
'4 ) O
OH
.,2A)
..2A=
2A
4 )'
' (
6
B K E L K
BLs EL
s ∆V
∆t
= 15 L
)s3ALBB×BB Q4sDL!
"'0 3A.DL!4 -
L
R
KALPLKA!
LsQL
sQLR
'4
2AF
2A$
2AF
2A
4 $ ( 8 D
$
-
'1
'
$ '
'1
2A
4 $ ( s # ⋅
T6U⋅T:AU
(
A
'4 JsLDBR:A?
)2A6
+2A=
,2A=
.2A$
2A)
2A6
2A=
2A
4
'4 $ ' '
3(4 8
1
4 RH
2AF
2A)
)2A=
+2AF
,2AF
.2A$
2A=
2AF
2A$
2A=
2AF
2A=
)2ABD
+2A=
,2A$
.2A
" ( 'C
" 2 (
"(
'C
" 2 (
2A
, 8 1 $ 1 / ,
1
2A6
2A$
2A)
2A=
2A$
)2AF
+2A=
,2AF
.2A$
2A)
Download

Apostila – 3º E.M – Parte II - Marista Centro