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Guias e Dicas
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2.1 Derivadas Parciais, Notas de estudo de Cálculo

)$ #& t dt. Use o Teorema. Fundamental do Cálculo e a Regra da Cadeia e calcule as derivadas parciais f+, e g+,. 2.4B Se f x ...

Tipologia: Notas de estudo

2022

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/ E  C D 0 E  %' DC 1 E  CD )$ C D 

2 E  &" CD # C D^ 3 E  C D 4 E  %#& CD

  ; 1/2/ 1/A= 1/:1C:3 / 23@7D/2/ 7<271/2/ 2/ 4C<PO= E  1 C D

/ E  C %&" C  D  1 +     0 E  )$ CD & CD  1 ,   

1 E  C D  1 +,    3 1 ,+    2 E  CD " CD  1 ,   

  =<A723@3 / 4C<PO=      23G<72/ >=@

 C D 

)$ C  D  A3 C D    

 A3 C D     

/:1C:3 1/A= 3E7AB/; /A 23@7D/2/A >/@17/7A +     ,     +,    3 ,+    

  =<A723@3 / 4C<PO= 1     23G<72/ >=@

1 C D 

CD C  D 

C D ^ A3^ C D^  ^ ^ 

 A3 C D     

/ #=AB@3 ?C3 1 +,     1 ,+    

0 <D3AB75C3 / 1=<B7<C72/23 2/A 23@7D/2/A >/@17/7A 1 + 3 1 , </ =@753;

  )38/ 1 C D  C D /:1C:3 1C^ C D  D^3 C^  1 C D  D^ 

  #=AB@3 ?C3 / 4C<PO= E  C CDD A/B7A4/F L 3?C/PO= 2743@3<17/: CE+ DE,  E

  ,3@7G?C3 ?C3 / 4C<PO= @ C ?  ? )$  C6? ?  3 6 C;/ 1=<AB/<B3 <O= <C:/

A/B7A4/F /  $#   #    @(  6@++  

  %        3;  Q 23G<72= >=@   ++ ,, #=AB@3 ?C3 /A 4C<PV3A

@ C D  %'" DC 3 @ C D  0 +^ #& D A/B7A4/F3; / 3?C/PO= 23 "/>:/13 @  

  3B3@;7<3 1=<27PV3A A=0@3 /A 1=<AB/<B3A      3  >/@/ ?C3 / 4C<PO= @ C D 

C CD D C D  /B3<2/ L 3?C/PO= 23 "/>:/

  )3 @ C D 3 A C D AO= 4C<PV3A 1=; 23@7D/2/A >/@17/7A 1=<BS<C/A /BQ / ^ =@23; 3

A/B7A4/F3; LA 3?C/PV3A @+  A, 3 @,  A+ ;=AB@3 ?C3 @ 3 A /B3<23; L 3?C/PO= 23 "/>:/

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C D

C D  A3^ C D^  ^ ^ 

 A3 C D     

 &@=D3 ?C3

/ 1 Q 1=<BS<C/ </ =@753;

0 A 23@7D/2/A >/@17/7A 1 + 3 1 , 3E7AB3; 3; B=2=   ;/A <O= AO= 1=<BS<C/A 3;    

1 1 <O= Q 2743@3<17MD3: </ =@753; &=@ ?C3 7AA= <O= 1=<B@/27F = "3;/ C<2/;3<B/:

  /:A= =C D3@2/237@= CAB7G?C

/ )3 1 Q 2743@3<17MD3: 3; $ 3<BO= /A 23@7D/2/A >/@17/7A 1 + 3 1 , 3E7AB3; 3; $

0 *=2/ 4C<PO= 2743@3<17MD3: Q 1=<BS<C/

1 *=2/ 4C<PO= 1=<BS<C/ Q 2743@3<17MD3:

2 )3 E  1 C D B3; 23@7D/2/A >/@17/7A 1 + 3 1 , <= >=<B= $ 3<BO= 1 Q 1=<BS<C/ 3; $

3 )3 C;/ 4C<PO= E  1 C D B3; 23@7D/2/A >/@17/7A 1 + 3 1 , 1=<BS<C/A 3<BO= 1 Q 2743@3<17MD3:

4 *=2/ 4C<PO= 2743@3<17MD3: >=AAC7 23@7D/2/A >/@17/A 23 ^ =@23; 1=<BS<C/A

5 )3 /A 23@7D/2/A >/@17/7A 1 + 3 1 , 3E7AB3; 3; $ 3<BO= 1 Q 2743@3<17MD3: 3; $

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/ E  C D     0 E  " C D ^        

1 E  C CDD         2 E  )$  C CD D    C D    C  D

  +; B/<?C3 17:S<2@71= ;3BM:71= 1=; B/;>/ B3; /:BC@/ 23  8 3 @/7=  .8 3; AC/A

27;3<AV3A 7<B3@</A )3 / 3A>3AAC@/ 2/A >/@323A Q 23  88  1/:1C:/@ / ?C/<B72/23 />@=E7;/2/ 23

;3B/: CA/2= </ 4/0@71/PO= 2= B/<?C3 -@3A>   .8  1=; 3@@= 2/ =@23;  

  =7A :/2=A 23 C;/ M@3/ B@7/<5C:/@ ;323; C  8 3 D  8  1=; >=AASD37A 3@@=A

23 1; % N<5C:= 3<B@3 3AA3A :/2=A Q 23  # 1=; >=AASD3: 3@@= 23 #^ /:1C:3 = 3@@= />@=E7;/2=

2/ M@3/ B@7/<5C:/@ -@3A>   8 

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/ />@=E7;/PO= ^  =/ 3 1/:1C:3 / /:BC@/ />@=E7;/2/ 2/ B=@@3 3 A3C >=AASD3: 3@@= -@3A>

3  ^ 3@@=   8 

  A 27;3<AV3A 23 C;/ 1/7E/ @3B/<5C:/@ AO= 8  8 3  8  1=; >=AASD3: 23  8 3; 1/2/

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  C/A @3A7ABR<17/A = 3 = 3ABO= 1=<31B/2/A 3; >/@/:3:= 7AB= Q / @3A7ABR<17/ 3?C7D/:3<B3 &

Q 1/:1C:/2/ >=@   & &  )C>=<2= ?C3 =  :38> 3 /C;3<B/  :38> 3 ?C3 =   :38>

3 27;7<C7   :38> 23B3@;7<3 / D/@7/PO= />@=E7;/2/ 2/ @3A7ABR<17/ B=B/: & -   ^ :38>.

  % 1=;>@7;3<B= 7 3 = >3@S=2= ( 23 C; >R<2C:= A7;>:3A 3ABO= @3:/17=</2=A >3:/ 3?C/PO=

(  72 )3 = D/:=@ 23 7 Q 1/:1C:/2= ?C/<2= (  >02 3 2  ;0>>  23B3@;7<3 = 3@@=

1=;3B72= A3 </ @3/:72/23 (   >02 3 2   ;0>> -@3A> ^  

  +;/ 7<2WAB@7/ >@=2CF 23F ;7: 1/7E/A 23 >/>3:O= 4316/2/A 1=; 27;3<AV3A /8  /

3  /8 )3 = 1CAB= 2= >/>3:O= / A3@ CA/2= Q 23 &   >=@ /8 3 /A ;M?C7</A CA/2/A <= 1=@B

2= >/>3:O= 1=;3B3; 3@@= 23   /8 3; 1/2/ 27;3<AO= ?C/: = 3@@= />@=E7;/2= </ 3AB7;/B7D/ 2=

1CAB= 2= >/>3:O= -@3A> &  .

  +;/ 1/7E/ A3; B/;>/ D/7 A3@ 4/0@71/2/ 1=; ;/237@/ 23  .8 23 3A>3AAC@/ A

27;3<AV3A 7<B3@</A 2/ 1/7E/ A3<2=  .8 23 1=;>@7;3<B= .8 23 :/@5C@/ 3  .8 23 /:BC@/

1/:1C:3 / ?C/<B72/23 23 ;/237@/ CA/2/ </ 4/0@71/PO= 2/ 1/7E/

       

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C<2/;3<B/: 2= M:1C:= 3 / (35@/ 2/ /237/ 3 1/:1C:3 /A 23@7D/2/A >/@17/7A 1 +, 3 2 +,

  )3 1 C D  &" CD " DC ;=AB@3 ?C3 C1+ D1,  

  )38/ / 4C<PO= 1 C D  C^ C CDD D 23G<72/ 3;   C D    C D  

,3@7G?C3 ?C3 1 + 3 1 , AO= 723<B71/;3<B3 <C:/A 3;   ;/A 1 <O= Q 1=<AB/<B

  /2/ C;/ 4C<PO= @3/: 23@7DMD3: 1     ;=AB@3 ?C3 /A 4C<PV3A  C D  1 C  D 3

 C D  1 CD A/B7A4/F3; LA @3:/PV3A + ,  3 C+  D,  

  /:1C:3 /E/? <=A A35C7<B3A 1/A=A

/ E  D0+^ C0, C ? 3 D  &"? 0 E  " ^ C D ^  C  "? 3 D  0 (

1 E  C D  C ? 3 D  #&? 2 E  @ A AB @AB  @ ?  A ? 3 B ?

  /:1C:3 BC 3 BD <=A A35C7<B3A 1/A=A

/ B  @ A  @  C  D 3 A  C D 0 B  " ? > ^ ?  C D 3 >  CD

1 B  @ A @  C D 3 A  CD 2 B  #&     C D 3   CD

  =<A723@3 / 4C<PO= 1 C D  ^ +, )$ ? ^ /? /:1C:3 /A 23@7D/2/A >/@17/7A 1 ' 1& 3 1 &'

<= 1/A= 3; ?C3 C  => 3 D  = >

  )38/; F=  CF 4 DF 5 = D3B=@ >=A7PO= 2= >=<B= $ C D 3 =  F=  )3 1     Q C;/

4C<PO= 2C/A DRF3A 23@7DMD3: 3 E  1 = ;=AB@3 ?C3 E  E&& & E&

  =<A723@3 2C/A 4C<PV3A @3/7A 1     3 2     3 A38/; B  1 @ 3 @  2 C D

2;7B7<2= / 3E7ABR<17/ 2/A 23@7D/2/A 3<D=:D72/A 232CF/ ?C

B  1 ^ @  2 + 2 ,^ ^1 ^ @ 2

  +;/ 4C<PO= 1      Q 27B/ & 23 5@/C 9 ?C/<2= 1 ?C ?D 

?" 1 C D  ?    C D   #=AB@3 ?C3 ?C/:?C3@ 4C<PO= 6=;=5R<3/ 1 A/B7A4/F L @3:/PO=

C1+ C D D1, C D  91 C D 3; 

       

 )38/  / 1C@D/ 3;  23A1@7B/ >=@ C  &" ? D  &"? 3 E  #& ??  #=AB@ ?C3 / 1C@D/  3ABM 1=<B72/ <= >/@/0=:T723 C D E  3 23B3@;7<3 / @3B/ B/<53<B3 3 = >:/<= <=@;/: L 1C@D/ <= >=<B= 1=@@3A>=<23<B3 /?  

 /:1C:3  1 3 D3@7G?C3 3; 1/2/ 1/A= ?C3 3AB3 D3B=@ Q <=@;/: /A 1C@D/A =C AC>3@4S173A 23 <SD3:

/ 1 C D  C D 0 1 C D  )$ C D ^ 1 1 C D E  C D  CE

 )38/ 1 C D E  C D E 3 23<=B3 >=@ FA = 1/;>= 23 D3B=@3A <=@;/7A 3EB3@7=@3A L 3A43@/ C D E  & /:1C:3 / 23@7D/2/ 27@317=</: .* 1 C D E 

 /:1C:3 / 23@7D/2/ 27@317=</: <= >=<B= $     2/ 4C<PO= B  C D E  </ 27@3PO=

B/<53<B3 L 1C@D/

C D  E 

C D  E  

<= >=<B= $

 =<A723@3 / 4C<PO= 1 C D  C C^ DD  A3 C D     3 1     ,3@7G?C3 ?C3 1

B3; 23@7D/2/ 27@317=</: </ =@753; 3; ?C/:?C3@ 27@3PO= ;/A <O= Q /S 2743@3<17MD3:

 2;7B7<2= /A =>3@/PV3A >=AASD37A 3 1=<A723@/<2= 1=<AB/<B3 >@=D3 /A A35C7<B3A  

/   1 2    1  2 0  1 2  2  1 1  2 1  1 2  2 ^1 2 ^1 ^2

 )38/ F=  CF 4 DF 5 EF 6 = D3B=@ >=A7PO= 23 C; >=<B= $ C D E 2=  3 @3>@3A3=@ =

AC/ <=@;/ )3 1 ? Q C;/ 4C<PO= @3/: 23@7DMD3: ;=AB@3 ?C3

 1 =  1 ^ = F== 

+A/<2= 3AA/ 4T@;C:/ 1/:1C:3  =    = 3   " = 

 )38/;     3 1 C D  CD^  #=AB@3 ?C3

0 1 B3; 23@7D/2/ 27@317=</: </ =@753; />3</A </A 27@3PV3A F 4 3 F 5

            

 ;75/-EJ1>

 3B3@;7<3 / @3B/ B/<53<B3 L 1C@D/ <= >=<B= $ 7<271/2=

C D E  

C D E 

CD DE  

C  CE D E 

 /:1C:3 / 23@7D/2/ 27@317=</: <= >=<B= $     2/ 4C<PO= B  C D E  </ 27@3PO=

2/ @3B/ ?C3 >/AA/ <=A >=<B=A      3      

 =<A723@3 / 1C@D/ 23 3?C/PV3A >/@/;QB@71/A C  ? D ? 3 E ?   ?  

/ 3B3@;7<3 / @3B/ B/<53<B3 3 = >:/<= <=@;/: <= >=<B= $     

0 3B3@;7<3 / @3B/ B/<53<B3 ?C3 >/AA/ <= >=<B= $      

1 ,3@7G?C3 A3 3E7AB3 @3B/ B/<53/AA/<2= <= >=<B= %      

 )38/ 1     C;/ 4C<PO= 23@7DMD3: 1=; 1 ^ ?   ? )3 2 C D  1 C D 

;=AB@3 ?C3 / 23@7D/2/ 27@317=</: .* 2 C D A3@M ;ME7;/ ?C/<2= FA  CF 4 DF5

 )3 1     Q C;/ 4C<PO= 23@7DMD3: ;=AB@3 ?C3 =A >:/<=A B/<533@4S173 23

3?C/PO= E  D1 CD >/AA/; B=2=A >3:/ =@753;

 3B3@;7<3 = >:/<= B/<533@4S173 E  C D  CD >/@/:3:= /= >:/<= 23 3?C/PO=

C  D  E    -@3A> C  D  E  

 3B3@;7<3 C; >:/<= ?C3 >/AA/ <=A >=<B=A $    3 %     3 ?C3 A38/ B/<53<B3 L

AC>3@4S173 C D E   -@3A> C D E  .

 3B3@;7<3 =A >=<B=A 2/ AC>3@4S173 E    C  D <=A ?C/7A = >:/<= B/<53<B3 Q

>3@>3<271C:/@ L @3B/ C   ? D   ? E  ? -@3A> $           C D E 

 3B3@;7<3 = >=<B= 2/ AC>3@4S173 E  C  D =<23 = >:/<= B/<53/@/:3:= /= >:/<=

23 3?C/PO= C D E  -@3A> $        C D E  .

 3B3@;7<3 3; ?C3 >=<B=A 2/ AC>3@4S173 E  C  CD D  C D = >:/<= B/<53<B3 Q

6=@7F=<B/: -@3A> $     .

 ,3@7G?C3 ?C3 <= 2=;S<7=   C D    C  3 D  ^ / 4C<PO= 1 2= E3@1S17=

7 <O= B3; ;S<7;= 'C/: = ;/7=@ D/:=@ ?C3 1 /AAC;3 3;  =<AB@C/ C;/ 4C<PO= 1=<BS<C/

3;  ?C3 <O= >=AAC/ ;ME7;= <3; ;S<7;=

 3B3@;7<3 = ;ME7;= 3 = ;S<7;= /0A=:CB=A 23 E  1 C D <= 1=<8C<B=  7<271/2=

/ E  CD   C D 0 E  C D  Q = ?C/2@/2= 23 DQ@B713A    

1 E  C D    C   D 2 E  C0+^ #& D         

3 E  C D  C   C D 4 E  C D  C  D   C  D

 3B3@;7<3 =A >=<B=A 2/ 1C@D/ C  #& ? D  &" ? E  &"?  ;/7A 27AB/<B3A 2/

 'C/7A 2/A 4C<PV3A A35C7<B3A B3; C; ;ME7;= =C ;S<7;= 3; B=2= >:/<=  

/ E  0 +, 0 E  0 +, 1 E  C  C &" D #& D 

 3B3@;7<3 =A >=<B=A 2/ AC>3@4S173 E  CD ;/7A >@TE7;=A 2/ =@753;

 3B3@;7<3 / 27ABN<17/ ;S<7;/ 2/ =@753; L 1C@D/ D  C    &=@ ?C3 = #QB=2=

2=A #C:B7>:71/2=@3A 23 "/5@/<53 <O= A3 />:71/ <3AA3 1/A=

 =; /CES:7= 2= #QB=2= 2=A #C:B7>:71/2=@3A 23 "/5@/<53 @3A=:D/ =A A35C7@=0:3;/A

23 3EB@3;=A D7<1C:/2=A

/ E  C D C D  0 E  #& C #& D C  D   C

1 E  C D CD   C   D  2 B  CD DE CE C D E 

3 E  C D  C D  4 B  CDE CD DE CE   C  D  E

5 B  C D E C D E  6 B  C D E  C D E 

7 E  C D  C D  8 E  C D  C D  

 /:1C:3 / 27ABN<17/ 2= >=<B= $    L >/@M0=:/ D  C

 /:1C:3 / 27ABN<17/ 2/ =@753; L 1C@D/ C D CD  

 3B3@;7<3 =A >=<B=A 2/ 1C@D/ 7<B3@A3PO= 2= 3:7>AT723 C D E  1=; = >:/<=

C  D  E  ;/7A >@TE7;=A 2/ =@753;

 3B3@;7<3 <W;3@=A >=A7B7D=A 1C8/ A=;/ A38/  3 = >@=2CB= = ;/7=@ >=AASD3:

 )3 C D 3 E AO= <W;3@=A @3/7A <O= <35/B7D=A ;=AB@3 ?C3 CDE C D E 

 3B3@;7<3 = >=<B= 2= >/@/0=:T723 E  C D ;/7A >@TE7;= 2= >=<B=      

! 3B3@;7<3 = >=<B= 2/ 3:7>A3 C D   ;/7A >@TE7;= 2/ @3B/ C  D  

" /:1C:3 = ;/7=@ D/:=@ /AAC;72= >3:/ 4C<PO= 1 C D  &" C &" D &" C D </ @357O=

1=;>/1B/ &  C  D  C D 

# 3B3@;7<3 =A 3EB@3;=A 2/ 4C<PO= 1 C D  C  CD D <= ?C/2@/2= %        

$ /:1C:3 = ;/7=@ D/:=@ 2/ 3E>@3AAO= C D E ?C/<2= C D  3 CE  

% <B@3 B=2=A =A >=<B=A 2= >/@/0=:T723 E   C  D ?C3 3ABO= /17;/ 2= >:/<=

C D E   3<1=<B@3 /?C3:3 ;/7A /4/AB/2= 2= >:/<=

& 23<B7G?C3 =A >=<B=A 1@SB71=A 2/ 4C<PO= 1 C D E  C D E  AC837B=A L 1=<27PO=

C DE  

' /:1C:3 / 27ABN<17/ 2/ >/@M0=:/ D  C L @3B/ C  D  

( /:1C:3 / 27ABN<17/ 2= >/@/0=:T723 E  C D /= >:/<= C D  E  

) 'C/7A =A >=<B=A 2/ 3:7>A3 C CD D  ;/7A >@TE7;=A 3 ;/7A 27AB/<B3A 2/ =@753;

 "=:.718-> 01 DB58: 1 H958:

  B3;>3@/BC@/ ( <= 27A1= C D Q 2/2/ >=@ ( C D  C D  D ; ?C3 >=<B=

2= 27A1= / B3;>3@/BC@/ Q ;/7A /:B/ 3 3; ?C3 >=<B= 3:/ Q ;/7A 0/7E/

  B3;>3@/BC@/ ( 3; C; >=<B= $ C D E 2/ 3A43@/ C D E   Q 2/2/ >=@

( $   CD E ; ?C/: >=<B= 2/ 3A43@/ / B3;>3@/BC@/ Q ;ME7;/ 3 3; ?C/: >=<B= 3:/ Q

;S<7;/

 +;/ 1/7E/ @3B/<5C:/@ A3; B/;>/ 23D3 B3@ 8 23 D=:C;3 3B3@;7<3 AC/A 27;3<AV3A

23 ;=2= ?C3 AC/ M@3/ B=B/: A38/ ;S<7;/

 +;/ 8/<3:/ B3; = 4=@;/B= 23 C; @3BN<5C:= AC>3@>=AB=

>=@ C; B@7N<5C:= 7AT13:3A 1=;= ;=AB@/ / G5C@/ /= :/2= )3 =

>3@S;3B@= 2/ 8/<3:/ Q  1/:1C:3 C D 3  23 ;=2= ?C3 / M@3/

2/ 8/<3:/ A38/ / ;/7=@ >=AASD3:

 +;/ 7<2WAB@7/ 4/0@71/ 1/7E/A @3B/<5C:/@3A 23  8 23 D=:C;3 3B3@;7<3 /A 27;3<AV3A

?C3 B=@</; = 1CAB= ;S<7;= A3 = ;/B3@7/: >/@/ / B/;>/ 3 / 0/A3 1CAB/ = 2=0@= 2= ;/B3@7/: >/@/

=A :/2=A

 3B3@;7<3 = @3BN<5C:= 23 ;/7=@ >3@S;3B@= 7<A1@7B= </ 3:7>A3 C ,^ D -  

 +;/ B3<2/ Q >@=83B/2/ </ 4=@;/ 23 C; 17:7<2@= 17@1C:/@ @3B= 1=; B3B= 23 4=@;/ 1U<7

1/ 1=;= AC53@3 / G5C@/ /= :/2= )3 = 17:7<2@= B3; @/7= 75C/:

/  8 3 / M@3/ B=B/: 2/ AC>3@4S173 ?C3 3<D=:D3 / B3<2/ Q 8 

1/:1C:3 / /:BC@/  2= 17:7<2@= 3 / /:BC@/ 3 2= 1=<3 23 ;=2=

?C3 / B3<2/ B3<6/ = ;/7=@ 3A>/P= 7<B3@<= >=AASD3:

! *@RA 1=;>=<3<B3A 3:QB@71=A 23 C; 1=;>CB/2=@ 3ABO= :=1/:7F/2=A 3;        

3     3B3@;7<3 / >=A7PO= 23 C; ?C/@B= 1=;>=<3<B3 23 ;=2= ?C3 / 23;=@/ 2= A7</: A38/

;S<7;/

" *@RA 53<3A   3 % 23B3@;7</; =A ?C/B@= B7>=A A/<5XS<3=A 6C;/<=A  =C %

  =C % 3   :37 23    3AB/03:313 ?C3 / >@=>=@PO= 23 7<27DS2C=A 23 C;/

>=>C:/PO= ?C3 AO= >=@B/2=@3A 23 53<3A 2743@3<B3A Q 5=D3@</2/ >3:/ 4T@;C:/ $  ;< ;= =<

A3<2= ; < 3 = /A >@=>=@PV3A 23 53<3A   3  @3A>31B7D/;3<B3 </ >=>C:/PO= &@=D3 ?C3 $ <O=

3E1323  $=B3 ?C3 ; < 3 = AO= <O= <35/B7D=A 3 ; < =  

#  @3A7ABR<17/ 23 C;/ D75/ @3B/<5C:/@ D/@7/ 1=;= = >@=2CB= 23 AC/ :/@5C@/ >3:= ?C/2@/2=

23 AC/ >@=4C<272/23 3B3@;7<3 /A 27;3<AV3A 2/ D75/ 23 ;/7=@ @3A7ABR<17/ 1=@B/2/ 23 C; B=@=

17:S<2@71= 1=; A3PV3A 3:S>B71/A 23 37E=A ;/7=@ 3 ;3<=@ ;327<2=  3  @3A>31B7D/;3<B

71=?-  1=<27PO=  1  2  F^ Q <313AAM@7/ ;/A <O= ACG173/@/ 5/@/<B7@ / =1=@@R<17/

23 C; D/:=@ 3EB@3;= 23 1 C D AC837B= L @3AB@7PO= DS<1C:= 2 C D  &=@ 3E3;>:= 1=<A723@/<2=

1 C D  C D 3 / @3AB@7PO= CD   = ;QB=2= 2=A #C:B7>:71/2=@3A 23 "/5@/<53 <=A :3D/ /=A >=<B=A

$   3 $   1=;= 1/<272/B=A / >=<B=A 3EB@3;=A 7<2/ /AA7; 1 <O= B3; ;ME7;= </

67>Q@0=:3 CD   'C/<B= ;/7A 27AB/<B3 2/ =@753; 3AB7D3@ $ C D <3AA/ 67>Q@0=:3 <= #^ ?C/2@/<B

;/7=@ A3@M = D/:=@ 23 C D

 @9EJ1> 8;7H/5?-> 1 -/:.5-9:>

 ,3@7G?C3 / />:71/07:72/23 2= *3=@3;/ 2/ C<PO= ;>:S17B/ 3 1/:1C:3 D^ $ 3 D^ $ 

/ D  CD C    $     0 " CD CD    $    

1 C " C D0,^   $     2 " CD  CD   $    

 +A3 = *3=@3;/ 2/ C<PO= ;>:S17B/ 3 1/:1C:3 /C/D <= >=<B= $ 3A>317G1/2=

/ D C  #& CD   $     0 C D  " C D ^   $    

 /:1C:3 EC 3 DE  =<23 E  1 C D Q 23G<72/ >3:/ 3?C/PO=

/ C D E  0 CD C D  E  1 CE  DE #& E 

 (3A=:D/ = A7AB3;/

@ A &" CD 

@ A C D 

3 23B3@;7<3 @ 3 A 1=;= 4C<PV3A 23 C 3 D

 +; 5MA 723/: =032313 / A35C7<B3 :37 $ )  6(  A3<2= 6 1=<AB/<B3 3 $ ) 3 (  @3A>31B7

D/;3<B3 / >@3AAO= = D=:C;3 3 / B3;>3@/BC@/ 32CF/ / @3:/PO= $))(($   

 )3  C D E   A3<2= B   C D E C;/ 4C<PO= 2743@3<17MD3: B/: ?C3  $ 

 + $   , $  3 - $   ;=AB@3 ?C3 3; $ D/:3 / @3:/PO= CDDEEC   

            

 =<A723@3 / B@/<A4=@;/PO= ( @ A B  C @ A B  D @ A B  E @ A B 1=; 8/1=

07/<=  #=AB@3 ?C3 @+  (^)  A BD E 3 @,   (^)  A BC E 32CF/ 3E>@3AAV3A /<M:=5/A >/@/ /A

23@7D/2/A @-  A+ A, A-  B+ B, 3 B- 

! ,3@7G?C3 ?C3 / ;C2/<P/ 23 1==@23</2/A (      C .? C  .? B@/<A4=@;/ / 3?C/PO= 23 =<2/A @(( . @++  </ 3?C/PO= A7;>:7G1/2/ @ 

" #=AB@3 ?C3 / ;C2/<P/ 23 1==@23</2/A @ A  ,C -D .C /D B@/<A4=@;/ = ?C/2@/2=

23 DQ@B713A            3    3; C; >/@/:3:=5@/;= 23 M@3/  ^ @ AC D 

# ,3@7G?C3 ?C3 / ;C2/<P/ 23 1==@23</2/A (  @ A  C, D- B@/<A4=@;/ / 3:7>A3 C ,

D

- ^ <= 1S@1C:= C<7BM@7= 23 13<B@= </ =@753; 2= >:/<=^ @A^ 3G</ C;/ ;C2/<P/ 23 1==@23</2/A

(     ?C3 />:71/ = 3:7>AT723 C ,^ D -^ E.  </ 3A43@/ C<7BM@7/

$ 'C/: / 7;/53; 2/ 17@1C<43@R<17/ C D  , >3:/ B@/<A4=@;/PO= ( C D  C D

% 3B3@;7<3 / 7;/53; 2/ @3B/ C . >3:/ ;C2/<P/ ( C D   0 +^ #& D 0+^ &" D 

& A0=13 <= >:/<= CD / @357O= 23:7;7B/2/ >3:/A >/@M0=:/A C  D C  D D  C 3

D  C 3 23B3@;7<3 / 7;/53; 23AA/ @357O= >3:/ ;C2/<P/ 23 1==@23</2/A C  D@ D  CA

' 3B3@;7<3 / 7;/53; 2/ @357O= &  C D >3:/ ;C2/<P/ 23 1==@23</2/A ( 

@ A  C D C  D  'C/: / 7;/53; 2/ 67>Q@0=:3 CD  >=@ ( 

( )38/ 1     C;/ 4C<PO= 1=; 23@7D/2/ 1=<BS<C/ 3 >=A7B7D/ #=AB@3 ?C3 / B@/<A4=@

;/PO= (     23G<72/ >=@ ( @ A   1 @  A @1 @ B3; 8/1=07/<= <O= <C:= 3; ?C/:?C3@

>=<B= @ A A3<2= >=@B/<B= 7<D3@BSD3: ,3@7G?C3 ?C3 ( ^ C D   1 ^ C  D C1 ^ C^ 

) ; 1/2/ 1/A= Q 2/2/ C;/ ;C2/<P/ 23 1==@23</2/A @ A  ( C D 3A1@3D/ /A

@3B/A @  @ 3 A  A <=A 2=7A A7AB3;/A 23 1==@23</2/A >:/<= CD 3 >:/<= @A >/@/ =A D/:=@3A

6         3B3@;7<3 / B@/<A4=@;/PO= 7<D3@A/ ( ^ 

       

/ ( C D   C D 0 ( C D  C  D C D 1 ( C D  C  C D 2 ( C D  C   D ^ 3 ( C D   0 + 0, 4 ( C D   0 + 0^ ,

* ; 1/2/ 1/A= 3<1=<B@3 / 7;/53; 2/ 1C@D/. >3:/ ;C2/<P/ 23 1==@23</2/A ( C D /. Q = @3BN<5C:= 23 DQ@B713A            3     ( C D   C D 0. Q = 1S@1C:= C D   ( C D   C D 1. Q = B@7N<5C:= 23 DQ@B713A         3    ( C D  D  C  2. Q / @3B/ C  D   ( C D  D  C  3. Q / @3B/ C D   ( C D  C  D C D 4. Q = ?C/2@/2= 23 DQ@B713A             3     ( C D  C D C  D 5. Q = = 1S@1C:= C D   ( C D  C D C  D

+ )38/  / 1C@D/ 23A1@7B/ >=@   , C D ^ -C .D /   / =<A723@3 / >=AA707:72/23 23 , =C / A3@ F3@= 3 723<B7G?C3  1=;= A3<2= C; 1S@1C:= =C C;/ @3B/ 0 3B3@;7<3 / 7;/53; 2/ 1C@D/  >3:/ B@/<A4=@;/PO= ( C D  (^) C CD  (^) C DD 

 ::=019-0-> @=A57H91->

A ?C/<B72/23A =  E 23G<72/A <= E3@1S17= 5 AO= 23<=;7</2/A    ' 

3<?C/<B=    23G<72/A <= E3@1S17= 6 AO= /A    %  2= >=<B= $ C D E 

*3;=A

  =;>:3B3 / A35C7<B3 B/03:/ 23 1==@23</2/A

$1>;:>?-> 1 %@31>?J1>

B1=/H/5:> 

  *=2/A A/ 23@7D/2/A AO= 1/:1C:/2/A CA/<2= @35@/A 0MA71/A 23 23@7D/PO= &=@ A7;>:7172/23 <=

SB3; 1 AC0AB7BCS;=A / 3E>@3AAO= 0 +, >=@ 

E+ E, E++ E,, E,+

/ C D  D

0 C DD C CD C CDD  C ^ CDD  C^ D ^ DC 

1 ^ C ^ D ^ D ^ C  C ^ CD  D ^ CD ^ C D ^ 

2 D #& CD C C #& CD D D &" CD  C C &" CD  D #& CD  CD &" CD

3 C C D^ D C D^ D

 C D  

C

 C D  

CD

 C D  

4  CD D ^  CC D  ^ CCD D   ^ CC DD ^ C^ D ^ ^ CC^  D^ C^ D

  /  ^ 0 1 1 +,     1 ,+     2 

  )38/  C D  )$  C D  $= >=<B= C D     B3;=A +  C C C DD  3

,  D C C DD  $/ =@753; /A 23@7D/2/A AO= 1/:1C:/2/A >3:/ 23G<7PO= 3 4/B=

+     !^ ^ 3 3  !^0

  3 ^3 ,^ ^ ^  ^ ! 

6 ^ 

5=@/ +,     , +     ! ^ +^ ^  6^  6 +^ ^ ^   ! ^0

   6 ^ 3 ;=2= /<M:=5= ;=AB@/A3 ?C3 +,        +; 1M:1C:= 27@3B= <=A 2M

(^1) + 

C D  D C D

C D   A3^ C D^  ^ ^ 

 A3 C D    

C  CD  C D

C D   A3^ C D^  ^ ^ 

 A3 C D    

/ 1 +,     ! ^1 +^ ^  6^  6 1 +^ ^ ^  ! ^  66    = ;3A;= ;=2= 3<1=<B@/A3 1 ,+    

 0 #=AB@3;=A ?C3 / 23@7D/2/ >/@17/: 1 + Q 1=<BS<C/ </ =@753; &/@/ 7AB= CA/@3;=A = *3=@3;/

2= =<4@=<B= *3;=A

 1 + C D  C^ D^ C D^ D C^ D^ D^ D^ CD ^ C^ D  C^ D^ D  D 

=;= /A 3EB@3;72/23A  2  3 3   D BR; :7;7B3 F3@= </ =@753; 3<BO= !+, 1 + C D 

 1 +    = ?C3 23;=<AB@/ A3@ 1 + 1=<BW<C/ 3;    / ;3A;/ 4=@;/ 232CFA3 / 1=<B7<C72/

23 1 ,  1=<1:CAO= >=23 A3@ 3AB/03:3172/ B/;0Q; >3:= *3=@3;/ 2= =<4@=<B=

  1C^ C D  D^  C D ^3 C  1 C D  D  C C D ^   /:1C:3 CA/<2=

@35@/A 23 23@7D/PO= /A 23@7D/2/A E+ 3 E, 3 3; A35C72/ = D/:=@ 2/ 3E>@3AAO= CE+ DE, &@=1327;3<B=

A3;3:6/<B3 /2=B/A3 <=A E3@1S17=A  3    *3;=A ?C3 @++   3 @,,   3 >=@B/<B=

@  A3 3 A=;3<B3 A3      @+  A,  @++  A,+ 3 @,  A+  @,,  A+,

B1=/H/5:> 

  / 1 C D  =^ #&=^ ^ &"^   1=; =   7<23>3<23<B3 23  "=5= +,! 1 C D 

 1    3 1 Q 1=<BS<C/ 3;    0 A 23@7D/2/A >/@17/7A 1 + 3 1 , AO= 2/2/A >=@

CD

C D   A3^ C D^  ^ ^ 

 A3 C D    

C  C D

C D   A3^ C D^  ^ ^ 

 A3 C D    

3 1=<A723@/<2= =A 1/;7<6=A   D  3   D  C DRA3 ?C3 1 + 3 1 , <O= BR; :7;7B3 3;   

A3<2= 1=<A3?X3<B3;3<B3 23A1=<BS<C/A /S 1 $=B3 ?C3 <3AB3 1/A=  3  3 6 6    3 3 66   <O=

B3; :7;7B3 3;    3 2/S @3AC:B/ ?C3 1 <O= Q 2743@3<17MD3: 3;    AB= <O= 1=<B@/27F = "3;/

C<2/;3<B/: >=@?C3 <3AB3 1/A= 3:3 <O= A3 />:71/

  , ,   ,   (31=@23A3 ?C3 C;/ 4C<PO= E  1 C D Q 2743@3<17MD3: <= >=<B= $ C D

?C/<2= /A 23@7D/2/A >/@17/7A 1 + 3 1 , 3E7AB7@3; 3; $ 3

1 C 3 D 6   1 C D 1 + C D 3 1 , C D 6  3 6 

=<23  3  3 6 6    ?C/<2= 6 3     

/ =<A3?XR<17/ 27@3B/ 2/ 23G<7PO= 0 /P/ 3 6  3 232CF/ ?C3 1 C 3 D 6  

1 C D AB= Q / 1=<B7<C72/23 23 1 3; $ 1  4C<PO= 2= E3@1S17= Q C; 1=<B@/3E3;>:=

2  4C<PO= 1 C D  CD C D ^  C D     3 1     B3; 23@7D/2/A >/@17/7A 23 

=@23; </ =@753; ;/A <O= Q 2743@3<17MD3: /S 3 AB/ /G@;/PO= Q = "3;/ C<2/;3<B/: 4 /

4C<PO= 2= E3@1S17=  Q C; 1=<B@/3E3;>:= 5  3E7ABR<17/ 2/A 23@7D/2/A 1 + 3 1 , <O= 7;>:71/

A3?C3@ </ 1=<B7<C72/