thi thu dh mon toan lan 5 DHSP HN

7KஏLJLDQOjPEjLSK~WNK{QJN୵WKஏLJLDQJLDRÿ୳

&kXÿL୵P&KRKjPVஃ\  





−

+

[

[



.K୕RViWVணELୱQWKLrQYjY୯ÿஅWK୽&FஙDKjPVஃ

7uPNÿ୵WUrQÿஅWK୽&FyKDLÿL୵PSKkQEL୹W0[0\01[1\1WK஁DPmQ

⎩

⎨

⎧

= +

= +

N

\ [

N

\ [

1 1

0 0

&K஛QJPLQKUୣQJKDLÿL୵P01FQJWKX஋FP஋WQKiQKFஙDÿஅWK୽&

&kXÿL୵P



 VLQ





 VLQ





 VLQ

*L୕LSKŲţQJWUuQK[[±  [+

&kXÿL୵P

7tQKWtFKSKkQ

G[

[ [

[









FRV VLQ

VLQ

π

π



&kXÿL୵P

&KROăQJWU஗[LrQ$%&$¶%¶&¶Fyÿi\OjWDPJLiFÿ୳XF୓QKD+uQKFKLୱXFஙD$¶[XஃQJP୩WSK୥QJ $%&WUQJY஍LWkPÿŲஏQJWUzQQJR୓LWLୱSWDPJLiF$%&7tQKGL୹QWtFK[XQJTXDQKFஙDKuQKOăQJ WU஗ELୱW% $ Ö$ 

&kXÿL୵P

&K஛QJPLQKUୣQJK୹SKŲţQJWUuQKVDXFyÿ~QJKDLQJKL୹PSKkQEL୹W

⎪

⎩

⎪

⎨

⎧

= + +

= + +

= + +



















\ [ ]

[ ]

\

]

\ [

&kXÿL͋P

7URQJP୩WSK୥QJ2[\FKRHOLS(  





=

+ \

[

YjÿŲஏQJWK୥QJG[\± 

&K஛QJPLQKUୣQJÿŲஏQJWK୥QJGOX{QFୡW(W୓LKDLÿL୵PSKkQEL୹W$%7uPÿL୵P&WKX஋F(VDR FKRGL୹QWtFKWDPJLiF$%&EୣQJÿYGW

7URQJNK{QJJLDQ2[\]FKREஃQÿL୵P$%&'9LୱWSKŲţQJ WUuQKP୩WSK୥QJα WLୱS[~FY஍LP୩WF୙XQJR୓LWLୱSW஛GL୹Q$%&'YjVRQJVRQJY஍LPS%&'

&kXÿL୵P

7uPWୟSKகSFiFÿL୵PWUrQP୩WSK୥QJSK஛FEL୵XGL୷QFiFVஃSK஛F]WK஁DPmQ] __ ] 



 'ӵNLӃQNuWKLWKӱĈҥLKӑFOҫQWKӭVӁÿѭӧFWәFKӭFYjRQJj\





+'*,Ҧ,/63+1

&kX

7KHREjLUD01OjÿL୵PWKX஋FÿŲஏQJWK୥QJ[\ NKD\\ [N

;pWSKŲţQJWUuQKKRjQKÿ஋JLDRÿL୵P 





−

+

[

[

 [N ⎩⎨

⎧

+

−

−

= +

≠

−

⇔











N [ [

[ [

⇔ [

⇔ N±±N! ⇔ N±N±! ⇔









±N±W 9஍LÿL୳XNL୹QWUrQWKu

⇔ +DLÿL୵P01FQJWKX஋FP஋WQKiQKFஙDÿஅWK୽&

/ѭXê.K{QJVӱGөQJÿӏQKOêÿҧRYӅGҩXWDPWKӭFEұFKDLÿӇFKӭQJPLQK

&kX



 VLQ





 VLQ





 VLQ



3KŲţQJWUuQK

⇔



 FRV







 VLQ





>



@>VLQ[± 

π ±VLQ[ 

π @ 



⇔



 VLQ



 FRV







 VLQ



⇔

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

+

−

=

=

=

+

=

⇔

⎢

⎢

⎢

⎢

⎣

⎡

−

= +

=

−



























 FRV







 VLQ

π π

π π

π π π

π

N [

N [

N [

[ [

N

∈

=

*LҧLSKѭѫQJWUuQK[[±  [+

&iFKSKŲţQJWUuQK

⇔

[[± 



 +

[



⇔

[± [[   







+ +

−

[

[

⇔

[±>[[  



−

[

[

@ 

⇔

[± 

⇔

[ 

⎢

⎢

⎣

⎡

−

<

+

>













N N

9u[[  [ [ ++ !Y஍LP୿L[

&iFK;pWI[ [[±±



 +

[

OjKjPVஃ[iFÿ୽QKWUrQ5FyIµ[ [± [ + >

[

 9u



+

[

!_[_Y஍LP୿L[QrQ± [ + >

[

1KŲYୟ\I[OjKjPVஃÿஅQJELୱQWUrQ5

0୩WNKiFI 9ୟ\SKŲţQJWUuQKFyQJKL୹PGX\QKୗW

&kX7tQKWtFKSKkQ, 

G[

[ [

[









FRV VLQ

VLQ

π

π



;pWWKrP- 

G[

[ [

[









FRV VLQ

FRV

π

π

7DFy,- 

 π

1JRjLUD

-±, 

∫

∫

−

= +

















 VLQ







 FRV FRV

VLQ

VLQ FRV

π

π

π

π

G[

[

[ G[

[ [

[ [

 

G[

[

[

∫ −



 VLQ



 FRV



π

π

Ĉ୩WW VLQ[WDFyGW FRV[G[.KLÿy

-±, ∫ −

  W

GW









OQ













 OQ









 OQ







−

= +

−

= +

−

−

W

W



7஝ VX\UD





&kX

,   OQ  



π



[ 

$¶2 ⊥ PS$%&Pj$% ⊥ &2QrQ$% ⊥

6X\UD

Ö$

$

%



⇒

$$¶ $+

  



D

$+ $¶+  

D

'Rÿy6$$¶%¶%  



D

 9u$2 ⊥ %&QrQ$$¶ ⊥ %&⇒ %%¶ ⊥ %&

⇒ %%¶&¶&FNJQJOjKuQKFK஡QKୟW

⇒

6%%¶&¶& %%¶%&  





D

 9ୟ\





&kX

7DWKୗ\[\] OjQJKL୹PFஙDK୹1ୱXP஋WWURQJ୛Q[\]EୣQJWKuWKuF୕ÿ୳XEୣQJ

;pWWUŲஏQJKகS[\] ≠ 7஝K୹VX\UDF୕୛Q[\]ÿ୳XkP

'R[\]EuQKÿ୥QJWKHRKRiQY୽YzQJTXDQKQrQ[pWKDLWUŲஏQJKகS

7+[

≤

\

≤

].KLÿy ⎪⎩

⎪

⎨

⎧

≥

≥

≥

≥













]

\ [

]

\ [



⇒

 [\]

≤

][\ 

⇒

[ \ ]

7+\ ≤ [ ≤ ][pWWŲţQJWணYଢ଼QVX\UD[ \ ]

7KୱYjRP஋WSKŲţQJWUuQKFஙDK୹ÿŲகFI[ [[ Y஍L[7DFyIµ[ [!Y஍LP୿L [QrQKjPVஃOX{QÿஅQJELୱQPjI I !VX\UDSKŲţQJWUuQKFyGX\QKୗWP஋WQJKL୹PkP 9ୟ\K୹SKŲţQJWUuQKÿmFKRFyÿ~QJKDLQJKL୹PSKkQEL୹W

&kX



$¶&¶





%¶





$&



+2

%



6[T  





D



+୹SKŲţQJWUuQKW୿Dÿ஋JLDRÿL୵PFஙDGYj(⎪⎩+  =

\ [



⎣



9ୟ\GFୡW(W୓LKDLÿL୵PSKkQEL୹W$%*L୕Vட$Yj%

7DFy$% 6$%& QrQG&$%  



$%

6 $%F

7஝ÿyFyK୹SKŲţQJWUuQKW୿Dÿ஋ÿL୵P&

⎪

⎩

⎪

⎨

⎧

=

±

=

⇔

⎩

⎨

⎧

= +

=

− +

⇔

⎪

⎪

⎩

⎪⎪

⎨

⎧

=

+

= +

−

+

















_





 _













_







_













\

[

\ [

\ [

\

[

\

[

9ୟ\FyKDLÿL୵PWK஁DPmQÿ୳EjL

























 &

7DFy $ '  % '  & ' 

$ '

&

'

&

'

% '

%

'

$

'  =  = 

6X\UD'$'%'&W஝QJÿ{LP஋WYX{QJJyF

*୿L0OjWUXQJÿL୵PFஙD$&0[OjWU஗FFஙDWDPJLiF$%&WKuWkP

,FஙDP୩WF୙XQJR୓LWLୱSW஛GL୹Q$%&'OjJLDRJL஡D0[YjP୩W

SK୥QJWUXQJWUணFFஙD%'7஝ÿyVX\UD0 , ' %



=



;iFÿ୽QKÿŲகF, 



Yj5 ,'  



 3KŲţQJWUuQKP୩WF୙X6[± 

 \±]±  



 0୩WSK୥QJα PS%&'QrQFyYpFWţSKiSWX\ୱQOj ' $

KD\Q3KŲţQJWUuQKPSα \G 

0S

α

WLୱS[~FY஍L6

⇔

G,

α  5

⇔ _G_  





⇔



±



&yKDLP୩WSK୥QJWK஁DPmQÿ୳EjLOj













%

[





.

,



& '

0

$















±

 



⎢

⎢

⎢

⎣

⎡

−

=

=

=

=

=

=

⇔

⎪⎩

⎪

⎨

⎧

= + +

−

=























\ [

\ [

\ [

\ [

\ [

[\



9ୟ\WୟSKகSÿL୵PEL୵XGL୷QVஃSK஛F]JஅPÿL୵P



^`