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بسم اهللا الرحمن الرحيم والصالة والسالم على أشرف المخلوقين محمد سيد المرسلين وعلى آله وصحبه أجمعين
ملخصات مع تقنيات أما بعد ٬ يسرني أن أقدم لكم هذا العمل المتواضع وهو عبارة على
مجمعة في كتاب واحد بكالوريا علوم تجريبية ثانية الرياضيات لمستوى ال
وهي لألستاذ حميد بوعيون
sefroumaths.site.voila.fr
تجميع وترتيب
ALMOHANNAD
لا یت وا
I ( 1 ( :
2 ( :
( )
a
a
a
+∞ + = +∞−∞ + = −∞+∞ + ∞ = +∞ ∈−∞ − ∞ = −∞+∞ − ∞
( )0
0
a a× ∞ = ∞ ≠∞ × ∞ = ∞
× ∞
00
00
0
a a
a
∞ ≠= ∞ = = ∞∞
∞∞
3 ( !:
(a ( )( )lim
f x
g x∞
∞=∞
.
(b ( ) ( )( )lim f x g x∞
+ = +∞ − ∞
(* ( )f x ( )g x .
(* ( )f x ( )g x .
(c ( ) ( )( )0
00 lim
0 0x
f x a aa
g x
−≠ = = .
(d ( ) ( )( )0
0 0 00 lim
0 0x
f xa
g x
+≠ = = ! " .
e("# :2x x= $
2
2
; 0
; 0
x x x
x x x
= ≥
= − ≤
4 ( . ( )
( )( ) ( )
20 0 0
1 cos tan sin1lim lim 1 lim 1
2x x x
ax ax ax
ax axax→ → →
−= = =
5 (%. (a% & f ' 0x() & ( )
0
limx
f x &
%( ) ( )0
0limx
f x f x=* f ' 0x.
(b+& f ,' -. /0 1 2 3& 3" . /0 ' 3&* -.4 ' (
( .
6 (%! f 0x 5 % & f
'20x& () ( )0
limx
f x& ( )0
limx
f x l= ∈
*f g '2 0x 6
:( ) ( )( )
0
0
,g x f x x x
g x l
= ≠ =
7 ( & . (a ( ) ( )f x l g x− ≤ 0x
( )0
lim 0x
g x =
(b ( ) ( )f x g x≤ 0x ( )
0
limx
f x = +∞
(c ( ) ( )f x g x≤ 0x ( )
0
limx
g x = −∞
(d+& ( ) ( ) ( )g x f x h x≤ ≤ 0x ( ) ( )
0 0
lim limx x
g x h x l= =
II (% ! ' &%. 1 ((a 7 ' 8' .
(b' 9: 8' 9: 7 . 2 ((a+& f* . ' :
(*[ ]( ) ( ) ( ), ,f a b f a f b= (* ] ]( ) ( ), lim ,a
f a b f f b+
= .
(b +& f* ':& ' :
(*[ ]( ) ( ) ( ), ,f a b f b f a= (* ] [( ), lim , limb a
f a b f f− +
= .
3 ((& ) *! (a & +f/0 ' [ ],a b
λ ' 0 ( )f a ( )f b
(b & +f/0 ' [ ],a b
( ) ( ) 0f a f b⋅ ⟨ &( ) 0f x = ; ] [,a b
"#: (* ( ) ( ) 0f a f b⋅ ≤ * [ ],c a b∈. (*+& f 9: * c .
III ( 1( +& : (*f /0 ' I
(*f /0 9: I (*( )f I J=
f ) 0 1 :f J I− → & :
( )( ) ( ) ( )1:x J y I f x y f y x−∀ ∈ ∀ ∈ = ⇔ = 2 ((a 1f −/0 ' J
(b 1f −/0 9: J <"& 3 f. (c ..&& fC 1f
C − 6'& )& ;
( ) : y x∆ =.
*( )0
limx
f x l=
* ( )0
limx
g x = +∞
* ( )0
limx
f x = −∞
* ( )0
limx
f x l=
*[ ]( ) ( ), :c a b f c λ∃ ∈ =
*] [( ) ( ), : 0c a b f c∃ ∈ =
*f I& J
0
0
∞∞
0∞× +∞ − ∞
3 ( +1f −
. +& f /0 9: =; : I
( ) ( ): 0x I f x′∀ ∈ ≠* 1f −/0 =; : ( )J f I=
( ) ( ) ( ) ( )( )1
1
1:x J f x
f f x−
−′∀ ∈ =
′
IV (! , ' n ( )*n∈ 1 (-: x +
n x 7 y IR +
1 ny x=.
(* :4 16 2= 42 0 2 ≤و16 =.
(* ( )42 16− = 4 16 2≠ − 2 +− ∉
2 (%. (a n/0 +
(b ( ) : 0nx x+∀ ∈ ≥
c( ( ), : )*
)*
n n
n n
x y x y x y
x y x y
+∀ ∈ = ⇔ =
< ⇔ <
.
(d ( ), : )*
)*
n n
n n
x y x y x y
x y x y
+∀ ∈ = ⇔ =
⟨ ⇔ ⟨
.
(e n* 1 : ( ), : )*
)*
n n
n n
x y x y x y
x y x y
∀ ∈ = ⇔ =
⟨ ⇔ ⟨
(f n* . :( ), : )*
)*
n n
n n
x y x y x y
x y x y
∀ ∈ = ⇔ =
⟨ ⇔ ⟨
(g (* ( ) ( )0 .n
nn nx x x x∀ ≥ = =
(* n . ( ): n nx IR x x∀ ∈ =
(h n p *IN a b +
(*.n n na b ab=
(*( );pnp np pn na a a a= =
(*( ); 0n
p npn nn
a aa a b
bb= ⟩ =
(*.npp n pn a a a +=
(i (*( ) ( )* , 0 :p
pnnn p x x x∈ ∈ ∀ ⟩ =
(* p. :( ) :p
pn nx x x∀ ∈ =. "#:
1 ( 0xy ⟩* n n nx y x y= ⋅ n
nn
xx
y y=
(2 (* ( ) 330 :x x x∀ ≥ = ( )
( )
333 3
333 3
; 0
; 0
x x x x
x x x x
= = ≥ = − − = − − ≤
.
(* 3 3 3 3
2 2 2 2
a b a ba b a b
a ab b a ab b
+ −+ = − =− + + +
4 4
3 2 2 3
a ba b
a a b ab b
−− =+ + +
(j a b *IR+ r 'r
r r r ra a a′ ′+⋅ = ( )rr r ra a
′ ′=
r
r rr
aa
a′−
′ = ( )r r rab a b= ⋅
1 r
ra
a−=
r r
r
a a
b b =
ت ادیا
(I . 1 ( :
U I :
( ):U I
n u n
→→
2 ( : :
( )n n IU
∈:
(a M ( ) nn I U M∀ ∈ ≤.
(b "# m ( ) nn I U m∀ ∈ ≥.
(c$ "# % . mM ( ) : nn I m u M∀ ∈ ≤ ≤.
: ( )n n I
U∈
0k ≥ ( ) : nn I U k∀ ∈ ≤
3 ( : :
( )n nU
∈:
(a ( ) 1n nn U U +∀ ∈ ≤.
(b $& ( ) 1n nn U U +∀ ∈ ⟨.
(c #& ( ) 1n nn U U +∀ ∈ ≥.
(d $& #& ( ) 1n nn U U +∀ ∈ ⟩.
(e ' ( ) 1n nn U U +∀ ∈ =.
: (1 % ( )nU( p np n u u⟨ ⇒ ≤.
(2 % ( )nU( #& p np n u u⟨ ⇒ ≥.
(3 ) ( )nU * +
1n nu u+ −. (*% 1 0n nu u+ − ≥( ( )nU .
(*% 1 0n nu u+ − ⟩( ( )nU $& .
(*% 1 0n nu u+ − ≤( ( )nU#& .
(*% 1 0n nu u+ − ⟨( ( )nU $& #& .
(*% 1 0n nu u+ − =( ( )nU ' .
(II 1( : ( )n n
U∈
r
( ) 1: n nn U U r+∀ ∈ = + r . ) / . :
(a ( )nU % ' $ .. 0 1 ' 21 .
(b) ( )nU3 + 1n nu u+ − 1n nu u cte+ − =. 0 1 ' .
(cو 0 $ و 4' 3 1
2a c b+ = $ 2
a bb
+ =.
2 ( . ( )nu 5 ) r0 1 0u
0nU U nr= + ( )n∀ ∈
: 1 (1 0 1u1 + $ ( :
( )1 1nU U n r= + −.
2 (1 0 2u1 + $ ( : ( )2 2nU U n r= + −.
3 ( 6# : p nU Uو
5 )r( ( )n pU U n p r= + − ) 3pوn+5 8 .(
3 ( : ( )nU 5 ) r 0 1 0u :
( ) ( )00 1 2 ... 1
2n
n
u uS u u u u n
+= + + + + = +
0u 0 9: S
nu ;0 9: S 1n + 9 S .
:
1 (1
1 2 ....2
nn
u uu u u n
++ + + =.
2 (0 10 1 1....
2n
n
u uu u u n −
−+
+ + + =.
3( 6#
( )1 .... 12
p nP p n
u uu u u n p+
++ + + = − +.
(III . (1 :
( )n nu
∈ 1 q
: ( ) 1: .n nn U qU+∀ ∈ = $q . ) /
: (a )$ 8 1 ( 1
. 0 1 ' 21 % ' $ < ;. (b ) ( )nU3 1 1nU +
=nU 1n nu q u+ = ⋅. (c0 cوbوa $ 4' 3 1
1 2ac b=.
2 ( : ( )nu 5 ) 1 q0 1 0u
( ) 0: nnn u u q∀ ∈ = ⋅.
: 1 (1 0 1u1 + $ ( 1
n pnu u q −= ⋅
2 ( 6#: n pu 1 uو
5 )q ( n p
n pu u q −= ⋅ )3pوn +5 8 .(
3 ( : ( )nU 5 ) 1 q0 1 0U.
>( 1)q ≠.
1
0 1 0
1.... .
1
n
n
qS U U U U
q
+−= + + + =−
0u :0 9: S .
( )1n + : 9 S. :
1 ( 1q = ( ( )0 1 0.... 1nS u u u n u= + + + = +
2 (0 1 1 0
1.... : 1
1
n
n
qu u u u q
q−−+ + + = ⋅ ≠−.
1 2 1
1....
1
n
n
qu u u u
q
−+ + + = ⋅−
6#
1
1
1....
1
n p
p p n p
qu u u u
q
− +
+−+ + + =
−
(IV . 1 (lim nq
0 ; 1 1
1 ; 1lim
; 1
. 1
n
q
q
دة q
− ⟨ ⟨ == +∞ ⟩ ≤ −
2 ( !". (a ( )nU ( )nV n nU l V− ≤ ?# &4 .
lim 0 limn nV U l= ⇒ =
(b ( )nU ( )nV n nU V≤ ?# &4 lim limn nU V= +∞ ⇒ = +∞ lim limn nV U= −∞ ⇒ = −∞
(c ( )nU ( )nV ( )nW n n nV U W≤ ≤ &4 ?#.
lim lim limn n nV W l U l= = ⇒ =
3 ( ( )nU 5 5 % . @;0 %= 5 .
4 ( .
(a ( )nU ( )nV n nU V≤ ) )n nU V⟨ (
?# &4% ( )nU ( )nV
(lim limn nU V≤. (b . (c "# #& .
5 ( ( )1n nU f U+ =
f/: $ I $ ( )0
1n n
U I
U f U+
∈ =
(*% ( )f I I⊂$ ( .
(*% f/: :# I ( )nU 5 5 ( l
( )f l l=
I( . 1 ((a ( ) 2 2/ , 1z a ib a b =iو = + ∈ = −
(b z z a ib= + aوb ∈
i ( ) 2 1i i∉ = −
(c (* z a ib= + ! " ! # z. (*a % &'! # z ( )oR z a′ =
(*b % ( &'! # z ( )Im z b= (* )*0b =+, z a= ∈. (* )*0a =+, z ib= ∈* z- % ( . 2 ( a b a′ b′ . α ∈
a a
a ib a ibb b
′=′ ′+ = + ⇔ ′=
00
0
aa ib
b
=+ = ⇔ =
II ( . 1 ( - z a ib= + . aوb ∈
, %#z / ' ) z% - z a ib= −. 2 (0 (.
(a z z z
z z z i
= ⇔ ∈= − ⇔ ∈
(b z z z z
z z
′ ′= ⇔ ==
(c 1 2 1 2.... ....n n
z z z z
z z z z z z
′ ′⋅ = +
+ + + = + + + (d
( )1 2 1 2.... ....n n
n n
z z z z
z z z z z z
z z n
′ ′⋅ = ⋅
= ⋅ ⋅
= ∈
(e 1 1
z z =
z z
z z = ′ ′
(f z x iy= + ( )
( )2 2
2 2 Im
z z x Ré z
z z iy i z
+ = =
− = = 2 2zz x y= +
III ( 1 ( : z a ib= + . aوb ∈
%#z / ' ) ! % z -
% : 2 2z zz a b= = +
2 (":
(a )* z a= ∈ +, z a=
)* z ib= ∈ +, z b=
(b 2 2 2z zz a b= = + z z z= = −
(c 0 0z z= ⇔ = z z z z′ ′+ ≤ +
(d 1 2 1 2..... ....n n
zz z z
z z z z z z
′ ′=
=
nnz z=
(e 1 1
z z=
zz
z z=
′ ′
#$% :. # ! % :
(* 2'
z zz zz
z z z z
′ ′= =
′ ′ ′
(*( )( )( )( )
( )( )2 2
a ib c id a ib c ida ib
c id c id c id c d
+ − + −+ = =+ + − +
IV ( &'( ) . 2# " 34P56 5 * # ( )1 2, ,o e e
1 ( :
(a ( ),M x y P z a ib= + # M
( )aff M.
(b ( ),u x y
v
z a ib= +7! # u
( )aff u z=
(c z a ib= + ( ),M x y # z
%,P ( )M z.
(d z a ib= + 7! ( ),u x y
# z
%,2v ( )u z.
#$%: (a ( ) ( ) ( )2 1. 1 . 0aff e i aff e aff o= = =
(b
( ) ( )( ) [ )( ) ( ]
z M z x ox
z M z ox
z M z x o
+
−
′∈ ⇔ ∈
∈ ⇔ ∈
′∈ ⇔ ∈
(c
( ) ( )( ) [ )( ) ( ]
z i M z y oy
z i M z oy
z i M z y o
+
−
′∈ ⇔ ∈
∈ ⇔ ∈
′∈ ⇔ ∈
2 (".
(a
( ) ( )( ) ( ) ( )
( ) ( )
aff M aff M M M
aff MM aff M aff M
MM aff M aff M
′ ′= ⇔ =
′ ′= −
′ ′= −
(b
( ) ( )( ) ( ) ( )( ) ( )
( )
aff u aff v u v
aff u v aff u aff v
aff u aff u
u aff u
α α
= ⇔ =
+ = +
=
=
(c G .! ( )( ) , ,A Bα β
( ) ( ) ( )( )1aff G aff A aff Bα β
α β= +
+
(d I- [ ]AB ( ) ( ) ( )( )1
2aff I aff A aff B= +
(e A B C % 78 9: , ,c B Az z z
A B≠ A B C )* , )* #
C A
B A
z z
z z
− ∈−
.
V( * & )+, - . 1 (* ( *z ∈ ( )M z %#z ; 8
'( )1,e OM
. / ' argz
(* ( ) [ ]1arg , 2z e OM π≡
#$% :
[ ][ ]
*
*
*
arg 0 2
arg 2
arg
z z
z z
z z k
π
π π
π
+
−
∈ ⇔ ≡
∈ ⇔ ≡
∈ ⇔ =
[ ]
[ ]
*
*
*
arg 22
arg 22
arg2
z i z
z i z
z i z k
π π
π π
π π
+
−
∈ ⇔ ≡
∈ ⇔ ≡ −
∈ ⇔ ≡ +
2 ( z *
( )cos sinz r iθ θ= + z r= [ ]arg 2z θ π≡ <)=
# %::z [ ],z r θ=" iz reθ=.
3 (#$%: a( [ ] [ ] [ ], , 2r r r r θو θ θ θ π′ ′ ′ ′= ⇔ = ≡
(b %:: z a ib= +% .
( )
2 2
2 2 2 2
2 2 2 2cos sin ,
a bz a ib a b i
a b a b
a b i a bθ θ θ
= + = + + + +
= + + = +
(c cos sin cos( ) sin( )i iα α α α− = − + −
cos sin cos( ) sin( )i iα α π α π α− + = − + −
cos sin cos( ) sin( )i iα α π α π α− − = + + +
sin cos cos( ) sin( )2 2
i iπ πα α α α+ = − + −
4 (
[ ] [ ] [ ][ ][ ][ ]
[ ][ ]
, , ,
, ,
,,
,
1 1,
,
, ,
n n
r r rr
r r n
r r
r r
r r
r r
θ θ θ θ
θ θ
θθ θ
θ
θθ
θ θ
′ ′ ′ ′⋅ = +
=
′= − ′ ′ ′
= −
= −
( ) [ ][ ]
[ ]
[ ]
( ) [ ]
arg arg arg 2
arg arg 2
arg arg arg 2
1arg arg 2
arg arg 2
n
zz z z
z n z
zz z
z
zz
z z
π
π
π
π
π
′ ′≡ +
≡
′≡ − ′
≡ −
≡ −
( )ii ie e e θ θθ θ ′+′⋅ = ( )ni ine eθ θ= i ie eθ θ−=
( )i
i
i
ee
e
θθ θ
θ′−
′ = 1 ii
ee
θθ
−=
1 ie π− = 2
ii e
π
= 2
ii e
π−− =
5( ( ) ( )( )[ ]( ) ( )( ) ( )( )[ ]
1, arg 2
, arg arg 2
e u aff u
u v aff v aff u
π
π
≡
≡ −
( ) ( )[ ]
( ) [ ]
1, arg 2
, arg 2
B A
D C
B A
e AB z z
z zAB CD
z z
π
π
≡ −
−≡ −
#$% :
)*[ ]1,C A
B A
z z
z zθ−
=−
( ) [ ]
1
, arg 0 2
C A C A
B A B A
C A
B A
z z z zACAC AB
AB z z z z
z zAB AC
z zπ
− −= = = ⇒ =
− −
−≡ ≡ −
6 ( ."Moivre
( ) ( ) ( )cos sin cos sinn
i n i nθ θ θ θ+ = +
7( ."ulerE
( )
( )
1cos
21
sin2
ix ix
ix ix
x e e
x e ei
−
−
= +
= −
2cos
2 sin
ix ix
ix ix
e e x
e e i x
−
−
+ =− =
#$% : ;4 7 >! %:: ?=
/1. :: @ # .
2 1i iz e z eβ α= =
( )1 2 cos sin cos sin
cos cos sin sin
2cos cos 2sin cos2 2 2 2
2cos cos sin2 2 2
i iz z e e i i
i
i
i
α β α α β βα β α β
α β α β α β α β
α β α β α β
+ = + = + + += + + +
+ − + −= +
− + + = +
( )1 2 cos cos sin sin
2sin sin 2cos sin2 2 2 2
2sin cos sin2 2 2
z z i
i
i
α β α βα β α β α β α β
α β α β α β
− = − + −+ − + −= − +
− + + = +
/2 . %#A ' #.
2 2 21 2
2 2cos2
i i ii i
i
z z e e e e e
e
α β α β α βα β
α β α β
+ − −−
+
+ = + = +
−=
2 2 21 2
2 2 sin2
i i ii i
i
z z e e e e e
e i
α β α β α βα β
α β α β
+ − −−
+
− = − = +
−=
VI(- * 01 . 1( *z ∈ *n ∈ % ! %#z z
nz Z=.
2( nz a= )! %= a.
3( [ ],Z r θ= *)! Z A %=
2, / 0,1,...., 1n
k
kz r k n
n n
θ π = + ∈ − = n
4( )!1 A %= 2
1,k
kw
n
π =
0,1,..., 1k n∈ −
5 ( 01*
a( / : [ ],Z r θ= *
)! Z .[ ]2
, ,2
Z r rθθ = =
)* )! Z= : ,2
u rθ =
u−
b(1 / : 1( + 02 *Z a += ∈
: ( )2
Z a a= =
)* )! Z= u a= u− 2( + 02 ( )*Z a a += − ∈
( ) ( )2 22Z a i a i a= − = =
)* )! Z= u i a= u− 33+ 02 ( )*Z ib b += ∈
( ) ( )2 2
22 . 1 1
2 2 2
b b bZ ib i i i
= = = + = +
)! )* Z = ( )12
bu i= + u−
4(+ 02 ( )*Z ib b += − ∈
( ) ( )2 2
22 . 1 1
2 2 2
b b bZ ib i i i
= − = − = − = −
)! )* Z = ( )12
bu i= − u−
5( + 02 Z a ib= + . ( )0 0b ≠aو ≠
): )! :
3 4Z i= − + .B z x iy= + 2 2 2 2z x y ixy= − +
2 2 2z x y= + 5Z =
( )( )( )
2 22
22
2 2
3 1
2 4 2
5 3
x yz Z
z Z xyz Z
x y
− = −= = ⇔ ⇔ = = + =
)1 () +3 (" E#22 2x =% 1x = " 1x = − )1 (–) 3 (" E#22 8y =
% 2 4y = " 2y = " 2y = − 9( )2 ( 2 0xy = ⟩)* y xوG ;4 7
)* 1
2
x
y
= =
" 1
2
x
y
= − = −
)! )* Z= 1 2u i= + u−
VII( 1 4 II: ":
2 0az bz c+ + = . 0a ≠ .B 2 4b ac∆ = −
1 H )* 0∆ = 9 +, :2
bz
a= −.
2H )* 0∆ ≠ +, ∆ )! u −uو
9 :2
b uz
a
− += 2
b uz
a
− −=.
#$%: (* 2 0az bz c+ + = . 0a ≠
)*2 1z ,+zو % :
1 2
1 2
bz z
ac
z za
− + = =
(* 2 2 0az b z c′+ + = . 0a ≠ ( ' # !"
b ac′ ′∆ = −
1H )* 0′∆ = 7 bz
a
′= −
2H )* 0′∆ ≠9 7 :
1 2
b uz
a
′− += 2 2
b uz
a
′− −=. u. )! ′∆.
I
1 !" (a f 0x
( ) ( )0
0
0
limx x
f x f xl
x x→
−= ∈
− ( )0f x l′ =.
(b f 0x
( ) ( )0
0
0
limx x
f x f xl
x x+→
−= ∈
− ( )0df x l′ =.
(c f 0x
( ) ( )0
0
0
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