الرياضيات باك2 ملخص الدروس .pdf

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n →+∞

n →+∞

 W^7‫א‬b+↖
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D.
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D.
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
vn ≤ un ≤ wn 

lim vn = l  ⇒ lim un = l
 n →∞
n →+∞
lim vn = l 
n →+∞


un − l ≤ vn 
 ⇒ lim u = l
lim vn = 0 n →∞ n
n →+∞


un ≥ vn

un ≤ vn



⇒ lim u = +∞
lim vn = +∞ n →+∞ n

n →+∞



⇒ lim u = −∞
lim vn = −∞ n →+∞ n

n →+∞

 W un +1 = f (un ) ! ‫↖
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WD
,8
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 ‫
א‬Y

u0 = a



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=
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19

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NH



//$
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 I
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f
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•

20

 W
+J‫א‬4‫
<א(א‬Nc‫א‬4‫א(א‬4( ↖
f (x )

F (x )

a∈


ax + k
1
x² + k
2
−1
+k
x

x
1
x²
1
x

(r ∈ » * − {-1})

x

2 x +k
x r +1
+k
r +1

r

sin x

− cos x + k

cos x

sin x + k

1 + tan ²x =

1
cos ²x

tan x + k

1
x

ln x + k

ex

ex + k

(k ∈
)

 W
Nc‫א‬4‫<א(א‬8((%S7J‫א‬eN40‫↖א‬

(r ∈ » * − {-1})

f (x )

F (x )

u ' (x )
u (x )

2 u (x ) + k

−v ' (x )
[v (x )] ²

1
+k
v (x )

u ' (x ) ×[u (x )]

[u (x )] r +1

r

r +1

u ' (x )
u (x )

+k

ln u (x ) + k
( )

u ' (x ) ×eu x

eu x + k

(a ≠ 0)

cos (ax + b )

(a ≠ 0)

sin (ax + b )

1
sin (ax + b ) + k
a
1
− cos (ax + b ) + k
a

( )

21

(k ∈
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3 d‫!=
=
א‬4
b  a 

 




WD// ‫(
א‬#‫
א‬.b k& a = f ‫א‬#‫
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a

 WNH↖


W
9f‫א‬
a

a

b



∫ f (x ) dx = 0



∫ f (x ) dx = −∫ f (x ) dx

a
b
b
b
b
b


∫ [ f (x ) + g (x )]dx = ∫ f (x )dx + ∫ g (x )dx

(k ∈
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a
a
a
a
a
 W4
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c
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f
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a

 WB1C‫א‬/‫↖א‬


∀x ∈ [a,b ]

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WO 8
‫א‬S&
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b


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WOTU
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a



∀x ∈ [a,b ]

f (x ) ≥ 0 WO 8
‫א‬S&

b



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a

 W
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a
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(o, i, j )
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x = b x = a

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f
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23

 4‫(א‬05K2
0g ‫א‬4‫א(א‬
 

 
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 W:1
]0; +∞[ 3 d‫ ^
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1
x


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vK<‫
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א‬


ln
W!
C
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1
a
#-
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 WNH ‫א‬

∀x ∈ ]0; +∞[

∀y ∈ ]0; +∞[

ln 1 = 0

ln (xy ) = ln x + ln y

ln e = 1

∀x ∈ ]0; +∞[
∀y ∈ ]0; +∞[
ln x = ln y ⇔ x = y
ln x > ln y ⇔ x > y

(r ∈ ) ln (x r ) = r ln x
1
ln   = − ln x
x 
x 
ln   = ln x − ln y
 y 

∀x ∈ ]0; +∞[

∀y ∈


ln x = y ⇔ x = e y



∀x ∈
* ln (x n ) = n ln x


WOTU
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‫(א‬#
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O 8
‫א‬S&
 W:‫ 
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WD. f
‫א‬#‫
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/ x ∈ Du ‫ و‬u (x ) > 0}
D f = {x ∈
/ x ∈ Du



WD
,8
U!
f ‫א‬#‫א‬
f (x ) = ln [u (x )]
f (x ) = ln (u (x ))2 



‫ و‬u (x ) ≠ 0}





f (x ) = ln u (x )

W
A=

ln x
n =0
x →+∞ x
lim

( n ∈ » *)

lim (x n ln x ) = 0

x →0
>

ln (x + 1)
=1
x →0
x
lim

lim (ln x ) = +∞

x →+∞

lim (ln x ) = −∞

x →0
>

ln x
=1
x →1 x − 1
lim

W41J‫א‬



]0; +∞[
3 d‫
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 ln x
‫א‬#‫א‬
 



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 ln [u (x )]
‫א‬#‫
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< u c 8
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24

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3 2
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 g

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<
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I 3 d‫
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 g x
  ∀x ∈ I

'
)]

 ln [u (x )]

‫א‬#‫
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(ln [u (x ) =

u ' (x )
u (x )




W #



]0; +∞[
^
6 /%*
 g x

: #

 

1
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x

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ln x

-

WQ#‫א‬/0‫א‬
 

  0   1   +∞

 x

 ln x
‫א‬#‫א‬

+

a ∈ » *+ − {1} Wa)a UhOg ‫↖א(א
א‬




loga W!
C
!
r‫א
א‬#‫
א‬D.
a
q @…
o ?‫א
א‬#‫  א‬W:1
  ∀x ∈ ]0; +∞[
∀x ∈ ]0; +∞[

loga (x ) =

∀y ∈ ]0; +∞[

loga (xy ) = loga x + logay

(r ∈  )

ln x



WM$
ln a
 WNH ‫א‬
loga 1 = 0

logaa = 1

loga (x r ) = r loga x

∀x ∈ ]0; +∞[

1
loga   = −loga x
x 

l oga x = l oga y ⇔ x = y

x 
loga   = loga x − logay
 y 

 

∀y ∈ ]0; +∞[

∀r ∈ 

l oga x = r ⇔ x = a r

 :1&=
0 <a <1
loga x < logay ⇔ x < y
lim loga x = −∞

a >1
loga x > logay ⇔ x > y
lim loga x = +∞

x →+∞

x →+∞

lim loga x = +∞

lim loga x = −∞

x →0+

x →0+

 W
7X#‫א‬
∀x ∈ ]0, +∞[

( loga x ) ' =
25

1
x ln a





 4‫(א‬05K2
c‫א‬4‫א(א‬
 

 
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א‬, ?‫א
א‬#

‫א
א‬#‫
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exp
W!
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! 

  exp (x ) = e x



















= x +
HB
WNH ‫א‬
∀x ∈


∀x ∈
e x > 0
∀x ∈
ln ( e x ) = x

∀y ∈


e x ×ey = e x +y

∀x ∈


1
−x
x =e
e
ex
x −y
y =e
e





∀y ∈ ]0; +∞[



e x = y ⇔ x = ln y
∀ (x ; y ) ∈
²
e x = ey ⇔ x = y
e x > ey ⇔ x > y
W:‫ 
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WD. f
‫א‬#‫
א‬1!-
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e ln x = x

∀x ∈ ]0, +∞[

r
(e x ) = erx

(r ∈ )



WD
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U!
f ‫א‬#‫א‬

Df =






D f = {x ∈
/ x ∈ Du }





f (x ) = e

f (x ) = e

x

u (x )

W
A=
lim e x = +∞

x →+∞

  lim e x = 0

( n ∈ » *)

x →−∞
 x

e 

  lim  n  = +∞
x →+∞  x 
  lim (x ne x ) = 0
x →−∞

ex − 1
  lim
=1
x →0
x

 W41J‫א‬





^
4 x

e

x


‫א‬#‫א‬
 



I 3 2
^
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‫
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I
3 d‫
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א‬4 x

e

u (x )


‫א‬#‫
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^
א‬4
u c 8
‫א‬S&
26

 WS7J‫א‬

(ex )′ = ex W #



^
6 /%*  g x



∀x ∈


e

x


‫א‬#‫א‬



I 3 2
^
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I 3 d‫
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א‬6 /%*
 g
x

e

u (x )


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3 d‫
^ א‬6 /%*
 g

u c 8
‫א‬S&
'

(eu x )
( )

∀x ∈ I

u (x )
= u ′ (x ) ×e









W #


 W ln
‫(א‬Q#‫א‬/0‫א‬

 

a ∈
∗+ − {1} Wa) a Uh
c‫↖א(א
א‬



expa
W!
C
! 
a
q @…
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א‬#‫^
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loga ‫א‬#

‫א
א‬#‫א‬

W:1
 

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= x +
HB
WNH ‫א‬
∀ (x ; y ) ∈
2
a x × a y = a x +y
r
(a x ) = a r x
1
−x
x =a
a
ax
x −y
y =a
a

(r ∈ )

a x = e x ln a
loga (a x ) = x

∀x ∈


∀x ∈ ]0; +∞[
∀ (x ; y ) ∈
2

a

loga (x )

=a

a x = ay ⇔ x = y

∀x ∈

∀y ∈ ]0; +∞[
x
a = y ⇔ x = loga (y )
W1&=

0 <a <1
a < ay ⇔ x < y

a >1
a > ay ⇔ x > y

lim a x = 0

lim a x = +∞

x

x

x →+∞

x →+∞

x

lim a x = 0

lim a = +∞

x →−∞

x →−∞
x

a −1
= ln a
x →0
x

lim

  (a x )′ = (ln a ) ×a x
27

 W
7X#‫א‬

4‫(א‬05K2


(7‫א‬+‫(א‬c‫א‬


 = {z = a + ib / (a;b ) ∈
²

i ² = −1} WD.
#/‫א(
א‬#‫
א‬,2

 Wi(7+(
P@‫
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(a;b ) ∈
²
WM$
#/
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=
z
v#/‫(
א‬#
Y;‫^
א 
א‬,- a + ib

•

Re (z )
W!
j
!

z
(#
D// ‫^
א;†
א‬,
a
(#‫א‬

•



Im (z ) W!
j
!

z
(#
D‡‫^
א;†
א‬,
b
(#‫• א‬
D//$
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z
OTU Im (z ) = 0 WO 8
‫א‬S& •  WjNHj)


U!>
ˆ
‫(א‬#
^, z
OTU Im (z ) ≠ 0 
Re (z ) = 0 WO 8
‫א‬S& •

 WE(7+(i1↖


L#/
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=


z = z ′ ⇔ Re (z ) = Re (z ′)
 Im (z ) = Im (z ′)

 Wi(7+(Q#‫א‬/0‫↖א‬
 

(o,e1,e2 )
o€J
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o
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=
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M (a, b )
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z
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z = Aff (OM ) POM (z ) Wt
OM "|‫
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 Wi(7+(bk‫↖א‬


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² WM$
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=

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•



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z + z = 2 Re (z )


z − z = 2i Im (z )


zz = [Re (z )]² + [ Im (z )]²

•
•
•



z + z ' = z + z ' •


z ×z ' = z ×z ' •



(n ∈  *)











z n = z n •
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1



  =

•
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z  z


(z ' ≠ 0)










  =
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z'

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z = zz = a ² + b ²
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28

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zn = z n

z = z

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z
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(n ∈ * )
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arg z = θ [2π ]
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W. z
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a = [−a, π ]

a = [a, 0 ]

π
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π
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z = r (cos θ + i sin θ ) = [r , θ ]




z = reiθ



WD.
z
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reiθ × r ' eiθ ' = rr ' ei (θ +θ ') • [r , θ ] × [r ', θ '] = [rr '; θ + θ ']
[r , θ ] = [r, −θ ]
reiθ = re−iθ •
−[r, θ ] = [r, π + θ ]
−reiθ = re i(π +θ ) •
n
[r , θ ]n = r n ; n θ 
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1
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( )

re

iθ

r 'e

iθ '

=

[r ; θ ]
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r i (θ−θ ')
e
•
r'

•

arg (zz ') ≡ (arg z + arg z ') [2π ] •

•

arg z ≡ − arg z [2π ] •
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•

•
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r
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( k ∈ )

arg z n ≡ n arg z [2π ] •
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z

•

‫ 
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π
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z
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∀k ∈ 

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(
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)
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e + e −iθ

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sin θ =
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2i
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W( ‫א‬

S = {− a ; a }
S = {0}
S = {−i −a ; i −a }

a>0
a=0
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29

z ∈  z² = a

  (a ≠ 0)
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W( ‫
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W( ‫א‬

∆>0
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(∆ = b2 − 4ac )

∆<0

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AB = z B − z A
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AB; AC ≡ arg  c
 z B − z A 
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∈

zB − zA
z −z
z −z
z −z
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D A × D C ∈

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r " %
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z − zA = z − zB



A a
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A a
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A a
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Lg ‫
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:*A‫
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zC − z A
π
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2 
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= [1; θ ]
zB − zA
zC − z A
π
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u "|‫א
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k j<
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z ′ − ω = ei θ (z − ω )



θ j‫א‬X
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30

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(α ∈
)



WA ‫ (
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y (x ) = αer1x + βer2x
(α, β ) ∈
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WM$


y (x ) = (αx + β )erx


(α, β ) ∈
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WM$



WA ‫א (
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b
a

y ' = ay + b
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W
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r2

r1 LŠ


r
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r ² + ar + b = 0

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WL/U‫א‬Q
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y (x ) = (α cos qx + β sin qx )e px


(α, β ) ∈
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WA ‫א (
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r2 = p + iq

31

y ''+ ay '+ by = 0

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•


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•


i a a'
b b'  a a'  a a' 


u ∧ v = j b b ' =
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k#‫↖א‬
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A
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WD.
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j( 
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=
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d (M , (Ρ)) =
 
AM ∧ u


d (A, (∆)) =

u

ax M + byM + cz M + d
a ² + b² + c²



WD.
∆ (A, u )
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U ‫
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n (a,b, c ) ⇔ (Ρ) : ax + by + cz + d = 0






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32


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WD ‫~
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33

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W
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A = {x ∈ E / x ∉ A}





























M$
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cardA = cardE − cardA

•
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 W+‫(א‬Qc‫א‬A(#‫↖א‬


(p ∈  *)

‫ א\ א‬p "| 
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n1 × n2 × n3 × ... × n p


W
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34

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W.
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W #

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37

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W
#‫*א‬M‫
א(א‬+J‫א‬O7‫א‬4( ↖

x

0

sin x

0

cosx

1

tan x

0

π

π

π

π

6
1
2
3
2
3
3

4
2
2
2
2

3
3
2
1
2

2

1

3

1

0

 W
#‫א‬B‫א‬E8T.‫↖א‬

−x

cos (x + 2k π ) = cos x
sin (x + 2k π ) = sin x
tan (x + k π ) = tan x

π − x π +x

π
2

−x

π
2

+x

sin

- sin x

sin x

- sin x

cos x

cos x

cos

cos x

- cos x

- cos x

sin x

− sin x

tan x =

sin x
cos x

1
1 + tan ²x =
cos ²x

-1 ≤ cos x ≤ 1
-1 ≤ sin x ≤ 1
cos ²x + sin ²x = 1

 W
A
J+↖
cos x = cos a ⇔ x = a + 2k π
P x = -a + 2k π
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tan x = tan a ⇔ x = a + k π
( k ∈ )
38

 W! 0;/ _eN↖
cos (a - b ) = cos a × cos b + sin a × sin b

cos (a + b ) = cos a × cos b - sin a × sin b

sin (a - b ) = sin a × cos b - cos a × sin b

sin (a + b ) = sin a × cos b + cos a × sin b

tan (a - b ) =

tan a - tan b
1 + tan a × tan b

tan (a + b ) =

tan a + tan b
1 - tan a × tan b

 WLM↖
t = tan

cos 2a = cos ² a - sin ² a

a
WHA
2

= 2 cos ² a - 1

2t
1 + t²
1 - t²
cos a =
1 + t²
2t
tan a =
1 - t²

= 1 - 2 sin ² a

sin a =

sin 2a = 2 sin a × cos a
2 tan a
1 - tan ² a
1 + cos 2a
cos ² a =
2
1 - cos 2a
sin ² a =
2

tan 2a =

 WF‫ (א‬r'! 0;/ _↖W! 0;r'F‫ (א‬/ _↖
 p + q 
 p − q 
cos p + cos q = 2 cos 
cos 

 2 
 2 
 p + q   p − q 
cos p − cos q = −2 sin 
sin 
 2   2 
 p + q 
 p − q 
sin p + sin q = 2 sin 
cos 

 2 
 2 
 p + q   p − q 
sin p − sin q = 2 cos 
sin 
 2   2 

1
[ cos (a + b ) + cos (a - b )]
2
1
sin a × sin b = − [ cos (a + b ) − cos (a − b )]
2
1
sin a × cos b = [ sin (a + b ) − sin (a − b )]
2
1
cos a × sin b = [ sin (a + b ) - sin (a − b )]
2
cos a × cos b =

 

  (a,b ) ≠ (0,0) a cos x + b sin x W/ _↖


a
b
a cos x + b sin x = a ² + b ² 
cos x +
sin x 
 a ² + b ²

a ² + b²
= a ² + b ² cos (x − α)



We/
D//$
(#
α
M$
cos α =

a
b



















sin α =
a ² + b²
a ² + b²
39



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