‫

‬
‫ ‬
‫

‬

‫
‪W‬‬

‫ ‬

‫א
‬
‫א א א
‬
‫ א א ‬

‫ א א ‬
‫ א א ‬
‫ א א א ‬
‫ ‬

‫ א א א ‬

‫ א א א ‬
‫ א א א "! ‬
‫ ‪$ %
& !"# :‬‬

2

‫@ @‬

‫א‪ W! " #‬‬

‫א &‪ W %‬‬

‫&‪#$ %‬א ‪ )*) + , - %& J‬א ‪ ( #‬‬

‫‪4‬‬

‫ ‪( 1 ! - ,2 J . / 0‬א ‬

‫‪5‬‬

‫א " ‬

‫‪6‬‬

‫א‪ 3 4-5‬‬

‫‪8‬‬

‫א‪ 6 / %5‬‬

‫‪10‬‬

‫‪ 7‬א ‪ 8! J+) ,‬א ‪ 0/ J+) ,‬א‪ 9 0 5‬‬

‫‪12‬‬

‫א ! ‪ :‬א * " ‬

‫‪13‬‬

‫א ‪#‬א א ‬

‫‪14‬‬

‫(א א;‪ = #‬א !‪ n <-‬‬

‫‪16‬‬

‫א א ‪ (#‬‬

‫‪18‬‬

‫א ‪ #‬א‪ 3‬א > ‬

‫‪20‬‬

‫א ‪ +‬‬

‫‪22‬‬

‫א ‪ #‬א‪ 3‬א ? ‪ ,‬‬

‫‪24‬‬

‫א ‪ #‬א‪ 3‬א @ ‬

‫‪26‬‬

‫א ‪#‬א( א ‪ #/‬‬

‫‪28‬‬

‫א (‪ 5‬א ‪ A‬‬

‫‪31‬‬

‫א‪ @# C‬א ‪ B‬‬

‫‪32‬‬

‫א ‪#‬א( ‬

‫‪34‬‬

‫א‪ 5 , $5‬‬

‫‪36‬‬

‫א ‪ F‬א ‪ DE E‬‬

‫‪38‬‬
‫‪3‬‬

‫' * )(א‬
4 ‫( א‬05 K2 + (,‫ א‬-.- / 0 1 * '
(a ≠ 0) ax + b W ‫(א‬,‫↖' * א‬
−∞

x


−

a %& G

ax + b

b
a

+∞

a %&



(a ≠ 0) ax ² + bx + c W+ (,‫ א‬-.- / 0 1 * '↖
Ρ (x ) = ax ² + bx + c W HB

Ρ (x ) + , = J K?

x

L ‫א‬#$ 0@‫ א‬

Ρ (x )




Ρ (x ) = a x

b ²
+ 
2a 

W ( ‫ א‬+$



Ρ (x ) %&

x∈»

S = ∅

a %&


b
x −∞ − +∞
a
Ρ (x )



+∞

−∞

%&
a

%&





, ‫א‬

Ρ (x ) = 0

S =

{ }
−b
2a

∆<0


∆=0


a
∆ = b² - 4ac

S = {x1; x2 }

Ρ (x )

= a (x − x1)(x − x 2 )



x1

x

Ρ (x )


%&
a



WM $

x 2 +∞
G



−∞ x1 =
a

E x1 < x2 WOP Q F

−b − ∆
2a
∆>0



%&

%&
a



x2 =



−b + ∆
2a


(a ≠ 0)

ax ² + bx + c = 0 W ( ‫ א‬D $ x 2 x1 O 8 ‫א‬S&
c
−b
x1 × x2 = x1 + x2 = WOTU
a
a
x∈

4

6 78 9
4 ‫( א‬05 K2



+( ‫א‬+ : 1 0;

W 6 78 9 ↖
b a L / /$ = (# +
(a + b )2 = a 2 + 2ab + b 2
(a − b )2 = a 2 − 2ab + b 2
a 2 − b 2 = (a − b )(a + b )
(a + b )3 = a 3 + 3a 2b + 3ab 2 + b 3
(a − b )3 = a 3 − 3a 2b + 3ab 2 − b 3
a 3 − b 3 = (a − b )(a 2 + ab + b 2 )
a 3 + b 3 = (a + b )(a 2 − ab + b 2 )

W +( ‫ א‬4‫ < א ( א‬8 : 1 0;↖
L ( #$Q P =
WD. f ‫א‬# ‫ א‬1 ! - ,2

WD W U! x D/ /$ KV (# ‫ (א‬f

Df =

f (x ) = Ρ (x )
Ρ (x )
Q (x )

D f = {x ∈ /Q (x ) ≠ 0}

f (x ) =

D f = {x ∈ / Ρ (x ) ≥ 0}

f (x ) = Ρ (x )

D f = {x ∈ /Q (x ) > 0}

f (x ) =

Ρ (x )
Q (x )

f (x ) =

Ρ (x )
Q (x )

f (x ) =

Ρ (x )
Q (x )

Df = {x ∈ / Ρ (x ) ≥ 0 Q (x ) > 0}


Df = x ∈ /



Ρ (x )
≥ 0 Q (x ) ≠ 0}
Q (x )

5

4 ‫( א‬05 K2

(>
?1) = ‫א‬
W =1 8 7 x x (n ∈ *)x x n 4‫↖ = א ( א‬

lim x = 0

lim x n = 0

x →0
>

lim

x →+∞

lim

x →+∞

x →0

1
n = 0
x →−∞ x
1
lim n = 0
x →+∞ x

x = +∞

lim

1
=0
x

WOTU (!U ‫(א‬# n O 8 ‫א‬S&

WOTU X ‫(א‬# n O 8 ‫א‬S&
lim x n = +∞

lim x n = +∞

x →+∞

x →+∞

lim x n = +∞

n

lim x = −∞

x →−∞

x →−∞

1
n = +∞
x →0 x

1
n = +∞
x →0 x

1
lim n = −∞
x →0 x

1
n = +∞
x →0 x

lim

lim

>

>

lim

<

<

W −∞ ( A +∞ ( ?@‫ א‬4‫ א ( א‬+ (,‫ א‬4‫↖ = א ( א‬
−∞ # P +∞ # Z ‫ " (א‬

−∞ # P +∞ # ( #$ "

( Y8 ‫ " א‬#$ [ \ " D.

( Y8 ‫ א‬.#$ " D.
W #‫ א‬4‫↖ = א ( א‬

1 − cos x
1
=
x →0
x²
2
lim

tan x
=1
x →0 x
lim

sin x
=1
x →0 x
lim

x u (x ) W! ‫ א‬4‫↖ = א ( א‬
lim
x →x0

lim u (x )

u (x )

x →x0

l≥0

l
+∞

+∞

−∞ # P +∞ # P ‫ ^ א‬x 0 # P L, ‫ ^ א‬x 0 # > ^/<- " ‫] א‬Z.
6

WB 1C ‫↖א = א‬

u (x ) ≤ f (x ) ≤ V (x )

lim u (x ) = l
 ⇒ lim f (x ) = l
 x →x 0
x →x 0

lim V (x ) = l

x →x 0


f (x ) − l ≤ V (x )

⇒ lim f (x ) = l
(
)
lim V x = 0  x →x 0

x →x 0


 ⇒ lim f (x ) = −∞
lim V (x ) = −∞ x →x 0

x →x 0


 ⇒ lim f (x ) = +∞
lim u (x ) = +∞
x →x 0

x →x 0

u (x ) ≤ V (x )

u (x ) ≤ f (x )

−∞ # P +∞ # P ‫ ^ א‬x 0 # P L, ‫ ^ א‬x 0 # > ^/<- " ‫] א‬Z.
W = ‫ א‬D 0 ‫↖א‬
WE ‫א‬+ ! 0; =
lim f (x )

l

l

l

−∞

+∞

+∞

lim g (x )

l'

−∞

+∞

−∞

+∞

−∞

lim [g (x ) + f (x )]

l + l'

−∞

+∞

−∞

+∞ ` _

x →x 0
x →x 0
x →x 0

WE ‫א‬+ F‫ = (א‬
lim f (x )

l

lim g (x )

l'

−∞

+∞

−∞

lim [g (x ) × f (x )]

l × l'

+∞

−∞

−∞

x →x 0
x →x 0
x →x 0

l<0

l>0

−∞

−∞

+∞

0

+∞

−∞

+∞

+∞

±∞

+∞

+∞

−∞

+∞ ` _

WE ‫א‬+ G H =
lim f (x )

x →x 0

l

l

lim g (x ) l' ≠ 0 ±∞

x →x 0

g (x )
x →x 0 f (x )
lim

l
l'

0

l<0

0−

0+

l>0

0−

0+

−∞
0−

+∞

0+

0−

0+

+∞ −∞ −∞ +∞ +∞ −∞ −∞ +∞

0

±∞

0

±∞

` _ ` _





W I).

−∞ # P +∞ # P ‫ ^ א‬x 0 # P L, ‫ ^ א‬x 0 # > ^/<- " ‫] א‬Z.
7

4 ‫( א‬05 K2 4 1J‫א‬
W 97 K 4 1J‫↖א‬
W: 1

x 0 a 4 f ⇔ lim f (x ) = f (x 0 )
x →x 0

W ‫ א‬D 4 1J‫ – א‬E0 ‫ א‬D 4 1J‫א‬
x 0 a L, ‫ ^ א‬4 f ⇔ lim f (x ) = f (x 0 ) •
x → x0
>

x 0 a ‫ ^ א‬4 f ⇔ lim f (x ) = f (x 0 )

•

x → x0
<

x 0 a 4 f ⇔ x 0 a ‫ ^ א‬L, ‫ ^ א‬4 f

W4 ; D 4 1J‫↖א‬
]a,b[ 3 d‫! = א‬4 +8 a 4 f c 8 ‫א‬S& ]a,b[ b 3 2 ^ 4 ‫ (א‬f O ]a,b[ b ‫ א‬3 d‫ ^ א‬4 f c 8 ‫א‬S&[a,b ] e V 3 2 ^ 4 ‫ (א‬f O b a ‫ ^ א‬4 a a L, ‫ ^ א‬4
W #‫ א‬4‫ א ( א‬D 0 ‫↖א‬

D/ /$ (# k I 3 2 ^ L 4 L ‫ (א‬g f =
I 3 d‫ ^ א‬4 kf f × g f + g 3‫ א‬# ‫• א‬
f

1

g

g

I 3 d‫ ^ א‬L 4 L ‫א‬# ‫ א‬OTU I ^ # - 5 g c 8 ‫א‬S&

•

^ 4 ( #$ ‫ (א‬+8 •

" ! - ,2 ^ 4 Z ‫ (א‬+8

WLM

•

» + ^ 4 x x ‫א‬# ‫• א‬
» ^ O 4 x cos x x sin x O ‫א‬# ‫• א‬

−

{π2 + k π / k ∈ } " ! - ,2 ^ 4 x tan x ‫א‬# ‫א‬

•

WE ‫א‬+ B
4 1‫↖א‬
f (I ) ⊂ J WM f J 3 2 ^ 4 g I 3 2 ^ 4 ‫ (א‬f c 8 ‫א‬S&
I 3 d‫ ^ א‬4 g ο f WOTU

W ‫(א‬8 4 ; * N↖
0g D. 4 ‫א‬# 0g >

•

3 2 D. 4 ‫א‬# 3 2 >

•

I 3 2 ^ 0g < - 4 ‫ (א‬f = W N H J )
f (I ) 3 d‫< א‬h iA D ‫ א‬3 #;‫א‬
8

f (I ) 3 d‫א‬
I ^ 0g 4g - f
[ f (b ); f (a )]

I ^ 0g # ‫ א‬- f
[ f (a ); f (b )]



 lim− f (x ) ; f (a )
 x →b



 f (b ); lim+ f (x )
x →a





 f (a ) ; lim− f (x )


x →b


 lim+ f (x ) ; f (b )
 x →a




 lim− f (x ) ; lim+ f (x )
 x →b

x →a


 lim f (x ) ; f (a )
 x →+∞




 lim+ f (x ) ; lim− f (x )
 x →a

x →b


 f (a ) ; lim f (x )
x →+∞





 lim f (x ) ; lim+ f (x )
 x →+∞

x →a
 ( )

 f a ; lim f (x )
x →−∞




 lim− f (x ) ; lim f (x )
x →−∞
 x →a



 lim f (x ) ; lim f (x )
 x →+∞

x →−∞



 lim+ f (x ) ; lim f (x )
 x →a
x →+∞



 lim f (x ) ; f (a )
 x →−∞



 lim f (x ) ; lim− f (x )
x →a
 x →−∞



 lim f (x ) ; lim f (x )
 x →−∞

x →+∞

I 3 d‫א‬
[a, b ]
[a, b[
]a, b ]
]a, b[
[a, +∞[
]a, +∞[
]−∞,a ]
]−∞,a[


W 9 ‫ א‬O 7 ‫ א‬6P ↖

f (b ) f (a ) = (# ‫ א‬L 47 β D/ /$ (# + j TU [a,b ] 3 2 ^ 4 f c 8 ‫א‬S&
f (α) = β WM f [a,b ] 3 d‫ = א‬α D/ /$ (# +g ‫ ^ א‬#


f (a ) × f (b ) < 0 [a,b ] 3 2 ^ 4 f c 8 ‫א‬S&

W

[a,b ] 3 d‫ א‬k& D, α *$ +g ‫ ^ א‬+</- f (x ) = 0 ( ‫ א‬OTU



f (a ) × f (b ) < 0 [a,b ] 3 2 ^ 0g < - 4 ‫ (א‬f c 8 ‫א‬S&



[a,b ] 3 d‫ א‬k& D, α ‫א‬# $ *$ +</- f (x ) = 0 ( ‫ א‬OTU
WQM ‫ א & ! א‬7 R↖
f (a ) × f (b ) < 0 WM f [a,b ] 3 2 ^ 0g < - 4 ‫ (א‬f =
[a, b ] 3 d‫ א‬a f (x ) = 0 ( , # $ ‫ א‬+ ‫ א‬α =
a + b 
a + b 

 < 0 WO 8 ‫א‬S&
 < 0 WO 8 ‫א‬S&
f (b )× f 
f (a )× f 

 2 
 2 
b −a
a +b
b −a
a +b

j @ Khl ‫א א‬Z.
< α < b WOTU
j @ Khl ‫א א‬Z. a < α <
WOTU
2
2
2
2
a + b 
 a +b 
3 4n 
;b  3 d‫ ^ א‬/ !0 ‫] א‬Z. ( & = m
3 4n a;
 3 d‫ ^ א‬/ !0 ‫] א‬Z. ( & = m
 2


2 
α (# 6(P Khl- ^
α (# 6(P Khl- ^

" U F ?! j @ α (# Khl- ^ 3 4 ‫ א‬o OP k& / !0 ‫] א‬Z. ( & = m ‫א ( א‬Z . W I).
9

4 ‫( א‬05 K2

S 7 J‫א‬


W+( K S 7 J‫ א‬8 T↖

" lim

x →x 0

f (x ) − f (x 0 )
x − x0

W " ‫ א‬c 8 ‫א‬S& x 0 (# ‫ א‬a 6 / %* g f ‫ (א‬O& 3 /

f ' (x 0 ) W ! j ! x 0 a f ‫א‬# e p ‫( א‬# ‫^ א‬, - " ‫] א‬Z.

W ‫א‬+ D % # 0#‫ & א‬W ‫ א (א א‬J ‫א‬+ D % # U 0#‫ א‬4+ ↖
x 0 a 6 / %* g ‫ (א‬f =
y = f ' (x 0 )(x − x 0 ) + f (x 0 ) WD. x 0 C 4UP r ‫ א‬0/ ‫ א‬a f ‫א‬# ‫ ^ א‬n q , ‫ ( א‬
u (x ) = f ' (x 0 )(x − x 0 ) + f (x 0 ) WD ,8 ^ U! ‫ א‬u ‫א‬# ‫א‬
x 0 ‫ א‬u f ‫א‬# t !/- D. x 0 C 4UP r ‫ א‬0/ ‫ א‬a f ‫א‬# ‫ ^ א‬n @ , ‫ א‬s ‫א א‬# ‫^ א‬, -

W E0 ‫ א‬D S 7 J‫ א‬8 T JE0 ‫ א‬D S 7 J‫ א‬8 T↖
" lim

x → x0
>

f (x ) − f (x 0 )
" ‫ א‬c 8 ‫א‬S& x 0 a L, ‫ ^ א‬6 / %* g f
x − x0

‫ (א‬O& 3 /

f 'd (x 0 ) W ! j ! x 0 a L, ‫ ^ א‬f ‫א‬# e p ‫( א‬# ‫^ א‬, - " ‫] א‬Z.
" lim

x → x0
<

f (x ) − f (x 0 )
x − x0

W " ‫ א‬c 8 ‫א‬S& x 0 a ‫ ^ א‬6 / %* g f ‫ (א‬O& 3 /

f 'g (x 0 ) W ! j ! x 0 a ‫ ^ א‬f ‫א‬# e p ‫( א‬# ‫^ א‬, - " ‫] א‬Z.
f 'g (x 0 ) = f 'd (x 0 ) x 0 a ‫ ^ א‬L, ‫ ^ א‬6 / %* g f c 8 ‫א‬S& x 0 a 6 / %* g f ‫ (א‬O -

W4 1J‫ א‬S 7 J‫↖א‬
x 0 a 4 O - f OTU x 0 (# a 6 / %* g f ‫ (א‬c 8 ‫א‬S&

W + J‫ א‬4‫ < א ( א‬8 7 X 4 ( ↖
f (x )

f ′ (x )

(k ∈ )

k

0

(r ∈ * − {1})

x
1
x
xr

1
−1
x²
rx r −1

x

sin x
cos x

1
2 x
cos x
− sin x

tan x

1 + tan2 x =

10

1
cos2 x

W ?@‫א א‬+ 7 X JE ‫א‬+ B
7 X - 7 X#‫ א‬4‫ א ( א‬D 0 ‫↖א‬
(k ∈ )

(ku )′ = k (u )′

u − v )′ = u ′ − v ′

(u

(

(un )′ = nu ′.u n −1

+ v )′ = u ′ + v ′

uv )′ = u ′v + uv ′

(

()

1 ′ −v ′
=
v
v²

u ′ u ′v − uv ′
=
v
v²
u′
( u )′ =
2 u

()

(u οv )′

= u ′οv  × v ′

W ‫א‬+ ‫>א‬Y1 S 7 J‫↖א‬
I 3 2 ^ 6 / %* g ‫ (א‬f =
I

3 d‫ ^ א‬# ‫ א‬- f ⇔ ∀x ∈ I

f ′ (x ) ≥ 0

I

3 d‫ ^ א‬4g - f

⇔ ∀x ∈ I

f ' (x ) ≤ 0

I 3 d‫ ) ^ א‬f

⇔ ∀x ∈ I

f ' (x ) = 0

WQ ( Z‫ א‬/ [ ‫ א‬S 7 J‫↖א‬
W+</ (C f ) ^ n , D@# C‫ א‬+ l ‫א‬
a . j ‫ א‬j A (x 0 ; f (x 0 )) 0/ ‫ א‬a @ J
A (x 0 ; f (x 0 )) 0/ ‫ א‬a /UP @ J
A (x 0 ; f (x 0 )) 0/ ‫ א‬a L, ‫ ^ א‬q J 14
a . j ‫ א‬j
A (x 0 ; f (x 0 )) 0/ ‫ א‬a L, ‫ ^ א‬D/UP q J 14
L, ‫ ^ א‬v( , q J 14

+ @ ‫ א‬w j A (x 0 ; f (x 0 )) 0/ ‫ א‬a
L, ‫ ^ א‬v( , q J 14

^ ‫ א‬w j A (x 0 ; f (x 0 )) 0/ ‫ א‬a
A (x 0 ; f (x 0 )) 0/ ‫ א‬a ‫ ^ א‬q J 14
a . j ‫ א‬j
A (x 0 ; f (x 0 )) 0/ ‫ א‬a ‫ ^ א‬D/UP q J 14
a ‫ ^ א‬v( , q J 14

^ ‫ א‬w j A (x 0 ; f (x 0 )) 0/ ‫א‬
‫ ^ א‬v( , q J 14

+ @ ‫ א‬w j A (x 0 ; f (x 0 )) 0/ ‫ א‬a

[ @‫א‬
a 6 / %* g f
x0

" ‫א‬
lim

x →x 0

f (x ) − f (x 0 )
= a
x − x0
(a ≠ 0)

lim

x →x 0

6 / %* g f
x 0 Lm ^
g K? f
6 / %*
x 0 Lm ^
6 / %* g f
x 0 ^
g K? f
6 / %*
x 0 ^
11

f (x ) − f (x 0 )
=0
x − x0

f (x ) − f (x 0 )
= a
x − x0
x →x 0
(a ≠ 0)

lim +

f (x ) − f (x 0 )
=0
x − x0
x →x 0

lim +

f (x ) − f (x 0 )
= −∞
x − x0
x →x 0

lim +

f (x ) − f (x 0 )
= +∞
x − x0
x →x 0

lim +

f (x ) − f (x 0 )
= a
x →x 0
x − x0
(a ≠ 0)

lim −

f (x ) − f (x 0 )
=0
x →x 0
x − x0

lim −

f (x ) − f (x 0 )
= −∞
x →x 0
x − x0

lim −

f (x ) − f (x 0 )
= +∞
x →x 0
x − x0

lim −

/- 0 ‫ –
\ א‬/- 0 ‫ א‬5
4 ‫( א‬05 K2

] 9 J‫ א‬97
W/- 0 ‫ א‬5↖

(C f ) ^ n , +) x 7 x = a j ( vZ ‫ א‬o / ‫ א‬O
WO ‫ א‬O h!p ‫ א‬e/y ‫א‬S&
∀x ∈ D f

∀x ∈ D f

(2a − x ) ∈ Df

•

f (2a − x ) = f (x )

•

W/- 0 ‫↖
\ א‬
(C f ) ^ n , +) x 8! I (a,b ) 0/ ‫ א‬O WO ‫ א‬O h!p ‫ א‬e/y ‫א‬S&
∀x ∈ D f

∀x ∈ D f

(2a − x ) ∈ D f

•

f (2a − x ) + f (x ) = 2b

•

W] 9 J‫ א‬97 - ^(% ‫ – א‬7 ‫↖א‬
# O 8 ‫א‬S& 3 2 ^ ‫ !א‬/ ‫ ^ (א‬n O
3 d‫א א‬Z. ^ j- @ J H z cy
∀x ∈ I


f ′′ (x ) ≤ 0 WO 8 ‫א‬S&

I 3 d‫ !א ^ א‬/ O (C f ) ^ n ‫א‬WOTU
# O 8 ‫א‬S& 3 2 ^ #7 ‫ ^ (א‬n O
3 d‫א א‬Z. ^ j- @ J H z 6 U
∀x ∈ I

f ′′ (x ) ≥ 0 WO 8 ‫א‬S&

I 3 d‫ ^ א‬#7 O (C f ) ^ n ‫ א‬WOTU
.# r ‫ ^ א‬n ‫ = א‬0/ D. ‫ ^ (א‬n 9 0 ‫ א‬0/
^ n ‫א א‬Z. ! /- KV
%{‫ א‬K V- H x 0 a # - f ′′ c 8 ‫א‬S&
x 0 C 4UP 9 0 ‫ א‬0/ +</ (C f ) ^ n ‫ א‬OTU
%{‫ א‬K V- O ( x 0 a # - f ′ c 8 ‫א‬S&
x 0 C 4UP 9 0 ‫ א‬0/ +</ (C f ) ^ n ‫ א‬OTU
12

lim f (x ) = ∞

lim f (x ) = a

lim [ f (x ) − (ax + b )] = 0

4 ‫( א‬05 K2

M = . ‫א & ! א‬

x →∞

f (x )
= a
(a ≠0)
x →∞ x
lim

lim [ f (x ) − ax ] = b

x →∞



W +</ (C f )



W +</ (C f )
* /

/UP /
Wj (


y =a

∞ ‫ א‬u

lim f (x ) = ∞

x →∞

x →∞

Wj (
y = ax + b
∞ ‫ א‬u


x →a

f (x )
=∞
x →∞ x
lim

f (x )
=0
x →∞ x
lim

lim [ f (x ) − ax ] = ∞

x →∞

W +</ (C f )
,| % !U
o / ‫ א‬j. }‫א‬
j ( vZ ‫א‬
y = ax
∞ ‫ א‬u


13





W +</ (C f )

W +</ (C f )

,| % !U

,| % !U

j. }‫א‬

j. }‫א‬

t -‫ א א‬7
∞ ‫ א‬u

+ > U ‫ א‬7
∞ ‫ א‬u



W +</ (C f )
( , /
Wj (
x

=a

4 ‫( א‬05 K2



‫א (א א‬

I 3 2 ^ 0g < - 4 ‫ (א‬f c 8 ‫א‬S&

W N H ↖

I 3 d‫ א‬w f (I ) 3 d‫ = א‬U! ‫ (א‬+</- f OTU



f −1 W ! C !
 f −1 (y ) = x

f (x ) = y





⇔



x
∈
I
y ∈ f (I )





∀x ∈ I
( f −1ο f )(x ) = x
∀y ∈ f (I )

WLM
•
•

( f ο f −1 )(y ) = y

•

W ‫ א (א א‬Y N ( (_ ↖
I 3 2 ^ 0g < - 4 ‫ (א‬f =
I 3 d‫!א = א‬4 y f (I ) 3 d‫!א = א‬4 x =
f −1 (x ) = y ⇔ f (y ) = x WD ‫~ א‬U @5
f (I ) = x !4 + f −1 (x ) V >  x 5# y # #n
W ‫ א (א א‬4 1‫↖א‬
I 3 2 ^ 0g < - 4 ‫ (א‬f c 8 ‫א‬S&
f (I ) 3 d‫ ^ א‬4 f −1 ‫א א‬# ‫ א‬OTU
W ‫ א (א א‬S 7 ‫↖א‬
I 3 2 ^ 0g < - 4 ‫ (א‬f =
y0 = f (x 0 ) f (I ) 3 d‫!א = א‬4 x 0 =
f ' (x 0 ) ≠ 0 x 0 a 6 / %* g f c 8 ‫א‬S&
y 0 a 6 / %* g f −1 ‫א א‬# ‫ א‬OTU
'

( f −1 ) (y 0 ) =

1

f ' (x 0 )

W #

I 3 2 ^ 0g < - 4 ‫ (א‬f =
I 3 d‫ ^ א‬# - 5 f ′ / p ‫ (א " א‬I 3 d‫ א‬+ 6 / %* g f c 8 ‫א‬S&
f (I ) 3 d‫ ^ א‬6 / %* g f −1 ‫א א‬# ‫ א‬OTU
∀x ∈ f (I )

'

( f −1 ) (x ) = f '  f −11 (x )


14



W #

W ‫ א (א א‬8 1 ↖
I 3 2 ^ 0g < - 4 ‫ (א‬f =
f ‫א‬# ‫ א‬KV- ^n G C f −1 ‫א א‬# ‫א‬
W ‫ (א א‬Q #‫ א‬/ 0 ‫↖א‬
I 3 2 ^ 0g < - 4 ‫ (א‬f =
o€ J # o a f −1 f L ‫א‬# O < ‫ א‬O* E, ‫א‬
o , 3 ‫ א‬14 , < O*) ,

W 6 I). ↖

(

)

C f −1 ^ n ‫א‬

(

A ' (b, a ) ∈ C f −1

)



/UP / +</
y = a W j (
( , / +</
x = b W j (
1
a

b
a

( )

C f ^ n ‫א‬




y = x + W j ( * / +</
W g* ‫ = א‬g*0 ‫ א ( א‬# #y o
x = ay + b
Eq J 14 PF @ J +</
/UP
Eq J 14 PF @ J +</
( ,






15

A (a,b ) ∈ (C f )
( , / +</
x = a W j (
/UP / +</
y = b W j (
* / +</
y = ax + b W j (
Eq J 14 PF @ J +</
( ,
Eq J 14 PF @ J +</
/UP

(n ∈ *)n 1 ‫א א@? א‬+
4 ‫( א‬05 K2



?@‫ ` א‬7 ‫א‬

W: 1 N H↖
n <-! ‫ = א‬Z;‫^ (א א‬, - ‫ (א‬+</- + ^ U! ‫ א‬x x n W ‫א‬# ‫א‬


n

: + → +
n

x x

∀ (x ; y ) ∈ 2+

n



W ! C !

n

x = y ⇔ x = yn

W N H J )
x = 2x
•
x 3 t ‫ א‬Z;‫^ א‬, 3 x W(# ‫א‬
•



( )

∀ (x ; y ) ∈ 2+
n

∀ (m; n ) ∈ *

2

∀ (x ; y ) ∈ 2+

x × n y = n x ×y

n

(n x )m = n x m
x
x
=n
ny
y
=

∀n ∈ *

xn = x

(n x )n = x

n

nmx

W N H↖

(y ≠ 0)

n×m x

n

x =ny ⇔x =y

n

x >ny ⇔x >y

W 6 I).
3

x −y
3
x ² + 3 x 3 y + 3 y²

x −3y =

x− y =

x −y
x+ y

W: ‫ א‬0;↖
WD. f ‫א‬# ‫ א‬1 ! - ,2

WD ,8 U! f ‫א‬# ‫א‬

D f = [0; +∞[

f (x ) = n x

D f = {x ∈ / x ∈ Du u (x ) ≥ 0}

f (x ) = n u (x )

W = ‫↖א‬
lim
x →x0

lim u (x )

n u (x )

x →x0

l≥0
+∞

n

l
+∞

−∞ # P +∞ # P ‫ ^ א‬x 0 # P L, ‫ ^ א‬x 0 # > ^/<- " ‫] א‬Z.
16



W4 1J‫↖א‬
+ ^ 4 x n x ‫א‬# ‫א‬


I 3 2 ^ U! ‫ (א‬u =
I 3 d‫ ^ א‬4 x n u (x ) ‫א‬# ‫ א‬OTU I 3 2 ^ 4 < ‫ (א‬u c 8 ‫א‬S&

WS 7 J‫↖א‬
I 3 2 ^ U! ‫ (א‬u =

]0;+∞[ 3 d‫ ^ א‬6 / %* g x n x ‫א‬# ‫א‬

I 3 2 ^ 6 / %* g 0g < ‫ (א‬u c 8 ‫א‬S&
I 3 d‫ ^ א‬6 / %* g x n u (x ) ‫א‬# ‫ א‬OTU
∀x ∈ I

(

n

u (x )

)′ =

u ′ (x )
n n [u (x )]n −1

W #



(n x )′ =

∀x ∈ ]0; +∞[

W #

1
n

n x

(a ∈ ) x ∈
D X (# n

n −1

x n = a W + #‫ א‬/)↖

v(!U (# n

S = {−n a ; n a }
S = {0}

S = {n a }
S = {0}

S =∅

S = {−n a }

a >0
a=0
a <0

W 9T B Q7 7) +( ?@‫ ` א‬7 ‫↖א‬
q ∈ * p ∈ * WM $ # K? Z ‫(א‬# r =

p
=
q

p

∀x ∈ ]0, +∞[

q

r

x = xq = xp

W I).
1
xn

(r ∈ *)

nx =
∀x ∈ ]0; +∞[
WD ,8 U! x D/ /$ KV f (# ‫ (א‬1 ! - ,2

f (x ) = [u (x )]r

•
•

Df = {x ∈ / x ∈ Du u (x ) > 0} WD.


1 ′





(n u (x ) )′ = (u (x ))n 

=

1
1
× u ' (x ) × [u (x )]n −1 •
n

* = r ′ r = !4 + *+ = y x = !4 +
x r × x r ' = x r +r '
(x × y ) = x × y
r


 x

r

r 

r

 = x r −r ′
 x r ′ 

r'

•

(x r ) = x r×r '

•

 x  = x r
y
 y 

 r



•

17

1
x

r'

•

r

= x −r '

•
•



4 ‫( א‬05 K2

+( ‫ א‬#‫א‬



W ( Z‫ א‬#‫ – א‬8 ,‫ א‬#‫↖א‬
@# .

$

un +1 = q × un

un +1 = un + r

q @ ‫ א‬. q
un = u p × q n −p



q @ ‫ א‬. r
un = u p + (n − p )r

(p ≤ n )

1 ! ‫ א‬# ‫א‬

(p ≤ n )

q n −p +1 − 1

u p + ... + un = u p × 
 n −p +1 − 1 ( #$ : ,2
 q − 1  u + ... + u = u × q

p
n
p
 q − 1 

(q

≠ 1)

c b a
( #$ )*)

2b = a + c

b ² = a × c



:* Y #‫ א‬#‫ * – א‬#‫ א‬#‫↖א‬
(# (un )n ∈I =
M (# < (un )n ∈I ⇔ ∀n ∈ I

un ≤ M

•

m (# V4 (un )n ∈I ⇔ ∀n ∈ I

un ≥ m

•

( #7 (un )n ∈I ⇔ V4 < (un )n ∈I

•

W +( 8 1 ↖
(# (un )n ∈I =
4g - (un )n ∈I ⇔ ∀n ∈ I

un +1 ≤ un

•

# ‫ א‬- (un )n ∈I ⇔ ∀n ∈ I

un +1 ≥ un

•

) (un )n ∈I ⇔ ∀n ∈ I

un +1

•

18

= un

W = ↖
W α ∈ * Wa ) (n α ) #‫ = א‬

α<0

α>0

lim n α = 0

lim n α = +∞

n →+∞

n →+∞

Wq ∈ Wa ) (q n ) ( Z‫ א‬#‫ = א‬
q ≤ −1

(q n ) ‫א‬
" C G

−1 < q < 1

lim q n = 0
n →+∞

q =1

q >1

lim q n = 1 lim q n = +∞
n →+∞

n →+∞

W^ 7 ‫ א‬b + ↖
/ D. < # ‫ א‬- +8

•

/ D. V4 4g - +8

•


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 ) ! ‫↖ א‬
WD ,8 U! ‫( א‬un ) ‫ א‬Y

u0 = a

un +1 = f (un )

I = ‫!א‬4 a f (I ) ⊂ I M f I 3 2 ^ 4 ‫ (א‬f M $

f (x ) = x W ( , +$ l " " OTU / (un ) c 8 ‫א‬S&


19

4 ‫( א‬05 K2

Nc‫ א‬4‫א ( א‬



W4 ; D ‫ (א‬Nc‫ א‬4‫↖א ( א‬
W: 1

I 3 2 ^ U! (# ‫ (א‬f =
I 3 d‫ ^ א‬f ‫א‬# >P ‫ (א‬D. F OP 3 /
WO ‫ א‬O h!p ‫ א‬e/y ‫א‬S&
I 3 d‫ ^ א‬6 / %* g F

∀x ∈ I

•

F ' (x ) = f (x ) •

W N H

3 d‫א א‬Z. ^ >P ‫ (א‬+</- 3 2 ^ 4 ‫ (א‬+8
I 3 2 ^ U! (# ‫ (א‬f =
WOTU I 3 d‫ ^ א‬f ‫א‬# >P ‫ (א‬F c 8 ‫א‬S&
WD W I ^ U! f ‫א‬# > ‫ א‬3‫ א‬# ‫ א‬H z
x F (x ) + k

(k ∈ )

I 3 2 ^ >P ‫ (א‬+</- (# ‫ (א‬f =
» = ‫!א‬4 y0 I = ‫!א‬4 x 0 =
I 3 d‫ ^ א‬f ‫א‬# F # $ >P ‫ (א‬# -

F (x 0 ) = y0 WD #< ‫! א‬p ‫ א‬e/y
WQ7 7) +( ‫א‬+ F ‫ @(א‬-E ‫א‬+ ! 0d W Nc‫ א‬4‫↖א ( א‬
W N H

/ /$ ‫(א‬# k I 3 2 ^ L U! L (# L ‫ (א‬g f =
WOTU D ‫ ^ א א‬I 3 d‫ ^ א‬g f L ‫א‬# L >P L ‫(א‬G F c 8 ‫א‬S&
I 3 d‫ ^ א‬f + g ‫א‬# >P ‫ (א‬F + G

•

I 3 d‫ ^ א‬kf ‫א‬# >P ‫ (א‬kF

•

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 ( (% S 7 J‫ א‬e N 4 0 ‫↖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 ∈ )

4 ‫( א‬05 K2 Q ‫ ^ א‬,‫א‬


W 9T D ‫א‬+ / 1↖
I 3 d‫ ^ א‬f ‫א‬# >P ‫ (א‬F I 3 2 ^ 4 ‫ (א‬f = W: 1

I 3 d‫! = = א‬4 b a



WD/ / ‫( א‬# ‫ א‬.b k& a = f ‫א‬# ‫ א‬+ b
∫ f (x ) dx = [F (x )]ba = F (b ) − F (a )
a

W N H↖
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 ∈ ) ∫ kf (x ) dx = k ∫ f (x ) dx
a
a
a
a
a
W4 T.
b
c
b
(x ) dx =
(x ) dx +
f
f
∫a
∫a
∫c f (x ) dx
b

a

WB 1C ‫ א‬/ ‫↖א‬
∀x ∈ [a,b ]

f (x ) ≤ g (x ) WO 8 ‫א‬S&
b
b
∫ f (x ) dx ≤ ∫ g (x ) dx WOTU
a
a

∀x ∈ [a,b ]

f (x ) ≥ 0 WO 8 ‫א‬S&

b

∫ f (x ) dx ≥ 0 WOTU
a

W 9 #‫ א‬0 7 ‫↖א‬
[a,b ] 3 2 ^ 4 ‫ (א‬f =


b
1
f (x ) dx WD/ / ‫( א‬# ‫ א‬D. 3 d‫א ^ א‬# 0@ ‫ א‬, / ‫א‬
b − a ∫a

WF‫ \א‬c 8 #‫↖א‬
I 3 d‫ ^ א‬L 4 v ′ u ′ L ‫א‬# ‫ א‬M f I 3 2 ^ 6 / %* L g L ‫ (א‬v u =
I 3 d‫! = = א‬4 b a
b
b
∫ u (x )v ′ (x )dx = [u (x ) v (x )]ba − ∫ u ′ (x )v (x )dx
a
a

\ ) ) ↖



(o, i, j ) # o k& ‚ ‫ = א‬



j i L "| ‫ א‬o 0/ (#ƒ‫ א‬+ 0 ‫ א‬$ D. u.A $ ‫ א‬#$






1.u . A = i × j
22

[a,b ] 3 2 ^ L 4 L ‫ (א‬g f =

[a,b ] 3 2 ^ 4 ‫ (א‬f =

7 C g C f L n ‫ א‬L 4ƒ‫ א א‬$

+ „> U ‫ א‬7 C f ^ n ‫ א‬L 4ƒ‫ א א‬$

W ,. ( = Z ‫ א‬L, / ‫ א‬+ > U ‫א‬

W ,. ( = Z ‫ א‬L, / ‫ א‬

WD. x = b x = a

x = b x = a

 b


(x ) − g (x ) dx 
f
.u.A
 ∫a




 b

∫ f (x ) dx .u.A

WD.

 a



WD.

W N H J )
WD. o@! ‫ א‬a D| < ‫ א א‬$

€$*

 b

∫ f (x ) dx .u.A

< f

 a

[a,b ] 3 d‫ ^ א‬



 b



< @ f

 a



[a,b ] 3 d‫ ^ א‬

∫ −f (x ) dx .u.A


Dn A - o@





< f •
 c

b



 a

c



[a, c ] 3 d‫ ^ א‬

∫ f (x ) dx + ∫ −f (x ) dx .u.A


< @ f •
[c,b ] 3 d‫ ^ א‬

 b

∫ ( f (x ) − g (x )) dx .u.A
 a

(C g ) 6 U # (C f )





[a,b ] 3 d‫ ^ א‬

(C g ) 6 U (C f ) •

 c

b
 ∫ ( f (x ) − g (x )) dx + ∫ (g (x ) − f (x ))dx .u.A
c
 a


[a, c ] 3 d‫ ^ א‬

(C f ) 6 U (C g ) •



[c,b ] 3 d‫ ^ א‬



:O ) ^ )↖

7 3 $ (C f ) ^ n ‫ א‬O‫ א‬# # ‫ א‬o d‫ א‬o|$
[a;b ] 3 2 a 8 ( + > U ‫א‬
 b

V =  ∫ π ( f (x ))²dx  u.v
 a


W .

| ‫ א‬#$ W uv

23

4 ‫( א‬05 K2 0 g ‫ א‬4‫א ( א‬


> ‫ א‬0 g ‫↖א (א א‬
W: 1
]0; +∞[ 3 d‫ ^ א‬x

1
x

‫א‬# > ‫א א‬# ‫ א‬D. vK< ‫ א‬o ? ‫(א א‬
ln W ! C ! 1 a # - r ‫ א‬
W N H ‫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 X ‫(א‬# n O 8 ‫א‬S&
W: ‫ א‬0;

WD. f ‫א‬# ‫ א‬1 ! - ,2
D f = {x ∈ / 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

W4 1J‫א‬

]0; +∞[ 3 d‫ ^ א‬4 x

ln x ‫א‬# ‫א‬


I 3 2 ^ U! ‫ (א‬u =

I 3 d‫ ^ א‬4 x

ln [u (x )] ‫א‬# ‫ א‬OTU I 3 2 ^ 4 0g < u c 8 ‫א‬S&
24

WS 7 J‫א‬
I 3 2 ^ U! ‫ (א‬u =

I 3 2 ^ 6 / %* g 0g < ‫ (א‬u c 8 ‫א‬S&
I 3 d‫ ^ א‬6 / %* g x
∀x ∈ I

'
)]

ln [u (x )] ‫א‬# ‫ א‬WOTU

(ln [u (x ) =

u ' (x )
u (x )



W #

]0; +∞[ ^ 6 / %* g x

: #



1
∀x ∈ ]0; +∞[ (ln x )′ =
x

W ln * '

ln x

-

WQ #‫ א‬/ 0 ‫א‬


0 1 +∞

x

ln x ‫א‬# ‫א‬

+

a ∈ » *+ − {1} Wa ) a U h O g ‫↖א (א א‬

loga W ! C ! r ‫א 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
W N H ‫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 7 X#‫א‬
∀x ∈ ]0, +∞[

( loga x ) ' =
25

1
x ln a



4 ‫( א‬05 K2 c‫ א‬4‫א ( א‬


> ‫ א‬0 g ‫↖א (א א‬
W: 1

K< ‫ א‬, ? ‫א א‬# ‫א א‬# ‫ א‬D. K< ‫א א @ א‬# ‫א‬
exp W ! C !
exp (x ) = e x = x + HB
W N H ‫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: ‫ א‬0;

WD. f ‫א‬# ‫ א‬1 ! - ,2


e ln x = x

∀x ∈ ]0, +∞[

r
(e x ) = erx

(r ∈ )

WD ,8 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

W4 1J‫א‬

^ 4 x

e

x

‫א‬# ‫א‬


I 3 2 ^ U! ‫ (א‬u =

I 3 d‫ ^ א‬4 x

e

u (x )

‫א‬# ‫ א‬OTU I 3 d‫ ^ א‬4 u c 8 ‫א‬S&
26

WS 7 J‫א‬

(ex )′ = ex W # ^ 6 / %* g x

∀x ∈

e

x

‫א‬# ‫א‬

I 3 2 ^ U! ‫ (א‬u =

I 3 d‫ ^ א‬6 / %* g x

e

u (x )

‫א‬# ‫ א‬WOTU I 3 d‫ ^ א‬6 / %* g u c 8 ‫א‬S&
'

(eu x )
( )

∀x ∈ I

u (x )
= u ′ (x ) ×e


W #

W ln ‫ (א‬Q #‫ א‬/ 0 ‫א‬



a ∈ ∗+ − {1} Wa ) a U h c‫↖א (א א‬

expa W ! C ! a q @… @ ‫א א‬# ‫^ א‬, - loga ‫א‬# ‫א א‬# ‫א‬

W: 1


expa (x ) = a x = x + HB
W N H ‫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 )
W 1 & =

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 7 X#‫א‬

4 ‫( א‬05 K2

(7 ‫ א‬+‫ (א‬c‫א‬
= {z = a + ib / (a;b ) ∈ ² i ² = −1} WD. #/ ‫א( א‬# ‫ א‬,2

Wi(7 +( P@‫ א‬8 ‫↖א‬
(a;b ) ∈ ² WM $ #/ ‫(א‬# z = a + ib =
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/ /$ (# . z OTU Im (z ) = 0 WO 8 ‫א‬S& • Wj N H j )
U!> ˆ ‫(א‬# ^, z OTU Im (z ) ≠ 0 Re (z ) = 0 WO 8 ‫א‬S& •

WE (7 +( i 1↖
L #/ = (# z ′ z =
z = z ′ ⇔ Re (z ) = Re (z ′) Im (z ) = Im (z ′)

Wi(7 +( Q #‫ א‬/ 0 ‫↖א‬


(o,e1,e2 ) o€ J # o k& v#/ ‫ = א ‚ א‬


(a;b ) ∈ ² WM $ #/ ‫(א‬# z = a + ib =
M (a, b ) 0/ z v#/ ‫( א‬# ‫ ! ‰ א‬
M (z ) Wt z (# ‫^ > א‬, - M 0/ ‫ א‬M 0/ ‫ א‬e ^, z (# ‫• א‬


z = Aff (OM ) POM (z ) Wt OM "| ‫ א‬e Z8 ^, z (# ‫• א‬


Wi(7 +( bk‫↖ א‬
(a;b ) ∈ ² WM $ #/ ‫(א‬# z = a + ib =

z = a − ib Wv#/ ‫( א‬# ‫ א‬. z (# ‫ א‬eU‫ !א‬


D/ / ‫ א‬n, < O*) , M ′ (z ) M (z )



D/ /$ (# z ⇔ z = z

•

9!> D ˆ (# z ⇔ z = −z
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 •
1

1

  = •
z ' z '

•

z  z
(z ' ≠ 0)   =
z '

z'

•

Wi(7 +( ↖
(a;b ) ∈ ² WM $ #/ ‫(א‬# z = a + ib =
z = zz = a ² + b ² Wt ‫ א‬D/ / ‫( א‬# ‫ א‬. z v#/ ‫( א‬# ‫ א‬
28

z ×z′ = z × z′

zn = z n

z = z

−z = z

1
1
=
z′
z′

z
z
=
z′
z′

(n ∈ * )
(z ' ≠ 0)

W ( >g i(7 +( c‫ א‬8 ‫ א‬Q #‫ א‬/ X ‫↖א‬
M j- > # K? #/ ‫(א‬# z =

(e
1, OM ) W " ‫ @ א א א‬g #$P θ . z v#/ ‫( א‬# ‫ א‬#,


arg z W ! j !
arg z = θ [2π ] Wt


W N H J )

# K? #/ ‫(א‬# z =

# K? a D/ /$ (# E E ‫א א‬
a <0
a >0

arg z = θ [2π ] r = z HB





W . z v#/ ‫( א‬# DE E ‫ א‬+ p ‫• א‬



a = [−a, π ]

a = [a, 0 ]

π

ai = −a, − 
2 


π

ai = a, + 
2 




z = r (cos θ + i sin θ ) = [r , θ ]

z = reiθ WD. z v#/ ‫( א‬# @ ‫• א א‬


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 
rei θ = r nein θ •
1
1

1
1 −iθ '
=  ; −θ '
=
e
•
[r '; θ ']  r '

r'
r ' eiθ '

( )

re

iθ

r 'e

iθ '

=

[r ; θ ]
=
[r '; θ ']

r i (θ−θ ')
e
•
r'

•

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

•

arg z ≡ − arg z [2π ] •
− arg z ≡ (π + arg z )[2π ] •

•

•
arg

r

 ; θ − θ ' •
 r '


‫
د‬z ⇔ arg z = k π •

( k ∈ )

arg z n ≡ n arg z [2π ] •
1
arg ≡ − arg z [2π ] •
z

•

‫
د ف‬z ⇔ arg z =

π
+ kπ •
2

z
≡ (arg z − arg z ') [2π ] •
z'

∀k ∈

[r, θ + 2k π ] = [r, θ ]


W> A Y N↖
∀θ ∈

(
(

W k‫ א‬Y N↖

)
)

1 iθ
e + e −iθ ∀n ∈
2
1 iθ
(cos θ + i sin n θ )n = cos (n θ ) + i sin (n θ )
sin θ =
e − e−i θ
2i
cos θ =

W (a ∈ ) a ) z ∈ z ² = a + #‫ א‬/)↖
W ( ‫ א‬3 $ ,2

W ( ‫א‬

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

a>0
a=0
a <0

29

z ∈ z² = a

(a ≠ 0) 7 7) +‫ (א‬Ac b a Wa ) z ∈
W ( ‫ א‬3 $ ,2
−b − ∆ −b + ∆ 
S = 
;

 2a

2a
−b
S=
2a
−b − i −∆ −b + i −∆ 
S = 
;



2a
2a

{ }

az ² + bz + c = 0 W + #‫ א‬/)↖
W ( ‫א‬

∆>0
z ∈
∆=0

az 2 + bz + c = 0

(∆ = b2 − 4ac )

∆<0

W (7 ‫ א‬+‫ (א‬c‫ א‬% 9 ( 6 O 6 & ↖
#/ ‫ א‬g* ‫א‬

D@# C‫א " א‬

AB = z B − z A
z + zB
zI = A
2

 z − z A 
[2π ]
AB; AC ≡ arg  c
 z B − z A 
zC − z A
∈
zB − zA
z −z
z −z
z −z
z −z
D A × D C ∈ P D A × B C ∈
z B − z A z B − zC
z B − z A z D − zC

AB U ‫א‬

(

)

D@# C‫א " א‬

[A; B ] 0/ ‫ א‬14 I

(

)




AB
; AC ‫ א א‬q g

, / ‰/ C B A
‫א‬# ‰/ D C B A
#/ ‫ א‬g* ‫א‬

AM = r
r " % A . 8! r ‫א ! א‬# ‫ א‬k& D, - M

•
•

z − zA = r
(r > 0)

AM = BM
[AB ] ‰@‫ א‬k& D, - M

•
•

z − zA = z − zB

A a ‫ א א‬o g M E ABC
A a Lg ‫ א‬v M E ABC
A a ‫ א א‬o g Lg ‫ א‬v M E ABC
:*A ‫ א‬v M E ABC

zC − z A
π

= r ; ± 

zB − zA
2 
zC − z A
= [1; θ ]
zB − zA
zC − z A
π

= 1; ± 

zB − zA
2 

zC − z A
π

= 1; ± 

zB − zA
3 

W + J‫ א‬. % ‫( < א‬7 . l↖
W . v#/ ‫ א‬j Ex

+ n ‫א‬





u "| ‫ א‬e b M $ z ′ = z + b

u "| ‫א א‬S t $‫א‬X{‫א‬

Ω 0/ ‫ א‬e ω M $ z ′ − ω = k (z − ω )

k j < Ω ] 8! vZ ‫ א‬h D8 n ‫א‬

Ω 0/ ‫ א‬e ω M $ z ′ − ω = ei θ (z − ω )

θ j ‫א‬X Ω ] 8! vZ ‫ א‬r O‫ א‬# ‫א‬

30

y (x ) = αeax −

(α ∈ )

W A ‫ ( א‬, ‫ א‬+ ‫א‬
y (x ) = αer1x + βer2x
(α, β ) ∈ ² WM $
y (x ) = (αx + β )erx
(α, β ) ∈ ² WM $

W A ‫א ( א‬

b
a

y ' = ay + b
(a ≠ 0)

W +</- , ‫א ( א‬
L / /$ L $
r2 r1 L Š
r ‫א‬# $ / /$ *$

W , ‫ ( " א‬

∆>0
∆=0

r1 = p − iq


r ² + ar + b = 0

(∆ = a ² − 4b )

WL/U‫א‬Q L #/ L $
y (x ) = (α cos qx + β sin qx )e px
(α, β ) ∈ ² WM $

W A ‫א ( א‬

∆<0

r2 = p + iq

31

y ''+ ay '+ by = 0

4 ‫( א‬05 K2 " & ‫ א‬J+ #‫א‬

W A ‫ ( א‬, ‫ א‬+ ‫א‬

4 ‫( א‬05 K2

M m& ‫ ( א‬Z‫א‬



(o, i , j , k ) !% < o€ J # o k& † B ‫א א ‡‹ = א‬Z. 6 @ a
Q= #‫ א‬F‫ א@(א‬J = OI JQ0 ‫ א‬F‫ א@(א‬W4 % ‫ א‬Y ‫↖א‬




ϑ3 = L "| v (a ',b ', c ') u (a,b, c ) =




u .v = aa '+ bb '+ cc '

•


u = a ² + b ² + c ²

•


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

u ∧ v = j b b ' =
i −
j +
k
c
c
'
c
c
'
b
b
'

k c c'

•

W k #‫↖א‬
WD. B A L 0/ L U ‫א‬
AB = (x B − x A ) ² + (yB − yA ) ² + (z B − z A ) ²

WD. ax + by + cz + d = 0 j ( (P ) ‚ = M 0/ U ‫א‬
d (M , (Ρ)) =

AM ∧ u
d (A, (∆)) =

u

ax M + byM + cz M + d
a ² + b² + c²



WD. ∆ (A, u ) o / = M 0/ U ‫ א‬
W` + ↖


(P ) ‚ ‫ ^ א‬,€ "| n (a,b, c ) ⇔ (Ρ) : ax + by + cz + d = 0




(ABC ) ‚ ‫ ^ א‬,€ "| AB ∧ AC OTU , / K? ‰/ C B A c 8 ‫א‬S&
WD ‫~ א‬U @5 (ABC ) ‚ ‫ ( א‬# #y = m




M ∈ (ABC ) ⇔ AM .(AB ∧ AC ) = 0

W k + ↖


WD. R " % Ω (a,b, c ) . 8! U (


(x − a ) ² + (y − b ) ² + (z − c ) ² = R²


32

@5 .# #y = m [AB ] . 0gP #$P (S ) U (







M ∈ (S ) ⇔ AM . BM = 0 WD ‫~ א‬U

AB
" % [AB ] 14 Ω . 8! (S ) ‫ א‬W €$*
2

(Ρ) : ax + by + cz + d = 0 ` S (Ω, R) k nR 71↖
(Ρ) ‚ ‫ ^ א‬Ω 8!, v( , ‫‰ א‬/ ‫ א‬H =
d = ΩH = d (Ω; (Ρ)) WHB



(S ) ‫ א‬H0/ (P ) ‚ ‫ א‬





(C ) ! ‫ (א‬eU
H W . 8!

(S ) q J (P ) ‚ ‫א‬

(S ) ‫( א‬P ) ‚ ‫א‬

H 0/ ‫ א‬a

O h / 5

r = R2 − d 2 W " %





W (∆) O 7 S (Ω, R) k nR 71↖
(∆) o / ‫ ^ א‬Ω 8!, v( , ‫‰ א‬/ ‫ א‬H =
d = ΩH = d (Ω; (∆)) WHB



(S ) ‫ א‬H0/ (∆) o / ‫א‬

(S ) q J (∆) o / ‫א‬

(S ) ‫ )∆( א‬o / ‫א‬

L Š L 0/ a

H 0/ ‫ א‬a

O h / 5

33

4 ‫( א‬05 K2 +‫א (א‬


W 0; Q M ↖
W: 1

CardE W ! j ! E ,d‫( >! א‬# . E " ,2 D
Card∅ = 0

W N H )

W N H
O " O ,2 B A
Card (A ∪ B ) = CardA + CardB − Card (A ∩ B )

W 0; O0 ↖
W: 1

E " ,2 = ‫ †א‬A =
A W ! C ! r ‫ א‬,d‫ א‬D. E ,|, < A o,
A = {x ∈ E / x ∉ A} M $
W I).
A∩A = ∅
A∪A = E
cardA = cardE − cardA

•
•
•

W+‫ (א‬Q c‫ א‬A( #‫↖א‬
(p ∈ *)

‫ א\ א‬p "| t 0 - !} Y

Š 8 n1 F o 3 ‫\ א‬5‫ א‬O 8 ‫א‬S&
Š 8 n2 F o D E ‫\ א‬5‫ א‬O 8
.........................................
Š 8 n p F o p \5‫ א‬O 8
n1 × n2 × n3 × ... × n p

W †‫א‬#;‫ א‬. , ‫( א  א‬# OTU
W ‫ א‬8 j (8 1C ‫ א‬- ‫ א‬8 1C ‫↖א‬
W ‫ א‬8 1C ‫א‬

(p ≤ n ) » * = = !4 p n =
n p W . !4 n L = !4 p 3 ‫ < !א‬-Q ‫( א‬#
34

W ‫ א‬8 j (8 1C ‫א‬

(p ≤ n ) » * = = !4 p n =
W . !4 n L = !4 p 3 ‫ !א‬- O # < -Q ‫( א‬#
Anp = n × (n − 1) × (n − 2) × ... × (n − p + 1)



+ ‫ = א א‬p
W N H )

!4 n 3 #<- Z8 ^, - !4 n L = !4 n 3 ‫ !א‬- O # < -!- +8
n ! = n × (n − 1) × (n − 2) × ... × 2 × 1 W .(#
W & [ ‫↖א‬
n .!> (# " ,2 E =
(p ≤ n ) p ]!> (# E = A † +8
!4 n L = !4 p 3 l- ^,
C np =

Anp
W . l ‫] א‬Z. (#
p!

C np Anp n ! W+‫ (א‬c‫↖א‬
n ∈ ∗

n ! = n × (n − 1) × (n − 2) × ... × 2 × 1
0! = 1
n!
n!
Anp =
C np =
p ! (n − p ) !
(n − p ) !
n −1
0
1
C nn = 1
Cn = n
Cn = 1
Cn = n
C np = C nn −p
C np−1 + C np = C np+1

WB% ‫ א! א‬A < 8↖
(p ≤ n ) !4 n L = !4 p tn
WD ‫ א‬3 #;‫ א‬a  ‫ ‡‹ א‬
t -Q ‫א‬

W . , ‫< א‬n ‫( א‬#

Wtn ‫ א‬:

o" K?

C np

D Œ

o"

np
Anp

3*$T H

o"

35

3*$& O # H

4 ‫( א‬05 K2 J 0 )J‫א‬


% 9 ↖
W]

WD , $5‫ א‬i 04 ‫א‬

| = !E8P +</- !} +8

‫ א‬p !}

‫ א‬p !| , ‫ א{ א‬,2 D.

{‫ א‬O 8 Ω

Ω {‫ א‬O 8 = ‫ †א‬A

A #$

‫א‬# $ ‫!א‬4 =,B #$ +8

D ‫א‬# ‫ א‬#$

#$‫ א‬OŒ a B A O )# ‫ א‬e/y ‫א‬S&

A ∩ B # ‫ א‬e/y

,. P B P A e/y ‫א‬S&

A ∪ B # ‫ א‬e/y

(A ∩ A = ∅‫و‬A ∪ A = Ω) A # ‫ א‬.

A #n ( B ‫ א‬# ‫א‬

A∩B = ∅

L,| K? O )#$ B A

Wo() 4 0 )‫ א‬J o() ‫ א‬7 ‫↖א‬
‫ א‬p !} & O 8 Ω = W: 1


pi W . {ωi } # ‫ א‬3 , $‫ א‬OP 3 / pi j , g a {ωi } D ‫א‬# ‫  א‬#$ 3 , $‫! א‬/ # •
P ({ωi }) = pi Wt
# ‫א א‬Z. O - r ‫א א‬# 5‫ א‬5 , $5‫ א‬: ,2 . #$ 3 , $‫• א‬
W . A # ‫ א‬3 , $‫ א‬OTU Ω = )#$ A = {ω1; ω2 ; ω3 ;...; ωn } O 8 ‫א‬S& vP
p (A) = p (ω1 ) + p (ω2 ) + p (ω3 ) + ... + p (ωn )
‫ א‬p !} & O 8 Ω =

W N H


p (Ω) = 1 p (∅) = 0 •
Ω = A #$ + 0 ≤ p (A) ≤ 1 •

WL)#$ ( y‫ א‬3 , $‫• א‬
Ω = B A L)#$ +
p (A ∪ B ) = p (A) + p (B ) − p (A ∩ B )
L,| K? B A O 8 ‫א‬S& p (A ∪ B ) = p (A) + p (B )
W( B ‫ א‬# ‫ א‬3 , $‫• א‬

p (A) = 1 − p (A)

W Ω = A #$ +

W J 0 )J‫ א‬i 1 " k↖
Ω " & O 8 ‫ א‬p !} a 3 , $5‫א א‬# 5‫א א‬#$ ‫ א‬H z c 8 ‫א‬S&
cardA
p (A) =

W . Ω = A #$ +8 3 , $‫ א‬OTU
card Ω
36

W: 1


WE-() .7 ‫ א‬JQR X ‫ א‬4 0 )J‫↖א‬
p (A) ≠ 0 WM f ‫ א‬p ‫ א |! א‬G L0<-! L)#$ B A = W: 1
p (A ∩ B )
p (B ) = p (B A) =
W(# ‫ א‬. e/7 A # ‫ א‬OP , B #$ 3 , $‫א‬
p
A
(
)
A
p (A) × p (B ) ≠ 0 WM f ‫ א‬p ‫ א |! א‬G L0<-! B A L)#$ +
p (A ∩ B ) = p (A) × p (B A) = p (B ) × p (A B ) W #

W

W: 1

‫ א‬p ‫ א |! א‬G L0<-! B A L)#$ +
O*/ O )#$ B A ⇔ p (A ∩ B ) = p (A) × p (B )



Ω 3 Ž } Ω2 Ω1 ‫ א‬p !} & O 8 Ω =

W N H

(Ω1 ∩ Ω2 = ∅ Ω1 ∪ Ω2 = Ω)

( )



( )

p (A) = p (Ω1 ) × p A Ω + p (Ω2 ) × p A Ω W Ω = A #$ +
1
2

W* #‫ א א‬HJ‫↖ א‬
p j , $‫ א א‬p !} a )#$ A =
W . ‰<B ! k , A # ‫ א‬e/y 3 , $‫ א‬O U ! n !| ‫] א‬Z. # P ‫א‬S&
C nk (p )k (1 − p )n −k

(k ≤ n )

WQM‫ א‬X >Y 4 0 )‫ א‬j T↖
‫ א‬p !} & O 8 Ω ^ ‫ א‬p ‫א‬KV =
WL ‫ א‬L $! ‫ א‬H< X D ‫ א‬p ‫ א‬KV ‫ א‬3 , $‫ א‬O g # #n
X KV ‫ א‬.Z\l r ‫ א‬o / ‫ א‬,2 W X (Ω) = {x1; x 2; x 3 ;...; x n } # #y •

{1;2;...;n } ,d‫ = א‬i + p (X = x i ) 3 , $5‫ א‬t w •

WQM‫ א‬X >Y # ip‫ א‬9 ‫ א] א‬qJ‫ א‬J* Y#‫ א‬JQ" ‫ א‬/ c‫↖א‬
j g ‫ א‬p ‫א‬KV X =

xi

x1 x 2 x 3 ... x n
p (X = x i ) p1 p2 p3 ... pn

WD ‫ א‬3 #; 9!

E (X ) = x1 × p1 + x 2 × p2 + x 3 × p3 + ... + x n × pn W X KV , DA ! ‫ א‬+ ‫א‬
V (X ) = E (X ²) − [E (X )] ²

W X KV , ! V ‫א‬

σ (X ) = V (X )

W X KV , vX‫!א‬0 ‫ א‬9‫!א‬w5‫א‬

W: 1


WQ ‫(א‬,‫ א‬j 7 ‫↖א‬
! n !| ‫] א‬Z. # { ‫ א‬p !} a A #$ 3 , $‫ א‬p =
p n ] 0 @ ‫א‬#$ X - ^, A # ‫ " א‬U e/n r ‫( א !א א‬# | +8 ‰ ! vZ ‫ א‬X D ‫ א‬p ‫ א‬KV ‫א‬
∀k ∈ {0;1;2;...; n }
V (X ) = np (1 − p )

p (X = k ) = C nk × p k × (1 − p )n −k



E (X ) = n × p

#



37

4 ‫( א‬05 K2 (>
?1) Q #‫ ^ א‬,‫א‬


W #‫ * א‬M‫ א (א‬+ J‫ א‬O 7 ‫ 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
sin x = sin a ⇔ x = a + 2k π P x = (π - a ) + 2k π
tan x = tan a ⇔ x = a + k π
( k ∈ )
38

W! 0; / _ e N↖
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