ﻦﻳﺮﻤﺗ  9  ﻦﻳﺮﻤﺗ  8  ﻦﻳﺮﻤﺗ  7  ﻦﻳﺮﻤﺗ  6  ﻦﻳﺮﻤﺗ  5  ﻦﻳﺮﻤﺗ  4  ﻦﻳﺮﻤﺗ  3  ﻦﻳﺮﻤﺗ  2  ﻦﻳﺮﻤﺗ  1  ﻞﺤﻟﺍ ﻞﺤﻟﺍ ﻞﺤﻟﺍ ﻞﺤﻟﺍ ﻞﺤﻟﺍ ﻞﺤﻟﺍ ﻞﺤﻟﺍ ﻞﺤﻟﺍ ﻞﺤﻟﺍ ﻦﻳﺮﻤﺗ  18  ﻦﻳﺮﻤﺗ  17  ﻦﻳﺮﻤﺗ  16  ﻦﻳﺮﻤﺗ  15  ﻦﻳﺮﻤﺗ  14  ﻦﻳﺮﻤﺗ  13  ﻦﻳﺮﻤﺗ  12  ﻦﻳﺮﻤﺗ  11  ﻦﻳﺮﻤﺗ  10  ﻞﺤﻟﺍ ﻞﺤﻟﺍ ﻞﺤﻟﺍ ﻞﺤﻟﺍ ﻞ

ﺱﺮﻬﻔﻟﺍ 

ﻦﻳﺮﻤﺗ 

9  8 ﻦﻳﺮﻤﺗ  7 ﻦﻳﺮﻤﺗ  6 ﻦﻳﺮﻤﺗ  5 ﻦﻳﺮﻤﺗ  4 ﻦﻳﺮﻤﺗ  3 ﻦﻳﺮﻤﺗ  2 ﻦﻳﺮﻤﺗ  1 ﻦﻳﺮﻤﺗ 

ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ 

ﻦﻳﺮﻤﺗ 

18  17 ﻦﻳﺮﻤﺗ  16 ﻦﻳﺮﻤﺗ  15 ﻦﻳﺮﻤﺗ  14 ﻦﻳﺮﻤﺗ  13 ﻦﻳﺮﻤﺗ  12 ﻦﻳﺮﻤﺗ  11 ﻦﻳﺮﻤﺗ  10 ﻦﻳﺮﻤﺗ 

ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ  ﻞﺤﻟﺍ

 ﻦﻳﺮﻤﺗ 1 :

 ﻦﻳﺮﻤﺗ

2

:

 ﻦﻳﺮﻤﺗ 3 :

 ﻦﻳﺮﻤﺗ

4

:

 ﻦﻳﺮﻤﺗ 5 :

 ﻦﻳﺮﻤﺗ

6

:

 ﻦﻳﺮﻤﺗ 7 :

 ﻦﻳﺮﻤﺗ

8

:

ﻦﻜﻴﻟو (C ) ﺪﻣﺎﻌﺘﻣ ﻢﻠﻌﻣ ﻲﻓ ﺎهﺎﻨﺤﻨﻣ ( ,

o i j, ) ﻢﻈﻨﻤﻣ G G

1 - a - دﺪﺣ D ﺔﻟاﺪﻟا ﻒﻳﺮﻌﺗ ﺰﻴﺣ f

.

b - ﺔﻴﻟﺎﺘﻟا تﺎﻳﺎﻬﻨﻟا ﻦﻣ ﻼآ ﺐﺴﺣا :

lim ( )

x f x

و →+∞

و و

lim (

x f x)

) →−∞

lim0 (

x

+ f x

→

lim0 ( )

x

− f x

→

2 - a – نأ ﻦﻣ ﻖﻘﺤﺗ :

2

1 1 27

: ( ) ( ) ( )

2 2 27

x x

x D f x

x x x x

+ +

∀ ∈ − =

+ +

b - ﻢﻴﻘﺘﺴﻤﻟا نأ ﺞﺘﻨﺘﺳا ل ﻞﺋﺎﻣ برﺎﻘﻣ

(C ) راﻮﺠﺑ

( )

1

: 1 2 y x+

Δ =

+∞

( )

^{نأ ﻦﻴﺑ–C}

ل ﻞﺋﺎﻣ برﺎﻘﻣ ( C )

راﻮﺠﺑ

₂ : ¹

2 y x+ Δ = −

−∞

3 - a - نأ ﻦﻴﺑ ﻞﻜﻟ

x ﻦﻣ D .

3

2 2

'( ) 27

2 2

x x 7

x x

= − f +

b - نأ ﻦﻣ ﻖﻘﺤﺗ f

لﺎﺠﻤﻟا ﻰﻠﻋ ﺔﻳﺪﻳاﺰﺗ

[

ﻦﻴﻟﺎﺠﻤﻟا ﻦﻣ ﻞآ ﻰﻠﻋ ﺔﻴﺼﻗﺎﻨﺗو

]

و .

] ]

^{0,3 −∞,0}

[

^3,+∞

[

0 y0) 5, 2

c - ﺔﻟاﺪﻟا تاﺮﻴﻐﺗ لوﺪﺟ ﻂﻋا f

.

4 - a – ﻊﻃﺎﻘﺗ دﺪﺣ (C)

ﻞﻴﺻﺎﻓﻷا رﻮﺤﻣ ﻊﻣ .

b - أ ﻞﺒﻘﻧ ﺚﻴﺣ

و ﻰﻨﺤﻨﻤﻠﻟ ةﺪﻴﺣﻮﻟا فﺎﻄﻌﻧﻻا ﺔﻄﻘﻧ ﻲه

(C ) نأو f’(x) ﺔﺒﻟﺎﺳ

ﻟا ﻰﻠﻋ

[ [

ﻦﻴﻟﺎﺠﻤﻟا ﻦﻣ ﻞآ ﻰﻠﻋ ﺔﺒﺟﻮﻣو

]

و

0 2,9

y ≈ x₀ ≈ − A x( , ن لﺎﺠﻤ

0,0

]

x ,x0

]

^0,+∞

[

−∞

. ﺬﺧﺄﻧ 1 i = j =

G G

cm

ﺊﺸﻧأ C .

 ﻦﻳﺮﻤﺗ 9 :

 ﻦﻳﺮﻤﺗ

: 10

ﺔﻳدﺪﻌﻟا ﺔﻟاﺪﻟا ﺮﺒﺘﻌﻧ f

ﻲﻘﻴﻘﺤﻟا ﺮﻴﻐﺘﻤﻠﻟ x

ﻲﻟﺎﺘﻟﺎآ ﺔﻓﺮﻌﻤﻟا :

2 2

3

2 1

( ) x . 1

f x x

x

= − +

ﻦﻜﻴﻟو ﺔﻟاﺪﻠﻟ ﻞﺜﻤﻤﻟا ﻰﻨﺤﻨﻤﻟا

f ﺪﻣﺎﻌﺘﻣ ﻢﻠﻌﻣ ﻲﻓ

( , , )o i j ﻢﻈﻨﻤﻣ

( )

ζ

G G

)

1 - دﺪﺣ D ﺔﻟاﺪﻟا ﻒﻳﺮﻌﺗ ﺔﻋﻮﻤﺠﻣ f

نأ ﻦﻣ ﻖﻘﺤﺗو ، f

ﺔﻳدﺮﻓ .

2 - ﺐﺴﺣا و

ﺎﻤﻬﻴﻠﻋ ﻞﺼﺤﻤﻟا ﻦﻴﺘﺠﻴﺘﻨﻟا ﺎﻴﺳﺪﻨه لوأو .

lim (

x f x

00

lim ( )

x

x f x →+∞

→;

3 - ﻞﻜﻟ ﻪﻧأ ﻦﻴﺑ x

ﻦﻣ D ﺎﻨﻳﺪﻟ : تاﺮﻴﻐﺗ ﺞﺘﻨﺘﺳاو

f .

4 2

'( ) 3 .

f x . 1

x x

= +

4 -

( )

ﺸﻧأ .

ζ ﺊ

5 - ﻦﻜﻴﻟ g ﺔﻟاﺪﻟا رﻮﺼﻗ f

لﺎﺠﻤﻟا ﻰﻠﻋ .

]

^0,+∞

[

أ - نأ ﻦﻴﺑ g ﻦﻣ ﻞﺑﺎﻘﺗ لﺎﺠﻣ ﻮﺤﻧ

J ﻩﺪﻳﺪﺤﺗ ﻲﻐﺒﻨﻳ .

]

^0,+∞

[

g⁻

g 1) . J ﻰﻠﻋ قﺎﻘﺘﺷﻼﻟ ﺔﻠﺑﺎﻗ نأ ﻦﻴﺑ -ب ﺔﻟاﺪﻠﻟ ﺔﻴﺴﻜﻌﻟا ﺔﻟاﺪﻟا ﻲه

g (

^{− 1}

ج - ﺊﺸﻧأ

(

ﻤﻟا ﻰﻨﺤﻨﻤﻟا ﺔﻟاﺪﻠﻟ ﻞﺜﻤ

.

^{g−1 ζ'}

)

ﺔﻳدﺪﻌﻟا ﺔﻟاﺪﻟا ﺮﺒﺘﻌﻧ f

ﻰﻠﻋ ﺔﻓﺮﻌﻤﻟا ^{f x( ) 6= x23 −4x : ب}

[

^0,+∞

[

1 - ﺐﺴﺣا .

ﺔﺠﻴﺘﻨﻟا ﺎﻴﺳﺪﻨه لوأ .

0

0

lim ( )

x x

f x

→ x

;

2 - ل ﻲﺋﺎﻬﻧﻼﻟا عﺮﻔﻟا دﺪﺣ ﻰﻨﺤﻨﻣ

f .

( )

^ζ

3 - ﺐﺴﺣا f’(x) ﻞﻜﻟ x ﻦﻣ تاﺮﻴﻐﺗ لوﺪﺟ دﺪﺣ ﻢﺛ

f .

]

^0,+∞

[

4 - أ - ﻊﻃﺎﻘﺗ ﻲﺘﻄﻘﻧ دﺪﺣ

(

ﻞﻴﺻﺎﻓﻷا رﻮﺤﻣ ﻊﻣ .

^ζ

)

ب - ﺔﻟدﺎﻌﻣ دﺪﺣ

( ) (

سﺎﻤﻣ

لﻮﺼﻓﻷا تاذ ﺔﻄﻘﻨﻟا ﻲﻓ

^{27 ζ}

)

^Δ

8 ^{( , , )o i jG G ﻢ}ﻈﻨﻤﻣ ﺪﻣﺎﻌﺘﻣ ﻢﻠﻌﻣ ﻲﻓ

( )

^{ζ و}

( )

^Δ ﺊ^{ﺸﻧأ -ج} ) سﺎﻴﻘﻟا ةﺪﺣو :

1 cm .(

5 - g رﻮﺼﻗ f لﺎﺠﻤﻟا ﻰﻠﻋ ^{I = +∞}

[

^1,

[

نأ ﻦﻴﺑ g ﻦﻣ ﻞﺑﺎﻘﺗ I

ﻩﺪﻳﺪﺤﺗ ﻢﺘﻳ لﺎﺠﻣ ﻮﺤﻧ .

ﺣا

( )

^{.g−1 ' 0}

^{( )}

^ﺐﺴ

 ﻦﻳﺮﻤﺗ 11 :

 ﻦﻳﺮﻤﺗ

: 12

ﻦﻜﺘﻟ f ﻲﻘﻴﻘﺤﻟا ﺮﻴﻐﺘﻤﻠﻟ ﺔﻳدﺪﻌﻟا ﺔﻟاﺪﻟا x

ﻰﻠﻋ ﺔﻓﺮﻌﻤﻟا ﻲﻠﻳ ﺎﻤﺑ

:

IR

2 2

( ) ( 1 )

f x = +x −x

(

ﺔﻟاﺪﻟا ﻰﻨﺤﻨﻣ ﻮه f

ﺪﻣﺎﻌﺘﻣ ﻢﻠﻌﻣ ﻲﻓ

( , , )o i j ﻢﻈﻨﻤﻣ ζ

)

G G

lim ( )

x ﺔﻳﺎﻬﻨﻟا ﺐﺴﺣا -أ-1

f x

→−∞

lim ( ) 0

x f x

→+∞ = : نأ ﻦﻴﺑ -ب

2 - أ - ﻞﻜﻟ نأ ﻦﻴﺑ x

ﻦﻣ :

IR

2

2 ( ) '( ) 1 f x = − f x

+x

ب - نأ ﺖﺒﺛأ ﻞﻜﻟ

x ﻦﻣ ﺔﻟاﺪﻟا تاﺮﻴﻐﺗ لوﺪﺟ ﻊﺿ ﻢﺛ f

.

IR f '( ) 0x ≠

3 - نأ ﻦﻴﺑ ل ﻲﺋﺎﻬﻧﻼﻟا عﺮﻔﻟا ﺞﺘﻨﺘﺳا ﻢﺛ

(

راﻮﺟ

lim ^{( )}

n

)

f x

ζ ^→−∞ x = −∞

−∞

4 - أ - ﻢﻴﻘﺘﺴﻤﻠﻟ ﺔﻴﺗرﺎﻜﻳد ﺔﻟدﺎﻌﻣ ﺐﺘآا (T)

ﻰﻨﺤﻨﻤﻟا سﺎﻤﻣ لﻮﺼﻓﻷا تاذ ﺔﻄﻘﻨﻟا ﻲﻓ

0 .

( )

^ζ

ب - ﻢﻴﻘﺘﺴﻤﻟا ﺊﺸﻧأ (T)

ﻰﻨﺤﻨﻤﻟاو

(

) ةﺪﺣﻮﻟا 2cm (

ζ

)

5 - أ - نأ ﻦﻴﺑ f ﻦﻣ ﻞﺑﺎﻘﺗ لﺎﺠﻣ ﻮﺤﻧ

J ﻩﺪﻳﺪﺤﺗ ﻢﺘﻳ .

IR

( )

^{f−1 ' 1}

( )

^{ﺐﺴﺣا -ب}

ج - ﺐﺴﺣا f’(x) ﻞﻜﻟ x ﻦﻣ J .

ﺔﻳدﺪﻌﻟا ﺔﻟاﺪﻟا ﺮﺒﺘﻌﻧ f

ﻰﻠﻋ ﺔﻓﺮﻌﻤﻟا

[

ﻲﻠﻳ ﺎﻤﺑ :

^0,+∞

[

3 2

( ) 2 1

f x = − +x x +

ﻦﻜﻴﻟو ﺪﻣﺎﻌﺘﻣ ﻢﻠﻌﻣ ﻲﻓ ﻲﻧﺎﻴﺒﻤﻟا ﺎﻬﻠﻴﺜﻤﺗ

^{( , , )o i j ﻢﻈﻨﻤﻣ}

( )

^ζ

lim ( )

x ﺐﺴﺣا -أ-1

f x

→+∞

ب - ﻼﻟا عﺮﻔﻟا سردا ﻰﻨﺤﻨﻤﻠﻟ ﻲﺋﺎﻬﻧ

( )

^ζ

2 - أ - نأ ﻦﻴﺑ :

2 2

3

2 2

3

3 ( 1) 2

'( ) 3 ( 1)

x x

f x

x + +

= +

[ [

, (∀ ∈x 0,+∞ )

ب - ﺔﻟاﺪﻟا تاﺮﻴﻐﺗ لوﺪﺟ ﻂﻋا f

.

3 - نأ ﻦﻴﺑ f ﻦﻣ ﻞﺑﺎﻘﺗ

[

لﺎﺠﻣ ﻮﺤﻧ I

ﻩﺪﻳﺪﺤﺗ ﺐﺠﻳ

^0,+∞

[

4 - أ - ﺔﻟدﺎﻌﻤﻟا نأ ﻦﻴﺑ f(x) =0

ﻘﺗ اﺪﻴﺣو ﻼﺣ ﻞﺒ )

بﺎﺴﺣ نود .(

¹ 1 ﺚﻴﺤﺑ α

2≺α ≺ α

ب - ﺔﻄﻘﻧ دﺪﺣ ﻢﻴﻘﺘﺴﻤﻟاو

(

ﻪﺘﻟدﺎﻌﻣ يﺬﻟا y =x

.

^Δ

) ( )

^{ζ ﻊﻃﺎﻘﺗ}

ج - ﺸﻧأ ﺔﻟاﺪﻟا ﻰﻨﺤﻨﻣ

ﺔﻟاﺪﻠﻟ ﻲﺴﻜﻌﻟا ﻞﺑﺎﻘﺘﻟا f

.

^f−1

( )

^{ζ ' ﻢﺛ}

( )

^ζ ﺊ

 ﻦﻳﺮﻤﺗ 13 :

 ﻦﻳﺮﻤﺗ

: 14

ﺔﻳدﺪﻌﻟا ﺔﻟاﺪﻟا ﺮﺒﺘﻌﻧ f

ﻲﻘﻴﻘﺤﻟا ﺮﻴﻐﺘﻤﻠﻟ x

ﻰﻠﻋ ﺔﻓﺮﻌﻤﻟا ﻲﻠﻳ ﺎﻤﺑ

:

IR^*

] [ ] [

[ [

( ) 2 ( ,0 0,1 )

( ) 1 ( 1,

2

f x x x

x

f x x x

x

⎧ = − + ∈ −∞ ∪

⎪⎪⎨ +

⎪ = ∈ +∞

⎪⎩

1 - ﺔﻟاﺪﻟا نأ ﻦﻴﺑ f

ﺔﻄﻘﻨﻟا ﻲﻓ ﺔﻠﺼﺘﻣ x₀ =1

2 - أ - ﺔﻟاﺪﻟا نأ ﻦﻴﺑ f

ﺔﻄﻘﻨﻟا ﺪﻨﻋ قﺎﻘﺘﺷﻼﻟ ﺔﻠﺑﺎﻗ رﺎﺴﻴﻟا ﻰﻠﻋ

.

x₀ =1

ب - ﺔﻟاﺪﻟا نأ ﻦﻴﺑ f

ﺔﻄﻘﻨﻟا ﺪﻨﻋ قﺎﻘﺘﺷﻼﻟ ﺔﻠﺑﺎﻗ ﻦﻴﻤﻴﻟا ﻰﻠﻋ

) نأ ﻆﺣﻻ

(1+ −x 2 x =( x−1)² x₀ =1

3 - أ - نأ ﻦﻴﺑ ^{(∀ ∈ −∞x}

]

^,0

[ ] [

^{∪ 0,1 ) f '( ) 0x ≺}

ب - نأ ﻦﻴﺑ

1

'( ) 4 f x x

x

= −

] [

(∀ ∈ +∞x 1, )

( )

^-ج

ﺔﻟاﺪﻟا تاﺮﻴﻐﺗ لوﺪﺟ ﻂﻋا f

.

4 - ﻜﻴﻟ ζ ﻦ ﺔﻟاﺪﻠﻟ ﻞﺜﻤﻤﻟا ﻰﻨﺤﻨﻤﻟا f

ﺪﻣﺎﻌﺘﻣ ﻢﻠﻌﻣ ﻲﻓ ( , , )o i j ﻢﻈﻨﻤﻣ

أ - ﻰﻨﺤﻨﻤﻠﻟ ﺔﻴﺋﺎﻬﻧﻼﻟا عوﺮﻔﻟا سردا

(

. ζ

)

ب - ﻨﺤﻨﻤﻟا ﺊﺸﻧأ )

ﻰﻨﺤﻨﻤﻠﻟ نأ ﻞﺒﻘﻧ

(

ﺎﻬﻟﻮﺼﻓأ ةﺪﻴﺣو فﺎﻄﻌﻧا ﺔﻄﻘﻧ 3

(

^ζ

) ( )

^{ζ ﻰ}

ﺔﻳدﺪﻌﻟا ﺔﻟاﺪﻟا ﺮﺒﺘﻌﻧ f

ﻰﻠﻋ ﺔﻓﺮﻌﻤﻟا ﻲﻠﻳ ﺎﻤﺑ

:

IR

3 3

( ) 1 2 1 ; 1

( ) 1 ; 1

3

f x x x x

f x x x

x

⎧ = − + − ≤

⎪⎨ −

= ≥

⎪ +

⎩

ﻦﻜﻴﻟو ( C) ﺪﻣﺎﻌﺘﻣ ﻢﻠﻌﻣ ﻲﻓ ﺎهﺎﻨﺤﻨﻣ ( ,

o i j, ) ﻢﻈﻨﻤﻣ G G

) ﺐﺴﺣا -1 نأ ﻦﻴﺑو

:

lim ( )

x f x

→−∞ lim (

x f x

= −∞ →+∞

2 - ﺑﺎﻗ سردا قﺎﻘﺘﺷا ﺔﻴﻠ

f ﻲﻓ رﺎﺴﻴﻟا ﻰﻠﻋو ﻦﻴﻤﻴﻟا ﻰﻠﻋ 1

ﺎﻤﻬﻴﻠﻋ ﻞﺼﺤﻤﻟا ﻦﻴﺘﺠﻴﺘﻨﻠﻟ ﺎﻴﺳﺪﻨه ﻼﻳوﺄﺗ ﻂﻋا ﻢﺛ .

3 - أ - ﺔﻟاﺪﻟا نأ ﻦﻴﺑ f

لﺎﺠﻤﻟا ﻰﻠﻋ ﺎﻌﻄﻗ ﺔﻳﺪﻳاﺰﺗ

[

^1,+∞

[

نأ ﻦﻴﺑ ﻞﻜﻟ

x ﻦﻣ

'( )

1 (1 1

f x x

)

x x

= −

− + −

]

^−∞,1

[

ج - ﻟا تاﺮﻴﻐﺗ لوﺪﺟ ﻂﻋا ﺔﻟاﺪ

f .

4 - أ - ﻰﻨﺤﻨﻤﻠﻟ ﻦﻴﻴﺋﺎﻬﻧﻼﻟا ﻦﻴﻋﺮﻔﻟا سردا (C )

ب - ﻰﻨﺤﻨﻤﻟا ﻢﺳرا ( C )

) نأ ﻆﺣﻻ f(-3)=0

5 - ﻦﻜﺘﻟ g ﺔﻟاﺪﻟا رﻮﺼﻗ f

لﺎﺠﻤﻟا ﻰﻠﻋ

[

^1,+∞

[

أ - نأ ﻦﻴﺑ g ﻦﻣ ﻞﺑﺎﻘﺗ

[

لﺎﺠﻣ ﻮﺤﻧ I

ﻩﺪﻳﺪﺤﺗ ﻲﻐﺒﻨﻳ .

^1,+∞

[

( )

g⁻1 x ﺣ -ب ﻞﻜﻟ

x لﺎﺠﻤﻟا ﻦﻣ I

.

دﺪ

ج - نأ ﻦﻴﺑ ﺔﻟاﺪﻠﻟ ﺔﻴﻠﺻأ ﺔﻟاد ﻲه

لﺎﺠﻤﻟا ﻰﻠﻋ g⁻¹

3 2 2

3

2

3 ( 1)

x

x x x

→ − − +

 ﻦﻳﺮﻤﺗ 15 :

 ﻦﻳﺮﻤﺗ

16

:

ﺔﻳدﺪﻌﻟا ﺔﻟاﺪﻟا ﺮﺒﺘﻌﻧ f

لﺎﺠﻤﻟا ﻰﻠﻋ ﺔﻓﺮﻌﻤﻟا ﻲﻠﻳ ﺎﻤﺑ

:

]

^0,+∞

[

( ) 1

f x x x

= − + x

ﻦﻜﻴﻟو ( C ) ﺪﻣﺎﻌﺘﻣ ﻢﻠﻌﻣ ﻲﻓ ﺎهﺎﻨﺤﻨﻣ ( , , )o i j ﻢﻈﻨﻤﻣ

G G

( ) ﺐﺴﺣا -أ-1 و

li

lim

x f x

00

m ( )

x x

f x →+∞

→;

ب - ﻔﻟا دﺪﺣ ﻰﻨﺤﻨﻤﻠﻟ ﻦﻴﻴﺋﺎﻬﻧﻼﻟا ﻦﻴﻋﺮ (C )

2 1

'( ) ( 1)

2 )

x x

2 - نأ ﻦﻴﺑ :

x x

x x

⎛ + + ⎞

]

^0,+∞

[

ﻦﻣ x ﻞﻜﻟ f =⎜⎜⎝ ⎟⎟⎠ −

ب - ﺔﻟاﺪﻟا تاﺮﻴﻐﺗ لوﺪﺟ ﻂﻋا f

.

3 - أ - ﻰﻨﺤﻨﻤﻠﻟ ﻲﺒﺴﻨﻟا ﻊﺿﻮﻟا سردا (C)

ﻢﻴﻘﺘﺴﻤﻟاو ﺔﻟدﺎﻌﻤﻟا يذ

.

^y=x

( )

^Δ

ب - ﻰﻨﺤﻨﻤﻟا ﻢﺳرا ( C)

(

⁵)

(4) 2

f =

و

1 7

( )4 4

f =

4 - ﻦﻜﺘﻟ g ﺔﻟاﺪﻟا رﻮﺼﻗ f

لﺎﺠﻤﻟا ﻰﻠﻋ

[

^1,+∞

[

g ﺔﻴﺴﻜﻋﺔﻟاد ﻞﺒﻘﺗ g نأ ﻦﻴﺑ -أ ﺔﻟاﺪﻟا ﺔﻋﻮﻤﺠﻣ دﺪﺣو

g^{−1 −1}

ب - ﻠﻌﻤﻟا ﺲﻔﻧ ﻲﻓ ﻢﺳرا ﺔﻟاﺪﻠﻟ ﻞﺜﻤﻤﻟا ﻰﻨﺤﻨﻤﻟا ،

.

( , , )o i j ﻢ

1 G G g⁻

5 - ﺔﻳدﺪﻌﻟا ﺔﻴﻟﺎﺘﺘﻤﻟا ﺮﺒﺘﻌﻧ ( )

ﻲﻠﻳ ﺎﻤﺑ ﺔﻓﺮﻌﻤﻟا :

⁰ a_{n n IN∈}

1

2

n ( )n

a

a₊ f a

⎧ =

⎨ =

⎩

أ - نا ﻦﻴﺑ

(

:

∀ ∈n IN

)

an ;¹

n n IN

a _∈

( )a_{n n IN∈}

ب - ﺔﻴﻟﺎﺘﺘﻤﻟا نأ ﻦﻴﺑ ( )

ﺔﻴﺼﻗﺎﻨﺗ .

ج - ﺔﻴﻟﺎﺘﺘﻤﻟا نأ ﺞﺘﻨﺘﺳا ﺎﻬﺘﻳﺎﻬﻧ دﺪﺣ ﻢﺛ ﺔﺑرﺎﻘﺘﻣ

.

ﻦﻜﺘﻟ f ﻰﻠﻋ ﺔﻓﺮﻌﻤﻟا ﺔﻳدﺪﻌﻟا ﺔﻟاﺪﻟا IR

ﻲﻠﻳ ﺎﻤﺑ :

( ) 2 1 ; 1

( ) 2 ; 1

f x x x x

f x x x x x

⎧ = − − ≥

⎪⎨

= + −

⎪⎩ ≺

ﻦﻜﻴﻟو C) ( م ﻲﻓ ﺎهﺎﻨﺤﻨﻣ .

م . ( , ,o i j) م

) ﺐﺴﺣا -1 و

lim (

x f x

lim ( )

x f x →+∞

→−∞

2 - أ - لﺎﺼﺗا سردا f

ﻲﻓ 1 .

ب - قﺎﻘﺘﺷا ﺔﻴﻠﺑﺎﻗ سردا f

ﻦﻴﻤﻴﻟا ﻰﻠﻋ ﻲﻓ رﺎﺴﻴﻟا ﻰﻠﻋو

1 ﻢﺛ ، ﺎﻤﻬﻴﻠﻋ ﻞﺼﺤﻤﻟا ﻦﻴﺘﺠﻴﺘﻨﻠﻟ ﺎﻴﺳﺪﻨه ﻼﻳوﺄﺗ ﻂﻋا .

3 - أ - ﺐﺴﺣا f’(x) ﻞﻜﻟ x ﻦﻣ ^IR−

{ }

¹

ب - تاﺮﻴﻐﺗ لوﺪﺟ ﻂﻋا f

.

4 - أ - ﻰﻨﺤﻨﻤﻠﻟ ﻦﻴﻴﺋﺎﻬﻧﻼﻟا ﻦﻴﻋﺮﻔﻟا دﺪﺣ (C )

.

ب - ﻰﻨﺤﻨﻤﻟا ﻊﻃﺎﻘﺗ دﺪﺣ (C )

ﻞﻴﺻﺎﻓﻷا رﻮﺤﻣ ﻊﻣ .

ج - ﻰﻨﺤﻨﻤﻟا ﻢﺳرا (C )

5 - ﻦﻜﺘﻟ g ﺔﻟاﺪﻠﻟ ﺔﻴﻠﺻﻷا ﺔﻟاﺪﻟا f

لﺎﺠﻤﻟا ﻰﻠﻋ

[

ﻖﻘﺤﺗ ﻲﺘﻟاو ^2,+∞

[

(2) 2 g =3

أ - ﺐﺘآا g(x) ﺔﻟﻻﺪﺑ x

ب - ﺔﻟاﺪﻟا تاﺮﻴﻐﺗ لوﺪﺟ ﻂﻋا g

.

 ﻦﻳﺮﻤﺗ 17 :

 ﻦﻳﺮﻤﺗ

: 18

≥0)

( 1

1 - أ - ( (x+1)(x−1) و

x∈Df ⇔ x≠

]

1) x≤ − وأ (

و ^⇔

[

^{x≠1 x≥1}

]

^{, 1}

] ]

^1,

[

⇔ ∈ −∞ − ∪ +∞x

نذإ

]

^{, 1}

] ]

^1,

[

Df = −∞ − ∪ +∞

ب - (

lim ¹ 1

1

x

x+ = x−

→+∞

نذإ و

lim ( )

x f x

→+∞ = +∞

lim ( )

x f x

→−∞ = −∞

نأ ﺎﻤﺑ 1

lim 1 1

x

x

+ x

→

+ = +∞

− lim ( )1

x ₊ f x

→ = +∞

( 1) 0 f − =

نﺈﻓ

و

2 - أ - (

1 1

( ) ( 1) 1

lim lim 0

1 1

x x

f x f x

x x

− −

→− →−

− − +

= =

+ −

'( 1) fg − =

{

نذإ f ﻲﻓ رﺎﺴﻴﻟا ﻞﻋ قﺎﻘﺘﺷﻼﻟ ﺔﻠﺑﺎﻗ 1

- .

و 0

ب - x∈Df − −¹

}

⁽

2

2

1 ( 1)

'( ) ( 1)

1 1

2 1

x x

f x x

x x

x

−

+ −

= + +

− +

−

2

2

( 1) 1 ( 1)

1 ( 1) 1

1

x x x

x x x

x

− + −

= −

− +

− +

نذإ

ﻞﻜﻟ x

{ }

¹ ﻦﻣ Df − −

2

( 1)( 2)

'( ) 1

( 1)

1

x x

f x x

x x

+ −

= − +

−

ج - '( ) ةرﺎﺷإ ( f x

ةرﺎﺷإ ﻲه (x+1)(x−2)

f(2) 3 3=

 ﻦﻳﺮﻤﺗ

1

:

3 - أ - (

[ ]

1

lim ( ) ( 2) lim ( 1) ( 2)

1

x x

f x x x x x

→+∞ →+∞ x

− + = + + − +

−

2 1 2

( 1) ( 2)

lim 1

( 1) 1 ( 2)

1

x

x x x

x

x x x

x

→+∞

+ + − +

= − +

+ + +

−

2

3 5

lim 1

( 1) ( 2)( 1)

1

x

x

x x x x

x

→+∞

= +

− + + + −

−

3 5

lim 0

1 1 2

( ) (1 )( 1)

1

x

x

x x x

x x x

→+∞

+

= =

− + + + −

− 2

y x ﻪﺘﻟدﺎﻌﻣ يﺬﻟا (D) ﻢﻴﻘﺘﺴﻤﻟا نذإ

= +

ل برﺎﻘﻣ (C )

راﻮﺠﺑ و

−∞ +∞

ب ( ءﺎﺸﻧإ (C )

1 2 -

lim ( ) lim ( 4) 1

4

x f x x x

→−∞ →−∞ x

⎡ ⎤

= − ⎢⎣− + − ⎥⎦= −∞

2 -

4 4

4 4

( ) (4) 4 2 4

lim lim

4 4

x x

x x

f x f x x

x x

→ →

− = − + −

− −

≺ ≺

44

lim 1 2 4

x

x^→ x

⎛ ⎞

= ≺ ⎜⎝ − − ⎟⎠= −∞

f ﻲﻓ رﺎﺴﻴﻟا ﻰﻠﻋ قﺎﻘﺘﺷﻼﻟ ﺔﻠﺑﺎﻗ ﺮﻴﻏ 4

ﻞﺒﻘﻳ (C) ﺔﻄﻘﻨﻟا ﻲﻓ سﺎﻤﻣ ﻒﺼﻧ A(4,0)

ﺐﻴﺗارﻷا رﻮﺤﻣ يزاﻮﻳ .

3 - أ - ( ﻞﻜﻟ x ﻦﻣ

]

^{−∞, 4}

[

'( ) 1 2 1

f x 2 4−

= + −

x

1 4 1

1 4 4

x

x x

= − = − −

− −

ةرﺎﺷإ

ب - ( ةرﺎﺷإ f’(x) ﻲه 4− −x 1

4 1 3

4 1

4 1 4 1

x x

x

x x

− − −

− − = =

− + − +

ﺎﻨﻳﺪﻟ : 4− +x 1 0 x 4

∀ ≺

ةرﺎﺷإ نذإ f’(x)

ةرﺎﺷإ ﻲه 3

−x

3 '( ) x≺ ⇔ f x 0

0

3≺ ≺x 4⇔ f '( )x ≺

(3) 1 2 1 f = − + =

( ) 4 2 4

lim lim 1

x x

f ^{x x} -4

x x

→−∞ →−∞

⎛ − ⎞

= ⎜⎜⎝ + + x ⎟⎟⎠

2

4 4 1

lim 1 2 1

x^→−∞ x x x

⎛ ⎞

= ⎜⎜ + − − ⎟⎟=

⎝ ⎠

( )

lim ( ( ) ) lim 4 2 4

x f x x x x

→−∞ − = →−∞ − + − =

نذإ ﻞﺒﻘﻳ (C) ﻟا يذ ﻢﻴﻘﺘﺴﻤﻟا ﻪهﺎﺠﺗا ﺎﻴﻤﺠﻠﺷ ﺎﻋﺮﻓ

+∞

ﻪﺘﻟدﺎﻌﻤ y=x

4

x≺

-5

f x( ) 0= ⇔ − +x 4 2 4− =x 0

 ﻦﻳﺮﻤﺗ

2

:

2 4 x 4 x

⇔ − = − 4 4 x

⇔ = − x 0

⇔ =

0

نذإ (C) ﺤﻣ ﻊﻄﻘﻳ ﻢﻠﻌﻤﻟا ﻞﺻأ ﻲﻓ ﻞﻴﺻﺎﻓﻷا رﻮ

.

6 - (0)

f =

و '(0) ¹

f = 2

( )

^{: 1}

T y= 2x

7 - f( 5)− = −3

2 0 x ∈Df ⇔x +x

( 1)

⇔x x + 1 - (

0

]

^{, 1}

[ ]

^0,

[

x ∈⇔x −∞ − ∪ +∞

نذإ Df = −∞ − ∪

]

^{, 1}

[ ]

^0,+∞

[

2 - ( ﻢ ﻠﻌﻧ نأ lim 2

x x x

→+∞ + = +∞

و

lim ₂¹

x^→+∞=x x

+

نذإ lim ( )

x f x

→+∞ = −∞

نأ ﺎﻤﺑو + و

=

m₊ x²+ =x 0

li 0 x→ 2 ∞

0

lim 1

x^{→ +} x +x

نﺈﻓ

lim ( )

0 f x

x → + = +∞

3 - (

2 1 ( 1 )

2 x f x

⎡ ⎛ ⎞ ⎤

f ⎢⎣ ⎜⎝− ⎟⎠− ⎥⎦= − −

2 2

1 ( 1) 1

( 1) 1 x x

x x

= − + − −

+ − −

2 2

1 2 1 1 (

2 1 x x x f x)

x x x

= − + + − −

+ − − =

1 نذإ

2 2 ( )

f ⎡⎢⎣ x⎤⎥⎦=f x (∀x ∈D_f ) ⎛⎜⎝− ⎞⎟⎠−

Δ

ﻢﻴﻘﺘﺴﻤﻟا

( )

ﻞﺛﺎﻤﺗ رﻮﺤﻣ (C )

.

Df

ﻦﻣ x ﻦﻜﻴﻟ (-4

2 2 2

2 1 2 1

'( ) ( ) 2

x x

f x

x x

+

x

= − − +

+ +

x

2 2 2

1 1

(2 1)

( ) 2

x x x x x

⎡ ⎤

= − + ⎢⎣ + + + ⎥⎦

] [

2 2 2 ^(-5

1 1

( 0, )

( ) 2

x Df

x x x x

∀ ∈ +∞ +

+ +

'( ) 0

ﻞﺒﻘﺗ نذإ ﺷإ ﺲﻜﻋ ةرﺎﺷإ

ﻰﻠﻋ

]

^0,+∞

[

^{2x +1 ةرﺎ f x}

: 0

( 0 نأ ﺎﻤﺑ

∀ 〉x ) 2x +1

0 : نﺈﻓ ∀x 0 f '( )x ≺

+∞

0

x

-

f’(x)

+∞

−∞

f(x)

 ﻦﻳﺮﻤﺗ

3

:

6 -

3 2

( ) 1 1

lim lim 1 1

x x

f x

x x x x

→+∞ →+∞

⎛ ⎞

= ⎜⎜⎝ + − − ⎟⎟⎠= −

2 2

lim ( ( ) ) lim 1

x f x x x x x x

x x

→+∞ →+∞

⎛ ⎡ ⎤⎞

+ = ⎜⎝ + −⎣ + − ⎦⎟⎠

2

lim ( ) lim 2

x x

x x x x

x x x

→+∞ + − = →+∞

+ +

1 1

lim 1 2

1 1

x

x

→+∞ =

+ + نذإ lim ( ( ) ) ¹

2

x f x x

→+∞ + = −

ﻢﻴﻘﺘﺴﻤﻟا نﺈﻓ ﻲﻟﺎﺘﻟﺎﺑو (D)

ﻪﺘﻟدﺎﻌﻣ يﺬﻟا ل برﺎﻘﻣ

(c )

راﻮﺠﺑ ¹

y = − −x 2 +∞

0 7 - ( ) ( f x = x 0

2 2

( ) 0 1

f x x x

=

x x

⇔ = +

+

2 2

(x x) 1

⇔ + =

2 1 0

x x

⇔ + − = 1 4 5

Δ = + = نذإ

^{1 5}

x =− +2

) ﻞﺤﻟا 1

2 ﺐﻟﺎﺳ ﻪﻧﻷ لﻮﺒﻘﻣ ﺮﻴﻏ − − (

⁵

1 5

S ⎧⎪− +2 ⎫⎪

= ⎨ ⎬

⎪ ⎪

⎩ ⎭

1 5

2 , 0

⎛A =− + ⎞

⎜ ⎟

⎜ ⎟

⎝ ⎠

ﺔﻄﻘﻧ ﻲه ﻊﻃﺎﻘﺗ

(C ) ﻞﻴﺻﺎﻓﻷا رﻮﺤﻣو

* ﻰﻠﻋ ⁺ .

2 1 x ∈ ⇔D x + 0 -1

1,

x ⎤ 2 ⎡

⇔ ∈ −⎥⎦ +∞⎢⎣

1, D = −⎤⎥⎦ 2 +∞⎡⎢⎣

ﺎﻨﻳﺪﻟ

1 2

lim ( 1) 1

x 2

+ x

→−

+ = و

⁺

1 2

lim 2 1 0

x

+ x + =

→−

نذإ 1

2

lim ( )

x

+f x

→−

= +∞

2

1 1

lim ( ) lim

2 1

x x

f x x

x x

→+∞ →+∞

= + = +∞

+

2 1 - ( ) :

D y = −2 ل برﺎﻘﻣ

(C )

1 1

lim ( ) lim 0

2 1

x x

f x x

x x

→+∞ →+∞

= + =

+ نذإ (C ) ﻞﻴﺻﺎﻓﻷا رﻮﺤﻣ ﻩﺎﺠﺗا ﻲﻓ ﺎﻴﻤﺠﻠﺷ ﺎﻋﺮﻓ ﻞﺒﻘﻳ

3 - أ - ( ﻦﻜﻴﻟ x ﻦﻣ D .

2 1 ( 1) 1

2 1

'( ) (2 1)

x x

f x x

x + − +

= + +

2 1 1

(2 1) 2 1

x x

x x

+ − −

= + +

3 2 3

2

(2 1) (2 1)

x x x

x

−

= = +

+

ب - (

4 - أ - ( ﻦﻜﻴﻟ x ﻦﻣ D .

3 5

2 3 2

''( ) (2 1) (2 1) 2

f x x 2 x

= + − × + − ×

 ﻦﻳﺮﻤﺗ

4

:

5

(1 x)(2x 1)2

−

= − +

5

1 2

( ) ''( ) (1 )(2 1)

x 2 f x x x

−

∀ − = − +

ب - (

¹

x −2 x 1

⇔ ≺ ''

''( ) 0 1 0

f x ≥ ⇔ −x

نذإ ﻲﻓ ةرﺎﺷﻹا ﺮﻴﻐﺗو مﺪﻌﻨﺗ

x₀ =1 f

2 2 3

(1) 3 3

f = =

نﺈﻓ ﻪﻨﻣ

و ل فﺎﻄﻌﻧا ﺔﻄﻘﻧ

(C ) .

1,^{2 3}

A⎛ 3 ⎞

⎜ ⎟

⎜ ⎟

⎝ ⎠

6 - أ - ( g ﻰﻠﻋ ﺎﻌﻄﻗ ﺔﻴﺼﻗﺎﻨﺗو ﺔﻠﺼﺘﻣ ﺔﻟاد I

.

^{J =g I( )= +∞}

[

^1,

[

^و

نﺈﻓ ﻪﻨﻣو g

ﻦﻣ ﻞﺑﺎﻘﺗ I

ﻮﺤﻧ J .

ب - ( ﻦﻜﻴﻟ x ﻦﻣ I و y ﻦﻣ J .

( ) 1

2 1

y g x x y

x

= ⇔ + =

+

2 .

2 2 1

2 1

x x

y x

+ +

⇔ =

+

2 2 (1 2) 1 0

x x y y²

⇔ + − + − =

2 2

' y (y 1)

Δ = − ≥0

2 2 نذإ

1 1 1

x =y − −y y −

2 2 و

2 1 1

x =y − +y y −

ﻞﺤﻟا

2 1 2 1

y − −y y − x1

ﺐﻟﺎﺳ ﻪﻧﻷ لﻮﺒﻘﻣ ﺮﻴﻏ .

=

2 2 نذإ

2 1 1

x=x = y − + y y − :

نﺈﻓ ﻪﻨﻣو g⁻¹( )x =x²− +1 x x²−1

I - 1 - '( )

h x =3(1 2− x) x 0

∀

h x'( ) 0⇔ −1 2 x 0 1 x 2

⇔ ≺

0 1

x 4

⇔ ≺ ≺

2 1 - h⎛ ⎞4

⎜ ⎟⎝ ⎠ ﺔﻟاﺪﻠﻟ ﺔﻳﻮﺼﻗ ﺔﻤﻴﻗ h

.

نذإ

( )

¹

4

⎛ ⎞⎜ ⎟ (∀ ∈x ⁺) ⎝ ⎠

(∀ ∈x ⁺)

h x ≤h

نأ يأ

( )

⁰

h x ≤

II 1 -

0 0

( ) (0) 4 1

lim lim 4

x x

f x f x

x x x

+ +

→ →

− = ⎛⎜ − − ⎞⎟= −

⎝ ⎠ ∞

*+

ﺔﻟاﺪﻟا نذإ f

ﺮﻔﺼﻟا ﻲﻓ ﻦﻴﻤﻴﻟا ﻰﻠﻋ قﺎﻘﺘﺷﻼﻟ ﺔﻠﺑﺎﻗ ﺮﻴﻏ .

ﻞﺒﻘﻳ (C ) لﻮﺼﻓﻷا تاذ ﺔﻄﻘﻨﻟا ﻲﻓ يدﻮﻤﻋ سﺎﻤﻣ ﻒﺼﻧ 0

2 - أ - ( ﻞﻜﻟ x ﻦﻣ

'( ) 4 (4 1) 1 8

f x x x 2

x

= + − × − x

8 4 1 16

2

x x x

x

− − −

= x

4 3 4 1

12 16 1 4

2 2

x x x

x x x

x x

⎛ − − ⎞

⎜ ⎟

− − ⎝ ⎠

= =

نﺈﻓ ﻲﻟﺎﺗﺎﺑو : *+ ﻦﻣ x ﻞﻜﻟ

2 ( )

'( ) h x

f x

x

=

ب - (

 ﻦﻳﺮﻤﺗ

5

:

2

4 1 1

lim lim lim 0

2

x^→+∞ x =x^→+∞x x =x^→+∞ x =

نذإ lim ( )

x f x

→+∞ = −∞

ج - (

+∞

0 x

-

f’(x)

1/2

2

( ) 4 1 1

lim lim 4

2

x x

f x

x x x x x

→+∞ →+∞

⎛ ⎞

= ⎜⎝ − − + ⎟⎠= −∞

ﻞﺒﻘﻳ (C ) ﺐﻴﺗارﻷا رﻮﺤﻣ ﻩﺎﺠﺗا ﻲﻓ ﺎﻴﻤﺠﻠﺷ ﺎﻋﺮﻓ .

3 - أ - ( g ﻰﻠﻋ ﺎﻌﻄﻗ ﺔﻴﺼﻗﺎﻨﺗو ﺔﻠﺼﺘﻣ ﺔﻟاد I

.

نذإ g ﻦﻣ ﻞﺑﺎﻘﺗ I

ﻮﺤﻧ J .

(1) lim ( ), 1 4

J g x g x g

→+∞

⎤ ⎛ ⎞⎤

= ⎥⎦ ⎜ ⎟⎝ ⎠⎥⎦

ﻪﻨﻣو ,1

J = −∞⎤⎥⎦ 4⎤⎥⎦

0 J

ب - ( ﺎﻨﻳﺪﻟ g ﻦﻣ ﻞﺑﺎﻘﺗ I

ﻮﺤﻧ J و

∈

نذإ 0 ﻲﻓ ﺪﻴﺣو ﻖﺑﺎﺳ ﻞﺒﻘﻳ I

.

ﺔﻟدﺎﻌﻤﻟا نأ ﻲﻨﻌﻳ اﺪﻴﺣو ﻼﺣ ﻞﺒﻘﺗ x

α ∈I g x( )=0

ﺎﻨﻳﺪﻟ 1 2 2 1

( )2 4

g = +

و 0

3 7

( ) 3 0

4 4

g = − ≺

1 3 نذإ 2 4, α_{∈ ⎥}^{⎤ ⎡}_⎢

⎦ ⎣

4 -

+∞

fx)

(

²

)

^{2 ﺎﻨﻳﺪﻟ -1}

lim 1 lim

x x x x

→+∞ − = →+∞ = +∞

نأ ﻲﻨﻌﻳ

lim ² 1

x x

→+∞ − = +∞

lim ( )

x f x

→+∞ = +∞

نذإ

2 2

2 2

( 1) 1

lim ( ) lim lim

1 1

x x x

x x

f x

x x x x

→−∞ →−∞ →−∞

− −

= =

− − − −

ن

أ ﺎﻤﺑو

lim ² 1

x x

→−∞ − = +∞

2 نﺈﻓ lim(x − x − =1

lim ( ) 0

x f x

→−∞ =

−∞

ﻪﻨﻣو

2 - أ - (

( ) (1) lim 2 1

lim 1 1

f x −f x x 1

x x

+ −

− = −

−

2

2

1 1

1 1

1 1

lim 1 lim 1

1 1

x x

x x

x x

x x

→ →

⎛ + − ⎞= ⎛ + + ⎞= +

⎜ ⎟ ⎜ ⎟

⎜ − ⎟ ⎝ − ⎠

⎝ ⎠ ∞

نﻷ و

²

11

lim 1 0

x x

x ⁺

lim(1 1) 2

x x

→ + =

→ − =

11

( ) (1) lim^x_x 1

f x f

→ x

− = +∞

−

ﺔﻟاﺪﻟا نﺈﻓ ﻲﻟﺎﺘﻟﺎﺑو f

ﻦﻴﻤﻳ ﻰﻠﻋ قﺎﻘﺘﺷﻼﻟ ﺔﻠﺑﺎﻗ ﺮﻴﻏ 1

و (C ) ﺪﻨﻋ ﺔﺒﺟﻮﻤﻟا ﺐﻴﺗارﻷا ﻮﺤﻧ ﻪﺟﻮﻣ يدﻮﻤﻋ سﺎﻤﻣ ﻒﺼﻧ ﻞﺒﻘﻳ

ﺔﻄﻘﻨﻟا A(1,1)

2

1 1

1 1

( ) ( 1) 1 1

lim lim

1 1

x x

x x

f x f x x

x x

→− →−

− −

− − + − +

+ = +

≺ ≺

1 2 1

lim 1 1

1

x x

x

→− x

−

⎛ − ⎞

= ⎜ + ⎟= −∞

⎝ − ⎠

≺

−2 نﻷ lim (1 1)

x x

→− − =

2 و

11

lim 1 0

x x

x − ⁺

→−−

=

≺

11

( ) ( 1)

lim 1

x x

f x f

→− x

−

− − = −∞

≺ +

ﺔﻟاﺪﻟا f ﻰﻠﻋ قﺎﻘﺘﺷﻼﻟ ﺔﻠﺑﺎﻗ ﺮﻴﻏ رﺎﺴﻳ

1 و (C ) ﺔﻄﻘﻨﻟا ﺪﻨﻋ ﺔﺒﺟﻮﻤﻟا ﺐﻴﺗارﻷا رﻮﺤﻣ ﻮﺤﻧ ﻪﺟﻮﻣ يدﻮﻤﻋ سﺎﻤﻣ ﻒﺼﻧ ﻞﺒﻘﻳ

( 1, 1) B − −

]

^,1

[ ]

^1,

[

x ∈ −∞ ∪ +∞

(-ب

 ﻦﻳﺮﻤﺗ

6

:

'( ) 1 2

1 f x x

x

= + −

] [ ] [

² ₂¹

( ,1 1, ) '( )

1

x x

x f x

x

∀ ∈ −∞ ∪ +∞ = − + −

] [

^{2 : ﺎﻨﻳﺪﻟ ( -ج}

(∀ ∈ + +∞x 1, ) x − +1

نﺈﻓ ﻲﻟ 0 x

ﺎﺘﻟﺎﺑو ^{(∀ ∈ +∞x}

]

^1,

[

^{) '( ) 0f x}

نذإ f ﺔﻟاد ﻰﻠﻋ ﺔﻳﺪﻳاﺰﺗ

]

^1,+∞

[

] [ ] [

(∀ ∈ −∞ − ∪ +∞x , 1 1, )

2 2

2 2

'( ) 1

1( 1 )

x x

f x

x x x

= − −

− − −

2 2

1

1( 1 )

x x x

= −

− − −

] [

(

²

)

(∀ ∈ −∞ −x , 1 ) x − −1 x 0

] [

^نذإ

(∀ ∈ −∞ −x , 1 ) f '(x) 0≺

نﺈﻓ ﻲﻟﺎﺘﻟﺎﺑو f

ﻰﻠﻋ ﺔﻴﺼﻗﺎﻨﺗ

]

^{−∞ −, 1}

[

3 - أ - ^{x ∈ +∞}

]

^1,

[

⁽

[ ] (

²

)

lim ( ) 2 lim 1

x f x x x x x

→+∞ − = →+∞ − −

2

lim 1 0

1

x^→+∞ x x

− =

− +

)

(

^{2 نﻷ}

lim 1

x x x

→+∞ − + = +

نذإ ﻰﻨﺤﻨﻤﻟا (C ) ﻞﺒﻘﻳ برﺎﻘﻣ

∞ راﻮﺠﺑ

+∞

2

y = x ﻪﺘﻟدﺎﻌﻣ

ب - (

1 - g ﻰﻠﻋ ﺎﻌﻄﻗ ﺔﻳﺪﻳاﺰﺗو ﺔﻠﺼﺘﻣ ﺔﻟاد I

.

و g I( )=I

نذإ g ﻦﻣ ﻞﺑﺎﻘﺗ I

ﻮﺤﻧ I .

نﺈﻓ ﻪﻨﻣو g

ﺔﻴﺴﻜﻋ ﺔﻟاد ﻞﺒﻘﺗ

g⁻1

ﻰﻠﻋ ﺔﻓﺮﻌﻣ I

.

1 - أ - ( Df = ^* = −∞

]

^,0

[ ]

∪ ^0,+∞

[

ب - ( نﺎآ اذإ نﺈﻓ

− ∈x ^* x ∈ ^*

و

2 3

( ) 2 x

f x x

x

− = − − +

−

2 3

2 x (

)

x f x

x

= − + + = −

* f ( x) f x( ) نذإ

∀∈ − = −

نذإ f ﺔﻳدﺮﻓ ﺔﻟاد .

2 -

2

2

1 3

lim ( ) lim 2

x x

x x

f x x

→+∞ →+∞ x

⎛ ⎛ + ⎞⎞

⎜ ⎜⎝ ⎟⎠⎟

⎜ ⎟

= ⎜ − ⎟

⎜ ⎟

⎜ ⎟

⎝ ⎠

2

lim 2 1 3

x x

→+∞ x

⎛ ⎞

= ⎜⎜⎝ − + ⎟⎟⎠= +∞

lim ( )

x f x

→+∞ = +∞ نذإ

3 - 1 - ( x ∈I

2 3

( ) (2 1) 2 x 2 1

f x x x x

x

− − = − + − +

2 3 2 3

1 x x x

x x

+ − +

= − =

ب - (

( ) ( )

2 2

2 2

( 3) 3

( ) (2 1)

3 3

x x

f x x

x x x x x x

− + −

− − = =

+ + + +

(

²

)

lim 3

x x x x

→+∞ نأ ﺎﻤﺑ

+ + = +∞

نﺈﻓ ^{xlim→+∞x x}

(

^+−3x2+3

)

⁼⁰

نذإ

^lim

[

^{( ) (2 1)}

]

⁰

x f x x

→+∞ − − =

( )

^Δ نﺈﻓ ﻲﻟﺎﺘﻟﺎﺑو ﻰﻨﺤﻨﻤﻠﻟ برﺎﻘﻣ

(C ) راﻮﺠﺑ +

∞

ج - ( ﺎﻨﻳﺪﻟ

(

^{3 2}

)

( )

x I 3

x x x

∀ ∈ −

+ +

(∀ ∈x I) f x( ) 2≺ x−1 نأ يأ

ﻰﻨﺤﻨﻤﻟا نذإ (C )

ﺖﺤﺗ ﺪﺟﻮﻳ

]

^0,+∞

[

^{لﺎﺠﻤﻟا ﻰﻠﻋ}

( )

^{Δ ﻢﻴﻘﺘﺴﻤﻟا}

 ﻦﻳﺮﻤﺗ

7

:

4 - أ - (

2 2

2 2

3 3 0; '( ) 2

x x

x f x x

x

− +

≠ = − +

2 2

2 2

( 3

2 3

x x

x x

− +

= − +

)

2 2

2 3

3 x x

= + +

2 2

( ) '( ) 2 3 x I f x 3

x x

∀ ∈ = +

+

ب - +∞ (

0

x

+

f’(x)

+∞

−∞

f(x)

(-أ-5 ﻊﻃﺎﻘﺗ (C ) ﻰﻠﻋ ﻞﻴﺻﺎﻓﻷا رﻮﺤﻣ ﻊﻣ I

.

2 2 3 0

x I

x x

⎧⎪ ∈

⇔ ⎨⎪⎩ − + = ( ) 0

x I f x

⎧ ∈

⎨ =

⎩

2 2

3

x x

x I

⎧⎪ = +

⇔ ⎨⎪⎩ ∈

4 2

4x x 3 0

x I

⎧ − − =

⇔ ⎨⎩ ∈

ﺔﻟدﺎﻌﻤﻟا

4 2 0

x −x − =

4 3

ﺔﻴﻧﺎﺜﻟا ﺔﺟرﺪﻟا ﻦﻣ ﺔﻟدﺎﻌﻣ ﻰﻟإ ﺎﻬﻠﺣ لوﺆﻳ .

2 3

4) x = − أ و

4 2 2 1

4x −x − = ⇔3 0 (x =

1) x = − (x 1 وأ

⇔ =

( ) 0 f x 1 x I x

= ⎫⎬⇔ =

∈ ⎭

ﻰﻨﺤﻨﻤﻟا نذإ (C )

ﺔﻄﻘﻨﻟا ﻲﻓ ﻞﻴﺻﺎﻓﻷا رﻮﺤﻣ ﻊﻄﻘﻳ لﺎﺠﻤﻟا ﻰﻠﻋ

I

(1, 0) A

T

3

= x − ﺔﻟدﺎﻌﻣ

( )

سﺎﻤﻣ (C ) ﻘﻨﻟا ﺪﻨﻋ ﺔﻄ A ﻲه :

'(1)( 1) (1) y =f x − +f

: 3

T y

ب -

ﺎﻨﻳﺪﻟ f' ﺔﻳدﺮﻓ ﺔﻟاد

ﺎهﺎﻨﺤﻨﻣ نذإ (C )

ﺔﻄﻘﻨﻠﻟ ﺔﺒﺴﻨﻟﺎﺑ ﻞﺛﺎﻤﺘﻣ O

ﻢﻠﻌﻤﻟا ﻞﺻأ .

6 -

g لﺎﺠﻤﻟا ﻰﻠﻋ ﺎﻌﻄﻗ ﺔﻳﺪﻳاﺰﺗو ﺔﻠﺼﺘﻣ ﺔﻟاد I

.

و g(1)=

نذإ g ﻦﻣ ﻞﺑﺎﻘﺗ I

ﻮﺤﻧ

lim ( 2)( 3) lim 2

x x x x x

→+∞ →+∞

1 - أ - (

+ − = = +∞ ﺎﻨﻳﺪﻟ

lim ( )

x f x

→+∞ = +∞

نذإ

ب -

2 2

2 2

2 ( 2)( 3

lim ( ) lim

2 2

x x

x x

x x

f x

x x

→− →−

− −

+ +

+ = +

)

2 2

2 2

2( 2)( 3) 2( 3)

lim lim

( 2) ( 2)( 3) ( 2)( 3)

x x

x x

x x x

x x x x x

→− →−

− −

+ + +

= =

+ + + + +

lim 2(32 ) 10

x x

→− − = نأ ﺎﻤﺑ

+ 22

lim ( 2)( 3) 0

x x

x x

→−−

+ + =

نﺈﻓ 2

2

lim ( ) 2

x x

f x

→− x

−

+ = +∞

3 3

3 3

2 ( 2)( 3) lim ( ) lim

3 3

x x

x x

x x

f x

x x

→ →

+ −

− = −

33

2( 2) lim^x_x ( 2)( 3)

x

x x

→

= + = +∞

+ +

33

lim ( ) 3

x x

f x

→ x = +∞

−

3 3

3 3

2 ( 2)(3 ) lim ( ) lim

3 3

x x

x x

x x

f x

x x

→ →

+ −

− = −

≺ ≺

33

2( 2) lim^x_x ( 2)(3 )

x

x x

→

− + + − = −∞

≺

lim ( ) 3 3 3

f x x x

x

− = −∞

→

≺

2 - أ - (

( ) 2 ( 2)(3 ); 2 3

( ) 2 ( 2)( 3); 3

f x x x x

f x x x x

⎧ = + − −

⎪⎨

= + −

⎪⎩

≺ ≺

نﺎآ اذإ ^{x ∈ −}

]

^2,3

[

[

_نﺈﻓ

_{'( )} ^{2 ( 2)(3 )}

]

2 ( 2)(3 )

x x

f x

x x

+ −

= + −

] [

^{1 2 نأ يأ}

( 2,3 ) '( )

( 2)(3 )

x f x x

x x

∀ ∈ − = −

+ −

ﻲﻟﺎﺘﻟﺎﺑو

]

^−2,3

[

'( ) ةرﺎﺷإ ﻲه 1 2x−

)

نﻷ ﻰﻠﻋ f x ةرﺎﺷإ نﺈﻓ

( (x−2)(3−x)

0

]

^3,

[

^{نﺎآ اذإ}

x ∈ +∞

 ﻦﻳﺮﻤﺗ

8

:

[

_نﺈﻓ

_{'( ) 2} ^{( 2)( 3)}

]

^'

2 ( 2)( 3

x x

f x

x x

+ −

= + −

] [

)

2 1

( 3, ) '( )

( 2)( 3)

x f x x

x x

∀ ∈ +∞ = −

+ −

3 1 2

x ⇒x 2⇒ x −1 0

نذإ

'( ) 0 f x

⇒

]

^3,

[

^{'( ) 0}

x f x

∀ ∈ +∞

ب -

3 - أ ( 2 ( 2)( 3) lim ( ) lim

x x

x x

f x

x x

→+∞ →+∞

+ −

=

2 2 6

lim

x

x x

→+∞ x

= − −

2

2

1 6

2 1

xlim

x x x

→+∞ x

⎛ − − ⎞

⎜ ⎟

⎝ ⎠

=

2

2

1 6

2 1

xlim

x x

x x

→+∞ x

− −

=

2

1 6

lim 2 1 2

x^→+∞ x x

= − − =

2 نﻷ

1 6

lim lim 0

x x

^→+∞x = ^→+∞x =

[ ]

lim ( ) 2 lim 2 ( 2)( 3) 2

x f x x x x x x

→+∞ − = →+∞⎡⎣ + − − ⎤⎦

( 2)( 3) 2

2 lim

( 2)( 3)

x

x x x

x x x

→+∞

+ − −

= + − +

2 lim 6

( 2)( 3)

x

x

x x x

→+∞

= −

+ − +

6 1

2 lim 1

1 6

1 1

x

x x x

→+∞

= − = −

− − +

ﻢﻴﻘﺘﺴﻤﻟا نذإ (D )

ﻰﻨﺤﻨﻤﻠﻟ برﺎﻘﻣ (C )

راﻮﺠﺑ +∞

ب - ( ﻦﻜﻴﻟ x ﻦﻣ اﺮﺼﻨﻋ

]

^3,+∞

[

( ) (2 1) 2 ( 2)( 3) (2 1)

f x − x − = x + x − − x −

4( 2)( 3) (2 1)2

2 ( 2)( 3) (2 1)

x x x

x x x

+ − − −

= + − + −

25 0

2 (x 2)(x 3) (2x 1)

= −

+ − + − ≺

] [

( ) 2 1; 3,

f x ≺ x − +∞ ﻦﻣ x ﻞﻜﻟ

ﻢﻴﻘﺘﺴﻤﻟا نذإ (D)

ﻰﻨﺤﻨﻤﻟا ﺖﺤﺗ ﺪﺟﻮﻳ (C)

لﺎﺠﻤﻟا ﻰﻠﻋ .

]

^3,+∞

[