سلسلة تمارين في النهايات والإتصال السنة الثانية بكالوريا مسلك العلوم التجريبية من إنجاز الأستاذ حميد

Page 1

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1 : ‫أ ا ت ا‬ lim ( 9 x 2 + x + 1 + 3x) (2 lim ( 9 x 2 + x + 1 − 2 x) (1

x → −∞

x → +∞

lim ( 1 − 2 x − − x + x + 1)(4 3

3

x →−∞

lim−

x → −2

x2 − 4 (9 lim+ x→2 x+2

x2 − 4 (8 x−2 lim

x →0

lim

x → −1

1 − 3x − 2 (7 x +1

x+ x x 2 +x −x

(12

lim

x → −1

lim

x → −∞

x +1+ − x +1 x2 − x2 + 2

x − x2 + x − 1

lim

(6

x → +∞

x2 − x4 − 1

(5

x 2 − x − 6 + 3x − x 2 2+ x + 3− x −3 (10 (11 lim− x →3 x+3 x +1

lim x →0

limπ

x→

lim ( x 2 − x + 2 )(3

x → +∞

4

1 (13 3 cos x − sin x − 3

tan x − 1 cos x − 3 sin x (15 limπ (14 6x − π 2 cos x − 2 x→ 6 2

1  ( x − 1) sin( x − 1) ; x < 1 f ( x) =  x 2 − 2 x − 8  ; x ≥1  x − 2 − 2 lim f ( x)

x → +∞

‫ ا ا‬

‫( ا‬2 x0=1 ! f ‫ Ùˆادرس ا ل‬Df ‫( د‬1 4 )* ‫ ا‬+, ‫ ل‬- . ‫ ا‬/ ' * f ‫( Ù‡' ا ا‬3 3

 x + 2x − a ;x > 2  x − 2 f ( x) =  2  2 x + b − a ;x ≤ 2  x 2

2 )* ‫ ا‬+, 1 :

+1 /. , / ‫ا‬

f

f

‫ ا ا‬

‫ن ا ا‬67 8 9. b‫ و‬a 2 ‫ د ا د‬ 4

x − 4x + 4x ‫ ا ا‬ x 2 −4 D f ‫ ات‬9: ! ‫ وأ ا ت‬D f ‫( د‬1 ‫ Ø&#x;‬-2 ‫ و‬2 2: '‫ آ‬+, ‫ < ل‬. ‫ ا‬/ ' * f ‫( Ù‡' ا ا‬2 f (x ) =

3

2

5

f (x ) =

1 + sin x − 1 ‫ ا ا‬ x2 2

D f ‫( د‬1 . 0 +, ‫ < ل‬. ‫ ا‬/ ' * f ‫ أن ا ا‬2 . (2

6

 f  f 

3 cos x − sin x π ;x ≠2 cos x − 1 3 π 2 ( )= 3 3

(x ) =

: +1 /. , / ‫ ا‬f ‫ ا ا‬

.

Ï€ 3

+, 1 : f ‫ أن‬2 . 7

 2 f ( x ) = x sin   , x ≠0 : f ‫ ا ا‬ x    f ( 0) = 0  . lim f ( x ) ‫( ا‬2 .0 +, f ‫( ادرس ا ل‬1 +∞

8 IR +, @ 'A-‫ ا‬B1! ' * x + x − x + x + 1 = 0 ‫ د‬/ ‫ أن ا‬2 . (1 IR +, ‫ * ' @ Ùˆ ا‬3 x 7 + 2 x 5 + x − 10 6 4 3 = 0 ‫ د‬/ ‫ أن ا‬2 . (2 ‫ ل‬E/ ‫ ا‬+, ' F ,G‫ر ا‬69: H)* C f B 9 / ‫ أن ا‬2 . . f (x ) = x 4 + x − 1 ‫( ا ا‬3 5

[0,1]

3

2

g ( x) = − x 3 ‫ و‬f ( x) = x + 1 : 2 ‫( ا ا‬4 7 3 − <α < − 8 4

: 8 9. α 6 ,‫ *) Ùˆ Ø© أ‬+, ‫ ن‬J * C g ‫ و‬C f

2 9 / ‫ أن ا‬2 .

9 f(1)=1‫ و‬f(0)=0 8 9. [0,1] B1! 1 : ‫ دا‬f 27 1− c '( ∃c ∈ ]0,1[): f (c) = ‫ أن‬2 . 1+ c 10 8 9. [a , b ] B1! 1 : ‫ دا‬f 27

(a < b ) H: f (b ) > b 2 ‫ و‬f (a ) < ab f (c ) = bc 8 9. [a , b ] 2: c +* * ‫ ! د‬L6 ‫ Ø£ ﻩ‬2 . 11 ∃k ∈ IR ; ∀( x, y ) ∈ IR ² : f ( x) − f ( y ) ≤ k × x − y N*9 ‫ دا ! د‬f 27 * +

IR B1! 1 : ‫ دا‬f ‫ أن‬2 . 12 . ( ∀x ∈[0,1]): f ( x) ≥ 0 ‫ و‬f(1)=f(0)=0 8 9. [0,1] B1! 1 : ‫ دا‬f 27 1 ( ∀n ∈ IN * )( ∃c ∈[0,1]): f ( c) = f ( c + ) ‫ أن‬2 .

n

13 (∀x ∈ IR ) : f ( x) < x ‫ و‬IR B1! 1 : f : 8 9. IR 69 IR 2: , : ‫ دا‬f 27 . f (0) = 0 ‫ أن‬2 . (1 (∀(a, b) ∈ ( IR*+2 ))(∃M ∈ [0,1[)(∀x ∈ [a, b]) : f ( x) ≤ Mx : ‫ أن‬2 . (2 +

+

+

+

14 . ]0,1[ ‫ ل‬E/ ‫ ا‬+, a n ‫ * ' @ Ùˆ ا‬Arc cos( x) − x = 0 ‫ د‬/ ‫ ا‬IN 2: n '7 ‫ Ø£ ﻩ‬2 . (1 1 . ‫ و‬a n 2 ‫ رÙ† ا د‬A (2 2 . (∀n ∈ IN * ) : a n +1 > a n : ‫ Ø£ ﻩ‬2 . (3 n

*

15 . ∀x ∈ [a, b] : f ( x) > 0 8 [a; b] B1! 1 : ‫ دا ! د‬f 27

∃m > 0, f ( x) ≥ m ‫ أن‬P Q‫أ‬

16

A= 3+ 9 +

125 125 et B = −3 + 9 + 27 27

2 ‫ ا د‬ 3

AB

A − B ‫( أ‬1

‫و‬

125 3 125 − −3 + 9 + ‫( ا د‬2 27 27 . x = 1 ‫ أن‬R S‫( ا‬b . x - . x 3 ‫( أ‬a

x = 3 3+ 9 +

17 : ‫ت ا‬-‫ د‬/ ‫ ا‬IR +, '

1− 3 x 3 ( ) + 8 = 0 (5 3− 3 x

x 4 = −2 (4

x 6 = 6 (3 ( x + 1) 3 = −27 (2 (

t=

6

1+ x 1− x

HT‫ و‬27/ )

3

( 2 x − 1) 5 = 32 (1

1 + x − 3 1 − x = 6 1 − x 2 (6

18 f (x ) = 2x − 4x + 1 ‫ ا ا‬ . ‫ ه‬9 : VW ‫ Ùˆا‬f ‫ ات‬U ‫( ادرس‬1 I = [1, +∞[ ‫ ل‬E/ ‫ ا‬B1! f ‫ر ا ا‬6 A g 27 (2 2

Z 9 E J ‫ ل‬E: 69 I ‫ ل‬E/ ‫ ا‬2: '. * g ‫ أن‬2 . (a C g −1 [S‫ Ùˆار‬g −1 ( x) ‫( د‬b 19

f

−1

4 x ‫ ا ا‬ f ( x) = 2 x +1

( x) ‫[ د‬Q Z 9 E ‫ ل‬E: 69 [− 1,1] 2: '. * f ‫ أن‬2 .

20 f (x ) =

2+ 4−x x

2

‫ ا ا‬

D f ‫( د‬1 I = ]0, 2] ‫ ل‬E/ ‫ ا‬B1! f ‫ر ا ا‬6 A g 27 (2

Z 9 E J ‫ ل‬E: 69 I ‫ ل‬E/ ‫ ا‬2: '. * g ‫ أن‬2 . (a g −1 ( x) ‫( د‬b 21 : +1 /. , / ‫ ا‬f ‫ ا ا‬ f ( x) = ( x + 1 − 1) 3 . f ‫( د ] \ ا ا‬1 Z 9 E J ‫ ل‬E: 69 [− 1,+∞[ ‫ ل‬E/ ‫ ا‬2: '. * f ‫ أن ا ا‬2 . (a (2 . J 2: x '7 f

−1

( x) ‫( د‬b

22

f ( x) = x − 3 x + 3 x 3

2

3

‫ ا ا‬ . f ‫( د ] \ ا ا‬1 f ( x ) = x ‫ د‬/ ‫ ا‬IR + +, ' (2

f ( x) = (3 x − 1) 3 + 1 ‫ أن‬2 . (a (3 [0,+∞[ ‫ ل‬E/ ‫ ا‬2: )A ‫ ]ا‬f ‫ أن ا ا‬2 . (b Z 9 E J ‫ ل‬E: 69 [0,+∞[ ‫ ل‬E/ ‫ ا‬2: '. * f ‫ أن ا ا‬2 . (c . J 2: x '7 f

−1

( x ) ‫( د‬d

23 +1 /. , / ‫ ا‬f ‫ ا ا‬

f (x ) = −x − 3 3 (1 − x ) 2 + 3 3 1 − x + 1 . f ‫( د ] \ ا ا‬1 −1 . f ( x) ‫[ د‬Q Z 9 E ‫ ل‬E: 69 ]−∞,1] 2: '. * f ‫ أن‬2 . (2 . f (x ) = 1 ‫ د‬/ ‫] ا‬−∞,1] +, ' (3

24 x f ( x) = 1+ x (∀x ∈ ]−1, +∞[) : f (x ) = x + 1 −

f

−1

‫ ا ا‬ 1 : ‫ أن‬2 . (a (1 x +1

( x) ‫[ د‬Q Z 9 [ J ‫ ل‬E: 69 ]−1, +∞[ 2: '. * f ‫ أن‬2 . (b

f −1 ( x) = f ( x) : ‫ د‬/ ‫ ا‬R +, ' (2

25 : f (x ) = (4 − 3 x 2 )3 :‫ ب‬, / ‫ ا‬f ‫ ا ا‬ D f : f ‫! \ ا ا‬6/E: ‫( د‬1

I = [ 0,8] ‫ ل‬E/ ‫ ا‬B1! f ‫ر ا ا‬6 A g 27 (2

−1

g ( x) ‫[ د‬Q Z 9 E J ‫ ل‬E: 69 I 2: '. * g ‫ أن‬2 . . f ( x) = x ‫ د‬/ ‫ ا‬R +, ' (3

26

f (x ) =

3

x 2 −1 3 x

+1 /. , / ‫ ا‬f ‫ ا ا‬

Z 9 E J ‫ ل‬E: 69 ]0, +∞[ 2: '. * f ‫ أن‬2 . (1 (∀x ∈ IR ): x − x 2 + 4 < 0 ‫ أن‬2 . (2

f −1 ( x) ‫( د‬3 ‫ د‬/ ‫] ا‬0, +∞[ +, ' (4

. J 2: x '7

f (x ) = 5

27 : ‫أ ا ت ا‬

lim ( 3 −x 3 + 2x 2 − x − 2x 2 + 1 )

x →+∞

lim ( 3 −3x 3 − 1 + x

x →−∞

3

3)

(6

(3 lim ( 3 1 − x 3 + 2x ) (2 lim ( 3 −8x 3 + x 2 + 1 + 2x ) 3 x →−∞ 4

lim−

x →−2

(x + 2) 2 x +2

x 2 +1 − 3 1− x

− x + 4x + 1 − 2 − x 3

3

(11

x → +∞

(5

x →−∞

lim

3

lim (3 8 x 3 − x + 1 − x) (1

x →−∞

lim−

x →−2

x 2 −4 x +2

lim ( 3 8x 3 − 1 − 2x ) (4

x →+∞

(8

3

lim

x →−∞ 3

3

(10 3

lim−

x →1

lim x →0

x +1 −1 x

(7

(9

x 2 −1 + x 2 + x − 2 x −1

(12

(x + 1) 2 + x 2 + x x +1

(13

3

lim

x 2 +1 + x 1− x − x 2

x →−1−

28 : ‫ Ùˆ ت ا‬/ ‫ ا‬P Q‫أ‬ 1 2 Ï€

Arc tan( ) + Arc tan( ) = (1 5 3 4 1 1 1 π A rc tan + A rc tan + A rc tan = (2 2 5 8 4 1 1 1 A rc tan + A rc tan − A rc tan = 0 (3 3 7 2 1 π

(∀x > 0) : A rc tan( x ) + A rc tan( ) = (4 x 2 1 π (∀x < 0) : A rc tan( x ) + A rc tan( ) = − (5 x 2

(∀x ∈ IR ) : cos(Arc tan x ) =

1 π 0 ≤ arctan( ) ≤ ‫ أن‬a -) 5 8

1

(6

1+ x 2

1 1 π 4 arctan − arctan = 5 239 4

(∀x > 0) : Arc tan(x + 1) − Arc tan x = Arc tan(

(7

1 ) x 2 + x +1

(8

29 arctan 2 + arctan 3

‫أ‬

30 : / ‫ ا‬IR +, ' x −8 Ï€ Arc tan( ) − Arc tan( x) = 8 2

A rc tan(

31 . ‫ت ا‬-‫ د‬/ ‫ ا‬IR +, ' Ï€ Arc tan 2x + Arc tan(3x ) = (1 4

x 2 −1 π ) + A rc tan( x ) = (2 2 x 2

32 ‫ د‬/ ‫ ا‬IR +,

(E ) : arctan(x − 1) + arctan x + arctan(x + 1) =

Ï€

2 . ]0,1[ B ‫ إ‬+/ '9 ‫ا ا‬c‫ وأن ه‬IR +, ‫( * ' @ Ùˆ ا‬E ) ‫ د‬/ ‫ ا‬2 . (1 . (E ) ‫ د‬/ ‫( ' ا‬2

Arc tan x −

Ï€

6 3 x− 3 1 π Arc tan − x 2 lim− x →0 x

lim

3 x→ 3

1 π lim− x (arctan( ) + ) x →0 x 2

(6 .

33 : ‫أ ا ت‬ arctan x lim (1 (2 x →0 x

(4

lim A rc tan( x →1

x −1 π

lim x (Arc tan x − ) 2

x →+∞

lim (x arctan(

x →+∞

x 2

x +1 π )− x ) x 4

)

(3 (5

(7

34

x +1 ) ‫ ا ا‬ x . D f ‫( د‬1 . ]0, +∞[ B1! f ‫ر‬6 A g 27 (2

f (x ) = arctan(

. Z 9 E ‫ ل‬E: 69 ]0, +∞[ 2: '. * g ‫ أن‬2 . (a . f

−1

(x ) ‫( د‬b

35 : 2 / ‫ ا‬IR +, ' Arc tan(

x 2 −1 π ) + Arc tan(x ) = x2 2

(2

Arc tan 2x + Arc tan(3x ) =

(E ) : arctan(x − 1) + arctan x + arctan(x + 1) =

Ï€

Ï€ 4

(1

36

‫ د‬/ ‫ ا‬IR +, 2 . ]0,1[ B ‫ إ‬+/ '9 ‫ا ا‬c‫( * ' @ Ùˆ ا وأن ه‬E ) ‫ د‬/ ‫ ا‬2 . (1 . (E ) ‫ د‬/ ‫( ' ا‬2

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