**************************************

2 3 4 5-1-2-3-4-5

2

3

4

5

-1

-2

-3

-4

0 1

1

x

y

2 3 4 5-1-2-3-4-5

2

3

4

5

-1

-2

-3

-4

0 1

1

x

y

BAC2021

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Page 1 of 28101010

Page 2 of 28101010

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1،،2

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Page 3 of 28101010

Page 4 of 28101010

0202 2

f

13f (x) x 3x 1

fD 2 2x

f (x)x² 4

fD 33x² 4x 5

f (x)(x 1)²

fD 1

4 4x 8

f (x)x² 4x 3

fD 1;3 5

3 2

2

2x 2xf (x)

(2x 1)

f

1D

2

f (x) x x² 1(6 fD

7 x 1

f (x) 2x 3x² 4

fD 2;2 8

xf (x)

x² 1

fD ; 1 1;

Ig1 2x² 5x 4

f (x)x 1

)C( f

1abcx 1 c

f (x) ax bx 1

2f

3)C( f

4)C( f

fCf D 1;1

3 2

2

x x 1f x

x 1

1

D

2

3f (x)f

4f ( x) f (x)

I

23x 1

x² 3x 2lim

x 1

0

2

xlim (x x 1)

3x 2

x 1 1lim

x 2

4

xlim ( 4x² 4x 3 (2x 1)

II

22

20

23

24

Page 5 of 28101010

0202 0

23

2x 2

x 4x² 5x 2 1lim

x x 2 3

0

x 0

sin xlim 2

x 1 1

3

x 1

x 1 1lim

x² x 2 6

4

xlim ( x² 3 x 2) 2

f D ;0 2; 2f (x) x 2x 2x 1

1xlim f (x)x

lim f (x)

2 xlim f (x) x

f(C )

3 xlim f (x) ( 3x) 2

f D 1; 2x sin x

f (x)x 1

2x 1 2x 1

f (x)x 1 x 1

xlim f (x)

x 0

sin xlim 1

x

x 0

f (x)lim 1

x

x 1

lim f (x)

f

1x 02x 3x x 2

xlim(3x x 3x)

2x 1>2x 1 x 2x 3 2

xlim(3x x 2x 3)

f2f (x) x 5 f

C

1x 2 2

f (x) f (2) x 2

x 2 x 5 3

2x 2

f (x) f (2)lim

x 2

f

32

x

x 5 3lim 1

x 2

2

x

x 5 3lim 1

x 2

4y xf

C

5 ff

C

26

27

28

25

Page 6 of 28101010

0202 3

f3f (x) x 3x 1

1f ،

2

f (x) 0f (x)

f (x) 0 1;1 0,1

f (x) 0 2;3

fa

1f (x) x² 3x a 1 2f (x) x² x 1 a 13f (x) 4 x² a 2

fC gC

f g 2;3

1

f 1 g 1

f 2

g 2

2x 2;3 h x f 2x 1

h 2;3

h 0 h 2TDL

29

22

22

(T) (D)

L

Page 7 of 28101010

0202 4

f

f (x) x 1 (x 1) fC

1 fCf1

2

3m

f (x) m² f (0)

g(C )g

g(C )A( 1;0)B(0;1)C(1;0)

I1

g( 1)g(0)g(1)g '( 1)g '(0)g '(1)

2 AB

3 2;2g(x) 0

g'(x) 1 g'(x) 0

II3g(x) ax bx² cx 1

abc

I1

f*

2x² 3f (x) : x 2

x

f (x) x² 2x : x 2

)C( f

1f2

219

f

)C( f( )( )

(T)2

)C( f

23

24

20

Page 8 of 28101010

0202 5

f

4x 4f (x) : x 1

x² 2x 3

f (x) x² 3 x : x 1

)C( f

1x 1 x 1

lim f (x) lim f (x)

21

x

lim f (x)x

lim f (x)

h 0

f (h 1) f (1)lim

h

h 0

f (h 1) f (1)lim

h

1

g(C )g 1; 3g(x) ax bx c g(C )

1

g(0)g(1)g '(1)abc

g

33x 3x 4 0 2;2,25

g(x)

fD 1

311 x

6

2

f(x)

1ff '(x)f (x)D

2ghkg(x) f (x²)1

h(x) f ( )x

k(x) f ( 2x)

ghk

3REV R(x) f (x) ² 3

E(x) f (x)1

V(x)f (x)

R '(x)E'(x)V'(x)f (x)f '(x)

REV

25

26

27

Page 9 of 28101010

0202 6

f ;2 2;

320 x

0

2

4

1

f(x)

(C)

1f1

fx

f (x) 2 0;2

2g31

g(x) ;x 2f (x)

g(2) 0

g3g

f1 f

(C )

521 x

0 f'(x)

1442

2021

f (x)

1f fC

2f

3f (x) 0f (x)

4g ; 1;2 g(x) f (x)

g

fgg

Dg

O,i, j4cm

f 1;1f (x) x 1 x² f

(C )

1f

28

0 0

29

02

3-

Page 10 of 28101010

0202 7

2x 1

f (x)lim

x 1 x 1

f (x)lim

x 1 3f

4(T)f

(C )

(T)f

(C )

5(T)f

(C )

f(C )f

1

fDff

2xlim(f (x) x 1

3f (0)f '( 2)f '(0)f '(0)

f0

4 f (x) 0f '(x) 0

5m f (x) x m

fCf1

( )( ') fCA(3;0)B( 1;0)

1f

f (x)f (x)

f

f (3)f ( 1) ( )( ')

2 x² ax b

f x(x 1)²

1ab

3h 2

h x f (x)

h (x)f (x)f (x)h (x)

h

02

00( ') ( )

A B

Page 11 of 28101010

0202 8

f 2 x² x

f (x)x 2

)C( f

1)C( fA(1; 3)2

3

23 7

)C( fy x 1

)C( f4y x 0

)C( f

If3g(x) x 5x² 5x 4

g(C )

(T)A(0;4)

1g(x)

g(C )(T)

2g

xg '(x)g

3g(x) 0 1;0110

xg(x)

4(T)8

(C )(T)

( )f*( )

x=0y=1

04

03

05

C A

Page 12 of 28101010

0202 9

1 xlimf (x)x

limf (x)x

f (x)lim

x

xlim f (x) x

x 0limf (x)

xlim fof (x) f (x)

2f

3 f (x) 1 f (x) 3

4 f ( 0; ) f ( ;0 ) f ( 0;4 )

g 3 1 x²

g(x)x 3

1

2g2cm

3hh(x) (fog)(x)

h 3

xlimh(x)x 3

lim h(x)

x 3

lim h(x)

fx² ax b

f (x)cx² dx 2

abcd

)C( f

Iabcd)C( f

x 2

A(1; 2)5

IIx2x² x

f (x)x 2

1f

xlim f (x) x 3

2x2f '(x)f

3x 2 f (x) f (4 x) 10

06

Page 13 of 28101010

0202 22

)C( f

mx² (m 1)x 2m

g(C )g

( )g

(C )

g(C )

1

g( 3)g( 3)g'( 3)

xlim g(x)x

lim g(x)x

g(x)lim

x

xlim g(x) 3x

2ff

f '(x) g(x)f

(C )

f '(x) g(x)

f(C )

07

Page 14 of 28101010

0202 22

Ig3g(x) 2x 4x² 7x 4

1xlim g(x)x

lim g(x)

g

2g(x) 00,7 0,8

xg(x)

IIf3x 2x 1

f (x)2x² 2x 1

f(C ) O, i , j

1xlim f (x)x

lim f (x)

2x1 1 3x

f (x) (x 1)2 2(2x² 2x 1)

f(C )( )

f(C )( )

3xx.g(x)

f '(x)(2x² 2x 1)²

f 'f

f '(x)xff ( ) 0,1

4f (1)f (x) 05( )f

(C )

6h3x 4x² 2x 1

h(x)2x² 2x 1

h

(C )

xh(x) f (x) 2

h(C )

f(C )

h(C )

080224

Page 15 of 28101010

0202 20

2-1

2

3

-1

-2

0 1

1

x

y

If I ; 1 1;0

4f (x) x

x 1

f

(C )

(O ;i ; j )

1fI

f

2g 0;4

g(x) xx 1

g(C )

g

g(C )( )

g

IIk 1 4

k(x) xx 1

1h 0

k(h) k(0)lim

h

h 0

k(h) k(0)lim

h

21

( )2

( )0

x 0

31

( )2

( )k

(C )

C)g

I 1; 3g(x) x 3x² 3x 1

1gg(0)g(0,5)

0;0,5g( ) 0

g(x)I

2f 1;

3x 3x² 3x 2f (x)

(x 1)²

( )

x I3

g(x)f '(x)

(x 1)

x

f (x) f ( )lim

x

x 1

lim f x

xlim[f x (x 1)]

f

30,26 f ( )10-2

( )

090229

320228

2-1-2-3-4-5-6-7

2

3

4

5

6

7

-1

-2

-3

-4

-5

0 1

1

x

y

Page 16 of 28101010

0202 23

Ig3g(x) x 6x 12

1g

2g(x) 0 1,48; 1,47

xg(x)

IIf3x 6

f (x)x² 2

f(C ) O, i , j

1xlim f (x)x

lim f (x)

xx.g(x)

f '(x)(x² 2)²

f

2( )y xf

(C ).

f

(C )( )

33

f ( )2

f ( )

4( )f(C )

f 1; 2

f (x) xx 1

f(C )f

1f

2f(C )(D)y x

f(C )(D)

3f(C )0

x0

1,3 x 1,4

( )f(C )f(C )

( )f(C )

4 g 1; g(x) f (x)g(C )g

g(C )f(C )

m2g(x) m

320227

300229

Page 17 of 28101010

0202 24

f 1

f (x) x 1x² 1

f

(C )

1f

2x2

1f '(x) 1

(x 1) x² 1

3f

4(T)f

(C )0

5f

(C )(T)f

(C )

6(D)y=x+1f

(C )

(D')

7(D)(D')f

(C )

8 g1

g(x) x 1x² 1

gf

(C )g

(C )

f 1;3 2

2

x 2x 15f x

x 2x 3

Cf ; ;O i j

1fC

2C5

3x 1CC

4mf 2

m 2

x mx 15f x

x mx 3

m

mfmC

mC

4;1

f1

;22

2

2

4x 5xf x

2x 5x 2

f(C )

330228

342982

352997

Page 18 of 28101010

0202 25

1abc

1x ;2

2

b bf x a

2x 1 x 2

3ffC

4f

(C )0

5f

(C )6

1f 1 1

f (x) 2x 3(x 1)²

C(O ;i ; j )

1x 1 3

2(x 2)(x² x 1)f '(x)

(x 1)

2f3C

4C 0x 0,37; 0,25

5C0

6C

732x (7 m)x² 2(4 m)x 2 m 0....(e) m x

x(e)f (x) m

Cm(e)

f 1 x²(x 2)

f (x)(x 1)²

f(C )(O ;i ; j )

1،، x 1

f (x) x(x 1) (x 1)²

2f3f(C )

f(C )

f(C )

4 f(C )1

5f(C )

362997

372997

Page 19 of 28101010

0202 26 Page 20 of 28101010

0202 27

I gCg

3g(x) x ax b

gC A 0;3

1

g( 1)g(1)g (0)g ''(0)g

2g '(x)a 3 b 3

3g(x) 0 2,2; 2,1 g(x)

IIf4 2f (x) x 6x 12x

1xf (x) 4g(x)

f

2f ( ) 3 ( 3) f ( )

IIIkk(x) f ( x ) k

(C )

2k

0k3k0

f

012 x

0 0 f'(x)

2

2

f(x)

f(x)c

f (x) ax bx 1

abc

2f '(x)ac0f

a bc

x 1

lim f (x)x 1

lim f (x)

39

38

Page 21 of 28101010

0202 28

1

2

3

4f

3a b c 1 )C( f

x( )( ))C( f( )y x 1

)C( f( )

( 1;0) )C( f

)C( f)C( f

mf (x) m 2

Ig33x

1bax)x(g

abC

1ab

CA(2 ;1)

2)ba3;3(I C

IIf 3R 3x

7x5x)x(f

2

)C( f(O ;i ; j )

1x 3R f(x) = g(x)

2f3 )2x()x(flimx

4)C( f)(

5)C( f3

6)()C( f

7)C( fmf(x)=3x+m

f ; 1 1; f (x) x x² 1

(C)(O;i; j)

1ff

2 xlim f (x) 2x 0

3f 1

42

42

Page 22 of 28101010

0202 29

4f 1; ; 1

5(C)2y 51 2

6(C)

f2

x 2f (x) x

x 1

f

(C )(O ;i ; j )

1xlim f (x)x

lim f (x)

2f2

2 2

x(x 1)(x x 4)f '(x)

(x 1)

3f

4(D) : y x f

(C )f

(C )(D)

5f

(C )1f ( 1.25)

6(D)f

(C )

7gg

(C )g

(C )f

(C )

1

1v g(x)

g(C )

f 2ax² bx c

f (x)x 2

f(C )

Iabcf(C )y x 3

3

IIa 1b 5 c 7

1f

2f(C )1

(D )2

(D )3

1(D )

2(D )

31

(D )2

(D )f(C )

4mf (x) 3x m 0

5 f 2;2 g(x) f ( x )

g0

g(C )gf(C )g(C )

40

43

Page 23 of 28101010

0202 02

I g3g(x) x 3x 3 g

(C )

1 g

2g(x) 0 2,1;2,2g(x)

3xg( x) g(x)

IIf 1; 1 3

2

2x 3f (x)

x 1

f

(C )

1x 1; 1 2 2

2x.g(x)f (x)

(x 1)

2 f3f ( ) 3 f ( )

4( )y 2xf

(C )

f(C )( )

5f

(C )( )

6f

(C )

7m3 22x mx m 3 0

IIIh 1; 1

3

2

2 x 3h(x)

x 1

1h

2h

(C )f

(C )

f 1;1 2

xf x | x 1|

x 1

fC

1

f x

2 f ' xf

3( ) : y x 1 ( ) : y x 1 fC

fC( )( )

fC

0,5;1 2; 1,5

44

45

Page 24 of 28101010

0202 02

fC

4 mDy mx 1 m

m mD

m2

x| x 1| mx 1

x 1

f 2f x x 1 x fC ; ,O i j

1lim ( )x

f x

lim ( ) 0x

f x

2x

2

2

x x 1f (x)

x 1

x( ) 0f x

3f

xlim f x 2x 0

4 fC f

C

I)g 1; 3g(x) ax 3x b

g(C )

1abg(C )y 6 1

2g g

33x 3x 4 0 2;2,25g(x)

IIf 1;1 1, x²(x 2)

f (x)x² 1

f

(C )

1f

2f 1;1 1, f '(x)

2

x.g(x)f '(x)

(x² 1)

f

33

f ( ) 22

f ( )

5xlim(f (x) (x 2))

f

(C )( )

6f

(C )( )

7( )f

(C )

46

47

Page 25 of 28101010

0202 00

2 3 4-1-2-3-4-5-6-7-8

2

3

4

5

6

7

8

-1

-2

-3

-4

-5

-6

-7

0 1

1

x

y

f2

2

(x a)f (x)

x b

ab

f(C ) O,i, j

1ab( )0y 2x 1

2y 1f

3a b 1 x2

2 2

2(x 1)f (x)

(x 1)

f

f( )A(0,1)

A(0,1)f

(C )

f(C )( )

4g1

g(x) fx

gf

f 2 c

f (x) ax bx 2

abc

, ,o i jfCf

( )( )fC04-

1-(0)f ( 4)f ( )( )

2-m( ) 2 1 f x x m

3-f

4-g( ) ( )g x f x

( )fCg

g

5-abc

48

49

Page 26 of 28101010

0202 03

mf1m

x² mxf (x)

x 1

m

m(C )

mf O,i, j

1m

(C )m 1 (

2m

(C )

3abc m

cf (x) ax b

x 1

m 1 (

4m(1;m 2)

m(C )

mm 1

5mf ' (x)mf

mmf

mmf

52

Page 27 of 28101010

0202 04

0202تمنياتنا لكم بالتوفيق التام في بكالوريا

BEY MATHS

Page 28 of 28101010