: جـئاـــــــــتـن

Omar Abidar -AGADIR

2 BAC SM A et B

Arctan

Arctan

2

;

2

tan

06.06.26.59.07 :

2

;

2

:

Arctan

Arctan

Arctan

x Arctanx Arctan x

Arctan

x

y

2

;

2

Arctan x y x tan y

x tanArctan x x

x

2

;

2

Arctan tan x x

x, y Arctanx Arctan y x y

x, y Arctanx Arctan y x y

x

Arctan

x

'

1

x

' x2

1



Omar Abidar -AGADIR

2 BAC SM A et B

3 2

;

2

2

;

3 2

Arctan

f x Arctantan x :

06.06.26.59.07 : 01

C

Arctan

tan

2009 4

,

B=

tan

arctan

2

,

A=

Arctan

tan

101 4

:

: 02

D arctan 1 arctan 1 arctan 1 , C= arctan 2 arctan 3 arctan 2 3

2

5

8

lim

x0

1 x

Arctan

1 x

2

: 03

Arctan x2 2x

,

lim x

x

2

Arctan

x

, lim x0

x

Arctan x

, lim

x0

x

Arctan x

lim

4

x1

x 1

,

lim x Arctan x2 1 x ,

x

lim

x

x

1

x

Arctan

1 x

:

: 04

x 0,

arctan

x

arctan

1 x

2

x 0,

arctan

x

arctan

1 x

2

Arctan

4 3

2

Arctan

1 2

arctan

1 7

2

arctan

1 3

4

x 0

Arctan

x

1

Arctan

x

Arctan

x2

1 x

1

x

cos Arctan x 1 1 x2

x sin Arctan x

ab 1 : arctan a arctan b arctan a b

1 ab

S 2arctan 1 arctan 1 2arctan 1

4

7

13

: 05

x

-1

1 x2

-2

:

-3

E2

:

arctan

2

x

arctan

x

4

E2 4

:

Arctan

x

1

Arctan

x

Arctan

x

1

2

: 06

E1 : arctan x arctan

3x 7 12

E3

:

arctan

hxttpa:r/c/atadn2mx a3th.c7om/

12

Omar Abidar -AGADIR

2 BAC SM A et B

Arctan

06.06.26.59.07

E : arctan 2x arctan x

4

E

E

: 07 1 2

x, y

: 08

y 0 Arctanx y Arctanx Arctan y

(1

y 0 Arctanx y Arctanx Arctan y

(2

x : Arctan x Arctan x

(3

Arctan x y Arctan x Arctan y

(4

John MACHIN (1680-1751)

: 09

4

4 arctan

1 5

arctan

1 239

tan4 tan2

arctan

1 5

1

4

arctan

1 5

4

0

arctan

1 5

6

2

John MACHIN

3

f x 2arctan 1 x2 x arctanx

x

f

f 0 f 'x

: 10

1 2 3

f

x

Arctan

x 1 1 x

x 1; x ;1

f

: 11

Df x

f 'x (1

f x Arctan x

(2

4

f x Arctan x 3

(2

4

3



Omar Abidar -AGADIR

2 BAC SM A et B

Arctan

06.06.26.59.07

x : f x Arctan 1 x2 x

f

: 12

x : 0 f x

1

2

x

2

1 tan2 f x 2x tan f x

a

x

tan

2

2

f

x

b

x : f x 1 Arctan x

c

42

x 1;1:

f

x

2 Arctan

1 x 1 x

f

x1;1: 0 f x

1;1

x

1

tan

2

f

x

2

x

1

tan2

f

x

2

x

sin

2

f

x

: 13 3 4

d

e

f

x

Arctan x

Arctan

1 1

x x

:

f x

1

f f

: 14

( m * 4

x

x x3 Arctan x x x3 x5

3

35

Arctan x x

lim

x0

x3

(1 : 15 (2

m

)

f

m

x

Arctan

xm 1 mx

f

: 16

D fm

x

fm ' x (1

x

\

1 m

fm x fm x

(2

m *



Omar Abidar -AGADIR

2 BAC SM A et B

Arctan

06.06.26.59.07

x

1 m

,

fm x Arctan x Arctan m

(3

Arctan

m

Arctan

1 m

2

(4

,

1 m

fm x Arctan x Arctan m

(5

1,0,1

x, y x y

Arctan

x y 1 xy

Arctan

x

Arctan

y

(6

x 0;1

f

x

x3

Arctan

1

f

1 x4

:17

D f

(1

x0 0 f

(2

x Arctan x x

(3

1 x2

x

x3 f x x

(4

1 x4

x

xf

'

x

3

f

x

2 x5 1 x4

(5

x

x2 1 x4

3

1 x4 2x2

f 'x

(6

f

(7

f

x

x

Arctan

1 x

f 0 0

, x * f

: 18

0 f

1

lim f x

2

x

x *

x 2 x2

Arctan

1 x

3

f

4

O f

5

Cf 6

5



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