g - Madariss
ﺍﻷﻭﻟﻰ ﺒﺎﻜﺎﻟﻭﺭﻴﺎ ﻋﻠﻭﻡ ﺭﻴﺎﻀﻴﺔ ﺘﻤﺎﺭﻴﻥ ﻋﻤﻭﻤﻴﺎﺕ ﺤﻭل ﺍﻟﺩﻭﺍل ﺍﻟﻌﺩﺩﻴﺔ Prof : BEN ELKHATIR
•
ﺗﻤﺮﻳﻦ:01
ﻨﻌﺘﺒﺭ ﺍﻟﺩﺍﻟﺘﻴﻥ ﺍﻟﻌﺩﺩﻴﺘﻴﻥ fﻭ gﺍﻟﻤﻌﺭﻓﺘﻴﻥ ﺒﻤﺎ ﻴﻠﻲ :
1− x
= ) f ( xﻭ g ( x ) = 3 − 2x − x 2
1+ x
G JG
ﻭ ﻟﻴﻜﻥ ) (C fﻭ ) (C gﻤﻨﺤﻨﻴﻴﻬﻤﺎ ﻓﻲ ﻤﻌﻠﻡ ﻤﺘﻌﺎﻤﺩ ﻭ ﻤﻤﻨﻅﻡ O , i , j
-(1ﺤﺩﺩ ﺃﻓﺎﺼﻴل ﻨﻘﻁ ﺘﻘﺎﻁﻊ ) (Cﻭ ) (C
f
g
)
(
•
ﻟﺘﻜﻥ fﺍﻟﺩﺍﻟﺔ ﺍﻟﻌﺩﺩﻴﺔ ﺍﻟﻤﻌﺭﻓﺔ ﺒﻤﺎ ﻴﻠﻲ ، f ( x ) = x 2 − x + 1 :ﻭ ) (C fﻤﻨﺤﻨﺎﻫﺎ
G JG
ﻓﻲ ﻤﻌﻠﻡ ﻤﺘﻌﺎﻤﺩ ﻭ ﻤﻤﻨﻅﻡ . O , i , j
)
)
3
-(1ﺘﺤﻘﻕ ﻤﻥ ﺃﻥ \ = ، D fﺜﻡ ﺒﻴﻥ ﺃﻥ ﺍﻟﺩﺍﻟﺔ fﻤﺼﻐﻭﺭﺓ ﺒﺎﻟﻌﺩﺩ
2
1
1 1
⎡1
⎡
-(2ﺃﺜﺒﺕ ﺃﻥ :
، ∀x ∈ ⎢ , +∞ ⎢ : x − < f ( x ) ≤ x − +ﺜﻡ ﺇﺴﺘﻨﺘﺞ ﺃﻥ ﻤﻨﺤﻨﻰ
2
2 2x
⎣2
⎣
⎡1
⎡
ﻗﺼﻭﺭ fﻋﻠﻰ ﺍﻟﻤﺠﺎل ⎢ ∞ ⎢ , +ﻴﻭﺠﺩ ﻀﻤﻥ ﺍﻟﺸﺭﻴﻁ ) ( ∆1ﺍﻟﻤﺤﺩﻭﺩ ﺒﺎﻟﻤﺴﺘﻘﻴﻤﻴﻥ :
⎣2
⎣
1
1
( D1 ) : y = x −ﻭ . ( D 2 ) : y = x +
2
2
-(3ﺒﻴﻥ ﺃﻥ ، ∀x ∈ \ : f (1 − x ) = f ( x ) :ﺜﻡ ﺇﺴﺘﻨﺘﺞ ﺃﻥ ﻤﻨﺤﻨﻰ ﻗﺼﻭﺭ ﺍﻟﺩﺍﻟﺔ fﻋﻠﻰ
.
.
x 3 + 3x 2
ﺍﻟﻤﺘﺭﺍﺠﺤﺔ ≥ 2 :
x +1
-(3ﻨﻌﺘﺒﺭ ﺍﻟﺩﺍﻟﺘﻴﻥ ﺍﻟﻌﺩﺩﻴﺘﻴﻥ hﻭ kﺍﻟﻤﻌﺭﻓﺘﻴﻥ ﺒﻤﺎ ﻴﻠﻲ :
2
x
= ) . k (x
= ) h (xﻭ
1− x
x −2
(I ) :
.
أ -ﺒﻴﻥ ﺃﻥ hﺩﺍﻟﺔ ﻓﺭﺩﻴﺔ ﻭ ﺃﻥ ، ∀x ∈ D h ∩ \ − : h ( x ) = f ( x + 1) :ﺜﻡ ﺃﻨﺸﻰﺀ
G JG
) (C hﻓﻲ ﺍﻟﻤﻌﻠﻡ O , i , jﺇﻨﻁﻼﻗﺎ ﻤﻥ ) . (C f
)
(
ب -ﺒﻴﻥ ﺃﻥ kﺩﺍﻟﺔ ﺯﻭﺠﻴﺔ ﻭ ﺃﻥ ، ∀x ∈ D k ∩ \ − : k ( x ) = 1 + f ( x ) :ﺜﻡ ﺃﻨﺸﻰﺀ
G JG
) (C kﻓﻲ ﺍﻟﻤﻌﻠﻡ O , i , jﺇﻨﻁﻼﻗﺎ ﻤﻥ ) . (C f
)
•
(
•
ﺗﻤﺮﻳﻦ:02
ﻨﻌﺘﺒﺭ ﺍﻟﺩﺍﻟﺘﻴﻥ ﺍﻟﻌﺩﺩﻴﺘﻴﻥ fﻭ gﺍﻟﻤﻌﺭﻓﺘﻴﻥ ﺒﻤﺎ ﻴﻠﻲ :
2
x
f (x ) = x 2 − +1ﻭ
x
x +x −2
3
)
= ) . g (x
2
ﺍﻟﻤﺠﺎل
⎤1
⎤
ﺍﻟﻤﺠﺎل ⎥ ⎥ −∞,ﻴﻭﺠﺩ ﻀﻤﻥ ﺍﻟﺸﺭﻴﻁ ) ( ∆ 2ﺍﻟﻤﺤﺩﻭﺩ ﺒﺎﻟﻤﺴﺘﻘﻴﻤﻴﻥ :
⎦2
⎦
3
1
( D1' ) : y = −x +ﻭ . ( D 2' ) : y = −x +
2
2
ﺗﻤﺮﻳﻦ:04
ﻨﻌﺘﺒﺭ ﺍﻟﺩﻭﺍل ﺍﻟﻌﺩﺩﻴﺔ fﻭ gﻭ hﺍﻟﻤﻌﺭﻓﺔ ﻋﻠﻰ \ ﺒﻤﺎ ﻴﻠﻲ :
) f ( x ) = (1 + x )( 2 − xﻭ ) g ( x ) = f ( xﻭ ) h ( x ) = f ( x
G JG
-(1ﺃﻨﺸﻰﺀ ﺍﻟﻤﻨﺤﻨﻰ ) (C fﻓﻲ ﻤﻌﻠﻡ ﻤﺘﻌﺎﻤﺩ ﻭ ﻤﻤﻨﻅﻡ . O , i , j
G JG
-(2ﺒﻴﻥ ﺃﻥ gﺩﺍﻟﺔ ﺯﻭﺠﻴﺔ ،ﺜﻡ ﺃﻨﺸﻰﺀ ) (C gﻓﻲ ﺍﻟﻤﻌﻠﻡ O , i , jﺇﻨﻁﻼﻗﺎ ﻤﻥ ) . (C f
G JG
-(3ﺃﻜﺘﺏ ) h ( xﺒﺩﻭﻥ ﻗﻴﻤﺔ ﻤﻁﻠﻘﺔ ،ﺜﻡ ﺃﻨﺸﻰﺀ ) (C hﻓﻲ ﺍﻟﻤﻌﻠﻡ O , i , jﺇﻨﻁﻼﻗﺎ ﻤﻥ ) . (C f
G JG
-(4ﺇﻨﻁﻼﻗﺎ ﻤﻥ ) (C fﺃﻨﺸﻰﺀ ﻓﻲ ﺍﻟﻤﻌﻠﻡ O , i , jﻤﻨﺤﻨﻰ ﻜل ﺩﺍﻟﺔ ﻤﻥ ﺍﻟﺩﻭﺍل ﺍﻟﺘﺎﻟﻴﺔ :
⎞ ⎛ 1
⎜⎜ . ∀x ∈ D g − {0} :
-(1ﺤﺩﺩ D fﻭ ، D gﺜﻡ ﺒﻴﻥ ﺃﻥ ⎟⎟ = f ( x ) :
⎠ ) ⎝ g (x
-(2ﺒﻴﻥ ﺃﻥ ﺍﻟﺩﺍﻟﺔ fﺘﺯﺍﻴﺩﻴﺔ ﻗﻁﻌﺎ ﻋﻠﻰ ﺍﻟﻤﺠﺎﻟﻴﻥ [∞ ]0, +ﻭ [ [ −1, 0ﻭ ﺘﻨﺎﻗﺼﻴﺔ ﻗﻁﻌﺎ ﻋﻠﻰ
]]−∞, −1
(
.
G JG
-(2ﺃﻨﺸﻰﺀ ﻓﻲ ﺍﻟﻤﻌﻠﻡ O , i , jﺍﻟﻤﻨﺤﻨﻴﻴﻥ ) (C fﻭ ) ، (C gﺜﻡ ﺤل ﻤﺒﻴﺎﻨﻴﺎ ﻓﻲ ﺍﻟﻤﺠﻤﻭﻋﺔ \
(
ﺗﻤﺮﻳﻦ:03
)
(
)
(
)
(
(
) f 1 : x 6 −f ( xﻭ ) f 2 : x 6 f ( − xﻭ ) f 3 : x 6 −f ( − x
.
)
-(3ﺇﺴﺘﻨﺘﺞ ﺭﺘﺎﺒﺔ ﺍﻟﺩﺍﻟﺔ gﺜﻡ ﺃﻨﺸﻰﺀ ﺠﺩﻭل ﺘﻐﻴﺭﺍﺘﻬﺎ .
-1-
f 4 : x 6 f (− x
ﻭ ) f 5 : x 6 f ( −x
.
ﺍﻷﻭﻟﻰ ﺒﺎﻜﺎﻟﻭﺭﻴﺎ ﻋﻠﻭﻡ ﺭﻴﺎﻀﻴﺔ ﺘﻤﺎﺭﻴﻥ ﻋﻤﻭﻤﻴﺎﺕ ﺤﻭل ﺍﻟﺩﻭﺍل ﺍﻟﻌﺩﺩﻴﺔ Prof : BEN ELKHATIR
•
-(3ﺒﻴﻥ ﺃﻥ hﺩﺍﻟﺔ ﻓﺭﺩﻴﺔ ﻭ ﺃﻥ . ∀x ∈ \ − : h ( x ) = f ( − x ) :
G JG
-(4ﺃﻨﺸﻰﺀ ) (C gﻭ ) (C hﻓﻲ ﺍﻟﻤﻌﻠﻡ O , i , jﺇﻨﻁﻼﻗﺎ ﻤﻥ ) . (C f
ﺗﻤﺮﻳﻦ:05
2x + 3
ﻨﻌﺘﺒﺭ ﺍﻟﺩﺍﻟﺔ ﺍﻟﻌﺩﺩﻴﺔ fﺍﻟﻤﻌﺭﻓﺔ ﻋﻠﻰ \ ﺒﻤﺎ ﻴﻠﻲ :
x 2 +4
-(1ﻟﻴﻜﻥ ) t ( x , yﻤﻌﺩل ﺘﻐﻴﺭ fﺒﻴﻥ xﻭ yﺤﻴﺙ xﻭ yﻋﻨﺼﺭﻴﻥ ﻤﻥ \ ﻭ . x ≠ y
)
= ) . f (x
ﺒﻴﻥ ﺃﻥ :
) ( x + 4 )(1 − y ) + ( y + 4 )(1 − x
) + 4 )( y 2 + 4
2
(x
-(5ﺃﻨﺸﻰﺀ ﺠﺩﻭل ﺘﻐﻴﺭﺍﺕ ﻜل ﻤﻥ fﻭ gﻭ ، hﺜﻡ ﺇﺴﺘﻨﺘﺞ ﺠﺩﺍﻭل ﺘﻐﻴﺭﺍﺕ ﺍﻟﺩﻭﺍل ﺍﻟﺘﺎﻟﻴﺔ :
f Dfﻭ g Dfﻭ . h Df
• ﺗﻤﺮﻳﻦ:08
ﻨﻌﺘﺒﺭ ﺍﻟﺩﺍﻟﺘﻴﻥ ﺍﻟﻌﺩﺩﻴﺘﻴﻥ gﻭ hﺍﻟﻤﻌﺭﻓﺘﻴﻥ ﺒﻤﺎ ﻴﻠﻲ :
⎞ 1
2
⎛⎞ ⎛ 1
. h ( x ) = ⎜1 + ⎟⎜ 1 +
g (x ) = 1+ﻭ ⎟
x
⎠ ⎝ x ⎠⎝ 1 − x
-(1ﺒﻴﻥ ﺃﻥ ، h = g D f :ﺤﻴﺙ fﺤﺩﻭﺩﻴﺔ ﻤﻥ ﺍﻟﺩﺭﺠﺔ ﺍﻟﺜﺎﻨﻴﺔ ﻴﻨﺒﻐﻲ ﺘﺤﺩﻴﺩﻫﺎ .
-(2ﺃﻨﺸﻰﺀ ﺠﺩﻭﻟﻲ ﺘﻐﻴﺭﺍﺕ ﺍﻟﺩﺍﻟﺘﻴﻥ ﺍﻟﻌﺩﺩﻴﺘﻴﻥ fﻭ ، gﺜﻡ ﺇﺴﺘﻨﺘﺞ ﺭﺘﺎﺒﺔ ﺍﻟﺩﺍﻟﺔ ﺍﻟﻌﺩﺩﻴﺔ h
ﻭ ﺃﻨﺸﻰﺀ ﺠﺩﻭل ﺘﻐﻴﺭﺍﺘﻬﺎ .
-(3ﺃﺜﺒﺕ ﺃﻥ ﺍﻟﺩﺍﻟﺔ hﻤﺼﻐﻭﺭﺓ ﺒﺎﻟﻌﺩﺩ 9ﻋﻠﻰ ﺍﻟﻤﺠﺎل [ ]0,1ﻭ ﻤﻜﺒﻭﺭﺓ ﺒﻨﻔﺱ ﺍﻟﻌﺩﺩ ﻋﻠﻰ
= ) t (x , y
] ]−∞, −4ﻭ ] [ −4,1ﻭ [∞[1, +
ﺜﻡ ﺇﺴﺘﻨﺘﺞ ﺭﺘﺎﺒﺔ ﺍﻟﺩﺍﻟﺔ fﻋﻠﻰ ﻜل ﺍﻟﻤﺠﺎﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ :
ﻭ ﺃﻨﺸﻰﺀ ﺠﺩﻭل ﺘﻐﻴﺭﺍﺘﻬﺎ .
1
-(2ﺒﻴﻥ ﺃﻥ ﺍﻟﺩﺍﻟﺔ fﻤﻜﺒﻭﺭﺓ ﺒﺎﻟﻌﺩﺩ 1ﻭ ﻤﺼﻐﻭﺭﺓ ﺒﺎﻟﻌﺩﺩ . −
4
2x + 1
= ) . g (x
-(3ﻟﺘﻜﻥ gﺍﻟﺩﺍﻟﺔ ﺍﻟﻌﺩﺩﻴﺔ ﺍﻟﻤﻌﺭﻓﺔ ﺒﻤﺎ ﻴﻠﻲ :
x −2
2
. ∀x ∈ \ : −3 ≤ g D f ( x ) ≤ −
ﺒﻴﻥ ﺃﻥ \ = ، D g Dfﻭ ﺃﻥ :
5
-(4ﺒﻴﻥ ﺃﻥ ﺍﻟﺩﺍﻟﺔ gﺘﻨﺎﻗﺼﻴﺔ ﻋﻠﻰ ﺍﻟﻤﺠﺎﻟﻴﻥ [ ]−∞, 2ﻭ [∞ ، ]2, +ﺜﻡ ﺇﺴﺘﻨﺘﺞ ﺭﺘﺎﺒﺔ g D f
ﻭ ﺃﻨﺸﻰﺀ ﺠﺩﻭل ﺘﻐﻴﺭﺍﺘﻬﺎ .
• ﺗﻤﺮﻳﻦ :06ﻨﻌﺘﺒﺭ ﺍﻟﺩﺍﻟﺘﻴﻥ ﺍﻟﻌﺩﺩﻴﺘﻴﻥ fﻭ gﺍﻟﻤﻌﺭﻓﺘﻴﻥ ﺒﻤﺎ ﻴﻠﻲ :
[∞]−∞, 0[ ∪ ]1, +
•
4
−2
1− x
1 + 2x 2
= ) f (xﻭ
1 − 2x 2
1+ x
-(1ﻗﺎﺭﻥ ﻋﻠﻰ D = D f ∩ D gﺍﻟﺩﺍﻟﺘﻴﻥ fﻭ ، gﺜﻡ ﺤﺩﺩ ﺍﻟﻭﻀﻊ ﺍﻟﻨﺴﺒﻲ ل ) (C fﻭ ) . (C g
= ) g (x
-(2ﻗﺎﺭﻥ ﺍﻟﻌﺩﺩﻴﻥ Aﻭ Bﺤﻴﺙ :
•
999000
1000002
= Aﻭ
999998
1001000
.
) f ( x ) = −x ( x + 2ﻭ ) g ( x ) = x ( 2 − xﻭ
G JG
-(1ﺃﻨﺸﻰﺀ ﺍﻟﻤﻨﺤﻨﻰ ) (C fﻓﻲ ﻤﻌﻠﻡ ﻤﺘﻌﺎﻤﺩ ﻭ ﻤﻤﻨﻅﻡ O , i , j
)
-(2ﺒﻴﻥ ﺃﻥ gﺩﺍﻟﺔ ﺯﻭﺠﻴﺔ ﻭ ﺃﻥ :
(
.
ﺗﻤﺮﻳﻦ:09
ﺇﻨﻁﻼﻗﺎ ﻤﻥ ﺠﺩﻭل ﺘﻐﻴﺭﺍﺕ ﺍﻟﺩﺍﻟﺔ ﺍﻟﻌﺩﺩﻴﺔ fﺍﻟﺘﺎﻟﻲ ﻋﻠﻰ ﺍﻟﻤﺠﺎل ]: [ −4, 4
3
2
0
3
−2
0
= . B
−4
4
x
f
−3
ﺗﻤﺮﻳﻦ:07
ﻨﻌﺘﺒﺭ ﺍﻟﺩﻭﺍل ﺍﻟﻌﺩﺩﻴﺔ fﻭ gﻭ hﺍﻟﻤﻌﺭﻓﺔ ﻋﻠﻰ \ ﺒﻤﺎ ﻴﻠﻲ :
)
(
1
ﺃﻨﺸﻰﺀ ﺠﺩﻭل ﺘﻐﻴﺭﺍﺕ ﺍﻟﺩﺍﻟﺔ gﻓﻲ ﻜل ﺤﺎﻟﺔ ﻤﻥ ﺍﻟﺤﺎﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ (1) : g ( x ) = −f ( x ) + 1 :
) 1− f (x
2
1
= ) ( 3) : g ( xﻭ
⎦⎤ ) ( 2 ) : g ( x ) = ⎡⎣ f ( xﻭ
= ) ( 4) : g ( x
) f (x
) 3 + f (x
. h (x ) = x (2 − x
.
⎞⎛1
⎟ ⎜ ( 5) : g ( x ) = fﻭ
⎠ ⎝x
) . ∀x ∈ \ − : g ( x ) = f ( x
-2-
) (x
2
( 6) : g ( x ) = f
ﻭ ) . (7) : g ( x ) = f (3 − x
ﺍﻷﻭﻟﻰ ﺒﺎﻜﺎﻟﻭﺭﻴﺎ ﻋﻠﻭﻡ ﺭﻴﺎﻀﻴﺔ ﺘﻤﺎﺭﻴﻥ ﻋﻤﻭﻤﻴﺎﺕ ﺤﻭل ﺍﻟﺩﻭﺍل ﺍﻟﻌﺩﺩﻴﺔ Prof : BEN ELKHATIR
• ﺗﻤﺮﻳﻦ:10
G JG
ﻤﺜل ﻓﻲ ﻤﻌﻠﻡ ﻤﺘﻌﺎﻤﺩ ﻭ ﻤﻤﻨﻅﻡ O , i , jﻤﻨﺤﻨﻰ ﺍﻟﺩﺍﻟﺔ ﺍﻟﻌﺩﺩﻴﺔ fﻓﻲ ﻜل ﺤﺎﻟﺔ ﻤﻥ
(
)
ﺍﻟﺤﺎﻻﺕ ﺍﻟﺘﺎﻟﻴﺔ :
1
⎞⎟(1) : f ( x ) = E ⎜⎛ x + 1
⎝2
⎠
) ( 2 ) : f ( x ) = E ( 2x − 1ﻭ ) ( 3) : f ( x ) = E ( x 2ﻭ
•
ﺗﻤﺮﻳﻦ:11
ﻨﻌﺘﺒﺭ ﺍﻟﺩﺍﻟﺔ ﺍﻟﻌﺩﺩﻴﺔ fﺍﻟﻤﻌﺭﻓﺔ ﻋﻠﻰ \ ﺒﻤﺎ ﻴﻠﻲ :
) E (x
)( 4 ) : f ( x ) = ( −1
) . f (x ) = x − E (x
-(1ﺒﻴﻥ ﺃﻥ fﺩﺍﻟﺔ ﺩﻭﺭﻴﺔ ﺩﻭﺭﻫﺎ . T = 1
G JG
-(2ﻤﺜل ﻤﻨﺤﻨﻰ ﺍﻟﺩﺍﻟﺔ fﻋﻠﻰ ﺍﻟﻤﺠﺎل [ [ −1, 2ﻓﻲ ﻤﻌﻠﻡ ﻤﺘﻌﺎﻤﺩ ﻭ ﻤﻤﻨﻅﻡ O , i , j
)
(
.
-(3ﺤل ﻓﻲ ﺍﻟﻤﺠﻤﻭﻋﺔ \ ﻜل ﻤﻌﺎﺩﻟﺔ ﻤﻥ ﺍﻟﻤﻌﺎﺩﻟﺘﻴﻥ :
x
x
( 2 ) : f ( x ) = 1 −ﺤﻴﺙ *` ∈ . n
(1) : f ( x ) = 1 −ﻭ
5
n
•
ﺗﻤﺮﻳﻦ:12
⎛
⎞⎞ ⎛ x
ﻨﻌﺘﺒﺭ ﺍﻟﺩﺍﻟﺔ ﺍﻟﻌﺩﺩﻴﺔ fﺍﻟﻤﻌﺭﻓﺔ ﻋﻠﻰ \ ﺒﻤﺎ ﻴﻠﻲ . f ( x ) = E ⎜ x − 2E ⎜ ⎟ ⎟ :
⎠⎠ ⎝ 2
⎝
-(1ﺒﻴﻥ ﺃﻥ fﺩﺍﻟﺔ ﺩﻭﺭﻴﺔ ﺩﻭﺭﻫﺎ . T = 2
-(2ﻋﺭﻑ fﻋﻠﻰ ﺍﻟﻤﺠﺎل [ ، [ 0, 2ﺜﻡ ﺃﻨﺸﻰﺀ ﻤﻨﺤﻨﺎﻫﺎ ﻋﻠﻰ ﺍﻟﻤﺠﺎل [ [ −2, 4ﻓﻲ ﻤﻌﻠﻡ ﻤﺘﻌﺎﻤﺩ
G JG
ﻭ ﻤﻤﻨﻅﻡ . O , i , j
)
(
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Abdellah BEN ELKHATIR – Lycée alfath – khémisset
-3-
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