2

‫ﺗﻤﺎرﯾﻦ ﻣﺤﻠﻮﻟﺔ‪:‬اﻟﺪوال اﻷﺻﻠﯿﺔ‬
‫أﻛﺎدﻳﻤﯿﺔ‬
‫اﻟﺠﮫﺔ‬
‫اﻟﺸﺮﻗﯿﺔ‬
‫اﻟﻤﺴﺘﻮى ‪ :‬اﻟﺜﺎﻧﯿﺔ ﺑﺎك ﻋﻠﻮم ﻓﯿﺰﯾﺎﺋﯿﺔ وﻋﻠﻮم اﻟﺤﯿﺎة‬
‫واﻷرض واﻟﻌﻠﻮم اﻟﺰراﻋﯿﺔ‬
‫ﺗﻤﺮﯾﻦ‪ :1‬ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ ‪ f‬اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ ¡ ﻛﺎﻟﺘﺎﻟﻲ‪:‬‬
‫اﻟﺪاﻟﺔ ‪f‬‬
‫ﺣﺪد داﻟﺔ ‪ F‬ﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق‬
‫ﻋﻠﻰ ¡ ﺑﺤﯿﺚ ) ‪( "x Î ¡ ) ; F ¢ ( x ) = f ( x‬‬
‫ﺗﻮﺟﺪ‬
‫‪.2‬‬
‫ھﻞ‬
‫‪.3‬‬
‫ﻛﻢ ﺗﻮﺟﺪ ﻣﻦ داﻟﺔ ‪ F‬ﺑﺤﯿﺚ ) ‪ ( "x Î ¡ ) ; F ¢ ( x ) = f ( x‬؟‬
‫) ‪( "x Î ¡ ) ; G ¢ ( x ) = f ( x‬‬
‫داﻟﺔ‬
‫أﺧﺮى‬
‫ﺑﺤﯿﺚ‬
‫‪G‬‬
‫اﻷﺟﻮﺑﺔ ‪(1:‬اﻟﺪاﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻛﺎﻟﺘﺎﻟﻲ ‪:‬‬
‫‪ F ( x ) = 1 x3 + x 2 + 3 x‬ﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق‬
‫‪3‬‬
‫ﻋﻠﻰ ¡ وﺗﺤﻘﻖ ) ‪( "x Î ¡ ) ; F ¢ ( x ) = f ( x‬‬
‫ﻧﻘﻮل أن ‪ F :‬داﻟﺔ أﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ ‪ f‬ﻋﻠﻰ ¡‬
‫‪ (2‬اﻟﺪاﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻛﺎﻟﺘﺎﻟﻲ ‪:‬‬
‫‪ G ( x ) = 1 x 3 + x 2 + 3 x + 2‬ﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق ﻋﻠﻰ ¡‬
‫¡ ‪x a k; k Î‬‬
‫‪xax‬‬
‫‪2‬‬
‫‪x‬‬
‫¡ ‪+ c; c Î‬‬
‫‪2‬‬
‫اﻟﺪاﻟﺔ ‪f‬‬
‫*‪x a x n ; n Î ¥‬‬
‫‪1 n+1‬‬
‫¡‪x + c; c Î‬‬
‫‪n +1‬‬
‫‪1‬‬
‫¡ ‪x a - + c; c Î‬‬
‫‪x‬‬
‫‪1 -n+1‬‬
‫‪xa‬‬
‫¡‪x +c; cÎ‬‬
‫‪-n +1‬‬
‫‪xa‬‬
‫(‬
‫‪1‬‬
‫}‪; n Î ¥* - {1‬‬
‫‪xn‬‬
‫‪xa‬‬
‫‪1‬‬
‫‪xa‬‬
‫‪x‬‬
‫)}‪x a xr ; r Î( ¤* -{-1‬‬
‫اذن أﯾﻀﺎ ‪ G :‬داﻟﺔ أﺻﻠﯿﺔ أﺧﺮى ﻟﻠﺪاﻟﺔ ‪ f‬ﻋﻠﻰ ¡‬
‫‪ (3‬ھﻨﺎك ﻋﺪد ﻻﻣﻨﺘﮫ ﻣﻦ اﻟﺪول اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ ‪f‬‬
‫وﻧﻘﻮل ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ ‪ f‬ﻋﻠﻰ ¡ ھﻲ‬
‫‪xa‬‬
‫ﺣﯿﺚ ‪ k‬ﻋﺪد ﺣﻘﯿﻘﻲ‪.‬‬
‫ﺗﻤﺮﯾﻦ‪ :2‬ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ ‪ f‬اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [‪ ]0; +¥‬ﻛﺎﻟﺘﺎﻟﻲ‪:‬‬
‫‪f ( x ) = 2x2 + x + 1 +‬‬
‫‪.1‬‬
‫ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ ‪ f‬ﻋﻠﻰ [‪]0; +¥‬‬
‫‪.2‬‬
‫ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ ‪ F‬ﻟﻠﺪاﻟﺔ ‪ f‬ﺑﺤﯿﺚ ‪F (1) = 3‬‬
‫‪1‬‬
‫اﻷﺟﻮﺑﺔ ‪f ( x ) = 2 x + x + 1 + 1:‬‬
‫‪2‬‬
‫‪x2‬‬
‫‪1‬‬
‫‪1‬‬
‫‪1‬‬
‫اذن ‪F ( x ) = 2 ´ x2+1 + x1+1 +1x - 2 + k :‬‬
‫‪3‬‬
‫‪2‬‬
‫‪x‬‬
‫‪2‬‬
‫‪1‬‬
‫‪3‬‬
‫‪2‬‬
‫‪ F ( x) = x + x + x - 1 + k‬ﺣﯿﺚ ¡ ‪k Î‬‬
‫‪3‬‬
‫‪2‬‬
‫‪x‬‬
‫‪2 3 1 2‬‬
‫‪1‬‬
‫‪ F (1) = 3 (2‬ﯾﻌﻨﻲ ‪´1 + ´1 +1 - + k = 3‬‬
‫‪3‬‬
‫‪2‬‬
‫‪1‬‬
‫‪7‬‬
‫‪2 1‬‬
‫ﯾﻌﻨﻲ ‪ + +1 -1 + k = 3‬ﯾﻌﻨﻲ ‪+ k = 3‬‬
‫‪6‬‬
‫‪3 2‬‬
‫ﯾﻌﻨﻲ ‪11‬‬
‫‪7‬‬
‫=‪k‬‬
‫ﯾﻌﻨﻲ ‪k = 3 -‬‬
‫‪6‬‬
‫‪6‬‬
‫وﻣﻨﮫ اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ ‪ F‬ﻟﻠﺪاﻟﺔ ‪ f‬ﺑﺤﯿﺚ ‪F (1) = 3‬‬
‫‪xa‬‬
‫اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ ‪f‬‬
‫ﻋﻠﻰ ﻣﺠﺎل ‪I‬‬
‫)‬
‫‪3‬‬
‫‪1‬‬
‫‪x2‬‬
‫اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ ‪f‬‬
‫ﻋﻠﻰ ﻣﺠﺎل ‪I‬‬
‫¡ ‪x a kx + c; c Î‬‬
‫‪1‬‬
‫‪xa 2‬‬
‫‪x‬‬
‫وﺗﺤﻘﻖ أﯾﻀﺎ ) ‪( "x Î ¡ ) ; G ¢ ( x ) = f ( x‬‬
‫‪1 3‬‬
‫اﻟﺪوال اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ ¡ ﺑﻤﺎ ﯾﻠﻲ ‪x + x 2 + 3 x + k :‬‬
‫‪3‬‬
‫ﻧﺠﯿﺐ‬
‫ﻋﺜﻤﺎﻧﻲ‬
‫ﺟﺪاول ﺗﻤﻜﻨﻨﺎ ﻣﻦ اﻟﺒﺤﺚ ﻋﻦ اﻟﺪوال اﻷﺻﻠﯿﺔ‬
‫‪f ( x ) = x2 + 2x + 3‬‬
‫‪.1‬‬
‫اﻷﺳﺘﺎذ‪:‬‬
‫) ‪x a cos ( x‬‬
‫) ‪x a sin ( x‬‬
‫‪1‬‬
‫)‪cos2 ( x‬‬
‫= )‪x a1+ tan2 ( x‬‬
‫¡ ‪x a 2 x + c; c Î‬‬
‫‪1 r +1‬‬
‫‪xa‬‬
‫¡ ‪x + c; c Î‬‬
‫‪r +1‬‬
‫¡ ‪x a sin ( x ) + c; c Î‬‬
‫¡‪x a -cos ( x) + c;c Î‬‬
‫¡ ‪x a tan ( x ) + c; c Î‬‬
‫اﻟﺪاﻟﺔ ‪ f‬ﻣﻌﺮﻓﺔ ﻋﻠﻰ‬
‫ﻣﺠﺎل ‪I‬‬
‫داﻟﺔ أﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ ‪f‬‬
‫ﻋﻠﻰ اﻟﻤﺠﺎل ‪I‬‬
‫‪u+v‬‬
‫‪uv¢ + vu ¢‬‬
‫‪uv‬‬
‫*‪u¢u n ; n Î ¥‬‬
‫‪1 n+1‬‬
‫‪u‬‬
‫‪n +1‬‬
‫‪u ¢ + v¢‬‬
‫‪u¢‬‬
‫‪u2‬‬
‫}‪¢ r ;r Î ¤* -{-1‬‬
‫‪uu‬‬
‫)‬
‫‪u¢‬‬
‫‪u‬‬
‫‪u ¢v - uv¢‬‬
‫‪v2‬‬
‫(‬
‫‪1‬‬
‫‪u‬‬
‫‪-‬‬
‫‪1 r +1‬‬
‫‪u‬‬
‫‪r +1‬‬
‫‪2 u‬‬
‫‪u‬‬
‫‪v‬‬
‫‪2‬‬
‫‪1‬‬
‫‪1 11‬‬
‫ھﻲ اﻟﺪاﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻛﺎﻟﺘﺎﻟﻲ ‪F ( x) = x3 + x2 + x - + :‬‬
‫‪3‬‬
‫‪2‬‬
‫‪x 6‬‬
‫‪1‬‬
‫اﻷﺳﺘﺎذ‪ :‬ﻧﺠﯿﺐ ﻋﺜﻤﺎﻧﻲ‬
‫‪http:// xyzmath.e-monsite.com‬‬
k Î ¡ ‫ ﺣﯿﺚ‬F ( x ) = - 1 1 + k ‫اذن‬
3
3 x +2
: ‫ ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ‬:3‫ﺗﻤﺮﯾﻦ‬
4
1
f ( x) = + cos x + sin x -1 (2 f ( x ) = 5x + 3x +1 (1
: ‫ ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ‬:5‫ﺗﻤﺮﯾﻦ‬
x (2 f x = 2 2 x + 1 (1
( )
f ( x) =
x
x2 + 1
f ( x ) = ( 2 x - 1) (4 f ( x) = sin x + x cos x (3
3
1
2
f ( x ) = 2 2 x + 1 = ( 2 x + 1)¢ ( 2 x + 1) (1:
k Î ¡ ‫ﺣﯿﺚ‬
x au¢( ax +b) ; aΡ*;bΡ
1
x a u ( ax + b )
a
1
F ( x) =
1
+1
2
( 2x +1)
1
+1
2
‫أﺟﻮﺑﺔ‬
f ( x) =
‫اذن‬
+k
- 1)
(5
2
f ( x) = 5x + 3x +1 (1: ‫أﺟﻮﺑﺔ‬
4
k Î ¡ ‫ ﺣﯿﺚ‬F ( x) = 2 ( 2x +1) 2 + k ‫وﻣﻨﮫ‬
3
k Î ¡ ‫ ﺣﯿﺚ‬F ( x ) = 5 ´ 1 x5 + 3´ 1 x2 + 1x + k ‫اذن‬
3
k Î ¡ ‫ ﺣﯿﺚ‬F ( x) = 2 ( 2x +1) = 2 ( 2x + 1) + k ‫وﻣﻨﮫ‬
3
2
5
3
3
(x
x
2
f ( x) =
3
( x + 1)¢
2
2
1
x
+ cos x + sin x -1(2
(2
k Î ¡ ‫ ﺣﯿﺚ‬F ( x ) = 2 x + sin x - cos x - x + k ‫اذن‬
k Î ¡ ‫ ﺣﯿﺚ‬F ( x) = x2 + 1 + k ‫اذن‬
f ( x) = sin x + x cos x = x¢sin x + x (sin x)¢ (3
x
f ( x) =
x2 + 1
=
2 x2 + 1
: ‫ ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ‬:6‫ﺗﻤﺮﯾﻦ‬
2
(2 f ( x ) = x x 2 + 1 (1
f ( x ) = sin ( 4 x - 1) (3 f ( x ) = x
k Î ¡ ‫ ﺣﯿﺚ‬F ( x) = x ´ sin x + k ‫اذن‬
1
3
3
f ( x ) = ( 2 x - 1) = ( 2 x - 1)¢ ( 2 x - 1) (4
f ( x ) = ( sin x) cos x (5 f ( x ) = cos ( 2 x + 8 ) (4
k Î ¡ ‫ ﺣﯿﺚ‬F ( x) = 1 ´ 1 ( 2x -1)3+1 + k ‫اذن‬
8 + x3
2
2
1
1
f ( x ) = x x + 1 = ( x2 + 1)¢ ( x2 + 1) 2 (1:
2
2
F ( x) =
1 1
( x2 +1)
21
+1
2
1
+1
2
+k =
2 3 +1
k Î ¡ ‫ ﺣﯿﺚ‬F ( x) = 1 ( 2x -1)4 + k ‫وﻣﻨﮫ‬
‫أﺟﻮﺑﺔ‬
8
‫اذن‬
1 2
x + 1) + k
(
3
3
2
(
3
)
f ( x) = -
( x - 1)¢
2
(x
2
x2
8 + x3
=
3
3 ¢
2 (8 + x )
3 2 8 + x3
3
- 1)
(5
f ( x) =
x2
( x + 2)
: ‫أﺟﻮﺑﺔ‬
2
3
f ( x) = 8x3 + 4x2 + x + 6 (1
1
2+1
( sin x) + k ‫وﻣﻨﮫ‬
2 +1
k Î ¡ F ( x) = 1 ( sin x)3 + k ‫ﯾﻌﻨﻲ‬
3
1
1
1
4
1
F ( x) = 8´ x4 + 4´ x3 + x2 + 6x + k = 2x4 + x3 + x2 + 6x + k ‫اذن‬
4
3
2
3
2
[0; +¥[ ‫ اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ‬f ‫ ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ‬:7‫ﺗﻤﺮﯾﻦ‬
k Î ¡ ‫ ﺣﯿﺚ‬f ( x) = 2sinx +cos x -3x +k
F ( x) =
x + 2x
: ‫ﻛﺎﻟﺘﺎﻟﻲ‬
2
( x + 1)
.1
:‫ ﺑﺤﯿﺚ‬b ‫ و‬a ‫ﺣﺪد اﻟﻌﺪدﯾﻦ اﻟﺤﻘﯿﻘﯿﯿﻦ‬
f ( x) =
2
"x Î [ 0; +¥ [ f ( x ) = a +
F (1) =
(x
2
: ‫ ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ‬:4‫ﺗﻤﺮﯾﻦ‬
3
2
f ( x) = 2cos x -sin x -3 (2 f ( x) = 8x + 4x + x + 6 (1
2
2
(3
f ( x ) = ( 4 x + 5 ) (4 f ( x) = 2xsinx+ x cosx
k Î ¡ F ( x) = - 1 cos ( 4x - 1) + k ‫ اذن‬f ( x ) = sin ( 4 x - 1) (3
4
k Î ¡ F ( x) = 1 sin ( 2x + 8) + k ‫ اذن‬f ( x ) = cos ( 2 x + 8) (4
2
2
2
f ( x ) = ( sin x ) cos x (5
f ( x ) = ( sin x )¢ ( sin x) ‫ﯾﻌﻨﻲ‬
(5
x
2
x -1
(2
k Î ¡ ‫ ﺣﯿﺚ‬F ( x) = 2 8 + x3 + k ‫اذن‬
‫ ﯾﻌﻨﻲ‬f ( x ) = -
k Î ¡ ‫ ﺣﯿﺚ‬F ( x) = 1 + k ‫اذن‬
2
k Î ¡ ‫ ﺣﯿﺚ‬F ( x) = 1 x2 +1 + k ‫وﻣﻨﮫ‬
f ( x ) ==
- 1)
2
5 ‫ﺑﺤﯿﺚ‬
2
f ‫ ﻟﻠﺪاﻟﺔ‬F ‫ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ‬
2
f ( x) = a +
b
( x + 1)
a ( x + 1) + b
2
=
f ( x) =
( x + 1)
x + 2x
2
( x + 1)
2
=
.2
ax + 2ax + a + b
2
( x + 1)
: ‫ﺑﺎﻟﻤﻘﺎرﻧﺔ ﻣﻊ اﻟﻜﺘﺎﺑﺔ‬
2
http:// xyzmath.e-monsite.com
2
k Î ¡ ‫ ﺣﯿﺚ‬F ( x) = x2 ´ sin x + k ‫اذن‬
f ( x ) = ( 4 x + 5 ) (4
1
2
( 4 x + 5)¢ ( 4 x + 5 )
4
1 1
2+1
F ( x) = ´
( 4x + 5) + k ‫اذن‬
4 2 +1
f ( x ) = ( 4 x + 5) =
2
(1:‫أﺟﻮﺑﺔ‬
2
f ( x) = 2x sin x + x2 cos x = ( x2 )¢ sin x + x2 ( sin x)¢ (3
2
b
( x + 1)
k Î ¡ ‫ﺣﯿﺚ‬
f ( x) = 2cos x -sin x -3(2
k Î ¡ ‫ﺣﯿﺚ‬
k Î ¡ ‫ ﺣﯿﺚ‬F ( x) = 1 ( 4x + 5)3 + k ‫وﻣﻨﮫ‬
12
æ
¢
3
1 ç ( x + 2)
f (x) = - ç 3 ç ( x3 + 2 )2
è
ö
÷
÷
÷
ø
‫ ﯾﻌﻨﻲ‬f ( x ) =
‫ ﻧﺠﯿﺐ ﻋﺜﻤﺎﻧﻲ‬:‫اﻷﺳﺘﺎذ‬
x2
( x3 + 2 )
(5
2
2
:
c
ì=
ï=
‫ وﻣﻨﮫ‬ïc
í
ïa=
ïî=
b
ìc = 5
‫ ﯾﻌﻨﻲ‬ïï8c = 40
í
ïa = 20
ïîb + 16c = 80
5
5
20
0
:
f ( x=
)
f ( x ) = 10
( x 2 + 4 )¢
(x
"x Î [0; +¥[
15
k=
2
‫ﯾﻌﻨﻲ‬
2
+ 4)
2
+5
‫ ﯾﻌﻨﻲ‬f ( x=)
(x
‫أن‬
‫ﻧﺠﺪ‬
20 x
+5
2
+ 4)
20 x
(x
2
+ 4)
2
2
f ( x) = 1-
( x + 1)
2
: ‫ﻧﺠﺪ أن‬
‫ ﯾﻌﻨﻲ‬f ( x ) = 1 -
(2
( x + 1)2
+ k : ‫وﻣﻨﮫ‬
í
ï b = -1
î
( x + 1)¢
( x + 1)
"x Î [0; +¥[
+5
(2
ìa = 1
: ‫ وﻣﻨﮫ‬ï a = 1 ‫ ﯾﻌﻨﻲ‬ï 2a = 2
f ( x) = 1-
2
í
ïa + b = 0
î
k Î ¡ F ( x) = x + 1
x +1
1
f ‫ ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ‬:8‫ﺗﻤﺮﯾﻦ‬
f ( x ) = x x - 1 : ‫[ ﻛﺎﻟﺘﺎﻟﻲ‬1; +¥ [ ‫اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ‬
k Î ¡ F ( x) = - 10 + 5x + k : ‫وﻣﻨﮫ‬
2
"x Î [1; +¥ [
x +4
10
- + k = 5 ‫ ﯾﻌﻨﻲ‬F ( 0 ) = 5 (3
4
"x Î [0; +¥[ F ( x) = - 210 + 5x + 15 : ‫وﻣﻨﮫ‬
x +4
2
5
k = 5+
2
ìa = 1
1
3
( x - 1) + x - 1 : ‫ﺑﯿﻦ أن‬
F ( 2 ) = 1 ‫ ﺑﺤﯿﺚ‬f ‫ ﻟﻠﺪاﻟﺔ‬F ‫ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ‬
‫ﯾﻌﻨﻲ‬
f ( x) =
.1
.2
:‫أﺟﻮﺑﺔ‬
( x -1)
+ x -1 = ( x -1) ´ x -1+ x -1 = x -1 ´ x -1+ x -1(1
3
2
x - 1 ³ 0 ‫ ﯾﻌﻨﻲ‬x ³ 1 : ‫ اذن‬x Î [1; +¥[ : ‫ﻧﻌﻠﻢ أن‬
x -1 = x -1 : ‫وﻣﻨﮫ‬
‫اذن‬
:
=
+ x-1 x x-1-1 x-1+ x-1 x x-1
( x-1=
) + x-1 ( x-1) ´ x-1=
3
‫ ﯾﻌﻨﻲ‬f ( x )
=
(
)
1
1
3
1
3
+ x - 1 (2
3
1
( x -1)¢ ( x -1)2 + ( x -1)¢ ( x -1) 2
=
=
f ( x) ( x -1=
) 2 + ( x -1) 2 ( x -1) 2 + ( x -1) 2
3
( x - 1)
‫اذن‬
3
1
5
3
1
1
2
2
+1
+1
=
F ( x)
=
x -1) 2 +
x -1) 2 + k
x -1) 2 + ( x -1) 2 + k
(
(
(
3
1
5
3
+1
+1
2
2
k Î ¡ ‫ ﺣﯿﺚ‬F ( x)
=
2
5
(
)
5
x -1 +
2
3
(
)
x - 1 + k ‫وﻣﻨﮫ‬
3
:‫ اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ ¡ ﻛﺎﻟﺘﺎﻟﻲ‬f ‫ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ‬:9‫ﺗﻤﺮﯾﻦ‬
5 x 4 + 40 x 2 + 20 x + 80
f ( x) =
(x
+ 4)
2
c ‫ و‬b ‫ و‬a ‫ﺣﺪد اﻷﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ‬
"x Î [ 0; +¥[
f ( x) =
ax + b
(x
2
+ 4)
2
+c
2
.1
: ‫ﺑﺤﯿﺚ‬
f ‫ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ‬
F ( 0 ) = c ‫ ﺑﺤﯿﺚ‬f ‫ ﻟﻠﺪاﻟﺔ‬F ‫ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ‬
f ( x) =
ax + b
(x
2
+ 4)
2
+c =
« c’est en forgeant que l’on devient
forgeron » dit un proverbe.
c’est en s’entraînant régulièrement
aux calculs et exercices que l’on
ax + b + c ( x + 4)
2
(x
2
+ 4)
: ‫أﺟﻮﺑﺔ‬
2
2
f ( x) =
f ( x) =
=
ax + b + cx + 8cx2 + 16c
4
(x
2
+ 4)
2
cx4 + 8cx2 + ax + ( b +16c )
5 x 4 + 40 x 2 + 20 x + 80
( x2 + 4)
.2
.3
(x
2
+ 4)
2
: ‫ﺑﺎﻟﻤﻘﺎرﻧﺔ ﻣﻊ اﻟﻜﺘﺎﺑﺔ‬
2
devient un mathématicien
http:// xyzmath.e-monsite.com
‫ ﻧﺠﯿﺐ ﻋﺜﻤﺎﻧﻲ‬:‫اﻷﺳﺘﺎذ‬
3