أﻛﺎدﯾﻤﯿﺔ اﻟﺠﮭﺔ اﻟﺸﺮﻗﯿﺔ
ﻣﺎدة اﻟﺮﯾﺎﺿﯿﺎت
اﻷﺳﺘﺎذ :ﻋﺜﻤﺎﻧﻲ ﻧﺠﯿﺐ
ﻣﺬﻛﺮة رﻗﻢ5/
ﻧﯿﺎﺑﺔ وﺟﺪة
ﻣﺳﺗوى :اﻟﺳﻧﺔ اﻟﺛﺎﻧﯾﺔ ﻣن ﺳﻠك اﻟﺑﺎﻛﺎﻟورﯾﺎ
ﺷﻌﺑﺔ اﻟﻌﻠوم اﻟﺗﺟرﯾﺑﯾﺔ
· ﻣﺳﻠك ﻋﻠوم اﻟﺣﯾﺎة و اﻷرض
· ﻣﺳﻠك اﻟﻌﻠوم اﻟﻔﯾزﯾﺎﺋﯾﺔ
· ﻣﺳﻠك اﻟﻌﻠوم اﻟزراﻋﯾﺔ
ﻣﺬﻛﺮة رﻗﻢ 5ﻓﻲ درس اﻟﺪوال اﻷﺻﻠﻴﺔ
ﻣﺤﺘﻮى اﻟﺒﺮﻧﺎﻣﺞ
اﻟدوال اﻷﺻﻠﯾﺔ ﻟداﻟﺔ ﻣﺗﺻﻠﺔﻋﻠﻰ ﻣﺟﺎل اﻟدوال اﻷﺻﻠﯾﺔ ﻟﻣﺟﻣوع داﻟﺗﯾت اﻟدوال اﻷﺻﻠﯾﺔ ﻟﺟداء داﻟﺔ وﻋدد ﺣﻘﯾﻘﻲاﻟﻘﺪرات اﻟﻤﻨﺘﻈﺮة
ﺗﺣدﯾد اﻟدوال اﻷﺻﻠﯾﺔ ﻟﻠدوال اﻻﻋﺗﯾﺎدﯾﺔ -اﺳﺗﻌﻣﺎل ﺻﯾﻎ اﻻﺷﺗﻘﺎق ﻟﺗﺣدﯾد اﻟدوال اﻷﺻﻠﯾﺔ ﻟداﻟﺔ ﻋﻠﻰ ﻣﺟﺎل
.Iاﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﺪاﻟﺔ:
ﺧﺎﺻﯿﺔ :2ﻟﺘﻜﻦ fداﻟﺔ ﻋﺪدﯾﺔ ﻣﻌﺮﻓﺔ ﻋﻠﻰ ﻣﺠﺎل Iو x0ﻋﻨﺼﺮا ﻣﻦ
Iو y0ﻋﺪدا ﺣﻘﯿﻘﯿﺎ ﻣﻌﻠﻮﻣﺎ.
إذا ﻛﺎﻧﺖ fداﻟﺔ ﺗﻘﺒﻞ داﻟﺔ أﺻﻠﯿﺔ ﻋﻠﻰ Iﻓﺎﻧﮫ ﺗﻮﺟﺪ داﻟﺔ أﺻﻠﯿﺔ وﺣﯿﺪة
Gﻟﻠﺪاﻟﺔ fﻋﻠﻰ IﺑﺤﯿﺚG ( x0 ) = y0 :
(1داﻟﺔ أﺻﻠﯿﺔ ﻟﺪاﻟﺔ ﻋﻠﻰ ﻣﺠﺎل:
ﻧﺸﺎط :ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ fاﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ ¡ ﻛﺎﻟﺘﺎﻟﻲ:
f ( x ) = x2 + 2x + 3
.1ﺣﺪد داﻟﺔ Fﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق
ﻋﻠﻰ ¡ ﺑﺤﯿﺚ ) ( "x Î ¡ ) ; F ¢ ( x ) = f ( x
.2ھﻞ ﺗﻮﺟﺪ داﻟﺔ أﺧﺮى Gﺑﺤﯿﺚ ) ( "x Î ¡ ) ; G ¢ ( x ) = f ( x
.3ﻛﻢ ﺗﻮﺟﺪ ﻣﻦ داﻟﺔ Fﺑﺤﯿﺚ ) ( "x Î ¡ ) ; F ¢ ( x ) = f ( x؟
اﻟﺒﺮھﺎن:إذا ﻛﺎﻧﺖ Fداﻟﺔ أﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ , Iﻓﺎن ﺟﻤﯿﻊ اﻟﺪوال
اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ fﻣﻌﺮﻓﺔ ﻋﻠﻰ Iﺑﻤﺎ ﯾﻠﻲG ( x ) = F ( x ) + k :
ﺣﯿﺚ kﻋﺪد ﺣﻘﯿﻘﻲ.اﻟﺸﺮط G ( x0 ) = y0ﯾﻌﻨﻲ F ( x0 ) + k = y0أي
) k = y0 - F ( x0
ﻧﺸﺎط(1:اﻟﺪاﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻛﺎﻟﺘﺎﻟﻲ :
إذن ﺗﻮﺟﺪ داﻟﺔ أﺻﻠﯿﺔ وﺣﯿﺪة Gﻟﻠﺪاﻟﺔ fﻋﻠﻰ Iﻣﻌﺮﻓﺔ ﺑﻤﺎ ﯾﻠﻲ:
F ( x ) = 1 x3 + x 2 + 3 xﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق
3
) G ( x ) = F ( x ) + y0 - F ( x0
¢
وﺗﺤﻘﻖ
ﻋﻠﻰ ¡
"
Î
=
x
¡
;
F
x
f
x
(
) ( ) ( )
ﺧﺎﺻﯿﺔ :3ﻛﻞ داﻟﺔ ﻣﺘﺼﻠﺔ ﻋﻠﻰ ﻣﺠﺎل Iﺗﻘﯿﻞ داﻟﺔ أﺻﻠﯿﺔ ﻋﻠﻰ . I
ﻧﻘﻮل أن F :داﻟﺔ أﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ ¡
ﺧﺎﺻﯿﺔ :4ﻟﺘﻜﻦ fو gداﻟﺘﯿﻦ ﻋﺪدﯾﺘﯿﻦ ﻣﻌﺮﻓﺘﯿﻦ ﻋﻠﻰ ﻣﺠﺎل , Iو k
(2اﻟﺪاﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻛﺎﻟﺘﺎﻟﻲ :
ﻋﺪدا ﺣﻘﯿﻘﯿﺎ.
G ( x ) = 1 x 3 + x 2 + 3 x + 2ﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق ﻋﻠﻰ ¡ وﺗﺤﻘﻖ أﯾﻀﺎ إذا ﻛﺎﻧﺖ Fو Gداﻟﺘﯿﻦ أﺻﻠﯿﺘﯿﻦ,ﻋﻠﻰ اﻟﺘﻮاﻟﻲ ﻟﻠﺪاﻟﺘﯿﻦ fو gﻋﻠﻰ I
3
,ﻓﺎن:
"
Î
x
¡
(
)) ; G¢( x) = f ( x
§ اﻟﺪاﻟﺔ F + Gداﻟﺔ أﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ f + gﻋﻠﻰ . I
ﻧﻘﻮل أن G :ھﻲ داﻟﺔ أﺻﻠﯿﺔ أﺧﺮى ﻟﻠﺪاﻟﺔ fﻋﻠﻰ ¡
§ اﻟﺪاﻟﺔ kFداﻟﺔ أﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ kfﻋﻠﻰ.
(3ھﻨﺎك ﻋﺪد ﻻﻣﻨﺘﮫ ﻣﻦ اﻟﺪول اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ f
ﻣﺜﺎل :ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ fاﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [ ]0; +¥ﻛﺎﻟﺘﺎﻟﻲ:
وﻧﻘﻮل ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ ¡ ھﻲ
1
f ( x ) = 2x2 + x + 1 + 2
1
x
اﻟﺪوال اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ ¡ ﺑﻤﺎ ﯾﻠﻲ x a x 3 + x 2 + 3 x + k :
f
3
.1ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ ﻋﻠﻰ []0; +¥
ﺣﯿﺚ kﻋﺪد ﺣﻘﯿﻘﻲ.
.2ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ Fﻟﻠﺪاﻟﺔ fﺑﺤﯿﺚ F (1) = 3
ﺗﻌﺮﯾﻒ:ﻟﺘﻜﻦ fداﻟﺔ ﻋﺪدﯾﺔ ﻣﻌﺮﻓﺔ ﻋﻠﻰ ﻣﺠﺎل I
1
أﺟﻮﺑﺔ :اﻷﺟﻮﺑﺔ f ( x ) = 2 x 2 + x + 1 + 2 1:
ﻧﺴﻤﻲ داﻟﺔ أﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ , Iﻛﻞ داﻟﺔ Fﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق ﻋﻠﻰ I
x
1 2+1 1 1+1
1
,و ﻣﺸﺘﻘﺘﮭﺎ fھﻲ ,أي ) ( "x Î I ) ; F ¢ ( x ) = f ( x
اذن F ( x ) = 2 ´ x + x +1x - + k :
ﺧﺎﺻﯿﺔ :1ﻟﺘﻜﻦ fداﻟﺔ ﻋﺪدﯾﺔ ﻣﻌﺮﻓﺔ ﻋﻠﻰ ﻣﺠﺎل , Iو Fداﻟﺔ أﺻﻠﯿﺔ
ﻟﻠﺪاﻟﺔ ﻋﻠﻰ , I
اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ Iھﻲ اﻟﺪوال اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ Iﺑﻤﺎ ﯾﻠﻲ :
, x a F ( x ) + kﺣﯿﺚ kﻋﺪد ﺣﻘﯿﻘﻲ.
ص1
x2
2
3
F ( x) = 2 x3 + 1 x2 + 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
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اﻷﺳﺘﺎذ :ﻋﺜﻤﺎﻧﻲ ﻧﺠﯿﺐ
ﯾﻌﻨﻲ 11
7
ﯾﻌﻨﻲ k = 3 -
6
6
أﻣﺜﻠﺔ :ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ :
4
1
f ( x) = + cos x + sin x -1 (2 f ( x ) = 5x + 3x +1 (1
=k
وﻣﻨﮫ اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ Fﻟﻠﺪاﻟﺔ fﺑﺤﯿﺚ F (1) = 3
x
2
1
1 11
ھﻲ اﻟﺪاﻟﺔ اﻟﻤﻌﺮﻓﺔ ﻛﺎﻟﺘﺎﻟﻲ F ( x) = x3 + x2 + x - + :
3
2
x 6
.IIﺟﺪول دوال أﺻﻠﯿﺔ ﻟﺪوال اﻋﺘﯿﺎدﯾﺔ:
اﻧﻄﻼﻗﺎ ﻣﻦ اﻟﻘﺮاءة اﻟﻌﻜﺴﯿﺔ ﻟﺠﺪول ﻣﺸﺘﻘﺎت اﻟﺪوال اﻻﻋﺘﯿﺎدﯾﺔ ﻧﺤﺼﻞ
ﻋﻠﻰ اﻟﺠﺪول اﻟﺘﺎﻟﻲ:
اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ f
اﻟﺪاﻟﺔ f
ﻋﻠﻰ ﻣﺠﺎل I
¡ x a kx + c; c Î
¡ x a k; k Î
2
xax
x
¡ + c; c Î
اﻟﺪاﻟﺔ f
اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ f
ﻋﻠﻰ ﻣﺠﺎل I
*x a x n ; n Î ¥
1
x2
)
(
xa
1
x
r
*
)}x a x ; r Î( ¤ -{-1
xa
) x a cos ( x
¡ x a 2 x + c; c Î
1 r +1
¡ x + c; c Î
r +1
xa
¡ x a sin ( x ) + c; c Î
) x a sin ( x
1
)cos2 ( x
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
xa
1
}; n Î ¥* - {1
xn
2
xa
¡x a -cos ( x) + c;c Î
= )x a1+ tan2 ( x
¡ x a tan ( x ) + c; c Î
اﻧﻄﻼﻗﺎ ﻣﻦ اﻟﻘﺮاءة اﻟﻌﻜﺴﯿﺔ ﻟﻠﻌﻤﻠﯿﺎت ﻋﻠﻰ اﻟﺪوال اﻟﻤﺸﺘﻘﺔ
ﺣﺼﻠﻨﺎ ﻋﻠﻰ اﻟﺠﺪول أﺳﻔﻠﮫ:
اﻟﺪاﻟﺔ fﻣﻌﺮﻓﺔ ﻋﻠﻰ ﻣﺠﺎل Iداﻟﺔ أﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ fﻋﻠﻰ
اﻟﻤﺠﺎل I
u+v
u ¢ + v¢
uv
uv¢ + vu ¢
1 n+1
*u¢u n ; n Î ¥
u
n +1
u¢
u2
}¢ r ;r Î ¤* -{-1
uu
)
(
-
u 1 u r +1
r +1
u¢
u
2 u
u ¢v - uv¢
v2
u
v
¡x au¢( ax +b) ; aΡ*;bÎ
2
)- 1
2
أﺟﻮﺑﺔ f ( x) = 5x4 + 3x +1 (1:
اذن F ( x ) = 5 ´ 1 x5 + 3´ 1 x2 + 1x + kﺣﯿﺚ ¡ k Î
5
2
+ cos x + sin x -1(2
1
x
= )f ( x
اذن F ( x ) = 2 x + sin x - cos x - x + kﺣﯿﺚ ¡ k Î
f ( x) = sin x + x cos x = x¢sin x + x (sin x)¢ (3
اذن F ( x) = x ´ sin x + kﺣﯿﺚ ¡ k Î
1
3
3
f ( x ) = ( 2 x - 1) = ( 2 x - 1)¢ ( 2 x - 1) (4
2
1 1
3+1
´ = )F ( x
اذن ( 2x -1) + k
2 3 +1
ﺣﯿﺚ ¡ k Î
وﻣﻨﮫ F ( x) = 1 ( 2x -1)4 + kﺣﯿﺚ ¡ k Î
8
(5
x
2
)( x 2 - 1
f ( x ) = -ﯾﻌﻨﻲ
( x - 1)¢
2
2
)- 1
2
(x
f ( x) = -
اذن F ( x) = 1 + kﺣﯿﺚ ¡ k Î
2
x -1
ﺗﻤﺮﯾﻦ :1ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ :
3
2
f ( x) = 2cos x -sin x -3 (2 f ( x) = 8x + 4x + x + 6 (1
x2
(5 f ( x ) = ( 4 x + 5 )2 (4 f ( x) =2xsinx+x2 cosx (3
2
)+ 2
3
(x
= )f ( x
أﺟﻮﺑﺔ :
3
2
f ( x) = 8x + 4x + x + 6 (1
1
) x a u ( ax + b
a
ﺣﯿﺚ ¡ k Î
f ( x) = 2cos x -sin x -3(2
f ( x) = 2sinx +cos x -3x +kﺣﯿﺚ ¡ k Î
f ( x) = 2x sin x + x2 cos x = ( x2 )¢ sin x + x2 ( sin x)¢ (3
اذن F ( x) = x2 ´ sin x + kﺣﯿﺚ ¡ k Î
f ( x ) = ( 4 x + 5 ) (4
2
1
2
) ( 4 x + 5)¢ ( 4 x + 5
4
1 1
2+1
´ = )F ( x
اذن ( 4x + 5) + k
4 2 +1
= )f ( x ) = ( 4 x + 5
2
ﺣﯿﺚ ¡ k Î
وﻣﻨﮫ F ( x) = 1 ( 4x + 5)3 + kﺣﯿﺚ ¡ k Î
12
2
(5
2
x
)+ 2
3
(x
= ) f ( xﯾﻌﻨﻲ
ö
÷
÷
÷
ø
æ
¢
3
)1 ç ( x + 2
f (x) = - ç3 ç ( x3 + 2 )2
è
اذن F ( x ) = - 1 1 + kﺣﯿﺚ ¡ k Î
3
3 x +2
ﻣﺜﺎل :ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ :
x (2 f x = 2 2 x + 1 (1
) (
= )f ( x
x2 + 1
1
أﺟﻮﺑﺔ f ( x ) = 2 2 x + 1 = ( 2 x + 1)¢ ( 2 x + 1) 2 (1:
2
(x
= )f ( x
1
1
1
4
1
اذن F ( x) = 8´ x4 + 4´ x3 + x2 + 6x + k = 2x4 + x3 + x2 + 6x + k
4
3
2
3
2
.IIIاﻟﺪوال اﻷﺻﻠﯿﺔ و اﻟﻌﻤﻠﯿﺎت:
1
u
(5 f ( x ) = ( 2 x - 1) (4 f ( x) = sin x + x cos x (3
x
3
اﻷﺳﺘﺎذ :ﻧﺠﯿﺐ ﻋﺜﻤﺎﻧﻲ
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1
اذن
1
+1
( 2x +1) 2 + k
1
+1
2
ﺣﯿﺚ ¡ k Î
= )F ( x
ﺗﻤﺮﯾﻦ :4ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ f
اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [ [1; +¥ﻛﺎﻟﺘﺎﻟﻲ f ( x ) = x x - 1 :
وﻣﻨﮫ F ( x) = 2 ( 2x +1) 2 + kﺣﯿﺚ ¡ k Î
3
3
وﻣﻨﮫ F ( x) = 2 ( 2x +1) 2 = 2 ( 2x + 1)3 + kﺣﯿﺚ ¡ k Î
(2
+ 1)¢
2
(x
2 x2 + 1
3
x
=
= )f ( x
x2 + 1
ﻧﻌﻠﻢ أن x Î [1; +¥[ :اذن x ³ 1 :ﯾﻌﻨﻲ x - 1 ³ 0
اذن F ( x) = x2 + 1 + kﺣﯿﺚ ¡ k Î
ﺗﻤﺮﯾﻦ :2ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ :
2
(2 f ( x ) = x x 2 + 1 (1
f ( x ) = sin ( 4 x - 1) (3 f ( x ) = x
8 + x3
f ( x ) = ( sin x) cos x (5 f ( x ) = cos ( 2 x + 8 ) (4
= f ( x ) = x x2 + 1
1
3
+1
اذن
1 1
1
x2 + 1) 2 + k = ( x2 + 1) 2 + k
(
21
3
+1
2
)
= )F ( x
(
3
(2
3 ¢
) 2 (8 + x
3 2 8 + x3
=
x2
1
2+1
وﻣﻨﮫ ( sin x) + k
2 +1
ﯾﻌﻨﻲ k Î ¡ F ( x) = 1 ( sin x)3 + k
3
= )F ( x
( x -1) 2
1
5
3
اذن
2
2
( x -1) 2 + ( x -1) 2 + k
5
3
ax + 2ax + a + b
a ( x + 1) + b
=
)( x + 1
ﺑﺎﻟﻤﻘﺎرﻧﺔ ﻣﻊ اﻟﻜﺘﺎﺑﺔ :
2
2
)( x + 1
x2 + 2x
2
ìa = 1
)( x + 1
b
)( x + 1
x +1
)+ 4
2
(x
=
)ax + b + c ( x 2 + 4
2
= )F (1
(2
+5
20 x
2
)( x2 + 4
)+ 4
2
(x
1
( x + 1)2
f ( x) = 1 -
= +c
ax + b
2
)+ 4
2
(x
5 x + 40 x + 20 x + 80
2
)+ 4
4
2
(x
= )f ( x
ìc = 5
ïïc = 5وﻣﻨﮫ :
+5
í
ïa = 20
ïîb = 0
= ) f ( xﯾﻌﻨﻲ
+5
+ 4 )¢
2
)+ 4
2
20 x
2
)+ 4
2
(x
= )f ( x
(x
f ( x ) = 10
(x
2
["x Î [0; +¥
F ( 0 ) = 5 (3ﯾﻌﻨﻲ - 10 + k = 5ﯾﻌﻨﻲ
4
وﻣﻨﮫ "x Î [0; +¥[ F ( x) = - 210 + 5x + 15 :
x +4
2
5
k = 5+
2
ﯾﻌﻨﻲ
15
=k
2
f ( x) = 1 -
["x Î [0; +¥
اﻷﺳﺘﺎذ :ﻧﺠﯿﺐ ﻋﺜﻤﺎﻧﻲ
= )f ( x
= )f ( x
ìc = 5
ïï8c = 40ﯾﻌﻨﻲ
í
ïa = 20
îïb + 16c = 80
x +4
1
(2
( x + 1)2
وﻣﻨﮫ + k :
k Î ¡ F ( x) = x + 1
2
وﻣﻨﮫ k Î ¡ F ( x) = - 10 + 5x + k :
2
f ( x ) = 1 -ﯾﻌﻨﻲ
)( x + 1
2
(x
ﺑﺎﻟﻤﻘﺎرﻧﺔ ﻣﻊ اﻟﻜﺘﺎﺑﺔ :
f ( x) = a +
ﻧﺠﺪ أن :
í
ï b = -1
î
( x + 1)¢
2
)+ 4
2
(x
2
= )f ( x
ìa = 1
)+ 4
2
ﺑﺤﯿﺚ 5
2
ï 2a = 2ﯾﻌﻨﻲ ï a = 1وﻣﻨﮫ :
í
ï
îa + b = 0
2
.2ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ f
.3ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ Fﻟﻠﺪاﻟﺔ fﺑﺤﯿﺚ F ( 0 ) = c
أﺟﻮﺑﺔ :
أﺟﻮﺑﺔ(1:
=
1
+1
2
3
+1
2
= )F ( x
= )f ( x
(x
2
ﻧﺠﺪ أن :
2
1
( x -1) 2
1
+
+1
3
(
)f ( x) = ( x -1
.1ﺣﺪد اﻷﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ aو bو c
ﺑﺤﯿﺚ "x Î [ 0; +¥[ f ( x ) = ax + b + c :
"x Î [ 0; +¥ [ f ( x ) = a +
2
= +k
3
2
+1
1
2
5
) cx4 + 8cx2 + ax + ( b +16c
= )f ( x
.2ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ Fﻟﻠﺪاﻟﺔ f
3
)+ 4
2
ﻛﺎﻟﺘﺎﻟﻲ :
2
)( x + 1
.1ﺣﺪد اﻟﻌﺪدﯾﻦ اﻟﺤﻘﯿﻘﯿﯿﻦ aو bﺑﺤﯿﺚ:
2
1
2
ax + b + cx4 + 8cx2 + 16c
ﺗﻤﺮﯾﻦ :3ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ fاﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [[0; +¥
2
)+ ( x -1) = ( x -1) + ( x -1) = ( x -1)¢ ( x -1) + ( x -1)¢ ( x -1
3
2
2
3
b
)
1
3 2
5 x 4 + 40 x 2 + 20 x + 80
f ( x ) = sin ( 4 x - 1) (3اذن k Î ¡ F ( x) = - 1 cos ( 4x - 1) + k
4
1
f ( x ) = cos ( 2 x + 8) (4اذن k Î ¡ F ( x) = sin ( 2x + 8) + k
2
2
2
f ( x ) = ( sin x ) cos x (5
ﯾﻌﻨﻲ )f ( x ) = ( sin x )¢ ( sin x
)( x + 1
3
)( x-1
ﺗﻤﺮﯾﻦ7ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ fاﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ ¡ ﻛﺎﻟﺘﺎﻟﻲ:
اذن F ( x) = 2 8 + x3 + kﺣﯿﺚ ¡ k Î
x2 + 2x
1
2
)( x - 1
3
ﯾﻌﻨﻲ
= )f ( x
3
== ) f ( x
8 + x3
اذن + x -1=( x-1) ´ x-1+ x-1= x x-1-1 x-1+ x-1 = x x-1 :
وﻣﻨﮫ F ( x) = 2 ( x -1)5 + 2 ( x -1)3 + kﺣﯿﺚ ¡ k Î
وﻣﻨﮫ F ( x) = 1 x2 +1 + kﺣﯿﺚ ¡ k Î
3
وﻣﻨﮫ x -1 = x -1 :
+ x - 1 (2
2
1
1 2
أﺟﻮﺑﺔ x + 1)¢ ( x2 + 1) 2 (1:
(
2
3
.1ﺑﯿﻦ أن ( x - 1) + x - 1 :
.2ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ Fﻟﻠﺪاﻟﺔ fﺑﺤﯿﺚ F ( 2 ) = 1
أﺟﻮﺑﺔ:
3
2
( x -1) + x -1 = ( x -1) ´ x -1+ x -1 = x -1 ´ x -1+ x -1(1
= )f ( x
3
3
[ "x Î [1; +¥
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