مــــسـألة
لتكن
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F1@
الدالة
العددية للمتغير الحقيقي
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadIhaaaa@3703@
المعرفة على
ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabl2riHcaa@3776@
بما يلي:
f(
x
)={
3
x
2
−4x
x
,x≥0
1−
1−x
3
,x<0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgadaqada
qaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaGabaabaeqabaGaaG4m
aiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI0aGaamiEam
aakaaabaGaamiEaaWcbeaakiaacYcacaWG4bGaeyyzImRaaGimaaqa
aiaaigdacqGHsisldaGcbaqaaiaaigdacqGHsislcaWG4baaleaaca
aIZaaaaOGaaiilaiaadIhacqGH8aapcaaIWaaaaiaawUhaaaaa@4EDC@
، و ليكن
(
C
f
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4qamaaBaaaleaacaWGMbaabeaaaOGaayjkaiaawMcaaaaa@3978@
منحناها في م. م .م.
(
O,
i
→
,
j
→
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4taiaacYcadaWhcaqaaiaadMgaaiaawEniaiaacYcadaWhcaqaaiaa
dQgaaiaawEniaaGaayjkaiaawMcaaaaa@3F08@
.
I)- الجزء
الأول:(7.5ن)
(1ن) 1)- أ- أحسب النهايتين :
lim
x→+∞
f(
x
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWG4bGaeyOKH4Qaey4kaSIaeyOhIuka
beaakiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaa@41C7@
و
lim
x→−∞
f(
x
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWG4bGaeyOKH4QaeyOeI0IaeyOhIuka
beaakiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaaaaa@41D2@
.
(1ن) ب- بين أن الدالة
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F1@
متصلة في الصفر.
(1ن) 2)-
أ- أدرس قابلية اشتقاق
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F1@
في
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaaicdaaaa@36C0@
على اليمين ثم أول
النتيجة هندسيا .
(1ن) ب- أدرس قابلية اشتقاق
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F1@
في
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaaicdaaaa@36C0@
على اليسار ثم أول
النتيجة هندسيا .
(1ن) 3)- أ- بين أن:
f'(
x
)=6
x
(
x
−1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgacaGGNa
WaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaaGOnamaakaaa
baGaamiEaaWcbeaakmaabmaabaWaaOaaaeaacaWG4baaleqaaOGaey
OeI0IaaGymaaGaayjkaiaawMcaaaaa@415D@
لكل
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadIhaaaa@3703@
من المجال
]
0,+∞ [
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaajmcabaGaaG
imaiaacYcacqGHRaWkcqGHEisPaiaaw2facaGLBbaaaaa@3BDB@
.
(1ن) ب- استنتج رتابة الدالة
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F1@
على كل من المجالين
[
0,1 ]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaadmaabaGaaG
imaiaacYcacaaIXaaacaGLBbGaayzxaaaaaa@3A1D@
و
[
1,+∞ [
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaajicabaGaaG
ymaiaacYcacqGHRaWkcqGHEisPaiaawUfacaGLBbaaaaa@3BD9@
.
(1ن) 4)- أ- أحسب
f'(
x
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgacaGGNa
WaaeWaaeaacaWG4baacaGLOaGaayzkaaaaaa@3A22@
لكل
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadIhaaaa@3703@
من المجال
]
−∞,0 [
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaajmcabaGaey
OeI0IaeyOhIuQaaiilaiaaicdaaiaaw2facaGLBbaaaaa@3BE6@
.
(0.5ن) ب-
استنتج أن الدالة
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F1@
تزايدية قطعا على المجال
]
−∞,0 ]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaajmdabaGaey
OeI0IaeyOhIuQaaiilaiaaicdaaiaaw2facaGLDbaaaaa@3BF8@
.
II)- الجزء
الثاني:(6.5ن)
(1ن) 1)- أ- حل في المجال
[
0,+∞ [
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaajicabaGaaG
imaiaacYcacqGHRaWkcqGHEisPaiaawUfacaGLBbaaaaa@3BD8@
المعادلة
f(
x
)=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgadaqada
qaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@3B37@
.
(1ن) ب- بين أن
(
C
f
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4qamaaBaaaleaacaWGMbaabeaaaOGaayjkaiaawMcaaaaa@3978@
يقبل نقطة انعطاف
I
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadMeaaaa@36D4@
أفصولها موجب و حدد
إحداثيتيها.
(1ن) 2)- أ- أدرس الفرعين اللانهائيين للمنحنى
(
C
f
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4qamaaBaaaleaacaWGMbaabeaaaOGaayjkaiaawMcaaaaa@3978@
.
(1.5ن) ب- أنشئ منحنى الدالة
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F1@
في المعلم
(
O,
i
→
,
j
→
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4taiaacYcadaWhcaqaaiaadMgaaiaawEniaiaacYcadaWhcaqaaiaa
dQgaaiaawEniaaGaayjkaiaawMcaaaaa@3F08@
.
(1.5ن) 3)- أ- تحقق من الدالة
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F1@
تقبل دوال أصلية على
ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabl2riHcaa@3776@
ثم حددها.
(0.5ن) ب- استنتج الدالة الأصلية
G
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadEeaaaa@36D2@
للدالة
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F1@
على
ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabl2riHcaa@3776@
التي تحقق
G(
1
)=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadEeadaqada
qaaiaaigdaaiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@3AD6@
.
III)- الجزء
الثالث:(6ن)
لتكن
g
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadEgaaaa@36F2@
قصور الدالة
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F1@
على المجال
I=]
−∞,0 ]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadMeacqGH9a
qpdaqcZaqaaiabgkHiTiabg6HiLkaacYcacaaIWaaacaGLDbGaayzx
aaaaaa@3DCC@
.
(1ن) 1)-
بين أن
g
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadEgaaaa@36F2@
تقبل دالة عكسية
g
−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadEgadaahaa
WcbeqaaiabgkHiTiaaigdaaaaaaa@38C7@
معرفة على مجال
J
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadQeaaaa@36D5@
يجب تحديده.
(1ن) 2)-
أنشئ منحنى الدالة
g
−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadEgadaahaa
WcbeqaaiabgkHiTiaaigdaaaaaaa@38C7@
في نفس المعلم السابق
(
O,
i
→
,
j
→
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4taiaacYcadaWhcaqaaiaadMgaaiaawEniaiaacYcadaWhcaqaaiaa
dQgaaiaawEniaaGaayjkaiaawMcaaaaa@3F08@
.
(1ن) 3)- أ- حل في المجال
]
−∞,0 ]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaajmdabaGaey
OeI0IaeyOhIuQaaiilaiaaicdaaiaaw2facaGLDbaaaaa@3BF8@
المعادلة
g(
x
)=−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadEgadaqada
qaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcqGHsislcaaIXaaaaa@3C26@
، ثم
استنتج
g
−1
(
−1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadEgadaahaa
WcbeqaaiabgkHiTiaaigdaaaGcdaqadaqaaiabgkHiTiaaigdaaiaa
wIcacaGLPaaaaaa@3C02@
(1ن) ب- بين أن الدالة
g
−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadEgadaahaa
WcbeqaaiabgkHiTiaaigdaaaaaaa@38C7@
قابلة للاشتقاق في
−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabgkHiTiaaig
daaaa@37AE@
،ثم
أحسب
(
g
−1
)
′
(
−1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4zamaaCaaaleqabaGaeyOeI0IaaGymaaaaaOGaayjkaiaawMcaamaa
CaaaleqabaGccWaGGBOmGikaamaabmaabaGaeyOeI0IaaGymaaGaay
jkaiaawMcaaaaa@40AA@
.
(1ن) 4)-
حدد التعبير
g
−1
(
x
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadEgadaahaa
WcbeqaaiabgkHiTiaaigdaaaGcdaqadaqaaiaadIhaaiaawIcacaGL
Paaaaaa@3B57@
لكل
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadIhaaaa@3703@
من المجال
J
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadQeaaaa@36D5@
.
(1ن) 5)- أحسب
(
g
−1
)
′
(
x
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4zamaaCaaaleqabaGaeyOeI0IaaGymaaaaaOGaayjkaiaawMcaamaa
CaaaleqabaGccWaGGBOmGikaamaabmaabaGaamiEaaGaayjkaiaawM
caaaaa@3FFF@
لكل
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadIhaaaa@3703@
من
J
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadQeaaaa@36D5@
ثم أحسب بطريقة أخرى
(
g
−1
)
′
(
−1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaam
4zamaaCaaaleqabaGaeyOeI0IaaGymaaaaaOGaayjkaiaawMcaamaa
CaaaleqabaGccWaGGBOmGikaamaabmaabaGaeyOeI0IaaGymaaGaay
jkaiaawMcaaaaa@40AA@
.
Aguelmous le 02/12/2008 Mohamed _iaalou@Yahoo.fr
|