·
التمرين الأول:
لتكن
f
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F0@
الدالة العددية المعرفة
على
ℝ
∗
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabl2riHoaaCa
aaleqabaGaey4fIOcaaaaa@3891@
بما يلي:
f(
x
)=
x
2
E(
1
x
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfKCHjwAJbqefeetbDvsTmtlXaWexLMBbX
gBcf2CPn2qVrwzqf2zLnharyavP1wzZbItLDhis9wBH5garmWu51My
VXgaryWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC
0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi
0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeWabmGadmWadmWabi
WacmabdiWafmaakeaacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzk
aaGaeyypa0JaamiEamaaCaaaleqabaGaaGOmaaaakiaadweadaqada
qaamaalaaabaGaaGymaaqaaiaadIhaaaaacaGLOaGaayzkaaaaaa@4657@
حيث
E(
x
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadweadaqada
qaaiaadIhaaiaawIcacaGLPaaaaaa@3956@
هو الجزء الصحيح للعدد
الحقيقي
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadIhaaaa@3703@
.
1)- بين أن:
∀x∈]
−1,1 [−{ 0 },|
f(
x
) |≤2| x |
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabgcGiIiaadI
hacqGHiiIZdaqcJaqaaiabgkHiTiaaigdacaGGSaGaaGymaaGaayzx
aiaawUfaaiabgkHiTmaacmaabaGaaGimaaGaay5Eaiaaw2haaiaacY
cadaabdaqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaaaiaa
wEa7caGLiWoacqGHKjYOcaaIYaWaaqWaaeaacaWG4baacaGLhWUaay
jcSdaaaa@502B@
.
2)- استنتج
أن الدالة
f
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F0@
تقبل تمديدا بالاتصال في
الصفر.
·
التمرين الثاني:
A)-1)- بين
انه إذا كانت
lim
x→
x
0
f(
x
)=l≠0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWG4bGaeyOKH4QaamiEamaaBaaameaa
caaIWaaabeaaaSqabaGccaWGMbWaaeWaaeaacaWG4baacaGLOaGaay
zkaaGaeyypa0JaamiBaiabgcMi5kaaicdaaaa@45D9@
فإنه:
∃r>0/∀x]
x
0
−r,
x
0
+r [−{
x
0
}:f(
x
).l>0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabgoGiKiaadk
hacqGH+aGpcaaIWaGaai4laiabgcGiIiaadIhadaqcJaqaaiaadIha
daWgaaWcbaGaaGimaaqabaGccqGHsislcaWGYbGaaiilaiaadIhada
WgaaWcbaGaaGimaaqabaGccqGHRaWkcaWGYbaacaGLDbGaay5waaGa
eyOeI0YaaiWaaeaacaWG4bWaaSbaaSqaaiaaicdaaeqaaaGccaGL7b
GaayzFaaGaaiOoaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaa
caGGUaGaamiBaiabg6da+iaaicdaaaa@5310@
2)- استنتج
أنه إذا كانت
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgaaaa@36EF@
متصلة في
x
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadIhadaWgaa
WcbaGaaGimaaqabaaaaa@37E7@
فإنه:
∃r>0/∀x]
x
0
−r,
x
0
+r [:f(
x
)f(
x
0
)>0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabgoGiKiaadk
hacqGH+aGpcaaIWaGaai4laiabgcGiIiaadIhadaqcJaqaaiaadIha
daWgaaWcbaGaaGimaaqabaGccqGHsislcaWGYbGaaiilaiaadIhada
WgaaWcbaGaaGimaaqabaGccqGHRaWkcaWGYbaacaGLDbGaay5waaGa
aiOoaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacaWGMbWaae
WaaeaacaWG4bWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaGa
eyOpa4JaaGimaaaa@50C3@
B)- لتكن
f
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F0@
دالة عددية معرفة على
ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabl2riHcaa@3775@
بحيث:
∀(
x,y
)∈
ℝ
2
,f(
x+y
)=f(
x
)+f(
y
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabgcGiImaabm
aabaGaamiEaiaacYcacaWG5baacaGLOaGaayzkaaGaeyicI4SaeSyh
He6aaWbaaSqabeaacaaIYaaaaOGaaiilaiaadAgadaqadaqaaiaadI
hacqGHRaWkcaWG5baacaGLOaGaayzkaaGaeyypa0JaamOzamaabmaa
baGaamiEaaGaayjkaiaawMcaaiabgUcaRiaadAgadaqadaqaaiaadM
haaiaawIcacaGLPaaaaaa@4DBC@
.
1)- أحسب
f(
0
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgadaqada
qaaiaaicdaaiaawIcacaGLPaaaaaa@3932@
،و بين أن الدالة
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgaaaa@36EF@
دالة فردية.
2)- أ-
بين بالترجع أن :
∀n∈ℕ,f(
n
)=nf(
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabgcGiIiaad6
gacqGHiiIZcqWIvesPcaGGSaGaamOzamaabmaabaGaamOBaaGaayjk
aiaawMcaaiabg2da9iaad6gacaWGMbWaaeWaaeaacaaIXaaacaGLOa
Gaayzkaaaaaa@43F6@
و استنتج أن
∀n∈ℤ,f(
n
)=nf(
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabgcGiIiaad6
gacqGHiiIZcqWIKeIOcaGGSaGaamOzamaabmaabaGaamOBaaGaayjk
aiaawMcaaiabg2da9iaad6gacaWGMbWaaeWaaeaacaaIXaaacaGLOa
Gaayzkaaaaaa@4402@
.
ب-
بين أن:
∀n∈
ℕ
∗
,f(
1
n
)=
1
n
f(
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabgcGiIiaad6
gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaakiaacYcacaWG
MbWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWGUbaaaaGaayjkaiaawM
caaiabg2da9maalaaabaGaaGymaaqaaiaad6gaaaGaamOzamaabmaa
baGaaGymaaGaayjkaiaawMcaaaaa@46B2@
، و استنتج
أن
∀r∈ℚ,f(
r
)=rf(
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabgcGiIiaadk
hacqGHiiIZcqWIAecPcaGGSaGaamOzamaabmaabaGaamOCaaGaayjk
aiaawMcaaiabg2da9iaadkhacaWGMbWaaeWaaeaacaaIXaaacaGLOa
Gaayzkaaaaaa@4406@
.
3)-
نفترض أن
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgaaaa@36EF@
متصلة في الصفر، و لتكن
g
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadEgaaaa@36F0@
الدالة المعرفة على
ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabl2riHcaa@3775@
بما يلي:
g(
x
)=f(
x
)−xf(
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadEgadaqada
qaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaWGMbWaaeWaaeaacaWG
4baacaGLOaGaayzkaaGaeyOeI0IaamiEaiaadAgadaqadaqaaiaaig
daaiaawIcacaGLPaaaaaa@4307@
.
أ- بين
أن الدالة
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgaaaa@36EF@
متصلة على
ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabl2riHcaa@3774@
.
ب-
تحقق أن :
g(
r
)=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadEgadaqada
qaaiaadkhaaiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@3B30@
،لكل
r
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadkhaaaa@36FB@
من
ℚ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiablQriKcaa@3774@
.
ج-
برهن بالخلف أن
g(
x
)=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadEgadaqada
qaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaaIWaaaaa@3B36@
،لكل
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadIhaaaa@3701@
من
ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabl2riHcaa@3774@
، و استنتج
تعبير الدالة
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgaaaa@36EF@
.
4)-
حدد جميع الدوال
f
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgaaaa@36EF@
المعرفة على
ℝ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiabl2riHcaa@3774@
و المتصلة في الصفر و التي
تحقق:
∀(
x,y
)∈
ℝ
2
,f(
x+y
)=f(
x
)+f(
y
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabgcGiImaabm
aabaGaamiEaiaacYcacaWG5baacaGLOaGaayzkaaGaeyicI4SaeSyh
He6aaWbaaSqabeaacaaIYaaaaOGaaiilaiaadAgadaqadaqaaiaadI
hacqGHRaWkcaWG5baacaGLOaGaayzkaaGaeyypa0JaamOzamaabmaa
baGaamiEaaGaayjkaiaawMcaaiabgUcaRiaadAgadaqadaqaaiaadM
haaiaawIcacaGLPaaaaaa@4DBC@
و
f(
2012
)=
2012
2011
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqabeWadmqadqWaaOqaaiaadAgadaqada
qaaiaaikdacaaIWaGaaGymaiaaikdaaiaawIcacaGLPaaacqGH9aqp
caaIYaGaaGimaiaaigdacaaIYaWaaWbaaSqabeaacaaIYaGaaGimai
aaigdacaaIXaaaaaaa@4271@
.
·
التمرين الثالث:
لتكن
f
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgaaaa@36F0@
الدالة العددية
المعرفة بما يلي:
f
n
(
x
)=tan(
π
2
x
)−
π
2nx
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaadAgadaWgaa
WcbaGaamOBaaqabaGcdaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH
9aqpciGG0bGaaiyyaiaac6gadaqadaqaamaalaaabaGaeqiWdahaba
GaaGOmaaaacaWG4baacaGLOaGaayzkaaGaeyOeI0YaaSaaaeaacqaH
apaCaeaacaaIYaGaamOBaiaadIhaaaaaaa@48EB@
،حيث
n∈
ℕ
∗
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaad6gacqGHii
IZcqWIvesPdaahaaWcbeqaaiabgEHiQaaaaaa@3B05@
.
1)- بين أن
∀n∈
ℕ
∗
,∃!
α
n
∈]
0,1 [/
f
n
(
α
n
)=0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiabgcGiIiaad6
gacqGHiiIZcqWIvesPdaahaaWcbeqaaiabgEHiQaaakiaacYcacqGH
dicjcaGGHaGaeqySde2aaSbaaSqaaiaad6gaaeqaaOGaeyicI48aaK
WiaeaacaaIWaGaaiilaiaaigdaaiaaw2facaGLBbaacaGGVaGaamOz
amaaBaaaleaacaWGUbaabeaakmaabmaabaGaeqySde2aaSbaaSqaai
aad6gaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaaGimaaaa@4F6A@
.
2)- بين أن
المتتالية
(
α
n
)
n>0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaeq
ySde2aaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqa
aiaad6gacqGH+aGpcaaIWaaabeaaaaa@3D38@
تناقصية.
3)- بين أن
المتتالية
(
α
n
)
n>0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaabmaabaGaeq
ySde2aaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaWaaSbaaSqa
aiaad6gacqGH+aGpcaaIWaaabeaaaaa@3D38@
متقاربة ثم أحسب
lim
n→+∞
α
n
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4Qaey4kaSIaeyOhIuka
beaakiabeg7aHnaaBaaaleaacaWGUbaabeaaaaa@410A@
.
4)- حدد
lim
n→+∞
(
n
α
n
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4Qaey4kaSIaeyOhIuka
beaakmaabmaabaWaaOaaaeaacaWGUbaaleqaaOGaeqySde2aaSbaaS
qaaiaad6gaaeqaaaGccaGLOaGaayzkaaaaaa@43B5@
.
·
التمرين الرابع:
أحسب النهايات
التالية:
·
lim
x→1
x
n
+
x
n−1
+
x
n−2
+...+x−n
(
2−x
)
n
−1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWG4bGaeyOKH4QaaGymaaqabaGcdaWc
aaqaaiaadIhadaahaaWcbeqaaiaad6gaaaGccqGHRaWkcaWG4bWaaW
baaSqabeaacaWGUbGaeyOeI0IaaGymaaaakiabgUcaRiaadIhadaah
aaWcbeqaaiaad6gacqGHsislcaaIYaaaaOGaey4kaSIaaiOlaiaac6
cacaGGUaGaey4kaSIaamiEaiabgkHiTiaad6gaaeaadaqadaqaaiaa
ikdacqGHsislcaWG4baacaGLOaGaayzkaaWaaWbaaSqabeaacaWGUb
aaaOGaeyOeI0IaaGymaaaaaaa@5610@
، حيث
n∈
ℕ
∗
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaad6gacqGHii
IZcqWIvesPdaahaaWcbeqaaiabgEHiQaaaaaa@3B05@
.
·
lim
x→
π
2
(
1−sinx
)(
1−
sin
2
x
)×...(
1−
sin
n
x
)
cos
2n
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaamaaxababaGaci
iBaiaacMgacaGGTbaaleaacaWG4bGaeyOKH46aaSaaaeaacqaHapaC
aeaacaaIYaaaaaqabaGcdaWcaaqaamaabmaabaGaaGymaiabgkHiTi
GacohacaGGPbGaaiOBaiaadIhaaiaawIcacaGLPaaadaqadaqaaiaa
igdacqGHsislciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaa
GccaWG4baacaGLOaGaayzkaaGaey41aqRaaiOlaiaac6cacaGGUaWa
aeWaaeaacaaIXaGaeyOeI0Iaci4CaiaacMgacaGGUbWaaWbaaSqabe
aacaWGUbaaaOGaamiEaaGaayjkaiaawMcaaaqaaiGacogacaGGVbGa
ai4CamaaCaaaleqabaGaaGOmaiaad6gaaaGccaWG4baaaaaa@5FAE@
، حيث
n∈
ℕ
∗
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb
a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr
0=vqpWqadeWabmGadiWaceqadeWadmqadqWaaOqaaiaad6gacqGHii
IZcqWIvesPdaahaaWcbeqaaiabgEHiQaaaaaa@3B05@
.