الدوال اللوغاريتمية
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اﻟﺪوال اﻟﻠﻮﻏﺎرﻳﺘﻤﻴﺔ
اﻟﺜﺎﻧﻴﺔ ﺑﻜﺎﻟﻮرﻳﺎ ﻋﻠﻮم ﺗﺠﺮﻳﺒﻴﺔ أﺣﺴﺐ اﻟﻨﻬﺎﻳﺎت اﻟﺘﺎﻟﻴﺔ : اﻟﺘﻤﺮﻳﻦ : 1 (a (c (e (g (i
lim 3x − 5ln x
∞x →+
(q
;
lim x − ln (1 + x 2 ) (f
;
(h
) lim+ x ( ln x
;
(j
;
(l
x →0
2
x →0
ln x x →+∞ x 2 lim
2
(o
;
(d
ln x x →+∞ 1 + ln x
1 − ln x x →0 ln x 1 lim+ + ln x x →0 x lim+ x 2 ln x lim+
(k (m
;
(b
lim ( ln x ) + ln x
) ( ln x
lim
∞x →+
x ) ln (1 + ln x ; lim x →1 x 2 −1 ) ln ( ln x ∞x →+ x lim
اﻟﺘﻤﺮﻳﻦ : 2
2
∞x →+
lim x 3 ln x
x → 0+
x →0
ln x x →+∞ x 3 lim
;
(n
ﻡﻌﻠﻢ ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ ) . (O, i, j أ -أدرس اﻟﻔﺮوع اﻟﻼﻥﻬﺎﺋﻴﺔ ﻟﻠﻤﻨﺤﻨﻰ
lim
∞x →+
x ) ln (1 − sin x lim ) x →0 ln (1 + tan x
(p
ln x x
(r
ب -أدرس اﻟﻮﺿﻊ اﻟﻨﺴﺒﻲ ﻟﻠﻤﻨﺤﻨﻰ
⎞ ⎛ x + 2x + 1 ⎜ lim ln ⎟ (a 3 ∞x →+ x 5 + ⎝ ⎠ ) lim x + ln ( x 2
∞x →−
(e
⎞ 2 ⎛ 1 − 2x ⎜ ln ⎟ ⎠ x ⎝ 1 + 2x
(g
1 + 3ln x x
lim x →0
;
(b
;
(d
;
(f
1 ﺟـ -ﺏﻴﻦ أﻥﻪ ﻳﻮﺟﺪ ﻋﺪد ﺣﻘﻴﻘﻲ αﺣﻴﺚ< α < 1 : 2 3 ﺏﻴﻦ أﻥﻪ ﻳﻮﺟﺪ ﻋﺪد ﺣﻘﻴﻘﻲ βﺣﻴﺚ − < β < −1 :و f ( β ) = 02 د -أرﺱﻢ ) . ( Cf
lim+
x →0
; (h
lim
x →0+
) ﻥﺄﺧﺬ 0, 69 < ln(2) < 0, 70 :و ( 1, 09 < ln(3) < 1,10
اﻟﺘﻤﺮﻳﻦ :5
⎞⎛ x +2 ⎜ lim x ln ⎟ ⎠ ⎝ x )ln(x − 2 lim x →3 x −3
ﻥﻌﺘﺒﺮ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ fﻟﻤﺘﻐﻴﺮ ﺣﻘﻴﻘﻲ xﺣﻴﺚ :
∞x →+
2
Arc tan ( ln x ) + x
ﻟﻴﻜﻦ ) ( Cf
lim x →0 x >0
) ln ( cos x (i ∞x →+ x →0 x2 )ln ( x 2 − 2x + 2 ) ln (1 − x 3 lim ; (l lim (k ∞x →+ ∞x →− x x أﺣﺴﺐ اﻟﻨﻬﺎﻳﺎت اﻟﺘﺎﻟﻴﺔ : اﻟﺘﻤﺮﻳﻦ : 3 2 2 lim+ ln ( x − x − 2 ) − ln ( x + 5x − 14 ) (a ;
lim
) lim Arc tan ( ln x
(j
x →2
lim ln (ex 2 − x − 2 ) − ln ( x 2 + x + 7 ) (b
∞x →+
2 1 + (c x +2 4
lim ln(x + 2) − ln x −
∞x →+
⎞ 1 ⎛ 1− x 2 ⎜ ln ⎟ (d x →0 x 2 ⎠ ⎝ cos x
; lim
(f
lim x 3 − 2x − ln x
ﻣﺤﻤﺪ اﻟﺤﻴﺎن
∞x →+
(e ;
(g
x −1 x
ln
( Cfواﻟﻤﺴﺘﻘﻴﻢ ذي اﻟﻤﻌﺎدﻟﺔ :
و . f (α ) = 0
⎞ ⎛ 1 ⎟ lim ln ⎜ 2 ∞x →+ ⎠ ⎝ x +1
π
)
)
. ( Cf
. y = −x
أﺣﺴﺐ اﻟﻨﻬﺎﻳﺎت اﻟﺘﺎﻟﻴﺔ : 3
(c
) lim+ x ( ln x
) ( ln x
(i
)
(
lim ln x − x 2 − 3x + 2
∞x →+
اﻟﺘﻤﺮﻳﻦ : 4ﻥﻌﺘﺒﺮ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ fﻟﻤﺘﻐﻴﺮ ﺣﻘﻴﻘﻲ xﺣﻴﺚ : ⎞⎛ 1 ⎟ f ( x) = − x + ln ⎜1 + ⎠⎝ x .1ﺣﺪد Dﺣﻴﺰ ﺕﻌﺮﻳﻒ اﻟﺪاﻟﺔ . f .2أﺣﺴﺐ ﻥﻬﺎﻳﺎت fﻋﻨﺪ ﻡﺤﺪات . D .3أ -أﺣﺴﺐ ) f ′( xﻟﻜﻞ xﻡﻦ . D ب -أﻋﻂ ﺟﺪول ﺕﻐﻴﺮات اﻟﺪاﻟﺔ . f .4ﻟﻴﻜﻦ ) ( Cfاﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ اﻟﻤﺴﺘﻮى اﻟﻤﻨﺴﻮب إﻟﻰ
lim
3
;
ln x +1 1 ln x − ; lim (h x →0 x x >0
x → 0+
3
اﻷﺳﺘﺎذ :اﻟﺤﻴﺎن
⎞ ⎛ x2 + 3 ⎜ f ( x) = x + 2 ln ⎟ ⎠ ⎝ 4x اﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ اﻟﻤﺴﺘﻮى اﻟﻤﻨﺴﻮب إﻟﻰ
ﻡﻌﻠﻢ ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ ) . (O, i, j .1ﺣﺪد Dﻡﺠﻤﻮﻋﺔ ﺕﻌﺮﻳﻒ اﻟﺪاﻟﺔ . f .2أﺣﺴﺐ : و )lim f ( x ∞x →+
x →0 x >0
.3أ -ﺏﻴﻦ أﻥﻪ ﻟﻜﻞ xﻡﻦ D؛ ﻟﺪﻳﻨﺎ : )( x − 1)( x 2 + 3 x + 6 ′ = )f ( x )x( x 2 + 3 ب -أﻋﻂ ﺟﺪول ﺕﻐﻴﺮات اﻟﺪاﻟﺔ . f .4أ -ﺕﺤﻘﻖ ﻡﻦ أﻥﻪ ﻟﻜﻞ xﻡﻦ D؛ ﻟﺪﻳﻨﺎ : ⎞3 ⎛ )f ( x) = x + 2 ln( x) + 2 ln ⎜1 + 2 ⎟ − 4 ln(2 ⎠ ⎝ x ب -أدرس اﻟﻔﺮوع اﻟﻼﻥﻬﺎﺋﻴﺔ ﻟﻠﻤﻨﺤﻨﻰ ) . ( Cf
ﺟـ -أدرس اﻟﻮﺿﻊ اﻟﻨﺴﺒﻲ ﻟﻠﻤﻨﺤﻨﻰ ) ( Cf
lim
). lim f ( x
واﻟﻤﺴﺘﻘﻴﻢ ذي اﻟﻤﻌﺎدﻟﺔ :
.y=x د -أرﺱﻢ اﻟﻤﻨﺤﻨﻰ ) . ( Cf
x →0 x >0
lim sin x ln x
x → 0+
-1-
.2ب.ع.ت
2
اﻟﺘﻤﺮﻳﻦ : 6ﻟﺘﻜﻦ fاﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ ﻟﻤﺘﻐﻴﺮ ﺣﻘﻴﻘﻲ xاﻟﻤﻌﺮﻓﺔ ﺏﻤﺎﻳﻠﻲ: ⎞ ⎛ x −1 2 ⎜ f ( x) = x + ln ⎟ ) − (1 + ln x ⎠⎝ x +1 : أن ﺏﻴﻦ أ . 2 = ) ∀x ∈ ]0, +∞[ : f ′( x 2 2 .1ﺏﻴﻦ أن ﺣﻴﺰ ﺕﻌﺮﻳﻒ fهﻮ . D = ]−∞, −1[ ∪ ]1, +∞[ : ⎤ ) x 2 ⎡1 + ( ln x ⎣ ⎦ .2ﺏﻴﻦ أن fداﻟﺔ ﻓﺮدﻳﺔ . ⎞⎛1 ب -أﺣﺴﺐ . f ′ ⎜ ⎟ : .3أﺣﺴﺐ lim f ( x) :و ). lim f ( x x →1 ∞x →+ ⎠ ⎝e x >1 ﺟـ -أﻋﻂ ﺟﺪول ﺕﻐﻴﺮات اﻟﺪاﻟﺔ . f .4ﺕﺤﻘﻖ ﻡﻦ أن اﻟﻤﺴﺘﻘﻴﻢ اﻟﺬي ﻡﻌﺎدﻟﺘﻪ y = xﻡﻘﺎرب ﻡﺎﺋﻞ ﻟﻠﻤﻨﺤﻨﻰ .3أ -ﺏﻴﻦ أن : ) ( Cfاﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ ﻓﻲ اﻟﻤﺴﺘﻮى اﻟﻤﻨﺴﻮب إﻟﻰ ﻡﻌﻠﻢ ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ 2 2 (1 + ln x ) ⎡2 + (1 + ln x ) ⎤ ln x ⎣ ⎦ = ) ∀x ∈ ]0, +∞[ : f ′′(x ) . (O, i, j 2 3 ⎤ ) x 3 ⎡1 + ( ln x 2 x +1 ⎣ ⎦ .5أ -ﺏﻴﻦ أن : . ∀x ∈ D : f ′(x ) = 2 ب -اﺱﺘﻨﺘﺞ أن اﻟﻤﻨﺤﻨﻰ ) ( Cfﻳﻘﺒﻞ ﻥﻘﻄﺘﻲ اﻥﻌﻄﺎف Aو Bﻳﻨﺒﻐﻲ x −1 ب -ﺣﺪد ﺟﺪول ﺕﻐﻴﺮات اﻟﺪاﻟﺔ fﻋﻠﻰ اﻟﻤﺠﺎل [∞. ]1, + ﺕﺤﺪﻳﺪ إﺣﺪاﺙﻴﺘﻴﻬﻤﺎ . ﺟـ -ﺣﺪد ﻡﻌﺎدﻟﺘﻲ اﻟﻤﻤﺎﺱﻴﻦ ﻟﻠﻤﻨﺤﻨﻰ ) ( Cfﻓﻲ اﻟﻨﻘﻄﺘﻴﻦ Aو . B .6ﺏﻴﻦ أن اﻟﻤﻨﺤﻨﻰ ) ( Cfﻳﻘﻄﻊ ﻡﺤﻮر اﻷﻓﺎﺻﻴﻞ ﻓﻲ ﻥﻘﻄﺔ أﻓﺼﻮﻟﻬﺎ ب -أﺣﺴﺐ ) lim f (x
و ) . lim f (x
x →0 x >0
3 ﻡﺤﺼﻮر ﺏﻴﻦ 2 .7أﻥﺸﺊ اﻟﻤﻨﺤﻨﻰ ) ) . ( Cfﻥﺄﺧﺬ ln(3) ≈ 1,1 :و ( ln(5) ≈ 1, 6 و .2
اﻟﺘﻤﺮﻳﻦ : 7ﻟﺘﻜﻦ fاﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ ﻟﻤﺘﻐﻴﺮ ﺣﻘﻴﻘﻲ xاﻟﻤﻌﺮﻓﺔ ﺏﻤﺎﻳﻠﻲ: ⎞ ⎛ x2 − 2x + 2 ⎜ f ( x) = 2 x − 2 + ln ⎟ x2 ⎝ ⎠ * . .1ﺏﻴﻦ أن ﻡﺠﻤﻮﻋﺔ ﺕﻌﺮﻳﻒ اﻟﺪاﻟﺔ fهﻲ : .2أﺣﺴﺐ ﻥﻬﺎﻳﺎت fﻋﻨﺪ ﻡﺤﺪات * . )2( x − 1)( x 2 − x + 2 .3أ -ﺏﻴﻦ أﻥﻪ ﻟﻜﻞ xﻡﻦ * ؛ ﻟﺪﻳﻨﺎ : 2 )x( x − 2 x + 2 ب -أﻋﻂ ﺟﺪول ﺕﻐﻴﺮات اﻟﺪاﻟﺔ . f .4ﻟﻴﻜﻦ ) ( Cfاﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ اﻟﻤﺴﺘﻮى اﻟﻤﻨﺴﻮب إﻟﻰ ﻡﻌﻠﻢ = )f ′( x
ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ ) . (O, i, j أ -أدرس اﻟﻔﺮوع اﻟﻼﻥﻬﺎﺋﻴﺔ ﻟﻠﻤﻨﺤﻨﻰ ) . ( Cf
ب -أدرس اﻟﻮﺿﻊ اﻟﻨﺴﺒﻲ ﻟﻠﻤﻨﺤﻨﻰ ) ( Cf
واﻟﻤﺴﺘﻘﻴﻢ ذي اﻟﻤﻌﺎدﻟﺔ :
ﻣﺤﻤﺪ اﻟﺤﻴﺎن
ﺕﻐﻴﺮات اﻟﺪاﻟﺔ . g ب -اﺱﺘﻨﺘﺞ أن . ∀x ∈ ]0, +∞[ : g (x ) ≥ 0 : .2أ -ﺏﻴﻦ أن : ∀x ∈ ]0, +∞[ : h (x ) = 1 + g ( x ) + ( x − 1) ln x ب -ﺏﻴﻦ أن :
∀x ∈ ]0, +∞[ : (x − 1) ln x > 0
.3اﺱﺘﻨﺘﺞ أن :
∀x ∈ ]0, +∞[ : h (x ) > 0
.IIﻥﻌﺘﺒﺮ اﻟﺪاﻟﺔ fاﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [∞ ]0, +ﺏﻤﺎ ﻳﻠﻲ : ) f (x ) = 1 + x ln x − ( ln x
ب -أﺣﺴﺐ ) lim f (x
∞x →+
ﺙﻢ ﺣﺪد اﻟﻔﺮع اﻟﻼﻥﻬﺎﺋﻲ ﻟﻠﻤﻨﺤﻨﻰ ) ( C
ﺏﺠﻮار ∞. + ) h (x = ) . ∀x ∈ ]0, +∞[ : f ′(x .2أ -ﺏﻴﻦ أن : x ب -اﺱﺘﻨﺘﺞ أن اﻟﺪاﻟﺔ fﺕﺰاﻳﺪﻳﺔ ﻗﻄﻌﺎ ﻋﻠﻰ اﻟﻤﺠﺎل [∞. ]0, +
.3أ -ﺣﺪد ﻡﻌﺎدﻟﺔ دﻳﻜﺎرﺕﻴﺔ ﻟﻠﻤﻤﺎس ) ∆ ( ل ) ( Cﻓﻲ اﻟﻨﻘﻄﺔ )A (1,1 ب -ﺕﺤﻘﻖ ﻡﻦ أن : ) ∀x ∈ ]0, +∞[ : f (x ) − x = (ln x − 1) g ( x ﺟـ -ﺣﺪد اﻟﻮﺿﻊ اﻟﻨﺴﺒﻲ ﻟﻠﻤﻨﺤﻨﻰ ) ( Cواﻟﻤﺴﺘﻘﻴﻢ ) ∆ ( .
(
ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ i = 2cm . O , i , j x →0 x >0
g (x ) = x − 1 − ln xو . h (x ) = x + (x − 2) ln x .1أ -أﺣﺴﺐ ) g ′(xﻟﻜﻞ xﻡﻦ اﻟﻤﺠﺎل [∞ ]0, +ﺙﻢ أدرس ﻡﻨﺤﻰ
x →0 x >0
2 ⎤ ) x ⎡1 + ( ln x ⎣ ⎦ وﻟﻴﻜﻦ ) ( Cfاﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ اﻟﻤﺴﺘﻮى اﻟﻤﻨﺴﻮب إﻟﻰ ﻡﻌﻠﻢ
2
.Iﻥﻌﺘﺒﺮ اﻟﺪاﻟﺘﻴﻦ اﻟﻤﻌﺮﻓﺘﻴﻦ ﻋﻠﻰ اﻟﻤﺠﺎل [∞ ]0, +ﺏﻤﺎ ﻳﻠﻲ :
.1أ -أﺣﺴﺐ ) lim f (xﺙﻢ أول اﻟﻨﺘﻴﺠﺔ ﻡﺒﻴﺎﻥﻴﺎ .
= ) f (x
.1أ -ﺏﻴﻦ أن lim x ( ln x ) = 0 :
اﻟﺘﻤﺮﻳﻦ : 9
وﻟﻴﻜﻦ ) ( Cاﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ ﻡﻌﻠﻢ ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ .
اﻟﺘﻤﺮﻳﻦ : 8ﻥﻌﺘﺒﺮ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ fاﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [∞ ]0, +ﺏﻤﺎﻳﻠﻲ :
)
.4أﻥﺸﺊ اﻟﻤﻨﺤﻨﻰ ) . ( Cf
2
. y = 2x − 2 ﺟـ -ﺏﻴﻦ أﻥﻪ ﻳﻮﺟﺪ ﻋﺪد ﺣﻘﻴﻘﻲ وﺣﻴﺪ αﺣﻴﺚ f (α ) = 0 :و 8 ⎡⎤ 1 1 ⎢ α ∈ ⎥ − , −؛ ﻋﻠﻤﺎ أن ln(13) < 3 :و > ). ln(25 3 ⎣⎦ 2 3 د -أرﺱﻢ اﻟﻤﻨﺤﻨﻰ ) ) . ( Cfﺕﺤﺪﻳﺪ ﻥﻘﻄﺔ اﻹﻥﻌﻄﺎف ﻏﻴﺮ ﻡﻄﻠﻮب ( 1
∞x →+
.4أﻥﺸﺊ اﻟﻤﻨﺤﻨﻰ ) ( Cواﻟﻤﺴﺘﻘﻴﻢ ) ∆ ( ﻓﻲ ﻥﻔﺲ اﻟﻤﻌﻠﻢ ) .ﻥﻘﺒﻞ أن
) ﻳﻤﻜﻦ وﺿﻊ . ( x = t 2
اﻟﻤﻨﺤﻨﻰ ) ( Cﻳﻘﺒﻞ ﻥﻘﻄﺔ اﻥﻌﻄﺎف أﻓﺼﻮﻟﻬﺎ ﻡﺤﺼﻮر ﺏﻴﻦ 1و (1,5
-2-
.2ب.ع.ت
2
اﻟﺜﺎﻧﻴﺔ ﺑﻜﺎﻟﻮرﻳﺎ ﻋﻠﻮم ﺗﺠﺮﻳﺒﻴﺔ أﺣﺴﺐ اﻟﻨﻬﺎﻳﺎت اﻟﺘﺎﻟﻴﺔ : اﻟﺘﻤﺮﻳﻦ : 1 (a (c (e (g (i
lim 3x − 5ln x
∞x →+
(q
;
lim x − ln (1 + x 2 ) (f
;
(h
) lim+ x ( ln x
;
(j
;
(l
x →0
2
x →0
ln x x →+∞ x 2 lim
2
(o
;
(d
ln x x →+∞ 1 + ln x
1 − ln x x →0 ln x 1 lim+ + ln x x →0 x lim+ x 2 ln x lim+
(k (m
;
(b
lim ( ln x ) + ln x
) ( ln x
lim
∞x →+
x ) ln (1 + ln x ; lim x →1 x 2 −1 ) ln ( ln x ∞x →+ x lim
اﻟﺘﻤﺮﻳﻦ : 2
2
∞x →+
lim x 3 ln x
x → 0+
x →0
ln x x →+∞ x 3 lim
;
(n
ﻡﻌﻠﻢ ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ ) . (O, i, j أ -أدرس اﻟﻔﺮوع اﻟﻼﻥﻬﺎﺋﻴﺔ ﻟﻠﻤﻨﺤﻨﻰ
lim
∞x →+
x ) ln (1 − sin x lim ) x →0 ln (1 + tan x
(p
ln x x
(r
ب -أدرس اﻟﻮﺿﻊ اﻟﻨﺴﺒﻲ ﻟﻠﻤﻨﺤﻨﻰ
⎞ ⎛ x + 2x + 1 ⎜ lim ln ⎟ (a 3 ∞x →+ x 5 + ⎝ ⎠ ) lim x + ln ( x 2
∞x →−
(e
⎞ 2 ⎛ 1 − 2x ⎜ ln ⎟ ⎠ x ⎝ 1 + 2x
(g
1 + 3ln x x
lim x →0
;
(b
;
(d
;
(f
1 ﺟـ -ﺏﻴﻦ أﻥﻪ ﻳﻮﺟﺪ ﻋﺪد ﺣﻘﻴﻘﻲ αﺣﻴﺚ< α < 1 : 2 3 ﺏﻴﻦ أﻥﻪ ﻳﻮﺟﺪ ﻋﺪد ﺣﻘﻴﻘﻲ βﺣﻴﺚ − < β < −1 :و f ( β ) = 02 د -أرﺱﻢ ) . ( Cf
lim+
x →0
; (h
lim
x →0+
) ﻥﺄﺧﺬ 0, 69 < ln(2) < 0, 70 :و ( 1, 09 < ln(3) < 1,10
اﻟﺘﻤﺮﻳﻦ :5
⎞⎛ x +2 ⎜ lim x ln ⎟ ⎠ ⎝ x )ln(x − 2 lim x →3 x −3
ﻥﻌﺘﺒﺮ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ fﻟﻤﺘﻐﻴﺮ ﺣﻘﻴﻘﻲ xﺣﻴﺚ :
∞x →+
2
Arc tan ( ln x ) + x
ﻟﻴﻜﻦ ) ( Cf
lim x →0 x >0
) ln ( cos x (i ∞x →+ x →0 x2 )ln ( x 2 − 2x + 2 ) ln (1 − x 3 lim ; (l lim (k ∞x →+ ∞x →− x x أﺣﺴﺐ اﻟﻨﻬﺎﻳﺎت اﻟﺘﺎﻟﻴﺔ : اﻟﺘﻤﺮﻳﻦ : 3 2 2 lim+ ln ( x − x − 2 ) − ln ( x + 5x − 14 ) (a ;
lim
) lim Arc tan ( ln x
(j
x →2
lim ln (ex 2 − x − 2 ) − ln ( x 2 + x + 7 ) (b
∞x →+
2 1 + (c x +2 4
lim ln(x + 2) − ln x −
∞x →+
⎞ 1 ⎛ 1− x 2 ⎜ ln ⎟ (d x →0 x 2 ⎠ ⎝ cos x
; lim
(f
lim x 3 − 2x − ln x
ﻣﺤﻤﺪ اﻟﺤﻴﺎن
∞x →+
(e ;
(g
x −1 x
ln
( Cfواﻟﻤﺴﺘﻘﻴﻢ ذي اﻟﻤﻌﺎدﻟﺔ :
و . f (α ) = 0
⎞ ⎛ 1 ⎟ lim ln ⎜ 2 ∞x →+ ⎠ ⎝ x +1
π
)
)
. ( Cf
. y = −x
أﺣﺴﺐ اﻟﻨﻬﺎﻳﺎت اﻟﺘﺎﻟﻴﺔ : 3
(c
) lim+ x ( ln x
) ( ln x
(i
)
(
lim ln x − x 2 − 3x + 2
∞x →+
اﻟﺘﻤﺮﻳﻦ : 4ﻥﻌﺘﺒﺮ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ fﻟﻤﺘﻐﻴﺮ ﺣﻘﻴﻘﻲ xﺣﻴﺚ : ⎞⎛ 1 ⎟ f ( x) = − x + ln ⎜1 + ⎠⎝ x .1ﺣﺪد Dﺣﻴﺰ ﺕﻌﺮﻳﻒ اﻟﺪاﻟﺔ . f .2أﺣﺴﺐ ﻥﻬﺎﻳﺎت fﻋﻨﺪ ﻡﺤﺪات . D .3أ -أﺣﺴﺐ ) f ′( xﻟﻜﻞ xﻡﻦ . D ب -أﻋﻂ ﺟﺪول ﺕﻐﻴﺮات اﻟﺪاﻟﺔ . f .4ﻟﻴﻜﻦ ) ( Cfاﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ اﻟﻤﺴﺘﻮى اﻟﻤﻨﺴﻮب إﻟﻰ
lim
3
;
ln x +1 1 ln x − ; lim (h x →0 x x >0
x → 0+
3
اﻷﺳﺘﺎذ :اﻟﺤﻴﺎن
⎞ ⎛ x2 + 3 ⎜ f ( x) = x + 2 ln ⎟ ⎠ ⎝ 4x اﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ اﻟﻤﺴﺘﻮى اﻟﻤﻨﺴﻮب إﻟﻰ
ﻡﻌﻠﻢ ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ ) . (O, i, j .1ﺣﺪد Dﻡﺠﻤﻮﻋﺔ ﺕﻌﺮﻳﻒ اﻟﺪاﻟﺔ . f .2أﺣﺴﺐ : و )lim f ( x ∞x →+
x →0 x >0
.3أ -ﺏﻴﻦ أﻥﻪ ﻟﻜﻞ xﻡﻦ D؛ ﻟﺪﻳﻨﺎ : )( x − 1)( x 2 + 3 x + 6 ′ = )f ( x )x( x 2 + 3 ب -أﻋﻂ ﺟﺪول ﺕﻐﻴﺮات اﻟﺪاﻟﺔ . f .4أ -ﺕﺤﻘﻖ ﻡﻦ أﻥﻪ ﻟﻜﻞ xﻡﻦ D؛ ﻟﺪﻳﻨﺎ : ⎞3 ⎛ )f ( x) = x + 2 ln( x) + 2 ln ⎜1 + 2 ⎟ − 4 ln(2 ⎠ ⎝ x ب -أدرس اﻟﻔﺮوع اﻟﻼﻥﻬﺎﺋﻴﺔ ﻟﻠﻤﻨﺤﻨﻰ ) . ( Cf
ﺟـ -أدرس اﻟﻮﺿﻊ اﻟﻨﺴﺒﻲ ﻟﻠﻤﻨﺤﻨﻰ ) ( Cf
lim
). lim f ( x
واﻟﻤﺴﺘﻘﻴﻢ ذي اﻟﻤﻌﺎدﻟﺔ :
.y=x د -أرﺱﻢ اﻟﻤﻨﺤﻨﻰ ) . ( Cf
x →0 x >0
lim sin x ln x
x → 0+
-1-
.2ب.ع.ت
2
اﻟﺘﻤﺮﻳﻦ : 6ﻟﺘﻜﻦ fاﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ ﻟﻤﺘﻐﻴﺮ ﺣﻘﻴﻘﻲ xاﻟﻤﻌﺮﻓﺔ ﺏﻤﺎﻳﻠﻲ: ⎞ ⎛ x −1 2 ⎜ f ( x) = x + ln ⎟ ) − (1 + ln x ⎠⎝ x +1 : أن ﺏﻴﻦ أ . 2 = ) ∀x ∈ ]0, +∞[ : f ′( x 2 2 .1ﺏﻴﻦ أن ﺣﻴﺰ ﺕﻌﺮﻳﻒ fهﻮ . D = ]−∞, −1[ ∪ ]1, +∞[ : ⎤ ) x 2 ⎡1 + ( ln x ⎣ ⎦ .2ﺏﻴﻦ أن fداﻟﺔ ﻓﺮدﻳﺔ . ⎞⎛1 ب -أﺣﺴﺐ . f ′ ⎜ ⎟ : .3أﺣﺴﺐ lim f ( x) :و ). lim f ( x x →1 ∞x →+ ⎠ ⎝e x >1 ﺟـ -أﻋﻂ ﺟﺪول ﺕﻐﻴﺮات اﻟﺪاﻟﺔ . f .4ﺕﺤﻘﻖ ﻡﻦ أن اﻟﻤﺴﺘﻘﻴﻢ اﻟﺬي ﻡﻌﺎدﻟﺘﻪ y = xﻡﻘﺎرب ﻡﺎﺋﻞ ﻟﻠﻤﻨﺤﻨﻰ .3أ -ﺏﻴﻦ أن : ) ( Cfاﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ ﻓﻲ اﻟﻤﺴﺘﻮى اﻟﻤﻨﺴﻮب إﻟﻰ ﻡﻌﻠﻢ ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ 2 2 (1 + ln x ) ⎡2 + (1 + ln x ) ⎤ ln x ⎣ ⎦ = ) ∀x ∈ ]0, +∞[ : f ′′(x ) . (O, i, j 2 3 ⎤ ) x 3 ⎡1 + ( ln x 2 x +1 ⎣ ⎦ .5أ -ﺏﻴﻦ أن : . ∀x ∈ D : f ′(x ) = 2 ب -اﺱﺘﻨﺘﺞ أن اﻟﻤﻨﺤﻨﻰ ) ( Cfﻳﻘﺒﻞ ﻥﻘﻄﺘﻲ اﻥﻌﻄﺎف Aو Bﻳﻨﺒﻐﻲ x −1 ب -ﺣﺪد ﺟﺪول ﺕﻐﻴﺮات اﻟﺪاﻟﺔ fﻋﻠﻰ اﻟﻤﺠﺎل [∞. ]1, + ﺕﺤﺪﻳﺪ إﺣﺪاﺙﻴﺘﻴﻬﻤﺎ . ﺟـ -ﺣﺪد ﻡﻌﺎدﻟﺘﻲ اﻟﻤﻤﺎﺱﻴﻦ ﻟﻠﻤﻨﺤﻨﻰ ) ( Cfﻓﻲ اﻟﻨﻘﻄﺘﻴﻦ Aو . B .6ﺏﻴﻦ أن اﻟﻤﻨﺤﻨﻰ ) ( Cfﻳﻘﻄﻊ ﻡﺤﻮر اﻷﻓﺎﺻﻴﻞ ﻓﻲ ﻥﻘﻄﺔ أﻓﺼﻮﻟﻬﺎ ب -أﺣﺴﺐ ) lim f (x
و ) . lim f (x
x →0 x >0
3 ﻡﺤﺼﻮر ﺏﻴﻦ 2 .7أﻥﺸﺊ اﻟﻤﻨﺤﻨﻰ ) ) . ( Cfﻥﺄﺧﺬ ln(3) ≈ 1,1 :و ( ln(5) ≈ 1, 6 و .2
اﻟﺘﻤﺮﻳﻦ : 7ﻟﺘﻜﻦ fاﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ ﻟﻤﺘﻐﻴﺮ ﺣﻘﻴﻘﻲ xاﻟﻤﻌﺮﻓﺔ ﺏﻤﺎﻳﻠﻲ: ⎞ ⎛ x2 − 2x + 2 ⎜ f ( x) = 2 x − 2 + ln ⎟ x2 ⎝ ⎠ * . .1ﺏﻴﻦ أن ﻡﺠﻤﻮﻋﺔ ﺕﻌﺮﻳﻒ اﻟﺪاﻟﺔ fهﻲ : .2أﺣﺴﺐ ﻥﻬﺎﻳﺎت fﻋﻨﺪ ﻡﺤﺪات * . )2( x − 1)( x 2 − x + 2 .3أ -ﺏﻴﻦ أﻥﻪ ﻟﻜﻞ xﻡﻦ * ؛ ﻟﺪﻳﻨﺎ : 2 )x( x − 2 x + 2 ب -أﻋﻂ ﺟﺪول ﺕﻐﻴﺮات اﻟﺪاﻟﺔ . f .4ﻟﻴﻜﻦ ) ( Cfاﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ اﻟﻤﺴﺘﻮى اﻟﻤﻨﺴﻮب إﻟﻰ ﻡﻌﻠﻢ = )f ′( x
ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ ) . (O, i, j أ -أدرس اﻟﻔﺮوع اﻟﻼﻥﻬﺎﺋﻴﺔ ﻟﻠﻤﻨﺤﻨﻰ ) . ( Cf
ب -أدرس اﻟﻮﺿﻊ اﻟﻨﺴﺒﻲ ﻟﻠﻤﻨﺤﻨﻰ ) ( Cf
واﻟﻤﺴﺘﻘﻴﻢ ذي اﻟﻤﻌﺎدﻟﺔ :
ﻣﺤﻤﺪ اﻟﺤﻴﺎن
ﺕﻐﻴﺮات اﻟﺪاﻟﺔ . g ب -اﺱﺘﻨﺘﺞ أن . ∀x ∈ ]0, +∞[ : g (x ) ≥ 0 : .2أ -ﺏﻴﻦ أن : ∀x ∈ ]0, +∞[ : h (x ) = 1 + g ( x ) + ( x − 1) ln x ب -ﺏﻴﻦ أن :
∀x ∈ ]0, +∞[ : (x − 1) ln x > 0
.3اﺱﺘﻨﺘﺞ أن :
∀x ∈ ]0, +∞[ : h (x ) > 0
.IIﻥﻌﺘﺒﺮ اﻟﺪاﻟﺔ fاﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [∞ ]0, +ﺏﻤﺎ ﻳﻠﻲ : ) f (x ) = 1 + x ln x − ( ln x
ب -أﺣﺴﺐ ) lim f (x
∞x →+
ﺙﻢ ﺣﺪد اﻟﻔﺮع اﻟﻼﻥﻬﺎﺋﻲ ﻟﻠﻤﻨﺤﻨﻰ ) ( C
ﺏﺠﻮار ∞. + ) h (x = ) . ∀x ∈ ]0, +∞[ : f ′(x .2أ -ﺏﻴﻦ أن : x ب -اﺱﺘﻨﺘﺞ أن اﻟﺪاﻟﺔ fﺕﺰاﻳﺪﻳﺔ ﻗﻄﻌﺎ ﻋﻠﻰ اﻟﻤﺠﺎل [∞. ]0, +
.3أ -ﺣﺪد ﻡﻌﺎدﻟﺔ دﻳﻜﺎرﺕﻴﺔ ﻟﻠﻤﻤﺎس ) ∆ ( ل ) ( Cﻓﻲ اﻟﻨﻘﻄﺔ )A (1,1 ب -ﺕﺤﻘﻖ ﻡﻦ أن : ) ∀x ∈ ]0, +∞[ : f (x ) − x = (ln x − 1) g ( x ﺟـ -ﺣﺪد اﻟﻮﺿﻊ اﻟﻨﺴﺒﻲ ﻟﻠﻤﻨﺤﻨﻰ ) ( Cواﻟﻤﺴﺘﻘﻴﻢ ) ∆ ( .
(
ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ i = 2cm . O , i , j x →0 x >0
g (x ) = x − 1 − ln xو . h (x ) = x + (x − 2) ln x .1أ -أﺣﺴﺐ ) g ′(xﻟﻜﻞ xﻡﻦ اﻟﻤﺠﺎل [∞ ]0, +ﺙﻢ أدرس ﻡﻨﺤﻰ
x →0 x >0
2 ⎤ ) x ⎡1 + ( ln x ⎣ ⎦ وﻟﻴﻜﻦ ) ( Cfاﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ اﻟﻤﺴﺘﻮى اﻟﻤﻨﺴﻮب إﻟﻰ ﻡﻌﻠﻢ
2
.Iﻥﻌﺘﺒﺮ اﻟﺪاﻟﺘﻴﻦ اﻟﻤﻌﺮﻓﺘﻴﻦ ﻋﻠﻰ اﻟﻤﺠﺎل [∞ ]0, +ﺏﻤﺎ ﻳﻠﻲ :
.1أ -أﺣﺴﺐ ) lim f (xﺙﻢ أول اﻟﻨﺘﻴﺠﺔ ﻡﺒﻴﺎﻥﻴﺎ .
= ) f (x
.1أ -ﺏﻴﻦ أن lim x ( ln x ) = 0 :
اﻟﺘﻤﺮﻳﻦ : 9
وﻟﻴﻜﻦ ) ( Cاﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ ﻡﻌﻠﻢ ﻡﺘﻌﺎﻡﺪ ﻡﻤﻨﻈﻢ .
اﻟﺘﻤﺮﻳﻦ : 8ﻥﻌﺘﺒﺮ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ fاﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [∞ ]0, +ﺏﻤﺎﻳﻠﻲ :
)
.4أﻥﺸﺊ اﻟﻤﻨﺤﻨﻰ ) . ( Cf
2
. y = 2x − 2 ﺟـ -ﺏﻴﻦ أﻥﻪ ﻳﻮﺟﺪ ﻋﺪد ﺣﻘﻴﻘﻲ وﺣﻴﺪ αﺣﻴﺚ f (α ) = 0 :و 8 ⎡⎤ 1 1 ⎢ α ∈ ⎥ − , −؛ ﻋﻠﻤﺎ أن ln(13) < 3 :و > ). ln(25 3 ⎣⎦ 2 3 د -أرﺱﻢ اﻟﻤﻨﺤﻨﻰ ) ) . ( Cfﺕﺤﺪﻳﺪ ﻥﻘﻄﺔ اﻹﻥﻌﻄﺎف ﻏﻴﺮ ﻡﻄﻠﻮب ( 1
∞x →+
.4أﻥﺸﺊ اﻟﻤﻨﺤﻨﻰ ) ( Cواﻟﻤﺴﺘﻘﻴﻢ ) ∆ ( ﻓﻲ ﻥﻔﺲ اﻟﻤﻌﻠﻢ ) .ﻥﻘﺒﻞ أن
) ﻳﻤﻜﻦ وﺿﻊ . ( x = t 2
اﻟﻤﻨﺤﻨﻰ ) ( Cﻳﻘﺒﻞ ﻥﻘﻄﺔ اﻥﻌﻄﺎف أﻓﺼﻮﻟﻬﺎ ﻡﺤﺼﻮر ﺏﻴﻦ 1و (1,5
-2-
.2ب.ع.ت
2
