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ايرولاكبلا عوضوم 6102
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يبرعلا دمحم يديبعلاب:ذاتسلأا دادعإ 0
6102
10
0 (P)
(P ')
(P) (P ')
nP
nP '
n (2;1; 1)P n (1; 2;1)P'
1.2 2.1
6 ( )
d(M,(P))d(M,(P')) 2x y z 1 x 2y z 2
2² 1² ( 1)² 1² ( 2)² (1)²
2x y z 1 x 2y z 2 2x y z 1 x 2y z 2
2x y z 1 (x 2y z 2)
x 3y 2z 3 0....(1) 3x y 1 0...(2)
( )
(P )1
(P )2
1 2
3 A(1;2;0)
( )
A(1;2;0) ( )
d(A,(P))d(A,(P'))
2(1) 2 0 1 5 d(A,(P))
2² 1² ( 1)² 6
(1) 2(2) 1(0) 2 5
d(A,(P'))
2² 1² ( 1)² 6
4 (AH)
(AH ')
(AH) A(1;2;0)
n (2;1; 1)P
M(x;y;z) (AH) AM t.nP
x 2t 1 y t 2 z t
t
(AH ') A(1;2;0)
n (1; 2;1)P'
M(x;y;z) (AH') AM t '.nP'
x t ' 1 y 2t ' 2 z t '
t
H H '
1 H (s ) 2x y z 1 0 x 2t 1
y t 2 z t
(s )1
2(2t 1) t 2 t 1 0 t 5
6 t
4 7 5 (AH) H( ; ; )
6 6 6
2 H ' (s ) x 2y z 2 0
x t ' 1 y 2t ' 2 z t '
(s )2
t ' 1 2( 2t ' 2) t ' 2 0 t ' 5
6 t
(AH ') 11 2 5
H '( ; ; ) 6 6 6
5 I
AHH '
HH '
I IH H '
I
x x 7
x 2 12
H H '
I
y y 9
y 2 12
H H '
I
z z 10
z 2 12
AHH ' AHAH '
AI.HH ' S 2
7 9 10 5 14
AI ( 1)² ( 2)² ( )²
12 12 12 12
11 4 7 2 5 5 5 10
HH ' ( )² ( )² ( )²
6 6 6 6 6 6 6
AI.HH ' 5 10.5 14 25 35
S u.a
2 144 72
15
0I lim f (x)x
0;
f f (x) 2x8x x
limf (x) lim 2x
f
f '(x)
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ايرولاكبلا عوضوم 6102
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يبرعلا دمحم يديبعلاب:ذاتسلأا دادعإ 3
0;
f (2x 8) ' 1f '(x)
2 2x 8 2x 8
f '(x) 0 f
0
x f '(x)
f (0)
f (x)
6 (C)
( )
(C) ( )
f (x) x
f (x)x
f (x) ²
x²x²2x 8 0
x²2x 8 0 x 4
x 2
(C) ( )
(4;4)
3 (C)
( )
0 II u0
u1
u2
u3 n 6 (u )
u0
u1
u2
u3
(u )n
4
3
n n 0u 4
P(0)
0 n=0 0u 4 u0 0
n P(n) 0u 4
P(n+1) 0un 1 4
0un 4 f (0)f (u )n f (4)
n 1 f 8 u 4
0un 1 4 8 0
0un 4
(u )n
(u )n
n 1 n
u u
2
n n n n n n
n 1 n n n
n n n n
( 2u 8 u )( 2u 8 u ) u 2u 8
u u 2u 8 u
2u 8 u 2u 8 u
2 3 4 5 6
2 3 4 5
0 1
1
x y
u0 u1 u2u3
2
n n
u 2u 8
2
n n
u 2u 8 0
un 4
un 2 0un 4
2
n n
u 2u 8
4
n 0 u
0
(u )n
(u )n
0un 4 (u )n
n 1 n n
4 u 1(4 u )
2
n n n n
n 1 n
n n n
(4 2u 8)(4 2u 8) (4)² 2u 8 2(4 u )
4 u 4 2u 8
4 2u 8 4 2u 8 4 2u 8
n n
2(4 u )
4 2u 8
n
n
1 4
(4 u )
2 4 2u 8
n
0 4 1
4 2u 8
n n
n
1 4 1
(4 u ) (4 u )
2 4 2u 8 2
n 1 n
4 u 1(4 u )
2
n
n n 0 n
4 u 1 (4 u )
2
n 1 n
4 u 1(4 u )
2
1 0 n=0
4 u 1(4 u )
2 1
2 1 n=
4 u 1(4 u )
2
1
n n 1 n
4 u 1(4 u )
2
n
n 0
4 u 1 (4 u )
2
n n 0 n
4 u 1 (4 u )
2
xlim un
n n
4 u 4
2 4 u n 0
n n
0 4 u 4
2
x n
lim 4 0
2
n x
lim(4 u ) 0
xlim un 4
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ايرولاكبلا عوضوم 6102
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يبرعلا دمحم يديبعلاب:ذاتسلأا دادعإ 5
15
0 z 'z
z 'z z 2
z 1 z
z²2z 2 0
z 1 i z 1 i
6
2
1
z z
2 2i
1
z 1 i (1 i)²
i e
z 1 i 2
B A
R O
2 2i
1
z e
z
2i
B O A O
z z e (z z )...(*)
(*) B
A R
2 O
3 ( )
( )
( ) M(z)
M '(z ')
M '(z ') z '
z ' z 2 z 1
M C
M D
z z z ' z z
(MC;MD) k
2
( )
CD0 S
R 2 O
H 2 O
S 2 O
2
S
i i S
z 'ke z (1 ke )z
k 2 2
z 0
2i
z ' 2e z
z '2iz
( ')
( ') ( )
S
( ')
C'D'
S(C)C' S(D)D'
S(C)C'
C' C
z 2iz 4i S(D)D'
D' D
z 2iz 2i
2 3
-1 -2
2 3 4
-1
0 1
1
x y
D C C'
D'
( ')
( )
2.5
0 I g
0;
g g(x) x² 1 ln x g '(x) g
1 2x² 1 g '(x) 2x
x x
g '(x)0 2x² 1 0
x 2
2
g '(x) 0 0 x 2
g 2 0; 2
2
g '(x) 0 x 2
g 2 2;
2
2 2 g( ) g(x) 0 2
x 0;
2 1 2 1 3 1
g( ) 1 ln ln 2
2 2 2 2 2
2 g
g( ) 0 2
g(x) 0
x 0;
0 II
xlim f (x)0
limf (x)x
0;
f f (x) ln x x 1 x
x 0
lim 1.ln x
x
xlim ln x0
x 0
lim 1
x
x x
limf (x) lim(x 1)
x
limln x 0
x
6x 0;
f '(x) g(x)
x²
1.x 1ln x
ln x x x² 1 ln x g(x)
f '(x) ' 1 1
x x² x² x²
3 f (
f '(x) g(x)
x² f
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ايرولاكبلا عوضوم 6102
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يبرعلا دمحم يديبعلاب:ذاتسلأا دادعإ 7
f
x 0 f '(x)
f (x)
3 (T )
0 0 0 (T )
yf '(x )(xx )f (x )
yf '(1)(x 1) f (1) y2x2
0 (C)
( ) y x 1
y x 1
(C)lim f (x)x y 0
x x
lim f (x) y limln x 0
x
x
limln x 0
x
(C) ( )
(C) ( )
f (x) y ln x
x
ln x 1
x 0 f (x)y 0
(C) ( )
(C) ( )
(C) ( )
5 (T )
( ) (C)
(T ) A(1;0)
u(1;2)
( ) A(1;0)
v(1;1)
2 3 4 5 6
-1 -2
2 3
-1
-2
-3
0 1
1
x y
(T)
( ) (C)
2 A(1;0)
(m)
A(1;0) ( m) (1;0)
ymxm
ymxm ym(x 1)
(m)
A(1;0) f (x)mxm
f (x)mxm y f (x)
y mx m
f (x) mxm
m (C) ( )
A(1;0)
1 m 1 2
1 m 2
3 m2 4
m 2
ln x 7 x x
0;
u(x)ln x u '(x) 1
x x ln x
x u(x).u '(x)
2x 1 u(x) c
2
x ln x
x
2x 1 ln(x) c
2
In
n n
1 1
ln x 1 1
f (x) (x 1) dx dx (ln n)² (ln1)² (ln n)²
x 2 2
In
n0
In 2
In 2
0
1(ln n )² 2
0 2 (ln n )² 4
(ln n )²0 4 (ln n )0 2
2
n0 e n0 8
