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ÚÖæ ÌÏíÏ
ÇáãÚÇÏáÇÊ ÇáãÑÊÈØÉ ÈÇáÒãä ( ááËÇäí ÚÔÑ)
ÇäÇ ÚÖæÉ ÌÏíÏÉ
åÐå ãÌãæÚÉ ãä ÇáÇãËáÉ æÇáÊãÇÑíä ãäÞæá æãÚÏá ÇÑÌæ ÇáÇÓÊÝÇÏÉ ãäåÇ
ÇáÓÄÇá ÇáÃæá :
یÊäÇÞÕ Øæá ãÓÊØیá ÈãÚÏá 4 Óã ÈیäãÇ یÊÒÇیÏ ÚÑÖª ÈãÞÏÇÑ
2Óã Ýí ÇáËÇäیª Ýí ÇááÍÙª ÇáÊí یßæä ÝیªÇ Øæá ÇáãÓÊØیá
30 Óã æÚÑÖª 20 Óã ÇæÌÏ
1) ãÚÏá ÇáÊÛیÑ Ýí ãÍیØ ÇáãÓÊØیá )
2) ãÚÏá ÇáÊÛیÑ Ýí ãÓÇÍʪ¿ )
Y X
Y = 20 X = 30
dx / dt = -4 cm / s
dy / dt = 2 cm / s
2C
C = 2 ( X + Y )
t :
dc/dt = 2 ( dx/dt + dy/dt )
dc/dt = 2 ( -4 + 2 ) = 2 ( -2 ) = -4
4
__________________________________________________
XA
A = X . Y
( t ) :
dA/dt = X . dy/dt + Y . dx/dt
Da/dt = ( 30 x 2 ) + ( 20 x -4 ) = -20
2^20
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ÇáÓÄÇá ÇáËÇäí :
Óáã Øæ᪠15 ã یÑÊßÒ Úáì ÌÏÇÑ ÑÇÓí æÇÑÖ ÇÝÞیª ÝÇÐÇ
ÇäÒáÞ ÇáÓáã ãÈÊÚÏÇ Úä ÇáÍÇÆØ ÈãÚÏá 1 ã ßá ËÇäیª
ÝÌÏ ãÚÏá ÇáÒÇæیª Çáã̪æ᪠ãÇÈیä ÇáÓáã æÇáÍÇÆØ ÚäÏãÇ
یßæä ÈÚÏ ÇáÓáã 9 ã ãä ÇáÍÇÆØ
Ɵ
h
L
dL/dt = 1 m/s
Ɵ
1SIN ( Ɵ ) = L / 15
t
2COS ( Ɵ ) dƟ/dt = ( 1 / 15 ) x dL/dt
COS ( Ɵ ) dƟ/dt
: >XXX;h L = 9
L 2 + h 2 = (15 ) 2
(9) 2 + h 2 = (15 ) 2
h 2 = (15 ) 2 - (9)2 = 144
h = 12
COS ( Ɵ ) = 12 / 15
( 2 )
12 / 15 x dƟ/dt = ( 1 / 15 ) x 1
dƟ/dt12 m / s
/12
9
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یÓیÑ ÞØÇÑÇä ãä ÇÍÏì ãÍØÇÊ ÇáÞØÇÑ ÈÍیË یÊ̪ ÇáÇæá äÍæ
ÇáÛÑÈ ÈÓÑÚÉ ËÇÈʪ ãÞÏÇÑªÇ 65 ßã/ÓÇÚÉ ÈیäãÇ یÊ̪ ÇáÂÎÑ
äÍæ ÇáÌäæÈ ÈÓÑÚÉ ËÇÈʪ ãÞÏÇÑªÇ 75 ßã/ÓÇÚÉ . Ýí ÊãÇã
ÇáÓÇÚÉ ÇáËÇáËÉ ÈÚÏ áÙªÑ ÞØÚ ÇáÞØÇÑ ÇáÇæá ãÓÇÝÉ 120 ßã
ÈیäãÇ ÞØÚ ÇáÞØÇÑ ÇáËÇäí ãÓÇÝÉ 130 ßã . ÇæÌÏ ãÚÏá
ÇáÊÛیÑ Ýí ÇáãÓÇÝÉ Èیä ÇáÞØÇÑیä
ÈÇÓÊÎÏÇã äÙÑیÉ ÝیËÇÛæÑÓ
S
2
= X
2
+ Y
2
……………(1)
2S ds/dt = 2x dx/dt + 2y dy/dt …………..(2)
2S1
S2 = (120)2 + (130)2 = 31300
S ≈ 177
2 x 177 ds/dt = ( 2 x 120 x 65 ) + ( 2 x 130 x 75 )
ãÓÊØیá یÒÏÇÏ Øæ᪠ÈãÚÏá 6 ã/Ë æیÒÏÇÏ ÚÑÖª ÈãÚÏá 2 ã/Ë
ÇæÌÏ ãÚÏá ÇáÊÛیÑ Ýí ãÓÇÍʪ ÚäÏãÇ یÕá ÇáØæá Çáì 25 ã
æÇáÚÑÖ 15 ã
A = X . Y
dA/dt = X dy/dt + y dx/dt
dA/dt =( 25 x 2 )+( 15 x 6 )
dA/dt = 140 m2/s
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ÊÒÏÇÏ ãÓÇÍÉ ÏÇÆÑÉ ÈãÚÏá 24 Óã 2/Ë ÇæÌÏ ãÚÏá ÊÛیÑ (ÒیÇÏÉ)
Óã 2 äÕÝ ÇáÞØÑ ÚäÏãÇ Êßæä ãÓÇÍÉ ÇáÏÇÆÑÉ 36
A = r2 -------- (1)
dA/dt = 2 r dr/dt ------(2)
( ãä ÇáãÚÇÏáÉ ÑÞã ( 1) æäÚæÖªÇ Ýí ( 2 r äÌÏ ÞیãÉ
36 = r2 r2 = 36 r = 6 cm
24 = 2 . . 6 dr/dt dr/dt = 24/12 = 2/
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BA 5
0.6 A
3B
x2 + y2 = 25
, x = 3
5
y
x
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2
6
31.2
0.54 3C
C4
r = 2 , h = 6 , , , h = 4
( 1 ) V = r2h >XXX;
h r
1
V = ( )2h = 1/27 h3
6
2
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31540
R = 3.14 7
2 3 4
R = 7 cm ,
dv
/dt = 1540 cm
3
/s
A = 4 r2
dA/dt = 8 r
dr/dt ……………..( 1 )
dr/dt 1
V = 4/3 r3 dv/dt = 4 r2 dr/dt
1540 = 4 ( 3.14) (7)2 dr/dt dr/dt = 2.5
1
dA/dt = 8 r
dr/dt
dA/dt = 8 ( 3.14 ) ( 7 ) ( 2.5 ) = 440 cm/s
dA/dt = 440 cm/s
dr/dt = 2.5
dA/dt = 440 cm/s
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0.9 0.6
15
30
r
2
h 4
/3 r
3
12
18
12
>XXX;
2
31 2
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1.8
.4>XXX; 5.4
.
.
F(X) = X
3
– X
2
0.2
( -1 , -2 )
10
60
0 >XXX;
>XXX; 2
5
10
>XXX;
2
8
5.4
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AB ABC
F(X) = 5 – X
2 C
2
1B ABC
0.3
0.2
30
37
10>XXX;
41
ÇáÊÚÏíá ÇáÃÎíÑ Êã ÈæÇÓØÉ ÞáÈ ÇáÔÇã ; 14-02-2011 ÇáÓÇÚÉ 03:13 PM
ÖæÇÈØ ÇáãÔÇÑßÉ
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- áÇ ÊÓÊØíÚ ÅÑÝÇÞ ãáÝÇÊ
- áÇ ÊÓÊØíÚ ÊÚÏíá ãÔÇÑßÇÊß
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