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سلسلة ديماديما الباكالوريا علوم تجريبية
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(,*'Wax b+( )0a
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a x
%&a %&Ga ax b+
+(,-.-/01*'Wax bx c+ +( )0a
HBW( ) x ax bx c = + +
,,,, (+$(+$(+$(+$WWWW
( ) 0x x =
%&%&%&%&( )x +,-+,-+,-+,-( )x
0 < S =
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0 = { }2
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a
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x
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= b - 4ac
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5
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=(#+L//$ab ( )2 2 22a b a ab b+ = + +
( )2 2 22a b a ab b = + ( )( )2 2a b a b a b = +
( )3 3 2 2 33 3a b a a b ab b+ = + + +
( )3 3 2 2 33 3a b a a b ab b = +
( )( )3 3 2 2a b a b a ab b = + + ( )( )3 3 2 2a b a b a ab b+ = + +
(>>>
( )( )
( )
xf x
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= ( ) }0Q x >>>>( ){ / 0fD x x=
( )( )
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6
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4(=( )* nn x x x x 7=18W
0lim 0nx
x
= 0
lim 0x
x
=
>>>>
1lim 0n
x x=
1lim 0n
x x+=
limx
x+
= +
1lim 0
x x+=
O8S&O8S&O8S&O8S&nOTUX(#OTUX(#OTUX(#OTUX(#WWWW O8S&O8S&O8S&O8S&nOTU(!U(#OTU(!U(#OTU(!U(#OTU(!U(#WWWW lim n
xx
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lim nx
x
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7
B1C=W
( ) ( )
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00
limlim 0 x x
x x
f x V x
f xV x
==
l
l
( ) ( ) ( )
( )
( )
( )0 0
0
lim lim
lim
x x x x
x x
u x f x V x
u x f x
V x
= ==
l l
l
( ) ( )
( )( )
00
limlim x x
x x
u x f x
f xu x
= += +
( ) ( )
( )( )
00
limlim x x
x x
u x V x
f xV x
= =
]Z."#>^/>>>l 0>>>l 0
8
41J2K4(05
97K41JW :1W ( ) ( )
00lim
x xf f x f x
=a40x
E0D41JD41JW
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0limx x
f f x f x>
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f f x f x 32D.4# 32>
NHJ)W=f32^0g
9
3d3d3d3d( )f I 3d3d3d3dI
f^0g#-^0g#-^0g#-^0g#-I f^0g4g-^0g4g-^0g4g-^0g4g-I [ ],a b ( ) ( )[ ];f a f b ( ) ( )[ ];f b f a
[ [,a b ( ) ( ); limx b
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x bf x f a
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+
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[ [,a + ( ) ( ); limx
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xf x f a
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xx af x f x
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+
9O76PW c8S&f32^4[ ],a bD//$(#+jTU=(#L 47( )f a( )f b
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W
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10
S7J 2K4(05
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3/&(Of(#a6/%* g0x"c8S&W ( ) ( )0
0 0
limx x
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#ep(#^,-"]Z.fa0x! j!W( )0'f x
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=fa6/%* g(0x #^nq,(fC4UPr0/a0xD.W ( )( ) ( )0 0 0'y f x x x f x= + #u^U!D,8W ( ) ( )( ) ( )0 0 0'u x f x x x f x= +
#^n@,s#^,-fC4UPr0/a0x#t!/-D.fu0x
E0DS7J8TJE0DS7J8TW
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0
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>
"
#ep(#^,-"]Z.f^L,a0x! j!W ( )0'f xd
3/&(Of^6/%* g a0x"c8S&W ( ) ( )
0
0
0
limx x
f x f x
x x
0 <
limn
n
+= + lim 0
nn
+=
(Z#=( )nqa)W q W
1q > 1q = 1 1q < < 1q
lim nn
q+
= + lim 1nn
q+
= lim 0nn
q+
= ( )nq
"CG
^7b+W
20
4(Nc 2K4(05
4;D(Nc4(W
:1W
=f32^U!(#(I
OP3/F#>P(D.f3d^I
OOh!pe/yS&W
F/%* g3d^6I ( ) ( )'x I F x f x =
NHW
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#>3#Hzf^U!IDWW ( ) ( )x F x k k+
=f+PI
=0x=!4I0y=!4
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F G+#>P(f g+^3dI
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21
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22
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#+-f=ak&bD//(#.W
( ) ( )[ ] ( ) ( )ba
bf x dx F x F b F a
a= =
NHW 9fW
( ) 0a
f x dxa
= ( ) ( )a b
f x dx f x dxb a
=
( ) ( ) ( )b b
k kf x dx k f x dxa a
= ( ) ( )[ ] ( ) ( )b b b
f x g x dx f x dx g x dxa a a
+ = + 4T.W
( ) ( ) ( )b c b
f x dx f x dx f x dxa a c
= + B1C/W
O8S&W [ ] ( ), 0x a b f x
OTUW( ) 0b f x dxa
O8S&W[ ] ( ) ( ),x a b f x g x
OTUW( ) ( )b bf x dx g x dxa a
9#07W
=f32^4([ ],a b
3d^#0@,/D.D//(#W ( )1 b f x dxab a
F\ c8#W =uv32^6/%*L gL(IL#Mfuvd^L43I
ab3d==!4I
( ) ( ) ( ) ( )[ ] ( ) ( )bab bu x v x dx u x v x u x v x dx
a a =
\))
#ok& =( ), ,o i j
$#$ .u A(#+0$D.0/ oL"|i
j
1. .u A i j=
23
=f32^4([ ],a b ^ nL 4$fC+>U7
,.(=ZL,/W x a=x b=
D.W ( ) . .b f x dx u Aa
=fg32^L4L([ ],a b LnL 4$fCgC7
>U,.(=ZL,/+Wx a=x b=D.W
D.W ( ) ( ) . .b f x g x dx u Aa
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24
0g4(2K4(05
0g(> :1W
(#>#D.vK >
] [0;
ln y
x y
x y x e
+
= =
] [ ] [
( )
( )
0; 0;
ln ln ln
ln ln
1ln ln
ln ln ln
r
x y
xy x y
x r x
xx
xx y
y
+ +
= +
=
= =
( )r
O8S&nX(# OTUW( )* ln lnnx x n x = ;:0W
####fD,8U!D,8U!D,8U!D,8U!WWWW #1!-,2#1!-,2#1!-,2#1!-,2fD.D.D.D.WWWW ( ) ( )[ ]lnf x u x= ( ) }0u x > { /f uD x x D=
( ) ( )( )2lnf x u x=
( ) ( )lnf x u x= ( ) }0u x { /f uD x x D=
A=W
( )lim lnx
x+
= + lnlim 0nx
x
x+=
( )0
lim lnx
x
= >
( )0
lim ln 0nx
x x
=>
1
lnlim 1
1x
x
x=
( )0
ln 1lim 1x
x
x
+=
( )n *
41JW
#lnx x3d^4] [0;+
=uU!(^32I c8S&u32^40g
25
S7JW
#lnx x6/%* g^] [0;+
#:
] [ ( )1
0; lnx xx
+ =
=uU!(^32I c8S&u32^6/%* g0g
26
c4(2K4(05
0g(> :1W
#@K > > > > > > >
x y x y x ye e e + =
( )r ( )rx rxe e= 1 xx ee
=
xx y
y
ee
e
=
:0;W
####fD,8U!D,8U!D,8U!D,8U!WWWW #1!-,2#1!-,2#1!-,2#1!-,2fD.D.D.D.WWWW ( ) xf x e= fD =
( )( )u x
f x e= { }/f uD x x D= A=W
lim xx
e+
= +
lim 0xx
e
=
limx
nx
e
x+
= +
( )lim 0n xx
x e
=
0
1lim 1
x
x
e
x
=
( )n *
41JW
#xx e^4
=u(U!^32I c8S&u^4d3I#OTU( )u xx e3d^4I
27
S7JW
#xx e g 6/%*^#W( )x xx e e =
=uU!(^32I c8S&u^6/%* g d3IOTUW#( )u xx e3d^6/%* gI
#W ( )( ) ( ) ( )'u x u xx I ue x e = (Q#/0lnW
(Uhcaa)W}{1a +
:1W
##aog lq@@#^,-a ! C!Wexpa
+HBx=( )exp xa x a= NHW
lnx x ax a e = ( )xaog a x=l
] [ ( )0; og xax a a + = l ( ) 2; x yx y a a x y = =
] [0;x y + ( )aog y l
xa y x= =
( ) 2;x y x y x ya a a + =
( )r ( )rx rxa a= 1 xx aa
= x
x yy
aa
a
=
=1&W
1a > 0 1a< < x ya a x y> > x ya a x y< <
lim xx
a+
= +
lim 0xx
a
=
lim 0xx
a+
=
lim xx
a
= +
0
1lim ln
x
x
aa
x
=
7X#W ( ) ( )lnx xa a a =
28
(7+(c 2K4(05
D.#/(#,2W } 1i = ( ){ / ; z a ib a b= = + i(7+(P@8W
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jNHj)W O8S&W ( )Im 0z =OTUzD//$(#. O8S&W ( )Re 0z =( )Im 0z OTUzU!>(#^,
E(7+(i1W =zz L#/=(#
( ) ( )Im Imz z =( ) ( )Re Rez z z z = = #/0i(7+(QW oJ#ok& v#/=( )1 2, ,o e e
i(7+(bkW =z a ib= +M$#/(#W( ); a b
(#eU!zv#/(#.Wz a ib= ( )M z( )M zD//n,
29
( )
( )
*
' 0
n nz z n
z z
z zz
z z
=
=
=
1 1
z z z z
z z
z z
=
=
=
/XQ#(>gi(7+(c8W
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! j!Wargz tW[ ]arg 2z pi=
=z#K?#/(# HBr z=[ ]arg 2z pi=
#DEE+pv#/(z.W ( ) [ ]cos sin ,z r i r = + =
v#/(#@ zD.Wiz re =
NHJ)W D//$(#EE a#K?
0a > 0a AYNW
( ) ( ) ( )cos sin cos sinnn
i n n i n
+ = +
( )1cos2
i ie e = +
( )1sin2
i ie ei
=
+#/)z z a =
a)( )a W ((((WWWW 22223$,3$,3$,3$,((((WWWW
0a > { };S a a= 0a = { }0S = z z a = 0a < { };S i a i a=
30
+#/)W 0z az bz c + + =
a)Wab c77)+(A( )0a
((((WWWW (3$,2(3$,2(3$,2(3$,2WWWW
0 > ;2 2
b bS
a a
+ =
0 = { }2b
Sa
=
( )
2
2
0
4
z az bz c
b ac
+ + =
=
0 < ;2 2
b i b iS
a a
+ =
(7+(c%9(6O6&W
D@#C"D@#C"D@#C"D@#C" #/g*#/g*#/g*#/g* UAB B AAB z z=
I0/14[ ];A B 2
A BI
z zz
+=
qg( );AB AC ( ) [ ]; arg 2c AB A
z zAB AC
z zpi
ABC,// C AB A
z z
z z
ABCD#/ D A B CB A D C
z z z z
z z z z
PD A D C
B A B C
z z z z
z z z z
#/g*#/g*#/g*#/g* D@#C"D@#C"D@#C"D@#C"
Az z r = ( )0r >
AM r= M.8!r!#k&D,-A"%r
A Bz z z z = AM BM= M@k&D,-[ ]AB
;2
C A
B A
z zr
z z
pi =
ABCaogMEA
[ ]1;C AB A
z z
z z
=
ABCvMEaLgA
1;2
C A
B A
z z
z z
pi =
ABCaogLgvMEA
1;3
C A
B A
z z
z z
pi =
ABC:*AvME
31
"&J+#
2K
4(05
A(A(A(A(WWWW A(,+A(,+A(,+A(,+WWWW
'y ay b= + ( )0a
( ) axb
y x ea
=
( )
A(A(A(A(WWWW ,"(,"(,"(,"(WWWW +
32
Mm&(Z 2K4(05
oJ#ok& B=Z.6@a!%
33
U(( )S.0gP#$P[ ]AB@5 .##y=mD~U W ( ) . 0M S AM BM =
$*W( )S.8!14[ ]AB"%2
AB
knR71( ),S R`( ) : 0ax by cz d + + + =
=H,/8!,v(^( ) HBW( )( );d H d= =
( )P( )S 5Oh/
( )PqJ( )S 0/aH
( )PH0/( )S !(eU( )C
.8!WH
"%W2 2r R d=
knR71( ),S RO7( )W =H8!,v(,/o/^( ) HBW( )( );d H d= =
o/( )( )S
5Oh/ o/( )qJ( )S
0/aH o/( )H0/( )S
LL0/a
34
+(2K4(05
0;QMW
:1W
",2DE,d!>(#.E! j!WCardE NH)W 0Card =
NHW
ABO"O,2 ( ) ( )Card A B CardA CardB Card A B = +
0;O0W
:1W
=A",2=E o,A ,|,
35
(81Cj8W
=np==!4*( )p n
36
J0)J2K4(05
%9
i04i04i04i04D,$5D,$5D,$5D,$5WWWW ]]]]WWWW p !} +
37
QRX40)JJE-().7W :1W =ABMfp !|G L0
38
Q#^,)>?1(2K4(05
#*M(+JO74( W
#BE8T.W
-1 cos 1
-1 sin 1
cos sin 1
x
x
x x
+ =
sintan
cos1
1 tan cos
xx
x
xx
=
+ =
( )
( )
( )
cos 2 cos
sin 2 sin
tan tan
x k x
x k x
x k x
pi
pi
pi
+ =
+ =
+ =
AJ+W
- 2x a kpi= + Pcos cos 2x a x a kpi= = + ( )- 2x a kpi pi= + P sin sin 2x a x a kpi= = +
( ) tan tanx a x a k kpi= = +
2
pi
3
pi
4
pi
6
pi 0 x
1 3
2
2
2
1
2 0 sinx
0 1
2
2
2
3
2 1 cosx
3 1 3
3 0 tanx
2x
pi+
2x
pi +xpi xpi x
cosx cosx -sinx sinx -sinx sin
sinx sinx - cosx - cosx cosx cos
39
!0;/_eNW ( )
( )
( )
cos cos cos - sin sin
sin sin cos cos sin
tan tantan
1 - tan tan
a b a b a b
a b a b a b
a ba b
a b
+ =
+ = +
++ =
( )
( )
( )
cos - cos cos sin sin
sin - sin cos - cos sin
tan - tantan -
1 tan tan
a b a b a b
a b a b a b
a ba b
a b
= +
=
=+
LMW
cos 2 cos - sin
2 cos - 1
1 - 2 sin
sin 2 2 sin cos
2 tantan 2
1 - tan
a a a
a
a
a a a
aa
a
=
=
=
=
=
1 cos 2cos
21 - cos 2
sin 2
aa
aa
+=
=
HA W tan2
at =
2sin
1
1 - cos
1 2
tan1 -
ta
t
ta
t
ta
t
=+
=+
=
!0;r'F( /_WF( r'!0;/_W
( ) ( )[ ]
( ) ( )[ ]
( ) ( )[ ]
( ) ( )[ ]
1cos cos cos cos -
21
sin sin cos cos2
1sin cos sin sin
21
cos sin sin - sin2
a b a b a b
a b a b a b
a b a b a b
a b a b a b
= + +
= +
= +
= +
cos cos 2cos cos2 2
cos cos 2 sin sin2 2
sin sin 2 sin cos2 2
sin sin 2cos sin2 2
p q p qp q
p q p qp q
p q p qp q
p q p qp q
+ + = + =
+ + = + =
/_Wcos sina x b x+( ) ( ), 0,0a b
( )
cos sin cos sin
cos
a ba x b x a b x x
a b a b
a b x
+ = + + + +
= +
M$e/D//$(#W
sin
b
a b =
+cos
a
a b =
+