الرياضيات باك2 ملخص الدروس .pdf
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∀ (x ; y ) ∈ 2+
x × n y = n x ×y
n
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∀n ∈ *
xn = x
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n
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x −y
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WD. f א#א1!-,2
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D f = [0; +∞[
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f (x ) = n u (x )
W=↖א
lim
x →x0
lim u (x )
n u (x )
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l≥0
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l
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−∞ #P +∞ #P ^אx 0 #PL, ^אx 0 # >^/<- "]אZ.
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]0;+∞[ 3 d^א6 /%* g x n x א#א
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I 3 d^א6 /%* g x n u (x ) א#אOTU
∀x ∈ I
(
n
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)′ =
u ′ (x )
n n [u (x )]n −1
W #
(n x )′ =
∀x ∈ ]0; +∞[
W #
1
n
n x
(a ∈ ) x ∈
DX(# n
n −1
x n = a W+#א/)↖
v(!U(# n
S = {−n a ; n a }
S = {0}
S = {n a }
S = {0}
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a <0
W9TB Q77)+(?@ `א7↖א
q ∈ * p ∈ * WM$#K? Z(א# r =
p
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q
p
∀x ∈ ]0, +∞[
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r
x = xq = xp
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xn
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∀x ∈ ]0; +∞[
WD ,8U! x D//$KV f (#(א1!-,2
f (x ) = [u (x )]r
•
•
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1 ′
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=
1
1
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n
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r
x
r
r
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x r ′
r'
•
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4(א05K2
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un +1 = q × un
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1! א# א
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q n −p +1 − 1
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n
p
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M (# < (un )n ∈I ⇔ ∀n ∈ I
un ≤ M
•
m (# V4 (un )n ∈I ⇔ ∀n ∈ I
un ≥ m
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•
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4g - (un )n ∈I ⇔ ∀n ∈ I
un +1 ≤ un
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un +1 ≥ un
•
) (un )n ∈I ⇔ ∀n ∈ I
un +1
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18
= un
W=↖
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α<0
α>0
lim n α = 0
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n →+∞
n →+∞
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q ≤ −1
(q n ) א
" CG
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n →+∞
n →+∞
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/ D.<#א- +8
•
/ D.V44g - +8
•
vn ≤ un ≤ wn
lim vn = l ⇒ lim un = l
n →∞
n →+∞
lim vn = l
n →+∞
un − l ≤ vn
⇒ lim u = l
lim vn = 0 n →∞ n
n →+∞
un ≥ vn
un ≤ vn
⇒ lim u = +∞
lim vn = +∞ n →+∞ n
n →+∞
⇒ lim u = −∞
lim vn = −∞ n →+∞ n
n →+∞
W un +1 = f (un ) ! ↖א
WD ,8U!( אun ) אY
u0 = a
un +1 = f (un )
I =!א4 a f (I ) ⊂ I Mf I 3 2^4 (אf M$
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19
4(א05K2
Ncא4א(א
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W:1
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∀x ∈ I
•
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WNH
3 dאאZ.^>P(א+</-3 2^4(א+8
I 3 2^U!(# (אf =
WOTU I 3 d ^אf א#>P (אF c 8אS&
WD W I ^U! f א#>א3א#אHz
x F (x ) + k
(k ∈ )
I 3 2^>P(א+</-(# (אf =
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I 3 d ^אf א# F #$>P(א#-
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WQ77)+(א+F@(א-Eא+! 0dWNcא4↖א(א
WNH
//$(א# k I 3 2^LU!L(#L (אg f =
WOTUD ^אאI 3 d ^אg f Lא#L>PL(אG F c 8אS&
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•
I 3 d ^אkf א#>P (אkF
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20
W+Jא4<א(אNcא4א(א4( ↖
f (x )
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1
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2
−1
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x
x
1
x²
1
x
(r ∈ » * − {-1})
x
2 x +k
x r +1
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r +1
r
sin x
− cos x + k
cos x
sin x + k
1 + tan ²x =
1
cos ²x
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1
x
ln x + k
ex
ex + k
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WNcא4<א(א8((%S7JאeN40↖א
(r ∈ » * − {-1})
f (x )
F (x )
u ' (x )
u (x )
2 u (x ) + k
−v ' (x )
[v (x )] ²
1
+k
v (x )
u ' (x ) ×[u (x )]
[u (x )] r +1
r
r +1
u ' (x )
u (x )
+k
ln u (x ) + k
( )
u ' (x ) ×eu x
eu x + k
(a ≠ 0)
cos (ax + b )
(a ≠ 0)
sin (ax + b )
1
sin (ax + b ) + k
a
1
− cos (ax + b ) + k
a
( )
21
(k ∈ )
4(א05K2Q^א,א
W9TDא+/1↖
I 3 d ^אf א#>P (אF I 3 2^4 (אf = W:1
I 3 d!==א4b a
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∫ f (x ) dx = [F (x )]ba = F (b ) − F (a )
a
WNH↖
W9fא
a
a
b
∫ f (x ) dx = 0
∫ f (x ) dx = −∫ f (x ) dx
a
b
b
b
b
b
∫ [ f (x ) + g (x )]dx = ∫ f (x )dx + ∫ g (x )dx (k ∈ ) ∫ kf (x ) dx = k ∫ f (x ) dx
a
a
a
a
a
W4T.
b
c
b
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f
f
∫a
∫a
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b
a
WB1Cא/↖א
∀x ∈ [a,b ]
f (x ) ≤ g (x ) WO 8אS&
b
b
∫ f (x ) dx ≤ ∫ g (x ) dx WOTU
a
a
∀x ∈ [a,b ]
f (x ) ≥ 0 WO 8אS&
b
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a
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b
1
f (x ) dx WD// (א#אD.3 dא^א#0@א,/א
b − a ∫a
WF \אc8#↖א
I 3 d^אL4 v ′ u ′ Lא#אMf I 3 2^6 /%*L gL (אv u =
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b
b
∫ u (x )v ′ (x )dx = [u (x ) v (x )]ba − ∫ u ′ (x )v (x )dx
a
a
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(o, i, j ) # ok& =א
j i L"| אo 0/ (#א+0א$ D. u.A $ א#$
1.u . A = i × j
22
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[a,b ]3 2^4 (אf =
7C g C f LnאL 4א א$
+> Uא7C f ^nאL 4א א$
W ,. ( =ZאL,/א+> Uא
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WD. x = b x = a
x = b x = a
b
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f
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b
∫ f (x ) dx .u.A
WD.
a
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WNHJ)
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a
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a
[a,b ]3 d^א
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DnA- o@
< f •
c
b
a
c
[a, c ] 3 d^א
∫ f (x ) dx + ∫ −f (x ) dx .u.A
< @ f •
[c,b ] 3 d^א
b
∫ ( f (x ) − g (x )) dx .u.A
a
(C g ) 6U# (C f )
[a,b ]3 d^א
(C g ) 6U (C f ) •
c
b
∫ ( f (x ) − g (x )) dx + ∫ (g (x ) − f (x ))dx .u.A
c
a
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[a;b ]3 2a 8(+> Uא
b
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a
W.
| א#$W uv
23
4(א05K20g א4א(א
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W:1
]0; +∞[ 3 d ^אx
1
x
א#>אא#אD.vK<אo ?(אא
ln W! C! 1a#-rא
WNH א
∀x ∈ ]0; +∞[
∀y ∈ ]0; +∞[
ln 1 = 0
ln (xy ) = ln x + ln y
ln e = 1
∀x ∈ ]0; +∞[
∀y ∈ ]0; +∞[
ln x = ln y ⇔ x = y
ln x > ln y ⇔ x > y
(r ∈ ) ln (x r ) = r ln x
1
ln = − ln x
x
x
ln = ln x − ln y
y
∀x ∈ ]0; +∞[
∀y ∈
ln x = y ⇔ x = e y
∀x ∈ * ln (x n ) = n ln x WOTU X(א# n O 8אS&
W: א0;
WD. f א#א1!-,2
D f = {x ∈ / x ∈ Du وu (x ) > 0}
D f = {x ∈ / x ∈ Du
WD ,8U! f א#א
f (x ) = ln [u (x )]
f (x ) = ln (u (x ))2
وu (x ) ≠ 0}
f (x ) = ln u (x )
WA=
ln x
n =0
x →+∞ x
lim
( n ∈ » *)
lim (x n ln x ) = 0
x →0
>
ln (x + 1)
=1
x →0
x
lim
lim (ln x ) = +∞
x →+∞
lim (ln x ) = −∞
x →0
>
ln x
=1
x →1 x − 1
lim
W41Jא
]0; +∞[ 3 d^א4 x
ln x א#א
I 3 2^U! (אu =
I 3 d^א4 x
ln [u (x )] א#אOTU I 3 2^4 0g< u c 8אS&
24
WS7Jא
I 3 2^U! (אu =
I 3 2^6 /%* g 0g< (אu c 8אS&
I 3 d^א6 /%* g x
∀x ∈ I
'
)]
ln [u (x )] א#אWOTU
(ln [u (x ) =
u ' (x )
u (x )
W #
]0; +∞[ ^6 /%* g x
: #
1
∀x ∈ ]0; +∞[ (ln x )′ =
x
W ln *'
ln x
-
WQ#א/0א
0 1 +∞
x
ln x א#א
+
a ∈ » *+ − {1} Wa)a UhOg ↖א(אא
loga W! C!rאא#אD.a q @ o ?אא# אW:1
∀x ∈ ]0; +∞[
∀x ∈ ]0; +∞[
loga (x ) =
∀y ∈ ]0; +∞[
loga (xy ) = loga x + logay
(r ∈ )
ln x
WM$
ln a
WNH א
loga 1 = 0
logaa = 1
loga (x r ) = r loga x
∀x ∈ ]0; +∞[
1
loga = −loga x
x
l oga x = l oga y ⇔ x = y
x
loga = loga x − logay
y
∀y ∈ ]0; +∞[
∀r ∈
l oga x = r ⇔ x = a r
:1&=
0 <a <1
loga x < logay ⇔ x < y
lim loga x = −∞
a >1
loga x > logay ⇔ x > y
lim loga x = +∞
x →+∞
x →+∞
lim loga x = +∞
lim loga x = −∞
x →0+
x →0+
W7X#א
∀x ∈ ]0, +∞[
( loga x ) ' =
25
1
x ln a
4(א05K2cא4א(א
>א0g ↖א(אא
W:1
K<א, ?אא#אא#אD.K<אא@א#א
exp W! C!
exp (x ) = e x = x +HB
WNH א
∀x ∈
∀x ∈ e x > 0
∀x ∈ ln ( e x ) = x
∀y ∈
e x ×ey = e x +y
∀x ∈
1
−x
x =e
e
ex
x −y
y =e
e
∀y ∈ ]0; +∞[
e x = y ⇔ x = ln y
∀ (x ; y ) ∈ ²
e x = ey ⇔ x = y
e x > ey ⇔ x > y
W: א0;
WD. f א#א1!-,2
e ln x = x
∀x ∈ ]0, +∞[
r
(e x ) = erx
(r ∈ )
WD ,8U! f א#א
Df =
D f = {x ∈ / x ∈ Du }
f (x ) = e
f (x ) = e
x
u (x )
WA=
lim e x = +∞
x →+∞
lim e x = 0
( n ∈ » *)
x →−∞
x
e
lim n = +∞
x →+∞ x
lim (x ne x ) = 0
x →−∞
ex − 1
lim
=1
x →0
x
W41Jא
^4 x
e
x
א#א
I 3 2^U! (אu =
I 3 d^א4 x
e
u (x )
א#אOTU I 3 d^א4 u c 8אS&
26
WS7Jא
(ex )′ = ex W # ^6 /%* g x
∀x ∈
e
x
א#א
I 3 2^U! (אu =
I 3 d^א6 /%* g x
e
u (x )
א#אWOTU I 3 d^ א6 /%* g u c 8אS&
'
(eu x )
( )
∀x ∈ I
u (x )
= u ′ (x ) ×e
W #
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a ∈ ∗+ − {1} Wa) a Uhc↖א(אא
expa W! C! a q @ @אא#^א,- loga א#אא#א
W:1
expa (x ) = a x = x +HB
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∀ (x ; y ) ∈ 2
a x × a y = a x +y
r
(a x ) = a r x
1
−x
x =a
a
ax
x −y
y =a
a
(r ∈ )
a x = e x ln a
loga (a x ) = x
∀x ∈
∀x ∈ ]0; +∞[
∀ (x ; y ) ∈ 2
a
loga (x )
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∀x ∈
∀y ∈ ]0; +∞[
x
a = y ⇔ x = loga (y )
W1&=
0 <a <1
a < ay ⇔ x < y
a >1
a > ay ⇔ x > y
lim a x = 0
lim a x = +∞
x
x
x →+∞
x →+∞
x
lim a x = 0
lim a = +∞
x →−∞
x →−∞
x
a −1
= ln a
x →0
x
lim
(a x )′ = (ln a ) ×a x
27
W7X#א
4(א05K2
(7א+(אcא
= {z = a + ib / (a;b ) ∈ ² i ² = −1} WD.#/א(א#א,2
Wi(7+(P@א8↖א
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z v#/(א#Y;^א א,- a + ib
•
Re (z ) W! j! z (#D// ^א;א, a (#א
•
Im (z ) W! j! z (#D^א;א, b (#• א
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z = z ′ ⇔ Re (z ) = Re (z ′) Im (z ) = Im (z ′)
Wi(7+(Q#א/0↖א
(o,e1,e2 ) oJ# ok& v#/=אא
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M (a, b ) 0/ z v#/(א#! א
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z = Aff (OM ) POM (z ) WtOM "|אe Z8^, z (#• א
Wi(7+(bk↖א
(a;b ) ∈ ² WM$ #/(א# z = a + ib =
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D// אn,< O*) , M ′ (z ) M (z )
D//$(# z ⇔ z = z
•
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z + z = 2 Re (z )
z − z = 2i Im (z )
zz = [Re (z )]² + [ Im (z )]²
•
•
•
z + z ' = z + z ' •
z ×z ' = z ×z ' •
(n ∈ *) z n = z n •
1
1
= •
z ' z '
•
z z
(z ' ≠ 0) =
z '
z'
•
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(a;b ) ∈ ² WM$ #/(א# z = a + ib =
z = zz = a ² + b ² WtאD// (א#א. z v#/(א# א
28
z ×z′ = z × z′
zn = z n
z = z
−z = z
1
1
=
z′
z′
z
z
=
z′
z′
(n ∈ * )
(z ' ≠ 0)
W(>gi(7+(cא8אQ#א/X↖א
M j->#K? #/(א# z =
(e
1, OM ) W" @ אאאg#$P θ . z v#/(א#א#,
arg z W! j!
arg z = θ [2π ] Wt
WNHJ)
#K? #/(א# z =
#K? a D//$(#EEא א
a <0
a >0
arg z = θ [2π ] r = z HB
W. z v#/(א#DEEא+p• א
a = [−a, π ]
a = [a, 0 ]
π
ai = −a, −
2
π
ai = a, +
2
z = r (cos θ + i sin θ ) = [r , θ ]
z = reiθ WD. z v#/(א#@• א א
reiθ × r ' eiθ ' = rr ' ei (θ +θ ') • [r , θ ] × [r ', θ '] = [rr '; θ + θ ']
[r , θ ] = [r, −θ ]
reiθ = re−iθ •
−[r, θ ] = [r, π + θ ]
−reiθ = re i(π +θ ) •
n
[r , θ ]n = r n ; n θ
rei θ = r nein θ •
1
1
1
1 −iθ '
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=
e
•
[r '; θ '] r '
r'
r ' eiθ '
( )
re
iθ
r 'e
iθ '
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[r ; θ ]
=
[r '; θ ']
r i (θ−θ ')
e
•
r'
•
arg (zz ') ≡ (arg z + arg z ') [2π ] •
•
arg z ≡ − arg z [2π ] •
− arg z ≡ (π + arg z )[2π ] •
•
•
arg
r
; θ − θ ' •
r '
دz ⇔ arg z = k π •
( k ∈ )
arg z n ≡ n arg z [2π ] •
1
arg ≡ − arg z [2π ] •
z
•
د فz ⇔ arg z =
π
+ kπ •
2
z
≡ (arg z − arg z ') [2π ] •
z'
∀k ∈
[r, θ + 2k π ] = [r, θ ]
W>AYN↖
∀θ ∈
(
(
Wk אYN↖
)
)
1 iθ
e + e −iθ ∀n ∈
2
1 iθ
(cos θ + i sin n θ )n = cos (n θ ) + i sin (n θ )
sin θ =
e − e−i θ
2i
cos θ =
W (a ∈ ) a) z ∈ z ² = a +#א/)↖
W( א3$,2
W( א
S = {− a ; a }
S = {0}
S = {−i −a ; i −a }
a>0
a=0
a <0
29
z ∈ z² = a
(a ≠ 0) 77)+(אAc b a Wa) z ∈
W( א3$,2
−b − ∆ −b + ∆
S =
;
2a
2a
−b
S=
2a
−b − i −∆ −b + i −∆
S =
;
2a
2a
{ }
az ² + bz + c = 0 W+#א/)↖
W( א
∆>0
z ∈
∆=0
az 2 + bz + c = 0
(∆ = b2 − 4ac )
∆<0
W(7א+(אcא%9(6O6&↖
#/אg*א
D@#Cא"א
AB = z B − z A
z + zB
zI = A
2
z − z A
[2π ]
AB; AC ≡ arg c
z B − z A
zC − z A
∈
zB − zA
z −z
z −z
z −z
z −z
D A × D C ∈ P D A × B C ∈
z B − z A z B − zC
z B − z A z D − zC
AB U א
(
)
D@#Cא"א
[A; B ] 0/א14 I
(
)
AB
; AC אאq g
,//C B A
א#/ D C B A
#/אg*א
AM = r
r " % A .8!rא!א#אk&D,- M
•
•
z − zA = r
(r > 0)
AM = BM
[AB ] @אk&D,- M
•
•
z − zA = z − zB
A aאאo gME ABC
A aLg אv ME ABC
A aאאo gLg אv ME ABC
:*Aאv ME ABC
zC − z A
π
= r ; ±
zB − zA
2
zC − z A
= [1; θ ]
zB − zA
zC − z A
π
= 1; ±
zB − zA
2
zC − z A
π
= 1; ±
zB − zA
3
W+Jא. %(<א7.l↖
W.v#/אjEx
+nא
u "|אe b M$ z ′ = z + b
u "|אאS t $אX{א
Ω 0/אe ω M$ z ′ − ω = k (z − ω )
k j< Ω ]8!vZ אh D8 nא
Ω 0/אe ω M$ z ′ − ω = ei θ (z − ω )
θ jאX Ω ]8!vZ אr Oא#א
30
y (x ) = αeax −
(α ∈ )
WA (א, א+ א
y (x ) = αer1x + βer2x
(α, β ) ∈ ² WM$
y (x ) = (αx + β )erx
(α, β ) ∈ ² WM$
WA א (א
b
a
y ' = ay + b
(a ≠ 0)
W+</-,א (א
L//$L$
r2 r1 L
r א#$ //$*$
W, (" א
∆>0
∆=0
r1 = p − iq
r ² + ar + b = 0
(∆ = a ² − 4b )
WL/UאQL#/L$
y (x ) = (α cos qx + β sin qx )e px
(α, β ) ∈ ² WM$
WA א (א
∆<0
r2 = p + iq
31
y ''+ ay '+ by = 0
4(א05K2 "&אJ+#א
WA (א, א+ א
4(א05K2
Mm&(אZא
(o, i , j , k ) !% <oJ# ok& Bאא=אZ.6 @a
Q=#אFא@(אJ=OIJQ0אFא@(אW4%אY↖א
ϑ3 =L"| v (a ',b ', c ') u (a,b, c ) =
u .v = aa '+ bb '+ cc '
•
u = a ² + b ² + c ²
•
i a a'
b b' a a' a a'
u ∧ v = j b b ' =
i −
j +
k
c
c
'
c
c
'
b
b
'
k c c'
•
Wk#↖א
WD. B A L0/L U א
AB = (x B − x A ) ² + (yB − yA ) ² + (z B − z A ) ²
WD. ax + by + cz + d = 0 j( (P ) = M 0/U א
d (M , (Ρ)) =
AM ∧ u
d (A, (∆)) =
u
ax M + byM + cz M + d
a ² + b² + c²
WD. ∆ (A, u ) o/= M 0/U א
W` +↖
(P ) ^א,"| n (a,b, c ) ⇔ (Ρ) : ax + by + cz + d = 0
(ABC ) ^א,"| AB ∧ AC OTU,/K?/ C B A c 8אS&
WD ~אU @5 (ABC ) (א##y=m
M ∈ (ABC ) ⇔ AM .(AB ∧ AC ) = 0
Wk+↖
WD. R " % Ω (a,b, c ) .8!U(
(x − a ) ² + (y − b ) ² + (z − c ) ² = R²
32
@5 .##y=m [AB ] . 0gP#$P (S ) U(
M ∈ (S ) ⇔ AM . BM = 0 WD ~אU
AB
" %[AB ] 14 Ω .8! (S ) אW$*
2
(Ρ) : ax + by + cz + d = 0 ` S (Ω, R) knR71↖
(Ρ) ^אΩ 8!,v(,א/ אH =
d = ΩH = d (Ω; (Ρ)) WHB
(S ) אH0/ (P ) א
(C ) !(אeU
H W .8!
(S ) q J (P ) א
(S ) ( אP ) א
H 0/אa
O h /5
r = R2 − d 2 W " %
W (∆) O7 S (Ω, R) knR71↖
(∆) o/ ^אΩ 8!,v(,א/ אH =
d = ΩH = d (Ω; (∆)) WHB
(S ) אH0/ (∆) o/א
(S ) q J (∆) o/א
(S ) )∆( אo/א
LL0/a
H 0/אa
O h /5
33
4(א05K2+א(א
W 0;QM↖
W:1
CardE W! j! E ,d( >!א#. E ",2D
Card∅ = 0
WNH)
WNH
O "O ,2 B A
Card (A ∪ B ) = CardA + CardB − Card (A ∩ B )
W 0;O0↖
W:1
E ",2= אA =
A W! C!rא,dאD. E ,|,< A o,
A = {x ∈ E / x ∉ A} M$
WI).
A∩A = ∅
A∪A = E
cardA = cardE − cardA
•
•
•
W+(אQcאA(#↖א
(p ∈ *)
א\ אp "| t0- !}Y
8 n1 Fo3\ א5אO 8אS&
8 n2 FoD E\ א5אO 8
.........................................
8 n p Fo p \5אO 8
n1 × n2 × n3 × ... × n p
Wא#;א.,(א א#OTU
Wא8j(81Cא- א81C↖א
Wא81Cא
(p ≤ n ) » * ==!4 p n =
n p W.!4 n L =!4 p 3< !א-Q(א#
34
Wא8j(81Cא
(p ≤ n ) » * ==!4 p n =
W.!4 n L =!4 p 3!א-O# <-Q(א#
Anp = n × (n − 1) × (n − 2) × ... × (n − p + 1)
+ =אאp
WNH)
!4 n 3#<-Z8^,-!4 n L =!4 n 3!א-O# <-!-+8
n ! = n × (n − 1) × (n − 2) × ... × 2 × 1 W .(#
W&[↖א
n .!> (#",2 E =
(p ≤ n ) p ]!> (# E = A +8
!4 n L =!4 p 3l-^,
C np =
Anp
W. l]אZ.(#
p!
C np Anp n ! W+(אc↖א
n ∈ ∗
n ! = n × (n − 1) × (n − 2) × ... × 2 × 1
0! = 1
n!
n!
Anp =
C np =
p ! (n − p ) !
(n − p ) !
n −1
0
1
C nn = 1
Cn = n
Cn = 1
Cn = n
C np = C nn −p
C np−1 + C np = C np+1
WB% א!אA<8↖
(p ≤ n ) !4 n L =!4 p tn
WD א3#;אa א
t-Qא
W.,< אn(א#
Wtnא:
o"K?
C np
D
o"
np
Anp
3*$T H
o"
35
3*$&O# H
4(א05K2J0)Jא
%9↖
W]
WD ,$5אi04א
|=!E8P+</- !}+8
אp !}
אp !|,א{ א,2D.
{אO8 Ω
Ω {אO8= אA
A #$
א#$!א4=,B#$+8
Dא# א#$
#$אOa B A O )# אe/yאS&
A ∩ B # אe/y
,.P B P A e/yאS&
A ∪ B # אe/y
(A ∩ A = ∅وA ∪ A = Ω) A # א.
A #n( Bא# א
A∩B = ∅
L,|K?O )#$ B A
Wo()40)אJo()א7↖א
אp !} &O8 Ω = W:1
pi W. {ωi } # א3 ,$אOP3/ pi j,ga {ωi } Dא# א#$3 ,$!א/ # •
P ({ωi }) = pi Wt
# אאZ.O-rאא# 5א5 ,$5א:,2.#$3 ,$• א
W. A # א3 ,$אOTU Ω = )#$ A = {ω1; ω2 ; ω3 ;...; ωn } O 8אS&vP
p (A) = p (ω1 ) + p (ω2 ) + p (ω3 ) + ... + p (ωn )
אp !} &O8 Ω =
WNH
p (Ω) = 1 p (∅) = 0 •
Ω = A #$+ 0 ≤ p (A) ≤ 1 •
WL)#$( yא3 ,$• א
Ω = B A L)#$+
p (A ∪ B ) = p (A) + p (B ) − p (A ∩ B )
L,|K? B A O 8אS& p (A ∪ B ) = p (A) + p (B )
W( Bא# א3 ,$• א
p (A) = 1 − p (A)
W Ω = A #$+
WJ0)Jאi1"k↖
Ω " &O8אp !}a3 ,$5א א# 5אא#$אHzc 8אS&
cardA
p (A) =
W. Ω = A #$+83 ,$אOTU
card Ω
36
W:1
WE-().7אJQRXא40)J↖א
p (A) ≠ 0 WMfאpא|! אG L0<-!L)#$ B A = W:1
p (A ∩ B )
p (B ) = p (B A) =
W(#א.e/7 A # אOP , B #$3 ,$א
p
A
(
)
A
p (A) × p (B ) ≠ 0 WMfאpא|! אG L0<-! B A L)#$+
p (A ∩ B ) = p (A) × p (B A) = p (B ) × p (A B ) W #
W
W:1
אpא|! אG L0<-! B A L)#$+
O*/O )#$ B A ⇔ p (A ∩ B ) = p (A) × p (B )
Ω 3 } Ω2 Ω1 אp !} &O8 Ω =
WNH
(Ω1 ∩ Ω2 = ∅ Ω1 ∪ Ω2 = Ω)
( )
( )
p (A) = p (Ω1 ) × p A Ω + p (Ω2 ) × p A Ω W Ω = A #$+
1
2
W*#אאHJ↖א
p j ,$אאp !}a )#$ A =
W.<B ! k , A # אe/y3 ,$אO U! n !|]אZ.#PאS&
C nk (p )k (1 − p )n −k
(k ≤ n )
WQM אX>Y40)אj T↖
אp !} &O8 Ω ^ אpאKV=
WL אL$!אH< X DאpאKVא3 ,$אO g##n
X KV א.Z\lrאo/א,2W X (Ω) = {x1; x 2; x 3 ;...; x n } ##y •
{1;2;...;n } ,d =אi + p (X = x i ) 3 ,$5אtw •
WQM אX>Y#ipא9א]אqJאJ*Y#אJQ"א/c↖א
j g אpאKV X =
xi
x1 x 2 x 3 ... x n
p (X = x i ) p1 p2 p3 ... pn
WD א3#; 9!
E (X ) = x1 × p1 + x 2 × p2 + x 3 × p3 + ... + x n × pn W X KV,DA !א+א
V (X ) = E (X ²) − [E (X )] ²
W X KV,! Vא
σ (X ) = V (X )
W X KV,vX!א0א9!אw5א
W:1
WQ(א,אj 7↖א
! n !|]אZ.#{אp !}a A #$3 ,$ אp =
p n ] 0@ א#$ X-^, A # " אUe/nr(א!אא# |+8 !vZ אX DאpאKVא
∀k ∈ {0;1;2;...; n }
V (X ) = np (1 − p )
p (X = k ) = C nk × p k × (1 − p )n −k
E (X ) = n × p
#
37
4(א05K2(>
?1)Q#^א,א
W#*אMא(א+JאO7א4( ↖
x
0
sin x
0
cosx
1
tan x
0
π
π
π
π
6
1
2
3
2
3
3
4
2
2
2
2
3
3
2
1
2
2
1
3
1
0
W#אBאE8T.↖א
−x
cos (x + 2k π ) = cos x
sin (x + 2k π ) = sin x
tan (x + k π ) = tan x
π − x π +x
π
2
−x
π
2
+x
sin
- sin x
sin x
- sin x
cos x
cos x
cos
cos x
- cos x
- cos x
sin x
− sin x
tan x =
sin x
cos x
1
1 + tan ²x =
cos ²x
-1 ≤ cos x ≤ 1
-1 ≤ sin x ≤ 1
cos ²x + sin ²x = 1
WAJ+↖
cos x = cos a ⇔ x = a + 2k π P x = -a + 2k π
sin x = sin a ⇔ x = a + 2k π P x = (π - a ) + 2k π
tan x = tan a ⇔ x = a + k π
( k ∈ )
38
W! 0;/ _eN↖
cos (a - b ) = cos a × cos b + sin a × sin b
cos (a + b ) = cos a × cos b - sin a × sin b
sin (a - b ) = sin a × cos b - cos a × sin b
sin (a + b ) = sin a × cos b + cos a × sin b
tan (a - b ) =
tan a - tan b
1 + tan a × tan b
tan (a + b ) =
tan a + tan b
1 - tan a × tan b
WLM↖
t = tan
cos 2a = cos ² a - sin ² a
a
WHA
2
= 2 cos ² a - 1
2t
1 + t²
1 - t²
cos a =
1 + t²
2t
tan a =
1 - t²
= 1 - 2 sin ² a
sin a =
sin 2a = 2 sin a × cos a
2 tan a
1 - tan ² a
1 + cos 2a
cos ² a =
2
1 - cos 2a
sin ² a =
2
tan 2a =
WF (אr'! 0;/ _↖W! 0;r'F (א/ _↖
p + q
p − q
cos p + cos q = 2 cos
cos
2
2
p + q p − q
cos p − cos q = −2 sin
sin
2 2
p + q
p − q
sin p + sin q = 2 sin
cos
2
2
p + q p − q
sin p − sin q = 2 cos
sin
2 2
1
[ cos (a + b ) + cos (a - b )]
2
1
sin a × sin b = − [ cos (a + b ) − cos (a − b )]
2
1
sin a × cos b = [ sin (a + b ) − sin (a − b )]
2
1
cos a × sin b = [ sin (a + b ) - sin (a − b )]
2
cos a × cos b =
(a,b ) ≠ (0,0) a cos x + b sin x W/ _↖
a
b
a cos x + b sin x = a ² + b ²
cos x +
sin x
a ² + b ²
a ² + b²
= a ² + b ² cos (x − α)
We/D//$(# α M$
cos α =
a
b
sin α =
a ² + b²
a ² + b²
39