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. . .
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pp p p
p q p q
pq p q
= p=q p p q
pq p q
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p q p p q p q p=q pq
1 1 0 1 1 1 11 0 0 0 1 0 00 1 1 0 1 1 00 0 1 0 0 1 1
A={x1, x2, . . . , xn}={x|P(x)}
A B={x|xA xB}
A B ={x|xA xB}
A \ B={x|xA (xB)}
AB= A \ B B\ A
U Ac =cA= A={x|xU\ A}={x|(xA)}
P(A) = 2A ={E|EA}|P(A)|= 2|A|
(x, y) =x, y={{x}, {x, y}} AB={(x, y)|xAyB}A1 A2 An ={(x1, x2, , xn)|xiAii={1, 2, , n}}
(E F)c =Ec Fc(E F)c =Ec Fc
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AA A A A BB A (A B) CA (B C) A BB A (A B) CA (B C) A AA A
A
A
A BA AA B A (B C)(A B) (A C) A (B C)(A B) (A C) A (A B)A A (A B)A (A B) A B (A B) A B A A (A A) (AB)(BA) (AB)(B A) A A A(BA) A(AB) ((AB) B) A (A A) A (AA)A ((AB)A)A ((AB) (BA)) A((AB)B) (A(BC))(B(AC)) ((AC) (BC))(A BC) ((AB) (AB))B (A
(B
C))
((A
B)
(A
C))
((AB) (BC))(AC) (A(BC))((A B)C)
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[R, +, ] [R, +] [R\ {0}, ]
R
0
x R x
1
x R\{0} 1x = 1/x=x1
xyx + zy + zz R
xyxzyzz R, z0
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A B a A, bB, ab
cR :abc
R = R{}
(a b)n =n
k=0
nk ankbk
(a b)2 =a2 2ab + b2 (a b)3 =a3 3a2b + 3ab2 b3 (a b c)2 =a2 2ab + b2 2bc + c2 2ca
P(x) =ann
i=0(x xi) =
n
i=0ai xi
P(x) = a(x x1) (x x2) (x xn)nN :xn yn = (x y) (xn1 + xn.2y+ + xyn2 + yn1)n= 2k +1, k Z :xn + yn = (x+ y) (xn1xn2y + xyn2+ yn1)
a2 b2 = (a b)(a + b)a3 b3 = (a b)(a2 ab + b2)
P(x) = ax2 + bx + c x1,2= b
b24ac2a
x1,2=
2ac
a := b2
A B=
A+A2B
2
AA2B
2
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f(x)g(x)
g(x)0f(x)g2(x)
g(x)< 0f(x)0
f(x)< g(x)
f(x)0g(x)> 0f(x)< g2(x)
y 0, n N n y ny= y1/n =sup{xR :xn < y}= inf{x R :xn > y}
a0 = 1a1 =a
am+n =am anamn = a
m
an
(am)n =amn
am =emlog a
a1n =n
a
am = 1am
a= logb cba
=c
loga1 = 0loga a= 1loga(m n) = loga m + loga nloga
mn = loga m loga n
loga m = loga m R
loga b= logcblog
ca
loga b= 1logba
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f : R R+0 = [0, +]f(x) =|x|= (x) = {x, x}=
x2
aR, |a| 0
|a|= 0a = 0
R, aR, | a|=|| |a|
a, b R, |a + b| |a| + |b|
|
n
i=1
ai|
n
i=1 |
ai|
||a| |b|| |a b|
nN, n! =f att(n) =n
i=1i=
= 1 2 (n 1) n
0! = 1n! =n (n 1)!
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k N, (2k+ 1)!! =
ki=0(2i + 1)
(2k)!! =
1 k= 0k
i=1 k >0
(1)!! = 1
x R \ {0}, (x) = (x) = (x) =x|x| =
|x|x =
1 x >01 x
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sinh : R Rsinh x= x= e
xex2
cosh : R [1, +)cosh x= x= e
x+ex
2
sinh2 x cosh2 x= 1
tanh : R Rtanh x= x= sinhxcoshx
coth : R \ {0} Rcoth x=
x= coshx
sinhx
:R R y= y = log(y+
y2 + 1)
: R R y= y = log(y+
y2 1)
: R R x= y= 1
2 log 1+x1x
ex = (x)
! f : R R
x1, x2 R, f(x1+ x2) = f(x1) f(x2)
f(1) =e e
xR f(x) = ex = (x)
f(x) =ax
=ex loga
an = a1+ (n 1)d
Sn =n
i=1
ai =n(a1+ an)
2 =
n(2a1+ (n 1)d)2
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an = a1 qn1
Sn =n
i=1
ai = a11 qn
1 q = 1
q= 1 Sn =n
i=1
ai = n a1
Sk,n =n
i=k
ai = qk 1 q
nk+1
1 q = 1
x[2 , 2 ], | sin x| |x|
nN, a 1, (1+ a)n 1 +an
xR, ex x + 1
x >1, log(x + 1)x
a, b >0,p,q >1 : 1p + 1q = 1, a b 1p ap + 1q bq
ni=1
i= n(n + 1)
2n
i=1
i2 = n(n + 1)(2n + 1)
6n
i=1
i3 = (n(n + 1)
2 )2
ni=0
ni = (1 + 1)n = 2nn
i=0
(1)i
ni
= 0
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sin2 x + cos2 x= 1
tan x= sinxcosx
x
=
2 + k
cot x= cosxsinxx=k
cot x= 1
tanxx=k 2
sec = 1cossec : R
\ {pi2
+ k,k
Z
} R
csc = 1sin
csc : R \ {k,k Z} R
sin( ) = sin cos cos sin cos( ) = cos cos sin sin tan( ) = tantan1tan tancot( ) = cot cot1cotcot
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sin(2) = 2sin cos cos(2) = cos2 sin2 = 1 2sin2 = 2 cos2 1tan(2) = 2tan1tan2 sin(3) = 3sin 4sin3 cos(3) = 4 cos3 3cos tan(3) = 3tantan
3 13tan2
sin(
2 ) =1cos2cos(
2) =
1+cos
2
tan(2
) =
1cos1+cos
= sin1+cos
= 1cossin
tdef
= tan 2
sin = 2t1+t2
cos = 1t2
1+t2
tan = 2t1t2
sinp + sin q= 2 sin p+q2 cospq2
sinp sin q= 2 cos p+q2 sin pq2cosp + cos q= 2 cos p+q2 cos
pq2
cosp cos q=2sin p+q2 sin pq2
cosp sin q= 12 [sin(p + q) sin(qp)]sinp
sin q= 1
2
[sin(p
q)
cos(p + q)]cosp cos q= 12 [cos(p + q) + cos(p q)]
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sin sin 1 cos2 tan1+tan2
cos 1 sin2 cos 11+tan2
tan sin1sin2
1cos2 cos
tan
cot
1sin2 sin cos1cos2 1tan
sec 11sin2
1cos
1 + tan2
csc 1sin 11cos2 1+tan2 tan
12
15624
6+2
4 2 3 2 + 3
8
2230
222
2+2
2
2 1 2 + 1
6
30 12
32
33
3
4
4522
22
1 13
6032
12
3
33
38 6730 2+22 222 2 + 1 2 1512
756+2
4
624
2 +
3 2 32
90 1 0 0
sin x cos x tan x cot x
x sin x
cos x
tan x
cot x
+ x sin x cos x tan x cot xx sin x cos x tan x cot x
2 x sin x cos x tan x cot x2 x cos x sin x cot x tan x2
+ x cos x sin x cot x tan x32
x cos x sin x cot x tan x32
+ x cos x sin x cot x tan x
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S= 12
ab sin = 12
bc sin = 12
ac sin
S= 12a2sin sin sin(+) =
12b
2sin sin sin(+) =
12c
2sin sinsin(+)
S=
p(p a)(p b)(p c)
a,b,c,d p= a+b+c+d
2 S=
(p a)(p b)(p c)(p d)
d= 0
AB= 2r sin
a
sin = a
sin = a
sin = 2R= abc
4S
a= b cos + c cos b= a cos + c cos c= a cos + b cos
a2 =b2 + c2 2bc cos b2 =a2 + c2 2ac cos c2 =a2 + b2 2ab cos
b= a sin = a cos c= a sin = a cos b= c tan = c cot c= b tan = b cot
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r y= mx + n y= ax + by+ c m=ba n=ca
P1(x1, y1)P2(x2, y2)P3(x3, y3)
rP1 (y y1) = m(x x1)srP1 y1= mconstx1+ nrP1P2 :
y
y1
y2y1 = x
x1x2x1
mr = y2y1x2x1
rsmr =msrsmr = 1ms
P1P2= dist(P1, P2) =
(x1 x2)2 + (y1 y2)2dist(P1, r) =
|ax1+by1+c|a2+b2
P1P2= (x1+x2
2 ; y1+y2
2 )
P1P2P3= (x1+x2+x3
3 ; y1+y2+y3
3 )
: x2 + y2 + ax + by+ c= 0
C(x0, y0) r2 =x20+ y
20 c0
P(x, y)(P C) = r: (x x0)2 + (y y0)2 =r2
a=2x0b=2y0c= x20+ y
20 r2
x0=a2y0=b2r=
a2
4 + b
2
4 c
P :y = ax2 + bx + c
F(x0, y0); d: y = kP(x, y) P dist(P, F) = dist(P, d)
y0> k a >0 y0< k a
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F( b2a , 14a )d: y =
1
4a )F( b2a , 4a )a: x = b2a =b2 4ac
P :x = ay2 + by+ c
F(x0, y0); d: y = kP(x, y)
P dist(P, F) = dist(P, d) x0> k a >0
x0< k a b e= ca a e= cb 1
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I: x2
a2 y
2
b2 =1
c2 =a2 + b2, c0I x V1(0, b) V2(0, b)F1(0, c) F2(0, c); F1, F2yP(x, y) I |P F1 P F2|= 2aa1: y =
bax
a2: y =baxe= cb >1
I :x2 y2 =a2
c= a
2 a= bI x V1(a, 0) V2(a, 0)F1(a
2, 0) F2(a
2, 0); F1, F2x
P(x, y) I |P F1 P F2|= 2aa1: y = xa2: y =xe= 2
I: x2 y2 =a2
c= a
2 a= bI y V1(0, a) V2(0, a)F1(0, a
2) F2(0, a
2); F1, F2y
P(x, y) I |P F1 P F2|= 2aa1: y = xa2: y =xe=
2
I :xy = k
k >0 I bI,III :y = x V1(
k,
k) V2(
k, k)k >0 I bII,IV :y =x V1(
|k|,
|k|) V2(
|k|,
|k|)
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leftrightarrow |P F1 P F2|= 2aa1: x = 0a2: y = 0e=
2
I :y = ax + bcx + d
I c= 0ad
bc
= 0
a1: x =dca2: y =
ac
e=
2
C : a11x2 + a22y2 + a12xy+ a13x + a23y+ a33 = 0
A= a11
12a21
12a13
12a12 a22
12a231
2a1312a23 a33
A=
a11
12
a2112
a12 a22
|A|= 0
|A| = 0
|A|> 0 |A|= 0 |A|< 0
|A
|= 0
|A| = 0
T :R R R R
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(x, y)(x, y)
x= ax + by+py= cx + dy+ q
A=
a bc d
|A| = 0
T1 :R R R R(x, y)(x, y)
S
S =|A| x= d|A|x
+ b|A|y+ d|A|p +
b|A|q
y= c|A|x+ a|A|y
+ c|A|p +a|A|q
A1 =
d|A|
b|A|
c|A|
a|A|
T T T T T T
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x= x +py= y + q
A=
1 00 1
|A|= 1
x= x cos y sin y= x sin + y cos
x= xcos + ysin y=xsin + ycos
A=
cos sin sin cos
|A|= cos2 + sin2 = 1
A1 =
cos sin sin cos
|A1
|= cos2 + sin2 = 1
x= ax byy= bx + ay a
2 + b2 = 1
:{ x
= x cos y sin y = x sin + y cos
:{ x = x +p
y= y + q
:{ x= x cos y sin +p
y= x sin + y cos + q
x= 2x0 xy= 2y0 y
(x0, y0)
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y= y0
x= xy= 2y0 y
|A|= 1 00 1
=1
x= x0 x= 2x0 xy= y
|A|= 1 00 1
=1
y= mx + q
x = 11+m2
[(1 m2)x + 2my 2mq]y= 1
1+m2[2mx + (m2 1)x + 2q]
|A|=
1m21+m2
2m1+m2
2m1+m2
m211+m2
=1
x= ax + hy= ay + k
|A|= a 00 a
=a2 >0
x= ax by+py= bx + ay+ q
|A|= a bb a
=a2 + b2 >0
|A|=
a bb a
=a2 b2
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|k|> 1 |k|< 1
x = kxy= y
y
x = xy= ky
x = x + kyy= y
x = xy= y + kx
P T(P)
P Q
T(P)T(Q) P Q
T(P)T(Q) S S
|det(A)|
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Mm A aA, M ama MA ={M R :aA, Ma}mA={mR :aA, ma}
aA,supAa >0, aA : supA asupA aA,infAa
>0, aA : infAainfA + A supA= +infA=
A
xR, aA : axxR, aA : ax
AR A M 0 :xA|x| M
aA,maxAa
maxAA aA, minAa
minAA
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A R x, y : x < y
z : x < z < y, zA
A
(a, b) =]a, b[={xR :a < x < b}, a, bR
(a, b] =]a, b] ={xR : a < xb}, a
R, b R
[a, b) = [a, b[={xR : ax < b}, a
R, bR
[a, b] ={xR :axb}, a, bR
A A, A + B A A={y R :yA} A R A={ x: xA} A B A + B={xR :x = a + b, aA, bB}
sup(A+B) = supA+supB, inf(A+B) = infA + infB
sup(A) =infA, inf(A) =supA0 :sup( A) = supA, inf( A) = infA0 :sup( A) = infA,inf( A) = supA
a > 0, b N , n N :na > b
A R A
Q
Q
R
a, b R, a < b, (a, b)Q= (a, b)Q {x Q : a 0 x0 (x0 , x0+ ) 2
A R x0 A x0 A >0 : (x0 , x0+ )A (x0 , x0+ ) A=
A =intAA A
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A R x0 R x0
A
>0, (x0 , x0+ ) A= (x0 , x0+ ) Ac
=
AR x0 R x0 A >0, (x0 , x0+ ) A \ {x} = r >0, yA : y=x : y(x r, x + r)
A A A
AR x0 R x0 A x0 A >0 : (x0 , x0+) A \ {x}=
A A = AA
A A= A x0A A x0 x0A, r >0 : (x0 r, x0+ r)A
A Ac A= A
AR A
R
A, B
R A
B
R x, y R x y xRy R
xA, xRx
xA, yB, xRyyRx
xA, yB, (xRy) (yRx)x = y
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xA, y(A B), zB, (xRy) (yRz)(xRz)
R R
xA, yB, (xRy) (yRx)
R
P
(xP) (xP)
(x, yP)(x + y, x yP)
xyx yP xyyx > x > y
(x
y)
(x
=y) <
x < y(yx) (x=y)
f f
aA, !bB : af b
f :AB f :aAbB af b b a f b= f(a) a b f a= f1(b)
A B f A B
f bB, !!aA : f(a) = b a1, a2 A, f(a1) = f(a2) a1 = a2 a1, a2A : a1=a2f(a1)=f(a2)
f bB, aA : f(a) = b
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f
f
b B, !a A : f(a) = b f1 f f1 =I dB, f1 f =I dA IdA IdB A B IdA : AA, xx IdB :BB, xx
f : AB
x, y A , x < y f(x) f(y)
x, y A, x < yf(x)f(y)
f :I R
I
f
f+ g f g fg f f g f g AR R
(f+ g)(x) : AR, xf(x) + g(x)
(f g)(x) :AR, xf(x) g(x)
(fg )(x) :
{x
A : g(x)
= 0
} R, x
f(x)g(x)
R, ( f)(x) : AR, x f(x)
f :AB g : BC
(g f)(x) : AC, xg(f(x))
g f g f
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f : I R :L > 0 :x, yI, |f(x) f(y)| L |x y|
L
f L
f : I R : L 0 : x, y I, |f(x)f(y)| L |xy| > 0 fC0,(I)
limxx0
f(x) = def
= >0,I(x0)
> 0 x, |x x0|< |f(x) |<
limxx0
f(x) =def=M >0,IM(x0)
M >0 x, |x x0|< M |f(x)> M
limx
f(x) =def
=
>0,
I()
N > 0 x, |x|> N |f(x) |<
limx f(x) =
def=M >0,
IM() NM>0 x, |x|> NM |f(x)|> M
(x0, x0+ )(x0 , x0)
limxx0
f(x) = limxx+0
f(x) = limxx0
f(x)
000 + 00
01
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limx0
sin x
x = 1
limx+
(1 +1
x)x =e
limxP(n)Q(n) =
+ p > q ab >0 p > q ab
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limx0
log(1 + x)
x = 1
limx0
loga(1 + x)
x =
1
log a
limx0
log(1 + x)
x =
limx0ex
1
x = 1
limx0
ax 1x
= log a
limx0
(1 + x) 1x
=, R
limx+
x
x = + > 1 , R
limx0+
x log x= 0 >0 R
limx+
(1 +n
x)x =en nR
limx+
log x
x = 0 >0 R
limx+ x
x
x = + R
limx+
n!
np = +, p R
limx+
n!
an = +, aR+0
limx+
n!
nn = 0
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limx+
n!
nn en 2 n = 1
limx+
(2n)!!
(2n 1)!! 2 n = 1
+
+
= +
= =
R
+ = + = = = 0 = 0 = 0 = 0
+ = + > 0+ = 0 1= + 0< 1
f :A = (a, b) R R, x0 A, = limxx0f(x); =limxx0f(x); 1= limxx0g(x); 2= limxx0h(x) ,
, 1, 2 R
= !
A= (a, b), x0
[a, b]
=
0 I(x0) f(x) x0
( > 0 :x (x0, x0+ )\ {x0}, f(x) = g(x)) (1 = 1)
x(a, b) \ {x0}, f(x)g(x)h(x) = 21= = 2
x(a, b) \ {x0}, f(x)g(x) = +1= 1= +=
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gf limy g(y) =1
>0 :x(x0 , x0+ ) \ {x0}, f(x)=
g() =1 g limxx0g f(x) = 1
f, g: (a, b)R, x0[a, b]
f, g (a, b)
x(a, b) \ {x0}, g(x)= 0
f(x0) =g(x0) = 0
limxx0 f(x)
g(x) =R
limxx0
f(x)g(x)
=
f, g : (a, b)R, a , bR
f, g (a, b)
x(a, b), g(x)= 0
limxa+f(x) =; limxa+g(x) =
limxa+ f(x)
g(x) =R
limxa+
f(x)
g(x) =
xb
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= lim xx0f(x); 1 = lim xx0g(x); 2 = lim xx0h(x) , 1, 2 R ,,R aR+0\ {1} bR+0; nN x0= R x0=
limxx0 [f(x) + g(x)] = + 1
limxx0 [f(x) + g(x)] = + 1
limxx0 [f(x)]n =n >0
limxx01
f(x) = 1 = 0
limxx01
f(x) = 0limxx0f(x) =
limxx0f(x)g(x)
= 1 1= 0
limxx0 |f(x)|=||
limxx0loga f(x) = loga
limxx0bf(x) = b
limxx0 [f(x)]g(x) =1 >0
f : (a, b) R, a , b R,a < b, s0(a, b)
limx
x0
f(x) = {f(y) : y < x0}= {f(y) : y < x0}= f((a, x0))= f((a, x0))
limxx+0
f(x) = {f(y) : y > x0}= {f(y) : y > x0}= f((x0, b))= f((x0, b))
x0(a, b)
x0 R f, g x0 x0 limxx0f(x) = limxx0g(x) = 0
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f g xx0 f
g
xx0
f=o(g, x0)
f=o(g)
limxx0
f(x)
g(x) = 0
0 xx0 o(1, x0)
limxx0f(x) = 0 >0 : limxx0
f(x)(xx0) = R \ {0} f
xx0 (x x0)
f1 = o(g, x0), f2 = o(g, x0)
f1+ f2= o(g, x)
k R, kf =o(kg,x0) =o(g, x0)
f1= o(g1, x0), f2= o(g2, x0)
f1+g1
f2+g2= g1
g2 x0
x,l >0, xk =o(xl, x0) =o(xl+k, xo)
f1 f2= o(g1 g2, x0)
f=o(g, x0), g= 0o(h, x0)
f=o(h, x0)
x0 R, f x0 x0 f(x) = f(x0) +a(xx0) + o(xxo, x0), a= 0 f1(y) =x0+
1a (y y0) + o(y y0, y0) y0= f(x0)
f, g x0 R : f, g= 0 x0 f g xx0 limxx0 f(x)g(x) =R \ {0} f g
fgf
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fg ghfh
f g( limxx0f(x) limxx0g(x))
y= f(x) x0 lim xx0f(x) =f(x0) >0 >0 x, |x x0|< |f(x) f(x0)|<
f, g x0 R f+g, f g, f
x0 g f x0 f1 x0
f :I R f(x0)> 0, >0 :x(x0 , x0+ )
[a, b] a, b R
f([a, b]) = [inf
x[a, b]f(x),sup
x[a, b]f(x)] c1, c2[a, b] :
f(c1) =inf
x[a, b]f(x), f(c2) =sup
x[a, b]f(x)
I
m
M x, y : f(x) < f(y), R : f(x) < < f (y) , zI :f(z) =
x 1 x2 ]x1, x2[ f(x) = 0 f : [a, b) R [a, b] a, b R, a < b :f(a) f(b)< 0 c(a, b) : f(c) = 0
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I x0 I x0
Rx0f(x) = f(x) f(x0)
x x0 I\ {0} f x0 lim xx0Rx0f(x)R
f(x) = D[f(x)] = dfdx
= f(x) = limh0 f(x + h) f(x)h = limxx0 Rx0f(x).
f : I R x0 f x0
f(x)f(x0)xx0 0f
)x0) + o(1, x0)f(x) f(x0) = f(x0) (x x0) + (x x0) o(1, x0)
f(x) =f(x0) + f(x0) (x x0) + o(x x0, x0) fI R, x0 I f x0 xI , L > 0 :f(x) = f(x=) + L (x x0) + o(x x0, x0)
k f(x0) = d2
dx2 = f(x) = ddxf
(x)f(k)(x0) =
dk
dxk = d
dxf(k1)
Ck f Ck, k N, k1 f k I I f(k) C0(I) =C(I) I C+(I) I k, dk
dxkfC(I)
C+(R) dkdxk
P =0 k > Pex C+(R) f :I R x0 I x0 L= f
(x)
f, g : I R, x0 I fg x0 f x0 g f
(x0) = g (x0)
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f :I R x0 x0 f(x0)0 0
f :I R x0I x0 f xI (x0 , x0+), >0 :f(x)f(x0) f(x0)
f : I R x0 I x0
f
x
I
(x0
, x0+ ),
>0 :f(x)< f(x0)
>f(x0)
x0 f xI , f(x) f(x0)
x0 f(xo) x0
f
f : I R x0 x0 I f(x0) = 0
f : IR
I I f
f : I R I I f
> 0 : f(x) 0() (x0
, x0) f(x)0(0) (x0, x0+) x0
f x0
f :I RC2, f(x0) = 0, f(x0)> 0(
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f f
f I x > x0, f(x)f(x0) + f+(x0) (x x0) x < x0, f(x)f(x0) + f(x0) (x x0)
f : (a, b)R x: 0 I :f (a, x0) (x0, b) x0 f x0 f x0
f : (a, +) R + limx+=
limx+
f(x)x
=a R \ {0}
limx+
f(x) a x= bR
y: a x + b
f, g If+ g I
f : [a, b] R [a, b] (a, b) f(a) =f(b) x0(a, b) f(x0) = 0
f : [a, b]R [a, b] (a, b) x0]a, b[ f(b)f(a)ba =f(x0)
f(x) g(x) [a, b]
]a, b[ g(x)= 0x[a, b] x0]a, b[ f(b)f(a)g(b)g(a) =f(x0)
g
(x0)
f : I R x0, x1 I : x0 < x1 x[x0, x1], f(x)f(x0) + f(x1)f(x0)x1x0 (x x0)
def= rx0x1(x);
x[x0, x1], f(x)rx0x1(x)
f : I R I r x0 < x1 0 I = 1a Ax
1
dx+ 1a Bx
2
dx = Aa log |x 1| +Ba log |x 2| + c
P2(x) = 0 I = 1a
Axdx +
1a
B(x)2dx =
1aA log |x |
A+Ba(x) + c
P2(x)< 0 I=
gx+hax2+bx+c
dx =
=gs d(ax2+bx+c)
x2+bx+c + ht
dx(kx+j)2+1 =
=gs log |ax2 + bx + c| + ht arctan(kx +j) + c
x= g(t) f(x) dx= f(g(t)) g(t) dt
f(x)g(x) dx= f(x)g(x) f(x)g(x) dx
n 0 e 0
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b
a
f(x) dx
b a
n [f(x0) + f(x1) +
+ f(xn
1)]
e (b a)2
2n M |f(x)| M
n 0 e 0 ba
f(x) dx b a2n
[f(x0) + f(xn) + 2 [f(x1) + + f(xn1)]]
e (b a)3
12n2 M |f(x)| M
ba
f(x) dx b a3n
[f(x0) + f(xn) + 4 [f(x1) + f(x3) + ] + 2 [f(x2) + f(x4) + ]]e (b a)
5
180n4 M |fiv(x)| M
=
x2
x1 1 + [f(x)]2 dx
x= x(t)y= y(t)
= t2t1
[x(t)]2 + [y(t)]2 dt
V= ba
[f(x)]2 dx
S S S
23
Slaterale = 2 ba
f(x)
1 + [f(x)]2 dx
n N, P = Q = n : f(x) = P(x) + o((x x0)n, x0) =Q(x)o((x x0)n, x0)P Q
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f : (a, b)R, x0(a, b)
f
n1
(a, b)
n
x0
f
f(x) =n
k=0
f(k)(x0)
k! (x x0)k + o((x x0)n, x0)
n x0
Pn,x0f=n
k=0
f(k)(x0)
k! (x x0)k
x0= 0
f : (a, b)R, x0(a, b), f n+1 x0
c= c(x)( {x0, x}, {x0, x}) :
f(x) = Pn,x0f(x) +f(n+1)(c(x))
(n + 1)! (x x0)n+1
f :I R, x0I , fCn(I)
f(x) Px0n f(x) = 1
n! xx0
f(n+1)(t)(xt)n dt
f(x) = Pn,0f+ o(xn, 0) =
nk=0
f(k)(x0)
k! (x x0)k + o(xn, 0)
ex = 1 + x +x2
2 + + x
n
n! + o(xn, 0)
log(1 + x) =x x2
2 +x3
3 + + (1)n1 xn
n + o(xn, 0)
sin x= x x3
3! +
x5
5! + + (1)n x
2n+1
(2n + 1)!+ o(x2n+2, 0)
cos x= 1 x2
2 +
x4
4! + + (1)n x
2n
(2n)!+ o(x2n+1, 0)
tan x= P6,0tan +o(x6, 0) = x +
x3
3 +
2 x515
+ o(x6, 0)
arcsin x= x +1
2 x
3
3 +
3
8 x
5
5 + +(2n 1)!!
(2n)!!
x2n+1
2n + 1+ o(x2n+2, 0)
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arccos x=
2x
1
2
x3
3
+
(2n 1)!!
(2n)!!
x2n+1
2n + 1
+ o(x2n+2, 0)
arctan x= x x3
3 +
x5
5 + + (1)n x
2n+1
2n + 1+ o(x2n+2, 0)
sinh x= x +x3
3! +
x5
5! + + x
2n+1
(2n + 1)!+ o(x2n+2, 0)
cosh x= 1 +x2
2 +
x4
4! + + x
2n
(2n)!+ o(x2n+1, 0)
tanh x= P6,0tanh +o(x6, 0) = x x
3
3 +
2 x515
+ o(x6, 0)
x= x 1
2x3
3 +
3
8x5
5 + + (1)n
(2n
1)!!
(2n)!!
x2n+1
2n + 1+ o(x
2n+2
, 0)
x= 1
2 log1 + x
1 x =x +x3
3 +
x5
5 + + x
2n+1
2n + 1+ o(x2n+2, 0)
(1 + x) = 1 + x + ( 1)2
x2 + + !( n)! n! x
n + o(xn, 0)
1
1 + x = 1 x + x2 + + (1)n xn + o(xn, 0)
limxx0f(x) =
limx
f(x) = den= num
limx f(x) = den= num 1 m= limx
f(x)x
n= limx[f(x) mx]A.Ob.: y = mx + n
+
0
+
0
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f(x) [a, b] f(a)f(b)< 0 f(x) x0]a, b[ f(a+b
2 )
]a, a+b2
[[ f(a) a+b2
< 0 ]a+b2
, b[[ a+b2 f(b) < 0
f(x) [a, b] f(a) f(b)< 0 f(x) x0]a, b[
(a, f(a)) (b, f(b)) x: y = 0 c
f(c)
]a, c[[f(a) f(c)< 0
]c, b[[f(c) f(b) < 0
f(x) [a, b] f(a) f(b)< 0 f(x) x0]a, b[
(a, f(a)) (b, f(b)) x : y = 0 c]a, b[ f(c) ]a, c[[f(a) f(c)< 0 ]c, b[[f(c) f(b)< 0
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n! = 1 2 (n 1) n=n
i=1
i
0 ! = 1n ! =n (n 1) ! n1
n
k
=
n !
(n k) ! k !
nk = n
n kn
k
=
n 1k 1
+
n 1
k
Pn,k = n!(nk)!k! =nk
Cn,k =
n+k1
k
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Pn = n!Dnk1,k2,,kn = n!k1!k2!kn!
Dn,k =Cn,k Pk = n!(nk)!
Dn,k
= nk
p= fn
p= limn
f
n
p() = 0
p() = 1
0fn0 fn 10p1
p(Ac) = p(A) = 1 p(A)
p(A \ B) = p(AB)p(B)
A B = p(A B) =p(A) +p(B)
A B= p(A B) = p(A) +p(B) p(A B)
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p(A) =p(A\B p(A B) = p(A) p(B)
p(A)=p(A \ B p(A B) = p(A) p(B\ A)
p(Hi\ E) = p(Hi) p(E\ Hi)n
1p(Hi) p(E\ Hi)
pn,k =
n
k
pk (1 p)nk
M(X) =n1
xipi
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N
N=anbn + an1bn1 + . . . + a1b + a0
0
ai < b i= 1, . . . , n
R
R= anbn + an1bn1 + . . . + a1b + a0+
i=1
aibi
0ai < b in
c= b x
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b|c
a= 0
b= aq+ r,0r 1
n > 1
n= pn11 pn22 pnmm
p1, p2, . . . pm n1, n2, . . . nm 1 n
a b c
c|a c|b
a b
D|a D|b x|a x|b x|D
a, b D= (a, b)
a b
h A + k B
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a b D = (a, b) m n
ma + nb= D
a b m
a|m b|m n a|n b|n=m|n
a b m m|(a b)
ab mod m
m
a aa mod m
ab mod m ba mod m
ab mod m bc mod m ac mod m
m m m m a
[a]m
[a]m+ [b]m = [a + b]m [a]m [b]m = [a b]m
a[0]m
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a m a0 modm
p
(p 1)! 1 modp
(m) m m r
1r < m (r, m) = 1
(b, m) = 1 b(m) 1 modm
ap1 1 modp
ap a mod p
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Pn n
P0
nN Pn n+1 Pn nN
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=b2 4ac
106
3, 141592653589793238462643383279... 1, 618...
1, 618... (n) (V)
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ex = (x)
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