ΕΛΛΗΝΙΚΟ ΑΝΟΙΚΤΟ...
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Transcript of ΕΛΛΗΝΙΚΟ ΑΝΟΙΚΤΟ...
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( 12)
17 2013 (: 3 30 .)
5
1
1) (10.) 3{( , , ) 0}RV x y z x y z= + = 3{( , , ) 2 2 0}RW x y z x y z= + + = . R3. i) V. ii) W. iii) W V. W+V = R3, WV = R3 ; 1) (10.) R4 {(1, 1,1, 1), (1,0,1,0)}U span= R4. i) U. ii) U. iii) w = 1 2 3 4( , , , )x x x x R
4 U.
1) i) 0x y z+ = x y z= + V ( , , ) ( , ,0) ( ,0, ) ( 1,1,0) (1,0,1)y z y z y y z z y z + = + = + . V 1={( 1, 1, 0), (1, 0, 1)} 1. 2. ii) W {( -2,1,0), (-2,0,1)} W 2.
iii) : W V= 3{( , , ) 0 2 2 0}Rx y z x y z x y z + = + + = .
: 1 1 1 1 1 1 1 0 4
1 2 2 0 1 3 0 1 3
{( 4, - 3, 1) } .
1. dim (W +V)=dim W +dimV- dim (W V)=2+2-1=3, W +V= R3 W V {0} . 1) i) T U , 1 2(1, 1,1, 1), (1,0,1,0)u u= = (1, 1,1, 1) (1,0,1,0) (0,0,0,0)x y + = ( , , , ) (0,0,0,0)x y x x y x+ + = 0x y= = .
U 1 2{ , }u u .
ii) 1 2{ , }u u Gram-Schmidt ()
( 2 1, 2u u = ) 1 1, 4u u = :
( )2 11 1 2 2 121
, 2 1(1, 1,1, 1), (1,0,1,0) (1, 1,1, 1) 1,1,1,1
4 2
u vv u v u v
v= = = = =
1 21 2
1 2
1 1(1, 1,1, 1), (1,1,1,1)
2 2
v vb b
v v= = = = , U : ={b1, b2}.
iii) u , 1 2 3 4( , , , )w x x x x= U, : 1 1 2 3 4 11
, ( )2
w b x x x x s< >= + ,
2 1 2 3 4 2
1, ( )
2w b x x x x s< >= + + + , 1 21 1 2 2 (1, 1,1, 1) (1,1,1,1)2 2
s su s b s b= + = +
1 2 1 2 1 2 1 2, , ,2 2 2 2
s s s s s s s s+ + + + =
( )1 3 2 4 1 3 2 41
, , ,2
x x x x x x x x= + + + + .
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2 2) (10.) ( , , ) ( , 2 , 2 )f x y z x y z x y z x y= + + + + R3. i) f , R3. ii) Kerf Imf. iii) . iv) R3 = Kerf Imf ;
2) (10.) = 1/ 2 1
1/ 2 0
.
i) = 1. ii) n n. iii) n n ; 2)
1 1 1
2 1 1
1 2 0
A
=
. :
1 1 1 1 1 1 1 0 2 / 3
0 3 1 0 1 1/ 3 0 1 1/ 3
0 3 1 0 0 0 0 0 0
A
: A =0 = ( -2 z/3 z /3 z)T = z ( -2 1 3)T/3, Kerf { (-2 , 1, 3)} dim Kerf =1, A 1 2 , Imf : { (1 , 2, 1), (-1 , 1, 2) } dim Imf =2. , . ( ) dimR3 = dim Kerf + dim Imf. R3 = Kerf Imf , (-2 , 1, 3), Imf.
1 1 2 1 1 2 1 1 2
2 1 1 0 3 5 0 3 5
1 2 3 0 3 5 0 0 0
. R3 f. 2)
i) 1/ 2 1
1/ 2 0A
=
:
21/ 2 1det (1/ 2 )( ) 1/ 2 / 2 1/ 2 ( 1)( 1/ 2)1/ 2
= = = +
, ( ): 1, -1/2 .
= 1: A = 1/ 2 1 1 1/ 2 1 1 2 1 2
1/ 2 1 1/ 2 1 1 2 0 0
, 2
1
.
= -1/2: A+/2= 1 1 1 1
1/ 2 1/ 2 0 0
=
( ),
1
1
.
n 1A P P=
12 1 1 1 1 01, ,1 1 1 2 0 1/ 23
P P
= = = ,
( )( )( )
11 0 2 1/ 22 1 1 1 1 11 1
1 1 1 2 1 23 30 1/ 2 1 1/ 2
nnn n
n nA P P
= = = =
( ) ( )( ) ( )
2 1/ 2 2 2 1/ 21
3 1 1/ 2 1 2 1/ 2
n n
n n
+ = +
, , n ,
2 21
1 13
.
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3 3) (8.) ( LHpital
):
(i) 3
1
n
n
na
n
+ = + , (ii)
2n nn
b = , (iii) 21
20nn n
cn
+=
+ , (iv) 2
ln
1nn n
dn
+=
+ .
3) (12.) ( x,
):
(i) 1 5
n
nn
nx+
= , (ii) 1
1
1( 1)n
n n
+
=
, (iii) 21 1n
n
n n n
= + + , (iv)
0
1 1
2nn n
+
=
+
.
3)
(i) 3
2
33 11311 11 1
nn
n
n
n enn en e
n n
++ + = = = + + +
,
(ii) . 1
1
11 12 1
2 22
nn
nn
n
nb n
nb n
++
++
= = < , nb .
(iii) 22
2 2
1 1(1 ) (1 )
1 10
20 2020 1 1n
n nn n n n n nc
n nnn n
+ ++
= = = + + +
,
(iv) 2ln
1nn n
dn
+=
+,
( )
( )2
1 11 1ln 1
02 21
x
x x x xx xx
+
+ ++= =
+ , L Hpital :
2
lnlim 0
1xx x
x++
=+
, lim 0nd = .
3) (i) 1 5
n
nn
nx+
= ,
( )( )
1
1
11 115
5 5 55
n
n
n
n
n x xn x xn
nx n
+
+
+ + + = = ,
15
x< 5x < x >5. x
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(-5,5) x (- , 5) (5, + ).
[-5 5,]: x=5 1n
n+
= ( ) x =5
1
( 1)nn
n+
=
( ). : x (-5,5)
.
(ii) 1
1
1( 1)n
n n
+
=
, .
1
n
1 10 1 0 1
1n n n n
n n< < + < < + 1, .
(iv)0
1 1
2nn n
+
=
+
: 0
1
2nn
+
= . .
0
1
n n
+
= ( ,
p- p=1). .
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4 4) (12.) 65 56)( xxxf = . i) 1, 2 f . ii) f : , , , . iii) : , (, ), f . iv) lim ( ), lim ( )x xf x f x + f.
v) 1
( )3
f x = () [0, 1].
(: ).
4) (8.)
i) 0
sin( )x n x dx
, n1.
ii) 2
0
xx e dx+ .
4)
i) 5 6( ) 6 5f x x x= : 4 5 3 4( ) 30 30 , ( ) 120 150f x x x f x x x = = :
5 4 3( ) (6 5 ), ( ) 30 (1 ), ( ) 30 (4 5 )f x x x f x x x f x x x = = = . :
x 0 4/5 1 6/5 + 1 x + + + 0 x5 0 + + + + + x4 + 0 + + + + + x3 0 + + + + +
65 x + + + + 0 45 x + + 0
f + 0 + + 0 f 0 + 0 f 0
.. + .. + f(1)=1
.. + 0
: ii) f (- , 1], [1, +), [0, 4/5] (- , 0] [4/5, +), iii) f 0 6/5, ( ). 1 1 x = 0 x = 4/5,
iv) 66
lim ( ) lim ( 5) ( )( 5)x xf x x x = = + = . 6
6lim ( ) lim ( 5)x xf x x x+ +
= = .
( )( 5)= + = . f , f (- , 1].
v) [0,1] f , ,
1-1. ,
. f(0)=0 f(0)=1
0 < 1/3
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(0,1) f 1/3.
1
( )3
f x =
0 1.
-0.5 0.5 1.0 1.5
-1.0
-0.5
0.5
1.0
4) i)
cos( ) cos( ) cos( ) cos( ) 1sin( ) cos( )
nx x nx nx x nxx nx dx x dx x dx nx dx
n n n n n
= = = +
2
cos( ) sin( )x nx nx
n n= + .
2 200
cos( ) sin( ) cos( ) sin( ) ( 1)sin( ) (0) =
nx nx nx n nx nx dx
n n n n n
= + = +
.
ii) 2xx e dx [0, )x + , 2 , 2u x du xdx= = ,
: 2
2x u dux e dx e = =
21 1 1
2 2 2u u xe du e C e C = + = + .
2
0
xx e dx+ = ( ) ( )2 2 2 00 0
1 1lim lim lim
2 2
aa x x aa a ax e dx e e e
+ + +
= =
( ) ( )21 1 1lim 1 0 12 2 2
aa e
+
= = = .
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5 5) (6.) , . . , 2% 5%. , : i) , ii) , . 5) (14 ) 20 2 . i) 18 22 ; ii) 15 ; iii) q q 0.9 ; iv) 4 2 15 ; (: ). : (1) = 0.8413, (2.5) = 0.9938, (1.28) = 0.9. 5) : E , , A , , B , . ( ) 0.5P A = , ( ) 0.5P B = :
1) ( ) ( )E E A E B=
( ) ( ) ( )P E P E A P E B= + = ( ) ( ) ( ) ( )P E A P A P E B P B + =2 5 5 5 35
100 10 100 10 1000+ = .
2) ( )P A E Bayes : ( ) ( ) (0.02) (0.5) 10
( )( ) 0.035 35
P E A P AP A E
P E
= = = .
5) , X 20= 2.= ,
20
2
XZ
= (0,1)N :
i) 18 22
18 20 20 22 20(18 22) ( 1 1)
2 2 2
XP X P P Z
= = =
(1) ( 1) (1) [1 (1)] 2 (1) 1 2 0.8413 1 1.6826 1 0.6826.= = = = = =
ii) 15
20 15 20( 15) ( 2.5) 1 ( 2.5)
2 2
XP X P P Z P Z
= = = < =
1 [1 ( 2.5)] ( 2.5) (2.5) 0.9938.P Z P Z= < = < = =
iii) q
20 20 20 20 20( ) 1 1
2 2 2 2 2
X q q q qP X q P P Z P Z
> = > = > = =
.
( ) 0.9P X q> = , 20 20
1 0.9 0.12 2
q q = =
,
20 20 200.9 (1.28) 1.28 20 2.56 17.44
2 2 2
q q qq q
= = = + = =
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iv) Y ( 4) 15 . , Y 4n = (), ( 15 ) ( 15) 0.9938p P X= = ( ii) 1 .q p= ~ ( , ) (4,0.9938)Y B n p B= . , 0,1, 2,3,4k =
4 44 4( ) (1 )k k k kP Y k p q p pk k
= = =
. 2 4
0 4 0 1 4 1 2 4 24 4 4( 2) ( 0) ( 1) ( 2) (1 ) (1 ) (1 )0 1 2
P Y P Y P Y P Y p p p p p p
= = + = + = = + +
.
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m n
[ ]ijA a= [ ]T
jiA a= , (,
).
: ( )T TA A= ( )T T TA B A B+ = +
( )T TA A = , R ( )T T TAB B A=
m n [ ]ijA a=
ij jia a= . TA A=
n n
[ ]ijA a= ( ) 1A
1 1 nAA AA I = = .
: , nn
1 1( )A A = 1 1( ) ( )T TA A =
1 1 1( )AB B A = 1 1( ) ( )k kA A = k Z
Laplace [ ]ijA a= i j
: det( )A A= =
11 12 1
21 22 2
1 1
1 2
n
n nn
i k ik k j k jk k
n n nn
a a a
a a aa A a A
a a a= =
= = =
L
L
M M M
L
( 1)i jij ijA M+= ijM
ij-. n n A :
det( ) det( )TA A=
det( ) det( )nA A = , R
det( ) det( )det( )AB A B=
[ ]det( ) det( ) kkA A= , \{0}k Z A det( ) 0A
11
( )det( )
A adj AA
=
( )adj A
A .
* * * * * * * * * * * * U V ..
V ,k R 1 2, U u u
1 2k U+ u u .
1 2, , , kv v vK
1 1 2 2 1 2 0.0k k k + + + = = = = =L Lv v v
1 2{ , , , }kv v vK .. V
V
I. 1 2, , , kv v vK
I. .. V 1 2, , , kv v vK V dimV k= .
={ 1 2, , , kKu u u } ()
V x V , 1
k
i iix a u
== ,
ia R . [a1 a2 ak ]T
x B [ ]Bx .
V .. ,U W .V :
dim( ) dim dim dim( )U W U W U W+ = +
,U W V .. V V U W=
( V U W= + { }0U W = )
( V U W= + dim dim dimV U W= + ). * * * * * * * * * * * *
nR
( , ) n nx y R R ,
ox y :
. ( ) ( ) ( )k k + = +o o ox y z x z y z ,
, , n x y z R , ,k R
. =o ox y y x , , n x y R
. 0ox x 0= = 0ox x x To x
= ox x x .
[0, ] , \{ }nx y 0R
: cos=ox y
x y.
, nx y R (
) 0=ox y .
, nx y R :
. 2 2 2
0= + = +ox y x y x y
. + +x y x y
I. =x x , R
IV. x y x yo (Cauchy-Schwarz)
p x
y 2 .=x y
p yy
o
nE R
{ }: 0,nE E = = y x y xR . , nE E = R , ( )E E = .
1 2, , ,n
k Ku u u R
(.
0i j =ou u i j , 1i =u ) .
1 2, , , nK nR ,
1 1=
1 2 11 2 1
1 1 2 2 1 1
j j j jj j j
j j
= o o o
Lo o o
2,3, ,j n= K ,
11
1
=u
, 222
=u
, , nnn
=K u
nR . n n A
T TA A A A I= = , 1 TA A = , .
, : I. ( )
,nR
II. det 1,A =
III. ,A =x x
IV. A A =o ox y x y
V. .
* * * * * * * * * * * * () :f U V ( ,U V
) ( ) ( ) ( )f k k f f + = +x y x y , , U x y
,k R . ( U V= .
U ). ker { : ( ) }f U f U= = 0x x
f U .
Im { : ( ) , }f V f U V= = y x y x
f V .
:f U V -- (1-1)
, U x y
( ) ( )f f= =x y x y .
:f U V ( )f U V= .
:f U V :
. dim dim ker dim ImU f f= +
. f 1-1 ker { }f = 0 .
. 1={ 1 2, , , nKu u u }
U 2={ 1 2, , , mKv v v }
V ,
1 11 1 21 2 1
2 12 1 22 2 2
1 1 2 2
( )
( )
( )
m m
m m
n n n mn m
f a a a
f a a a
f a a a
= + + +
= + + +
= + + +
L
L
M
L
u v v v
u v v v
u v v v
m n f
11 12 1
21 22 2
1 2
n
n
m m mn
a a a
a a aA
a a a
=
L
L
M M M
L
,
[ ] [ ]1 2
( )B B
A f=x x , Ux .
dim dimU V n= = , :f U V .
. f ( 1f )
II. f 1-1
III. ker { }f = 0
IV. f * * * * * * * * * * * *
n n A i
n
11 12 1
21 22 2
1 2
1
1 1 0
( ) det
n
n
n n nn
n n
n
a a a
a a ap
a a a
a a a
=
= + + + +
L
L
M M M
L
L
A , . i , 1, 2, ,i n= K , -
-
[ ]1 2T
nx x x= Kx
11 1 12 2 1
21 1 22 2 2
1 1 2 2
( ) 0
( ) 0
( ) 0
i n n
i n n
n n nn i n
a x a x a x
a x a x a x
a x a x a x
+ + + =
+ + + =
+ + + =
L
L
M
L
A :
1 2 0det ( 1)nnA a = = L
1 2 1n ntrA a = + + + = L ,
0 1, na a
( )p .
i ix
A , ,ki i x kA .
, . n n A , D , . P ,
1A PDP= . D A P -
nR . : k k , , , ( ) n . . T Q ,
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1 2( , , , )T
nA Q diag Q= K .
( )f ,
( )1 11 2( ) ( ) ( ), ( ), , ( )nf A P f D P Pdiag f f f P = = K
A 1
1 1 0( )n n
np A A a A a A a I
= + + + + =L O .
( ) ( )f ( )p , ( ) ( )f A A= .
* * * * * * * * * * * *
1 2, , , nx x xK ( )TF A=x x x ,
[ ]1 2T
nx x x= Kx A n n
, .
1 2( , , , )T
nA Qdiag Q = K , ( )F x
2 2 2
1 1 2 2( ) n nF y y y = + + +Ly ,
[ ]1 2T T
ny y y Q= =Ky x .
( )1 2, , , 0 0n > , 1 2,x x A 1 2x x< .
:f A R
s ( f )
: ( )f x s , x A . ( ). . 1-1 :f A R
1 2,x x A 1 2x x ,
1 2( ) ( )f x f x ,
: 1 2( ) ( )f x f x= , 1 2x x= .
:f A R :g B R ,
( )( ) ( ( ))g f x g f x=o , x A
( )f x B .
1-1 f
1 : ( )f f A A ,
( )y f A x ,
( )y f x= , . 1( ) ( )f y x f x y = = .
0x -
0
lim ( )x x
f x
= l 0 0
lim ( ) lim ( )x x x x
f x f x+
= = l
: ( ) ( ) ( )g x f x h x 0x
0 0
lim ( ) lim ( )x x x x
h x g x
= = l , 0
lim ( )x x
f x
= l .
x + , x .
0
sinlim 1x
x
x= ,
0
cos 1lim 0x
x
x
=
:f A R
0x A 0
0lim ( ) ( )x x
f x f x
= .
* * * * * * * * * * * *
( ( , )A a b= R)
:f A R
0x A
0
00
0
( ) ( )lim ( )x x
f x f xf x
x x
=
R
fC
0 0( , ( ))x f x 0 0 0( ) ( )( )y f x f x x x =
f f
f f
. : f, g
( ) ( )( ) ( ) ,cf x c f x c = R
( ) ( ) ( )( ) ( ) ( ) ( )f x g x f x g x =
( )( ) ( ) ( ) ( ) ( ) ( )f x g x f x g x f x g x = +
2
( ) ( ) ( ) ( ) ( ), ( ) 0
( ) ( )
f x f x g x f x g xg x
g x g x
=
A 0f f
1f
( )1 1 ,ff
=
( ( ))f g x
( ) ( ( )) ( ) ( )( ( )) df g x df g dg xf g xdx dg dx
= =
( ) ' 0c = , cR ( )1'k kx k x = ,
k R
( )sin ' cos( )x x= ( )cos ' sin( )x x=
( ) 21
tan 'cos
xx
= ( ) 'x xe e=
( ) 1ln 'xx
= ( ) ' ln( )x xa a a= ,
1 0a >
( )2
1arcsin '
1x
x=
( ) 21
arctan '1
xx
=+
l Hospital : ( ) ( ) 0f a g a= =
( ), ( )f a g a ( ) 0g a ,
( ) ( ) ( )lim lim
( ) ( ) ( )x a x a
f x f x f a
g x g x g a
= =
: 0 0( ) ( ) 0f x g x= = ,
( ), ( )f x g x ( , )a b ,
( ) 0g x , 0 ( , )x a b ,
0 0
( ) ( )lim lim
( ) ( )x x x xf x f x
g x g x
=
( ), ( )f x g x .
, , 0
m
1 /
/ :1 /
f g
g f =
( )0 :1/
ffg
g =
1 / 1 /
:1 /
g ff g
fg
=
0 00 , , 1+
lim ( ( ) ln ( ))( )lim ( ) x a
g x f xg x
x af x e >
= ,
( ) ( ) ln ( )( )g x g x f xf x e=
fC :f A R .
( ) 0,f x > x I A , f . ( ) 0,f x < x I A , f . 0( ) 0f x = , 0x A
>0 : ( ) 0f x > , 0 0x x x < x I A , f . ( ) 0,f x < x I A , f . >0 ( ) 0f x > 0 0x x x < ,
0x .
) 0( ) 0f x =
0( ) 0f x < ,
0x .
x a= R ,
lim ( )x a
f x
= lim ( )x a
f x +
=
y b= , bR ,
lim ( )x
f x b
= lim ( )x
f x b
=
fC
y ax b= + , ( )lim ( ) 0x
f x ax b
=
( )lim lim( ( ) )x x
f xa f x ax b
x = = R R
* * * * * * * * * * * * : [ , ]f a b R .
Bolzano: f [ , ]a b
( ) ( ) 0f a f b < ,
0 ( , )x a b 0( ) 0f x = .
: f
[ , ]a b ( ) ( )f a f b , ,
( )f a ( )f b
0 ( , )x a b 0( )f x = .
- : f [ , ]a b , f
[ , ]a b . 1 2, [ , ]x x a b
1 2( ) ( ) ( )f x f x f x , [ , ]x a b .
(): f [ , ]a b
( , )a b , ( , )a b
: ( ) ( )
( ).f b f a
fb a
=
Rolle: f [ , ]a b ,
( , )a b , ( ) ( )f a f b= ,
( , )a b :
( ) 0f = .
f ( , )a b ,
( ) 0,f x = ( , )x a b , ( ) .f x c=
Cauchy: ( ), ( )f x g x
[ , ]a b , ( , )a b
( ) 0g x , ( , )x a b ,
( , )c a b :( ) ( ) ( )
( ) ( ) ( )
f b f a f c
g b g a g c
=
Darboux: f [ , ]a b
( ) ( )f a f b > cR ( ) ( )f b c f a < < ,
( , )a b ( ) .f c =
(, ( ) ( )f a f b < ).
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( )x f x=
a , f
[ ], ,a h a h +
'( ) 1f x m< < , [ ],x a h a h + , [ ]0 ,x a h a h +
1( ),n nx f x = 1, 2,...,n = a .
* * * * * * * * * * * *
f
( ) ( )b a
a b
f x dx f x dx= ( ) ( ) ( )
b c b
a a c
f x dx f x dx f x dx= + ( ) ( )
b b
a acf x dx c f x dx=
( )( ) ( ) ( ) ( )b b b
a a a
f x g x dx f x dx g x dx+ = + ( ) ( ) ( ) ( )
b b
a a
f x g x f x dx g x dx : f , [ , ]a b
( ) ( )( )b
a
f x dx f b a= ()
( )( ) ( ) ( ) ( )F x c f x dx F x c f x+ = + =
( ) ( )df x f x c= +
( )1 2 1 2( ) ( ) ( ) ( )c f x c h x dx c f x dx c h x dx+ = +
( )x g t= , ( ( )) '( ) ( )f g t g t dt f x dx= o
( ) ( ) ( ) ( ) ( ) ( )f x g x dx f x g x f x g x dx =
kdx kx c= +
1
, { 1}1
aa xx dx c a
a
+
= + + R
1
lndx x cx
= +
cos sinx dx x c= + sin cosxdx x c= +
1
2 2tan ( ) tan( )
adx x xc arc c
x a a a= + = +
+
1
2 2sin ( ) sin( )
dx x xc arc c
a aa x
= + = +
x xe dx e c= +
( )
ln | ( ) |( )
f x dxf x c
f x
= +
. f
[ , ]a b F
f , ( ) ( ) ( )b
a
f x dx F b F a= . f [ , ]a b ,
( ) ( )x
a
dF df t dt f x
dx dx= =
* * * * * * * * * * * *
( ) ( ) ( )limb
ba a
f x dx f x dx+
+=
( ) ( )limb b
aa
f x dx f x dx
=
( ) ( ) ( )0
limb b
a af x dx f x dx
+
=
( b )
( ) ( )0
limb b
ea ea
f x dx f x dx+
+
+
=
( a ) ( ) = ,
( ) ( ) ( )0 0
lim limb c b e
e ea e ca
f x dx f x dx f x dx
+ ++
+
= +
a c b< < ( ,a b )
( ) ( ) ( )lim limc b
a ba c
f x dx f x dx f x dx+
+
= +
( ) ( ) ( )0
lim lima c a e
b eb c
f x dx f x dx f x dx
+
= +
( a )
( ) ( ) ( )0
lim limc b
bea e ca
f x dx f x dx f x dx+
+
+
++
= +
( a )
( ),c a b
( ) ( ) ( )0 0
lim limb c e b
e ea a c e
f x dx f x dx f x dx+ +
+
= +
Cauchy
( ) ( ) ( )0
limb c e b
ea a c e
f x dx f x dx f x dx+
+
= +
(c ) Laplace : [0, )f + R
0
{ ( )}( ) ( )xtL f t x e f t dt+
= ,
x .
* * * * * * * * * * * *
( ) , ( ) 0b
a
E f x dx f x=
( )2
1b
a
S f x dx= +
( ) ( )2
2 1b
ox
a
E f x f x dx = +
( )2
b
ox
a
V f x dx=
( ) ( )2 1b
a
E f x f x dx=
( ) ( )2 22 1b
ox
a
V f x f x dx = * * * * * * * * * * * *
- . : ( )na a n= .
: 1n na a+ = + , 1 ( 1) n a n = +
n ..: 1[2 ( 1) ]
2nn a n
S +
=
: 1n na a+ = 1 1nna a =
n ..: 11
1
n
nS a
=
,
1 . : , ,a b c 3
.. 2b a c= . xR n ( x )
1lim 0n n
= lim 1n
nn
= lim 0, 1n
nx x
= <
lnlim 0n
n
n = lim !
n
nn
= lim 1, 0n
nx x
= >
lim 1n
x
n
xe
n + =
, lim 0!
n
n
x
n= .
: MR: ,na M n N.
: mR: ,nm a n N . : , . ,m M R : nm a M , n N .
. . na , nN
, 1n na a + , n N .
, 1n na a + , n N .
, . - . R . . lim 0nn = ,n na n N
lim 0nna
= .
1lim /n nn a a+ = : 1p :
) :1
nn
a
= , 1n n na b b +=
lim nnb
.
: 1 lim nnb b
.
) : ( )0
1n
nn
a
=
, 0na >
0na < 0 1 2n , , ,...= ) Taylor: f
(1) (2) ( ), ,..., nf f f
[ , ]a b ( )nf
( , )a b , ( ),a x
( ) ( )
( )
(1) (2)2
( )
( ) ( )( ) ( )
1! 2!( )
( )!
nn
n
f a f af x f a x a x a
f ax a R x
n
= + + +
+ +
L
L
( )( 1)
1( )( )
( 1)!
nn
n
fR x x a
n
+ +=
+
n-. 0a = , Maclaurin. Taylor ( 0a = ) xR
2
12! !
nx x xe x
n= + + + + +L L
3 5 2 1
sin ( 1)3! 5! (2 1)!
nnx x xx x
n
+
= + + ++
L L 2 4 2
cos 1 ( 1)2! 4! (2 )!
nnx x xx
n= + + +L L
-
12 2012-2013
-1< x < 1 2 3 1
ln(1 ) ( 1)2 3 1
nnx x xx x
n
+
+ = + + ++
L L
3 5 2 1
arctan 1 ( 1)3 5 2 1
nnx x xx
n
+
= + + ++
L L ) Fourier: :[ , ]f L L R 2L . Fourier f
0
( ) ~ ( cos sin )n nn
n x n xf x a b
L L
=
+
01
( )2
L
L
a f x dxL
=
1( )cos
L
n
L
n xa f x dx
L L
= , 1, 2,n = K
1( )sin
L
n
L
n xb f x dx
L L
= , 1, 2,n = K
. lim 0nna
,
0n
n
a
= .
. ) 0
nn
a
= ,
0n
n
b
= ,
,k R
0 0 0
( )n n n nn n n
ka b k a b
= = =
+ = + .
) 0
nn
a
=
0n
n
b
= ,
0
( )n nn
a b
=
+ .
I. 0
nn
| a |
= ,
0n
n
a
=
. . V. ( ) 0 n na b .
0
nn
b
= ,
0n
n
a
=
0
nn
a
= ,
0n
n
b
= .
V. ( )
0 na , 0 nb< , lim 0n
nn
ac
b= > .
0n
n
a
=
0n
n
b
=
. VI. ( - d Alembert) 0na
0n n 1lim n
nn
a
a+
= . :
1 < , 0
nn
a
=
1 > , 0
nn
a
=
1 = , .
VI. ( - Cauchy). 0na >
lim n nna
=
1 < , 0
nn
a
=
1 > , 0
nn
a
=
1 = , .
VII. ( Leibnitz) ( )0
1n
nn
a
=
.
( )na , lim 0,nn a = .
IX. ( ) - :[1, )f + R
( )1
I f x dx+
= ( )1n
S f n
=
=
: (1)I S I f< < + .
:( )
!! !
nr
n nC
r r n r
= =
( ) ( )( )
PP /
P
A BA B
B
=
: ( ) ( ) ( )P P P .A B A B = i jA A = , i j 1 2 ... ,nA A A =
:
( )P B = 1 1P( )P( / ) P( )P( / )n nA B A A B A+ +L
Bayes: P( )P( / )
P( / )P( )
k kk
A B AA B
B=
(..)
( )X . .. ( )E X
X : ( ) = ( )x
E X xf x
.., : ( ) = ( ) E X xf x dx
. .., ( )f x (..) ( ..) (...) ( ..). H .. :
( ) ( )2 2var( ) = = ( )x xx
X E X x f x
.. :
( ) ( )2 2var( ) = = ( ) x xX E X x f x dx
: [ ]( )22var( ) = X E X E X . ..
X ()
, : = var( )X X .
X .. ( ). Y aX b= + :
( ) ( ) ( )E Y E aX b aE X b= + = + 2( ) ( ) ( )Var Y Var aX b a Var X= + =
: ( , ) : ( ) (1 )k n kn
B n p f k p pk
=
0,1,...,k n=
( ) ,E X np= ( ) (1 ).Var X np p=
Poisson ( )P : ( ) ,!
k
f k ek
= 0,1,...k =
( )E X = , ( )Var X = :
1(1 ) 1,2,...( ) : ( )
0
kp p kG p f k
==
( ) 1 / ,E X p=
2( ) (1 ) /Var X p p= :
1( ) (1 ) ,
1kkf k p p
=
, 1,...k = +
( ) /E X p= , 2( ) (1 ) /Var X p p=
:
1 2
( )
N N
k n kf k
N
n
=
10,..., min( , )k n N= , 1 2N N N+ =
1( )N
E X nN
= , 1 2( )1
N N N nVar X n
N N N
=
: ( , )U a b
1( )
0
a x bf x b a
=
( ) ( ) / 2E X a b= + , 2( ) ( ) /12Var X b a=
( )2,N : 2
1
21( )2
x
f x e
=
x < <
( )E X = , 2( )Var X =
0
( ) : ( )0
a xa e xE a f x
=
( ) 1/E X a= , 2( ) 1/Var X a= 1 2, ,..., nX X X ( )iE X = .
2( )iVar X = , ( )
~ (0,1)n X
2
1
~ ( , )n
ii
X N n n = 30n .
* * * * * * * * * * * * :
( ) 1 ... ...1
n n n n r r nn na b a a b a b br
+ = + + + + +
2 2 2( ) 2a b a ab b = + 3 3 2 2 3( ) 3 3a b a a b ab b = +
2 2 ( )( ) a b a b a b = + 3 3 2 2( )( ) a b a b a ab b = +m
1 2 3 2
2 3 2 1( )( ...
), 1, 2, 3, ...
n n n n n
n n na b a b a a b a b
a b ab b n
= + + +
+ + + =
(1 ) 1 , 0, 1, 2,3,...na na a n+ + > = * * * * * * * * * * * *
( xR ) sin( ) sin( ), cos( ) cos( )x x x x= =
2 2sin cos 1x x+ = , sin
tancos
xx
x=
sin( ) sin cos sin cosx y x y y x =
cos( ) cos cos sin sinx y x y x y = m
( ) tan tantan1 tan tan
x yx y
x y
=
m
2
2 tansin 2 2sin cos
1 tan
xx x x
x= =
+
22 2 2
2
1 tancos 2 cos sin 2cos 1
1 tanx
x x x xx
= = =
+
2
2 tantan 2
1 tan
xx
x=
, 2
2
11 tan
cos
+ =
sin sin 2sin cos2 2
x y x yx y
=
m
cos cos 2cos cos2 2
x y x yx y
+ + =
cos cos 2sin sin2 2
x y x yx y
+ =
sin(0) cos( / 2) 0, cos(0) sin( / 2) 1 = = = = sin( / 6) cos( / 3) 1/ 2 = =
2 3sin( ) cos( ) , sin( ) cos( )
4 4 2 3 6 2= = = =
* * * * * * * * * * * * { }| ,z x i y x y= = + C R : z x iy=
: 1 21 z
zz z
= =
: 2 2r z x y= = + 2 2| |r z z z= =
(cos sin )z r i = + , .
De Moivre
( ) ( )( )cos sinn n in nz r e r n i n = = + , n n
nx z= , nN , ( n - z ),
2 2cos sin , 0,1, 1nkk k
z r i k nn n
+ + = + =
K .
