Match II (armand)

1 ‫آ‬ :‫اﺳﺘﺎد‬‫ﻗﺎي‬‫آرﻣﻨﺪ‬: ‫ﺻﻔﺤﻪ‬ ‫ﺷﻤﺎره‬
‫دو‬ ‫ﻋﻤﻮﻣﯽ‬ ‫رﯾﺎﺿﯽ‬‫دو‬ ‫ﻋﻤﻮﻣﯽ‬ ‫رﯾﺎﺿﯽ‬
))‫ﭘﺎﯾﻪ‬ ‫ﻋﻠﻮم‬ ‫داﻧﺸﺠﻮﯾﺎن‬ ‫ﺑﺮاي‬‫ﭘﺎﯾﻪ‬ ‫ﻋﻠﻮم‬ ‫داﻧﺸﺠﻮﯾﺎن‬ ‫ﺑﺮاي‬((
:‫ﻣﻮﻟﻒ‬:‫ﻣﻮﻟﻒ‬
‫آرﻣﻨﺪ‬ ‫آﻻش‬‫آرﻣﻨﺪ‬ ‫آﻻش‬
‫ﮔﺮﮔﺎن‬ ‫واﺣﺪ‬ ‫اﺳﻼﻣﯽ‬ ‫آزاد‬ ‫داﻧﺸﮕﺎه‬‫ﮔﺮﮔﺎن‬ ‫واﺣﺪ‬ ‫اﺳﻼﻣﯽ‬ ‫آزاد‬ ‫داﻧﺸﮕﺎه‬
‫ﺗﺎﺑﺴﺘﺎن‬‫ﺗﺎﺑﺴﺘﺎن‬13871387

2 ‫آ‬ :‫اﺳﺘﺎد‬‫ﻗﺎي‬‫آرﻣﻨﺪ‬: ‫ﺻﻔﺤﻪ‬ ‫ﺷﻤﺎره‬
‫ﻓﻬﺮﺳﺖ‬‫ﻓﻬﺮﺳﺖ‬
‫ﻓﺼﻞ‬‫اول‬
‫دﻧﺒﺎﻟﻪ‬‫ﻧﺎﻣﺘﻨﺎﻫﯽ‬ ‫ﻫﺎي‬ ‫ﺳﺮي‬ ‫و‬..........................................................................1
‫ﻓﺼﻞ‬‫دو‬‫م‬
‫ﻗﻀﺎﯾﺎي‬‫ﺳﺮي‬ ‫ﻫﻤﮕﺮاﯾﯽ‬‫ﺗﻮاﻧﯽ‬ ‫ﻫﺎي‬ ‫ﺳﺮي‬ ‫و‬ ‫ﻫﺎ‬..............................................7
‫ﻓﺼﻞ‬‫ﺳﻮم‬
‫ﺑﺴﻂ‬‫ﺗﻮاﺑﻊ‬ ‫ﺗﯿﻠﻮر‬ ‫و‬ ‫ﻣﮑﻠﻮرن‬.........................................................................33
‫ﻓﺼﻞ‬‫ﭼﻬﺎرم‬
‫ﻣﻘﺎﻃﻊ‬‫و‬ ‫ﻣﺨﺮوﻃﯽ‬‫ﻗﻄﺒﯽ‬ ‫و‬ ‫دﮐﺎرﺗﯽ‬ ‫ﻣﺨﺘﺼﺎت‬ ‫در‬ ‫آﻧﻬﺎ‬ ‫ﻣﻌﺎدﻻت‬......................44
‫ﻓﺼﻞ‬‫ﭘﻨﺠﻢ‬
‫ﺑﺮدارﻫﺎ‬،‫ﺗﺤﻠﯿﻠﯽ‬ ‫ﻫﻨﺪﺳﻪ‬‫ﺑﺮداري‬ ‫ﺗﻮاﺑﻊ‬ ،‫دﺳﺘﮕﺎه‬ ‫و‬TNB.............................55
‫ﻓ‬‫ﺼ‬‫ﻞ‬‫ﺷﺸﻢ‬
‫روﯾﻪ‬،‫دوم‬ ‫درﺟﻪ‬ ‫ﻫﺎي‬‫ﻣﺨﺘﺼﺎت‬‫اﺳﺘﻮاﻧﻪ‬‫و‬ ‫اي‬‫ﮐﺮوي‬‫ﻣﺘﻐﯿﺮه‬ ‫ﭼﻨﺪ‬ ‫دﺳﺘﮕﺎه‬ ،......65
»‫ﮐﺘﺎب‬ ‫ﺷﻨﺎﺳﻨﺎﻣﻪ‬«
‫ﮐﺘﺎب‬ ‫ﻧﺎم‬:‫دو‬ ‫ﻋﻤﻮﻣﯽ‬ ‫رﯾﺎﺿﯽ‬
‫ﻣﻮﻟﻒ‬:‫آرﻣﻨﺪ‬ ‫آﻻش‬
‫ﻧﺎﺷﺮ‬:‫ﮔﺮﮔﺎن‬ ‫واﺣﺪ‬ ‫اﺳﻼﻣﯽ‬ ‫آزاد‬ ‫داﻧﺸﮕﺎه‬
‫ﻃﺮاﺣ‬ ‫و‬ ‫وﯾﺮاﯾﺶ‬‫ﺟﻠﺪ‬ ‫ﯽ‬:‫ﻗﺎﺳﻤﯽ‬ ‫ﻣﺤﻤﺪ‬
‫ﻧﺸﺮ‬ ‫ﺳﺎل‬:‫ﺗﺎﺑﺴﺘﺎن‬1387
‫ﺗﯿﺘﺮاژ‬:1000‫ﻧﺴﺨﻪ‬
‫ﻗﯿﻤﺖ‬:
‫ﺷﺎﺑﮏ‬:

1 ::1
:
.!" #{0}$ " %
&'('.: {0}na
( )na f n=
( !" +0{ }n na =,( & -.
.:
1
1 1 1
{ } {1, , ,...}
2 3
n
n
= =
0{( 1) } {1, 1,1, 1,...}n
n = =
0
1 2 3
{ } {0, , , ,...}
1 2 3 4
n
n
n
= =
+
( !&/-0$ "/1" % &2 3 2 4 " & '/.
1 1a =%0 1a =%1 6n na a+ = +
{1, 6 1, 6 7 ,...}+ +
( !5 & + 6" 7 % $ " 8 0 9 # :4 & ( ; &< (=.
2 1 0 1, 1, 1n n na a a a a+ += + = =
{1,1,2,3,5,8,...}
.{ }na" +l0 ,( 0 >lim n
x
a l=
.:
1
1
{3 } 3n
n
=+1
1
{ } 0n
n
=
( 0, 0: )nN n N a l> > <
?(( " @1, 0,{ }np
c
c p
n
=>> :A ".

::2
! " ! # ! $ %"
)/1 % <(( 0, 0: p
p
c c
N n N n
n
> > < <lim 0pn
c
n
=
1lim 0 { } 0p p
np pn
c c c c
n N
n n
=> =
.{ }na5 > 0 0 %) .5 " /5 4% C &"( .
/1 0 % $:
1{ }nn =%1{( 1) }n
n =%1{sin }nn =
2
2 1 2
1
2 : lim lim lim(1 ) 1
2
1
2 1: lim lim lim 0
(2 1)
n k
n n n
n k
n n n
n k a a
k
n k a a
k
+
= = = =
= + = = =
+2
1
1 ,
1
,
n
n E
n
a
n O
n
=
.
1({ }na0 ,( 0 + D( E $ +na l:l n
+ C +( | : )nk n k a l
2({ }na0 ,( 0 + G " $ +na u:u n
+ G " $ C +( | : )nk n k a u
3({ }na0 ,( 0 + +:| : nM n a M
4({ }na0 ,( 0 A +:1: n nn a a +
5({ }naK +0 ,( 0 &%:1: n nn a a +
6({ }na5 " &%K A 0 0 M +.
7({ }na0 0 O% / +:1: 0n nn a a +× <
2
0
1
( )
2 n =
1 1 1 1
{1,2,5, , , , ,...}
2 4 8 16
(P3.0{ }na%{ }nb5 " >.
lim( ) lim limn n n n
n n n
a b a b± = ±
lim( ) lim limn n n n
n n n
a b a b× = ×
( : 0) lim 0)n n
n
n b b
lim
lim( )
lim
n
nn
n
n n
n
aa
b b
=
: lim limn n n n
n n
n a b a b

3 ::3
+ -= (P3)Q %. (
{ }na%{ }nb%{ }nc+ R" 4: n n nn b a c0 S+ T +
{ }nb%{ }nc"l, + 5 " >:lim n
n
a l=
(P3.0lim ( )
n
f x l=%( )na f n=7 >lim n
n
a l=.
?(( D(( + $ + /=+.
2
2
1
3 1
:
5 3 n
n n
n n =
+
+
2
2 2
2
2
1 1
(3
3 1 3
lim lim
15 3 5(5 )
n n
n
n n n n
n n n
n
+
+
= =
+ +
1
cos
:
n
n
n =
1
lim 0
1 cos 1
1 cos 1
1
lim 0
n
n
n nn
n n n
n
=
=
U
cos
lim 0
n
n
n
=
{ }2
1
2 :
n
n n n
=
+
2 2
2
2 2
2 2
lim( 2 ) lim
2 2n n
n n n n n n
n n n
n n n n n n
+ + +
+ × =
+ + + +
2
2
2 2 2 2
lim 1
22 22
(1 ) 2 1 1
n
n n n
n n n
n n
n n
= = = =
+ +
+ + + +
1
1
1
( ) :
n
nTan
n =
1
1 1
1
(lim )
1 1
lim (lim )(lim )
1
n
n n n
Tan
nnTan n Tan
n n
n
= =
2 2
4 42 22
2 2
2 22 2
2
1
11 1(1 )
1
1 1 1(1 ) 1 0( ( 1))
H
n n
n nn nn
n n n nn n n
+ +
= = = = = =
+ ++

::4
! " ! # ! $ %"
(P3.0{ }na> 7 > 5 " A % + G " $.
(P3.0{ }na> 5 " &%K % + D( E $> 7.
?(+ > " + ' % > V = ( - W% : $
( (E.
1 0 16 , 1, 7 :n na a a a+ = + = =
" A S @1 00,k a a= >
1, n nk n a a+= >
1 2 16 6n n n na a a a+ + ++ > + >2 11, n nk n a a+ += + >
" + G " $ S @
00, 1 3k a= = <
, 3nk n a= <
1 16 6 3 3 3n n na a a+ += + + =1,?k n= +
, +2
1lim lim 6 6 6n n
n n
a a l l l l+ = + = + = +lim n
n
a l=
2 3
6 0 ( 3)( 2) 0
2
l
l l l l
l
=
= + =
= ×
1 lim 0:n
n
x x< =
lim lim( ) lim( ) 0
n Ln x nLn xn
n n n
x e e e= = = =%lim lim
n
n n
Ln x nLn x= =
1
n nn
x x x x x x x< < < < <
lim 0
1: lim 0
lim 0
n
n n
n n
n
x
x x
x
=
< =
=
| 0:
!
n
x
x
n
,( & - /"lim 0
!
n
n
x
n
=$:
1
... ... ( ... )( )
! 1 2 1 1 2 1
n
n Nx x x x x x x x x x
n N N n N N
= × × × × × × < × × ×
1
lim lim( ... )( ) 0
! 1 2 1
n
n N
n n
x x x x x
n N N
< × × × =

5 ::5
lim 0
!
0
! ! ! !
lim 0
!
n
n nn n
n
n
n
x
x xx xn
n n n nx
n
=
=
=
?(( & + " + $ & 0 % & >.
2
1
2n
n
n
=
2 2
2
2 2
lim lim lim lim 0
2 2 ( 2)2 ( 2) 2
H H
n x x xn x x x
n x x x
Ln Ln
= = = =
{ } 43
1
:1, 2, 3, 4,...n
n
n
=
1
0
1
1
( ) ( ) lim lim 0 lim 1
1
H
x x
x x x
Lnx xy x Lny Ln x Ln x Lny y e
x x
= = = = = = = =
1
1
(1 ) :n
nn =
+
1 1 1 1
lim(1 ) 1 (1 ) (1 ) lim lim (1 )n x
n x x
Lny Ln xLn Lny xLn
n x x x
+ = = + = + = +
1
lim 1 lim
x x
Lny y e e= = =%
2
1
1
(1 ) 11 1
lim lim lim 1
1 1 11
H
x x x
x
Ln
x xLny
x
x x
+ +
= = = =
+
1
1
( ) :
1
n
n
n
n ++
1 1
lim lim ( )
1 1x x
n n
Lny nLn Lny n Ln
n n
= =
+ +
2
2
( 1) ( 1)
( 1) 01
1( )
0( 1)( 1)1 1lim lim 0
1 1 1 ( 1)( 1)x x
n n
n
n
nLn
n nn n
n n
nn n
+
+
++ += = = =
+
0
lim 1
x
y e= =

::6
! " ! # ! $ %"
0
:
n
n
n
e
=
( ) ( ) lim lim ( ) lim 0
1
n
x x x
e
Ln
e e e
Lny Ln Lny nLn Lny nLn
n
= = = = =
lim( ) 0,( 1)n
x
e e
= <
1
( ) :
2 n
n
nCos
=
lim ( )
2x
n
nCos
{ }2
1
4 :
n
n n n
=
2 2 2
2
2 2 2
4 4 4 4
lim( 4 ) lim
4 44 4 (1 ) (1 )
n n
n n n n n n n n
n n n
n n n n n n n n n n
n n
+ +
× = = =
+ + + +
4 4
2
24(1 1 )
n n
nn
n
= = +
+
1 01 2 , 1:{1, 3, 1 2 3,...}n na a a+ = + = +
2 2
1lim lim 1 2 1 2 1 2 2 1 0n n
n n
a a l l l l l l+ = + = + = + =
2
1
( !)
:
(2 )! n
n
n =
(1 2 ... )(1 2 ... ) (1 2 ... )
lim lim
(1 2 ... ) ( 1) ... ( ) ( 1) ... ( )n n
n n n
n n n n n n n
× × × × × × × × ×
= =
× × × × + × × + + × × +
1 2 1 1 1 1
... ( )( )...( ) ( ) 0
1 2 2 2 2 2 2
nn
n n n
= × × × =
+ +

7 ::7
.:
{ }na1 2
1
...n
n
a a a
=
= + +!)(
$% &.'
1
n
n k
k
S a
=
=()* + ,-n. & / 0.
1(.% ' 2 3 / 4,.
1
1
:
( 1)n n n= +
1 1
1 1 1
( )
( 1) 1
n n
n
k k
S
k k k k= =
= =
+ +
1 1 1 1 1 1 1
(1 ) ( ) ... ( ) ( )
2 2 3 1 1
nS
n n n n
= + + + +
+
1
1
lim 1 1
( 1)
n
n
n
S
n n=
= =
+
1
1
1
nS
n
=
+
1
( ):
1n
n
Ln
n= +
1 1
( ) ( ( 1)
1
n n
n
k k
k
S Ln Lnk Ln k
k= =
= = +
+
( 1 2) ( 2 3) ... ( ( ) ( 1)) 1 ( 1) ( 1)nS Ln Ln Ln Ln Ln n Ln n Ln Ln n Ln n= + + + + = + = +
lim lim( ( 1))n
n n
S Ln n= + =
.:
0
( , ), n
n
a r ar
=
5 62 . & . !:
1
1
0
...
n
k n
n
k
S ar a ar ar
=
= = + + +
1
2 1
0
...
n
k n n
n
k
rS r ar ar ar ar ar
=
= = + + + +
(1 )
(1 ) (1 )
1
n
n n
n n n n
a r
S rS a ar S r a r S
r
= = =

::8
! " # $ " "% $ &
, 1:(1 )
lim lim 1
1
1:
n
n
n n
a
ra r
S r
r
r
<
=
>
7 8.3/ 3 3/333...=
0
3 3 1 1 1 3 3 30 10
3/ 3 3 ... 3(1 ...) 3
1 910 10 10 100 10 9 31
10 10
n
n=
= + + + = + + + = = = = =
!%:
0
1
n n=
/ &:
1
1 1 1
1 ...
2
n
n
k
S
k n=
= = + + +
2
12
1
1 1 1 1 1 1 1 1 1 1
1 ( ) ( ) ... ( ... )
2 3 4 5 6 7 8 2 2
n
n n n
k
S
k=
= = + + + + + + + + + + + =
1
1 1 2 1
...
2 2 2 2
n
n n n
> + + = =
1 1 1 1 1
8 8 8 8 2
> + + + =
1 1 1
4 4 2
> + =
2 2
1 lim lim(1 ) lim
2 2
n n N
n n n N
n n
S S S> + > + = = =
4, ! / . && 4, $ / 2 $ 2 &.
%9:).4, 0/; < =:(
$% ' >
1
n
n
a
=
.= 2 4,lim 0n
n
a =
$ 3:
1
n
n k
k
S a
=
=5 /
1
n
n
a
=
2 2 4,:( )L R
lim n
n
S L=
1 1: lim lim lim limn n n n n n n
n n n n
n N S S a S S S a= + = = +
lim lim 0n n
n n
L L a a= + =
p q q p
-%:&lim 0n
n
a? 4 @
1
n
n
a
=
&.
7 8:/ . A & /:
lim 1
1n
n
n
=
+
:&
1 1n
n
n= +
2
2
2 ( 2)2 ( 2) 2
lim
2 2
n n nH H
n
Ln Ln
n n
= = =:&2
1
2n
n n=

9 ::9
1
lim 1n
n
n
n
=
=:&
1
n
n
n
=
* 7 ,C:
naDnb. *.
1 1
: n n
n n
c R c a ca
= =
=)1
&=,4±&
1 1 1
( )n n n n
n n n
a b a b
= = =
± = ±)2
8 H,* 2:
1
n
n
a
=
$% & 8 H,* 2 !)$% & 8(&
)I(: : 0nk N n k a>, 0nn a
A J K /@:
8nbna5 62 . A:
(&nb? 4 @ .= 2 4,na4,.
L(&na? 4 @ .= 2 &nb&.
.M K /@:
8nanb5 62 . N:lim n
n
n
a
l
b
=
(0 )llim (0 )n
n
n
a
l l
b
= <
O:
(&0 l <nb? 4 @ .= 2 4,na4,.
L(&0 l<nb? 4 @ .= 2 &na&.
7 8:
1 1
n n
&
&
1
n
& A J K /@
1 1
: :n
nn1
1
n n=
22
2
1
lim lim 1
1
( 1)
n n
n nn
n
n n
+
= = <
+
4,
1
( 1)
n
n n +2
1
1
n n=

::10
! " # $ " "% $ &
4,2
1
n
.M K /@
&
2
2
1
lim
n
n
n n
+
=
2
2
2
1
:
n
n
n n=
+
2
2 2 22
2
2
1
( 1) ( 1) ( 1)
1 1 1
(1 ) 1
n
n n n n n nn n
n n
n n
n n n
+
+ + +
= = = =
2
21
1
1
1
n
n
n
+
= = + =
7 4 K /@:
$% ' >( )f x?/ 2 2 %P 8 Q2 ![ )1,5 62 ) .= 2( )na f n=n N
1
n
n
a
=
& RJ> & 4,
1
( )f x dx.= 2 4,.
3:
1
2 3 1
1
11
1 1
2 1
1 1
1
... ( )
( )
lim ( )
n
n
nn
n k
k
n
n
a a a f x dx
S a a f x dx
S a f x dx
+
+
++
+
=
+
+ + + >
= >
>
1
1 2
1
1
1 1
1
1 1
... ( )
( )
lim lim ( ) ( )
n
n
nn
n k
k
n
n
n n
a a a f x dx
S a f x dx
S f x dx f x dx
+
+
=
+
+ + + >
= >
> =
7 8:2 / >.% '.
7 4 K /@
2
1
:
lnn n n=
1
ln ,
ln
2 2 ln 2
2 ln2
ln
1 1 1
lim lim
ln ln
v u dv dx
xu u
x u
x v
x u v u
dx dx dv
x x x x v
= =
= =
= =
= =
&
ln
lim(ln ) lim(ln(ln ) ln(ln 2)
ln 2u u
u
v u= = =
3
1
1
:
n n=
4,
2
3
3 2
1 1
1 1 1 1
lim lim( ) lim( )
12 2 2 2
u
ux
dx x du
x u
= = = + =

11 ::11
:
1
1
p
n
p
n
+
=
!p-$ ' $% &:
(&1p >.= 24,.
L(&1p& .= 2.
/:T;' M I .21p =0p.= 2p-K U > ' 7 M &
$%p-20 1p< <2 &1p >4,.? 7 4 K /@ / V 2
$% '.
8 D %PD )[ ]1,+0>p
1
( ) p
f x
x
=
1 1
1 1
0 1
1 1
lim lim( ) lim
11 1 1
1 1
1
x p p
p
p
p
ux u
dx x dx
x p p p
p !"#
p
+
< <
= = = =
+
>
1(.% ' 2 / >.
.M K /@
1
1
sin :
n n=
&
1
0
0
1
sin
sin
lim lim 1
1
x
n
nn x
x
xn
x
n
=
= =
.M K /@
0
2 1
:
3 1
n
n
n=
+
+
2 1
3 (2 1)3 1lim lim
2 2 (3 1)
3
n
n nn
n n n
n
+
++ =
+
1 1
2 (1 ) 1
22 2lim lim( ) lim 0 1 0
1 133 (1 ) 1
3 3
n
n n
n
n
n n
+ +
= × = × =
+ +
4,
1
1
3 2 1 2 3 2lim( ) lim lim( ) lim( ) lim 1 1
12 3 1 3 2 1
3
n n
n n n
nn n n
n
+
+
= × = × × =
+ +

% ::12
! " # $ " "% $ &
1
1
2 sin :
3
n
n
n=
4,
1 1
2 sin 2
3 3lim lim 1 0
2 2
3 3
n n
n n
n n
n n
×
= = >
0 0
limsin lim
x x
x x#0×
1
1
1 1
:n
n nn n
n n=
=
.M K /@
1
lim lim 1
1
n
n
nn n
n n
n
= =
1 1
1
1
lim ln ln lim ln 0
1
H
n n n nn n y y n n
n
= = = = = =
0
lim 1n
n e= =
4,
1
( 1)n n +2
ln n
n2
1
ln
:
n
n
n=
2
1
ln
n
n
n=
4, X >
2
3
2
ln 1
ln
lim lim lim 0
1 1
2
H
n
nn n
n
n
n
= = =.M K /@
1
1
:
( 1)( 2)n n n n= + +
4,)&!p1p >(
3
2
1
( 1)( 2)
lim 1
1
n n n
n
+ +
=
7 4 K /@2
1
arctan
:
1n
n
n= +
2
1
arctan ,
1 arctan
1
41 arctan(1)
4
arctan
arctan
1
lim lim
2
4
x v dv dx
x u
x v
x u v u
u
vdv v$
$
$
= =
+
= = =
= =
==2 21 1
arctan arctan
lim
1 1
ux x
dx
x x
=
+ +
4,
2 2 2
1 2 21 3
(tan )
2 32 8 32 32
u
$ $ $
$= = =

13 ::13
3
2
1
:
(ln )n n n=
3 3 31 2 2
1 1
lim lim
(ln ) (ln )
u u dv
dx dx
x x x x v
= =
4,2
1
ln , 2 2
l
2 2 2 2ln
ln
2 ln2
ln 1 1 1 1
lim lim lim( ) 0
ln 22 2 2(ln ) 2(ln ) 2(ln ) 2(ln )
v x dv
x
n
x u u
x
nv v
u u u u
= =
=
=
= = + = + ==
3
1
:
(ln )(ln(ln ))n n n n=
1
ln(ln ),
ln ln(ln )
ln(ln3)
, 33 3
n(ln )1 1
lim lim lim(ln )
ln(ln3)(ln )(ln(ln )) (ln )(ln(ln ))
xu x du
x u
x
x u xx x
udv
dx dx v
x x x x x x v
= =
= == =
= ==
lim(ln(ln(ln )) ln(ln(ln(3)))u= =
7 4 K /@
2
1
:
lnn n n=
{
1
ln ,
ln ln
1ln2 ln2
ln1 2 2
2 ln 2
1 1
lim lim lim
ln ln
v x dvu x u u
x u v u
x v
dv dv
dx dx
x x x x v
v
= =
= =
= =
= = = ==
1
2 ln
lim lim(2 ln 2 ln 2)
1 ln 2
2
u
uv
u= = =
2
1
( !)
:
2 !n
n
n=
2
2
(( 1)!)
( 1) !( 1) !(2 )! ( 1)( 1) 1(2 2)!
lim lim lim 1
( !) (2 2)(2 1)(2 )! ! ! 2( 1)(2 1) 4 4
2 !
n
n n n n n n n nn
n n n n n n n n n
n
+
+ + + ++
= = = <
+ + + +
2
2
ln
( ) :
n
n
n=
2
2 2 2
2
2 2 2
2
ln( 1)
( )
(ln( 1)) (ln( 1))( 1)
.lim lim lim( ) lim
ln ( 1) (ln ) 1 (ln )( )
n
n n n nn
N
n n n n n
n
+
+ ++
= = × =
+ +
2
2 2
2
1
lim( ) (ln lim( )) 1 lim(ln1) 1 0 0
2 1
H n n
n n n
+
= × = × = × =
+ +

+ ::14
! " # $ " "% $ &
2
3
2 222
2
3
2
(ln )
(ln ) (ln )
.lim lim lim
1
n
n n nnM
n n
n
= = =
1 4
2( )ln
4ln 8
lim lim lim 0
1 1
2 2
H Hn
nn n
n n
n n
= = = =
2
1
:n
n
n
e=
2
2 2( 1)
1
2 2 1 2
( 1)
( 1) 1
lim lim lim lim 1
nn
n
n
n
n
n
a n e ne
na n e n e e
e
+
+
+
+
+
= = = = <
2
1
sin1
:
n
n
n=
2 2
2
2
sin1 1
sin
1
lim lim 2sin 1
1 1
n
n nn
n
n n
= = =
ln(ln )
10
2
:
ln
n
n n n=
ln(ln ) ln(ln ) ln(ln ) ln(ln ) ln(ln )
1 ln(ln10)10 ln(ln10)
10 10 101
ln ln
ln(ln )2 2 1
lim lim lim 2 lim 2 lim 2
ln(ln10)ln ln ln 2
u ux x v x x u v u u
v v v
x v
xdv
x x nx
u
dv dv
x x x x
= = =
= =
= =
= = = =
ln(ln ) ln(ln10) ln(ln10)1 1
lim (2 2 ) ( 2 )
ln 2 ln 2
u
= =
L:
8{ } 1n n
a =
1
1
( 1)n
n
n
a
=
. & L !.
L K /@:
L1
1
( 1)n
n
n
a
=
? & 4,:
(8{ }na.= 2 ).
L(lim 0n
n
a =

15 ::15
7 8 2:
1
1
1 1
1 1
( 1)
1
lim 0
n
n
n
n n
n
n
=
>
+
=
2 2
1
2
1
2
1 1
1 ( 1)
( 1)
1
lim 0
n
n
n
n n
n
n
=
>
+
=
1(>1.% ' 2 /
))[ / U%2 .=(
2
1
1
( 1) :n
n
n
n
e=
2
2 2
lim 0
H H
n n nn
n n n
e e e
= = = =
1
1
1
( 1) ln( ) :n
n
n
n
+
=
+
)
2 1 2 1
ln( ) ln( ) ( 2) ( 1) 0 1
1 1
n n n n
n n n
n n n n
+ + + +
< < + < + <
+ +
1 1
limln( ) 0 ln lim( ) ln1 0
n n
n n
n n
+ +
= = =
) / U%2 [ .=1
2
ln
( 1) :n
n
n
n
+
=
1
ln 2 2 2 1 1
lim lim lim 0
1 1
2
H H Hn n nn
nn n
n
×
= = = = = =
:
1
n
n
a
=
& $% & 4, ]^6 62
1
n
n
a
=
.= 2 4,.
(!&
1
n
n
a
=
]^6 62 D.= 2 4,)
1
n
n
a
=
.= 4,(
U 4,$% & <.
7 8:
&
1 1 1
1 1 1
( 1) ( 1)n n
n n nn n n= = =
=
4, ]^6 622 2
1 1
1 1
n nn n= =
=

, ::16
! " # $ " "% $ &
%9:.4, )% ` .= 2 4, ]^6 62 &) .4,
1
n
n
a
=
4,
1
n
n
a
=
(
:Q2 .% ' >{ } { }: 0 0g( ) kg k n=.= 2
0
nk
k
a
=
a @ . .- !
0
n
n
a
=
.= 2.
0 1 100 101
0 0
... ...n nk
n k
a a a a a a
= =
= + + + + + =
1($%1
1
1
( 1)n
n n=
b8 .C 2S. @ .2 X > / U @ . .- 4,
.C 2 '
2
S
.= 2 4,.
1
1
1 1 1 1 1 1 1
( 1) 1 0 0 0 ...
2 3 4 5 6 7
n
n
S
n=
= = + + + + + + +
1 1 1 1 1 1
0 0 0 0 ...
2 2 4 6 8 10 12
S
= + + + + + + +
1 1 1 1 1 1 1 1
1 ...
2 2 4 3 8 5 10 12 7
S
S = + + + +
1
1
1 1 1 1 1 1 1 1 1 1
( 1) 1 ...
2 3 4 5 6 7 8 9 10
n
n
S
n=
= = + + + + +
C 0 ,
2
S
S$V 62
2
S
S= &
%9:.&
1
n
n
a
=
K , 2 X > / a @ . .- .= 2 4, ]^6 62
4, .C.
&na2 ' ` U @ . .- K J%JM .C 2 .= 2 4, < U 62
.= 2 4, K@.
A K /@:
2
0
n
n
a
=
&1
lim n
n
n
a
l
a
+
=)K@ 'l =)%.= 2 .(:
(&0 1l <na4, ]^6 62.
L(&n1l >l =na&.

17 ::17
[(&1l =-% 2 K /@.
U K /@:
2
0
n
n
a
=
&lim n
na l=K@ 'L.
.= 2 I 2 .C.:
(&0 1l <na4, ]^6 62.
L(&1l >l =na&.
[(&1l =-% 2 K /@.
( !.= 2 A K /@ / : U K /@.
7 8:.% ' 2 / >.
0
1
:
!n n=
4,
1
!( 1)!
lim lim 0 1
1 ( 1) !
!
nn
n n
n
+
= = <
+
A K /@
1
1
( )
!
n
y
n
=$I
1 0
lim
! 0
n
n n
=U K /@
1
2
1 1 1 1 1 1
ln ln( ) ln ln limln lim ln (ln1 ln !)
! !
n
y y y n
n n n n n n
= = = =
0e =
1
ln ! ! 1 2 ...
lim lnlim ln( 1)
1
ln !
n n nn n
n n n
n
× × ×
= = = = =
1
!
( 1) :n
n
n
n
n=
4,
1 ( 1)!
( 1)
!( 1)( 1)( 1)
lim lim lim( ) 1
! ( 1) !( 1) 1( 1)
n
n
n
nn nn
n
n
n n n nn n
n n n n n
n
+ +
++ +
= = =
+ + +
11 1 1
lim lim 1
1 1
( ) (1 )n n
e
n e
n n
= = = = <
+
+

::18
! " # $ " "% $ &
! 1 ! 1
?)lim ! lim lim (*)
n
n n
nn n
n n
n
n e n e
= = ' =
1
! 1 1.2... 1 1 2
(*): ( ) ln ln( ) ln( . ... )
. ...
n
n
n n n
y y
n n n n n n n n n
= = =
1
1
0
1
11 1
limln lim ln( ) ln ln (0 1) (0 0) 1 lim
0
n
i
i
y ndx x x x y e
n n e=
= = = = = = =
( 1)( 2)...2 4
?)lim lim (*)
...
n
n n
n n n
n n e
+ +
=
1
( 1)...2 1 1 2
(*) ( ) ln ln(1 )(1 )...(1 )
...
n
n n n
y y
n n n n n n
+
= = + + +
{
]
1 ,1 2 2
10 10 1
1 1 2
1
limln lim ln(1 ) ln(1 ) ln ( ln )
n r x dr dx
x r
i x r
i
y x dx rdr r r r
n n
= + =
= =
= = =
= + = + =
ln4
ln 4 1 4
(2ln 2 2) ( 1) 2ln 2 1 ln 4 1 lim
e
y e
e e
= = = = = =
lim (2 1)(2 2)...(2 ) :n n n n+ + +
27 (2 1)(2 2)...3 27
lim lim
4 4
n
n
n n n n
e n e
+ +
= = =
5:
2
3
4 4
lim (2 )! lim ! ( 1)...(2 )nn n
n n n
n n n n
e e e
= × + = × =
2 3
2 3
4 27 27
lim (3 )! lim 2 ! (2 1)...(3 )
4
nn n
n n n
n n n n
e e e
= × + = × =
1(.% ' 2 / >.
2
1
:
(2 1)!n
n
n=
2
2 2
1
2 2 2
( 1)
( 1) (2 1)! ( 1) (2 1)!(2 1)!
lim lim lim lim
(2 1)! (2 1)2 (2 1)!
(2 1)!
n
n
n
a n n n nn
na n n n n n n
n
+
+
+ ++
= = = =
+ +
2
2 2
2 1
lim 0
(4 2 )
n n
n n n
+ +
= =
+

19 ::19
1
( 1) :
2 !
n
n n
n
1
1
2
( 1)
( 1) 2 ! ( 1) ( 1) ( 1) 1(2 2)!
lim lim lim lim
(2 2)! (2 2)(2 1) 2(2 1)
(2 1)!
n
n n
n n
n
n n n n n nn
n n n n n n n n
n
+
+
+
+ + + ++
= = = × =
+ + +
0 0 1e= × = <
2
1
10
:
n
n
n n$=
1
2
1 1
1 2 2 2
12 2 2
10
10 10 10 10( 1)
lim lim lim lim
( 1)
10 10 ( 1) 10 ( 1)
n
n n
n nn
n n n
n n
n
n n nn
n
n n
n
$ $$
$
$ $ $
$
+
+
+
+
+
= = = =
+
+ +
10 10 3.16
lim 1
3.14
n
n$ $ $
= = = >
+
1
!(2 !)
:
3 !n
n n
n=
( 1)!(2 2)!
( 1)!(2 2)!3 ! ( 1)(2 2)(2 1)(3 3)!
lim lim lim
!(2 !) (3 3)! !(2 !) (3 3)(3 2)(3 1)
3 !
n n
n n n n n nn
n n n n n n n n
n
+ +
+ + + + ++
= = =
+ + + +
3
3
4 4
lim 1
27 27
n
n
= = <
1
1.3.5...(2 1)
:
3.6...(3 )n
n
n=
1.3.5...(2 1)
1.3.5...(2 1)3.6...(3 ) 2 1 23.6...(3 3)
lim lim lim
1.3.5...(2 1) 3.6...(3 3)1.3.5...(2 1) 3 3 3
3.6...(3 )
n
n n nn
n n n n
n
+
+ ++
= = =
+ +
1
1
( 1)
:
n
n
n
n
n +
=
+
-% 2
( 1) ( 1)
lim lim 1
n
n
n n
n n
n n n n
+ +
= =

% ::20
! " # $ " "% $ &
K c
1
n
& dP &
1
1
( 1)
( 1) ( 1)
lim lim lim
1
n
n nn
n n
n
n n n nn e
n n n
n
+
+
+
+ +
= = =
1
1
:
1
n
n
n
n= +
1 1 1
lim lim 1
1 1 1
n
n
n n
n n
= = =
+ + +
n n n
n n1-n (-1)(n-1) 1-n n+1-2
lim =lim =lim(-1) = lim(-1)n( ) =
1+n 1+n 1+n n+1
2
n
2
2
lim( 1) lim(1 )
-2 1
=lim(-1)n(1+ )
2n+1
lim( 1) lim(1 ) ( )
1
n n
n n
e
n
e
n
+ =
+
=
+ =
+
. .M dP A% > 2 Oe .M& dP A%) .lim(1 )n aa
e
n
+ =(
1
! 19
( ) :
7
n
n
n
n
n=
1
1
1
( 1)! 19 19 19 19( ) ( 1)
19( 1) 7 7 7 7lim lim lim lim 0/99 1
! 19 1 1( 1) 7( ) ( ) (1 )
7
n n
n
n
n n n
n
n
n n
n
n nn e
n n n
+
+
+
+
+
+
= = = = <
++ +
0
( )n
n
n
a x c
=
.= 2 [ Q2 ! &2
2
( 1)
:
(ln )
n
n n=
4, L K /@ 2 22
1
lim 0
(ln )n
=
) 82
1
(ln )
na
n
=

21 ::21
:
.%fx{ }naO
0
n
n
n
a x
=
!
$% &)$% & 6J 7 M !.(
6 %,
0
( )n
n
n
a x c
=
6J 7 M !c$% &.
4, + =
1
1
0 1
lim 1 lim
n
n n n
n n n n
n n
a x a
a x x R
a x a
+
+
= +
< < =
] [
[ [
] ]
[ ]
,
,
,
,
R R
R R
R R
R R
0
0
( )
n
n
n
n
n
n
x R a R
x R a R
=
=
=
=
4, + =
1
lim 1 limn
n
n
n
a x x R
a
< < =U K /@
( !4, K@ / 2 ' ?/ 2 4, ?/ 2.
7 8:4, ?/ 21. @ .2 /
@A K /
1
:
1
n
n
x
n= +
1
1 22lim 1 lim 1 lim
2 1
1
n
n
x
n nn x x
x n n
n
+
+ ++ < < <
+ +
+
4, L K /@ 2 2
1
( 1)
1
1
n
n
x
n=
=
+
& .M K /@ 2 2
1
1
(1) 11 : 1
11
n
n
nx
n
n
=
+= =
+
U K /@ln
1
:n n
n
n x
=
( ) ( )
1
ln
ln ln 1
lim 1 lim 1 lim 1
n
n
n
n n nn n x x n x R
n
< < < = =

%% ::22
! " # $ " "% $ &
( )ln
2
1
2ln 2lnln ln
ln ln limln lim 2 lim 0
1 1
n
n
H Hnnn nn ny n y n y
n n n
×
= = = = = = =
0
lim 1y e= =
&ln
1
1 ( 1)n n
n
x n
=
=
&ln
1
1 n
n
x n
=
=
ln 2
ln ln (ln ) (ln ) lim ( )n
y n y n n n y e+
= = = = = =
U K /@
2
1
1
(1 ) :n n
n
x
n=
+
21 1 1 1
lim (1 ) 1 lim (1 ) 1 lim
1
(1 )
n n nn
n n
x x x x R
n n e
n
+ < + < < < =
+
&
2
2
1 1 1
11 (1 )(1 )
1 1 1
(1 ) ( ) 1
n
nn
n
n n
n n n
n n n
ennx
e n e e e e= = =
++
= + = = = =
A% *
2
2
1 1 1
11 (1 )(1 )
1 1 1
(1 ) ( ) ( 1) ( 1)
n
nn
n n n n
n n
n n n
nnx
e n e e e= = =
++
= + = = = (
A K /@
3
3
1
( !)
:
(3 )!
n
n
n
x
n=
3
3 3
3 3
3 3
3 3 3
3
(( 1)!)
(( 1)!) (3 )! (3 3)!( !)(3 3)!
lim 1 lim 1 lim
( !) (3 3)!( !) (( 1)!) (3 )!
(3 )!
n
n
n
x
n n n nn
x x
n n n n n
x
n
++
+ ++
< < <
+ +
3 2
3 3 2 2
(3 3)!( !) (3 3)(3 2)(3 1) 3(3 2)(3 1) 27
lim lim lim lim 27
(( 1)!) (3 )! ( 1) ( 1)
Hn n n n n n n n
n n n n n
+ + + + + +
= = = =
+ + +
3 3
27 27 3x x x< < <
1(. @ .2 / 4, ?/ 2.
2
1
3 2
( ) :
n n
n
n
x
n n=
+

23 ::23
2
1
( 4)
:
!
n
n
x
n=
2
1
:
2
n
n
n
x
=
1
1 1
(1 ... ) :
2
n
n
x
n=
+ + +
4
1
2 4 ... (2 )
:
1 2 ... (2 1)
n
n
n
x
n=
× × ×
× × ×
1
1
10 ( ) :
5
n n
n
x
=
1
:
n
n n
n
x
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2
( 1)
:
ln
n
n
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n n=
+
2
1
!
:
3
n
n
n
n x
=
1
1 1
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n
x
n=
+ + +
1
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:
ln
n
n
x
n=
2
1
1 4 ... (3 1)
:
1 5 9 ... (4 1)
n
n
n
x
n=
× × × +
× × × × +
1
:
1
n
n
x
n= +
.
0
n
n
n
a x
=
4, ?/ 2 2IQ2f?/ 2I/ O2
?.=:

%+ ::24
! " # $ " "% $ &
0
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n
n
x I f x a x
=
=
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0
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n
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=
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& 6J 7 M..
.
0
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n
n
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=
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?.= /:
0
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n
n
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n
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f6J 7 Mc. &.
( !4, + = .% ' >
0
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n
n
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=
2 .CD.=:
U K /@:
1
lim 1 limnn
n
n
n
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a
< < =
A K /@:
1
1
1
lim 1 lim
n
n n
n
n n
a x a
x R
a x a
+
+
+
< < =
/ . ' , %%f 4, + = X > / ^,* 2 ^,* %& 7 4 %& ] U 2:
] U:1 1
0 1
n n
n n
n n
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=
/@A K:1
1
1 1
( 1)
lim 1 lim lim
1
n
n n n
n
n n n
n a x n a a
x R
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+
+ +
+
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+
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1
0 1
n
n
n
a x
n
+
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2
1
1
1 1
( 2) 22lim 1 lim lim
( 1) 1
1
n
n
n n
n
n n n
a x
n a n an x R
a x a n n a
n
+
+
+
+ +
+ ++ < < = × =
+ +
+
( !/ %& 7 4 %& ] U 2
0
n
n
n
a x
=
. ' %%f 5, 4, ?/ 2
2 3 T , M 4, + =.

25 ::25
7 8.7 ] U 4, ?/ 2 4, ?/ 2 D /
.% ' e 7 4 0 ] U.
U K /@2
1
:
n
n
x
n=
[ ]
2
12
2
2
1
1
1
lim 1 lim 1,1
1
1 ( 1)
n
nn
n
n
n
x
nx
x n
n
x
n
=
=
=
< <
=
1 1
2
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: , 1
n n
n n
nx x
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n n= =
= =] U
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1
1,1
1
1
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n
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n
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*
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+
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0 2
( 1) ( 1)
: , 1
n n
n n
n x n x
R
n n= =
= =0 ] U
( )
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2
2
( 1)( 1) 1
1 lim( ) lim( 1)
1,1
( 1)(1) 1
1 lim 1
n
n
n
n n
x
n n
n n
x
n n
)
= ×
*
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+
1
2
0
: , 1
( 1)
n
n
x
R
n n
+
=
=
+
7 4
[ ]
1
2
2
( 1)
1
( 1)
1,1
1
1
( 1)
n
x
n n
x
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+
)
=
+
*
=
+ +
.% @ . @ .2 K@ 4, ?/ 2 /-,+
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0
: lim 1 lim ! lim
! !
n n
n
n
n
x x n
x n R
n n e=
< < = = =
K c= 4, ?/ 2 e K .2 K A% .C:( , )I = +
1 1
0 0 1
( , ) : ( ) ( )
! ! ( 1)!
n n n
n n n
x nx x
x f x f x
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,+ = = =
0
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1: ( ) ( )
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k
k
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k n f x f x
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= = )
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%, ::26
! " # $ " "% $ &
( )
( ) ( )
( , ) : ( ) ( ) ( ) 0 1 1
( ) ( )
f x f x
x f x f x f x dx dx
f x f x
, ,
,+ = = = =
( )
ln ( )
( )
ln ln ( )
u f u
f x x c
du f u du
du
x c u x c f x x c e e
u
=
+
,=
= + = + = + =
0
( ) ( ) (0) 1 1 ( )
x c x c x
e e e ke
x c u x x
f x e ke f u ke f ke k k f u e
+
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+
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( )
0
, :
!
n
n
x
x ex
n=
+ =
%9:..% ' >
0
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n
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0
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n
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=
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( ) 1
1
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n
n
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( )
1
0
1
, : ( )
1
n
x
n
n
a x
x R R f t dt
n
+
=
=
+
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lim (1 )n
n
a
n L
a
+
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0
n
n
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L(& 4, < U i 2 &1L <
j(&1L == , kU K /@ 2 +.
A% OU b2 : A K /@ + ' 52 : K /@ l^m.

33 ::33
( )2
0
1
1) 1 ... 1,1
1
n
n
x x x x
x =
= = + + +
( )
( )1 1
2
0 1
1
2) 1,1
1
n n
n n
nx nx x
x = =
= =
( )
( )2 2
3
0 2
1
3) ( 1) ( 1) 1,1
1
n n
n n
n n x n n x x
x = =
= =
( )2
0
1
4) ( ) 1 ... ( 1) 1,1
1
n n n
n
x x x x x
x =
= = + =
+
( )
1
0 0 00 0 0
1 ( 1)
5) ( ) ( 1) ln(1 ) 1,1
1 1
x x x n n
n n n
n n n
x
dt t dt t dt x x
t n
+
= = =
= = + =
+ +
( ) ( ) ( )
22
2
0 0
1
6) ( 1) ( 1) 1,1
1
n nn n
n n
x x x
x = =
= = =
+
( ) ( )
2 1
2 1
2
0 00 0
1
7) ( 1) tan ( 1) 1,1
1 2 1
x x n
nn n
n n
x
dt x dt x x
t n
+
= =
= =
+ +
..
( )
2
1
:
1 x+
( )
( )
( )
1 1 1
2
0 1 1
1 1
( 1) 1,1 ( 1) ( 1)
1 1
n n n n n n
n n n
x x nx nx
x x
+
= = =
= = =
+ +
2
2 2(1 ) 1
2 2
x
x x
x xx
= =:
2
x
x
( ) ( )1
0 0
2 1
1,1 1,1
1 1
2 1
Xn n
n n
x X
X
X x X xx X X X
x X
× +
= =
=
= =
=
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1 1
1
0 0
1
2,2
2 2 2
n n
n
n n
x x
x x
x
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3
:
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":#:34
%# & #'
( )
( )
( )1
2
0 1
1 1
1,1 1,1
1 1
n n
n n
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x x= =
= =
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( )
( )
( )
2
2
2 2
3 3
2 2
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( 1) 1,1 ( 1) 1,1
21 1
x
n n
n n
x
n n x x n n x x
x x
×
= =
= =
(.
( )
22
:
1
x
x
2
1
:
3 2x x +
1
ln( ) :
1
x
x
+
2
ln(1 ) :x x+
sinh :
2
x x
e e
x =
:
1 2
x
x+
!."
1
( 1)...( 1)
1
!
n
n
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x
n=
+
+# $ %& '( ) %*+, $x
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35 ::35
1
( 1)...( 1)
( ) 1
!
n
n
r r r n
f x x
n=
+
= +
1 1
1 1
( 1)...( 1) ( 1)...( 1)
( )
! ( 1)!
n n
n n
r r r n r r r n
f x nx x
n n= =
+ +
= × = ×
1
1 1
( 1)...( 1) ( 1)...( 1)
( )
! ( 1)!
n n
n n
r r r n r r r n
xf x x nx x
n n= =
+ +
= × = ×
1
( 1)...( 1) ( 1)...( 1)( )
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n
n
r r r n r r r n r n
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n n=
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( ) ( ) (1 )
( 1)! !
n n
n n
r r r n r n r r r n
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!
n
n
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n=
+
+ = +
0 0
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( ) 1 ( ) 1
x x
x xf x r f t r
xf x f x rf x dt dt f t r t
f x x f t t
+ = = = = +
+ +
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r
r rf u x r
f u x e e f u u u
+
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(78%.3(4" ( 9:
(-& ; <f2 $I%=9" > 5a? @- $ A 5 B. CDE:
( ): ( ) ( ) ( )( ) ,x I f x f a f t x a t a x= +
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( )
1
1( ) ( )
: ( ) ( ) ( )( ) ... ( ) ( ) ,
! ( 1)!
n n
n nx a x a
x I f x f a f a x a f a f t t a x
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+
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+
( )
0
1
1
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kn
k
n
k
n n n
n
n
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k
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":#:36
%# & #'
.*( %(78 2$ GD0
cos47%, $ H 92n =I # &
(-& %J K H 9 3 =.
0
0
1
180 180
47
4 90
45
4
2
R
R
x
a
n
= =
= = +
= =
=
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( ) sin
( ) cos
( ) sin
( ) cos
f x x
f x x
f x x
f x x
f x x
=
=
=
=
=
2
2 2
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2!
x a
f x P x f a f a x a f a R= + + +
2
0
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4 90 4 4 90 4 2!
= + + +
3 33
(3)
2 2
( ) ( )( ) 90 90( ) ( ) ( ) sin( )
3! 4 90 3! 3!
x a
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; <(-&f2 $I%=9" > 5a-D " ? @- $ 5 B. CDE M"
0
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n
n
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n
n
n
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n
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3 5 7 3 5 7
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1
0
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n n
n n
n n
n
n
n
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n n
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( ) sin
( ) cos
( ) sin
f x x
f x x
f x x
f x x
f x x
=
=
=
=
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37 ::37
E< %(78 CJQ
1 1
1
lim ( ) lim (sin ) lim 0
( 1)! ( 1)!
n n
n
n x tn n n
x x
R x x
n n
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+
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n
n
n
x
x x
n
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x n
x
x
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n
nx
x
x
e
n=
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1
lim ( ) lim( ) lim 0 0
( 1)! ( 1)! !
n n n
nx n t x x
n x t x t
x x x
R x e e e e
n n n
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+
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(-D 4+ 3 $ % %& (-& S # *L.
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f x x
x
=
1
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::44
! " # $" %
:
.)"($%)&' ( $%(
)*+ , -.
PF PQ=
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2
2
4
4
x
x cy y
c
= =
". / .:0' . 1 " %
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c>0
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y=-c
c<0
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x = -c c>0
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x = -c
c<0

45
& ' ( )
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::46
! " # $" %
c
D
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C
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b
K=:
' / "A L - M)K= & ( "(&. )
)+.
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2 2 2 2 2 2
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1
b a c x y
b x a y a b
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x y y x
O
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47
& ' ( )
*:+,:47
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11 ( )
2
x y
x x y y
+
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( , ) ( 2,1) ( 2 ,1) ( 2 ,1)
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A
F F
B
=
= = = +
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1
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C
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?
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9 9X(:
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1Q
2Q
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2 ( ,0)F c1( ,0)F c

::48
! " # $" %
9 9X( & =;N:
•". / & (.
•
( !9' : .' ;
2 2
2 2
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2 2
2 2
1
y x
a b
=&' 9 9X(
' +.
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*: )*+ D E4 & (. & ". / & (. ; <= ; '. ) 4 . 9 9X(:
&'( )
2 2
2 2
( ) ( )
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x y
a b
=
I G.& 9' : 9 9X( . '2 2
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2 2
( 2 1) 4( 4 4) 1x x y y+ + =
( ) ( )
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( 1) ( 2) ( 2) ( 1)
1 1
1 11 1
2 2
x y y x
= =
1 2
5 5
(1,2 ) , (1,2 )
4 4
F F +
1 1
(1,2 ) , (1,2 )
2 2
A B +
( !9 9X( &< . ' K=.
c
e
a
=R U %
1e =K=1e >9 9X(0 1e< <
] %& &' (9 9X( K=:
2 2
2 2
( ,0)
1
( ,0)
A ax y
B aa b
+ =
1 :
a
L x
e
=" =^ &' ( $%1 ( ,0)F c=
2 :
a
L x
e
=" =^ &' ( $%2 ( ,0)F c=
a
d c
e
=' =^ " / &' ( $% , -K= .
2F
1F
1L2L
BA O1F
2Fa
c
a
c

49
& ' ( )
*:+,:49
2 2
2 2
( ,0)
1
( ,0)
A ax y
B aa b
=
1 :
a
L x
e
=" =^ &' ( $%1 ( ,0)F c=
2 :
a
L x
e
=" =^ &' ( $%2 ( ,0)F c=
a
d c
e
=K= .' =^ " / &' ( $% , -
" = %–&' (:
;P5 & . &e. *= ( )+ "A R U %PF, -
P5 /PQ, -P< . ' D. E* .' )*+ " =^ &' ( $% /.
PF e PQ=
) '. .'.< . ' )+ K= L5+ ;:
2 2 2 2
2 2 2 2
1 1
x y y x
a b b a
+ = =
( )
2
2 2 2
2
b
y a x
a
=
2 2
1
2
2 2 2 2
1 2
( )
2 ( )
PF x c y
b
PF x cx c a x
a
= ± +
= ± + +
2 2 2 2
2 2 2 2 2 2
1 2 2 2
(1 ) 2 ( ) 2 2
b a b c
PF x cx a x cx a x eax a
a a a
= ± + = ± + = ± +
( )
22 2 2
1 2PF e x eax a ex a ex a ex a= ± + = ± = + = +
12 12 12 12( , ) ,
a a a
Q y PQ x x PF e PQ
e e e
= ± = + = + =
)+ 9 9X( L5+ ;.
( )
2 2 2 2 2
2 2 2
2 2 2 2 2
1 1
x y y x b
y x a
a b b a a
= = =
2 2 2
1 2, ( ,0) , ( ,0)c b a F c F c= + = =
B
2F1F
2L1L
A
c
a
c
a
1L 2L
BA O1F
2F
1Q ( , )P x y

! ::50
! " # $" %
2 2
2
2
2 2 2 2
2
( )
2 ( )
PF x c y
b
x cx c x a
a
= ± +
= ± + +
( )
2 2
2 2 2
2
( ) 2
a b
x cx c b
a
+
= ± +
( )
2
22 2
1 2
2
c
PF x eax a ex a
a
= ± + = ±
( ) ( )
1 12 12( , ) ,
a ax xa ae e
PF ex a Q y PQ x
a ae ex x
e e
= ± = ± = ±
± ±
12 12PF e PQ=
" = % `5–&' (:
1G ')e=[4 , - 0)+ 0' 'P/F)"(, - & HP
$%)&' ((L + =aFD. E* .':
P9.P; & .1e =
b.P; K= & .0 1e< <
U.P; 9 9X( & .1e >
D 1:P9 9 Z-0) A ) . *= ( 9' : d " S ).
b 9 Z-K=
0 1
PF e PQ
P
e
=
< <
2 2
2 2 2 2
( )
( )
PF x d y
x d y e x
PQ x
= +
+ =
=
2 2 2 2
(1 ) 2 0e x dx y d+ + =
2 2 2
2 2 2 2 2 2
2 2 2 2 2
2
(1 ) (1 ) ( )
(1 ) (1 ) (1 ) (1 ) (1 )
dx y d y d
e x d e x d
e e e e e
+ = + = +! !
" # " #
2
2 20 1 2
2 2
2 2 2
2 2
2 2
( )
1( ) ( ) 1
1 1 1 ( ) ( )
1 1
e
d
x
d y ed yex
ed ede e e
e e
< <
+ = + =
B
2F1F
2L1L
A
( , )P x y
2Q1Q
( ,0)F d
( , )P x yQ

51
& ' ( )
*:+,:51
x
( , )P x y
x
x
y
y
y
y
x
$
U 9 Z-
2
2 2
2 2 2
( ) ( )
1 1 1
d y ed
x
e e e
+ =
2
2 21 2
2 2
2 2 2
2 2
2 2
( )
11:( ) ( ) 1
1 1 1 ( ) ( )
1 1
e
d
x
d y ed yee x
ed ede e e
e e
> +
> + = =
" . ':
cos( )
:
sin( )
x op
op r
y op
oxp
$
$
% = +
=
= +
cos( )
:
sin( )
x r
y r
ox p
$
$
%
= +
= +
cos( ) cos cos sin sin
sin( ) sin cos cos sin
x r r r
y r r r
$ $ $
$ $ $
= + =
= + = +
" . ' ` /
cos sin cos sin
sin cos sin cos
x x y x x
y x y y y
$ $ $ $
$ $ $ $
=
=! ! != + " # " # " #
f ' 7.' & ( * *:
2 2
0Ax Bxy cy Dx Ey F+ + + + + =
;0B &"A < . & )+ '. ) 4 =a ) / 8 - 9' : )+
<=* L g ..
2 2
( cos sin ) ( cos sin )( sin cos ) ( sin cos )
( cos sin ) ( sin cos ) 0
A x y B x y x y C x y
D x y E x y F
$ $ $ $ $ $ $ $
$ $ $ $
+ + + +
+ + + + =
2 2
0A x B x y C y D x E y F+ + + + + =
2 2
cos sin cos sinA A B C$ $ $ $= + +
2 2
(cos sin ) 2( )sin cosB B C A$ $ $= +
2 2
sin sin cos cosC A B C$ $ $ $= +
cos sinD D E$ $= +
sin cosE D E$ $= +
F F=

& ::52
! " # $" %
0 cos2 ( )sin 2 cot 2 2 arctan
A C A C
B B A C
B B
$ $ $= = = =
GI.)=* < . . . & 9' :.2 2
2 8 8 0x xy y x y+ =
'. ) 4 =a:1 , 2 , 1 , 8 , 8 , 0A B C D E F= = = = = =
1 1
2 arctan 0 2
2 2 4
' '
$ $ $= = = =
' 2 2 2 ' ' 2 2 22 2 2 2 2 2
( ) 2( ) ( ) 0 , 0 , ( ) 2( ) ( ) 2
2 2 2 2 2 2
A B C= + = = = + + =
' ' '2 2 2 2
8 8 8 2 , 8 8 0 , 0
2 2 2 2
D E F= = = = =
2
2 2
2
4
2 8 2 0 4 2
4
y cx
y x y x
x cy
=
= =
=
( ) ( )2 0 , , 0,0c = > =
( )
2
2 1
2
2,0
2
2 1
2
x
F F
y
= × =
=
= × =
1
1, 1 1( 1) 2 2
1
Lm y x y x x= = + = + = ) =
=K3.R U %e=^ &' ( $% / " , -d.' 9' : & . '
.' B 3 1 3 D E4&' ( 1 3 . "L% 4 7 'U.L)+ 4+ '.
1 cos
ed
r
e $
=
:PF e PQ P=
( )cos cosr e d r ed er$ $= + = +
(1 cos )
1 cos
ed
r e ed r
e
$
$
= =
I G.
9
5
41 cos
5
r
$
=
29
95
4 41
5
ed a
d c
c ce
a
=
= =
= = <
F
Q ( , )P r $
Q
d
O F=
r
$
$
L
cosr $

53
& ' ( )
*:+,:53
2 2
9 25 9 94
5 4 16 16 4
a c
a c c c
c c
= = × =
4
25 16 3
5
c
b
a
=
= =
=
2 2
( 4)
1
25 9
x y
+ =
<( " ' , -2c)+
*:8 2 4= ×T.R 3 2 . ' , -2aS 32b)+.
0
0
F7 / 1 3 R " (
4c =' + C==:/ R.
( !0 ) . ;2cD. C .' <=(' " . ' & ( ij% " ' .
)+ D. "A 9' :.
1 cos
ed
r
e $
=
( )2 1 sin1 cos
2
ed ed
r
ee
'
' $$
= = =
( )1 cos 1 cos
ed ed
r
e e
'
$ ' $
= = =
+
( )2 1 sin1 cos
2
ed ed
r
ee
'
' $$
= = =
++
I G.)=* < . . "A 0'. A ) . /. ' 9' :.
10
:
1 sin
r
$
=
+
0 ) " . '
2
'< . ':
( )
10 10
2 1 cos1 cos ( )
2
r r
'
' $$
= = =
1 : 10 10e ed d= = =
. & . " " Sx. . `O - `O ()+.
2 2
4 5( 5) 20( 5)y x y x= × + = +
2 2
: 4( 5)( 5) 20( 5)x y x y= =4- " . ')C= O(
L
( )9,0B( )1,0A =
1(0,0)O F=
2 (8,0)F(4,0)
(-5,0)
L: x = -10
x
y
O=F
d=10

55 ::55
TNB
.
op p xi yj zk= = + +
:
2 2 2
P x y z= + +
:
P
UP
P
=
1
PP
UP
P P
= = =.
1 2 3A a i a j a k= + +1 2 3B bi b j b k= + +
1 1 2 2 3 3( ) ( ) ( )A B a b i a b j a b k± = ± + ± + ±
1 2 3:c cA ca i ca j ca k= + +
2) ( ) ( )A B C A B C+ + = + +1)A B B A+ = +
! " #)%&: (
. cosA B A B=
( )*+,-(AB.0 < </ --01 cos 1< </ --0.
. . . 0i j i k j k= = =,. 1 . .i i j j k k= = =,
2
1) .A A A=
2
A B= = =2) . .Ab B A=
3) .( ) . .A B C A B AC+ = +
2 2 2 2
1 2 3.A A A a a a= = + +
1 1 2 2 3 3.A B a b a b a b= + +1 1 2 2 3 3. ( . ) ( . ) ( . )A B a b i i a b j j a b k k= + +
A
B
i y
z
x
j
k
P(x,y.z)

::56
" # $ %& '& # ( ( ) # ) %&TNB
,1:
,1BA2B
AOH proj=
0 < <
2
:/ 34A ,B
Aproj
< <
2
: / 34 56& A , B
Aproj
0
2
B
Aproj= =
cos2cos
cos
2
B
A
B
proj OH B
B
< =
= =
=<
cosB
Acomp B=7 89,BA
( ) . .
cos
.
B B
A A
A A B A A B
proj B proj A
A A A A A
= = =
.
.
A
B
B A
proj B
B B
=
: ;.2 3B i j k= +, * 34 ,<7 7,<= > ,1
3A i j=.
. 5 3 1
( ) (4 )
. 10 2 2
B
A
A B
C proj A i j i j
A A
= = = =
3 1 1 3
(2 3 ) ( ) 3
2 2 2 2
D B C i j k i j i j k= = + = +
. 0D C D C =
: ;.?8@ , ,<7 A B ,% 4.
?
,
D A
D
D B C
*/ C,D B C& D- / 2,B C:
( 2 ) ( 2 ) (1 ) (2 ) (1 2 )
. 0
D B tC D i j k ti tj tk D t i t j t k
D A D A
= + = + + + = + + + +
=
2(1 ) (2 ) (1 2 ) 0 1 1t t t t t D j k+ + + = = = = +
O
A
B
H
B
A
H
A
B
B
AprojC =
D

57 ::57
& " #:
( )sinA B A B n× =
(-,B A,n*( 1)n =2F , G
/ - H F.
,FA B n×,FB A n×
( ) , ( ) , ( )i j k j i j k i k j k i j k i× = = × × = = × × = = ×
0 ( )i i j j k k A B B A× = × = × = × = ×
( !.2 2 2 1 1 1,B a i b j c k A a i b j c k= + + = + +3 > ,1/ 2 /:
( ) ( )1 1 1 2 2 2A B a i b j c k a i b j c k× = + + × + +
1 2 (
k
a b i j= × 1 2) (
j
a c i k+ × 1 2) (
k
b a j i+ × 1 2) (
i
b c j k× 1 2) (
j
c a k i+ × 1 2) (
i
c b k j+ × )
1 2 1 2 1 2 1 2 1 2 1 2( ) ( ) ( )b c c b i a c c a j a b b a k= + +
1 1 1 1 1 1
2 2 2 2 2 2
b c a c a b
i j k
b c a c a b
= + +
1 1 1
2 2 2
i j k
A B a b c
a b c
× =
( !4,B A76#J ,) A B > ,1/ /)B4 K 8( LF,
, M-*N,B A2:
:sin sin
h
h B
BOBH = =
sinS A h A B A B= = = ×A BO6P9 ,)
( )
( ) ( ) ,
A B B A A B C A B A C
A B C A B C A B A A B B
× = × × + = × + ×
× × × × × ×
O
A
B
H
A
B
O

% ::58
" # $ %& '& # ( ( ) # ) %&TNB
: ;.42 2 ,B i j k A i j k= = + +B ,% 2 /# 8:
9(A B×"(( )A B
A BPorj +
×Q(, ?
2
C A B
C
C
×
=
=
9-
1 1 1 1 1 1
1 1 1
2 1 2 1 2 2
2 2 1
i j k
A B i j k× = = +
( 1 2) ( 1 2) ( 2 2) 3 4i j k i j k= + + = +
"–3 , 3 4A B i j AB i j k+ = = +
.
.
B
A
A B
Porj A
A A
=
0
( )
( )
( ).( )A B
A BPorj
A B A B+
× =
+ ×
( ) 0.( ) 0
( ).( )
A B A B
A B A B
× = × =
× ×
Q–
3 4 3 4
2( ) 2
1 9 16 26
CA C A
A B A B
A B i j k i j k
u u C C
A B
=
× ×
× + +
= = = = =
× + +
PS ?8@ 9:
0 0 0 0A A x x i y y j z z k= + +
0 0A P A A P n A A
0 0 0 0.( ) 0 ( ) ( ) ( ) 0n A A a x x b y y c z z= + + =
0 0 0
d
ax by cz ax by cz ax by cz d+ + = + + + + =
PS L& 9:
> # 80 0 0 0( , , )
:
A x y z L
v L v ai bj ck
=
= + +
0 0 :A L A A v A A tv t=
T D # * /)B4 ,
.
P
( , , )A x y z
n ai bj ck= + +
0 0 0 0( , , )A x y z
( , , )A x y z
0 0 0 0( , , )A x y z
L
v

59 ::59
V 9L& ):
0 0
0 0
0 0
:
x x ta x x ta
L y y tb y y tb
z z tc z z tc
= = +
= = +
= = +
0 0 0 0( ) ( ) ( )A A x x i y y j z z k
tv tai tbj tck
= + +
= + +
, 9:0 0 0x x y y z z
a b c
= =
: ;.C ! W. ?8@ 9(0, 1,1), (2,0,2), (1,1, 1)C B AF.
- : ! ?8@ 9 () ,3:
B/ / ,H7 ?8@ ! F */ C
H ).
B/ / ,H7 ?8@ /- ! F */ C
).
3 , 3 2AB i j k AC i j k= + = +
1 1 3 ( 2 9) (2 3) ( 3 1)
1 3 2
i j k
n AB AC i j k= × = = + + +
7 5 4 : 7( 1) 5( 1) 4( 1) 0 7 5 4 6n i j k A P x y z x y z= + = =
: ;.?8@ 9C ! W.,B AF ?8@ ,<7.
(1,2,3) (3,2,1) : 4 2 7A B P x y z= = + =
?8@ */P?8@ ,<7QXV FPQ n
F.D #, ,z y xF : > 1)Y.
Q P
P
AB Q
n AB n Q
n Q
= ×
, ?8@ )G.M& ,
<4 Z/B<4 : !) ?8@
2 0 2 2 12 2
4 1 2
Q
i j k
n i j k= =
AB AC n× =
B
P
C
A
PnQn
Q
P
B
A Qn

* ::60
" # $ %& '& # ( ( ) # ) %&TNB
2 2
2( 1) 12( 2) 2( 3)
4 2P
AB i k
A Q Q x y z
n i j k
=
=
= +
: ;.! %@ S(2, 3,4)A?8@: 2 2 13P x y z+ + =F.
.A,P7P> 1)YAP/ [ @2.
[ @APF 8@ %@ S F:
(1,1,4) , 2 2 , 4PB n i j k AB i j k= + + = + +
B- 8@ %@ S
!B?8@ F -)& !PXV G
1)YP/ [ @.AH, F
Pn2AH,1ABF.
. 1 8 2 9
3
31 4 4P
PAB
n
P
AB n
d Porj
n
+ +
= = = = =
+ +
! %@ SA?8@
.. . .
.
A A
B B
A BA B A B B A B
Porj A Porj
B B B B B B
= = = =
2 2( 3) 2(4) 13 9
3
31 4 4
d
+ +
= = =
+ +
](2 -/ ^%@ S!0 0 0 0( , , )A x y z?8@: 0P Ax By Cz D+ + =
F.0 0 0
2 2 2
Ax By Cz D
d
A B C
+ +
=
+ +
.. PB
A
P
AB nA B
Porj
B n
= =
0 0 0
2 2 2
( ) ( ) ( )A x x B y y C z z
A B C
+ +
=
+ +
0 0 0
2 2 2
( )Ax By Cz Ax By Cz
A B C
+ + + +
=
+ +
0 0 0
2 2 2
Ax By Cz D
d
A B C
+ +
=
+ +
P
Pn
Pn
H
B
A
P
( , , )Pn A B C
H
( , , )B x y z
0 0 0 0( , , )A x y z
:P Ax By Cz D+ + =

61 ::61
: ;.%@ S(1,2,3)AF L&.
G .A LtC ! <4 3-4 !
A*/ ,HL> 1)Y XV F
L/ [ @.0 0 0 0: (1 ,3 2 , 2 2 )t H t t t! +
0 0 0 02 2 : (1 ,3 2 , 2 2 )v i j k t H t t t= + + ! +
0 0(1 2 ) ( 5 2 )AH ti t j t k= + + +
0 0 0 0 0
4
. 0 2(1 2 ) 1( 5 2 ) 0 9 12
3
AH v AH v t t t t t= + + = = =
0
4 4 5 7 16 25 49
, 10
3 3 3 3 9
t AH i j k d AH
+ +
= = = = =
: ;.L& %@ S
1
: 2
3 2
x t
L y t
z t
=
=
= +
?8@: 3P x y z+ + =F.
(1,1,1) ( 1, 1,2)P L Pn v n AH= = =
(0,1,2)A P
0 0 0:(1 ,2 ,3 2 )H t t t L+
0 0 0(1 ,1 ,1 2 )AH t t t=
(1,1,1) 3AH AH= =
0 0 0 0 0 0 0 0(1 )( 1) ( 1)(1 ) 2(1 ) 1 1 2 4 6 0 0AH v t t t t t t t t+ + + = + + + + = = =
](&C,2 1,L L. -/ ^ /# 8
1 2 1 2
1 2 1 2
1 2 1 2
0
a a x x
b b y y
c c z z
C, &2 1,L LS /).
2 2 1 1
2 2 2 1 1 1
2 2 1 1
,
x x a t x x a t
L y y b t L y y bt
z z c t z z c t
= + = +
= = + = = +
= + = +
: ;.>JF ?8@ a )N M1S ) V.
2
,
1
x y
x t
x y z
+ =
=
+ + =
V
L
H
A ?
P
Pn
A
HL Ln

, ::62
" # $ %& '& # ( ( ) # ) %&TNB
0 0
2
: 2 2
1 1 (2 ) 1
1 1
x t
y t
L y t A L v i j
z x y t t
z t
= +
=
= =
= = =
= +
: ;.>JF F L& ( (- %@ S -/ (-- ,Y9 S /) L& ) V
.
1 2 3
x y z
= =
1 2 1 2 3
2 1 1 2 1 1
x y z
= =
" #7 . F:
K S-/, ,A B C> ,1/ / F( ).A B C×/ ,. 7 . F " #.
( ). cos ,A B C A B C A B C× = × =< × >
1 1 1 3 3 3 1 1 1
2 2 2 1 1 1 2 2 2
2 2 2 3 3 33 3 3
.( ) ( ).
A a i b j c k a b c a b c
B a i b j c k C A B a b c a b c A B C
a b c a b cC a i b j c k
= + +
= + + × = = = ×
= + +
,1O6#J ,):
,)O6#JPQRSPS
O6#J ,) ( . F K 8
?8@P* :
n( 1, )n n P=.
> ,1/3-4 ,&
O6#J ,) ,1 A B
P,) A B DBA
O6#JPQRS( 3 F
,b/:
( )
( )
A PQ
n P PQRS A B
B PS
A P Q
n P Q R S P Q R S A B
B P S
=
= ×
=
=
= ×
=
x
y
z
Q
S
P
R
Q
S
P
R
A
B
A
B
A
B
A
B
xyn

63 ::63
1 2 1 2( )( ) ( )( ) ( ) ( )A B SS A R R PP B S S t n A t n B t n B t A n A B× = + + + + = + + = × + × + ×
( ) ( )1 2SS RR n SS RR t n PP S S n PP S S t n+ + = + + =
1( ). (A B n t n B× = × 2) (t A n+ × ). ( ). ( ). cos0n A B n A B n A B n A B+ × = × = × = ×
( ).A B n= ×,1 O6#J ,) A BP Q R S
: ;.(1,20,), (1,0, 1), (2, 1,4)C B A,1 A B 2 O6#J ,) d F( )ABCD
, ,yz xz xyF.
AD AC CD AD AC AB= + = +
( ) ( )
( 2) ( 1) ( 4)
3 4 3 5 2 4 9
AD x i y j z k
AC CD i j k i j k i j k
= + + +
+ = + + + = +
2 2 0
1 4 3
4 9 5
x x
AD AC CD y y
z z
= =
= + + = =
= =
1 1 5
( ) ( ). | 1 3 4 | 3 1 2
0 0 1
n k
AB CD kS =
= × = = + =
1 1 5
( ) ( ). | 1 3 4 | 4 5 1
0 0 1
n j
AC CD jS =
= × = = =
1 1 5
( ) ( ). | 1 3 4 | 4 15 11
0 0 1
n i
AC CD iS =
= × = = + =

::65
:
3
:
( ) ( ) ( ) ( )
F
f t x t i y t j z t k= + +
( )
( ) ( ) ( ) ( ) ( )
df t
v t f t x t i y t j z t k
dt
= = = + +
2
2
( )
( ) ( ) ( ) ( ) ( )
d f t
a t f t x t i y t j z t k
dt
= = = + +
( !( )f t!" # $ !" % &'
( ) * # &' + , %( )f t-(. / ' + ,( )R t
01 2.( ) ( ) ( ) ( ) ( )f t R t x t i y t j z t k= = + +
4 5( )R t4 6 $ ! 78 9 :( )f t;$ 9 : !7 ' 5 9 :( )f t9 : $
1 2 <$.
!(+4
2
2
( )
( ) ( )
d R t
a t a t
dt
=$+4
( )
( ) ( )
dR t
v t v t
dt
=
( !2 1,F F) * # 0 =1 /$ >:
1 1 1 1 2 2 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )F t x t i y t j z t k F t x t i y t j z t k= + + = + +
? @*1 2 1 2 1 2( ( ) ( )) ( ). ( ) ( ). ( )
d d d
F t F t F t F t F t F t
dt dt dt
+ = +
@*1 2 1 2 1 2( ( ) ( )) ( ) ( ) ( ) ( )
d d d
F t F t F t F t F t F t
dt dt dt
× = × + ×
; A.+ , 70 / 'B C>*)( ) ( sin ) ( cos )R t t t i i t j tk= + +
$- DC.
( ) (1 cos ) (sin ) ( ) (sin ) (cos )v t t i t j k a t t i t j= + + = +
( ). ( ) 0 (1 cos )(sin ) (sin )(cos ) 0 sin 0a t v t t t t t t t k k= + = = =

12
67
!"# $% & %# '# # ( ) * $+ , % -
#:. /0.:67
; A.+ , 70 / '+4 ! 7 $ ! G H 4 ) B C>*
I 1- DC.( ) cos 2sinR t ti tj= +
&7
2 2
2 2
cos
1
1 2sin
2
x t
x y
y
t
=
+ =
=
( ) sin 2cos , ( ) cos 2sinv t ti j a t ti tj= + =
2 2 2
2
6sin cos
( ) cos 4sin 1 3sin ( ) ( ) 0 6sin cos 0
1 3sin
t t
a t t t t f x a t t t
t
= + = + = = = =
+
2
3(sin ) 0 2 ( )
2
k
t t k t k= = =
2
2
2
3sin 2
2cos2 1 3sin sin 2
3 2 1 3sin( )
2 1 3sin
t
t t t
ta t
t
× + ×
+= ×
+
5k!71 H...,6,4,2,01 J$4 $( ) 3a t =G5 ! 7 KL $ MA
!71 ' H...,5,3,1! G >0 1 ':
: min : max
2 2
k k
k E t k O t= =
&' !" N 6 ; O:
!"( ) ( ) ( ) ( )R t x t i y t j z t k= + +PH 4 !" $ 5 70 / ' /$ >AB
!70a t b4 N 6 ; O !71 " 0 77QAB! $-:
2 2 2
1 ( ) ( ) ( )k k k k kP P x y z+ + +
1 1
2 2 2
1 ( ) ( ) ( )
0 0
:
n n
k k k k k
k k
V L P P x y z+
= =
= = + +
0 @7 07 @ M $ = 41( )k kP P+ =
( )A R a=x
z
y
( )B R b=
1P
2P
1nP
( , , )k k k kP x y z
1 1 1 1( , , )k k k kP x y z+ + + +

::65
( )x t
( )y t
2 ( )R t
1
2 2 2 2 2 2
( ) ( ) ( )
0
: lim ( ) ( ) ( ) ( )
b bn
k k k k k k
n
k t a t a
dx dy dz
V L x y z L v t dt
dt dt dt= = =
= + + = + + =
R.( ) ( )
b
t a
s t v t dt
=
=!70 R ) B0:
0 0 0( ) 0 , ( ) 0 , ( ) 0s t t t s t t t s t t t= = < < > >
( )s t!7 5 !" $ S T* '.( ) 0 ( )
ds
v t v t
dt
= =
$ N R( )T:
( ) ( ) ( ) ( )R t x t i y t j z t k= + +
2 1
1 2
2 1
( ) ( )
( ) ( )
R R t R t
R t R R t
t t t
=
+ =
=1
1( ) lim ( ) t t
R R
v t R t
t t =
= =
( )
( )
dR
dR v tdtT
dsdt v t
dt
= = =
C>* !" 5 7 ":
( ) ( ) ( )R t x t i y t j= +
$ N !70 C>* 6 $ UL 9 : !" % $ 6$( )T. !" 6$ 1
" I !".2" I V01-( )T75 +4$ 4 77Q)$ 4Ti
4(! 75 +4.PH 1P!" 6 $
df
ds
- !" 5 7 "(!7 5 PHS
)L(!71 2.
d
dt
=
( )y t
( )x t
1( )R t

12
69
!"# $% & %# '# # ( ) * $+ , % -
#:. /0.:69
( ) ( ) ( ) ( )R t x t i y t j z t k= + +
1
tan tan
dy
dy ydt y x
dydx x
dt
= = = = =
1
2
(tan )
1
u
u
u
=
+
2
2
1
2 2 2
1
( )
1 ( )
( )
y x x y
yd x
d dt x
dsds
dt x y
×
+
= =
+
22
2
2
( ) ( )
d y x x y
ds x y
x
x
=
+
1
2
1
2
2 2
2 2
( )
( )
d x y y x
ds x y
x y
= =
+
+
( ) ( ) ( )
( ) ( ) 0 ( )
( ) ( ) ( )
( ) ( ) 0
i j k
v t x t i y t j
v a x t y t xy yx k
a t x t i y t j
x t y t
= +
× = =
= +
2 2 3
, ( )
v a
v a x y y x v x y v
v
×
× = = =
; A.PH 4 !"
2
t =$-.( ) cos sinR t a ti a tj= +
( ) sin cos , ( ) cos sinv t a ti a tj a t a ti a tj= + =
2 2 2
( ) cos sin , sin cos 0 (sin cos )
cos sin 0
i j k
a t a ti a tj v a a t a t a t t k
a t a t
= × = = +
2
0 3
1v a a
v a a
×
= = =
( !Y + 5 7 "aZ H PH 1
1
a
.
[(>* PH 1 " % 5 7 " 70 Z.
, ( ) ( )y mx b x t y mt b R t ti mt b j= + = = + = + +
3 2 2
( ) 0
0
( ) 0 (1 )3
v t i mj v a
a t v m
= + ×
= = =
= +
( )R t P=
T
T

::65
x
z
y
( !7 &' 6 $ 1 !"P 4 PH 1 5 7 " G3
v a
v
×
-.
; A:3
v a
v
×
!" 5 7 "
( ) cos sinR i ti tj tk= + +$-.
( ) sin cos ( ) cos sinv t ti tj k a t ti tj= + + =
sin cos 1
cos sin 0
i j ki
v a t t
t t
× =
( sin ) (cos ) (sin 2 cos4)i t j t k t= + +
sin cosv a ti tj k× = +
2 2
2 2
sin cos 1 2
sin cos 1 2
v a t t
v t t
× = + + =
= + + =
3 3
2 1
2( 2)
v a
v
×
= = =
( +:
( I PHv a×PH # C>* !" % 6 $v a×!" ]>* +
? P:
R-PHv a×N !".
-PH !" 5 7 "v a×+ 5 7 "
.
J-!" H S + # G.
2 2
: ( )y x R t ti t j= = +
1
=5 7 " Y)+ 5 7 "(
5 7 " G:G( +.
C>* $ !_ 6:
NTH S (- S
% (- +4 $.
1
P
k
=
N T

12
71
!"# $% & %# '# # ( ) * $+ , % -
#:. /0.:71
( ) ( ) ( )R t x t i y t j= +
2 2
cos sin sin cos cos sin 1
dT dT
T i j i j T T
d d
= + = + = + =
dT dT d dT dT d dT d
dS d dS dS d dS dS dS
= = = =
!(N- S $ +»H)!"(.
R.$ !_ 6:
dT dtdT dT dt
dt dsds dt dsN
dT dt
ds ds
= = =
dT dt
dt ds
dT
dtN
dT
dt
=
!(5( )R t( !" $ PH 1 ) B0 &' !" %N)$ !_ 6(
? P R:
dT dT
dt dtN N
dT
dt
= =
R.B T N= ×4 % (- +4 $ !7 <$ !_ 6:
sin 1
2
B T N= =
R.
dB
dS
=!"!7 5.
C>* $ " $B77Q
>* (- $ 0.
( ) ( ) ( ) ( )R t x t i y t j z t k= + +
2
0
x y z
x y z
x y z
v a
= ±
×
x
z
yT
B
N
( )R t
(
8
!_ 6

::65
R.70 / 'P!" 6 $ PH( )R t) B0:
R-@ C>*P(- ; $ +B!7 5 ( C>*.
–@ C>*P(- ; $ +NC>*!7 5 (.
J–@ C>*P(- ; $ +T!7 5 ( C>*.
; A.4 !"/$ >( ) cos sin 2t t t
R t e ti e tj e k= + +1, ,B N T
5 7 " H $-!_ 6 $ 8 . ( ) C>* #70b 1 $ $- DC
2
t =$-.
, ( ) (cos sin ) (sin cos ) 2t t tv
T v t e t t i e t t j e k
v
= = + + +
2 4
( ) [(1 2sin cos ) (1 2sin cos ) 2] 4 2t t
v t e t t t t e et= + + + = =
1
[(cos sin ) (sin cos ) 2
2
T t t i t t j k= + + +
( )
1
( sin cos ) (cos sin)
2
1 2
1 2sin cos (1 2sin cos
2 2
dT t t i t jdT dtdtN
dT
dTdc t t t t
dt
= + =
=
= + + =
1
( sin cos ) (cos sin )
2
N t t i t t j!= +" #
1 1
cos sin sin cos 2
3 2
sin cos cos sin 0
i j k
B T N B t t t t
t t t t
= × = × +
1
2(sin cos ) 2(sin cos ) (1 1)
2 2
t t i t t j k != + + +" #
[ ] [ ]3
( ) (cos sin ) (sin cos (sin cos )(cos sin ) 2t t tv a
a t e t t t t i e t t t t j e k
v
×
= = + + + +
0 0 0
2 , 2 2 , 2 ,t t t
v i j a j k v= = =
= + + = + =
0
1 1 2 ( 2) 2
0 2 2
t
i j k
v a i j=
× = = +
( )( ) 2 sin 2 cos 2 , ,t t
a t e t e t etk x= + + &

12
73
!"# $% & %# '# # ( ) * $+ , % -
#:. /0.:73
Gd:
0
( ) ,
t
t
dR
dR dR dS dR vdtv v T St v u du T
dtdt dS dt dS v
dR
= = = = = = =
dT dT
dT ds dT dtdS dSN kn
dTdt dt ds ds
ds
= × = = =
2 2
2 2
( ) ( ) ( ) ( ) ( )
dv d d dS d dS dT d S dT
a v T T T T
dt dt dt dt dt dt dt dt dt
= = = = + = +
2
2
2
( ) ( )
d S dS
a T N
dt dt
= +
<$ S 1 $:
1.$ &7:
2 2 2
2 2 2
( , , ) : 1
x y z
a b c
a b c
+ + =
2 2
2 2
0 1 :
x y
z
a b
= + =
2 2
2 2
0 1 :
y z
x
b c
= + =
2 2
2 2
0 1 :
x y
y
a c
= + =
2 2 2
2 2 2
2 2 2 2
2 2 2
1
:
k x y
z k
c a b
k c x y
c a b
= =
=
2 2 2 2
2 2 2
:
: (0,0, )
:
c
k c
k c t
x y c k
k c
a b c
> = '
=
< = + =
x
z
y
N
T
B
x
z
y
(0,0, )c
(0,0, )c
(0, ,0)b
( ,0,0)a(0, ,0)b
( ,0,0)a

::65
2.&7 $:
2 2
2 2
( , 0, , , , , , 0): 1
x y z
a b a b c a b c
a b c
> ( + + =
2 2
2 2
0 0 :
x y
z
a b
= + =
2
2
0 :
c
x z y
b
= =
2
2
0
c
y z x
a
= =
2 2 2
2 2 2
2 2 2 2
2 2 2
:
x y k
z k
a b c
k c x y
c a b
= + =
=
?:5c0 5 $ $ #.
3.&7 f$ g:
2 2 2
2 2 2
( , , ):
x y z
a b c
a b c
+ + =
2 2
2 2
0 0 :
x y
z
a b
= + =
2 2
2 2
0 :
y z c
x z y
b c b
= = = ±
2 2
2 2
0 :
y x c
y z y
c a a
= = = ±
2 2 2
2 2 2
k x y
z k
c a b
= = +
4.I i? $ U1:
2 2 2
2 2 2
( , , ) : 1
x y z
a b c
a b c
+ =
2 2
2 2
0 1 :
x y
z
a b
= + =
2 2
2 2
0 1 :
y z
x
b c
= =
2 2
2 2
0 1 :
z x
y
c a
= =
x
z
y
0e <
0e >
0z k= >
0z k= >
z k=
z k=
x
z
y
x
z
y
0z k= >
0z k= <

12
75
!"# $% & %# '# # ( ) * $+ , % -
#:. /0.:75
2 2 2
2 2 2
1
x y k
z k
a b c
= + = +
5.I L $ $ U1:
2 2 2
2 2 2
( , , 0) : 1
x y z
a b c
a b c
> + =
2 2
2 2
0 1 :
z y
x
c b
= =
2 2
2 2
0 1 :
z x
y
c a
= =
2 2 2 2 2
2 2 2 2
2 2
2 2
2 2 2
2 2 2
1
0 0
1 0
x y k k c
k c
a b c c
x y
z k k c k c x y
a b
x y r
k c
a b c
> + = =
= = = = = =
= = >
6.U1 $:
2 2 2
2 2 2
( , , 0) : 1
x y z
a b c
a b c
> + =
2
2
0 :
c
x z y
b
= =
2
2
0 :
c
y z x
a
= =
z k=
x
z
y
(0,0, )c
(0,0, )c
z k=
x
z
y
0 4 PH

::65
. ) B g&' $ $:
) B g:
) B g:
, 0
, ,0 2
,
r op r
op i
z z z
) )
= >
< >
=
: cos cos
:sin sin
x
oxp x r
r
y
oyp y r
r
)
) )
=
= =
B g @ M + 8):
2 2
arctan( ) sin
r x y
y
r
x
z z
)
= +
=
=
cos
sin
x r
y r
z z
)
)
=
=
=
x
z
y
( , , )P x y z
( , )P x y
( , , )P * )
y
x
z
P
)
r
*

12
77
!"# $% & %# '# # ( ) * $+ , % -
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B 1L
2L
A
C
D
1v
2v
d

Match II (armand)

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