Upload
dangtuyen
View
212
Download
0
Embed Size (px)
Citation preview
riTrauorfitdfiu f @) LLavlFr c
f (*) tugrJroror4n:l rir#r6
rur'Lri' r:rfrornr:lfielu
"n^JonL
7 - (-R,R) riu6o?{SJtJFr?1t
.f(x) - Z6^.-c)",=riKo*"ril -- If lo u"eUe;f s . . ' ' " 'y le , , €e*r j (o- c= o l
@
f (x) -Zon*" uudrtn=0
f (x) - g+ 1E+ =2 + gs!-3 + o.4*4 +... + anxn + . ..
f'(x) - oi+ Terx * \e.r { (( , c I ' c r ' c e " ' : - S- : " - -
t: It ?f'(x) = 7i, )- \.'z(t
:Layx { { ; \{ ; \ eu x- { - . " -6
:) f(0)_ a"G{ x=o
+ f'(01 - a,"3
f'Q) -z..z\ = o=> f'(0) -31 ar
:\=o
, r(n\ rr ' . , , I- J
'w) =v ' t \ Q,a
vo , .
t/
t
( 7 L t i3
@u,r l t Y
f ' ( * ) = !21 . I q . . \ . 2 c * v 1 . . . .lt,
7@74 - . { . \ .2 { + s . . { . \ Z cgx t . . . * -3
f@e)- , {Jo.+:==--
. = g l a . t
{ ( ,^rr) n C*-r? - 'Z
+ . . .
s i+ r=L
22
t:14{Lfln?1
ao - €co)alrlt r
C l a = \ L 9 /
\ t .
s o i u ,!9luVl? LU an =
'Luliruorrdrn-u fr'r@
lv[!LA? an =
C,GI
f cu/ro)
cl f.
n=0,1,2,3, . . . r iu6o
;Ce^la-
o r = U =L
a l =
Q 4 =
C'(o)
z l
f ( x ) - l an@-c) " qn 1 xe (c -R ,c+R)n=o
;;-'co"'lo'',r") g)
n =0,1,2,3,... r iun'o
23
6l ,2a n 4t e,{ de! 4. er d a.t v V etA114 J rUU?\{ntu{{141oql\Uf,ilnoUF|ULFlVt!Fl c
oun:ilri1a'{tu:rjq U
d t 6 d ^r:un?104n:ilrvrsma:r0{ J souno c
freaeirrfi rz a{uloqnturyeiraoirot f (x) - 2t
c-=2
1x
:OUAFIc
r t r t l ,
LLAuUllt?{ ttu{n1tA Lt1U
ael o
?0v11
f (x) -x- r =
f '(x)- - f z
f '(x)- 2;). - t 6 d- '->
. f ' (*)-(-s)2x'= x-; f*(2)- - ;--.-t
f(4) (O= (-qX-\) ?;t - f(4) e)= q I
7_)! = q l x s 2 : \ z
f(") (*)=Q ,)'. n I i "-'
,rD p)- (r )1 ,n I
.l.X
I= --;-
x'L
x )
Sal r: Z
tf(2)-
f'(2)=
"f'(2) =
II
_ L
tr(
t3
^ /--)t- "f'(2)
2 t
"f'(2)
I= 8
3 !
J'@Q)4 t
tt") p)
\za r 4
t r )T
J-I (
I
Ln*'= (.,)n ^t, ;G,) 2n !
24
= *-* (* -z)
6 6 d
0un:ilrvruLa0TFlotn( . l \ f ln ) f . l \
f (2 )+ f ' (2 ) (x - \+13) - ( * -2 )2 + . . .+ r , \ ' ) G -2 ) " + . . .2 l n !
+ * (" -z)'-,{ f,.-,7t r $ C*-allv d r r 9 , r d
:6il L!14{fl1TAtt1Fl0lJ
t lt n I
R - lim l:-;n+@ lan+tl
roo
zq- - O
qznn'
( ' - , ) '
l i , ^t4 -)94
'/rn ,
I l2^'t'
o q + 2r Cliu.r
," -: "O 2t*'
- 1- L
:t t t 9 , r 4
ttau{?.r t!14{ n 1ta tfl 1Fl 0qJ
I - ( c -R ,c+R)
J;,uc..,1e.l
- ( 2 - L , L+ z ) = ( o , , 4 )
(4lnt)ew{e5 .1- dJ"rC,,tre-!
25
| | ^ dt o.l d <tt 4 d ,
t:1n41??'l I t!UYhlnUU?tn:1v14 (anaLytic\ . J d t d - l ^
tunctron) vll9l c flF)oLtJo t:141:Ja:fltlluuttvlu I (x)v I t , v 6 ( o i 4
LFl9l?U0UnTil tylUtA0::0!4f l c UUFI0l c
1450
iilnilvrn x tudrr
Most functions which are infiniteLy differentiable on
their domain are ana[ytic at every point c in the
domain, and can be represented by the Taylor series
centered at c, on the whote interval of convergence
of the Tay[or series.
There exists, however,
this is not true; we wil.L
some 'stranqe' functions where
not deal with such functions.
I - (c -r,c + r) drtufilf ' ,
- -
L vraat be. Swrcl lc,, *ho.n *Lei . . , le-n:.( oT fg6 vQwgQ.*c q-
S- .ag Co-,,,r 4JScsr-r e"
( o r e , r
I e q.. o'lsL o"., {1".
-fi..,-.J" o,,^ f Cu1i ls Tay lo- SQv"oS
\ -nJq-r l r )q" f o{ Co,. ,vQ,, \Qqcg2 6
a-l o
TEdr f(x)-e*d:$\- '+
.f'(x) - f"(x) - .f"(x) = .. . = .f\') (r) -(ol l=.ao' 3" f Q) - f'(0) - 1'(o)= ... = f\n)(g) -
d 6 d
0un:ilrvruLa0:Fl01 " ,
h r .2.
+ x * 2 12 ' .
a , r d t t 7 t 4
:fiilttil{n1:aLt1a0IJ
t lt n I
R - lim l-3-ln-+a lAr*11
fi'radrrfi rg
\ao
(_
h ? ( 2
r . . . . ' r ro r l l
A = l
r a \t x
= 1_ e-1
_h r g__l^._
. ! v (+ L r % { . . - .
\ L q :
- limn-+@
r , qnr r) ,! l l t a n
h - ) \ F 1 4 :I
( ^ r 1 1 l
. hX
I , . ( " ' t i ' l . ' (f - l r . -
- t ' ' - ' : 9 @I
- n-r.oo .,b'. ::
vX '. - - + . . .q ' ,
( a o
Xa = y l .
C r O6 6
a{u1oun53J!iluLao:flot "f(x) - e^ :o!qa 0q r ' \ /
@zh ? o
= l
J_!A
t.
L1\Tlu'jl f @) -e' u{JufirrirYuiun:Tvr,i 6'rriu
27
30
� � � � � � � � � � � � � � � � �0
2 3 41
1
1n
nx x xx
xx
∞
=
= = + +−
+ + +…∑ ( 1 1)x− < <
1 2 3 4
1
( 1)ln(1 ) ( 1 1)
2 3 4
n n
n
x x x xx x x
n
−∞
=
−+ = = − + − +… − < <∑
2 3 4
0
1 ( )! 2! 3! 4!
nx
n
x x x xe x x
n
∞
=
= = + + + + +… −∞ < < ∞∑
2 2 4 6
0
( 1)cos 1 ( )
(2 )! 2! 4! 6!
n n
n
x x x xx x
n
∞
=
−= = − + − +… −∞ < < ∞∑
2 1 3 5 7
0
( 1)sin ( )
(2 1)! 3! 5! 7!
n n
n
x x x xx x x
n
+∞
=
−= = − + − +… −∞ < < ∞
+∑
2 1 3 5 7
0
1 ( 1)(
2tan 1 1)
5 71 3
n n
n
x x xx x x
x
n
∞
=
+−
= = − + − +…−
− < <+
∑
c+ do t r q+ dv iilht ntr%Gt t[6t e1^|,,J ntr%a(u
s )oe) r9 tnl f Gr) - f (*) 6l't?1TU ULn ay r l,% tq rlJ%
?JO{ f ur,firu:ruTun f 'jrril.rfifi'uqi (even function)
Gxr){ is Q\€rr'. Y = fc*i = Sf-x/
(rv)
nmrnlt o'rrffrriffunfi a N N'tnrrfi uu rTu unu aq,
ffrodr.r f@):5r4-3r2*1
Chc.-qa X {o -X :
{(-xl =s'(-*)'t \(-t)t + I
= S.XY-\'xrt1 : f Gl ll
-) e \rtn C*^ - {Jo'^ .
Y
;u * "* ; ' c r -ve
f Cx? = Sxu +Jxz+ l.r
Qu e\^ \
)v
oNoln'
Fvo-yl" fCx) e Cls )<
{C-*) = c.,s ( -x/ = Q>s Cu) 42
=) g:g f,**. |,o,^
9)
il'l fc d r w d o t I
tu1,[v{'t nfl1't91U
9)
tra?
A,: f t recr
O.rr9cr, A
Oa( - . (
) frx) dx + )-CL fO
SeL g= -)(
d'*= -dx
0
( € (-..) CJ.-) +)
o,o
5 {(.) d.. +a-
&
S f(*) t* |o t , )
CL.c^'nac ,-r- ltl xa,
z S Q(*l dvo
&-aq 4 ' [1 lAg
{ rS Qvon
Au
=A,{Ae= A,f A,=2A,
I r@)d*-a
2l0
f (") d*-0 '
Adv\|6[0%
U
L-W":e!=
Avec.
Tol ot
lgrtUa-(
) SCx) d,x =-o
t is .e-ven
QC-*) = QC.-l =
€.xchartg liv':l-s -o! i* \ eXvofJe-
-
f(x/ dv
{ ('a {v
{C"l {'.
fCxl t '<
a
5o
a
too.
5o
43
9)
nl fG")- -f(")v
?r0{ f [ra? tT? rTU n f
cfrvrYu uusinv r tuTo' uJl,r'j r ril'r fl trud (odd function)
Y.€(xl
X
.f(-;)I- Kv)
a-
S ;s oU.9C-xl
I a / de r d . sn T1 V{ ?J 0.t 1^l { n t 1,[ 9t N 6l N N 1 gt T t14 U U n U q O n 1 tU O
q
a-, I
ffrotirt f @) : *3 - 5r
CL',e^op: Y lo - >< :
{ (-x) - Qx)t - s(-x)a
= x +SxT ] \ = f,Cxl= f x -sx J
--.-*- - ( (*1
=+ odl f o.^ c'["o'1
ct \\ P or^r QvS
- \ ,3 F l
{Cvt = x -s'x
t(x) = Sin x
= Sin C-t ) =-Si",C>./
dJ +".t. ":
= - fCvJ
t r t | r& -? Ogd
:
N o\e.,
Exer.plrr (-><t
_\-r/
44
s ) 6 r w du inl f rul,[?{{nfil,[el
A d
vr6[01,[(U
t t -t. tPo* €- V.L
9)
Ira?
o
=) I tc
d(x) dr +t
Se\ t t=-Xd,^= -Av
((-*l [ -J.^) +
f (,^) J,*-
S(*) [,C\te^1p ,^ \.r v
t( vl d.v 'V
fI f@)dr - o
,l-4 ,
'ra /I //'/,
d rs odC
a
Ar=
d.x =
{x= -
/ \
N.N{zo
9+o
F ) . ^ tc. Lo wr .p u,J-e..
a/ A .
\ t(xl dx-&
Cx"C a_
a
t f(xlo
o
( d c><l' a
clr
x? Cr + )o
&'€o-
A,
Az=
$Cxl d x
A,
=-A,fA,=p
=
t^
\ is odl,
!(.^) f qc*t =l l
€.)x che*ce li-lt5l < t
o\ i n{exvo.\Jo*
QC"t cv
q,
5 €Cv) croq
t fcxJ dvo
q
S QC r.l {,.tJ
K)4 dx =o
e
tD
oC
-a-
o(
&o(
- )
@
lr
q
so
a,
)o
o(
so
45
L)dn = (.f - \x'+ r) JxL
- r (
L\
o)-x
L]+r+
L\
+ xJa
- \ \Lt-o)=
) , 1L
2' I xs
z" ( rli: s2
-f ,>r 3-F / I 'r.; If |\ !3
f,'
1
lrldn - 1 (
/ L "
\,/o
I-Yl=\xl:) Qyeul f.^.rtk ''-rt.
lxl C,. = ?'/
\x[ = X;S o.x2\
]( r
)x cxo.)
t - ' l )= x \o
=1!A
i -4 1.?8a J I
i ' t / sini . o . - |
-r .28n
/ . )sin (Qx;/
a odJ
=O-A
/ '
-(s\= ! .J .^ \ -X / =
f - r\ cx-,^g. { -a^
("t) o*
- Sc\ ("tl
a S S"r'' C- e) = - Stt" g
10
tJ
-10
5r dr /-1- \-,
:
s@ = ry -x\s,<
- \ l l- ) / ' \AA- \-/v\*::
d;rf ;...
46