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