CHAPTER 05
TAYLOR SERIES & STATIONARY POINTS
•
"
%)
Recall : f- ( x ) :
⇐ ☐
an /✗ -
has a
powers series
expansion about xo and
Radius of R f. Gc)
convergence obtained
> 0
a
,
then
may
be by differentiating
each term
' 's
got a. ( x ) (x xD
"
f- ( x ) = -
Xo + az - + a ]
Gc - Xo ) t - - -
t an Ix -
Xo ) t - - -
2 l
) (x
"
2oz ( x ) (x
-
f (a) 3A )
'
+ + Xo t
=
a Xo + Man Xo t
- - - - -
- - -
-
, ,
N
{
" '
'
f /x) )
'
=
nanlx -
Xo
n :c
" "
(a) )
"
f- =
Zaz +
Gaz (x -
%) + . - - + nln -
1) an Ix -
Xo
1- - - -
N
" -2
f '( x)
'
: { Mln -
1) an / ✗ -
%)
n=z
•
f' ( )
"
x : E nln -
1) In 2) -
an lx -
xo )
" -2
etc . . .
N =3
calculated if Xo
Derivatives
easily X
giving
:
=
f ( Xo ) : 90 > .
:
coefficients in the
power series can be defined by
( Xo )
'
f- : A ,
f / Xo)
"
( Xo )
"
f- =
Zaz an
=
( xo) 3×292
' ' '
f- =
:
( xo)
"
f- = n ! an
Definition ( for for C- ( x))
: a
Taylor series
If f- (x) has series representation
:
a
power
A
§
"
flu) =
an / x xD -
,
R then f- ( xo) for
with radius of convergence
> 0
,
exists
every positive integer
hr and
,
= f^( Xo)
an
Therefore :
f"§?÷( "n, (
f
flx) flxo) %) 't
"
= + f' lxollx -
Xo) + x -
- -
-
+ x -
xo ) + . . -
NOTE :O Maclaurin Series
Taylor series with Xo a
:
is
When
question asks to find Taylor series check for convergence with Ratio test
,
, TAYLOR POLYNOMIAL
Definition '
Let xo be a real number and f- Gc) be a function that has n derivatives
at ✗ =
Xo .
The nth degree Taylor Polynomial ,
Pnlx) of flx) about ✗ =
Xo is :
"¥%;÷;;---+f;¥a-%#
Pn / ) flxo) f- ( xo) (x )
'
x = + -
Xo +
I
Therefore Pack ) has ntl terms
P ( x)
,
=
f- ( Xo) + f' ( Xo ) ( x -
Xo ) P / xD
,
=
f- ( xo) because ✗ -
Xo =D
Pino)= f' (xo)
f"§ (
"
f- ( Xo) f / xo) ( ) %)
pz ( x )
'
+ " -
Xo
p, (xo) fcxo)
= x
+ -
=
piled =
f' Coco)
Pnlx
pilxo) f ( Xo)
"
=
Generally :
dd÷m ( Pnlx))×=×
.
=
fmlxo)
ERROR ASSOCIATED WITH Pulse)
since truncated Taylor series
only an
approximation for fcx) there will a difference /
,
error between Pnlx) and f- ( x) . Error estimated using Taylor Remainder Rn (a)
Suppose C- ( x ) is continuous in [a ,
b ] and has Inti ) derivatives in ( a. b) .
Then
There exists C cb such that :
point acc
a
, ,
f- Gc) =
Pnlx ) + Rn ( x )
1 I
remainder of
Taylor polynomial of Taylor
order n about ✗ order n about Xo
o
Remainder given by :
" + '
Xo x c- (a. b)
f- (c)
,
""
Rnlx ) = -
(x -
Xo)
and
( ntc ) ! C lies between ✗ ☐
x
TAYLOR SERIES & STATIONARY POINTS
•
"
%)
Recall : f- ( x ) :
⇐ ☐
an /✗ -
has a
powers series
expansion about xo and
Radius of R f. Gc)
convergence obtained
> 0
a
,
then
may
be by differentiating
each term
' 's
got a. ( x ) (x xD
"
f- ( x ) = -
Xo + az - + a ]
Gc - Xo ) t - - -
t an Ix -
Xo ) t - - -
2 l
) (x
"
2oz ( x ) (x
-
f (a) 3A )
'
+ + Xo t
=
a Xo + Man Xo t
- - - - -
- - -
-
, ,
N
{
" '
'
f /x) )
'
=
nanlx -
Xo
n :c
" "
(a) )
"
f- =
Zaz +
Gaz (x -
%) + . - - + nln -
1) an Ix -
Xo
1- - - -
N
" -2
f '( x)
'
: { Mln -
1) an / ✗ -
%)
n=z
•
f' ( )
"
x : E nln -
1) In 2) -
an lx -
xo )
" -2
etc . . .
N =3
calculated if Xo
Derivatives
easily X
giving
:
=
f ( Xo ) : 90 > .
:
coefficients in the
power series can be defined by
( Xo )
'
f- : A ,
f / Xo)
"
( Xo )
"
f- =
Zaz an
=
( xo) 3×292
' ' '
f- =
:
( xo)
"
f- = n ! an
Definition ( for for C- ( x))
: a
Taylor series
If f- (x) has series representation
:
a
power
A
§
"
flu) =
an / x xD -
,
R then f- ( xo) for
with radius of convergence
> 0
,
exists
every positive integer
hr and
,
= f^( Xo)
an
Therefore :
f"§?÷( "n, (
f
flx) flxo) %) 't
"
= + f' lxollx -
Xo) + x -
- -
-
+ x -
xo ) + . . -
NOTE :O Maclaurin Series
Taylor series with Xo a
:
is
When
question asks to find Taylor series check for convergence with Ratio test
,
, TAYLOR POLYNOMIAL
Definition '
Let xo be a real number and f- Gc) be a function that has n derivatives
at ✗ =
Xo .
The nth degree Taylor Polynomial ,
Pnlx) of flx) about ✗ =
Xo is :
"¥%;÷;;---+f;¥a-%#
Pn / ) flxo) f- ( xo) (x )
'
x = + -
Xo +
I
Therefore Pack ) has ntl terms
P ( x)
,
=
f- ( Xo) + f' ( Xo ) ( x -
Xo ) P / xD
,
=
f- ( xo) because ✗ -
Xo =D
Pino)= f' (xo)
f"§ (
"
f- ( Xo) f / xo) ( ) %)
pz ( x )
'
+ " -
Xo
p, (xo) fcxo)
= x
+ -
=
piled =
f' Coco)
Pnlx
pilxo) f ( Xo)
"
=
Generally :
dd÷m ( Pnlx))×=×
.
=
fmlxo)
ERROR ASSOCIATED WITH Pulse)
since truncated Taylor series
only an
approximation for fcx) there will a difference /
,
error between Pnlx) and f- ( x) . Error estimated using Taylor Remainder Rn (a)
Suppose C- ( x ) is continuous in [a ,
b ] and has Inti ) derivatives in ( a. b) .
Then
There exists C cb such that :
point acc
a
, ,
f- Gc) =
Pnlx ) + Rn ( x )
1 I
remainder of
Taylor polynomial of Taylor
order n about ✗ order n about Xo
o
Remainder given by :
" + '
Xo x c- (a. b)
f- (c)
,
""
Rnlx ) = -
(x -
Xo)
and
( ntc ) ! C lies between ✗ ☐
x
