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Taylo= r's Remainder Theorem I
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Taylo= r's Remainder Theorem I
In would be great if that the Taylor power series of a = function, if it exists, would be the same as the function itself. 
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Taylo= r's Remainder Theorem I
In would be great if that the Taylor power series of a = function, if it exists, would be the same as the function itself.  Such is not the case= by the following example.
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Taylo= r's Remainder Theorem I
Example: <= /div>
In would be great if that the Taylor power series of a = function, if it exists, would be the same as the function itself.  Such is not the case= by the following example.
Let f(x) =3D
{
e-1/x
2
if x =3D 0=
0 if x =3D 0
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Taylo= r's Remainder Theorem I
Example: <= /div>
In would be great if that the Taylor power series of a = function, if it exists, would be the same as the function itself.  Such is not the case= by the following example.
Let f(x) =3D
{
e-1/x
2
if x =3D 0=
0 if x =3D 0
We leave it as an exercise that it's derivatives of all = the orders are 0, i.e. f(n)(0) =3D 0 for all n. 
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Taylo= r's Remainder Theorem I
Example: <= /div>
In would be great if that the Taylor power series of a = function, if it exists, would be the same as the function itself.  Such is not the case= by the following example.
Let f(x) =3D
{
e-1/x
2
if x =3D 0=
0 if x =3D 0
We leave it as an exercise that it's derivatives of all the orders are 0, i.e. f(n)(0) =3D 0 for all n.  Therefore <= span style=3D'position:absolute;top:65.75%;left:5.24%;width:91.57%;height:6.0%'= >it's Maclaurin se= ries P(x) =3D 0.
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Taylo= r's Remainder Theorem I
Example: <= /div>
In would be great if that the Taylor power series of a = function, if it exists, would be the same as the function itself.  Such is not the case= by the following example.
Let f(x) =3D
{
e-1/x
2
if x =3D 0=
0 if x =3D 0
We leave it as an exercise that it's derivatives of all the orders are 0, i.e. f(n)(0) =3D 0 for all n.  Therefore <= span style=3D'position:absolute;top:65.75%;left:5.24%;width:106.17%;height:6.0%= '>it's Maclaurin se= ries P(x) =3D 0. Clearly P(x) is not the same as f(x).=
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Taylo= r's Remainder Theorem I
Example: <= /div>
In = the following discussion, we will assume all the functions are infinitely differentiable over an open interval unless otherwise stated.<= /span>
In would be great if that the Taylor power series of a = function, if it exists, would be the same as the function itself.  Such is not the case= by the following example.
Let f(x) =3D
{
e-1/x
2
if x =3D 0=
0 if x =3D 0
We leave it as an exercise that it's derivatives of all the orders are 0, i.e. f(n)(0) =3D 0 for all n.  Therefore <= span style=3D'position:absolute;top:65.75%;left:5.24%;width:106.17%;height:6.0%= '>it's Maclaurin se= ries P(x) =3D 0. Clearly P(x) is not the same as f(x).=
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Taylo= r's Remainder Theorem I
Tay= lor's Remainder Theorem gives a formula for the difference between function value f(b) and the pn(b) where pn is the n'th Taylor polynomial  = (expanded about some point a).<= span style=3D'mso-spacerun:yes'> 
------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0361.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Taylo= r's Remainder Theorem I
Tay= lor's Remainder Theorem gives a formula for the difference between function value f(b) and the pn(b) where pn is the n'th Taylor polynomial  = (expanded about some point a).<= span style=3D'mso-spacerun:yes'> 
We = will give the theorem where pn is the Maclaurin polynomial first (i.e. expanded about 0).
------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0362.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Taylo= r's Remainder Theorem I
Tay= lor's Remainder Theorem:  Let f(x) = be an infinitely differentiable function over some open interval that contains [0, b] and pn(x)= be the n'th Mac-poly .
Tay= lor's Remainder Theorem gives a formula for the difference between function value f(b) and the pn(b) where pn is the n'th Taylor polynomial  = (expanded about some point a).<= span style=3D'mso-spacerun:yes'> 
We = will give the theorem where pn is the Maclaurin polynomial first (i.e. expanded about 0).
------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0363.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Taylo= r's Remainder Theorem I
Tay= lor's Remainder Theorem:  Let f(x) = be an infinitely differentiable function over some open interval that contains [0, b] and pn(x)= be the n'th Mac-poly .
Tay= lor's Remainder Theorem gives a formula for the difference between function value f(b) and the pn(b) where pn is the n'th Taylor polynomial  = (expanded about some point a).<= span style=3D'mso-spacerun:yes'> 
We = will give the theorem where pn is the Maclaurin polynomial first (i.e. expanded about 0).
0
(
)
[
]
b
f(x) is infinitely differentiable in here
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Taylo= r's Remainder Theorem I
Tay= lor's Remainder Theorem:  Let f(x) = be an infinitely differentiable function over some open interval that contains [0, b] and pn(x)= be the n'th Mac-poly .
Tay= lor's Remainder Theorem gives a formula for the difference between function value f(b) and the pn(b) where pn is the n'th Taylor polynomial  = (expanded about some point a).<= span style=3D'mso-spacerun:yes'> 
Then there exists a "c" between b and 0 such that
<= /div>
f(b) =3D pn(b)= +
We = will give the theorem where pn is the Maclaurin polynomial first (i.e. expanded about 0).
bn+= 1
(n += 1)!
f(= n+1)(= c)
0
(
)
[
]
b
c
f(x) is infinitely differentiable in here
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Taylor= 's Remainder Theorem I
or in full detail,
f '(= 0)b
f(2= )(0)
+
2!
=3D
f(0)<= /span>
+
b2<= /span>
f(b)
+..
f(n= )(0)
n!
bn<= /span>
+ 
bn+1
(n + 1)!
f(n+1)(c)
+
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Taylor= 's Remainder Theorem I
or in full detail,
The difference term
denoted as Rn(b)
bn+1
(n + 1)!
f(n+1)(c)
is the Lagrange form of the Taylor-remainder <= /span>
(or error) term (there are other forms).<= /span>
f '(= 0)b
f(2= )(0)
+
2!
=3D
f(0)<= /span>
+
b2<= /span>
f(b)
+..
f(n= )(0)
n!
bn<= /span>
+ 
bn+1
(n + 1)!
f(n+1)(c)
+
------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0365_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBM AAEAgAAAAAAAAAIHjI+py+3vCgA7 ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0365_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhnQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0365_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0366.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Taylor= 's Remainder Theorem I
or in full detail,
The difference term
denoted as Rn(b)
bn+1
(n + 1)!
f(n+1)(c)
is the Lagrange form of the Taylor-remainder <= /span>
(or error) term (there are other forms).<= /span>
f '(= 0)b
f(2= )(0)
+
2!
=3D
f(0)<= /span>
+
b2<= /span>
f(b)
+..
f(n= )(0)
n!
bn<= /span>
+ 
bn+1
(n + 1)!
f(n+1)(c)
+
Remarks:
------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0366_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBM AAEAgAAAAAAAAAIHjI+py+3vCgA7 ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0366_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhnQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0366_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0367.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Taylor= 's Remainder Theorem I
or in full detail,
The difference term
denoted as Rn(b)
bn+1
(n + 1)!
f(n+1)(c)
is the Lagrange form of the Taylor-remainder <= /span>
(or error) term (there are other forms).<= /span>
f '(= 0)b
f(2= )(0)
+
2!
=3D
f(0)<= /span>
+
b2<= /span>
f(b)
+..
f(n= )(0)
n!
bn<= /span>
+ 
bn+1
(n + 1)!
f(n+1)(c)
+
Remarks:
* the theorem also works for the interval [b, 0]
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Taylor= 's Remainder Theorem I
or in full detail,
The difference term
denoted as Rn(b)
bn+1
(n + 1)!
f(n+1)(c)
is the Lagrange form of the Taylor-remainder <= /span>
(or error) term (there are other forms).<= /span>
f '(= 0)b
f(2= )(0)
+
2!
=3D
f(0)<= /span>
+
b2<= /span>
f(b)
+..
f(n= )(0)
n!
bn<= /span>
+ 
bn+1
(n + 1)!
f(n+1)(c)
+
Remarks:
* the theorem also works for the interval [b, 0]
* the value c changes if the va= lue of b or n changes
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Taylor= 's Remainder Theorem I
or in full detail,
The difference term
denoted as Rn(b)
bn+1
(n + 1)!
f(n+1)(c)
is the Lagrange form of the Taylor-remainder <= /span>
(or error) term (there are other forms).<= /span>
f '(= 0)b
f(2= )(0)
+
2!
=3D
f(0)<= /span>
+
b2<= /span>
f(b)
+..
f(n= )(0)
n!
bn<= /span>
+ 
bn+1
(n + 1)!
f(n+1)(c)
+
Remarks:
* the theorem also works for the interval [b, 0]
* the value c can't be easily determined,= we just know   =
  there is at least one c that fits the description 
* the value c changes if the va= lue of b or n changes
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Example: f(x) =3D= ex
Taylor= 's Remainder Theorem I
 is infinitely differentiable every= where.
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Example: f(x) =3D= ex
Mac-poly of ex is
Pn(x) =3D
Σ
k=3D0
xk
k!
n
 =3D 1 + x +
x2
2!
+ .. +
xn
n!
Taylor= 's Remainder Theorem I
 is infinitely differentiable every= where.
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Example: f(x) =3D= ex
The Mac-poly of e= x is
Pn(x) =3D
Σ
k=3D0
xk
k!
n
 =3D 1 + x +
x2
2!
+ .. +
xn
n!
At x =3D b, by the theorem we have
f(b) =3D e= b =3D 1 + b +
b2
2!
+ .. +
bn
n!
+
f(n+1)(c)
(n+1)!
for some c that is between 0 and b.
bn+1
Taylor= 's Remainder Theorem I
 is infinitely differentiable every= where.
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Example: f(x) =3D= ex
Mac-poly of ex is
Pn(x) =3D
Σ
k=3D0
xk
k!
n
 =3D 1 + x +
x2
2!
+ .. +
xn
n!
At x =3D b, by the theorem we have
f(b) =3D e= b =3D 1 + b +
b2
2!
+ .. +
bn
n!
+
f(n+1)(c)
(n+1)!
for some c that is between 0 and b.
bn+1
Since f(n+1)(x) =3D ex so f(n+1)(c) =3D ec.  
Taylor= 's Remainder Theorem I
 is infinitely differentiable every= where.
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Example: f(x) =3D= ex
Mac-poly of ex is
Pn(x) =3D
Σ
k=3D0
xk
k!
n
 =3D 1 + x +
x2
2!
+ .. +
xn
n!
At x =3D b, by the theorem we have
f(b) =3D e= b =3D 1 + b +
b2
2!
+ .. +
bn
n!
+
f(n+1)(c)
(n+1)!
for some c that is between 0 and b.
bn+1
Since f(n+1)(x) =3D ex so f(n+1)(c) =3D ec.  Hence 
f(b) =3D e= b =3D 1 + b +
b2
2!
+ .. +
bn
n!
+
ec
(n+1)!
bn+1
for some c that is between 0 and b.
Taylor= 's Remainder Theorem I
 is infinitely differentiable every= where.
Hence 
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Example: f(x) =3D= ex
Mac-poly of ex is
Pn(x) =3D
Σ
k=3D0
xk
k!
n
 =3D 1 + x +
x2
2!
+ .. +
xn
n!
At x =3D b, by the theorem we have
f(b) =3D e= b =3D 1 + b +
b2
2!
+ .. +
bn
n!
+
f(n+1)(c)
(n+1)!
for some c that is between 0 and b.
bn+1
Since f(n+1)(x) =3D ex so f(n+1)(c) =3D ec.  Hence 
f(b) =3D e= b =3D 1 + b +
b2
2!
+ .. +
bn
n!
+
ec
(n+1)!
bn+1
for some c that is between 0 and b.
Taylor= 's Remainder Theorem I
 is infinitely differentiable every= where.
Hence 
The point here is not to find c but to use the formula to <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;mso-fareast-language:ZH-CN'>calculate the behavior of the e= rror as n à    .
∞
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Example: Show that in the above example, the remainder term goes to 0 as= n à     
∞.
Taylor= 's Remainder Theorem I
------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0376.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Show that in the a= bove example, the remainder term goes to 0 as n = à     Hen= ce f(b) =3D P(b) for all values b
where P(x) is the Mac-serie= s.
∞.
Taylor= 's Remainder Theorem I
------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0377.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Show that in the a= bove example, the remainder term goes to 0 as n = à     Hen= ce f(b) =3D P(b) for all values b
where P(x) is the Mac-serie= s.
f(b) =3D e= b =3D 1 + b +
b2
2!
+ .. +
bn
n!
+
ec
(n+1)!
bn+1
where b is a fixed value and c is between 0 and b.
∞.
Taylor= 's Remainder Theorem I
We have that
------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0377_image063.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAc AAEAgAAAAAAAAAIEjI+pWwA7 ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0377_image064.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0377_image065.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0378.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Show that in the a= bove example, the remainder term goes to 0 as n = à     Hen= ce f(b) =3D P(b) for all values b
where P(x) is the Mac-serie= s.
f(b) =3D e= b =3D 1 + b +
b2
2!
+ .. +
bn
n!
+
ec
(n+1)!
bn+1
where b is a fixed value and c is between 0 and b.
∞.
The maximum possi= ble value of ec is eb. 
Taylor= 's Remainder Theorem I
We have that
------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0378_image066.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAc AAEAgAAAAAAAAAIEjI+pWwA7 ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0378_image067.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAb AAEAgAAAAAAAAAIEjI+pWgA7 ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0378_image068.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C97CE6.BC2F0450 Content-Location: file:///C:/255438E9/33Taylor'sRemainderTheoremI_files/slide0379.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example: Show that in the a= bove example, the remainder term goes to 0 as n = à     Hen= ce f(b) =3D P(b) for all values b
where P(x) is the Mac-serie= s.
f(b) =3D e= b =3D 1 + b +
b2
2!
+ .. +
b