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Taylor= 's Remainder Theorem II
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0444.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Taylor= 's Remainder Theorem II
We state the general form of the Taylor's remainder = formula.
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0420.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Taylor= 's Remainder Theorem II
We state the general form of the Taylor's remainder = formula.
Taylor's Remainder Theorem (General Form):
Let f(x) be an infinitely differentiable function over some open interval that contains [a, b]
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0421.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Taylor= 's Remainder Theorem II
We state the general form of the Taylor's remainder = formula.
Taylor's Remainder Theorem (General Form):
Let f(x) be an infinitely differentiable function over some open interval that contains [a, b]
a
(
)
[
]
b
f(x) is infinitely differentiable in here
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0421_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAEEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQBo AQMAgAAAAAAAAAIuhI+py+0PYZi02ouz3rz7D4biSJbmiabqyrbuC8fyTNf2jU/Rzvf+DwwKh0RH AQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0421_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQC7 AAoAgAAAAP8AAAKqhG+Be6Hs2IoSvZqu1ZJODoGAh4kflnWmpaKjSKprisZ0adsf6Myvq9vwgo0h EObrHZeho7GWI94q0tcw+cS1oFjmidsE77xU7JU83X5xXai6nIWH2lqxEP1zh9n49ca5JzemV8c3 WFjUV6VEWPYmEwh56NgoeVeZh5hoR6mZNhcJOvkYOmqFt4gEiOm3afjHmUHXKXjpOeKSm+uDq+vb 8QtBFUwsDAxUUQAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0421_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMQAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAcAAQAo ABgAgAAAAAAAAAJChI+pyxYN4wpUWkjzu/zo3HFfFVrjVkZnKq2sOr5YLDNurdw4ou/e58vRaijH kEjqIU/FHbPpZAZ5x+lzapRiq9gCADs= ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0422.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Taylor= 's Remainder Theorem II
We state the general form of the Taylor's remainder = formula.
Taylor's Remainder Theorem (General Form):
Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and
 
pn(x) =3D
<= /div>
be the n'th Taylor-poly expanded around a= t a.
a
(
)
[
]
b
f(x) is infinitely differentiable in here
Σ
k=3D0
n
(x &= #8211; a)k
k!
f(k= )(a)
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0422_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAEEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQBo AQMAgAAAAAAAAAIuhI+py+0PYZi02ouz3rz7D4biSJbmiabqyrbuC8fyTNf2jU/Rzvf+DwwKh0RH AQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0422_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQC7 AAoAgAAAAP8AAAKqhG+Be6Hs2IoSvZqu1ZJODoGAh4kflnWmpaKjSKprisZ0adsf6Myvq9vwgo0h EObrHZeho7GWI94q0tcw+cS1oFjmidsE77xU7JU83X5xXai6nIWH2lqxEP1zh9n49ca5JzemV8c3 WFjUV6VEWPYmEwh56NgoeVeZh5hoR6mZNhcJOvkYOmqFt4gEiOm3afjHmUHXKXjpOeKSm+uDq+vb 8QtBFUwsDAxUUQAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0422_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABm AAEAgAAAAAAAAAIIjI+py+0P4ysAOy== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0422_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMQAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAcAAQAo ABgAgAAAAAAAAAJChI+pyxYN4wpUWkjzu/zo3HFfFVrjVkZnKq2sOr5YLDNurdw4ou/e58vRaijH kEjqIU/FHbPpZAZ5x+lzapRiq9gCADs= ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0424.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Taylor= 's Remainder Theorem II
We state the general form of the Taylor's remainder = formula.
Taylor's Remainder Theorem (General Form):
Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and
 
pn(x) =3D
<= /div>
be the n'th Taylor-poly expanded around a= t a.
Then there exists a "c" between a and b such that
<= /div>
f(b) =3D pn(b)= +
(b &#= 8211; a)n+1
(n += 1)!
f(= n+1)(= c)
a
(
)
[
]
b
c
f(x) is infinitely differentiable in here
Σ
k=3D0
n
(x &= #8211; a)k
k!
f(k= )(a)
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0424_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQBM AAEAgAAAAAAAAAIHjI+py+3vCgA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0424_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAEEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAQBo AQMAgAAAAAAAAAIuhI+py+0PYZi02ouz3rz7D4biSJbmiabqyrbuC8fyTNf2jU/Rzvf+DwwKh0RH AQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0424_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQC7 AAoAgQAAAAAAAP8AAAECAwLFhG+Ce6Ls2IoSvZqu1ZLOF4TB13Ed1pgkpE4lCngb7LQprcqlGYW1 EMrwdCwcbmjLFFHEWNL5bM6WmNoy8HOMrJ+o98j6rqo5r5j8YhIp2Kx2Lb2pwV3juNKMC+1oKh5+ 0OaGFcb3R7dxJtd3t2OY1lh4J5g1UjeHSYb06MTZqaYIOaXpB0BZY5noOfq3mbl4WCrJGBt5yRo4 eFvL67grqvQK60sLHExqayAi8iubnOJanABDTf0UU52Npw2xzf3tckJTUQAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0424_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhaAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABm AAEAgAAAAAAAAAIIjI+py+0P4ysAOy== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0424_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMQAhAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAcAAQAo ABgAgAAAAAAAAAJChI+pyxYN4wpUWkjzu/zo3HFfFVrjVkZnKq2sOr5YLDNurdw4ou/e58vRaijH kEjqIU/FHbPpZAZ5x+lzapRiq9gCADs= ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0369.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
or in full detail,
f '(= a)(b – a)
f(2= )(a)
+
2!
=3D
f(a)<= /span>
+
(b &= #8211; a)2
f(b)
..
f(n= )(a)
n!
(b &= #8211; a)n
+
Taylor= 's Remainder Theorem II
+ Rn(b)<= /span>
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0369_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAp AAEAgAAAAAAAAAIFjI+py1kAO1== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0369_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAp AAEAgAAAAAAAAAIFjI+py1kAO1== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0425.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
or in full detail,
where Rn(b) =3D
(b – a)n+1
(n + 1)!
f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(= a)(b – a)
f(2= )(a)
+
2!
=3D
f(a)<= /span>
+
(b &= #8211; a)2
f(b)
..
f(n= )(a)
n!
(b &= #8211; a)n
+
Taylor= 's Remainder Theorem II
+ Rn(b)<= /span>
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0425_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABL AAEAgAAAAAAAAAIHjI+py+3fCgA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0425_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAp AAEAgAAAAAAAAAIFjI+py1kAO1== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0425_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAp AAEAgAAAAAAAAAIFjI+py1kAO1== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0426.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
or in full detail,
where Rn(b) =3D
(b – a)n+1
(n + 1)!
f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(= a)(b – a)
f(2= )(a)
+
2!
=3D
f(a)<= /span>
+
(b &= #8211; a)2
f(b)
..
f(n= )(a)
n!
(b &= #8211; a)n
+
Again, we note the following
Taylor= 's Remainder Theorem II
+ Rn(b)<= /span>
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0426_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABL AAEAgAAAAAAAAAIHjI+py+3fCgA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0426_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAp AAEAgAAAAAAAAAIFjI+py1kAO1== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0426_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAp AAEAgAAAAAAAAAIFjI+py1kAO1== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0427.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
or in full detail,
where Rn(b) =3D
(b – a)n+1
(n + 1)!
f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(= a)(b – a)
f(2= )(a)
+
2!
=3D
f(a)<= /span>
+
(b &= #8211; a)2
f(b)
..
f(n= )(a)
n!
(b &= #8211; a)n
+
Again, we note the following
* the theorem also works for the interval [b, a]
Taylor= 's Remainder Theorem II
+ Rn(b)<= /span>
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0427_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABL AAEAgAAAAAAAAAIHjI+py+3fCgA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0427_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAp AAEAgAAAAAAAAAIFjI+py1kAO1== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0427_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAp AAEAgAAAAAAAAAIFjI+py1kAO1== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0428.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
or in full detail,
where Rn(b) =3D
(b – a)n+1
(n + 1)!
f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(= a)(b – a)
f(2= )(a)
+
2!
=3D
f(a)<= /span>
+
(b &= #8211; a)2
f(b)
..
f(n= )(a)
n!
(b &= #8211; a)n
+
Again, we note the following
* the theorem also works for the interval [b, a]
* the value c changes if the va= lue of b or n changes
Taylor= 's Remainder Theorem II
+ Rn(b)<= /span>
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0428_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABL AAEAgAAAAAAAAAIHjI+py+3fCgA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0428_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAp AAEAgAAAAAAAAAIFjI+py1kAO1== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0428_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAp AAEAgAAAAAAAAAIFjI+py1kAO1== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0429.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
or in full detail,
where Rn(b) =3D
(b – a)n+1
(n + 1)!
f(n+1)(c)
is the Lagrange form of the Taylor-remainder.
f '(= a)(b – a)
f(2= )(a)
+
2!
=3D
f(a)<= /span>
+
(b &= #8211; a)2
f(b)
..
f(n= )(a)
n!
(b &= #8211; a)n
+
Again, we note the following
* the theorem also works for the interval [b, a]
* 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
Taylor= 's Remainder Theorem II
+ Rn(b)<= /span>
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Example:
A. Find the Taylor-series of f(x) =3D sin(x) at x =3D       .
`= 0;
2 =
Taylor= 's Remainder Theorem II
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0417_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0430.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example:
A. Find the Taylor-series of f(x) =3D sin(x) at x =3D       .
`= 0;
2 =
sin(x)
cos(x)
-sin(x)
-cos(x)
Taylor= 's Remainder Theorem II
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Example:
A. Find the Taylor-series of f(x) =3D sin(x) at x =3D       .
`= 0;
2 =
sin(x)
cos(x)
-sin(x)
-cos(x)
At x =3D <= /div>
`= 0;
2 =
, we get the sequ= ence of coefficients
1, 0, -1, 0, 1, 0, -1, …
Taylor= 's Remainder Theorem II
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0431_image034.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0431_image035.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgBQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAAAa AEsAgAAAAAAAAAJ0DI6pm2EMo4t0zYrBzXRz6H3WIzJhiZyoRq5p67JxA7sqepe5uH9OjetlfrPD T9ghxo7AIbOoLEqn1Kr1is1qt9yu9wsOi6VNHjIJLXPOErUz7cawQfHK3FRHz6K2fDsNuOdHN4hn Y1RIc6dwJMhntkgjVQAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0431_image036.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgBRAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAc AEwAgAAAAAAAAAJ0hI+pGLG/mnSwnmkzwBq23T2SQYXMZ0Zomqzs+bZuPMbX/OJ5adf2pkv5ejzi jxQ0DX/Fo/MJjUqn1Kr1is1qt9yu9wsOi79L2qQpLLM4R0dyjTZH3+l5fGfPQ+nKO9zu1xcYorYz 2FEId6iRmLaY0agEUAAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0431_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0432.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example:
A. Find the Taylor-series of f(x) =3D sin(x) at x =3D       .
`= 0;
2 =
sin(x)
cos(x)
-sin(x)
-cos(x)
At x =3D <= /div>
`= 0;
2 =
, we get the sequ= ence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
=3D
1
P(x)
Taylor= 's Remainder Theorem II
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0432_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0432_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgBQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAAAa AEsAgAAAAAAAAAJ0DI6pm2EMo4t0zYrBzXRz6H3WIzJhiZyoRq5p67JxA7sqepe5uH9OjetlfrPD T9ghxo7AIbOoLEqn1Kr1is1qt9yu9wsOi6VNHjIJLXPOErUz7cawQfHK3FRHz6K2fDsNuOdHN4hn Y1RIc6dwJMhntkgjVQAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0432_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgBRAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAc AEwAgAAAAAAAAAJ0hI+pGLG/mnSwnmkzwBq23T2SQYXMZ0Zomqzs+bZuPMbX/OJ5adf2pkv5ejzi jxQ0DX/Fo/MJjUqn1Kr1is1qt9yu9wsOi79L2qQpLLM4R0dyjTZH3+l5fGfPQ+nKO9zu1xcYorYz 2FEId6iRmLaY0agEUAAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0432_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0433.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example:
A. Find the Taylor-series of f(x) =3D sin(x) at x =3D       .
`= 0;
2 =
sin(x)
cos(x)
-sin(x)
-cos(x)
At x =3D <= /div>
`= 0;
2 =
, we get the sequ= ence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
–
=3D
1
P(x)
(x – π/2)2
2!
Taylor= 's Remainder Theorem II
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0433_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0433_image043.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgBQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAAAa AEsAgAAAAAAAAAJ0DI6pm2EMo4t0zYrBzXRz6H3WIzJhiZyoRq5p67JxA7sqepe5uH9OjetlfrPD T9ghxo7AIbOoLEqn1Kr1is1qt9yu9wsOi6VNHjIJLXPOErUz7cawQfHK3FRHz6K2fDsNuOdHN4hn Y1RIc6dwJMhntkgjVQAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0433_image044.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgBRAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAc AEwAgAAAAAAAAAJ0hI+pGLG/mnSwnmkzwBq23T2SQYXMZ0Zomqzs+bZuPMbX/OJ5adf2pkv5ejzi jxQ0DX/Fo/MJjUqn1Kr1is1qt9yu9wsOi79L2qQpLLM4R0dyjTZH3+l5fGfPQ+nKO9zu1xcYorYz 2FEId6iRmLaY0agEUAAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0433_image045.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0433_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhNwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA1 AAEAgAAAAAAAAAIGjI+py80FADs= ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0434.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example:
A. Find the Taylor-series of f(x) =3D sin(x) at x =3D       .
`= 0;
2 =
sin(x)
cos(x)
-sin(x)
-cos(x)
At x =3D <= /div>
`= 0;
2 =
, we get the sequ= ence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
–
=3D
1
P(x)
(x – π/2)2
2!
1(x – π/2)4
+
4!
Taylor= 's Remainder Theorem II
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0434_image047.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0434_image048.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgBQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAAAa AEsAgAAAAAAAAAJ0DI6pm2EMo4t0zYrBzXRz6H3WIzJhiZyoRq5p67JxA7sqepe5uH9OjetlfrPD T9ghxo7AIbOoLEqn1Kr1is1qt9yu9wsOi6VNHjIJLXPOErUz7cawQfHK3FRHz6K2fDsNuOdHN4hn Y1RIc6dwJMhntkgjVQAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0434_image049.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgBRAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAc AEwAgAAAAAAAAAJ0hI+pGLG/mnSwnmkzwBq23T2SQYXMZ0Zomqzs+bZuPMbX/OJ5adf2pkv5ejzi jxQ0DX/Fo/MJjUqn1Kr1is1qt9yu9wsOi79L2qQpLLM4R0dyjTZH3+l5fGfPQ+nKO9zu1xcYorYz 2FEId6iRmLaY0agEUAAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0434_image050.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0434_image051.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhkQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACO AAIAgAAAAAAAAAIWhI+py+0PVZi02ouz3rz7f0XiSJZiAQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0435.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example:
A. Find the Taylor-series of f(x) =3D sin(x) at x =3D       .
`= 0;
2 =
sin(x)
cos(x)
-sin(x)
-cos(x)
At x =3D <= /div>
`= 0;
2 =
, we get the sequ= ence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
–
=3D
1
P(x)
(x – π/2)2
2!
1(x – π/2)4
+
4!
–
1(x – π/2)6
6!
..
1(x – π/2)8
+
8!
Taylor= 's Remainder Theorem II
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0435_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0435_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgBQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAAAa AEsAgAAAAAAAAAJ0DI6pm2EMo4t0zYrBzXRz6H3WIzJhiZyoRq5p67JxA7sqepe5uH9OjetlfrPD T9ghxo7AIbOoLEqn1Kr1is1qt9yu9wsOi6VNHjIJLXPOErUz7cawQfHK3FRHz6K2fDsNuOdHN4hn Y1RIc6dwJMhntkgjVQAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0435_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIgBRAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAc AEwAgAAAAAAAAAJ0hI+pGLG/mnSwnmkzwBq23T2SQYXMZ0Zomqzs+bZuPMbX/OJ5adf2pkv5ejzi jxQ0DX/Fo/MJjUqn1Kr1is1qt9yu9wsOi79L2qQpLLM4R0dyjTZH3+l5fGfPQ+nKO9zu1xcYorYz 2FEId6iRmLaY0agEUAAAOw== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0435_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAT AAEAgAAAAAAAAAIEjI+JBQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0435_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOgEEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA4 AQMAgAAAAAAAAAIshI+py+0Po5y0WhOy3rz7D4bbhYnmiX7kyq7pC4ftTNf2XcX6i/f+DwwKhQUA Ow== ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0436.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example:
A. Find the Taylor-series of f(x) =3D sin(x) at x =3D       .
`= 0;
2 =
sin(x)
cos(x)
-sin(x)
-cos(x)
At x =3D <= /div>
`= 0;
2 =
, we get the sequ= ence of coefficients
1, 0, -1, 0, 1, 0, -1, … So the Taylor expansions is
–
=3D
1
P(x)
(x – π/2)2
2!
1(x – π/2)4
+
4!
–
1(x – π/2)6
6!
..
or P(x) =3D Σ
(-1)n(x – π/2)2n
(2n)!
n=3D0
n =3D∞ <= /div>
1(x – π/2)8
+
8!
Taylor= 's Remainder Theorem II
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Example:
 
B. Describe the Taylor remainder Rn(b) and show that
f(b) =3D P(b) for all value= s b. (a =3D      )
`= 0;/2
Taylor= 's Remainder Theorem II
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0437.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example:
 
B. Describe the Taylor remainder Rn(b) and show that
f(b) =3D P(b) for all value= s b. (a =3D      )
The remainder Rn(b) =3D
(b – a)n+1
(n + 1)!
f(n+1)(c)
`= 0;/2
Taylor= 's Remainder Theorem II
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0437_image064.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABM AAEAgAAAAAAAAAIHjI+py+3vCgA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0438.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example:
 
B. Describe the Taylor remainder Rn(b) and show that
f(b) =3D P(b) for all value= s b. (a =3D      )
The remainder Rn(b) =3D
(b – a)n+1
(n + 1)!
f(n+1)(c)
 =3D
`= 0;/2
(b –      )n+1
(n + 1)!
f(n+1)(c)
`= 0;
2 =
Taylor= 's Remainder Theorem II
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0438_image065.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABM AAEAgAAAAAAAAAIHjI+py+3vCgA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0438_image066.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABM AAEAgAAAAAAAAAIHjI+py+3vCgA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0438_image067.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0439.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example:
 
B. Describe the Taylor remainder Rn(b) and show that
f(b) =3D P(b) for all value= s b. (a =3D      )
The remainder Rn(b) =3D
(b – a)n+1
(n + 1)!
f(n+1)(c)
 =3D
`= 0;/2
(b –      )n+1
(n + 1)!
f(n+1)(c)
`= 0;
2 =
where f(n+1) may be ±sin(x) or = ±cos(x) and for some c that is = between a and b.
Taylor= 's Remainder Theorem II
------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0439_image068.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABM AAEAgAAAAAAAAAIHjI+py+3vCgA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0439_image069.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABM AAEAgAAAAAAAAAIHjI+py+3vCgA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0439_image070.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.C685A760 Content-Location: file:///C:/AA871D69/34Taylor'sRemainderTheoremII_files/slide0440.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Example:
 
B. Describe the Taylor remainder Rn(b) and show that
f(b) =3D P(b) for all value= s b. (a =3D      )
The remainder Rn(b) =3D
(b – a)n+1
(n + 1)!
f(n+1)(c)
 =3D
`= 0;/2
(b –      )n+1
(n + 1)!
f(n+1)(c)
`= 0;
2 =
where f(n+1) may be ±sin(x) or = ±cos(x) and for some c that is = between a and b.
Since |f(n+1)(c)| < 1,  <= span style=3D'mso-bidi-font-family:Arial'>
Taylor= 's Remainder Theorem II
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Example:
 
B. Describe the Taylor remainder Rn(b) and show that
f(b) =3D P(b) for all value= s b. (a =3D      )
The remainder Rn(b) =3D
(b – a)n+1
(n + 1)!
f(n+1)(c)
 =3D
`= 0;/2
(b –      )n+1
(n + 1)!
f(n+1)(c)
`= 0;
2 =
where f(n+1) may be ±sin(x) or = ±cos(x) and for some c that is = between a and b.
Since |f(n+1)(c)| < 1, we've 
(b –      )n+1
(n + 1)!
f(n+1)(c)
`= 0;
2 =
<
(n + 1)!
(b –      )n+1
`= 0;
2 =
Taylor= 's Remainder Theorem II
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Example:
 
B. Describe the Taylor remainder Rn(b) and show that
f(b) =3D P(b) for all value= s b. (a =3D      )
The remainder Rn(b) =3D
(b – a)n+1
(n + 1)!
f(n+1)(c)
 =3D
`= 0;/2
(b –      )n+1
(n + 1)!
f(n+1)(c)
`= 0;
2 =
where f(n+1) may be ±sin(x) or = ±cos(x) and for some c that is = between a and b.
Since |f(n+1)(c)| < 1, we've