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Approxi= mations Using Taylor Expansions
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0314.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
We have the Taylor's Remainder Theorem:
Approxi= mations Using Taylor Expansions
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0304.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
We have the Taylor's Remainder Theorem:
Approxi= mations Using Taylor Expansions
Let f(x) be an infinitely differentiable function over = some open interval that contains [a, b]
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0309.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
We have the Taylor's Remainder Theorem:
Approxi= mations Using Taylor Expansions
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.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0309_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAEEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABp AQQAgAAAAAAAAAI5hI+py+0Po5y02ouzrqH7D4biSJbmiabqyrbuC8fyTNf2jef6zvf+DwwKU5fg 5ohMKpfMpvMJnRQAADs= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0309_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC8 AAoAgAAAAP8AAAKjhG+By+3/FJy0QmkXztxuFISTyJDRV5qOerBp42puDNItKsN4jYb7rRv9EjZi 8IIrAnwv5HD5jCmZsmjSKsQ2T8dTz6rlblez6VWYRY+7tXTOqRa/valwG+mW3/VA+Jxsx9ZHVxW3 R5aHuLYIlWjkx8do9hX4GFmIx3g4CKh5U/nHqUNpKOpJBGomWHcKBbpp2dkq9bREodpqmyt1+wqy 0utQAAA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0309_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMAAgAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAYAAQAo ABcAgAAAAAAAAAJBhI+pyxDRolzvzdtqxfzo3WGfFV4jVJpjOp2s5L7ZKjNxTdF4cu+e7jP0ZCQH sAYaIk9FHxMVNB6DSiczKnxiAQUAOw== ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0310.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
We have the Taylor's Remainder Theorem:
Approxi= mations Using Taylor Expansions
Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and pn(x) be the n'th Taylor-poly expanded around at a= ,
a
(
)
[
]
b
f(x) is infinitely differentiable in here
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0310_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAEEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABp AQQAgAAAAAAAAAI5hI+py+0Po5y02ouzrqH7D4biSJbmiabqyrbuC8fyTNf2jef6zvf+DwwKU5fg 5ohMKpfMpvMJnRQAADs= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0310_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC8 AAoAgAAAAP8AAAKjhG+By+3/FJy0QmkXztxuFISTyJDRV5qOerBp42puDNItKsN4jYb7rRv9EjZi 8IIrAnwv5HD5jCmZsmjSKsQ2T8dTz6rlblez6VWYRY+7tXTOqRa/valwG+mW3/VA+Jxsx9ZHVxW3 R5aHuLYIlWjkx8do9hX4GFmIx3g4CKh5U/nHqUNpKOpJBGomWHcKBbpp2dkq9bREodpqmyt1+wqy 0utQAAA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0310_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMAAgAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAYAAQAo ABcAgAAAAAAAAAJBhI+pyxDRolzvzdtqxfzo3WGfFV4jVJpjOp2s5L7ZKjNxTdF4cu+e7jP0ZCQH sAYaIk9FHxMVNB6DSiczKnxiAQUAOw== ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0311.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
We have the Taylor's Remainder Theorem:
Approxi= mations Using Taylor Expansions
Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and pn(x) be the n'th Taylor-poly expanded around at a= ,
then there exists a "c" between a and b
a
(
)
[
]
b
c
f(x) is infinitely differentiable in here
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0311_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAEEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABp AQQAgAAAAAAAAAI5hI+py+0Po5y02ouzrqH7D4biSJbmiabqyrbuC8fyTNf2jef6zvf+DwwKU5fg 5ohMKpfMpvMJnRQAADs= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0311_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC8 AAoAgQAAAAAAAP8AAAECAwK9hG+Cy+3/FJy0ghCatGvzb3mIQE4lc0Yimjrt8bKagpHYGMedjpsr wIPlgokfEfjDJYUzXzNAigoyyNUx6oTorsNlsan1VrM7cu8ijUKHPjEXrLK+ZeG2mVmnr9NTdtgN COfSdfeVF1eIlJiylwZFh3g4KDgjFwg5+bd4WXbQKEV1FqmyabfDKZo5qppKaYjwOYVZmdj6OmhZ Kkm7e+uq2NB4Y3tai9e5esyq7EIxhxzx64kROuKM2qxx7VAAADs= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0311_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMAAgAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAYAAQAo ABcAgAAAAAAAAAJBhI+pyxDRolzvzdtqxfzo3WGfFV4jVJpjOp2s5L7ZKjNxTdF4cu+e7jP0ZCQH sAYaIk9FHxMVNB6DSiczKnxiAQUAOw== ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0312.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
We have the Taylor's Remainder Theorem:
Approxi= mations Using Taylor Expansions
Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and pn(x) be the n'th Taylor-poly expanded around at 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
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0312_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhTgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAABM AAEAgAAAAAAAAAIHjI+py+3vCgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0312_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhbAEEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABp AQQAgAAAAAAAAAI5hI+py+0Po5y02ouzrqH7D4biSJbmiabqyrbuC8fyTNf2jef6zvf+DwwKU5fg 5ohMKpfMpvMJnRQAADs= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0312_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhvgAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC8 AAoAgQAAAAAAAP8AAAECAwK9hG+Cy+3/FJy0ghCatGvzb3mIQE4lc0Yimjrt8bKagpHYGMedjpsr wIPlgokfEfjDJYUzXzNAigoyyNUx6oTorsNlsan1VrM7cu8ijUKHPjEXrLK+ZeG2mVmnr9NTdtgN COfSdfeVF1eIlJiylwZFh3g4KDgjFwg5+bd4WXbQKEV1FqmyabfDKZo5qppKaYjwOYVZmdj6OmhZ Kkm7e+uq2NB4Y3tai9e5esyq7EIxhxzx64kROuKM2qxx7VAAADs= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0312_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMAAgAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAYAAQAo ABcAgAAAAAAAAAJBhI+pyxDRolzvzdtqxfzo3WGfFV4jVJpjOp2s5L7ZKjNxTdF4cu+e7jP0ZCQH sAYaIk9FHxMVNB6DSiczKnxiAQUAOw== ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0313.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
We have the Taylor's Remainder Theorem:
Approxi= mations Using Taylor Expansions
Let f(x) be an infinitely differentiable function over some open interval that contains [a, b] and pn(x) be the n'th Taylor-poly expanded around at 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
We = use the remainder formula to control the error when we use Taylor polynomials = to approximate f(b).
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An approximation is said = to be accurate to n decimal places if the error is less than 0.5 x 10-n. Therefore,
Approxi= mations Using Taylor Expansions
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An approximation is said = to be accurate to n decimal places if the error is less than 0.5 x 10-n. Therefore,
1 decimal place of accuracy ßà error < 0.05
Approxi= mations Using Taylor Expansions
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0316.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
An approximation is said = to be accurate to n decimal places if the error is less than 0.5 x 10-n. Therefore,
1 decimal place of accuracy ßà error < 0.05
Approxi= mations Using Taylor Expansions
2 decimal places of accuracy ßà error < 0.005
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An approximation is said = to be accurate to n decimal places if the error is less than 0.5 x 10-n. Therefore,
1 decimal place of accuracy ßà error < 0.05
Approxi= mations Using Taylor Expansions
2 decimal places of accuracy ßà error < 0.005
3 decimal places of accuracy ßà error < 0.0005<= /div>
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An approximation is said = to be accurate to n decimal places if the error is less than 0.5 x 10-n. Therefore,
1 decimal place of accuracy ßà error < 0.05
Approxi= mations Using Taylor Expansions
2 decimal places of accuracy ßà error < 0.005
3 decimal places of accuracy ßà error < 0.0005<= /div>
Example: The Mac-polynomial of ex is
x
+
2!
1
+
x2
+ ..
+
+
3!
x3
n!
xn
pn(x) =3D<= /div>
Find an n such that pn(1) approximates e with an accuracy to 4 decimal places.  
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0321_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0321_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0321_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0346.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
An approximation is said = to be accurate to n decimal places if the error is less than 0.5 x 10-n. Therefore,
1 decimal place of accuracy ßà error < 0.05
Approxi= mations Using Taylor Expansions
2 decimal places of accuracy ßà error < 0.005
3 decimal places of accuracy ßà error < 0.0005<= /div>
Example: The Mac-polynomial of ex is
x
+
2!
1
+
x2
+ ..
+
+
3!
x3
n!
xn
pn(x) =3D<= /div>
Find an n such that pn(1) approximates e1 with an <= span lang=3DEN-US style=3D'mso-fareast-font-family:SimSun;mso-hansi-font-family= :Arial; font-size:156%;color:#006600;mso-fareast-language:ZH-CN'>accuracy to 4 dec= imal places.   =
f(n+1)(c)
(n+1)!
bn+1  is
The error term R<= /span>n(b) =3D
ec
(n+1)!
1n+1
for some c betwee= n 0 and b =3D 1.
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0346_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0346_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0346_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhFAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAS AAEAgAAAAAAAAAIEjI95BQA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0346_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0346_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0331.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Approxi= mations Using Taylor Expansions
Assuming we know = that e < 3, then the error
ec
(n+1)!
Rn(1) =3D
<
3
(n+1)!
For n large enough such that
we will have pn(1)  accurate to 4 dec= imal places.
0.00005 >
3
(n+1)!
By trying different values = for n (trial and error),
we found 0.00005 > 3/9!,= or n =3D 8 is large enough.
 
So p8(1) » e =3D 1 + 1 + 1/2! + 1/3! + .. + 1/8! » 2.7183 approximates e accurately to at least 4 places.
 >
Rn(1)
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0331_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0331_image027.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0331_image028.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0308.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Approxi= mations Using Taylor Expansions
Example: Approximate cos(3o) to an accuracy = of 3 decimal places.
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0336.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Approxi= mations Using Taylor Expansions
Example: Approximate cos(3o) to an accuracy = of 3 decimal places.
Set b =3D 3o as 
π/60.
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0337.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Approxi= mations Using Taylor Expansions
Example: Approximate cos(3o) to an accuracy = of 3 decimal places.
1
–
pn(x) =3D<= /div>
Expanding around = 0, the Mac-poly of cos(x) is
+
4!
x4
6!
x6
8!
x8
+
–
2!
x2
..
Set b =3D 3o as 
π/60.
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0337_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0337_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0337_image031.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0337_image032.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0338.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Approxi= mations Using Taylor Expansions
Example: Approximate cos(3o) to an accuracy = of 3 decimal places.
1
–
pn(x) =3D<= /div>
with the error te= rm Rn(   ) =3D<= /div>
(n+1)!
Expanding around = 0, the Mac-poly of cos(x) is
+
4!
x4
6!
x6
8!
x8
+
–
2!
x2
..
Set b =3D 3o as 
π/60.
π =
60
(
)n+1,
cos(n+1)(c)
π =
60
,and at b =3D 
π/60,
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0338_image033.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0338_image034.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0338_image035.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0338_image036.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0338_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0338_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0338_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0339.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Approxi= mations Using Taylor Expansions
Example: Approximate cos(3o) to an accuracy = of 3 decimal places.
1
–
pn(x) =3D<= /div>
with the error te= rm Rn(   ) =3D<= /div>
(n+1)!
Expanding around = 0, the Mac-poly of cos(x) is
+
4!
x4
6!
x6
8!
x8
+
–
2!
x2
..
Set b =3D 3o as 
π/60.
π =
60
(
)n+1,
where c is some number between 0 and
π/60.
cos(n+1)(c)
π =
60
,and at b =3D 
π/60,
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0339_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0339_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0339_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0339_image043.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0339_image044.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0339_image045.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0339_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0340.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Approxi= mations Using Taylor Expansions
Example: Approximate cos(3o) to an accuracy = of 3 decimal places.
1
–
pn(x) =3D<= /div>
with the error te= rm Rn(   ) =3D<= /div>
(n+1)!
Expanding around = 0, the Mac-poly of cos(x) is
+
4!
x4
6!
x6
8!
x8
+
–
2!
x2
..
Set b =3D 3o as 
π/60.
π =
60
(
)n+1,
where c is some number between 0 and
π/60.
|cosn+1(c)| < 1,
cos(n+1)(c)
π =
60
,and at b =3D 
π/60,
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0340_image047.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0340_image048.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0340_image049.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0340_image050.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0340_image051.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0340_image052.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0340_image053.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0340_image054.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0341.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Approxi= mations Using Taylor Expansions
Example: Approximate cos(3o) to an accuracy = of 3 decimal places.
1
–
pn(x) =3D<= /div>
with the error te= rm Rn(   ) =3D<= /div>
(n+1)!
Expanding around = 0, the Mac-poly of cos(x) is
+
4!
x4
6!
x6
8!
x8
+
–
2!
x2
..
Set b =3D 3o as 
π/60.
π =
60
(
)n+1,
where c is some number between 0 and
π/60.
|cosn+1(c)| < 1, hence |Rn(    )| <
cos(n+1)(c)
π =
60
,and at b =3D 
π/60,
π =
60
(n+1)!
(π/60)n+1
.
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0341_image055.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0341_image056.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0341_image057.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0341_image058.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0341_image059.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0341_image060.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0341_image061.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0341_image062.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0341_image063.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0341_image064.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhVwAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABV AAIAgAAAAAAAAAIRhI+pGu0Po5y02riy3rx7UAAAOw== ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0342.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Approxi= mations Using Taylor Expansions
Example: Approximate cos(3o) to an accuracy = of 3 decimal places.
1
–
pn(x) =3D<= /div>
with the error te= rm Rn(   ) =3D<= /div>
(n+1)!
Expanding around = 0, the Mac-poly of cos(x) is
+
4!
x4
6!
x6
8!
x8
+
–
2!
x2
..
Set b =3D 3o as 
π/60.
π =
60
(
)n+1,
where c is some number between 0 and
π/60.
|cosn+1(c)| < 1, hence |Rn(    )| <
cos(n+1)(c)
π =
60
,and at b =3D 
π/60,
π =
60
=
(n+1)!
(π/60)n+1
.
We want 0.0005 &g= t;
(n+1)!
(π/60)n+1
.
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Approxi= mations Using Taylor Expansions
Example: Approximate cos(3o) to an accuracy = of 3 decimal places.
1
–
pn(x) =3D<= /div>
with the error te= rm Rn(   ) =3D<= /div>
(n+1)!
Expanding around = 0, the Mac-poly of cos(x) is
+
4!
x4
6!
x6
8!
x8
+
–
2!
x2
..
Set b =3D 3o as 
π/60.
π =
60
(
)n+1,
where c is some number between 0 and
π/60.
|cosn+1(c)| < 1, hence |Rn(    )| <
cos(n+1)(c)
π =
60
,and at b =3D 
π/60,
π =
60
=
(n+1)!
(π/60)n+1
.
We want 0.0005 &g= t;
(n+1)!
(π/60)n+1
.
By trial and erro= r n =3D 2 is sufficient.  =
------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343_image076.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343_image077.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343_image078.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAY AAEAgAAAAAAAAAIEjI+pVwA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343_image079.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343_image080.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343_image081.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343_image082.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343_image083.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAN AAEAgAAAAAAAAAIDjI9XADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343_image084.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAO AAEAgAAAAAAAAAIDjI9YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343_image085.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhVwAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABV AAIAgAAAAAAAAAIRhI+pGu0Po5y02riy3rx7UAAAOw== ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0343_image086.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA7 AAEAgAAAAAAAAAIGjI+py+1YADt= ------=_NextPart_01C97CE6.E719F490 Content-Location: file:///C:/23664A33/35ApproximationusingTaylorExpansions_files/slide0344.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
Approxi= mations Using Taylor Expansions
Example: Approximate cos(3o) to an accuracy = of 3 decimal places.
1
–
pn(x) =3D<= /div>
with the error te= rm Rn(   ) =3D<= /div>
(n+1)!
Expanding around = 0, the Mac-poly of cos(x) is
+
4!
x4
6!
x6
8!
x8
+
–
2!
x2
..
Set b =3D 3o as 
π/60.
π =
60
(
)n+1,
where c is some number between 0 and
π/60.
|cosn+1(c)| < 1, hence |Rn(    )| <
cos(n+1)(c)
π =
60
,and at b =3D 
π/60,
π =
60
=
(n+1)!
(π/60)n+1
.
We want 0.0005 &g= t;
(n+1)!
(π/60)n+1
.
By trial and erro= r n =3D 2 is sufficient.  =
So p2(     )=
π/60
=3D 1 – (π= ;/60)<= /span>2/2
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