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Max-estimate version of Lagrange remainder formula - Revision history
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https://calculus.subwiki.org/w/index.php?title=Max-estimate_version_of_Lagrange_remainder_formula&diff=2088&oldid=prev
Vipul at 20:17, 12 July 2012
2012-07-12T20:17:22Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:17, 12 July 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td>
<td colspan="2" class="diff-lineno">Line 11:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If <math>f^{(n+1)})</math> is continuous on <math>J</math>, the <math>\sup</math> can be replaced by <math>\max</math>:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If <math>f^{(n+1)})</math> is continuous on <math>J</math>, the <math>\sup</math> can be replaced by <math>\max</math>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>|R_n(f;x_0)(x)| \le \left( \<del style="font-weight: bold; text-decoration: none;">sup_</del>{t \in J} |f^{(n+1)}(t)|\right) \frac{|x - x_0|^{n+1}}{(n+1)!}</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>|R_n(f;x_0)(x)| \le \left( \<ins style="font-weight: bold; text-decoration: none;">max_</ins>{t \in J} |f^{(n+1)}(t)|\right) \frac{|x - x_0|^{n+1}}{(n+1)!}</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===About the point 0===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===About the point 0===</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17">Line 17:</td>
<td colspan="2" class="diff-lineno">Line 17:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any <math>x</math>, let <math>J</math> is the interval between <math>x</math> and <math>0</math> (it might be the interval <math>[x,0]</math> or <math>[0,x]</math> depending on whether <math>x < 0</math> or <math>x > 0</math>). If <math>f</math> is <math>n + 1</math> times differentiable everywhere on <math>J</math>, then we have:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any <math>x</math>, let <math>J</math> is the interval between <math>x</math> and <math>0</math> (it might be the interval <math>[x,0]</math> or <math>[0,x]</math> depending on whether <math>x < 0</math> or <math>x > 0</math>). If <math>f</math> is <math>n + 1</math> times differentiable everywhere on <math>J</math>, then we have:</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math>|R_n(f;0)(x)| \le \left( \sup_{t \in J} |f^{(n+1)}(t)|\right) \frac{|x|^{n+1}}{(n+1)!}</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If <math>f^{(n+1)})</math> is continuous on <math>J</math>, the <math>\sup</math> can be replaced by <math>\max</math>:</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>|R_n(f;0)(x)| \le \left( \max_{t \in J} |f^{(n+1)}(t)|\right) \frac{|x|^{n+1}}{(n+1)!}</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>|R_n(f;0)(x)| \le \left( \max_{t \in J} |f^{(n+1)}(t)|\right) \frac{|x|^{n+1}}{(n+1)!}</math></div></td></tr>
</table>
Vipul
https://calculus.subwiki.org/w/index.php?title=Max-estimate_version_of_Lagrange_remainder_formula&diff=2087&oldid=prev
Vipul at 20:16, 12 July 2012
2012-07-12T20:16:44Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:16, 12 July 2012</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td>
<td colspan="2" class="diff-lineno">Line 7:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any <math>x</math>, let <math>J</math> is the interval between <math>x</math> and <math>x_0</math> (it might be the interval <math>[x,x_0]</math> or <math>[x_0,x]</math> depending on whether <math>x < x_0</math> or <math>x > x_0</math>). If <math>f</math> is <math>n + 1</math> times differentiable everywhere on <math>J</math>, then we have:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any <math>x</math>, let <math>J</math> is the interval between <math>x</math> and <math>x_0</math> (it might be the interval <math>[x,x_0]</math> or <math>[x_0,x]</math> depending on whether <math>x < x_0</math> or <math>x > x_0</math>). If <math>f</math> is <math>n + 1</math> times differentiable everywhere on <math>J</math>, then we have:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>|R_n(f;x_0)(x)| \le \left( \<del style="font-weight: bold; text-decoration: none;">max_</del>{t \in J} |f^{(n+1)}(t)|\right) \frac{|x - x_0|^{n+1}}{(n+1)!}</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>|R_n(f;x_0)(x)| \le \left( \<ins style="font-weight: bold; text-decoration: none;">sup_</ins>{t \in J} |f^{(n+1)}(t)|\right) \frac{|x - x_0|^{n+1}}{(n+1)!}</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If <math>f^{(n+1)})</math> is continuous on <math>J</math>, the <math>\sup</math> can be replaced by <math>\max</math>:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math>|R_n(f;x_0)(x)| \le \left( \sup_{t \in J} |f^{(n+1)}(t)|\right) \frac{|x - x_0|^{n+1}}{(n+1)!}</math></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===About the point 0===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===About the point 0===</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
</table>
Vipul
https://calculus.subwiki.org/w/index.php?title=Max-estimate_version_of_Lagrange_remainder_formula&diff=2085&oldid=prev
Vipul: Vipul moved page Max-estimate version of Lagrange formula to Max-estimate version of Lagrange remainder formula
2012-07-12T20:04:46Z
<p>Vipul moved page <a href="/wiki/Max-estimate_version_of_Lagrange_formula" class="mw-redirect" title="Max-estimate version of Lagrange formula">Max-estimate version of Lagrange formula</a> to <a href="/wiki/Max-estimate_version_of_Lagrange_remainder_formula" title="Max-estimate version of Lagrange remainder formula">Max-estimate version of Lagrange remainder formula</a></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<tr class="diff-title" lang="en">
<td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:04, 12 July 2012</td>
</tr><tr><td colspan="2" class="diff-notice" lang="en"><div class="mw-diff-empty">(No difference)</div>
</td></tr></table>
Vipul
https://calculus.subwiki.org/w/index.php?title=Max-estimate_version_of_Lagrange_remainder_formula&diff=1984&oldid=prev
Vipul: Created page with "==Statement== ===About a general point=== Suppose <math>f</math> is a function of one variable and <math>x_0</math> is a point in the domain such that <math>f</math> is <mat..."
2012-07-07T14:46:57Z
<p>Created page with "==Statement== ===About a general point=== Suppose <math>f</math> is a function of one variable and <math>x_0</math> is a point in the domain such that <math>f</math> is <mat..."</p>
<p><b>New page</b></p><div>==Statement==<br />
<br />
===About a general point===<br />
<br />
Suppose <math>f</math> is a function of one variable and <math>x_0</math> is a point in the domain such that <math>f</math> is <math>(n + 1)</math> times differentiable at <math>x_0</math>. Denote by <math>R_n(f;x_0)</math> the function of <math>x</math> given by <math>x \mapsto f(x) - P_n(f;x_0)(x)</math>, i.e., <math>R_n(f;x_0)</math> is the '''remainder''' when we subtract from <math>f</math> its <math>n^{th}</math> [[Taylor polynomial]] at <math>x_0</math>. <br />
<br />
For any <math>x</math>, let <math>J</math> is the interval between <math>x</math> and <math>x_0</math> (it might be the interval <math>[x,x_0]</math> or <math>[x_0,x]</math> depending on whether <math>x < x_0</math> or <math>x > x_0</math>). If <math>f</math> is <math>n + 1</math> times differentiable everywhere on <math>J</math>, then we have:<br />
<br />
<math>|R_n(f;x_0)(x)| \le \left( \max_{t \in J} |f^{(n+1)}(t)|\right) \frac{|x - x_0|^{n+1}}{(n+1)!}</math><br />
<br />
===About the point 0===<br />
<br />
Suppose <math>f</math> is a function of one variable such that <math>f</math> is <math>(n + 1)</math> times differentiable at <math>0</math>. Denote by <math>R_n(f;0)</math> the function of <math>x</math> given by <math>x \mapsto f(x) - P_n(f;0)(x)</math>, i.e., <math>R_n(f;0)</math> is the '''remainder''' when we subtract from <math>f</math> its <math>n^{th}</math> [[Taylor polynomial]] at <math>0</math>.<br />
<br />
For any <math>x</math>, let <math>J</math> is the interval between <math>x</math> and <math>0</math> (it might be the interval <math>[x,0]</math> or <math>[0,x]</math> depending on whether <math>x < 0</math> or <math>x > 0</math>). If <math>f</math> is <math>n + 1</math> times differentiable everywhere on <math>J</math>, then we have:<br />
<br />
<math>|R_n(f;0)(x)| \le \left( \max_{t \in J} |f^{(n+1)}(t)|\right) \frac{|x|^{n+1}}{(n+1)!}</math></div>
Vipul