https://wiki.math.ucr.edu/index.php?action=history&feed=atom&title=009C_Sample_Final_1%2C_Problem_10
009C Sample Final 1, Problem 10 - Revision history
2026-04-22T13:53:30Z
Revision history for this page on the wiki
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https://wiki.math.ucr.edu/index.php?title=009C_Sample_Final_1,_Problem_10&diff=1468&oldid=prev
MathAdmin at 04:14, 1 April 2017
2017-04-01T04:14:17Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 04:14, 1 April 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
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<tr><td class='diff-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><span class="exam">A curve is given <del class="diffchange diffchange-inline">in polar </del>parametrically by </div></td><td class='diff-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><span class="exam">A curve is given parametrically by </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>::::::<span class="exam"><math>x(t)=3\sin t</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>::::::<span class="exam"><math>x(t)=3\sin t</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>::::::<span class="exam"><math>y(t)=4\cos t</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>::::::<span class="exam"><math>y(t)=4\cos t</math></div></td></tr>
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MathAdmin
https://wiki.math.ucr.edu/index.php?title=009C_Sample_Final_1,_Problem_10&diff=1467&oldid=prev
MathAdmin at 04:07, 1 April 2017
2017-04-01T04:07:36Z
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<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;"></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;"></td></tr>
<tr><td class='diff-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>'''(b)'''</div></td><td class='diff-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 class="diffchange diffchange-inline">&thinsp;</ins>'''(b)'''</div></td></tr>
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<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>{| class="mw-collapsible mw-collapsed" style = "text-align:left;"</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>{| class="mw-collapsible mw-collapsed" style = "text-align:left;"</div></td></tr>
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MathAdmin
https://wiki.math.ucr.edu/index.php?title=009C_Sample_Final_1,_Problem_10&diff=1466&oldid=prev
MathAdmin at 04:03, 1 April 2017
2017-04-01T04:03:47Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 04:03, 1 April 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l24" >Line 24:</td>
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<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>'''Solution:'''</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>'''Solution:'''</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;"></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;"></td></tr>
<tr><td class='diff-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><del class="diffchange diffchange-inline">'''(a)'''</del></div></td><td class='diff-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> </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;"></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;"></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>{| class="mw-collapsible mw-collapsed" style = "text-align:left;"</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>{| class="mw-collapsible mw-collapsed" style = "text-align:left;"</div></td></tr>
<tr><td class='diff-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>!<del class="diffchange diffchange-inline">Step 1: </del>&nbsp; </div></td><td class='diff-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 class="diffchange diffchange-inline">(a) </ins>&nbsp; </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>|-</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>|-</div></td></tr>
<tr><td class='diff-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>|[[File:009C_SF1_10_GP.<del class="diffchange diffchange-inline">jpg</del>|<del class="diffchange diffchange-inline">350px</del>]]</div></td><td class='diff-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>|[[File:009C_SF1_10_GP.<ins class="diffchange diffchange-inline">png</ins>|<ins class="diffchange diffchange-inline">500px</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>|}</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>|}</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;"></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;"></td></tr>
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MathAdmin
https://wiki.math.ucr.edu/index.php?title=009C_Sample_Final_1,_Problem_10&diff=1464&oldid=prev
MathAdmin at 16:23, 28 March 2017
2017-03-28T16:23:04Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:23, 28 March 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l29" >Line 29:</td>
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<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>!Step 1: &nbsp; </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>!Step 1: &nbsp; </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>|-</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>|-</div></td></tr>
<tr><td class='diff-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>|[[File:009C_SF1_10_GP.jpg|<del class="diffchange diffchange-inline">300px</del>]]</div></td><td class='diff-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>|[[File:009C_SF1_10_GP.jpg|<ins class="diffchange diffchange-inline">350px</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>|}</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>|}</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;"></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;"></td></tr>
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MathAdmin
https://wiki.math.ucr.edu/index.php?title=009C_Sample_Final_1,_Problem_10&diff=1407&oldid=prev
MathAdmin at 20:08, 13 May 2016
2016-05-13T20:08:35Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:08, 13 May 2016</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l29" >Line 29:</td>
<td colspan="2" class="diff-lineno">Line 29:</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>!Step 1: &nbsp; </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>!Step 1: &nbsp; </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>|-</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>|-</div></td></tr>
<tr><td class='diff-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>|<del class="diffchange diffchange-inline">Insert sketch of curve</del></div></td><td class='diff-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 class="diffchange diffchange-inline">[[File:009C_SF1_10_GP.jpg|300px]]</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>|}</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>|}</div></td></tr>
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MathAdmin
https://wiki.math.ucr.edu/index.php?title=009C_Sample_Final_1,_Problem_10&diff=1394&oldid=prev
MathAdmin: Created page with "<span class="exam">A curve is given in polar parametrically by ::::::<span class="exam"><math>x(t)=3\sin t</math> ::::::<span class="exam"><math>y(t)=4\cos t</math> ::::::<sp..."
2016-04-19T01:30:27Z
<p>Created page with "<span class="exam">A curve is given in polar parametrically by ::::::<span class="exam"><math>x(t)=3\sin t</math> ::::::<span class="exam"><math>y(t)=4\cos t</math> ::::::<sp..."</p>
<p><b>New page</b></p><div><span class="exam">A curve is given in polar parametrically by <br />
::::::<span class="exam"><math>x(t)=3\sin t</math><br />
::::::<span class="exam"><math>y(t)=4\cos t</math><br />
::::::<span class="exam"><math>0\leq t \leq 2\pi</math><br />
<br />
::<span class="exam">a) Sketch the curve.<br />
<br />
::<span class="exam">b) Compute the equation of the tangent line at <math>t_0=\frac{\pi}{4}</math>.<br />
<br />
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"<br />
!Foundations: &nbsp; <br />
|-<br />
|'''1.''' What two pieces of information do you need to write the equation of a line?<br />
|-<br />
|<br />
::You need the slope of the line and a point on the line.<br />
|-<br />
|'''2.''' What is the slope of the tangent line of a parametric curve?<br />
|-<br />
|<br />
::The slope is <math style="vertical-align: -21px">m=\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}.</math><br />
|}<br />
<br />
'''Solution:'''<br />
<br />
'''(a)'''<br />
<br />
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"<br />
!Step 1: &nbsp; <br />
|-<br />
|Insert sketch of curve<br />
|}<br />
<br />
'''(b)'''<br />
<br />
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"<br />
!Step 1: &nbsp; <br />
|-<br />
|First, we need to find the slope of the tangent line. <br />
|-<br />
|Since <math style="vertical-align: -14px">\frac{dy}{dt}=-4\sin t</math> and <math style="vertical-align: -14px">\frac{dx}{dt}=3\cos t,</math> we have<br />
|-<br />
|<br />
::<math>\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{-4\sin t}{3\cos t}.</math><br />
|-<br />
|So, at <math>t_0=\frac{\pi}{4},</math> the slope of the tangent line is <br />
|-<br />
|<br />
::<math>m=\frac{-4\sin\bigg(\frac{\pi}{4}\bigg)}{3\cos\bigg(\frac{\pi}{4}\bigg)}=\frac{-4}{3}.</math><br />
|}<br />
<br />
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"<br />
!Step 2: &nbsp;<br />
|-<br />
|Since we have the slope of the tangent line, we just need a find a point on the line in order to write the equation.<br />
|-<br />
|If we plug in <math>t_0=\frac{\pi}{4}</math> into the equations for <math style="vertical-align: -5px">x(t)</math> and <math style="vertical-align: -5px">y(t),</math> we get<br />
|-<br />
|<br />
::<math>x\bigg(\frac{\pi}{4}\bigg)=3\sin\bigg(\frac{\pi}{4}\bigg)=\frac{3\sqrt{2}}{2}</math> and<br />
|-<br />
|<br />
::<math>y\bigg(\frac{\pi}{4}\bigg)=4\cos\bigg(\frac{\pi}{4}\bigg)=2\sqrt{2}.</math><br />
|-<br />
|Thus, the point <math>\bigg(\frac{3\sqrt{2}}{2},2\sqrt{2}\bigg)</math> is on the tangent line.<br />
|}<br />
<br />
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"<br />
!Step 3: &nbsp;<br />
|-<br />
|Using the point found in Step 2, the equation of the tangent line at <math>t_0=\frac{\pi}{4}</math> is <br />
|-<br />
|<br />
::<math>y=\frac{-4}{3}\bigg(x-\frac{3\sqrt{2}}{2}\bigg)+2\sqrt{2}.</math><br />
|}<br />
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"<br />
!Final Answer: &nbsp; <br />
|-<br />
|&nbsp;&nbsp; '''(a)''' See Step 1 above for the graph. <br />
|-<br />
|&nbsp;&nbsp; '''(b)''' <math style="vertical-align: -14px">y=\frac{-4}{3}\bigg(x-\frac{3\sqrt{2}}{2}\bigg)+2\sqrt{2}</math> <br />
|}<br />
[[009C_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]</div>
MathAdmin