<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_10_ex07_3" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="a8403a6ffdf811ee9b5616fff75c5923">
<h2 class="hd hd-2 unit-title">Fourier Decomposition Continued</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@lect_10_07_3">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@lect_10_07_3" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="a8403a6ffdf811ee9b5616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><h2>FUNCTIONS OVER THE INTERVAL [0,L]</h2> The general solution for an arbitrary waveform on a string is a summation over all possible normal modes: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\psi (x,t) = \sum _{m} A_{m}\sin \left(\omega _{m}t + \beta _{m}\right)\sin \left(k_{m}x + \alpha _{m}\right)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>1</span>)</span></td></tr></table><p>
In the case where the string is deformed to a particular shape and released from rest, we have: </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\frac{\partial }{\partial t}{\psi }(x,t=0)=0[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table><p>
from which [mathjaxinline]\cos \left(\beta _{m}\right)=0[/mathjaxinline], and therefore [mathjaxinline]\beta _{m}=\frac{\pi }{2}[/mathjaxinline]. </p><p>
Previously, we considered functions defined over the interval [mathjaxinline][0,L][/mathjaxinline]. For instance, a system with fixed ends over this interval has the following constraints: [mathjaxinline]\alpha _{m}=0[/mathjaxinline] and [mathjaxinline]k_{m}=\frac{m\pi }{L}[/mathjaxinline]. Thus, the normal modes of a system with fixed ends over this interval have the following form: </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\psi _{m}(x,0) = A_{m} \sin \left(\dfrac {m\pi }{L}x\right)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>3</span>)</span></td></tr></table><p>
In this case, the amplitudes are calculated using the following integral: </p><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]A_{m}=\frac{2}{L}\int _{0}^{L}\psi (x,0)\sin \left(\frac{m\pi }{L}x\right)dx[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>4</span>)</span></td></tr></table><p><h2>FUNCTIONS OVER THE INTERVAL [-L,L]</h2></p><p>
Now we will see how to perform this analysis for functions defined over the interval [mathjaxinline][-L,L][/mathjaxinline]. Again, we assume that the string is released from rest, so [mathjaxinline]\beta _{m}=\frac{\pi }{2}[/mathjaxinline]. Let's again consider the case with fixed ends. </p><p>
The normal modes of this system are identical (qualitatively) to the normal modes of the system over the interval [mathjaxinline][0,L][/mathjaxinline]. They look like sine functions, starting from [mathjaxinline]x=-L[/mathjaxinline], but one can see that the lowest normal mode should actually be a <b class="bfseries">cosine function</b>, since it should peak at [mathjaxinline]x=0[/mathjaxinline] (see below). </p><center><img src="/assets/courseware/v1/1896264b8dedbf9ae0eff170ce1b39f5/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_w6_psp1a.png" width="440"/></center><p>
The second lowest normal mode is a <b class="bfseries">sine function</b>, and it will have a node at [mathjaxinline]x=0[/mathjaxinline] (see below). </p><center><img src="/assets/courseware/v1/fa326bd5e3b3c92514b3eaa695794e3d/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_w6_psp1b.png" width="440"/></center><p>
The third normal mode is a <b class="bfseries">cosine function</b> again (see below). So what's going on here? </p><center><img src="/assets/courseware/v1/8a7e34537f90ed883b14f2282391dcba/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_w6_psp1c.png" width="440"/></center><p>
To understand the functional form of the normal modes in this system, we can transform the normal modes from the [mathjaxinline][0,L][/mathjaxinline] system. Let the new normal modes have the following functional form: </p><table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\psi _{m}(x,0) = A_{m} \sin \left(\dfrac {m\pi }{L}x'\right)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>5</span>)</span></td></tr></table><p>
We can see that, over the interval [mathjaxinline][-L,L][/mathjaxinline], the relation [mathjaxinline]x'=\frac{x+L}{2}[/mathjaxinline] will map to the behavior of the normal modes from the [mathjaxinline][0,L][/mathjaxinline] system—see that [mathjaxinline]x=-L[/mathjaxinline] corresponds to [mathjaxinline]x'=0[/mathjaxinline], and [mathjaxinline]x=L[/mathjaxinline] corresponds to [mathjaxinline]x'=L[/mathjaxinline]. </p><p>
Therefore, the actual normal modes of the system are: </p><table id="a0000000007" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000008"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \psi _{m}(x,0)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = A_{m} \sin \left(\dfrac {m\pi }{2L}(x+L)\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>6</span>)</span></td></tr><tr id="a0000000009"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = A_{m}\sin \left(\dfrac {m\pi x}{2L} + \dfrac {m\pi }{2}\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>7</span>)</span></td></tr></table><p><b class="bfseries">NOTE:</b> we can conclude that [mathjaxinline]k_{m}=\frac{m\pi }{2L}[/mathjaxinline] and [mathjaxinline]\alpha _{m}=\frac{m\pi }{2}[/mathjaxinline]. </p><p><h2>ODD AND EVEN NORMAL MODES</h2></p><p>
This is a bit strange. We would like to express the normal modes in the form of [mathjaxinline]\sin (k_{m}x)[/mathjaxinline] and/or [mathjaxinline]\cos (k_{m}x)[/mathjaxinline] in order to perform easy integration. Let's expand the expression above: </p><table id="a0000000010" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\sin \left(\dfrac {m\pi x}{2L} + \dfrac {m\pi }{2}\right) = \sin \left(\dfrac {m\pi x}{2L}\right)\cos \left(\dfrac {m\pi }{2}\right) + \cos \left(\dfrac {m\pi x}{2L}\right)\sin \left(\dfrac {m\pi }{2}\right)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>8</span>)</span></td></tr></table><p>
Now we're getting in good shape. We can see that [mathjaxinline]\cos \left(\frac{m\pi }{2}\right)=0[/mathjaxinline] for odd values of [mathjaxinline]m[/mathjaxinline] and [mathjaxinline]\sin \left(\frac{m\pi }{2}\right)=0[/mathjaxinline] for even values of [mathjaxinline]m[/mathjaxinline]. Consequently, the normal modes are either [mathjaxinline]\cos \left(\frac{m\pi x}{2L}\right)[/mathjaxinline] for odd [mathjaxinline]m[/mathjaxinline] or [mathjaxinline]\sin \left(\frac{m\pi x}{2L}\right)[/mathjaxinline] for even [mathjaxinline]m[/mathjaxinline]. </p><p>
Thus, we can redefine the general solution to be the following: </p><table id="a0000000011" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\psi (x,t) = \sum _{m=\mathrm{even}} B_{m}\cos \left(\omega _{m}t\right)\sin \left(\dfrac {m\pi x}{2L}\right) + \sum _{m=\mathrm{odd}} C_{m}\cos \left(\omega _{m}t\right)\cos \left(\dfrac {m\pi x}{2L}\right)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>9</span>)</span></td></tr></table><p>
The amplitudes [mathjaxinline]B_{m}[/mathjaxinline] and [mathjaxinline]C_{m}[/mathjaxinline] are found by the following integrals, and are defined for even and odd values of [mathjaxinline]m[/mathjaxinline], respectively: </p><table id="a0000000012" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]B_{m}=\frac{1}{L}\int _{-L}^{L}\psi (x,0) \sin \left(\dfrac {m\pi x}{2L}\right)dx[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>10</span>)</span></td></tr></table><table id="a0000000013" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]C_{m}=\frac{1}{L}\int _{-L}^{L}\psi (x,0) \cos \left(\dfrac {m\pi x}{2L}\right)dx[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>11</span>)</span></td></tr></table><p><h2>RE-INDEXING THE NORMAL MODES</h2></p><p>
Alternatively, if we wanted all [mathjaxinline]m[/mathjaxinline] on the same footing, we could define [mathjaxinline]m\, \rightarrow \, m'[/mathjaxinline] such that the solution has the following form: </p><table id="a0000000014" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\psi (x,0) = \sum _{m'} B_{m'} \sin \left(\dfrac {m' \pi x}{L}\right) + \sum _{m'} C_{m'} \cos \left(\dfrac {(2m'-1)\pi x}{2L}\right)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>12</span>)</span></td></tr></table><p>
Where [mathjaxinline]m'[/mathjaxinline] does not index the normal modes, rather, it indexes the terms in the Fourier series. For this formulation, the amplitudes of the system are: </p><table id="a0000000015" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]B_{m'}=\frac{1}{L}\int _{-L}^{L}\psi (x,0)\sin \left(\dfrac {m'\pi x}{L}\right)dx[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>13</span>)</span></td></tr></table><table id="a0000000016" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]C_{m'}=\frac{1}{L}\int _{-L}^{L}\psi (x,0)\cos \left(\dfrac {(2m'-1)\pi x}{2L}\right)dx[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>14</span>)</span></td></tr></table><p>
Again, note that in this case [mathjaxinline]m'=1, 2, 3, \ldots[/mathjaxinline] do not necessarily correspond to the normal modes of the system that satisfy only the boundary conditions. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_10_ex07_2" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="a8403a6ffdf811ee9b5616fff75c5923">
<h2 class="hd hd-2 unit-title">Fourier Decomposition Practice</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2a" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="a8403a6ffdf811ee9b5616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_10_07_2a" class="problems-wrapper" role="group"
aria-labelledby="lect_10_07_2a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2a/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_10_07_2a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2a-problem-progress" tabindex="-1">
Fourier Decomposition Practice
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2a-problem-progress"></div>
<div class="problem">
<div>
<p>
In this problem, we will calculate [mathjaxinline]A_{m}[/mathjaxinline] given the following initial shape [mathjaxinline]\psi (x,0)[/mathjaxinline] for a string with fixed endpoints and extending over the range [mathjaxinline]x=-L[/mathjaxinline] to [mathjaxinline]x=L[/mathjaxinline] ([mathjaxinline]\psi (x,0)=0[/mathjaxinline] otherwise): </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\psi (x,0)= \begin{cases} A\frac{x}{L} + A &amp; -L \leq x \leq 0 \\ -A\frac{x}{L} + A &amp; 0 \lt x \leq L \\ \end{cases}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> In the preceding text, we showed that we can express the normal modes of this system as: </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\psi (x,t) = \sum _{m=\mathrm{even}} B_{m}\cos \left(\omega _{m}t\right)\sin \left(\dfrac {m\pi x}{2L}\right) + \sum _{m=\mathrm{odd}} C_{m}\cos \left(\omega _{m}t\right)\cos \left(\dfrac {m\pi x}{2L}\right)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
What modes will have nonzero amplitude? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_10_07_2a_2_1">
<fieldset aria-describedby="status_lect_10_07_2a_2_1">
<div class="field">
<input type="radio" name="input_lect_10_07_2a_2_1" id="input_lect_10_07_2a_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_10_07_2a_2_1-choice_1-label" for="input_lect_10_07_2a_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_10_07_2a_2_1">
<text> a) Even [mathjaxinline]m[/mathjaxinline] ([mathjaxinline]B_{m}\neq 0[/mathjaxinline])</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_10_07_2a_2_1" id="input_lect_10_07_2a_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_10_07_2a_2_1-choice_2-label" for="input_lect_10_07_2a_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_10_07_2a_2_1">
<text> b) Odd [mathjaxinline]m[/mathjaxinline] ([mathjaxinline]C_{m}\neq 0[/mathjaxinline])</text>
</label>
</div>
<span id="answer_lect_10_07_2a_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_10_07_2a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_10_07_2a_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Fourier Decomposition Practice" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_10_07_2a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_10_07_2a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_10_07_2a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_10_07_2a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_10_07_2a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2b" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="a8403a6ffdf811ee9b5616fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_10_07_2b" class="problems-wrapper" role="group"
aria-labelledby="lect_10_07_2b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2b/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_10_07_2b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2b-problem-progress" tabindex="-1">
Fourier Decomposition Practice
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_10_07_2b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]B_{m}[/mathjaxinline] and [mathjaxinline]C_{m}[/mathjaxinline]. We will presume that [mathjaxinline]B_{m}[/mathjaxinline] is defined for [mathjaxinline]m_{\mathrm{even}}[/mathjaxinline] and [mathjaxinline]C_{m}[/mathjaxinline] is defined for [mathjaxinline]m_{\mathrm{odd}}[/mathjaxinline]. Express your answer in terms of <code>A</code>, <code>L</code>, <code>m</code>, and <code>pi</code> for [mathjaxinline]\pi[/mathjaxinline], as needed. </p>
<p>
Hint: You may find the integration "trick" in a Lesson 15 question (or a variation of it) useful. </p>
<p>
<p style="display:inline">[mathjaxinline]B_{m} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_10_07_2b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_10_07_2b_2_1" id="input_lect_10_07_2b_2_1" aria-describedby="status_lect_10_07_2b_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_10_07_2b_2_1"/>
<span class="status unanswered" id="status_lect_10_07_2b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_10_07_2b_2_1" class="answer"/>
<div id="display_lect_10_07_2b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_10_07_2b_2_1_dynamath" name="input_lect_10_07_2b_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]C_{m} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_lect_10_07_2b_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_10_07_2b_3_1" id="input_lect_10_07_2b_3_1" aria-describedby="status_lect_10_07_2b_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_10_07_2b_3_1"/>
<span class="status unanswered" id="status_lect_10_07_2b_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_10_07_2b_3_1" class="answer"/>
<div id="display_lect_10_07_2b_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_10_07_2b_3_1_dynamath" name="input_lect_10_07_2b_3_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_10_07_2b_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Fourier Decomposition Practice" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_10_07_2b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_10_07_2b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_10_07_2b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_10_07_2b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_10_07_2b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>