<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@vertical+block@ps4-tab1" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">PS4.1. ps6</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+18.031+2019_Spring+type@html+block@ps4-tab1-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@html+block@ps4-tab1-text1" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bf">Note: there is no problem set this week but this activity should be good practice on feedback and convolutions. <br /></b></p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:OCW+18.031+2019_Spring+type@vertical+block@ps4-tab2">
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@vertical+block@ps4-tab2" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem1" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps4-tab2-problem1" class="problems-wrapper" role="group"
aria-labelledby="ps4-tab2-problem1-problem-title"
data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem1" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="3"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps4-tab2-problem1-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem1-problem-progress" tabindex="-1">
1(a)
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
A certain amplifier has system function [mathjaxinline]\displaystyle \frac{ca}{s+a}[/mathjaxinline], where [mathjaxinline]c[/mathjaxinline] and [mathjaxinline]a[/mathjaxinline] are positive real constants. </p>
<ul class="itemize">
<li>
<p>
Is this system stable? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ps4-tab2-problem1_2_1">
<fieldset aria-describedby="status_ps4-tab2-problem1_2_1">
<div class="field">
<input type="radio" name="input_ps4-tab2-problem1_2_1" id="input_ps4-tab2-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ps4-tab2-problem1_2_1-choice_1-label" for="input_ps4-tab2-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem1_2_1"> <text> yes</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps4-tab2-problem1_2_1" id="input_ps4-tab2-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="ps4-tab2-problem1_2_1-choice_2-label" for="input_ps4-tab2-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem1_2_1"> <text> no</text>
</label>
</div>
<span id="answer_ps4-tab2-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ps4-tab2-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div> </p>
</li>
<li>
<p>
How many independent transients are there for this system? (Enter a non-negative integer.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps4-tab2-problem1_3_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_ps4-tab2-problem1_3_1" id="input_ps4-tab2-problem1_3_1" aria-describedby="status_ps4-tab2-problem1_3_1" value=""/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem1_3_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem1_3_1" class="answer"/>
</div>
</div></div>
</p>
</li>
<li>
<p>
You can express the transients in terms of exponentials written as [mathjaxinline]e^{-t/t_0}[/mathjaxinline] (perhaps times sinusoids); [mathjaxinline]t_0[/mathjaxinline] is the "characteristic time" of this decaying exponential. Find [mathjaxinline]t_0[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]t_0=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_ps4-tab2-problem1_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem1_4_1" id="input_ps4-tab2-problem1_4_1" data-input-id="ps4-tab2-problem1_4_1" value="" aria-describedby="status_ps4-tab2-problem1_4_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem1_4_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem1_4_1" class="answer"/>
<div id="input_ps4-tab2-problem1_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
</li>
</ul>
<span>
<link href="/assets/courseware/v1/2d585b0f4e8978427e0be0805091392e/asset-v1:OCW+18.031+2019_Spring+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a href="#mymodal-one" class="mymodal-btn mymodal-btn-big">FORMULA 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>Formula Input Guide</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">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> 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 <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="mymodal-btn">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_ps4-tab2-problem1_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="1(a)" />
<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_ps4-tab2-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps4-tab2-problem1">
<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="ps4-tab2-problem1-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="ps4-tab2-problem1-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="ps4-tab2-problem1-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="False">
<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:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem2" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps4-tab2-problem2" class="problems-wrapper" role="group"
aria-labelledby="ps4-tab2-problem2-problem-title"
data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem2" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem2/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="4"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps4-tab2-problem2-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem2-problem-progress" tabindex="-1">
1(b)
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem2-problem-progress"></div>
<div class="problem">
<div>
<ul class="itemize">
<li>
<p>
Compute the gain function of this system. At what frequency [mathjaxinline]\omega _{\mathrm{max}}[/mathjaxinline] is the gain maximal? </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _{\mathrm{max}}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_ps4-tab2-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem2_2_1" id="input_ps4-tab2-problem2_2_1" data-input-id="ps4-tab2-problem2_2_1" value="" aria-describedby="status_ps4-tab2-problem2_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem2_2_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem2_2_1" class="answer"/>
<div id="input_ps4-tab2-problem2_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
</li>
<li>
<p>
What is the gain at [mathjaxinline]\omega =\omega _{\mathrm{max}}[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]g(\omega _{\mathrm{max}})=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_ps4-tab2-problem2_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem2_3_1" id="input_ps4-tab2-problem2_3_1" data-input-id="ps4-tab2-problem2_3_1" value="" aria-describedby="status_ps4-tab2-problem2_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem2_3_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem2_3_1" class="answer"/>
<div id="input_ps4-tab2-problem2_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
</li>
<li>
<p>
Estimate the gain for large [mathjaxinline]\omega[/mathjaxinline] (as a function of [mathjaxinline]\omega[/mathjaxinline]). </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \lim _{\omega \rightarrow \infty }g(\omega )=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_ps4-tab2-problem2_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem2_4_1" id="input_ps4-tab2-problem2_4_1" data-input-id="ps4-tab2-problem2_4_1" value="" aria-describedby="status_ps4-tab2-problem2_4_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem2_4_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem2_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem2_4_1" class="answer"/>
<div id="input_ps4-tab2-problem2_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
</li>
<li>
<p>
This system may be regarded as a low-pass filter. The "half-power point" is the frequency [mathjaxinline]\omega _{1/2}[/mathjaxinline] at which the gain is [mathjaxinline]\displaystyle \frac{1}{\sqrt {2}}[/mathjaxinline] times the maximal gain (because the power scales as the square of the amplitude). What is the half power frequency of this system? </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _{1/2}=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="formulaequationinput_ps4-tab2-problem2_5_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem2_5_1" id="input_ps4-tab2-problem2_5_1" data-input-id="ps4-tab2-problem2_5_1" value="" aria-describedby="status_ps4-tab2-problem2_5_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem2_5_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem2_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem2_5_1" class="answer"/>
<div id="input_ps4-tab2-problem2_5_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
</li>
</ul>
<span>
<link href="/assets/courseware/v1/2d585b0f4e8978427e0be0805091392e/asset-v1:OCW+18.031+2019_Spring+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a href="#mymodal-one" class="mymodal-btn mymodal-btn-big">FORMULA 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>Formula Input Guide</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">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> 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 <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="mymodal-btn">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_ps4-tab2-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="1(b)" />
<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_ps4-tab2-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps4-tab2-problem2">
<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="ps4-tab2-problem2-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="ps4-tab2-problem2-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="ps4-tab2-problem2-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="False">
<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-2" data-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem3" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps4-tab2-problem3" class="problems-wrapper" role="group"
aria-labelledby="ps4-tab2-problem3-problem-title"
data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem3" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem3/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="2"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps4-tab2-problem3-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem3-problem-progress" tabindex="-1">
1(c)
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
<div class="hideshowbox">
<h4 onclick="hideshow(this);" style="margin: 0px">Laplace table<span class="icon-caret-down toggleimage"/></h4>
<div class="hideshowcontent">
<p>
<h3>Calculations</h3>
</p>
<table id="a0000001199" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000001200">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \delta (t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 1[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.6)</td>
</tr>
<tr id="a0000001201">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac1{s}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.7)</td>
</tr>
<tr id="a0000001202">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)e^{rt}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac1{s-r}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.8)</td>
</tr>
<tr id="a0000001203">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\cos \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{s}{s^2+\omega ^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.9)</td>
</tr>
<tr id="a0000001204">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\sin \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{\omega }{s^2+\omega ^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.10)</td>
</tr>
<tr id="a0000001205">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac1{s^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.11)</td>
</tr>
<tr id="a0000001206">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t^ n[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{n!}{s^{n+1}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.12)</td>
</tr>
<tr id="a0000001207">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{2\omega s}{(s^2+\omega ^2)^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.13)</td>
</tr>
<tr id="a0000001208">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\cos (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.14)</td>
</tr>
<tr id="a0000001209">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega }t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{s}{(s^2+\omega ^2)^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.15)</td>
</tr>
<tr id="a0000001210">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega ^2}\left(\frac{1}{\omega }\sin (\omega t)-t\cos (\omega t)\right)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{(s^2+\omega ^2)^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.16)</td>
</tr>
</table>
<p>
<h3>Rules</h3>
</p>
<table id="a0000001211" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000001212">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle sF(s) - f(0^-), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.17)</td>
</tr>
<tr id="a0000001213">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle tf(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle -F'(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.18)</td>
</tr>
<tr id="a0000001214">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{at}f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle F(s-a), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-shift rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.19)</td>
</tr>
<tr id="a0000001215">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t-a)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}F(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, first form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.20)</td>
</tr>
<tr id="a0000001216">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}\mathcal{L}(f(t+a);s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, second form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.21)</td>
</tr>
</table>
</div>
<p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px">
<a href="javascript: {return false;}">Show</a>
</p>
</div>
</p>
<p>
Recall that certain amplifier has system function [mathjaxinline]\displaystyle \frac{ca}{s+a}[/mathjaxinline], where [mathjaxinline]c[/mathjaxinline] and [mathjaxinline]a[/mathjaxinline] are positive real constants. </p>
<p>
Compute the unit step response and the unit impulse response. </p>
<p>
<p style="display:inline">Impulse response for [mathjaxinline]t&gt;0[/mathjaxinline]: </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_ps4-tab2-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem3_2_1" id="input_ps4-tab2-problem3_2_1" data-input-id="ps4-tab2-problem3_2_1" value="" aria-describedby="status_ps4-tab2-problem3_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem3_2_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem3_2_1" class="answer"/>
<div id="input_ps4-tab2-problem3_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p>
<p style="display:inline">Step response for [mathjaxinline]t&gt;0[/mathjaxinline]: </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_ps4-tab2-problem3_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem3_3_1" id="input_ps4-tab2-problem3_3_1" data-input-id="ps4-tab2-problem3_3_1" value="" aria-describedby="status_ps4-tab2-problem3_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem3_3_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem3_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem3_3_1" class="answer"/>
<div id="input_ps4-tab2-problem3_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/2d585b0f4e8978427e0be0805091392e/asset-v1:OCW+18.031+2019_Spring+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a href="#mymodal-one" class="mymodal-btn mymodal-btn-big">FORMULA 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>Formula Input Guide</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">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> 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 <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="mymodal-btn">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_ps4-tab2-problem3_solution_1"/>
</div></p>
<SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:OCW+18.031+2019_Spring+type@asset+block/latex2edx.js" type="text/javascript"/>
<LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:OCW+18.031+2019_Spring+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="1(c)" />
<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_ps4-tab2-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps4-tab2-problem3">
<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="ps4-tab2-problem3-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="ps4-tab2-problem3-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="ps4-tab2-problem3-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="False">
<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-3" data-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem4">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem4" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps4-tab2-problem4" class="problems-wrapper" role="group"
aria-labelledby="ps4-tab2-problem4-problem-title"
data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem4" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem4/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps4-tab2-problem4-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem4-problem-progress" tabindex="-1">
1(d) Adding feedback
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem4-problem-progress"></div>
<div class="problem">
<div>
<p>
Now add a negative feedback branch with feedback a constant multiple [mathjaxinline]k[/mathjaxinline]. </p>
<center>
<img src="/assets/courseware/v1/3b90971b25b85abacaa85df22df3fe4b/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_ps3-3-block.png" width="360"/>
</center>
<div>
<br/>
</div>
<p>
What is the new system function? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_ps4-tab2-problem4_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem4_2_1" id="input_ps4-tab2-problem4_2_1" data-input-id="ps4-tab2-problem4_2_1" value="" aria-describedby="status_ps4-tab2-problem4_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem4_2_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem4_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem4_2_1" class="answer"/>
<div id="input_ps4-tab2-problem4_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/2d585b0f4e8978427e0be0805091392e/asset-v1:OCW+18.031+2019_Spring+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a href="#mymodal-one" class="mymodal-btn mymodal-btn-big">FORMULA 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>Formula Input Guide</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">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> 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 <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="mymodal-btn">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_ps4-tab2-problem4_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="1(d) Adding feedback" />
<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_ps4-tab2-problem4" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps4-tab2-problem4">
<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="ps4-tab2-problem4-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="ps4-tab2-problem4-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="ps4-tab2-problem4-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="False">
<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-4" data-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem5">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem5" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps4-tab2-problem5" class="problems-wrapper" role="group"
aria-labelledby="ps4-tab2-problem5-problem-title"
data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem5" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem5/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="9"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps4-tab2-problem5-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem5-problem-progress" tabindex="-1">
1(e)
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem5-problem-progress"></div>
<div class="problem">
<div>
<p>
Carry out steps (a)&#8211;(c) for this new system with negative feedback. </p>
<center>
<img src="/assets/courseware/v1/3b90971b25b85abacaa85df22df3fe4b/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_ps3-3-block.png" width="360"/>
</center>
<div>
<br/>
</div>
<p>
(a) </p>
<ul class="itemize">
<li>
<p>
Is this system with negative feedback stable? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ps4-tab2-problem5_2_1">
<fieldset aria-describedby="status_ps4-tab2-problem5_2_1">
<div class="field">
<input type="radio" name="input_ps4-tab2-problem5_2_1" id="input_ps4-tab2-problem5_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ps4-tab2-problem5_2_1-choice_1-label" for="input_ps4-tab2-problem5_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem5_2_1"> <text> yes</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps4-tab2-problem5_2_1" id="input_ps4-tab2-problem5_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="ps4-tab2-problem5_2_1-choice_2-label" for="input_ps4-tab2-problem5_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem5_2_1"> <text> no</text>
</label>
</div>
<span id="answer_ps4-tab2-problem5_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ps4-tab2-problem5_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div> </p>
</li>
<li>
<p>
How many independent transients are there for this system? (Enter a non-negative integer.) <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_ps4-tab2-problem5_3_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem5_3_1" id="input_ps4-tab2-problem5_3_1" data-input-id="ps4-tab2-problem5_3_1" value="" aria-describedby="status_ps4-tab2-problem5_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem5_3_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem5_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem5_3_1" class="answer"/>
<div id="input_ps4-tab2-problem5_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div> </p>
</li>
<li>
<p>
You can express the transients in terms of exponentials written as [mathjaxinline]e^{-t/t_0}[/mathjaxinline] (perhaps times sinusoids); [mathjaxinline]t_0[/mathjaxinline] is the &#8220;characteristic time" of this decaying exponential. Find [mathjaxinline]t_0[/mathjaxinline]. <p style="display:inline">[mathjaxinline]t_0=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_ps4-tab2-problem5_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem5_4_1" id="input_ps4-tab2-problem5_4_1" data-input-id="ps4-tab2-problem5_4_1" value="" aria-describedby="status_ps4-tab2-problem5_4_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem5_4_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem5_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem5_4_1" class="answer"/>
<div id="input_ps4-tab2-problem5_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div> </p>
</li>
</ul>
<p>
(b) </p>
<ul class="itemize">
<li>
<p>
Compute the gain function of this system. At what frequency [mathjaxinline]\omega _{\mathrm{max}}[/mathjaxinline] is the gain maximal? <p style="display:inline">[mathjaxinline]\omega _{\mathrm{max}}=[/mathjaxinline] </p><div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="formulaequationinput_ps4-tab2-problem5_5_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem5_5_1" id="input_ps4-tab2-problem5_5_1" data-input-id="ps4-tab2-problem5_5_1" value="" aria-describedby="status_ps4-tab2-problem5_5_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem5_5_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem5_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem5_5_1" class="answer"/>
<div id="input_ps4-tab2-problem5_5_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div> </p>
</li>
<li>
<p>
What is the gain at [mathjaxinline]\omega =\omega _{\mathrm{max}}[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]g(\omega _{\mathrm{max}})=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="formulaequationinput_ps4-tab2-problem5_6_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem5_6_1" id="input_ps4-tab2-problem5_6_1" data-input-id="ps4-tab2-problem5_6_1" value="" aria-describedby="status_ps4-tab2-problem5_6_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem5_6_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem5_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem5_6_1" class="answer"/>
<div id="input_ps4-tab2-problem5_6_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
</li>
<li>
<p>
Estimate the gain for large [mathjaxinline]\omega[/mathjaxinline] (as a function of [mathjaxinline]\omega[/mathjaxinline]). </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \lim _{\omega \rightarrow \infty }g(\omega )=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="formulaequationinput_ps4-tab2-problem5_7_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem5_7_1" id="input_ps4-tab2-problem5_7_1" data-input-id="ps4-tab2-problem5_7_1" value="" aria-describedby="status_ps4-tab2-problem5_7_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem5_7_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem5_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem5_7_1" class="answer"/>
<div id="input_ps4-tab2-problem5_7_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
</li>
<li>
<p>
This system may be regarded as a low-pass filter. The "half-power point" is the frequency [mathjaxinline]\omega _{1/2}[/mathjaxinline] at which the gain is [mathjaxinline]\frac{1}{\sqrt {2}}[/mathjaxinline] times the maximal gain (because the power scales as the square of the amplitude). What is the half power frequency of this system? </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _{1/2}=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 7" role="group"><div id="formulaequationinput_ps4-tab2-problem5_8_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem5_8_1" id="input_ps4-tab2-problem5_8_1" data-input-id="ps4-tab2-problem5_8_1" value="" aria-describedby="status_ps4-tab2-problem5_8_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem5_8_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem5_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem5_8_1" class="answer"/>
<div id="input_ps4-tab2-problem5_8_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
</li>
</ul>
<p>
(c) Compute the unit step response and unit impulse response for [mathjaxinline]t&gt;0[/mathjaxinline]. </p>
<p>
<div class="hideshowbox">
<h4 onclick="hideshow(this);" style="margin: 0px">Laplace table<span class="icon-caret-down toggleimage"/></h4>
<div class="hideshowcontent">
<p>
<h3>Calculations</h3>
</p>
<table id="a0000001226" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000001227">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \delta (t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 1[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.22)</td>
</tr>
<tr id="a0000001228">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac1{s}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.23)</td>
</tr>
<tr id="a0000001229">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)e^{rt}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac1{s-r}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.24)</td>
</tr>
<tr id="a0000001230">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\cos \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{s}{s^2+\omega ^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.25)</td>
</tr>
<tr id="a0000001231">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\sin \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{\omega }{s^2+\omega ^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.26)</td>
</tr>
<tr id="a0000001232">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac1{s^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.27)</td>
</tr>
<tr id="a0000001233">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t^ n[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{n!}{s^{n+1}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.28)</td>
</tr>
<tr id="a0000001234">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{2\omega s}{(s^2+\omega ^2)^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.29)</td>
</tr>
<tr id="a0000001235">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\cos (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.30)</td>
</tr>
<tr id="a0000001236">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega }t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{s}{(s^2+\omega ^2)^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.31)</td>
</tr>
<tr id="a0000001237">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega ^2}\left(\frac{1}{\omega }\sin (\omega t)-t\cos (\omega t)\right)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{(s^2+\omega ^2)^2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.32)</td>
</tr>
</table>
<p>
<h3>Rules</h3>
</p>
<table id="a0000001238" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000001239">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle sF(s) - f(0^-), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.33)</td>
</tr>
<tr id="a0000001240">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle tf(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle -F'(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.34)</td>
</tr>
<tr id="a0000001241">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{at}f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle F(s-a), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-shift rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.35)</td>
</tr>
<tr id="a0000001242">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t-a)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}F(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, first form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.36)</td>
</tr>
<tr id="a0000001243">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}\mathcal{L}(f(t+a);s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, second form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(8.37)</td>
</tr>
</table>
</div>
<p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px">
<a href="javascript: {return false;}">Show</a>
</p>
</div>
</p>
<p>
<p style="display:inline">Step response for [mathjaxinline]t&gt;0[/mathjaxinline]: </p>
<div class="inline" tabindex="-1" aria-label="Question 8" role="group"><div id="formulaequationinput_ps4-tab2-problem5_9_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem5_9_1" id="input_ps4-tab2-problem5_9_1" data-input-id="ps4-tab2-problem5_9_1" value="" aria-describedby="status_ps4-tab2-problem5_9_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem5_9_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem5_9_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem5_9_1" class="answer"/>
<div id="input_ps4-tab2-problem5_9_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p>
<p style="display:inline">Impulse response for [mathjaxinline]t&gt;0[/mathjaxinline]: </p>
<div class="inline" tabindex="-1" aria-label="Question 9" role="group"><div id="formulaequationinput_ps4-tab2-problem5_10_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_ps4-tab2-problem5_10_1" id="input_ps4-tab2-problem5_10_1" data-input-id="ps4-tab2-problem5_10_1" value="" aria-describedby="status_ps4-tab2-problem5_10_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab2-problem5_10_1"/>
<span class="status unanswered" id="status_ps4-tab2-problem5_10_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab2-problem5_10_1" class="answer"/>
<div id="input_ps4-tab2-problem5_10_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/2d585b0f4e8978427e0be0805091392e/asset-v1:OCW+18.031+2019_Spring+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a href="#mymodal-one" class="mymodal-btn mymodal-btn-big">FORMULA 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>Formula Input Guide</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">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> 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 <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="mymodal-btn">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_ps4-tab2-problem5_solution_1"/>
</div></p>
<SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:OCW+18.031+2019_Spring+type@asset+block/latex2edx.js" type="text/javascript"/>
<LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:OCW+18.031+2019_Spring+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="1(e)" />
<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_ps4-tab2-problem5" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps4-tab2-problem5">
<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">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</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="ps4-tab2-problem5-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="ps4-tab2-problem5-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="ps4-tab2-problem5-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="False">
<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-5" data-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem6">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem6" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps4-tab2-problem6" class="problems-wrapper" role="group"
aria-labelledby="ps4-tab2-problem6-problem-title"
data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem6" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem6/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps4-tab2-problem6-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem6-problem-progress" tabindex="-1">
1(f) Analyze results
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab2-problem6-problem-progress"></div>
<div class="problem">
<div>
<p>
What effect does adding the feedback have on the pole(s) of the system? Has it improved stability? What other consequences are there to adding feedback to this system? (Choose all that apply.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ps4-tab2-problem6_2_1">
<fieldset aria-describedby="status_ps4-tab2-problem6_2_1">
<div class="field">
<input type="checkbox" name="input_ps4-tab2-problem6_2_1[]" id="input_ps4-tab2-problem6_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="ps4-tab2-problem6_2_1-choice_0-label" for="input_ps4-tab2-problem6_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem6_2_1">
<text>increases the rate at which the unforced system returns to equilibrium</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_ps4-tab2-problem6_2_1[]" id="input_ps4-tab2-problem6_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="ps4-tab2-problem6_2_1-choice_1-label" for="input_ps4-tab2-problem6_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem6_2_1">
<text>decreases the rate at which the unforced system returns to equilibrium</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_ps4-tab2-problem6_2_1[]" id="input_ps4-tab2-problem6_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="ps4-tab2-problem6_2_1-choice_2-label" for="input_ps4-tab2-problem6_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem6_2_1">
<text>increases the characteristic time</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_ps4-tab2-problem6_2_1[]" id="input_ps4-tab2-problem6_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="ps4-tab2-problem6_2_1-choice_3-label" for="input_ps4-tab2-problem6_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem6_2_1">
<text>decreases the characteristic time</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_ps4-tab2-problem6_2_1[]" id="input_ps4-tab2-problem6_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="ps4-tab2-problem6_2_1-choice_4-label" for="input_ps4-tab2-problem6_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem6_2_1">
<text>maximal gain increases</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_ps4-tab2-problem6_2_1[]" id="input_ps4-tab2-problem6_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="ps4-tab2-problem6_2_1-choice_5-label" for="input_ps4-tab2-problem6_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem6_2_1">
<text>maximal gain decreases</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_ps4-tab2-problem6_2_1[]" id="input_ps4-tab2-problem6_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="ps4-tab2-problem6_2_1-choice_6-label" for="input_ps4-tab2-problem6_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem6_2_1">
<text>pass-band broadens</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_ps4-tab2-problem6_2_1[]" id="input_ps4-tab2-problem6_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="ps4-tab2-problem6_2_1-choice_7-label" for="input_ps4-tab2-problem6_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_ps4-tab2-problem6_2_1">
<text>pass-band narrows</text>
</label>
</div>
<span id="answer_ps4-tab2-problem6_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ps4-tab2-problem6_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_ps4-tab2-problem6_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="1(f) Analyze results" />
<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_ps4-tab2-problem6" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps4-tab2-problem6">
<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="ps4-tab2-problem6-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="ps4-tab2-problem6-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="ps4-tab2-problem6-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="False">
<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>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@vertical+block@ps4-tab3" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">PS4.3. Convolution.</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem1" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps4-tab3-problem1" class="problems-wrapper" role="group"
aria-labelledby="ps4-tab3-problem1-problem-title"
data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem1" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="2"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps4-tab3-problem1-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem1-problem-progress" tabindex="-1">
2(a)
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Compute [mathjaxinline]f(t)*u(t)[/mathjaxinline] (where [mathjaxinline]u(t)[/mathjaxinline] denotes the Heaviside step function). Explain your result by regarding [mathjaxinline]f(t)[/mathjaxinline] as the input signal for a system with weight function [mathjaxinline]u(t)[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ps4-tab3-problem1_2_1">
<fieldset aria-describedby="status_ps4-tab3-problem1_2_1">
<div class="field">
<input type="radio" name="input_ps4-tab3-problem1_2_1" id="input_ps4-tab3-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ps4-tab3-problem1_2_1-choice_1-label" for="input_ps4-tab3-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ps4-tab3-problem1_2_1"> <text> [mathjaxinline]f(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps4-tab3-problem1_2_1" id="input_ps4-tab3-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="ps4-tab3-problem1_2_1-choice_2-label" for="input_ps4-tab3-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ps4-tab3-problem1_2_1"> <text> [mathjaxinline]tf(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps4-tab3-problem1_2_1" id="input_ps4-tab3-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="ps4-tab3-problem1_2_1-choice_3-label" for="input_ps4-tab3-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_ps4-tab3-problem1_2_1"> <text> [mathjaxinline]f(t)u(t-1)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps4-tab3-problem1_2_1" id="input_ps4-tab3-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="ps4-tab3-problem1_2_1-choice_4-label" for="input_ps4-tab3-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_ps4-tab3-problem1_2_1"> <text> [mathjaxinline]\displaystyle \int _0^ t f(\tau )d\tau[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps4-tab3-problem1_2_1" id="input_ps4-tab3-problem1_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="ps4-tab3-problem1_2_1-choice_5-label" for="input_ps4-tab3-problem1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_ps4-tab3-problem1_2_1"> <text> [mathjaxinline]\displaystyle -\int _0^ t f(\tau )d\tau[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps4-tab3-problem1_2_1" id="input_ps4-tab3-problem1_2_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="ps4-tab3-problem1_2_1-choice_6-label" for="input_ps4-tab3-problem1_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_ps4-tab3-problem1_2_1"> <text> [mathjaxinline]\displaystyle \int _0^ t u(t-\tau ) d\tau[/mathjaxinline]</text>
</label>
</div>
<span id="answer_ps4-tab3-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ps4-tab3-problem1_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>
By the way, what is the differential operator with weight function [mathjaxinline]u(t)[/mathjaxinline]? </p>
<p>
(Enter as a function of [mathjaxinline]D[/mathjaxinline] and [mathjaxinline]I[/mathjaxinline].) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps4-tab3-problem1_3_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_ps4-tab3-problem1_3_1" id="input_ps4-tab3-problem1_3_1" aria-describedby="status_ps4-tab3-problem1_3_1" value=""/>
<span class="trailing_text" id="trailing_text_ps4-tab3-problem1_3_1"/>
<span class="status unanswered" id="status_ps4-tab3-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab3-problem1_3_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ps4-tab3-problem1_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="2(a)" />
<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_ps4-tab3-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps4-tab3-problem1">
<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="ps4-tab3-problem1-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="ps4-tab3-problem1-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="ps4-tab3-problem1-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="False">
<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:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem2" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps4-tab3-problem2" class="problems-wrapper" role="group"
aria-labelledby="ps4-tab3-problem2-problem-title"
data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem2" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem2/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps4-tab3-problem2-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem2-problem-progress" tabindex="-1">
2(b)
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Laplace transform can be used to verify many properties of convolution; for example, it is associative and commutative, and [mathjaxinline]\delta (t)[/mathjaxinline] serves as a unit (so [mathjaxinline]\delta *f=f[/mathjaxinline]). Use the Laplace transform to verify the surprising fact that if [mathjaxinline]f(0^-)=g(0^-)=0[/mathjaxinline] then </p>
<table id="a0000001254" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]f'*g=f*g'=(f*g)'[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Check the first of these equalities by integration. </p>
<p>
If [mathjaxinline]f(0^-)=g(0^-)=0[/mathjaxinline] then [mathjaxinline]f'*g-f*g'=0[/mathjaxinline]. What's the formula for [mathjaxinline]f'*g-f*g'[/mathjaxinline] when [mathjaxinline]f(0^-)[/mathjaxinline] and [mathjaxinline]g(0^-)[/mathjaxinline] are not necessarily zero? </p>
<span>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps4-tab3-problem2_2_1" class="capa_inputtype">
<div class="drag_and_drop_problem_div" id="drag_and_drop_div_ps4-tab3-problem2_2_1" data-plain-id="ps4-tab3-problem2_2_1">
</div>
<div class="drag_and_drop_problem_json" id="drag_and_drop_json_ps4-tab3-problem2_2_1" style="display:none;">{"target_outline": "false", "draggables": [{"target_fields": [], "can_reuse": "", "icon": "/assets/courseware/v1/c7e2ef9256354b9707438ec290d663af/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_laplace_ps3_3_3_laplace_ps3_3_3_dnd_label1.png", "label": "", "id": "ft"}, {"target_fields": [], "can_reuse": "", "icon": "/assets/courseware/v1/e36912e53a93f16e918c5da32fc4b092/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_laplace_ps3_3_3_laplace_ps3_3_3_dnd_label2.png", "label": "", "id": "gt"}, {"target_fields": [], "can_reuse": "", "icon": "/assets/courseware/v1/8703095ff106b520fc844938cc0e8669/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_laplace_ps3_3_3_laplace_ps3_3_3_dnd_label3.png", "label": "", "id": "ftprime"}, {"target_fields": [], "can_reuse": "", "icon": "/assets/courseware/v1/4c16149020702c39d2f44ddfdbc6b9a9/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_laplace_ps3_3_3_laplace_ps3_3_3_dnd_label4.png", "label": "", "id": "gtprime"}, {"target_fields": [], "can_reuse": "", "icon": "/assets/courseware/v1/a71e65ded0bf0ad0a4118394625166eb/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_laplace_ps3_3_3_laplace_ps3_3_3_dnd_label5.png", "label": "", "id": "fminus"}, {"target_fields": [], "can_reuse": "", "icon": "/assets/courseware/v1/90ee402de317ff5eb4c36dd478cee83c/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_laplace_ps3_3_3_laplace_ps3_3_3_dnd_label6.png", "label": "", "id": "gminus"}, {"target_fields": [], "can_reuse": "", "icon": "/assets/courseware/v1/538e650982b8c8b07be06ee961692ba7/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_laplace_ps3_3_3_laplace_ps3_3_3_dnd_label7.png", "label": "", "id": "fplus"}, {"target_fields": [], "can_reuse": "", "icon": "/assets/courseware/v1/33ed6f6c2ec7d07e8da5d19943445b4d/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_laplace_ps3_3_3_laplace_ps3_3_3_dnd_label8.png", "label": "", "id": "gplus"}, {"target_fields": [], "can_reuse": "", "icon": "/assets/courseware/v1/8e69162eea40f316cbd59ea1c795d6eb/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_laplace_ps3_3_3_laplace_ps3_3_3_dnd_label9.png", "label": "", "id": "fprime"}, {"target_fields": [], "can_reuse": "", "icon": "/assets/courseware/v1/6da0f7c31394f2dd89f75fc7d5a9b312/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_laplace_ps3_3_3_laplace_ps3_3_3_dnd_label10.png", "label": "", "id": "gprime"}], "label_bg_color": "rgb(222, 139, 238)", "one_per_target": "true", "targets": [{"y": "2", "id": "1", "h": "66", "w": "93", "x": "202"}, {"y": "2", "id": "2", "h": "66", "w": "93", "x": "333"}, {"y": "2", "id": "3", "h": "66", "w": "93", "x": "477"}, {"y": "2", "id": "4", "h": "66", "w": "93", "x": "608"}], "base_image": "/assets/courseware/v1/dd9d3ec74203abe9b78ecaa6d84a9efd/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_laplace_ps3_3_3_laplace_ps3_3_3_dnd.png"}</div>
<div class="script_placeholder" data-src="/static/js/capa/drag_and_drop.a31124208b9b.js"/>
<div class="unanswered" id="status_ps4-tab3-problem2_2_1">
<input type="text" name="input_ps4-tab3-problem2_2_1" id="input_ps4-tab3-problem2_2_1" aria-describedby="answer_ps4-tab3-problem2_2_1" value="" style="display:none;"/>
<p class="indicator-container drag-and-drop--status" aria-describedby="input_ps4-tab3-problem2_2_1">
<span class="status unanswered" id="status_ps4-tab3-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</p>
<p id="answer_ps4-tab3-problem2_2_1" class="answer"/>
</div>
</div></div>
</span>
<p>
<div class="solution-span">
<span id="solution_ps4-tab3-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="2(b)" />
<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_ps4-tab3-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps4-tab3-problem2">
<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="ps4-tab3-problem2-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="ps4-tab3-problem2-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="ps4-tab3-problem2-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="False">
<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-2" data-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem3" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps4-tab3-problem3" class="problems-wrapper" role="group"
aria-labelledby="ps4-tab3-problem3-problem-title"
data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem3" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem3/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps4-tab3-problem3-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem3-problem-progress" tabindex="-1">
2(c)
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab3-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
Use Laplace transform to compute [mathjaxinline]\displaystyle \frac{t^{22}}{22!}*\frac{t^{101}}{101!}[/mathjaxinline]. </p>
<p>
(You may type factorial expressions in the answer box below; e.g. type factorial(5) for [mathjaxinline]5![/mathjaxinline].) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_ps4-tab3-problem3_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_ps4-tab3-problem3_2_1" id="input_ps4-tab3-problem3_2_1" data-input-id="ps4-tab3-problem3_2_1" value="" aria-describedby="status_ps4-tab3-problem3_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_ps4-tab3-problem3_2_1"/>
<span class="status unanswered" id="status_ps4-tab3-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps4-tab3-problem3_2_1" class="answer"/>
<div id="input_ps4-tab3-problem3_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/2d585b0f4e8978427e0be0805091392e/asset-v1:OCW+18.031+2019_Spring+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a href="#mymodal-one" class="mymodal-btn mymodal-btn-big">FORMULA 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>Formula Input Guide</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">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> 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 <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="mymodal-btn">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_ps4-tab3-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="2(c)" />
<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_ps4-tab3-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps4-tab3-problem3">
<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="ps4-tab3-problem3-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="ps4-tab3-problem3-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="ps4-tab3-problem3-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="False">
<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>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@vertical+block@ps4-tab4" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">PS4.4. Sketching convolutions.</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem1" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps4-tab4-problem1" class="problems-wrapper" role="group"
aria-labelledby="ps4-tab4-problem1-problem-title"
data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem1" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps4-tab4-problem1-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem1-problem-progress" tabindex="-1">
2(d)i
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Use the fact that the rest initial condition (ZSR) response to an LTI system with weight function [mathjaxinline]w(t)[/mathjaxinline] to an input signal [mathjaxinline]f(t)[/mathjaxinline] is given by [mathjaxinline]f(t)*w(t)[/mathjaxinline] to sketch [mathjaxinline]f(t)*w(t)[/mathjaxinline] for the following pair of functions. </p>
<table id="a0000001261" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]f(t)=2u(t) , \qquad w(t)=e^{-t}.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps4-tab4-problem1_2_1" class="jsinput" data="getGrade" data-getstate="getState" data-setstate="setState" data-sop="false" data-processed="false">
<div class="script_placeholder" data-src="/static/js/capa/src/jschannel.7433acd0217e.js"/>
<div class="script_placeholder" data-src="/static/js/capa/src/jsinput.d14d79904ed5.js"/>
<div class="unanswered">
<iframe name="iframe_ps4-tab4-problem1_2_1" id="iframe_ps4-tab4-problem1_2_1" sandbox="allow-scripts allow-popups allow-same-origin allow-forms allow-pointer-lock" seamless="seamless" frameborder="0" src="https://sketch-response.vercel.app/sketch-tool/v1.5/#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" height="550" width="800" title="Problem Remote Content"/>
<input type="hidden" name="input_ps4-tab4-problem1_2_1" id="input_ps4-tab4-problem1_2_1" waitfor="" value=""/>
<br/>
<p id="answer_ps4-tab4-problem1_2_1" class="answer"/>
<div class="indicator-container">
<span class="status unanswered" id="status_ps4-tab4-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<div class="error_message" style="padding: 5px 5px 5px 5px; background-color:#FA6666; height:60px;width:400px; display: none"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ps4-tab4-problem1_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="2(d)i" />
<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_ps4-tab4-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps4-tab4-problem1">
<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="ps4-tab4-problem1-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="ps4-tab4-problem1-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="ps4-tab4-problem1-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="False">
<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:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem2" data-graded="False" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps4-tab4-problem2" class="problems-wrapper" role="group"
aria-labelledby="ps4-tab4-problem2-problem-title"
data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem2" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem2/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps4-tab4-problem2-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem2-problem-progress" tabindex="-1">
2(d)iii
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps4-tab4-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Use the fact that the rest initial condition (ZSR) response to an LTI system with weight function [mathjaxinline]w(t)[/mathjaxinline] to an input signal [mathjaxinline]f(t)[/mathjaxinline] is given by [mathjaxinline]f(t)*w(t)[/mathjaxinline] to sketch [mathjaxinline]f(t)*w(t)[/mathjaxinline] for the following pair of functions. </p>
<p>
[mathjaxinline]f(t)=u(t)\sin (t)[/mathjaxinline], [mathjaxinline]w(t)=u(t)\sin (t)[/mathjaxinline]. </p>
<p>
You must sketch at least two full oscillations. (i.e., 4 peaks, 2 positive and 2 negative), and label the zeros using the zero point tool for [mathjaxinline]t&gt;0[/mathjaxinline]. (Your sketch need only be qualitatively correct.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps4-tab4-problem2_2_1" class="jsinput" data="getGrade" data-getstate="getState" data-setstate="setState" data-sop="false" data-processed="false">
<div class="script_placeholder" data-src="/static/js/capa/src/jschannel.7433acd0217e.js"/>
<div class="script_placeholder" data-src="/static/js/capa/src/jsinput.d14d79904ed5.js"/>
<div class="unanswered">
<iframe name="iframe_ps4-tab4-problem2_2_1" id="iframe_ps4-tab4-problem2_2_1" sandbox="allow-scripts allow-popups allow-same-origin allow-forms allow-pointer-lock" seamless="seamless" frameborder="0" src="https://sketch-response.vercel.app/sketch-tool/v1.5/#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" height="550" width="800" title="Problem Remote Content"/>
<input type="hidden" name="input_ps4-tab4-problem2_2_1" id="input_ps4-tab4-problem2_2_1" waitfor="" value=""/>
<br/>
<p id="answer_ps4-tab4-problem2_2_1" class="answer"/>
<div class="indicator-container">
<span class="status unanswered" id="status_ps4-tab4-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
<div class="error_message" style="padding: 5px 5px 5px 5px; background-color:#FA6666; height:60px;width:400px; display: none"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ps4-tab4-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="2(d)iii" />
<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_ps4-tab4-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps4-tab4-problem2">
<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="ps4-tab4-problem2-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="ps4-tab4-problem2-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="ps4-tab4-problem2-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="False">
<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-2" data-id="block-v1:OCW+18.031+2019_Spring+type@html+block@ps4-tab4-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="30787212ea7411efb8100e0e3c45b88f" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@html+block@ps4-tab4-text1" data-graded="False" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:OCW+18.031+2019_Spring" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bf">There is no checker for 2(d)ii the sketch of [mathjaxinline]f(t)*w(t)[/mathjaxinline] for the pair: [mathjaxinline]f(t)[/mathjaxinline] the “square wave" and [mathjaxinline]w(t) = e^{-t}[/mathjaxinline].</b></p>
</div>
</div>
</div>
</div>