<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="False" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@vertical+block@ps5-checkvertical" data-init="VerticalStudentView" data-graded="False" data-runtime-version="1" data-block-type="vertical" data-request-token="e30aa394383c11f0982f12f8c5a42579">
<h2 class="hd hd-2 unit-title">Pset 5 checker</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+ES.1803+2023_Fall+type@html+block@8dd581605e2a41448ca710b93695d9ee">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="False" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@html+block@8dd581605e2a41448ca710b93695d9ee" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="html" data-request-token="e30aa394383c11f0982f12f8c5a42579">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><a href="/assets/courseware/v1/d61ed8727ce5edfc3d6c5f54c439c463/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/mites_1803_s24_ps5.pdf" target="[object Object]">Problem set 5 (PDF)</a></p>
<p></p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-1-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-1-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="e30aa394383c11f0982f12f8c5a42579">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps5-1-check" class="problems-wrapper" role="group"
aria-labelledby="ps5-1-check-problem-title"
data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-1-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-1-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps5-1-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-1-check-problem-progress" tabindex="-1">
Problem 1
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-1-check-problem-progress"></div>
<div class="problem">
<div>
<p>
To check your answers put them in the appropriate box and click the &#8216;Check' button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. </p>
<p>
Unless otherwise stated the checker expects numerical answers to 3 significant digits. </p>
<p>
For formulas you should use Matlab notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] <b class="bf">NOT </b> [mathjaxinline]3x[/mathjaxinline]. </p>
<p>
The checker knows many of the standard functions. Use exp(x) and ln(x) for the exponential and natural log functions. </p>
<p>
The formula checker expects a specific letter for the variables, so be sure to use the one specified. </p>
<p>
<b class="bf">Problem 1</b>
<em>No answer checker given.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 1" />
<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_ps5-1-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps5-1-check">
<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="ps5-1-check-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="ps5-1-check-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="ps5-1-check-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:MITx+ES.1803+2023_Fall+type@problem+block@ps5-2-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-2-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="e30aa394383c11f0982f12f8c5a42579">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps5-2-check" class="problems-wrapper" role="group"
aria-labelledby="ps5-2-check-problem-title"
data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-2-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-2-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps5-2-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-2-check-problem-progress" tabindex="-1">
Problem 2
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-2-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">Problem 2(i)</b> Give the value of [mathjaxinline]K[/mathjaxinline] accurate to 0.02, </p>
<p>
<p style="display:inline">[mathjaxinline]K =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps5-2-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-2-check_2_1" id="input_ps5-2-check_2_1" aria-describedby="status_ps5-2-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-2-check_2_1"/>
<span class="status unanswered" id="status_ps5-2-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-2-check_2_1" class="answer"/>
<div id="display_ps5-2-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-2-check_2_1_dynamath" name="input_ps5-2-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">Problem 2(ii)</b> Give the estimate for [mathjaxinline]y(100)[/mathjaxinline] to the nearest whole number <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps5-2-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-2-check_3_1" id="input_ps5-2-check_3_1" aria-describedby="status_ps5-2-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-2-check_3_1"/>
<span class="status unanswered" id="status_ps5-2-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-2-check_3_1" class="answer"/>
<div id="display_ps5-2-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-2-check_3_1_dynamath" name="input_ps5-2-check_3_1_dynamath"/>
</div>
</div></div> </p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 2" />
<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_ps5-2-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps5-2-check">
<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="ps5-2-check-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="ps5-2-check-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="ps5-2-check-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:MITx+ES.1803+2023_Fall+type@problem+block@ps5-3-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-3-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="e30aa394383c11f0982f12f8c5a42579">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps5-3-check" class="problems-wrapper" role="group"
aria-labelledby="ps5-3-check-problem-title"
data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-3-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-3-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps5-3-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-3-check-problem-progress" tabindex="-1">
Problem 3
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-3-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">3(a)</b> Give the values of [mathjaxinline]x_8[/mathjaxinline] and [mathjaxinline]y_8[/mathjaxinline] correct to 3 decimal places </p>
<p>
<p style="display:inline">[mathjaxinline]x_8 =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps5-3-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-3-check_2_1" id="input_ps5-3-check_2_1" aria-describedby="status_ps5-3-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-3-check_2_1"/>
<span class="status unanswered" id="status_ps5-3-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-3-check_2_1" class="answer"/>
<div id="display_ps5-3-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-3-check_2_1_dynamath" name="input_ps5-3-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]y_8 =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps5-3-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-3-check_3_1" id="input_ps5-3-check_3_1" aria-describedby="status_ps5-3-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-3-check_3_1"/>
<span class="status unanswered" id="status_ps5-3-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-3-check_3_1" class="answer"/>
<div id="display_ps5-3-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-3-check_3_1_dynamath" name="input_ps5-3-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">3(b)</b> Give the error between [mathjaxinline]y_8[/mathjaxinline] and the exact value correct to 3 decimal places. </p>
<p>
<p style="display:inline">Error[mathjaxinline]_8 =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps5-3-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-3-check_4_1" id="input_ps5-3-check_4_1" aria-describedby="status_ps5-3-check_4_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-3-check_4_1"/>
<span class="status unanswered" id="status_ps5-3-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-3-check_4_1" class="answer"/>
<div id="display_ps5-3-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-3-check_4_1_dynamath" name="input_ps5-3-check_4_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<b class="bf">3(c)</b>
<em>No answer checker given.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 3" />
<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_ps5-3-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps5-3-check">
<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="ps5-3-check-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="ps5-3-check-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="ps5-3-check-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:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4a-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4a-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="e30aa394383c11f0982f12f8c5a42579">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps5-4a-check" class="problems-wrapper" role="group"
aria-labelledby="ps5-4a-check-problem-title"
data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4a-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4a-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps5-4a-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4a-check-problem-progress" tabindex="-1">
Problem 4a
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4a-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">4a(i)</b> Give the bifurcation points as a comma separated list. Answer none if there are none. </p>
<p>
<p style="display:inline">Bifurcation points: </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps5-4a-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-4a-check_2_1" id="input_ps5-4a-check_2_1" aria-describedby="status_ps5-4a-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-4a-check_2_1"/>
<span class="status unanswered" id="status_ps5-4a-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-4a-check_2_1" class="answer"/>
<div id="display_ps5-4a-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-4a-check_2_1_dynamath" name="input_ps5-4a-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">4a(ii)</b> Enter the number of critical points as a comma separated list. Use the following order: The bifurcation points split the [mathjaxinline]a[/mathjaxinline]-axis into intervals. Starting at the left, give the number of critical points for values of [mathjaxinline]a[/mathjaxinline] in each interval. </p>
<p>
<p style="display:inline">Number of critical points: </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps5-4a-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-4a-check_3_1" id="input_ps5-4a-check_3_1" aria-describedby="status_ps5-4a-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-4a-check_3_1"/>
<span class="status unanswered" id="status_ps5-4a-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-4a-check_3_1" class="answer"/>
<div id="display_ps5-4a-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-4a-check_3_1_dynamath" name="input_ps5-4a-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<b class="bf">4a(iii)</b>
<em>No answer checker given.</em>
</p>
<p>
<b class="bf">4a(iv)</b>
<em>No answer checker given.</em>
</p>
<p>
<b class="bf">4a(v)</b>
<em>No answer checker given.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 4a" />
<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_ps5-4a-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps5-4a-check">
<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="ps5-4a-check-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="ps5-4a-check-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="ps5-4a-check-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:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4b-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4b-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="e30aa394383c11f0982f12f8c5a42579">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps5-4b-check" class="problems-wrapper" role="group"
aria-labelledby="ps5-4b-check-problem-title"
data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4b-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4b-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps5-4b-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4b-check-problem-progress" tabindex="-1">
Problem 4b
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4b-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">4b(i)</b> Give the bifurcation points as a comma separated list. Answer none if there are none. </p>
<p>
<p style="display:inline">Bifurcation points: </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps5-4b-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-4b-check_2_1" id="input_ps5-4b-check_2_1" aria-describedby="status_ps5-4b-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-4b-check_2_1"/>
<span class="status unanswered" id="status_ps5-4b-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-4b-check_2_1" class="answer"/>
<div id="display_ps5-4b-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-4b-check_2_1_dynamath" name="input_ps5-4b-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">4b(ii)</b> Enter the number of critical points as a comma separated list. Use the following order: The bifurcation points split the [mathjaxinline]a[/mathjaxinline]-axis into intervals. Starting at the left, give the number of critical points for values of [mathjaxinline]a[/mathjaxinline] in each interval. </p>
<p>
<p style="display:inline">Number of critical points: </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps5-4b-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-4b-check_3_1" id="input_ps5-4b-check_3_1" aria-describedby="status_ps5-4b-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-4b-check_3_1"/>
<span class="status unanswered" id="status_ps5-4b-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-4b-check_3_1" class="answer"/>
<div id="display_ps5-4b-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-4b-check_3_1_dynamath" name="input_ps5-4b-check_3_1_dynamath"/>
</div>
</div></div>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 4b" />
<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_ps5-4b-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps5-4b-check">
<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="ps5-4b-check-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="ps5-4b-check-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="ps5-4b-check-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-6" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4c-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4c-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="e30aa394383c11f0982f12f8c5a42579">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps5-4c-check" class="problems-wrapper" role="group"
aria-labelledby="ps5-4c-check-problem-title"
data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4c-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4c-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps5-4c-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4c-check-problem-progress" tabindex="-1">
Problem 4c
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-4c-check-problem-progress"></div>
<div class="problem">
<div>
<b class="bf">4(c)</b>
<em>No answer checker given.</em>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 4c" />
<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_ps5-4c-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps5-4c-check">
<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="ps5-4c-check-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="ps5-4c-check-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="ps5-4c-check-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-7" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-5-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-5-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="e30aa394383c11f0982f12f8c5a42579">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps5-5-check" class="problems-wrapper" role="group"
aria-labelledby="ps5-5-check-problem-title"
data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-5-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-5-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps5-5-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-5-check-problem-progress" tabindex="-1">
Problem 5
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps5-5-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">5(a)</b> Give the general solution [mathjaxinline]x(t)[/mathjaxinline] and the gain [mathjaxinline]g(w)[/mathjaxinline]. </p>
<p>
Use the letters, w, t, k, B, C in your answer. Here, C is the &#8216;constant of integration' in the homogeneous solution. Use the function arctan2(x,y) for the phase angle. </p>
<p>
<p style="display:inline">[mathjaxinline]x(t)[/mathjaxinline] = </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps5-5-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-5-check_2_1" id="input_ps5-5-check_2_1" aria-describedby="status_ps5-5-check_2_1" value="" class="math" size="70"/>
<span class="trailing_text" id="trailing_text_ps5-5-check_2_1"/>
<span class="status unanswered" id="status_ps5-5-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-5-check_2_1" class="answer"/>
<div id="display_ps5-5-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-5-check_2_1_dynamath" name="input_ps5-5-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]g(w)[/mathjaxinline] = </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps5-5-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-5-check_3_1" id="input_ps5-5-check_3_1" aria-describedby="status_ps5-5-check_3_1" value="" class="math" size="70"/>
<span class="trailing_text" id="trailing_text_ps5-5-check_3_1"/>
<span class="status unanswered" id="status_ps5-5-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-5-check_3_1" class="answer"/>
<div id="display_ps5-5-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-5-check_3_1_dynamath" name="input_ps5-5-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
On your pset, don't forget to answer the question about the transient. </p>
<p><b class="bf">5(b)</b> Give the general solution [mathjaxinline]x(t)[/mathjaxinline]. </p>
<p>
Use the letters, w, t, k, B, R, C in your answer. Again, C is the &#8216;constant of integration' in the homogeneous solution. Use the function arctan2(x,y) for the phase angle. </p>
<p>
<p style="display:inline">[mathjaxinline]x(t)[/mathjaxinline] = </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps5-5-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-5-check_4_1" id="input_ps5-5-check_4_1" aria-describedby="status_ps5-5-check_4_1" value="" class="math" size="70"/>
<span class="trailing_text" id="trailing_text_ps5-5-check_4_1"/>
<span class="status unanswered" id="status_ps5-5-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-5-check_4_1" class="answer"/>
<div id="display_ps5-5-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-5-check_4_1_dynamath" name="input_ps5-5-check_4_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<b class="bf">
<p>
5(c) <em>No answer checker given.</em></p>
<p>
<b class="bf">
<p>
5(d) <em>No answer checker given.</em></p>
<p>
<b class="bf">5(e) <em>No answer checker given.</em></b>
</p>
</b>
</p>
</b>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 5" />
<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_ps5-5-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps5-5-check">
<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="ps5-5-check-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="ps5-5-check-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="ps5-5-check-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>