<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">1. Motivation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab1-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab1-video1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Measurement error</h3>
<div
id="video_approx_1-tab1-video1"
class="video closed"
data-metadata='{"captionDataDir": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab1-video1/handler/transcript/available_translations", "start": 0.0, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "saveStateEnabled": false, "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab1-video1/handler/publish_completion", "lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab1-video1/handler/xmodule_handler/save_user_state", "completionEnabled": false, "prioritizeHls": false, "sources": [], "completionPercentage": 0.95, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab1-video1/handler/transcript/translation/__lang__", "transcriptLanguages": {"en": "English"}, "streams": "1.00:5dHVBiN0hsE", "showCaptions": "true", "speed": null, "ytMetadataEndpoint": "", "autohideHtml5": false, "duration": 0.0, "poster": null, "recordedYoutubeIsAvailable": true, "end": 0.0, "ytTestTimeout": 1500, "autoplay": false}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="approx_1-tab1-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_approx_1-tab1-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_approx_1-tab1-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab1-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab1-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab2" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">2. Linear Approximation and Measurement Error</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab2-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab2-text1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><span style="color:#99182C"><b class="bf">products</b></span><br/><span style="color:#99182C"><b class="bf">compositions</b></span><br/><span style="color:#99182C"><b class="bf">error</b></span><br/></p><p><b class="bfseries">Objectives</b></p><p>
At the end of this sequence, and after some practice, you should be able to: </p><ul class="itemize"><li><p>
Find <span style="color:#27408C"><b class="bf">linear approximations</b></span> of any function. </p></li><li><p>
Recognize linearizations of basic functions at 0. </p></li><li><p>
Find linear approximations of <span style="color:#27408C"><b class="bf">compositions and products</b></span> using basic approximations. </p></li><li><p>
Use the linear approximation to <span style="color:#27408C"><b class="bf">estimate error</b></span>. </p></li></ul><p><b class="bfseries">Contents: 14 pages</b></p><p>
8 videos (32 minutes 1x speed) 23 questions </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab3" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">3. A problem with a zipline</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab3-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab3-text1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Linear approximations revisited</b></p><p>
You already know how to take linear approximations. This was the main tool for finding the derivatives of all of the basic functions in the previous unit. </p><p>
We return again to linear approximations because they are the main tool that we use in finding approximations in science and engineering. Here we develop some more tools for using linear approximations. We will also change our perspective and use linear approximations to estimate how measurement error propogates through a system, and understand the system's sensitivity to small perturbations. </p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab3-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab3-video1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Zipline design problem setup</h3>
<div
id="video_approx_1-tab3-video1"
class="video closed"
data-metadata='{"captionDataDir": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab3-video1/handler/transcript/available_translations", "start": 0.0, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "saveStateEnabled": false, "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab3-video1/handler/publish_completion", "lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab3-video1/handler/xmodule_handler/save_user_state", "completionEnabled": false, "prioritizeHls": false, "sources": [], "completionPercentage": 0.95, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab3-video1/handler/transcript/translation/__lang__", "transcriptLanguages": {"en": "English"}, "streams": "1.00:-vszqFLuJYY", "showCaptions": "true", "speed": null, "ytMetadataEndpoint": "", "autohideHtml5": false, "duration": 0.0, "poster": null, "recordedYoutubeIsAvailable": true, "end": 0.0, "ytTestTimeout": 1500, "autoplay": false}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="approx_1-tab3-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab3-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab3-problem1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab3-problem1" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab3-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab3-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab3-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="approx_1-tab3-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab3-problem1-problem-progress" tabindex="-1">
Find the maximum velocity
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab3-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Find an expression for the maximum velocity of a rider on the zipline in terms of the total system energy [mathjaxinline]E[/mathjaxinline] and the rider mass [mathjaxinline]m[/mathjaxinline]. </p>
<p>
(Enter expression in terms of E and m. Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]v=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab3-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_approx_1-tab3-problem1_2_1" id="input_approx_1-tab3-problem1_2_1" data-input-id="approx_1-tab3-problem1_2_1" value="" aria-describedby="status_approx_1-tab3-problem1_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_approx_1-tab3-problem1_2_1"/>
<span class="status unanswered" id="status_approx_1-tab3-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab3-problem1_2_1" class="answer"/>
<div id="input_approx_1-tab3-problem1_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:MITx+18.01.1x+2T2019+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 class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#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" rowspan="3" scope="row">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" rowspan="4" scope="row">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" rowspan="3" scope="row">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>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab3-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Find the maximum velocity" />
<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_approx_1-tab3-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-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="approx_1-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="approx_1-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="approx_1-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab4" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">4. Solving the problem</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab4-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab4-problem1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab4-problem1" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab4-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab4-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-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="approx_1-tab4-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab4-problem1-problem-progress" tabindex="-1">
Review problem
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab4-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Recall that the linear approximation of a function [mathjaxinline]\ f(x)[/mathjaxinline] for [mathjaxinline]x[/mathjaxinline] near [mathjaxinline]x_0[/mathjaxinline] is given by the formula [mathjaxinline]\ f(x) \approx f(x_0) + f'(x_0)(x-x_0)[/mathjaxinline]. </p>
<p>
Find the linear approximation of the function [mathjaxinline](1+u)^ r[/mathjaxinline] near [mathjaxinline]u=0[/mathjaxinline]. </p>
<p>
(Note that this answer will involve both [mathjaxinline]u[/mathjaxinline] and [mathjaxinline]r[/mathjaxinline]. Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline](1+u)^ r \approx[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab4-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_approx_1-tab4-problem1_2_1" id="input_approx_1-tab4-problem1_2_1" data-input-id="approx_1-tab4-problem1_2_1" value="" aria-describedby="status_approx_1-tab4-problem1_2_1" size="40"/>
<span class="trailing_text" id="trailing_text_approx_1-tab4-problem1_2_1"/>
<span class="status unanswered" id="status_approx_1-tab4-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab4-problem1_2_1" class="answer"/>
<div id="input_approx_1-tab4-problem1_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:MITx+18.01.1x+2T2019+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 class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#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" rowspan="3" scope="row">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" rowspan="4" scope="row">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" rowspan="3" scope="row">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>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab4-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Review problem" />
<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_approx_1-tab4-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-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="approx_1-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="approx_1-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="approx_1-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab4-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab4-video1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Zipline example solved</h3>
<div
id="video_approx_1-tab4-video1"
class="video closed"
data-metadata='{"captionDataDir": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab4-video1/handler/transcript/available_translations", "start": 0.0, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "saveStateEnabled": false, "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab4-video1/handler/publish_completion", "lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab4-video1/handler/xmodule_handler/save_user_state", "completionEnabled": false, "prioritizeHls": false, "sources": [], "completionPercentage": 0.95, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab4-video1/handler/transcript/translation/__lang__", "transcriptLanguages": {"en": "English"}, "streams": "1.00:eB4HCt0OFIw", "showCaptions": "true", "speed": null, "ytMetadataEndpoint": "", "autohideHtml5": false, "duration": 0.0, "poster": null, "recordedYoutubeIsAvailable": true, "end": 0.0, "ytTestTimeout": 1500, "autoplay": false}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="approx_1-tab4-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_approx_1-tab4-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_approx_1-tab4-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab4-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab4-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab4-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab4-problem2" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab4-problem2" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab4-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab4-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-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="approx_1-tab4-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab4-problem2-problem-progress" tabindex="-1">
Simplified problem
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab4-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Let's make another simplifying assumption. Suppose that rider starts completely at rest. This means that initially, the kinetic energy is zero, so the total energy is given by [mathjaxinline]E = mgh[/mathjaxinline]. </p>
<p>
With this added assumption, suppose we change the height displaced during the ride by [mathjaxinline]2\%[/mathjaxinline], i.e. [mathjaxinline]\displaystyle \frac{\Delta h}{h} = .02[/mathjaxinline]. What percentage did [mathjaxinline]v[/mathjaxinline] change by? </p>
<p>
(Find an approximation for [mathjaxinline]\displaystyle \frac{\Delta v}{v}[/mathjaxinline]. Enter your answer as a decimal with 2 decimal digits rather than as a percentage.) </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{\Delta v}{v}=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_approx_1-tab4-problem2_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_approx_1-tab4-problem2_2_1" id="input_approx_1-tab4-problem2_2_1" aria-describedby="status_approx_1-tab4-problem2_2_1" value="" size="40"/>
<span class="trailing_text" id="trailing_text_approx_1-tab4-problem2_2_1"/>
<span class="status unanswered" id="status_approx_1-tab4-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab4-problem2_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab4-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Simplified problem" />
<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_approx_1-tab4-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-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="approx_1-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="approx_1-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="approx_1-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab5" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">5. Linear approximations near zero</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab5-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab5-text1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
Recall that the linear approximation of a function [mathjaxinline]\ f(x)[/mathjaxinline] for [mathjaxinline]x[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline] is given by the formula [mathjaxinline]\ f(x) \approx f(0) + f'(0)x[/mathjaxinline]. Find linear approximations of all of the basic functions near [mathjaxinline]x=0[/mathjaxinline]. We are going to be using these formulas a lot! So you may even want to memorize them. </p><p>
We call the linear approximation the <span style="color:#27408C"><b class="bf">linearization</b></span> of a function. </p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab5-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab5-problem1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab5-problem1" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab5-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab5-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab5-problem1/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="approx_1-tab5-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab5-problem1-problem-progress" tabindex="-1">
Linear approximations of basic functions
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab5-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Find the linear approximation of [mathjaxinline]\ \sin (x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab5-problem1_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_approx_1-tab5-problem1_2_1" id="input_approx_1-tab5-problem1_2_1" data-input-id="approx_1-tab5-problem1_2_1" value="" aria-describedby="status_approx_1-tab5-problem1_2_1" size="40"/>
<span class="trailing_text" id="trailing_text_approx_1-tab5-problem1_2_1"/>
<span class="status unanswered" id="status_approx_1-tab5-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab5-problem1_2_1" class="answer"/>
<div id="input_approx_1-tab5-problem1_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>
Find the linear approximation of [mathjaxinline]\ \cos (x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_approx_1-tab5-problem1_3_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_approx_1-tab5-problem1_3_1" id="input_approx_1-tab5-problem1_3_1" data-input-id="approx_1-tab5-problem1_3_1" value="" aria-describedby="status_approx_1-tab5-problem1_3_1" size="40"/>
<span class="trailing_text" id="trailing_text_approx_1-tab5-problem1_3_1"/>
<span class="status unanswered" id="status_approx_1-tab5-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab5-problem1_3_1" class="answer"/>
<div id="input_approx_1-tab5-problem1_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>
<p>
Find the linear approximation of [mathjaxinline]\ e^ x[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_approx_1-tab5-problem1_4_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_approx_1-tab5-problem1_4_1" id="input_approx_1-tab5-problem1_4_1" data-input-id="approx_1-tab5-problem1_4_1" value="" aria-describedby="status_approx_1-tab5-problem1_4_1" size="40"/>
<span class="trailing_text" id="trailing_text_approx_1-tab5-problem1_4_1"/>
<span class="status unanswered" id="status_approx_1-tab5-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab5-problem1_4_1" class="answer"/>
<div id="input_approx_1-tab5-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>
<p>
Find the linear approximation of [mathjaxinline]\ \ln (1+x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div id="formulaequationinput_approx_1-tab5-problem1_5_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_approx_1-tab5-problem1_5_1" id="input_approx_1-tab5-problem1_5_1" data-input-id="approx_1-tab5-problem1_5_1" value="" aria-describedby="status_approx_1-tab5-problem1_5_1" size="40"/>
<span class="trailing_text" id="trailing_text_approx_1-tab5-problem1_5_1"/>
<span class="status unanswered" id="status_approx_1-tab5-problem1_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab5-problem1_5_1" class="answer"/>
<div id="input_approx_1-tab5-problem1_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>
<span>
<link href="/assets/courseware/v1/2d585b0f4e8978427e0be0805091392e/asset-v1:MITx+18.01.1x+2T2019+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 class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#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" rowspan="3" scope="row">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" rowspan="4" scope="row">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" rowspan="3" scope="row">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>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab5-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Linear approximations of basic functions" />
<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_approx_1-tab5-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab5-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="approx_1-tab5-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="approx_1-tab5-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="approx_1-tab5-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab6" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">6. Combining approximations</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab6-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab6-video1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Combining approximations of sums, products, compositions</h3>
<div
id="video_approx_1-tab6-video1"
class="video closed"
data-metadata='{"captionDataDir": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab6-video1/handler/transcript/available_translations", "start": 0.0, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "saveStateEnabled": false, "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab6-video1/handler/publish_completion", "lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab6-video1/handler/xmodule_handler/save_user_state", "completionEnabled": false, "prioritizeHls": false, "sources": [], "completionPercentage": 0.95, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab6-video1/handler/transcript/translation/__lang__", "transcriptLanguages": {"en": "English"}, "streams": "1.00:-NFsbM3mCu8", "showCaptions": "true", "speed": null, "ytMetadataEndpoint": "", "autohideHtml5": false, "duration": 0.0, "poster": null, "recordedYoutubeIsAvailable": true, "end": 0.0, "ytTestTimeout": 1500, "autoplay": false}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="approx_1-tab6-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_approx_1-tab6-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_approx_1-tab6-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab6-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab6-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab6-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab6-text1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Approximations (likely nonlinear) of compositions</b></p><p>
Suppose that [mathjaxinline]g(x)[/mathjaxinline] is a function such that [mathjaxinline]g(0) =0[/mathjaxinline]. To find an approximation of a function [mathjaxinline]f(g(x))[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline], we can take a linear approximation for [mathjaxinline]f(u)[/mathjaxinline] and then substitute [mathjaxinline]g(x)[/mathjaxinline] in for [mathjaxinline]u[/mathjaxinline]. The resulting approximation is likely nonlinear, but it is still an approximation! <b class="bfseries"><span style="color:#FF7F00">Warning:</span></b> this only works if [mathjaxinline]g(0) = 0[/mathjaxinline]. </p><table id="a0000000470" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000471"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f(u) \approx f(0) + f'(0)u \qquad \Longrightarrow \qquad f\left(g(x)\right) \approx f(g(0)) + f'(g(0))g(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.6)</td></tr></table>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab6-problem1" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab6-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-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="approx_1-tab6-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem1-problem-progress" tabindex="-1">
Exponential
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
What is the linear approximation of [mathjaxinline]e^{-x}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline].? </p>
<p>
(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab6-problem1_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_approx_1-tab6-problem1_2_1" id="input_approx_1-tab6-problem1_2_1" data-input-id="approx_1-tab6-problem1_2_1" value="" aria-describedby="status_approx_1-tab6-problem1_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab6-problem1_2_1"/>
<span class="status unanswered" id="status_approx_1-tab6-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab6-problem1_2_1" class="answer"/>
<div id="input_approx_1-tab6-problem1_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:MITx+18.01.1x+2T2019+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 class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#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" rowspan="3" scope="row">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" rowspan="4" scope="row">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" rowspan="3" scope="row">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>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab6-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Exponential" />
<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_approx_1-tab6-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab6-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="approx_1-tab6-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="approx_1-tab6-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="approx_1-tab6-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem2" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab6-problem2" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab6-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-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="approx_1-tab6-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem2-problem-progress" tabindex="-1">
Powers
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
What is the linear approximation to [mathjaxinline](1+3x)^{-1/2}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline].? </p>
<p>
(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab6-problem2_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_approx_1-tab6-problem2_2_1" id="input_approx_1-tab6-problem2_2_1" data-input-id="approx_1-tab6-problem2_2_1" value="" aria-describedby="status_approx_1-tab6-problem2_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab6-problem2_2_1"/>
<span class="status unanswered" id="status_approx_1-tab6-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab6-problem2_2_1" class="answer"/>
<div id="input_approx_1-tab6-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>
<span>
<link href="/assets/courseware/v1/2d585b0f4e8978427e0be0805091392e/asset-v1:MITx+18.01.1x+2T2019+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 class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#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" rowspan="3" scope="row">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" rowspan="4" scope="row">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" rowspan="3" scope="row">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>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab6-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Powers" />
<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_approx_1-tab6-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab6-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="approx_1-tab6-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="approx_1-tab6-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="approx_1-tab6-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem3" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab6-problem3" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab6-problem3-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem3" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-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="approx_1-tab6-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem3-problem-progress" tabindex="-1">
Approximation and compositions
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
Use a linear approximation for [mathjaxinline]e^ u[/mathjaxinline] to find a (nonlinear in [mathjaxinline]x[/mathjaxinline] but still polynomial) approximation to the function [mathjaxinline]e^{x^3}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. </p>
<p>
(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]e^{x^3}\approx[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab6-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_approx_1-tab6-problem3_2_1" id="input_approx_1-tab6-problem3_2_1" data-input-id="approx_1-tab6-problem3_2_1" value="" aria-describedby="status_approx_1-tab6-problem3_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab6-problem3_2_1"/>
<span class="status unanswered" id="status_approx_1-tab6-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab6-problem3_2_1" class="answer"/>
<div id="input_approx_1-tab6-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:MITx+18.01.1x+2T2019+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 class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#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" rowspan="3" scope="row">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" rowspan="4" scope="row">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" rowspan="3" scope="row">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>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab6-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Approximation and compositions " />
<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_approx_1-tab6-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab6-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="approx_1-tab6-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="approx_1-tab6-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="approx_1-tab6-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-5" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem4">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem4" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab6-problem4" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab6-problem4-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem4" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-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="approx_1-tab6-problem4-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem4-problem-progress" tabindex="-1">
Approximation and compositions
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab6-problem4-problem-progress"></div>
<div class="problem">
<div>
<p>
Combine linear approximations for the natural logarithm and square root functions to find a (nonlinear in [mathjaxinline]x[/mathjaxinline] but still polynomial) approximation of the function [mathjaxinline]\ln {\sqrt {1+x^2}}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. </p>
<p>
(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]\ln {\sqrt {1+x^2}} \approx[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab6-problem4_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_approx_1-tab6-problem4_2_1" id="input_approx_1-tab6-problem4_2_1" data-input-id="approx_1-tab6-problem4_2_1" value="" aria-describedby="status_approx_1-tab6-problem4_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab6-problem4_2_1"/>
<span class="status unanswered" id="status_approx_1-tab6-problem4_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab6-problem4_2_1" class="answer"/>
<div id="input_approx_1-tab6-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:MITx+18.01.1x+2T2019+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 class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#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" rowspan="3" scope="row">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" rowspan="4" scope="row">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" rowspan="3" scope="row">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>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab6-problem4_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Approximation and compositions" />
<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_approx_1-tab6-problem4" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab6-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="approx_1-tab6-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="approx_1-tab6-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="approx_1-tab6-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab7" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">7. Linear to nonlinear approximations</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab7-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab7-video1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Time dilation example</h3>
<div
id="video_approx_1-tab7-video1"
class="video closed"
data-metadata='{"captionDataDir": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab7-video1/handler/transcript/available_translations", "start": 0.0, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "saveStateEnabled": false, "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab7-video1/handler/publish_completion", "lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab7-video1/handler/xmodule_handler/save_user_state", "completionEnabled": false, "prioritizeHls": false, "sources": [], "completionPercentage": 0.95, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab7-video1/handler/transcript/translation/__lang__", "transcriptLanguages": {"en": "English"}, "streams": "1.00:t2ymPxm3DnU", "showCaptions": "true", "speed": null, "ytMetadataEndpoint": "", "autohideHtml5": false, "duration": 0.0, "poster": null, "recordedYoutubeIsAvailable": true, "end": 0.0, "ytTestTimeout": 1500, "autoplay": false}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="approx_1-tab7-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_approx_1-tab7-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_approx_1-tab7-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab7-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab7-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab7-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab7-text1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
That last video mentioned quadratic approximations. We haven't quite gotten there yet, but that is coming next! Don't worry, the students in this class didn't know about quadratic approximations yet either. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab8" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">8. Products of functions</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab8-problem1" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab8-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-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="approx_1-tab8-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem1-problem-progress" tabindex="-1">
Complicated example
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Find the linear approximation of the function [mathjaxinline]\ \displaystyle \frac{e^{-3x}}{\sqrt {1+x}}[/mathjaxinline] at [mathjaxinline]x=0[/mathjaxinline]. </p>
<p>
(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division. Type [mathjaxinline]{}^{\wedge }[/mathjaxinline] for exponents. Type sqrt(x) for [mathjaxinline]\sqrt {x}[/mathjaxinline].) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab8-problem1_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_approx_1-tab8-problem1_2_1" id="input_approx_1-tab8-problem1_2_1" data-input-id="approx_1-tab8-problem1_2_1" value="" aria-describedby="status_approx_1-tab8-problem1_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab8-problem1_2_1"/>
<span class="status unanswered" id="status_approx_1-tab8-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab8-problem1_2_1" class="answer"/>
<div id="input_approx_1-tab8-problem1_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:MITx+18.01.1x+2T2019+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 class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#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" rowspan="3" scope="row">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" rowspan="4" scope="row">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" rowspan="3" scope="row">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>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab8-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Complicated example" />
<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_approx_1-tab8-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab8-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="approx_1-tab8-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="approx_1-tab8-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="approx_1-tab8-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab8-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab8-video1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Approximation of products</h3>
<div
id="video_approx_1-tab8-video1"
class="video closed"
data-metadata='{"captionDataDir": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab8-video1/handler/transcript/available_translations", "start": 0.0, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "saveStateEnabled": false, "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab8-video1/handler/publish_completion", "lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab8-video1/handler/xmodule_handler/save_user_state", "completionEnabled": false, "prioritizeHls": false, "sources": [], "completionPercentage": 0.95, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab8-video1/handler/transcript/translation/__lang__", "transcriptLanguages": {"en": "English"}, "streams": "1.00:wO2PKbb6zLI", "showCaptions": "true", "speed": null, "ytMetadataEndpoint": "", "autohideHtml5": false, "duration": 0.0, "poster": null, "recordedYoutubeIsAvailable": true, "end": 0.0, "ytTestTimeout": 1500, "autoplay": false}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="approx_1-tab8-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_approx_1-tab8-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_approx_1-tab8-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab8-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab8-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab8-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab8-text1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
To find the linear approximation of a function [mathjaxinline]h(x) = f(x)g(x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline], it suffices to find a linear approximation for [mathjaxinline]f(x)[/mathjaxinline], find a linear approximation for [mathjaxinline]g(x)[/mathjaxinline], and then the linear approximation for [mathjaxinline]h(x)[/mathjaxinline] is the product of these two approximations where we cancel all of the terms that are quadratic (or higher for more products). </p><table id="a0000000475" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000476"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle h(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(x)g(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.7)</td></tr><tr id="a0000000477"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(0) + f'(0)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.8)</td></tr><tr id="a0000000478"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle g(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle g(0) + g'(0)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.9)</td></tr><tr id="a0000000479"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle h(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(f(0) + f'(0)x\right)\left(g(0) + g'(0)x\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.10)</td></tr><tr id="a0000000480"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(0)g(0) + \left(f'(0)g(0) +f(0)g'(0)\right)x + f'(0)g'(0)x^2[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.11)</td></tr><tr id="a0000000481"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(0)g(0) + \left(f'(0)g(0) +f(0)g'(0)\right)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.12)</td></tr></table>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem2" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab8-problem2" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab8-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-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="approx_1-tab8-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem2-problem-progress" tabindex="-1">
Practice with products 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Find the linear approximation for [mathjaxinline]\displaystyle \frac{\ln (1-x)}{(1+x)^2}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. </p>
<p>
What is the linear approximation near [mathjaxinline]x=0[/mathjaxinline]? </p>
<p>
(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division.) </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{\ln (1-x)}{(1+x)^2} \approx[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab8-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_approx_1-tab8-problem2_2_1" id="input_approx_1-tab8-problem2_2_1" data-input-id="approx_1-tab8-problem2_2_1" value="" aria-describedby="status_approx_1-tab8-problem2_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab8-problem2_2_1"/>
<span class="status unanswered" id="status_approx_1-tab8-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab8-problem2_2_1" class="answer"/>
<div id="input_approx_1-tab8-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>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab8-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice with products 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_approx_1-tab8-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab8-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="approx_1-tab8-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="approx_1-tab8-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="approx_1-tab8-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem3" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab8-problem3" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab8-problem3-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem3" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-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="approx_1-tab8-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem3-problem-progress" tabindex="-1">
Practice with products 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
Find the linear approximation of [mathjaxinline]\ \displaystyle \frac{\sin (x)}{\cos ^2(x)}[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline] using the known approximations of [mathjaxinline]\sin (x)[/mathjaxinline] and [mathjaxinline]\cos (x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. </p>
<p>
(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab8-problem3_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_approx_1-tab8-problem3_2_1" id="input_approx_1-tab8-problem3_2_1" data-input-id="approx_1-tab8-problem3_2_1" value="" aria-describedby="status_approx_1-tab8-problem3_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab8-problem3_2_1"/>
<span class="status unanswered" id="status_approx_1-tab8-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab8-problem3_2_1" class="answer"/>
<div id="input_approx_1-tab8-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>
<div class="solution-span">
<span id="solution_approx_1-tab8-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice with products 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_approx_1-tab8-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab8-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="approx_1-tab8-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="approx_1-tab8-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="approx_1-tab8-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-5" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem4">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem4" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab8-problem4" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab8-problem4-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem4" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-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="approx_1-tab8-problem4-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem4-problem-progress" tabindex="-1">
Practice with products 3
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab8-problem4-problem-progress"></div>
<div class="problem">
<div>
<p>
Find the linear approximation for [mathjaxinline]\ln (x)\sin (x-1)[/mathjaxinline] near [mathjaxinline]x=1[/mathjaxinline]. </p>
<p>
(Type [mathjaxinline]{}^*[/mathjaxinline] for multiplication. Type / for division.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab8-problem4_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_approx_1-tab8-problem4_2_1" id="input_approx_1-tab8-problem4_2_1" data-input-id="approx_1-tab8-problem4_2_1" value="" aria-describedby="status_approx_1-tab8-problem4_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab8-problem4_2_1"/>
<span class="status unanswered" id="status_approx_1-tab8-problem4_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab8-problem4_2_1" class="answer"/>
<div id="input_approx_1-tab8-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>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab8-problem4_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice with products 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_approx_1-tab8-problem4" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab8-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="approx_1-tab8-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="approx_1-tab8-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="approx_1-tab8-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab9" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">9. Measurement error</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab9-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab9-video1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Height of the Green Building</h3>
<div
id="video_approx_1-tab9-video1"
class="video closed"
data-metadata='{"captionDataDir": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab9-video1/handler/transcript/available_translations", "start": 0.0, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "saveStateEnabled": false, "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab9-video1/handler/publish_completion", "lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab9-video1/handler/xmodule_handler/save_user_state", "completionEnabled": false, "prioritizeHls": false, "sources": [], "completionPercentage": 0.95, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab9-video1/handler/transcript/translation/__lang__", "transcriptLanguages": {"en": "English"}, "streams": "1.00:IQ_V4muj8qs", "showCaptions": "true", "speed": null, "ytMetadataEndpoint": "", "autohideHtml5": false, "duration": 0.0, "poster": null, "recordedYoutubeIsAvailable": true, "end": 0.0, "ytTestTimeout": 1500, "autoplay": false}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="approx_1-tab9-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_approx_1-tab9-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_approx_1-tab9-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab9-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab9-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab9-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab9-text1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Estimate of the height</b></p><p>
Here is a diagram of the measurements we just carried out. </p><center><img src="/assets/courseware/v1/4fd3055c881f14b6ee4b57d09c758fb4/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_greenbld_diagram.svg" width="350px" alt="See table below." style="margin: 10px 25px 25px 25px"/></center><p>
The height [mathjaxinline]y[/mathjaxinline] of the Green building is given by </p><table id="a0000000489" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]y = h+x\tan (\theta ).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Below is a table of the measurements made, as well as the error in each term. </p><table style="table-layout:auto"><tr><th> Quantity </th><th> Variable </th><th> Measurement </th><th> Error </th></tr><tr><td style="border:none"> Height to Jen's eyeball </td><td style="border:none"> h </td><td style="border:none"> 4.9 feet </td><td style="border:none"> [mathjaxinline]\pm [/mathjaxinline] .01 feet </td></tr><tr><td style="border:none"> Distance to Green Building </td><td style="border:none"> x</td><td style="border:none"> 175 feet </td><td style="border:none"> [mathjaxinline]\pm [/mathjaxinline] 3 feet </td></tr><tr><td style="border:none"> Angle </td><td style="border:none"> [mathjaxinline]\theta [/mathjaxinline]</td><td style="border:none"> 57 degrees </td><td style="border:none"> [mathjaxinline]\pm [/mathjaxinline] 3 degrees </td></tr></table><p>
Where is the error coming from? </p><ul class="itemize"><li><p>
The error in the measurement of Jen's height comes from rounding error and inherent error in using a tape measure to enough accuracy. </p></li><li><p>
The error in the distance to the Green building is coming from a few places: error in the actual length of the rope which may differ by a few inches, error in our walking in a perfectly straight line as we measure, and error coming from the fact that the rope is stretching when we pull on it, lengthening the rope. So in fact, we expect our measurement is likely an overestimate. </p></li><li><p>
The error in the angle is coming from wind and the string not hanging straight down, error in the construction of the device, and error in holding the device to read the measurement. </p></li></ul>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-problem1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab9-problem1" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab9-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-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="approx_1-tab9-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-problem1-problem-progress" tabindex="-1">
Height estimate
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
What is the estimated height of the Green Building? </p>
<p>
(Enter an integer with 3 significant figures.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_approx_1-tab9-problem1_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_approx_1-tab9-problem1_2_1" id="input_approx_1-tab9-problem1_2_1" aria-describedby="trailing_text_approx_1-tab9-problem1_2_1 status_approx_1-tab9-problem1_2_1" value=""/>
<span class="trailing_text" id="trailing_text_approx_1-tab9-problem1_2_1">feet</span>
<span class="status unanswered" id="status_approx_1-tab9-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab9-problem1_2_1" class="answer"/>
</div>
</div></div>
</p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Height estimate" />
<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_approx_1-tab9-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab9-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="approx_1-tab9-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="approx_1-tab9-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="approx_1-tab9-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab9-text2">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab9-text2" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Error in estimate</b></p><p>
What is [mathjaxinline]\Delta y[/mathjaxinline], which is an error term representing the difference between the height that we calculated for the building from our measurements and its true height? </p><p>
First note that what we really have is the following. </p><center><img src="/assets/courseware/v1/d37cb727913f5ab0ae7c0a4fb3dc4b91/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_greenbld_diagram2.svg" width="355px" alt="Each variable includes a small error." style="margin: 10px 25px 25px 25px"/></center><p>
And </p><table id="a0000000490" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000491"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \tan (\theta +\Delta \theta )[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{(y+\Delta y)-(h+\Delta h)}{x+\Delta x}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.15)</td></tr><tr id="a0000000492"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{\Delta y -\Delta h + (y-h)}{x+\Delta x}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.16)</td></tr><tr id="a0000000493"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{\Delta y -\Delta h + x \tan \theta }{x+\Delta x}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.17)</td></tr></table><p>
The last line uses the fact that [mathjaxinline]\displaystyle \tan (\theta ) = \frac{y-h}{x}[/mathjaxinline]. </p><p>
To determine the error, we solve for [mathjaxinline]\Delta y[/mathjaxinline]: </p><table id="a0000000494" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\Delta y = \Delta h + (x+\Delta x)\left(\tan (\theta + \Delta \theta )\right) - x\tan (\theta ).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
To get a sense of the order of this error term, let's find a linear approximation of [mathjaxinline]\tan (\theta + \Delta \theta )[/mathjaxinline] near [mathjaxinline]\theta = 57[/mathjaxinline] degrees. </p>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-problem2" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab9-problem2" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab9-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-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="approx_1-tab9-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-problem2-problem-progress" tabindex="-1">
Linear approximation of tan
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab9-problem2-problem-progress"></div>
<div class="problem">
<div>
<span>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_approx_1-tab9-problem2_2_1" class="capa_inputtype">
<div class="drag_and_drop_problem_div" id="drag_and_drop_div_approx_1-tab9-problem2_2_1" data-plain-id="approx_1-tab9-problem2_2_1">
</div>
<div class="drag_and_drop_problem_json" id="drag_and_drop_json_approx_1-tab9-problem2_2_1" style="display:none;">{"base_image": "/assets/courseware/v1/717bf3958a3e565e3ac0a6e3e652b351/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_green_bld_tanapprox_green_bld_tanapprox_dnd.png", "targets": [{"w": "77", "h": "72", "y": "11", "id": "1", "x": "83"}, {"w": "141", "h": "72", "y": "11", "id": "2", "x": "321"}, {"w": "141", "h": "72", "y": "11", "id": "3", "x": "518"}], "one_per_target": "true", "draggables": [{"label": "", "icon": "/assets/courseware/v1/ddc9a76173e427bd494bae66f55273fa/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_green_bld_tanapprox_green_bld_tanapprox_dnd_label1.png", "can_reuse": "", "target_fields": [], "id": "deg"}, {"label": "", "icon": "/assets/courseware/v1/b653c5e8b02e5d5d46de14cd5cf50692/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_green_bld_tanapprox_green_bld_tanapprox_dnd_label2.png", "can_reuse": "", "target_fields": [], "id": "tand"}, {"label": "", "icon": "/assets/courseware/v1/6e4acb677fda6493ef5c78d2e0bb5c63/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_green_bld_tanapprox_green_bld_tanapprox_dnd_label3.png", "can_reuse": "", "target_fields": [], "id": "secd"}, {"label": "", "icon": "/assets/courseware/v1/4f0bd7008bdab75fa0d0939fcb13e2de/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_green_bld_tanapprox_green_bld_tanapprox_dnd_label4.png", "can_reuse": "", "target_fields": [], "id": "sec2d"}, {"label": "", "icon": "/assets/courseware/v1/d7747cff23f516135771283395cfd563/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_green_bld_tanapprox_green_bld_tanapprox_dnd_label5.png", "can_reuse": "", "target_fields": [], "id": "rad"}, {"label": "", "icon": "/assets/courseware/v1/91a22c4698f3df1354df1cb3a4192810/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_green_bld_tanapprox_green_bld_tanapprox_dnd_label6.png", "can_reuse": "", "target_fields": [], "id": "tan"}, {"label": "", "icon": "/assets/courseware/v1/3b83c2590505787457bfc7f88799a83c/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_green_bld_tanapprox_green_bld_tanapprox_dnd_label7.png", "can_reuse": "", "target_fields": [], "id": "sec"}, {"label": "", "icon": "/assets/courseware/v1/07b5ba2931b25b1dbc9ef36d2b61f049/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_green_bld_tanapprox_green_bld_tanapprox_dnd_label8.png", "can_reuse": "", "target_fields": [], "id": "sec2"}], "target_outline": "false", "label_bg_color": "rgb(222, 139, 238)"}</div>
<div class="script_placeholder" data-src="/static/js/capa/drag_and_drop.a31124208b9b.js"/>
<div class="unanswered" id="status_approx_1-tab9-problem2_2_1">
<input type="text" name="input_approx_1-tab9-problem2_2_1" id="input_approx_1-tab9-problem2_2_1" aria-describedby="answer_approx_1-tab9-problem2_2_1" value="" style="display:none;"/>
<p class="indicator-container drag-and-drop--status" aria-describedby="input_approx_1-tab9-problem2_2_1">
<span class="status unanswered" id="status_approx_1-tab9-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_approx_1-tab9-problem2_2_1" class="answer"/>
</div>
</div></div>
<script src="/assets/courseware/v1/c5a72cdba617092eb404736ef314d71b/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/dragDropTouchSupport2.js" type="text/javascript"/>
</span>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab9-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Linear approximation of tan" />
<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_approx_1-tab9-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab9-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="approx_1-tab9-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="approx_1-tab9-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="approx_1-tab9-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab10" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">10. Continuing the error estimate</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab10-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab10-text1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
To estimate the error term, we find: </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \Delta y = \Delta h + (x+\Delta x)\left(\tan (\theta + \Delta \theta )\right) - x\tan (\theta )[/mathjaxinline] </td><td style="text-align:left; border:none">
This is the term from the previous page.</td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \quad \approx \Delta h + (x+\Delta x)\left(\tan (\theta ) + \sec ^2(\theta )\Delta \theta \right)-x\tan (\theta )[/mathjaxinline] </td><td style="text-align:left; border:none">
Here we use the linear approximation for [mathjaxinline]\tan (\theta + \Delta \theta )[/mathjaxinline]. </td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\quad \displaystyle = \Delta h + \Delta x\tan (\theta ) + x\sec ^2(\theta )\Delta \theta + \Delta x \Delta \theta \sec ^2(\theta )[/mathjaxinline] </td><td style="text-align:left; border:none">
Note this is equal to the previous line. We multiplied out the middle term in the previous line, and canceled the [mathjaxinline]x\tan (\theta )[/mathjaxinline] terms.</td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \quad \approx \Delta h + \Delta x\tan (\theta ) + x\sec ^2(\theta )\Delta \theta .[/mathjaxinline] </td><td style="text-align:left; border:none">
This last is an approximate equality. We canceled the quadratic term as it is not linear. </td></tr></table>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-problem1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab10-problem1" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab10-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-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="approx_1-tab10-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-problem1-problem-progress" tabindex="-1">
Identify largest error term
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
In the error term [mathjaxinline]\Delta y \approx \Delta h + \Delta x\tan (\theta ) + x\sec ^2(\theta )\Delta \theta[/mathjaxinline], which term has the largest contribution to the error? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_approx_1-tab10-problem1_2_1">
<fieldset aria-describedby="status_approx_1-tab10-problem1_2_1">
<div class="field">
<input type="radio" name="input_approx_1-tab10-problem1_2_1" id="input_approx_1-tab10-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="approx_1-tab10-problem1_2_1-choice_1-label" for="input_approx_1-tab10-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_approx_1-tab10-problem1_2_1">
<text> [mathjaxinline]\Delta h[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_approx_1-tab10-problem1_2_1" id="input_approx_1-tab10-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="approx_1-tab10-problem1_2_1-choice_2-label" for="input_approx_1-tab10-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_approx_1-tab10-problem1_2_1">
<text> [mathjaxinline]\Delta x\tan (\theta )[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_approx_1-tab10-problem1_2_1" id="input_approx_1-tab10-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="approx_1-tab10-problem1_2_1-choice_3-label" for="input_approx_1-tab10-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_approx_1-tab10-problem1_2_1">
<text> [mathjaxinline]x\sec ^2(\theta )\Delta \theta[/mathjaxinline]</text>
</label>
</div>
<span id="answer_approx_1-tab10-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_approx_1-tab10-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>
<div class="solution-span">
<span id="solution_approx_1-tab10-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Identify largest error term" />
<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_approx_1-tab10-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab10-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="approx_1-tab10-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="approx_1-tab10-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="approx_1-tab10-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-problem2" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab10-problem2" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab10-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-problem2/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="approx_1-tab10-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-problem2-problem-progress" tabindex="-1">
The error
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab10-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
What is the magnitude of the error of our height measurement? </p>
<p>
(Enter an integer.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_approx_1-tab10-problem2_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_approx_1-tab10-problem2_2_1" id="input_approx_1-tab10-problem2_2_1" aria-describedby="trailing_text_approx_1-tab10-problem2_2_1 status_approx_1-tab10-problem2_2_1" value=""/>
<span class="trailing_text" id="trailing_text_approx_1-tab10-problem2_2_1">feet</span>
<span class="status unanswered" id="status_approx_1-tab10-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab10-problem2_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
Compare this to the actual error given that the Green Building is actually 295 feet tall. Did our estimate fall within the error bounds? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_approx_1-tab10-problem2_3_1">
<fieldset aria-describedby="status_approx_1-tab10-problem2_3_1">
<div class="field">
<input type="radio" name="input_approx_1-tab10-problem2_3_1" id="input_approx_1-tab10-problem2_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="approx_1-tab10-problem2_3_1-choice_1-label" for="input_approx_1-tab10-problem2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_approx_1-tab10-problem2_3_1">
<text> yes</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_approx_1-tab10-problem2_3_1" id="input_approx_1-tab10-problem2_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="approx_1-tab10-problem2_3_1-choice_2-label" for="input_approx_1-tab10-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_approx_1-tab10-problem2_3_1">
<text> no</text>
</label>
</div>
<span id="answer_approx_1-tab10-problem2_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_approx_1-tab10-problem2_3_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_approx_1-tab10-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="The error" />
<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_approx_1-tab10-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab10-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="approx_1-tab10-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="approx_1-tab10-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="approx_1-tab10-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab11" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">11. Project</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab11-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab11-video1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Altimeter (or Astrolabe)</h3>
<div
id="video_approx_1-tab11-video1"
class="video closed"
data-metadata='{"captionDataDir": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab11-video1/handler/transcript/available_translations", "start": 0.0, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "saveStateEnabled": false, "transcriptLanguage": "en", "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab11-video1/handler/publish_completion", "lmsRootURL": "https://openlearninglibrary.mit.edu", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab11-video1/handler/xmodule_handler/save_user_state", "completionEnabled": false, "prioritizeHls": false, "sources": [], "completionPercentage": 0.95, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab11-video1/handler/transcript/translation/__lang__", "transcriptLanguages": {"en": "English"}, "streams": "1.00:TXgbP8ZxcIk", "showCaptions": "true", "speed": null, "ytMetadataEndpoint": "", "autohideHtml5": false, "duration": 0.0, "poster": null, "recordedYoutubeIsAvailable": true, "end": 0.0, "ytTestTimeout": 1500, "autoplay": false}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="approx_1-tab11-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_approx_1-tab11-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_approx_1-tab11-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab11-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@approx_1-tab11-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab11-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab11-text1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
Build your own altitude measuring equipment and find the heights of objects, including the heights of your friends, and other known quantities. Make sure to answer the following questions and test your hypothesis! </p><ol class="enumerate"><li value="1"><p>
What do you predict to be the measurement error in each measurement that you make? </p></li><li value="2"><p>
What is the error bound for the height? </p></li><li value="3"><p>
Does your height measurement fall within the bounds of your expected error? Why or why not? </p></li></ol>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab12" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">12. Error approximation practice</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-problem1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab12-problem1" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab12-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-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="approx_1-tab12-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-problem1-problem-progress" tabindex="-1">
Velocity measurement
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Suppose that you measure the velocity of an object by measuring that it takes 1 second to travel 1.2 meters. The measurement error is .001 meters in distance, and the error in time is .01 second. What is the absolute value of the error in the linear approximation for the velocity? </p>
<p>
(Enter decimal answer to three decimal digits. Enter the units of the error using [mathjaxinline]m[/mathjaxinline] to denote meters and [mathjaxinline]s[/mathjaxinline] to denote seconds.) </p>
<p>
<p style="display:inline">[mathjaxinline]\left| \Delta v \right| \leq[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab12-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_approx_1-tab12-problem1_2_1" id="input_approx_1-tab12-problem1_2_1" data-input-id="approx_1-tab12-problem1_2_1" value="" aria-describedby="status_approx_1-tab12-problem1_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_approx_1-tab12-problem1_2_1"/>
<span class="status unanswered" id="status_approx_1-tab12-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab12-problem1_2_1" class="answer"/>
<div id="input_approx_1-tab12-problem1_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>
<br/>
<p style="display:inline">Units: </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_approx_1-tab12-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_approx_1-tab12-problem1_3_1" id="input_approx_1-tab12-problem1_3_1" data-input-id="approx_1-tab12-problem1_3_1" value="" aria-describedby="status_approx_1-tab12-problem1_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_approx_1-tab12-problem1_3_1"/>
<span class="status unanswered" id="status_approx_1-tab12-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab12-problem1_3_1" class="answer"/>
<div id="input_approx_1-tab12-problem1_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>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab12-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Velocity measurement" />
<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_approx_1-tab12-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab12-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="approx_1-tab12-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="approx_1-tab12-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="approx_1-tab12-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-problem2" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab12-problem2" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab12-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-problem2/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="approx_1-tab12-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-problem2-problem-progress" tabindex="-1">
Kinetic energy
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab12-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Suppose you measure the velocity of a mosquito to be [mathjaxinline]1.3[/mathjaxinline] m/s with an error of [mathjaxinline]0.2[/mathjaxinline] m/s. We approximate the mosquito to weigh approximately equal to a grain of rice, which is on the order of magnitude of [mathjaxinline]10^{-6}[/mathjaxinline] kg. What is the error in the kinetic energy? Assume there is no rotational velocity. In the second answer box, enter units involving m, s, and kg to denote meters, seconds, and kilograms respectively. (Do <b class="bf">not</b> use J for Joules or N for Newtons.) </p>
<p>
(Enter answer with 2 significant figures of accuracy, i.e.with 2 nonzero numbers.) </p>
<p>
<p style="display:inline">[mathjaxinline]\left| \Delta K \right| \leq[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab12-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_approx_1-tab12-problem2_2_1" id="input_approx_1-tab12-problem2_2_1" data-input-id="approx_1-tab12-problem2_2_1" value="" aria-describedby="status_approx_1-tab12-problem2_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_approx_1-tab12-problem2_2_1"/>
<span class="status unanswered" id="status_approx_1-tab12-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab12-problem2_2_1" class="answer"/>
<div id="input_approx_1-tab12-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>
<p>
<p style="display:inline">Units: </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_approx_1-tab12-problem2_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_approx_1-tab12-problem2_3_1" id="input_approx_1-tab12-problem2_3_1" data-input-id="approx_1-tab12-problem2_3_1" value="" aria-describedby="status_approx_1-tab12-problem2_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_approx_1-tab12-problem2_3_1"/>
<span class="status unanswered" id="status_approx_1-tab12-problem2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab12-problem2_3_1" class="answer"/>
<div id="input_approx_1-tab12-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>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab12-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Kinetic energy" />
<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_approx_1-tab12-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab12-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="approx_1-tab12-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="approx_1-tab12-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="approx_1-tab12-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab13" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">13. Practice</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab13-problem1" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab13-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-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="approx_1-tab13-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem1-problem-progress" tabindex="-1">
Practice problem 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
What is the linear approximation to [mathjaxinline]\cos (x/2)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab13-problem1_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_approx_1-tab13-problem1_2_1" id="input_approx_1-tab13-problem1_2_1" data-input-id="approx_1-tab13-problem1_2_1" value="" aria-describedby="status_approx_1-tab13-problem1_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab13-problem1_2_1"/>
<span class="status unanswered" id="status_approx_1-tab13-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab13-problem1_2_1" class="answer"/>
<div id="input_approx_1-tab13-problem1_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>
<div class="solution-span">
<span id="solution_approx_1-tab13-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice 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_approx_1-tab13-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab13-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="approx_1-tab13-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="approx_1-tab13-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="approx_1-tab13-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem2" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab13-problem2" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab13-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-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="approx_1-tab13-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem2-problem-progress" tabindex="-1">
Practice problem 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
What is the linear approximation to [mathjaxinline]\ln \left((1+ax)^ r\right)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab13-problem2_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_approx_1-tab13-problem2_2_1" id="input_approx_1-tab13-problem2_2_1" data-input-id="approx_1-tab13-problem2_2_1" value="" aria-describedby="status_approx_1-tab13-problem2_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab13-problem2_2_1"/>
<span class="status unanswered" id="status_approx_1-tab13-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab13-problem2_2_1" class="answer"/>
<div id="input_approx_1-tab13-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>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab13-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice 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_approx_1-tab13-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab13-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="approx_1-tab13-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="approx_1-tab13-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="approx_1-tab13-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem3" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab13-problem3" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab13-problem3-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem3" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-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="approx_1-tab13-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem3-problem-progress" tabindex="-1">
Practice problem 3
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
Use a linear approximation for [mathjaxinline]\sin[/mathjaxinline] to find an approximation (this will not be linear) to the function [mathjaxinline]\sin (x^3)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\sin (x^3)\approx[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab13-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_approx_1-tab13-problem3_2_1" id="input_approx_1-tab13-problem3_2_1" data-input-id="approx_1-tab13-problem3_2_1" value="" aria-describedby="status_approx_1-tab13-problem3_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab13-problem3_2_1"/>
<span class="status unanswered" id="status_approx_1-tab13-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab13-problem3_2_1" class="answer"/>
<div id="input_approx_1-tab13-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>
<div class="solution-span">
<span id="solution_approx_1-tab13-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice 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_approx_1-tab13-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab13-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="approx_1-tab13-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="approx_1-tab13-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="approx_1-tab13-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem4">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem4" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab13-problem4" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab13-problem4-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem4" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-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="approx_1-tab13-problem4-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem4-problem-progress" tabindex="-1">
Complicated problem (*)
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem4-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider [mathjaxinline]\cos \left(\frac{\pi }{2}(1+x)^{3/2}\right)[/mathjaxinline]. </p>
<p>
What is the linear approximation near [mathjaxinline]x=0[/mathjaxinline]? <p style="display:inline">[mathjaxinline]\cos \left(\frac{\pi }{2}(1+x)^{3/2}\right)\approx[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_approx_1-tab13-problem4_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_approx_1-tab13-problem4_2_1" id="input_approx_1-tab13-problem4_2_1" data-input-id="approx_1-tab13-problem4_2_1" value="" aria-describedby="status_approx_1-tab13-problem4_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_approx_1-tab13-problem4_2_1"/>
<span class="status unanswered" id="status_approx_1-tab13-problem4_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab13-problem4_2_1" class="answer"/>
<div id="input_approx_1-tab13-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>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab13-problem4_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Complicated problem (*)" />
<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_approx_1-tab13-problem4" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab13-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="approx_1-tab13-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="approx_1-tab13-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="approx_1-tab13-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem5">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-has-score="True" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem5" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_approx_1-tab13-problem5" class="problems-wrapper" role="group"
aria-labelledby="approx_1-tab13-problem5-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem5" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem5/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="approx_1-tab13-problem5-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem5-problem-progress" tabindex="-1">
Implicit differentiation and linear approximation (*)
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@approx_1-tab13-problem5-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider the implicit function [mathjaxinline](w(x) + 1)e^{w(x)} = x[/mathjaxinline]. Approximate [mathjaxinline]w(1.1)[/mathjaxinline] using that [mathjaxinline]w(1) =0[/mathjaxinline]. </p>
<p>
(Enter answer to 2 decimal places.) </p>
<p>
<p style="display:inline">[mathjaxinline]w(1.1)\approx[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_approx_1-tab13-problem5_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_approx_1-tab13-problem5_2_1" id="input_approx_1-tab13-problem5_2_1" aria-describedby="status_approx_1-tab13-problem5_2_1" value="" size="40"/>
<span class="trailing_text" id="trailing_text_approx_1-tab13-problem5_2_1"/>
<span class="status unanswered" id="status_approx_1-tab13-problem5_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_approx_1-tab13-problem5_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_approx_1-tab13-problem5_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Implicit differentiation and linear approximation (*)" />
<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_approx_1-tab13-problem5" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_approx_1-tab13-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">
</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="approx_1-tab13-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="approx_1-tab13-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="approx_1-tab13-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="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-has-score="False" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@vertical+block@approx_1-tab14" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<h2 class="hd hd-2 unit-title">14. Summary</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab14-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-has-score="False" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-version="1" data-usage-id="block-v1:MITx+18.01.1x+2T2019+type@html+block@approx_1-tab14-text1" data-course-id="course-v1:MITx+18.01.1x+2T2019" data-request-token="71f223985aa311ef94de022c6f26e6af" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Linear approximations revisited</b></p><p>
You already know how to take linear approximations. This was the main tool for finding the derivatives of all of the basic functions in the previous unit. </p><p>
We return again to linear approximations because they are the main tool that we use in finding approximations in science and engineering. Here we develop some more tools for using linear approximations, as well as changing our perspective to use linear approximations to estimate how measurement error perpetuates through a system, and understand the system's sensitivity to small perturbations. </p><p><b class="bfseries">Linear Approximations near [mathjaxinline]x=0[/mathjaxinline]</b></p><p>
Recall that the linear approximation of a function [mathjaxinline]\ f(x)[/mathjaxinline] for [mathjaxinline]x[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline] is given by the formula [mathjaxinline]\ f(x) \approx f(0) + f'(0)x[/mathjaxinline]. Find linear approximations of all of the basic functions near [mathjaxinline]x=0[/mathjaxinline]. We are going to be using these formulas a lot! So you may even want to memorize them. </p><p>
We call the linear approximation the <span style="color:#27408C"><p><b class="bf">linearization</b> of a function. </p><p><b class="bfseries">Approximations (likely nonlinear) of compositions</b></p><p>
Suppose that [mathjaxinline]g(x)[/mathjaxinline] is a function such that [mathjaxinline]g(0) =0[/mathjaxinline]. To find an approximation of a function [mathjaxinline]f(g(x))[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline], we can take a linear approximation for [mathjaxinline]f(u)[/mathjaxinline] and then substitute [mathjaxinline]g(x)[/mathjaxinline] in for [mathjaxinline]u[/mathjaxinline]. The resulting approximation is likely nonlinear, but it is still an approximation! <span style="color:#99182C"><p>
Warning: this only works if [mathjaxinline]g(0) = 0[/mathjaxinline]. </p><table id="a0000000529" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000530"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f(u) \approx f(0) + f'(0)u \qquad \Longrightarrow \qquad f\left(g(x)\right) \approx f(g(0)) + f'(g(0))g(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.28)</td></tr></table><p><b class="bfseries">Linear approximations of products</b></p><p>
To find the linear approximation of a function [mathjaxinline]h(x) = f(x)g(x)[/mathjaxinline] near [mathjaxinline]x=0[/mathjaxinline], it suffices to find a linear approximation for [mathjaxinline]f(x)[/mathjaxinline], find a linear approximation for [mathjaxinline]g(x)[/mathjaxinline], and then the linear approximation for [mathjaxinline]h(x)[/mathjaxinline] is the product of these two approximations where we cancel all of the terms that are quadratic (or higher for more products). </p><table id="a0000000531" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000532"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle h(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(x)g(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.29)</td></tr><tr id="a0000000533"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(0) + f'(0)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.30)</td></tr><tr id="a0000000534"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle g(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle g(0) + g'(0)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.31)</td></tr><tr id="a0000000535"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle h(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(f(0) + f'(0)x\right)\left(g(0) + g'(0)x\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.32)</td></tr><tr id="a0000000536"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(0)g(0) + \left(f'(0)g(0) +f(0)g'(0)\right)x + f'(0)g'(0)x^2[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.33)</td></tr><tr id="a0000000537"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(0)g(0) + \left(f'(0)g(0) +f(0)g'(0)\right)x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.34)</td></tr></table></span></p></span></p>
</div>
</div>
</div>
</div>
© All Rights Reserved