<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab1" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">1. Motivation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab1-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab1-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">The Second Fundamental Theorem of Calculus</h3>
<div
id="video_theory3-tab1-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab1-video1/handler/publish_completion", "streams": "1.00:AjBRTOGSwRs", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab1-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab1-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab1-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-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_theory3-tab1-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-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-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab2" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">2. The second fundamental theorem of calculus</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab2-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab2-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Know the statement of the <span style="color:#27408C"><b class="bf">second fundamental theorem of calculus</b></span>. </p></li><li><p>
Compute the derivatives of functions defined by definite integrals, including using the <span style="color:#27408C"><b class="bf">chain rule</b></span>. </p></li><li><p>
Analyze functions defined by definite integrals with the usual techniques, such as linear and quadratic approximations, and graphing. </p></li></ul><p><b class="bfseries">Broader Goals</b></p><ul class="itemize"><li><p>
Understand the proofs of the two fundamental theorems of calculus. </p></li><li><p>
Understand the relationship between differentiation and integration. </p></li></ul><p><b class="bfseries">Contents: 18 pages</b></p><p>
14 videos (64 minutes 1x speed) 20 questions </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab3" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">3. Exploration</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab3-problem1" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab3-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="2"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="theory3-tab3-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem1-problem-progress" tabindex="-1">
Differentiating the integral of a constant
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Let us start again with the definite integral [mathjaxinline]\displaystyle \int _{a}^{b} m \, dt[/mathjaxinline] of a constant [mathjaxinline]m&gt;0[/mathjaxinline]. </p>
<p>
But this time, we will let the upper limit of the integral vary and rename it [mathjaxinline]x[/mathjaxinline] (in place of [mathjaxinline]b[/mathjaxinline]). In other words, define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001101" style="table-layout:auto" width="100%">
<tr id="a0000001102">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle F(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 \int _{a}^{x} m\, dt,\qquad m&gt;0.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.206)</td>
</tr>
</table>
<p>
As the upper limit [mathjaxinline]x[/mathjaxinline] of the integral changes, the value of the integral changes.<br/></p>
<center><img alt="Go to text below image" src="/assets/courseware/v1/26f5d1d65b809ef9c015f958407b2a92/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_constant.svg" style="margin: 10px 25px 25px 25px" width="300px"/>The horizontal line given by y equals m is plotted in the first quadrant of the y t plane. The region beneath the line y equals m and between the vertical lines t equals a and t equals x is shaded in blue. This region is a rectangle with base x minus a and height m. The shaded region is labeled capital F of x.<br/>Geometrically, [mathjaxinline]F(x)[/mathjaxinline] is the area of the shaded rectangle. This area changes as the upper limit [mathjaxinline]x[/mathjaxinline] changes, and hence is a function of [mathjaxinline]x[/mathjaxinline].<br/></center>
<p>
Evaluate [mathjaxinline]F(x)[/mathjaxinline]. Which of the following is the correct graph of [mathjaxinline]F(x)[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab3-problem1_2_1">
<fieldset aria-describedby="status_theory3-tab3-problem1_2_1">
<div class="field">
<input type="radio" name="input_theory3-tab3-problem1_2_1" id="input_theory3-tab3-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory3-tab3-problem1_2_1-choice_1-label" for="input_theory3-tab3-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab3-problem1_2_1">
<text>
<img alt="The line F of x equals ma" src="/assets/courseware/v1/f55ceaf32a0d7c12c15152b692919241/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_horilinepos.svg" style="margin: 10px 25px 25px 25px" width="320px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab3-problem1_2_1" id="input_theory3-tab3-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory3-tab3-problem1_2_1-choice_2-label" for="input_theory3-tab3-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab3-problem1_2_1">
<text>
<img alt="The line F of x equals negative ma" src="/assets/courseware/v1/c93e0fabe80c0e2aee53916dc6b88e94/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_horilineneg.svg" style="margin: 10px 25px 25px 25px" width="320px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab3-problem1_2_1" id="input_theory3-tab3-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory3-tab3-problem1_2_1-choice_3-label" for="input_theory3-tab3-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab3-problem1_2_1">
<text>
<img alt="The line F of x equals mx" src="/assets/courseware/v1/96f2887910f1984eb3e18b6b40d84c77/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_linethroughorigin.svg" style="margin: 10px 25px 25px 25px" width="260px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab3-problem1_2_1" id="input_theory3-tab3-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="theory3-tab3-problem1_2_1-choice_4-label" for="input_theory3-tab3-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory3-tab3-problem1_2_1">
<text>
<img alt="The line F of x equals mx minus ma" src="/assets/courseware/v1/ea1cf4867af896f30408a3e91308769f/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_line.svg" style="margin: 10px 25px 25px 25px" width="260px"/>
</text>
</label>
</div>
<span id="answer_theory3-tab3-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab3-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
Now, differentiate [mathjaxinline]F(x)[/mathjaxinline].<br/><p style="display:inline">[mathjaxinline]\displaystyle F'(x)\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_theory3-tab3-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab3-problem1_3_1" id="input_theory3-tab3-problem1_3_1" data-input-id="theory3-tab3-problem1_3_1" value="" aria-describedby="status_theory3-tab3-problem1_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab3-problem1_3_1"/>
<span class="status unanswered" id="status_theory3-tab3-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab3-problem1_3_1" class="answer"/>
<div id="input_theory3-tab3-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>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab3-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Differentiating the integral of a constant" />
<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_theory3-tab3-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-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="theory3-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="theory3-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="theory3-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 class="vert vert-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab3-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab3-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">The Second Fundamental Theorem of Calculus</h3>
<div
id="video_theory3-tab3-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab3-video1/handler/publish_completion", "streams": "1.00:-qM5jLUwpTY", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab3-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab3-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab3-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-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>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_theory3-tab3-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-tab3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab3-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+18.01.2x+3T2019+type@html+block@theory3-tab3-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab3-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Second Fundamental Theorem of Calculus</b></p><p>
The <span style="color:#99182C"><b class="bf">Second Fundamental Theorem of Calculus</b></span> states the following.<br/>Given a continuous function [mathjaxinline]f(x)[/mathjaxinline]. If </p><table id="a0000001109" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle G(x)=\int _{a}^{x} f(t)\, dt \qquad (\, \, t\, \, \text {between}\, \, a\, \, \text {and}\, x),[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
then </p><table id="a0000001110" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]G'(x)=f(x).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><center><img src="/assets/courseware/v1/b0086bdea7eaee80c86ce42e733ce366/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_G.svg" width="350px" alt="See caption" style="margin: 10px 25px 25px 25px"/><br/>Geometrically, [mathjaxinline]G(x)[/mathjaxinline] is the area under the curve [mathjaxinline]y=f(t)[/mathjaxinline] between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]x[/mathjaxinline]. This area varies as [mathjaxinline]x[/mathjaxinline] varies.<br/></center><p><br/></p><p>
We will abbreviate this theorem by <span style="color:#99182C"><b class="bf">FTC2</b></span>.<br/></p><p>
In terms of differential equations, FTC2 says that [mathjaxinline]G(x)[/mathjaxinline] is the solution to the following differential equation and initial condition: </p><table id="a0000001111" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001112"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y'[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle f[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {differential equation})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.211)</td></tr><tr id="a0000001113"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y(a)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle 0[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {initial condition}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.212)</td></tr></table><p>
The initial condition [mathjaxinline]y(a)=0[/mathjaxinline] is satisfied because [mathjaxinline]\, \, \displaystyle G(a)=\int _{a}^{a} f(t)\, dt = 0[/mathjaxinline].<br/></p><p>
Recall that any function [mathjaxinline]G[/mathjaxinline] such that [mathjaxinline]G'=f[/mathjaxinline] is an antiderivative of [mathjaxinline]f[/mathjaxinline]. Hence, FTC2 gives us a formula for an antiderivative of [mathjaxinline]f(x)[/mathjaxinline]. This formula is different from the formulas you have seen. It is in terms of a definite integral, and is called an <span style="color:#27408C"><b class="bf">integral formula</b></span>. </p><p><b class="bfseries">Note:</b> The integrand [mathjaxinline]f(x)[/mathjaxinline] can be any continuous function, not just the ones whose antiderivative we know how to find. These integral formulas are still useful because there are numerical methods that allow us to compute them.</p><p>
You have just verified FTC2 for a constant function [mathjaxinline]f(x)=m[/mathjaxinline] in the previous problem.<br/></p>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem2" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab3-problem2" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab3-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem2" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem2/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="theory3-tab3-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem2-problem-progress" tabindex="-1">
When the upper limit is less than a
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
The FTC2 works not only for [mathjaxinline]\, x\geq a,\, \,[/mathjaxinline] but also for [mathjaxinline]\, \, x\leq a[/mathjaxinline].<br/></p>
<p>
Recall that </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001114" style="table-layout:auto" width="100%">
<tr id="a0000001115">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle G(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 \int _{a}^{x} f(t)\, dt.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.213)</td>
</tr>
</table>
<center>
<img alt="Area under curve f of xx between t equals a and t equals x for x less than a." src="/assets/courseware/v1/d5f969539dd6a079d7446a71392fbf90/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_Gneg.svg" style="margin: 10px 25px 25px 25px" width="380px"/>
<br/>
</center>
<p>
Consider [mathjaxinline]x\leq a[/mathjaxinline]. Let [mathjaxinline]A&gt;0[/mathjaxinline] be the area of the shaded region above the interval [mathjaxinline][x,a][/mathjaxinline].<br/></p>
<p>
Find [mathjaxinline]G(x)[/mathjaxinline] in terms of [mathjaxinline]A[/mathjaxinline].<br/>(Enter your answer to the following in terms of [mathjaxinline]A[/mathjaxinline].)<br/><p style="display:inline">For [mathjaxinline]x\leq a[/mathjaxinline], [mathjaxinline]\, \, G(x)\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab3-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab3-problem2_2_1" id="input_theory3-tab3-problem2_2_1" data-input-id="theory3-tab3-problem2_2_1" value="" aria-describedby="status_theory3-tab3-problem2_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab3-problem2_2_1"/>
<span class="status unanswered" id="status_theory3-tab3-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab3-problem2_2_1" class="answer"/>
<div id="input_theory3-tab3-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/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab3-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="When the upper limit is less than a" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_theory3-tab3-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-tab3-problem2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="theory3-tab3-problem2-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="theory3-tab3-problem2-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="theory3-tab3-problem2-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="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+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem3" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab3-problem3" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab3-problem3-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem3" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem3/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="3"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="theory3-tab3-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem3-problem-progress" tabindex="-1">
Quiz
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab3-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
Give the solution to the differential equation and initial condition below in terms of an integral.<br/></p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001120" style="table-layout:auto" width="100%">
<tr id="a0000001121">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \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 x^2[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.217)</td>
</tr>
<tr id="a0000001122">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle H(0)[/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 0.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.218)</td>
</tr>
</table>
<p>
(Note all three answer boxes are graded together. You get all correct, or all wrong. The integration variable is [mathjaxinline]t[/mathjaxinline], the [mathjaxinline]dt[/mathjaxinline] is provided for you.) </p>
<span>
<style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax {
display: inline-block !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax_Preview {
display: inline-block !important;
}
</style>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="multi-inputs-group" role="group"><table>
<tbody>
<tr>
<td/>
<td colspan="3">
<div id="inputtype_theory3-tab3-problem3_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_theory3-tab3-problem3_2_1" id="input_theory3-tab3-problem3_2_1" aria-describedby="status_theory3-tab3-problem3_2_1" value="" size="5"/>
<span class="trailing_text" id="trailing_text_theory3-tab3-problem3_2_1"/>
<span class="status unanswered" id="status_theory3-tab3-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab3-problem3_2_1" class="answer"/>
</div>
</div></td>
</tr>
<tr>
<td>
<p style="text-align:left"> \( \Large{H(x)} = \)</p>
</td>
<td>
<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
</td>
<td>
<br/>
<div id="formulaequationinput_theory3-tab3-problem3_2_2" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab3-problem3_2_2" id="input_theory3-tab3-problem3_2_2" data-input-id="theory3-tab3-problem3_2_2" value="" aria-describedby="trailing_text_theory3-tab3-problem3_2_2 status_theory3-tab3-problem3_2_2" size="10"/>
<span class="trailing_text" id="trailing_text_theory3-tab3-problem3_2_2"> [mathjaxinline] dt [/mathjaxinline]</span>
<span class="status unanswered" id="status_theory3-tab3-problem3_2_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab3-problem3_2_2" class="answer"/>
<div id="input_theory3-tab3-problem3_2_2_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></td>
</tr>
<tr>
<td/>
<td colspan="2">
<div id="inputtype_theory3-tab3-problem3_2_3" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_theory3-tab3-problem3_2_3" id="input_theory3-tab3-problem3_2_3" aria-describedby="status_theory3-tab3-problem3_2_3" value="" size="5"/>
<span class="trailing_text" id="trailing_text_theory3-tab3-problem3_2_3"/>
<span class="status unanswered" id="status_theory3-tab3-problem3_2_3" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab3-problem3_2_3" class="answer"/>
</div>
</div></td>
</tr>
</tbody>
</table>
</div></div>
</span>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab3-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Quiz" />
<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_theory3-tab3-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-tab3-problem3">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="theory3-tab3-problem3-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="theory3-tab3-problem3-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="theory3-tab3-problem3-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="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-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab4" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">4. Practice</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab4-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab4-problem1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab4-problem1" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab4-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab4-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab4-problem1/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="2.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="theory3-tab4-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab4-problem1-problem-progress" tabindex="-1">
The difference of two antiderivatives
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab4-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Let [mathjaxinline]x&gt;0[/mathjaxinline].<br/></p>
<p>
Find an integral formula for the antiderivative [mathjaxinline]F_1[/mathjaxinline] of [mathjaxinline]\ln (x)[/mathjaxinline] so that [mathjaxinline]F_1[/mathjaxinline] crosses the [mathjaxinline]x[/mathjaxinline]-axis at [mathjaxinline]1[/mathjaxinline]:<br/></p>
<p>
(Note all three answer boxes are graded together. You get all correct, or all wrong.)<br/></p>
<span>
<style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax {
display: inline-block !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax_Preview {
display: inline-block !important;
}
</style>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="multi-inputs-group" role="group"><table>
<tbody>
<tr>
<td/>
<td colspan="3">
<div id="inputtype_theory3-tab4-problem1_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_theory3-tab4-problem1_2_1" id="input_theory3-tab4-problem1_2_1" aria-describedby="status_theory3-tab4-problem1_2_1" value="" size="5"/>
<span class="trailing_text" id="trailing_text_theory3-tab4-problem1_2_1"/>
<span class="status unanswered" id="status_theory3-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_theory3-tab4-problem1_2_1" class="answer"/>
</div>
</div></td>
</tr>
<tr>
<td>
<p style="text-align:left"> \( \Large{F_1(x)} = \)</p>
</td>
<td>
<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
</td>
<td>
<br/>
<div id="formulaequationinput_theory3-tab4-problem1_2_2" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab4-problem1_2_2" id="input_theory3-tab4-problem1_2_2" data-input-id="theory3-tab4-problem1_2_2" value="" aria-describedby="trailing_text_theory3-tab4-problem1_2_2 status_theory3-tab4-problem1_2_2" size="10"/>
<span class="trailing_text" id="trailing_text_theory3-tab4-problem1_2_2"> [mathjaxinline] dt [/mathjaxinline]</span>
<span class="status unanswered" id="status_theory3-tab4-problem1_2_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab4-problem1_2_2" class="answer"/>
<div id="input_theory3-tab4-problem1_2_2_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></td>
</tr>
<tr>
<td/>
<td colspan="2">
<div id="inputtype_theory3-tab4-problem1_2_3" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_theory3-tab4-problem1_2_3" id="input_theory3-tab4-problem1_2_3" aria-describedby="status_theory3-tab4-problem1_2_3" value="" size="5"/>
<span class="trailing_text" id="trailing_text_theory3-tab4-problem1_2_3"/>
<span class="status unanswered" id="status_theory3-tab4-problem1_2_3" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab4-problem1_2_3" class="answer"/>
</div>
</div></td>
</tr>
</tbody>
</table>
</div></div>
</span>
<p>
Find an integral formula for the antiderivative [mathjaxinline]F_{10}[/mathjaxinline] of [mathjaxinline]\ln (x)[/mathjaxinline] so that [mathjaxinline]F_{10}[/mathjaxinline] crosses the [mathjaxinline]x[/mathjaxinline]-axis at [mathjaxinline]10[/mathjaxinline]:<br/></p>
<p>
(Note all three answer boxes are graded together. You get all correct, or all wrong.)<br/></p>
<span>
<style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax {
display: inline-block !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax_Preview {
display: inline-block !important;
}
</style>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="multi-inputs-group" role="group"><table>
<tbody>
<tr>
<td/>
<td colspan="3">
<div id="inputtype_theory3-tab4-problem1_3_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_theory3-tab4-problem1_3_1" id="input_theory3-tab4-problem1_3_1" aria-describedby="status_theory3-tab4-problem1_3_1" value="" size="5"/>
<span class="trailing_text" id="trailing_text_theory3-tab4-problem1_3_1"/>
<span class="status unanswered" id="status_theory3-tab4-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab4-problem1_3_1" class="answer"/>
</div>
</div></td>
</tr>
<tr>
<td>
<p style="text-align:left"> \( \Large{F_{10}(x)} = \)</p>
</td>
<td>
<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
</td>
<td>
<br/>
<div id="formulaequationinput_theory3-tab4-problem1_3_2" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab4-problem1_3_2" id="input_theory3-tab4-problem1_3_2" data-input-id="theory3-tab4-problem1_3_2" value="" aria-describedby="trailing_text_theory3-tab4-problem1_3_2 status_theory3-tab4-problem1_3_2" size="10"/>
<span class="trailing_text" id="trailing_text_theory3-tab4-problem1_3_2"> [mathjaxinline] dt [/mathjaxinline]</span>
<span class="status unanswered" id="status_theory3-tab4-problem1_3_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab4-problem1_3_2" class="answer"/>
<div id="input_theory3-tab4-problem1_3_2_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></td>
</tr>
<tr>
<td/>
<td colspan="2">
<div id="inputtype_theory3-tab4-problem1_3_3" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_theory3-tab4-problem1_3_3" id="input_theory3-tab4-problem1_3_3" aria-describedby="status_theory3-tab4-problem1_3_3" value="" size="5"/>
<span class="trailing_text" id="trailing_text_theory3-tab4-problem1_3_3"/>
<span class="status unanswered" id="status_theory3-tab4-problem1_3_3" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab4-problem1_3_3" class="answer"/>
</div>
</div></td>
</tr>
</tbody>
</table>
</div></div>
</span>
<p>
Therefore, <br/>(Note all three answer boxes are graded together. You get all correct, or all wrong.)<br/></p>
<span>
<style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax {
display: inline-block !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax_Preview {
display: inline-block !important;
}
</style>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="multi-inputs-group" role="group"><table>
<tbody>
<tr>
<td/>
<td colspan="3">
<div id="inputtype_theory3-tab4-problem1_4_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_theory3-tab4-problem1_4_1" id="input_theory3-tab4-problem1_4_1" aria-describedby="status_theory3-tab4-problem1_4_1" value="" size="5"/>
<span class="trailing_text" id="trailing_text_theory3-tab4-problem1_4_1"/>
<span class="status unanswered" id="status_theory3-tab4-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab4-problem1_4_1" class="answer"/>
</div>
</div></td>
</tr>
<tr>
<td>
<p style="text-align:left"> \( \Large{F_{10}(x)-F_1(x)} = \)</p>
</td>
<td>
<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
</td>
<td>
<br/>
<div id="formulaequationinput_theory3-tab4-problem1_4_2" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab4-problem1_4_2" id="input_theory3-tab4-problem1_4_2" data-input-id="theory3-tab4-problem1_4_2" value="" aria-describedby="trailing_text_theory3-tab4-problem1_4_2 status_theory3-tab4-problem1_4_2" size="10"/>
<span class="trailing_text" id="trailing_text_theory3-tab4-problem1_4_2"> [mathjaxinline] dt [/mathjaxinline]</span>
<span class="status unanswered" id="status_theory3-tab4-problem1_4_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab4-problem1_4_2" class="answer"/>
<div id="input_theory3-tab4-problem1_4_2_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></td>
</tr>
<tr>
<td/>
<td colspan="2">
<div id="inputtype_theory3-tab4-problem1_4_3" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_theory3-tab4-problem1_4_3" id="input_theory3-tab4-problem1_4_3" aria-describedby="status_theory3-tab4-problem1_4_3" value="" size="5"/>
<span class="trailing_text" id="trailing_text_theory3-tab4-problem1_4_3"/>
<span class="status unanswered" id="status_theory3-tab4-problem1_4_3" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab4-problem1_4_3" class="answer"/>
</div>
</div></td>
</tr>
</tbody>
</table>
</div></div>
</span>
<p>
Is this consistent with the fact that any two antiderivatives of a function can differ only by a constant? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab4-problem1_5_1">
<fieldset aria-describedby="status_theory3-tab4-problem1_5_1">
<div class="field">
<input type="radio" name="input_theory3-tab4-problem1_5_1" id="input_theory3-tab4-problem1_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory3-tab4-problem1_5_1-choice_1-label" for="input_theory3-tab4-problem1_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab4-problem1_5_1">
<text> No, we do not know any antiderivative of [mathjaxinline]\ln (x)[/mathjaxinline], so we do not know if two of its antiderivatives differ by a constant.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab4-problem1_5_1" id="input_theory3-tab4-problem1_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory3-tab4-problem1_5_1-choice_2-label" for="input_theory3-tab4-problem1_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab4-problem1_5_1">
<text> No, FTC2 gives a different kind of antiderivative, so two antiderivatives do not have to differ by a constant.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab4-problem1_5_1" id="input_theory3-tab4-problem1_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory3-tab4-problem1_5_1-choice_3-label" for="input_theory3-tab4-problem1_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab4-problem1_5_1">
<text> Yes, since both the upper and lower limits of the definite integral for [mathjaxinline]F_{10}(x)-F_1(x)[/mathjaxinline] are specific numbers, [mathjaxinline]F_{10}(x)-F_1(x)[/mathjaxinline] is a constant.</text>
</label>
</div>
<span id="answer_theory3-tab4-problem1_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab4-problem1_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab4-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="The difference of two antiderivatives" />
<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_theory3-tab4-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-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="theory3-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="theory3-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="theory3-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+18.01.2x+3T2019+type@video+block@theory3-tab4-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab4-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">The quiz question</h3>
<div
id="video_theory3-tab4-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab4-video1/handler/publish_completion", "streams": "1.00:RsGBxtmuC00", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab4-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab4-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab4-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-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_theory3-tab4-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-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+18.01.2x+3T2019+type@problem+block@theory3-tab4-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab4-problem2" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab4-problem2" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab4-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab4-problem2" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-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="theory3-tab4-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab4-problem2-problem-progress" tabindex="-1">
You would not want to integrate
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab4-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Let us verify FTC2 on a more complicated integral.<br/>Using FTC2, <br/><p style="display:inline">[mathjaxinline]\displaystyle \frac{d}{dy} \int _{0}^{y} \left( \, xe^{x^2}+ \frac{\sqrt [5]{x}}{6}-2\right) \, dx\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab4-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab4-problem2_2_1" id="input_theory3-tab4-problem2_2_1" data-input-id="theory3-tab4-problem2_2_1" value="" aria-describedby="status_theory3-tab4-problem2_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab4-problem2_2_1"/>
<span class="status unanswered" id="status_theory3-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_theory3-tab4-problem2_2_1" class="answer"/>
<div id="input_theory3-tab4-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/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab4-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="You would not want to integrate" />
<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_theory3-tab4-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-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="theory3-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="theory3-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="theory3-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-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab5" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">5. Applications of FTC2</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Recitation video: an application of FTC2</h3>
<div
id="video_theory3-tab5-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video1/handler/publish_completion", "streams": "1.00:qVhRZfueTPY", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-tab5-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_theory3-tab5-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-tab5-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-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+18.01.2x+3T2019+type@video+block@theory3-tab5-video2">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video2" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Recitation video: quadratic approximation and the FTC2</h3>
<div
id="video_theory3-tab5-video2"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video2/handler/publish_completion", "streams": "1.00:aEy5Zg7DClk", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video2/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video2/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video2/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-tab5-video2"></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_theory3-tab5-video2">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-tab5-video2">
<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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video2/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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab5-video2/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-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab6" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">6. FTC2 and the chain rule</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab6-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab6-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Worked example: FTC2 and the chain rule</h3>
<div
id="video_theory3-tab6-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab6-video1/handler/publish_completion", "streams": "1.00:2msynAIKE8k", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab6-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab6-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab6-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-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_theory3-tab6-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-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+18.01.2x+3T2019+type@html+block@theory3-tab6-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab6-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
We have evaluated in the video an example of the following. </p><table id="a0000001138" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001139"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d}{dx}\int _{a}^{u(x)} f(t)\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.228)</td></tr></table><p>
(In the video, [mathjaxinline]u(x)=\sin (x)[/mathjaxinline].) In general, we can evaluate this using the following procedure.<br/>Let </p><table id="a0000001140" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001141"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle G(u)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\int _{a}^{u} f(t)\, dt,[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.229)</td></tr></table><p>
then </p><table id="a0000001142" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001143"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle G(u(x))[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\int _{a}^{u(x)} f(t)\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.230)</td></tr></table><p>
That is, decompose the definite integral with [mathjaxinline]u(x)[/mathjaxinline] as its upper limit as the composition of the two functions [mathjaxinline]G(u)[/mathjaxinline] and [mathjaxinline]u(x)[/mathjaxinline]. Now, let us apply the chain rule to find its derivative.<br/></p><table id="a0000001144" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001145"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx} G(u(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 G'(u(x)) \cdot u'(x)\qquad (\text {chain rule});[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.231)</td></tr><tr id="a0000001146"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {or equivalently}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{dG}{dx}[/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 \left.\frac{dG}{du}\right|_{u=u(x)} \cdot \frac{du}{dx},[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.232)</td></tr></table><p>
where [mathjaxinline]G'(u(x))[/mathjaxinline] is the derivative of [mathjaxinline]G[/mathjaxinline] with respect to [mathjaxinline]u[/mathjaxinline] evaluated at [mathjaxinline]u(x)[/mathjaxinline]. Another notation for [mathjaxinline]G'(u(x))[/mathjaxinline] is [mathjaxinline]\displaystyle \left.\frac{dG}{du}\right|_{u=u(x)}[/mathjaxinline]. By FTC2, </p><table id="a0000001147" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001148"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle G'(u)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = f(u)\qquad (\text {FTC2}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.233)</td></tr></table><p>
Therefore, </p><center><table id="a0000001149" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001150"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{d}{dx}G(u(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(u(x)) \cdot u'(x),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.234)</td></tr><tr id="a0000001151"><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 \frac{d}{dx} \int _{a}^{u(x)} f(t)\, dt[/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(u(x)) \cdot u'(x).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.235)</td></tr></table></center>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab6-problem1" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab6-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-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="theory3-tab6-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem1-problem-progress" tabindex="-1">
Practice 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{d}{dx} \int _{1}^{x^3} \tan ^2(\theta ) \, d\theta \, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab6-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab6-problem1_2_1" id="input_theory3-tab6-problem1_2_1" data-input-id="theory3-tab6-problem1_2_1" value="" aria-describedby="status_theory3-tab6-problem1_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab6-problem1_2_1"/>
<span class="status unanswered" id="status_theory3-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_theory3-tab6-problem1_2_1" class="answer"/>
<div id="input_theory3-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/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab6-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice 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_theory3-tab6-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-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="theory3-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="theory3-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="theory3-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+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem2" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab6-problem2" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab6-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem2" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-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="theory3-tab6-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem2-problem-progress" tabindex="-1">
Practice 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Let [mathjaxinline]-\pi /2\leq t\leq \pi /2[/mathjaxinline]. Reverse the limits of the integral, and use the chain rule to evaluate the following. <p style="display:inline">[mathjaxinline]\displaystyle \frac{d}{dt} \int _{\sin (t)}^{0} \arcsin (x) dx\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab6-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab6-problem2_2_1" id="input_theory3-tab6-problem2_2_1" data-input-id="theory3-tab6-problem2_2_1" value="" aria-describedby="status_theory3-tab6-problem2_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab6-problem2_2_1"/>
<span class="status unanswered" id="status_theory3-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_theory3-tab6-problem2_2_1" class="answer"/>
<div id="input_theory3-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/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab6-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice 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_theory3-tab6-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-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="theory3-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="theory3-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="theory3-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+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem3" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab6-problem3" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab6-problem3-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem3" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-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="theory3-tab6-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem3-problem-progress" tabindex="-1">
Practice 3
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab6-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
Let [mathjaxinline]y&gt;0[/mathjaxinline]. Decompose the integral below into a sum of two integrals, and then evaluate the following.<br/><p style="display:inline">[mathjaxinline]\displaystyle \frac{d}{dy} \int _{y}^{\sqrt {y}} \ln (t)\, dt \, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab6-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab6-problem3_2_1" id="input_theory3-tab6-problem3_2_1" data-input-id="theory3-tab6-problem3_2_1" value="" aria-describedby="status_theory3-tab6-problem3_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab6-problem3_2_1"/>
<span class="status unanswered" id="status_theory3-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_theory3-tab6-problem3_2_1" class="answer"/>
<div id="input_theory3-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/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab6-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice 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_theory3-tab6-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-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="theory3-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="theory3-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="theory3-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>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab7" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">7. Proofs of FTC1 and FTC2</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab7-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab7-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">FTC1 and FTC2 side by side</b></p><p>
Let us recall the two fundamental theorems of calculus. </p><dl class="description"><dt>FTC1:</dt><dd><p>
Given a differentiable function [mathjaxinline]F[/mathjaxinline] with continuous derivative [mathjaxinline]F'[/mathjaxinline], </p><table id="a0000001176" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \left.\int _{a}^{b }F'(t) \, dt \, =\, F(x)\, \, \right|_ a^ b.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd><dt>FTC2:</dt><dd><p>
Given a continuous function [mathjaxinline]f[/mathjaxinline], </p><table id="a0000001177" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{d}{dx}\int _{a}^{x} f(t) \, dt = f(x).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd></dl><p>
We will now prove FTC2 and then FTC1. </p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab7-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab7-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Proof of FTC2</h3>
<div
id="video_theory3-tab7-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab7-video1/handler/publish_completion", "streams": "1.00:_i1bdX8CS1s", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab7-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab7-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab7-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-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_theory3-tab7-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab7-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+18.01.2x+3T2019+type@html+block@theory3-tab7-text2">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab7-text2" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Proof of FTC2</b></p><p>
Recall that FTC2 states that given a continuous function [mathjaxinline]f[/mathjaxinline], </p><table id="a0000001178" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001179"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \int _{a}^{x} f(t)\, dt[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \Rightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle F'(x)=f(x).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.252)</td></tr></table><p>
To prove this theorem, we will compute the derivative of [mathjaxinline]F[/mathjaxinline] by using the geometric picture of [mathjaxinline]F[/mathjaxinline]. We will prove the case for [mathjaxinline]f>0[/mathjaxinline] and you will verify that the same argument works for any [mathjaxinline]f[/mathjaxinline].<br/></p><p>
Recall the definition of the derivative: </p><table id="a0000001180" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001181"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle F'(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\lim _{\Delta x\rightarrow 0} \frac{\Delta F}{\Delta x} \qquad \text {where} \, \, \Delta F\, =\, F(x+\Delta x)-F(x).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.253)</td></tr></table><center><img src="/assets/courseware/v1/ec5ad3ffea88f8183191ba112abb085f/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_proofFTC2.svg" width="350px" alt="The area under a curve y equals f of t between t equals a and t equals x is F. The area under the curve y equals f of t between t equals x and t equals x plus Delta x is Delta F" style="margin: 10px 25px 25px 25px"/><br/><table id="a0000001182" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001183"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \text {Geometrically,}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle F(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \int _{a}^{x} f(t)\, dt\, \,[/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 \text {Area between}\, \, a\, \, \text {and}\, \, x,[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.254)</td></tr><tr id="a0000001184"><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 \Delta F[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \int _{x}^{x+\Delta x} f(t)\, dt\, \,[/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 \text {Area between}\, \, x\, \, \text {and}\, \, x+\Delta x.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.255)</td></tr></table></center><p>
We can approximate [mathjaxinline]\Delta F[/mathjaxinline] by the area of the rectangle with base [mathjaxinline]\Delta x[/mathjaxinline] and height [mathjaxinline]\, f(x)[/mathjaxinline]. This gives </p><table id="a0000001185" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001186"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{\Delta F}{\Delta 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:center; border:none">
[mathjaxinline]\displaystyle \frac{f(x)\cdot \Delta x}{\Delta 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).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.256)</td></tr></table><p>
Now since [mathjaxinline]f[/mathjaxinline] is continuous, </p><table id="a0000001187" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001188"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \lim _{\Delta x\rightarrow 0} \frac{\Delta F}{\Delta x}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \lim _{\Delta x\rightarrow 0} f(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).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.257)</td></tr></table><p>
This is equivalent to [mathjaxinline]F'(x)=f(x)[/mathjaxinline], which is what we needed to prove. </p>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab7-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab7-problem1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab7-problem1" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab7-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab7-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab7-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="3"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="theory3-tab7-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab7-problem1-problem-progress" tabindex="-1">
When the integrand is negative
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab7-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
As above, let </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001189" style="table-layout:auto" width="100%">
<tr id="a0000001190">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle F(x)\, =\int _{a}^{x} f(t)\, dt[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.258)</td>
</tr>
</table>
<p>
The graph of [mathjaxinline]f[/mathjaxinline] is as in the figure below. Let us check whether [mathjaxinline]F'(3)=f(3)[/mathjaxinline]. This is the statement of FTC2 at a point where the integrand is negative. <br/></p>
<center>
<img alt="The function f of x is negative. The signed area between x equals a and x equals 3 for a less than 3 is F(3). The area between x equals a and x equals 3+h is depicted for h greater than 0." src="/assets/courseware/v1/0fc48e4e2d85e7d1ad58ce6ab03ecf1f/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_proofFTC2neg.svg" style="margin: 10px 25px 25px 25px" width="350px"/>
</center>
<p>
Approximate [mathjaxinline]\Delta F= F(3+h)-F(3)[/mathjaxinline] using the area of a rectangle.<br/>(Enter your answer in terms of [mathjaxinline]h[/mathjaxinline] and the numerical values on the graph.)<br/><p style="display:inline">[mathjaxinline]\Delta F= F(3+h)-F(3)\, \approx \,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab7-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab7-problem1_2_1" id="input_theory3-tab7-problem1_2_1" data-input-id="theory3-tab7-problem1_2_1" value="" aria-describedby="status_theory3-tab7-problem1_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab7-problem1_2_1"/>
<span class="status unanswered" id="status_theory3-tab7-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab7-problem1_2_1" class="answer"/>
<div id="input_theory3-tab7-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>
<p>
Evaluate [mathjaxinline]F'(3)[/mathjaxinline] using the definition of derivative.<br/><p style="display:inline">As [mathjaxinline]h\longrightarrow 0[/mathjaxinline], [mathjaxinline]\, \, \displaystyle \frac{F(3+h)-F(3)}{h} \longrightarrow[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_theory3-tab7-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab7-problem1_3_1" id="input_theory3-tab7-problem1_3_1" data-input-id="theory3-tab7-problem1_3_1" value="" aria-describedby="status_theory3-tab7-problem1_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab7-problem1_3_1"/>
<span class="status unanswered" id="status_theory3-tab7-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab7-problem1_3_1" class="answer"/>
<div id="input_theory3-tab7-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><br/></p>
<p>
<p style="display:inline">On the other hand, [mathjaxinline]\, \, f(3)\, \, =\, \,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_theory3-tab7-problem1_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_theory3-tab7-problem1_4_1" id="input_theory3-tab7-problem1_4_1" aria-describedby="status_theory3-tab7-problem1_4_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_theory3-tab7-problem1_4_1"/>
<span class="status unanswered" id="status_theory3-tab7-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab7-problem1_4_1" class="answer"/>
<div id="display_theory3-tab7-problem1_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_theory3-tab7-problem1_4_1_dynamath" name="input_theory3-tab7-problem1_4_1_dynamath"/>
</div>
</div></div>
<br/>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab7-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="When the integrand is negative" />
<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_theory3-tab7-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-tab7-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="theory3-tab7-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="theory3-tab7-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="theory3-tab7-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-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab8" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">8. Proof of FTC1</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab8-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab8-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Proof of FTC1</h3>
<div
id="video_theory3-tab8-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab8-video1/handler/publish_completion", "streams": "1.00:AvtX2fInY6g", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab8-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab8-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab8-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-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_theory3-tab8-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab8-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+18.01.2x+3T2019+type@html+block@theory3-tab8-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab8-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
We will now review the proof from the video.<br/></p><p>
Recall the statement of FTC1:<br/>Given a differentiable function [mathjaxinline]F[/mathjaxinline] with continuous derivative [mathjaxinline]F'=f[/mathjaxinline], </p><table id="a0000001198" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{b} f(x) \, dx \, =\, F(b)- F(a).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
First, define </p><table id="a0000001199" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001200"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle G(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 \int _{a}^{x} f(t)\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.263)</td></tr></table><p>
Note that [mathjaxinline]G(x)[/mathjaxinline] makes sense as the limit of Riemann sums.</p><p>
Second, since [mathjaxinline]f[/mathjaxinline] is continuous, the FTC2 says that </p><table id="a0000001201" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001202"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle G'=f.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.264)</td></tr></table><p>
Now, since [mathjaxinline]F'=f[/mathjaxinline], and [mathjaxinline]G'=f[/mathjaxinline], both [mathjaxinline]F[/mathjaxinline] and [mathjaxinline]G[/mathjaxinline] are antiderivatives of [mathjaxinline]f[/mathjaxinline]. Therefore, as a consequence of the Mean Value Theorem, </p><table id="a0000001203" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001204"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle F'(x)\, =\, G'(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \Rightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle F(x)\, =\, G(x)+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.265)</td></tr></table><p>
where [mathjaxinline]C[/mathjaxinline] is a constant.<br/></p><p>
Finally, we compute [mathjaxinline]F(b)-F(a)[/mathjaxinline] in terms of [mathjaxinline]G[/mathjaxinline]: </p><table id="a0000001205" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001206"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle F(b)-F(a)[/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 (G(b)+C)\, -\, (G(a)+C)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.266)</td></tr><tr id="a0000001207"><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 G(b)-G(a)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.267)</td></tr><tr id="a0000001208"><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 \int _{a}^{b} f(t)\, dt- \int _{a}^{a} f(t)\, dt\qquad (\text {Definition of}\, \, G)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.268)</td></tr><tr id="a0000001209"><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 \int _{a}^{b} f(t)\, dt - 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.269)</td></tr><tr id="a0000001210"><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 \int _{a}^{b} f(t)\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.270)</td></tr></table><p>
Therefore, </p><table id="a0000001211" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001212"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _{a}^{b} f(t)\, dt[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =F(b)-F(a).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.271)</td></tr></table><p>
This is the statement of FTC1, even though the integration variable is renamed [mathjaxinline]t[/mathjaxinline]. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab9" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">9. The fundamental theorems of calculus</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab9-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab9-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
Here is another point of view of the two fundamental theorems of calculus together. This is what we mentioned in the introductory (very first) video of this section. (This video is at the bottom of the page again here.)<br/></p><dl class="description"><dt>FTC1:</dt><dd><p>
Given a differentiable function [mathjaxinline]F[/mathjaxinline] with continuous derivative [mathjaxinline]F'[/mathjaxinline], </p><table id="a0000001213" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{x}F'(t) \, dt \, =\, F(x)-F(a).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd><dt>FTC2:</dt><dd><p>
Given a continuous function [mathjaxinline]f[/mathjaxinline], </p><table id="a0000001214" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{d}{dx}\int _{a}^{x} f(t) \, dt = f(x).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd></dl><p>
In other words, FTC1 says that if we start with a function [mathjaxinline]F[/mathjaxinline], and first differentiate and then integrate (from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]x[/mathjaxinline]), we will get back the function [mathjaxinline]F[/mathjaxinline] up to a constant, which is given by [mathjaxinline]F(a)[/mathjaxinline].<br/></p><p>
On the other hand, FTC2 says that if we start with a function [mathjaxinline]f[/mathjaxinline], and first integrate (from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]x[/mathjaxinline]) and then differentiate, we will get back to [mathjaxinline]f[/mathjaxinline]. Notice there is no ambiguity of a constant here.<br/></p><p>
So, the two fundamental theorems together say that differentiation and integration (from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]x[/mathjaxinline]) are “inverse" operations of one another, up to a constant.</p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab9-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab9-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">The Second Fundamental Theorem of Calculus</h3>
<div
id="video_theory3-tab9-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab9-video1/handler/publish_completion", "streams": "1.00:AjBRTOGSwRs", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab9-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab9-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab9-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-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_theory3-tab9-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab9-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-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab10" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">10. Constructing new functions using integrals</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab10-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab10-problem1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab10-problem1" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab10-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab10-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab10-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="theory3-tab10-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab10-problem1-problem-progress" tabindex="-1">
Applying FTC2 again
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab10-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
For [mathjaxinline]x&gt;0[/mathjaxinline], define the function </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001215" style="table-layout:auto" width="100%">
<tr id="a0000001216">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle L(x)\, =\, \int _{1}^{x} \frac{dt}{t} \qquad (x&gt;0).[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.272)</td>
</tr>
</table>
<center><img alt="See caption" src="/assets/courseware/v1/80ba300f0266465664ef76717dbcf78f/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_L.svg" style="margin: 10px 25px 25px 25px" width="300px"/><br/>Geometrically, [mathjaxinline]L(x)[/mathjaxinline] is the signed area of the shaded region. </center>
<p>
According to FTC2, what differential equation and initial condition does [mathjaxinline]L(x)[/mathjaxinline] solve? In other words,<br/></p>
<table cellspacing="0" class="tabular" style="table-layout:auto">
<tr>
<td style="text-align:right; border:none">
[mathjaxinline]L'(x)\, =\,[/mathjaxinline]</td>
<td style="text-align:left; border:none">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab10-problem1_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_theory3-tab10-problem1_2_1" id="input_theory3-tab10-problem1_2_1" data-input-id="theory3-tab10-problem1_2_1" value="" aria-describedby="status_theory3-tab10-problem1_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab10-problem1_2_1"/>
<span class="status unanswered" id="status_theory3-tab10-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab10-problem1_2_1" class="answer"/>
<div id="input_theory3-tab10-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>
</td>
<td style="text-align:left; border:none">
(differential equation)</td>
</tr>
<tr>
<td style="text-align:right; border:none">
[mathjaxinline]L(a)=0\qquad[/mathjaxinline] for [mathjaxinline]a=\, \,[/mathjaxinline]</td>
<td style="text-align:left; border:none">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_theory3-tab10-problem1_3_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_theory3-tab10-problem1_3_1" id="input_theory3-tab10-problem1_3_1" data-input-id="theory3-tab10-problem1_3_1" value="" aria-describedby="status_theory3-tab10-problem1_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab10-problem1_3_1"/>
<span class="status unanswered" id="status_theory3-tab10-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab10-problem1_3_1" class="answer"/>
<div id="input_theory3-tab10-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>
</td>
<td style="text-align:left; border:none">
(initial condition). </td>
</tr>
</table>
<p>
On the other hand, for [mathjaxinline]x&gt;0[/mathjaxinline], which function do you already know that solves the same differential equation and initial condition? <br/>(Enter your answer as a function of [mathjaxinline]x[/mathjaxinline].)<br/><div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_theory3-tab10-problem1_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab10-problem1_4_1" id="input_theory3-tab10-problem1_4_1" data-input-id="theory3-tab10-problem1_4_1" value="" aria-describedby="status_theory3-tab10-problem1_4_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab10-problem1_4_1"/>
<span class="status unanswered" id="status_theory3-tab10-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab10-problem1_4_1" class="answer"/>
<div id="input_theory3-tab10-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><br/></p>
<p>
Does [mathjaxinline]L(x)[/mathjaxinline] equal the function you enter above? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab10-problem1_5_1">
<fieldset aria-describedby="status_theory3-tab10-problem1_5_1">
<div class="field">
<input type="radio" name="input_theory3-tab10-problem1_5_1" id="input_theory3-tab10-problem1_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory3-tab10-problem1_5_1-choice_1-label" for="input_theory3-tab10-problem1_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab10-problem1_5_1"> <text> Yes</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab10-problem1_5_1" id="input_theory3-tab10-problem1_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory3-tab10-problem1_5_1-choice_2-label" for="input_theory3-tab10-problem1_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab10-problem1_5_1"> <text> No</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab10-problem1_5_1" id="input_theory3-tab10-problem1_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory3-tab10-problem1_5_1-choice_3-label" for="input_theory3-tab10-problem1_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab10-problem1_5_1"> <text> Cannot be determined</text>
</label>
</div>
<span id="answer_theory3-tab10-problem1_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab10-problem1_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div> </p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab10-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Applying FTC2 again" />
<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_theory3-tab10-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-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="theory3-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="theory3-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="theory3-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-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab10-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab10-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Constructing new functions</h3>
<div
id="video_theory3-tab10-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab10-video1/handler/publish_completion", "streams": "1.00:CVGLAmMPpYY", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab10-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab10-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab10-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-tab10-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_theory3-tab10-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-tab10-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab10-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab10-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+18.01.2x+3T2019+type@html+block@theory3-tab10-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab10-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Constructing functions using integrals</b></p><p>
Recall the FTC2 gives the integral formula </p><table id="a0000001222" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle G(x) = \int _{a}^{x} f(t) \, dt,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
as the solution to the following differential equation and initial condition: </p><table id="a0000001223" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001224"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y'[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle f[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {differential equation})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.275)</td></tr><tr id="a0000001225"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y(a)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle 0[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {initial condition}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.276)</td></tr></table><p>
Depending on what the differential equation is, [mathjaxinline]G(x)[/mathjaxinline] can be a function we already know. An example is the function we have just seen in the previous problem.</p><table id="a0000001226" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001227"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle L(x)\,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \, \int _{1}^{x} \frac{dt}{t}[/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 \ln (x) \qquad (x>0)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.277)</td></tr></table><p>
Notice that while the integrand [mathjaxinline]\displaystyle \frac{1}{t}[/mathjaxinline] is a ratio of two polynomials, [mathjaxinline]L(x)[/mathjaxinline] cannot be written in terms of polynomials using algebraic operations [mathjaxinline]+[/mathjaxinline],[mathjaxinline]-[/mathjaxinline],[mathjaxinline]\cdot[/mathjaxinline],[mathjaxinline]/[/mathjaxinline],[mathjaxinline]\sqrt [n]{\phantom{2}}[/mathjaxinline]. It is an example of a “transcendental" function, a function that “transcends" algebra.<br/></p><p>
On the other hand, the solution that FTC2 gives for another differential equation may be a function that cannot be expressed (without an integral) in terms of any function we already know. For example, </p><table id="a0000001228" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001229"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle F(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \int _{0}^{x} e^{-t^2} \, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.278)</td></tr></table><p>
is the solution to </p><table id="a0000001230" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001231"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y'[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-x^2}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {differential equation})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.279)</td></tr><tr id="a0000001232"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y(0)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle 0[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {initial condition}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.280)</td></tr></table><center><img src="/assets/courseware/v1/de8844c0c083459e6c1853d6ee10e401/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_bell.svg" width="450px" alt="Go to text below image" style="margin: 10px 25px 25px 25px"/><br/>Geometrically, [mathjaxinline]\displaystyle F(x) = \int _{0}^{x} e^{-t^2} \, dt = \text {Area of shaded region}.[/mathjaxinline]<br/></center><p><br/></p><p>
This function is used extensively in probability.<br/></p><p>
Finally, even though we may not have explicit formulas for these functions defined by integral formulas, we can apply FTC2 to find their derivatives. Therefore, we have all of our usual tools at our disposal. For example, we can graph them, or approximate them by linear and quadratic approximations.<br/></p><p>
Let us now discuss the two examples above in more detail.<br/></p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab11" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">11. Properties of Log</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab11-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab11-problem1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab11-problem1" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab11-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab11-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab11-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="3"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="theory3-tab11-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab11-problem1-problem-progress" tabindex="-1">
First derivative of L
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab11-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
As above, define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001233" style="table-layout:auto" width="100%">
<tr id="a0000001234">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle L(x)\, =\, \int _{1}^{x} \frac{dt}{t} \qquad \text {for} \, \, x&gt;0.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.281)</td>
</tr>
</table>
<p>
Let us prepare for sketching the graph of [mathjaxinline]L(x)[/mathjaxinline] by computing its derivatives. </p>
<p>
<p style="display:inline">By FTC2, [mathjaxinline]\displaystyle \frac{dL}{dx}\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab11-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab11-problem1_2_1" id="input_theory3-tab11-problem1_2_1" data-input-id="theory3-tab11-problem1_2_1" value="" aria-describedby="status_theory3-tab11-problem1_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab11-problem1_2_1"/>
<span class="status unanswered" id="status_theory3-tab11-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab11-problem1_2_1" class="answer"/>
<div id="input_theory3-tab11-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>
<p>
<p style="display:inline">Hence, find the critical points of [mathjaxinline]L(x)[/mathjaxinline] for [mathjaxinline]x&gt;0[/mathjaxinline].</p>
</p>
<p>
(Enter your answer separated by commas, e.g. "1.2, -5, e". Enter &#8220;none" if there are no critical points.)<br/></p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_theory3-tab11-problem1_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_theory3-tab11-problem1_3_1" id="input_theory3-tab11-problem1_3_1" aria-describedby="status_theory3-tab11-problem1_3_1" value=""/>
<span class="trailing_text" id="trailing_text_theory3-tab11-problem1_3_1"/>
<span class="status unanswered" id="status_theory3-tab11-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab11-problem1_3_1" class="answer"/>
</div>
</div></div>
<br/>
</p>
<p>
Therefore, [mathjaxinline]L(x)[/mathjaxinline] is<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab11-problem1_4_1">
<fieldset aria-describedby="status_theory3-tab11-problem1_4_1">
<div class="field">
<input type="radio" name="input_theory3-tab11-problem1_4_1" id="input_theory3-tab11-problem1_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory3-tab11-problem1_4_1-choice_1-label" for="input_theory3-tab11-problem1_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab11-problem1_4_1"> <text> always increasing</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab11-problem1_4_1" id="input_theory3-tab11-problem1_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory3-tab11-problem1_4_1-choice_2-label" for="input_theory3-tab11-problem1_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab11-problem1_4_1"> <text> always decreasing</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab11-problem1_4_1" id="input_theory3-tab11-problem1_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory3-tab11-problem1_4_1-choice_3-label" for="input_theory3-tab11-problem1_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab11-problem1_4_1"> <text> sometimes increasing and sometimes decreasing</text>
</label>
</div>
<span id="answer_theory3-tab11-problem1_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab11-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div><br/></p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab11-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="First derivative of L" />
<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_theory3-tab11-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-tab11-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="theory3-tab11-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="theory3-tab11-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="theory3-tab11-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+18.01.2x+3T2019+type@problem+block@theory3-tab11-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab11-problem2" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab11-problem2" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab11-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab11-problem2" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab11-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="theory3-tab11-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab11-problem2-problem-progress" tabindex="-1">
Second derivative of L
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab11-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
As above, define the function </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001236" style="table-layout:auto" width="100%">
<tr id="a0000001237">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle L(x)\, =\, \int _{1}^{x} \frac{dt}{t} \qquad \text {for} \, \, x&gt;0.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.282)</td>
</tr>
</table>
<p>
Compute the second derivative of [mathjaxinline]L[/mathjaxinline] by differentiating your answer to the previous problem.<br/><p style="display:inline">[mathjaxinline]\displaystyle \frac{d^2}{dx^2} L(x)\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab11-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab11-problem2_2_1" id="input_theory3-tab11-problem2_2_1" data-input-id="theory3-tab11-problem2_2_1" value="" aria-describedby="status_theory3-tab11-problem2_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab11-problem2_2_1"/>
<span class="status unanswered" id="status_theory3-tab11-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab11-problem2_2_1" class="answer"/>
<div id="input_theory3-tab11-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><br/></p>
<p>
Hence, [mathjaxinline]L(x)[/mathjaxinline] is <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab11-problem2_3_1">
<fieldset aria-describedby="status_theory3-tab11-problem2_3_1">
<div class="field">
<input type="radio" name="input_theory3-tab11-problem2_3_1" id="input_theory3-tab11-problem2_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory3-tab11-problem2_3_1-choice_1-label" for="input_theory3-tab11-problem2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab11-problem2_3_1"> <text> always concave up</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab11-problem2_3_1" id="input_theory3-tab11-problem2_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory3-tab11-problem2_3_1-choice_2-label" for="input_theory3-tab11-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab11-problem2_3_1"> <text> always concave down</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab11-problem2_3_1" id="input_theory3-tab11-problem2_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory3-tab11-problem2_3_1-choice_3-label" for="input_theory3-tab11-problem2_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab11-problem2_3_1"> <text> sometimes concave up and sometimes concave down</text>
</label>
</div>
<span id="answer_theory3-tab11-problem2_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab11-problem2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div><br/></p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab11-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Second derivative of L" />
<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_theory3-tab11-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-tab11-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="theory3-tab11-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="theory3-tab11-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="theory3-tab11-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+18.01.2x+3T2019+type@video+block@theory3-tab11-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab11-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">The logarithm as an integral</h3>
<div
id="video_theory3-tab11-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab11-video1/handler/publish_completion", "streams": "1.00:druz4O6QKu0", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab11-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab11-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab11-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-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_theory3-tab11-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab11-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-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab12" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">12. The graph of L</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab12-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab12-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
As above, the definition of [mathjaxinline]L(x)[/mathjaxinline] is </p><table id="a0000001240" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001241"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle L(x)\, =\, \int _{1}^{x} \frac{dt}{t} \qquad \text {where}\, \, x>0.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.284)</td></tr></table><center><img src="/assets/courseware/v1/80ba300f0266465664ef76717dbcf78f/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_L.svg" width="350px" alt="Go to text below image" style="margin: 10px 25px 25px 25px"/><br/>Geometrically, [mathjaxinline]L(x)[/mathjaxinline] is the area of the shaded region.<br/></center><p><br/><br/>Using the first and second derivatives, we have just sketched [mathjaxinline]L(x)[/mathjaxinline] in the video: </p><center><img src="/assets/courseware/v1/9e1bdf2c93bfed6b16013fd9183563bc/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_Log.svg" width="400px" alt="Go to text below image" style="margin: 10px 25px 25px 25px"/><br/>Graph of [mathjaxinline]L(x)[/mathjaxinline]<br/></center><p><h3>The number e</h3></p><p>
We define [mathjaxinline]e[/mathjaxinline] to be the number such that [mathjaxinline]L(e)=1[/mathjaxinline]. Since [mathjaxinline]L[/mathjaxinline] is always increasing, there is only one value at which [mathjaxinline]L(x)=1[/mathjaxinline]. </p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab12-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab12-problem1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab12-problem1" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab12-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab12-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab12-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="theory3-tab12-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab12-problem1-problem-progress" tabindex="-1">
Inverse of L
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab12-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
As above, define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001242" style="table-layout:auto" width="100%">
<tr id="a0000001243">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle L(x)\, =\, \int _{1}^{x} \frac{dt}{t} \qquad \text {for}\, \, x&gt;0,[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.285)</td>
</tr>
</table>
<p>
whose graph is given below. </p>
<center>
<img alt="Graph of L of x" src="/assets/courseware/v1/9e1bdf2c93bfed6b16013fd9183563bc/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_Log.svg" style="margin: 10px 25px 25px 25px" width="340px"/>
<br/>
</center>
<p>
Does the inverse function of [mathjaxinline]L[/mathjaxinline] exist?<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab12-problem1_2_1">
<fieldset aria-describedby="status_theory3-tab12-problem1_2_1">
<div class="field">
<input type="radio" name="input_theory3-tab12-problem1_2_1" id="input_theory3-tab12-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory3-tab12-problem1_2_1-choice_1-label" for="input_theory3-tab12-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab12-problem1_2_1"> <text> No, the domain of [mathjaxinline]L[/mathjaxinline] is [mathjaxinline]x&gt;0[/mathjaxinline] only.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab12-problem1_2_1" id="input_theory3-tab12-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory3-tab12-problem1_2_1-choice_2-label" for="input_theory3-tab12-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab12-problem1_2_1"> <text> No, the graph of [mathjaxinline]L[/mathjaxinline] has a vertical asymptote at [mathjaxinline]x=0[/mathjaxinline].</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab12-problem1_2_1" id="input_theory3-tab12-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory3-tab12-problem1_2_1-choice_3-label" for="input_theory3-tab12-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab12-problem1_2_1"> <text> Yes</text>
</label>
</div>
<span id="answer_theory3-tab12-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab12-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div><br/></p>
<p>
<div class="solution-span">
<span id="solution_theory3-tab12-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Inverse of L" />
<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_theory3-tab12-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-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="theory3-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="theory3-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="theory3-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-2" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab12-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab12-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">An identity of log</h3>
<div
id="video_theory3-tab12-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab12-video1/handler/publish_completion", "streams": "1.00:RoXt1S8SauM", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab12-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab12-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab12-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-tab12-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_theory3-tab12-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-tab12-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab12-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab12-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-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab13" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">13. Log identities</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab13-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab13-problem1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab13-problem1" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab13-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab13-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab13-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="theory3-tab13-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab13-problem1-problem-progress" tabindex="-1">
The logarithm of reciprocals
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab13-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
As above, define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001246" style="table-layout:auto" width="100%">
<tr id="a0000001247">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle L(x)\, =\, \int _{1}^{x} \frac{dt}{t} \qquad \text {for} \, x&gt;0.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.287)</td>
</tr>
</table>
<p>
Our goal is to show the following identity of the logarithm:<br/></p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001248" style="table-layout:auto" width="100%">
<tr id="a0000001249">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle L\left(\frac{1}{x}\right)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =-L(x).[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.288)</td>
</tr>
</table>
<p>
Start by making a change of variable [mathjaxinline]\displaystyle u=\frac{1}{t}[/mathjaxinline] for the integral [mathjaxinline]L(x)[/mathjaxinline]. Find [mathjaxinline]t[/mathjaxinline] and [mathjaxinline]dt[/mathjaxinline] in terms of [mathjaxinline]u[/mathjaxinline] and [mathjaxinline]du[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]t=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab13-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab13-problem1_2_1" id="input_theory3-tab13-problem1_2_1" data-input-id="theory3-tab13-problem1_2_1" value="" aria-describedby="status_theory3-tab13-problem1_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab13-problem1_2_1"/>
<span class="status unanswered" id="status_theory3-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_theory3-tab13-problem1_2_1" class="answer"/>
<div id="input_theory3-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>
<p style="display:inline">[mathjaxinline]dt=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_theory3-tab13-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab13-problem1_3_1" id="input_theory3-tab13-problem1_3_1" data-input-id="theory3-tab13-problem1_3_1" value="" aria-describedby="status_theory3-tab13-problem1_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab13-problem1_3_1"/>
<span class="status unanswered" id="status_theory3-tab13-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab13-problem1_3_1" class="answer"/>
<div id="input_theory3-tab13-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>
So after the change of variable, the property is [mathjaxinline]L(x)=?[/mathjaxinline] </p>
<span>
<style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
}
</style>
<table>
<tbody>
<tr>
<td/>
<td colspan="2">
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_theory3-tab13-problem1_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_theory3-tab13-problem1_4_1" id="input_theory3-tab13-problem1_4_1" aria-describedby="status_theory3-tab13-problem1_4_1" value="" size="5"/>
<span class="trailing_text" id="trailing_text_theory3-tab13-problem1_4_1"/>
<span class="status unanswered" id="status_theory3-tab13-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab13-problem1_4_1" class="answer"/>
</div>
</div></div>
</td>
</tr>
<tr>
<td>
<p style="text-align:left"> \( \Large{L(x)} = \)</p>
</td>
<td>
<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int_1 }\)</p>
</td>
<td>
<br/>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_theory3-tab13-problem1_5_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_theory3-tab13-problem1_5_1" id="input_theory3-tab13-problem1_5_1" aria-describedby="trailing_text_theory3-tab13-problem1_5_1 status_theory3-tab13-problem1_5_1" value="" size="5"/>
<span class="trailing_text" id="trailing_text_theory3-tab13-problem1_5_1"> [mathjaxinline] \Large{du} [/mathjaxinline]</span>
<span class="status unanswered" id="status_theory3-tab13-problem1_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab13-problem1_5_1" class="answer"/>
</div>
</div></div>
</td>
<td>
<p style="display:inline"> \( = \Large{-L(} \)</p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_theory3-tab13-problem1_6_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_theory3-tab13-problem1_6_1" id="input_theory3-tab13-problem1_6_1" aria-describedby="trailing_text_theory3-tab13-problem1_6_1 status_theory3-tab13-problem1_6_1" value="" size="5"/>
<span class="trailing_text" id="trailing_text_theory3-tab13-problem1_6_1"> [mathjaxinline] \Large) [/mathjaxinline]</span>
<span class="status unanswered" id="status_theory3-tab13-problem1_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab13-problem1_6_1" class="answer"/>
</div>
</div></div>
</td>
</tr>
</tbody>
</table>
</span>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab13-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="The logarithm of reciprocals" />
<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_theory3-tab13-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-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="theory3-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="theory3-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="theory3-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>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab14" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">14. The bell curve</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab14-problem1" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab14-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="3"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="theory3-tab14-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem1-problem-progress" tabindex="-1">
Applying FTC2 to the integral of the bell curve
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Let us now apply FTC2 to another function that is defined by an integral formula.<br/></p>
<p>
Let </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001255" style="table-layout:auto" width="100%">
<tr id="a0000001256">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle F(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 \int _{0}^{x} e^{-t^2}\, dt.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.293)</td>
</tr>
</table>
<p>
The graph of the integrand [mathjaxinline]e^{-t^2}[/mathjaxinline] is called the <span style="color:#27408C"><b class="bf">bell curve</b></span>.<br/></p>
<center><img alt="Go to text below image" src="/assets/courseware/v1/de8844c0c083459e6c1853d6ee10e401/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_bell.svg" style="margin: 10px 25px 25px 25px" width="450px"/><br/>Geometrically, [mathjaxinline]\displaystyle F(x) \, =\, \int _{0}^{x} e^{-t^2} \, dt\, = \, \text {Area of shaded region.}[/mathjaxinline]<br/></center>
<p>
<p style="display:inline">By FTC2, [mathjaxinline]F'(x)\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab14-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab14-problem1_2_1" id="input_theory3-tab14-problem1_2_1" data-input-id="theory3-tab14-problem1_2_1" value="" aria-describedby="status_theory3-tab14-problem1_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab14-problem1_2_1"/>
<span class="status unanswered" id="status_theory3-tab14-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab14-problem1_2_1" class="answer"/>
<div id="input_theory3-tab14-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>
<p>
Find all critical points of [mathjaxinline]F(x)[/mathjaxinline].<br/>(Enter your answer separated by commas, e.g. "1.2, -5, e". Enter &#8220;none" if there are no critical points.)<br/><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_theory3-tab14-problem1_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_theory3-tab14-problem1_3_1" id="input_theory3-tab14-problem1_3_1" aria-describedby="status_theory3-tab14-problem1_3_1" value=""/>
<span class="trailing_text" id="trailing_text_theory3-tab14-problem1_3_1"/>
<span class="status unanswered" id="status_theory3-tab14-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab14-problem1_3_1" class="answer"/>
</div>
</div></div><br/></p>
<p>
Therefore, [mathjaxinline]F(x)[/mathjaxinline] is<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab14-problem1_4_1">
<fieldset aria-describedby="status_theory3-tab14-problem1_4_1">
<div class="field">
<input type="radio" name="input_theory3-tab14-problem1_4_1" id="input_theory3-tab14-problem1_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory3-tab14-problem1_4_1-choice_1-label" for="input_theory3-tab14-problem1_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab14-problem1_4_1"> <text> always increasing</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab14-problem1_4_1" id="input_theory3-tab14-problem1_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory3-tab14-problem1_4_1-choice_2-label" for="input_theory3-tab14-problem1_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab14-problem1_4_1"> <text> always decreasing</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab14-problem1_4_1" id="input_theory3-tab14-problem1_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory3-tab14-problem1_4_1-choice_3-label" for="input_theory3-tab14-problem1_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab14-problem1_4_1"> <text> sometimes increasing and sometimes decreasing</text>
</label>
</div>
<span id="answer_theory3-tab14-problem1_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab14-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div> </p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab14-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Applying FTC2 to the integral of the bell curve" />
<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_theory3-tab14-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-tab14-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="theory3-tab14-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="theory3-tab14-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="theory3-tab14-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+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem2" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab14-problem2" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab14-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem2" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-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="theory3-tab14-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem2-problem-progress" tabindex="-1">
The second derivative
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
As above, let </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001259" style="table-layout:auto" width="100%">
<tr id="a0000001260">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle F(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 \int _{0}^{x} e^{-t^2}\, dt.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.295)</td>
</tr>
</table>
<p>
<p style="display:inline">[mathjaxinline]F^{\prime \prime }(x)\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab14-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab14-problem2_2_1" id="input_theory3-tab14-problem2_2_1" data-input-id="theory3-tab14-problem2_2_1" value="" aria-describedby="status_theory3-tab14-problem2_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab14-problem2_2_1"/>
<span class="status unanswered" id="status_theory3-tab14-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab14-problem2_2_1" class="answer"/>
<div id="input_theory3-tab14-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>
<br/>
</p>
<p>
Hence, [mathjaxinline]F(x)[/mathjaxinline] is <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab14-problem2_3_1">
<fieldset aria-describedby="status_theory3-tab14-problem2_3_1">
<div class="field">
<input type="radio" name="input_theory3-tab14-problem2_3_1" id="input_theory3-tab14-problem2_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory3-tab14-problem2_3_1-choice_1-label" for="input_theory3-tab14-problem2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab14-problem2_3_1"> <text> always concave up</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab14-problem2_3_1" id="input_theory3-tab14-problem2_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory3-tab14-problem2_3_1-choice_2-label" for="input_theory3-tab14-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab14-problem2_3_1"> <text> always concave down</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab14-problem2_3_1" id="input_theory3-tab14-problem2_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory3-tab14-problem2_3_1-choice_3-label" for="input_theory3-tab14-problem2_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab14-problem2_3_1"> <text> sometimes concave up and sometimes concave down</text>
</label>
</div>
<span id="answer_theory3-tab14-problem2_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab14-problem2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div><br/></p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab14-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="The second derivative" />
<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_theory3-tab14-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-tab14-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="theory3-tab14-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="theory3-tab14-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="theory3-tab14-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+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem3" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab14-problem3" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab14-problem3-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem3" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-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="theory3-tab14-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem3-problem-progress" tabindex="-1">
Review: Integral of even functions
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab14-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
The bell curve is an even function, so let us review the following property of the integrals of even functions.<br/></p>
<p>
Let [mathjaxinline]g(t)[/mathjaxinline] be an even function. That is, [mathjaxinline]\, g(-t)=g(t)[/mathjaxinline].<br/></p>
<p>
For problems like this, it is convenient to consider separately the cases [mathjaxinline]x&gt;0[/mathjaxinline] and [mathjaxinline]x&lt;0[/mathjaxinline]. It is easier to first look at the case [mathjaxinline]\, x&gt;0[/mathjaxinline].</p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \int _{0}^{x} g(t) \, dt = c \left( \int _{0}^{-x} g(t) \, dt \right) \, \, \, \,[/mathjaxinline] for [mathjaxinline]\, \, \, c\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab14-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab14-problem3_2_1" id="input_theory3-tab14-problem3_2_1" data-input-id="theory3-tab14-problem3_2_1" value="" aria-describedby="status_theory3-tab14-problem3_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab14-problem3_2_1"/>
<span class="status unanswered" id="status_theory3-tab14-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab14-problem3_2_1" class="answer"/>
<div id="input_theory3-tab14-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_theory3-tab14-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Review: Integral of even 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_theory3-tab14-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-tab14-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="theory3-tab14-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="theory3-tab14-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="theory3-tab14-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>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab15" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">15. The integral of the bell curve</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab15-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab15-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">The integral of the bell curve</h3>
<div
id="video_theory3-tab15-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab15-video1/handler/publish_completion", "streams": "1.00:HtsHtJ-9yEc", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab15-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab15-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab15-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-tab15-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_theory3-tab15-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-tab15-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab15-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab15-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+18.01.2x+3T2019+type@html+block@theory3-tab15-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab15-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The integral of the bell curve: summary</b></p><p>
The integral of the bell curve is an example of a function that cannot be expressed (without using an integral) in terms of functions that we already know.<br/></p><p>
Recall that the <span style="color:#27408C"><b class="bf">bell curve</b></span> is the graph of [mathjaxinline]e^{-t^2}[/mathjaxinline]. As above, let </p><table id="a0000001266" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001267"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle F(x)= \int _{0}^{x} e^{-t^2}\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.299)</td></tr></table><center><img src="/assets/courseware/v1/de8844c0c083459e6c1853d6ee10e401/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_bell.svg" width="450px" alt="Go to text below image" style="margin: 10px 25px 25px 25px"/><br/>[mathjaxinline]\displaystyle F(x) = \int _{0}^{x} e^{-t^2} \, dt = \text {Area of shaded region.}.[/mathjaxinline]<br/></center><p><br/>Using the first and second derivatives of [mathjaxinline]F[/mathjaxinline], we sketched the graph of [mathjaxinline]F[/mathjaxinline] as shown below.<br/></p><center><img src="/assets/courseware/v1/6d3adf77cfb2e51b39490f7e1bba80de/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_bellintegral.svg" width="450px" alt="Go to text below image" style="margin: 10px 25px 25px 25px"/><br/>Graph of [mathjaxinline]\displaystyle F(x)[/mathjaxinline] <br/></center><p><b class="bfseries">Properties</b></p><p>
Here are two properties of [mathjaxinline]F[/mathjaxinline].<br/></p><ul class="itemize"><li><p>
[mathjaxinline]F[/mathjaxinline] is odd: [mathjaxinline]F(-x)\, =\, - F(x)[/mathjaxinline]<br/></p></li><li><p>
[mathjaxinline]\displaystyle \lim _{x\rightarrow \infty } F(x)= \frac{\sqrt {\pi }}{2}[/mathjaxinline];[mathjaxinline]\, \, \displaystyle \lim _{x\rightarrow -\infty } F(x)= \frac{-\sqrt {\pi }}{2}[/mathjaxinline]. </p></li></ul><center><img src="/assets/courseware/v1/b54dde00b8f3959b40fb6c82d4f3c706/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_bellinfinite.svg" width="350px" alt="Go to text below image" style="margin: 10px 25px 25px 25px"/><br/>Geometrically, [mathjaxinline]\displaystyle \lim _{x\rightarrow \infty } F(x)[/mathjaxinline] is the area of the shaded infinite stretch. </center><p><br/>The graph of [mathjaxinline]F(x)[/mathjaxinline] has two horizontal asymptotes at [mathjaxinline]\displaystyle y=\frac{\sqrt {\pi }}{2}[/mathjaxinline] and [mathjaxinline]\displaystyle y=-\frac{\sqrt {\pi }}{2}[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/f7414794d93aaec68bfffa7d759da3a8/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_bellintegraly.svg" width="450px" alt="Go to text above image" style="margin: 10px 25px 25px 25px"/><br/></center><p><br/><b class="bfseries">The error function</b></p><p>
The <span style="color:#99182C"><b class="bf">error function</b></span>, denoted by [mathjaxinline]\, \text {erf}(x)[/mathjaxinline], is the renormalized version of [mathjaxinline]\displaystyle F(x) = \int _{0}^{x} e^{-t^2} \, dt[/mathjaxinline] that has horizontal asymptotes at [mathjaxinline]y=\pm 1[/mathjaxinline]. </p><table id="a0000001268" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001269"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \text {erf}(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{2}{\sqrt {\pi }}\, F(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 \frac{2}{\sqrt {\pi }}\, \int _{0}^{x} e^{-t^2}\, dt\qquad (\text {Error function}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.300)</td></tr></table><p>
The error function is a new transcendental function to add to your repertoire of functions along with [mathjaxinline]\, \, e^ x[/mathjaxinline],[mathjaxinline]\, \, \ln (x)[/mathjaxinline], and trigonometric functions. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab16" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">16. More new functions</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab16-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab16-video1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">More new functions</h3>
<div
id="video_theory3-tab16-video1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab16-video1/handler/publish_completion", "streams": "1.00:lnttPKyFegI", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab16-video1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab16-video1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab16-video1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
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="theory3-tab16-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_theory3-tab16-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_theory3-tab16-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab16-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+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@theory3-tab16-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+18.01.2x+3T2019+type@html+block@theory3-tab16-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab16-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
There are many more new functions that can be defined using integrals, but cannot be expressed in terms of any elementary functions. Here are four more examples. </p><table id="a0000001270" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001271"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle C(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 \int _{0}^{x} \cos \left(t^2\right)\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.301)</td></tr><tr id="a0000001272"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle S(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 \int _{0}^{x} \sin \left(t^2\right)\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.302)</td></tr></table><p>
These are called <span style="color:#27408C"><b class="bf">Fresnel Integrals</b></span>. You will explore [mathjaxinline]C(x)[/mathjaxinline] in your Part B problem set. </p><p>
Here is another function that is used in Fourier analysis. </p><table id="a0000001273" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001274"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; 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:center; border:none">
[mathjaxinline]\displaystyle \int _{0}^{x} h(t)\, dt \qquad \text {where}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle h(t)[/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 \begin{cases} 1 & \mbox{if } t=0 \\ \frac{\sin (t)}{t} & \mbox{if } t\neq 0 \end{cases}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.303)</td></tr></table><p>
You will look at [mathjaxinline]H(x)[/mathjaxinline] in more detail on the next page.<br/></p><p>
Last, but not least, the function </p><table id="a0000001275" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001276"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle Li(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 \int _{2}^{x} \frac{dt}{\ln (t)}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.304)</td></tr></table><p>
approximates the number of prime numbers smaller than [mathjaxinline]x[/mathjaxinline] and is related to what is known as the Riemann Hypothesis. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab17" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">17. Properties of H</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab17-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab17-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
In these final problems, we will apply our usual tools along with FTC2 to find properties of [mathjaxinline]H(x)[/mathjaxinline]. Recall </p><table id="a0000001277" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001278"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; 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:center; border:none">
[mathjaxinline]\displaystyle \int _{0}^{x} h(t)\, dt \qquad \text {where}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle h(t)[/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 \begin{cases} 1 & \mbox{if } t=0 \\ \frac{\sin (t)}{t} & \mbox{if } t\neq 0. \end{cases}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.305)</td></tr></table><p>
Recall that the integrand [mathjaxinline]h(t)[/mathjaxinline] is continuous since [mathjaxinline]\displaystyle \lim _{t\rightarrow 0} \frac{\sin (t)}{t} =1[/mathjaxinline].<br/></p><center><img src="/assets/courseware/v1/02aea99d8d5f2c0eb47d5e548b9c792a/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_Harea.svg" width="800px" alt="Go to text below image" style="margin: 10px 25px 25px 25px"/><br/>[mathjaxinline]\displaystyle H(x) = \int _{0}^{x}\frac{\sin (t)}{t} \, dt = \text {Signed area from} \, \, \, 0\, \, \, \text {to}\, \, \, x.[/mathjaxinline]<br/></center>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab17-problem1" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab17-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-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="theory3-tab17-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem1-problem-progress" tabindex="-1">
Odd or even
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Let </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001279" style="table-layout:auto" width="100%">
<tr id="a0000001280">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle h(t)[/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 \begin{cases} 1 &amp; \mbox{if } t=0 \\ \frac{\sin (t)}{t} &amp; \mbox{if } t\neq 0. \end{cases}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.306)</td>
</tr>
</table>
<p>
<p style="display:inline">For [mathjaxinline]\, \, t\neq 0[/mathjaxinline], [mathjaxinline]\, \, \, \, \displaystyle h(-t)\, =\, c \cdot h(t)\, \, \, \,[/mathjaxinline] for [mathjaxinline]\, \, \, c=\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab17-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab17-problem1_2_1" id="input_theory3-tab17-problem1_2_1" data-input-id="theory3-tab17-problem1_2_1" value="" aria-describedby="status_theory3-tab17-problem1_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab17-problem1_2_1"/>
<span class="status unanswered" id="status_theory3-tab17-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab17-problem1_2_1" class="answer"/>
<div id="input_theory3-tab17-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>
As above, define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001281" style="table-layout:auto" width="100%">
<tr id="a0000001282">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \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 \int _{0}^{x} h(t)\, dt.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.307)</td>
</tr>
</table>
<p>
[mathjaxinline]\displaystyle H(x)[/mathjaxinline] is <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab17-problem1_3_1">
<fieldset aria-describedby="status_theory3-tab17-problem1_3_1">
<div class="field">
<input type="radio" name="input_theory3-tab17-problem1_3_1" id="input_theory3-tab17-problem1_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="theory3-tab17-problem1_3_1-choice_1-label" for="input_theory3-tab17-problem1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem1_3_1"> <text> odd</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab17-problem1_3_1" id="input_theory3-tab17-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="theory3-tab17-problem1_3_1-choice_2-label" for="input_theory3-tab17-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem1_3_1"> <text> even</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_theory3-tab17-problem1_3_1" id="input_theory3-tab17-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="theory3-tab17-problem1_3_1-choice_3-label" for="input_theory3-tab17-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem1_3_1"> <text> not odd and not even</text>
</label>
</div>
<span id="answer_theory3-tab17-problem1_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab17-problem1_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_theory3-tab17-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Odd or even" />
<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_theory3-tab17-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-tab17-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="theory3-tab17-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="theory3-tab17-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="theory3-tab17-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+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem2" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab17-problem2" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab17-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem2" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-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="theory3-tab17-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem2-problem-progress" tabindex="-1">
Linear approximation of H
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
As above, let </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001287" style="table-layout:auto" width="100%">
<tr id="a0000001288">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; 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:center; border:none">
[mathjaxinline]\displaystyle \int _{0}^{x} h(t)\, dt \qquad \text {where}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle h(t)[/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 \begin{cases} 1 &amp; \mbox{if } t=0 \\ \frac{\sin (t)}{t} &amp; \mbox{if } t\neq 0. \end{cases}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.310)</td>
</tr>
</table>
<p>
Give the linear approximation of [mathjaxinline]H(x)[/mathjaxinline] at [mathjaxinline]x=0[/mathjaxinline].<br/><p style="display:inline">[mathjaxinline]H(x)\approx \, \,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_theory3-tab17-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_theory3-tab17-problem2_2_1" id="input_theory3-tab17-problem2_2_1" data-input-id="theory3-tab17-problem2_2_1" value="" aria-describedby="status_theory3-tab17-problem2_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_theory3-tab17-problem2_2_1"/>
<span class="status unanswered" id="status_theory3-tab17-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab17-problem2_2_1" class="answer"/>
<div id="input_theory3-tab17-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/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab17-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Linear approximation of H" />
<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_theory3-tab17-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-tab17-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="theory3-tab17-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="theory3-tab17-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="theory3-tab17-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-3" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem3" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_theory3-tab17-problem3" class="problems-wrapper" role="group"
aria-labelledby="theory3-tab17-problem3-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem3" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem3/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="theory3-tab17-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem3-problem-progress" tabindex="-1">
Shape of H
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@theory3-tab17-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
As above, let </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000001292" style="table-layout:auto" width="100%">
<tr id="a0000001293">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; 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:center; border:none">
[mathjaxinline]\displaystyle \int _{0}^{x} h(t)\, dt \qquad \text {where}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle h(t)[/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 \begin{cases} 1 &amp; \mbox{if } t=0 \\ \frac{\sin (t)}{t} &amp; \mbox{if } t\neq 0. \end{cases}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.313)</td>
</tr>
</table>
<p>
Use the graph of [mathjaxinline]h(t)[/mathjaxinline] below to find all critical points of [mathjaxinline]H[/mathjaxinline] in [mathjaxinline][0,3\pi ][/mathjaxinline].<br/>(Enter your answer separated by commas, e.g. "12, -5, 3*e". Enter "pi" for [mathjaxinline]\pi[/mathjaxinline]. Enter &#8220;none" if there are no critical points.)<br/><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_theory3-tab17-problem3_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_theory3-tab17-problem3_2_1" id="input_theory3-tab17-problem3_2_1" aria-describedby="status_theory3-tab17-problem3_2_1" value=""/>
<span class="trailing_text" id="trailing_text_theory3-tab17-problem3_2_1"/>
<span class="status unanswered" id="status_theory3-tab17-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_theory3-tab17-problem3_2_1" class="answer"/>
</div>
</div></div><br/></p>
<center><img alt="Go to text below image" src="/assets/courseware/v1/48aa95d2f71c3e25f45589e5c9a33fd9/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_hnew.svg" style="margin: 10px 25px 25px 25px" width="500px"/><br/>Graph of [mathjaxinline]h(t)[/mathjaxinline] </center>
<p>
Referring to the graph of [mathjaxinline]h(t)[/mathjaxinline] above, in which subintervals of [mathjaxinline][0,3\pi ][/mathjaxinline] is [mathjaxinline]\displaystyle H(x)=\int _{0}^{x} h(t)\, dt[/mathjaxinline] </p>
<table cellspacing="0" class="tabular" style="table-layout:auto">
<tr>
<td style="text-align:left; border:none">
increasing?</td>
<td style="text-align:left; border:none">
decreasing?</td>
<td style="text-align:left; border:none">
concave up?</td>
<td style="text-align:left; border:none">
concave down?</td>
</tr>
<tr>
<td style="text-align:left; border:none">
(Check all that apply.)</td>
<td style="text-align:left; border:none">&#160;</td>
<td style="text-align:left; border:none">&#160;</td>
<td style="text-align:left; border:none">&#160;</td>
</tr>
<tr>
<td style="text-align:left; border:none">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab17-problem3_3_1">
<fieldset aria-describedby="status_theory3-tab17-problem3_3_1">
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_3_1[]" id="input_theory3-tab17-problem3_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="theory3-tab17-problem3_3_1-choice_0-label" for="input_theory3-tab17-problem3_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_3_1">
<text>[mathjaxinline][0,\pi ][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_3_1[]" id="input_theory3-tab17-problem3_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="theory3-tab17-problem3_3_1-choice_1-label" for="input_theory3-tab17-problem3_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_3_1">
<text>[mathjaxinline][\pi ,A][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_3_1[]" id="input_theory3-tab17-problem3_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="theory3-tab17-problem3_3_1-choice_2-label" for="input_theory3-tab17-problem3_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_3_1">
<text>[mathjaxinline][A,2\pi ][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_3_1[]" id="input_theory3-tab17-problem3_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="theory3-tab17-problem3_3_1-choice_3-label" for="input_theory3-tab17-problem3_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_3_1">
<text>[mathjaxinline][2\pi ,B][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_3_1[]" id="input_theory3-tab17-problem3_3_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="theory3-tab17-problem3_3_1-choice_4-label" for="input_theory3-tab17-problem3_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_3_1">
<text>[mathjaxinline][B,3\pi ][/mathjaxinline]</text>
</label>
</div>
<span id="answer_theory3-tab17-problem3_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab17-problem3_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</td>
<td style="text-align:left; border:none">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab17-problem3_4_1">
<fieldset aria-describedby="status_theory3-tab17-problem3_4_1">
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_4_1[]" id="input_theory3-tab17-problem3_4_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="theory3-tab17-problem3_4_1-choice_0-label" for="input_theory3-tab17-problem3_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_4_1">
<text>[mathjaxinline][0,\pi ][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_4_1[]" id="input_theory3-tab17-problem3_4_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="theory3-tab17-problem3_4_1-choice_1-label" for="input_theory3-tab17-problem3_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_4_1">
<text>[mathjaxinline][\pi ,A][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_4_1[]" id="input_theory3-tab17-problem3_4_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="theory3-tab17-problem3_4_1-choice_2-label" for="input_theory3-tab17-problem3_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_4_1">
<text>[mathjaxinline][A,2\pi ][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_4_1[]" id="input_theory3-tab17-problem3_4_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="theory3-tab17-problem3_4_1-choice_3-label" for="input_theory3-tab17-problem3_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_4_1">
<text>[mathjaxinline][2\pi ,B][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_4_1[]" id="input_theory3-tab17-problem3_4_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="theory3-tab17-problem3_4_1-choice_4-label" for="input_theory3-tab17-problem3_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_4_1">
<text>[mathjaxinline][B,3\pi ][/mathjaxinline]</text>
</label>
</div>
<span id="answer_theory3-tab17-problem3_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab17-problem3_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</td>
<td style="text-align:left; border:none">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab17-problem3_5_1">
<fieldset aria-describedby="status_theory3-tab17-problem3_5_1">
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_5_1[]" id="input_theory3-tab17-problem3_5_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="theory3-tab17-problem3_5_1-choice_0-label" for="input_theory3-tab17-problem3_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_5_1">
<text>[mathjaxinline][0,\pi ][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_5_1[]" id="input_theory3-tab17-problem3_5_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="theory3-tab17-problem3_5_1-choice_1-label" for="input_theory3-tab17-problem3_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_5_1">
<text>[mathjaxinline][\pi ,A][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_5_1[]" id="input_theory3-tab17-problem3_5_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="theory3-tab17-problem3_5_1-choice_2-label" for="input_theory3-tab17-problem3_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_5_1">
<text>[mathjaxinline][A,2\pi ][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_5_1[]" id="input_theory3-tab17-problem3_5_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="theory3-tab17-problem3_5_1-choice_3-label" for="input_theory3-tab17-problem3_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_5_1">
<text>[mathjaxinline][2\pi ,B][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_5_1[]" id="input_theory3-tab17-problem3_5_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="theory3-tab17-problem3_5_1-choice_4-label" for="input_theory3-tab17-problem3_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_5_1">
<text>[mathjaxinline][B,3\pi ][/mathjaxinline]</text>
</label>
</div>
<span id="answer_theory3-tab17-problem3_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab17-problem3_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</td>
<td style="text-align:left; border:none">
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_theory3-tab17-problem3_6_1">
<fieldset aria-describedby="status_theory3-tab17-problem3_6_1">
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_6_1[]" id="input_theory3-tab17-problem3_6_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="theory3-tab17-problem3_6_1-choice_0-label" for="input_theory3-tab17-problem3_6_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_6_1">
<text>[mathjaxinline][0,\pi ][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_6_1[]" id="input_theory3-tab17-problem3_6_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="theory3-tab17-problem3_6_1-choice_1-label" for="input_theory3-tab17-problem3_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_6_1">
<text>[mathjaxinline][\pi ,A][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_6_1[]" id="input_theory3-tab17-problem3_6_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="theory3-tab17-problem3_6_1-choice_2-label" for="input_theory3-tab17-problem3_6_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_6_1">
<text>[mathjaxinline][A,2\pi ][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_6_1[]" id="input_theory3-tab17-problem3_6_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="theory3-tab17-problem3_6_1-choice_3-label" for="input_theory3-tab17-problem3_6_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_6_1">
<text>[mathjaxinline][2\pi ,B][/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_theory3-tab17-problem3_6_1[]" id="input_theory3-tab17-problem3_6_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="theory3-tab17-problem3_6_1-choice_4-label" for="input_theory3-tab17-problem3_6_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_theory3-tab17-problem3_6_1">
<text>[mathjaxinline][B,3\pi ][/mathjaxinline]</text>
</label>
</div>
<span id="answer_theory3-tab17-problem3_6_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_theory3-tab17-problem3_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</td>
</tr>
</table>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+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_theory3-tab17-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Shape of H" />
<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_theory3-tab17-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_theory3-tab17-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="theory3-tab17-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="theory3-tab17-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="theory3-tab17-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>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@theory3-tab18" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<h2 class="hd hd-2 unit-title">18. Summary</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab18-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@theory3-tab18-text1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="44aaa24601e011ef9cf916fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">The Second Fundamental Theorem of Calculus</b></p><p>
The <span style="color:#99182C"><b class="bf">Second Fundamental Theorem of Calculus</b></span> states the following.<br/>Given a continuous function [mathjaxinline]f(x)[/mathjaxinline], if </p><table id="a0000001300" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle G(x)=\int _{a}^{x} f(t)\, dt \qquad (\, \, t\, \, \text {between}\, \, a\, \, \text {and}\, x),[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
then </p><table id="a0000001301" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]G'(x)=f(x).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><center><img src="/assets/courseware/v1/b0086bdea7eaee80c86ce42e733ce366/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_ftc2_G.svg" width="350px" alt="Go to text below image" style="margin: 10px 25px 25px 25px"/><br/>Geometrically, [mathjaxinline]G(x)[/mathjaxinline] is the area between [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]x[/mathjaxinline]. This area varies as [mathjaxinline]x[/mathjaxinline] varies.<br/></center><p><br/></p><p>
We will abbreviate this theorem by <span style="color:#99182C"><b class="bf">FTC2</b></span>.<br/></p><p>
In terms of differential equations, FTC2 says that [mathjaxinline]G(x)[/mathjaxinline] is the solution to the following differential equation and initial condition: </p><table id="a0000001302" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001303"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y'[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle f[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {differential equation})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.318)</td></tr><tr id="a0000001304"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y(a)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle 0[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {initial condition}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.319)</td></tr></table><p>
The initial condition [mathjaxinline]y(a)=0[/mathjaxinline] is satisfied because [mathjaxinline]\, \, \displaystyle G(a)=\int _{a}^{a} f(t)\, dt = 0[/mathjaxinline].<br/></p><p>
Recall that any function [mathjaxinline]G[/mathjaxinline] such that [mathjaxinline]G'=f[/mathjaxinline] is an antiderivative of [mathjaxinline]f[/mathjaxinline]. Hence, FTC2 gives us a formula for an antiderivative of [mathjaxinline]f(x)[/mathjaxinline]. This formula is different from the formulas you have seen. It is in terms of a definite integral, and is called an <span style="color:#27408C"><b class="bf">integral formula</b></span>. </p><p><b class="bfseries">Note:</b> The integrand [mathjaxinline]f(x)[/mathjaxinline] can be any continuous function, not just the ones whose antiderivative we know how to find. These integral formulas are still useful there are numerical methods that allow us to compute them.</p><p><b class="bfseries">FTC2 and the chain rule</b></p><p>
For any continuous function [mathjaxinline]f[/mathjaxinline], and differentiable function [mathjaxinline]u(x)[/mathjaxinline],<br/></p><table id="a0000001305" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001306"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d}{dx}\int _{a}^{u(x)} f(t)\, dt[/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(u(x))u'(x).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.320)</td></tr></table><p>
This is because </p><table id="a0000001307" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001308"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _{a}^{u(x)} f(t)\, dt[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle G(u(x))[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \text {where}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle G(y)[/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 \int _{a}^{y} f(t)\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.321)</td></tr></table><p>
In other words, the integral [mathjaxinline]\, \, \displaystyle \int _{a}^{u(x)} f(t)\, dt\, \,[/mathjaxinline] can be written as a composition of the two functions [mathjaxinline]G[/mathjaxinline] and [mathjaxinline]u[/mathjaxinline]. We can then apply the chain rule and FTC2 to get the derivative.</p><p><b class="bfseries">The two fundamental theorems of calculus</b></p><p>
Here is another point of view of the two fundamental theorems of calculus together. This is what we mentioned in the introductory (very first) video of this section.<br/></p><dl class="description"><dt>FTC1:</dt><dd><p>
Given a differentiable function [mathjaxinline]F[/mathjaxinline] with continuous derivative [mathjaxinline]F'[/mathjaxinline], </p><table id="a0000001309" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _{a}^{x}F'(t) \, dt \, =\, F(x)-F(a),[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd><dt>FTC2:</dt><dd><p>
Given a continuous function [mathjaxinline]f[/mathjaxinline], </p><table id="a0000001310" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{d}{dx}\int _{a}^{x} f(t) \, dt = f(x),[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></dd></dl><p>
In other words, FTC1 says that if we start with a function [mathjaxinline]F[/mathjaxinline], and first differentiate and then integrate (from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]x[/mathjaxinline]), we will get back the function [mathjaxinline]F[/mathjaxinline] up to a constant, which is given by [mathjaxinline]F(a)[/mathjaxinline].<br/></p><p>
On the other hand, FTC2 says that if we start with a function [mathjaxinline]f[/mathjaxinline], and first integrate (from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]x[/mathjaxinline]) and then differentiate, we will get back to [mathjaxinline]f[/mathjaxinline]. Notice there is no ambiguity of a constant here.<br/></p><p>
So, the two fundamental theorems together say that differentiation and integration (from [mathjaxinline]a[/mathjaxinline] to [mathjaxinline]x[/mathjaxinline]) are “inverse" operations of one another, up to a constant.</p><p><b class="bfseries">Constructing functions using integrals</b></p><p>
Recall the FTC2 gives the integral formula </p><table id="a0000001311" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle G(x) = \int _{a}^{x} f(t) \, dt[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
as the solution to the following differential equation and initial condition: </p><table id="a0000001312" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001313"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y'[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle f[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {differential equation})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.322)</td></tr><tr id="a0000001314"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y(a)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle 0[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {initial condition}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.323)</td></tr></table><p>
Depending on what the differential equation is, [mathjaxinline]G(x)[/mathjaxinline] can be a function we already know, or one that cannot be expressed (without an integral) in terms of any function we already know. There are many functions of either kind. </p><p>
For example, the logarithmic function can be defined using an integral:</p><table id="a0000001315" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001316"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle L(x)\,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \, \int _{1}^{x} \frac{dt}{t}[/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 \ln (x) \qquad \text {where}\, \, x>0.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.324)</td></tr></table><p>
On the other hand, the integral of [mathjaxinline]\, \, e^{-t^2},\, \,[/mathjaxinline] the <span style="color:#27408C"><b class="bf">bell curve</b></span>, is a function defined by an integral and cannot be expressed in terms of functions we already know: </p><table id="a0000001317" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001318"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle F(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \int _{0}^{x} e^{-t^2} \, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.325)</td></tr></table><p>
This function is central in probability and will be discussed more in the next unit.<br/></p><p>
Finally, even though we may not have explicit formulas for the functions defined by integral formulas, we can apply FTC2 to find their derivatives. Therefore, we have all of our usual tools at our disposal. For example, we can graph them, or approximate them by linear and quadratic approximations.<br/></p><p><b class="bfseries">Summary of the logarithm</b></p><p>
We have now seen the exponential and the logarithmic functions in two different ways. Let us summarize.<br/></p><p>
In <i class="itshape">Calculus 1A</i>, we defined the exponential function [mathjaxinline]E(y)[/mathjaxinline] to be the solution to the differential equation and initial condition: </p><table id="a0000001319" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001320"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle E'(y)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle E(y)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {differential equation})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.326)</td></tr><tr id="a0000001321"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle E(0)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle 1[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {initial condition} ).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.327)</td></tr></table><p>
We then let[mathjaxinline]\, e=E(1)\,[/mathjaxinline] and denoted [mathjaxinline]\, E(y)[/mathjaxinline] by [mathjaxinline]\, \, e^ y[/mathjaxinline]. <br/></p><p>
We defined the logarithm as the inverse of [mathjaxinline]E(y)[/mathjaxinline] </p><table id="a0000001322" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001323"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Definition of the logarithm}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \ln (x)\, \, =\, \, E^{-1}(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.328)</td></tr></table><p>
The properties of the logarithm were derived from the properties of the exponential. </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none">
Product: </td><td style="text-align:left; border:none">
[mathjaxinline]e^ a\cdot e^ b\, \, =\, \, e^{a+b}[/mathjaxinline]</td><td style="text-align:left; border:none">
[mathjaxinline]\Longrightarrow[/mathjaxinline]</td><td style="text-align:left; border:none">
[mathjaxinline]\ln (A\cdot B)\, \, =\, \, \ln (A)+ \ln (B)[/mathjaxinline],</td></tr><tr><td style="text-align:left; border:none">
Reciprocal:</td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{e^ a}\, \, =\, \, e^{-a}[/mathjaxinline]</td><td style="text-align:left; border:none">
[mathjaxinline]\Longrightarrow[/mathjaxinline]</td><td style="text-align:left; border:none">
[mathjaxinline]\ln \left(\frac{1}{A}\right)\, \, =\, \, -\ln (A)[/mathjaxinline]. </td></tr></table><p>
On the other hand, in this section, we first define the logarithm to be the solution to the differential equation and initial condition below: </p><table id="a0000001324" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001325"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle L'(x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{1}{x}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {differential equation})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.329)</td></tr><tr id="a0000001326"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle L(1)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle 0[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (\text {initial condition} ).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.330)</td></tr></table><p>
The FTC2 gives the integral formula for [mathjaxinline]L[/mathjaxinline]. </p><table id="a0000001327" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001328"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \text {Definition of the logarithm}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle L(x)\, \, =\, \, \int _{1}^{x}\, \frac{dt}{t}.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.331)</td></tr></table><p>
We then define [mathjaxinline]e[/mathjaxinline] to be the number such that [mathjaxinline]L(e)=1[/mathjaxinline], and the exponential function to be the inverse of [mathjaxinline]L[/mathjaxinline]. </p><table id="a0000001329" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001330"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Definition of the exponential}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle e^ y\, \, =\, \, L^{-1}(y).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.332)</td></tr></table><p>
To derive the properties of logarithm, we perform different changes of variable to the integral formula [mathjaxinline]\, \, \displaystyle L(x)= \int _{1}^{x}\, \frac{dt}{t}.[/mathjaxinline] </p>
</div>
</div>
</div>
</div>
© All Rights Reserved