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<h2 class="hd hd-2 unit-title">1. Getting to know you</h2>
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<p>
You are interested in learning integration, but we don't know very much about you or what you already know. So to help you learn best, please take the following diagnostics. These diagnostics will help us understand who you are, and how best to support you in this course. </p><p>
We want you to succeed, so make sure that you have the basic precalculus skills so you won't be frustrated! The first page tests your readiness for this calculus class. If you aren't ready yet, don't worry, you can take this course later. Or you can refresh your memory from the MIT OCW course 18.01 available online. (See link on the Course Info page under “Related Links".) </p><p>
Some of you may already know a lot of calculus. So the students in the course can benefit from your experience, this diagnostic exam helps us to identify you! The second page is a diagnostic on integration. We encourage you to take a look at these even if you don't know any integration yet. But don't worry; we designed this course for people with varying backgrounds, including those with no calculus experience. </p>
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<h2 class="hd hd-2 unit-title">2. Prerequisite Knowledge</h2>
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Prior Experience with Limits and Derivatives
</h3>
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<text> I have never seen derivatives before.</text>
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<input type="radio" name="input_diag-tab2-problem1_2_1" id="input_diag-tab2-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab2-problem1_2_1-choice_2-label" for="input_diag-tab2-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_2_1">
<text> I have learned derivatives in a face-to-face class.</text>
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<input type="radio" name="input_diag-tab2-problem1_2_1" id="input_diag-tab2-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab2-problem1_2_1-choice_3-label" for="input_diag-tab2-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_2_1">
<text> I took 18.01.1x Calculus 1A on edX.</text>
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<text> I have learned calculus before online, but not on edX.</text>
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<p>
If you get 0.75 or above, you have a good handle the differential calculus concepts needed to succeed in this course. </p><p>
If you get between 0.5 and 0.75 points, you may need to review derivatives and limits while you are taking this course to learn the content deeply. </p><p>
Otherwise, you may not be prepared for this course yet. </p>
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Prerequisite Knowledge
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<p><b class="bf">D1 &#8195;</b> Which of the following statements about the function [mathjaxinline]\ f(x) = x^4-2x^3[/mathjaxinline] is true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab2-problem2_2_1">
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<input type="radio" name="input_diag-tab2-problem2_2_1" id="input_diag-tab2-problem2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab2-problem2_2_1-choice_1-label" for="input_diag-tab2-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem2_2_1">
<text> The function has no relative extremum.</text>
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<input type="radio" name="input_diag-tab2-problem2_2_1" id="input_diag-tab2-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab2-problem2_2_1-choice_2-label" for="input_diag-tab2-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem2_2_1">
<text> The graph of the function has one point of inflection and the function has two relative extrema.</text>
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<input type="radio" name="input_diag-tab2-problem2_2_1" id="input_diag-tab2-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab2-problem2_2_1-choice_3-label" for="input_diag-tab2-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem2_2_1">
<text> The graph of the function has two points of inflection and the function has one relative extremum.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem2_2_1" id="input_diag-tab2-problem2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab2-problem2_2_1-choice_4-label" for="input_diag-tab2-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem2_2_1">
<text> The graph of the function has two points of inflection and the function has two relative extremum.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem2_2_1" id="input_diag-tab2-problem2_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab2-problem2_2_1-choice_5-label" for="input_diag-tab2-problem2_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem2_2_1">
<text> The graph of the function has two points of inflection and the function has three relative extrema.</text>
</label>
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<span id="answer_diag-tab2-problem2_2_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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</p>
<p><b class="bf">D2 &#8195;</b> What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow 3} \frac{6/x - 2}{3-4x + x^2} }[/mathjaxinline]? </p>
<p>
(If the limit does not exist, enter DNE.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_diag-tab2-problem2_3_1" class="inputtype formulaequationinput">
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<span class="status unanswered" id="status_diag-tab2-problem2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_diag-tab2-problem2_3_1" class="answer"/>
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\(\)
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</div>
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<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
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</p>
<p><b class="bf">D3 &#8195;</b> What is [mathjaxinline]\ \displaystyle \lim _{h \rightarrow 0} \frac{\cos \left(\frac{\pi }6 + h\right) - \cos \left(\frac{\pi }6\right)}{h}[/mathjaxinline]? </p>
<p>
(Enter the answer as a decimal. If the limit does not exist, enter DNE.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_diag-tab2-problem2_4_1" class="inputtype formulaequationinput">
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<span class="status unanswered" id="status_diag-tab2-problem2_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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\(\)
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<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
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</p>
<p><b class="bf">D4 &#8195;</b> Which of the following functions have a <em>removable discontinuity</em> at [mathjaxinline]x=2[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab2-problem2_5_1">
<fieldset aria-describedby="status_diag-tab2-problem2_5_1">
<div class="field">
<input type="checkbox" name="input_diag-tab2-problem2_5_1[]" id="input_diag-tab2-problem2_5_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="diag-tab2-problem2_5_1-choice_0-label" for="input_diag-tab2-problem2_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem2_5_1">
<text>[mathjaxinline]\displaystyle f(x) = \frac{x^2-x-2}{x-2}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_diag-tab2-problem2_5_1[]" id="input_diag-tab2-problem2_5_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="diag-tab2-problem2_5_1-choice_1-label" for="input_diag-tab2-problem2_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem2_5_1">
<text>[mathjaxinline]f(x) = \frac{1}{(x-2)^2}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_diag-tab2-problem2_5_1[]" id="input_diag-tab2-problem2_5_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="diag-tab2-problem2_5_1-choice_2-label" for="input_diag-tab2-problem2_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem2_5_1">
<text>[mathjaxinline]\displaystyle f(x) = \left\{ \begin{array}{ll} \frac{x^2-x-2}{x-2} &amp; x \neq 2\\ 3 &amp; x = 2 \end{array} \right.[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_diag-tab2-problem2_5_1[]" id="input_diag-tab2-problem2_5_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="diag-tab2-problem2_5_1-choice_3-label" for="input_diag-tab2-problem2_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem2_5_1">
<text>[mathjaxinline]\displaystyle f(x) = \left\{ \begin{array}{ll} x^3-1 &amp; x &gt; 2\\ -x^2 &amp; x \leq 2 \end{array} \right.[/mathjaxinline]</text>
</label>
</div>
<span id="answer_diag-tab2-problem2_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diag-tab2-problem2_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><b class="bf">D5 &#8195;</b> What is the average rate of change of the function [mathjaxinline]\ f(x) = x^4 - 5x[/mathjaxinline] between [mathjaxinline]x=0[/mathjaxinline] and [mathjaxinline]x=3[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_diag-tab2-problem2_6_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_diag-tab2-problem2_6_1" id="input_diag-tab2-problem2_6_1" aria-describedby="status_diag-tab2-problem2_6_1" value=""/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem2_6_1"/>
<span class="status unanswered" id="status_diag-tab2-problem2_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_diag-tab2-problem2_6_1" class="answer"/>
</div>
</div></div>
</p>
<p><b class="bf">D6 &#8195;</b> If [mathjaxinline]f[/mathjaxinline] is differentiable at [mathjaxinline]x=a[/mathjaxinline], which of the following must be true? </p>
<p>
Choose all of the following that must be true. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab2-problem2_7_1">
<fieldset aria-describedby="status_diag-tab2-problem2_7_1">
<div class="field">
<input type="checkbox" name="input_diag-tab2-problem2_7_1[]" id="input_diag-tab2-problem2_7_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="diag-tab2-problem2_7_1-choice_0-label" for="input_diag-tab2-problem2_7_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem2_7_1">
<text>[mathjaxinline]f[/mathjaxinline] is continuous at [mathjaxinline]x=a[/mathjaxinline].</text>
</label>
</div>
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<text>[mathjaxinline]\displaystyle \lim _{x\rightarrow a} f(x)[/mathjaxinline] exists.</text>
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<text>[mathjaxinline]\displaystyle \lim _{x\rightarrow a} \frac{f(x) - f(a)}{x-a}[/mathjaxinline] exists.</text>
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<text>[mathjaxinline]f'(a)[/mathjaxinline] is defined.</text>
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<text>[mathjaxinline]f^{\prime \prime }(a)[/mathjaxinline] is defined.</text>
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<p><b class="bf">D7 &#8195;</b> Identify the left-hand limit [mathjaxinline]\ \displaystyle \lim _{x \rightarrow 1^{-}} f(x)[/mathjaxinline] based on the graph of [mathjaxinline]f(x)[/mathjaxinline] shown below. </p>
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<img src="/assets/courseware/v1/65e1e31228eaba2bb5af941e94ee387b/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_diag0-img1.svg" style="margin: 5px 5px 5px 5px; border:0px" width="350px px"/>
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<text> 2</text>
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<text> 1</text>
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<text> 0</text>
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<text> -1.5</text>
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<text> Does not exist.</text>
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<p><b class="bf">D8 &#8195;</b> Let [mathjaxinline]\ \displaystyle f(t) = \frac{\ln (5t+1)}{\sqrt {t+1}}[/mathjaxinline]. What is [mathjaxinline]\ f'(0)[/mathjaxinline]? </p>
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<p style="display:inline">[mathjaxinline]f'(0)=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">3. Integration diagnostics</h2>
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Integration Diagnostic
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<p><b class="bf">I1</b> Which of the following statements about the function [mathjaxinline]\ f(x) = x^4-2x^3[/mathjaxinline] is true? </p>
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<text> The function has no relative extremum.</text>
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<text> The graph of the function has one point of inflection and the function has two relative extrema.</text>
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<text> The graph of the function has two points of inflection and the function has one relative extremum.</text>
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<text> The graph of the function has two points of inflection and the function has two relative extremum.</text>
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<text> The graph of the function has two points of inflection and the function has three relative extrema.</text>
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<p><b class="bf">I2 &#8195;</b> Compute the indefinite integral [mathjaxinline]\displaystyle \int (x-1)\sqrt {x} dx[/mathjaxinline]. </p>
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<text> [mathjaxinline]\displaystyle \frac32\sqrt {x}-\frac{1}{\sqrt {x}} + C[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac12 x^2-x + C[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac23 x^{3/2} + \frac12 x^{1/2} + C[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_3_1" id="input_diag-tab3-problem2_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab3-problem2_3_1-choice_4-label" for="input_diag-tab3-problem2_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_3_1">
<text> [mathjaxinline]\displaystyle \frac25 x^{5/2} - \frac23 x^{3/2} + C[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_3_1" id="input_diag-tab3-problem2_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab3-problem2_3_1-choice_5-label" for="input_diag-tab3-problem2_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_3_1">
<text> [mathjaxinline]\displaystyle \frac12 x^{2} + 2x^{3/2} -x + C[/mathjaxinline]</text>
</label>
</div>
<span id="answer_diag-tab3-problem2_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diag-tab3-problem2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><b class="bf">I3 &#8195;</b> The figure below shows the region bounded by the graphs of [mathjaxinline]y=4x-2x^2[/mathjaxinline] and the line [mathjaxinline]y=x[/mathjaxinline]. What is the area of the shaded region? </p>
<center>
<img src="/assets/courseware/v1/620a04dadce76ea7f84fa115171d04b5/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_diag_i1.svg" style="margin: 5px 5px 5px 5px; border:0px" width="300 px"/>
</center>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_diag-tab3-problem2_4_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_diag-tab3-problem2_4_1" id="input_diag-tab3-problem2_4_1" aria-describedby="status_diag-tab3-problem2_4_1" value=""/>
<span class="trailing_text" id="trailing_text_diag-tab3-problem2_4_1"/>
<span class="status unanswered" id="status_diag-tab3-problem2_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_diag-tab3-problem2_4_1" class="answer"/>
</div>
</div></div>
</p>
<p><b class="bf">I4 &#8195;</b> If [mathjaxinline]\displaystyle y = 5+\int _2^{2x} e^{-t^2}dt[/mathjaxinline], which of the following is true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab3-problem2_5_1">
<fieldset aria-describedby="status_diag-tab3-problem2_5_1">
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_5_1" id="input_diag-tab3-problem2_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab3-problem2_5_1-choice_1-label" for="input_diag-tab3-problem2_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_5_1">
<text> [mathjaxinline]\displaystyle \frac{dy}{dx} = e^{-x^2}[/mathjaxinline] and [mathjaxinline]y(0) = 5[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_5_1" id="input_diag-tab3-problem2_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab3-problem2_5_1-choice_2-label" for="input_diag-tab3-problem2_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_5_1">
<text> [mathjaxinline]\displaystyle \frac{dy}{dx} = e^{-x^2}[/mathjaxinline] and [mathjaxinline]y(1) = 5[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_5_1" id="input_diag-tab3-problem2_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab3-problem2_5_1-choice_3-label" for="input_diag-tab3-problem2_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_5_1">
<text> [mathjaxinline]\displaystyle \frac{dy}{dx} = e^{-4x^2}[/mathjaxinline] and [mathjaxinline]y(1) = 5[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_5_1" id="input_diag-tab3-problem2_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab3-problem2_5_1-choice_4-label" for="input_diag-tab3-problem2_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_5_1">
<text> [mathjaxinline]\displaystyle \frac{dy}{dx} = 2e^{-4x^2}[/mathjaxinline] and [mathjaxinline]y(0) = 5[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_5_1" id="input_diag-tab3-problem2_5_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab3-problem2_5_1-choice_5-label" for="input_diag-tab3-problem2_5_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_5_1">
<text> [mathjaxinline]\displaystyle \frac{dy}{dx} = 2e^{-4x^2}[/mathjaxinline] and [mathjaxinline]y(1) = 5[/mathjaxinline]</text>
</label>
</div>
<span id="answer_diag-tab3-problem2_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diag-tab3-problem2_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><b class="bf">I5 &#8195;</b> Which of the following is a slope field for the differential equation [mathjaxinline]\displaystyle \frac{dy}{dx} = \frac{x}{y}[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab3-problem2_6_1">
<fieldset aria-describedby="status_diag-tab3-problem2_6_1">
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_6_1" id="input_diag-tab3-problem2_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab3-problem2_6_1-choice_1-label" for="input_diag-tab3-problem2_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_6_1">
<text>
<img src="/assets/courseware/v1/4cf26650e7fe9f4dc787a8d2f51298b0/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3_slopeplot5.svg" style="margin: 5px 5px 5px 5px; border:0px" width="375 px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_6_1" id="input_diag-tab3-problem2_6_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab3-problem2_6_1-choice_2-label" for="input_diag-tab3-problem2_6_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_6_1">
<text>
<img src="/assets/courseware/v1/9d215bd61355412079eda96f2a6a313e/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3_slopeplot1.svg" style="margin: 5px 5px 5px 5px; border:0px" width="375 px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_6_1" id="input_diag-tab3-problem2_6_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab3-problem2_6_1-choice_3-label" for="input_diag-tab3-problem2_6_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_6_1">
<text>
<img src="/assets/courseware/v1/1916b8f6a2dbd497bfb552405713dac0/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3_slopeplot3.svg" style="margin: 5px 5px 5px 5px; border:0px" width="375 px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_6_1" id="input_diag-tab3-problem2_6_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab3-problem2_6_1-choice_4-label" for="input_diag-tab3-problem2_6_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_6_1">
<text>
<img src="/assets/courseware/v1/87202337af75a114e69211c5b66c7041/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3_slopeplot6.svg" style="margin: 5px 5px 5px 5px; border:0px" width="375 px"/>
</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_6_1" id="input_diag-tab3-problem2_6_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab3-problem2_6_1-choice_5-label" for="input_diag-tab3-problem2_6_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_6_1">
<text>
<img src="/assets/courseware/v1/c420675df37ca619afbb4cdcd95b2996/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_u5s3_slopeplot7.svg" style="margin: 5px 5px 5px 5px; border:0px" width="375 px"/>
</text>
</label>
</div>
<span id="answer_diag-tab3-problem2_6_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diag-tab3-problem2_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><b class="bf">I6 &#8195;</b> A particle travels along a straight line with velocity of [mathjaxinline]v(t) = \cos (\pi t)[/mathjaxinline] meters per second. What is the total displacement, in meters, traveled by the particle during the time interval [mathjaxinline]0 \leq t \leq 2[/mathjaxinline] seconds? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_diag-tab3-problem2_7_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_diag-tab3-problem2_7_1" id="input_diag-tab3-problem2_7_1" aria-describedby="status_diag-tab3-problem2_7_1" value=""/>
<span class="trailing_text" id="trailing_text_diag-tab3-problem2_7_1"/>
<span class="status unanswered" id="status_diag-tab3-problem2_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_diag-tab3-problem2_7_1" class="answer"/>
</div>
</div></div>
</p>
<p><b class="bf">I7 &#8195;</b> The graph of the function [mathjaxinline]\ f[/mathjaxinline] shown below consists of three line segments. If [mathjaxinline]\ g[/mathjaxinline] is the function defined by [mathjaxinline]\displaystyle g(x) = \int _0^ x f(t) dt[/mathjaxinline], then what is [mathjaxinline]g(-1)[/mathjaxinline]? </p>
<center>
<img src="/assets/courseware/v1/a79946a1fd69d316572fee91af92e40c/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_diag_i2.svg" style="margin: 5px 5px 5px 5px; border:0px" width="300 px"/>
</center>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 7" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab3-problem2_8_1">
<fieldset aria-describedby="status_diag-tab3-problem2_8_1">
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_8_1" id="input_diag-tab3-problem2_8_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab3-problem2_8_1-choice_1-label" for="input_diag-tab3-problem2_8_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_8_1">
<text> -2</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_8_1" id="input_diag-tab3-problem2_8_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab3-problem2_8_1-choice_2-label" for="input_diag-tab3-problem2_8_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_8_1">
<text> -1</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_8_1" id="input_diag-tab3-problem2_8_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab3-problem2_8_1-choice_3-label" for="input_diag-tab3-problem2_8_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_8_1">
<text> 0</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_8_1" id="input_diag-tab3-problem2_8_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab3-problem2_8_1-choice_4-label" for="input_diag-tab3-problem2_8_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_8_1">
<text> 1</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_8_1" id="input_diag-tab3-problem2_8_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab3-problem2_8_1-choice_5-label" for="input_diag-tab3-problem2_8_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_8_1">
<text> 2</text>
</label>
</div>
<span id="answer_diag-tab3-problem2_8_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diag-tab3-problem2_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><b class="bf">I8 &#8195;</b> Which of the functions pictured below has a nonzero average value over the closed interval [mathjaxinline][-1, 1][/mathjaxinline]? (Choose all with nonzero average value.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 8" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab3-problem2_9_1">
<fieldset aria-describedby="status_diag-tab3-problem2_9_1">
<div class="field">
<input type="checkbox" name="input_diag-tab3-problem2_9_1[]" id="input_diag-tab3-problem2_9_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="diag-tab3-problem2_9_1-choice_0-label" for="input_diag-tab3-problem2_9_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_9_1">
<text>
<img src="/assets/courseware/v1/21deaecfa25be7d891afe3a32474f0f3/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_diag_i3A.svg" style="margin: 5px 5px 5px 5px; border:0px" width="300 px"/>
</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_diag-tab3-problem2_9_1[]" id="input_diag-tab3-problem2_9_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="diag-tab3-problem2_9_1-choice_1-label" for="input_diag-tab3-problem2_9_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_9_1">
<text>
<img src="/assets/courseware/v1/b0e7c55ca92ec68ffd5f70a8870cb83f/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_diag_i3B.svg" style="margin: 5px 5px 5px 5px; border:0px" width="300 px"/>
</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_diag-tab3-problem2_9_1[]" id="input_diag-tab3-problem2_9_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="diag-tab3-problem2_9_1-choice_2-label" for="input_diag-tab3-problem2_9_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_9_1">
<text>
<img src="/assets/courseware/v1/f9a142c5ea093f6386adbb85305e8a3b/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_diag_i3C.svg" style="margin: 5px 5px 5px 5px; border:0px" width="300 px"/>
</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_diag-tab3-problem2_9_1[]" id="input_diag-tab3-problem2_9_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="diag-tab3-problem2_9_1-choice_3-label" for="input_diag-tab3-problem2_9_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_9_1">
<text>
<img src="/assets/courseware/v1/fc736d4504acbfdd6a98e6d3c019d95d/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_diag_i3D.svg" style="margin: 5px 5px 5px 5px; border:0px" width="300 px"/>
</text>
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<input type="checkbox" name="input_diag-tab3-problem2_9_1[]" id="input_diag-tab3-problem2_9_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="diag-tab3-problem2_9_1-choice_4-label" for="input_diag-tab3-problem2_9_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_9_1">
<text>
<img src="/assets/courseware/v1/22d10ee4b3269fb768ff1b7f672db0d9/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_diag_i3E.svg" style="margin: 5px 5px 5px 5px; border:0px" width="300 px"/>
</text>
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<span id="answer_diag-tab3-problem2_9_1"/>
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