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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
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab2-problem1_2_1">
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<input type="radio" name="input_diag-tab2-problem1_2_1" id="input_diag-tab2-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab2-problem1_2_1-choice_1-label" for="input_diag-tab2-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_2_1">
<text> I have seen derivatives before only.</text>
</label>
</div>
<div class="field">
<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 seen integrals before only.</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 have seen derivatives and integrals before.</text>
</label>
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<input type="radio" name="input_diag-tab2-problem1_2_1" id="input_diag-tab2-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab2-problem1_2_1-choice_4-label" for="input_diag-tab2-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_2_1">
<text> I have not taken any calculus before.</text>
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<p>
If you get 0.7 or above, you have a good handle the calculus concepts needed to succeed in this course. </p><p>
If you get between 0.45 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">T1 &#8195;</b> To estimate the height of a skyscraper 1km in the distance, Jenny finds that if her friend Steve stands 2.5 meters away, the top of his head just lines up with the top of the building. Steve is 2 meters tall, and Jenny's eye is 1.5 meters from the ground. How high is the building? </p>
<p>
(The dotted lines may help you.) </p>
<center>
<img src="/assets/courseware/v1/ee463ec0fd40fd3c5ccd25b8b7d1fcc6/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_diagnostic_G3.svg" style="margin: 5px 5px 5px 5px; border:0px" width="400 px"/>
</center>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_diag-tab2-problem2_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_diag-tab2-problem2_2_1" id="input_diag-tab2-problem2_2_1" aria-describedby="status_diag-tab2-problem2_2_1" value=""/>
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<p><b class="bf">T2 &#8195;</b> In the given right triangle, what is [mathjaxinline]\tan y[/mathjaxinline]? </p>
<center>
<img src="/assets/courseware/v1/ed707564534c3e5551a544ec87599351/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_diagnostic_T0.svg" style="margin: 5px 5px 5px 5px; border:0px" width="250 px"/>
</center>
<p>
(Enter as a fraction or a decimal to 2 decimal places.) </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">
<div class="unanswered">
<input type="text" name="input_diag-tab2-problem2_3_1" id="input_diag-tab2-problem2_3_1" data-input-id="diag-tab2-problem2_3_1" value="" aria-describedby="status_diag-tab2-problem2_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem2_3_1"/>
<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"/>
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<p id="answer_diag-tab2-problem2_3_1" class="answer"/>
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\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
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</p>
<p><b class="bf">T3 &#8195;</b> By using the trigonometric identity [mathjaxinline]\cos 2x = \cos ^2 x - \sin ^2 x[/mathjaxinline], and other identities, find the <b class="bf">positive</b> expression for [mathjaxinline]\sin (A/2)[/mathjaxinline] in terms of [mathjaxinline]\cos (A)[/mathjaxinline]. </p>
<p>
(Type sqrt( ) for [mathjaxinline]\sqrt { \qquad }[/mathjaxinline]; type cos(A) for [mathjaxinline]\cos A[/mathjaxinline]. Type [mathjaxinline]*[/mathjaxinline] to denote multiplication, e.g. 2*cos(A). Type [mathjaxinline]\wedge[/mathjaxinline] for exponents, e.g. (1+cos(A))[mathjaxinline]\wedge[/mathjaxinline](1/2) = [mathjaxinline]\displaystyle \left(1 + \cos A\right)^{\frac12}.[/mathjaxinline]) </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">
<div class="unanswered">
<input type="text" name="input_diag-tab2-problem2_4_1" id="input_diag-tab2-problem2_4_1" data-input-id="diag-tab2-problem2_4_1" value="" aria-describedby="status_diag-tab2-problem2_4_1" size="20"/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem2_4_1"/>
<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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<div id="input_diag-tab2-problem2_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p><b class="bf">T4 &#8195;</b> Find the smallest positive solution to the equation [mathjaxinline]\ \displaystyle \sin 2x = \frac12[/mathjaxinline]; here [mathjaxinline]x[/mathjaxinline] is in radians. </p>
<p>
(Enter as a fraction or a decimal to 2 places. If needed, you can use pi for [mathjaxinline]\pi[/mathjaxinline], and sqrt(5) for [mathjaxinline]\sqrt {5}[/mathjaxinline]. You can enter fractions using the forward slash / ; e.g. pi/2 for [mathjaxinline]\frac{\pi }{2}[/mathjaxinline]. ) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div id="formulaequationinput_diag-tab2-problem2_5_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_diag-tab2-problem2_5_1" id="input_diag-tab2-problem2_5_1" data-input-id="diag-tab2-problem2_5_1" value="" aria-describedby="status_diag-tab2-problem2_5_1" size="20"/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem2_5_1"/>
<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>
<p id="answer_diag-tab2-problem2_5_1" class="answer"/>
<div id="input_diag-tab2-problem2_5_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p><b class="bf">T5 &#8195;</b> What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow 3} \frac{6/x - 2}{3-4x + x^2} }[/mathjaxinline]? </p>
<p>
(Enter the answer as a fraction or as a decimal up to 2 places. If the limit does not exist, enter DNE.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div id="formulaequationinput_diag-tab2-problem2_6_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_diag-tab2-problem2_6_1" id="input_diag-tab2-problem2_6_1" data-input-id="diag-tab2-problem2_6_1" value="" aria-describedby="status_diag-tab2-problem2_6_1" size="20"/>
<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 id="input_diag-tab2-problem2_6_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p><b class="bf">T6 &#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 as a fraction or a decimal up to 2 places. If the limit does not exist, enter DNE.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div id="formulaequationinput_diag-tab2-problem2_7_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_diag-tab2-problem2_7_1" id="input_diag-tab2-problem2_7_1" data-input-id="diag-tab2-problem2_7_1" value="" aria-describedby="status_diag-tab2-problem2_7_1" size="20"/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem2_7_1"/>
<span class="status unanswered" id="status_diag-tab2-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-tab2-problem2_7_1" class="answer"/>
<div id="input_diag-tab2-problem2_7_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p><b class="bf">T7 &#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>
(Enter as a fraction or a decimal up to 2 places.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 7" role="group"><div id="inputtype_diag-tab2-problem2_8_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_diag-tab2-problem2_8_1" id="input_diag-tab2-problem2_8_1" aria-describedby="status_diag-tab2-problem2_8_1" value=""/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem2_8_1"/>
<span class="status unanswered" id="status_diag-tab2-problem2_8_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_8_1" class="answer"/>
</div>
</div></div>
</p>
<p><b class="bf">T8 &#8195;</b> Which of the following statements about the function [mathjaxinline]\ \displaystyle f(x) = e^{x} - e^{-x}[/mathjaxinline] is true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 8" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab2-problem2_9_1">
<fieldset aria-describedby="status_diag-tab2-problem2_9_1">
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<input type="radio" name="input_diag-tab2-problem2_9_1" id="input_diag-tab2-problem2_9_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab2-problem2_9_1-choice_1-label" for="input_diag-tab2-problem2_9_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem2_9_1">
<text> The function has no relative extremum.</text>
</label>
</div>
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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">T9 &#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>
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<img src="/assets/courseware/v1/620a04dadce76ea7f84fa115171d04b5/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_diag_i1.svg" style="margin: 5px 5px 5px 5px; border:0px" width="300 px"/>
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(Enter as a fraction or a decimal up to 2 places.) </p>
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<p><b class="bf">T10 &#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>
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<text> 1</text>
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<h2 class="hd hd-2 unit-title">3. Prior</h2>
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Prior Experience
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I have seen parametric curves before. </p>
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I have seen polar curves before. </p>
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I have seen infinite series and Taylor series before. </p>
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I have not studied any of this content before. </p>
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(1) &#8195;What are all of the values of [mathjaxinline]x[/mathjaxinline] for which the series [mathjaxinline]\displaystyle \sum _{n=1}^{\infty } \frac{n4^ n}{x^ n}[/mathjaxinline] converges? </p>
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<input type="radio" name="input_diag-tab3-problem2_2_1" id="input_diag-tab3-problem2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab3-problem2_2_1-choice_4-label" for="input_diag-tab3-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_2_1">
<text> [mathjaxinline]|x|&gt;4[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_2_1" id="input_diag-tab3-problem2_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab3-problem2_2_1-choice_5-label" for="input_diag-tab3-problem2_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_2_1">
<text> The series diverges for all [mathjaxinline]x[/mathjaxinline]</text>
</label>
</div>
<span id="answer_diag-tab3-problem2_2_1"/>
</fieldset>
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<span class="status unanswered" id="status_diag-tab3-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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</div></div>
</p>
<p>
(2) &#8195;Which of the following series converge to [mathjaxinline]2[/mathjaxinline]? </p>
<ol class="enumerate">
<li value="1">
<p>
[mathjaxinline]\displaystyle \sum _{n=1}^{\infty } \frac{2n}{n+3}[/mathjaxinline] </p>
</li>
<li value="2">
<p>
[mathjaxinline]\displaystyle \sum _{n=1}^{\infty } \frac{-8}{(-3)^ n}[/mathjaxinline] </p>
</li>
<li value="3">
<p>
[mathjaxinline]\displaystyle \sum _{n=0}^{\infty } \frac{1}{2^ n}[/mathjaxinline] </p>
</li>
</ol>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab3-problem2_3_1">
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<input type="radio" name="input_diag-tab3-problem2_3_1" id="input_diag-tab3-problem2_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab3-problem2_3_1-choice_1-label" for="input_diag-tab3-problem2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_3_1">
<text> 1 only</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_3_1" id="input_diag-tab3-problem2_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab3-problem2_3_1-choice_2-label" for="input_diag-tab3-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_3_1">
<text> 2 only</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_3_1" id="input_diag-tab3-problem2_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab3-problem2_3_1-choice_3-label" for="input_diag-tab3-problem2_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_3_1">
<text> 3 only</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> 1 and 3 only</text>
</label>
</div>
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<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> 2 and 3 only</text>
</label>
</div>
<span id="answer_diag-tab3-problem2_3_1"/>
</fieldset>
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<span class="status unanswered" id="status_diag-tab3-problem2_3_1" data-tooltip="Not yet answered.">
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<p>
(3) &#8195;Which of the following integrals represents the area enclosed by the smaller loop of the graph of [mathjaxinline]r = 1+2\sin \theta[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab3-problem2_4_1">
<fieldset aria-describedby="status_diag-tab3-problem2_4_1">
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_4_1" id="input_diag-tab3-problem2_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab3-problem2_4_1-choice_1-label" for="input_diag-tab3-problem2_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_4_1">
<text> [mathjaxinline]\displaystyle \frac12 \int _{7\pi /6}^{11\pi /6} (1+2\sin \theta )^2 \, d\theta[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_4_1" id="input_diag-tab3-problem2_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab3-problem2_4_1-choice_2-label" for="input_diag-tab3-problem2_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_4_1">
<text> [mathjaxinline]\displaystyle \frac12 \int _{7\pi /6}^{11\pi /6} (1+2\sin \theta ) \, d\theta[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_4_1" id="input_diag-tab3-problem2_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab3-problem2_4_1-choice_3-label" for="input_diag-tab3-problem2_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_4_1">
<text> [mathjaxinline]\displaystyle \frac12 \int _{-\pi /6}^{7\pi /6} (1+2\sin \theta )^2 \, d\theta[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_4_1" id="input_diag-tab3-problem2_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab3-problem2_4_1-choice_4-label" for="input_diag-tab3-problem2_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_4_1">
<text> [mathjaxinline]\displaystyle \int _{-\pi /6}^{7\pi /6} (1+2\sin \theta )^2 \, d\theta[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_4_1" id="input_diag-tab3-problem2_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab3-problem2_4_1-choice_5-label" for="input_diag-tab3-problem2_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_4_1">
<text> [mathjaxinline]\displaystyle \int _{7\pi /6}^{-\pi /6} (1+2\sin \theta ) \, d\theta[/mathjaxinline]</text>
</label>
</div>
<span id="answer_diag-tab3-problem2_4_1"/>
</fieldset>
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<span class="status unanswered" id="status_diag-tab3-problem2_4_1" data-tooltip="Not yet answered.">
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</p>
<p>
(4) &#8195;Which of the following integrals gives the length of the curve </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000003" style="table-layout:auto" width="100%">
<tr id="a0000000004">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle 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 \sin 2t[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2)</td>
</tr>
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</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:left; border:none">
[mathjaxinline]\displaystyle \cos t[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(3)</td>
</tr>
</table>
<p>
between [mathjaxinline]t=a[/mathjaxinline] and [mathjaxinline]t=b[/mathjaxinline], where [mathjaxinline]0 &lt; a &lt; b[/mathjaxinline]? </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 \int _ a^ b \sqrt {2} \, dt[/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 \int _ a^ b \sqrt {\sin 2t + \cos t} \, dt[/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 \int _ a^ b \sqrt {\sin ^2 2t + \cos ^2 t} \, dt[/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 \int _ a^ b \sqrt {4\sin ^2 2t + \cos ^2 t} \, dt[/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 \int _ a^ b \sqrt {4\cos ^2 2t + \sin ^2 t} \, dt[/mathjaxinline]</text>
</label>
</div>
<span id="answer_diag-tab3-problem2_5_1"/>
</fieldset>
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<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"/>
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<p>
(5) &#8195;The fourth degree Taylor polynomial about [mathjaxinline]x=0[/mathjaxinline] of [mathjaxinline]\sin x[/mathjaxinline] is <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> [mathjaxinline]\displaystyle 1 + x + \frac{x^2}2 + \frac{x^3}{3} + \frac{x^4}{4}[/mathjaxinline]</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> [mathjaxinline]\displaystyle 1 + x + \frac{x^2}2 + \frac{x^3}{6} + \frac{x^4}{24}[/mathjaxinline]</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> [mathjaxinline]\displaystyle 1 + \frac{x^2}2 + \frac{x^4}{24}[/mathjaxinline]</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> [mathjaxinline]\displaystyle 1 - \frac{x^2}2 + \frac{x^4}{24}[/mathjaxinline]</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> [mathjaxinline]\displaystyle x + \frac{x^3}{6}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab3-problem2_6_1" id="input_diag-tab3-problem2_6_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="diag-tab3-problem2_6_1-choice_6-label" for="input_diag-tab3-problem2_6_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_diag-tab3-problem2_6_1"> <text> [mathjaxinline]\displaystyle x - \frac{x^3}{6}[/mathjaxinline]</text>
</label>
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<span id="answer_diag-tab3-problem2_6_1"/>
</fieldset>
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