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<h2 class="hd hd-2 unit-title">1. Choose Your Own Calculus Adventure</h2>
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<p>
You are interested in learning calculus, 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 you choose a path through the content that makes the most sense for you. </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 4 pages test your readiness for this calculus class. If you aren't ready yet, don't worry, you can take this course later. </p><p>
Some of you may already know a lot of calculus. To help you get started in the right place, we have further diagnostics. On pages 5 and 6, you can take the limits and derivatives diagnostics. We encourage you to take a look at these even if you don't know any calculus 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. Algebra Problems</h2>
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Algebra
</h3>
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<div class="problem">
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<p><b class="bf">A1 &#8195;</b> What is the slope of the line through the points [mathjaxinline](3, -5)[/mathjaxinline] and [mathjaxinline](1, -1)[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_diag-tab2-problem1_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_diag-tab2-problem1_2_1" id="input_diag-tab2-problem1_2_1" data-input-id="diag-tab2-problem1_2_1" value="" aria-describedby="status_diag-tab2-problem1_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem1_2_1"/>
<span class="status unanswered" id="status_diag-tab2-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_diag-tab2-problem1_2_1" class="answer"/>
<div id="input_diag-tab2-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><b class="bf">A2 &#8195;</b> The lines [mathjaxinline]\ 3x + 2y = 7[/mathjaxinline] and [mathjaxinline]\ x - 3y = 6[/mathjaxinline] intersect in a point with what coordinates? </p>
<p>
<p style="display:inline">[mathjaxinline]x=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_diag-tab2-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_diag-tab2-problem1_3_1" id="input_diag-tab2-problem1_3_1" data-input-id="diag-tab2-problem1_3_1" value="" aria-describedby="status_diag-tab2-problem1_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem1_3_1"/>
<span class="status unanswered" id="status_diag-tab2-problem1_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-problem1_3_1" class="answer"/>
<div id="input_diag-tab2-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 style="display:inline">[mathjaxinline]y=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_diag-tab2-problem1_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_diag-tab2-problem1_4_1" id="input_diag-tab2-problem1_4_1" data-input-id="diag-tab2-problem1_4_1" value="" aria-describedby="status_diag-tab2-problem1_4_1" size="20"/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem1_4_1"/>
<span class="status unanswered" id="status_diag-tab2-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_diag-tab2-problem1_4_1" class="answer"/>
<div id="input_diag-tab2-problem1_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
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</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
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</p>
<p><b class="bf">A3 &#8195;</b> Which of the following expressions is equivalent to [mathjaxinline]\displaystyle {\left(\frac{1}{x} + \frac{1}{y}\right)^{-1}}[/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-problem1_5_1">
<fieldset aria-describedby="status_diag-tab2-problem1_5_1">
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_5_1" id="input_diag-tab2-problem1_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab2-problem1_5_1-choice_1-label" for="input_diag-tab2-problem1_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_5_1">
<text> [mathjaxinline]\displaystyle x-y[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_5_1" id="input_diag-tab2-problem1_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab2-problem1_5_1-choice_2-label" for="input_diag-tab2-problem1_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_5_1">
<text> [mathjaxinline]\displaystyle x+y[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_5_1" id="input_diag-tab2-problem1_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab2-problem1_5_1-choice_3-label" for="input_diag-tab2-problem1_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_5_1">
<text> [mathjaxinline]\displaystyle x^2+y^2[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_5_1" id="input_diag-tab2-problem1_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab2-problem1_5_1-choice_4-label" for="input_diag-tab2-problem1_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_5_1">
<text> [mathjaxinline]\displaystyle \frac{1}{x}+\frac{1}{y}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_5_1" id="input_diag-tab2-problem1_5_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab2-problem1_5_1-choice_5-label" for="input_diag-tab2-problem1_5_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_5_1">
<text> [mathjaxinline]\displaystyle \frac{1}{x^2}+\frac{1}{y^2}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_5_1" id="input_diag-tab2-problem1_5_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="diag-tab2-problem1_5_1-choice_6-label" for="input_diag-tab2-problem1_5_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_5_1">
<text> [mathjaxinline]\displaystyle \frac{xy}{x+y}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_5_1" id="input_diag-tab2-problem1_5_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="diag-tab2-problem1_5_1-choice_7-label" for="input_diag-tab2-problem1_5_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_5_1">
<text> [mathjaxinline]\displaystyle \frac{x+y}{xy}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_5_1" id="input_diag-tab2-problem1_5_1_choice_8" class="field-input input-radio" value="choice_8"/><label id="diag-tab2-problem1_5_1-choice_8-label" for="input_diag-tab2-problem1_5_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_5_1">
<text> None of the above</text>
</label>
</div>
<span id="answer_diag-tab2-problem1_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diag-tab2-problem1_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
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</p>
<p><b class="bf">A4 &#8195;</b> List all possible solutions to the equation [mathjaxinline]x^3 - x^2 - 2x = 0[/mathjaxinline]. </p>
<p>
(Use decimals only, not fractions, and separate answers with commas.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_diag-tab2-problem1_6_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_diag-tab2-problem1_6_1" id="input_diag-tab2-problem1_6_1" aria-describedby="status_diag-tab2-problem1_6_1" value=""/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem1_6_1"/>
<span class="status unanswered" id="status_diag-tab2-problem1_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-problem1_6_1" class="answer"/>
</div>
</div></div>
</p>
<p><b class="bf">A5 &#8195;</b> A [mathjaxinline]0.25[/mathjaxinline] mL sample of water drawn from a [mathjaxinline]5[/mathjaxinline] liter flask contains [mathjaxinline]1.25\times 10^8[/mathjaxinline] bacteria. Give the approximate number of bacteria in the flask, expressing your answer in scientific notation. </p>
<p>
(Scientific Notation: find a real number [mathjaxinline]a[/mathjaxinline] between 1 and 10, and an integer [mathjaxinline]n[/mathjaxinline], such that [mathjaxinline]x = a \times 10^{n}[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]a=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="formulaequationinput_diag-tab2-problem1_7_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_diag-tab2-problem1_7_1" id="input_diag-tab2-problem1_7_1" data-input-id="diag-tab2-problem1_7_1" value="" aria-describedby="status_diag-tab2-problem1_7_1" size="20"/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem1_7_1"/>
<span class="status unanswered" id="status_diag-tab2-problem1_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-problem1_7_1" class="answer"/>
<div id="input_diag-tab2-problem1_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 style="display:inline">[mathjaxinline]n=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 7" role="group"><div id="formulaequationinput_diag-tab2-problem1_8_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_diag-tab2-problem1_8_1" id="input_diag-tab2-problem1_8_1" data-input-id="diag-tab2-problem1_8_1" value="" aria-describedby="status_diag-tab2-problem1_8_1" size="20"/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem1_8_1"/>
<span class="status unanswered" id="status_diag-tab2-problem1_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-problem1_8_1" class="answer"/>
<div id="input_diag-tab2-problem1_8_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">A6 &#8195;</b> For what value of the constant [mathjaxinline]a[/mathjaxinline] will the system of linear equations </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 6x -5 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 3[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.2)</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 3x+ay[/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 1[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.3)</td>
</tr>
</table>
<p>
have no solution? </p>
<p>
<p style="display:inline">[mathjaxinline]a=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 8" role="group"><div id="formulaequationinput_diag-tab2-problem1_9_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_diag-tab2-problem1_9_1" id="input_diag-tab2-problem1_9_1" data-input-id="diag-tab2-problem1_9_1" value="" aria-describedby="status_diag-tab2-problem1_9_1" size="20"/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem1_9_1"/>
<span class="status unanswered" id="status_diag-tab2-problem1_9_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_diag-tab2-problem1_9_1" class="answer"/>
<div id="input_diag-tab2-problem1_9_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">A7 &#8195;</b> Find the value of the constant [mathjaxinline]a[/mathjaxinline] for which the polynomial [mathjaxinline]x^3+ax^2-1[/mathjaxinline] will have [mathjaxinline]-1[/mathjaxinline] as a zero. </p>
<p>
<p style="display:inline">[mathjaxinline]a=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 9" role="group"><div id="inputtype_diag-tab2-problem1_10_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_diag-tab2-problem1_10_1" id="input_diag-tab2-problem1_10_1" aria-describedby="status_diag-tab2-problem1_10_1" value=""/>
<span class="trailing_text" id="trailing_text_diag-tab2-problem1_10_1"/>
<span class="status unanswered" id="status_diag-tab2-problem1_10_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_diag-tab2-problem1_10_1" class="answer"/>
</div>
</div></div>
<br/>
</p>
<p><b class="bf">A8 &#8195;</b> If [mathjaxinline]\displaystyle a_ n = \frac{x^ n}{2^ nn!}[/mathjaxinline], find [mathjaxinline]\displaystyle \frac{a_{n+1}}{a_ n}[/mathjaxinline]. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 10" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab2-problem1_11_1">
<fieldset aria-describedby="status_diag-tab2-problem1_11_1">
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_11_1" id="input_diag-tab2-problem1_11_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab2-problem1_11_1-choice_1-label" for="input_diag-tab2-problem1_11_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_11_1">
<text> [mathjaxinline]\displaystyle \frac{x^{n+1}}{2^{n+1}(n+1)}![/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_11_1" id="input_diag-tab2-problem1_11_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab2-problem1_11_1-choice_2-label" for="input_diag-tab2-problem1_11_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_11_1">
<text> [mathjaxinline]\displaystyle \frac{nx^{n}}{2^{n}(n+1)}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_11_1" id="input_diag-tab2-problem1_11_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab2-problem1_11_1-choice_3-label" for="input_diag-tab2-problem1_11_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_11_1">
<text> [mathjaxinline]\displaystyle \frac{nx}{2^{n}(n+1)}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab2-problem1_11_1" id="input_diag-tab2-problem1_11_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab2-problem1_11_1-choice_4-label" for="input_diag-tab2-problem1_11_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab2-problem1_11_1">
<text> [mathjaxinline]\displaystyle \frac{x}{2^ n(n+1)}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
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<text> [mathjaxinline]\displaystyle \frac{x}{2(n+1)}[/mathjaxinline]</text>
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If you score 0.6 or above, you have a good grasp of algebraic manipulations, and can do them accurately enough to succeed in this class! </p><p>
Otherwise, this course will be very difficult for you. We recommend taking an algebra/trigonometry class to solidify your familiarity and accuracy before attempting this course. </p>
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<h2 class="hd hd-2 unit-title">3. Geometry</h2>
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Geometry
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<p><b class="bf">G1 &#8195;</b> A bed that is [mathjaxinline]4[/mathjaxinline] feet wide must enter through a door (that is just over [mathjaxinline]4[/mathjaxinline] feet wide) along the [mathjaxinline]8[/mathjaxinline] foot wall of a [mathjaxinline]8[/mathjaxinline] by [mathjaxinline]20[/mathjaxinline] foot room. What is the largest length of a bed that can be rotated to fit with longest side along the [mathjaxinline]8[/mathjaxinline] foot long wall opposite the door. </p>
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(Note that the bed must be rotated [mathjaxinline]90[/mathjaxinline] degrees to fit, and the diagonal length of the bed must be small enough to undergo the rotation.) </p>
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<img alt="See text above" src="/assets/courseware/v1/d2c4566eeb57b190cc17be6169f4a420/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_diagnostic_G1.svg" style="margin: 10px 25px 25px 25px" width="300px"/>
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(Picture is not to scale. Enter your answer as a decimal number to 2 decimal places.) </p>
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<p><b class="bf">G2 &#8195;</b> The four-sided solid shown is the part of the solid sphere (of radius 2, centered at the origin) in the first octant. Find its total surface area. </p>
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<p>
(Type <b class="bf">pi</b> for [mathjaxinline]\pi[/mathjaxinline]. Use [mathjaxinline]*[/mathjaxinline] to denote multiplication; e.g. type 2*pi for [mathjaxinline]2\pi[/mathjaxinline]. Use [mathjaxinline]\wedge[/mathjaxinline] for powers; e.g. type pi[mathjaxinline]\wedge[/mathjaxinline]2 for [mathjaxinline]\pi ^2[/mathjaxinline].) </p>
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<p><b class="bf">G3 &#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>
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<h2 class="hd hd-2 unit-title">4. Trigonometry</h2>
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Trigonometry
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<p><b class="bf">T1 &#8195;</b> In the given right triangle, what is [mathjaxinline]\tan y[/mathjaxinline]? </p>
<center>
<img alt="The angle y in a right triangle has adjacent side of length 3 and hypotenuse of length 5" src="/assets/courseware/v1/ed707564534c3e5551a544ec87599351/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_diagnostic_T0.svg" style="margin: 10px 25px 25px 25px" width="250px"/>
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(Enter as decimal to 2 decimal places.) </p>
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<p><b class="bf">T2 &#8195;</b> A horse runs counterclockwise (anticlockwise) around the circular track of radius [mathjaxinline]400[/mathjaxinline]m at a constant speed, starting at the marked point. It completes one lap in three minutes. What is its [mathjaxinline]y[/mathjaxinline] coordinate after one minute? </p>
<p>
(If needed, you can use pi for [mathjaxinline]\pi[/mathjaxinline], and sqrt(5) for [mathjaxinline]\sqrt {5}[/mathjaxinline].) </p>
<center>
<img alt="A circle centered at the origin with marked point where the circle intersects the positive y axis." src="/assets/courseware/v1/60164cf65b8af5ee3580840a6f0759d1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_diag_racetrack.svg" style="margin: 10px 25px 25px 25px" width="200pxpx"/>
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<p><b class="bf">T3 &#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>
(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>
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<p><b class="bf">T4 &#8195;</b> A line with slope 1/2 makes an acute angle [mathjaxinline]\theta[/mathjaxinline] with the [mathjaxinline]x[/mathjaxinline] axis. What is [mathjaxinline]\sin \theta[/mathjaxinline]? </p>
<p>
(If needed, you can use pi for [mathjaxinline]\pi[/mathjaxinline], and sqrt(5) for [mathjaxinline]\sqrt {5}[/mathjaxinline]. You may enter your answer as a decimal number.) </p>
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<p><b class="bf">T5 &#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>
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<p><b class="bf">T6 &#8195;</b> The graph below represents which of these functions? </p>
<center>
<img alt="A function that oscillates between values one and negative one. It achieves maxima at x equals 0 and pi, and a minimum at pi/2." src="/assets/courseware/v1/139a72327d0b1cc84cf21a31fbb16fa0/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_diagnostic_T5.svg" style="margin: 10px 25px 25px 25px" width="300px"/>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab4-problem1_7_1">
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<text> [mathjaxinline]\displaystyle \sin x[/mathjaxinline]</text>
</label>
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<text> [mathjaxinline]\displaystyle \cos x[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \sin (x/2)[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \cos (x/2)[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \sin 2x[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \cos 2x[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">5. Logarithms and Exponentials</h2>
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Logarithms and Exponentials
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<p><b class="bf">E1 &#8195;</b> If [mathjaxinline]\log _{10} a = 4.2[/mathjaxinline] and [mathjaxinline]\log _{10} b = 0.5[/mathjaxinline], what is [mathjaxinline]\log _{10} ab[/mathjaxinline]? </p>
<p>
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<p><b class="bf">E2 &#8195;</b> If [mathjaxinline]2^ a = \frac{\sqrt {8}}{4^3}[/mathjaxinline], what is [mathjaxinline]a[/mathjaxinline]? </p>
<p>
(Enter answer as a decimal with up to 2 decimal places.) </p>
<p>
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<p><b class="bf">E3 &#8195;</b> Which of the following is equal to [mathjaxinline]\sqrt {\frac{x^{16}(1+x^{2})}{9}}[/mathjaxinline]? </p>
<p>
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<text> [mathjaxinline]\displaystyle \frac{x^4(1+x)}{3}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{x^8(1+x)}{3}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{x^4(1+x^2)^{0.5}}{3}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab5-problem1_4_1" id="input_diag-tab5-problem1_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab5-problem1_4_1-choice_4-label" for="input_diag-tab5-problem1_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab5-problem1_4_1">
<text> [mathjaxinline]\displaystyle \frac{x^8(1+x^2)^{0.5}}{3}[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_diag-tab5-problem1_4_1" id="input_diag-tab5-problem1_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab5-problem1_4_1-choice_5-label" for="input_diag-tab5-problem1_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab5-problem1_4_1">
<text> [mathjaxinline]\displaystyle \frac{x^4(1+x^2)}{3}[/mathjaxinline]</text>
</label>
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<text> [mathjaxinline]\displaystyle \frac{x^8(1+x^2)}{3}[/mathjaxinline]</text>
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<p><b class="bf">E4 &#8195;</b> Solve for [mathjaxinline]x[/mathjaxinline]: [mathjaxinline]\ \ \displaystyle \log _{10}\left[ \left(x+1 \right)^2\right] = 2.[/mathjaxinline] </p>
<p>
(Enter your answer as a list of [mathjaxinline]x-[/mathjaxinline]values, separated by commas.) </p>
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<p><b class="bf">E5 &#8195;</b> A pot of water (at sea level) is boiling; the heat is turned off at time [mathjaxinline]t=0[/mathjaxinline], and [mathjaxinline]2[/mathjaxinline] minutes later the water temperature has fallen to [mathjaxinline]80^{\circ }[/mathjaxinline]C. If the temperature [mathjaxinline]T[/mathjaxinline] (in [mathjaxinline]{}^{\circ }[/mathjaxinline] C) is expressed in terms of time [mathjaxinline]t[/mathjaxinline] (in minutes) by the law </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000006" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]T = Ae^{-kt},[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
find the values of the constants [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]k[/mathjaxinline]. </p>
<p>
(Enter answers as decimal numbers to two decimal places.) </p>
<p>
<p style="display:inline">[mathjaxinline]A=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]k=[/mathjaxinline]</p>
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If you got a 0.7 or higher, congratulations! You have an excellent understanding of logarithms and exponents! Good work. </p><p>
You can still succeed in this course if you got a 0.7 or lower, but we strongly recommend that you review logarithms and exponents before you get to the end of Unit 2: Differentiation where logarithms and exponents begin to take on a prominent role in the course. </p>
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<h2 class="hd hd-2 unit-title">6. Limits Diagnostic</h2>
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Prior Limits Experience
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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-tab6-problem1_2_1">
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<text> I have never seen limits before.</text>
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<text> I have learned about limits before.</text>
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You will <b class="bf">NOT</b> see which problems you get correct and incorrect. Sorry if this is frustrating. We will be using these problems to assess whether or not our content teaches you about limits. </p>
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Limits
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<p><b class="bf">L1 &#8195;</b> What is [mathjaxinline]\displaystyle \lim _{x \rightarrow \infty } \frac{x^2-4}{2+x-4x^2}[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab6-problem2_2_1">
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<text> [mathjaxinline]-2[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_diag-tab6-problem2_2_1" id="input_diag-tab6-problem2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab6-problem2_2_1-choice_2-label" for="input_diag-tab6-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-problem2_2_1">
<text> [mathjaxinline]\displaystyle \frac{-1}{4}[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_diag-tab6-problem2_2_1" id="input_diag-tab6-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab6-problem2_2_1-choice_3-label" for="input_diag-tab6-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-problem2_2_1">
<text> [mathjaxinline]\displaystyle \frac12[/mathjaxinline]</text>
</label>
</div>
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<input type="radio" name="input_diag-tab6-problem2_2_1" id="input_diag-tab6-problem2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab6-problem2_2_1-choice_4-label" for="input_diag-tab6-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-problem2_2_1">
<text> [mathjaxinline]1[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab6-problem2_2_1" id="input_diag-tab6-problem2_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab6-problem2_2_1-choice_5-label" for="input_diag-tab6-problem2_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-problem2_2_1">
<text> The limit does not exist.</text>
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<span id="answer_diag-tab6-problem2_2_1"/>
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<span class="status unanswered" id="status_diag-tab6-problem2_2_1" data-tooltip="Not yet answered.">
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<p><b class="bf">L2 &#8195;</b> The graph of a function [mathjaxinline]f[/mathjaxinline] is shown below. If the limit as [mathjaxinline]x\rightarrow b[/mathjaxinline] exists and [mathjaxinline]f[/mathjaxinline] is not continuous at [mathjaxinline]b[/mathjaxinline], then [mathjaxinline]b=[/mathjaxinline]? </p>
<center>
<img alt="At negative 1.1 the function has value 1.8; at negative 1 value 1.9; at negative 0.9, value 1.7. At negative 0.1 the function has value 1.01; at 0, value 2; at 0.1 value 1.1. At 0.9 the function has value 1.9; at 1, value -1.5; at -1.1 value -1.4. At 1.9 the function has value 0.8; at 2, value 1; at 2.1 value 0.8. At 3.1 the function takes value negative 500; at 2.9, value negative 200, and is undefined at 3. At 3.9 the function takes value 0.4; at 4, value 0.3; at 4.1 value 0.25. " src="/assets/courseware/v1/65e1e31228eaba2bb5af941e94ee387b/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_diag0-img1.svg" style="margin: 10px 25px 25px 25px" width="350pxpx"/>
</center>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab6-problem2_3_1">
<fieldset aria-describedby="status_diag-tab6-problem2_3_1">
<div class="field">
<input type="radio" name="input_diag-tab6-problem2_3_1" id="input_diag-tab6-problem2_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab6-problem2_3_1-choice_1-label" for="input_diag-tab6-problem2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-problem2_3_1">
<text> [mathjaxinline]-1[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab6-problem2_3_1" id="input_diag-tab6-problem2_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab6-problem2_3_1-choice_2-label" for="input_diag-tab6-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-problem2_3_1">
<text> [mathjaxinline]0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab6-problem2_3_1" id="input_diag-tab6-problem2_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab6-problem2_3_1-choice_3-label" for="input_diag-tab6-problem2_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-problem2_3_1">
<text> [mathjaxinline]1[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab6-problem2_3_1" id="input_diag-tab6-problem2_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab6-problem2_3_1-choice_4-label" for="input_diag-tab6-problem2_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-problem2_3_1">
<text> [mathjaxinline]2[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab6-problem2_3_1" id="input_diag-tab6-problem2_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab6-problem2_3_1-choice_5-label" for="input_diag-tab6-problem2_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-problem2_3_1">
<text> [mathjaxinline]3[/mathjaxinline]</text>
</label>
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<p><b class="bf">L3 &#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 3" role="group"><div id="formulaequationinput_diag-tab6-problem2_4_1" class="inputtype formulaequationinput">
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\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
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<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
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<p><b class="bf">L4 &#8195;</b> Which of the following functions have a <em>removable discontinuity</em> at [mathjaxinline]x=2[/mathjaxinline]? </p>
<p>
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<text>[mathjaxinline]\displaystyle f(x) = \frac{x^2-x-2}{x-2}[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_diag-tab6-problem2_5_1[]" id="input_diag-tab6-problem2_5_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="diag-tab6-problem2_5_1-choice_1-label" for="input_diag-tab6-problem2_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-problem2_5_1">
<text>[mathjaxinline]f(x) = \frac{1}{(x-2)^2}[/mathjaxinline]</text>
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<input type="checkbox" name="input_diag-tab6-problem2_5_1[]" id="input_diag-tab6-problem2_5_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="diag-tab6-problem2_5_1-choice_2-label" for="input_diag-tab6-problem2_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-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>
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<input type="checkbox" name="input_diag-tab6-problem2_5_1[]" id="input_diag-tab6-problem2_5_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="diag-tab6-problem2_5_1-choice_3-label" for="input_diag-tab6-problem2_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab6-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>
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<p><b class="bf">L5 &#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>
<center>
<img alt="At negative 1.1 the function has value 1.8; at negative 1 value 1.9; at negative 0.9, value 1.7. At negative 0.1 the function has value 1.01; at 0, value 2; at 0.1 value 1.1. At 0.9 the function has value 1.9; at 1, value -1.5; at -1.1 value -1.4. At 1.9 the function has value 0.8; at 2, value 1; at 2.1 value 0.8. At 3.1 the function takes value negative 500; at 2.9, value negative 200, and is undefined at 3. At 3.9 the function takes value 0.4; at 4, value 0.3; at 4.1 value 0.25. " src="/assets/courseware/v1/65e1e31228eaba2bb5af941e94ee387b/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_diag0-img1.svg" style="margin: 10px 25px 25px 25px" width="350pxpx"/>
</center>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab6-problem2_6_1">
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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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<p><b class="bf">L6 &#8195;</b> Identify the right-handed limit. </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000007" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \lim _{x \rightarrow -1^{+}} \frac{x^2-1}{\left|x+1\right|}.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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(Enter DNE if the limit does not exist.) </p>
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<ul class="itemize"><li><p>
If you got 0.65 or above, you have a good handle on limits. Move on to Unit 1. You can go back to Unit 0 at any time to fill any gaps in your understanding of limits. </p></li><li><p>
If you got between .35-.65 points, we recommend that you start by doing the in-lecture problems in Unit 0. You may be able to skip the video tutorials. </p></li><li><p>
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<h2 class="hd hd-2 unit-title">7. Derivatives Diagnostic</h2>
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Prior Calculus Experience
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<text> I have never seen derivatives before.</text>
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<text> I have learned about derivatives before.</text>
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You will not see which problems you get correct and incorrect. Sorry if this is frustrating. We will be using these problems to assess whether or not our content teaches you about derivatives. </p>
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Calculus Diagnostic
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<p><b class="bf">C1 &#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>
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<p><b class="bf">C2 &#8195;</b> At which of the five points on the graph are [mathjaxinline]\displaystyle \frac{dy}{dx}[/mathjaxinline] and [mathjaxinline]\displaystyle \frac{d^2y}{dx^2}[/mathjaxinline] both negative? </p>
<center>
<img alt="At point A the function is negative, decreasing, and concave up. At point B the function is negative, increasing, and concave up. At point C the function is positive, increasing, and concave down. At point D the function is positive, neither increasing or decreasing, and is concave down. At point E the function is positive, decreasing, and concave down. " src="/assets/courseware/v1/330421bf2c1643789c5355e6c13b3d91/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_diag1-img1.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<p>
Click all that apply. </p>
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<text>D</text>
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<text>E</text>
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</p>
<p><b class="bf">C3 &#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>
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<p><b class="bf">C4 &#8195;</b> The position of a particle moving along a line is [mathjaxinline]p(t) = 2t^3-24t^2+90t+7[/mathjaxinline] for [mathjaxinline]t \ge 0[/mathjaxinline]. For what values of [mathjaxinline]t[/mathjaxinline] is the speed of the particle increasing? </p>
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<text> [mathjaxinline]3&lt;t&lt;4[/mathjaxinline] only</text>
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<input type="radio" name="input_diag-tab7-problem2_5_1" id="input_diag-tab7-problem2_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab7-problem2_5_1-choice_4-label" for="input_diag-tab7-problem2_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_5_1">
<text> [mathjaxinline]0&lt;t&lt;3[/mathjaxinline] and [mathjaxinline]t&gt;5[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_5_1" id="input_diag-tab7-problem2_5_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab7-problem2_5_1-choice_5-label" for="input_diag-tab7-problem2_5_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_5_1">
<text> [mathjaxinline]3&lt;t&lt;4[/mathjaxinline] and [mathjaxinline]t&gt;5[/mathjaxinline]</text>
</label>
</div>
<span id="answer_diag-tab7-problem2_5_1"/>
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<span class="status unanswered" id="status_diag-tab7-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>
<p><b class="bf">C5 &#8195;</b> Evaluate the limit [mathjaxinline]\ \displaystyle \lim _{x\rightarrow \infty } \frac{\ln (x)}{x^2}.[/mathjaxinline] </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab7-problem2_6_1">
<fieldset aria-describedby="status_diag-tab7-problem2_6_1">
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_6_1" id="input_diag-tab7-problem2_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="diag-tab7-problem2_6_1-choice_1-label" for="input_diag-tab7-problem2_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_6_1">
<text> [mathjaxinline]0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_6_1" id="input_diag-tab7-problem2_6_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab7-problem2_6_1-choice_2-label" for="input_diag-tab7-problem2_6_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_6_1">
<text> [mathjaxinline]1[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_6_1" id="input_diag-tab7-problem2_6_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab7-problem2_6_1-choice_3-label" for="input_diag-tab7-problem2_6_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_6_1">
<text> [mathjaxinline]-1[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_6_1" id="input_diag-tab7-problem2_6_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab7-problem2_6_1-choice_4-label" for="input_diag-tab7-problem2_6_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_6_1">
<text> [mathjaxinline]\infty[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_6_1" id="input_diag-tab7-problem2_6_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab7-problem2_6_1-choice_5-label" for="input_diag-tab7-problem2_6_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_6_1">
<text> [mathjaxinline]-\infty[/mathjaxinline]</text>
</label>
</div>
<span id="answer_diag-tab7-problem2_6_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_diag-tab7-problem2_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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</p>
<p><b class="bf">C6 &#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-tab7-problem2_7_1">
<fieldset aria-describedby="status_diag-tab7-problem2_7_1">
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<text>[mathjaxinline]f[/mathjaxinline] is continuous at [mathjaxinline]x=a[/mathjaxinline].</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_diag-tab7-problem2_7_1[]" id="input_diag-tab7-problem2_7_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="diag-tab7-problem2_7_1-choice_1-label" for="input_diag-tab7-problem2_7_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_7_1">
<text>[mathjaxinline]\displaystyle \lim _{x\rightarrow a} f(x)[/mathjaxinline] exists.</text>
</label>
</div>
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<input type="checkbox" name="input_diag-tab7-problem2_7_1[]" id="input_diag-tab7-problem2_7_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="diag-tab7-problem2_7_1-choice_2-label" for="input_diag-tab7-problem2_7_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_7_1">
<text>[mathjaxinline]\displaystyle \lim _{x\rightarrow a} \frac{f(x) - f(a)}{x-a}[/mathjaxinline] exists.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_diag-tab7-problem2_7_1[]" id="input_diag-tab7-problem2_7_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="diag-tab7-problem2_7_1-choice_3-label" for="input_diag-tab7-problem2_7_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_7_1">
<text>[mathjaxinline]f'(a)[/mathjaxinline] is defined.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_diag-tab7-problem2_7_1[]" id="input_diag-tab7-problem2_7_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="diag-tab7-problem2_7_1-choice_4-label" for="input_diag-tab7-problem2_7_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_7_1">
<text>[mathjaxinline]f^{\prime \prime }(a)[/mathjaxinline] is defined.</text>
</label>
</div>
<span id="answer_diag-tab7-problem2_7_1"/>
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<span class="status unanswered" id="status_diag-tab7-problem2_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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</div></div>
</p>
<p><b class="bf">C7 &#8195;</b> Let [mathjaxinline]\ f(x) = x^3 + 5x^2 - 7x - 1[/mathjaxinline]. What is [mathjaxinline]\ f'(1)[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 7" role="group"><div id="inputtype_diag-tab7-problem2_8_1" class=" capa_inputtype textline">
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</p>
<p><b class="bf">C8 &#8195;</b> Let [mathjaxinline]\ \displaystyle g(x) = x^2 e^ x[/mathjaxinline]. What is [mathjaxinline]g'(1)[/mathjaxinline]? </p>
<p>
(You may enter as decimal number with 2 decimal places, or you can type e for [mathjaxinline]e[/mathjaxinline]. Use [mathjaxinline]*[/mathjaxinline] to denote multiplication; e.g. 7[mathjaxinline]*[/mathjaxinline]e for [mathjaxinline]7e[/mathjaxinline].) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 8" role="group"><div id="formulaequationinput_diag-tab7-problem2_9_1" class="inputtype formulaequationinput">
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<p><b class="bf">C9 &#8195;</b> Suppose that [mathjaxinline]\ f(x) = g(5x)[/mathjaxinline] for all [mathjaxinline]x[/mathjaxinline], and that both functions are differentiable. Which of the following is necessarily true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 9" role="group"><div class="choicegroup capa_inputtype" id="inputtype_diag-tab7-problem2_10_1">
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<text> [mathjaxinline]f'(1) = g'(1)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_10_1" id="input_diag-tab7-problem2_10_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="diag-tab7-problem2_10_1-choice_2-label" for="input_diag-tab7-problem2_10_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_10_1">
<text> [mathjaxinline]f'(5) = g'(1)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_10_1" id="input_diag-tab7-problem2_10_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="diag-tab7-problem2_10_1-choice_3-label" for="input_diag-tab7-problem2_10_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_10_1">
<text> [mathjaxinline]f'(1) = g'(5)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_10_1" id="input_diag-tab7-problem2_10_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="diag-tab7-problem2_10_1-choice_4-label" for="input_diag-tab7-problem2_10_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_10_1">
<text> [mathjaxinline]5f'(1) = g'(1)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_10_1" id="input_diag-tab7-problem2_10_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="diag-tab7-problem2_10_1-choice_5-label" for="input_diag-tab7-problem2_10_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_10_1">
<text> [mathjaxinline]5f'(1) = g'(5)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_10_1" id="input_diag-tab7-problem2_10_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="diag-tab7-problem2_10_1-choice_6-label" for="input_diag-tab7-problem2_10_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_10_1">
<text> [mathjaxinline]f'(1) = 5g'(1)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_10_1" id="input_diag-tab7-problem2_10_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="diag-tab7-problem2_10_1-choice_7-label" for="input_diag-tab7-problem2_10_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_10_1">
<text> [mathjaxinline]f'(1) = 5g'(5)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_diag-tab7-problem2_10_1" id="input_diag-tab7-problem2_10_1_choice_8" class="field-input input-radio" value="choice_8"/><label id="diag-tab7-problem2_10_1-choice_8-label" for="input_diag-tab7-problem2_10_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_diag-tab7-problem2_10_1">
<text> None of the above</text>
</label>
</div>
<span id="answer_diag-tab7-problem2_10_1"/>
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<span class="status unanswered" id="status_diag-tab7-problem2_10_1" data-tooltip="Not yet answered.">
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</p>
<p><b class="bf">C10 &#8195;</b> Let [mathjaxinline]\ \displaystyle f(t) = \frac{\ln (5t+1)}{\sqrt {t+1}}[/mathjaxinline]. What is [mathjaxinline]\ f'(0)[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]f'(0)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 10" role="group"><div id="inputtype_diag-tab7-problem2_11_1" class=" capa_inputtype inline textline">
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<p>
If you got 0.8 or above, you have a good handle on the derivative. There is likely no material in Unit 0 or Unit 1 that you are not familiar with. You may choose to do the in-lecture exercises, but may wish to skip the video tutorials. </p><p>
If you got less than 0.8, that is to be expected! We assume you are here to learn about differentiation after all. </p>
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