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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/0bf22c50b106e8316e376698054a56fc/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session24.pdf" target="[object Object]">Chapter 13.7 (PDF)</a> of <em>Mathematics for Computer Science</em> for 3.2 Asymptotics.</p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Asymptotic Notation</h2>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/308fa78c2f078dc00e1423344aa613ba/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS16_AsymNotation.pdf" target="[object Object]">Asymptotic Notation (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Asymptotics as Relations</h2>
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Asymptotics as Relations
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<legend id="7d3fc55c1e784c2a81a0e93b58200740_2_1-legend" class="response-fieldset-legend field-group-hd">1. Which of the following relations is/are symmetric?</legend>
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<legend id="7d3fc55c1e784c2a81a0e93b58200740_3_1-legend" class="response-fieldset-legend field-group-hd">2. Which of the following relations indicate(s) the same order of growth?</legend>
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<h2 class="hd hd-2 unit-title">Lecture Video | Asymptotic Properties</h2>
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<h3 class="hd hd-2">Asymptotic Properties</h3>
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<h2 class="hd hd-2 unit-title">Exercise | Little oh Big Oh</h2>
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<h3 class="hd hd-3 problem-header" id="0cc93810738b4b8aad76e66bf74a90fe-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@0cc93810738b4b8aad76e66bf74a90fe-problem-progress" tabindex="-1">
Little oh Big Oh
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@0cc93810738b4b8aad76e66bf74a90fe-problem-progress"></div>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_0cc93810738b4b8aad76e66bf74a90fe_2_1">
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<legend id="0cc93810738b4b8aad76e66bf74a90fe_2_1-legend" class="response-fieldset-legend field-group-hd">1. Which one of the following is true about the little oh and Big Oh?</legend>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_2_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="0cc93810738b4b8aad76e66bf74a90fe_2_1-choice_0-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_2_1"> If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>, then <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_2_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="0cc93810738b4b8aad76e66bf74a90fe_2_1-choice_1-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_2_1"> If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>, then <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>g</mi>
<mo>&#8800;<!-- &#8800; --></mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_2_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="0cc93810738b4b8aad76e66bf74a90fe_2_1-choice_2-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_2_1"> When <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>&#8764;<!-- &#8764; --></mo>
<mi>g</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>&#8800;<!-- &#8800; --></mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_2_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="0cc93810738b4b8aad76e66bf74a90fe_2_1-choice_3-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_2_1"> When <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>&#8769;<!-- &#8769; --></mo>
<mi>g</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_2_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="0cc93810738b4b8aad76e66bf74a90fe_2_1-choice_4-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_2_1"> If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>, then <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>.
</label>
</div>
<span id="answer_0cc93810738b4b8aad76e66bf74a90fe_2_1"/>
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<span class="status unanswered" id="status_0cc93810738b4b8aad76e66bf74a90fe_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
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<fieldset aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_3_1">
<legend id="0cc93810738b4b8aad76e66bf74a90fe_3_1-legend" class="response-fieldset-legend field-group-hd">2. Which of the following is NOT correct about the little oh?</legend>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_3_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="0cc93810738b4b8aad76e66bf74a90fe_3_1-choice_0-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>x</mi>
<mi>a</mi>
</msup>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mi>b</mi>
</msup>
<mo stretchy="false">)</mo>
</math> for all integers <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>b</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_3_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="0cc93810738b4b8aad76e66bf74a90fe_3_1-choice_1-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>b</mi>
<mo>&#8901;<!-- &#8901; --></mo>
<msup>
<mi>x</mi>
<mi>a</mi>
</msup>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mi>b</mi>
</msup>
<mo stretchy="false">)</mo>
</math> for all nonnegative constants <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>b</mi>
<mo>&gt;</mo>
<mi>a</mi>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_3_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="0cc93810738b4b8aad76e66bf74a90fe_3_1-choice_2-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mi>&#949;<!-- &#949; --></mi>
</msup>
<mo stretchy="false">)</mo>
</math> for all <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>&#949;<!-- &#949; --></mi>
<mo>&gt;</mo>
<mn>0</mn>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_3_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="0cc93810738b4b8aad76e66bf74a90fe_3_1-choice_3-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>x</mi>
<mi>c</mi>
</msup>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>a</mi>
<mi>x</mi>
</msup>
<mo stretchy="false">)</mo>
</math> for any <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
<mo>,</mo>
<mi>c</mi>
<mo>&#8712;<!-- &#8712; --></mo>
<mi mathvariant="double-struck">R</mi>
</math> with <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>a</mi>
<mo>&gt;</mo>
<mn>1</mn>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_3_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_3_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="0cc93810738b4b8aad76e66bf74a90fe_3_1-choice_4-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>o</mi>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mi>&#1013;<!-- &#1013; --></mi>
</msup>
<mo stretchy="false">)</mo>
</math> for all <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>&#1013;<!-- &#1013; --></mi>
<mo>&gt;</mo>
<mn>0</mn>
</math>
</label>
</div>
<span id="answer_0cc93810738b4b8aad76e66bf74a90fe_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_0cc93810738b4b8aad76e66bf74a90fe_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_0cc93810738b4b8aad76e66bf74a90fe_4_1">
<fieldset aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_4_1">
<legend id="0cc93810738b4b8aad76e66bf74a90fe_4_1-legend" class="response-fieldset-legend field-group-hd">3. Let <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>. Why do we say <math xmlns="http://www.w3.org/1998/Math/MathML">
<munder>
<mo form="prefix">lim&#8198;sup</mo>
<mrow>
<mi>x</mi>
<mo stretchy="false">&#8594;<!-- &#8594; --></mo>
<mi mathvariant="normal">&#8734;<!-- &#8734; --></mi>
</mrow>
</munder>
<mfrac>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math><math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&lt;</mo>
</math> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#8734;<!-- &#8734; --></mi>
</math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>i</mi>
<msub>
<mi>m</mi>
<mrow>
<mi>x</mi>
<mo stretchy="false">&#8594;<!-- &#8594; --></mo>
<mi mathvariant="normal">&#8734;<!-- &#8734; --></mi>
</mrow>
</msub>
<mfrac>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mo>&lt;</mo>
</math> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#8734;<!-- &#8734; --></mi>
</math>?</legend>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_4_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_4_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="0cc93810738b4b8aad76e66bf74a90fe_4_1-choice_0-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_4_1"> It accommodates cases in which a limit of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>/</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math> does not exist.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_4_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_4_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="0cc93810738b4b8aad76e66bf74a90fe_4_1-choice_1-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_4_1"> The limit superior, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>i</mi>
<mi>m</mi>
<mi>s</mi>
<mi>u</mi>
<mi>p</mi>
</math>, is a more strict limit compared to the ordinary limit, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>l</mi>
<mi>i</mi>
<mi>m</mi>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_4_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_4_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="0cc93810738b4b8aad76e66bf74a90fe_4_1-choice_2-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math> gives an upper bound instead of a lower bound on the growth of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_0cc93810738b4b8aad76e66bf74a90fe_4_1[]" id="input_0cc93810738b4b8aad76e66bf74a90fe_4_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="0cc93810738b4b8aad76e66bf74a90fe_4_1-choice_3-label" for="input_0cc93810738b4b8aad76e66bf74a90fe_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_0cc93810738b4b8aad76e66bf74a90fe_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math> can oscillate wildly as long as it never gets bigger than <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>c</mi>
<mo>&#8901;<!-- &#8901; --></mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math> for <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>x</mi>
<mo>&gt;</mo>
<mi>n</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>c</mi>
<mo>&gt;</mo>
<mn>0</mn>
</math>.
</label>
</div>
<span id="answer_0cc93810738b4b8aad76e66bf74a90fe_4_1"/>
</fieldset>
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<span class="status unanswered" id="status_0cc93810738b4b8aad76e66bf74a90fe_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 class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-runtime-version="1" data-usage-id="block-v1:OCW+6.042J+2T2019+type@vertical+block@b9180cdb346542eea472bdbfc769d2ad" data-graded="False" data-runtime-class="LmsRuntime" data-course-id="course-v1:OCW+6.042J+2T2019" data-has-score="False" data-init="VerticalStudentView" data-request-token="ea1dc276dbe711efb040029b57e3cc89">
<h2 class="hd hd-2 unit-title">Exercise | Theta</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:OCW+6.042J+2T2019+type@problem+block@786ff66aec1741db9f4885c6a5bf3efa">
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{"xmodule-type": "Problem"}
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<div id="problem_786ff66aec1741db9f4885c6a5bf3efa" class="problems-wrapper" role="group"
aria-labelledby="786ff66aec1741db9f4885c6a5bf3efa-problem-title"
data-problem-id="block-v1:OCW+6.042J+2T2019+type@problem+block@786ff66aec1741db9f4885c6a5bf3efa" data-url="/courses/course-v1:OCW+6.042J+2T2019/xblock/block-v1:OCW+6.042J+2T2019+type@problem+block@786ff66aec1741db9f4885c6a5bf3efa/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="2"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="786ff66aec1741db9f4885c6a5bf3efa-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@786ff66aec1741db9f4885c6a5bf3efa-problem-progress" tabindex="-1">
Theta
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@786ff66aec1741db9f4885c6a5bf3efa-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_786ff66aec1741db9f4885c6a5bf3efa_2_1">
<fieldset aria-describedby="status_786ff66aec1741db9f4885c6a5bf3efa_2_1">
<legend id="786ff66aec1741db9f4885c6a5bf3efa_2_1-legend" class="response-fieldset-legend field-group-hd">1. If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi mathvariant="normal">&#920;<!-- &#920; --></mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>, which of the following MUST be true?</legend>
<div class="field">
<input type="checkbox" name="input_786ff66aec1741db9f4885c6a5bf3efa_2_1[]" id="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="786ff66aec1741db9f4885c6a5bf3efa_2_1-choice_0-label" for="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_786ff66aec1741db9f4885c6a5bf3efa_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>g</mi>
<mo>=</mo>
<mi mathvariant="normal">&#920;<!-- &#920; --></mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_786ff66aec1741db9f4885c6a5bf3efa_2_1[]" id="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="786ff66aec1741db9f4885c6a5bf3efa_2_1-choice_1-label" for="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_786ff66aec1741db9f4885c6a5bf3efa_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_786ff66aec1741db9f4885c6a5bf3efa_2_1[]" id="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="786ff66aec1741db9f4885c6a5bf3efa_2_1-choice_2-label" for="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_786ff66aec1741db9f4885c6a5bf3efa_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>g</mi>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_786ff66aec1741db9f4885c6a5bf3efa_2_1[]" id="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="786ff66aec1741db9f4885c6a5bf3efa_2_1-choice_3-label" for="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_786ff66aec1741db9f4885c6a5bf3efa_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_786ff66aec1741db9f4885c6a5bf3efa_2_1[]" id="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="786ff66aec1741db9f4885c6a5bf3efa_2_1-choice_4-label" for="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_786ff66aec1741db9f4885c6a5bf3efa_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>g</mi>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_786ff66aec1741db9f4885c6a5bf3efa_2_1[]" id="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="786ff66aec1741db9f4885c6a5bf3efa_2_1-choice_5-label" for="input_786ff66aec1741db9f4885c6a5bf3efa_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_786ff66aec1741db9f4885c6a5bf3efa_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>&#8764;<!-- &#8764; --></mo>
<mi>g</mi>
</math>
</label>
</div>
<span id="answer_786ff66aec1741db9f4885c6a5bf3efa_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_786ff66aec1741db9f4885c6a5bf3efa_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_786ff66aec1741db9f4885c6a5bf3efa_3_1">
<fieldset aria-describedby="status_786ff66aec1741db9f4885c6a5bf3efa_3_1">
<legend id="786ff66aec1741db9f4885c6a5bf3efa_3_1-legend" class="response-fieldset-legend field-group-hd">2. If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi mathvariant="normal">&#920;<!-- &#920; --></mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>, which of the following CAN be true?</legend>
<div class="field">
<input type="checkbox" name="input_786ff66aec1741db9f4885c6a5bf3efa_3_1[]" id="input_786ff66aec1741db9f4885c6a5bf3efa_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="786ff66aec1741db9f4885c6a5bf3efa_3_1-choice_0-label" for="input_786ff66aec1741db9f4885c6a5bf3efa_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_786ff66aec1741db9f4885c6a5bf3efa_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>g</mi>
<mo>=</mo>
<mi mathvariant="normal">&#920;<!-- &#920; --></mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_786ff66aec1741db9f4885c6a5bf3efa_3_1[]" id="input_786ff66aec1741db9f4885c6a5bf3efa_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="786ff66aec1741db9f4885c6a5bf3efa_3_1-choice_1-label" for="input_786ff66aec1741db9f4885c6a5bf3efa_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_786ff66aec1741db9f4885c6a5bf3efa_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_786ff66aec1741db9f4885c6a5bf3efa_3_1[]" id="input_786ff66aec1741db9f4885c6a5bf3efa_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="786ff66aec1741db9f4885c6a5bf3efa_3_1-choice_2-label" for="input_786ff66aec1741db9f4885c6a5bf3efa_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_786ff66aec1741db9f4885c6a5bf3efa_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>g</mi>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
</math>
</label>
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<h2 class="hd hd-2 unit-title">Lecture Video | Asymptotic Blunders</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Asymptotics the Right Way</h2>
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<h3 class="hd hd-3 problem-header" id="39d89682cffa4588afa76acb57b8e768-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@39d89682cffa4588afa76acb57b8e768-problem-progress" tabindex="-1">
Asymptotics The Right Way
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<legend id="39d89682cffa4588afa76acb57b8e768_2_1-legend" class="response-fieldset-legend field-group-hd">1. On which side of the equal sign should the asymptotic notations <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>O</mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">)</mo>
</math>, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>o</mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">)</mo>
</math>, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi mathvariant="normal">&#920;<!-- &#920; --></mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">)</mo>
</math></legend>
<div class="field">
<input type="radio" name="input_39d89682cffa4588afa76acb57b8e768_2_1" id="input_39d89682cffa4588afa76acb57b8e768_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="39d89682cffa4588afa76acb57b8e768_2_1-choice_0-label" for="input_39d89682cffa4588afa76acb57b8e768_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_39d89682cffa4588afa76acb57b8e768_2_1"> left
</label>
</div>
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<input type="radio" name="input_39d89682cffa4588afa76acb57b8e768_2_1" id="input_39d89682cffa4588afa76acb57b8e768_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="39d89682cffa4588afa76acb57b8e768_2_1-choice_1-label" for="input_39d89682cffa4588afa76acb57b8e768_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_39d89682cffa4588afa76acb57b8e768_2_1"> right
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</label>
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<span id="answer_39d89682cffa4588afa76acb57b8e768_2_1"/>
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<legend id="39d89682cffa4588afa76acb57b8e768_3_1-legend" class="response-fieldset-legend field-group-hd">2. Why is the statement "<math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
</math> is at least <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">)</mo>
</math>" a blunder?</legend>
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<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">)</mo>
</math> is not a quantity.
</label>
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<input type="checkbox" name="input_39d89682cffa4588afa76acb57b8e768_3_1[]" id="input_39d89682cffa4588afa76acb57b8e768_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="39d89682cffa4588afa76acb57b8e768_3_1-choice_1-label" for="input_39d89682cffa4588afa76acb57b8e768_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_39d89682cffa4588afa76acb57b8e768_3_1"> Big Oh gives an upper bound on the growth of a function.
</label>
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<input type="checkbox" name="input_39d89682cffa4588afa76acb57b8e768_3_1[]" id="input_39d89682cffa4588afa76acb57b8e768_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="39d89682cffa4588afa76acb57b8e768_3_1-choice_2-label" for="input_39d89682cffa4588afa76acb57b8e768_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_39d89682cffa4588afa76acb57b8e768_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">)</mo>
</math> is a relation.
</label>
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<div class="field">
<input type="checkbox" name="input_39d89682cffa4588afa76acb57b8e768_3_1[]" id="input_39d89682cffa4588afa76acb57b8e768_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="39d89682cffa4588afa76acb57b8e768_3_1-choice_3-label" for="input_39d89682cffa4588afa76acb57b8e768_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_39d89682cffa4588afa76acb57b8e768_3_1"> If <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
</math> is at least <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</math>, then <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
</math>.
</label>
</div>
<span id="answer_39d89682cffa4588afa76acb57b8e768_3_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<legend id="39d89682cffa4588afa76acb57b8e768_4_1-legend" class="response-fieldset-legend field-group-hd">3. Why is the statement <math xmlns="http://www.w3.org/1998/Math/MathML">
<munderover>
<mo>&#8721;<!-- &#8721; --></mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<mi>i</mi>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math> a blunder?</legend>
<div class="field">
<input type="checkbox" name="input_39d89682cffa4588afa76acb57b8e768_4_1[]" id="input_39d89682cffa4588afa76acb57b8e768_4_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="39d89682cffa4588afa76acb57b8e768_4_1-choice_0-label" for="input_39d89682cffa4588afa76acb57b8e768_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_39d89682cffa4588afa76acb57b8e768_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<munderover>
<mo>&#8721;<!-- &#8721; --></mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<mi>i</mi>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_39d89682cffa4588afa76acb57b8e768_4_1[]" id="input_39d89682cffa4588afa76acb57b8e768_4_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="39d89682cffa4588afa76acb57b8e768_4_1-choice_1-label" for="input_39d89682cffa4588afa76acb57b8e768_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_39d89682cffa4588afa76acb57b8e768_4_1"> The Big Oh relation on individual terms of a sum does not necessarily apply to the sum.
</label>
</div>
<div class="field">
<input type="checkbox" name="input_39d89682cffa4588afa76acb57b8e768_4_1[]" id="input_39d89682cffa4588afa76acb57b8e768_4_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="39d89682cffa4588afa76acb57b8e768_4_1-choice_2-label" for="input_39d89682cffa4588afa76acb57b8e768_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_39d89682cffa4588afa76acb57b8e768_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<munderover>
<mo>&#8721;<!-- &#8721; --></mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<mi>i</mi>
<mo>=</mo>
<mi mathvariant="normal">&#920;<!-- &#920; --></mi>
<mo stretchy="false">(</mo>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_39d89682cffa4588afa76acb57b8e768_4_1[]" id="input_39d89682cffa4588afa76acb57b8e768_4_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="39d89682cffa4588afa76acb57b8e768_4_1-choice_3-label" for="input_39d89682cffa4588afa76acb57b8e768_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_39d89682cffa4588afa76acb57b8e768_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>O</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math> should not be treated as a number.
</label>
</div>
<span id="answer_39d89682cffa4588afa76acb57b8e768_4_1"/>
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<h2 class="hd hd-2 unit-title">Exercise | Practice with Big O</h2>
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<h3 class="hd hd-3 problem-header" id="2a5b2db37ebe4843b8091b8e4766bdd7-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@2a5b2db37ebe4843b8091b8e4766bdd7-problem-progress" tabindex="-1">
Practice with Big O
</h3>
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<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_2a5b2db37ebe4843b8091b8e4766bdd7_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_2a5b2db37ebe4843b8091b8e4766bdd7_2_1" id="label_2a5b2db37ebe4843b8091b8e4766bdd7_2_1">Find the least nonnegative integer, <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>n</mi>
</math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
</math> is <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>O</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</math> when <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
</math> is defined by each of the expressions below.
<br/>
<br/>
<em>If there is none, write "none."</em>
<br/>
<br/>
1. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
<mo>+</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mi>log</mi>
<mo>&#8289;<!-- --></mo>
<mi>x</mi>
</math></label>
<input type="text" name="input_2a5b2db37ebe4843b8091b8e4766bdd7_2_1" id="input_2a5b2db37ebe4843b8091b8e4766bdd7_2_1" aria-describedby="status_2a5b2db37ebe4843b8091b8e4766bdd7_2_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_2a5b2db37ebe4843b8091b8e4766bdd7_2_1"/>
<span class="status unanswered" id="status_2a5b2db37ebe4843b8091b8e4766bdd7_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_2a5b2db37ebe4843b8091b8e4766bdd7_2_1" class="answer"/>
</div>
</div><div class="solution-span">
<span id="solution_2a5b2db37ebe4843b8091b8e4766bdd7_solution_1"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_2a5b2db37ebe4843b8091b8e4766bdd7_3_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_2a5b2db37ebe4843b8091b8e4766bdd7_3_1" id="label_2a5b2db37ebe4843b8091b8e4766bdd7_3_1">2. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
<mi>log</mi>
<mo>&#8289;<!-- --></mo>
<mi>x</mi>
</math></label>
<input type="text" name="input_2a5b2db37ebe4843b8091b8e4766bdd7_3_1" id="input_2a5b2db37ebe4843b8091b8e4766bdd7_3_1" aria-describedby="status_2a5b2db37ebe4843b8091b8e4766bdd7_3_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_2a5b2db37ebe4843b8091b8e4766bdd7_3_1"/>
<span class="status unanswered" id="status_2a5b2db37ebe4843b8091b8e4766bdd7_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_2a5b2db37ebe4843b8091b8e4766bdd7_3_1" class="answer"/>
</div>
</div><div class="solution-span">
<span id="solution_2a5b2db37ebe4843b8091b8e4766bdd7_solution_2"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_2a5b2db37ebe4843b8091b8e4766bdd7_4_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_2a5b2db37ebe4843b8091b8e4766bdd7_4_1" id="label_2a5b2db37ebe4843b8091b8e4766bdd7_4_1">3. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1.1</mn>
<msup>
<mo stretchy="false">)</mo>
<mi>x</mi>
</msup>
</math></label>
<input type="text" name="input_2a5b2db37ebe4843b8091b8e4766bdd7_4_1" id="input_2a5b2db37ebe4843b8091b8e4766bdd7_4_1" aria-describedby="status_2a5b2db37ebe4843b8091b8e4766bdd7_4_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_2a5b2db37ebe4843b8091b8e4766bdd7_4_1"/>
<span class="status unanswered" id="status_2a5b2db37ebe4843b8091b8e4766bdd7_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_2a5b2db37ebe4843b8091b8e4766bdd7_4_1" class="answer"/>
</div>
</div><div class="solution-span">
<span id="solution_2a5b2db37ebe4843b8091b8e4766bdd7_solution_3"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_2a5b2db37ebe4843b8091b8e4766bdd7_5_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_2a5b2db37ebe4843b8091b8e4766bdd7_5_1" id="label_2a5b2db37ebe4843b8091b8e4766bdd7_5_1">4. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>0.1</mn>
<msup>
<mo stretchy="false">)</mo>
<mi>x</mi>
</msup>
</math></label>
<input type="text" name="input_2a5b2db37ebe4843b8091b8e4766bdd7_5_1" id="input_2a5b2db37ebe4843b8091b8e4766bdd7_5_1" aria-describedby="status_2a5b2db37ebe4843b8091b8e4766bdd7_5_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_2a5b2db37ebe4843b8091b8e4766bdd7_5_1"/>
<span class="status unanswered" id="status_2a5b2db37ebe4843b8091b8e4766bdd7_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_2a5b2db37ebe4843b8091b8e4766bdd7_5_1" class="answer"/>
</div>
</div><div class="solution-span">
<span id="solution_2a5b2db37ebe4843b8091b8e4766bdd7_solution_4"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_2a5b2db37ebe4843b8091b8e4766bdd7_6_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_2a5b2db37ebe4843b8091b8e4766bdd7_6_1" id="label_2a5b2db37ebe4843b8091b8e4766bdd7_6_1">5. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mstyle displaystyle="true" scriptlevel="0">
<mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mstyle>
</math></label>
<input type="text" name="input_2a5b2db37ebe4843b8091b8e4766bdd7_6_1" id="input_2a5b2db37ebe4843b8091b8e4766bdd7_6_1" aria-describedby="status_2a5b2db37ebe4843b8091b8e4766bdd7_6_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_2a5b2db37ebe4843b8091b8e4766bdd7_6_1"/>
<span class="status unanswered" id="status_2a5b2db37ebe4843b8091b8e4766bdd7_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_2a5b2db37ebe4843b8091b8e4766bdd7_6_1" class="answer"/>
</div>
</div><div class="solution-span">
<span id="solution_2a5b2db37ebe4843b8091b8e4766bdd7_solution_5"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_2a5b2db37ebe4843b8091b8e4766bdd7_7_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_2a5b2db37ebe4843b8091b8e4766bdd7_7_1" id="label_2a5b2db37ebe4843b8091b8e4766bdd7_7_1">6. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mstyle displaystyle="true" scriptlevel="0">
<mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<mn>5</mn>
<mi>log</mi>
<mo>&#8289;<!-- --></mo>
<mi>x</mi>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mstyle>
</math></label>
<input type="text" name="input_2a5b2db37ebe4843b8091b8e4766bdd7_7_1" id="input_2a5b2db37ebe4843b8091b8e4766bdd7_7_1" aria-describedby="status_2a5b2db37ebe4843b8091b8e4766bdd7_7_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_2a5b2db37ebe4843b8091b8e4766bdd7_7_1"/>
<span class="status unanswered" id="status_2a5b2db37ebe4843b8091b8e4766bdd7_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_2a5b2db37ebe4843b8091b8e4766bdd7_7_1" class="answer"/>
</div>
</div><div class="solution-span">
<span id="solution_2a5b2db37ebe4843b8091b8e4766bdd7_solution_6"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 7" role="group"><div id="inputtype_2a5b2db37ebe4843b8091b8e4766bdd7_8_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_2a5b2db37ebe4843b8091b8e4766bdd7_8_1" id="label_2a5b2db37ebe4843b8091b8e4766bdd7_8_1">7. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mn>2</mn>
<mrow>
<mn>3</mn>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
<mo>&#8289;<!-- --></mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msup>
</math></label>
<input type="text" name="input_2a5b2db37ebe4843b8091b8e4766bdd7_8_1" id="input_2a5b2db37ebe4843b8091b8e4766bdd7_8_1" aria-describedby="status_2a5b2db37ebe4843b8091b8e4766bdd7_8_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_2a5b2db37ebe4843b8091b8e4766bdd7_8_1"/>
<span class="status unanswered" id="status_2a5b2db37ebe4843b8091b8e4766bdd7_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_2a5b2db37ebe4843b8091b8e4766bdd7_8_1" class="answer"/>
</div>
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<span id="solution_2a5b2db37ebe4843b8091b8e4766bdd7_solution_7"/>
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<h2 class="hd hd-2 unit-title">Exercise | Practice with Order of Growth</h2>
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<h3 class="hd hd-3 problem-header" id="843a869ff5fb4028a24d033e996429d8-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@843a869ff5fb4028a24d033e996429d8-problem-progress" tabindex="-1">
Practice with Order of Growth
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@843a869ff5fb4028a24d033e996429d8-problem-progress"></div>
<div class="problem">
<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_843a869ff5fb4028a24d033e996429d8_2_1">
<fieldset aria-describedby="status_843a869ff5fb4028a24d033e996429d8_2_1">
<legend id="843a869ff5fb4028a24d033e996429d8_2_1-legend" class="response-fieldset-legend field-group-hd">For each pair of <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math> below, determine which of the listed relations apply.
<br/>
<br/>
1. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>l</mi>
<mi>o</mi>
<msub>
<mi>g</mi>
<mn>3</mn>
</msub>
<mi>n</mi>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>l</mi>
<mi>o</mi>
<msub>
<mi>g</mi>
<mn>7</mn>
</msub>
<mi>n</mi>
</math></legend>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_2_1[]" id="input_843a869ff5fb4028a24d033e996429d8_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="843a869ff5fb4028a24d033e996429d8_2_1-choice_0-label" for="input_843a869ff5fb4028a24d033e996429d8_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>&#8764;<!-- &#8764; --></mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_2_1[]" id="input_843a869ff5fb4028a24d033e996429d8_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="843a869ff5fb4028a24d033e996429d8_2_1-choice_1-label" for="input_843a869ff5fb4028a24d033e996429d8_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_2_1[]" id="input_843a869ff5fb4028a24d033e996429d8_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="843a869ff5fb4028a24d033e996429d8_2_1-choice_2-label" for="input_843a869ff5fb4028a24d033e996429d8_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_2_1[]" id="input_843a869ff5fb4028a24d033e996429d8_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="843a869ff5fb4028a24d033e996429d8_2_1-choice_3-label" for="input_843a869ff5fb4028a24d033e996429d8_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_2_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">&#920;<!-- &#920; --></mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_2_1[]" id="input_843a869ff5fb4028a24d033e996429d8_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="843a869ff5fb4028a24d033e996429d8_2_1-choice_4-label" for="input_843a869ff5fb4028a24d033e996429d8_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_2_1"> None of the above
</label>
</div>
<span id="answer_843a869ff5fb4028a24d033e996429d8_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_843a869ff5fb4028a24d033e996429d8_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div><div class="solution-span">
<span id="solution_843a869ff5fb4028a24d033e996429d8_solution_1"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_843a869ff5fb4028a24d033e996429d8_3_1">
<fieldset aria-describedby="status_843a869ff5fb4028a24d033e996429d8_3_1">
<legend id="843a869ff5fb4028a24d033e996429d8_3_1-legend" class="response-fieldset-legend field-group-hd">2. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>33</mn>
</math>.</legend>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_3_1[]" id="input_843a869ff5fb4028a24d033e996429d8_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="843a869ff5fb4028a24d033e996429d8_3_1-choice_0-label" for="input_843a869ff5fb4028a24d033e996429d8_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>&#8764;<!-- &#8764; --></mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_3_1[]" id="input_843a869ff5fb4028a24d033e996429d8_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="843a869ff5fb4028a24d033e996429d8_3_1-choice_1-label" for="input_843a869ff5fb4028a24d033e996429d8_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_3_1[]" id="input_843a869ff5fb4028a24d033e996429d8_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="843a869ff5fb4028a24d033e996429d8_3_1-choice_2-label" for="input_843a869ff5fb4028a24d033e996429d8_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_3_1[]" id="input_843a869ff5fb4028a24d033e996429d8_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="843a869ff5fb4028a24d033e996429d8_3_1-choice_3-label" for="input_843a869ff5fb4028a24d033e996429d8_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">&#920;<!-- &#920; --></mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_3_1[]" id="input_843a869ff5fb4028a24d033e996429d8_3_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="843a869ff5fb4028a24d033e996429d8_3_1-choice_4-label" for="input_843a869ff5fb4028a24d033e996429d8_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_3_1"> None of the above
</label>
</div>
<span id="answer_843a869ff5fb4028a24d033e996429d8_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_843a869ff5fb4028a24d033e996429d8_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div><div class="solution-span">
<span id="solution_843a869ff5fb4028a24d033e996429d8_solution_2"/>
</div></div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_843a869ff5fb4028a24d033e996429d8_4_1">
<fieldset aria-describedby="status_843a869ff5fb4028a24d033e996429d8_4_1">
<legend id="843a869ff5fb4028a24d033e996429d8_4_1-legend" class="response-fieldset-legend field-group-hd">3. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mo stretchy="false">(</mo>
<mfrac>
<mrow>
<mi>&#960;<!-- &#960; --></mi>
<mi>n</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo stretchy="false">)</mo>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mi>s</mi>
<mi>i</mi>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mfrac>
<mrow>
<mi>&#960;<!-- &#960; --></mi>
<mi>n</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo stretchy="false">)</mo>
</math></legend>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_4_1[]" id="input_843a869ff5fb4028a24d033e996429d8_4_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="843a869ff5fb4028a24d033e996429d8_4_1-choice_0-label" for="input_843a869ff5fb4028a24d033e996429d8_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>&#8764;<!-- &#8764; --></mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_4_1[]" id="input_843a869ff5fb4028a24d033e996429d8_4_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="843a869ff5fb4028a24d033e996429d8_4_1-choice_1-label" for="input_843a869ff5fb4028a24d033e996429d8_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_4_1[]" id="input_843a869ff5fb4028a24d033e996429d8_4_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="843a869ff5fb4028a24d033e996429d8_4_1-choice_2-label" for="input_843a869ff5fb4028a24d033e996429d8_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_4_1[]" id="input_843a869ff5fb4028a24d033e996429d8_4_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="843a869ff5fb4028a24d033e996429d8_4_1-choice_3-label" for="input_843a869ff5fb4028a24d033e996429d8_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">&#920;<!-- &#920; --></mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
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<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_4_1[]" id="input_843a869ff5fb4028a24d033e996429d8_4_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="843a869ff5fb4028a24d033e996429d8_4_1-choice_4-label" for="input_843a869ff5fb4028a24d033e996429d8_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_4_1"> None of the above
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_843a869ff5fb4028a24d033e996429d8_5_1">
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<legend id="843a869ff5fb4028a24d033e996429d8_5_1-legend" class="response-fieldset-legend field-group-hd">4. <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mn>1.01</mn>
<mi>n</mi>
</msup>
</math> and <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>n</mi>
<mn>100</mn>
</msup>
</math>.</legend>
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<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_5_1[]" id="input_843a869ff5fb4028a24d033e996429d8_5_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="843a869ff5fb4028a24d033e996429d8_5_1-choice_0-label" for="input_843a869ff5fb4028a24d033e996429d8_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>&#8764;<!-- &#8764; --></mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math>
</label>
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<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_5_1[]" id="input_843a869ff5fb4028a24d033e996429d8_5_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="843a869ff5fb4028a24d033e996429d8_5_1-choice_1-label" for="input_843a869ff5fb4028a24d033e996429d8_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>o</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_5_1[]" id="input_843a869ff5fb4028a24d033e996429d8_5_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="843a869ff5fb4028a24d033e996429d8_5_1-choice_2-label" for="input_843a869ff5fb4028a24d033e996429d8_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>O</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_5_1[]" id="input_843a869ff5fb4028a24d033e996429d8_5_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="843a869ff5fb4028a24d033e996429d8_5_1-choice_3-label" for="input_843a869ff5fb4028a24d033e996429d8_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_5_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">&#920;<!-- &#920; --></mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</math>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_843a869ff5fb4028a24d033e996429d8_5_1[]" id="input_843a869ff5fb4028a24d033e996429d8_5_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="843a869ff5fb4028a24d033e996429d8_5_1-choice_4-label" for="input_843a869ff5fb4028a24d033e996429d8_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_843a869ff5fb4028a24d033e996429d8_5_1"> None of the above
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<span id="answer_843a869ff5fb4028a24d033e996429d8_5_1"/>
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