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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Optimization: Max Min</h3>
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<h2 class="hd hd-2 unit-title">2. Optimization: Max Min Problems</h2>
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<p><b class="bfseries">Objectives</b></p><p>
At the end of this sequence, and after some practice, you should be able to: </p><ul class="itemize"><li><p>
Find maxima and minima of functions on intervals. </p></li><li><p>
Set up and solve optimization problems. </p></li></ul><p><b class="bfseries">Contents: 16 pages</b></p><p>
9 videos (60 minutes 1x speed) 31 questions </p>
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<h2 class="hd hd-2 unit-title">3. Preliminaries</h2>
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<h3 class="hd hd-2">Attaining Extrema</h3>
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<h2 class="hd hd-2 unit-title">4. Preliminary Exercises</h2>
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Attaining extremes?
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Given the graph above, does [mathjaxinline]\ f[/mathjaxinline] attain a maximum on the interval [mathjaxinline][1,4][/mathjaxinline]? </p>
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Given the graph above, does [mathjaxinline]\ f[/mathjaxinline] attain a minimum on the interval [mathjaxinline][1,4][/mathjaxinline]? </p>
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Attaining extremes? 2
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<img alt="The function f of x is decreasing when x is less than 1.5, increasing when x is between 1.5 and 2, decreasing when x is between 2 and 2.5, increasing when x is between 2.5 and 3, and decreasing when x is greater than 3. When x equals 2 and 3, f of x has sharp corners. The function is concave up for x less than 2 and for x between 2 and 3. The function is linear when x is greater than 3." src="/assets/courseware/v1/d6677876d267ff580fe31e20a8c64e20/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app3_maxmin2.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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Given the graph above, does [mathjaxinline]\ f[/mathjaxinline] attain a maximum on the interval [mathjaxinline][1,4][/mathjaxinline]? </p>
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</div></div>
</p>
<p>
Given the graph above, does [mathjaxinline]\ f[/mathjaxinline] attain a minimum on the interval [mathjaxinline][1,4][/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_3-tab4-problem2_3_1">
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<text> Yes</text>
</label>
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<text> No</text>
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Attaining extremes? 3
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<img alt="When x is less than 2, f is equal to the linear function x. When x is greater than or equal to 2 and less than 3, f is equal to 6.5 minus 2 x. When x is greater than or equal to 3, f is equal to x minus 2." src="/assets/courseware/v1/7a9f174817e90c24908ebed376ddc90c/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app3_maxmin3.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
<p>
Given the graph above, does [mathjaxinline]\ f[/mathjaxinline] attain a maximum on the interval [mathjaxinline][1,4][/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_3-tab4-problem3_2_1">
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<text> Yes</text>
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<text> No</text>
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</span>
</div>
</div></div>
</p>
<p>
Given the graph above, does [mathjaxinline]\ f[/mathjaxinline] attain a minimum on the interval [mathjaxinline][1,4][/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_3-tab4-problem3_3_1">
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<text> Yes</text>
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<text> No</text>
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What is the commonality?
</h3>
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<p>
Functions that don't attain either their maximum or minimum are problematic. Based on the examples that you have seen, what defect did these functions have that prevented them from attaining their maximum or minimum value? </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_3-tab4-problem4_2_1">
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<text> They were not differentiable</text>
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<text> They were not continuous</text>
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<h2 class="hd hd-2 unit-title">5. Extreme Value Theorem</h2>
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<p>
If [mathjaxinline]\ f[/mathjaxinline] is continuous on a closed interval [mathjaxinline][a,b][/mathjaxinline], then there are points at which [mathjaxinline]\ f[/mathjaxinline] attains its maximum and its minimum on [mathjaxinline][a,b][/mathjaxinline]. </p><p>
It is important to note that this is only guaranteed if the interval is finite and closed. We will talk about infinite intervals and open intervals a little later. </p><p>
The proof of this theorem is beyond the scope of this course, so we will not present it here. </p>
</div>
</div>
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<h2 class="hd hd-2 unit-title">6. Finding Extrema</h2>
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Critical Points
</h3>
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<div>
<p>
Let's remind ourselves what critical points mean exactly. True or false: Critical points of [mathjaxinline]\ f[/mathjaxinline] are precisely those points where the derivative [mathjaxinline]\ f'[/mathjaxinline] is equal to zero. </p>
<p>
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Where might extrema be?
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<p>
Let [mathjaxinline]\ f[/mathjaxinline] be a continuous function on the interval [mathjaxinline][0,10][/mathjaxinline]. </p>
<p>
Suppose [mathjaxinline]\ f[/mathjaxinline] does <span style="color:#99182C"><b class="bf">not</b></span> have a critical point at [mathjaxinline]x=3[/mathjaxinline]. Can [mathjaxinline]\ f[/mathjaxinline] attain its minimum or maximum on [mathjaxinline][0,10][/mathjaxinline] at [mathjaxinline]x=3[/mathjaxinline]? </p>
<p>
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</p>
<p>
Suppose [mathjaxinline]\ f[/mathjaxinline] does <span style="color:#99182C"><b class="bf">not</b></span> have a critical point at [mathjaxinline]x=10[/mathjaxinline]. Can [mathjaxinline]\ f[/mathjaxinline] attain its minimum or maximum on [mathjaxinline][0,10][/mathjaxinline] at [mathjaxinline]x=10[/mathjaxinline]? </p>
<p>
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<h2 class="hd hd-2 unit-title">7. Candidates for Extrema</h2>
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<p>
We want to find the maximum or minimum value of a continuous function on a closed interval. The Extreme Value Theorem guarantees that they will be attained. The maximum and minimum can only be attained at critical points or endpoints, so we just need to run through all of those candidates to find the largest and smallest values of [mathjaxinline]\ f[/mathjaxinline]. </p><p>
We'll walk through an example together! </p>
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Extrema example 1
</h3>
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<p>
We're going to find the maximum and minimum values of the function [mathjaxinline]\ f(x) = x^3 - 6x^2 - 15x + 10[/mathjaxinline] over the interval [mathjaxinline][-4,6][/mathjaxinline]. </p>
<p>
First of all, are we guaranteed that [mathjaxinline]\ f[/mathjaxinline] will attain its maximum and minimum over this interval? </p>
<p>
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Critical Points
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<p>
Locate all [mathjaxinline]x-[/mathjaxinline]coordinates of the critical points of [mathjaxinline]\ f[/mathjaxinline] on [mathjaxinline][-4,6][/mathjaxinline]. </p>
<p>
(If there is more than one, separate them with a comma; e.g pi/2 , 0.3 . If there aren't any, type none.) </p>
<p>
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<h3 class="hd hd-3 problem-header" id="app_3-tab7-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab7-problem3-problem-progress" tabindex="-1">
Additional Candidates for Extrema
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab7-problem3-problem-progress"></div>
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<p>
Other than critical points, what additional [mathjaxinline]x-[/mathjaxinline]coordinates do we need to check? </p>
<p>
(If there is more than one, separate them with a comma; e.g pi , 2 . If there aren't any, type none.) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_app_3-tab7-problem3_2_1" class=" capa_inputtype textline">
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<h3 class="hd hd-3 problem-header" id="app_3-tab7-problem4-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab7-problem4-problem-progress" tabindex="-1">
Putting it together
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab7-problem4-problem-progress"></div>
<div class="problem">
<div>
<p>
The Extreme Value Theorem guarantees that [mathjaxinline]\ f[/mathjaxinline] must attain its maximum and its minimum on [mathjaxinline][-4,6][/mathjaxinline] somewhere, and we now have the complete list of candidates where [mathjaxinline]\ f[/mathjaxinline] might do so - critical points of [mathjaxinline]\ f[/mathjaxinline], and the two endpoints. The easiest thing to do now is just plug them all into [mathjaxinline]\ f[/mathjaxinline] and figure out which one yields the largest value of [mathjaxinline]\ f[/mathjaxinline], and which one yields the smallest value of [mathjaxinline]\ f[/mathjaxinline]. </p>
<p>
Where are the maximum and minimum attained, and what are they? </p>
<p>
( If there is more than one [mathjaxinline]x[/mathjaxinline] value at which the maximum or minimum value is attained, separate with commas; e.g. 1, 3.5 .) </p>
<p>
<p style="display:inline">The maximum value is attained at [mathjaxinline]x=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_app_3-tab7-problem4_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_app_3-tab7-problem4_2_1" id="input_app_3-tab7-problem4_2_1" aria-describedby="status_app_3-tab7-problem4_2_1" value=""/>
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</span>
<p id="answer_app_3-tab7-problem4_2_1" class="answer"/>
</div>
</div></div>
<br/>
<p style="display:inline">The maximum value of [mathjaxinline]\ f[/mathjaxinline] over [mathjaxinline][-4,6][/mathjaxinline] is </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_app_3-tab7-problem4_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_app_3-tab7-problem4_3_1" id="input_app_3-tab7-problem4_3_1" aria-describedby="status_app_3-tab7-problem4_3_1" value=""/>
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<span class="status unanswered" id="status_app_3-tab7-problem4_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_app_3-tab7-problem4_3_1" class="answer"/>
</div>
</div></div>
<br/>
<br/>
<p style="display:inline">The minimum value is attained at [mathjaxinline]x=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_app_3-tab7-problem4_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_app_3-tab7-problem4_4_1" id="input_app_3-tab7-problem4_4_1" aria-describedby="status_app_3-tab7-problem4_4_1" value=""/>
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<span class="status unanswered" id="status_app_3-tab7-problem4_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_app_3-tab7-problem4_4_1" class="answer"/>
</div>
</div></div>
<br/>
<p style="display:inline">The minimum value of [mathjaxinline]\ f[/mathjaxinline] over [mathjaxinline][-4,6][/mathjaxinline] is </p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_app_3-tab7-problem4_5_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_app_3-tab7-problem4_5_1" class="answer"/>
</div>
</div></div>
<br/>
</p>
<p>
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<span id="solution_app_3-tab7-problem4_solution_1"/>
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<h2 class="hd hd-2 unit-title">8. Another Example</h2>
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<h3 class="hd hd-3 problem-header" id="app_3-tab8-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab8-problem1-problem-progress" tabindex="-1">
Extrema example 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab8-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
We're going to find the maximum and minimum values of the function [mathjaxinline]g(x) = (3/2)x^{2/3} + x + 1[/mathjaxinline] over the interval [mathjaxinline][-2,1][/mathjaxinline]. </p>
<p>
Are we guaranteed that [mathjaxinline]g[/mathjaxinline] will attain its maximum and minimum over this interval? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_app_3-tab8-problem1_2_1">
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<text> Yes</text>
</label>
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<div class="field">
<input type="radio" name="input_app_3-tab8-problem1_2_1" id="input_app_3-tab8-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="app_3-tab8-problem1_2_1-choice_2-label" for="input_app_3-tab8-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_app_3-tab8-problem1_2_1">
<text> No</text>
</label>
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</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_app_3-tab8-problem1_2_1" data-tooltip="Not yet answered.">
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</p>
<p>
<div class="solution-span">
<span id="solution_app_3-tab8-problem1_solution_1"/>
</div></p>
<script src="/assets/courseware/v1/14976dbaebca69b9b0e6989fe9838bb1/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/st.js" type="text/javascript"/>
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data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab8-problem2" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab8-problem2/handler/xmodule_handler"
data-problem-score="0"
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<h3 class="hd hd-3 problem-header" id="app_3-tab8-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab8-problem2-problem-progress" tabindex="-1">
Critical Points
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab8-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Locate all [mathjaxinline]x-[/mathjaxinline]coordinates of the critical points of [mathjaxinline]g[/mathjaxinline] in the given interval. </p>
<p>
(If there is more than one, separate them with a comma; e.g [mathjaxinline]x=[/mathjaxinline] pi , 2 . If there aren't any, type none. ) </p>
<p>
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Final Answer
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<p>
In addition to the critical points you found, we also need to check the endpoints of the interval. We now have the complete list of candidates where [mathjaxinline]g[/mathjaxinline] might attain its maximum or minimum over the interval [mathjaxinline][-2,1][/mathjaxinline]. </p>
<p>
At which [mathjaxinline]x[/mathjaxinline]-coordinate are the maximum and minimum attained, and what is the value of the function there? </p>
<p>
(If there is more than one [mathjaxinline]x[/mathjaxinline] value at which the maximum or minimum value is attained, separate with commas; e.g pi , 2 . If there aren't any, type none.) </p>
<p>
<p style="display:inline">The maximum value is attained at [mathjaxinline]\ x=[/mathjaxinline] </p>
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<p style="display:inline">The maximum value of [mathjaxinline]g[/mathjaxinline] over [mathjaxinline][-2,1][/mathjaxinline] is </p>
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<br/>
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<p>
<p style="display:inline">The minimum value is attained at [mathjaxinline]\ x=[/mathjaxinline] </p>
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<p style="display:inline">The minimum value of [mathjaxinline]g[/mathjaxinline] over [mathjaxinline][-2,1][/mathjaxinline] is </p>
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<h2 class="hd hd-2 unit-title">9. Review Questions</h2>
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No Critical Points
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If [mathjaxinline]\ f[/mathjaxinline] is a continuous function on the interval [mathjaxinline][1,5][/mathjaxinline] but has no critical points in the interval, then what can be said about [mathjaxinline]\ f[/mathjaxinline]? </p>
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<text> [mathjaxinline]\ f[/mathjaxinline] attains its maximum at [mathjaxinline]x=1[/mathjaxinline] and minimum at [mathjaxinline]x=5[/mathjaxinline], or vice versa</text>
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One Critical Point
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True or false: If [mathjaxinline]\ f[/mathjaxinline] is a continuous function on [mathjaxinline][1,5][/mathjaxinline] and the only critical point for [mathjaxinline]\ f[/mathjaxinline] within the interval is at [mathjaxinline]x=3[/mathjaxinline], then [mathjaxinline]\ f[/mathjaxinline] must attain either its maximum or its minimum on the interval at [mathjaxinline]x=3[/mathjaxinline]. </p>
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Attaining Extrema on Different Intervals
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Let [mathjaxinline]\ f(x) = \displaystyle {\frac{1}{x} + \sin x}.[/mathjaxinline] </p>
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On which of the following intervals is it guaranteed that [mathjaxinline]\ f[/mathjaxinline] will attain its maximum and minimum? Check all that apply. </p>
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<text>[mathjaxinline][-\pi ,\pi ][/mathjaxinline]</text>
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<text>[mathjaxinline][\pi , 2\pi ][/mathjaxinline]</text>
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<h3 class="hd hd-2">Non-Closed Intervals</h3>
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Extrema example 3
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<p>
We will find the maximum and minimum values of the function [mathjaxinline]g(x) = \displaystyle {\frac{x^2}{2x^3 + 1}}[/mathjaxinline] over the interval [mathjaxinline][0.5,\infty )[/mathjaxinline]. </p>
<p>
First, we will determine the function's behavior at the endpoints. What is [mathjaxinline]g(0.5)[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]g(0.5)=[/mathjaxinline]</p>
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<p>
What is [mathjaxinline]\displaystyle {\lim _{x\rightarrow \infty } g(x)}[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle {\lim _{x\rightarrow \infty } g(x)}=[/mathjaxinline]</p>
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Example 3 continued
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<p>
Now to determine what happens inside the interval! Recall [mathjaxinline]g(x) = \displaystyle {\frac{x^2}{2x^3 + 1}}[/mathjaxinline] over the interval [mathjaxinline][0.5,\infty )[/mathjaxinline]. </p>
<p>
Locate all [mathjaxinline]x-[/mathjaxinline]coordinates of the critical points of [mathjaxinline]g[/mathjaxinline] in the given interval. </p>
<p>
(If there is more than one, separate them with a comma; e.g pi , 2 . If there aren't any, type none.) </p>
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Final Answer
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<p>
List all [mathjaxinline]x-[/mathjaxinline]coordinates where [mathjaxinline]g(x) = \displaystyle {\frac{x^2}{2x^3 + 1}}[/mathjaxinline] attains its maximum value and minimum value on the interval [mathjaxinline][0.5,\infty )[/mathjaxinline]. </p>
<p>
(If there is more than one, separate them with a comma; e.g pi , 2.5 . If there aren't any, type none.) </p>
<p>
<p style="display:inline">The maximum value is attained at [mathjaxinline]x=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_app_3-tab10-problem4_2_1" class=" capa_inputtype inline textline">
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_app_3-tab10-problem4_2_1" class="answer"/>
</div>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">The minimum value is attained at [mathjaxinline]x=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_app_3-tab10-problem4_3_1" class=" capa_inputtype inline textline">
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<span class="status unanswered" id="status_app_3-tab10-problem4_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_app_3-tab10-problem4_3_1" class="answer"/>
</div>
</div></div>
</p>
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<span id="solution_app_3-tab10-problem4_solution_1"/>
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<h3 class="hd hd-3 problem-header" id="app_3-tab10-problem5-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab10-problem5-problem-progress" tabindex="-1">
Modified problem
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<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab10-problem5-problem-progress"></div>
<div class="problem">
<div>
<p>
Let's change the interval a bit. List all [mathjaxinline]x-[/mathjaxinline]coordinates where [mathjaxinline]g(x) = \displaystyle {\frac{x^2}{2x^3 + 1}}[/mathjaxinline] attains its maximum value and minimum value on the interval [mathjaxinline][0,\infty )[/mathjaxinline]. </p>
<p>
(If there is more than one, separate them with a comma; e.g pi/2 , 0.3 . If there aren't any, type none.) </p>
<p>
<p style="display:inline">The maximum value is attained when [mathjaxinline]x=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_app_3-tab10-problem5_2_1" class=" capa_inputtype inline textline">
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_app_3-tab10-problem5_2_1" class="answer"/>
</div>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">The minimum value is attained when [mathjaxinline]x=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_app_3-tab10-problem5_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_app_3-tab10-problem5_3_1" class="answer"/>
</div>
</div></div>
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<h2 class="hd hd-2 unit-title">11. Optimization Problems</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.1x+2T2019+type@video+block@app_3-tab11-video1">
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<script type="json/xblock-args" class="xblock-json-init-args">
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</script>
<h3 class="hd hd-2">Making a Box</h3>
<div
id="video_app_3-tab11-video1"
class="video closed"
data-metadata='{"saveStateUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_3-tab11-video1/handler/xmodule_handler/save_user_state", "completionEnabled": false, "ytTestTimeout": 1500, "autohideHtml5": false, "completionPercentage": 0.95, "poster": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "sources": [], "transcriptLanguages": {"en": "English"}, "saveStateEnabled": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "autoAdvance": false, "duration": 0.0, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_3-tab11-video1/handler/publish_completion", "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_3-tab11-video1/handler/transcript/translation/__lang__", "prioritizeHls": false, "autoplay": false, "savedVideoPosition": 0.0, "streams": "1.00:30d0SQA46V0", "speed": null, "generalSpeed": 1.0, "ytMetadataEndpoint": "", "showCaptions": "true", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "end": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_3-tab11-video1/handler/transcript/available_translations"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
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Your browser does not support this video format. Try using a different browser.
</h4>
</div>
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<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_app_3-tab11-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_app_3-tab11-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_3-tab11-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@video+block@app_3-tab11-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab11-problem1">
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aria-labelledby="app_3-tab11-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab11-problem1" data-url="/courses/course-v1:MITx+18.01.1x+2T2019/xblock/block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab11-problem1/handler/xmodule_handler"
data-problem-score="0"
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<h3 class="hd hd-3 problem-header" id="app_3-tab11-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab11-problem1-problem-progress" tabindex="-1">
Volume
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.1x+2T2019+type@problem+block@app_3-tab11-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
From the video, what is the volume [mathjaxinline]V(x)[/mathjaxinline] of the box in terms of [mathjaxinline]x[/mathjaxinline]? </p>
<p>
(Remember to type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponentiation.) </p>
<p>
<p style="display:inline">[mathjaxinline]V(x)=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_app_3-tab11-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_app_3-tab11-problem1_2_1" id="input_app_3-tab11-problem1_2_1" data-input-id="app_3-tab11-problem1_2_1" value="" aria-describedby="status_app_3-tab11-problem1_2_1" size="20"/>
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<span class="status unanswered" id="status_app_3-tab11-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_app_3-tab11-problem1_2_1" class="answer"/>
<div id="input_app_3-tab11-problem1_2_1_preview" class="equation">
\(\)
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<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Find the [mathjaxinline]x-[/mathjaxinline]coordinates of all critical points of [mathjaxinline]V(x)[/mathjaxinline] that lie within the interval of interest. </p>
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<h2 class="hd hd-2 unit-title">12. Optimization Problem 2</h2>
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The interval
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Let's say we have 100 inches of wire that we want to cut into two pieces. We will use one piece to make a circle and the other to make a square. (It is possible for one of the two shapes to have zero size.) </p>
<p>
The perimeter of the circle can be anywhere from 0 to 100 inches, and the perimeter of the square similarly can be anywhere from 0 to 100 inches. Are these two perimeters independent of one another? </p>
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Total area
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Let [mathjaxinline]c[/mathjaxinline] be the perimeter of the circle, and [mathjaxinline]s[/mathjaxinline] denote the perimeter of the square. </p>
<p>
Write [mathjaxinline]s[/mathjaxinline] in terms of [mathjaxinline]c[/mathjaxinline]. </p>
<p>
(Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponentiation.) </p>
<p>
<p style="display:inline">[mathjaxinline]s =[/mathjaxinline] </p>
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\(\)
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Find the total area of the two shapes [mathjaxinline]A(c)[/mathjaxinline] in terms of [mathjaxinline]c[/mathjaxinline]. </p>
<p>
(You can type pi for [mathjaxinline]\pi[/mathjaxinline]; e for Euler's number. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponentiation.) </p>
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<p style="display:inline">[mathjaxinline]A(c) =[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
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<font color="#0078b0">2520</font>
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<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Maximum
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To within [mathjaxinline]0.01[/mathjaxinline] inches, what should [mathjaxinline]c[/mathjaxinline] be in order to maximize the total area? </p>
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Minimum
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To within [mathjaxinline]0.01[/mathjaxinline] inches, what should [mathjaxinline]c[/mathjaxinline] be in order to minimize the total area? </p>
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<h2 class="hd hd-2 unit-title">13. Optimization Problem 3</h2>
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Building a Sheep Pen
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A farmer wants to build a rectangular pen for his sheep on the side of his barn. One side of the pen must be formed by part of the north wall of the barn, which measures 100 feet long, and the other three sides will be formed by fencing. The area of the pen should be 4000 square feet. </p>
<img alt="Sheep pen is shown as a rectangle with area 4000 square feet. One side lies along the barn. The side opposite and parallel to the barn is labeled a, and the two sides perpendicular to the barn are each labeled b." src="/assets/courseware/v1/bdf72222530328a5900e4911b79eb620/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u4app3_barn.svg" style="margin: 10px 25px 25px 25px" width="400pxpx"/>
<p>
If [mathjaxinline]a[/mathjaxinline] denotes the length of the pen and [mathjaxinline]b[/mathjaxinline] the width (in feet), express [mathjaxinline]b[/mathjaxinline] in terms of [mathjaxinline]a[/mathjaxinline]. </p>
<p>
(Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponentiation.) </p>
<p>
<p style="display:inline">[mathjaxinline]b =[/mathjaxinline] </p>
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<p>
Let [mathjaxinline]\ f[/mathjaxinline] denote the total amount of fence. Write [mathjaxinline]\ f[/mathjaxinline] as a function of [mathjaxinline]a[/mathjaxinline]. <p style="display:inline">[mathjaxinline]\ f(a) =[/mathjaxinline] </p><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_app_3-tab13-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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The Interval
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What is the allowable interval for [mathjaxinline]a[/mathjaxinline]? </p>
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<text> [mathjaxinline][1,100][/mathjaxinline]</text>
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<input type="radio" name="input_app_3-tab13-problem2_2_1" id="input_app_3-tab13-problem2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="app_3-tab13-problem2_2_1-choice_3-label" for="input_app_3-tab13-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_app_3-tab13-problem2_2_1">
<text> [mathjaxinline][1,4000][/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">14. Zipline</h2>
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<p>
In the next video, we work out a question posed at the very beginning of this section: </p><p>
What is the maximum vertical distance traversed during a zipline? </p><p>
To answer this question, we actually answer a related question: </p><p>
Given a mass on a weightless string, hold the two ends fixed. At what point will the mass settle? </p><p>
The point at which it settles is the minimum [mathjaxinline]y[/mathjaxinline]-coordinate of the path traversed by a zipline rider. And this minimum [mathjaxinline]y[/mathjaxinline]-coordinate corresponds exactly to the point at which the rider has traversed the greatest (maximal) vertical drop. </p>
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<p>
Let [mathjaxinline](x,y)[/mathjaxinline] be the minimum point of the elliptic path traversed by a zipline rider. Given that the rider starts at point [mathjaxinline](a,b)=(200,25)[/mathjaxinline], by how much is her potential energy reduced when she reaches the minimum height of her trajectory. (Which equals the maximum amount of kinetic energy of the rider during the trip, assuming he starts with zero velocity and there is zero friction etc.) </p>
<p>
Use the coordinate system set up in the video, and described in the image below. </p>
<center>
<img alt="Diagram of zipline coordinates. The leftmost point is the origin. Below and to the right of this is the points x comma y and is connected to the origin by a line segment. To the right of x comma y and above the origin is the point a comma 25. This point is connected to x comma y by a line segment." src="/assets/courseware/v1/4bfafa3249846a5ee67a454fd89a99df/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_ziplinecoordinates.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
</center>
<p>
Use the following data to compute your answer: </p>
<ul class="itemize">
<li>
<p>
The rider has a mass of [mathjaxinline]50[/mathjaxinline] kilograms. </p>
</li>
<li>
<p>
The zipline itself is massless. </p>
</li>
<li>
<p>
The horizontal span between the two fixed endpoints of the zipline ride is [mathjaxinline]200[/mathjaxinline] meters. </p>
</li>
<li>
<p>
The change in [mathjaxinline]y[/mathjaxinline]-coordinate between the two fixed end points is [mathjaxinline]25[/mathjaxinline] meters. </p>
</li>
</ul>
<p>
(Express your answer in terms of [mathjaxinline]g[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], and [mathjaxinline]y[/mathjaxinline] as needed. Type [mathjaxinline]*[/mathjaxinline] for multiplication; / for division; [mathjaxinline]\wedge[/mathjaxinline] for exponentiation.) </p>
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<h2 class="hd hd-2 unit-title">15. More practice problems: 3 recitation videos</h2>
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<h2 class="hd hd-2 unit-title">16. Summary</h2>
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<p><b class="bfseries">The Extreme Value Theorem</b></p><p>
If [mathjaxinline]\ f[/mathjaxinline] is continuous on a closed interval [mathjaxinline][a,b][/mathjaxinline], then there are points at which [mathjaxinline]\ f[/mathjaxinline] attains its maximum and its minimum on [mathjaxinline][a,b][/mathjaxinline]. </p><p>
It is important to note that this is only guaranteed if the interval is finite and closed. We will talk about infinite intervals and open intervals a little later. </p><p>
The proof of this theorem is beyond the scope of this course, so we will not present it here. </p><p><b class="bfseries">Candidates for Extrema</b></p><p>
We want to find the maximum or minimum value of a continuous function [mathjaxinline]\ f[/mathjaxinline] on a closed interval [mathjaxinline][a, b][/mathjaxinline]. The Extreme Value Theorem guarantees that they will be attained. </p><p>
The maximum and minimum can only be attained at critical points or endpoints, so we just need to run through all of those candidates to find the largest and smallest values of [mathjaxinline]\ f[/mathjaxinline]. </p>
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