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Merge pull request #386 from rhinopotamus/main
proteus chapter 4; favicons
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‎assets/favicon/favicon-16x16.png‎

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‎assets/favicon/favicon-32x32.png‎

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‎publication/publication-proteus.ptx‎

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<directories external="../assets" generated="../generated-assets-proteus"
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/>
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</source>
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<html>
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<html favicon="simple">
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<search google-cx="015103900096539427448:ngwuia10qci" />
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<css theme="default-modern" palette="blues" />
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<!-- Examples and exercises are knowlized by default -->

‎publication/publication.ptx‎

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<directories external="../assets" generated="../generated-assets-ed1" />
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<version include="ed1 production static" />
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</source>
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<html>
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<html favicon="simple">
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<css theme="default-modern" palette="blues" />
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<!-- Examples and exercises are knowlized by default -->
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<!-- We disable this behavior -->

‎source/proteus/proteus-4-1.xml‎

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<?xml version="1.0" encoding="UTF-8"?>
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<exercises component="proteus" xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="proteus-4-1">
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<title>PROTEUS exercises</title>
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<exercise component="proteus" label="proteus-multiply-units" xml:id="proteus-multiply-units" attachment="yes">
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<statement>
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<p>
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A rainfall rate of 0.30 inches or more per hour is considered heavy rain, according to the American Meteorological Society.
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<ol>
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<li>
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<p>
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What do you get if you multiply <m>0.30 \dfrac{0.30\text{ inches}}{\text{hour}} \cdot \dfrac16 \text{hour}</m>? Include units.
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</p>
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</li>
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<li>
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<p>
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What does your answer mean in the context of the thunderstorm?
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</p>
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</li>
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</ol>
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</p>
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</statement>
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</exercise>
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</exercises>

‎source/proteus/proteus-4-2.xml‎

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<?xml version="1.0" encoding="UTF-8"?>
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<exercises component="proteus" xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="proteus-4-2">
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<title>PROTEUS exercises</title>
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<exercise component="proteus" label="proteus-saturn-rocket" xml:id="proteus-saturn-rocket" attachment="yes">
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<statement>
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<p>
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In <xref ref="PA-4-2"/>, we used a formula for a velocity function to choose the heights of rectangles. However, in real-world scenarios, it is often the case that a formula is not given, and all we have is discrete data. For instance, here is <url href="https://spacemath.gsfc.nasa.gov/weekly/8Page2.pdf">velocity data</url> from the first 12 seconds after ignition of the Saturn V rocket.
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<sidebyside widths="47% 47%" valign="middle">
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<table xml:id="saturn-data">
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<title>Data for the Saturn V rocket.</title>
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<tabular halign="center">
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<row bottom="minor">
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<cell>seconds, <m>t</m></cell>
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<cell>Velocity in m/s, <m>v(t)</m></cell>
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</row>
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<row >
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<cell>0</cell>
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<cell>0</cell>
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</row>
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<row >
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<cell>2</cell>
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<cell>4</cell>
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</row>
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<row >
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<cell>4</cell>
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<cell>10</cell>
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</row>
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<row >
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<cell>6</cell>
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<cell>15</cell>
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</row>
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<row >
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<cell>8</cell>
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<cell>20</cell>
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</row>
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<row >
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<cell>10</cell>
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<cell>25</cell>
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</row>
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<row >
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<cell>12</cell>
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<cell>31</cell>
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</row>
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</tabular>
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</table>
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<figure xml:id="saturn-axes">
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<caption>Axes for plotting the data in <xref ref="saturn-data"/>.</caption>
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<image>
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<prefigure xmlns="https://prefigure.org"
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label="prefigure-saturn-axes">
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<diagram dimensions="(300,300)" margins="30">
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<coordinates bbox="(0,0,14, 32)">
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<grid-axes
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decorations="no"
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xlabel="t"
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ylabel="v"
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spacings="((0,2,14),(0,4,32))"
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hticks="(0,2,14)"
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hlabels="(0,4,12)"
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vticks="(0,4,32)"
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/>
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</coordinates>
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</diagram>
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</prefigure>
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</image>
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</figure>
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</sidebyside>
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Throughout this activity, we'll use four rectangles, each with a width of <m>\Delta t = 2</m> sec, to estimate the area under the velocity function <m>v(t)</m>.
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</p>
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<ol>
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<li>
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<p>
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Plot the given data on the set of axes provided in <xref ref="saturn-axes"/> with time on the horizontal axis and the velocity on the vertical axis.
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</p>
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</li>
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<li>
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<p>
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Create four rectangles in a consistent way, using the plotted data to provide the heights of each rectangle. Draw the four rectangles on your graph.
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</p>
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</li>
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<li>
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<p>
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Use the heights you determined for each rectangle to determine the area of each rectangle; include units in your answer. What does the area of each rectangle represent in the context of the rocket launch?
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</p>
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</li>
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<li>
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<p>
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Add up the areas of each of the four rectangles to produce an estimate for the <em>total</em> area under the velocity function <m>v(t)</m>. Give your answer as a number with units. What does this value represent in the context of the rocket launch?
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</p>
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</li>
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<li>
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<p>
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As illustrated in <xref ref="PA-4-2"/>, there are <em>multiple</em> consistent ways to use the plotted data points to determine the heights of rectangles. Try another one and produce a <em>different</em> estimate of the distance traveled by the rocket.
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</p>
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</li>
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</ol>
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</statement>
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<response />
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</exercise>
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</exercises>
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<!--
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<table>
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<title>Data for the Saturn V rocket.</title>
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<tabular top="minor" bottom="minor">
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<row >
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<cell>Time (Sec)</cell>
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<cell>Speed (m/s)</cell>
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</row>
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<row >
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<cell>0</cell>
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<cell>0</cell>
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</row>
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<row >
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<cell>1</cell>
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<cell>3</cell>
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</row>
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<row >
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<cell>2</cell>
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<cell>4</cell>
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</row>
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<row >
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<cell>3</cell>
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<cell>7</cell>
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</row>
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<row >
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<cell>4</cell>
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<cell>10</cell>
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</row>
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<row >
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<cell>5</cell>
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<cell>12</cell>
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</row>
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<row >
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<cell>6</cell>
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<cell>15</cell>
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</row>
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<row >
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<cell>7</cell>
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<cell>16</cell>
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</row>
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<row >
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<cell>8</cell>
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<cell>20</cell>
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</row>
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<row >
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<cell>9</cell>
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<cell>23</cell>
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</row>
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<row >
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<cell>10</cell>
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<cell>25</cell>
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</row>
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<row >
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<cell>11</cell>
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<cell>27</cell>
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</row>
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<row >
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<cell>12</cell>
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<cell>31</cell>
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</row>
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<row >
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<cell>13</cell>
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<cell>34</cell>
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</row>
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<row >
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<cell>14</cell>
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<cell>36</cell>
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</row>
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<row >
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<cell>15</cell>
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<cell>39</cell>
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</row>
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<row >
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<cell>16</cell>
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<cell>43</cell>
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</row>
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<row >
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<cell>17</cell>
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<cell>46</cell>
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</row>
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<row >
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<cell>18</cell>
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<cell>49</cell>
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</row>
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<row >
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<cell>19</cell>
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<cell>53</cell>
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</row>
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<row >
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<cell>20</cell>
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<cell>56</cell>
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</row>
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</tabular>
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</table>
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-->

‎source/proteus/proteus-4-3.xml‎

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<?xml version="1.0" encoding="UTF-8"?>
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<exercises component="proteus" xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="proteus-4-3">
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<title>PROTEUS exercises</title>
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<exercise component="proteus" label="proteus-area-reminders" xml:id="proteus-area-reminders" attachment="yes">
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<statement>
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Later in this section, we will learn several properties of the definite integral. In preparation for learning those properties, it will be useful to recall similar properties of simple areas.
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<ol>
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<li>
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<p>
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Suppose you have an original rectangle that is 5 units tall and 2 units wide. Draw a picture of such a rectangle, and find its area.
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</p>
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</li>
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<li>
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<p>
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Draw a new rectangle that is twice as tall as the original rectangle. What is the area of the new rectangle?
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</p>
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</li>
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<li>
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<p>
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Draw another rectangle that is 5 units tall but just 1 unit wide, and suppose it was glued to the side of the original rectangle. What is the area of the combined figure?
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</p>
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</li>
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<li>
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<p>
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Draw another rectangle that is 3 units tall and 2 units wide, and suppose it was glued to the top of the original rectangle. What is the area of the combined rectangle?
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</p>
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</li>
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</ol>
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</statement>
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<response />
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</exercise>
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</exercises>

‎source/sec-4-2-Riemann.xml‎

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<xi:include href="./previews/PA-4-2.xml" />
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</introduction>
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<xi:include href="./proteus/proteus-4-2.xml" />
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<subsection xml:id="sigma-notation" permid="jCJ">
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<title>Sigma Notation</title>
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<p permid="Tos">

‎source/sec-4-3-definite-integral.xml‎

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<xi:include href="./previews/PA-4-3.xml" />
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</introduction>
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<xi:include href="./proteus/proteus-4-3.xml" />
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<subsection permid="uqj">
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<title>The definition of the definite integral</title>
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<idx><h>definite integral</h><h>definition</h></idx>

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