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Merge pull request #387 from mattboelkins/proteus
lots of edits to new proteus questions
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‎source/proteus/proteus-1-3.xml‎

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<exercise component="proteus" label="proteus-AV-rise-run-1" xml:id="proteus-AV-rise-run-1" attachment="yes">
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<statement>
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<p>
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When we are talking about slope, students from American schools often say the words “rise” and “run.” Explain in your own words what “rise” and “run” mean. How do we combine “rise” and “run” to determine slope?
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When we are talking about slope, students from American schools often say the words <q>rise</q>and <q>run.</q> Explain in your own words what <q>rise</q>and <q>run</q> mean. How do we combine <q>rise</q>and <q>run</q> to determine slope?
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</p>
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</statement>
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<response />
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<exercise component="proteus" label="proteus-AV-rise-run-2" xml:id="proteus-AV-rise-run-2" attachment="yes">
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<statement>
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<p>
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In Definition 1.3.1, what does <m> h </m> represent in the computation of the average rate of change of <m>f </m>?
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In <xref ref="def-1-3-aroc">Definition</xref>, what does <m> h </m> represent in the computation of the average rate of change of <m>f </m>?
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</p>
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</statement>
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<response />
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<exercise component="proteus" label="proteus-AV-rise-run-3" xml:id="proteus-AV-rise-run-3" attachment="yes">
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<statement>
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<p>
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In Definition 1.3.1, what does the quantity <m>f(a + h) - f(a)</m> represent in the computation of average rate of change?
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In <xref ref="def-1-3-aroc">Definition</xref>, what does the quantity <m>f(a + h) - f(a)</m> represent in the computation of average rate of change?
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</p>
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</statement>
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<response />

‎source/proteus/proteus-1-6.xml‎

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<?xml version="1.0" encoding="UTF-8" ?>
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<!-- **********************************************************************-->
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<!-- Copyright 2012-2025 -->
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<!-- Matthew Boelkins -->
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<!-- -->
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<!-- This file is part of Active Calculus. -->
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<!-- -->
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<!-- Permission is granted to copy, distribute and/or modify this document -->
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<!-- under the terms of the Creative Commons BY-SA license. The work -->
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<!-- may be used for free by any party so long as attribution is given to -->
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<!-- the author(s), the work and its derivatives are used in the spirit of -->
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<!-- "share and share alike". All trademarks are the registered marks of -->
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<!-- their respective owners. -->
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<!-- **********************************************************************-->
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<exercises component="proteus" xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="proteus-1-6">
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<title>PROTEUS exercises</title>
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<p>
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As noted in the introduction to <xref ref="sec-1-6-second-d" text="type-global"/>, we are interested in learning more about how the the second derivative of a function and the behavior of the graph of the function are related.
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</p>
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<exercise>
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<statement>
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<p>
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In the <em>Desmos</em> window below, you will find a graph of a function that you can experiment with by clicking on the point <m>P</m> on the graph, and moving it along the graph of <m>f</m>. You will also see a green point representing the value of the second derivative of graph of <m>f</m> corresponding to the point <m>P</m>. As you move the point <m>P</m>, note that the tangent line to the graph moves along with the point. Then, do the following:
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</p>
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<p>
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<ol>
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<li>
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<p>Carefully observe the shape of the graph and where it opens up versus down. </p>
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</li>
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<li>
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<p>
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Investigate the relationship of the tangent line to the graph as you move the point. For example, at what points is the tangent line above or below the 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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Notice the sign of the second derivative at various points, by observing the green dot corresponding to <m>P</m>.
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</p>
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</li>
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<li>
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<p>
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In the <em>Desmos</em> command line, toggle on the graph of the derivative function and repeat the steps above, paying attention to where the derivative is increasing and decreasing.
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</p>
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</li>
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</ol>
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It will be useful to make a list of your observations as you experiment.
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</p>
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<p>
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<interactive desmos="cwu1h8unhh" width="74%" aspect="4:7" />
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</p>
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</statement>
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<matches>
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<match>
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<premise order="1"> <m>f'(x)</m> is increasing </premise>
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<premise> the tangent line lies below the graph near the point P </premise>
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<premise> the graph opens up (is shaped like a bowl that could hold water) </premise>
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<response> <m>f''(x)>0</m> </response>
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</match>
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<match>
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<premise order="2"> <m>f'(x)</m> switches between increasing and decreasing </premise>
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<premise> the tangent line lies both above and below the graph near the point P </premise>
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<premise> the graph switches from opening up to opening down </premise>
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<response> <m>f''(x)=0</m> </response>
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</match>
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<match>
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<premise order="3"> <m>f'(x)</m> is decreasing </premise>
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<premise> the tangent line lies above the graph near the point P </premise>
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<premise> the graph opens down (is shaped like an upside down bowl)</premise>
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<response> <m>f''(x) \lt 0 </m> </response>
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</match>
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</matches>
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</exercise>
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<exercise>
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<statement>
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<p>
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Finally, write a summary of your findings that will help you to understand how the sign of the second derivative relates to the shape of the graph.
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</p>
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</statement>
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<response />
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</exercise>
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</exercises>

‎source/proteus/proteus-2-8.xml‎

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<?xml version="1.0" encoding="UTF-8" ?>
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<!-- **********************************************************************-->
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<!-- Copyright 2012-2025 -->
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<!-- Matthew Boelkins -->
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<!-- -->
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<!-- This file is part of Active Calculus. -->
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<!-- -->
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<!-- Permission is granted to copy, distribute and/or modify this document -->
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<!-- under the terms of the Creative Commons BY-SA license. The work -->
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<!-- may be used for free by any party so long as attribution is given to -->
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<!-- the author(s), the work and its derivatives are used in the spirit of -->
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<!-- "share and share alike". All trademarks are the registered marks of -->
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<!-- their respective owners. -->
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<!-- **********************************************************************-->
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<exercises component="proteus" xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="proteus-2-8">
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<title>PROTEUS exercises</title>
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<p>
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In <xref ref="PA-2-8">Preview Activity</xref>, we saw a first example of how we can use the tangent line approximations of two functions to find the value of an indeterminate limit. Next, we explore several indeterminate limits involving linear functions.
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</p>
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<exercise component="proteus" label="proteus-2-8-L1" xml:id="proteus-2-8-L1" attachment="yes">
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<statement>
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<p>
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Explain why
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<me>
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\lim_{x \to 2} \frac{-3(x-2)}{7(x-2)}
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</me>
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is indeterminate, and then (if possible) find the numerical value of the limit. Write to explain your thinking (e.g. did you reason algebraically, numerically, or graphically?).
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</p>
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</statement>
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<response />
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</exercise>
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<exercise component="proteus" label="proteus-2-8-L2" xml:id="proteus-2-8-L2" attachment="yes">
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<statement>
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<p>
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Explain why
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<me>
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\lim_{x \to -3} \frac{4x+12}{3x+9}
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</me>
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is indeterminate, and then (if possible) find the numerical value of the limit. Write to explain your thinking.
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</p>
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</statement>
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<response />
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</exercise>
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<exercise component="proteus" label="proteus-2-8-L3" xml:id="proteus-2-8-L3" attachment="yes">
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<statement>
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<p>
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Explain why
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<me>
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\lim_{x \to 5} \frac{2(x-5)^2}{-9(x-5)}
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</me>
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is indeterminate, and then (if possible) find the numerical value of the limit. Write to explain your thinking.
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</p>
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</statement>
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<response />
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</exercise>
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</exercises>

‎source/proteus/proteus-4-1.xml‎

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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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What do you get if you multiply <m>0.30 \dfrac{\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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</ol>
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</p>
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</statement>
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<response />
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</exercise>
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‎source/proteus/proteus-4-2.xml‎

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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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In <xref ref="PA-4-2">Section</xref>, we used a formula for a velocity function to choose the heights of rectangles. However, in most realistic 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="bottom">
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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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</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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<caption>Axes for plotting.</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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<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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Plot the given data on the set of axes provided in <xref ref="saturn-axes">Figure</xref> 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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</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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As illustrated in <xref ref="PA-4-2">Preview Activity</xref>, there are <em>multiple</em> consistent ways to use the plotted data points to determine the heights of rectangles. Try another way 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>

‎source/proteus/proteus-4-3.xml‎

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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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Draw a new rectangle that is twice as tall as the original rectangle in part (a). What is the area of the new rectangle?
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</p>
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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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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 in part (a). What is the area of the combined figure?
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</p>
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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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Draw another rectangle that is 3 units tall and 2 units wide, and suppose it was glued to the top of the original rectangle in part (a). What is the area of the combined rectangle?
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</p>
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</li>
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</ol>

‎source/sec-1-3-derivative-pt.xml‎

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we make the following definition for an arbitrary function <m>y = f(x)</m>.
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</p>
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<definition permid="ipm">
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<definition xml:id="def-1-3-aroc" permid="ipm">
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<statement>
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<p permid="iZg">
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For a function <m>f</m>, the <term>average rate of change</term>
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<me permid="BME">
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AV_{[a,a+h]} = \frac{f(a+h)-f(a)}{h}
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</me>.
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Equivalently, the average rate of change of <m>f</m> on <m>[a,b]</m> is
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<me permid="hTN">
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AV_{[a,b]} = \frac{f(b)-f(a)}{b-a}
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</me>.
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</p>
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</statement>
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</definition>
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<p permid="TQi">
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Equivalently,
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if we want to consider the average rate of change of <m>f</m> on <m>[a,b]</m>,
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we compute
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<me permid="hTN">
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AV_{[a,b]} = \frac{f(b)-f(a)}{b-a}
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</me>.
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</p>
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<p permid="zXr">
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It is essential that you understand how the average rate of change of <m>f</m> on an interval is connected to its graph.
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</p>

‎source/sec-1-6-second-d.xml‎

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</p>
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<xi:include href="./previews/PA-1-6-WW.xml" />
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<xi:include href="./previews/PA-1-6.xml" />
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<xi:include href="./previews/PA-1-6.xml" />
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</introduction>
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<xi:include href="./proteus/proteus-1-6.xml" />
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<subsection permid="wlN">
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<title>Increasing or decreasing</title>
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<p permid="JjZ">

‎source/sec-2-8-LHR.xml‎

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</p>
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<xi:include href="./previews/PA-2-8-WW.xml" />
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<xi:include href="./previews/PA-2-8.xml" />
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<xi:include href="./previews/PA-2-8.xml" />
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</introduction>
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<xi:include href="./proteus/proteus-2-8.xml" />
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<subsection permid="SpQ">
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<title>Using derivatives to evaluate indeterminate limits of the form <m>\frac{0}{0}</m>.</title>
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<figure xml:id="F-2-8-LHR" permid="Lvl">

‎source/sec-4-1-velocity-distance.xml‎

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</p>
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<xi:include href="./previews/PA-4-1-WW.xml" />
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<xi:include href="./previews/PA-4-1.xml" />
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<xi:include href="./previews/PA-4-1.xml" />
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</introduction>
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<xi:include href="./proteus/proteus-4-1.xml" />
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<subsection permid="EWs">
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<title>Area under the graph of the velocity function</title>
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<idx><h>area</h><h>under velocity function</h></idx>

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