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Fix invalid xmls in LT3 #591

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StevenClontz merged 1 commit into from
Feb 9, 2025
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Fix invalid xmls in LT3 #591

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70 changes: 40 additions & 30 deletions source/calculus/source/01-LT/03.ptx
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Original file line number Diff line number Diff line change
Expand Up @@ -17,12 +17,14 @@
</remark>

<activity xml:id="activity-limits-analytically1">
<statement>
<p>Given <m>f(x)=3x^2-\dfrac{1}{2}x+4</m>, evaluate <m>f(2)</m> and approximate <m>\displaystyle \lim_{x\to2}f(x)</m> numerically (or graphically). What do you think is more likely?</p>
<ol marker="A." cols="2">
<li><m>\displaystyle \lim_{x \to 2}f(x)=f(2)</m></li>
<li><m>\displaystyle \lim_{x \to 2}f(x) \approx f(2)</m></li>
<li><m>\displaystyle \lim_{x \to 2}f(x) \neq f(2)</m></li>
</ol>
</statement>
<answer>
<p>
<m> f(2)= 15 </m> and
Expand All @@ -32,7 +34,7 @@
</activity>

<activity xml:id="activity-limits-analytically2">
<introduction>
<statement>
<p>The table below gives values of a few different functions.</p>
<table xml:id="table-activity-limits-analytically2">
<tabular>
Expand Down Expand Up @@ -140,13 +142,14 @@
</row>
</tabular>
</table>
<p>Using the table above, which of the following is <em>least</em> likely to be true?</p> </introduction>
<p>Using the table above, which of the following is <em>least</em> likely to be true?</p>
<ol marker="A.">
<li><m>\displaystyle \lim_{x\to 7}f(x)= 14</m> and <m>\displaystyle \lim_{x\to 7}g(x)= 23</m></li>
<li><m>\displaystyle \lim_{x \to 7}3f(x) = 3 \lim_{x\to 7}f(x)</m></li>
<li><m>\displaystyle \lim_{x\to 7}\left( f(x)+g(x) \right) = \lim_{x\to 7}f(x) + \lim_{x\to 7}g(x)</m></li>
<li><m>\displaystyle \lim_{x\to 7}\left( f(x)g(x) \right) =f(7) \left( \lim_{x\to 7}g(x) \right)</m></li>
</ol>
</statement>
<answer>
<p>
D. <m>\displaystyle \lim_{x\to 7}\left( f(x)g(x) \right) =f(7) \left( \lim_{x\to 7}g(x) \right)</m>
Expand Down Expand Up @@ -185,6 +188,7 @@
<li><m>\displaystyle \lim_{x\to 2} (f(x) - g(x)) = -2</m></li>
<li><m>\displaystyle \lim_{x\to 2} (f(x)/g(x)) = -2/3</m></li>
</ol>
</statement>
<answer>
<p>
A. <m>\displaystyle \lim_{x\to 2} (f(x) \cdot g(x)) = -6</m>
Expand All @@ -196,7 +200,6 @@
D. <m>\displaystyle \lim_{x\to 2} (f(x)/g(x)) = -2/3</m>
</p>
</answer>
</statement>
</activity>


Expand All @@ -207,12 +210,12 @@
<me>
\lim_{x \to c} f(x) = L, \, \, \, \, \lim_{x \to L} g(x) = K, \text{and} \, g(L)=K.
</me> Then we have the following limits as well.
</p>
<ol>
<li>Power Law: <m>\displaystyle \lim_{x \to c} f(x)^n = L^n</m>, for <m>n</m> a positive integer;</li>
<li>Root Law: <m>\displaystyle \lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L}</m>, for <m>n</m> a positive integer;</li>
<li>Composition Law: <m>\displaystyle \lim_{x \to c} g(f(x)) = K. </m></li>
</ol>
</p>
</statement>
</theorem>

Expand Down Expand Up @@ -264,41 +267,44 @@
</sidebyside>
</introduction>
<task> <statement> <p> <m>\displaystyle \lim_{x \to 1} f(x) + g(x) </m>. </p>
</statement>
<answer>
<p>
<m>\displaystyle \lim_{x \to 1} f(x) + g(x) = 2 </m>
</p>
</answer>
</statement> </task>
</task>
<task> <statement> <p> <m>\displaystyle \lim_{x \to 5^+} 3f(x) </m>. </p>
</statement>
<answer>
<p>
<m>\displaystyle \lim_{x \to 5^+} 3f(x)= 0 </m>
</p>
</answer>
</statement> </task>
</task>
<task> <statement> <p> <m>\displaystyle \lim_{x \to 0^+ } f(x)g(x) </m>. </p>
</statement>
<answer>
<p>
<m>\displaystyle \lim_{x \to 0^+ } f(x)g(x) = 0 </m>
</p>
</answer>
</statement> </task>
<task> <statement> <p> (Challenge) <m>\displaystyle \lim_{x \to 1} g(x) / f(x) </m>. </p>
</task>
<task> <statement> <p> (Challenge) <m>\displaystyle \lim_{x \to 1} g(x) / f(x) </m>. </p></statement>
<answer>
<p>
<m>\displaystyle \lim_{x \to 1} g(x) / f(x) </m> does not exist
</p>
</answer>
</statement> </task>
</task>

<task> <statement> <p> (Challenge) <m>\displaystyle \lim_{x \to 0^+} f(g(x)) </m>. </p>
<task> <statement> <p> (Challenge) <m>\displaystyle \lim_{x \to 0^+} f(g(x)) </m>. </p></statement>
<answer>
<p>
<m>\displaystyle \lim_{x \to 0^+} f(g(x)) </m> does not exist
</p>
</answer>
</statement> </task>
</task>
</activity>

<activity xml:id="activity-limits-analytically4">
Expand All @@ -312,6 +318,7 @@
<li>Power Law</li>
<li>Constant Law</li>
</ol>
</statement>
<answer>
<p>
A. Sums/Difference Law
Expand All @@ -327,7 +334,7 @@
</p>

</answer>
</statement>

</activity>


Expand All @@ -352,12 +359,13 @@
<li>Quotient and root law</li>

</ol>
</statement>
<answer>
<p>
B. <xref ref="theorem-limits-polynomials"/> and the quotient law.
</p>
</answer>
</statement>

</activity>

<theorem xml:id="theorem-limits-rationals">
Expand All @@ -375,12 +383,13 @@
<li>No, because if you graph <m>g(x)=\dfrac{x^2+1}{x-1}</m>, the value <m>g(1)</m> is not defined and the graph shows that the limit of <m>\displaystyle\lim_{x \to c}g(x)</m> does not exist.</li>
<li>Yes, because we can use <xref ref="theorem-limits-rationals"/>.</li>
</ol>
</statement>
<answer>
<p>
B. Yes, because if you graph <m>f(x)=\dfrac{x^2-1}{x-1}</m>, the value <m>f(1)</m> is not defined, but the graph shows that the limit of <m>f(x)</m> does exist as <m>x \to 1</m>.
</p>
</answer>
</statement>

</activity>


Expand All @@ -393,12 +402,13 @@
<li><m>\displaystyle \lim_{x\to 0} (f(x)/g(x))</m> cannot be determined </li>
<li><m>\displaystyle \lim_{x\to 0} (f(x)/g(x))</m> does not exist </li>
</ol>
</statement>
<answer>
<p>
B. <m>\displaystyle \lim_{x\to 0} (f(x)/g(x)) = 2 </m>
</p>
</answer>
</statement>

</activity>

<remark xml:id="zero-over-zero">
Expand All @@ -418,19 +428,19 @@
Determine the following limits and explain your reasoning.
</p>
</introduction>
<task><me>\lim_{x\to-6 } \dfrac{ x^{2} - 6 \, x + 5 }{ x^{2} - 3 \, x - 18 }</me>
<task><statement><p><me>\lim_{x\to-6 } \dfrac{ x^{2} - 6 \, x + 5 }{ x^{2} - 3 \, x - 18 }</me></p></statement>
<answer>
<p>
<m>\dfrac{77}{36}</m>
</p>
</answer></task>
<task><me>\lim_{x\to-1 } \dfrac{ x^{2} - 1 }{ x^{2} + 3 \, x + 2 }</me>
<task><statement><p><me>\lim_{x\to-1 } \dfrac{ x^{2} - 1 }{ x^{2} + 3 \, x + 2 }</me></p></statement>
<answer>
<p>
<m>-2</m>
</p>
</answer></task>
<task><me>\lim_{x\to5 } \dfrac{ x - 5 }{ \sqrt{x + 31} - 6 }</me>
<task><statement><p><me>\lim_{x\to5 } \dfrac{ x - 5 }{ \sqrt{x + 31} - 6 }</me></p></statement>
<answer>
<p>
<m>12</m>
Expand All @@ -447,8 +457,8 @@
<p> In <xref ref = "activity-bolt1"/> you studied the velocity of Usain Bolt in his Beijing 100 meters dash. We will now study this situation analytically. To make our computations simpler, we will approximate that he could run 100 meters in 10 seconds and we will consider the model <m>d=f(t)=t^2</m>, where <m>d</m> is the distance in meters and <m>t</m> is the time in seconds. </p>

<note xml:id="inst-velocity">
<p> The average velocity is the ratio distance covered over time elapsed. If we consider the interval that starts at <m>t=a</m> and has width <m>h</m>, written <m>[a,a+h]</m>, the average velocity on this interval is <m>\displaystyle \dfrac{f(a+h)-f(a)}{(a+h) - a} = \dfrac{f(a+h)-f(a)}{h}</m>. The instantaneous velocity at time <m>t=a</m> is given by: </p>
<me> \lim_{h \to 0} \dfrac{f(a+h)-f(a)}{h}</me>. </note>
<p> The average velocity is the ratio distance covered over time elapsed. If we consider the interval that starts at <m>t=a</m> and has width <m>h</m>, written <m>[a,a+h]</m>, the average velocity on this interval is <m>\displaystyle \dfrac{f(a+h)-f(a)}{(a+h) - a} = \dfrac{f(a+h)-f(a)}{h}</m>. The instantaneous velocity at time <m>t=a</m> is given by:
<me> \lim_{h \to 0} \dfrac{f(a+h)-f(a)}{h}</me>.</p> </note>
</introduction>
<task> <statement> <p> Compute the average velocity on the interval <m>[5,6]</m>. We think of this interval as <m>[5,5+h]</m> for the value of <m>h=1</m>.</p>
</statement>
Expand All @@ -458,41 +468,41 @@
</p>
</answer>
</task>
<task> <statement> <p> Compute the average velocity starting at 5 seconds, but now with <m>h=0.5</m> seconds. </p>
<task> <statement> <p> Compute the average velocity starting at 5 seconds, but now with <m>h=0.5</m> seconds. </p></statement>
<answer>
<p>
10.5
</p>
</answer>
</statement> </task>
<task> <statement> <p>We want to study the instantaneous velocity at <m>a=5</m> seconds. Find an expression for the average velocity on the interval <m>[5,5+h]</m>, where <m>h</m> is an unspecified value.</p>
</task>
<task> <statement> <p>We want to study the instantaneous velocity at <m>a=5</m> seconds. Find an expression for the average velocity on the interval <m>[5,5+h]</m>, where <m>h</m> is an unspecified value.</p></statement>
<answer>
<p>
<m> \dfrac{(5+h)^{2}-5^{2}}{h} </m>
</p>
</answer>
</statement> </task>
<task> <statement> <p>Expand your expression. When <m>h \neq 0</m>, you can simplify it!</p>
</task>
<task> <statement> <p>Expand your expression. When <m>h \neq 0</m>, you can simplify it!</p></statement>
<answer>
<p>
<m> 10+h </m>
</p>
</answer>
</statement> </task>
<task> <statement> <p>Recall that the instantaneous velocity is the limit of your expression as <m>h\to 0</m>. Find the instantaneous velocity given by this model at <m>t=5</m> seconds.</p>
</task>
<task> <statement> <p>Recall that the instantaneous velocity is the limit of your expression as <m>h\to 0</m>. Find the instantaneous velocity given by this model at <m>t=5</m> seconds.</p></statement>
<answer>
<p>
<m> 10 </m>
</p>
</answer>
</statement> </task>
<task> <statement> <p> The model <m>d=f(t)=t^2</m> does not really capture the real-world situation. Think of at least one reason why this model does not fit the scenario of Usain Bolt's 100 meters dash. </p>
</task>
<task> <statement> <p> The model <m>d=f(t)=t^2</m> does not really capture the real-world situation. Think of at least one reason why this model does not fit the scenario of Usain Bolt's 100 meters dash. </p></statement>
<answer>
<p>
With this model, instantaneous rate of change increases as time passes and it doesn't capture that he slows down at the end, which is not a real-world situation.
</p>
</answer>
</statement> </task>
</task>
</activity>

</subsection>
Expand Down