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Solution to 1.3 Limits-Analytically-(LT3) #481

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Feb 7, 2025
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Solution to 1.3 Limits-Analytically-(LT3) #481

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1439c3c
Solution added for 1.2 Limits Numerically (LT2) except the last one.
mlamich1 Dec 17, 2024
b03916b
Solution to 1.3 Limits-Analytically-(LT3)
mlamich1 Dec 17, 2024
9ebe25e
Update source/calculus/source/01-LT/03.ptx
mlamich1 Dec 18, 2024
e9d9420
Update source/calculus/source/01-LT/03.ptx
mlamich1 Dec 18, 2024
647617d
Update source/calculus/source/01-LT/03.ptx
mlamich1 Dec 18, 2024
b20b93a
Merge branch 'main' into Solution-to-1.3-Limits-Analytically-(LT3)
StevenClontz Feb 7, 2025
f258b71
revert LT2 source file
StevenClontz Feb 7, 2025
3c81d21
Update source/calculus/source/01-LT/02.ptx
StevenClontz Feb 7, 2025
ae4c4b3
Update source/calculus/source/01-LT/03.ptx
StevenClontz Feb 7, 2025
e39b831
Update source/calculus/source/01-LT/03.ptx
StevenClontz Feb 7, 2025
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140 changes: 133 additions & 7 deletions source/calculus/source/01-LT/03.ptx
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Original file line number Diff line number Diff line change
Expand Up @@ -23,6 +23,12 @@
<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>
<answer>
<p>
<m> f(2)= 15 </m> and
<m>\displaystyle \lim_{x \to 2}f(x)=f(2)</m>
</p>
</answer>
</activity>

<activity xml:id="activity-limits-analytically2">
Expand Down Expand Up @@ -141,6 +147,11 @@
<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>
<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>
</p>
</answer>
</activity>

<remark><p> In <xref ref="activity-limits-analytically2"/> we observed that limits seem to be "well-behaved" when combined with standard operations on functions. The next theorems, known as <term>Limit Laws</term>, tell us how limits interact with combinations of functions. </p></remark>
Expand Down Expand Up @@ -174,6 +185,17 @@
<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>
<answer>
<p>
A. <m>\displaystyle \lim_{x\to 2} (f(x) \cdot g(x)) = -6</m>
</p>
<p>
B. <m>\displaystyle \lim_{x\to 2} (f(x) + g(x)) = -1</m>
</p>
<p>
D. <m>\displaystyle \lim_{x\to 2} (f(x)/g(x)) = -2/3</m>
</p>
</answer>
</statement>
</activity>

Expand Down Expand Up @@ -242,14 +264,40 @@
</sidebyside>
</introduction>
<task> <statement> <p> <m>\displaystyle \lim_{x \to 1} f(x) + g(x) </m>. </p>
<answer>
<p>
<m>\displaystyle \lim_{x \to 1} f(x) + g(x) = 2 </m>
</p>
</answer>
</statement> </task>
<task> <statement> <p> <m>\displaystyle \lim_{x \to 5^+} 3f(x) </m>. </p>
<answer>
<p>
<m>\displaystyle \lim_{x \to 5^+} 3f(x)= 0 </m>
</p>
</answer>
</statement> </task>
<task> <statement> <p> <m>\displaystyle \lim_{x \to 0^+ } f(x)g(x) </m>. </p>
<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>
<answer>
<p>
<m>\displaystyle \lim_{x \to 1} g(x) / f(x) </m> does not exist
</p>
</answer>
</statement> </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>
<answer>
<p>
<m>\displaystyle \lim_{x \to 0^+} f(g(x)) </m> does not exist
</p>
</answer>
</statement> </task>
</activity>

Expand All @@ -264,6 +312,21 @@
<li>Power Law</li>
<li>Constant Law</li>
</ol>
<answer>
<p>
A. Sums/Difference Law
</p>
<p>
B. Scalar Multiple Law
</p>
<p>
D. Identity Law
</p>
<p>
F. Constant Law
</p>

</answer>
</statement>
</activity>

Expand All @@ -274,6 +337,7 @@
<statement>
<p>If <m>p(x)</m> is a polynomial and <m>c</m> is a real number, then <m>\displaystyle \lim_{x \to c} p(x) = p(c)</m>. This is also known as the <term>Direct Substitution Property</term><idx>Direct Substitution Property</idx> for polynomials.
</p>

</statement>
</theorem>

Expand All @@ -288,6 +352,11 @@
<li>Quotient and root law</li>

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

Expand All @@ -306,6 +375,11 @@
<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>
<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 @@ -319,6 +393,11 @@
<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>
<answer>
<p>
B. <m>\displaystyle \lim_{x\to 0} (f(x)/g(x)) = 2 </m>
</p>
</answer>
</statement>
</activity>

Expand All @@ -338,12 +417,28 @@
<p>
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>
<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>
<answer>
<p>
<m>-2</m>
</p>
</answer></task>
<task><me>\lim_{x\to5 } \dfrac{ x - 5 }{ \sqrt{x + 31} - 6 }</me>
<answer>
<p>
<m>12</m>
</p>
</answer>
</task>
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</introduction>
<task><statement><p><me>\lim_{x\to-6 } \dfrac{ x^{2} - 6 \, x + 5 }{ x^{2} - 3 \, x - 18 }</me></p></statement></task>
<task><statement><p><me>\lim_{x\to-1 } \dfrac{ x^{2} - 1 }{ x^{2} + 3 \, x + 2 }</me></p></statement></task>
<task><statement><p><me>\lim_{x\to5 } \dfrac{ x - 5 }{ \sqrt{x + 31} - 6 }</me></p></statement></task>
</activity>
</activity>



Expand All @@ -356,16 +451,47 @@
<me> \lim_{h \to 0} \dfrac{f(a+h)-f(a)}{h}</me>. </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> </task>
</statement>
<answer>
<p>
11
</p>
</answer>
</task>
<task> <statement> <p> Compute the average velocity starting at 5 seconds, but now with <m>h=0.5</m> seconds. </p>
<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>
<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>
<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>
<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>
<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>
</activity>

Expand Down