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answer added to1.4 Continuity (LT4) except 1.4.16 #500

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answer added to1.4 Continuity (LT4) except 1.4.16
mlamich1 Dec 18, 2024
f5e5d7d
Update source/calculus/source/01-LT/04.ptx
mlamich1 Dec 18, 2024
2e184fa
Update source/calculus/source/01-LT/04.ptx
mlamich1 Dec 18, 2024
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Update source/calculus/source/01-LT/04.ptx
mlamich1 Dec 18, 2024
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148 changes: 137 additions & 11 deletions source/calculus/source/01-LT/04.ptx
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Expand Up @@ -43,8 +43,13 @@
</p>
</li>
</ol>

</statement>
<answer>
<p>
C. The volume of water in a tank that is gradually filled over time
</p>
</answer>
</activity>

<remark xml:id="activity-continuity-def">
Expand Down Expand Up @@ -84,29 +89,71 @@
</figure>
</introduction>
<task>
<statement>

<p>
For each of the values <m>a = -3</m>, <m>-2</m>, <m>-1</m>, <m>0</m>, <m>1</m>, <m>2</m>, <m>3</m>, determine whether the limit <m>\displaystyle\lim_{x \to a} f(x)</m> exists. If the limit does not exist, be ready to explain why not.
</p>
</statement>
<answer>
<p>
Limit exists at <m> a= -3, -1 0, 1, 3 </m>
</p>
<p>
Limit does not exist at <m>a= -2 </m> (Limit from right is not equal to limit from left) and <m> a= 2 </m> ( the function oscillates at <m>a= 2 </m> )
</p>
</answer>
</task>
<task>
<statement>
<p>
For each of the values of <m>a</m> where the limit of <m>f</m> exists, determine the value of <m>f(a)</m> at each such point.
</p>
</statement>
<answer>
<p>
<m> f(-3) = 3</m> , <m> f(-1)= 1 </m> , <m> f(0)= -4 </m>, <m> f(1)= -2.5 </m> and <m>f(3) </m> DNE.
</p>
</answer>
</task>
<task>
<statement>
<p>
For each such <m>a</m> value, is <m>f(a)</m> equal to <m>\displaystyle\lim_{x \to a} f(x)</m>?
</p>
</statement>
<answer>
<p>
<m>f(-3)= 3 </m> and <m>\displaystyle\lim_{x \to -3} f(x)= 3</m>
</p>
<p>
<m>f(0)= -4 </m> and <m>\displaystyle\lim_{x \to 0} f(x)= -4 </m>
</p>
</answer>
</task>
<task>
<statement>
<p>
Use your understanding of continuity to determine whether <m>f</m> is continuous at each value of <m>a</m>.
</p>
</statement>
<answer>
<p>
<m>f</m> is continuous at those points where the value of function is equal to the limit of the function at the point.
</p>
</answer>
</task>
<task>
<statement>
<p>
Are there any revisions you would make to the definition of continuity that you arrived at toward the end of <xref ref="activity-continuity-def"/>?
</p>
</statement>
<answer>
<p>
<m> f </m> is continuous if at a point <m>a </m> if <m>\displaystyle\lim_{x \to a} f(x)= f(a) </m>
</p>
</answer>
</task>
</activity>

Expand Down Expand Up @@ -145,6 +192,7 @@
<p>
<m>\displaystyle\lim_{x \to -3^+} h(x)</m>
</p>

</li>
<li>
<p>
Expand All @@ -162,7 +210,13 @@
</p>
</li>
</ol>

</statement>
<answer>
<p>
All four choices are equal.
</p>
</answer>
</activity>

<activity xml:id="activity-continuous-graph2" estimated-time="10">
Expand Down Expand Up @@ -199,21 +253,44 @@
<p>
For which values of <m>a</m> do we have <m>\displaystyle\lim_{x \to a^-} f(x) \ne \lim_{x \to a^+} f(x)</m>?
</p>
<answer>
<p>
At <m>a = -2 </m> and <m>a = 2 </m>
</p>
</answer>
</task>
<task>
<p>
For which values of <m>a</m> is <m>f(a)</m> not defined?
</p>
<answer>
<p>
At <m>a = 0 </m>, <m>a= 2</m> and <m>a = 3 </m>
</p>
</answer>
</task>
<task>
<p>
For which values of <m>a</m> does <m>f</m> have a limit at <m>a</m>, yet <m>\displaystyle f(a) \ne \lim_{x \to a} f(x)</m>?
</p>
<answer>
<p>
At <m>a = -2 </m>, <m>a = -1 </m>, <m>a = 2 </m>, and <m>a = 3 </m>
</p>
</answer>
</task>
<task>
<p>
For which values of <m>a</m> does <m>f</m> fail to be continuous? Give a complete list of intervals on which <m>f</m> is continuous.
</p>
<answer>
<p>
<m>f</m> is not continuous at <m>a = -2 </m>, <m>a = -1 </m>, <m>a = 2 </m>, and <m>a = 3 </m>
</p>
<p>
<m>f</m> is continuous at <m>( -\infty, -2) </m>, <m>(-2,-1) </m>, <m>(-1,2) </m>, <m>(2,3 ) </m>, and <m>(3,\infty) </m>
</p>
</answer>
</task>
</activity>

Expand All @@ -234,7 +311,14 @@
</p>
</li>
</ol>

</statement>
<answer>
<p>
B. <m>f</m> is continuous at <m>x = a</m>
</p>

</answer>
</activity>

<activity xml:id="activity-continuity-graph-vs-table-vs-formula" estimated-time="5">
Expand Down Expand Up @@ -265,6 +349,11 @@
</li>
</ol>
</statement>
<answer>
<p>
C. Graphs and formulas only
</p>
</answer>
</activity>

<activity xml:id="activity-continuity-graphs-and-discontinuities" estimated-time="10">
Expand Down Expand Up @@ -297,6 +386,11 @@
Give a list of <m>x</m>-values where <m>f(x)</m> is not continuous. Be prepared to defend your answer based on <xref ref="definition-continuity"/>.
</p>
</statement>
<answer>
<p>
<m> x= 1, 5 </m> and <m> x \geq 7 </m>
</p>
</answer>
</activity>

<remark>
Expand Down Expand Up @@ -324,6 +418,11 @@
\end{cases}
</me>
</p>
<answer>

<p>To make <m>h(x)</m> continuous at <m>x=5</m>, <m>b=4</m>.</p>

</answer>
</task>
<task>
<p>
Expand All @@ -338,17 +437,15 @@ the function <m>f(x)</m>.</p>
\end{cases}
</me>
</p>
<answer>

<p>The function <m>f(x)</m> has a jump discontinuity.</p>


</answer>
</task>

<!-- <answer>
<ol>
<li>To make <m>h(x)</m> continuous at <m>x=5</m>, let <m>b=4</m>.</li>
<li>
The function <m>f(x)</m> has a
jump discontinuity.
</li>
</ol>
</answer>-->

</activity>


Expand Down Expand Up @@ -376,6 +473,11 @@ jump discontinuity.
\end{cases}
</me>
</p>
<answer>
<p>
<m> c = 2 </m>
</p>
</answer>
</task>
<task>
<p>
Expand All @@ -386,6 +488,11 @@ jump discontinuity.
\end{cases}
</me>
</p>
<answer>
<p>
<m> c = \pm 2 </m>
</p>
</answer>
</task>
<task>
<p>
Expand All @@ -396,6 +503,11 @@ jump discontinuity.
\end{cases}
</me>
</p>
<answer>
<p>
<m> c = 0 </m> and <m> c = 1 </m>
</p>
</answer>
</task>
</activity>

Expand Down Expand Up @@ -429,11 +541,25 @@ jump discontinuity.
<task>
<statement>
<p> The part of the theorem that starts with “Suppose…” forms the assumptions of the theorem, while the part of the theorem that starts with “Then…” is the conclusion of the theorem. What are the assumptions of the Intermediate Value Theorem? What is the conclusion?
</p> </statement> </task>
</p> </statement>
<answer>
<p>
Assumption: <p> The function <m>f</m> is continuous on the interval <m>[a,b]</m> ;</p>
<p> If <m>N</m> is the value between <m>f(a)</m> and <m>f(b)</m> such that <m>f(a)\leq N \leq f(b)</m> or <m>f(b)\leq N \leq f(a)</m>.</p>

Conclusion: There exists at least a number <m>c</m> within <m>a</m> to <m>b</m> such that <m> f(c) = k </m>
</p>
</answer>
</task>
<task>
<statement>
<p> Apply the Intermediate Value Theorem to show that the function <m>f(x) = x^3 +x -3</m> has a zero (so crosses the <m>x</m>-axis) at some point between <m>x=-1</m> and <m>x=2</m>. (Hint: What interval of <m>x</m> values is being considered here? What is <m>N</m>? Why is <m>N</m> between <m>f(a)</m> and <m>f(b)</m>?) </p>
</statement>
<answer>
<p>
<m>f(-1) = -5</m> and <m>f(2)= 7 </m> and <m>N = 0 </m>, since <m>-5 \leq 0 \leq 7</m>
</p>
</answer>
</task>
</activity>

Expand Down
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