CoCalc -- D3.qti

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  <objectbank ident="D3">
    <qtimetadata>
      <qtimetadatafield><fieldlabel>bank_title</fieldlabel><fieldentry>Differential Equations -- D3</fieldentry></qtimetadatafield>
    </qtimetadata>
  <item ident="D3-4847" title="D3 | Inverse Laplace transforms | ver. 4847"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20-%203%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}" alt="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}" title="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}" title="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5325" title="D3 | Inverse Laplace transforms | ver. 5325"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 1\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 1\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20+%201%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 1\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{2} \, e^{\left(-t\right)} - \frac{1}{2} \, e^{\left(-8 \, t\right)}" alt="\frac{1}{2} \, e^{\left(-t\right)} - \frac{1}{2} \, e^{\left(-8 \, t\right)}" title="\frac{1}{2} \, e^{\left(-t\right)} - \frac{1}{2} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{2} \, e^{\left(-t\right)} - \frac{1}{2} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B2%7D%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B2%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{1}{2} \, e^{\left(-t\right)} - \frac{1}{2} \, e^{\left(-8 \, t\right)}" title="\frac{1}{2} \, e^{\left(-t\right)} - \frac{1}{2} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{2} \, e^{\left(-t\right)} - \frac{1}{2} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7730" title="D3 | Inverse Laplace transforms | ver. 7730"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B7%20%5C,%20%7B%5Cleft(s%20-%205%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{2}{7} \, e^{\left(6 \, t\right)} - \frac{2}{7} \, e^{\left(5 \, t\right)}" alt="\frac{2}{7} \, e^{\left(6 \, t\right)} - \frac{2}{7} \, e^{\left(5 \, t\right)}" title="\frac{2}{7} \, e^{\left(6 \, t\right)} - \frac{2}{7} \, e^{\left(5 \, t\right)}" data-latex="\frac{2}{7} \, e^{\left(6 \, t\right)} - \frac{2}{7} \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B2%7D%7B7%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B2%7D%7B7%7D%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="\frac{2}{7} \, e^{\left(6 \, t\right)} - \frac{2}{7} \, e^{\left(5 \, t\right)}" title="\frac{2}{7} \, e^{\left(6 \, t\right)} - \frac{2}{7} \, e^{\left(5 \, t\right)}" data-latex="\frac{2}{7} \, e^{\left(6 \, t\right)} - \frac{2}{7} \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7994" title="D3 | Inverse Laplace transforms | ver. 7994"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%207%5Cright)%7D%20%7B%5Cleft(s%20-%204%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{55} \, e^{\left(4 \, t\right)} - \frac{3}{55} \, e^{\left(-7 \, t\right)}" alt="\frac{3}{55} \, e^{\left(4 \, t\right)} - \frac{3}{55} \, e^{\left(-7 \, t\right)}" title="\frac{3}{55} \, e^{\left(4 \, t\right)} - \frac{3}{55} \, e^{\left(-7 \, t\right)}" data-latex="\frac{3}{55} \, e^{\left(4 \, t\right)} - \frac{3}{55} \, e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B55%7D%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B55%7D%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="\frac{3}{55} \, e^{\left(4 \, t\right)} - \frac{3}{55} \, e^{\left(-7 \, t\right)}" title="\frac{3}{55} \, e^{\left(4 \, t\right)} - \frac{3}{55} \, e^{\left(-7 \, t\right)}" data-latex="\frac{3}{55} \, e^{\left(4 \, t\right)} - \frac{3}{55} \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-2880" title="D3 | Inverse Laplace transforms | ver. 2880"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B3%20%5C,%20%7B%5Cleft(s%20-%202%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{12} \, e^{\left(6 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}" alt="\frac{7}{12} \, e^{\left(6 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}" title="\frac{7}{12} \, e^{\left(6 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}" data-latex="\frac{7}{12} \, e^{\left(6 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B12%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B12%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="\frac{7}{12} \, e^{\left(6 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}" title="\frac{7}{12} \, e^{\left(6 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}" data-latex="\frac{7}{12} \, e^{\left(6 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-2588" title="D3 | Inverse Laplace transforms | ver. 2588"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20+%203%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{7} \, e^{\left(-3 \, t\right)} - \frac{1}{7} \, e^{\left(-8 \, t\right)}" alt="\frac{1}{7} \, e^{\left(-3 \, t\right)} - \frac{1}{7} \, e^{\left(-8 \, t\right)}" title="\frac{1}{7} \, e^{\left(-3 \, t\right)} - \frac{1}{7} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{7} \, e^{\left(-3 \, t\right)} - \frac{1}{7} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B7%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B7%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{1}{7} \, e^{\left(-3 \, t\right)} - \frac{1}{7} \, e^{\left(-8 \, t\right)}" title="\frac{1}{7} \, e^{\left(-3 \, t\right)} - \frac{1}{7} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{7} \, e^{\left(-3 \, t\right)} - \frac{1}{7} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5020" title="D3 | Inverse Laplace transforms | ver. 5020"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%206%5Cright)%7D%20%7B%5Cleft(s%20+%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{8} \, e^{\left(-2 \, t\right)} - \frac{3}{8} \, e^{\left(-6 \, t\right)}" alt="\frac{3}{8} \, e^{\left(-2 \, t\right)} - \frac{3}{8} \, e^{\left(-6 \, t\right)}" title="\frac{3}{8} \, e^{\left(-2 \, t\right)} - \frac{3}{8} \, e^{\left(-6 \, t\right)}" data-latex="\frac{3}{8} \, e^{\left(-2 \, t\right)} - \frac{3}{8} \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="\frac{3}{8} \, e^{\left(-2 \, t\right)} - \frac{3}{8} \, e^{\left(-6 \, t\right)}" title="\frac{3}{8} \, e^{\left(-2 \, t\right)} - \frac{3}{8} \, e^{\left(-6 \, t\right)}" data-latex="\frac{3}{8} \, e^{\left(-2 \, t\right)} - \frac{3}{8} \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-0968" title="D3 | Inverse Laplace transforms | ver. 0968"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%206%5Cright)%7D%20%7B%5Cleft(s%20+%205%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{5} \, e^{\left(-5 \, t\right)} - \frac{3}{5} \, e^{\left(-6 \, t\right)}" alt="\frac{3}{5} \, e^{\left(-5 \, t\right)} - \frac{3}{5} \, e^{\left(-6 \, t\right)}" title="\frac{3}{5} \, e^{\left(-5 \, t\right)} - \frac{3}{5} \, e^{\left(-6 \, t\right)}" data-latex="\frac{3}{5} \, e^{\left(-5 \, t\right)} - \frac{3}{5} \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B5%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B5%7D%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="\frac{3}{5} \, e^{\left(-5 \, t\right)} - \frac{3}{5} \, e^{\left(-6 \, t\right)}" title="\frac{3}{5} \, e^{\left(-5 \, t\right)} - \frac{3}{5} \, e^{\left(-6 \, t\right)}" data-latex="\frac{3}{5} \, e^{\left(-5 \, t\right)} - \frac{3}{5} \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7893" title="D3 | Inverse Laplace transforms | ver. 7893"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B3%20%5C,%20%7B%5Cleft(s%20-%205%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{3} \, e^{\left(6 \, t\right)} - \frac{7}{3} \, e^{\left(5 \, t\right)}" alt="\frac{7}{3} \, e^{\left(6 \, t\right)} - \frac{7}{3} \, e^{\left(5 \, t\right)}" title="\frac{7}{3} \, e^{\left(6 \, t\right)} - \frac{7}{3} \, e^{\left(5 \, t\right)}" data-latex="\frac{7}{3} \, e^{\left(6 \, t\right)} - \frac{7}{3} \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B3%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B3%7D%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="\frac{7}{3} \, e^{\left(6 \, t\right)} - \frac{7}{3} \, e^{\left(5 \, t\right)}" title="\frac{7}{3} \, e^{\left(6 \, t\right)} - \frac{7}{3} \, e^{\left(5 \, t\right)}" data-latex="\frac{7}{3} \, e^{\left(6 \, t\right)} - \frac{7}{3} \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-8222" title="D3 | Inverse Laplace transforms | ver. 8222"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%204%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{6} \, e^{\left(6 \, t\right)} - \frac{1}{6} \, e^{\left(-4 \, t\right)}" alt="\frac{1}{6} \, e^{\left(6 \, t\right)} - \frac{1}{6} \, e^{\left(-4 \, t\right)}" title="\frac{1}{6} \, e^{\left(6 \, t\right)} - \frac{1}{6} \, e^{\left(-4 \, t\right)}" data-latex="\frac{1}{6} \, e^{\left(6 \, t\right)} - \frac{1}{6} \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B6%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B6%7D%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="\frac{1}{6} \, e^{\left(6 \, t\right)} - \frac{1}{6} \, e^{\left(-4 \, t\right)}" title="\frac{1}{6} \, e^{\left(6 \, t\right)} - \frac{1}{6} \, e^{\left(-4 \, t\right)}" data-latex="\frac{1}{6} \, e^{\left(6 \, t\right)} - \frac{1}{6} \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-1182" title="D3 | Inverse Laplace transforms | ver. 1182"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%205%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{2}{33} \, e^{\left(6 \, t\right)} - \frac{2}{33} \, e^{\left(-5 \, t\right)}" alt="\frac{2}{33} \, e^{\left(6 \, t\right)} - \frac{2}{33} \, e^{\left(-5 \, t\right)}" title="\frac{2}{33} \, e^{\left(6 \, t\right)} - \frac{2}{33} \, e^{\left(-5 \, t\right)}" data-latex="\frac{2}{33} \, e^{\left(6 \, t\right)} - \frac{2}{33} \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B2%7D%7B33%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B2%7D%7B33%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="\frac{2}{33} \, e^{\left(6 \, t\right)} - \frac{2}{33} \, e^{\left(-5 \, t\right)}" title="\frac{2}{33} \, e^{\left(6 \, t\right)} - \frac{2}{33} \, e^{\left(-5 \, t\right)}" data-latex="\frac{2}{33} \, e^{\left(6 \, t\right)} - \frac{2}{33} \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-8007" title="D3 | Inverse Laplace transforms | ver. 8007"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B7%20%5C,%20%7B%5Cleft(s%20-%201%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{42} \, e^{\left(7 \, t\right)} - \frac{5}{42} \, e^{t}" alt="\frac{5}{42} \, e^{\left(7 \, t\right)} - \frac{5}{42} \, e^{t}" title="\frac{5}{42} \, e^{\left(7 \, t\right)} - \frac{5}{42} \, e^{t}" data-latex="\frac{5}{42} \, e^{\left(7 \, t\right)} - \frac{5}{42} \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B42%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B42%7D%20%5C,%20e%5E%7Bt%7D" alt="\frac{5}{42} \, e^{\left(7 \, t\right)} - \frac{5}{42} \, e^{t}" title="\frac{5}{42} \, e^{\left(7 \, t\right)} - \frac{5}{42} \, e^{t}" data-latex="\frac{5}{42} \, e^{\left(7 \, t\right)} - \frac{5}{42} \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-8959" title="D3 | Inverse Laplace transforms | ver. 8959"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%201%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{49} \, e^{\left(6 \, t\right)} - \frac{3}{49} \, e^{\left(-t\right)}" alt="\frac{3}{49} \, e^{\left(6 \, t\right)} - \frac{3}{49} \, e^{\left(-t\right)}" title="\frac{3}{49} \, e^{\left(6 \, t\right)} - \frac{3}{49} \, e^{\left(-t\right)}" data-latex="\frac{3}{49} \, e^{\left(6 \, t\right)} - \frac{3}{49} \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B49%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B49%7D%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="\frac{3}{49} \, e^{\left(6 \, t\right)} - \frac{3}{49} \, e^{\left(-t\right)}" title="\frac{3}{49} \, e^{\left(6 \, t\right)} - \frac{3}{49} \, e^{\left(-t\right)}" data-latex="\frac{3}{49} \, e^{\left(6 \, t\right)} - \frac{3}{49} \, e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7798" title="D3 | Inverse Laplace transforms | ver. 7798"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 9\right)} {\left(s + 4\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 9\right)} {\left(s + 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 9\right)} {\left(s + 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 9\right)} {\left(s + 4\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%209%5Cright)%7D%20%7B%5Cleft(s%20+%204%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 9\right)} {\left(s + 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 9\right)} {\left(s + 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 9\right)} {\left(s + 4\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{3} \, e^{\left(-4 \, t\right)} - \frac{1}{3} \, e^{\left(-9 \, t\right)}" alt="\frac{1}{3} \, e^{\left(-4 \, t\right)} - \frac{1}{3} \, e^{\left(-9 \, t\right)}" title="\frac{1}{3} \, e^{\left(-4 \, t\right)} - \frac{1}{3} \, e^{\left(-9 \, t\right)}" data-latex="\frac{1}{3} \, e^{\left(-4 \, t\right)} - \frac{1}{3} \, e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B3%7D%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="\frac{1}{3} \, e^{\left(-4 \, t\right)} - \frac{1}{3} \, e^{\left(-9 \, t\right)}" title="\frac{1}{3} \, e^{\left(-4 \, t\right)} - \frac{1}{3} \, e^{\left(-9 \, t\right)}" data-latex="\frac{1}{3} \, e^{\left(-4 \, t\right)} - \frac{1}{3} \, e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-9251" title="D3 | Inverse Laplace transforms | ver. 9251"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B7%20%5C,%20%7B%5Cleft(s%20-%202%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{21} \, e^{\left(8 \, t\right)} - \frac{1}{21} \, e^{\left(2 \, t\right)}" alt="\frac{1}{21} \, e^{\left(8 \, t\right)} - \frac{1}{21} \, e^{\left(2 \, t\right)}" title="\frac{1}{21} \, e^{\left(8 \, t\right)} - \frac{1}{21} \, e^{\left(2 \, t\right)}" data-latex="\frac{1}{21} \, e^{\left(8 \, t\right)} - \frac{1}{21} \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B21%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B21%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="\frac{1}{21} \, e^{\left(8 \, t\right)} - \frac{1}{21} \, e^{\left(2 \, t\right)}" title="\frac{1}{21} \, e^{\left(8 \, t\right)} - \frac{1}{21} \, e^{\left(2 \, t\right)}" data-latex="\frac{1}{21} \, e^{\left(8 \, t\right)} - \frac{1}{21} \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-0771" title="D3 | Inverse Laplace transforms | ver. 0771"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20+%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{4} \, e^{\left(-2 \, t\right)} - \frac{1}{4} \, e^{\left(-8 \, t\right)}" alt="\frac{1}{4} \, e^{\left(-2 \, t\right)} - \frac{1}{4} \, e^{\left(-8 \, t\right)}" title="\frac{1}{4} \, e^{\left(-2 \, t\right)} - \frac{1}{4} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{4} \, e^{\left(-2 \, t\right)} - \frac{1}{4} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B4%7D%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B4%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{1}{4} \, e^{\left(-2 \, t\right)} - \frac{1}{4} \, e^{\left(-8 \, t\right)}" title="\frac{1}{4} \, e^{\left(-2 \, t\right)} - \frac{1}{4} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{4} \, e^{\left(-2 \, t\right)} - \frac{1}{4} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-8454" title="D3 | Inverse Laplace transforms | ver. 8454"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20-%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" alt="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" title="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B20%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B20%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" title="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-2533" title="D3 | Inverse Laplace transforms | ver. 2533"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20-%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" alt="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" title="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B20%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B20%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" title="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{20} \, e^{\left(2 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7558" title="D3 | Inverse Laplace transforms | ver. 7558"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B7%20%5C,%20%7B%5Cleft(s%20-%204%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{28} \, e^{\left(8 \, t\right)} - \frac{3}{28} \, e^{\left(4 \, t\right)}" alt="\frac{3}{28} \, e^{\left(8 \, t\right)} - \frac{3}{28} \, e^{\left(4 \, t\right)}" title="\frac{3}{28} \, e^{\left(8 \, t\right)} - \frac{3}{28} \, e^{\left(4 \, t\right)}" data-latex="\frac{3}{28} \, e^{\left(8 \, t\right)} - \frac{3}{28} \, e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B28%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B28%7D%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="\frac{3}{28} \, e^{\left(8 \, t\right)} - \frac{3}{28} \, e^{\left(4 \, t\right)}" title="\frac{3}{28} \, e^{\left(8 \, t\right)} - \frac{3}{28} \, e^{\left(4 \, t\right)}" data-latex="\frac{3}{28} \, e^{\left(8 \, t\right)} - \frac{3}{28} \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-0082" title="D3 | Inverse Laplace transforms | ver. 0082"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%207%5Cright)%7D%20%7B%5Cleft(s%20+%201%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{15} \, e^{\left(-t\right)} - \frac{1}{15} \, e^{\left(-7 \, t\right)}" alt="\frac{1}{15} \, e^{\left(-t\right)} - \frac{1}{15} \, e^{\left(-7 \, t\right)}" title="\frac{1}{15} \, e^{\left(-t\right)} - \frac{1}{15} \, e^{\left(-7 \, t\right)}" data-latex="\frac{1}{15} \, e^{\left(-t\right)} - \frac{1}{15} \, e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B15%7D%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="\frac{1}{15} \, e^{\left(-t\right)} - \frac{1}{15} \, e^{\left(-7 \, t\right)}" title="\frac{1}{15} \, e^{\left(-t\right)} - \frac{1}{15} \, e^{\left(-7 \, t\right)}" data-latex="\frac{1}{15} \, e^{\left(-t\right)} - \frac{1}{15} \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5149" title="D3 | Inverse Laplace transforms | ver. 5149"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B5%20%5C,%20%7B%5Cleft(s%20-%202%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{25} \, e^{\left(7 \, t\right)} - \frac{7}{25} \, e^{\left(2 \, t\right)}" alt="\frac{7}{25} \, e^{\left(7 \, t\right)} - \frac{7}{25} \, e^{\left(2 \, t\right)}" title="\frac{7}{25} \, e^{\left(7 \, t\right)} - \frac{7}{25} \, e^{\left(2 \, t\right)}" data-latex="\frac{7}{25} \, e^{\left(7 \, t\right)} - \frac{7}{25} \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B25%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B25%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="\frac{7}{25} \, e^{\left(7 \, t\right)} - \frac{7}{25} \, e^{\left(2 \, t\right)}" title="\frac{7}{25} \, e^{\left(7 \, t\right)} - \frac{7}{25} \, e^{\left(2 \, t\right)}" data-latex="\frac{7}{25} \, e^{\left(7 \, t\right)} - \frac{7}{25} \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-0104" title="D3 | Inverse Laplace transforms | ver. 0104"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%205%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{91} \, e^{\left(8 \, t\right)} - \frac{5}{91} \, e^{\left(-5 \, t\right)}" alt="\frac{5}{91} \, e^{\left(8 \, t\right)} - \frac{5}{91} \, e^{\left(-5 \, t\right)}" title="\frac{5}{91} \, e^{\left(8 \, t\right)} - \frac{5}{91} \, e^{\left(-5 \, t\right)}" data-latex="\frac{5}{91} \, e^{\left(8 \, t\right)} - \frac{5}{91} \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B91%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B91%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="\frac{5}{91} \, e^{\left(8 \, t\right)} - \frac{5}{91} \, e^{\left(-5 \, t\right)}" title="\frac{5}{91} \, e^{\left(8 \, t\right)} - \frac{5}{91} \, e^{\left(-5 \, t\right)}" data-latex="\frac{5}{91} \, e^{\left(8 \, t\right)} - \frac{5}{91} \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-3224" title="D3 | Inverse Laplace transforms | ver. 3224"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%203%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{22} \, e^{\left(8 \, t\right)} - \frac{7}{22} \, e^{\left(-3 \, t\right)}" alt="\frac{7}{22} \, e^{\left(8 \, t\right)} - \frac{7}{22} \, e^{\left(-3 \, t\right)}" title="\frac{7}{22} \, e^{\left(8 \, t\right)} - \frac{7}{22} \, e^{\left(-3 \, t\right)}" data-latex="\frac{7}{22} \, e^{\left(8 \, t\right)} - \frac{7}{22} \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="\frac{7}{22} \, e^{\left(8 \, t\right)} - \frac{7}{22} \, e^{\left(-3 \, t\right)}" title="\frac{7}{22} \, e^{\left(8 \, t\right)} - \frac{7}{22} \, e^{\left(-3 \, t\right)}" data-latex="\frac{7}{22} \, e^{\left(8 \, t\right)} - \frac{7}{22} \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5784" title="D3 | Inverse Laplace transforms | ver. 5784"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 9\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 9\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B3%20%5C,%20%7B%5Cleft(s%20-%201%5Cright)%7D%20%7B%5Cleft(s%20-%209%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 9\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{24} \, e^{\left(9 \, t\right)} - \frac{5}{24} \, e^{t}" alt="\frac{5}{24} \, e^{\left(9 \, t\right)} - \frac{5}{24} \, e^{t}" title="\frac{5}{24} \, e^{\left(9 \, t\right)} - \frac{5}{24} \, e^{t}" data-latex="\frac{5}{24} \, e^{\left(9 \, t\right)} - \frac{5}{24} \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B24%7D%20%5C,%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B24%7D%20%5C,%20e%5E%7Bt%7D" alt="\frac{5}{24} \, e^{\left(9 \, t\right)} - \frac{5}{24} \, e^{t}" title="\frac{5}{24} \, e^{\left(9 \, t\right)} - \frac{5}{24} \, e^{t}" data-latex="\frac{5}{24} \, e^{\left(9 \, t\right)} - \frac{5}{24} \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-8756" title="D3 | Inverse Laplace transforms | ver. 8756"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20+%203%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s + 3\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{35} \, e^{\left(-3 \, t\right)} - \frac{3}{35} \, e^{\left(-8 \, t\right)}" alt="\frac{3}{35} \, e^{\left(-3 \, t\right)} - \frac{3}{35} \, e^{\left(-8 \, t\right)}" title="\frac{3}{35} \, e^{\left(-3 \, t\right)} - \frac{3}{35} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{35} \, e^{\left(-3 \, t\right)} - \frac{3}{35} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B35%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B35%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{3}{35} \, e^{\left(-3 \, t\right)} - \frac{3}{35} \, e^{\left(-8 \, t\right)}" title="\frac{3}{35} \, e^{\left(-3 \, t\right)} - \frac{3}{35} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{35} \, e^{\left(-3 \, t\right)} - \frac{3}{35} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-1054" title="D3 | Inverse Laplace transforms | ver. 1054"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%204%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{14} \, e^{\left(6 \, t\right)} - \frac{1}{14} \, e^{\left(-4 \, t\right)}" alt="\frac{1}{14} \, e^{\left(6 \, t\right)} - \frac{1}{14} \, e^{\left(-4 \, t\right)}" title="\frac{1}{14} \, e^{\left(6 \, t\right)} - \frac{1}{14} \, e^{\left(-4 \, t\right)}" data-latex="\frac{1}{14} \, e^{\left(6 \, t\right)} - \frac{1}{14} \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B14%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B14%7D%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="\frac{1}{14} \, e^{\left(6 \, t\right)} - \frac{1}{14} \, e^{\left(-4 \, t\right)}" title="\frac{1}{14} \, e^{\left(6 \, t\right)} - \frac{1}{14} \, e^{\left(-4 \, t\right)}" data-latex="\frac{1}{14} \, e^{\left(6 \, t\right)} - \frac{1}{14} \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6428" title="D3 | Inverse Laplace transforms | ver. 6428"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%202%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{24} \, e^{\left(6 \, t\right)} - \frac{7}{24} \, e^{\left(-2 \, t\right)}" alt="\frac{7}{24} \, e^{\left(6 \, t\right)} - \frac{7}{24} \, e^{\left(-2 \, t\right)}" title="\frac{7}{24} \, e^{\left(6 \, t\right)} - \frac{7}{24} \, e^{\left(-2 \, t\right)}" data-latex="\frac{7}{24} \, e^{\left(6 \, t\right)} - \frac{7}{24} \, e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B24%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B24%7D%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="\frac{7}{24} \, e^{\left(6 \, t\right)} - \frac{7}{24} \, e^{\left(-2 \, t\right)}" title="\frac{7}{24} \, e^{\left(6 \, t\right)} - \frac{7}{24} \, e^{\left(-2 \, t\right)}" data-latex="\frac{7}{24} \, e^{\left(6 \, t\right)} - \frac{7}{24} \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-3798" title="D3 | Inverse Laplace transforms | ver. 3798"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%205%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{18} \, e^{\left(7 \, t\right)} - \frac{1}{18} \, e^{\left(-5 \, t\right)}" alt="\frac{1}{18} \, e^{\left(7 \, t\right)} - \frac{1}{18} \, e^{\left(-5 \, t\right)}" title="\frac{1}{18} \, e^{\left(7 \, t\right)} - \frac{1}{18} \, e^{\left(-5 \, t\right)}" data-latex="\frac{1}{18} \, e^{\left(7 \, t\right)} - \frac{1}{18} \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B18%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B18%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="\frac{1}{18} \, e^{\left(7 \, t\right)} - \frac{1}{18} \, e^{\left(-5 \, t\right)}" title="\frac{1}{18} \, e^{\left(7 \, t\right)} - \frac{1}{18} \, e^{\left(-5 \, t\right)}" data-latex="\frac{1}{18} \, e^{\left(7 \, t\right)} - \frac{1}{18} \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5287" title="D3 | Inverse Laplace transforms | ver. 5287"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 6\right)} {\left(s + 1\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 6\right)} {\left(s + 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 6\right)} {\left(s + 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 6\right)} {\left(s + 1\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%206%5Cright)%7D%20%7B%5Cleft(s%20+%201%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 6\right)} {\left(s + 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 6\right)} {\left(s + 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 6\right)} {\left(s + 1\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{3} \, e^{\left(-t\right)} - \frac{1}{3} \, e^{\left(-6 \, t\right)}" alt="\frac{1}{3} \, e^{\left(-t\right)} - \frac{1}{3} \, e^{\left(-6 \, t\right)}" title="\frac{1}{3} \, e^{\left(-t\right)} - \frac{1}{3} \, e^{\left(-6 \, t\right)}" data-latex="\frac{1}{3} \, e^{\left(-t\right)} - \frac{1}{3} \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B3%7D%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="\frac{1}{3} \, e^{\left(-t\right)} - \frac{1}{3} \, e^{\left(-6 \, t\right)}" title="\frac{1}{3} \, e^{\left(-t\right)} - \frac{1}{3} \, e^{\left(-6 \, t\right)}" data-latex="\frac{1}{3} \, e^{\left(-t\right)} - \frac{1}{3} \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6775" title="D3 | Inverse Laplace transforms | ver. 6775"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20-%203%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{33} \, e^{\left(3 \, t\right)} - \frac{7}{33} \, e^{\left(-8 \, t\right)}" alt="\frac{7}{33} \, e^{\left(3 \, t\right)} - \frac{7}{33} \, e^{\left(-8 \, t\right)}" title="\frac{7}{33} \, e^{\left(3 \, t\right)} - \frac{7}{33} \, e^{\left(-8 \, t\right)}" data-latex="\frac{7}{33} \, e^{\left(3 \, t\right)} - \frac{7}{33} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B33%7D%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B33%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{7}{33} \, e^{\left(3 \, t\right)} - \frac{7}{33} \, e^{\left(-8 \, t\right)}" title="\frac{7}{33} \, e^{\left(3 \, t\right)} - \frac{7}{33} \, e^{\left(-8 \, t\right)}" data-latex="\frac{7}{33} \, e^{\left(3 \, t\right)} - \frac{7}{33} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7648" title="D3 | Inverse Laplace transforms | ver. 7648"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%205%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}" alt="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}" title="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}" data-latex="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B77%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B77%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}" title="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}" data-latex="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7696" title="D3 | Inverse Laplace transforms | ver. 7696"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s - 4\right)} {\left(s - 9\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s - 4\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s - 4\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s - 4\right)} {\left(s - 9\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B2%20%5C,%20%7B%5Cleft(s%20-%204%5Cright)%7D%20%7B%5Cleft(s%20-%209%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s - 4\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s - 4\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s - 4\right)} {\left(s - 9\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{10} \, e^{\left(9 \, t\right)} - \frac{3}{10} \, e^{\left(4 \, t\right)}" alt="\frac{3}{10} \, e^{\left(9 \, t\right)} - \frac{3}{10} \, e^{\left(4 \, t\right)}" title="\frac{3}{10} \, e^{\left(9 \, t\right)} - \frac{3}{10} \, e^{\left(4 \, t\right)}" data-latex="\frac{3}{10} \, e^{\left(9 \, t\right)} - \frac{3}{10} \, e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B10%7D%20%5C,%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B10%7D%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="\frac{3}{10} \, e^{\left(9 \, t\right)} - \frac{3}{10} \, e^{\left(4 \, t\right)}" title="\frac{3}{10} \, e^{\left(9 \, t\right)} - \frac{3}{10} \, e^{\left(4 \, t\right)}" data-latex="\frac{3}{10} \, e^{\left(9 \, t\right)} - \frac{3}{10} \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-1820" title="D3 | Inverse Laplace transforms | ver. 1820"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%204%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{77} \, e^{\left(7 \, t\right)} - \frac{5}{77} \, e^{\left(-4 \, t\right)}" alt="\frac{5}{77} \, e^{\left(7 \, t\right)} - \frac{5}{77} \, e^{\left(-4 \, t\right)}" title="\frac{5}{77} \, e^{\left(7 \, t\right)} - \frac{5}{77} \, e^{\left(-4 \, t\right)}" data-latex="\frac{5}{77} \, e^{\left(7 \, t\right)} - \frac{5}{77} \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B77%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B77%7D%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="\frac{5}{77} \, e^{\left(7 \, t\right)} - \frac{5}{77} \, e^{\left(-4 \, t\right)}" title="\frac{5}{77} \, e^{\left(7 \, t\right)} - \frac{5}{77} \, e^{\left(-4 \, t\right)}" data-latex="\frac{5}{77} \, e^{\left(7 \, t\right)} - \frac{5}{77} \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-9076" title="D3 | Inverse Laplace transforms | ver. 9076"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20-%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{70} \, e^{\left(2 \, t\right)} - \frac{3}{70} \, e^{\left(-8 \, t\right)}" alt="\frac{3}{70} \, e^{\left(2 \, t\right)} - \frac{3}{70} \, e^{\left(-8 \, t\right)}" title="\frac{3}{70} \, e^{\left(2 \, t\right)} - \frac{3}{70} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{70} \, e^{\left(2 \, t\right)} - \frac{3}{70} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B70%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B70%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{3}{70} \, e^{\left(2 \, t\right)} - \frac{3}{70} \, e^{\left(-8 \, t\right)}" title="\frac{3}{70} \, e^{\left(2 \, t\right)} - \frac{3}{70} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{70} \, e^{\left(2 \, t\right)} - \frac{3}{70} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6570" title="D3 | Inverse Laplace transforms | ver. 6570"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%201%5Cright)%7D%20%7B%5Cleft(s%20-%209%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}" alt="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}" title="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}" data-latex="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B50%7D%20%5C,%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B50%7D%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}" title="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}" data-latex="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-1034" title="D3 | Inverse Laplace transforms | ver. 1034"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%207%5Cright)%7D%20%7B%5Cleft(s%20-%201%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-\frac{1}{28} \, e^{\left(-7 \, t\right)} + \frac{1}{28} \, e^{t}" alt="-\frac{1}{28} \, e^{\left(-7 \, t\right)} + \frac{1}{28} \, e^{t}" title="-\frac{1}{28} \, e^{\left(-7 \, t\right)} + \frac{1}{28} \, e^{t}" data-latex="-\frac{1}{28} \, e^{\left(-7 \, t\right)} + \frac{1}{28} \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%5Cfrac%7B1%7D%7B28%7D%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D%20+%20%5Cfrac%7B1%7D%7B28%7D%20%5C,%20e%5E%7Bt%7D" alt="-\frac{1}{28} \, e^{\left(-7 \, t\right)} + \frac{1}{28} \, e^{t}" title="-\frac{1}{28} \, e^{\left(-7 \, t\right)} + \frac{1}{28} \, e^{t}" data-latex="-\frac{1}{28} \, e^{\left(-7 \, t\right)} + \frac{1}{28} \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6843" title="D3 | Inverse Laplace transforms | ver. 6843"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B3%20%5C,%20%7B%5Cleft(s%20-%201%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s - 1\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{18} \, e^{\left(7 \, t\right)} - \frac{5}{18} \, e^{t}" alt="\frac{5}{18} \, e^{\left(7 \, t\right)} - \frac{5}{18} \, e^{t}" title="\frac{5}{18} \, e^{\left(7 \, t\right)} - \frac{5}{18} \, e^{t}" data-latex="\frac{5}{18} \, e^{\left(7 \, t\right)} - \frac{5}{18} \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B18%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B18%7D%20%5C,%20e%5E%7Bt%7D" alt="\frac{5}{18} \, e^{\left(7 \, t\right)} - \frac{5}{18} \, e^{t}" title="\frac{5}{18} \, e^{\left(7 \, t\right)} - \frac{5}{18} \, e^{t}" data-latex="\frac{5}{18} \, e^{\left(7 \, t\right)} - \frac{5}{18} \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-4855" title="D3 | Inverse Laplace transforms | ver. 4855"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 9\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 9\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B7%20%5C,%20%7B%5Cleft(s%20-%202%5Cright)%7D%20%7B%5Cleft(s%20-%209%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 2\right)} {\left(s - 9\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{2}{49} \, e^{\left(9 \, t\right)} - \frac{2}{49} \, e^{\left(2 \, t\right)}" alt="\frac{2}{49} \, e^{\left(9 \, t\right)} - \frac{2}{49} \, e^{\left(2 \, t\right)}" title="\frac{2}{49} \, e^{\left(9 \, t\right)} - \frac{2}{49} \, e^{\left(2 \, t\right)}" data-latex="\frac{2}{49} \, e^{\left(9 \, t\right)} - \frac{2}{49} \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B2%7D%7B49%7D%20%5C,%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B2%7D%7B49%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="\frac{2}{49} \, e^{\left(9 \, t\right)} - \frac{2}{49} \, e^{\left(2 \, t\right)}" title="\frac{2}{49} \, e^{\left(9 \, t\right)} - \frac{2}{49} \, e^{\left(2 \, t\right)}" data-latex="\frac{2}{49} \, e^{\left(9 \, t\right)} - \frac{2}{49} \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-1342" title="D3 | Inverse Laplace transforms | ver. 1342"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B2%20%5C,%20%7B%5Cleft(s%20-%202%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{12} \, e^{\left(8 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}" alt="\frac{7}{12} \, e^{\left(8 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}" title="\frac{7}{12} \, e^{\left(8 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}" data-latex="\frac{7}{12} \, e^{\left(8 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B12%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B12%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="\frac{7}{12} \, e^{\left(8 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}" title="\frac{7}{12} \, e^{\left(8 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}" data-latex="\frac{7}{12} \, e^{\left(8 \, t\right)} - \frac{7}{12} \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-9892" title="D3 | Inverse Laplace transforms | ver. 9892"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B5%20%5C,%20%7B%5Cleft(s%20-%201%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{25} \, e^{\left(6 \, t\right)} - \frac{3}{25} \, e^{t}" alt="\frac{3}{25} \, e^{\left(6 \, t\right)} - \frac{3}{25} \, e^{t}" title="\frac{3}{25} \, e^{\left(6 \, t\right)} - \frac{3}{25} \, e^{t}" data-latex="\frac{3}{25} \, e^{\left(6 \, t\right)} - \frac{3}{25} \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B25%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B25%7D%20%5C,%20e%5E%7Bt%7D" alt="\frac{3}{25} \, e^{\left(6 \, t\right)} - \frac{3}{25} \, e^{t}" title="\frac{3}{25} \, e^{\left(6 \, t\right)} - \frac{3}{25} \, e^{t}" data-latex="\frac{3}{25} \, e^{\left(6 \, t\right)} - \frac{3}{25} \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6796" title="D3 | Inverse Laplace transforms | ver. 6796"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%206%5Cright)%7D%20%7B%5Cleft(s%20-%204%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{50} \, e^{\left(4 \, t\right)} - \frac{3}{50} \, e^{\left(-6 \, t\right)}" alt="\frac{3}{50} \, e^{\left(4 \, t\right)} - \frac{3}{50} \, e^{\left(-6 \, t\right)}" title="\frac{3}{50} \, e^{\left(4 \, t\right)} - \frac{3}{50} \, e^{\left(-6 \, t\right)}" data-latex="\frac{3}{50} \, e^{\left(4 \, t\right)} - \frac{3}{50} \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B50%7D%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B50%7D%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="\frac{3}{50} \, e^{\left(4 \, t\right)} - \frac{3}{50} \, e^{\left(-6 \, t\right)}" title="\frac{3}{50} \, e^{\left(4 \, t\right)} - \frac{3}{50} \, e^{\left(-6 \, t\right)}" data-latex="\frac{3}{50} \, e^{\left(4 \, t\right)} - \frac{3}{50} \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-9057" title="D3 | Inverse Laplace transforms | ver. 9057"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 4\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 4\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20-%204%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 4\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{8} \, e^{\left(4 \, t\right)} - \frac{1}{8} \, e^{\left(-8 \, t\right)}" alt="\frac{1}{8} \, e^{\left(4 \, t\right)} - \frac{1}{8} \, e^{\left(-8 \, t\right)}" title="\frac{1}{8} \, e^{\left(4 \, t\right)} - \frac{1}{8} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{8} \, e^{\left(4 \, t\right)} - \frac{1}{8} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{1}{8} \, e^{\left(4 \, t\right)} - \frac{1}{8} \, e^{\left(-8 \, t\right)}" title="\frac{1}{8} \, e^{\left(4 \, t\right)} - \frac{1}{8} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{8} \, e^{\left(4 \, t\right)} - \frac{1}{8} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-9419" title="D3 | Inverse Laplace transforms | ver. 9419"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%201%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 1\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{2}{21} \, e^{\left(6 \, t\right)} - \frac{2}{21} \, e^{\left(-t\right)}" alt="\frac{2}{21} \, e^{\left(6 \, t\right)} - \frac{2}{21} \, e^{\left(-t\right)}" title="\frac{2}{21} \, e^{\left(6 \, t\right)} - \frac{2}{21} \, e^{\left(-t\right)}" data-latex="\frac{2}{21} \, e^{\left(6 \, t\right)} - \frac{2}{21} \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B2%7D%7B21%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B2%7D%7B21%7D%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="\frac{2}{21} \, e^{\left(6 \, t\right)} - \frac{2}{21} \, e^{\left(-t\right)}" title="\frac{2}{21} \, e^{\left(6 \, t\right)} - \frac{2}{21} \, e^{\left(-t\right)}" data-latex="\frac{2}{21} \, e^{\left(6 \, t\right)} - \frac{2}{21} \, e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5049" title="D3 | Inverse Laplace transforms | ver. 5049"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%209%5Cright)%7D%20%7B%5Cleft(s%20-%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{55} \, e^{\left(2 \, t\right)} - \frac{3}{55} \, e^{\left(-9 \, t\right)}" alt="\frac{3}{55} \, e^{\left(2 \, t\right)} - \frac{3}{55} \, e^{\left(-9 \, t\right)}" title="\frac{3}{55} \, e^{\left(2 \, t\right)} - \frac{3}{55} \, e^{\left(-9 \, t\right)}" data-latex="\frac{3}{55} \, e^{\left(2 \, t\right)} - \frac{3}{55} \, e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B55%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B55%7D%20%5C,%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="\frac{3}{55} \, e^{\left(2 \, t\right)} - \frac{3}{55} \, e^{\left(-9 \, t\right)}" title="\frac{3}{55} \, e^{\left(2 \, t\right)} - \frac{3}{55} \, e^{\left(-9 \, t\right)}" data-latex="\frac{3}{55} \, e^{\left(2 \, t\right)} - \frac{3}{55} \, e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6854" title="D3 | Inverse Laplace transforms | ver. 6854"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 3\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 3\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 3\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 3\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B7%20%5C,%20%7B%5Cleft(s%20-%203%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 3\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 3\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 3\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{21} \, e^{\left(6 \, t\right)} - \frac{5}{21} \, e^{\left(3 \, t\right)}" alt="\frac{5}{21} \, e^{\left(6 \, t\right)} - \frac{5}{21} \, e^{\left(3 \, t\right)}" title="\frac{5}{21} \, e^{\left(6 \, t\right)} - \frac{5}{21} \, e^{\left(3 \, t\right)}" data-latex="\frac{5}{21} \, e^{\left(6 \, t\right)} - \frac{5}{21} \, e^{\left(3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B21%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B21%7D%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D" alt="\frac{5}{21} \, e^{\left(6 \, t\right)} - \frac{5}{21} \, e^{\left(3 \, t\right)}" title="\frac{5}{21} \, e^{\left(6 \, t\right)} - \frac{5}{21} \, e^{\left(3 \, t\right)}" data-latex="\frac{5}{21} \, e^{\left(6 \, t\right)} - \frac{5}{21} \, e^{\left(3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5463" title="D3 | Inverse Laplace transforms | ver. 5463"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%205%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}" alt="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}" title="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}" data-latex="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B77%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B77%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}" title="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}" data-latex="\frac{3}{77} \, e^{\left(6 \, t\right)} - \frac{3}{77} \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7915" title="D3 | Inverse Laplace transforms | ver. 7915"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20-%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}" alt="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}" title="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B15%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}" title="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-9946" title="D3 | Inverse Laplace transforms | ver. 9946"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 4\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 4\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B5%20%5C,%20%7B%5Cleft(s%20-%204%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 4\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{2}{15} \, e^{\left(7 \, t\right)} - \frac{2}{15} \, e^{\left(4 \, t\right)}" alt="\frac{2}{15} \, e^{\left(7 \, t\right)} - \frac{2}{15} \, e^{\left(4 \, t\right)}" title="\frac{2}{15} \, e^{\left(7 \, t\right)} - \frac{2}{15} \, e^{\left(4 \, t\right)}" data-latex="\frac{2}{15} \, e^{\left(7 \, t\right)} - \frac{2}{15} \, e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B2%7D%7B15%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B2%7D%7B15%7D%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="\frac{2}{15} \, e^{\left(7 \, t\right)} - \frac{2}{15} \, e^{\left(4 \, t\right)}" title="\frac{2}{15} \, e^{\left(7 \, t\right)} - \frac{2}{15} \, e^{\left(4 \, t\right)}" data-latex="\frac{2}{15} \, e^{\left(7 \, t\right)} - \frac{2}{15} \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-4651" title="D3 | Inverse Laplace transforms | ver. 4651"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%203%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{27} \, e^{\left(6 \, t\right)} - \frac{7}{27} \, e^{\left(-3 \, t\right)}" alt="\frac{7}{27} \, e^{\left(6 \, t\right)} - \frac{7}{27} \, e^{\left(-3 \, t\right)}" title="\frac{7}{27} \, e^{\left(6 \, t\right)} - \frac{7}{27} \, e^{\left(-3 \, t\right)}" data-latex="\frac{7}{27} \, e^{\left(6 \, t\right)} - \frac{7}{27} \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B27%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B27%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="\frac{7}{27} \, e^{\left(6 \, t\right)} - \frac{7}{27} \, e^{\left(-3 \, t\right)}" title="\frac{7}{27} \, e^{\left(6 \, t\right)} - \frac{7}{27} \, e^{\left(-3 \, t\right)}" data-latex="\frac{7}{27} \, e^{\left(6 \, t\right)} - \frac{7}{27} \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-9084" title="D3 | Inverse Laplace transforms | ver. 9084"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B3%20%5C,%20%7B%5Cleft(s%20-%204%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 4\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{6} \, e^{\left(8 \, t\right)} - \frac{1}{6} \, e^{\left(4 \, t\right)}" alt="\frac{1}{6} \, e^{\left(8 \, t\right)} - \frac{1}{6} \, e^{\left(4 \, t\right)}" title="\frac{1}{6} \, e^{\left(8 \, t\right)} - \frac{1}{6} \, e^{\left(4 \, t\right)}" data-latex="\frac{1}{6} \, e^{\left(8 \, t\right)} - \frac{1}{6} \, e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B6%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B6%7D%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="\frac{1}{6} \, e^{\left(8 \, t\right)} - \frac{1}{6} \, e^{\left(4 \, t\right)}" title="\frac{1}{6} \, e^{\left(8 \, t\right)} - \frac{1}{6} \, e^{\left(4 \, t\right)}" data-latex="\frac{1}{6} \, e^{\left(8 \, t\right)} - \frac{1}{6} \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7906" title="D3 | Inverse Laplace transforms | ver. 7906"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%204%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{22} \, e^{\left(7 \, t\right)} - \frac{3}{22} \, e^{\left(-4 \, t\right)}" alt="\frac{3}{22} \, e^{\left(7 \, t\right)} - \frac{3}{22} \, e^{\left(-4 \, t\right)}" title="\frac{3}{22} \, e^{\left(7 \, t\right)} - \frac{3}{22} \, e^{\left(-4 \, t\right)}" data-latex="\frac{3}{22} \, e^{\left(7 \, t\right)} - \frac{3}{22} \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="\frac{3}{22} \, e^{\left(7 \, t\right)} - \frac{3}{22} \, e^{\left(-4 \, t\right)}" title="\frac{3}{22} \, e^{\left(7 \, t\right)} - \frac{3}{22} \, e^{\left(-4 \, t\right)}" data-latex="\frac{3}{22} \, e^{\left(7 \, t\right)} - \frac{3}{22} \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5092" title="D3 | Inverse Laplace transforms | ver. 5092"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20+%204%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{20} \, e^{\left(-4 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" alt="\frac{3}{20} \, e^{\left(-4 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" title="\frac{3}{20} \, e^{\left(-4 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{20} \, e^{\left(-4 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B20%7D%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B20%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{3}{20} \, e^{\left(-4 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" title="\frac{3}{20} \, e^{\left(-4 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{20} \, e^{\left(-4 \, t\right)} - \frac{3}{20} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-2947" title="D3 | Inverse Laplace transforms | ver. 2947"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 4\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 4\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 4\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 4\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B5%20%5C,%20%7B%5Cleft(s%20-%204%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 4\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 4\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 4\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{\left(4 \, t\right)}" alt="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{\left(4 \, t\right)}" title="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{\left(4 \, t\right)}" data-latex="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{\left(4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B10%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B10%7D%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D" alt="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{\left(4 \, t\right)}" title="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{\left(4 \, t\right)}" data-latex="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{\left(4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6112" title="D3 | Inverse Laplace transforms | ver. 6112"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%207%5Cright)%7D%20%7B%5Cleft(s%20+%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{2} \, e^{\left(-2 \, t\right)} - \frac{1}{2} \, e^{\left(-7 \, t\right)}" alt="\frac{1}{2} \, e^{\left(-2 \, t\right)} - \frac{1}{2} \, e^{\left(-7 \, t\right)}" title="\frac{1}{2} \, e^{\left(-2 \, t\right)} - \frac{1}{2} \, e^{\left(-7 \, t\right)}" data-latex="\frac{1}{2} \, e^{\left(-2 \, t\right)} - \frac{1}{2} \, e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B2%7D%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B2%7D%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="\frac{1}{2} \, e^{\left(-2 \, t\right)} - \frac{1}{2} \, e^{\left(-7 \, t\right)}" title="\frac{1}{2} \, e^{\left(-2 \, t\right)} - \frac{1}{2} \, e^{\left(-7 \, t\right)}" data-latex="\frac{1}{2} \, e^{\left(-2 \, t\right)} - \frac{1}{2} \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-2064" title="D3 | Inverse Laplace transforms | ver. 2064"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s - 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s - 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%207%5Cright)%7D%20%7B%5Cleft(s%20-%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s - 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{45} \, e^{\left(2 \, t\right)} - \frac{7}{45} \, e^{\left(-7 \, t\right)}" alt="\frac{7}{45} \, e^{\left(2 \, t\right)} - \frac{7}{45} \, e^{\left(-7 \, t\right)}" title="\frac{7}{45} \, e^{\left(2 \, t\right)} - \frac{7}{45} \, e^{\left(-7 \, t\right)}" data-latex="\frac{7}{45} \, e^{\left(2 \, t\right)} - \frac{7}{45} \, e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B45%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B45%7D%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="\frac{7}{45} \, e^{\left(2 \, t\right)} - \frac{7}{45} \, e^{\left(-7 \, t\right)}" title="\frac{7}{45} \, e^{\left(2 \, t\right)} - \frac{7}{45} \, e^{\left(-7 \, t\right)}" data-latex="\frac{7}{45} \, e^{\left(2 \, t\right)} - \frac{7}{45} \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-4815" title="D3 | Inverse Laplace transforms | ver. 4815"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 9\right)} {\left(s + 3\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 9\right)} {\left(s + 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 9\right)} {\left(s + 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 9\right)} {\left(s + 3\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%209%5Cright)%7D%20%7B%5Cleft(s%20+%203%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 9\right)} {\left(s + 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 9\right)} {\left(s + 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 9\right)} {\left(s + 3\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{9} \, e^{\left(-3 \, t\right)} - \frac{1}{9} \, e^{\left(-9 \, t\right)}" alt="\frac{1}{9} \, e^{\left(-3 \, t\right)} - \frac{1}{9} \, e^{\left(-9 \, t\right)}" title="\frac{1}{9} \, e^{\left(-3 \, t\right)} - \frac{1}{9} \, e^{\left(-9 \, t\right)}" data-latex="\frac{1}{9} \, e^{\left(-3 \, t\right)} - \frac{1}{9} \, e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B9%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="\frac{1}{9} \, e^{\left(-3 \, t\right)} - \frac{1}{9} \, e^{\left(-9 \, t\right)}" title="\frac{1}{9} \, e^{\left(-3 \, t\right)} - \frac{1}{9} \, e^{\left(-9 \, t\right)}" data-latex="\frac{1}{9} \, e^{\left(-3 \, t\right)} - \frac{1}{9} \, e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6791" title="D3 | Inverse Laplace transforms | ver. 6791"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%209%5Cright)%7D%20%7B%5Cleft(s%20+%205%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{8} \, e^{\left(-5 \, t\right)} - \frac{7}{8} \, e^{\left(-9 \, t\right)}" alt="\frac{7}{8} \, e^{\left(-5 \, t\right)} - \frac{7}{8} \, e^{\left(-9 \, t\right)}" title="\frac{7}{8} \, e^{\left(-5 \, t\right)} - \frac{7}{8} \, e^{\left(-9 \, t\right)}" data-latex="\frac{7}{8} \, e^{\left(-5 \, t\right)} - \frac{7}{8} \, e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="\frac{7}{8} \, e^{\left(-5 \, t\right)} - \frac{7}{8} \, e^{\left(-9 \, t\right)}" title="\frac{7}{8} \, e^{\left(-5 \, t\right)} - \frac{7}{8} \, e^{\left(-9 \, t\right)}" data-latex="\frac{7}{8} \, e^{\left(-5 \, t\right)} - \frac{7}{8} \, e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-3505" title="D3 | Inverse Laplace transforms | ver. 3505"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%207%5Cright)%7D%20%7B%5Cleft(s%20+%201%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 7\right)} {\left(s + 1\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{9} \, e^{\left(-t\right)} - \frac{1}{9} \, e^{\left(-7 \, t\right)}" alt="\frac{1}{9} \, e^{\left(-t\right)} - \frac{1}{9} \, e^{\left(-7 \, t\right)}" title="\frac{1}{9} \, e^{\left(-t\right)} - \frac{1}{9} \, e^{\left(-7 \, t\right)}" data-latex="\frac{1}{9} \, e^{\left(-t\right)} - \frac{1}{9} \, e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B9%7D%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="\frac{1}{9} \, e^{\left(-t\right)} - \frac{1}{9} \, e^{\left(-7 \, t\right)}" title="\frac{1}{9} \, e^{\left(-t\right)} - \frac{1}{9} \, e^{\left(-7 \, t\right)}" data-latex="\frac{1}{9} \, e^{\left(-t\right)} - \frac{1}{9} \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-2631" title="D3 | Inverse Laplace transforms | ver. 2631"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%203%5Cright)%7D%20%7B%5Cleft(s%20-%209%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{18} \, e^{\left(9 \, t\right)} - \frac{1}{18} \, e^{\left(-3 \, t\right)}" alt="\frac{1}{18} \, e^{\left(9 \, t\right)} - \frac{1}{18} \, e^{\left(-3 \, t\right)}" title="\frac{1}{18} \, e^{\left(9 \, t\right)} - \frac{1}{18} \, e^{\left(-3 \, t\right)}" data-latex="\frac{1}{18} \, e^{\left(9 \, t\right)} - \frac{1}{18} \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B18%7D%20%5C,%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B18%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="\frac{1}{18} \, e^{\left(9 \, t\right)} - \frac{1}{18} \, e^{\left(-3 \, t\right)}" title="\frac{1}{18} \, e^{\left(9 \, t\right)} - \frac{1}{18} \, e^{\left(-3 \, t\right)}" data-latex="\frac{1}{18} \, e^{\left(9 \, t\right)} - \frac{1}{18} \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7832" title="D3 | Inverse Laplace transforms | ver. 7832"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 8\right)} {\left(s - 1\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 8\right)} {\left(s - 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 8\right)} {\left(s - 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 8\right)} {\left(s - 1\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20-%201%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 8\right)} {\left(s - 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 8\right)} {\left(s - 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 8\right)} {\left(s - 1\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-\frac{5}{27} \, e^{\left(-8 \, t\right)} + \frac{5}{27} \, e^{t}" alt="-\frac{5}{27} \, e^{\left(-8 \, t\right)} + \frac{5}{27} \, e^{t}" title="-\frac{5}{27} \, e^{\left(-8 \, t\right)} + \frac{5}{27} \, e^{t}" data-latex="-\frac{5}{27} \, e^{\left(-8 \, t\right)} + \frac{5}{27} \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%5Cfrac%7B5%7D%7B27%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D%20+%20%5Cfrac%7B5%7D%7B27%7D%20%5C,%20e%5E%7Bt%7D" alt="-\frac{5}{27} \, e^{\left(-8 \, t\right)} + \frac{5}{27} \, e^{t}" title="-\frac{5}{27} \, e^{\left(-8 \, t\right)} + \frac{5}{27} \, e^{t}" data-latex="-\frac{5}{27} \, e^{\left(-8 \, t\right)} + \frac{5}{27} \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5825" title="D3 | Inverse Laplace transforms | ver. 5825"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%204%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{22} \, e^{\left(7 \, t\right)} - \frac{5}{22} \, e^{\left(-4 \, t\right)}" alt="\frac{5}{22} \, e^{\left(7 \, t\right)} - \frac{5}{22} \, e^{\left(-4 \, t\right)}" title="\frac{5}{22} \, e^{\left(7 \, t\right)} - \frac{5}{22} \, e^{\left(-4 \, t\right)}" data-latex="\frac{5}{22} \, e^{\left(7 \, t\right)} - \frac{5}{22} \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="\frac{5}{22} \, e^{\left(7 \, t\right)} - \frac{5}{22} \, e^{\left(-4 \, t\right)}" title="\frac{5}{22} \, e^{\left(7 \, t\right)} - \frac{5}{22} \, e^{\left(-4 \, t\right)}" data-latex="\frac{5}{22} \, e^{\left(7 \, t\right)} - \frac{5}{22} \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6145" title="D3 | Inverse Laplace transforms | ver. 6145"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B5%20%5C,%20%7B%5Cleft(s%20-%202%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{30} \, e^{\left(8 \, t\right)} - \frac{7}{30} \, e^{\left(2 \, t\right)}" alt="\frac{7}{30} \, e^{\left(8 \, t\right)} - \frac{7}{30} \, e^{\left(2 \, t\right)}" title="\frac{7}{30} \, e^{\left(8 \, t\right)} - \frac{7}{30} \, e^{\left(2 \, t\right)}" data-latex="\frac{7}{30} \, e^{\left(8 \, t\right)} - \frac{7}{30} \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B30%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B30%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="\frac{7}{30} \, e^{\left(8 \, t\right)} - \frac{7}{30} \, e^{\left(2 \, t\right)}" title="\frac{7}{30} \, e^{\left(8 \, t\right)} - \frac{7}{30} \, e^{\left(2 \, t\right)}" data-latex="\frac{7}{30} \, e^{\left(8 \, t\right)} - \frac{7}{30} \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-0358" title="D3 | Inverse Laplace transforms | ver. 0358"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 1\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 1\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 1\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 1\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B3%20%5C,%20%7B%5Cleft(s%20-%201%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 1\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 1\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s - 1\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{3} \, e^{\left(8 \, t\right)} - \frac{1}{3} \, e^{t}" alt="\frac{1}{3} \, e^{\left(8 \, t\right)} - \frac{1}{3} \, e^{t}" title="\frac{1}{3} \, e^{\left(8 \, t\right)} - \frac{1}{3} \, e^{t}" data-latex="\frac{1}{3} \, e^{\left(8 \, t\right)} - \frac{1}{3} \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B3%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B3%7D%20%5C,%20e%5E%7Bt%7D" alt="\frac{1}{3} \, e^{\left(8 \, t\right)} - \frac{1}{3} \, e^{t}" title="\frac{1}{3} \, e^{\left(8 \, t\right)} - \frac{1}{3} \, e^{t}" data-latex="\frac{1}{3} \, e^{\left(8 \, t\right)} - \frac{1}{3} \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-8489" title="D3 | Inverse Laplace transforms | ver. 8489"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%203%5Cright)%7D%20%7B%5Cleft(s%20-%209%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 3\right)} {\left(s - 9\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{20} \, e^{\left(9 \, t\right)} - \frac{1}{20} \, e^{\left(-3 \, t\right)}" alt="\frac{1}{20} \, e^{\left(9 \, t\right)} - \frac{1}{20} \, e^{\left(-3 \, t\right)}" title="\frac{1}{20} \, e^{\left(9 \, t\right)} - \frac{1}{20} \, e^{\left(-3 \, t\right)}" data-latex="\frac{1}{20} \, e^{\left(9 \, t\right)} - \frac{1}{20} \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B20%7D%20%5C,%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B20%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="\frac{1}{20} \, e^{\left(9 \, t\right)} - \frac{1}{20} \, e^{\left(-3 \, t\right)}" title="\frac{1}{20} \, e^{\left(9 \, t\right)} - \frac{1}{20} \, e^{\left(-3 \, t\right)}" data-latex="\frac{1}{20} \, e^{\left(9 \, t\right)} - \frac{1}{20} \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-1479" title="D3 | Inverse Laplace transforms | ver. 1479"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%202%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 2\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{56} \, e^{\left(6 \, t\right)} - \frac{3}{56} \, e^{\left(-2 \, t\right)}" alt="\frac{3}{56} \, e^{\left(6 \, t\right)} - \frac{3}{56} \, e^{\left(-2 \, t\right)}" title="\frac{3}{56} \, e^{\left(6 \, t\right)} - \frac{3}{56} \, e^{\left(-2 \, t\right)}" data-latex="\frac{3}{56} \, e^{\left(6 \, t\right)} - \frac{3}{56} \, e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B56%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B56%7D%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="\frac{3}{56} \, e^{\left(6 \, t\right)} - \frac{3}{56} \, e^{\left(-2 \, t\right)}" title="\frac{3}{56} \, e^{\left(6 \, t\right)} - \frac{3}{56} \, e^{\left(-2 \, t\right)}" data-latex="\frac{3}{56} \, e^{\left(6 \, t\right)} - \frac{3}{56} \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-3814" title="D3 | Inverse Laplace transforms | ver. 3814"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s + 5\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s + 5\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s + 5\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s + 5\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%207%5Cright)%7D%20%7B%5Cleft(s%20+%205%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s + 5\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s + 5\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 7\right)} {\left(s + 5\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{10} \, e^{\left(-5 \, t\right)} - \frac{7}{10} \, e^{\left(-7 \, t\right)}" alt="\frac{7}{10} \, e^{\left(-5 \, t\right)} - \frac{7}{10} \, e^{\left(-7 \, t\right)}" title="\frac{7}{10} \, e^{\left(-5 \, t\right)} - \frac{7}{10} \, e^{\left(-7 \, t\right)}" data-latex="\frac{7}{10} \, e^{\left(-5 \, t\right)} - \frac{7}{10} \, e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B10%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B10%7D%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="\frac{7}{10} \, e^{\left(-5 \, t\right)} - \frac{7}{10} \, e^{\left(-7 \, t\right)}" title="\frac{7}{10} \, e^{\left(-5 \, t\right)} - \frac{7}{10} \, e^{\left(-7 \, t\right)}" data-latex="\frac{7}{10} \, e^{\left(-5 \, t\right)} - \frac{7}{10} \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-0415" title="D3 | Inverse Laplace transforms | ver. 0415"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B7%20%5C,%20%7B%5Cleft(s%20-%205%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s - 5\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{7} \, e^{\left(6 \, t\right)} - \frac{5}{7} \, e^{\left(5 \, t\right)}" alt="\frac{5}{7} \, e^{\left(6 \, t\right)} - \frac{5}{7} \, e^{\left(5 \, t\right)}" title="\frac{5}{7} \, e^{\left(6 \, t\right)} - \frac{5}{7} \, e^{\left(5 \, t\right)}" data-latex="\frac{5}{7} \, e^{\left(6 \, t\right)} - \frac{5}{7} \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B7%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B7%7D%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="\frac{5}{7} \, e^{\left(6 \, t\right)} - \frac{5}{7} \, e^{\left(5 \, t\right)}" title="\frac{5}{7} \, e^{\left(6 \, t\right)} - \frac{5}{7} \, e^{\left(5 \, t\right)}" data-latex="\frac{5}{7} \, e^{\left(6 \, t\right)} - \frac{5}{7} \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-4782" title="D3 | Inverse Laplace transforms | ver. 4782"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%207%5Cright)%7D%20%7B%5Cleft(s%20-%201%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 7\right)} {\left(s - 1\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-\frac{3}{40} \, e^{\left(-7 \, t\right)} + \frac{3}{40} \, e^{t}" alt="-\frac{3}{40} \, e^{\left(-7 \, t\right)} + \frac{3}{40} \, e^{t}" title="-\frac{3}{40} \, e^{\left(-7 \, t\right)} + \frac{3}{40} \, e^{t}" data-latex="-\frac{3}{40} \, e^{\left(-7 \, t\right)} + \frac{3}{40} \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%5Cfrac%7B3%7D%7B40%7D%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D%20+%20%5Cfrac%7B3%7D%7B40%7D%20%5C,%20e%5E%7Bt%7D" alt="-\frac{3}{40} \, e^{\left(-7 \, t\right)} + \frac{3}{40} \, e^{t}" title="-\frac{3}{40} \, e^{\left(-7 \, t\right)} + \frac{3}{40} \, e^{t}" data-latex="-\frac{3}{40} \, e^{\left(-7 \, t\right)} + \frac{3}{40} \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6104" title="D3 | Inverse Laplace transforms | ver. 6104"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 9\right)} {\left(s - 1\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 9\right)} {\left(s - 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 9\right)} {\left(s - 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 9\right)} {\left(s - 1\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%209%5Cright)%7D%20%7B%5Cleft(s%20-%201%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 9\right)} {\left(s - 1\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 9\right)} {\left(s - 1\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{7 \, {\left(s + 9\right)} {\left(s - 1\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-\frac{1}{14} \, e^{\left(-9 \, t\right)} + \frac{1}{14} \, e^{t}" alt="-\frac{1}{14} \, e^{\left(-9 \, t\right)} + \frac{1}{14} \, e^{t}" title="-\frac{1}{14} \, e^{\left(-9 \, t\right)} + \frac{1}{14} \, e^{t}" data-latex="-\frac{1}{14} \, e^{\left(-9 \, t\right)} + \frac{1}{14} \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-%5Cfrac%7B1%7D%7B14%7D%20%5C,%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D%20+%20%5Cfrac%7B1%7D%7B14%7D%20%5C,%20e%5E%7Bt%7D" alt="-\frac{1}{14} \, e^{\left(-9 \, t\right)} + \frac{1}{14} \, e^{t}" title="-\frac{1}{14} \, e^{\left(-9 \, t\right)} + \frac{1}{14} \, e^{t}" data-latex="-\frac{1}{14} \, e^{\left(-9 \, t\right)} + \frac{1}{14} \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6809" title="D3 | Inverse Laplace transforms | ver. 6809"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%206%5Cright)%7D%20%7B%5Cleft(s%20-%204%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 6\right)} {\left(s - 4\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{35} \, e^{\left(4 \, t\right)} - \frac{1}{35} \, e^{\left(-6 \, t\right)}" alt="\frac{1}{35} \, e^{\left(4 \, t\right)} - \frac{1}{35} \, e^{\left(-6 \, t\right)}" title="\frac{1}{35} \, e^{\left(4 \, t\right)} - \frac{1}{35} \, e^{\left(-6 \, t\right)}" data-latex="\frac{1}{35} \, e^{\left(4 \, t\right)} - \frac{1}{35} \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B35%7D%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B35%7D%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="\frac{1}{35} \, e^{\left(4 \, t\right)} - \frac{1}{35} \, e^{\left(-6 \, t\right)}" title="\frac{1}{35} \, e^{\left(4 \, t\right)} - \frac{1}{35} \, e^{\left(-6 \, t\right)}" data-latex="\frac{1}{35} \, e^{\left(4 \, t\right)} - \frac{1}{35} \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7227" title="D3 | Inverse Laplace transforms | ver. 7227"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%207%5Cright)%7D%20%7B%5Cleft(s%20-%204%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{33} \, e^{\left(4 \, t\right)} - \frac{5}{33} \, e^{\left(-7 \, t\right)}" alt="\frac{5}{33} \, e^{\left(4 \, t\right)} - \frac{5}{33} \, e^{\left(-7 \, t\right)}" title="\frac{5}{33} \, e^{\left(4 \, t\right)} - \frac{5}{33} \, e^{\left(-7 \, t\right)}" data-latex="\frac{5}{33} \, e^{\left(4 \, t\right)} - \frac{5}{33} \, e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B33%7D%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B33%7D%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="\frac{5}{33} \, e^{\left(4 \, t\right)} - \frac{5}{33} \, e^{\left(-7 \, t\right)}" title="\frac{5}{33} \, e^{\left(4 \, t\right)} - \frac{5}{33} \, e^{\left(-7 \, t\right)}" data-latex="\frac{5}{33} \, e^{\left(4 \, t\right)} - \frac{5}{33} \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6632" title="D3 | Inverse Laplace transforms | ver. 6632"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B2%20%5C,%20%7B%5Cleft(s%20-%201%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{t}" alt="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{t}" title="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{t}" data-latex="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B10%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B10%7D%20%5C,%20e%5E%7Bt%7D" alt="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{t}" title="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{t}" data-latex="\frac{7}{10} \, e^{\left(6 \, t\right)} - \frac{7}{10} \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5444" title="D3 | Inverse Laplace transforms | ver. 5444"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 9\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 9\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%205%5Cright)%7D%20%7B%5Cleft(s%20-%209%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 9\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{35} \, e^{\left(9 \, t\right)} - \frac{1}{35} \, e^{\left(-5 \, t\right)}" alt="\frac{1}{35} \, e^{\left(9 \, t\right)} - \frac{1}{35} \, e^{\left(-5 \, t\right)}" title="\frac{1}{35} \, e^{\left(9 \, t\right)} - \frac{1}{35} \, e^{\left(-5 \, t\right)}" data-latex="\frac{1}{35} \, e^{\left(9 \, t\right)} - \frac{1}{35} \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B35%7D%20%5C,%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B35%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="\frac{1}{35} \, e^{\left(9 \, t\right)} - \frac{1}{35} \, e^{\left(-5 \, t\right)}" title="\frac{1}{35} \, e^{\left(9 \, t\right)} - \frac{1}{35} \, e^{\left(-5 \, t\right)}" data-latex="\frac{1}{35} \, e^{\left(9 \, t\right)} - \frac{1}{35} \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-1934" title="D3 | Inverse Laplace transforms | ver. 1934"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 9\right)} {\left(s + 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 9\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 9\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 9\right)} {\left(s + 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%209%5Cright)%7D%20%7B%5Cleft(s%20+%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 9\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 9\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 9\right)} {\left(s + 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{2}{49} \, e^{\left(-2 \, t\right)} - \frac{2}{49} \, e^{\left(-9 \, t\right)}" alt="\frac{2}{49} \, e^{\left(-2 \, t\right)} - \frac{2}{49} \, e^{\left(-9 \, t\right)}" title="\frac{2}{49} \, e^{\left(-2 \, t\right)} - \frac{2}{49} \, e^{\left(-9 \, t\right)}" data-latex="\frac{2}{49} \, e^{\left(-2 \, t\right)} - \frac{2}{49} \, e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B2%7D%7B49%7D%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B2%7D%7B49%7D%20%5C,%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="\frac{2}{49} \, e^{\left(-2 \, t\right)} - \frac{2}{49} \, e^{\left(-9 \, t\right)}" title="\frac{2}{49} \, e^{\left(-2 \, t\right)} - \frac{2}{49} \, e^{\left(-9 \, t\right)}" data-latex="\frac{2}{49} \, e^{\left(-2 \, t\right)} - \frac{2}{49} \, e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6268" title="D3 | Inverse Laplace transforms | ver. 6268"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%203%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 3\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{27} \, e^{\left(6 \, t\right)} - \frac{5}{27} \, e^{\left(-3 \, t\right)}" alt="\frac{5}{27} \, e^{\left(6 \, t\right)} - \frac{5}{27} \, e^{\left(-3 \, t\right)}" title="\frac{5}{27} \, e^{\left(6 \, t\right)} - \frac{5}{27} \, e^{\left(-3 \, t\right)}" data-latex="\frac{5}{27} \, e^{\left(6 \, t\right)} - \frac{5}{27} \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B27%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B27%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="\frac{5}{27} \, e^{\left(6 \, t\right)} - \frac{5}{27} \, e^{\left(-3 \, t\right)}" title="\frac{5}{27} \, e^{\left(6 \, t\right)} - \frac{5}{27} \, e^{\left(-3 \, t\right)}" data-latex="\frac{5}{27} \, e^{\left(6 \, t\right)} - \frac{5}{27} \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6523" title="D3 | Inverse Laplace transforms | ver. 6523"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%207%5Cright)%7D%20%7B%5Cleft(s%20-%204%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s - 4\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{22} \, e^{\left(4 \, t\right)} - \frac{5}{22} \, e^{\left(-7 \, t\right)}" alt="\frac{5}{22} \, e^{\left(4 \, t\right)} - \frac{5}{22} \, e^{\left(-7 \, t\right)}" title="\frac{5}{22} \, e^{\left(4 \, t\right)} - \frac{5}{22} \, e^{\left(-7 \, t\right)}" data-latex="\frac{5}{22} \, e^{\left(4 \, t\right)} - \frac{5}{22} \, e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(4%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="\frac{5}{22} \, e^{\left(4 \, t\right)} - \frac{5}{22} \, e^{\left(-7 \, t\right)}" title="\frac{5}{22} \, e^{\left(4 \, t\right)} - \frac{5}{22} \, e^{\left(-7 \, t\right)}" data-latex="\frac{5}{22} \, e^{\left(4 \, t\right)} - \frac{5}{22} \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-3819" title="D3 | Inverse Laplace transforms | ver. 3819"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B7%20%5C,%20%7B%5Cleft(s%20-%201%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s - 1\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{2}{35} \, e^{\left(6 \, t\right)} - \frac{2}{35} \, e^{t}" alt="\frac{2}{35} \, e^{\left(6 \, t\right)} - \frac{2}{35} \, e^{t}" title="\frac{2}{35} \, e^{\left(6 \, t\right)} - \frac{2}{35} \, e^{t}" data-latex="\frac{2}{35} \, e^{\left(6 \, t\right)} - \frac{2}{35} \, e^{t}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B2%7D%7B35%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B2%7D%7B35%7D%20%5C,%20e%5E%7Bt%7D" alt="\frac{2}{35} \, e^{\left(6 \, t\right)} - \frac{2}{35} \, e^{t}" title="\frac{2}{35} \, e^{\left(6 \, t\right)} - \frac{2}{35} \, e^{t}" data-latex="\frac{2}{35} \, e^{\left(6 \, t\right)} - \frac{2}{35} \, e^{t}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-9081" title="D3 | Inverse Laplace transforms | ver. 9081"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 6\right)} {\left(s - 3\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 6\right)} {\left(s - 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 6\right)} {\left(s - 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 6\right)} {\left(s - 3\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%206%5Cright)%7D%20%7B%5Cleft(s%20-%203%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 6\right)} {\left(s - 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 6\right)} {\left(s - 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 6\right)} {\left(s - 3\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{27} \, e^{\left(3 \, t\right)} - \frac{7}{27} \, e^{\left(-6 \, t\right)}" alt="\frac{7}{27} \, e^{\left(3 \, t\right)} - \frac{7}{27} \, e^{\left(-6 \, t\right)}" title="\frac{7}{27} \, e^{\left(3 \, t\right)} - \frac{7}{27} \, e^{\left(-6 \, t\right)}" data-latex="\frac{7}{27} \, e^{\left(3 \, t\right)} - \frac{7}{27} \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B27%7D%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B27%7D%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="\frac{7}{27} \, e^{\left(3 \, t\right)} - \frac{7}{27} \, e^{\left(-6 \, t\right)}" title="\frac{7}{27} \, e^{\left(3 \, t\right)} - \frac{7}{27} \, e^{\left(-6 \, t\right)}" data-latex="\frac{7}{27} \, e^{\left(3 \, t\right)} - \frac{7}{27} \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-2198" title="D3 | Inverse Laplace transforms | ver. 2198"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 5\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 5\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 5\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 5\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B5%20%5C,%20%7B%5Cleft(s%20-%205%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 5\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 5\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s - 5\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{5} \, e^{\left(7 \, t\right)} - \frac{1}{5} \, e^{\left(5 \, t\right)}" alt="\frac{1}{5} \, e^{\left(7 \, t\right)} - \frac{1}{5} \, e^{\left(5 \, t\right)}" title="\frac{1}{5} \, e^{\left(7 \, t\right)} - \frac{1}{5} \, e^{\left(5 \, t\right)}" data-latex="\frac{1}{5} \, e^{\left(7 \, t\right)} - \frac{1}{5} \, e^{\left(5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B5%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B5%7D%20%5C,%20e%5E%7B%5Cleft(5%20%5C,%20t%5Cright)%7D" alt="\frac{1}{5} \, e^{\left(7 \, t\right)} - \frac{1}{5} \, e^{\left(5 \, t\right)}" title="\frac{1}{5} \, e^{\left(7 \, t\right)} - \frac{1}{5} \, e^{\left(5 \, t\right)}" data-latex="\frac{1}{5} \, e^{\left(7 \, t\right)} - \frac{1}{5} \, e^{\left(5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-0334" title="D3 | Inverse Laplace transforms | ver. 0334"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%209%5Cright)%7D%20%7B%5Cleft(s%20-%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 9\right)} {\left(s - 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{33} \, e^{\left(2 \, t\right)} - \frac{7}{33} \, e^{\left(-9 \, t\right)}" alt="\frac{7}{33} \, e^{\left(2 \, t\right)} - \frac{7}{33} \, e^{\left(-9 \, t\right)}" title="\frac{7}{33} \, e^{\left(2 \, t\right)} - \frac{7}{33} \, e^{\left(-9 \, t\right)}" data-latex="\frac{7}{33} \, e^{\left(2 \, t\right)} - \frac{7}{33} \, e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B33%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B33%7D%20%5C,%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="\frac{7}{33} \, e^{\left(2 \, t\right)} - \frac{7}{33} \, e^{\left(-9 \, t\right)}" title="\frac{7}{33} \, e^{\left(2 \, t\right)} - \frac{7}{33} \, e^{\left(-9 \, t\right)}" data-latex="\frac{7}{33} \, e^{\left(2 \, t\right)} - \frac{7}{33} \, e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-4963" title="D3 | Inverse Laplace transforms | ver. 4963"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20+%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{5 \, {\left(s + 8\right)} {\left(s + 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{10} \, e^{\left(-2 \, t\right)} - \frac{1}{10} \, e^{\left(-8 \, t\right)}" alt="\frac{1}{10} \, e^{\left(-2 \, t\right)} - \frac{1}{10} \, e^{\left(-8 \, t\right)}" title="\frac{1}{10} \, e^{\left(-2 \, t\right)} - \frac{1}{10} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{10} \, e^{\left(-2 \, t\right)} - \frac{1}{10} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B10%7D%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B10%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{1}{10} \, e^{\left(-2 \, t\right)} - \frac{1}{10} \, e^{\left(-8 \, t\right)}" title="\frac{1}{10} \, e^{\left(-2 \, t\right)} - \frac{1}{10} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{10} \, e^{\left(-2 \, t\right)} - \frac{1}{10} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-0337" title="D3 | Inverse Laplace transforms | ver. 0337"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%205%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 5\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{30} \, e^{\left(7 \, t\right)} - \frac{1}{30} \, e^{\left(-5 \, t\right)}" alt="\frac{1}{30} \, e^{\left(7 \, t\right)} - \frac{1}{30} \, e^{\left(-5 \, t\right)}" title="\frac{1}{30} \, e^{\left(7 \, t\right)} - \frac{1}{30} \, e^{\left(-5 \, t\right)}" data-latex="\frac{1}{30} \, e^{\left(7 \, t\right)} - \frac{1}{30} \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B30%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B30%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="\frac{1}{30} \, e^{\left(7 \, t\right)} - \frac{1}{30} \, e^{\left(-5 \, t\right)}" title="\frac{1}{30} \, e^{\left(7 \, t\right)} - \frac{1}{30} \, e^{\left(-5 \, t\right)}" data-latex="\frac{1}{30} \, e^{\left(7 \, t\right)} - \frac{1}{30} \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-3391" title="D3 | Inverse Laplace transforms | ver. 3391"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B2%20%5C,%20%7B%5Cleft(s%20-%202%5Cright)%7D%20%7B%5Cleft(s%20-%206%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s - 2\right)} {\left(s - 6\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{8} \, e^{\left(6 \, t\right)} - \frac{7}{8} \, e^{\left(2 \, t\right)}" alt="\frac{7}{8} \, e^{\left(6 \, t\right)} - \frac{7}{8} \, e^{\left(2 \, t\right)}" title="\frac{7}{8} \, e^{\left(6 \, t\right)} - \frac{7}{8} \, e^{\left(2 \, t\right)}" data-latex="\frac{7}{8} \, e^{\left(6 \, t\right)} - \frac{7}{8} \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(6%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="\frac{7}{8} \, e^{\left(6 \, t\right)} - \frac{7}{8} \, e^{\left(2 \, t\right)}" title="\frac{7}{8} \, e^{\left(6 \, t\right)} - \frac{7}{8} \, e^{\left(2 \, t\right)}" data-latex="\frac{7}{8} \, e^{\left(6 \, t\right)} - \frac{7}{8} \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-8147" title="D3 | Inverse Laplace transforms | ver. 8147"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%203%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}" alt="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}" title="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}" data-latex="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B2%7D%7B55%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B2%7D%7B55%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}" title="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}" data-latex="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-4367" title="D3 | Inverse Laplace transforms | ver. 4367"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 3\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 3\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%207%5Cright)%7D%20%7B%5Cleft(s%20+%203%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{2 \, {\left(s + 7\right)} {\left(s + 3\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{8} \, e^{\left(-3 \, t\right)} - \frac{5}{8} \, e^{\left(-7 \, t\right)}" alt="\frac{5}{8} \, e^{\left(-3 \, t\right)} - \frac{5}{8} \, e^{\left(-7 \, t\right)}" title="\frac{5}{8} \, e^{\left(-3 \, t\right)} - \frac{5}{8} \, e^{\left(-7 \, t\right)}" data-latex="\frac{5}{8} \, e^{\left(-3 \, t\right)} - \frac{5}{8} \, e^{\left(-7 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-7%20%5C,%20t%5Cright)%7D" alt="\frac{5}{8} \, e^{\left(-3 \, t\right)} - \frac{5}{8} \, e^{\left(-7 \, t\right)}" title="\frac{5}{8} \, e^{\left(-3 \, t\right)} - \frac{5}{8} \, e^{\left(-7 \, t\right)}" data-latex="\frac{5}{8} \, e^{\left(-3 \, t\right)} - \frac{5}{8} \, e^{\left(-7 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-0897" title="D3 | Inverse Laplace transforms | ver. 0897"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%209%5Cright)%7D%20%7B%5Cleft(s%20+%205%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 9\right)} {\left(s + 5\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{20} \, e^{\left(-5 \, t\right)} - \frac{7}{20} \, e^{\left(-9 \, t\right)}" alt="\frac{7}{20} \, e^{\left(-5 \, t\right)} - \frac{7}{20} \, e^{\left(-9 \, t\right)}" title="\frac{7}{20} \, e^{\left(-5 \, t\right)} - \frac{7}{20} \, e^{\left(-9 \, t\right)}" data-latex="\frac{7}{20} \, e^{\left(-5 \, t\right)} - \frac{7}{20} \, e^{\left(-9 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B20%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B20%7D%20%5C,%20e%5E%7B%5Cleft(-9%20%5C,%20t%5Cright)%7D" alt="\frac{7}{20} \, e^{\left(-5 \, t\right)} - \frac{7}{20} \, e^{\left(-9 \, t\right)}" title="\frac{7}{20} \, e^{\left(-5 \, t\right)} - \frac{7}{20} \, e^{\left(-9 \, t\right)}" data-latex="\frac{7}{20} \, e^{\left(-5 \, t\right)} - \frac{7}{20} \, e^{\left(-9 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5667" title="D3 | Inverse Laplace transforms | ver. 5667"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%201%5Cright)%7D%20%7B%5Cleft(s%20-%209%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{5 \, {\left(s + 1\right)} {\left(s - 9\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}" alt="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}" title="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}" data-latex="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B50%7D%20%5C,%20e%5E%7B%5Cleft(9%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B50%7D%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}" title="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}" data-latex="\frac{7}{50} \, e^{\left(9 \, t\right)} - \frac{7}{50} \, e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-3802" title="D3 | Inverse Laplace transforms | ver. 3802"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 2\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 2\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 2\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 2\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%202%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 2\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 2\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{7 \, {\left(s + 2\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{35} \, e^{\left(8 \, t\right)} - \frac{1}{35} \, e^{\left(-2 \, t\right)}" alt="\frac{1}{35} \, e^{\left(8 \, t\right)} - \frac{1}{35} \, e^{\left(-2 \, t\right)}" title="\frac{1}{35} \, e^{\left(8 \, t\right)} - \frac{1}{35} \, e^{\left(-2 \, t\right)}" data-latex="\frac{1}{35} \, e^{\left(8 \, t\right)} - \frac{1}{35} \, e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B35%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B35%7D%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="\frac{1}{35} \, e^{\left(8 \, t\right)} - \frac{1}{35} \, e^{\left(-2 \, t\right)}" title="\frac{1}{35} \, e^{\left(8 \, t\right)} - \frac{1}{35} \, e^{\left(-2 \, t\right)}" data-latex="\frac{1}{35} \, e^{\left(8 \, t\right)} - \frac{1}{35} \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-5705" title="D3 | Inverse Laplace transforms | ver. 5705"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%204%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{22} \, e^{\left(7 \, t\right)} - \frac{7}{22} \, e^{\left(-4 \, t\right)}" alt="\frac{7}{22} \, e^{\left(7 \, t\right)} - \frac{7}{22} \, e^{\left(-4 \, t\right)}" title="\frac{7}{22} \, e^{\left(7 \, t\right)} - \frac{7}{22} \, e^{\left(-4 \, t\right)}" data-latex="\frac{7}{22} \, e^{\left(7 \, t\right)} - \frac{7}{22} \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="\frac{7}{22} \, e^{\left(7 \, t\right)} - \frac{7}{22} \, e^{\left(-4 \, t\right)}" title="\frac{7}{22} \, e^{\left(7 \, t\right)} - \frac{7}{22} \, e^{\left(-4 \, t\right)}" data-latex="\frac{7}{22} \, e^{\left(7 \, t\right)} - \frac{7}{22} \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7078" title="D3 | Inverse Laplace transforms | ver. 7078"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20+%204%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{8} \, e^{\left(-4 \, t\right)} - \frac{3}{8} \, e^{\left(-8 \, t\right)}" alt="\frac{3}{8} \, e^{\left(-4 \, t\right)} - \frac{3}{8} \, e^{\left(-8 \, t\right)}" title="\frac{3}{8} \, e^{\left(-4 \, t\right)} - \frac{3}{8} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{8} \, e^{\left(-4 \, t\right)} - \frac{3}{8} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{3}{8} \, e^{\left(-4 \, t\right)} - \frac{3}{8} \, e^{\left(-8 \, t\right)}" title="\frac{3}{8} \, e^{\left(-4 \, t\right)} - \frac{3}{8} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{8} \, e^{\left(-4 \, t\right)} - \frac{3}{8} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-1298" title="D3 | Inverse Laplace transforms | ver. 1298"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B3%20%5C,%20%7B%5Cleft(s%20-%202%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s - 2\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{9} \, e^{\left(8 \, t\right)} - \frac{1}{9} \, e^{\left(2 \, t\right)}" alt="\frac{1}{9} \, e^{\left(8 \, t\right)} - \frac{1}{9} \, e^{\left(2 \, t\right)}" title="\frac{1}{9} \, e^{\left(8 \, t\right)} - \frac{1}{9} \, e^{\left(2 \, t\right)}" data-latex="\frac{1}{9} \, e^{\left(8 \, t\right)} - \frac{1}{9} \, e^{\left(2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B9%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B9%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="\frac{1}{9} \, e^{\left(8 \, t\right)} - \frac{1}{9} \, e^{\left(2 \, t\right)}" title="\frac{1}{9} \, e^{\left(8 \, t\right)} - \frac{1}{9} \, e^{\left(2 \, t\right)}" data-latex="\frac{1}{9} \, e^{\left(8 \, t\right)} - \frac{1}{9} \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6959" title="D3 | Inverse Laplace transforms | ver. 6959"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B7%20%5C,%20%7B%5Cleft(s%20+%205%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{7 \, {\left(s + 5\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{91} \, e^{\left(8 \, t\right)} - \frac{3}{91} \, e^{\left(-5 \, t\right)}" alt="\frac{3}{91} \, e^{\left(8 \, t\right)} - \frac{3}{91} \, e^{\left(-5 \, t\right)}" title="\frac{3}{91} \, e^{\left(8 \, t\right)} - \frac{3}{91} \, e^{\left(-5 \, t\right)}" data-latex="\frac{3}{91} \, e^{\left(8 \, t\right)} - \frac{3}{91} \, e^{\left(-5 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B3%7D%7B91%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B91%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D" alt="\frac{3}{91} \, e^{\left(8 \, t\right)} - \frac{3}{91} \, e^{\left(-5 \, t\right)}" title="\frac{3}{91} \, e^{\left(8 \, t\right)} - \frac{3}{91} \, e^{\left(-5 \, t\right)}" data-latex="\frac{3}{91} \, e^{\left(8 \, t\right)} - \frac{3}{91} \, e^{\left(-5 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-3766" title="D3 | Inverse Laplace transforms | ver. 3766"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%206%5Cright)%7D%20%7B%5Cleft(s%20+%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{8} \, e^{\left(-2 \, t\right)} - \frac{7}{8} \, e^{\left(-6 \, t\right)}" alt="\frac{7}{8} \, e^{\left(-2 \, t\right)} - \frac{7}{8} \, e^{\left(-6 \, t\right)}" title="\frac{7}{8} \, e^{\left(-2 \, t\right)} - \frac{7}{8} \, e^{\left(-6 \, t\right)}" data-latex="\frac{7}{8} \, e^{\left(-2 \, t\right)} - \frac{7}{8} \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="\frac{7}{8} \, e^{\left(-2 \, t\right)} - \frac{7}{8} \, e^{\left(-6 \, t\right)}" title="\frac{7}{8} \, e^{\left(-2 \, t\right)} - \frac{7}{8} \, e^{\left(-6 \, t\right)}" data-latex="\frac{7}{8} \, e^{\left(-2 \, t\right)} - \frac{7}{8} \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-1051" title="D3 | Inverse Laplace transforms | ver. 1051"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%206%5Cright)%7D%20%7B%5Cleft(s%20+%205%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 6\right)} {\left(s + 5\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{2} \, e^{\left(-5 \, t\right)} - \frac{7}{2} \, e^{\left(-6 \, t\right)}" alt="\frac{7}{2} \, e^{\left(-5 \, t\right)} - \frac{7}{2} \, e^{\left(-6 \, t\right)}" title="\frac{7}{2} \, e^{\left(-5 \, t\right)} - \frac{7}{2} \, e^{\left(-6 \, t\right)}" data-latex="\frac{7}{2} \, e^{\left(-5 \, t\right)} - \frac{7}{2} \, e^{\left(-6 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B2%7D%20%5C,%20e%5E%7B%5Cleft(-5%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B2%7D%20%5C,%20e%5E%7B%5Cleft(-6%20%5C,%20t%5Cright)%7D" alt="\frac{7}{2} \, e^{\left(-5 \, t\right)} - \frac{7}{2} \, e^{\left(-6 \, t\right)}" title="\frac{7}{2} \, e^{\left(-5 \, t\right)} - \frac{7}{2} \, e^{\left(-6 \, t\right)}" data-latex="\frac{7}{2} \, e^{\left(-5 \, t\right)} - \frac{7}{2} \, e^{\left(-6 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6594" title="D3 | Inverse Laplace transforms | ver. 6594"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%204%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{3 \, {\left(s + 4\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{33} \, e^{\left(7 \, t\right)} - \frac{7}{33} \, e^{\left(-4 \, t\right)}" alt="\frac{7}{33} \, e^{\left(7 \, t\right)} - \frac{7}{33} \, e^{\left(-4 \, t\right)}" title="\frac{7}{33} \, e^{\left(7 \, t\right)} - \frac{7}{33} \, e^{\left(-4 \, t\right)}" data-latex="\frac{7}{33} \, e^{\left(7 \, t\right)} - \frac{7}{33} \, e^{\left(-4 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B33%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B33%7D%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D" alt="\frac{7}{33} \, e^{\left(7 \, t\right)} - \frac{7}{33} \, e^{\left(-4 \, t\right)}" title="\frac{7}{33} \, e^{\left(7 \, t\right)} - \frac{7}{33} \, e^{\left(-4 \, t\right)}" data-latex="\frac{7}{33} \, e^{\left(7 \, t\right)} - \frac{7}{33} \, e^{\left(-4 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-8466" title="D3 | Inverse Laplace transforms | ver. 8466"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%203%5Cright)%7D%20%7B%5Cleft(s%20-%208%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 8\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}" alt="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}" title="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}" data-latex="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B2%7D%7B55%7D%20%5C,%20e%5E%7B%5Cleft(8%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B2%7D%7B55%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}" title="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}" data-latex="\frac{2}{55} \, e^{\left(8 \, t\right)} - \frac{2}{55} \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-8009" title="D3 | Inverse Laplace transforms | ver. 8009"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20-%202%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{3 \, {\left(s + 8\right)} {\left(s - 2\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}" alt="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}" title="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B15%7D%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B15%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}" title="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}" data-latex="\frac{1}{15} \, e^{\left(2 \, t\right)} - \frac{1}{15} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-6019" title="D3 | Inverse Laplace transforms | ver. 6019"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B2%7D%7B5%20%5C,%20%7B%5Cleft(s%20+%203%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{2}{5 \, {\left(s + 3\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{25} \, e^{\left(7 \, t\right)} - \frac{1}{25} \, e^{\left(-3 \, t\right)}" alt="\frac{1}{25} \, e^{\left(7 \, t\right)} - \frac{1}{25} \, e^{\left(-3 \, t\right)}" title="\frac{1}{25} \, e^{\left(7 \, t\right)} - \frac{1}{25} \, e^{\left(-3 \, t\right)}" data-latex="\frac{1}{25} \, e^{\left(7 \, t\right)} - \frac{1}{25} \, e^{\left(-3 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7B25%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B1%7D%7B25%7D%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="\frac{1}{25} \, e^{\left(7 \, t\right)} - \frac{1}{25} \, e^{\left(-3 \, t\right)}" title="\frac{1}{25} \, e^{\left(7 \, t\right)} - \frac{1}{25} \, e^{\left(-3 \, t\right)}" data-latex="\frac{1}{25} \, e^{\left(7 \, t\right)} - \frac{1}{25} \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-7211" title="D3 | Inverse Laplace transforms | ver. 7211"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 2\right)} {\left(s - 7\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 2\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 2\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 2\right)} {\left(s - 7\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B5%7D%7B3%20%5C,%20%7B%5Cleft(s%20+%202%5Cright)%7D%20%7B%5Cleft(s%20-%207%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 2\right)} {\left(s - 7\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 2\right)} {\left(s - 7\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{5}{3 \, {\left(s + 2\right)} {\left(s - 7\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{5}{27} \, e^{\left(7 \, t\right)} - \frac{5}{27} \, e^{\left(-2 \, t\right)}" alt="\frac{5}{27} \, e^{\left(7 \, t\right)} - \frac{5}{27} \, e^{\left(-2 \, t\right)}" title="\frac{5}{27} \, e^{\left(7 \, t\right)} - \frac{5}{27} \, e^{\left(-2 \, t\right)}" data-latex="\frac{5}{27} \, e^{\left(7 \, t\right)} - \frac{5}{27} \, e^{\left(-2 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B5%7D%7B27%7D%20%5C,%20e%5E%7B%5Cleft(7%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B5%7D%7B27%7D%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="\frac{5}{27} \, e^{\left(7 \, t\right)} - \frac{5}{27} \, e^{\left(-2 \, t\right)}" title="\frac{5}{27} \, e^{\left(7 \, t\right)} - \frac{5}{27} \, e^{\left(-2 \, t\right)}" data-latex="\frac{5}{27} \, e^{\left(7 \, t\right)} - \frac{5}{27} \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item><item ident="D3-4271" title="D3 | Inverse Laplace transforms | ver. 4271"><itemmetadata><qtimetadata><qtimetadatafield><fieldlabel>question_type</fieldlabel><fieldentry>essay_question</fieldentry></qtimetadatafield></qtimetadata></itemmetadata><presentation><material><mattextxml><div class="exercise-statement"><p><strong>D3.</strong></p><p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </p><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B7%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20+%204%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{7}{2 \, {\left(s + 8\right)} {\left(s + 4\right)}} \right\}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material><response_str ident="response1" rcardinality="Single"><render_fib><response_label ident="answer1" rshuffle="No"/></render_fib></response_str></presentation><itemfeedback ident="general_fb"><flow_mat><material><mattextxml><div class="exercise-answer"><h4>Partial Answer:</h4><p style="text-align:center;"><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{7}{8} \, e^{\left(-4 \, t\right)} - \frac{7}{8} \, e^{\left(-8 \, t\right)}" alt="\frac{7}{8} \, e^{\left(-4 \, t\right)} - \frac{7}{8} \, e^{\left(-8 \, t\right)}" title="\frac{7}{8} \, e^{\left(-4 \, t\right)} - \frac{7}{8} \, e^{\left(-8 \, t\right)}" data-latex="\frac{7}{8} \, e^{\left(-4 \, t\right)} - \frac{7}{8} \, e^{\left(-8 \, t\right)}"/></p></div></mattextxml><mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B7%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-4%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B7%7D%7B8%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{7}{8} \, e^{\left(-4 \, t\right)} - \frac{7}{8} \, e^{\left(-8 \, t\right)}" title="\frac{7}{8} \, e^{\left(-4 \, t\right)} - \frac{7}{8} \, e^{\left(-8 \, t\right)}" data-latex="\frac{7}{8} \, e^{\left(-4 \, t\right)} - \frac{7}{8} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext></material></flow_mat></itemfeedback></item></objectbank>
</questestinterop>
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