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    <name>
      <text>23.5.1.4 Task 1 cosine series</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
Obtain a half-range Fourier Cosine series to represent the function
\[ f(t) = 4 - t \hspace{1cm} 0 &lt; t &lt; 4 \]&nbsp;<p style="text-align: center;"><img src="@@PLUGINFILE@@/23-5-1-task-4-t.svg" alt="4-t" width="297" height="109" class="img-responsive atto_image_button_text-bottom"><br></p><p>&nbsp;Write you answers in the boxes below to correspond to the form 
\[ \dfrac{a_0}{2} + a_1 \cos c_1 t + a_2 \cos c_2 t  + a_3 \cos c_3 t + a_4 \cos c_4 t  + a_5 \cos c_5 t + \dots \]
You answers will be displayed below in the series as you fill in the boxes.
</p><p></p>
\(\dfrac{a_0}{2} =\) [[input:ans1]] \(a_1 =\) [[input:ans2]] \(c_1 =\) [[input:ans3]] \(a_2 =\) [[input:ans4]] \(c_2 =\) [[input:ans5]]  \(a_3 =\) [[input:ans6]] \(c_3 =\) [[input:ans7]] <p></p>
\(a_4 =\) [[input:ans8]] \(c_4 =\) [[input:ans9]] 
\(a_5 =\) [[input:ans10]] \(c_5 =\) [[input:ans11]] <p></p>
[[validation:ans1]] \(+\) [[validation:ans2]] \(\cos\) [[validation:ans3]] \(t\) \(+\) [[validation:ans4]] \(\cos\) [[validation:ans5]] \(t\) \(+\) [[validation:ans6]] \(\cos\) [[validation:ans7]] \(t\) \(+\) [[validation:ans8]] \(\cos\) [[validation:ans9]] \(t\)  \(+\) [[validation:ans10]] \(\cos\) [[validation:ans11]] \(t\) \(+ \dots\)
<p>[[feedback:prt1]][[feedback:prt2]][[feedback:prt3]][[feedback:prt4]][[feedback:prt5]][[feedback:prt6]][[feedback:prt7]][[feedback:prt8]][[feedback:prt9]][[feedback:prt10]][[feedback:prt11]]</p><p></p>]]></text>
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    <generalfeedback format="html">
      <text><![CDATA[First complete the definition to obtain an even periodic function \(F(t)\) of period \(8\) as
\[ F(t) =
\begin{cases}
t + 4 \hspace{1cm} -4 &lt; t &lt; 0 \\
4 - t \hspace{1.1cm} 0 &lt; t &lt; 4
\end{cases}
\]<p style="text-align: center;"><img src="@@PLUGINFILE@@/23-5-1-Ft.svg" alt="even function" width="297" height="128" class="img-responsive atto_image_button_text-bottom"><br></p>
Now formulate the integral from which the Fourier coefficients \(a_n\) can be calculated:
<p></p>
With \(T = 8\) 
\[ a_n = \dfrac{2}{8} \int_{-4}^4 F(t) \cos \left( \dfrac{2n\pi t}{8} \right) dt \]
Utilising the fact that the integrand here is even we get
\[ a_n = \dfrac{1}{2} \int_{0}^4 (4-t) \cos \left( \dfrac{n\pi t}{4} \right) dt \]
Using integration by parts we obtain for \(n = 1, 2, 3, \dots \)
\[ 
\begin{align}
a_n &amp;= \dfrac{1}{2} \left\{ \left[ (4-t) \dfrac{4}{n\pi} \sin \left( \dfrac{n\pi t}{4} \right) \right]_0^4 + \dfrac{4}{n\pi} \int _0^4 \sin \left( \dfrac{n\pi t}{4} \right) dt \right\} \\
&amp;= \dfrac{1}{2} \left( \dfrac{4}{n\pi} \right) \left( \dfrac{4}{n\pi} \right) \left[ -\cos \left( \dfrac{n\pi t}{4} \right) \right] _0^4 \\
&amp;= \dfrac{8}{n^2\pi^2} [ \cos (n\pi) + 1]
\end{align}
\]
i.e 
\[ a_n = 
\begin{cases}
0 \hspace{2cm} n = 2,4,6, \dots \\
\dfrac{16}{n^2\pi^2} \hspace{1.3cm} n = 1,3,5, \dots
\end{cases}
\]
Also, 
\[ a_0 = \dfrac{1}{2} \int _0^4 (4-t) dt = 4, \]
so the constant term is \( \dfrac{a_0}{2} = 2\).
<p></p>
Therefore we have 
\[ 2 + \dfrac{16}{\pi^2} \left\{ \cos \left( \dfrac{\pi t}{4} \right) + \dfrac{1}{9}\cos \left( \dfrac{3 \pi t}{4} \right) + \dfrac{1}{25}\cos \left( \dfrac{5 \pi t}{4} \right) + \dots \right\} \]]]></text>
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      <text>ta1:2;
ta2:16/pi^2;
ta3:pi/4;
ta4:0;
ta5:2*pi/4;
ta6:16/(9*pi^2);
ta7:3*pi/4;
ta8:0;
ta9:4*pi/4;
ta10:16/(25*pi^2);
ta11:5*pi/4;</text>
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        </falsefeedback>
      </node>
    </prt>
    <prt>
      <name>prt9</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans9</sans>
        <tans>ta9</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt9-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt9-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1</value>
      </testinput>
      <testinput>
        <name>ans10</name>
        <value>ta10</value>
      </testinput>
      <testinput>
        <name>ans11</name>
        <value>ta11</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ta2</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>ta3</value>
      </testinput>
      <testinput>
        <name>ans4</name>
        <value>ta4</value>
      </testinput>
      <testinput>
        <name>ans5</name>
        <value>ta5</value>
      </testinput>
      <testinput>
        <name>ans6</name>
        <value>ta6</value>
      </testinput>
      <testinput>
        <name>ans7</name>
        <value>ta7</value>
      </testinput>
      <testinput>
        <name>ans8</name>
        <value>ta8</value>
      </testinput>
      <testinput>
        <name>ans9</name>
        <value>ta9</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt10</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt10-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt11</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt11-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt2-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt3-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt4</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt4-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt5-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt6</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt6-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt7</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt7-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt8</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt8-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt9</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt9-1-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1+1</value>
      </testinput>
      <testinput>
        <name>ans10</name>
        <value>ta10+1</value>
      </testinput>
      <testinput>
        <name>ans11</name>
        <value>ta11+1</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ta2+1</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>ta3+1</value>
      </testinput>
      <testinput>
        <name>ans4</name>
        <value>ta4+1</value>
      </testinput>
      <testinput>
        <name>ans5</name>
        <value>ta5+1</value>
      </testinput>
      <testinput>
        <name>ans6</name>
        <value>ta6+1</value>
      </testinput>
      <testinput>
        <name>ans7</name>
        <value>ta7+1</value>
      </testinput>
      <testinput>
        <name>ans8</name>
        <value>ta8+1</value>
      </testinput>
      <testinput>
        <name>ans9</name>
        <value>ta9+1</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt10</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt10-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt11</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt11-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt2-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt3-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt4</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt4-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt5-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt6</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt6-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt7</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt7-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt8</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt8-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt9</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt9-1-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
