Skip to content

Commit

Permalink
Update question from CR with an alternative solution.
Browse files Browse the repository at this point in the history
  • Loading branch information
@sangwinc
sangwinc committed Feb 7, 2025
1 parent 1d9a97f commit be983c5
Showing 1 changed file with 96 additions and 39 deletions.
135 changes: 96 additions & 39 deletions samplequestions/stacklibrary/Calculus-Refresher/CR_Diff_07/CR-Diff-07-implicit-2-a.xml
View file
Original file line number Diff line number Diff line change
Expand Up @@ -5,22 +5,22 @@
<text>CR-Diff-07-implicit-2.a</text>
</name>
<questiontext format="html">
<text>Find \(\frac{dy}{dx}\) where \({@p@}=0\) using implicit differentiation.

[[input:ans1]] [[validation:ans1]]</text>
<text><![CDATA[<p>Find \(\frac{dy}{dx}\), where \({@p@}=0\), using implicit differentiation:</p>
<p>\(\displaystyle \frac{d{@v@}}{dx} = \) [[input:ans1]]. [[validation:ans1]]</p>]]></text>
</questiontext>
<generalfeedback format="html">
<text>Differentiating implicitly we obtain:
\[ -\frac{1}{2}{@sin(2*x)@}-\frac{1}{2}\frac{dy}{dx}{@sec(2*y)^2@} = 0. \]
Re-arranging we obtain:
\[ \frac{dy}{dx} = {@ans@}. \]</text>
<text>Differentiating implicitly, we obtain \[ -\frac{1}{2}{@sin(2*x)@}-\frac{1}{2}\frac{dy}{dx}{@sec(2*y)^2@} = 0.\]
Rearranging, we find \[ \frac{dy}{dx} = {@ans@}. \]
In this case we can use the general trig identity \(\sec^2(w)=\tan^2(w)+1\), and use the condition from the question \({\cos \left( 2\cdot x \right)=\tan \left( 2\cdot y \right)}\) as follows:
\[\sec^2(2\cdot y)=\tan^2(2\cdot y)+1=\cos^2 \left( 2\cdot x \right)+1.\]
Hence we can re-write the derivative purely as a function of \(x\) in the form \[ \frac{dy}{dx} = {@ans@} = {@tans1@}\]</text>
</generalfeedback>
<defaultgrade>1</defaultgrade>
<penalty>0.1</penalty>
<hidden>0</hidden>
<idnumber/>
<idnumber></idnumber>
<stackversion>
<text/>
<text>2025012100</text>
</stackversion>
<questionvariables>
<text>p:cos(2*x)-tan(2*y);
Expand All @@ -30,7 +30,8 @@ ta:subst(YP,('diff(yfun(x),x,1)),ta);
dum1:yfun(x) := y;
ta:ev(ta);
ta:solve(ta=0,YP);
ans:rhs(ta[1]);</text>
ans:rhs(ta[1]);
tans1:-2*sin(2*x)/(cos(2*x)^2+1);</text>
</questionvariables>
<specificfeedback format="html">
<text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
Expand All @@ -40,7 +41,7 @@ ans:rhs(ta[1]);</text>
\[\frac{dy}{dx} = {@ans@}\]</text>
</questionnote>
<questiondescription format="moodle_auto_format">
<text/>
<text></text>
</questiondescription>
<questionsimplify>1</questionsimplify>
<assumepositive>0</assumepositive>
Expand All @@ -56,95 +57,118 @@ ans:rhs(ta[1]);</text>
</prtincorrect>
<decimals>.</decimals>
<scientificnotation>*10</scientificnotation>
<multiplicationsign>dot</multiplicationsign>
<multiplicationsign>none</multiplicationsign>
<sqrtsign>1</sqrtsign>
<complexno>i</complexno>
<inversetrig>cos-1</inversetrig>
<logicsymbol>lang</logicsymbol>
<matrixparens>[</matrixparens>
<variantsselectionseed/>
<variantsselectionseed></variantsselectionseed>
<input>
<name>ans1</name>
<type>algebraic</type>
<tans>ans</tans>
<boxsize>15</boxsize>
<strictsyntax>1</strictsyntax>
<insertstars>0</insertstars>
<syntaxhint/>
<syntaxhint></syntaxhint>
<syntaxattribute>0</syntaxattribute>
<forbidwords/>
<allowwords/>
<forbidwords></forbidwords>
<allowwords></allowwords>
<forbidfloat>1</forbidfloat>
<requirelowestterms>1</requirelowestterms>
<checkanswertype>1</checkanswertype>
<mustverify>1</mustverify>
<showvalidation>1</showvalidation>
<options/>
<options></options>
</input>
<prt>
<name>Result</name>
<value>1.0000000</value>
<autosimplify>1</autosimplify>
<feedbackstyle>1</feedbackstyle>
<feedbackvariables>
<text/>
<text></text>
</feedbackvariables>
<node>
<name>0</name>
<description/>
<description></description>
<answertest>Diff</answertest>
<sans>ans1</sans>
<tans>ans</tans>
<testoptions>v</testoptions>
<quiet>0</quiet>
<truescoremode>=</truescoremode>
<truescore>1.0000000</truescore>
<truepenalty/>
<truescore>1</truescore>
<truepenalty></truepenalty>
<truenextnode>-1</truenextnode>
<trueanswernote>Result-0-T </trueanswernote>
<truefeedback format="html">
<text/>
<text></text>
</truefeedback>
<falsescoremode>=</falsescoremode>
<falsescore>0.0000000</falsescore>
<falsepenalty/>
<falsescore>0</falsescore>
<falsepenalty></falsepenalty>
<falsenextnode>1</falsenextnode>
<falseanswernote>Result-0-F </falseanswernote>
<falsefeedback format="html">
<text/>
<text></text>
</falsefeedback>
</node>
<node>
<name>1</name>
<description/>
<description></description>
<answertest>AlgEquiv</answertest>
<sans>ans1</sans>
<tans>-ans</tans>
<testoptions/>
<testoptions></testoptions>
<quiet>0</quiet>
<truescoremode>=</truescoremode>
<truescore>0.0000000</truescore>
<truepenalty/>
<truescore>0</truescore>
<truepenalty></truepenalty>
<truenextnode>-1</truenextnode>
<trueanswernote>Result-1-T </trueanswernote>
<truefeedback format="html">
<text>Your answer is the negative of what it should be.

Have you mislaid a sign?</text>
<text>Have you mislaid a sign?</text>
</truefeedback>
<falsescoremode>=</falsescoremode>
<falsescore>0.0000000</falsescore>
<falsepenalty/>
<falsenextnode>-1</falsenextnode>
<falsescore>0</falsescore>
<falsepenalty></falsepenalty>
<falsenextnode>2</falsenextnode>
<falseanswernote>Result-1-F </falseanswernote>
<falsefeedback format="html">
<text/>
<text></text>
</falsefeedback>
</node>
<node>
<name>2</name>
<description>Alternative solution not depending on \(y\)</description>
<answertest>Diff</answertest>
<sans>ans1</sans>
<tans>tans1</tans>
<testoptions>x</testoptions>
<quiet>0</quiet>
<truescoremode>=</truescoremode>
<truescore>1</truescore>
<truepenalty></truepenalty>
<truenextnode>-1</truenextnode>
<trueanswernote>Result-3-T</trueanswernote>
<truefeedback format="html">
<text></text>
</truefeedback>
<falsescoremode>-</falsescoremode>
<falsescore>0</falsescore>
<falsepenalty></falsepenalty>
<falsenextnode>-1</falsenextnode>
<falseanswernote>Result-3-F</falseanswernote>
<falsefeedback format="html">
<text></text>
</falsefeedback>
</node>
</prt>
<qtest>
<testcase>1</testcase>
<description/>
<description></description>
<testinput>
<name>ans1</name>
<value>ans</value>
Expand All @@ -158,17 +182,50 @@ Have you mislaid a sign?</text>
</qtest>
<qtest>
<testcase>2</testcase>
<description/>
<description></description>
<testinput>
<name>ans1</name>
<value>-ans</value>
</testinput>
<expected>
<name>Result</name>
<expectedscore>0.0000000</expectedscore>
<expectedpenalty>0.1000000</expectedpenalty>
<expectedpenalty>0.4000000</expectedpenalty>
<expectedanswernote>Result-1-T </expectedanswernote>
</expected>
</qtest>
<qtest>
<testcase>3</testcase>
<description></description>
<testinput>
<name>ans1</name>
<value>tans1</value>
</testinput>
<expected>
<name>Result</name>
<expectedscore>1.0000000</expectedscore>
<expectedpenalty>0.0000000</expectedpenalty>
<expectedanswernote>Result-3-T</expectedanswernote>
</expected>
</qtest>
<qtest>
<testcase>4</testcase>
<description>Simply wrong.</description>
<testinput>
<name>ans1</name>
<value>x^2</value>
</testinput>
<expected>
<name>Result</name>
<expectedscore>0.0000000</expectedscore>
<expectedpenalty></expectedpenalty>
<expectedanswernote>Result-3-F</expectedanswernote>
</expected>
</qtest>
<tags>
<tag><text>skill quiz</text>
</tag>
</tags>
</question>

</quiz>

0 comments on commit be983c5

Please sign in to comment.