Difference between revisions of "RecursivelyDefinedFunctions"
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− | <h2>Recursively Defined Functions</h2> |
+ | <h2>Recursively Defined Functions (Sequences)</h2> |
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<b>Setup:</b> |
<b>Setup:</b> |
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We define a new named function <code>f</code> as something the student is unlikely to guess. The named function <code>f</code> is, in some sense, just a placeholder since the student will enter expressions involving <code>f(n-1)</code>, WeBWorK will interpret it internally as <code>sin(pi^(n-1))+e</code>, and the only thing the student sees is <code>f(n-1)</code>. If the |
We define a new named function <code>f</code> as something the student is unlikely to guess. The named function <code>f</code> is, in some sense, just a placeholder since the student will enter expressions involving <code>f(n-1)</code>, WeBWorK will interpret it internally as <code>sin(pi^(n-1))+e</code>, and the only thing the student sees is <code>f(n-1)</code>. If the |
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− | recursion has an closed-form solution (e.g., the Fibonacci numbers are given by f(n) = (a^n - (1-a)^n)/sqrt(5) where a = (1+sqrt(5))/2), it would be good to define f using that explicit solution in case the student tries to answer the question by writing out the explicit solution (a^n - (1-a)^n)/sqrt(5) instead of using the shorthand f(n). |
+ | recursion has an closed-form solution (e.g., the Fibonacci numbers are given by f(n) = (a^n - (1-a)^n)/sqrt(5) where a = (1+sqrt(5))/2) and you want to allows students to enter the closed-form solution, it would be good to define f using that explicit solution in case the student tries to answer the question by writing out the explicit solution (a^n - (1-a)^n)/sqrt(5) instead of using the shorthand f(n). |
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</p> |
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</td> |
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<td style="background-color:#ffdddd;border:black 1px dashed;"> |
<td style="background-color:#ffdddd;border:black 1px dashed;"> |
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<pre> |
<pre> |
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+ | Context()->texStrings; |
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BEGIN_TEXT |
BEGIN_TEXT |
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The current value \( f(n) \) is three |
The current value \( f(n) \) is three |
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− | times the previous value plus two. Find |
+ | times the previous value, plus two. Find |
a recursive definition for \( f(n) \). |
a recursive definition for \( f(n) \). |
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Enter \( f_{n-1} \) as \( f(n-1) \). |
Enter \( f_{n-1} \) as \( f(n-1) \). |
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\( f(n) \) = \{ ans_rule(20) \} |
\( f(n) \) = \{ ans_rule(20) \} |
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END_TEXT |
END_TEXT |
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+ | Context()->normalStrings; |
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</pre> |
</pre> |
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<td style="background-color:#ffcccc;padding:7px;"> |
<td style="background-color:#ffcccc;padding:7px;"> |
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[[Category:Problem Techniques]] |
[[Category:Problem Techniques]] |
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+ | <ul> |
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+ | <li>POD documentation: [http://webwork.maa.org/pod/pg/macros/parserFunction.html parserFunction.pl]</li> |
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+ | <li>PG macro: [http://webwork.maa.org/viewvc/system/trunk/pg/macros/parserFunction.pl?view=log parserFunction.pl]</li> |
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+ | </ul> |
Revision as of 18:01, 7 April 2021
Recursively Defined Functions (Sequences)
This PG code shows how to check student answers that are recursively defined functions.
PG problem file | Explanation |
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DOCUMENT(); loadMacros( "PGstandard.pl", "MathObjects.pl", "parserFunction.pl", ); TEXT(beginproblem()); |
Initialization:
We will be defining a new named function and adding it to the context, and the easiest way to do this is using |
Context("Numeric")->variables->are(n=>"Real"); parserFunction(f => "sin(pi^n)+e"); $fn = Formula("3 f(n-1) + 2"); |
Setup:
We define a new named function |
Context()->texStrings; BEGIN_TEXT The current value \( f(n) \) is three times the previous value, plus two. Find a recursive definition for \( f(n) \). Enter \( f_{n-1} \) as \( f(n-1) \). $BR \( f(n) \) = \{ ans_rule(20) \} END_TEXT Context()->normalStrings; |
Main Text: The problem text section of the file is as we'd expect. We should tell students to use function notation rather than subscript notation so that they aren't confused about syntax. |
$showPartialCorrectAnswers=1; ANS( $fn->cmp() ); ENDDOCUMENT(); |
Answer Evaluation: As is the answer. |
- POD documentation: parserFunction.pl
- PG macro: parserFunction.pl