RecursivelyDefinedFunctions: Difference between revisions

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<h2>Recursively Defined Functions</h2>
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<p style="font-size: 120%;font-weight:bold">This problem has been replaced with [https://openwebwork.github.io/pg-docs/sample-problems/Sequences/RecursiveSequence.html a newer version of this problem]</p>
 
 
<h2>Recursively Defined Functions (Sequences)</h2>


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<b>Setup:</b>  
<b>Setup:</b>  
We define a new named function <code>f</code> as something the student is unlikely to guess.  The named function <code>f</codee> is, in some sense, just a placeholder since the student will enter things like <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  
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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<pre>
<pre>
Context()->texStrings;
BEGIN_TEXT
BEGIN_TEXT
The current value \( f(n) \) is three  
The current value \( f(n) \) is three  
times the previous value plus two.  Find
times the previous value, plus two.  Find
a recursive definition for \( f(n) \).   
a recursive definition for \( f(n) \).   
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) \}  
END_TEXT
END_TEXT
Context()->normalStrings;
</pre>
</pre>
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[[Category:Problem Techniques]]
[[Category:Problem Techniques]]
<ul>
<li>POD documentation: [http://webwork.maa.org/pod/pg/macros/parserFunction.html parserFunction.pl]</li>
<li>PG macro: [http://webwork.maa.org/viewvc/system/trunk/pg/macros/parserFunction.pl?view=log parserFunction.pl]</li>
</ul>

Latest revision as of 18:48, 16 July 2023

This article has been retained as a historical document. It is not up-to-date and the formatting may be lacking. Use the information herein with caution.

This problem has been replaced with a newer version of this problem


Recursively Defined Functions (Sequences)


This PG code shows how to check student answers that are recursively defined functions.

Problem Techniques Index

PG problem file Explanation 
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 parserFunction.pl. There is a more basic way to add functions to the context, which is explained in example 2 at AddingFunctions 

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 f as something the student is unlikely to guess. The named function f is, in some sense, just a placeholder since the student will enter expressions involving f(n-1), WeBWorK will interpret it internally as sin(pi^(n-1))+e, and the only thing the student sees is f(n-1). If the 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). 

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.

Problem Techniques Index


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