## PREP 2015 Question Authoring - Archived

### Scientific Notation Error in OPL problem

by tim Payer -
Number of replies: 2
Hello,

I am writing solutions to a number of selected OPL problems and came across a possible error in a problem that had the original file path of:

Library/Rochester/setDerivatives2Formulas/c2s5p3

The (algorithmic) problem answer has a number of zeros after the decimal before significant figures are encountered. As a hint to use scientific notation the student is given a prompt  in the wording of the problem that states:

"Tip: You can enter an answer such as 3.14e-1 for 0.314."

The problem is that when entered in this form the numerical value of e = 2.718241824.. is used in the calculation of the students answer.

I checked the course configuration and did not see a feature where scientific notation could be enabled for student answers in this way. Is there something else I should have checked to enable scientific notation?

Thanks,  tim

Below is the altered code for the problem with solutions:

## DESCRIPTION
##   The Chain Rule
## ENDDESCRIPTION

## Tagged by nhamblet

## DBsubject(Calculus - single variable)
## DBchapter(Differentiation)
## DBsection(Chain rule (without trigonometric functions))
## Institution(Rochester)
## Level(2)
## MO(1)
## KEYWORDS('Derivative', 'Polynomial', 'Chain')
## Library/Rochester/setDerivatives2Formulas/c2s5p3

DOCUMENT();        # This should be the first executable line in the problem.

"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGauxiliaryFunctions.pl",
"MathObjects.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1; Context("Numeric");$denom= 0;
while ($denom <= 0 ) {$t1 = random(-4,4,1);
$a= non_zero_random(-4,4,1);$a2 = $a*2;$b=   non_zero_random(-4,4,1);
$gcd = ($a2,$b);$a3 = $a2/$gcd;
$b3 =$b/$gcd;$gc8 = $gcd*(-8);$at3 = $a3*$t1;
$tsq =$t1*$t1;$bt1 = $b*$t1;
$abt3 =$at3+$b3;$atsq = $a*$tsq;
$c= non_zero_random(-4,4,1);$btc = $bt1 +$c;
$num =$gc8*$abt3;$den1 = $atsq+$btc;
$den2 =$den1**9;
$gcd = gcd($num,$den2);$num3 =$num/$gcd;
$den3 =$den2/$gcd;$ans = $num/$den1**9;
$d = random(-4,4,1);$denom = $a*$t1**2 +$b*$t1 +$c; }$fp= Compute("-8*$denom**(-9)*(2*$a*$t1 +$b)");

# Present the text.
TEXT(EV2(<<EOT));
Calculate $$f'(t1)$$ to 3 significant figures where
$f(t) = (a t^2 + b t + c )^{-8}$

$$f'( t1 ) =$$\{ ans_rule(20) \}

$PAR Tip: You can enter an answer such as 3.14e-1 for 0.314.$PAR
EOT

ANS($fp->cmp); SOLUTION(EV3(<<'END_SOLUTION'));$PAR SOLUTION $PAR Use the chain rule to find the derivative of composed functions.$BR
The function of  $$f(t) = (a t^2 + b t + c )^{-8}$$  is a composed function because the outer function of the negative exponent is "holding" the inner function of a quadratic equation. The chain rule applies the derivative on the outermost function first and then works inward applying the derivative to each successive inner function in a series of products. With each application of the derivative on the outer function the inner contents are left alone. In this way a "chain" of derivative products are formed. The chain rule can be expressed with notation as shown below: $BR$BR
Given that p(x) = f(g(x)), $BR$BR
The derivative is applied by the chain rule: $BR$BR
$$p'(x) = f'(g(x)) \cdot g'(x)$$   $BR$BR
Apply the derivative to f(t): $BR$BR
$$f(t) = (a t^2 + b t + c )^{-8}$$ $BR$BR
Apply the prime tics for the chain rule:$BR$BR
$$f'(t) = \left((a t^2 + b t + c )^{-8}\right)' \left(a t^2 + b t + c \right)'$$ $BR$BR
Apply the derivative: $BR$BR
$$f'(t) = -8(a t^2 + b t + c )^{-9} \left(a2 t + b \right)$$ $BR$BR
Push the quantity of the negative exponent down to the denominator: $BR$BR
$$\displaystyle{f'(t) = \frac{-8(a2 t + b) }{(a t^2 + b t + c )^{9}} }$$ $BR$BR
Pull any common factors from the numerator: $BR$BR
$$\displaystyle{f'(t) = \frac{-8 \cdot gcd(a3 t + b3) }{(a t^2 + b t + c )^{9}} }$$ $BR$BR
Combine factors for the reduced derivative: $BR$BR
$$\displaystyle{f'(t) = \frac{gc8(a3 t + b3) }{(a t^2 + b t + c )^{9}}}$$ $BR$BR
Evaluate the derivative at t = $t1:$BR
$BR $$\displaystyle{f'(t1) = \frac{gc8(a3 \cdot (t1) + b3) }{(a \cdot (t1)^2 + b \cdot (t1) + c )^{9}}}$$$BR
$BR Combine factors and squared values:$BR
$BR $$\displaystyle{f'(t1) = \frac{gc8(at3 + b3) }{(a \cdot tsq + bt1 + c )^{9}}}$$$BR
$BR Combine factors and terms:$BR
$BR $$\displaystyle{f'(t1) = \frac{gc8(abt3) }{(atsq + btc )^{9}}}$$$BR
$BR Combine factors and terms:$BR
$BR $$\displaystyle{f'(t1) = \frac{num }{(den1 )^{9}} =\frac{num }{den2} }$$$BR
$BR An exact value:$BR
$$\displaystyle{f'(t1) = \frac{num3 }{den3} }$$ $BR$BR
A decimal approximation:$BR$BR
$$\displaystyle{f'(t1) = ans}$$ $BR$BR

END_SOLUTION

ENDDOCUMENT();        # This should be the last executable line in the problem.

### Re: Scientific Notation Error in OPL problem

by Arnold Pizer -
Hi Tim,

The hint should read "Tip: You can enter an answer such as 3.14E-1 for 0.314." See http://webwork.maa.org/wiki/Available_Functions

I don't remember but ages ago before we had the constants pi and e and/or before the current parser, maybe 3.14e-1 did work for scientific notation or it could be a typo in the problem.

Arnie