#DESCRIPTION
##  Calculus: Taylor and Maclaurin series
##ENDDESCRIPTION

##KEYWORDS('calculus', 'series', taylor series', 'maclaurin series')

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Taylor and Maclaurin Series')
## Date('04/26/2010')
## Author('Justin Sukiennik')
## Institution('University of Minnesota')
## TitleText1('Calculus: Concepts and Contexts')
## EditionText1('4 Custom UMTYMP Ed.')
## AuthorText1('Stewart')
## Section1('11.10')
## Problem1('')

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

loadMacros(
  "PGstandard.pl",
  "MathObjects.pl",
  "weightedGrader.pl",
  "PGcourse.pl", 
);

#####################################################################
install_weighted_grader();

$showPartialCorrectAnswers = 0;

TEXT(beginproblem());

#####################################################################
Context("Numeric");
# Let students type "EmptySet" instead of "NONE"
Context()->constants->add(EmptySet => Set());
Context()->flags->set(noneWord=>"EmptySet");

$a = random(1,5);
$a2 = 2*$a;

$ans0 = Compute("$a");
$ans1 = Compute("-(1/(2*$a))*(x-$a)^2");
$ans2 = Compute("-(1/(8*$a^3))*(x-$a)^4");
$ans3 = Compute("-(1/(16*$a^5))*(x-$a)^6");
$ans4 = Compute("-(1/(128*$a^6))*(x-$a)^8");

Context("Numeric");
$ans5 = Compute("$a");
#####################################################################

Context()->texStrings;
BEGIN_TEXT
Find the first five non-zero terms of Taylor series centered at \(x = $a\) for the function below.
\[
f(x) = \sqrt{$a2 x - x^2}
\]
$HR
Answer: \(f(x) = \) \{ans_rule(8)\} \( + \) \{ans_rule(12)\} \( + \) \{ans_rule(12)\} \( + \) \{ans_rule(12)\} \( + \) \{ans_rule(14)\} \(+ \cdots\)
$HR
What is the radius of convergence? $BR
Answer: \(R = \) \{ans_rule(10)\} 
END_TEXT
Context()->normalStrings;

#####################################################################

WEIGHTED_ANS( ($ans0)->cmp(), 10 );
WEIGHTED_ANS( ($ans1)->cmp(), 10 );
WEIGHTED_ANS( ($ans2)->cmp(), 10 );
WEIGHTED_ANS( ($ans3)->cmp(), 10 );
WEIGHTED_ANS( ($ans4)->cmp(), 10 );
Context("Numeric");
WEIGHTED_ANS( ($ans5)->cmp(), 50 );

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