#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.