##DESCRIPTION ## Interval of convergence for power series ##ENDDESCRIPTION ##KEYWORDS('Power Series') ## tsch tagged and PAID on 3-22-2004 ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Power Series') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.8') ## Problem1('19') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); # No partial credit on this problem, so we say: install_problem_grader(~~&std_problem_grader); TEXT(&beginproblem); $showPartialCorrectAnswers = 0;$a = random(-10,-3,1); $b = random(1,11,1);$c = random(1,11,1); $d = random(1,11,1); TEXT(EV2(<["remove_whitespace","ignore_order","ignore_case"])); ##the correct answers are obtained by applying ##the inverse (adjoint) permutation to the captions. ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ #DESCRIPTION #Representation of function as a power series #ENDDESCRIPTION #KEYWORDS('Power Series' ) ## tsch tagged and PAID on 3-22-2004 ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Representations of Functions as Power Series') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.9') ## Problem1('20') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(2,10,1);$b = random(2,10,1); BEGIN_TEXT The function $$f(x) = a x^2 \arctan(x^{b})$$ is represented as a power series $BR $$\displaystyle f(x) = \sum_{n=0}^\infty c_n x^n .$$$BR What is the lowest term with a nonzero coefficient. $BR \{ans_rule(20)\}$BR Find the radius of convergence $$R$$ of the series. $BR $$R =$$ \{ans_rule(20)\} .$BR END_TEXT #@ans=(); $ans0 =$b+2 ; $ans1 = 1 ; #ANS(num_cmp(relTol=>@ans)); #ANS(ordered_num_cmp_list(@ans) ); ANS(num_cmp($ans0)); ANS(num_cmp($ans1)); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ #DESCRIPTION #Representation of function as a power series #ENDDESCRIPTION #KEYWORDS('Power Series' ) ## tsch tagged and PAID on 3-22-2004 ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Representations of Functions as Power Series') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.9') ## Problem1('14') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(1,10,1); BEGIN_TEXT The function $$f(x) = a x \ln(1 + 2x)$$ is represented as a power series$BR $$\displaystyle f(x) = \sum_{n=0}^\infty c_n x^n .$$ $BR Find the FOLLOWING coefficients in the power series.$BR $$c_0 =$$ \{ans_rule(20)\} $BR $$c_1 =$$ \{ans_rule(20)\}$BR $$c_2 =$$ \{ans_rule(20)\} $BR $$c_3 =$$ \{ans_rule(20)\}$BR $$c_4 =$$ \{ans_rule(20)\} $BR Find the radius of convergence $$R$$ of the series.$BR $$R =$$ \{ans_rule(20)\} . $BR END_TEXT #@ans=();$ans0 = 0; $ans1 = 0;$ans2 = $a*2;$ans3 = $a*(-2);$ans4 = $a*8/3;$ans5 = 1/2; #ANS(num_cmp(relTol=>@ans)); #ANS(ordered_num_cmp_list(@ans) ); ANS(num_cmp($ans0)); ANS(num_cmp($ans1)); ANS(num_cmp($ans2)); ANS(num_cmp($ans3)); ANS(num_cmp($ans4)); ANS(num_cmp($ans5)); ENDDOCUMENT(); # This should be the last executable line in the problem.