##KEYWORDS('Power Series', 'Representation of a function as a power series') ##DESCRIPTION ## Find the first few coefficients of a power series ##ENDDESCRIPTION ## Shotwell cleaned ## 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('5') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;$a = random(1,9,1); BEGIN_TEXT Suppose that $\frac{a}{(1 - x^3)} = \sum_{n=0}^\infty c_n x^n$ $BR Find the following coefficients of the power series.$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)\} $BR$BR Find the radius of convergence $$R$$ of the power series. $BR $$R =$$ \{ans_rule(20)\}$BR END_TEXT $ans0 =$a ; $ans1 = 0 ;$ans2 = 0 ; $ans3 =$a ; $ans4 = 0 ;$ans5 = 1 ; 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. ________________________________________________________________________________ #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 $$f(x) = \ln(1 - x^2)$$ is represented as a power series $BR $$f(x) = \sum_{n=0}^\infty c_n x^n .$$$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; $ans2 = -1;$ans3 = 0; $ans4 = -1/2;$ans5 = 1; #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. ________________________________________________________________________________ #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($ans5)); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##KEYWORDS('Power Series', 'Representation of a function as a power series') ##DESCRIPTION ## Find first few coefficients of power series and find radius of convergence ##ENDDESCRIPTION ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Representations of Functions as Power Series') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.9') ## Problem1('10') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(1,10,1);$c = random(1,7,1); $b =$a + $c;$b1 = -1/($b); BEGIN_TEXT Suppose that $\frac{a x}{x + b} = \sum_{n=0}^\infty c_n x^n .$ Find the following coefficients.$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)\} $BR$BR Find the radius of convergence $$R$$ of the power series. $BR $$R =$$ \{ans_rule(20)\}$BR END_TEXT $ans0 = 0 ;$ans1 = "-$a *$b1" ; $ans2 = "$ans1 * $b1" ;$ans3 = "$ans2 *$b1" ; $ans4 = "$ans3 * $b1" ;$ans5 = $b ; ANS(num_cmp($ans0) ); ANS(num_cmp($ans1) ); ANS(num_cmp($ans2) ); ANS(num_cmp($ans3) ); ANS(num_cmp($ans4) ); ANS(num_cmp($ans5) ); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##KEYWORDS('Power Series', 'Representation of a function as a power series') ##DESCRIPTION ## Find the first few coefficients of a power series ##ENDDESCRIPTION ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Representations of Functions as Power Series') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.9') ## Problem1('13') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(1,10,1);$b = random(2,10,1); $b1 = 1/$b; BEGIN_TEXT The function $$f(x) = \frac{a}{(1 - b x)^2}$$ is represented as a power series: $BR$BR $f(x) = \sum_{n=0}^\infty c_n x^n$ $BR Find the first few coefficients in the power series.$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)\} $BR$BR Find the radius of convergence $$R$$ of the series. $BR $$R =$$ \{ans_rule(20)\} .$BR END_TEXT $ans0 =$a ; $ans1 = "$a*2*$b" ;$ans2 = "$a*3*($b^2)"; $ans3 = "$a*4*($b^3)";$ans4 = "$a*5*($b^4)"; $ans5 =$b1; 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. ________________________________________________________________________________ ##KEYWORDS('Power Series', 'Representation of a function as a power series') ##DESCRIPTION ## Find the first few coefficients of a power series ##ENDDESCRIPTION ## Shotwell cleaned ## 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('17') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;$a = random(1,10,1); BEGIN_TEXT The function $$f(x) = \ln(a - x)$$ is represented as a power series: $BR $f(x) = \sum_{n=0}^\infty c_n x^n$$BR Find the first few coefficients in the power series. $BR $$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 =$$ \{ans_rule(20)\}$BR$BR Find the radius of convergence $$R$$ of the series.$BR $$R =$$ \{ans_rule(20)\} $BR END_TEXT #@ans=();$ans0 = "ln($a)" ;$ans1 = "-1/$a" ;$ans2 = "-1/(2*$a^2)";$ans3 = "-1/(3*$a^3)";$ans4 = "-1/(4*$a^4)";$ans5 = $a; ANS(num_cmp($ans0) ); ANS(num_cmp($ans1) ); ANS(num_cmp($ans2) ); ANS(num_cmp($ans3) ); ANS(num_cmp($ans4) ); ANS(num_cmp($ans5) ); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##KEYWORDS('Power Series', 'Representation of a function as a power series') ##DESCRIPTION ## Find the first few coefficients of a power series ##ENDDESCRIPTION ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Representations of Functions as Power Series') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.9') ## Problem1('20') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(2,10,1);$b = random(2,10,1); BEGIN_TEXT The function $$f(x) = a x \arctan (b x)$$ is represented as a power series: $f(x) = \sum_{n=0}^\infty c_n x^n$ Find the first few coefficients in the power series. $BR $$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 =$$ \{ans_rule(20)\}$BR$BR Find the radius of convergence $$R$$ of the series.$BR $$R =$$ \{ans_rule(20)\} . $BR END_TEXT$ans0 = 0 ; $ans1 = 0 ;$ans2 = "$a*$b"; $ans3 = 0 ;$ans4 = "-1*$a*$b^3/3"; $ans5 = "1/$b"; ANS(num_cmp($ans0) ); ANS(num_cmp($ans1) ); ANS(num_cmp($ans2) ); ANS(num_cmp($ans3) ); ANS(num_cmp($ans4) ); ANS(num_cmp($ans5) ); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##KEYWORDS('Power Series', 'Representation of a function as a power series') ##DESCRIPTION ## Find the first few coefficients of a power series ##ENDDESCRIPTION ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Representations of Functions as Power Series') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.9') ## Problem1('3') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;$a = random(1,10,1); $c = random(1,7,1);$b = $a +$c; $b1 = -1/($b); BEGIN_TEXT Suppose that $\frac{a}{(b + x)} = \sum_{n=0}^\infty c_n x^n$ $BR Find the following coefficients of the power series.$BR $$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 =$$ \{ans_rule(20)\} $BR$BR Find the radius of convergence $$R$$ of the power series. $BR $$R =$$ \{ans_rule(20)\} .$BR END_TEXT $ans0 = "-$a * $b1" ;$ans1 = "$ans0 *$b1" ; $ans2 = "$ans1 * $b1" ;$ans3 = "$ans2 *$b1" ; $ans4 = "$ans3 * $b1" ;$ans5 = $b ; ANS(num_cmp($ans0) ); ANS(num_cmp($ans1) ); ANS(num_cmp($ans2) ); ANS(num_cmp($ans3) ); ANS(num_cmp($ans4) ); ANS(num_cmp($ans5) ); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##KEYWORDS('Power Series', 'Representation of a function as a power series') ##DESCRIPTION ## Find the first few coefficients of a power series ##ENDDESCRIPTION ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Representations of Functions as Power Series') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.9') ## Problem1('9') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(2,10,1);$b = random(2,10,1); BEGIN_TEXT Represent the function $$f(x)= \frac{a}{(1 - b x)}$$ as a power series: $BR $f(x) = \sum_{n=0}^\infty c_n x^n$$BR Find the following coefficients: $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)\}$BR$BR Find the radius of convergence$BR $$R =$$ \{ans_rule(20)\} $BR END_TEXT$ans0 = $a;$ans1 = "$a *$b"; $ans2 = "$a * $b^2";$ans3 = "$a *$b^3"; $ans4 = "$a * $b^4";$ans5 = "1/$b"; ANS(num_cmp($ans0) ); ANS(num_cmp($ans1) ); ANS(num_cmp($ans2) ); ANS(num_cmp($ans3) ); ANS(num_cmp($ans4) ); ANS(num_cmp($ans5) ); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ #DESCRIPTION #Representation of function as a power series #ENDDESCRIPTION #KEYWORDS('Power Series' ) ## tsch tagged and PAID on 3-22-2004 ## DBchapter('Infinite Sequences and Series') ## DBsection('Representations of Functions as Power Series') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.9') ## Problem1('26') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(&beginproblem);$showPartialCorrectAnswers = 0; $b1 = random(2,6,1) ;$a = 5 ; $b =$b1*8 ; $c = random(6,12,1); BEGIN_TEXT (a) Evaluate the integral$BR $$\displaystyle \int_{0}^{2} \frac{b}{x^2+4} dx$$. $BR Your answer should be in the form $$k\pi$$, where $$k$$ is an integer. What is the value of $$k$$?$BR Hint: $$\frac{d \arctan(x)}{dx} = \frac{1}{x^2+1}$$ $BR $$k =$$ \{ans_rule(20)\}$BR (b) Now, lets evaluate the same integral using power series. First, find the power series for the function $$f(x) = \frac{b}{x^2+4}$$. Then, integrate it from 0 to 2, and call it S. S should be an infinite series. $BR What are the first few terms of S ?$BR $$a_0 =$$ \{ans_rule(20)\} $BR $$a_1 =$$ \{ans_rule(20)\}$BR $$a_2 =$$ \{ans_rule(20)\} $BR $$a_3 =$$ \{ans_rule(20)\}$BR $$a_4 =$$ \{ans_rule(20)\} $BR (c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by $$k$$ (the answer to (a)), you have found an estimate for the value of $$\pi$$ in terms of an infinite series. Approximate the value of $$\pi$$ by the first$a terms. $BR \{ans_rule(20)\} .$BR END_TEXT #@ans=(); $ans0 =$b1; $ans1 =$b/2; $ans2 = -1*$b/6; $ans3 =$b/10; $ans4 = -1*$b/14; $ans5 =$b/18; $ans6 = 4*(1-1/3+1/5-1/7+1/9); #&ANS(std_num_cmp_list(@ans) ); #&ANS(ordered_num_cmp_list(@ans) ); &ANS(std_num_cmp($ans0) ); &ANS(std_num_cmp($ans1) ); &ANS(std_num_cmp($ans2) ); &ANS(std_num_cmp($ans3) ); &ANS(std_num_cmp($ans4) ); &ANS(std_num_cmp($ans5) ); &ANS(std_num_cmp($ans6) ); ENDDOCUMENT(); # This should be the last executable line in the problem.