##KEYWORDS('Sequences') ##DESCRIPTION ## Find a formula for the given sequences ##ENDDESCRIPTION ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Sequences') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.1') ## Problem1('9') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 0;$showHints = 1; qa(~~@questions, ~~@answers, "$${ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, . . . }$$" , "1/(2^n)" , "$${ \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, . . . }$$" , "1/(2*n)" , "$${ \frac{3}{16}, \frac{4}{25}, \frac{5}{36}, \frac{6}{49}, . . . }$$", "(n+2)/((n+3)^2)", "$${ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5} , . . . }$$" , "n/(n+1)" ); # Now randomize the questions: @slice = &NchooseK(4,2); @shuffle = &shuffle(scalar(@slice)); ($ans1,$ans2) = @answers[@slice] ; BEGIN_TEXT For each sequence, find a formula for the general term, $$a_n$$. For example, answer $$n ^2$$ if given the sequence $$\lbrace 1, 4, 9, 16, 25, 36, . . . \rbrace$$$BR END_TEXT TEXT(EV2(match_questions_list(@questions[@slice]))); ANS(fun_cmp($ans1, vars=> ['n']), fun_cmp($ans2, vars=> ['n'])); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##DESCRIPTION ##KEYWORDS('algebra','geometric sequences') ## ## tsch tagged and PAID on 3-22-2004 ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Sequences') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.1') ## Problem1('6') ##ENDDESCRIPTION DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGgraphmacros.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(-10,10,1);$r = non_zero_random(2,6,.5); $n = random(4,7,1);$a2 = $a*$r; $a3 =$a*$r**2;$an = $a*$r**($n-1); BEGIN_TEXT Find the $$n^{th}$$ term of the geometric sequence$PAR $$a , a2 , a3 , ...$$ Answer: \{ans_rule(10)\} END_TEXT ANS(num_cmp($an, mode=>"arith")); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##DESCRIPTION ##KEYWORDS('sequences', 'limits') ## tsuda tagged and PAID on 2-20-2004 ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Sequences') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.1') ## Problem1('43') ##ENDDESCRIPTION DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a=non_zero_random(-8,8,2);$b=random(2,4,1); if ($a>0) {$soln=0; } else { $soln='INF'; } TEXT(EV2(<['INF', 'MINF', 'DIV'])); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ #DESCRIPTION #Sequences. Find the limit of Convergence #ENDDESCRIPTION #Keywords('Sequences', 'convergence') ## tsuda tagged and PAID on 2-20-2004 ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Sequences') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.1') ## Problem1('23') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(1,6,.1); TEXT(EV3(<<'EOT')); Find the limit of the sequence whose terms are given by$BR $$a_n = (n^2)(1- \cos (\frac{a}{n}) ) .$$ $BR \{ans_rule(40)\} EOT$answer = $a*$a/2 ; ANS(num_cmp($answer)); &ENDDOCUMENT; ________________________________________________________________________________ ##KEYWORDS('Sequences', 'convergence') ##DESCRIPTION ## Determine the limit of a sequence. ##ENDDESCRIPTION ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Sequences') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.1') ## Problem1('17') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(1,9,1);$b = random(1,9,1); $c = random(1,9,1);$d = random(1,9,1); $e = random(1,9,1);$f = random(1,9,1); $num = nicestring([$a, $b,$c], ['n^2', 'n', '']); $den= nicestring([$d, $e,$f], ['n^2', 'n', '']); BEGIN_TEXT Find the limit of the sequence: $$a_n = \frac{num }{den }$$ $BR$BR Answer: \{ans_rule(20)\} END_TEXT ANS(num_cmp("$a/$d")); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##KEYWORDS('sequences', 'convergent', 'divergent') ##DESCRIPTION ## Determine if a sequence converges and compute its limit ##ENDDESCRIPTION ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Sequences') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.1') ## Problem1('19') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;$a=random(2,5,1); $b=random(2,5,1);$c=random(4,19,1); $d=random(3,18,1);$f=random(3,18,1); if ($a>$b) { $soln='inf'; } else { if ($a==$b) {$soln="$d/$f"; } else { $soln=0; } } BEGIN_TEXT Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit.$BR$BR (If it diverges to infinity, state your answer as$BITALIC inf $EITALIC . If it diverges to negative infinity, state your answer as$BITALIC -inf $EITALIC . If it diverges without being infinity or negative infinity, state your answer as$BITALIC div $EITALIC .)$BR$BR $\lim_{n\rightarrow\infty} \frac{d (a^n) + c}{f (b^n)}$$BR$BR Answer: \{ans_rule( 30) \} END_TEXT ANS(num_cmp($soln, strings=>['inf', '-inf', 'div'])); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##KEYWORDS('sequences', 'limits') ##DESCRIPTION ## Determine if a sequence converges and compute its limit ##ENDDESCRIPTION ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Sequences') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.1') ## Problem1('24') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;$a=random(2,20,1); $b=random(2,30,1);$c=random(4,35,1); $d=random(2,7,1);$soln = "$b*pi/2"; BEGIN_TEXT Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit.$BR$BR (If it diverges to infinity, state your answer as$BITALIC inf $EITALIC . If it diverges to negative infinity, state your answer as$BITALIC -inf $EITALIC . If it diverges without being infinity or negative infinity, state your answer as$BITALIC div $EITALIC .$BR$BR $\lim_{n\rightarrow\infty} \frac{c}{a^n} + b \arctan (n^{d})$$BR$BR Answer: \{ans_rule( 30) \} END_TEXT ANS(num_cmp($soln, strings=>['inf', '-inf', 'div'] ) ); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##KEYWORDS('sequences', 'convergent', 'divergent') ##DESCRIPTION ## Determine if a sequence is convergent and compute its limit ##ENDDESCRIPTION ##Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Sequences') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.1') ## Problem1('25') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;$a=random(4,19,1); BEGIN_TEXT Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit. $BR$BR (If it diverges to infinity, state your answer as $BITALIC inf$EITALIC . If it diverges to negative infinity, state your answer as $BITALIC -inf$EITALIC . If it diverges without being infinity or negative infinity, state your answer as $BITALIC div$EITALIC .) $BR$BR $\lim_{n\rightarrow\infty} \frac{a + (-1)^n}{n^2}$ $BR$BR Answer: \{ans_rule( 30) \} END_TEXT ANS(num_cmp(0, strings=>['inf', '-inf', 'div'] ) ); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ #DESCRIPTION # Polar Coordinates # Arc Length. #ENDDESCRIPTION #KEYWORDS('Integration', 'Arc length', 'parametric') ## tsch tagged and PAID on 3-22-2004 ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Sequences') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.1') ## Problem1('68') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;$d = random(1,9,1); $k = random(2,7,1);$p = random(-1,1,2); $e =$p + $k +$k*$k;$w = $e**.5; TEXT(EV3(<<'EOT')); Let $$\displaystyle f(x) = \frac{x}{x^2 +d x +e}.$$$BR A. Find the smallest real number $$r$$ such that $$f(x)$$ is decreasing for all $$x$$ greater than $$r.$$$BR $$r =$$ \{ans_rule(20)\}$BR B. Find the smallest integer $$s$$ such that $$f(n)$$ is decreasing for all integers $$n$$ greater than or equal to $$s$$. $BR $$s =$$ \{ans_rule(20)\}$BR EOT $ans0 =$w; $ans1 =$k + .5 +.5*$p; #$ans2 = ($a +$b*cos( ($b/(2*$a) +1)*$pi) )*sin( ($b/(2*$a) +1)*$pi ); #$ans3 = 0 ; ANS(num_cmp($ans0)); ANS(num_cmp($ans1)); #ANS(num_cmp($ans2)); #ANS(num_cmp($ans3)); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##KEYWORDS('Sequences', 'limits') ##DESCRIPTION ##Sequences. Increasing or Decreasing? ##ENDDESCRIPTION ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Infinite Sequences and Series') ## DBsection('Sequences') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('11.1') ## Problem1('52 53 54 58') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); # No partial credit on this problem, so we say: install_problem_grader(~~&std_problem_grader); TEXT(beginproblem());$showPartialCorrectAnswers = 0; $a = random(2,5,1);$b = random(6,9,1); qa(~~@questions, ~~@answers, "$$a_n = \frac{\cos n}{a^n}$$", "N" , "$$a_n = \frac{n - a}{n + a}$$", "I" , "$$a_n = \frac{1}{a n + b}$$", "D" , "$$a_n = \frac{\sqrt{n + a}}{b n + a}$$", "D" ); # Now randomize the questions: @slice = &NchooseK(4,4); @shuffle = &shuffle(scalar(@slice)); ($ans1,$ans2, $ans3,$ans4) = @answers[@slice] ; BEGIN_TEXT Determine whether the sequences are increasing, decreasing, or not monotonic. If increasing, enter $BITALIC I$EITALIC as your answer. If decreasing, enter $BITALIC D$EITALIC as your answer. If not monotonic, enter $BITALIC N$EITALIC as your answer. $BR END_TEXT TEXT(EV2(match_questions_list(@questions[@slice]))); ANS(str_cmp($ans1), str_cmp($ans2), str_cmp($ans3), str_cmp(\$ans4)); ENDDOCUMENT(); # This should be the last executable line in the problem.