#DESCRIPTION
# Calculation of integrals using power series.
#ENDDESCRIPTION

#KEYWORDS('Taylor Series' , 'Integrals' )
## tsch tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Taylor and Maclaurin Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.10')
## Problem1('37')


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(0.6,0.8,0.01);
$c = random(2,8,1);

BEGIN_TEXT
Assume that \( \sin(x) \) equals its Maclaurin series for all x. $BR
Use the Maclaurin series for \( \sin($c x^2) \)
to evaluate the integral
\[ \int_0^{$a} \sin($c x^2) \  dx \]. 

Your answer will be an infinite series.  Use the first two terms to estimate its value.

\{ans_rule(40)\}
END_TEXT


$soln1 = "$c * x^3 / 3 - $c^3 * x^7 / 42";
$soln2 = $c * $a**3 / 3 - $c**3 * $a**7 / 42;


ANS(num_cmp($soln2, relTol=>1E-7));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________#DESCRIPTION
# Calculation of integrals using power series.
#ENDDESCRIPTION

#KEYWORDS('Taylor Series' , 'Integrals' )
## tsch tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Taylor and Maclaurin Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.10')
## Problem1('40')


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(0.1,0.2,0.01);
$b = non_zero_random(2,5,1);

BEGIN_TEXT

Assume that \( e^x \) equals its Maclaurin series for all x. $BR
Use the Maclaurin series for  \( e^{-$b x^4} \)
to evaluate the integral

\[ \int_0^{$a} e^{-$b x^4} \  dx \]

Your answer will be an infinite series.  Use the first two terms to estimate its value.

\{ans_rule(40)\}
END_TEXT


$soln1 = "x - $b * x**5 / 5";
$soln2 = $a - $b * $a**5 / 5;


ANS(num_cmp($soln2, relTol=>1E-7));


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('Taylor and Maclaurin Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.10')
## Problem1('21')


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);
$c = random(3,8,1);
$b1 = 1/($b);

BEGIN_TEXT

Find the Maclaurin series of the function \( f(x) = $a \cos ($b x^2) 
\)$BR
  \( (f(x) = \displaystyle \sum_{n=0}^\infty c_n x^n ) \) $BR


\( c_0 = \) \{ans_rule(20)\}  $BR

\( c_2 = \) \{ans_rule(20)\}  $BR

\( c_4 = \) \{ans_rule(20)\}  $BR

\( c_6 = \) \{ans_rule(20)\}  $BR

\( c_8 = \) \{ans_rule(20)\}  $BR

END_TEXT

$ans0 =  $a ;
$ans1 =  0 ;
$ans2 =  -($a * $b * $b) / 2 ;
$ans3 =  0 ;
$ans4 =  ($a * ($b ** 4)) / 24 ;

ANS(num_cmp($ans0));
ANS(num_cmp($ans1));
ANS(num_cmp($ans2));
ANS(num_cmp($ans3));
ANS(num_cmp($ans4));

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

#DESCRIPTION
#Taylor_series
#ENDDESCRIPTION

#KEYWORDS('Taylor Series')
## tsch tagged and PAID on 3-22-2004

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Taylor and Maclaurin Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.10')
## Problem1('55 53 54')


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;

$a = random(2,11,1);
$b = random(2,11,1);
$a1 =2*($a);

$questStr1 = EV2( " \( \displaystyle   \sum_{n=0}^\infty (-1)^n
\frac{x^{2n+1}}{(2n+1)!} \) " );

$ansStr1 =EV2(  " \( \sin(x) \) " );

$questStr2 = EV2( " \( \displaystyle   \sum_{n=0}^\infty \frac{x^n}{n!} \) " );

$ansStr2 =EV2(  " \( e^x \) " );

$questStr3 = EV2( " \(  \displaystyle  \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} \)
" );

$ansStr3 =EV2(  " \(  \cos(x) \) " );

$questStr4 = EV2( " \(  \displaystyle  \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1} \)
" );

$ansStr4 =EV2(   " \( \arctan(x) \) " );

@questions =( $questStr1,$questStr2,$questStr3,$questStr4);
@answers =( $ansStr1,$ansStr2,$ansStr3,$ansStr4);

# Now randomize the questions:
@slice = &NchooseK(4,4);
@shuffle = &shuffle(scalar(@slice));

TEXT(EV2(<<EOT));

Match each of the Maclaurin series with right function. $BR

EOT
TEXT(
&match_questions_list(@questions[@slice]),
&OL(@answers[@slice[@shuffle]]));
ANS(str_cmp([@ALPHABET[&invert(@shuffle)]], filters=>["remove_whitespace","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.
________________________________________________________________________________

#KEYWORDS('Power Series', 'Taylor Series'  )
##DESCRIPTION
##Representation of function as a Taylor series
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Taylor and Maclaurin Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.10')
## Problem1('6')

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

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;



BEGIN_TEXT

The Taylor 
series for \( f(x) = \ln(\sec(x)) \) at \( a = 0 \)  is \( \sum_{n=0}^\infty
c_n\, x^n. \) $BR$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
END_TEXT


ANS(num_cmp(0 ) );
ANS(num_cmp(0 ) );
ANS(num_cmp(.5 ) );
ANS(num_cmp(0 ) );
ANS(num_cmp("1/12" ) );

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

##KEYWORDS('Power Series'  )
##DESCRIPTION
##Representation of function as a power series
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Taylor and Maclaurin Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.10')
## Problem1('10')

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

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = non_zero_random(-4,4);

BEGIN_TEXT

The Taylor series for \(f(x) = x^3\) at \(a=$a\) is \(\sum_{n=0}^\infty c_n( x- $a )^n. \) $BR$BR

Find the first few 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)\}
END_TEXT

ANS(num_cmp( "$a^3" ) );
ANS(num_cmp( "3*($a)^2" ) );
ANS(num_cmp( "3*$a" ) );
ANS(num_cmp( 1 ) );
ANS(num_cmp( 0 ) );

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

##KEYWORDS('Power Series', 'Taylor Series')
##DESCRIPTION
## Determine coefficients in a Taylor series representation
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Taylor and Maclaurin Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.10')
## Problem1('12')

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

loadMacros(
"extraAnswerEvaluators.pl",
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,10,1);
$b = random(2,10,1);
$c = random(3,8,1);
$b1 = 1/$b;

BEGIN_TEXT

The Taylor series of function \( f(x)=\ln(x) \) at \( a = $b \) is given by: 

 \[ f(x) =\sum_{n=0}^\infty c_n (x- $b)^n \] $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

END_TEXT

$ans0 = "ln($b)" ;
$ans1 =  $b1 ;
$ans2 =  "-($b1^2)/2";
$ans3 =  "($b1^3)/3";
$ans4 =  "-($b1^4)/4";

ANS(num_cmp($ans0) );
ANS(num_cmp($ans1) );
ANS(num_cmp($ans2) );
ANS(num_cmp($ans3) );
ANS(num_cmp($ans4) );

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

##KEYWORDS('Power Series'  )
##DESCRIPTION
##Representation of function as a power series
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Taylor and Maclaurin Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.10')
## Problem1('14')

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

loadMacros(
"extraAnswerEvaluators.pl",
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,9,1);
$b = random(.2,.9,.1);
$c = random(3,8,1);
$b1 = 1/$b;

BEGIN_TEXT

Represent the function \( f(x) = x^{$b} \) as a power series: 
\[\sum_{n=0}^\infty c_n (x - $a )^n \] $BR

Find the following coefficients: $BR
\( c_0 = \) \{ans_rule(35)\}  $BR

\( c_1 = \) \{ans_rule(35)\}  $BR

\( c_2 = \) \{ans_rule(35)\}  $BR

\( c_3 = \) \{ans_rule(35)\}  $BR$BR

END_TEXT

$ans0 =  "$a^$b" ;
$ans1 =  "$ans0 * $b / $a";
$ans2 =  "$ans1 * ($b -1)/(2*$a)";
$ans3 =  "$ans2 * ($b -2)/(3*$a)";

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('Power Series'  )
##DESCRIPTION
## Find given coefficients of power series representation
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Taylor and Maclaurin Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.10')
## Problem1('15')

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

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;


BEGIN_TEXT

The Taylor 
series for \( f(x) = \sin(x) \) at \(a= \frac{\pi}{2} \)
  is \( \sum_{n=0}^\infty c_n( x-
\frac{\pi}{2} )^n. \) $BR$BR
Find the first few 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

END_TEXT

$ans0 =  1 ;
$ans1 =  0 ;
$ans2 =  -0.5 ;
$ans3 =  0 ;
$ans4 =  "1 / 24" ;

ANS(num_cmp($ans0 ) );
ANS(num_cmp($ans1 ) );
ANS(num_cmp($ans2 ) );
ANS(num_cmp($ans3 ) );
ANS(num_cmp($ans4 ) );
ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

##KEYWORDS('Taylor Series','cos')
##DESCRIPTION
##  Evaluate limit with Taylor series expansion
##ENDDESCRIPTION

## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Infinite Sequences and Series')
## DBsection('Taylor and Maclaurin Series')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('11.10')
## Problem1('45')

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

loadMacros(
"PGbasicmacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$b = random(2,15,1) ;
$a = "$b*3";
$c = -1/($a*3);

BEGIN_TEXT
Evaluate $BR
\[ \lim_{x \to 0} \frac{\ln (1-x) + x + \frac{x^2}{2}}{$b x^3} \]
$BR $BITALIC Hint: Use power series. $EITALIC
$BR
$BR
Answer: \{ans_rule(30)\}
END_TEXT

ANS(num_cmp($c));

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