chainrule.pg

## Template for calling Sage from within a WebWork pg file

## BEGIN_DESCRIPTION
## Sample problem embedding Sage in WW
## END_DESCRIPTION

DOCUMENT();


loadMacros(
  "PGstandard.pl",
  "MathObjects.pl",
);

###############################################
##
##  pg initializations and regular WeBWorK code


Context("Matrix");
Context()->variables->add(u => 'Real');
Context()->variables->add(v => 'Real');
Context()->variables->add(t => 'Real');

$SageCode1=<<END;
def func(arg):
    f = choice([sin,cos])
    return f(arg)
def term(*vars):
    varchoices = [[v^k for k in range(3)] for v in vars]
    varchoices.append(range(1,4)+range(-4,0))
    return prod(choice(i) for i in varchoices)

h(t) = (term(t), term(t), term(t))
g(x,y,z) = (term(y,z), term(x,z)+term(x,y))
f(u,v) = (func(term(u,v)), term(u,v)+term(u,v), term(u,v))
compose(t) = list(f(*g(*h(t))))
WEBWORK['f']=f(u,v)
WEBWORK['fdiff']=f.diff()(u,v)
WEBWORK['g']=g(x,y,z)
WEBWORK['gdiff']=g.diff()(x,y,z)
WEBWORK['h']=h(t)
WEBWORK['hdiff']=h.diff()(t)
WEBWORK['composediff'] = compose.diff()(t=1.).n()
END

# accepted_tos=>true indicates that you agree to the terms of service:
# https://sagecell.sagemath.org/tos.html
$reply = AskSage($SageCode1,{seed=>$problemSeed, accepted_tos=>true,debug=>0});
$f = Formula($reply->{f});
$g = Formula($reply->{g});
$h = Formula($reply->{h});
$fdiff = Matrix($reply->{fdiff});
$gdiff = Matrix($reply->{gdiff});
$hdiff = Matrix($reply->{hdiff});
$ans = Matrix($reply->{composediff});

Context()->texStrings;
BEGIN_TEXT
Let \(f(u,v)=$f\), \(g(x,y,z)=$g\), and \(h(t)=$h\).  

$BR$BR

\(Df(u,v)=\)\{ $fdiff->ans_array(30)\}

$BR$BR

\(Dg(x,y,z)=\)\{ $gdiff->ans_array(30)\}
$BR$BR
\(Dh(t)=\)\{ $hdiff->ans_array(30)\}
$BR$BR
Let \(\vec M(t)=f(g(h(t)))\).  Use the chain rule to find \(D\vec M\) at \(t=1\).
$BR
\(D\vec M(1)=\)\{ $ans->ans_array(20)\}

END_TEXT

Context()->normalStrings;
ANS($fdiff->cmp());
ANS($gdiff->cmp());
ANS($hdiff->cmp());
ANS($ans->cmp());
ENDDOCUMENT();