## DBsubject('Calculus')
## DBchapter('Differentiation in Several Variables')
## DBsection('The Gradient and Directional Derivatives')
## KEYWORDS('calculus')
## TitleText1('Calculus: Early Transcendentals')
## EditionText1('2')
## AuthorText1('Rogawski')
## Section1('14.5')
## Problem1('23')
## Author('JustAsk - Vladimir Finkelshtein')
## Institution('W.H.Freeman')

DOCUMENT();

loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
loadMacros("Parser.pl");
loadMacros("freemanMacros.pl");
loadMacros("PGauxiliaryFunctions.pl");
loadMacros("PGgraphmacros.pl");
loadMacros("PGchoicemacros.pl");

TEXT(beginproblem());

Context()->texStrings;
$x=non_zero_random(-2,2,1);
$y=non_zero_random(-2,2,1);
$xpow=random(1,3,1);
$ypow=random(1,3,1);
$i=non_zero_random(-3,3,1);
$j=non_zero_random(-3,3,1);
$i1=$i;
$j1=$j;
if($i1==1){$i1=''};
if($i1==-1){$i1='-'};
if($j1==1){$j1=''};
if($j1==-1){$j1='-'};

$norm2=($i)**2+($j)**2;
$context = Context();
$context->variables->add(y=>'Real');

$f=Formula("x^($xpow)*y^($ypow)")->reduce();
$fx=Formula("$xpow*x^($xpow-1)*y^($ypow)")->reduce();
$fy=Formula("$ypow*x^($xpow)*y^($ypow-1)")->reduce();
$gradx=$fx->eval(x=>$x, y=>$y);
$grady=$fy->eval(x=>$x, y=>$y);

$dernum=$gradx*$i+$grady*$j;
$der=($gradx*$i+$grady*$j)/sqrt($norm2);


BEGIN_TEXT
\{ textbook_ref_exact("Rogawski ET 2e", "14.5","23") \}
$PAR
Calculate the directional derivative of \(f(x,y)=$f\) in the direction of \(\mathbf{v}=$i1 \mathbf{i}+$j1\mathbf{j}\) at the point \(P=($x,$y)\). Remember to normalize the direction vector.
$PAR
\( D_uf($x,$y)=\) \{ans_rule()\} 
$BR
END_TEXT 

Context()->normalStrings;
ANS($der->cmp);
Context()->texStrings;

SOLUTION(EV3(<<'END_SOLUTION'));
$PAR
$SOL
$BR
We normalize \(\mathbf{v}\) to obtain a unit vector \(\mathbf{u}\) in the direction of \(\mathbf{v}\):
\[\mathbf{u}=\frac{\mathbf{v}}{||\mathbf{v}||}=\frac{1}{\sqrt{$norm2}}($i1\mathbf{i} +$j1 \mathbf{j})\]
We compute the gradient of \(f(x,y)=$f\) at the point \(P\):
\[\nabla f=\left< \frac{\partial{f}}{\partial{x}}, \frac{\partial{f}}{\partial{y}} \right>=\left<$fx,$fy\right>\quad \Rightarrow \quad \nabla f($x,$y)=\left<$gradx, $grady\right>\]
The directional derivative in the direction \(\mathbf{v}\) is thus
\[D_uf($x,$y)=\nabla f($x,$y) \cdot \mathbf{u}=\left< $gradx, $grady \right> \cdot \frac{1}{\sqrt{$norm2}} \left<$i, $j\right>=\frac{$dernum}{\sqrt{$norm2}}\approx $der\]
$BR
END_SOLUTION

ENDDOCUMENT();