## 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();