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| 1 : | aubreyja | 2584 | ## DBsubject('Calculus') |
| 2 : | ## DBchapter('Differentiation in Several Variables') | ||
| 3 : | ## DBsection('The Gradient and Directional Derivatives') | ||
| 4 : | ## KEYWORDS('calculus') | ||
| 5 : | ## TitleText1('Calculus: Early Transcendentals') | ||
| 6 : | ## EditionText1('2') | ||
| 7 : | ## AuthorText1('Rogawski') | ||
| 8 : | ## Section1('14.5') | ||
| 9 : | ## Problem1('23') | ||
| 10 : | ## Author('JustAsk - Vladimir Finkelshtein') | ||
| 11 : | ## Institution('W.H.Freeman') | ||
| 12 : | |||
| 13 : | DOCUMENT(); | ||
| 14 : | |||
| 15 : | loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); | ||
| 16 : | loadMacros("Parser.pl"); | ||
| 17 : | loadMacros("freemanMacros.pl"); | ||
| 18 : | loadMacros("PGauxiliaryFunctions.pl"); | ||
| 19 : | loadMacros("PGgraphmacros.pl"); | ||
| 20 : | loadMacros("PGchoicemacros.pl"); | ||
| 21 : | |||
| 22 : | TEXT(beginproblem()); | ||
| 23 : | |||
| 24 : | Context()->texStrings; | ||
| 25 : | $x=non_zero_random(-2,2,1); | ||
| 26 : | $y=non_zero_random(-2,2,1); | ||
| 27 : | $xpow=random(1,3,1); | ||
| 28 : | $ypow=random(1,3,1); | ||
| 29 : | $i=non_zero_random(-3,3,1); | ||
| 30 : | $j=non_zero_random(-3,3,1); | ||
| 31 : | $i1=$i; | ||
| 32 : | $j1=$j; | ||
| 33 : | if($i1==1){$i1=''}; | ||
| 34 : | if($i1==-1){$i1='-'}; | ||
| 35 : | if($j1==1){$j1=''}; | ||
| 36 : | if($j1==-1){$j1='-'}; | ||
| 37 : | |||
| 38 : | $norm2=($i)**2+($j)**2; | ||
| 39 : | $context = Context(); | ||
| 40 : | $context->variables->add(y=>'Real'); | ||
| 41 : | |||
| 42 : | $f=Formula("x^($xpow)*y^($ypow)")->reduce(); | ||
| 43 : | $fx=Formula("$xpow*x^($xpow-1)*y^($ypow)")->reduce(); | ||
| 44 : | $fy=Formula("$ypow*x^($xpow)*y^($ypow-1)")->reduce(); | ||
| 45 : | $gradx=$fx->eval(x=>$x, y=>$y); | ||
| 46 : | $grady=$fy->eval(x=>$x, y=>$y); | ||
| 47 : | |||
| 48 : | $dernum=$gradx*$i+$grady*$j; | ||
| 49 : | $der=($gradx*$i+$grady*$j)/sqrt($norm2); | ||
| 50 : | |||
| 51 : | |||
| 52 : | BEGIN_TEXT | ||
| 53 : | \{ textbook_ref_exact("Rogawski ET 2e", "14.5","23") \} | ||
| 54 : | $PAR | ||
| 55 : | 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. | ||
| 56 : | $PAR | ||
| 57 : | \( D_uf($x,$y)=\) \{ans_rule()\} | ||
| 58 : | $BR | ||
| 59 : | END_TEXT | ||
| 60 : | |||
| 61 : | Context()->normalStrings; | ||
| 62 : | ANS($der->cmp); | ||
| 63 : | Context()->texStrings; | ||
| 64 : | |||
| 65 : | SOLUTION(EV3(<<'END_SOLUTION')); | ||
| 66 : | $PAR | ||
| 67 : | $SOL | ||
| 68 : | $BR | ||
| 69 : | We normalize \(\mathbf{v}\) to obtain a unit vector \(\mathbf{u}\) in the direction of \(\mathbf{v}\): | ||
| 70 : | \[\mathbf{u}=\frac{\mathbf{v}}{||\mathbf{v}||}=\frac{1}{\sqrt{$norm2}}($i1\mathbf{i} +$j1 \mathbf{j})\] | ||
| 71 : | We compute the gradient of \(f(x,y)=$f\) at the point \(P\): | ||
| 72 : | \[\nabla f=\left< \frac{\partial{f}}{\partial{x}}, \frac{\partial{f}}{\partial{y}} \right>=\left<$fx,$fy\right>\quad \Rightarrow \quad \nabla f($x,$y)=\left<$gradx, $grady\right>\] | ||
| 73 : | The directional derivative in the direction \(\mathbf{v}\) is thus | ||
| 74 : | \[D_uf($x,$y)=\nabla f($x,$y) \cdot \mathbf{u}=\left< $gradx, $grady \right> \cdot \frac{1}{\sqrt{$norm2}} \left<$i, $j\right>=\frac{$dernum}{\sqrt{$norm2}}\approx $der\] | ||
| 75 : | $BR | ||
| 76 : | END_SOLUTION | ||
| 77 : | |||
| 78 : | ENDDOCUMENT(); |
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