View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/14_Differentiation_in_Several_Variables/14.5_The_Gradient_and_Directional_Derivatives/14.5.1.pg

    1 ## 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('1')
   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 
   26 $xpow=random(1,3,1);
   27 $ypow=random(1,3,1);
   28 $a=random(1,2,1);
   29 $b=random(1,2,1);
   30 $t=non_zero_random(-1,1);
   31 
   32 $context = Context();
   33 $context->variables->add(y=>'Real');
   34 $context->variables->add(t=>'Real');
   35 
   36 $f=Formula("x^($xpow)*y^($ypow)")->reduce();
   37 $cx=Formula("$a*t^2")->reduce();
   38 $cy=Formula("$b*t^3")->reduce();
   39 $fx=Formula("$xpow*x^($xpow-1)*y^($ypow)")->reduce();
   40 $fy=Formula("$ypow*x^($xpow)*y^($ypow-1)")->reduce();
   41 $cxx=Formula("2*$a*t")->reduce();
   42 $cyy=Formula("3*$b*t^2")->reduce();
   43 $answer=Formula("(2*$a*$xpow)*x^($xpow-1)*y^($ypow)*t+(3*$b*$ypow)*x^($xpow)*y^($ypow-1)*t^2")->reduce();
   44 
   45 $fc=Formula("($a^($xpow))*t^(2*$xpow)*($b^($ypow))*t^(3*$ypow)")->reduce();
   46 $fc1=Formula("$a^($xpow)*$b^($ypow)*t^(2*$xpow+3*$ypow)")->reduce();
   47 $fct=Formula("(2*$xpow+3*$ypow)*$a^($xpow)*$b^($ypow)*t^(2*$xpow+3*$ypow-1)")->reduce();
   48 $answer2=$fct->eval(t=>$t);
   49 
   50 BEGIN_TEXT
   51 \{ textbook_ref_exact("Rogawski ET 2e", "14.5","1") \}
   52 $PAR
   53 Let \(f(x,y)=$f\) and \(c(t)=\left($cx,$cy\right)\)
   54 $PAR
   55 (a) Calculate:
   56 $BR
   57 \(\nabla f \cdot c'(t)=\) \{ans_rule()\}
   58 $PAR
   59 (b) Use the Chain Rule for Paths to evaluate \( \frac{d}{dt}f(c(t))\) at \(t=$t\).
   60 $BR
   61 \( \frac{d}{dt}f(c($t))=\)\{ans_rule()\}
   62 $BR
   63 END_TEXT
   64 
   65 Context()->normalStrings;
   66 ANS($answer->cmp);
   67 ANS($answer2->cmp);
   68 Context()->texStrings;
   69 
   70 SOLUTION(EV3(<<'END_SOLUTION'));
   71 $PAR
   72 $SOL
   73 $BR
   74 (a) We compute the partial derivatives of \(f(x,y)=$f\)
   75 \[\frac{\partial{f}}{\partial{x}}=$fx, \quad \frac{\partial{f}}{\partial{y}}=$fy\]
   76 The gradient vector is thus
   77 \[\nabla f=\left<$fx,$fy\right>\]
   78 Also,
   79 \[c'(t)=\left< \left(cx\right)', \left(cy\right)' \right>=\left<$cxx, $cyy\right> \]
   80 \[ \nabla f \cdot c'(t)=\left<$fx,$fy\right>\cdot\left<$cxx, $cyy\right>=$answer\]
   81 $PAR
   82 (b) Using the Chain Rule and substituting \(x=$cx, y=$cy\) gives
   83 \[\frac{d}{dt}f(c(t))=\frac{d}{dt}\left($fc\right)=\frac{d}{dt}\left($fc1\right)=$fct\]
   84 At the point \(t=$t\), we obtain
   85 \[\left.\frac{d}{dt}f(c(t))\right|_{t=$t}=\left.$fct\right|_{t=$t}=$answer2\]
   86 $BR
   87 END_SOLUTION
   88 
   89 ENDDOCUMENT();