View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/14_Differentiation_in_Several_Variables/14.4_Differentiability_and_Tangent_Planes/14.4.7.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Differentiation in Several Variables')
    3 ## DBsection('Differentiability, Linear Approximation, and Tangent Planes')
    4 ## KEYWORDS('calculus')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('14.4')
    9 ## Problem1('7')
   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 $r=non_zero_random(-2,2,1);
   27 $s=1;
   28 $a=non_zero_random(-3,3,1);
   29 $b=non_zero_random(-3,3,1);
   30 $rpow=random(2,4,1);
   31 $spow=random(-4,-2,1);
   32 
   33 $context = Context();
   34 $context->variables->add(r=>'Real');
   35 $context->variables->add(s=>'Real');
   36 
   37 $f=Formula("$a*r^($rpow)*s^(-1/2)+$b*s^($spow)")->reduce();
   38 $fr=Formula("$a*$rpow*r^($rpow-1)*s^(-1/2)")->reduce();
   39 $fs=Formula("-1/2*$a*r^($rpow)*s^(-3/2)+$spow*$b*s^($spow-1)")->reduce();
   40 
   41 $f0=$f->eval(r=>$r, s=>$s);
   42 $fr0=$fr->eval(r=>$r, s=>$s);
   43 $fs0=$fs->eval(r=>$r, s=>$s);
   44 
   45 $z=Formula("$f0+$fr0*(r-$r)+$fs0*(s-$s)")->reduce();
   46 $coef=$f0-$fr0*$r-$fs0*$s;
   47 $z1=Formula("$fr0*r+$fs0*s+$coef")->reduce();
   48 
   49 BEGIN_TEXT
   50 \{ textbook_ref_exact("Rogawski ET 2e", "14.4","7") \}
   51 $PAR
   52 Find an equation of the tangent plane at the given point: $BR
   53 \(F(r,s)=$f, \qquad ($r,$s) \) $PAR
   54 \(z=\)\{ans_rule()\}
   55 $BR
   56 END_TEXT
   57 
   58 Context()->normalStrings;
   59 ANS($z1->cmp);
   60 Context()->texStrings;
   61 
   62 SOLUTION(EV3(<<'END_SOLUTION'));
   63 $PAR
   64 $SOL
   65 $BR
   66 The equation of the tangent plane at \(($r,$s)\) is
   67 \[z=f($r,$s)+f_r($r,$s)(r-$r)+f_s($r,$s)(s-$s)\]
   68 
   69 We compute the value of the function and its partial derivatives at the point \(($r,$s)\):
   70 \[
   71 \left. \begin{array}{lcl}
   72 f(r,s)=$f &  & f($r,$s)=$f0 \\
   73 f_r(r,s)=$fr & \Rightarrow & f_r($r,$s)=$fr0 \\
   74 f_s(r,s)=$fs &  & f_s($r,$s)=$fs0 \end{array} \right.
   75 \]
   76 
   77 We substitute these values to obtain the following equation of the tangent plane:
   78 \[z=$z=$z1\]
   79 $BR
   80 END_SOLUTION
   81 
   82 ENDDOCUMENT();