## DESCRIPTION ## Calculus ## ENDDESCRIPTION ## KEYWORDS('tangent' 'slope') ## Tagged by tda2d ## DBsubject('Calculus') ## DBchapter('Partial Derivatives') ## DBsection('Directional Derivatives and the Gradient Vector') ## Date('') ## Author('') ## Institution('Dartmouth') ## TitleText1('Basic Multivariable Calculus') ## EditionText1('3') ## AuthorText1('Marsden, Tromba, Weinstein') ## Section1('2.6') ## Problem1('') DOCUMENT(); loadMacros("PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl", "PGgraphmacros.pl", "Dartmouthmacros.pl"); ## Do NOT show partial correct answers $showPartialCorrectAnswers = 1; ## Lots of set up goes here$x1 = random(1,8,1); $y1 = random(1,8,1);$a = random(1,4,1); $a_x = clean_scalar_string($a, "x"); $b = random(1,4,1);$b_y = clean_scalar_string($b, "y");$c = random(1,4,1); $c_xy = clean_scalar_string($c, "xy"); #$d = sqrt($a*$x1+$b*$y1) + sqrt($c*$x1*$y1); $num =$a/(2*sqrt($a*$x1 + $b*$y1) ) + $c*$y1/(2*sqrt($c*$x1*$y1) );$denom = $b/(2*sqrt($a*$x1+$b*$y1) ) +$c*$x1/(2*sqrt($c*$x1*$y1) ); $yp = -$num/$denom; # ## Ok, we are ready to begin the problem... ## TEXT(beginproblem()); BEGIN_TEXT$BR Find the slope of the tangent line to the curve $BR $$\displaystyle \sqrt{a_x +b_y} + \sqrt{c_xy} = \sqrt{\{a*x1+b*y1\}} + \sqrt{\{c*x1*y1\}}$$ at the point $$( x1,y1 )$$.$BR The slope is \{ ans_rule(70) \}. END_TEXT ANS(num_cmp(\$yp)); ENDDOCUMENT();