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| 1 : | ted shifri | 1457 | ## DESCRIPTION |
| 2 : | ## Calculus: Functions | ||
| 3 : | ## ENDDESCRIPTION | ||
| 4 : | |||
| 5 : | ## KEYWORDS('gradient') | ||
| 6 : | |||
| 7 : | ## DBsubject('Calculus') | ||
| 8 : | ## DBchapter('Differentiation') | ||
| 9 : | ## DBsection('Differentiation Rules') | ||
| 10 : | ## Date('9/23/2009') | ||
| 11 : | ## Author('Ted Shifrin') | ||
| 12 : | ## Institution('UGA') | ||
| 13 : | |||
| 14 : | |||
| 15 : | DOCUMENT(); # This should be the first executable line in the problem. | ||
| 16 : | |||
| 17 : | loadMacros("PG.pl", | ||
| 18 : | "PGbasicmacros.pl", | ||
| 19 : | "PGchoicemacros.pl", | ||
| 20 : | "PGanswermacros.pl", | ||
| 21 : | "PGauxiliaryFunctions.pl", | ||
| 22 : | # "extraAnswerEvaluators.pl", | ||
| 23 : | # "compoundProblem.pl", | ||
| 24 : | "Parser.pl", | ||
| 25 : | "PGmatrixmacros.pl" | ||
| 26 : | ); | ||
| 27 : | |||
| 28 : | $a = non_zero_random(-5,5,1); | ||
| 29 : | do{$b = non_zero_random(-5,5,1)} until $b!=$a; | ||
| 30 : | |||
| 31 : | $P = '\left[\begin{array}{c}\mathbf{a}\\c\end{array}\right]'; | ||
| 32 : | |||
| 33 : | $i = random(0,3,1); | ||
| 34 : | $j = random(0,3,1); | ||
| 35 : | |||
| 36 : | @direction1 = ("west","north","east","south"); | ||
| 37 : | @direction2 = ("northwest","northeast","southeast","southwest"); | ||
| 38 : | |||
| 39 : | $ans1=-$a/$b; | ||
| 40 : | $ans2=0; | ||
| 41 : | $ans3=$a/($a**2+$b**2); | ||
| 42 : | $ans4=$b/($a**2+$b**2); | ||
| 43 : | |||
| 44 : | |||
| 45 : | if($i==0){$c=-$a;} elsif($i==2){$c=$a;} elsif($i==1){$c=$b;} else {$c=-$b;} | ||
| 46 : | |||
| 47 : | $ans5=arctan($c); | ||
| 48 : | |||
| 49 : | if($j==0){$d=($b-$a)/sqrt(2);} elsif($j==2){$d=($a-$b)/sqrt(2);} | ||
| 50 : | elsif($j==1){$d=($a+$b)/sqrt(2);} else {$d=-($a+$b)/sqrt(2);} | ||
| 51 : | |||
| 52 : | $ans6=arctan($d); | ||
| 53 : | |||
| 54 : | |||
| 55 : | TEXT(beginproblem()); | ||
| 56 : | # BEGIN_PROBLEM(); | ||
| 57 : | $showPartialCorrectAnswers=1; | ||
| 58 : | Context()->texStrings; | ||
| 59 : | |||
| 60 : | BEGIN_TEXT | ||
| 61 : | |||
| 62 : | Suppose the terrain of a hillside is represented as the graph \(z=f(\mathbf{x})\). Suppose \(f(\mathbf{a})=c\) and | ||
| 63 : | \{ mbox( '\(\nabla f(\mathbf{a}) = \)', display_matrix([['\($a\)'],['\($b\)']]), '.') \} | ||
| 64 : | |||
| 65 : | $PAR | ||
| 66 : | Give the vector in \(\mathbb{R}^3\) with first coordinate \(1\) tangent to the curve through \($P\) along which the elevation stays constant. (Be prepared to explain your answer to the class.) | ||
| 67 : | $BR | ||
| 68 : | \{ mbox(display_matrix([['\(1\)'],[ans_rule(10)],[ans_rule(10)]]))\} | ||
| 69 : | |||
| 70 : | $PAR | ||
| 71 : | Give the vector in \(\mathbb{R}^3\) with third coordinate \(1\) tangent to the curve through \($P\) that goes uphill as steeply as possible. (Be prepared to explain your answer to the class.) | ||
| 72 : | $BR | ||
| 73 : | \{ mbox(display_matrix([[ans_rule(10)],[ans_rule(10)],['\(1\)']])) \} | ||
| 74 : | |||
| 75 : | $PAR | ||
| 76 : | Consider a typical map, in which east is depicted in the \(\mathbf{e}_1\) direction and north is depicted in the \(\mathbf{e}_2\) direction. $BR | ||
| 77 : | If you walk along a path at \($P\) heading due @direction1[$i], at what angle (in radians) are you walking uphill at that instant? (Be prepared to explain your answer to the class.) You may give your answer in terms of an inverse trigonometric function. | ||
| 78 : | $BR | ||
| 79 : | \{ ans_rule(20) \} | ||
| 80 : | |||
| 81 : | $PAR | ||
| 82 : | If you now walk along a path at \($P\) heading @direction2[$j], at what angle (in radians) are you walking uphill at that instant? (Be prepared to explain your answer to the class.) You may give your answer in terms of an inverse trigonometric function. | ||
| 83 : | $BR | ||
| 84 : | \{ ans_rule(20) \} | ||
| 85 : | |||
| 86 : | END_TEXT | ||
| 87 : | |||
| 88 : | ANS(num_cmp($ans1)); | ||
| 89 : | ANS(num_cmp($ans2)); | ||
| 90 : | ANS(num_cmp($ans3)); | ||
| 91 : | ANS(num_cmp($ans4)); | ||
| 92 : | ANS(num_cmp($ans5)); | ||
| 93 : | ANS(num_cmp($ans6)); | ||
| 94 : | |||
| 95 : | #END_PROBLEM(); | ||
| 96 : | ENDDOCUMENT(); # This should be the last executable line in the problem. |
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