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Partial cleanup of problem.
1 #DESCRIPTION 2 #KEYWORDS('limits', 'sequences') 3 # Find limits (intuitively or by experimental calculations). 4 #ENDDESCRIPTION 5 6 ##KEYWORDS('Calculus') 7 ##Tagged by ynw2d 8 9 ##DBsubject('Calculus') 10 ## DBchapter('Limits and Derivatives') 11 ##DBsection('Continuity') 12 13 14 DOCUMENT(); # This should be the first executable line in the problem. 15 16 loadMacros("PG.pl", 17 "PGbasicmacros.pl", 18 "PGchoicemacros.pl", 19 "PGanswermacros.pl", 20 "PGauxiliaryFunctions.pl", 21 "PGgraphmacros.pl"); 22 23 $showPartialCorrectAnswers=1; 24 # define a broken graph f and a broken graph g: 25 26 # first define the parameters (value and first derivatives) 27 foreach my $i (0..5) { 28 $f1y[$i] = random(0, 3, 1); 29 $f1yp[$i] = random(-1,1, 2); 30 $f2y[$i] = random(0, 3, 1); 31 $f2yp[$i] = random(-1,1, 2); 32 $g1y[$i] = random(0, 3, 1); 33 $g1yp[$i] = random(-1,1, 2); 34 $g2y[$i] = random(0, 3, 1); 35 $g2yp[$i] = random(-1,1, 2); 36 } 37 38 # define the functions 39 $hermite_f1 = new Hermite( [ -1 , 0, 1, 2 , 3, 4 ], 40 [ @f1y ], 41 [ @f1yp ]); 42 $hermite_f2 = new Hermite( [ -1 , 0, 1, 2 , 3, 4 ], 43 [ @f2y ], 44 [ @f2yp ]); 45 $hermite_g1 = new Hermite( [ -1 , 0, 1, 2 , 3, 4 ], 46 [ @g1y ], 47 [ @g1yp ]); 48 $hermite_g2 = new Hermite( [ -1 , 0, 1, 2 , 3, 4 ], 49 [ @g2y ], 50 [ @g2yp ]); 51 52 # define the break points where we switch from one hemite function to the other 53 # and the value of the function at that point 54 55 $xf = random(0,3); 56 $yf = random(-1,4); 57 do {$xg = random(0,3);} until ($xg != $xf); 58 #do {$xg = random(0,3);} until (1 == 1); 59 $yg = random(-1,4); 60 61 62 # define the broken functions themselves -- we'll need them to answer questions. 63 sub f_rule { 64 my $x = shift; 65 my $out; 66 if ($x < $xf ) { 67 $out = $hermite_f1->rf_f->($x); 68 } elsif ($x == $xf) { 69 $out = $yf; 70 } else { 71 $out = $hermite_f2->rf_f->($x); 72 } 73 $out; 74 }; 75 sub g_rule { 76 my $x = shift; 77 my $out; 78 if ($x < $xg ) { 79 $out = $hermite_g1->rf_f->($x); 80 } elsif ($x == $xg) { 81 $out = $yg; 82 } else { 83 $out = $hermite_g2->rf_f->($x); 84 } 85 $out; 86 }; 87 88 89 # plot the f graph 90 91 $graphf = init_graph(-2,-2,5,5,grid =>[7,7], axes => [0,0]); 92 $graphg = init_graph(-2,-2,5,5,,grid =>[7,7], axes => [0,0]); 93 $f1 = new Fun($hermite_f1->rf_f,$graphf); 94 $f2 = new Fun($hermite_f2->rf_f,$graphf); 95 $graphf->stamps(open_circle($xf, $hermite_f1->rf_f->($xf), 'red') ); 96 $graphf->stamps(open_circle($xf, $hermite_f2->rf_f->($xf), 'red') ); 97 $graphf ->stamps( closed_circle($xf, $yf, 'red') ); # value at $xf; 98 $f1->color('red'); 99 $f2->color('red'); 100 # restrict the two partial graph domains so that only the 'active' one is drawn 101 $f1->domain(-1, $xf); 102 $f2->domain($xf, 4); 103 104 #plot the g graph 105 $g1 = new Fun($hermite_g1->rf_f,$graphg); 106 $g2 = new Fun($hermite_g2->rf_f,$graphg); 107 $graphg->stamps(open_circle($xg, $hermite_g1->rf_f->($xg), 'blue') ); 108 $graphg->stamps(open_circle($xg, $hermite_g2->rf_f->($xg), 'blue') ); 109 $graphg ->stamps(closed_circle($xg, $yg, 'blue') ); # value at $xg; 110 $g2->color('blue'); 111 $g2->color('blue'); 112 # restrict the two partial graph domains so that only the 'active' one is drawn 113 $g1->domain(-1, $xg); 114 $g2->domain($xg, 4); 115 116 TEXT(beginproblem()); 117 118 119 # draw the graphs 120 BEGIN_TEXT 121 \{ begintable(2) \} 122 \{ row( image( insertGraph($graphf), tex_size =>400), image( insertGraph($graphg) , tex_size =>400 ) )\} 123 \{ row( '\(f(x)\)', '\(g(x)\)') \} 124 \{ endtable() \} 125 END_TEXT 126 127 sub rnd { 128 my $x = shift; 129 my $places = shift; 130 $places = 0 unless defined($places); 131 my $sign = ($x <0 ) ? -1 : 1; 132 $x = abs($x); 133 134 $x = int( $x*10 **(-$places) +.5) *10 **($places); 135 $sign*$x; 136 } 137 # now construct the questions 138 @questions = (); 139 @answers = (); 140 141 qa( ~~@questions, ~~@answers, 142 '\( \displaystyle \lim_{x\to $xf^-} [f(x) + g(x) ] \) ', 143 num_cmp(rnd( f_rule($xf-.00001) +g_rule($xf-.00001) ), strings=>['DNE'] ), # we are expecting integer answers 144 '\( \displaystyle \lim_{x\to $xf^+} [f(x) + g(x) ] \) ', 145 num_cmp(rnd( f_rule($xf+.00001) +g_rule($xf+.00001) ), strings=>['DNE'] ), # we are expecting integer answers 146 '\( f($xf) + g($xf) \)', 147 num_cmp(f_rule($xf) + g_rule($xf), strings=>['DNE'] ), 148 149 '\( \displaystyle \lim_{x\to $xf^-} [f(x)g(x) ] \) ', 150 num_cmp(rnd( f_rule($xf-.00001)*g_rule($xf-.00001) ), strings=>['DNE'] ), # we are expecting integer answers 151 '\( \displaystyle \lim_{x\to $xf^+} [f(x)g(x) ] \) ', 152 num_cmp(rnd( f_rule($xf+.00001)*g_rule($xf+.00001) ), strings=>['DNE'] ), # we are expecting integer answers 153 '\( f($xf)g($xf) \)', 154 num_cmp(f_rule($xf)*g_rule($xf), strings=>['DNE'] ), 155 156 '\( \displaystyle \lim_{x\to $xf^-} [f( g(x) ) ] \) ', 157 num_cmp(rnd( f_rule(g_rule($xf-.00001) ) ), strings=>['DNE'] ), # we are expecting integer answers 158 '\( \displaystyle \lim_{x\to $xf^+} [f( g(x) ) ] \) ', 159 num_cmp(rnd( f_rule(g_rule($xf+.00001) ) ), strings=>['DNE'] ), # we are expecting integer answers 160 '\( f( g($xf) ) \)', 161 num_cmp(f_rule( g_rule($xf) ), strings=>['DNE'] ), 162 163 #check for zeros in answering the division limits 164 '\( \displaystyle \lim_{x\to $xf^-} [f(x)/g(x) ] \) ', 165 ( not rnd(g_rule($xf-.00001) ) and rnd(f_rule($xf-.00001) ) ) ? str_cmp('DNE') # if g is zero, but f is not 166 : num_cmp(rnd( f_rule($xf-.00001)/g_rule($xf-.00001) , -3 ), tol =>.002, strings=>['DNE'] ), # the chances of g being identically zero are small since the derivatives at defined points are never zero. 167 # we are expecting integer answers 168 '\( \displaystyle \lim_{x\to $xf^+} [f(x)/g(x) ] \) ', 169 ( not rnd(g_rule($xf+.00001) ) and rnd(f_rule($xf+.00001) ) ) ? str_cmp('DNE') 170 : num_cmp(rnd( f_rule($xf+.00001)/g_rule($xf+.00001) , -3 ), tol =>.002, strings=>['DNE'] ) , 171 # we are expecting integer answers 172 '\( f($xf)/g($xf) \)', 173 (rnd(g_rule($xf)) ) ? num_cmp( rnd(f_rule($xf)/g_rule($xf), -3), tol =>.002, strings=>['DNE'] ) : str_cmp('DNE'), 174 175 '\( \displaystyle \lim_{x\to $xg^-} [f(x) + g(x) ] \) ', 176 num_cmp(rnd( f_rule($xg-.00001) +g_rule($xg-.00001) ), strings=>['DNE'] ), # we are expecting integer answers 177 '\( \displaystyle \lim_{x\to $xg^+} [f(x) + g(x) ] \) ', 178 num_cmp(rnd( f_rule($xg+.00001) +g_rule($xg+.00001) ), strings=>['DNE'] ), # we are expecting integer answers 179 '\( f($xg) + g($xg) \)', 180 num_cmp(f_rule($xg) + g_rule($xg), strings=>['DNE'] ), 181 182 '\( \displaystyle \lim_{x\to $xg^-} [f(x)g(x) ] \) ', 183 num_cmp(rnd( f_rule($xg-.00001)*g_rule($xg-.00001) ), strings=>['DNE'] ), # we are expecting integer answers 184 '\( \displaystyle \lim_{x\to $xg^+} [f(x)g(x) ] \) ', 185 num_cmp(rnd( f_rule($xg+.00001)*g_rule($xg+.00001) ), strings=>['DNE'] ), # we are expecting integer answers 186 '\( f($xg)g($xg) \)', 187 num_cmp(f_rule($xg)*g_rule($xg), strings=>['DNE'] ), 188 189 '\( \displaystyle \lim_{x\to $xg^-} [f( g(x) ) ] \) ', 190 num_cmp(rnd( f_rule( g_rule($xg-.00001) )), strings=>['DNE'] ) , # we are expecting integer answers 191 '\( \displaystyle \lim_{x\to $xg^+} [f( g(x) ) ] \) ', 192 num_cmp(rnd( f_rule( g_rule( $xg + .00001 ) ) ), strings=>['DNE'] ), # we are expecting integer answers 193 '\( f( g($xg) ) \)', 194 num_cmp(f_rule( g_rule($xg) ), strings=>['DNE'] ), 195 196 #check for zeros in answering the division limits 197 '\( \displaystyle \lim_{x\to $xg^-} [f(x)/g(x) ] \) ', 198 ( not rnd(g_rule($xg-.00001) ) and rnd(f_rule($xg-.00001) ) ) ? str_cmp('DNE') # if g is zero, but f is not 199 : num_cmp(rnd( f_rule($xg-.00001)/g_rule($xg-.00001) , -3 ), tol =>.002, strings=>['DNE'] ), # the chances of g being identically zero are small since the derivatives at defined points are never zero. 200 # we are expecting integer answers 201 '\( \displaystyle \lim_{x\to $xg^+} [f(x)/g(x) ] \) ', 202 ( not rnd(g_rule($xg+.00001) ) and rnd(f_rule($xg+.00001) ) ) ? str_cmp('DNE') 203 : num_cmp(rnd( f_rule($xg+.00001)/g_rule($xg+.00001) , -3 ), tol =>.002, strings=>['DNE'] ) , 204 # we are expecting integer answers 205 '\( f($xg)/g($xg) \)', 206 (rnd(g_rule($xg)) ) ? num_cmp( rnd(f_rule($xg)/g_rule($xg), -3), tol =>.002, strings=>['DNE'] ) : str_cmp('DNE'), 207 208 ); 209 210 @slice = NchooseK(scalar(@questions) ,4); 211 212 $vf1 = f_rule(1.1); 213 $vg1 = g_rule(1.1); 214 $jjs=''; 215 for $jj (0..5) { 216 $jjs .= ' ('. ($jj-1) .', '. $f2y[$jj] . ', '. $f2yp[$jj] .') '; 217 } 218 $jj2 =''; 219 for $jj (0..20) { 220 $jj2 .= (f_rule(1 + $jj/20)) . ', '; 221 } 222 223 BEGIN_TEXT 224 $BR 225 The graphs of \( f(x) \) and \( g(x) \) are given above. Use them to evaluate each quantity below. 226 Write $BITALIC DNE$EITALIC if the limit or value does not exist (or if it's infinity). 227 $BR 228 \{ match_questions_list(@questions[@slice] ) \} 229 END_TEXT 230 231 #$BR 232 #$vf1 and $vg1 and $jjs 233 #$BR$BR 234 #$jj2 235 #END_TEXT 236 237 ANS(@answers[@slice] ); 238 ENDDOCUMENT(); # This should be the last executable line in the problem.
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