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Fixed one typesetting issue.
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('The Limit of a Function') 12 ## TitleText1('Calculus: Early Transcendentals') 13 ## EditionText1('1') 14 ## AuthorText1('Rogawski') 15 ## Section1('2.2') 16 ## Problem1('37') 17 18 ## TitleText2('Calculus: Early Transcendentals') 19 ## EditionText2('6') 20 ## AuthorText2('Stewart') 21 ## Section2('2.3') 22 ## Problem2('2') 23 24 25 26 DOCUMENT(); # This should be the first executable line in the problem. 27 28 loadMacros("PG.pl", 29 "PGbasicmacros.pl", 30 "PGchoicemacros.pl", 31 "PGanswermacros.pl", 32 "PGauxiliaryFunctions.pl", 33 "PGgraphmacros.pl"); 34 35 $showPartialCorrectAnswers=1; 36 37 # utility 38 sub rnd { 39 my $x = shift; 40 my $places = shift; 41 $places = 0 unless defined($places); 42 my $sign = ($x <0 ) ? -1 : 1; 43 $x = abs($x); 44 45 $x = int( $x*10 **(-$places) +.5) *10 **($places); 46 $sign*$x; 47 } 48 49 # define a broken graph f and a broken graph g: 50 51 # do .. until to protect against L'Hopital type questions 52 do { 53 $ok = 1; 54 55 # first define the parameters (value and first derivatives) 56 foreach my $i (0..5) { 57 $f1y[$i] = random(0, 3, 1); 58 $f1yp[$i] = random(-1,1, 2); 59 $f2y[$i] = random(0, 3, 1); 60 $f2yp[$i] = random(-1,1, 2); 61 $g1y[$i] = random(0, 3, 1); 62 $g1yp[$i] = random(-1,1, 2); 63 $g2y[$i] = random(0, 3, 1); 64 $g2yp[$i] = random(-1,1, 2); 65 } 66 67 # define the functions 68 $hermite_f1 = new Hermite( [ -1 , 0, 1, 2 , 3, 4 ], 69 [ @f1y ], 70 [ @f1yp ]); 71 $hermite_f2 = new Hermite( [ -1 , 0, 1, 2 , 3, 4 ], 72 [ @f2y ], 73 [ @f2yp ]); 74 $hermite_g1 = new Hermite( [ -1 , 0, 1, 2 , 3, 4 ], 75 [ @g1y ], 76 [ @g1yp ]); 77 $hermite_g2 = new Hermite( [ -1 , 0, 1, 2 , 3, 4 ], 78 [ @g2y ], 79 [ @g2yp ]); 80 81 # define the break points where we switch from one hemite function to the other 82 # and the value of the function at that point 83 84 $xf = random(0,3); 85 $yf = random(-1,4); 86 do { $xg = random(0,3); } until ($xf != $xg); 87 $yg = random(-1,4); 88 89 # Now check to see if we might produce a L'Hoptital type question, 90 # which would be much harder for some students than others 91 92 $ok = 0 if(($xf<$xg) and (rnd($hermite_f1->rf_f->($xf)) == 0 or rnd($hermite_f2->rf_f->($xf)) == 0) 93 and (rnd($hermite_g1->rf_f->($xf)) == 0) ); 94 $ok = 0 if(($xf>$xg) and (rnd($hermite_f2->rf_f->($xg)) == 0) 95 and (rnd($hermite_g1->rf_f->($xg)) == 0 or rnd($hermite_g2->rf_f->($xg)) == 0)); 96 $ok = 0 if(($xf==$xg) and (rnd($hermite_f1->rf_f->($xg)) == 0 or rnd($hermite_f2->rf_f->($xg)) == 0) 97 and (rnd($hermite_g1->rf_f->($xg)) == 0 or rnd($hermite_g2->rf_f->($xg)) == 0)); 98 99 } until ($ok == 1); 100 101 # define the broken functions themselves -- we'll need them to answer questions. 102 sub f_rule { 103 my $x = shift; 104 my $out; 105 if ($x < $xf ) { 106 $out = $hermite_f1->rf_f->($x); 107 } elsif ($x == $xf) { 108 $out = $yf; 109 } else { 110 $out = $hermite_f2->rf_f->($x); 111 } 112 $out; 113 }; 114 sub g_rule { 115 my $x = shift; 116 my $out; 117 if ($x < $xg ) { 118 $out = $hermite_g1->rf_f->($x); 119 } elsif ($x == $xg) { 120 $out = $yg; 121 } else { 122 $out = $hermite_g2->rf_f->($x); 123 } 124 $out; 125 }; 126 127 128 # plot the f graph 129 130 $graphf = init_graph(-2,-2,5,5,grid =>[7,7], axes => [0,0], size =>[500,500]); 131 $graphg = init_graph(-2,-2,5,5,,grid =>[7,7], axes => [0,0], size =>[500,500]); 132 $f1 = new Fun($hermite_f1->rf_f,$graphf); 133 $f2 = new Fun($hermite_f2->rf_f,$graphf); 134 $graphf->stamps(open_circle($xf, $hermite_f1->rf_f->($xf), 'red') ); 135 $graphf->stamps(open_circle($xf, $hermite_f2->rf_f->($xf), 'red') ); 136 $graphf ->stamps( closed_circle($xf, $yf, 'red') ); # value at $xf; 137 $f1->color('red'); 138 $f2->color('red'); 139 # restrict the two partial graph domains so that only the 'active' one is drawn 140 $f1->domain(-1, $xf); 141 $f2->domain($xf, 4); 142 143 #plot the g graph 144 $g1 = new Fun($hermite_g1->rf_f,$graphg); 145 $g2 = new Fun($hermite_g2->rf_f,$graphg); 146 $graphg->stamps(open_circle($xg, $hermite_g1->rf_f->($xg), 'blue') ); 147 $graphg->stamps(open_circle($xg, $hermite_g2->rf_f->($xg), 'blue') ); 148 $graphg ->stamps(closed_circle($xg, $yg, 'blue') ); # value at $xg; 149 $g2->color('blue'); 150 $g2->color('blue'); 151 # restrict the two partial graph domains so that only the 'active' one is drawn 152 $g1->domain(-1, $xg); 153 $g2->domain($xg, 4); 154 155 TEXT(beginproblem()); 156 157 158 # draw the graphs 159 TEXT( 160 begintable(2), 161 row( image( insertGraph($graphf), tex_size =>400), image( insertGraph($graphg) , tex_size =>400 ) ), 162 row(EV2('\(f(x)\)'), EV2('\(g(x)\)')), 163 endtable(), 164 ); 165 # now construct the questions 166 @questions = (); 167 @answers = (); 168 169 qa( ~~@questions, ~~@answers, 170 '\( \displaystyle \lim_{x\to $xf^-} [f(x) + g(x) ] \) ', 171 num_cmp(rnd( f_rule($xf-.00001) +g_rule($xf-.00001) ), strings => ['DNE'] ), # we are expecting integer answers 172 '\( \displaystyle \lim_{x\to $xf^+} [f(x) + g(x) ] \) ', 173 num_cmp(rnd( f_rule($xf+.00001) +g_rule($xf+.00001) ), strings => ['DNE'] ), # we are expecting integer answers 174 '\( f($xf) + g($xf) \)', 175 num_cmp(f_rule($xf) + g_rule($xf), strings => ['DNE'] ), 176 177 '\( \displaystyle \lim_{x\to $xf^-} [f(x)g(x) ] \) ', 178 num_cmp(rnd( f_rule($xf-.00001)*g_rule($xf-.00001) ), strings => ['DNE'] ), # we are expecting integer answers 179 '\( \displaystyle \lim_{x\to $xf^+} [f(x)g(x) ] \) ', 180 num_cmp(rnd( f_rule($xf+.00001)*g_rule($xf+.00001) ), strings => ['DNE'] ), # we are expecting integer answers 181 '\( f($xf)g($xf) \)', 182 num_cmp(f_rule($xf)*g_rule($xf), strings => ['DNE'] ), 183 184 '\( \displaystyle \lim_{x\to $xf^-} [f( g(x) ) ] \) ', 185 num_cmp(rnd( f_rule(g_rule($xf-.00001) ) ), strings => ['DNE'] ), # we are expecting integer answers 186 '\( \displaystyle \lim_{x\to $xf^+} [f( g(x) ) ] \) ', 187 num_cmp(rnd( f_rule(g_rule($xf+.00001) ) ), strings => ['DNE'] ), # we are expecting integer answers 188 '\( f( g($xf) ) \)', 189 num_cmp(f_rule( g_rule($xf) ), strings => ['DNE'] ), 190 191 #check for zeros in answering the division limits 192 '\( \displaystyle \lim_{x\to $xf^-} [f(x)/g(x) ] \) ', 193 ( not rnd(g_rule($xf-.00001) ) and rnd(f_rule($xf-.00001) ) ) ? num_cmp('DNE', strings => ['DNE'] ) # if g is zero, but f is not 194 : 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. 195 # we are expecting integer answers 196 '\( \displaystyle \lim_{x\to $xf^+} [f(x)/g(x) ] \) ', 197 ( not rnd(g_rule($xf+.00001) ) and rnd(f_rule($xf+.00001) ) ) ? num_cmp('DNE', strings => ['DNE'] ) 198 : num_cmp(rnd( f_rule($xf+.00001)/g_rule($xf+.00001) , -3 ), tol =>.002, strings => ['DNE'] ) , 199 # we are expecting integer answers 200 '\( f($xf)/g($xf) \)', 201 (rnd(g_rule($xf)) ) ? num_cmp( rnd(f_rule($xf)/g_rule($xf), -3), tol =>.002, strings => ['DNE'] ) : num_cmp('DNE', strings => ['DNE'] ), 202 203 '\( \displaystyle \lim_{x\to $xg^-} [f(x) + g(x) ] \) ', 204 num_cmp(rnd( f_rule($xg-.00001) +g_rule($xg-.00001) ), strings => ['DNE'] ), # we are expecting integer answers 205 '\( \displaystyle \lim_{x\to $xg^+} [f(x) + g(x) ] \) ', 206 num_cmp(rnd( f_rule($xg+.00001) +g_rule($xg+.00001) ), strings => ['DNE'] ), # we are expecting integer answers 207 '\( f($xg) + g($xg) \)', 208 num_cmp(f_rule($xg) + g_rule($xg), strings => ['DNE'] ), 209 210 '\( \displaystyle \lim_{x\to $xg^-} [f(x)g(x) ] \) ', 211 num_cmp(rnd( f_rule($xg-.00001)*g_rule($xg-.00001) ), strings => ['DNE'] ), # we are expecting integer answers 212 '\( \displaystyle \lim_{x\to $xg^+} [f(x)g(x) ] \) ', 213 num_cmp(int( f_rule($xg+.00001)*g_rule($xg+.00001) +.4 ), strings => ['DNE'] ), # we are expecting integer answers 214 '\( f($xg)g($xg) \)', 215 num_cmp(f_rule($xg)*g_rule($xg), strings => ['DNE'] ), 216 217 '\( \displaystyle \lim_{x\to $xg^-} [f( g(x) ) ] \) ', 218 num_cmp(rnd( f_rule( g_rule($xg-.00001) )), strings => ['DNE'] ) , # we are expecting integer answers 219 '\( \displaystyle \lim_{x\to $xg^+} [f( g(x) ) ] \) ', 220 num_cmp(rnd( f_rule( g_rule( $xg + .00001 ) ) ), strings => ['DNE'] ), # we are expecting integer answers 221 '\( f( g($xg) ) \)', 222 num_cmp(f_rule( g_rule($xg) ), strings => ['DNE'] ), 223 224 #check for zeros in answering the division limits 225 '\( \displaystyle \lim_{x\to $xg^-} [f(x)/g(x) ] \) ', 226 ( not rnd(g_rule($xg-.00001) ) and rnd(f_rule($xg-.00001) ) ) ? num_cmp('DNE', strings => ['DNE'] ) # if g is zero, but f is not 227 : 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. 228 # we are expecting integer answers 229 '\( \displaystyle \lim_{x\to $xg^+} [f(x)/g(x) ] \) ', 230 ( not rnd(g_rule($xg+.00001) ) and rnd(f_rule($xg+.00001) ) ) ? num_cmp('DNE', strings => ['DNE'] ) 231 : num_cmp(rnd( f_rule($xg+.00001)/g_rule($xg+.00001) , -3 ), tol =>.002, strings => ['DNE'] ) , 232 # we are expecting integer answers 233 '\( f($xg)/g($xg) \)', 234 (rnd(g_rule($xg)) ) ? num_cmp( rnd(f_rule($xg)/g_rule($xg), -3), tol =>.002, strings => ['DNE'] ) : num_cmp('DNE', strings => ['DNE'] ), 235 236 ); 237 238 @slice = NchooseK(scalar(@questions) ,4); 239 240 BEGIN_TEXT 241 $BR 242 The graphs of \( f \) and \( g \) are given above. Use them to evaluate each quantity below. 243 Write $BITALIC DNE$EITALIC if the limit or value does not exist 244 (or if it's infinity). 245 $BR 246 \{ match_questions_list(@questions[@slice] ) \} 247 END_TEXT 248 249 ANS(@answers[@slice] ); 250 ENDDOCUMENT(); # This should be the last executable line in the problem.
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