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| 1 : | jj | 143 | #DESCRIPTION |
| 2 : | #KEYWORDS('limits', 'sequences') | ||
| 3 : | # Find limits (intuitively or by experimental calculations). | ||
| 4 : | #ENDDESCRIPTION | ||
| 5 : | |||
| 6 : | jjholt | 314 | ##KEYWORDS('Calculus') |
| 7 : | ##Tagged by ynw2d | ||
| 8 : | |||
| 9 : | ##DBsubject('Calculus') | ||
| 10 : | ##DBchapter('Limits and Derivatives') | ||
| 11 : | ##DBsection('The Limit of a Function') | ||
| 12 : | |||
| 13 : | |||
| 14 : | jj | 143 | 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(0,3,1); | ||
| 56 : | $yf = random(-1,4,1); | ||
| 57 : | $xg = random(0,3,1); | ||
| 58 : | $yg = random(-1,4,1); | ||
| 59 : | |||
| 60 : | |||
| 61 : | # define the broken functions themselves -- we'll need them to answer questions. | ||
| 62 : | sub f_rule { | ||
| 63 : | my $x = shift; | ||
| 64 : | my $out; | ||
| 65 : | if ($x < $xf ) { | ||
| 66 : | $out = $hermite_f1->rf_f->($x); | ||
| 67 : | } elsif ($x == $xf) { | ||
| 68 : | $out = $yf; | ||
| 69 : | } else { | ||
| 70 : | $out = $hermite_f2->rf_f->($x); | ||
| 71 : | } | ||
| 72 : | $out; | ||
| 73 : | }; | ||
| 74 : | sub g_rule { | ||
| 75 : | my $x = shift; | ||
| 76 : | my $out; | ||
| 77 : | if ($x < $xg ) { | ||
| 78 : | $out = $hermite_g1->rf_f->($x); | ||
| 79 : | } elsif ($x == $xg) { | ||
| 80 : | $out = $yg; | ||
| 81 : | } else { | ||
| 82 : | $out = $hermite_g2->rf_f->($x); | ||
| 83 : | } | ||
| 84 : | $out; | ||
| 85 : | }; | ||
| 86 : | |||
| 87 : | |||
| 88 : | # plot the f graph | ||
| 89 : | |||
| 90 : | $graphf = init_graph(-2,-2,5,5,grid =>[7,7], axes => [0,0], size =>[500,500]); | ||
| 91 : | $graphg = init_graph(-2,-2,5,5,,grid =>[7,7], axes => [0,0], size =>[500,500]); | ||
| 92 : | $f1 = new Fun($hermite_f1->rf_f,$graphf); | ||
| 93 : | $f2 = new Fun($hermite_f2->rf_f,$graphf); | ||
| 94 : | $graphf->stamps(open_circle($xf, $hermite_f1->rf_f->($xf), 'red') ); | ||
| 95 : | $graphf->stamps(open_circle($xf, $hermite_f2->rf_f->($xf), 'red') ); | ||
| 96 : | $graphf ->stamps( closed_circle($xf, $yf, 'red') ); # value at $xf; | ||
| 97 : | $f1->color('red'); | ||
| 98 : | $f2->color('red'); | ||
| 99 : | # restrict the two partial graph domains so that only the 'active' one is drawn | ||
| 100 : | $f1->domain(-1, $xf); | ||
| 101 : | $f2->domain($xf, 4); | ||
| 102 : | |||
| 103 : | #plot the g graph | ||
| 104 : | $g1 = new Fun($hermite_g1->rf_f,$graphg); | ||
| 105 : | $g2 = new Fun($hermite_g2->rf_f,$graphg); | ||
| 106 : | $graphg->stamps(open_circle($xg, $hermite_g1->rf_f->($xg), 'blue') ); | ||
| 107 : | $graphg->stamps(open_circle($xg, $hermite_g2->rf_f->($xg), 'blue') ); | ||
| 108 : | $graphg ->stamps(closed_circle($xg, $yg, 'blue') ); # value at $xg; | ||
| 109 : | $g2->color('blue'); | ||
| 110 : | $g2->color('blue'); | ||
| 111 : | # restrict the two partial graph domains so that only the 'active' one is drawn | ||
| 112 : | $g1->domain(-1, $xg); | ||
| 113 : | $g2->domain($xg, 4); | ||
| 114 : | |||
| 115 : | TEXT(beginproblem()); | ||
| 116 : | |||
| 117 : | |||
| 118 : | # draw the graphs | ||
| 119 : | TEXT( | ||
| 120 : | begintable(2), | ||
| 121 : | row( image( insertGraph($graphf), tex_size =>400), image( insertGraph($graphg) , tex_size =>400 ) ), | ||
| 122 : | row( 'f(x)', 'g(x)'), | ||
| 123 : | endtable(), | ||
| 124 : | ); | ||
| 125 : | sub rnd { | ||
| 126 : | my $x = shift; | ||
| 127 : | my $places = shift; | ||
| 128 : | $places = 0 unless defined($places); | ||
| 129 : | my $sign = ($x <0 ) ? -1 : 1; | ||
| 130 : | $x = abs($x); | ||
| 131 : | |||
| 132 : | $x = int( $x*10 **(-$places) +.5) *10 **($places); | ||
| 133 : | $sign*$x; | ||
| 134 : | } | ||
| 135 : | # now construct the questions | ||
| 136 : | @questions = (); | ||
| 137 : | @answers = (); | ||
| 138 : | |||
| 139 : | qa( ~~@questions, ~~@answers, | ||
| 140 : | '\( \lim_{x\to $xf^-} [f(x) + g(x) ] \) ', | ||
| 141 : | num_cmp(rnd( f_rule($xf-.00001) +g_rule($xf-.00001) ), strings => ['DNE'] ), # we are expecting integer answers | ||
| 142 : | '\( \lim_{x\to $xf^+} [f(x) + g(x) ] \) ', | ||
| 143 : | num_cmp(rnd( f_rule($xf+.00001) +g_rule($xf+.00001) ), strings => ['DNE'] ), # we are expecting integer answers | ||
| 144 : | '\( f($xf) + g($xf) \)', | ||
| 145 : | num_cmp(f_rule($xf) + g_rule($xf), strings => ['DNE'] ), | ||
| 146 : | |||
| 147 : | '\( \lim_{x\to $xf^-} [f(x)g(x) ] \) ', | ||
| 148 : | num_cmp(rnd( f_rule($xf-.00001)*g_rule($xf-.00001) ), strings => ['DNE'] ), # we are expecting integer answers | ||
| 149 : | '\( \lim_{x\to $xf^+} [f(x)g(x) ] \) ', | ||
| 150 : | num_cmp(rnd( f_rule($xf+.00001)*g_rule($xf+.00001) ), strings => ['DNE'] ), # we are expecting integer answers | ||
| 151 : | '\( f($xf)g($xf) \)', | ||
| 152 : | num_cmp(f_rule($xf)*g_rule($xf), strings => ['DNE'] ), | ||
| 153 : | |||
| 154 : | '\( \lim_{x\to $xf^-} [f( g(x) ) ] \) ', | ||
| 155 : | num_cmp(rnd( f_rule(g_rule($xf-.00001) ) ), strings => ['DNE'] ), # we are expecting integer answers | ||
| 156 : | '\( \lim_{x\to $xf^+} [f( g(x) ) ] \) ', | ||
| 157 : | num_cmp(rnd( f_rule(g_rule($xf+.00001) ) ), strings => ['DNE'] ), # we are expecting integer answers | ||
| 158 : | '\( f( g($xf) ) \)', | ||
| 159 : | num_cmp(f_rule( g_rule($xf) ), strings => ['DNE'] ), | ||
| 160 : | |||
| 161 : | #check for zeros in answering the division limits | ||
| 162 : | '\( \lim_{x\to $xf^-} [f(x)/g(x) ] \) ', | ||
| 163 : | ( 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 | ||
| 164 : | : 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. | ||
| 165 : | # we are expecting integer answers | ||
| 166 : | '\( \lim_{x\to $xf^+} [f(x)/g(x) ] \) ', | ||
| 167 : | ( not rnd(g_rule($xf+.00001) ) and rnd(f_rule($xf+.00001) ) ) ? num_cmp('DNE', strings => ['DNE'] ) | ||
| 168 : | : num_cmp(rnd( f_rule($xf+.00001)/g_rule($xf+.00001) , -3 ), tol =>.002, strings => ['DNE'] ) , | ||
| 169 : | # we are expecting integer answers | ||
| 170 : | '\( f($xf)/g($xf) \)', | ||
| 171 : | (rnd(g_rule($xf)) ) ? num_cmp( rnd(f_rule($xf)/g_rule($xf), -3), tol =>.002, strings => ['DNE'] ) : num_cmp('DNE', strings => ['DNE'] ), | ||
| 172 : | |||
| 173 : | '\( \lim_{x\to $xg^-} [f(x) + g(x) ] \) ', | ||
| 174 : | num_cmp(rnd( f_rule($xg-.00001) +g_rule($xg-.00001) ), strings => ['DNE'] ), # we are expecting integer answers | ||
| 175 : | '\( \lim_{x\to $xg^+} [f(x) + g(x) ] \) ', | ||
| 176 : | num_cmp(rnd( f_rule($xg+.00001) +g_rule($xg+.00001) ), strings => ['DNE'] ), # we are expecting integer answers | ||
| 177 : | '\( f($xg) + g($xg) \)', | ||
| 178 : | num_cmp(f_rule($xg) + g_rule($xg), strings => ['DNE'] ), | ||
| 179 : | |||
| 180 : | '\( \lim_{x\to $xg^-} [f(x)g(x) ] \) ', | ||
| 181 : | num_cmp(rnd( f_rule($xg-.00001)*g_rule($xg-.00001) +.4 ), strings => ['DNE'] ), # we are expecting integer answers | ||
| 182 : | '\( \lim_{x\to $xg^+} [f(x)g(x) ] \) ', | ||
| 183 : | num_cmp(int( f_rule($xg+.00001)*g_rule($xg+.00001) +.4 ), strings => ['DNE'] ), # we are expecting integer answers | ||
| 184 : | '\( f($xg)g($xg) \)', | ||
| 185 : | num_cmp(f_rule($xg)*g_rule($xg), strings => ['DNE'] ), | ||
| 186 : | |||
| 187 : | '\( \lim_{x\to $xg^-} [f( g(x) ) ] \) ', | ||
| 188 : | num_cmp(int( f_rule( g_rule($xg-.00001) )), strings => ['DNE'] ) , # we are expecting integer answers | ||
| 189 : | '\( \lim_{x\to $xg^+} [f( g(x) ) ] \) ', | ||
| 190 : | num_cmp(rnd( f_rule( g_rule( $xg + .00001 ) ) ), strings => ['DNE'] ), # we are expecting integer answers | ||
| 191 : | '\( f( g($xg) ) \)', | ||
| 192 : | num_cmp(f_rule( g_rule($xg) ), strings => ['DNE'] ), | ||
| 193 : | |||
| 194 : | #check for zeros in answering the division limits | ||
| 195 : | '\( \lim_{x\to $xg^-} [f(x)/g(x) ] \) ', | ||
| 196 : | ( 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 | ||
| 197 : | : 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. | ||
| 198 : | # we are expecting integer answers | ||
| 199 : | '\( \lim_{x\to $xg^+} [f(x)/g(x) ] \) ', | ||
| 200 : | ( not rnd(g_rule($xg+.00001) ) and rnd(f_rule($xg+.00001) ) ) ? num_cmp('DNE', strings => ['DNE'] ) | ||
| 201 : | : num_cmp(rnd( f_rule($xg+.00001)/g_rule($xg+.00001) , -3 ), tol =>.002, strings => ['DNE'] ) , | ||
| 202 : | # we are expecting integer answers | ||
| 203 : | '\( f($xg)/g($xg) \)', | ||
| 204 : | (rnd(g_rule($xg)) ) ? num_cmp( rnd(f_rule($xg)/g_rule($xg), -3), tol =>.002, strings => ['DNE'] ) : num_cmp('DNE', strings => ['DNE'] ), | ||
| 205 : | |||
| 206 : | ); | ||
| 207 : | |||
| 208 : | @slice = NchooseK(scalar(@questions) ,4); | ||
| 209 : | |||
| 210 : | BEGIN_TEXT | ||
| 211 : | $BR | ||
| 212 : | The graphs of \( f \) and \( g \) are given above. Use them to evaluate each quantity below. | ||
| 213 : | Write 'DNE' if the limit or value does not exist (or if it's infinity). | ||
| 214 : | $BR | ||
| 215 : | \{ match_questions_list(@questions[@slice] ) \} | ||
| 216 : | |||
| 217 : | $BR | ||
| 218 : | |||
| 219 : | END_TEXT | ||
| 220 : | |||
| 221 : | ANS(@answers[@slice] ); | ||
| 222 : | ENDDOCUMENT(); # This should be the last executable line in the problem. |
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