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Fixed bug 1415, made problem look nicer.
1 ##DESCRIPTION 2 ##KEYWORDS('limit','definition', 'left limit') 3 ## Find the derivative of an oracle function using 4 ## the definition of the derivative. 5 ##ENDDESCRIPTION 6 7 ##KEYWORDS('Calculus') 8 ##Tagged by ynw2d 9 10 ##DBsubject('Calculus') 11 ## DBchapter('Limits and Derivatives') 12 ##DBsection('Continuity') 13 14 DOCUMENT(); 15 16 loadMacros("PG.pl", 17 "PGbasicmacros.pl", 18 "PGchoicemacros.pl", 19 "PGanswermacros.pl", 20 "PGnumericalmacros.pl", 21 ); 22 23 24 # define function1 to be evaluated 25 $a1= random(-3,3,0.1); 26 $b1= random(-3,3,0.1); 27 $c1 = random(-3,3,0.1); 28 29 30 $f1_str = FEQ(" ${a1}x^2+${b1}x +$c1 "); 31 $f1 = sub { 32 my $x = shift; 33 $a1*$x*$x + $b1*$x +$c1; 34 }; 35 36 # define function2 to be evaluated 37 $a2= random(-3,3,0.1); 38 $b2= random(-3,3,0.1); 39 $c2 = random(-3,3,0.1); 40 41 42 $f2_str = FEQ(" ${a2}x^2+${b2}x +$c2 "); 43 $f2 = sub { 44 my $x = shift; 45 $a2*$x*$x + $b2*$x +$c2; 46 }; 47 48 # define function3 to be evaluated 49 $a3= random(-3,3,0.1); 50 $b3= random(-3,3,0.1); 51 $c3 = random(-3,3,0.1); 52 53 54 $f3_str = FEQ(" ${a3}x^2+${b3}x +$c3 "); 55 $f3 = sub { 56 my $x = shift; 57 $a3*$x*$x + $b3*$x +$c3; 58 }; 59 60 61 62 $llimit = -5; $rlimit = 5; 63 $del = .05; 64 $x0=random($llimit+$del,$rlimit-$del , .01); 65 66 # Calculate the cubic spline approximation 67 68 my $num_of_intervals = 500; # number of interpolation points 69 my $delta = ($rlimit-$llimit)/($num_of_intervals); 70 my (@x_values, @y_values); 71 foreach my $i (0..$num_of_intervals) { 72 my $x = $llimit + $delta*$i; 73 my $y1 = &$f1($x); 74 my $y2 = &$f2($x); 75 my $y3 = &$f3($x); 76 push(@x_values, $x); push(@y1_values,$y1); push(@y2_values,$y2); push(@y3_values,$y3) 77 } 78 79 $one = random(1,3,1); 80 $two = random(1,3,1); 81 $three = random(1,3,1); 82 83 HEADER_TEXT(javaScript_cubic_spline(~~@x_values,~~@y1_values, name => 'func1', llimit => $llimit, rlimit => $rlimit) ); 84 HEADER_TEXT(javaScript_cubic_spline(~~@x_values,~~@y2_values, name => 'func2', llimit => $llimit, rlimit => $rlimit) ); 85 HEADER_TEXT(javaScript_cubic_spline(~~@x_values,~~@y3_values, name => 'func3', llimit => $llimit, rlimit => $rlimit) ); 86 87 $f1_approx = cubic_spline(~~@x_values, ~~@y1_values); 88 $f2_approx = cubic_spline(~~@x_values, ~~@y2_values); 89 $f3_approx = cubic_spline(~~@x_values, ~~@y3_values); 90 91 $left_limit = PG_restricted_eval( qq{ ~~&~~$f${one}_approx($x0) }); 92 $value = PG_restricted_eval( qq{ ~~&~~$f${two}_approx($x0) }); 93 $right_limit = PG_restricted_eval( qq{ ~~&~~$f${three}_approx($x0) }); 94 95 HEADER_TEXT(<<EOF); 96 <SCRIPT LANGUAGE="JavaScript"> 97 <!-- Begin 98 function func(x) { 99 if ( x < $x0 ) { //determines whether the function is left or right continuous. 100 return( func$one(x) ); 101 } else if ( x == $x0 ) { 102 return( func$two(x) ); 103 } else if ( x > $x0 ) { 104 return( func$three(x) ); 105 } else { 106 // do nothing if error. 107 } 108 } 109 --> 110 </SCRIPT> 111 112 EOF 113 TEXT(beginproblem()); 114 115 116 BEGIN_TEXT 117 Below is an "oracle" function. An oracle function is a function presented interactively. 118 When you type in an \(x\) value, and press the --f--\(>\) button and 119 the value \(f(x)\) appears in the right hand window. There are three lines, so you can easily calculate 120 three different values of the function at one time. 121 $BR 122 $BCENTER 123 END_TEXT 124 125 # A warning that we are using javaScript 126 TEXT(MODES( 127 TeX => "", 128 Latex2HTML => "\begin{rawhtml} 129 <NOSCRIPT>$BBOLD This problem requires that Java Script be enabled $BR$EBOLD</NOSCRIPT> \end{rawhtml} 130 ", 131 HTML => "$PAR<NOSCRIPT> <B>This problem requires that Java Script be enabled </NOSCRIPT></B>$PAR" 132 )); 133 134 $javaScript2 = begintable(3) . 135 MODES(HTML =>q{<TR> 136 <TH>x</TH><TH> </TH><TH>f(x)</TH> 137 <TR>}, 138 TeX => "x & \(\rightarrow\) & f(x)\\" 139 ) . 140 row(MODES(HTML => qq{<INPUT TYPE="text" NAME="Input" Value = "$x0" Size="16">}, TeX => '\fbox{Enter \(x\)}' ), 141 MODES(HTML => qq{<INPUT TYPE="button" VALUE="---f-->" 142 OnClick="this.form.Output.value=func(this.form.Input.value);">}, TeX => '\(\rightarrow\)' ), 143 MODES(HTML => qq{<INPUT TYPE="text" NAME="Output" Size="30">}, TeX => 'result: \(f(x)\)') 144 ) . 145 row(MODES(HTML => qq{<INPUT TYPE="text" NAME="Input2" Value = "$x0" Size="16">}, TeX => '\fbox{Enter \(x\)}' ), 146 MODES(HTML => qq{<INPUT TYPE="button" VALUE="---f-->" 147 OnClick="this.form.Output2.value=func(this.form.Input2.value);">}, TeX => '\(\rightarrow\)' ), 148 MODES(HTML => qq{<INPUT TYPE="text" NAME="Output2" Size="30">}, TeX => 'result: \(f(x)\)') 149 ) . 150 row(MODES(HTML => qq{<INPUT TYPE="text" NAME="Input3" Value = "$x0" Size="16">}, TeX => '\fbox{Enter \(x\)}' ), 151 MODES(HTML => qq{<INPUT TYPE="button" VALUE="---f-->" 152 OnClick="this.form.Output3.value=func(this.form.Input3.value);">}, TeX => '\(\rightarrow\)' ), 153 MODES(HTML => qq{<INPUT TYPE="text" NAME="Output3" Size="30">}, TeX => 'result: \(f(x)\)') 154 ) . endtable(); 155 156 TEXT($javaScript2); 157 158 159 BEGIN_TEXT 160 $ECENTER 161 $BR 162 Determine the limits for the function \( f(x)\) at \($x0\). 163 $BR$BR 164 \( \displaystyle \lim_{x\to $x0^{-} } f(x) \) = \{ans_rule(20)\} 165 $BR $BR 166 \( f($x0) \) = \{ ans_rule(10) \} 167 $BR $BR 168 \( \displaystyle \lim_{x\to $x0^{+} } f(x) \) = \{ ans_rule(20) \} 169 $BR $BR 170 Are all of these values the same? \{ pop_up_list(['?', 'Yes', 'No']) \}. 171 If so then the function 172 is $BBOLD continuous $EBOLD at \($x0\) $PAR 173 174 Are the left and right limits the same at \($x0 \)? \{ pop_up_list(['?', 'Yes', 'No']) \}. If so then this 175 function is almost continuous and could be made continuous by redefining one value of the function 176 namely \( f($x0) \). 177 END_TEXT 178 179 $continuousQ = 'No'; 180 $continuousQ = 'Yes' if ($left_limit == $value and $value == $right_limit); 181 $madeContinuousQ = 'No'; 182 $madeContinuousQ = 'Yes' if ($left_limit == $right_limit); 183 184 185 ANS(num_cmp([ $left_limit,$value,$right_limit ]), 186 #We are allowing 1 percent error for the answer. 187 strict_str_cmp($continuousQ), 188 strict_str_cmp($madeContinuousQ)); 189 190 191 &ENDDOCUMENT; 192
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