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LoyolaChicago/2.3: 4e tagging, cleanup.
1 ## DESCRIPTION 2 ## Functions: Piecewise defined functions 3 ## ENDDESCRIPTION 4 5 ## KEYWORDS('functions','piecwise') 6 7 ## DBsubject('Precalculus') 8 ## DBchapter('Functions') 9 ## DBsection('Piecewise Defined Functions') 10 ## Date('01/01/10') 11 ## Author('Paul Pearson') 12 ## Institution('Fort Lewis College') 13 ## TitleText1('Functions Modeling Change') 14 ## EditionText1('3') 15 ## AuthorText1('Connally') 16 ## Section1('2.3') 17 ## Problem1('19') 18 ## EditionText2('4') 19 ## AuthorText2('Connally') 20 ## Section2('2.Review') 21 ## Problem2('70') 22 23 DOCUMENT(); 24 25 loadMacros( 26 "PGstandard.pl", 27 "MathObjects.pl", 28 "PGunion.pl", 29 "PGcourse.pl", 30 "PGgraphmacros.pl", 31 ); 32 33 TEXT(beginproblem()); 34 35 ############################################## 36 # Setup 37 38 Context("Numeric")->variables->are(p=>"Real"); 39 40 $pagespersig = random(8,16,4); 41 $costpersig = sprintf("%0.2f", random(0.1,0.2,0.01) ); 42 $numsigs = random(7,12,1); 43 $pages = $pagespersig * $numsigs; 44 $pages1 = $pages + 1; 45 46 $costpages = Real("$numsigs * $costpersig"); 47 $costpages1 = Real("($numsigs + 1) * $costpersig"); 48 49 $costfunction = Formula("$costpersig * p / $pagespersig"); 50 51 52 ############################################## 53 # Main text 54 55 Context()->texStrings; 56 BEGIN_TEXT 57 58 Many printing presses are designed with large plates 59 that print a fixed number of pages as a unit. Each 60 unit is called a signature. A particular press 61 prints signatures of $pagespersig pages each. Suppose 62 \( C(p) \) is the cost of printing a book of \( p \) 63 pages, assuming each signature printed costs 64 ${DOLLAR}${costpersig}. 65 $PAR 66 (a) What is the cost of printing a book of $pages pages? 67 ${DOLLAR} \{ ans_rule(10) \} 68 $BR 69 What if there are $pages1 pages? 70 ${DOLLAR} \{ ans_rule(10) \} 71 $BR 72 What if there are \( p \) pages? 73 \( C(p) = \) 74 ${DOLLAR} \{ ans_rule(10) \} 75 where the cost is rounded (up/down) 76 \{ pop_up_list(["?","up","down"]) \} 77 for the next signature. 78 $PAR 79 (b) What are the domain and range of \( C(p) \)? 80 $BR 81 Domain: all nonnegative integer multiples of (1 through 16) 82 \{ pop_up_list(["?","1","2","3","4","5","6","7","8", 83 "9","10","11","12","13","14","15","16"]) \} 84 $BR 85 Range: $SPACE all nonnegative integer multiples of 86 (0.01 through 0.20) 87 \{ pop_up_list(["?","0.01","0.02","0.03","0.04","0.05", 88 "0.06","0.07","0.08","0.09","0.10","0.11","0.12","0.13", 89 "0.14","0.15","0.16","0.17","0.18","0.19","0.20"]) \} 90 $PAR 91 (c) On a piece of paper, sketch a graph of \( C(p) \). 92 93 END_TEXT 94 Context()->normalStrings; 95 96 ################################################## 97 # Answers 98 99 $showPartialCorrectAnswers = 1; 100 101 ANS($costpages->cmp()); 102 ANS($costpages1->cmp()); 103 ANS($costfunction->cmp()); 104 ANS(str_cmp("up")); 105 ANS(str_cmp("1")); 106 ANS(str_cmp("$costpersig")); 107 108 $numsigp1 = $numsigs + 1; 109 $pagespersigp1 = $pagespersig + 1; 110 $twopagespersig = 2*$pagespersig; 111 $twocostpersig = 2*$costpersig; 112 113 $solgr = init_graph( -$pagespersig, -$costpersig, $pages, $costpages+$costpersig, axes=>[0,0], 114 size=>[250,250], grid=>[$numsigs+1,$numsigs+2] ); 115 116 $datarows = ''; 117 for (my $i=0; $i<$numsigs; $i++ ) { 118 my $j = $i*$pagespersig + 1; 119 my $k = ($i+1)*$pagespersig; 120 $datarows .= row( "$j-$k", $costpersig*($i+1) ); 121 plot_functions( $solgr, 122 $costpersig*($i+1) . " for x in [$j,$k) using color:blue" . 123 " and weight:2" ); 124 } 125 126 127 128 SOLUTION(EV3(<<'END_SOLUTION')); 129 $PAR 130 $BBOLD SOLUTION $EBOLD 131 $PAR 132 133 (a) 134 Each signature printed costs ${DOLLAR}$costpersig, and in a book 135 of \(p\) pages, there are at least \(p/$pagespersig\) signatures. 136 In a book of $pages pages, there are 137 \[ \frac{$pages}{$pagespersig}=$numsigs \hbox{ signatures},\] 138 \[\hbox{Cost for $pages pages }=$costpersig($numsigs)=$costpages.\] 139 A book of pages1 pages requires $numsigp1 signatures, although the 140 ${numsig1}th signature is used to print only 1 page. Therefore, 141 \[\hbox{Cost for $pages2 pages }=$costpersig($numsigp1)=$costpages1.\] 142 $PAR 143 To find the cost of \(p\) pages, we first find the number of 144 signatures. If \(p\) is divisible by $pagespersig, then the number of signatures 145 is \(p/$pagespersig\) and the cost is 146 \[C(p)=$costpersig\left(\frac{p}{$pagespersig}\right).\] 147 If \(p\) is not divisible by $pagespersig, the number of signatures is \(p/$pagespersig\) 148 rounded up to the next highest integer and the cost is $costpersig times 149 that number. In this case, it is hard to write a formula for \(C(p)\) 150 without a symbol for ``rounding up.'' 151 152 $PAR 153 (b) 154 The number of pages, \(p\), 155 is greater than zero. Although it is possible to have a page which is 156 only half filled, we do not say that a book has 124 1/2 pages, so \(p\) 157 must be an integer. Therefore, the 158 domain of \(C(p)\) is \(p>0\), \(p\) an integer. Because the cost of a book 159 increases by multiples of ${DOLLAR}$costpersig (the cost of one signature), the range 160 of \(C(p)\) is \(C>0\), \(C\) an integer multiple of ${DOLLAR}$costpersig. 161 162 $PAR 163 (c) 164 For 1 to $pagespersig pages, the cost is ${DOLLAR}$costpersig, because only 1 165 signature is required. For $pagespersigp1 to $twopagespersig pages, 166 the cost is ${DOLLAR}$twocostpersig, because 2 signatures are required. These data are continued in the table below, and plotted in the following figure, 167 for \(0\leq p\leq $pages\). 168 A closed circle 169 represents a point included on the graph, and an open 170 circle indicates a point excluded from the graph. The unbroken lines 171 in the figure suggest, erroneously, that ${BITALIC}fractions$EITALIC of 172 pages can be printed. It would be more accurate to draw each step as 173 $pagespersig separate dots instead of as an unbroken line. 174 175 $PAR 176 \{ begintable(2) \} 177 \{ row( "\(p\) (pages)", "\(C\) (dollars)" ) \} 178 $datarows 179 \{ endtable() \} 180 181 $PAR 182 $BCENTER 183 \{ image(insertGraph($solgr), height=>250, width=>250, tex_size=>250, 184 extra_html_tags=>'alt="graph of a piecewise constant function ' . 185 'as described above."') \} 186 $ECENTER 187 188 END_SOLUTION 189 190 COMMENT('Uses MathObjects'); 191 ENDDOCUMENT();
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