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| 1 : | sam | 11 | #!/usr/local/bin/webwork-perl |
| 2 : | gage | 31 | sub _PGauxiliaryFunctions_init { |
| 3 : | sam | 2 | |
| 4 : | gage | 31 | } |
| 5 : | |||
| 6 : | sam | 2 | sub tan { |
| 7 : | sin($_[0])/cos($_[0]); | ||
| 8 : | } | ||
| 9 : | sub cot { | ||
| 10 : | cos($_[0])/sin($_[0]); | ||
| 11 : | } | ||
| 12 : | sub sec { | ||
| 13 : | 1/cos($_[0]); | ||
| 14 : | } | ||
| 15 : | sub csc { | ||
| 16 : | 1/sin($_[0]); | ||
| 17 : | } | ||
| 18 : | sub ln { | ||
| 19 : | log($_[0]); | ||
| 20 : | } | ||
| 21 : | sub logten { | ||
| 22 : | log($_[0])/log(10); | ||
| 23 : | } | ||
| 24 : | sub arcsin { | ||
| 25 : | atan2 ($_[0],sqrt(1-$_[0]*$_[0])); | ||
| 26 : | } | ||
| 27 : | sub asin { | ||
| 28 : | atan2 ($_[0],sqrt(1-$_[0]*$_[0])); | ||
| 29 : | } | ||
| 30 : | sub arccos { | ||
| 31 : | atan2 (sqrt(1-$_[0]*$_[0]),$_[0]); | ||
| 32 : | } | ||
| 33 : | sub acos { | ||
| 34 : | atan2 (sqrt(1-$_[0]*$_[0]),$_[0]); | ||
| 35 : | } | ||
| 36 : | sub arctan { | ||
| 37 : | atan2($_[0],1); | ||
| 38 : | } | ||
| 39 : | sub atan { | ||
| 40 : | atan2($_[0],1); | ||
| 41 : | } | ||
| 42 : | sub arccot { | ||
| 43 : | atan2(1,$_[0]); | ||
| 44 : | } | ||
| 45 : | sub acot { | ||
| 46 : | atan2(1,$_[0]); | ||
| 47 : | } | ||
| 48 : | sub sinh { | ||
| 49 : | (exp($_[0]) - exp(-$_[0]))/2; | ||
| 50 : | } | ||
| 51 : | sub cosh { | ||
| 52 : | (exp($_[0]) + exp(-$_[0]))/2; | ||
| 53 : | } | ||
| 54 : | sub tanh { | ||
| 55 : | (exp($_[0]) - exp(-$_[0]))/(exp($_[0]) + exp(-$_[0])); | ||
| 56 : | } | ||
| 57 : | sub sech { | ||
| 58 : | 2/(exp($_[0]) + exp(-$_[0])); | ||
| 59 : | } | ||
| 60 : | sub sgn { | ||
| 61 : | my $x = shift; | ||
| 62 : | my $out; | ||
| 63 : | $out = 1 if $x > 0; | ||
| 64 : | $out = 0 if $x == 0; | ||
| 65 : | $out = -1 if $x<0; | ||
| 66 : | $out; | ||
| 67 : | } | ||
| 68 : | gage | 5 | sub step { # heavyside function (1 or x>0) |
| 69 : | my $x = shift; | ||
| 70 : | ($x > 0 ) ? 1 : 0; | ||
| 71 : | } | ||
| 72 : | sam | 2 | sub ceil { |
| 73 : | my $x = shift; | ||
| 74 : | - floor(-$x); | ||
| 75 : | } | ||
| 76 : | sub floor { | ||
| 77 : | my $input = shift; | ||
| 78 : | my $out = int $input; | ||
| 79 : | $out -- if ( $out <= 0 and $out-$input); # does the right thing for negative numbers | ||
| 80 : | $out; | ||
| 81 : | } | ||
| 82 : | |||
| 83 : | sub max { | ||
| 84 : | |||
| 85 : | my $maxVal = shift; | ||
| 86 : | my @input = @_; | ||
| 87 : | |||
| 88 : | foreach my $num (@input) { | ||
| 89 : | $maxVal = $num if ($maxVal < $num); | ||
| 90 : | } | ||
| 91 : | |||
| 92 : | $maxVal; | ||
| 93 : | |||
| 94 : | } | ||
| 95 : | |||
| 96 : | sub min { | ||
| 97 : | |||
| 98 : | my $minVal = shift; | ||
| 99 : | my @input = @_; | ||
| 100 : | |||
| 101 : | foreach my $num (@input) { | ||
| 102 : | $minVal = $num if ($minVal > $num); | ||
| 103 : | } | ||
| 104 : | |||
| 105 : | $minVal; | ||
| 106 : | |||
| 107 : | } | ||
| 108 : | |||
| 109 : | #round added 6/12/2000 by David Etlinger | ||
| 110 : | sub round { | ||
| 111 : | my $input = shift; | ||
| 112 : | my $out; | ||
| 113 : | if( $input >= 0 ) { | ||
| 114 : | $out = int ($input + .5); | ||
| 115 : | } | ||
| 116 : | else { | ||
| 117 : | $out = ceil($input - .5); | ||
| 118 : | } | ||
| 119 : | $out; | ||
| 120 : | } | ||
| 121 : | |||
| 122 : | #least common multiple | ||
| 123 : | #VS 6/29/2000 | ||
| 124 : | sub lcm { | ||
| 125 : | my $a = shift; | ||
| 126 : | my $b = shift; | ||
| 127 : | |||
| 128 : | #reorder such that $a is the smaller number | ||
| 129 : | if ($a > $b) { | ||
| 130 : | my $temp = $a; | ||
| 131 : | $a = $b; | ||
| 132 : | $b = $temp; | ||
| 133 : | } | ||
| 134 : | |||
| 135 : | my $lcm = 0; | ||
| 136 : | my $curr = $b;; | ||
| 137 : | |||
| 138 : | while($lcm == 0) { | ||
| 139 : | $lcm = $curr if ($curr % $a == 0); | ||
| 140 : | $curr += $b; | ||
| 141 : | } | ||
| 142 : | |||
| 143 : | $lcm; | ||
| 144 : | |||
| 145 : | } | ||
| 146 : | |||
| 147 : | |||
| 148 : | # greatest common factor | ||
| 149 : | # takes in two scalar values and uses the Euclidean Algorithm to return the gcf | ||
| 150 : | #VS 6/29/2000 | ||
| 151 : | |||
| 152 : | sub gcf { | ||
| 153 : | my $a = abs(shift); # absolute values because this will yield the same gcd, | ||
| 154 : | my $b = abs(shift); # but allows use of the mod operation | ||
| 155 : | |||
| 156 : | # reorder such that a is the smaller number | ||
| 157 : | if ($a < $b) { | ||
| 158 : | my $temp = $a; | ||
| 159 : | $a = $b; | ||
| 160 : | $b = $temp; | ||
| 161 : | } | ||
| 162 : | |||
| 163 : | my $q = int($a/$b); # quotient | ||
| 164 : | my $r = $a % $b; # remainder | ||
| 165 : | |||
| 166 : | return $b if $r == 0; | ||
| 167 : | |||
| 168 : | my $tempR = $r; | ||
| 169 : | |||
| 170 : | while ($r != 0) { | ||
| 171 : | |||
| 172 : | #keep track of what $r was in the last loop, as this is the value | ||
| 173 : | #we will want when $r is set to 0 | ||
| 174 : | $tempR = $r; | ||
| 175 : | |||
| 176 : | $a = $b; | ||
| 177 : | $b = $r; | ||
| 178 : | $q = $a/$b; | ||
| 179 : | $r = $a % $b; | ||
| 180 : | |||
| 181 : | } | ||
| 182 : | |||
| 183 : | $tempR; | ||
| 184 : | } | ||
| 185 : | |||
| 186 : | |||
| 187 : | #greatest common factor. | ||
| 188 : | #same as gcf, but both names are sufficiently common names | ||
| 189 : | sub gcd { | ||
| 190 : | return gcf($_[0], $_[1]); | ||
| 191 : | } | ||
| 192 : | |||
| 193 : | #returns 1 for a prime number, else 0 | ||
| 194 : | #VS 6/30/2000 | ||
| 195 : | sub isPrime { | ||
| 196 : | my $num = shift; | ||
| 197 : | return 1 if ($num == 2 or $num == 3); | ||
| 198 : | return 0 if ($num == 1 or $num == 0); | ||
| 199 : | for (my $i = 3; $i <= $num/2; $i++) { return 0 if ($num % $i == 0); } | ||
| 200 : | return 1; | ||
| 201 : | } | ||
| 202 : | |||
| 203 : | #reduces a fraction, returning an array containing ($numerator, $denominator) | ||
| 204 : | #VS 7/10/2000 | ||
| 205 : | sub reduce { | ||
| 206 : | |||
| 207 : | my $num = shift; | ||
| 208 : | my $denom = shift; | ||
| 209 : | my $gcd = gcd($num, $denom); | ||
| 210 : | |||
| 211 : | $num = $num/$gcd; | ||
| 212 : | $denom = $denom/$gcd; | ||
| 213 : | |||
| 214 : | # formats such that only the numerator will be negative | ||
| 215 : | if ($num/$denom < 0) {$num = -abs($num); $denom = abs($denom);} | ||
| 216 : | else {$num = abs($num); $denom = abs($denom);} | ||
| 217 : | |||
| 218 : | my @frac = ($num, $denom); | ||
| 219 : | @frac; | ||
| 220 : | } | ||
| 221 : | |||
| 222 : | |||
| 223 : | # takes a number and fixed object, as in "$a x" and formats | ||
| 224 : | # to account for when $a = 0, 1, -1 | ||
| 225 : | # Usage: format($scalar, "quoted string"); | ||
| 226 : | # Example: format(-1, "\pi") returns "-\pi" | ||
| 227 : | # VS 8/1/2000 - slight adaption of code from T. Shemanske of Dartmouth College | ||
| 228 : | sub preformat { | ||
| 229 : | my $num = shift; | ||
| 230 : | my $obj = shift; | ||
| 231 : | my $out; | ||
| 232 : | |||
| 233 : | |||
| 234 : | if ($num == 0) { return 0; } | ||
| 235 : | elsif ($num == 1) { return $obj; } | ||
| 236 : | elsif ($num == -1) { return "-".$obj; } | ||
| 237 : | |||
| 238 : | return $num.$obj; | ||
| 239 : | } | ||
| 240 : | |||
| 241 : | |||
| 242 : | maria | 38 | #factorial |
| 243 : | |||
| 244 : | sub fact { | ||
| 245 : | my $num = shift; | ||
| 246 : | my $prod = 1; | ||
| 247 : | if ($num != int($num) or $num < 0) { | ||
| 248 : | warn 'Non-negative integer required.'; | ||
| 249 : | return; | ||
| 250 : | } | ||
| 251 : | |||
| 252 : | for (1..$num) { $prod *= $_ } | ||
| 253 : | |||
| 254 : | return $prod; | ||
| 255 : | } | ||
| 256 : | sam | 2 | # return 1 so that this file can be included with require |
| 257 : | 1 |
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