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| 1 : | lr003k | 244 | ##DESCRIPTION |
| 2 : | ad001h | 266 | ##KEYWORDS('Standard Example') |
| 3 : | lr003k | 244 | ##ENDDESCRIPTION |
| 4 : | |||
| 5 : | DOCUMENT(); # This should be the first executable line in the problem. | ||
| 6 : | |||
| 7 : | loadMacros( | ||
| 8 : | "PG.pl", | ||
| 9 : | "PGbasicmacros.pl", | ||
| 10 : | "PGchoicemacros.pl", | ||
| 11 : | "PGanswermacros.pl", | ||
| 12 : | "PGauxiliaryFunctions.pl" | ||
| 13 : | ); | ||
| 14 : | |||
| 15 : | TEXT(&beginproblem); | ||
| 16 : | ad001h | 266 | $showPartialCorrectAnswers=0; |
| 17 : | lr003k | 244 | |
| 18 : | ad001h | 266 | #Numerical Answer |
| 19 : | lr003k | 244 | |
| 20 : | ad001h | 266 | $a = random(2,9,1); |
| 21 : | $b = random(3,9,1); | ||
| 22 : | lr003k | 244 | |
| 23 : | BEGIN_TEXT | ||
| 24 : | ad001h | 266 | In this problem it will be useful to recall the following properties |
| 25 : | ad001h | 284 | of logarithms: \( \log(xy)=\log(x) + \log(y) \) and \( \log(x^a)=a\log(x) \). |
| 26 : | ad001h | 266 | $BR |
| 27 : | Find the least integer k such that f(n) is \( O(n^k) \) | ||
| 28 : | for each of the following functions: $BR | ||
| 29 : | ad001h | 284 | (a) \( f(n) = n\log(4^n) \) \{ ans_rule(10) \} |
| 30 : | lr003k | 244 | $PAR |
| 31 : | ad001h | 266 | (b) \( f(n) = 1^{$a} + 2^{$a} + \dots + n^{$a} \) \{ ans_rule(10) \} |
| 32 : | $PAR | ||
| 33 : | ad001h | 284 | (c) \( f(n) = \log(n!) \) \{ ans_rule(10) \} |
| 34 : | ad001h | 266 | $PAR |
| 35 : | ad001h | 284 | (d) \( f(n) = \frac {\log(n^n)}{n^{$b}+1} \) \{ ans_rule(10) \} |
| 36 : | ad001h | 266 | $PAR |
| 37 : | lr003k | 244 | END_TEXT |
| 38 : | |||
| 39 : | ad001h | 266 | $c = $a + 1; |
| 40 : | $d = 2 - $b; | ||
| 41 : | lr003k | 244 | |
| 42 : | ad001h | 266 | ANS( num_cmp( 2 ) ); |
| 43 : | ANS( num_cmp( $c ) ); | ||
| 44 : | ANS( num_cmp( 2 ) ); | ||
| 45 : | ANS( num_cmp( $d ) ); | ||
| 46 : | |||
| 47 : | lr003k | 244 | ENDDOCUMENT(); # This should be the last executable line in the problem.; |
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