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Annotation of /trunk/rochester_problib/setDiscrete4Functions/ur_dis_4_9.pg

 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.;