some_solns1 - CMPS 101 Midterm 1 Some solutions to review...

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CMPS 101 Midterm 1 Some solutions to review problems and one additional problem Problem 2 from the Midterm 1 review sheet State whether the following assertions are true or false. If any statements are false, give a related statement which is true. a. )) ( ( ) ( n g O n f = implies )) ( ( ) ( n g o n f = . False )) ( ( ) ( n g o n f = implies )) ( ( ) ( n g O n f = b. )) ( ( ) ( n g O n f = if and only if )) ( ( ) ( n f n g Ω = . True c. )) ( ( ) ( n g n f Θ = if and only if L n g n f n = )) ( / ) ( ( lim , where < < L 0 . False < < L 0 and L n g n f n = )) ( / ) ( ( lim implies )) ( ( ) ( n g n f Θ = Problem 3 from the Midterm 1 review sheet Prove that )) ( ) ( ( )) ( ( )) ( ( n g n f n g n f Θ = Θ Θ . In other words, if )) ( ( ) ( 1 n f n h Θ = and )) ( ( ) ( 2 n g n h Θ = , then )) ( ) ( ( ) ( ) ( 2 1 n g n f n h n h Θ = . Proof: By hypothesis there exist positive constants 1 n , 2 n , 1 a , 1 b , 2 a , and 2 b such that ) ( ) ( ) ( 0 : 1 1
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some_solns1 - CMPS 101 Midterm 1 Some solutions to review...

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