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Unformatted text preview: CMPS 101 Midterm 1 Review Problems 1. Let ) ( n f and ) ( n g be asymptotically non-negative functions which are defined on the positive integers. a. State the definition of )) ( ( ) ( n g O n f = . b. State the definition of )) ( ( ) ( n g n f = 2. 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 = . b. )) ( ( ) ( n g O n f = if and only if )) ( ( ) ( n f n g = . c. )) ( ( ) ( n g n f = if and only if L n g n f n = )) ( / ) ( ( lim , where < < L . 3. Prove that )) ( ) ( ( )) ( ( )) ( ( n g n f n g n f = . In other words, if )) ( ( ) ( 1 n f n h = and )) ( ( ) ( 2 n g n h = , then )) ( ) ( ( ) ( ) ( 2 1 n g n f n h n h = . 4. Use limits to prove the following (these are some of the exercises at the end of the asymptotic growth rates handout): a. If ) ( n P is a polynomial of degree k , then ) ( ) (...
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- Spring '09