MAT 243 Week 7/ 15 Written Homework . (3 points) Find the smallest integer n so that m5 + 3:3 logs: f(w)= $2\" is big-O of 1:". Show your work. . (2 points) Demonstrate that 32 ins: is big-O of 333. You may use a limit theorem given in the lecture, or make an argument based on the exact denition of the big-O relation ship. . (2 points) True or False? (In a)2 is big-O of n. Justify your answer. . (3 points) Find the smallest positive integer k such that 12+22+32+. . .+'n.2 is big-O of it". Show your work. . (2 points) Find the mistake in the following argument: We show that 12 + 22 + 32 + . . . + n2 is big-O of 11?. Since 2'2 S n2 for i = 1,2...n, each term in the sum 12+22+32+. ..+'n.2 is less than or equal to H2. Thus, the maximum of the big-O estimates of each term is 11.2, which implies that the sum 12 + 22 + 32 + . . . + n2 is big-O of oz. (a) (3 points) Is there a smallest real number a for which 3:269: is big-O of (1\"? Explain your answer. (b) (2 points) Is there a smallest integer number a for which 3:26\"c is big-O of (1\"? Explain your answer. . (3 points) Let f (x) = 3:1: + 4. We know that f (x) is big-O of 3:2. Find the lowest k that works for C 2 1 and justify your answer fully. Note that you have to Show two aspects: your 1: works and the no lower '9 value work. . (4 points) Find a simple function g of the smallest order so that f(:1:) is big-O of 9(32). In your estimation you can use theorems learned in lecture about the big-O estimates of sums/ products of functions. f(x) = (32-3 + 11:1nziz5)(l.1"""'1 + 1.2:\") + (9:2 +1.23)(x3 + 092$)