MAT 243 Practice for Test 3

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Practice for Test 3 Question 1 2.4/2.4pts If f(x) is O(x®) and g(x) is O(x*), then what is the best we can say about f(x)-g(x)? Find the "lowest" big-O estimate that is guaranteed by the given information. f(x)-g(x) is O(x%) f(x)-g(x) is O(x?) f(x)-g(x) is O(x"). f(x)-g(x) is O(x2). The example f(x) =x3 and g(x) = x* shows that we can't conclude that f(x)-g(x) = x is O(x®) or O(x*). We can conclude f(x)-g(x) is O(x) by the product rule. This means that f(x)-g(x) is also O(x'2), but the question asked for the "lowest" big-O estimate.

Question 2 2.4/2.4pts True or False? 5x2 is O(x?). True False If the limit of [f(x)I/Ig(x)| as x goes to infinity exists, then f(x) is 0(g(x)- In particular, if f(x)/g(x) is a constant, then f(x) is O(g(x)). Here, the quotient is a constant 5.

Question 3 2.4/2.4pts Find the lowest integer k so that 1+2°+3°+4° +5°+ . +n° is O(nk). Letuscall 1°+2°+3°+4° +5°+ ...+ n° = S(n). Since each base is at most n, we have the inequality S(n) £n°+n°+ ... +n° = n-n° = n®. Therefore, S(n) is O(n®). On the other hand, if n is even, then n/2 + 1 bases in the sum are at least n/2. For example, S(8) = 1°+2°+3°+4° +5°+ 6+ 7°+ 8°, which has 5 bases that are at least 4. Thus, for even n, S(n) > (n/2)° + (n/2)° + ... + (n/2)°, where we have more than n/2 terms (n/2)° on the right side. Therefore, S(n) > (n/2)° = n°/ 64. This means that S(n) cannot be O(n°). Another way to see that S(n) cannot be O(n°) is to use calculus. By making a Riemann sum diagram, you can see that S(n) is a left sum for the integral of f(x) = x° from x=1to x=n+1. Since f is decreasing, the left sum overestimates the integral. The value of the integral is 1/6 ((n+1)- 1), an order n® function.

Question 4 2.4/2.4pts Enter the smallest integer n so that the following function is O(x"). f(x)=x2(x® + 1)+x°log(x). The first term is order of x®, hence also big-O of x°. The second term is "between” x° and x® in order, hence big-O of x® but not x°. That makes the first term negligible, and the sum big-O of x® but not x°.

Question 5 2.4/2.4pts Check all functions f(x) that are Q(x?). f(x) = log(x?) (x) = x3/2 +x2 f(x) = [x+2] - [x] f(x)=2x"2 +2x719 (x) = (x*+2x+3)/(x2-2) (%) = (x+2x+3)/(x2-2) f(x) = %2 - log(x) f(x) = 2x+x2

Question 6 2.4/2.4pts If f(x) is O(x3) and g(x) is O(x*), then what is the best we can say about f(x)+g(x)? Find the "lowest" big-O estimate that is guaranteed by the given information. f(x)+g(x) is O() )+g(x) is O(x). F)+g(x) is O(x7). f(x)+g(x) is O(x™?) The example f(x) =x® and g(x) = x* shows that we can't conclude that f(x)+g(x) is 0(x3). We can conclude f(x)+g(x) is O(x*) by the sum rule. This means that f(x)+g(x) is also O(x”) and O(x"2), but the question asked for the "lowest" big-O estimate.

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