Ketone are oxidized with hno3 via enol form the product is oic acid
For each pair of functions determine if f(n) ? ?(g(n)) or f(n) ? ?(g(n)) or f(n) ? O(g(n)) and provide a proof as specified. For each of the following, give a proof using the definitions. 1. f(n) = log(n), g(n) = log(n + 1) 2. f(n) = n3 + nlog(n) ? n, g(n) = n4 + n 3. f(n) = log(n!), g(n) = nlog(n) 4. f(n) = log3(n), g(n) = log2(n) 5. f(n) = log(n), g(n) = log(log(n))
Prove that if f is a multiplicative arithmetic function then f([m, n])f((m, n)) = f(m)f(n) for all positive integers m and n. Hint: [m, n] is the least common multiple of m and n and (m, n) is the greatest common divisor of m and n.
Let S(n) = f(1) + f(2) + ... + f(n). For each f(n) below, derive the answer to the question S(n) = Simplify your answer as much as you can. (?). (i) f(n) = n log n
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...
i. f(n) 2(fn)) j. f(n) O(f(n))
The recursive definition of a Fibonacci Number is F(n) = F(n - 1) + F(n - 2), where F(0) = 1 and F(1) = 1. What is the value of Fib(3)?
3) Let N- 11,2,3,... and Nx N -(m,n) | m,n E N. Consider f NxN-N given by f(1,2)-3 | f(2,2)-6 | fa, 21-12 f (1,3)-5 f (2,3)10f (3,3)- 20 f (1,4) 7 f (2,4) 14 f (3,4) 28 2m-i (2n-1). Show, that f is one-to-one and In general "f(m, n) onto. 3) Let N- 11,2,3,... and Nx N -(m,n) | m,n E N. Consider f NxN-N given by f(1,2)-3 | f(2,2)-6 | fa, 21-12 f (1,3)-5 f (2,3)10f (3,3)- 20...
For each pair of functions f(n) and g(n), indicate whether f(n) = O(g(n)), f(n) = Ω(g(n)), and/or f(n) = Θ(g(n)), and provide a brief explanation of your reasoning. (Your explanation can be the same for all three; for example, “the two functions differ by only a multiplicative constant” could justify why f(n) = n, g(n) = 2n are related by big-O, big-Omega, and big-Theta.) i. f(n) = n^2 log n, g(n) = 100n^2 ii. f(n) = 100, g(n) = log(log(log...
Question 3 (15%) Function f(n) can be recursively defined as follows. f(n)- f(n -1)+4 f(n-2) f(0) 0 and f(1) = 1 (a) Write clear pseudo code to calculate f(n). (10 points
Write a recursive method to calculate F(N-1) = N + F(N+1); N is an interger, N < 10 and F(9)=1;