Please show all work for my understanding. Please show original work because I really want to learn this material. Thanks!

1a)

1b)

Suppose you have algorithms with the three running times listed below. (Assume these are the exact running times.) How much slower do each of these algorithms get when you (a) double the input size; or (b) increase the input size by one? (So you should give two answers for each of the three.) For some functions, you might show the answer in terms of a ratio; for others, in terms of a difference. Assume that all logs are base two. (1) f(n)=31n2+4 (2) f(n)=2n+n (3) f(n)=nlogn Suppose you have algorithms with the three running times listed below. (Assume these are the exact number of operations performed as a function of the input size n.) Suppose you have a computer that can perform 106 operations per second, and you need to compute a result in at most ten minutes of computation. For each of the algorithms, what is the largest input size n for which you would be able to get the result within ten minutes? (a) g(n)=2n(logn) (b) g(n)=(n!) (c) g(n)=(nk), for k=4; and then for k=6. Recall that (nk)=k!(nk)!n