CMSC 351 - Thought Questions #1

Solve each of the following to a specific number without the use 
of a calculator:

          (a) log2256

          (b) Given logab = 23 and logad = 15
                 compute loga(b/d)

          (c) log7(494)

          (d) log2(1/8)



Find the integrals of each of the following:
   (hint: think integration-by-parts and log rules for some of them)
   (note: loge is also known as ln)

          (e) x4
          (f) logex3
          (g) x logex





(h) Prove the following by mathematical induction:

               n           
               Σ (2i + 5/i) < (2n2+15)
              i=1              


          Show your Base Case(s):


          State your Inductive Hypothesis:


          State what you plan to prove in your Inductive Step:


          Prove it:





(i) Prove the following by mathematical induction:

               n           n(n+1)(n+2) 
               Σ i(i+1) = ------------
              i=1               3


          Show your Base Case(s):


          State your Inductive Hypothesis:


          State what you plan to prove in your Inductive Step:

          Prove it:





(j) Determine whether the following is true or false, and then prove that.

          3 | (n3+2n)





(k) Consider the following:
Assuming we are looking at comparisons, and working on the 
assumption that you can perform 220 comparisons per second, 
consider approximately how large n could be so that you 
could solve a problem of each of the following complexities 
in the specified time period.
 


1 Second
1 Minute
1 Hour
1 Day
1 Month
1 Year
1 Century

log2n
 
 
 
 
 
 
 
n
 
 
 
 
 
 
 
n log2n
 
 
 
 
 
 
 
n2
 
 
 
 
 
 
 
n3
 
 
 
 
 
 
 
2n