SLOG week 10

   I'm feeling more comfortable with this unit. The concepts of "big-O" and "big-Omega" and "big-Theta" are sinking in a little more. Basically, they are methods of determining the growth of a function. The definitions require an x-value B for which one function will predictably be larger, smaller, or equal to the other function when it is multiplied by a constant. I have to say that, because of my alienation with this topic that I discussed last week and which is only now starting to dissipate, I have been sort of confused when following the algebra of the "big-O", "big-Omega", and "big-Theta" proofs that we did this past week and last week.
   Because of this, I now have my new challenge: I have to figure this out. In class, nothing Danny did was too difficult, at least from a maths standpoint. It was algebra and simple manipulations of inequalities, mostly. But I just did not follow it well because I was preoccupied with trying to figure out the core of the concepts. Now, then, I have to review and go over it.
   Basically, what I understand so far (and I will use "big-O" specifically):

• To complete the proofs, you need to find the positive constants B and C.
• For polynomial functions in these proofs, it basically comes down to manipulating inequalities.
• The term in the polynomial of the highest degree is the most important part: all the other terms do not affect the growth of the function nearly as much. 
• Because of this (and this is my favourite part of the proofs so far), you can basically ignore the smaller terms and, as you need, replace them with multiples of the highest degree term to further the inequality. For example. if you have f(n) [less than or equal to] 3(n squared) + 16 then you can take out the +16 and replace it with (+n(squared)) for all n over 4. Then you can add up those terms since they are alike. 
• The other proofs, e.g. "Big-Omega" and "Big-Theta", are very similar, but the inequality is reversed.

That is a list of what I do as yet understand. I need practice to further my comfort with it all, but I can "see the light" as of now when it comes to these concepts. The biggest thing by which I am still confused is how to do these proofs with non-polynomials, e.g. log functions or exponential functions. That will have to come next for me.