Pure mathematics is a strict science that "approximation" is pointless in most of the proof, and some approximation tools like big O notation (O(f(x))) is bonded to give an accurate limits on the behavior of a function.

However, we can see that approximation in the field of physics and mathsmatics is just powerful, like the formula E=mc^2, is done by approximation. The following question is quite interesting in the sense that it is approximation does give a general view of the whole problem:

a)Define the term "score" in different perspectives. (15M)

b)Consider the following cases, select an appropiate function f(x), state its domain and co-domain as well as the brief explaination to the process f. You may select f as a overall score (insert x and directly takes f(x) as the score) and f as a partial score (f as a partial score, while the total score is equal to sum of f(x_i).)

i)Score = hits within a period of time

ii)RPG game final results

iii)Rhythm game, score from each hits is proportional to the combo and inversely proportional to the time offsets error. (12M)

c)With reference to your answer in (a), state the nature of score and what can score show. Furthermore discuss, with examples, the compatability of score and its meaning, i.e., the growth factor / f'(x) VS the number of plays, cumulative scores, etc. (30M)

d)Now extend the co-domain of the process into an n-space vector. Give examples that how f process and affect the coming decision of controlling invariate x. Discuss the feasibility to put a continous movement of controller (reality player) in the computer / processor, and after a strict process (i.e., f(x) is fixed), to give the expected results. (Hint: Chaos Theory may be involved.) (43M)

If possible I will present my answer (does not means that it is model answer) for part a,b,c.

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