Question 1 direct variations

a)For each of the following conditions, does it imply that x is directly proportional to y? Assume x,y are real.

i)x^2 varies directly to y^2. (4M)

ii)x^3 varies directly to y^2. (4M)

iii)log x varies directly to log y. (5M + 1 Bonus)

iv)tan x varies directly to tan y. (5M + 2 Bonus)

v)sin x varies directly to cos y. (5M + 2 Bonus)

b)In a more general way, when does x proportional to y if f(x) varies to f(y) is given? Explain your answer in terms of:

i)Algebrical method (11M)

ii)Geometry apporach (15M)

Question 2 partial variations

a)State the nature of functions that is used in partial variations. (3M)

b)If the function is

*symmetrical*, what's its special properties? Explain. (6M + 1 Bonus)

c)If the function is

*cyclic*, what's its special properties? Explain. (8M + 1 Bonus)

d)If the function is

*alternative*, what's its special properties? Explain. (6M + 1 Bonus)

e)If f'(x) is directly proportional to f'(y) where f is a polynomial, find the corresponding types of variation between x and y. (7M)

f)For the relationship x=f(y,z), where f is a polyomial, find the relationship between x and (δf/δy+δf/δz). (13M)

Full marks: 100 Time limits : 75 minutes or 1.5 hour

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