4 (a) Consider the function f(x, y) = (y + 1) (x + cos x).

i. Find the partial derivatives

af af a²fa²f a² f

and

дх’ ду’ дх2' дхду dy²

ii. Find the Taylor Series for f(x, y) about the point x = 0, y = 0.

iii. Find the Taylor Series for f(x, y) about the point x =

π

2

(b) The function g(x, y) = 4

X

has a first-order Taylor Series of

g(x, y)≈1-(x − 1) + (y − 1) + ...

which can also be written as

y = 1.

[3,2,3 marks]

g(x, y) 1+ y - x

With justification, explain which form is more useful for approximating

g(x, y) in the vicinity of x = 1, y = 1.

[2 marks]