Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 4 / c

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- Solution c

Solution

At the origin the needed partial derivatives are

f (0, 0) = 1, f_{x} (0, 0) = 0, f_{y} (0, 0) = - 3, f_{xx} (0, 0) = 2, f_{x y} (0, 0) = f_{yy} (0, 0) = 0.

The Taylor series through total degree two is therefore

f (0, 0) + f_{x} (0, 0) x + f_{y} (0, 0) y + \frac{1}{2} (f_{xx} (0, 0) x^{2} + 2 f_{x y} (0, 0) x y + f_{yy} (0, 0) y^{2}),

which gives

f (x, y) = 1 - 3 y + x^{2} + O (∥ (x, y) ∥^{3}) as (x, y) \to (0, 0) .

Here total degree is what matters:

x^{2} y^{2}

is quartic, not quadratic. The exact omitted polynomial is

y^{3} + x^{2} y^{2} + x^{2} y^{4}

, which verifies the stated Big O notation remainder directly.

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