Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 3 / a

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- Solution a

Solution

The Taylor expansions

sin t = t + O (t^{3})

and

e^{t} = 1 + t + t^{2} /2 + O (t^{3})

give

f (x, y) = - 3 + 6 x + 3 y + \frac{1}{2} (x + 2 y)^{2} + O ((∣ x ∣ + ∣ y ∣)^{3}) .

Hence the second-order Taylor polynomial is

T_{2} (x, y) = - 3 + 6 x + 3 y + \frac{1}{2} x^{2} + 2 x y + 2 y^{2} .

The sine contributes no quadratic term at zero.

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