Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 28K / b

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- Solution b

Solution

The Taylor theorem with Lagrange remainder supplies a point

ξ_{i}

between

x_{0}

and

X_{i} = x_{0} + h Z_{i}

such that

g (X_{i}) = g (x_{0}) + h Z_{i} g^{'} (x_{0}) + \frac{h ^{2} Z _{i}^{2}}{2} g^{''} (ξ_{i}) .

Since

Z_{i}^{2} = 1

and

Y_{i} = g (X_{i}) + ε_{i}

,

Y_{i} - g (x_{0}) = h Z_{i} g^{'} (x_{0}) + \frac{h ^{2}}{2} g^{''} (ξ_{i}) + ε_{i} .

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