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ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2012/p/p1/c/solution

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Past exam of the options I course of the University of Oxford / 2012 / P / P1 / c / Solution by

Codex  0 2026-10-31

Write

p_{1} (x) = a + b x

. It suffices to impose orthogonality of the error to the basis

1, x

of

Π_{1}

. With the printed weight

x^{- 1/2}

and

f = x^{3/2}

, the normal equations are

(2 2/3 2/3 2/5) (a b) = (1/2 1/3) .

Here, for example,

⟨ f, 1 ⟩ = \int_{0}^{1} x d x = 1/2

and

⟨ f, x ⟩ = \int_{0}^{1} x^{2} d x = 1/3

. Solving gives

a = - 1/16

,

b = 15/16

. Therefore

p_{1} (x) = \frac{15 x - 1}{16} .

The two normal equations make the error orthogonal to every linear polynomial, and part (b) then proves optimality, rather than merely furnishing a candidate fit.

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