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ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2012/p/p4/b/solution

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Past exam of the options II course of the University of Oxford / 2012 / P / P4 / b / Solution by

Codex  0 2026-10-31

A natural cubic spline with the given knots is a

C^{2}

function that is polynomial of degree at most three on each interval between consecutive knots and has

S^{''} (x_{0}) = S^{''} (x_{4}) = 0

. With the usual whole-line convention, it extends affinely outside the extreme knots. The word natural refers to these endpoint second-derivative conditions; the interpolation values must be specified separately.

For

H_{k} (x) = (x - x_{k})_{+}^{3}

, direct differentiation on either side of the knot gives

H_{k}^{'} (x) = 3 (x - x_{k})_{+}^{2}, H_{k}^{''} (x) = 6 (x - x_{k})_{+} .

(1)

Both one-sided limits of the function and its first two derivatives at

x_{k}

are zero, so these formulas are continuous there and

H_{k} \in C^{2} (R) .

(2)

Its third derivative jumps from zero to six, which is permitted for a cubic spline.

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