 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability/continuous-mathematics/5/c/solution

Apply the preceding Taylor's theorem at

t = h

and

t = - h

, with

h > 0

. The even-order terms cancel in the central finite difference, giving

d (x, h) - f^{'} (x) = \frac{R _{x} ( h ) - R _{x} ( - h )}{2 h} .

Both Lagrange remainders have magnitude at most

h^{3} /6

∣ d (x, h) - f^{'} (x) ∣ \leq \frac{h ^{2}}{6} .

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