 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/b/solution

t

f (x + t) = f (x) + t f^{'} (x) + \frac{t ^{2}}{2} f^{''} (x) + R_{x} (t), R_{x} (t) = \frac{t ^{3}}{6} f^{'''} (ξ_{t}),

where

ξ_{t}

lies between

x

and

x + t

. The given third derivative bound implies

∣ R_{x} (t) ∣ \leq ∣ t ∣^{3} /6

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