 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability/6/a/solution

f \in C^{n + 1}

on an interval containing

x

and

x + h

, there is a point

ξ

between them such that

f (x + h) = r = 0 \sum n \frac{f ^{(r)} ( x )}{r !} h^{r} + \frac{f ^{(n + 1)} ( ξ )}{( n + 1 )!} h^{n + 1} .

The Lagrange remainder makes this an exact equality, rather than just a truncated Taylor polynomial. For

h = 0

the equality is immediate.

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