 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability-resit/6/b/ii/solution

The relevant derivatives at

x = 3

are

g (3) = 32, g^{'} (3) = 0, g^{''} (3) = 0, g^{'''} (3) = 24.

Thus the cubic Taylor polynomial is

T_{3} (x) = 32 + 4 (x - 3)^{3} .

In fact the exact expansion is

g (x) = 32 + 4 (x - 3)^{3} + (x - 3)^{4}

, so the omitted term has order

O ((x - 3)^{4})

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