 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2002/paper-2/9/ii/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Let

g (x) = tan (x + π /4)

. The derivative identity for tangent gives

g^{'} = 1 + g^{2}

, so

g (0) = 1

g^{'} (0) = 2

, and

g^{''} (0) = 2 g (0) g^{'} (0) = 4

. Its first three nonzero Taylor expansion terms are therefore

tan (x + π /4) = 1 + 2 x + 2 x^{2} + O (x^{3}) .

The factor

1/2!

on the second derivative is what turns

g^{''} (0) = 4

2

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