 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/6/solution

Put

h = x - π /4

cos x sin x = \frac{1}{2} sin 2 x = \frac{1}{2} cos 2 h

. Expanding cosine at zero gives

f (x) = \frac{1}{2} - (x - \frac{π}{4})^{2} + O ((x - π /4)^{4}) .

The linear and cubic terms vanish, so these are the first two nonzero Taylor series terms.

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