 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-2/6/solution

Put

h = x - π /3

. The values

f (π /3) = 1/2

f^{'} (π /3) = - 3 /2

, and

f^{''} (π /3) = - 1/2

cos x = \frac{1}{2} - \frac{3}{2} (x - \frac{π}{3}) - \frac{1}{4} (x - \frac{π}{3})^{2} + O ((x - π /3)^{3}) .

The factorial in the quadratic term accounts for its coefficient

- 1/4

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