 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/15/b/ii/solution

f^{''} (x) = \frac{t ^{''} ( x )}{1 + t ( x )} - \frac{t ^{'} ( x ) ^{2}}{( 1 + t ( x ) ) ^{2}} .

With

t_{0} = 0

, the constant term vanishes,

f^{'} (0) = t_{1} \neq = 0

, and

f^{''} (0) = t_{2} - t_{1}^{2} \neq = 0

by the specified case. Thus the first two nonzero terms in the Taylor expansion are

f (x) = t_{1} x + \frac{t _{2} - t _{1}^{2}}{2} x^{2} + O (x^{3}) .

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