 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014/2/b/solution

Expand before taking the fourth derivative:

\frac{s i n ( x ^{2} )}{x} = x - \frac{x ^{5}}{6} + \frac{x ^{9}}{120} - \frac{x ^{13}}{5040} + O (x^{17}) .

The apparent singularity at zero is removable. The convergent power series may be differentiated termwise, giving

y (x) = - 20 x + \frac{126}{5} x^{5} - \frac{143}{42} x^{9} + O (x^{13}) .

x = 0.1

, the successive contributions are

- 2

0.000252

and approximately

- 3.405 \times 1 0^{- 9}

. Thus

y (0.1) ≃ - 1.9997480034.

Even the first two nonzero terms give

- 1.999748

; the next retained correction quantifies that estimate's accuracy.

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