 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/15/b/solution

Let

f (x) = cosh x

near

x = 2

. The chain rule gives

f^{'} (x) = \frac{s i n h x}{2 x}, f^{''} (x) = \frac{c o s h x}{4 x} - \frac{s i n h x}{4 x ^{3/2}} .

Substitute

x = 2

into the first three terms of the Taylor expansion:

cosh x = cosh 2 + \frac{sinh 2}{2 2} (x - 2) + (\frac{cosh 2}{16} - \frac{sinh 2}{16 2}) (x - 2)^{2} + O ((x - 2)^{3}) .

The quadratic coefficient is

f^{''} (2) /2

, not

f^{''} (2)

. The apparent square root creates no local difficulty at the positive expansion point.

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