 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/11/solution

The first four terms form the cubic Taylor polynomial about

x_{0}

f (x_{0}) + f^{'} (x_{0}) (x - x_{0}) + \frac{f ^{''} ( x _{0} )}{2 !} (x - x_{0})^{2} + \frac{f ^{(3)} ( x _{0} )}{3 !} (x - x_{0})^{3} .

With a continuous fourth derivative nearby, Taylor's theorem gives a remainder

O ((x - x_{0})^{4})

. Continuous derivatives through third order suffice for the weaker remainder

o ((x - x_{0})^{3})

. These are local approximations; equality with an infinite Taylor series requires additional convergence information.

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