Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 2 / 17 / b / Solution

The required derivatives are $f^{\prime }(x)=1/(2\sqrt{x})$, $f^{\prime \prime }(x)=-1/(4x^{3/2})$, $f^{(3)}(x)=3/(8x^{5/2})$, and $f^{(4)}(x)=-15/(16x^{7/2})$. The four-term Taylor polynomial about one, evaluated at two, isThe Taylor remainder is $R_3(2)=-5/(128{\xi }^{7/2})$ for some $1<\xi <2$. ThusIts negative sign also shows that this approximation is an overestimate. Bounding the fourth derivative throughout the interval establishes the requested error estimate without using a decimal value of $\sqrt{2}$.