Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 15 / b

For $f(x)=x^{1/3}$, the required derivatives areAt $a=8$ these give $f(8)=2$, $f^{\prime }(8)=1/12$, and $f^{\prime \prime }(8)=-1/144$. Setting $x-a=1$, the three-term Taylor polynomial yieldsBy the Taylor theorem with Lagrange remainder, the signed error isIt is positive, and ${\xi }^{-8/3}<8^{-8/3}=1/256$. ConsequentlyThe last comparison is exact because $5\cdot 4000=20000<20736$. Thus the estimate is an underestimate with the required rigorous absolute error bound.