Science:Math Exam Resources/Courses/MATH100/December 2010/Question 07/Hint 1

We want to know the value of (or at least an upper bound of)

$\displaystyle |T_{2}(1)-f(1)|$

Taylor's remainder formula says that there is a $c\in [0,1]$ such that

$\displaystyle |T_{2}(1)-f(1)|={\frac {|f'''(c)||x-a|^{3}}{3!}}\leq {\frac {M|1-0|^{3}}{6}}={\frac {M}{6}}$

where M is an upper bound for the absolute value of the third derivative on $\displaystyle [0,1]$.