We want to know the value of (or at least an upper bound of)
| T 2 ( 1 ) − f ( 1 ) | {\displaystyle \displaystyle |T_{2}(1)-f(1)|}
Taylor's remainder formula says that there is a c ∈ [ 0 , 1 ] {\displaystyle c\in [0,1]} such that
| T 2 ( 1 ) − f ( 1 ) | = | f ‴ ( c ) | | x − a | 3 3 ! ≤ M | 1 − 0 | 3 6 = M 6 {\displaystyle \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 [ 0 , 1 ] {\displaystyle \displaystyle [0,1]} .