Taylor series error estimation question

Question

Question is that Taylor series of cosx is restricted to only first two terms and permissible error is 0.54 × 10^(-2) then x can atmost be A) 0.6 B) 0.5 C) 0.4 D) 0.3

My atempt is as follows we need first two terms so expansion is as follows , cosx=1 - x^(2)\2! . I also know formula for remainder in taylor series but i am just confused that do i have to bound $R_2$ or$ R_3 $?

Answer 1

Using the Wiki formula for the interval $(-r,r)$ $$|R_3(x)| \le M \frac{r^4}{4!}$$

and the conservative estimate $M\le 1$, you get for the remainders in the case A: 0.54E-2, B: 0.26e-2, so the correct choice is A.

Edit: In your case you use $R_3$ because the Taylor polynom $P_3(x)$ is the same as $P_2(x) = 1 -\tfrac{1}{2}x^2.\;$ Note that even the $R_3$ term slighty overestimates the actual error, which is about $0.53356\cdot 10^{-2}\;$ in the intervall $(-0.6, +0.6).$