Chapter 11.1, Problem 27E

(a)

To determine

The number $M$, such that $\left|f^4\left(c\right)\right|\le M$ for all the values of $c$, for the function $f\left(x\right)=\cos x$ and the third remainder for $f\left(x\right)$ at $x=0$ is $R_3\left(x\right)=\frac{f^{\left(4\right)}\left(c\right)}{4!}{x}^4$

To determine

To prove: The error $R_3\left(0.12\right)$ in using $p_3\left(0.12\right)$ as an approximation to $f\left(0.12\right)=\cos .12$, does not exceed $8.64\times {10}^{-6}$