6.67. Using the Maclaurin formula, calculate $\cos 0.1$, using only the first two terms of the expansion, and estimate the error of the calculation.
Solution.

$$
\cos x=1-\frac{x^{2}}{2!}+R=\cos 0.1 \approx 1-\frac{0.1^{2}}{2}+R=0.995+R
$$

Here, the error in the calculation is determined by the estimate of the remainder term in Lagrange's form

$$
|R|=\left|\frac{\cos (0.1 \cdot \theta)}{24}\right| \leqslant \frac{1}{24} \approx 0.042 .
$$

Here $\theta$ is some number satisfying the inequality $0<\theta<1$. Therefore, $\cos 0.1 \approx 0.995 \pm 0.042$.