1. **Problem statement:** Find the error of the 4th degree Taylor polynomial approximation for $\ln(\sec(0.4))$. 2. **Formula and explanation:** The Taylor series expansion of a function $f(x)$ about $x=a$ is given by $$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$$ The error (remainder) after the 4th degree polynomial is $$R_4(x) = \frac{f^{(5)}(\xi)}{5!}(x-a)^5$$ for some $\xi$ between $a$ and $x$. 3. **Choosing expansion point:** We expand around $a=0$ (Maclaurin series) for $f(x) = \ln(\sec x)$. 4. **Calculate derivatives:** The 5th derivative $f^{(5)}(x)$ is complicated, but we can bound it on $[0,0.4]$. 5. **Bounding $f^{(5)}(x)$:** Numerically or by estimation, $|f^{(5)}(x)|$ is less than approximately 100 on $[0,0.4]$. 6. **Calculate error bound:** Using $$|R_4(0.4)| \leq \frac{100}{5!} |0.4|^5 = \frac{100}{120} \times 0.4^5 = \frac{100}{120} \times 0.01024 = 0.00853$$ 7. **Interpretation:** The error in approximating $\ln(\sec(0.4))$ by its 4th degree Taylor polynomial at 0 is at most approximately $0.00853$. **Final answer:** The error bound is approximately $0.0085$.