1. **Problem statement:** Find the second Taylor polynomial $P_2(x)$ for $f(x) = e^x \cos x$ about $x_0 = 0$. Use $P_2(0.5)$ to approximate $f(0.5)$, find an upper bound for the error $|f(0.5) - P_2(0.5)|$, and compare it to the actual error. Then find a bound for the error $|f(x) - P_2(x)|$ on $[0,1]$. 2. **Taylor polynomial formula:** The second Taylor polynomial about $x_0=0$ is $$P_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2}x^2$$ 3. **Calculate derivatives:** - $f(x) = e^x \cos x$ - $f'(x) = e^x \cos x - e^x \sin x = e^x(\cos x - \sin x)$ - $f''(x) = \frac{d}{dx}[e^x(\cos x - \sin x)] = e^x(\cos x - \sin x) + e^x(-\sin x - \cos x) = e^x[(\cos x - \sin x) + (-\sin x - \cos x)] = e^x(-2 \sin x)$ 4. **Evaluate at $x=0$:** - $f(0) = e^0 \cos 0 = 1 \cdot 1 = 1$ - $f'(0) = e^0(\cos 0 - \sin 0) = 1(1 - 0) = 1$ - $f''(0) = e^0(-2 \sin 0) = 1 \cdot 0 = 0$ 5. **Form $P_2(x)$:** $$P_2(x) = 1 + 1 \cdot x + \frac{0}{2} x^2 = 1 + x$$ 6. **Approximate $f(0.5)$:** $$P_2(0.5) = 1 + 0.5 = 1.5$$ 7. **Error bound formula:** The remainder term for the second degree Taylor polynomial is $$|R_2(x)| = \left| \frac{f^{(3)}(c)}{3!} x^3 \right|$$ for some $c$ between 0 and $x$. 8. **Find $f^{(3)}(x)$:** $$f^{(3)}(x) = \frac{d}{dx} f''(x) = \frac{d}{dx} [e^x(-2 \sin x)] = e^x(-2 \sin x) + e^x(-2 \cos x) = e^x(-2 \sin x - 2 \cos x) = -2 e^x (\sin x + \cos x)$$ 9. **Bound $|f^{(3)}(c)|$ on $[0,0.5]$:** - $e^x$ is increasing, max at $x=0.5$ is $e^{0.5} \approx 1.6487$ - $|\sin x + \cos x| \leq \sqrt{2}$ (max value of $\sin x + \cos x$ is $\sqrt{2}$) - So $|f^{(3)}(c)| \leq 2 \cdot 1.6487 \cdot \sqrt{2} \approx 2 \cdot 1.6487 \cdot 1.4142 = 4.66$ 10. **Calculate error bound at $x=0.5$:** $$|R_2(0.5)| \leq \frac{4.66}{6} (0.5)^3 = \frac{4.66}{6} \cdot 0.125 = 0.0971$$ 11. **Actual error:** - Calculate $f(0.5) = e^{0.5} \cos 0.5 \approx 1.6487 \times 0.8776 = 1.4467$ - Error $= |1.4467 - 1.5| = 0.0533$ 12. **Bound for error on $[0,1]$:** - Max of $|f^{(3)}(x)|$ on $[0,1]$: - $e^1 = 2.7183$ - $|\sin x + \cos x| \leq \sqrt{2} = 1.4142$ - So max $|f^{(3)}(x)| \leq 2 \times 2.7183 \times 1.4142 = 7.69$ 13. **Error bound on $[0,1]$:** $$|R_2(x)| \leq \frac{7.69}{6} |x|^3 \leq 1.282 |x|^3$$ **Final answers:** - $P_2(x) = 1 + x$ - Approximation $P_2(0.5) = 1.5$ - Error bound at $0.5$ is approximately $0.0971$ - Actual error at $0.5$ is approximately $0.0533$ - Error bound on $[0,1]$ is $|R_2(x)| \leq 1.282 |x|^3$