The function $f$ has derivatives of all orders for all real numbers. It is known that $f(0) = 2$, $f'(0) = 3$, $f''(x) = -f\left(x^{2}\right)$, and $f'''(x) = -2x \cdot f'\left(x^{2}\right)$. (a) Find $f^{(4)}(x)$, the fourth derivative of $f$ with respect to $x$. Write the fourth-degree Taylor polynomial for $f$ about $x = 0$. Show the work that leads to your answer. (b) The fourth-degree Taylor polynomial for $f$ about $x = 0$ is used to approximate $f(0.1)$. Given that $\left|f^{(5)}(x)\right| \leq 15$ for $0 \leq x \leq 0.5$, use the Lagrange error bound to show that this approximation is within $\frac{1}{10^{5}}$ of the exact value of $f(0.1)$. (c) Let $g$ be the function such that $g(0) = 4$ and $g'(x) = e^{x} f(x)$. Write the second-degree Taylor polynomial for $g$ about $x = 0$.
The function $f$ has derivatives of all orders for all real numbers. It is known that $f(0) = 2$, $f'(0) = 3$, $f''(x) = -f\left(x^{2}\right)$, and $f'''(x) = -2x \cdot f'\left(x^{2}\right)$.
(a) Find $f^{(4)}(x)$, the fourth derivative of $f$ with respect to $x$. Write the fourth-degree Taylor polynomial for $f$ about $x = 0$. Show the work that leads to your answer.
(b) The fourth-degree Taylor polynomial for $f$ about $x = 0$ is used to approximate $f(0.1)$. Given that $\left|f^{(5)}(x)\right| \leq 15$ for $0 \leq x \leq 0.5$, use the Lagrange error bound to show that this approximation is within $\frac{1}{10^{5}}$ of the exact value of $f(0.1)$.
(c) Let $g$ be the function such that $g(0) = 4$ and $g'(x) = e^{x} f(x)$. Write the second-degree Taylor polynomial for $g$ about $x = 0$.