Let $f$ be a function that has derivatives of all orders for all real numbers. Assume $f(0) = 5$, $f'(0) = -3$, $f''(0) = 1$, and $f'''(0) = 4$.
(a) Write the third-degree Taylor polynomial for $f$ about $x = 0$ and use it to approximate $f(0.2)$.
(b) Write the fourth-degree Taylor polynomial for $g$, where $g(x) = f\left(x^{2}\right)$, about $x = 0$.
(c) Write the third-degree Taylor polynomial for $h$, where $h(x) = \int_{0}^{x} f(t)\, dt$, about $x = 0$.
(d) Let $h$ be defined as in part (c). Given that $f(1) = 3$, either find the exact value of $h(1)$ or explain why it cannot be determined.