A function $f$ has derivatives of all orders for all real numbers $x$. A portion of the graph of $f$ is shown above, along with the line tangent to the graph of $f$ at $x = 0$. Selected derivatives of $f$ at $x = 0$ are given in the table below.
| $n$ | $f ^ { ( n ) } ( 0 )$ |
| 2 | 3 |
| 3 | $-\frac { 23 } { 2 }$ |
| 4 | 54 |
(a) Write the third-degree Taylor polynomial for $f$ about $x = 0$.
(b) Write the first three nonzero terms of the Maclaurin series for $e ^ { x }$. Write the second-degree Taylor polynomial for $e ^ { x } f ( x )$ about $x = 0$.
(c) Let $h$ be the function defined by $h ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$. Use the Taylor polynomial found in part (a) to find an approximation for $h ( 1 )$.
(d) It is known that the Maclaurin series for $h$ converges to $h ( x )$ for all real numbers $x$. It is also known that the individual terms of the series for $h ( 1 )$ alternate in sign and decrease in absolute value to 0. Use the alternating series error bound to show that the approximation found in part (c) differs from $h ( 1 )$ by at most 0.45.