Let $h$ be a function having derivatives of all orders for $x > 0$. Selected values of $h$ and its first four derivatives are indicated in the table below. The function $h$ and these four derivatives are increasing on the interval $1 \leq x \leq 3$.

\begin{center}
\begin{tabular}{ | c | | c | c | c | c | c | }
\hline
$x$ & $h ( x )$ & $h ^ { \prime } ( x )$ & $h ^ { \prime \prime } ( x )$ & $h ^ { \prime \prime \prime } ( x )$ & $h ^ { ( 4 ) } ( x )$ \\
\hline
1 & 11 & 30 & 42 & 99 & 18 \\
\hline
2 & 80 & 128 & $\frac { 488 } { 3 }$ & $\frac { 448 } { 3 }$ & $\frac { 584 } { 9 }$ \\
\hline
3 & 317 & $\frac { 753 } { 2 }$ & $\frac { 1383 } { 4 }$ & $\frac { 3483 } { 16 }$ & $\frac { 1125 } { 16 }$ \\
\hline
\end{tabular}
\end{center}

(a) Write the first-degree Taylor polynomial for $h$ about $x = 2$ and use it to approximate $h ( 1.9 )$. Is this approximation greater than or less than $h ( 1.9 )$ ? Explain your reasoning.

(b) Write the third-degree Taylor polynomial for $h$ about $x = 2$ and use it to approximate $h ( 1.9 )$.

(c) Use the Lagrange error bound to show that the third-degree Taylor polynomial for $h$ about $x = 2$ approximates $h ( 1.9 )$ with error less than $3 \times 10 ^ { - 4 }$.