The function $f$ is twice differentiable for $x > 0$ with $f(1) = 15$ and $f''(1) = 20$. Values of $f'$, the derivative of $f$, are given for selected values of $x$ in the table below.

\begin{center}
\begin{tabular}{ | c | | c | c | c | c | c | }
\hline
$x$ & 1 & 1.1 & 1.2 & 1.3 & 1.4 \\
\hline
$f'(x)$ & 8 & 10 & 12 & 13 & 14.5 \\
\hline
\end{tabular}
\end{center}

(a) Write an equation for the line tangent to the graph of $f$ at $x = 1$. Use this line to approximate $f(1.4)$.

(b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate $\int_{1}^{1.4} f'(x)\, dx$. Use the approximation for $\int_{1}^{1.4} f'(x)\, dx$ to estimate the value of $f(1.4)$. Show the computations that lead to your answer.

(c) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f(1.4)$. Show the computations that lead to your answer.

(d) Write the second-degree Taylor polynomial for $f$ about $x = 1$. Use the Taylor polynomial to approximate $f(1.4)$.