The function $f$ is twice differentiable for $x > 0$ with $f ( 1 ) = 15$ and $f ^ { \prime \prime } ( 1 ) = 20$. Values of $f ^ { \prime }$, the derivative of $f$, are given for selected values of $x$ in the table above.
| $x$ | 1 | 1.1 | 1.2 | 1.3 | 1.4 |
| $f ^ { \prime } ( x )$ | 8 | 10 | 12 | 13 | 14.5 |
(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 $\int _ { 1 } ^ { 1.4 } f ^ { \prime } ( x ) d x$. Use the approximation for $\int _ { 1 } ^ { 1.4 } f ^ { \prime } ( x ) d x$ 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 $f ( 1.4 )$.