A metal wire of length 8 centimeters (cm) is heated at one end. The table below gives selected values of the temperature $T ( x )$, in degrees Celsius $\left( {}^{\circ} \mathrm{C} \right)$, of the wire $x$ cm from the heated end. The function $T$ is decreasing and twice differentiable.
| \begin{tabular}{ c } Distance |
| $x ( \mathrm{~cm} )$ |
& 0 & 1 & 5 & 6 & 8 \hline
| Temperature |
| $T ( x ) \left( {}^{\circ} \mathrm{C} \right)$ |
& 100 & 93 & 70 & 62 & 55 \hline \end{tabular}
(a) Estimate $T ^ { \prime } ( 7 )$. Show the work that leads to your answer. Indicate units of measure.
(b) Write an integral expression in terms of $T ( x )$ for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure.
(c) Find $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) \, d x$, and indicate units of measure. Explain the meaning of $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) \, d x$ in terms of the temperature of the wire.
(d) Are the data in the table consistent with the assertion that $T ^ { \prime \prime } ( x ) > 0$ for every $x$ in the interval $0 < x < 8$ ? Explain your answer.