A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature $T ( x )$, in degrees Celsius ( ${ } ^ { \circ } \mathrm { C } $ ), of the wire $x \mathrm {~cm}$ from the heated end. The function $T$ is decreasing and twice differentiable. (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.
\ln \left( \frac { 5 } { 3 } \right)$.
A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature $T ( x )$, in degrees Celsius ( ${ } ^ { \circ } \mathrm { C } $ ), of the wire $x \mathrm {~cm}$ from the heated end. The function $T$ is decreasing and twice differentiable.
(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.