The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function $R$ of time $t$. The graph of $R$ and a table of selected values of $R(t)$, for the time interval $0 \leq t \leq 90$ minutes, are shown above.
(a) Use data from the table to find an approximation for $R'(45)$. Show the computations that lead to your answer. Indicate units of measure.
(b) The rate of fuel consumption is increasing fastest at time $t = 45$ minutes. What is the value of $R''(45)$? Explain your reasoning.
(c) Approximate the value of $\int_{0}^{90} R(t)\,dt$ using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation less than the value of $\int_{0}^{90} R(t)\,dt$? Explain your reasoning.
(d) For $0 < b \leq 90$ minutes, explain the meaning of $\int_{0}^{b} R(t)\,dt$ in terms of fuel consumption for the plane. Explain the meaning of $\frac{1}{b}\int_{0}^{b} R(t)\,dt$ in terms of fuel consumption for the plane. Indicate units of measure in both answers.