Chapter 3 Conceptual Project:

The following table shows the atmospheric pressure p at the altitude of k feet above sea level (pressure is measured in mm Hg; note that this unit of pressure is approximately the pressure generated by a column of mercury $1$ millimeter high).

1. Find the average rate of change of air pressure from sea level to $2000$ feet of altitude.

2. Find the average rate of change of air pressure between the altitudes of $4000$ and $\mathrm{10,000}$ feet.

3. Use a symmetric difference quotient

   ${\displaystyle \frac{p\left(c+h\right)-p\left(c-h\right)}{2h}}$

   to estimate the instantaneous rate of change of air pressure at $7000\mathrm{ft}$ by choosing $h=1000\mathrm{ft}$.

4. Tell whether you expect the answer to Question 2 or 3 to better approximate the instantaneous rate of change of air pressure at altitude $7000\mathrm{ft}$. Explain. (Hint: Plotting the data on paper may help.)

5. * Explain why you expect the symmetric difference quotient ${\displaystyle \frac{f\left(c+h\right)-f\left(c-h\right)}{2h}}$ in general to be a better approximation of the instantaneous rate of change of f at $x=c$ than the "regular" difference quotient ${\displaystyle \frac{f\left(c+h\right)-f\left(c\right)}{h}}$.

6. Use a graphing utility to find an exponential regression curve to the given data and plot the curve along with the data on the same screen.

7. Use the exponential function you found in Question 6 to estimate the instantaneous rate of change of air pressure at $7000\mathrm{ft}$, and compare with your estimate given in Question 3.

8. Is the instantaneous rate of change increasing or decreasing with altitude? Explain.