$[3/4$ Points$]$

SCALC$82\mathrm{.}7\mathrm{.}023.$

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: $PV=C.$

$($a$)$ Find the rate of change of volume with respect to pressure.

$\frac{dV}{dP}=$

$($b$)$ A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for $11$ minutes. Is the volume decreasing more

rapidly at the beginning or the end of the $11$ minutes? Explain.

From the formula for $\frac{dV}{dP},$ we see that as $P$ increases, the absolute value of $\frac{dV}{dP}$ decreases $,.$ Thus, the volume is

more rapidly at the beginning.

$($c$)$ Write the formula for isothermal compressibility $($see this example$)$ in terms of $P.$

$=$

$[6/8$ Points$]$

SCALC$82\mathrm{.}7\mathrm{.}025.$

A graphing calculator is recommended.

The table gives the population of the world $P(t),$ in millions, where $t$ is measured in years and $t=0$ corresponds to the year $1900.$

$($a$)$ Estimate the rate of population growth in $1920$ and in $1980$ by averaging the slopes of two secant lines.

growth in $1920$

million$/$year

growth in $1980$

million$/$year

$($b$)$ Use a graphing device to find a cubic function $($a third$-$degree polynomial$)$ that models the data. $($Let $t$ be the year. Round all numerical values to five

decimal places.$)$

$P(t)=-\mathrm{.}00028t^3+\mathrm{.}52243t^2-6\mathrm{.}89564t+1720\mathrm{.}58608$

ONLY NEED $6$ C $7$B