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\problemgrouptitle{Average and Instantaneous Rate of Change}

\item\label{itm:averagerategraph} Consider the function $f(x)=x^3+x+2$.
\begin{enumerate}
\item Find the slope of the secant line through points $(2,12)$ and $(1,4)$ on the graph of $f$.
\item\label{itm:estimateslope} From your graph, estimate the slope of the tangent line to the curve at the point $(2,12)$.  Next, numerically estimate the tangent slope by calculating the secant slope between the point $(2,12)$ and a point with $x$ value near $2$.
% We have to hard-code the reference (b) below for now because we don't want
% 1(b) printed and changing the enumeration style to (\alph*) will break
% a reference to this question from elsewhere.
\item What is the equation of the tangent line at $(2,12)$?  (Use your estimate from (b) for the slope.)
\end{enumerate}

\answer{ 
\begin{inparaenum}[(a)]
\item $8$
\item $m\approx 13$
\item Point-slope form: $y=13(x-2)+12$ , Slope-intercept form: $y=13x-14$
\end{inparaenum}
}

\item After $t$ seconds, a toy car moving along a straight track has position $s(t)$ measured from a fixed point of reference given by $s(t)=t^3+2t^2+1$~cm.
\begin{enumerate}
\item How far is the car initially from the reference point?
\item How far from the reference point is the car after 2~seconds?
\item What is the average velocity of the car during its first 2~seconds of motion?
\item\label{itm:estimatevelocity} By calculator, estimate the instantaneous velocity of the car at time $t=2$ by computing the average velocities over small time intervals near $t=2$.
\end{enumerate}

\answer{ 
\begin{inparaenum}[(a)]
\item $1$~cm
\item $17$~cm
\item $8$~cm/s
\item $v\approx 20$~cm/s
\end{inparaenum}
}

\item One mole of an ideal gas at a fixed temperature of $273$~K has a volume $V$ that is inversely proportional to the pressure $P$ (Boyle's Law) given by
\[V=\frac{22.4}{P}\ ,\]
where $V$ is in litres (L) and $P$ is in atmospheres (atm).
\begin{enumerate}
\item What is the average rate of change of $V$ with respect to $P$ as pressure varies from 1~atm to 3~atm?
\item\label{itm:estimatevolumechange} Use a calculator to estimate the instantaneous rate of change in the volume when the pressure is 3 atm  by computing the average rates of change over small intervals lying to the left and right of $P=3$~atm.
\end{enumerate}

\answer{ 
\begin{inparaenum}[(a)]
\item $\displaystyle\frac{\Delta V}{\Delta P}\approx-7.47$ L/atm
\item $-2.49$ L/atm
\end{inparaenum}
}
