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\markright{Chapter 3: Worksheet 2}
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\Bf{\Star\ Chapter 3: Worksheet 2  \hfill Jack K. Cohen \hfill Colorado School of Mines}

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\Cb{The Derivative as the Rate of Change Function}
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\noindent \Bf{Suggested Problems}
Section 3.1: 22, 24, 26

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\begin{enumerate}
\item Find the rate of change of the area of a circle with respect to its radius.

\item  A stone dropped into a pond causes an expanding circular ripple.   What is the rate of change of the area of a circle with respect to its radius when the radius is 3.5 meters?  Give the answer correct to two decimal places.

\item (3.1.31) Find the rate of change of the area of a circle with respect to its circumference.

\item  A stone dropped into a pond makes an expanding circular ripple.  What is the rate of change of the area of a circle with respect to its circumference when the circumference is 3.5 meters?

\item (3.1.32) A stone dropped into a pond causes a circular ripple that travels out from the point of impact at 5 ft/sec.  At what rate (in ft$^2$/sec) is the area within the circle increasing when $t = 10$.  Hint: Find a formula for $r$ in terms of $t$.

\item \Star\ (Generalization of Example 6) 
	\begin{enumerate}
	\item Find the maximum height attained by a ball that is thrown straight up from the ground with initial velocity $v_0$.  Assume a gravitational constant $g$.
	\item Find the velocity with which it hits the ground upon its return.
	\item  One benefit of using \Em{parameters} instead of specific numbers is the ability to deduce general laws.  Illustrate this by stating a general connection between the initial and final velocities.
	\item Another benefit of using parameters is the ability to check that answers have the correct dimensions.  For example, if somehow, we got the answer $s_{top} = \frac{v_0}{g}$, we could see that the dimensions of the purported answer were $\frac{L}{T} \div \frac{L}{T^2} = T$ when they \Em{should} have been $L$.  Dimensional checks can save a lot of embarrassment and every good scientist uses them to check calculations.  So don't \Em{you} be embarrassed---give a dimensional check for your answers to parts (a) and (b).
	\end{enumerate}

\item  The treatment in the last problem assumed that the force of gravity was a constant.  Actually it falls off like $1/r^2$ where $r$ is the distance of the body (ball) from the center of the Earth (this is called the ``inverse square law'') .    Taking this variable force law into account gives the equation \[s =  \frac{R(v_0^2 - v^2)}{2gR - (v_0^2 - v^2)}, \] where $R$ is the mean radius of the Earth and $s = r - R$ is the distance measured upward from the Earth's surface.
	\begin{enumerate}
	\item Is the final velocity still the same as the initial velocity when the inverse square law is taken into account?
	\item What is the maximum height attained when the inverse square law is taken into account?
	\item Using the numbers: $v_0 = 96$ ft/sec, $g = 32$ ft/sec$^2$, $R =3960$ miles, discuss whether it is worthwhile to consider the more complicated inverse square law model for this problem.  Remark: Air resistence is another effect we might consider.  Mathematical modeling is always a question of making things ``as simple as possible, but no simpler.''
	\end{enumerate}
	
\item (3.1.34) A water bucket containing 10 liters of water develops a leak at time $t = 0$, and the volume $V$ of water in the bucket $t$ seconds later is given by \[ V = 10\left( 1 -  \frac{t}{100} \right)^2 \] until the bucket is empty at time $t = 100$.
	\begin{enumerate}
	\item At what rate is water leaking from the bucket after exactly one minute has passed?
	\item When is the instantaneous rate of change of $V$ equal to the average change of $V$ from $t = 0$ to $t = 100$?
	\item Does the given formula for $V$ make sense when $t = 200$?
	\end{enumerate}

\item (3.1.36) The following data describe the growth of the population $P$ (in thousands) of a certain city during the 1970s.  Use the graphical method of Example 4 to estimates its rate of growth in 1975.

\begin{tabular}{|c|c|c|c|c|c|c|} \hline
Year & 1970 & 1972 & 1974 & 1976 & 1978 & 1980 \\ \hline
$P$ & 265 & 293 & 324 & 358 & 395 & 437  \\ \hline
\end{tabular}

	
\end{enumerate}
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