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\markright{Chapter 3: Worksheet 8}
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\Bf{\Star\ Chapter 3: Worksheet 8  \hfill Jack K. Cohen \hfill Colorado School of Mines}

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\Cb{Derivatives of Sine and Cosine; Applications}
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\noindent \Bf{Suggested Problems}
Section 3.7: 5, 7, 9, 21.  Problems 71 and 75 are honor problems (for those who want a challenge).

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\item Fill in the table:

\begin{tabular}{||l||l||}	\hline
$D \sin x = \qquad \qquad $ & $D \cos x = \qquad \qquad $      \\ \hline
$D \tan x = \qquad \qquad $ & $D \cot x = \qquad \qquad $      \\ \hline
$D \sec x = \qquad\qquad  $ & $D \csc x =\qquad \qquad $       \\ \hline
\end{tabular}

\item  Differentiate each function and check answers with \Mma:
	\begin{enumerate}
	\item $\sin x^2, \cos x^2, \tan x^2$
	\item $\sin^2 x, \cos^2 x, \tan^2 x$
	\item $\sin 2x, \cos 2x, \tan 2x$
	\item $\sin (\sin x), \cos (\cos x), \tan (\tan x)$
	\end{enumerate}

\item Compute {\Large $\frac{d}{d \theta}$} $(\cos^2 \theta + \sin^2 \theta)$ and explain the result.

\item \Star\ (3.7.76) A hiker starting at a point $P$ on a straight road wants to reach a forest cabin that is  2 km from the point $Q$ that itself is is 3 km down the road from $P$, as shown in Figure 1.  She can walk 8km/h along the road, but only 3km/h through the forest.  She wants to minimize the time required to reach the cabin.  Find out how far down the road she should walk first before setting off through the forest straight for the cabin by the methods requested below.
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	\centerline{\epsffile{ws8p4.eps}}
	\caption{Problem 4} 
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	\begin{enumerate}
	\item As suggested in the text, use the angle $\theta$ between the road and the path she takes through the forest as the independent variable.
	\item A very similar problem was formulated in the previous section (3.6.45) before we officially knew about derivatives of trig functions.  So repeat the solution using the algebraic variable $x$, the distance along the road from the take-off point to $Q$, as the independent variable.
	\item Which of methods (a) and (b) above is easier for this problem?
	\item To avoid letting your calculus instructor pose an unending number of ``cabin in the forest'' problems, \Em{generalize} and solve the above problem by using letters for the two distances and two rates.
	\item Under what circumstances is it best to immediately head straight for the cabin?
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\item Compute the following limits and check your results by using \Tt{LimitTable}.
	\begin{enumerate}
	\item (3.7.42) $\lim_{\theta \rightarrow 0}$ {\Large $\frac{\sin^2 \theta}{\theta^2}$}
	\item (3.7.44) $\lim_{\theta \rightarrow 0}$ {\Large $\frac{\tan \theta}{\theta}$}
	\item (3.7.48) $\lim_{x \rightarrow 0}$ {\Large $\frac{\sin 2x}{x \cos 3x}$}
	\item (3.7.50) $\lim_{x \rightarrow 0}$ {\Large $\frac{1 - \cos 2x}{x}$}
	\end{enumerate}

\item (3.7.65) A rocket is launched vertically upward from a point 2 mi west of an observer on the ground. 
	\begin{enumerate}
	\item What is the speed of the rocket when the angle of elevation (from the horizontal) of the observer's line of sight to the rocket is 50$^\circ$ and is increasing at 5$^\circ$ per second?
	\item This problem is similar to Illustrative Example 9 and others in this problem set, so generalize and solve the present problem.
	\end{enumerate}


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