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\markright{Chapter 3: Worksheet 4}
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\Bf{\Star\ Chapter 3: Worksheet 4  \hfill Jack K. Cohen \hfill Colorado School of Mines}

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\Cb{Techniques of Differentiation: Chain Rule}
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\noindent \Bf{Suggested Problems}
Section 3.3:  See below.

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\noindent  The chain rule is \Bf{vital}.  You absolutely must drill this until you can perform it on automatic pilot.  Again use \Mma\ to check your work.  Some additional commands that may be helpful are \Tt{Expand} and \Tt{Factor}.  Try \Tt{Expand[(1 + 2x)\^{}12]} followed by \Tt{Factor[\%]} to get the idea.  And don't forget \Tt{Simplify}.

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\begin{enumerate}
\item Do problems 2, 4, 6, 8,12, 14, 16, 22, 24, 38, 40, 42, 44.

\item  Use the fact that  $D \sin x = \cos x$ to find the derivatives of the following functions:
	\begin{enumerate}
	\item $\sin x^3$  (Recall that this notation means $\sin( x^3)$)
	\item $\sin^3 x$  (Recall that this notation means $(\sin x)^3$)
	\item $\sin( \sin x)$
	\item $\sin( \sin ( \sin x))$
	\end{enumerate}
\item Let $A$ be the area of a circle of radius $r$.
	\begin{enumerate}
	\item Express $\frac{dA}{dt}$ as a function of $r$ and $\frac{dr}{dt}$.
	\item (3.3.49)At what rate is the area of a circle increasing when its radius is 10 cm and is increasing at the rate of 2 cm/sec?
	\item Express $\frac{dr}{dt}$ as a function of $A$ and $\frac{dA}{dt}$.
	\item (3.3.50)At what rate is the radius of a circle decreasing when its area is 75$\pi$ cm$^2$ and is decreasing at the rate of 2$\pi$ cm$^2$/sec?
	\end{enumerate}
\item \Star\ Suppose that the rate of change of the volume of a sphere is proportional its surface area. 
	\begin{enumerate}
	\item Prove that the rate of change of its radius is a constant $c$.
	\item If the radius has length $r_0$ at $t = t_0$, what is the radius $r$ at other times $t$?
	\item Does your formula for $r$ ``work'' if $c < 0$ and $t$ is very large?
	\item (3.3.61) A spherical snowball is melting in such a way that the rate of decrease of its volume is proportional to its surface area.  At 10 {\sc a.m.} its volume was 500 cm$^3$, and at 11 {\sc a.m.} its volume was 250 cm$^3$.  When did the snowball finish melting?
	\end{enumerate}
\item Suppose that the rate of change of the volume of a cube is proportional its surface area. 
	\begin{enumerate}
	\item Prove that the rate of change of its edge is a constant $c$.
	\item If the edge has length $e_0$ at $t = t_0$, what is its length $e$ at other times $t$?
	\item (3.3.62) A cubical block of ice with edge 20 cm begins to melt at 8 {\sc a.m.} Each edge decreases steadily thereafter, and at 4 {\sc p.m.} is 8 cm. What was the rate of change of the volume of the block of ice at 12 noon?	\end{enumerate}
\item \Star\ (3.3.36) Use \Mma\ to compute the derivative of $G(x) = \{1 + [x + (x^2 + x^3)^4]^5\}^6$.  Try out each of \Tt{Simplify[G'[x]]}, \Tt{Factor[G'[x]]}, and \Tt{Expand[G'[x]]}.   \Em{Don't} hand in your results!	
\end{enumerate}
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