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\markright{Chapter 3: Worksheet 5}
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\Bf{\Star\ Chapter 3: Worksheet 5  \hfill Jack K. Cohen \hfill Colorado School of Mines}

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\Cb{Generalized Power Rule; Inverse Functions}
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\noindent \Bf{Suggested Problems}
Section 3.4: 2, 4, 7, 10, 14, 20.

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\begin{enumerate}
\item \Star\ (3.4.31) Differentiate \[h(y) = \frac{(1 + y)^{1/2} + (1 - y)^{1/2}}{y^{5/3}}. \]
Then use \Mma\ to check your work---you will probably need to do some additional hand simplifications to bring your answer into line with \Mma 's.  In particular, use the command \Tt{Together[h'[y]]} to collect terms into a single fraction.

\item Attempt to differentiate and check with \Mma .
	\begin{enumerate}
	\item  $y = x^{13/2}$
	\item  $y = x^{13.3}$
          \item $y = 13^x$
	\end{enumerate}

\item \Star\ For each of the following functions $y = f(x)$ on the stated domain, (1) find the range and derivative.  Then (2) find the \Em{inverse} function $y = g(x)$, (3) verify that $g$ has the correct domain and range and (4) check that $g'(x) = $ {\Large $\frac{1}{f'(g(x))}$}. 
	\begin{enumerate}
	\item $f(x) = x^2 - 1, \quad x \geq 0$
	\item $f(x) = 1 + x^2, \quad x \geq 0$
	\item $f(x) = \sqrt{x} - 1, \quad x \geq 0$
	\item $f(x) = \sqrt[3]{x} - 1, \quad -\infty < x < +\infty$
	\end{enumerate}


\item For each curve, find all points on the graph where the tangent line is either horizontal or vertical.  Check your work by using \Tt{Plot}.
	\begin{enumerate}
	\item (3.4.36) $y = x \sqrt{4 - x^2}$
	\item (3.4.38) $y = ${ \Large $\frac{1}{\sqrt{9 - x^2}}$}
	\end{enumerate}

\item In the following problems, cubic equations arise.  For the particular points mentioned, it is easy to guess a root and then use long division to reduce the problem to solving a quadratic.  If your guessing and/or dividing powers are weak today, think about using tools like \Tt{Plot, Table, Simplify} and \Tt{Factor} to help out. 
	\begin{enumerate}
	\item (3.4.45) Find a line through the point $P(18, 0)$ that is normal to the parabola $y = x^2$ at some point $Q(a, a^2)$.
	\item (3.4.46) Find three distinct lines through the point $P(3, 10)$ that are normal to the parabola $y = x^2$.
	\item \Em{Generalize} the above results.  For what regions of the plane are there respectively 0, 1, 2,  or 3 normals to $y = x^2$?
	\end{enumerate}


% Logistic growth
\end{enumerate}
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