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\title[Problem Sets 3]{Problem Sets 3.  Due Friday 8 September}

\begin{document}

\maketitle



\centerline{\sc Problem Set 3.  Problems from Lecture 3.}

\noindent {\bf Reading.} {\em Quick Calculus}, pp. 97--129.

\noindent {\bf Supplementary reading.} Simmons, Chapter 3.

\begin{enumerate}

\item Differentiate the following functions, using the rules we
learned in lecture today.

\begin{enumerate}
\item $y=3x^4+2x^3-x^2+4x-7$
\item $y=(2x^2+3)(3x^4-2x-5)$
\item $y=\frac{3x-7}{2x^2+4}$
\item $y=(10x-2)^5(3x^2-1)^2$
\item $y=\sec\theta\csc\theta$
\item $y=\tan\theta=\frac{\sin\theta}{\cos\theta}$
\item $y=e^{5x+7}$
\item $y=\ln (\frac{3x^2}{4x+2})$
\item $y^3=\sqrt{2xy-4xy^2}$
\item $y=(x^2+4)^{\frac{5}{2}}$
\end{enumerate}

\item Given a cubic equation $f(x)=ax^3+bx^2+cx+d$, for what constants
$a$, $b$, $c$, and $d$ does the graph of $f(x)$ have exactly
\begin{enumerate}
\item two horizontal tangents?
\item one horizontal tangent?
\item no horizontal tangents?
\end{enumerate}
(Hint: A horizontal tangent to the graph occurs when the derivative
$f'(x)=0$.)

\item Find the values of $x$ for which the graph $f(x)=x+2\sin(x)$ has 
a horizontal tangent.

\item Find the tangent line to
$$
f(x)=\frac{x^3+x}{x-1}
$$
at the point $(2,10)$.

\item We have talked about the tangent line to a graph at some point
$P$ on the graph.  The {\em normal line} to a graph at the point $P$
is the line that is perpindicular to the tangent line to the graph at
$P$.  Given a line $f(x)=mx+b$, the perpindicular line $g(x)$ to
$f(x)$ at $P$ is the line with slope $-\frac{1}{m}$, also going
through $P$. (See the figure on the next page.)
\begin{figure}[h]
\centerline{
\psfig{figure=normal.ps,width=2in}
}
\smallskip
\centerline{
\parbox{4.5in}{\caption[Normal line]{{\small This shows the line
$f(x)=mx+b$ and the perpindicular line $g(x)=-\frac{1}{m}+d$, where
$d$ is determined by our choice of $P$.}}}
}
\end{figure}
Find the tangent line and the normal line to the graph
$y=\frac{6}{x+2}$ at the point $(1,2)$.

\end{enumerate}





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