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{\Large TEST 2

Math 129 - R5}
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\vspace{.1in}
You have 50 minutes to complete the test.
Show all relevant work (or else no credit).  
Make your work legible, if I can't read it, I can't grade it.  
Each problem is worth 10 points, unless otherwise noted.
Use back of pages if necessary.
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Name \rule{5in}{.2pt}

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\begin{enumerate}
\item  True or False, if false either give an example why or explain why.

\begin{tabular}{lll}
T & F & If $f''(3)=-20$ and $f'(3)=1$ then $f$ has a local maximum at $x=3$ \\
\end{tabular}
\vspace{1in}

\begin{tabular}{lll}
T & F & Local maximum are absolute maximums \\
\end{tabular}
\vspace{1in}

\item  Find the derivative, $\frac{dy}{dx}$
\[
y=\ln(4e^{-x^2}+3x)
\]
%\vspace{3in}
\newpage

\item  The productivity of a worker can be modeled by the formula:
\[
P=25+60t+9t^2-2t^3
\]
where $t$ is the number of hours that a worker has been on the job 
$0 \leq t \leq 8$,
and $P$ is the number of units produced per hour.
After how many hours is the worker most productive, 
and how many items per hour is the worker producing at this time?
(When is the worker producing the most items per hour?)
\newpage

\item  Draw a graph of a function that matches the data below:
(label all local max and mins and all inflection points)
\[
\begin{array}{ll}
f'(x)=0 & \mbox{ if } x=1 \mbox{ or } x=3 \mbox{ or } x=5 \\
f'(x)<0 & \mbox{ if } x<1 \mbox{ or } 3<x<5 \\
f'(x)>0 & \mbox{ if } 1<x<3 \mbox{ or } 5<x \\
f''(x)=0 & \mbox{ if } x=2 \mbox{ or } x=4 \\
f''(x)<0 & \mbox{ if } 2<x<4 \\
f''(x)>0 & \mbox{ if } x<2 \mbox{ or } 4<x \\
\end{array}
\]
\newpage

\item  (30 points) Graph the given function.
Label all local maximums and minimums (if any),
all inflection points (if any),
and all asymptotes (if any).
\[
y = \frac{4x}{x^2+1}
\]
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\end{enumerate}


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