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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Quiz 8 -- December 3, 2015}
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\noindent
Your Name: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
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\noindent
{\it Directions}
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Do all work in the space provided below or on the back of the second sheet.  There are 30 total points possible.  You may use a calculator but not any graphing features.  
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\noindent
{\it Questions}
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\begin{enumerate}
\item[1)]  Both parts of this question refer to the function $f(x) = -x^3 + 3x + 1$
\begin{enumerate} 
\item[(a)] (5) Compute $f'(x)$ and find all of the critical points of $f$.
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\item[(b)] (5) Find the maximum and minimum values of $f$ on the closed interval $[-2,0]$.  
\end{enumerate}
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\item[2)]  Let $f(x) = xe^{-3x}$
\begin{enumerate}
\item[(a)] (5) Compute $f'(x)$ and $f''(x)$.
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\item[(b)] (5) Determine the interval(s) where $y = f(x)$ is \emph{concave up} and the
interval(s) and where $y = f(x)$ is \emph{concave down}.
\end{enumerate}
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\item[3)]  The graph above shows $y = f'(x)$ for some function $f$  (NOTE:  this is the 
graph of the derivative $y = f'(x)$, NOT $y = f(x)$).
\begin{figure}
\begin{center}
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\caption{Plot of $y = f'(x)$ for Problem 3}
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\begin{enumerate}
\item[(a)] (3) What are the critical points of $f$ in the interval $[-4,4]$? 
\item[] Answer: $x = \underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\item[(b)] (2)  At which $x$ value(s) in this interval does $f$ have a local maximum?
\item[] Answer: $x = \underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\item[(c)] (1) Explain briefly how you know your answer in (b) is correct.
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\item[(d)] (2)  On the interval $(-1,1)$, is the graph $y = f(x)$
concave up or concave down? 
\item[] Answer: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\item[(e)] (2)  How many points of inflection does $y = f(x)$ have in the interval
$[-4,4]$, and what are the approximate $x$-values where they are located? 
\item[] Answer: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\end{enumerate}
\end{enumerate}
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