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\begin{tabular*}{6in}{c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c}
  EXAM 1 &\huge TQS 211  &\normalsize Practice\\
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Note: This is a practice midterm (that may be a page shorter than it ought to be) and is intended only for study purposes.  The actual exam will contain different questions and perhaps a different layout. 


\begin{enumerate}
\item $[]$ TRUE/FALSE: Circle T in each of the following cases if the 
statement is \emph{always} true.  Otherwise, circle F. 

\begin{tabular}{lll}
~\\
T &F & $\frac{d}{dx}b^c=cb^{c-1}$ for a fixed $b$ and $c$\\
~\\
T &F & $(x+y)^2=x^2+y^2$\\
~\\
T &F & $\frac{d}{dx}2^x=x2^{x-1}$\\
~\\
T &F & No profit is made when $MR<MC$
~\\
\end{tabular}

\large
Show your work for the following problems.  The correct answer with 
no supporting work will receive NO credit (this includes multiple
choice questions).

\normalsize

\item Find the rule of a function whose second derivative is negative everywhere and whose graph passes through the point $(2,0)$.


\newpage
\item Let $f$ be the function whose graph is below.

\begin{center}
\includegraphics[height=2in]{Piece1}	
\end{center}

\begin{enumerate}
\item Find the following (if they exist):

$f(-4)$ \hspace{2.7in} $f'(-3)$ 
\vskip 1in

$(f\circ f)(-4)$ \hspace{2in} $(f\cdot f)'(-2.5)$
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\item \emph{Sketch} the graph of $f'$.
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\includegraphics[height=2in]{Blank5x4}	
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\end{enumerate}

\newpage

\item Assume you have a business making widgets.  Your have tracked your profits and found that the forth degree polynomial below approximates your marginal profit quite well.  Let $g$ be the function describing your profit when you produce $x$ widgets and $g'$ be the function with the graph below where $x$ is measured in hundreds of widgets and $y$ is measure in in hundreds of dollars.
\begin{center}
\includegraphics[height=2in]{DerivativeQuardic}	
\end{center}

\begin{enumerate}
\item Approximate $g'(3)$.  What are the units associated with $g'(3)$? What does this mean in terms of widgets and profit?
\vskip 1.5in
\item If you make \$1024 of profit when you make 400 widgets, find an approximation of your profit if you make 500 widgets.
\vskip 1in
\item How many widgets should be made to maximize profit? Explain how you know this.
\vskip 1in
\end{enumerate}



\newpage

\item $[]$ For each rule of $f$ given below, find $f'(x)$.

\begin{tabular*}{5.5in}{l@{\extracolsep{\fill}}l}
$\displaystyle{f(x) = \sin(x) + \frac{3}{x}}$  & $f(x) = 2x^2 + \ln(7x^2)$  \\
\end{tabular*}
\vskip 2in
\begin{tabular*}{5.5in}{l@{\extracolsep{\fill}}l}
$\displaystyle{f(x) = \frac{3^x\cos(x)}{\cos(\frac{1}{x})}}$  & $f(x) = (\sqrt{3x^4-x})(e^x - 4)$  \\
\end{tabular*}
\vskip 2in

\item  Given $f(x) = e^{3x-6} + x^2 - 4x + 6$ find an equation for the tangent line at $x = 0$. 


\end{enumerate}
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