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  EXAM 1 &\huge TQS 211  &\normalsize Fall 2010\\
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\begin{enumerate}
\item $[2]$ 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 & $\sqrt{x^2+y^2}=x+y$\\
~\\
T &F & $\frac{d}{dx}(e^1)=e$\\
~\\
T &F & Profit is equal to price minus revenue. \\
~\\
T &F & If the graph of $f'$ is increasing at $x=1$, then $f''(1)>0$.
~\\
\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 $[5]$ Draw a possible graph of $f(x)=y$ given the following information: %combo of 2.4 #12c and 3.3 #17
\begin{enumerate}
\item $f'(x)>0$ on $1<x<3$
\item $f'(x)<0$ when $x<1$ and $x>3$
\item $f''(x)<0$ on $1<x<3$
\item $f'(x)=0$ at $x=0$ and $x=3$
\end{enumerate}

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


\hspace{-.5in} \includegraphics[height=1.6in]{Piece1} 	\includegraphics[height=1.6in]{Piece2} 


\begin{enumerate}
\item $[10]$ Find the following (if they exist):

$g(-4)$ \hspace{2.7in} $g'(-4)$ 
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$(g\circ f)(-4)$ \hspace{2in} $(\frac{f}{g})'(-4)$
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\item $[3]$ \emph{Sketch} the graph of $g'$.
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\end{enumerate}

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\item The demand curve for a product is given by $q=300-3p$, where $p$ is the price of the product and $q$ is the quantity consumers will buy at that price.  %3.2 #50

\begin{enumerate}
\item $[2]$ Write the revenue as a function of price.
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\item $[3]$ Find the marginal revenue when the price is \$10, and interpret your answer in terms of revenue.
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\item $[4]$ If the marginal cost of making the product is \$20, and the business has the ability to set the price (by controlling $q$), what should the business set the price to so as to maximize profit? 
\vfill
\end{enumerate}

\item  $[4]$  Let $f(x)=x^2$ and $g(x)=x^2+3$. %\S 2.1 #25
What can you say about the slopes of the tangent lines to the two graphs at point $x=1$?  $x=a$, where $a$ is any value?  Justify your comments.
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\newpage

\item $[12]$ For each rule of $f$ given below, find $f'(x)$.

\begin{tabular*}{5.5in}{l@{\extracolsep{\fill}}l} %ex 3 in 3.1, 
$f(x) = 5\cdot3^x+\ln x$  &   $e^{5-2x}$\\
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\vfill
\begin{tabular*}{5.5in}{l@{\extracolsep{\fill}}l}%3.4 #18, 
$\displaystyle{f(x) = \frac{3x^7-x}{\sqrt{x}}}$  & $f(x) = (\sqrt{3x^4-x})e^{5-2x}$  \\
\end{tabular*}
\vfill

\item  $[5]$ Find the equation of the tangent line to the graph of $y=2x^3-5x^2+3x-5$ at $x=1$.  %3.1 #47
\vfill

\end{enumerate}
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