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\centerline{Mathematics 136, section 2 -- Advanced Placement Calculus}
\centerline{Exam 1 -- October 1, 2009}
\vskip 10pt
\noindent
{\it Directions}
\vskip 10pt
Do all work in the blue exam booklet.  There are 100 regular points,
and 10 Extra Credit points possible.  \emph{Please turn over this sheet
for the graph for question I, and the Extra Credit question}.

\begin{enumerate}
\item[I.]  All parts of this question refer to the graph $y = f(x)$ plotted in Figure 1 on the back of this sheet.
\begin{enumerate} 
\item[A.] (5)  Do $\displaystyle{\lim_{x\to 2^+} f(x)}$ and
$\displaystyle{\lim_{x\to 2^-} f(x)}$ exist?  If so, say what they are; if not say why not.
\item[B.] (5)  Does $f(x)$ appear to be differentiable at $x = 3$?   Why or why not? 
\item[C.] (10) On separate axes, plot the graphs $y = 2f(x+1)$ and 
$y = f(2x) + 1$.  Label which graph is which.
\end{enumerate}
\item[II.]  At time $t = 0$, a steak at temperature $0^\circ$ F
is taken out of the freezer and left to thaw in a kitchen heated to 
$70^\circ$ F.  The steak's temperature increases fastest at first, then 
at a slower and slower rate as it nears the room temperature; it never goes over
$70^\circ$ F.
\begin{enumerate}  
\item[A.] (10)  Sketch a graph of the temperature of the steak as a function of time.
(For full credit, your graph should show the proper slopes and concavities.)
\item[B.] (5)  A formula for the temperature as a function of $t = $ time in hours
is $T = 70 - 70 e^{-2t}$.  How long does it take for the temperature to reach $65^\circ$?
\end{enumerate}
\item[III.]
\begin{enumerate}  
\item[A.]  (5) What is the exact mathematical definition of the 
\emph{derivative} of a function $f(x)$ at $x = a$?  
\item[B.]  (10) Use the definition (not the ``shortcut rules'') 
to compute the derivative of 
$f(x) = \sqrt{x}$ at a general $x > 0$.
\item[C.]  (5) What is the equation of the tangent line to $y = \sqrt{x}$
at the point $(4,2)$?
\end{enumerate}
\item[IV.]  Compute derivatives using the ``shortcut rules'':
\begin{enumerate}
\item[A.]  (5) $f(x) = \sin(x) + x^{5/3} - e^x$.
\item[B.]  (5) $g(x) = \displaystyle{\frac{x^2 - x}{x^4 + 1}}$
\item[C.]  (10) $h(x) = \sin^{-1}(\ln(x))$, and give the domain of $h(x)$.
\item[D.]  (10) Find $\frac{dy}{dx}$ by implicit differentiation if  
$\cos(x + y^2) + \tan^{-1}(x) y = 4$.
\end{enumerate}
\item[V.]  (15)  100 cubic inches of cookie dough is being rolled out in a kitchen.
The dough has the shape of a circular cylinder at all times, but the length 
of the cylinder increases and the radius decreases as times goes on.  How fast is the radius
of the dough cylinder changing if the length is increasing at 1 inch per second when
the length is 10 inches?  
\end{enumerate}
\vskip 10pt
\noindent
{\it Extra Credit} (10)  The graph $y = e^x$ is shifted 1 unit to the left.  Can the same 
transformed graph be obtained by a vertical stretching or shrinking of $y = e^x$?  If you 
say yes, give the stretching or shrinking factor.  If you say no, explain why not.
\begin{figure}
\begin{center}
\includegraphics[width=5in]{I.ps}
\caption{Figure for Problem I}
\end{center}
\end{figure}
\end{document}
