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\centerline{Mathematics 136, section 2 -- Advanced Placement Calculus}
\centerline{Review Sheet for Exam 1}
\centerline{September 23, 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.
\begin{enumerate}
\item[I.]
\begin{enumerate} 
\item[A.] (5)  Plot the graph of the piecewise-defined function:
$$
\begin{cases} t + 1 & \text{ if } t < 0 \\
               -3   & \text{ if } t \ge 0.
\end{cases}
$$
\item[B.] (10) On one set of axes, plot the graphs $y = 2 f(t - 2)$ and 
$y = f(-t) + 1$.  Label which is which.
\item[C.] (5)  Is $f$ continuous at $t = 0$?  Explain why or why not.  
\item[D.] (5)  Is $f$ continuous at $t = 2$?  Explain why or why not.
\end{enumerate}
\item[II.] A population of wildebeest grows in a habitat that can support 
no more than $P_1$ animals.  At time $t = 0$, the number of 
wildebeest present is  $P_0 < P_1$.  The population grows more and
more rapidly at first, but then the rate of growth decreases and
the population approaches $P_1$, without exceeding it.
\begin{enumerate}  
\item[A.] (10) Sketch a graph
of the population of wildebeest as a function of time
that fits this description.   For full credit your graph should
show proper slope and concavity.  
\item[B.] (10)  Give a qualitative graph of the derivative function of your
population function.
\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 
$f(x) = 4x^2 - 2x + 2$ at a general $x$.
\item[C.]  (5) What is the equation of the tangent line to $y = 4x^2 - 2x + 2$
at the point $(1,4)$?
\end{enumerate}
\item[IV.]  Compute derivatives using the ``shortcut rules'':
\begin{enumerate}
\item[A.]  (5) $f(x) = 2x^3 + 6 x^{2/3} + e^\pi$.  For which $x$ is $f$ differentiable?
\item[B.]  (5) $f(x) = \frac{\sin(x)}{1 + \cos(x)}$
\item[C.]  (5) $g(x) = \tan^{-1}(e^x - \ln(x))$
\item[D.]  (5) Find $\frac{dy}{dx}$ by implicit differentiation if  $x y^6 + e^y = 1$.
\end{enumerate}
\item[V.]  (15)  A cube of ice is taken out of a freezer and left on a kitchen counter
to melt, losing
volume at a rate of $2$ cubic inches per minute.  At what rate is the side of the 
cube changing when the volume is 64 cubic inches?