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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 General Information}
\vskip 10pt
The first exam for the course will be given next Thursday, October 1 at 7:30pm \emph{in 
our regular classroom}.  You will have until
9:00pm to work on the exam if you need that much time. (Note:  This is one day later 
than announced in the course syllabus because of the Hanify-Howland Lecture).  
It will cover the material discussed in class from the start of the semester
{\it through and including\/} the material on implicit differentiation, derivatives of 
inverse functions and related rates problems from class on September 23.  There
will be 4 or 5 problems, some possibly with several parts.  Some
may ask for a graph or the result of a calculation; others may ask for 
a precise definition of a term or concept we have used, or a 
short description or explanation of some phenomenon.
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\noindent
{\it Calculators}
\vskip 10pt
As announced in the course syllabus, I will be providing a 
basic scientific numerical calculator (TI model TI-30Xa) for you to 
use on this exam.  These calculators can compute $+,-,\times,\div$
and values of all functions we have talked about, but have no
graphing or symbolic computation features.  
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\noindent
{\it Review Session}
\vskip 10pt
Since we are taking an extra day for the exam, class on Tuesday, September 29 or
Wednesday, September 30 will be an \emph{optional} review session (choice of which
day will be up to the class).
We will work some example problems, review the background for some
topics, and so forth.   I think you will find this to be valuable,
but you are not obligated to attend if you feel well-prepared.
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\noindent
{\it Suggested Practice Problems}
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\noindent
From the text:  
\begin{itemize}
\item  Chapter 1 Review problems:  5, 7, 9-15, 24-26 
\item  Chapter 2 Review problems:  1, limit problems like 3-18, 30, 32, 35, 37ab, 39, 41, 43
\item  Be able to differentiate any functions like p. 248-9/1 - 36 from
the Chapter 3 Review problems (you don't need to do all of these; just make sure
you can handle all the different functions involved, can spot which
derivative rule(s) apply, etc.  Also you might need to 
compute a derivative like one of these as part of a larger problem.)
\item  Chapter 3 Review problems:  41, 47ab, 53-60, 65
\item Section 4.1/3, 5, 11.
\end{itemize}
\vskip 10pt 
\noindent
{\it Sample Exam}  (The numbers next to each part are the possible
points on that part; they add up to 100.  Disclaimer:  The actual exam questions
may be posed differently, may combine these topics in somewhat different
ways, etc.)
\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?
\end{enumerate}
\end{document}
