\documentclass[12pt]{article}

\usepackage{latexsym,amsmath,epsfig,graphicx}
\setlength{\textwidth}{6.5in} \setlength{\textheight}{9in} \oddsidemargin = -.1 cm
\evensidemargin = -.1 cm \topmargin=-2cm
\begin{document}

\centerline{Mathematics 136 -- Advanced Placement Calculus}
\centerline{Exam 3 -- Review Sheet}
\centerline{November 23, 2009}

\vskip 10pt
\noindent
{\it General Information}
\vskip 10pt
As announced in the course syllabus, the third midterm exam of the semester
will be given at 7:00pm on Wednesday, December 2 (the Wednesday
after Thanksgiving break).  The format will be similar to that of the other midterms.

\vskip 10pt
\begin{itemize}
\item The exam will be designed to take an hour but you will have an extra 30 
minutes to work and check your solutions.
\item You will be given a TI-30 scientific calculator for the exam which
does NOT have graphing capabilities so be prepared to answer questions
without your personal calculator.  (Note: Some of you may have one of these
calculators purchased for use in Chemistry courses here.  That is also OK.)
\item Use of cell phones, I-pods, and all other electronic devices \emph{is not
allowed} during the exam.  Please leave such devices in your room
or put them away in your backpack  (make sure cell phones are turned off).
\end{itemize}
\vskip 10pt
\noindent
{\it What will be covered}
\vskip 10pt
The exam will cover the material since the last exam (Problem Sets 7 and 8), {\it plus}
the material on differential equations.  In other words, this is
the following material from sections 5.10, 6.1, 6.2, 6.3, 6.4, 6.6, and Chapter 7 of Stewart:
\vskip 10pt
\begin{enumerate}
\item Improper integrals (know how to tell what makes an integral improper, how
to set up the limit to determine if the integral converges correctly, and determine
the value if the integral does converge).
\item Areas between curves
\item Volumes of solids with known cross-sections and solids of revolution
\item Arc length of curves
\item Average value of $f(x)$ on an interval $[a,b]$.
\item Work done stretching/compressing a spring, emptying a tank, etc.
\item Computing centers of mass (centroids) for wires, thin plates.
\item Differential equations: solutions, slope fields (be prepared 
to sketch a simple one by hand and
then add qualitative graphs of solutions, separation of variables for analytic solutions, 
growth and decay problems.
\end{enumerate}
\vskip 10pt
\noindent
\emph{Important Notes:} 
\begin{enumerate}
\item Many of the problems on this exam will require you 
to set up and compute an integral to find the quantity that is asked for.
In addition to knowing how to set up the required integral, \emph{any} of
the methods of integration tested on the second exam (i.e. basic rules, 
$u$-substitution, integration by parts, trigonometric substitution, partial 
fractions, or consultation of a table of integrals) might be required
to evaluate the integral.  An ``extract'' of the table of integrals will 
be provided for you to use on the exam, but you may need to do a preliminary
substitution or transform your given integral in some
way. In other words, this exam is \emph{really a 
cumulative exam on the second two-thirds of the semester}.  Especially 
if you did poorly on the second exam, you will need to begin your review
for this exam by going back and looking at the material from 
sections 5.1 through 5.8 in Stewart(!)
\item Because of the timing of the Thanksgiving break, I have
decided not to give a mandatory Problem Set 9 before the exam.
However, you may submit the following problems (also listed below in the
Review Problems for the exam)
\vskip 10pt
\begin{itemize}
\item Section 7.3/1 - 17 (odd numbers).
\item Chapter 7 Review/1, 5-8 (these are all separable!), 11, 13, 15, 17.
\end{itemize}
as an {\it optional, Extra Credit problem set}, due Tuesday, December 1.
{\it All necessary work must be shown on these for credit.} 
{\it Note:  I will not have enough time to grade these before the exam, but
you will get them back graded before the final. Your score on this problem
set will be added on top of your regular problem set average, so this
can only help, even if you do not do all of them!} 
\end{enumerate}
\vskip 10pt
\noindent
There will be a review for the exam in class on Tuesday, December 1.  \emph{Please
do not postpone preparing for this exam until after you return from 
Thanksgiving break -- you may not have enough time to get ready!}

\vskip 10pt
\noindent
{\it Review Problems}
\vskip 10pt
\noindent
Section 5.10/1, 13, 15, 19 (use parts), 21, 23, 25, 29, 31, 55
\vskip 10pt
\noindent
Section 6.1/1, 3, 5, 7, 9, 11, 29 (place the origin at the center of 
the large circle, and note that the endpoints of the arc on the smaller
circle are at opposite ends of a diameter of that circle), 37
\vskip 10pt
\noindent
Section 6.2/1, 3, 5, 9, 21 (and evaluate), 29, 39 a (and evaluate)
\vskip 10pt
\noindent
Section 6.3/5, 7, 9 (look for an algebraic simplification to integrate), 
17, 19 (use the table for these)
\vskip 10pt
\noindent
Section 6.4/1, 3, 5, 13
\vskip 10pt
\noindent
Section 6.6/Any question I ask here will be similar to the
questions from Problem Set 8.
\vskip 10pt
\noindent
Section 7.3/1 - 17 (odd numbers).
\vskip 10pt
\noindent
Chapter 7 Review/1, 5-8 (these are all separable!), 11, 13, 15, 17. 
\vskip 10pt
\noindent
{\it Sample Exam Questions}
\vskip 10pt
\noindent
This list is {\it much longer} than the actual exam will be (to give you some
idea of the range of different questions that might be asked).  
Unless otherwise directed, you may use any entry of the Table of 
Integrals from the text that applies.
\vskip 10pt
\noindent
I.  For each of the following integrals, say why the integral is improper,
determine if the integral converges, and if so, find its value.
\vskip 10pt
\begin{enumerate}
\item[A)]  $\int_1^\infty \frac{1}{{\root 5 \of x}}\ dx$
\item[B)]  $\int_0^2 \frac{dx}{x^2 - 7x + 6}$
\item[C)]  $\int_0^\infty x e^{-3x}\ dx$
\item[D)]  For which values of $a$ is $\int_0^\infty e^{ax}\sin(x)\ dx$
convergent?  Evaluate the integral for those $a$. 
\end{enumerate}
\vskip 10pt
\noindent
II.
\begin{enumerate}  
\item[(A)]  Let $R$ be the region in the plane bounded by $y = 3 - x^2$ and the $x$-axis.
\begin{enumerate}
\item[(1)]  Sketch the region $R$.  
\item[(2)]  Find the area of $R$.
\item[(3)]  Find the volume of the solid generated by rotating $R$ about the
$x$-axis.
\item[(4)]  Find the coordinates $\overline{x},\overline{y}$ of its centroid.
\end{enumerate}
\item[(B)]  Let $R$ be the region in the plane bounded by $y = \cos(\pi x)$,
$x = 0$, $x = 1/2$, and $y = 0$.
\begin{enumerate}
\item[(1)]  Sketch the region $R$.  
\item[(2)]  Find the area of $R$.
\item[(3)]  Find the volume of the solid generated by rotating $R$ about the
$x$-axis.
\item[(4)]  Find the coordinates $\overline{x},\overline{y}$ of its centroid.
\end{enumerate}
\end{enumerate}
\vskip 10pt
\noindent
III.  The height of a monument is 20m.  The horizontal cross-section of
the monument at $x$ meters from the top is an isosceles right triangle
with legs $x/4$ meters.  Find the volume of the monument.  
\vskip 10pt
\noindent
IV. 
\begin{enumerate}
\item[(A)] Set up and evaluate the integral to compute the arclength of the 
curve $y = 3x^2$, $0 \le t \le 2$.
\item[(B)]  Set up and evaluate the integral to compute the arclength of the 
curve $y = \frac{1}{6}(x^2 + 4)^{3/2}$, $0 \le x \le 3$.  (Hint: the arclength
integral simplifies to a manageable form if you are careful with the algebra.)
\end{enumerate}
\vskip 10pt
\noindent
V.  
\begin{enumerate}
\item[(A)] Find the average value of $f(x) = \sqrt{1 - x^2}$ on the interval
$[0,1/2]$.  (Use trigonometric substitution, not the table.)
\item[(B)] Find the average value of $f(x) = x \sqrt{1 + x^4}$ on the interval $[0,2]$. 
\end{enumerate}
\vskip 10pt
\noindent
VI.  A cylindrical tank 10 meters tall and with base radius $3$ meters is
full of water (density 1000kg per cubic meter).  Find the work done in
pumping all the water out the top of the tank.
\vskip 10pt 
\noindent
VII.
\begin{enumerate}
\item[(A)]  Show that for any constant $c$, $y = x^2 + \frac{c}{x^2}$
is a solution of the differential equation
\[
y' = 4x - \frac{2}{x}y.
\]
\item[(B)] All parts of this question refer to the differential equation
\[
y' = y(4 - y)
\]
\begin{enumerate}
\item[(1)]  Sketch the slope field of this equation, showing 
the slopes at points on the lines $y = 0,1,2,3,4,5$
\item[(2)]  On your slope field, sketch the graph of the 
solution of the equation with $y(0) = 1$.
\item[(3)]  This is a separable equation; find the general solution
and determine the constant of integration from the initial condition $y(0) = 1$. 
\end{enumerate}
\item[(C)]  Find the general solutions of the following differential equations
\begin{enumerate}
\item[(1)]  ${\displaystyle y' = \frac{y}{x(x+1)} }$
\item[(2)]  ${\displaystyle y' = \frac{\sqrt{1 - x^2}}{e^{2y}}.}$
\end{enumerate}
\item[(D)]  Newton's Law of Cooling states that the 
rate at which the temperature of an object changes is 
proportional to the difference between the object's temperature
and the surrounding temperature.  A hot cup of tea with temperature
$100^\circ$C is placed on a counter in a room maintained at constant 
temperature $20^\circ$C.  Ten minutes later the tea has cooled to 
$76^\circ$C.  How long will it take to cool off to $45^\circ$C?  
(Express Newton's Law as a differential equation, solve it for the 
temperature function, then use that to answer the question.)
\end{enumerate} 
\end{document}  
