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\centerline{Mathematics 136 -- Calculus 2}
\centerline{Exam 3 -- Review Sheet}
\centerline{April 22, 2014}

\vskip 10pt
\noindent
{\it General Information}
\vskip 10pt
As announced in the course syllabus, the third midterm exam of the semester
will be given in class on Friday, May 2. 
The format will be similar to that of the first two midterms
and the exams from last semester.

\vskip 10pt
\begin{itemize}
\item You will be provided with (an extract from) the table of integrals
for use on this exam.  
\item You will not be given a calculator for the exam.  If you 
want to use one, you will need to supply it yourself.  Any calculator 
like the TI-30, which
does NOT have graphing capabilities, is OK.  (Note: Some of you may have one of these
calculators purchased for use in Chemistry courses here.  That is also OK.)
\item Use of phones, I-pods, and any electronic devices other than your
calculator \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, 8, and 9), namely
the following material from sections 6.4 - 6.6, 6.8, 7.1 - 7.5, and 8.2 - 8.5 of Stewart:
\vskip 10pt
\begin{enumerate}
\item Additional applications of integration (Chapter 6) 
\begin{enumerate}
\item Arc length of graphs and parametric curves
\item Average value of a function
\item Center of mass (centroid)
\item Probability applications
\end{enumerate}
\item Differential equations (Chapter 7)
\begin{enumerate}
\item Solutions of a differential equation (know how to determine whether a 
given function is or is not a solution of a given equation; for this type of
question it will not be necessary to derive a general solution), 
\item direction fields (be prepared to sketch a simple direction field by hand, 
and/or identify computer-generated plots), 
\item Euler's method for approximating solutions,
\item Separable equations and solution by separation of variables and integration, 
\item Exponential growth and decay problems; other growth 
and decay problems (be prepared to set up a differential equation matching
a verbal description, solve it, and use the solution to answer questions), 
\item Logistic equations -- know the form 
\[
y' = ky\left(1 - \frac{y}{M}\right)
\]
of the logistic equation, the 
properties of the solutions, and the analytic formula
\[
y = \frac{M}{1 + b e^{-kt}}
\]
for the solutions.
\end{enumerate}
\item Infinite Series (Chapter 8)
\begin{enumerate}
\item the concepts of convergence for an infinite series,
\item key classes of examples such as geometric series, telescoping series,
the harmonic series, $p$-series, etc.
\item general convergence tests:  the Integral Test, the Alternating Series Test, 
the $n$th Term Test for Divergence, the Ratio Test for absolute convergence
\item power series, radius of convergence, interval of convergence
\end{enumerate}
\end{enumerate}
\vskip 10pt
\noindent
\emph{Important Note:}  Some of the problems on this exam (in particular
anything from the sections in Chapter 6, plus
solving separable differential equations and applying the Integral Test
for series) will require you 
to set up and compute integrals 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 first and 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.  Again, there is a high degree of ``cumulativity''
built into this material.  
\vskip 10pt
\noindent
There will be a review for the exam in class on Wednesday, April 30.  

\vskip 10pt
\noindent
{\it Review Problems}
\begin{itemize}
\item Section 6.4/7,11
\item Section 6.5/7,11
\item Section 6.6/49,51
\item Section 6.8/3,5,7
\item Section 7.1/1,3,5,9,11
\item Section 7.2/1,3-6,9,23
\item Section 7.3/1,3,5,7,9,11,33,35
\item Section 7.4/3,9,11,13,17
\item Section 7.5/1,3,7
\item Section 8.2/11,13,15,19,29
\item Section 8.3/3,5,7,21 (use a comparison test)
\item Section 8.4/5,7, 21, 23, 25
\item Section 8.5/5,7,9
\end{itemize}
\vskip 10pt
\noindent
{\it Sample Exam Questions}
\vskip 10pt
\noindent
This list is much longer than the actual exam will be (to give you some
idea of the range of different questions that might be asked).  
\vskip 10pt
\noindent
\begin{enumerate}
\item[I.] 
\begin{enumerate}
\item[(A)] Set up and evaluate the integral to compute the arclength of the 
curve $x = 3t^2$, $y = 2t^3$, $0 \le t \le 2$.
\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
\item[II.]
\begin{enumerate}
\item[(A)] Find the average value of $f(x) = x \sqrt{1 + x^4}$ on the interval $[0,2]$. 
\item[(B)] Set up the integrals needed to compute 
the location of the center of mass of a thin plate of 
constant density in the shape of the 
region between the graph $y = f(x)$ from part (A) and the $x$-axis.  (Do not try 
to evaluate them.)
\item[(C)] For which $c$ is $f(x) = c x \sqrt{1 + x^4}$ a 
probability density function on the interval $[0,2]$ (i.e. take $f(x) = 0$ if
$x$ is outside $[0,2]$). 
\end{enumerate}
\item[III.]
\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)]  Use Euler's method to approximate the solution of 
this equation with $y(0) = 1$ for $0 \le x \le 1$ using
$n = 4$.  
\item[(4)]  This is a separable equation, find the general solution
and determine the constant of integration from the initial condition $y(0) = 1$. 
(Note: this is similar in form to a logistic equation.)
\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}
\item[IV.]
\begin{enumerate}
\item[(A)] 
Does the infinite series ${\displaystyle \sum_{n=1}^\infty n \ln(1 + n)}$ converge?
Why or why not?

\item[(B)]  Use the Integral Test to determine whether or not
  \[
  \sum_{k=1}^\infty\frac{k}{e^k}
  \]
  converges.
  
\item[(C)]  Use the Ratio Test to determine whether or not
  \[
  \sum_{k=0}^\infty \frac{3^n}{n!}
  \]
  converges.
  
\item[(D)] Determine (with justification!) whether or not the following
  series converge:
  \[
  \sum_{k=1}^\infty \frac{1}{\sqrt{k}},\qquad
  \sum_{n=0}^\infty (-1)^n\frac{3^n}{\pi^{2n}},\qquad
  \sum_{n=1}^\infty \frac{1}{n^{1.01}}.
  \]
  
\item[(E)] For each of the given power series, find the interval of
  convergence.  (In particular, give the radius of convergence, 
  and investigate convergence at the endpoints.)
\begin{eqnarray*}
f(x) &=& {\displaystyle\sum_{k=0}^\infty x^k = \frac{1}{1-x}},\\
g(x) &=& \sum_{n=1}^\infty\frac{(2x)^n}{\sqrt{n}},\\
h(x) &=& \sum_{n=1}^\infty(-1)^{n-1}\frac{(x-5)^n}{n\cdot3^n}.
\end{eqnarray*}
  
\end{enumerate}
\end{enumerate}

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