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\centerline{Mathematics 132 -- Calculus for Physical and Life Sciences 2}
\centerline{Exam 3 -- Review Sheet}
\centerline{April 15, 2008}

\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, April 23 (note different time). 
The format will be similar to that of the first midterm
and the exams from last semester.

\vskip 10pt
\begin{itemize}
\item {\bf Bring a photo ID to the exam}.
\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, 8, and 9), namely
the following material from sections 7.1 - 7.5, and 8.1 - 8.7 of Stewart:
\vskip 10pt
\begin{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 Sequences and Series (Chapter 8)
\begin{enumerate}
\item the concepts of sequences and series (and the difference between them!), the
concepts of convergence for sequences and for series (and the difference between them!)
\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 Test for Divergence, the Ratio Test for absolute convergence (in addition to using 
these on specific examples, be prepared to give careful \emph{statements}
of the theorems concerning each test -- what hypotheses are necessary, what 
conditions imply convergence or divergence, and the conclusion in each case; also
be prepared for questions like 8 from Problem Set 8, where you must give
a reason why a particular test does not apply for a particular example).
\item power series, radius of convergence, interval of convergence
\item power series representing functions, Taylor series and Taylor polynomials.
\end{enumerate}
\end{enumerate}
\vskip 10pt
\noindent
\emph{Important Note:}  Some of the problems on this exam (in particular
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 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 cumulativeness
built into this material.  
\vskip 10pt
\noindent
There will be a review for the exam in class on Tuesday, April 22.  

\vskip 10pt
\noindent
{\it Review Problems}
\begin{itemize}
\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 8.1/3,5,9,11,13
\item Section 8.2/9,11,13,15,19,21,25
\item Section 8.3/7,21 (use the integral test)
\item Section 8.4/19, 21, 23, 25
\item Section 8.5/5,7,9
\item Section 8.6/3,5,7,9,11,21
\item Section 8.7/1,3 (Maclaurin series $=$ Taylor series with $a = 0$),11,13,15,41,43
\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)]  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 $y(0) = 1$. 
\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[II.]
\begin{enumerate}
\item[(A)]  Does the sequence $a_n = n \ln(1 + n)$ converge?  Why or why not?
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)] Let $f(x)=\sqrt{1+x}=(1+x)^{1/2}$. Find the 4th degree Taylor
  polynomial of~$f$ centered at $a=0$.  Find a factorial
  expression for the general term of the Taylor series.

\item[(F)] Consider the geometric series
  $f(x)={\displaystyle\sum_{k=0}^\infty x^k = \frac{1}{1-x}}$.

  \begin{enumerate}
  \item[(1)] Use series manipulations to find the Taylor series of $xf'(x)$.

  \item[(2)] Use series manipulations to find the Taylor series
  of~$-\ln(1-x)$. 
  
  \item[(3)] Find the radius of convergence of the series in part~(2), and
  investigate convergence at the endpoints.

  \item[(4)] Use parts (1)~and (2) to \emph{evaluate} the sums of the series
  ${\displaystyle \sum_{k=1}^\infty\frac{1}{k\cdot 2^k}}$~and 
  ${\displaystyle \sum_{k=1}^\infty\frac{k}{2^k}}$.
  \end{enumerate}

\item[(G)] For each of the given power series, find the interval of
  convergence.
  \[
  f(x)=\sum_{n=1}^\infty\frac{(2x)^n}{\sqrt{n}},\qquad
  g(x)=\sum_{n=1}^\infty(-1)^{n-1}\frac{(x-5)^n}{n\cdot3^n}.
  \]
  (In particular, give the radius of convergence, 
  and investigate convergence at the endpoints.)

\item[(H)] The second degree Taylor polynomial of $f(x)$ at $a = 0$
is $p_2(x) = c + bx + ax^2$.  What can you say about the 
signs of $a,b,c$ if you know $f(x)$ is increasing and 
concave down at $x = 0$?
\end{enumerate}
\end{enumerate}

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